thesis_berceanu_words.txt

Superfluidity
the
ability
of
fluid
to
flow
without
apparent
viscosity
is
one
of
the
most
striking
consequences
of
collective
quantum
coherence
with
manifestations
ranging
from
metastability
of
supercurrents
in
multiply
connected
geometries
to
the
appearance
of
quantized
vortices
or
the
existence
of
critical
velocity
for
frictionless
flow
when
scattering
against
defect
While
traditionally
investigated
in
equilibrium
systems
like
liquid
He
and
ultracold
atomic
gases
experimental
advances
in
nonlinear
optics
in
particular
regarding
microcavity
exciton
polaritons
paved
the
way
for
studying
superfluid
related
phenomena
in
driven
dissipative
framework
Equally
exciting
is
the
possibility
of
realising
topological
phases
of
matter
such
as
the
integer
or
fractional
quantum
Hall
states
outside
of
traditional
electronic
systems
Photonics
experiments
in
driven
dissipative
resonator
arrays
in
particular
offer
high
degree
of
controllability
and
tunability
as
well
as
unprecedented
experimental
access
to
the
eigenstates
and
energy
spectrum
This
thesis
reports
on
hydrodynamic
effects
as
well
as
topological
properties
of
driven
dissipative
systems
In
particular
we
analyze
the
superfluid
like
behaviour
of
microcavity
exciton
polaritons
as
well
as
the
momentum
space
topology
of
coupled
resonator
arrays
Microcavity
exciton
polaritons
are
quasiparticles
resulting
from
the
mixing
of
excitons
bound
electron
hole
pairs
and
photons
confined
inside
semiconductor
microcavities
While
polariton
fluids
have
been
shown
to
display
collective
coherence
the
connection
between
the
various
manifestations
of
superfluid
behaviour
is
more
involved
compared
to
equilibrium
systems
In
this
manuscipt
we
consider
both
the
case
of
single
fluid
pump
only
configuration
as
well
as
the
three
fluid
optical
parametric
oscillator
regime
that
results
from
parametric
scattering
of
the
pump
to
the
signal
and
idler
states
In
both
cases
we
look
at
the
response
of
the
moving
polaritons
scattering
against
weak
static
defect
present
in
the
microcavity
For
the
single
fluid
we
evaluate
analytically
the
drag
exerted
by
the
fluid
on
the
defect
For
low
fluid
velocities
the
pump
frequency
classifies
the
collective
excitation
spectra
in
three
different
categories
linear
diffusive
like
and
gapped
We
show
that
both
the
linear
and
diffusive
like
cases
share
qualitatively
similar
crossover
of
the
drag
from
the
subsonic
to
the
supersonic
regime
as
function
of
the
fluid
velocity
with
critical
velocity
given
by
the
speed
of
sound
found
for
the
linear
regime
In
contrast
for
gapped
spectra
we
find
that
the
critical
velocity
exceeds
the
speed
of
sound
In
all
cases
we
show
that
the
residual
drag
in
the
subcritical
regime
is
caused
by
the
nonequilibrium
nature
of
the
system
Also
well
below
the
critical
velocity
the
drag
varies
linearly
with
the
polariton
lifetime
in
agreement
with
previous
numerical
studies
The
optical
parametric
oscillator
regime
presents
an
additional
challenge
as
one
is
dealing
with
three
coupled
fluids
The
spontaneous
macroscopic
coherence
following
the
phase
locking
of
the
signal
and
idler
fluids
has
been
already
shown
to
be
responsible
for
their
simultaneous
quantized
flow
metastability
We
find
that
the
modulations
generated
by
the
defect
in
each
fluid
are
not
only
determined
by
its
associated
scattering
ring
in
momentum
space
but
each
component
displays
additional
rings
because
of
the
cross
talk
with
the
other
components
imposed
by
nonlinear
and
parametric
processes
We
single
out
three
factors
determining
which
one
of
these
rings
has
the
biggest
influence
on
each
fluid
response
the
coupling
strength
between
the
three
fluids
the
resonance
of
the
ring
with
the
polariton
dispersion
and
the
values
of
each
fluid
group
velocity
and
lifetime
together
establishing
how
far
each
modulation
can
propagate
from
the
defect
For
the
typical
conditions
of
parametric
scattering
the
pump
is
in
the
supercritical
regime
so
the
signal
and
idler
will
show
the
modulations
of
the
pump
meaning
none
of
the
three
states
manifests
superfluid
behaviour
However
the
signal
appears
to
flow
without
friction
in
the
experimental
study
because
the
three
factors
mentioned
above
conspire
to
reduce
the
amplitude
of
its
modulations
below
currently
detectable
levels
Driven
dissipative
systems
can
show
interesting
phenomena
also
without
interactions
stemming
from
the
nontrivial
topology
of
their
energy
bands
In
the
final
part
of
this
thesis
we
present
realistic
proposal
for
an
optical
experiment
using
state
of
the
art
coupled
resonator
arrays
We
study
theoretically
the
driven
dissipative
Harper
Hofstadter
model
on
square
lattice
in
the
presence
of
weak
harmonic
trap
Without
pumping
and
losses
the
eigenstates
of
this
system
can
be
understood
under
certain
approximations
as
momentum
space
toroidal
Landau
levels
where
the
Berry
curvature
geometrical
property
of
an
energy
band
acts
like
momentum
space
magnetic
field
We
show
how
key
features
of
these
eigenstates
can
be
observed
in
the
steady
state
of
the
driven
dissipative
system
under
monochromatic
coherent
drive
We
also
show
that
momentum
space
Landau
levels
would
have
clear
signatures
in
spectroscopic
measurements
in
such
experiments
and
we
discuss
the
insights
gained
in
this
way
into
geometrical
energy
bands
and
particles
in
magnetic
fields
List
of
Publications
List
of
Publications
The
following
articles
have
been
published
in
the
context
of
this
thesis
Onset
and
dynamics
of
vortex
antivortex
pairs
in
polariton
optical
parametric
oscillator
superfluids
Tosi
Marchetti
Sanvitto
Ant
Szyma
ska
Berceanu
Tejedor
Marrucci
Lema
itre
Bloch
and
Vi
Phys
Rev
Lett
Drag
in
resonantly
driven
polariton
fluid
Berceanu
Cancellieri
and
Marchetti
Phys
Condens
Matter
Chapter
Multicomponent
polariton
superfluidity
in
the
optical
parametric
oscillator
regime
Berceanu
Dominici
Carusotto
Ballarini
Cancellieri
Gigli
Szyma
ska
Sanvitto
and
Marchetti
Phys
Rev
Chapter
Momentum
space
Landau
levels
in
driven
dissipative
cavity
arrays
Berceanu
Price
Ozawa
and
Carusotto
Phys
Rev
Chapter
Conference
proceedings
Momentum
space
Landau
levels
in
arrays
of
coupled
ring
resonators
Price
Berceanu
Ozawa
and
Carusotto
Proc
SPIE
Preface
Preface
How
birds
fly
together
Systems
far
from
thermal
equilibrium
frequently
show
novel
features
when
compared
to
their
equilibrium
counterparts
As
an
everyday
example
consider
group
of
birds
displaying
long
range
ordered
behaviour
manifested
by
forming
flock
under
certain
conditions
This
behaviour
can
be
modeled
by
introducing
time
step
rule
such
that
each
individual
bird
in
group
determines
its
next
direction
on
each
time
step
by
averaging
the
directions
of
its
neighbours
and
adding
some
random
noise
on
top
of
that
It
can
be
shown
that
in
the
limit
of
the
velocity
going
to
zero
the
model
reduces
to
the
XY
model
in
two
dimensions
where
the
spin
is
represented
by
the
bird
velocity
Since
the
XY
model
does
not
spontaneously
break
the
symmetry
at
any
finite
temperature
as
justified
by
the
Mermin
Wagner
theorem
one
can
show
that
the
appearance
of
the
long
range
ordered
phase
is
direct
consequence
of
nonequilibrium
aspects
of
the
model
In
nutshell
the
neighbours
of
one
particular
bird
will
be
different
at
different
times
depending
on
the
velocity
field
This
gives
rise
to
time
dependent
variable
ranged
interaction
which
can
stabilize
the
ordered
phase
Driven
dissipative
systems
Non
equilibrium
driven
dissipative
photonic
systems
such
as
polaritons
in
semiconductor
microcavities
or
arrays
of
coupled
optical
resonators
have
recently
attracted
lot
of
interest
due
to
the
possibility
of
observing
quantum
phenomena
which
normally
require
very
low
temperatures
and
or
intense
magnetic
fields
and
are
traditionally
restricted
to
the
domain
of
solid
state
systems
or
ultracold
atomic
gases
Besides
being
highly
tunable
these
optical
systems
facilitate
direct
experimental
access
to
observables
such
as
the
wavefunction
or
energy
spectrum
all
at
room
temperature
In
particular
microcavity
polaritons
have
allowed
the
observation
of
collective
hydrodynamic
phenomena
ranging
from
frictionless
flow
around
small
defect
to
the
formation
of
quantized
vortices
and
dark
solitons
at
the
surface
of
large
impenetrable
obstacles
while
ring
resonator
arrays
coupled
to
artificial
magnetic
fields
have
recently
allowed
engineering
topological
edge
states
robust
to
disorder
Polaritons
Microcavity
polaritons
are
quasiparticles
resulting
from
the
strong
coupling
of
cavity
photons
and
quantum
well
excitons

and
have
the
prerogative
of
being
both
easy
to
manipulate
via
an
external
laser
and
detect
via
the
light
escaping
from
the
cavity
The
finite
polariton
lifetime
establishes
the
system
as
intrinsically
out
of
equilibrium
an
external
pump
is
needed
to
continuously
replenish
the
cavity
of
polaritons
that
quickly
on
scale
of
tens
of
picoseconds
escape
The
pumping
can
be
done
resonantly
close
to
the
polariton
energy
dipersion
or
non
resonantly
Incoherent
pumping
polariton
BEC
The
landmark
observation
of
polariton
condensation
in
as
shown
in
Fig
was
achieved
using
non
resonant
experimental
setup
In
the
experiments
the
system
was
incoherently
excited
by
laser
beam
tuned
at
very
high
energy
Relaxation
of
the
excess
energy

lead
to
population
of
the
cavity
polariton
states
and
above
certain
laser
power
threshold
to
Bose
Einstein
condensation
into
the
lowest
polariton
state
Experimental
observation
of
polariton
Bose
Einstein
condensation
sharp
and
intense
peak
corresponding
to
the
lowest
momentum
state
is
formed
in
the
center
of
the
far
field
emission
top
panels
with
increasing
pump
power
left
to
right
The
corresponding
energy
resolved
emission
bottom
panels
show
that
above
the
condensation
threshold
the
emission
comes
almost
entirely
from
the
lowest
energy
state
situated
at
the
bottom
of
the
polariton
dispersion
From
Ref
Coherent
pumping
OPO
Due
to
their
energy
dispersion
and
strong
nonlinearity
inherited
from
the
excitonic
component
polaritons
resonantly
injected
by
the
external
laser
into
the
pump
state
with
suitable
wavevector
and
energy
can
undergo
coherent
stimulated
scattering
into
two
conjugate
states
called
the
signal
and
the
idler
in
process
known
as
optical
parametric
oscillator
OPO
Since
their
first
realisation

the
interest
in
microcavity
optical
parametric
phenomena
has
involved
several
fields
of
fundamental
and
applicative
research

Top
panels
Simulated
right
panel
and
observed
left
panel
edge
state
propagation
in
ring
resonator
array
The
light
enters
from
one
corner
and
exits
from
the
other
as
signaled
by
the
arrows
Adapted
from
Ref
Bottom
panels
Simulated
edge
state
propagation
in
polariton
system
Intensities
of
the
exciton
left
panel
and
photon
right
panel
fields
obtained
when
pumping
lattice
at
the
center
of
its
lower
edge
From
Ref
Superfluidity
The
superfluid
properties
of
resonantly
pumped
polariton
quantum
fluid
both
in
the
pump
only
configuration
without
parametric
scattering
as
well
as
the
OPO
regime
have
been
actively
investigated
experimentally
as
well
as
theoretically
In
particular
supression
of
scattering
in
the
pump
only
case
was
observed
below
critical
velocity
similar
to
what
has
been
predicted
by
the
Landau
criterion
for
equilibrium
superfluid
condensates
While
in
equilibrium
condensates
different
aspects
of
superfluidity
are
typically
closely
related
this
is
no
longer
true
in
non
equilibrium
context
Independent
of
the
pumping
scheme
the
driving
and
the
polariton
finite
lifetime
force
one
to
reconsider
the
meaning
of
superfluid
behaviour
when
the
spectrum
of
collective
excitations
is
complex
rather
than
real
raising
question
about
the
applicability
of
Landau
criterion
An
additional
complexity
characterises
the
OPO
regime
namely
the
simultaneous
presence
of
three
oscillation
frequencies
and
momenta
for
pump
signal
and
idler
correspondingly
increases
up
to
or
depending
on
the
approximation
used
to
describe
the
system
the
number
of
collective
excitation
branches
Note
that
from
the
experimental
point
of
view
pioneering
experiments
have
observed
ballistic
nonspreading
propagation
of
signal
idler
polariton
wavepackets
in
triggered
OPO
configuration
as
well
as
demonstrating
the
existence
and
metastability
of
vortex
configurations
in
the
signal
and
idler
Topology
in
polaritons
Very
recently
the
polariton
community
started
to
explore
topological
effects
in
polariton
lattices


The
idea
is
to
break
time
reversal
symmetry
by
the
application
of
strong
external
magnetic
field
giving
rise
to
energy
bands
with
nontrivial
topology
In
particular
Ref
proposes
an
exciton
photon
coupling
with
winding
phase
in
momentum
space
giving
rise
to
polaritonic
bands
with
chiral
edge
modes
that
allow
unidirectional
propagation
protected
against
backscattering
The
bottom
panels
of
Fig
show
the
exciton
and
photon
fields
traveling
together
as
unique
polaritonic
counter
clockwise
chiral
edge
mode
In
this
context
dissipation
in
the
form
of
photon
losses
provides
coupling
to
the
outside
continuum
of
modes
and
as
such
can
be
used
to
detect
or
inject
edge
modes
It
would
be
interesting
to
study
the
feasability
of
coupled
micropillars
for
this
type
of
physics
in
view
of
the
recent
experimental
findings
of
linear
graphene
like
dispersion
as
well
as
edge
states
in
honeycomb
micropillar
lattices
On
the
other
hand
topologically
protected
edge
states
have
already
been
experimentaly
observed
in
the
case
of
silicon
based
optical
resonator
arrays
see
top
panels
of
Fig
due
to
recent
advances
in
creating
synthetic
gauge
fields
which
have
opened
new
horizons
for
simulating
topological
phases
of
matter
also
with
neutral
particles
such
as
photons
or
ultracold
atoms

Rather
than
simply
replicating
previous
measurements
experiments
with
synthetic
gauge
fields
allow
for
unprecedented
access
to
properties
such
as
the
eigenstates
or
eigenspectrum
while
the
tunability
and
controllability
of
these
experiments
offer
the
prospect
of
simulating
novel
physics
Contents
of
this
thesis
Thesis
layout
This
manuscript
is
split
in
two
parts
part
is
detailed
review
of
the
basic
concepts
including
both
the
theoretical
formalism
as
well
as
the
relevant
experiments
necessary
for
understanding
part
II
which
presents
the
three
main
works
published
as
part
of
my
PhD
Chapters
and
The
two
systems
chosen
for
ilustrating
the
basic
physical
concepts
are
ultracold
atomic
gases
Chapter
and
microcavity
exciton
polaritons
Chapter
We
now
give
brief
description
of
the
content
of
each
of
the
chapters
In
Chapter
we
describe
the
phenomenon
of
Bose
Einstein
condensation
and
introduce
the
Gross
Pitaevskii
equation
which
is
extended
in
Chapter
to
the
case
of
polariton
condensates
We
then
review
the
linear
response
of
moving
atomic
condensate
to
weak
stationary
defect
leading
to
discussion
on
superfluidity
This
scattering
problem
is
the
equilibrium
conterpart
of
the
problems
studied
in
Chapters
and
in
the
context
of
polaritons
in
the
resonantly
pumped
and
optical
parametric
oscillator
regimes
respectively
Finally
we
introduce
the
Harper
Hofstadter
model
which
is
the
backbone
of
Chapter
In
Chapter
we
introduce
microcavity
exciton
polaritons
and
their
theoretical
description
in
terms
of
driven
dissipative
Gross
Pitaevskii
equation
We
discuss
both
the
resonantly
pumped
system
which
is
used
in
Chapter
as
well
as
the
optical
parametric
oscillator
of
Chapter
and
we
make
the
connection
to
the
superfluid
related
phenomena
previously
introduced
in
Chapter
In
Chapter
we
study
the
scattering
of
resonantly
pumped
polariton
condensate
against
static
defect
of
the
microcavity
for
the
simplified
case
of
small
fluid
velocities
when
the
polariton
dispersion
can
be
considered
quadratic
After
discussing
the
effects
of
dissipation
on
the
Bogoliubov
excitation
spectra
we
find
that
the
finite
polariton
lifetime
also
affects
the
drag
exerted
by
the
condensate
on
the
defect
In
particular
we
show
that
there
is
nonzero
drag
force
entirely
due
to
the
out
of
equilibrium
nature
of
the
system
even
in
the
"superfluid
regime
"Finally
we
characterise
the
behaviour
of
the
drag
as
function
of
the
condensate
velocity
and
polariton
lifetime
In
Chapter
we
present
theoretical
and
experimental
study
of
the
same
scattering
problem
now
in
the
context
of
an
optical
parametric
oscillator
consisting
of
three
coupled
condensates
the
pump
the
signal
and
the
idler
Apart
from
using
the
linear
response
analysis
first
introduced
in
Chapter
and
then
extended
to
the
case
of
pump
only
condensate
in
Chapter
we
also
numerically
solve
the
full
driven
dissipative
Gross
Pitaevskii
equation
We
show
that
while
the
modulation
patterns
are
present
in
all
three
condensates
their
relative
amplitudes
depend
on
various
factors
In
particular
for
the
typical
experimental
conditions
that
favour
parametric
scattering
the
signal
has
undetectable
modulations
while
the
pump
and
idler
show
the
same
response
In
Chapter
we
add
harmonic
trap
to
the
Harper
Hofstadter
lattice
model
of
Chapter
We
discuss
how
the
eigenstates
of
the
new
Hamiltonian
can
be
seen
as
momentum
space
Landau
levels
by
making
use
of
the
nontrivial
topology
of
the
Harper
Hofstadter
energy
bands
We
then
extend
the
model
to
include
driving
and
dissipation
and
finally
present
proposal
for
the
physical
implementation
of
the
model
in
state
of
the
art
driven
dissipative
photonic
systems
The
source
code
belonging
to
Chapters
and
is
available
online
Condensed
matter
systems
Ultracold
atomic
gases
Laser
cooling
allowed
achieving
temperatures
in
the
micro
Kelvin
regime
and
eventually
led
to
the
realization
of
optical
lattices
It
also
paved
the
way
for
more
powerful
cooling
techniques
such
as
evaporative
cooling
which
made
possible
the
Bose
Einstein
condensation
of
dilute
atomic
gases
Ultracold
gases
have
been
at
the
forefront
of
simulating
quantum
phenomena
with
analogs
throughout
physics
from
nonlinear
optics
to
condensed
matter
systems
In
particular
their
link
to
quantum
simulation
of
condensed
matter
phenomena
becomes
obvious
when
adding
optical
lattice
potentials
combined
with
synthetic
magnetic
fields
In
this
Chapter
we
review
the
physics
of
ultracold
atomic
gases
with
an
emphasis
on
their
link
to
hydrodynamic
effects
such
as
superfluidity
as
well
as
their
connection
to
traditional
solid
state
lattice
models
such
as
the
celebrated
Harper
Hofstadter
model
which
originally
describes
the
single
particle
physics
of
band
electrons
in
intense
magnetic
fields
Chapter
organization
This
Chapter
is
organized
as
follows
in
Section
we
present
short
history
of
Bose
Einstein
condensation
describe
its
main
features
and
state
its
formal
definition
in
terms
of
the
Penrose
Onsager
criterion
leading
to
the
weakly
interacting
Bose
gas
paradigm
and
the
Gross
Pitaevskii
equation
Sec
an
essential
theoretical
tool
for
the
mean
field
description
of
atomic
condensates
In
Section
we
introduce
the
linear
response
formalism
which
proves
useful
for
interpreting
experiments
where
weak
perturbation
is
applied
to
the
condensate
We
employ
this
formalism
in
order
to
study
scattering
problem
concerning
the
flow
of
BEC
in
the
presence
of
weak
static
defect
Sec
In
this
context
we
review
Bogoliubov's
excitation
spectrum
and
its
associated
Landau
criterion
for
superfluidity
followed
by
detailed
discussion
on
superfluidity
and
related
phenomena
such
as
quantized
vortices
in
Sec
We
briefly
touch
on
the
subject
of
synthetic
gauge
fields
for
neutral
atoms
before
investigating
the
properties
of
BECs
in
periodic
potentials
created
by
optical
lattices
Sec
Finally
we
combine
the
concepts
of
synthetic
gauge
fields
and
optical
lattices
in
Section
where
we
show
the
main
features
of
the
Harper
Hofstadter
model
which
was
recently
realized
using
atomic
gases
Bose
Einstein
condensation
BEC
in
noninteracting
gas
In
Albert
Einstein
prompted
by
the
earlier
work
of
the
indian
polyglot
Satyendra
Nath
Bose
considered
what
would
happen
to
non
interacting
bosonic
gas
of
non
relativistic
particles
in
the
thermodynamic
limit
as
one
lowers
the
temperature
He
predicted
the
phenomenon
we
now
call
Bose
Einstein
condensation
BEC
namely
phase
transition
to
new
state
of
matter
in
which
finite
fraction
of
all
the
particles
would
occupy
the
same
single
particle
state
The
transition
occurs
at
fixed
density
below
critical
temperature
or
alternatively
at
fixed
temperature
above
critical
density
In
particular
if
we
take
neutral
particles
in
cubic
box
of
volume
then
they
would
predominantly
occupy
the
zero
wavevector
state
and
the
critical
temperature
would
be
with
the
density
and
being
the
particle
mass
and
Boltzmann's
constant
respectively
At
its
core
BEC
is
paradigm
of
quantum
statistical
mechanics
stemming
from
the
indistinguishability
of
elementary
particles
and
the
Bose
Einstein
statistics
that
they
obey
One
can
hand
wavingly
deduce
the
critical
temperature
or
critical
density
where
quantum
degeneracy
would
start
playing
role
in
many
body
system
by
arguing
that
the
thermal
de
Broglie
wavelength
should
be
comparable
to
or
greater
than
the
inter
particle
distance
which
in
our
case
is
on
average
Apart
from
the
numerical
prefactor
we
get
the
same
answer
as
Eq
eq
Tc
While
one
may
argue
that
elementary
massive
bosons
do
not
exist
it
is
worth
emphasizing
that
indistinguishability
only
plays
role
when
there
is
finite
probability
for
exchange
processes
to
occur
between
the
particles
In
that
sense
all
odd
isotope
alkali
atoms
under
relevant
experimental
conditions
see
below
effectively
behave
as
bosons
their
many
body
wavefunction
is
symmetric
under
the
exchange
of
any
two
such
atoms
BEC
in
interacting
system
Interestingly
enough
BEC
was
considered
by
many
at
the
time
to
be
pathological
behaviour
of
the
non
interacting
gas
which
would
resolve
once
interactions
were
properly
accounted
for
In
fact
it
is
well
known
that
the
ideal
Bose
gas
has
infinite
compresibility
This
pathology
is
cured
by
introducing
weak
repulsive
interaction
between
bosons
regime
where
BEC
survives
as
we
will
see
next
One
body
density
matrix
Following
Leggett
we
characterise
each
of
the
particles
assumed
spinless
for
simplicity
by
position
vector
with
the
label
running
from
to
Any
pure
state
of
the
now
interacting
system
-which
can
be
also
subjected
to
an
external
potential
-can
be
described
at
time
by
the
many
body
wavefunction
Therefore
the
most
general
state
of
the
system
also
called
mixed
state
can
be
written
as
superposition
of
pure
orthonormal
states
with
different
weights
The
single
particle
density
matrix
represents
the
probability
amplitude
at
time
of
finding
specific
particle
at
position
multiplied
by
the
amplitude
of
finding
it
at
and
averaged
over
the
positions
of
all
the
other
particles
where
in
the
second
line
we
have
re
written
the
density
matrix
in
diagonal
form
introducing
its
eigenvalues
and
eigenvectors
which
form
complete
orthonormal
set
at
any
time
here
labels
good
quantum
number
of
the
problem
momentum
in
translationally
symmetric
situation
Penrose
Onsager
criterion
for
BEC
We
are
now
ready
to
state
the
Penrose
Onsager
criterion
for
condensation
first
formulated
in
if
at
any
given
time
it
is
possible
to
find
complete
orthonormal
basis
of
states
of
such
that
one
and
only
one
of
these
states
has
an
eigenvalue
of
order
the
rest
being
of
order
then
we
say
the
system
exhibits
BEC
One
should
note
that
this
definition
only
applies
to
"simple
"BEC
as
opposed
to
the
"fragmented
"case
of
no
concern
to
us
here
where
two
or
more
of
the
eigenvalues
of
the
one
body
density
matrix
are
of
order
Condensate
wavefunction
and
the
order
parameter
We
denote
the
single
macroscopic
eigenvalue
of
the
density
matrix
by
and
its
corresponding
eigenfunction
by
is
called
the
condensate
wavefunction
and
the
particles
occupying
it
the
condensate
while
the
ratio
is
the
condensate
fraction
It
is
not
necessarily
true
that
even
at
zero
temperature
Also
note
that
while
behaves
as
single
particle
Schr
dinger
wavefunction
it
is
generally
not
an
eigenfunction
of
the
single
particle
part
of
the
Hamiltonian
or
of
any
other
simple
operator
for
that
matter
other
than
Another
useful
quantity
frequently
found
in
the
literature
is
the
so
called
order
parameter
We
see
that
while
is
normalized
to
will
be
normalized
to
No
go
theorem
for
lower
dimensionality
It
is
worth
mentioning
the
existence
of
theorem
due
to
Hohenberg
stating
that
in
the
thermodynamic
limit
BEC
cannot
occur
at
finite
temperature
in
any
system
moving
freely
in
space
in
less
than
three
dimensions
irespective
of
the
existence
and
or
sign
of
the
interparticle
interactions
as
thermal
fluctuations
would
destroy
the
condensate
Note
that
this
theorem
however
only
applies
under
equilibrium
conditions
the
nonequilibrium
case
still
being
an
open
question
Furthermore
there
is
no
general
proof
that
realistic
system
of
interacting
bosonic
particles
must
show
BEC
even
at
zero
temperature
-the
solid
phase
of
He
constitutes
an
obvious
counter
example
Experimental
proof
Most
gases
with
the
notable
exception
of
He
are
solids
at
the
densities
and
temperatures
predicted
by
Eq
eq
Tc
That
is
why
it
took
no
less
than
years
between
Einstein's
original
paper
and
the
first
experimental
observation
of
BEC
in
an
atomic
gas
In
the
group
of
Eric
Cornell
and
Carl
Wieman
succesfully
condensed
cloud
of
Rb
atoms
closely
followed
by
the
group
of
Wolfgang
Ketterle
at
MIT
with
Na
atoms
by
first
bringing
the
system
to
very
low
density
and
then
cooling
it
fast
enough
to
prevent
any
recombination
processes
that
would
have
lead
to
the
formation
of
the
solid
phase
While
other
odd
isotope
alkali
elements
especially
Na
or
Li
are
also
routinely
used
in
experiments
the
first
non
alkali
atom
to
be
cooled
into
the
BEC
phase
was
hydrogen
Due
to
the
extreme
diluteness
of
these
systems
atoms
cm
the
typical
range
of
is
from
nK
to
few
Achieving
such
ultra
low
temperatures
stimulated
the
development
of
novel
experimental
techniques
such
as
magnetic
laser
trapping
and
evaporative
cooling
of
atoms
Diagnostic
techniques
The
very
low
densities
of
alkali
gases
also
limit
the
range
of
available
diagnostic
techniques
The
most
commonly
employed
method
in
BEC
experiments
is
optical
absorption
imaging
where
one
shines
laser
on
the
gas
and
detects
the
percentage
of
transmitted
power
The
image
is
usually
taken
after
removing
the
trap
and
allowing
the
gas
to
expand
This
gives
information
about
the
gas
density
as
function
of
coordinates
and
time
with
spatial
resolution
of
few
In
stark
contrast
to
liquid
He
density
related
information
seems
to
be
sufficient
for
most
practical
purposes
Diluteness
weak
interaction
condition
Gross
Pitaevskii
equation
Short
history
and
utility
of
GP
equation
As
it
turns
out
many
of
the
experimental
results
in
ultracold
gases
can
be
interpreted
on
the
basis
of
single
equation
for
the
condensate
wavefunction
This
equation
first
derived
in
independently
by
Eugene
Gross
and
Lev
Pitaevskii
was
originally
intended
as
phenomenological
description
of
quantum
vortices
in
the
superfluid
phase
of
liquid
He
below
the
lambda
point
Since
liquid
helium
is
strongly
interacting
system
however
the
GP
equation
turned
out
to
be
much
better
suited
to
alkali
gases
Before
giving
the
concrete
formulation
of
the
GP
equation
we
must
first
explore
the
nature
of
the
inter
atomic
interactions
wave
scattering
length
In
dilute
systems
the
inter
atomic
distance
is
on
the
order
of
while
the
range
of
the
inter
atomic
potential
namely
the
extent
of
the
last
bound
state
of
the
van
der
Waals
interaction
is
about
As
the
probability
of
three
atom
colissions
is
substantially
diminished
This
justifies
limiting
ourselves
to
binary
instead
of
three
body
or
more
scattering
problem
consider
two
atoms
separated
by
relative
distance
and
interacting
in
three
dimensions
through
potential
We
can
therefore
decouple
their
center
of
mass
motion
from
their
relative
one
and
write
Schr
dinger
equation
for
the
scattering
states
For
temperatures
below
the
thermal
de
Broglie
wavelength
as
mentioned
in
Sec
meaning
all
significantly
occupied
states
will
have
small
wavevectors
This
directly
translates
to
low
relative
kinetic
energy
and
hence
small
relative
wavevectors
for
the
scattering
problem
outlined
above
However
we
know
from
scattering
theory
that
the
probability
for
two
atoms
with
relative
angular
momentum
of
being
separated
by
distance
is
proportional
to
therefore
essentially
negligible
in
the
limit
That
is
of
course
unless
meaning
their
relative
state
is
wave
which
is
what
we
will
assume
from
now
on
Since
one
can
use
the
asymptotic
expression
for
which
only
depends
on
the
scattering
amplitude
At
small
wavevectors
this
amplitude
can
be
safely
replaced
by
the
wave
scattering
length
which
will
encapsulate
all
the
interaction
effects
on
macroscopic
properties
of
the
atomic
gas
One
can
now
replace
the
two
body
potential
with
an
effective
interaction
provided
it
gives
the
same
scattering
length
The
limit
of
small
wavevectors
prompts
us
to
only
consider
the
lowest
Fourier
component
of
equivalent
in
real
space
to
contact
interaction
(Technically
one
should
also
include
regularizing
part
in
order
to
remove
any
divergencies
of
the
wavefunction
)where
we
have
introduced
the
interaction
coupling
constant
whose
value
can
be
calculated
using
the
first
order
Born
approximation
The
scattering
length
therefore
becomes
the
small
parameter
of
the
theory
of
weakly
interacting
ultracold
gases
and
the
validity
of
the
Born
approximation
rests
on
the
following
two
conditions
Eq
eq
diluteness
condition
is
called
the
"diluteness
condition
"and
it
paves
the
way
to
various
mean
field
approaches
such
as
the
GP
equation
Derivation
of
Gross
Pitaevskii
equation
Formally
the
GP
equation
corresponds
to
the
lowest
order
expansion
in
of
the
more
exact
Bogoliubov
theory
However
we
will
try
to
give
hand
waving
justification
of
it
for
the
zero
temperature
case
At
all
particles
are
in
the
condensate
therefore
one
could
neglect
all
inter
particle
correlations
and
introduce
the
simplest
Hartree
Fock
ansatz
expressing
the
ground
state
many
body
wavefunction
in
the
symmetrized
form
As
mentioned
in
Sec
the
single
particle
state
now
occupied
by
all
the
bosons
obeys
Schr
dinger
like
equation
to
which
we
must
add
the
energy
of
the
effective
binary
interactions
In
mean
field
these
interactions
contribute
the
equivalent
of
one
particle
potential
term
proportional
to
Together
with
the
kinetic
part
this
results
in
the
nonlinear
equation
(We
have
tacitly
assumed
that
is
large
enough
such
that
)where
we
have
also
included
an
external
potential
normally
used
to
model
harmonic
trapping
of
the
gas
Note
that
Eq
eq
TDGP
is
valid
for
physics
occuring
over
distances
much
larger
than
the
scattering
length
which
in
turn
must
be
smaller
than
the
typical
range
of
the
potential
Caveats
of
TDGP
Eq
eq
TDGP
is
the
time
dependent
Gross
Pitaevskii
TDGP
equation
mean
field
result
where
the
condensate
wavefunction
must
be
calculated
self
consistently
It
is
important
to
emphasize
that
the
TDGP
equation
is
also
valid
at
nonzero
temperatures
provided
that
the
density
of
non
condensed
particles
is
much
smaller
than
the
condensate
density
In
that
case
the
condensate
number
is
smaller
but
still
on
the
order
of
the
total
particle
number
Finally
one
must
note
that
the
nonlinearity
of
Eq
eq
TDGP
builds
bridge
connecting
BEC
to
nonlinear
optics
where
similar
relation
is
used
under
the
name
of
nonlinear
Schr
dinger
equation
Time
independent
GP
In
case
the
external
potential
does
not
depend
explicitly
on
time
the
stationary
solutions
of
Eq
eq
TDGP
evolve
with
trivial
phase
factor
This
yields
time
independent
GP
equation
for
we
set
from
here
on
where
is
chemical
potential
of
the
gas
the
energy
required
to
add
one
more
particle
to
the
system
(Technically
it
is
the
Lagrange
multiplier
associated
to
the
conservation
of
particle
number
and
can
be
shown
to
be
very
close
to
the
actual
chemical
potential
in
the
thermodynamic
limit
)Linear
response
theory
Following
loosely
the
formalism
presented
in
Ref
we
now
let
be
the
steady
state
solution
to
the
GP
equation
in
the
time
independent
trapping
potential
with
the
GP
Hamiltonian
defined
as
This
Hamiltonian
describes
bosonic
condensate
of
particles
with
contact
interactions
quantified
by
and
chemical
potential
Now
consider
adding
small
time
dependent
perturbation
on
top
of
the
trap
giving
We
are
interested
in
the
response
of
the
condensate
to
this
perturbation
For
weak
perturbations
we
can
perform
linearization
of
the
GP
equation
Eq
eq
GP
atoms
around
the
stationary
solution
-an
approach
known
in
the
literature
as
the
"linear
response
"formalism
The
condensate
wavefunction
evolves
according
to
We
assume
small
deviation
of
the
wavefunction
from
its
initial
steady
state
such
that
we
can
expand
Eq
eq
GP
atoms
evolution
and
keep
only
linear
terms
in
and
We
get
Note
that
Eq
eq
GP
atoms
lin
is
not
strictly
linear
due
to
the
coupling
of
to
To
restore
linearity
we
consider
the
functions
and
as
being
independent
and
write
the
linear
system
where
we
have
introduced
the
linear
operator
and
the
source
term
Note
that
is
non
Hermitian
operator
We
now
consider
the
eigenvalue
equation
for
the
operator
with
being
the
right
eigenvector
and
its
corresponding
eigenvalue
Similarly
we
also
introduce
the
left
eigenvector
obeying
and
the
orthonormality
condition
Notice
that
and
are
connected
by
the
unitary
transformation
(Note
that
this
holds
as
long
as
the
Hamiltonian
only
contains
real
terms
)where
is
the
third
Pauli
matrix
We
say
that
is
Hermitian
meaning
that
one
can
define
new
scalar
product
with
different
signature
such
that
The
operator
is
usually
called
the
metric
operator
and
not
suprisingly
in
our
case
it
is
the
same
as
the
one
of
the
scalar
Klein
Gordon
equation
pseudo
Hermitian
operator
usually
also
posesses
antilinear
symmetries
and
as
we
will
see
below
this
is
also
the
case
for
Interestingly
for
operators
with
real
spectrum
it
can
be
shown
that
one
can
define
another
metric
which
guarantees
positive
definite
inner
product
or
in
other
words
provided
of
course
This
can
be
used
to
formulate
probabilistic
quantum
theory
for
the
new
wave
functions
and
For
the
general
theory
and
properties
of
pseudo
Hermitian
operators
we
point
the
interested
reader
to
Ref
Using
Eq
eq
symmetry
we
get
the
general
form
of
the
left
eigen
vectors
as
with
normalization
factor
We
can
chose
and
group
the
eigenvalues
of
into
families
according
to
the
quantity
We
therefore
have
the
""family
corresponding
to
the
""family
such
that
and
the
""family
with
We
are
now
ready
to
write
the
completeness
relation
Using
Eq
eq
completeness
we
can
decompose
any
column
vector
as
(The
modes
in
the
""family
do
not
appear
in
this
expansion
as
their
components
live
in
the
space
orthogonal
to
the
one
of
our
solution
)"family
"family
There
is
now
further
symmetry
of
that
we
can
exploit
in
our
problem
sort
of
time
reversal
"spin
"flip
symmetry
namely
where
with
the
first
Pauli
matrix
and
the
complex
conjugation
antilinear
operator
This
results
in
duality
between
the
""family
with
eigenvectors
and
energy
and
the
""family
with
eigenvectors
and
energy
We
can
now
finally
project
Eq
eq
GP
atoms
system
onto
the
eigenvectors
of
Using
the
above
mentioned
duality
and
Eq
eq
decomposition
we
get
with
the
complex
amplitudes
satisfying
where
we
introduced
Cherenkov
emission
of
Bogoliubov
excitations
We
now
turn
to
applying
the
formalism
developed
in
Sec
to
concrete
physical
example
namely
flowing
condensate
scattering
against
static
defect
The
BEC
(We
integrate
all
density
profiles
along
the
direction
resulting
in
an
effective
dimensional
description
)is
therefore
in
state
with
well
defined
momentum
described
by
the
plane
wave
and
chemical
potential
Since
we
have
no
trap
and
Eq
eq
GP
atoms
produces
the
equation
of
state
where
we
have
introduced
the
condensate
density
We
now
introduce
weak
perturbation
in
the
form
of
static
localized
defect
potential
which
can
represent
for
instance
laser
spot
depleting
small
area
of
the
condensate
as
shown
in
Fig
Density
profiles
of
an
expanding
BEC
hitting
stationary
defect
created
by
the
repulsive
potential
of
blue
detuned
laser
beam
The
condensate
has
different
speeds
in
the
two
panels
moving
roughly
twice
as
fast
in
the
right
panel
Notice
the
Mach
cone
formed
behind
the
defect
which
gets
narrower
as
the
condesate
moves
faster
From
Ref
Top
panels
Bogoliubov
dispersion
Eq
eq
bogoliubov
The
dotted
lines
indicate
the
plane
Middle
panels
Locus
of
intersection
of
the
dispersion
with
the
plane
Green
arrows
are
normal
to
while
the
dashed
lines
indicate
the
Cherenkov
cone
Bottom
panels
Real
space
density
modulation
with
-defect
at
Dashed
lines
show
the
Mach
cone
Left
column
panels
are
for
and
right
column
for
From
Ref
Using
Eq
eq
atom
MF
the
GP
Hamiltonian
becomes
and
the
source
term
We
now
get
the
linear
operator
for
our
problem
in
the
form
Notice
that
due
to
the
presence
of
the
off
diagonal
exponential
terms
does
not
commute
with
the
momentum
operator
which
is
the
generator
of
the
spatial
translation
group
Luckily
however
we
can
restore
translational
invariance
by
simple
unitary
transformation
as
shown
below
Using
the
standard
commutation
relations
one
can
show
that
for
constant
wavevector
the
unitary
operator
(The
hat
symbol
denotes
operators
in
the
relevant
Hilbert
space
)performs
translation
in
momentum
space
with
the
ket
representing
single
particle
state
with
wavevector
such
that
Using
the
definitions
above
one
can
easily
obtain
the
commutator
This
allows
us
to
rewrite
the
following
expressions
We
now
recognize
the
two
exponentials
in
Eq
eq
ourL
as
being
the
real
space
representation
of
and
its
hermitian
conjugate
This
motivates
us
to
define
the
following
unitary
operator
such
that
unitary
transformation
of
our
operator
now
restores
translational
symmetry
Indeed
one
can
see
that
where
we
have
made
use
of
Eqs
eq
products
and
we
have
written
in
base
independent
representation
In
the
subspace
of
momentum
eigenstates
we
can
write
the
right
eigenvalue
equation
corresponding
to
Eq
eq
translated
as
where
we
have
recovered
the
matrix
representation
of
Eq
eq
LGP
and
introduced
the
notation
Here
labels
the
different
eigenmodes
and
we
defined
the
condensate
speed
Notice
that
the
mode
has
only
one
eigenvector
However
one
can
safely
exclude
it
as
this
mode
does
not
imply
energy
or
momentum
transport
Excluding
the
point
one
can
then
solve
Eq
eq
right
eigen
obtaining
the
celebrated
Bogoliubov
excitation
spectrum
with
as
before
Here
represents
the
momentum
of
the
quasiparticle
excitation
with
respect
to
the
momentum
of
the
condensate
Note
that
the
complex
amplitudes
and
only
depend
on
the
absolute
value
of
while
the
real
spectrum
of
Eq
eq
translated
is
the
Bogoliubov
spectrum
with
an
additional
Galilean
boost
We
can
now
further
simplify
the
problem
As
can
be
seen
from
Eq
eq
symmetry
the
eigen
families
and
are
linked
by
duality
stemming
from
the
symmetry
(This
can
be
actually
formally
proven
after
defining
the
parity
and
time
reversal
operators
corresponding
to
our
problem
For
details
see
Ref
)of
the
Bogoliubov
operator
We
therefore
drop
the
subscript
and
make
the
convention
that
Bogoliubov
spectrum
discussion
The
Bogoliubov
spectrum
Eq
eq
bogoliubov
is
shown
in
the
top
row
of
Fig
for
two
different
values
of
which
sets
the
slope
of
the
dotted
lines
indicating
the
plane
In
the
non
interacting
case
and
the
spectrum
reduces
to
simple
parabola
characterising
free
particle
For
repulsive
interactions
and
we
can
distinguish
two
qualitatively
different
domains
after
first
introducing
the
typical
length
scale
of
the
problem
called
the
healing
length
The
healing
length
is
measure
of
the
distance
over
which
the
condensate
density
recovers
its
equilibrium
value
when
forced
to
vary
away
from
this
value
For
example
in
box
the
boundary
conditions
fix
the
density
to
zero
at
the
positions
of
the
walls
In
mathematical
terms
We
first
explore
the
domain
of
small
momenta
which
is
characterised
by
linear
behaviour
of
the
dispersion
that
implies
the
propagation
of
low
energy
excitations
in
the
form
of
sound
waves
with
velocity
(The
sound
velocity
here
is
measured
in
the
condensate
rest
frame
)given
by
The
Landau
criterion
for
superfluidity
determines
the
maximum
velocity
at
which
weak
impurity
can
travel
through
the
condensate
without
dissipating
energy
In
order
for
it
to
dissipate
energy
such
an
impurity
must
be
able
to
create
quasiparticle
excitations
in
the
condensate
Conservation
of
energy
and
momentum
then
results
in
critical
velocity
below
which
no
dissipation
can
occur
In
our
case
this
velocity
is
precisely
equal
to
the
speed
of
sound
Furthermore
the
two
situations
the
one
of
particle
moving
through
the
condensate
or
of
the
condensate
moving
against
fixed
defect
are
physically
equivalent
being
connected
by
Galilean
transformation
We
can
therefore
conclude
that
we
must
have
in
order
to
observe
any
propagating
perturbation
otherwise
for
the
superfluid
will
remain
unperturbed
Before
moving
on
it
is
worth
noting
that
the
Landau
criterion
has
some
asociated
caveats
One
is
assuming
that
the
only
excitations
are
density
excitations
phonons
Thus
one
is
for
example
neglecting
the
nucleation
of
vortices
by
macroscopic
defect
with
size
comparable
to
the
healing
length
which
would
lower
the
effective
critical
velocity
Vortices
would
furthermore
also
break
the
translational
invariance
along
the
transverse
directions
an
invariance
that
we
already
made
use
of
The
second
caveat
is
that
quantum
fluctuations
are
also
neglected
As
seen
in
Ref
they
could
lead
to
nonzero
dissipation
even
at
sub
sonic
speeds
The
second
domain
of
interest
is
the
one
of
large
momenta
Looking
at
Eq
eq
right
eigen
one
notices
that
the
only
dependent
parts
of
are
the
diagonal
terms
These
terms
have
opposite
sign
so
the
off
diagonal
coupling
between
and
becomes
highly
off
resonant
at
large
Completely
neglecting
it
gives
the
free
particle
like
shifted
parabola
with
and
The
discontinuity
in
the
spectrum
at
zero
momentum
can
be
explained
by
the
fact
that
the
diagonal
and
off
diagonal
parts
of
are
equal
in
absolute
value
In
order
to
obtain
the
defect
induced
density
perturbation
we
must
now
also
determine
the
eigenvectors
of
the
problem
Making
use
of
Eq
eq
symmetry
we
can
act
with
on
the
eigenstates
of
to
obtain
the
ones
of
This
finally
leads
us
to
biorthonormal
basis
containing
basis
vectors
which
fulfill
the
orthonormality
condition
and
the
completeness
relation
provided
of
course
that
we
normalize
in
such
way
that
In
this
basis
the
spectral
decomposition
of
Eq
eq
translated
is
the
diagonal
form
The
concrete
form
of
coupled
with
the
normalization
condition
Eq
eq
norms
determines
the
eigenvectors
of
the
""family
up
to
phase
factor
Indeed
one
can
choose
with
Furthermore
in
case
the
Hamiltonian
doesn't
contain
any
time
reversal
symmetry
breaking
terms
one
can
chose
and
to
be
real
quantities
without
loss
of
generality
It
is
now
straightforward
to
solve
the
linearized
evolution
equation
Eq
eq
GP
atoms
system
In
particular
for
localized
static
defect
potential
quasiparticle
modes
at
all
wavevectors
are
excited
(If
one
wishes
to
selectively
excite
pair
of
modes
one
can
use
periodic
potential
following
for
example
Ref
)The
source
term
is
time
independent
and
hence
the
quasi
particle
amplitudes
of
Eq
eq
amplitudes
bk
have
the
simple
form
Equivalently
we
can
obtain
the
defect
induced
perturbation
of
the
wavefunction
from
its
initial
steady
state
by
directly
inverting
Eq
eq
bidecomposition
and
then
reversing
the
unitary
transformation
that
was
applied
to
obtain
Eq
eq
translated
Whichever
route
we
take
it
is
clear
that
the
final
answer
will
have
resonant
structure
containing
in
the
denominator
hence
the
dominant
modes
will
be
the
ones
that
satisfy
Landau
causality
rule
We
must
also
mention
the
subject
of
adiabatic
switching
at
this
point
Adiabatic
switching
is
neccesary
to
causally
distinguish
the
past
from
the
future
making
sure
that
the
time
is
prior
to
that
of
the
occurence
of
any
cause
in
our
case
the
defect
giving
rise
to
the
effect
pertubation
of
the
condensate
density
The
way
to
achieve
this
in
practice
is
by
shifting
the
real
poles
of
the
Bogoliubov
dispersion
Eq
eq
bogoliubov
into
the
lower
half
of
the
complex
plane
by
an
infinitesimal
amount
This
corresponds
to
weak
damping
of
the
plane
wave
solution
and
ensures
that
no
Bogoliubov
excitations
were
present
at
Geometrical
construction
We
show
the
defect
induced
perturbation
of
the
condensate
density
in
the
bottom
row
of
Fig
for
two
distinct
values
of
the
condensate
speed
Rather
than
giving
its
full
analytical
expression
it
is
more
instructive
to
present
geometrical
construction
detailed
in
Ref
that
can
shed
light
on
the
main
features
of
the
condensate
response
We
have
already
identified
the
importance
of
the
poles
of
the
solutions
of
can
be
visualized
if
one
plots
the
intersection
of
the
Bogoliubov
dispersion
surface
with
the
plane
The
locus
of
this
intersection
is
closed
curve
that
we
will
denote
by
and
that
is
plotted
in
the
middle
row
of
Fig
Making
the
connection
with
the
Landau
criterion
presented
earlier
we
can
identify
two
regimes
For
small
velocities
we
have
the
sub
sonic
regime
where
is
single
point
at
the
origin
Consequently
one
can
observe
superfluid
like
behaviour
with
no
propagating
density
modulation
The
modulation
will
stay
localized
in
the
vecinity
of
the
defect
and
in
the
Galilean
equivalent
problem
of
particle
moving
through
the
condensate
it
would
renormalize
the
particle
mass
As
we
gradually
increases
the
condensate
speed
and
hence
the
slope
of
the
plane
this
plane
will
touch
the
surface
of
the
dispersion
relation
when
marking
the
entry
into
the
super
sonic
dissipative
regime
Further
increasing
the
speed
will
increase
the
size
of
as
can
be
seen
in
Fig
The
green
arrows
orthogonal
to
the
curve
at
each
point
represent
the
group
velocity
of
the
Bogoliubov
mode
at
that
particular
value
They
show
the
direction
of
propagation
of
the
density
perturbation
away
from
the
defect
up
to
infinity
It
is
the
interference
of
these
propagating
modes
that
we
see
as
the
real
space
density
pattern
Note
that
the
locus
has
two
distinct
regions
inherited
from
the
linear
and
quadratic
domains
of
the
dispersion
Eq
eq
bogoliubov
The
linear
region
of
close
to
the
origin
is
characterised
by
essentially
the
same
physics
as
the
Cherenkov
effect
in
non
dispersive
media
The
Charenkov
effect
consists
in
the
emission
of
electromagnetic
radiation
by
charged
particle
moving
relativistically
through
dielectric
medium
at
velocity
higher
than
the
phase
velocity
of
light
in
that
medium
The
emission
is
concentrated
into
Cherenkov
cone
in
momentum
space
of
aperture
where
The
higher
the
particle
speed
with
respect
to
the
speed
of
light
the
wider
the
angle
between
the
emission
and
the
direction
of
motion
The
Cherenkov
cone
is
depicted
by
the
dashed
lines
in
the
middle
panels
of
Fig
and
the
angle
in
our
case
is
of
course
given
by
Due
to
the
momentum
space
singularity
at
the
origin
the
group
velocity
has
jump
defining
whole
region
of
space
where
no
phonons
are
emitted
This
corresponds
in
real
space
to
the
so
called
Mach
cone
named
in
analogy
to
the
cone
that
is
created
by
super
sonic
aircraft
since
we
are
dealing
with
terminology
it
is
worth
noting
that
the
ratio
goes
by
the
name
of
Mach
number
The
Mach
cone
is
depicted
by
dashed
lines
in
the
bottom
panels
of
Fig
as
its
aperture
is
quantified
by
that
means
that
the
faster
the
fluid
the
narrower
the
cone
will
be
The
high
momentum
rounded
region
of
which
corresponds
to
the
quadratic
single
particle
like
dispersion
has
no
equivalent
in
Cherenkov
physics
It
is
responsible
for
the
hyperbolic
like
wavefronts
(The
wavefronts
are
actually
parabolic
in
the
non
interacting
limit
of
)emitted
in
the
positive
direction
upstream
The
physical
origin
of
these
rounded
waves
lies
in
the
interference
between
the
coherent
matter
wave
of
the
BEC
and
the
wave
scattered
off
the
defect
It
it
worth
noting
that
the
curve
also
helps
one
determine
the
spacing
between
the
emitted
wavefronts
which
is
inversely
proportional
to
the
value
of
the
momentum
at
the
particular
point
on
that
corresponds
to
the
wave
propagation
direction
In
practice
we
see
for
example
that
the
spacing
along
in
the
positive
direction
is
wider
in
the
left
bottom
panel
than
in
the
right
one
Aside
from
being
just
useful
geometrical
construction
the
locus
can
actually
be
observed
in
scattering
experiments
both
in
the
context
of
atomic
condensates
as
well
as
for
microcavity
polaritons
Chapter
where
it
takes
the
name
of
Rayleigh
scattering
ring
Before
closing
this
Section
it
is
worth
making
the
connection
between
the
Cherenkov
waves
shown
in
the
bottom
panels
of
Fig
and
the
more
mundane
example
of
surface
waves
created
at
the
interface
between
fluid
layer
and
gas
We
first
need
to
define
the
capillary
length
with
the
surface
tension
of
the
fluid
gas
interface
the
gravitational
constant
and
the
fluid
density
Now
if
the
height
of
the
fluid
layer
in
question
is
smaller
than
the
fluid
dispersion
is
dominated
by
capillary
effects
and
its
form
is
similar
to
Eq
eq
bogoliubov
see
Ref
But
we
have
just
seen
that
the
form
of
the
dispersion
more
precisely
its
intersection
with
the
plane
determines
the
shape
of
the
ougoing
waves
This
means
that
for
very
shallow
fluids
we
will
observe
similar
wavefronts
to
the
ones
of
Fig
More
concretely
for
the
water
air
interface
thickness
of
millimeter
and
speed
of
cm
should
do
the
trick
Superfluidity
Short
history
The
history
of
superfluidity
is
intrinsically
linked
to
that
of
helium
the
liquefaction
of
which
was
first
achieved
in
by
Kammerlingh
Onnes
This
ushered
in
new
era
of
low
temperature
physics
and
it
was
soon
realized
that
the
thermal
expansion
coefficient
of
liquid
He
had
discontinuity
at
the
so
called
lambda
point
This
lead
to
the
classification
of
liquid
helium
in
two
distinct
phases
above
He
and
below
He
II
the
critical
temperature
It
was
in
this
context
that
years
later
Kapitza
in
Moscow
and
Allen
and
Misener
in
Cambridge
investigated
the
flow
of
He
II
through
narrow
opening
between
two
large
containers
Observing
that
the
fluid
essentially
behaves
as
having
no
viscosity
(In
fact
formally
one
cannot
even
define
viscosity
in
this
case
as
the
ratio
of
the
mass
flow
to
the
pressure
differential
diverges
)Kapitza
introduced
the
term
"superfluidity
"in
analogy
to
the
superconductivity
observed
in
charged
systems
It
turned
out
to
be
an
inspired
analogy
as
the
prevalent
view
of
the
scientific
community
nowadays
is
that
the
two
are
essentially
the
same
phenomenon
stemming
from
BEC
in
the
neutral
case
respectively
Cooper
pairing
in
the
charged
case
arguably
kind
of
pseudo
BEC
The
connection
between
superfluidity
and
BEC
was
first
made
by
Fritz
London
who
noted
that
was
not
too
far
off
from
the
critical
temperature
one
would
predict
by
applying
Eq
eq
Tc
to
noninteracting
gas
of
helium
atoms
Of
course
liquid
helium
is
anything
but
noninteracting
gas
so
in
that
sense
it
is
rather
amusing
that
this
connection
was
made
The
modern
conception
of
superfluidity
is
that
it
represents
collection
of
phenomena
most
of
them
related
to
flow
properties
and
the
initial
observations
of
Kapitza
can
in
fact
be
broken
down
into
such
"elementary
"ingredients
as
we
shall
see
Two
fluid
model
great
leap
in
the
understanding
of
superfluidity
was
made
with
the
help
of
Lev
Landau's
phenomenological
two
fluid
model
In
nutshell
one
can
think
of
the
problem
in
terms
of
two
liquids
one
formed
by
the
condesate
occupying
the
single
particle
state
and
flowing
without
friction
and
the
other
behaving
as
normal
fluid
To
be
fair
it
must
be
said
that
Landau
actually
opposed
the
whole
notion
of
BEC
and
formulated
his
model
without
it
We
must
also
note
that
the
superfluid
and
normal
components
in
He
II
are
not
the
condensed
and
noncondensed
atoms
as
the
whole
liquid
is
superfluid
at
zero
temperature
while
only
around
of
the
atoms
are
in
the
BEC
state
That
being
said
the
model
describes
He
II
in
thermodynamic
equilibrium
by
introducing
two
independent
velocities
and
and
associated
densities
and
which
are
temperature
dependent
One
can
then
express
the
total
mass
current
density
and
kinetic
energy
of
flow
as
with
We
can
futher
define
the
superfluid
and
normal
fractions
as
and
with
Furthermore
we
have
the
limits
))))It
is
in
this
sense
that
one
is
dealing
with
an
intuitive
picture
of
He
II
as
mixture
of
two
independent
interpenetrating
components
each
associated
with
its
own
mass
current
and
flow
energy
Superfluid
velocity
)We
now
look
at
the
atoms
condensed
in
the
state
in
order
to
investigate
the
origin
of
the
superfluid
velocity
One
can
express
the
condesate
wavefunction
in
the
Madelung
form
(We
explicitly
re
introduced
in
the
equations
of
this
Section
to
show
their
quantum
nature
)with
being
the
particle
density
(We
again
emphasize
that
in
general
the
superfluid
density
is
not
equal
to
the
density
of
condensed
atoms
)and
the
phase
in
dimensions
of
an
action
both
real
functions
The
quantum
mechanical
definition
of
the
probability
current
density
of
the
condensate
yields
see
Appendix
im
with
as
before
representing
the
particle
mass
The
ratio
will
then
be
the
condensate
velocity
called
superfluid
velocity
in
the
literature
for
the
historical
reasons
detailed
above
Onsager
Feynman
quantization
condition
As
the
curl
of
gradient
is
zero
the
flow
given
by
Eq
eq
supervelocity
is
irrotational
in
other
words
the
vorticity
provided
of
course
that
is
defined
in
all
space
the
magnitude
of
is
finite
If
that
is
not
the
case
one
can
consider
contour
around
the
region
where
vanishes
and
the
condition
that
be
single
valued
then
gives
the
famous
Onsager
Feynman
quantization
with
being
the
windinding
number
the
number
of
turns
made
by
the
phase
in
the
complex
plane
as
we
go
around
the
integration
contour
Thus
the
circulation
of
the
irrotational
flow
is
quantized
with
the
quantum
of
circulation
being
cm
Two
effects
intro
We
are
now
in
the
position
to
discuss
the
two
elementary
ingredients
which
together
account
for
the
original
obsevation
of
superfluidity
by
Kapitza
Following
Ref
we
consider
multiply
connected
geometry
in
the
form
of
hollow
cylindrical
pipe
with
solid
walls
shown
in
Fig
The
multiply
connected
toroidal
geometry
of
radius
used
to
showcase
superfluid
behaviour
The
anullus
which
contains
He
atoms
rotates
at
an
angular
velocity
Assuming
the
pipe
of
radius
is
filled
with
liquid
helium
and
rotates
with
angular
velocity
one
can
then
obtain
the
temperature
dependent
orbital
angular
momentum
of
the
system
by
minimizing
the
effective
free
energy
Apart
from
the
already
defined
quantities
we
have
introduced
the
moment
of
inertia
and
defined
the
quantum
unit
of
angular
velocity
Finally
is
the
nearest
integer
to
expressed
as
One
needs
to
compare
Eq
eq
HF
angularmom
with
the
angular
momentum
of
normal
liquid
under
the
same
circumstances
which
is
given
by
Keeping
this
in
mind
we
can
now
introduce
the
two
qualitatively
different
experiments
Persistent
currents
The
first
experiment
starts
by
rotating
the
torus
at
large
angular
velocity
on
the
high
temperature
side
of
the
point
Once
the
fluid
equilibrates
with
the
motion
of
the
container
walls
we
then
cool
the
system
through
maintaining
the
same
angular
velocity
Finally
we
stop
the
container
from
rotating
and
measure
the
angular
momentum
After
short
while
proportional
to
the
viscosity
of
the
normal
component
its
velocity
becomes
zero
However
the
superfluid
part
of
the
liquid
experiences
no
friction
with
the
pipe
walls
and
will
therefore
continue
circulating
indefinately
Since
we
are
not
dealing
with
the
ground
state
of
the
system
but
with
an
excited
state
with
astronomical
lifetime
we
call
this
phenomenon
metastability
of
supercurrents
"persistent
currents
"is
another
commonly
found
term
However
metastability
of
superflow
is
not
direct
consequence
of
the
BEC
state
-the
underlying
mechanism
is
one
of
topological
origin
consider
the
contour
in
Eq
eq
feynman
to
run
around
the
torus
of
Fig
The
current
in
the
ring
will
be
proportional
to
the
circulation
Eq
eq
feynman
Since
the
latter
is
quantized
the
windinding
number
is
conserved
hence
so
is
the
super
current
Similar
arguments
explain
the
existence
of
so
called
topological
defects
in
superfluid
systems
Such
defects
are
singularities
of
the
condensate
wavefunction
for
instance
vortex
lines
or
rings
if
the
line
closes
in
on
itself
associated
with
the
superfluid
component
The
conservation
laws
presented
above
make
the
vortex
highly
stable
configuration
as
opposed
to
the
case
of
normal
fluid
where
it
is
short
lived
In
this
sense
the
circulating
currents
which
constitute
the
vortex
are
persistent
corresponding
to
metastable
supercurrents
Furthermoe
superfluid
system
cannot
sustain
bulk
vorticity
due
to
Eq
eq
vorticity
and
the
vortices
are
quantized
according
to
Eq
eq
feynman
In
the
final
state
with
the
container
at
rest
so
the
first
term
of
Eq
eq
HF
angularmom
dissapears
The
second
term
survives
as
the
system
cannot
change
its
winding
number
In
the
limit
of
large
the
angular
momentum
will
then
be
approximately
given
by
We
see
that
one
can
reversibly
increase
or
decrease
the
angular
momentum
by
varying
the
temperature
provided
of
course
that
it
is
always
kept
below
Since
the
superfluid
circulates
despite
any
roughness
of
the
container
walls
we
can
reasonably
expect
that
an
object
moving
though
stationary
fluid
also
experiences
reduced
friction
This
led
to
series
of
experiments
with
charged
ions
being
accelerated
at
various
speeds
by
an
applied
external
electric
field

It
was
seen
that
in
accordance
with
the
Landau
criterion
presented
in
Sec
there
exists
critical
speed
below
which
the
ions
experience
reduced
viscosity
and
actually
no
viscosity
at
all
in
the
limit
HF
effect
We
now
turn
to
the
second
experiment
which
is
arguably
one
of
the
fundamental
defining
properties
of
superfluidity
The
effect
was
first
predicted
by
Fritz
London
and
then
observed
by
Hess
and
Fairbank
in
in
the
context
of
liquid
helium
hence
the
name
Hess
Fairbank
HF
effect
As
before
we
cool
the
sample
while
rotating
with
lower
angular
velocity
We
wait
for
the
system
to
equilibrate
and
then
measure
the
angular
momentum
this
time
without
stopping
the
rotation
of
the
container
The
value
of
will
be
given
by
Eq
eq
HF
angularmom
instead
of
the
classical
that
is
why
the
HF
effect
is
also
called
nonclassical
rotational
inertia
NCRI
We
can
now
distinguish
two
different
regimes
The
first
one
for
implies
therefore
so
the
superfluid
component
contributes
nothing
to
the
current
At
and
we
get
the
spectacular
result
that
the
liquid
completely
ceases
to
rotate
along
with
the
container
In
the
general
case
the
second
term
of
Eq
eq
HF
angularmom
survives
If
we
again
consider
the
zero
temperature
limit
and
plot
the
angular
momentum
versus
angular
velocity
we
get
series
of
plateaus
of
width
at
positions
and
so
on
The
centers
of
these
plateaus
all
fall
on
the
"classical
"line
hand
waving
explanation
for
the
simplest
case
of
noninteracting
atomic
gas
goes
as
follows
In
noncondensed
system
rotating
with
angular
velocity
the
effective
single
particle
energies
are
shifted
by
an
amount
with
the
angular
momentum
quantum
number
of
state
The
Bose
Einstein
function
then
redistributes
the
particles
to
accomodate
this
shift
producing
"classical
"angular
momentum
In
BEC
however
all
atoms
must
be
in
the
same
state
and
posess
the
same
value
of
Their
only
allowed
contribution
to
is
then
For
small
the
condensate
is
in
the
state
and
so
it
does
not
rotate
with
the
container
This
argument
can
be
formalized
so
one
can
show
the
HF
effect
to
be
impossible
inside
the
framework
of
classical
statistical
mechanics
much
like
the
Bohr
Van
Leeuwen
theorem
shows
magnetism
to
be
purely
quantum
phenomenon
Note
that
while
the
HF
effect
can
be
at
least
hand
wavingly
explained
without
including
interactions
these
are
crucial
to
any
theory
of
metastable
currents
Meissner
effect
As
already
explained
in
the
introductory
paragraph
of
this
Section
there
is
deep
underlying
connection
between
superfluidity
in
neutral
system
and
superconductivity
in
charged
one
In
fact
the
analog
of
the
HF
effect
in
charged
system
is
the
well
known
Meissner
effect
We
now
consider
the
same
topology
of
Fig
and
replace
the
container
by
crystal
lattice
and
the
liquid
helium
or
atomic
gas
for
that
matter
by
the
electrons
inside
this
lattice
If
we
now
keep
the
crystal
at
rest
and
produce
tangential
vector
potential
this
will
generate
flux
inside
the
conducting
ring
usually
called
an
"Aharonov
Bohm
"flux
The
definition
of
the
"superfluid
"velocity
Eq
eq
supervelocity
now
needs
to
be
revised
according
to
the
gauge
coupling
recipe
which
consists
of
replacing
the
canonical
momentum
by
the
kinematic
one
As
result
we
get
where
the
factors
of
appear
because
the
macroscopic
wavefunction
is
now
that
of
the
center
of
mass
of
Cooper
pair
of
electrons
so
and
are
both
multiplied
by
The
resulting
Onsager
Feynman
quantization
Eq
eq
feynman
then
becomes
where
is
the
superconducting
flux
quantum
If
the
mass
supercurrent
is
given
as
before
by
then
this
new
quantization
condition
results
in
non
zero
current
even
for
Indeed
for
compare
to
the
regime
discussed
for
the
HF
effect
the
state
is
realized
and
the
electrical
current
is
given
by
This
proves
that
while
in
normal
conductor
the
flux
would
not
generate
any
circulating
current
superconducting
ring
would
respond
with
diamagnetic
note
the
minus
sign
in
Eq
eq
diamagnet
electrical
current
proportional
(The
proportionality
constant
is
called
the
susceptibility
)to
the
vector
potential
Coupling
this
with
Maxwell's
equations
immediately
results
in
the
well
known
Meissner
effect
namely
the
expulsion
of
an
applied
magnetic
field
inside
superconducting
sample
Synthetic
gauge
field
We
now
proceed
to
closer
analysis
of
the
connection
between
the
charged
and
neutral
case
The
Hamiltonian
for
particle
of
charge
and
mass
moving
in
vector
potential
and
in
the
presence
of
an
external
potential
is
given
by
Eq
eq
charged
ham
In
our
problem
is
imposed
by
the
toroidal
geometry
of
the
container
which
is
for
now
assumed
to
be
stationary
in
the
inertial
frame
of
the
stars
On
the
other
hand
the
Hamiltonian
of
neutral
particle
with
the
same
mass
but
this
time
in
the
frame
that
rotates
with
the
container
at
an
angular
velocity
is
shown
in
Eq
eq
neutral
ham
with
being
the
coordinate
and
the
canonical
momentum
in
the
rotating
frame
Making
use
of
the
cylindrical
symmetry
one
can
decompose
as
and
therefore
We
now
see
that
neutral
system
in
the
rotating
frame
apart
from
centrifugal
term
is
formally
identical
to
charged
system
in
the
rest
frame
with
the
correspondence
This
connection
can
be
used
to
make
neutral
particle
experience
an
"effective
"electromagnetic
vector
potential
Its
gauge
coupling
to
this
potential
is
what
eventually
led
to
the
celebrated
concept
of
synthetic
gauge
field
In
fact
such
artificial
rotation
induced
gauge
fields
have
already
been
proposed
in
the
context
of
ultracold
atomic
gases
in
order
to
asess
their
normal
and
superfluid
fractions
BEC
Experiments
Scanning
blue
detuned
laser
beam
with
velocity
though
the
cigar
shaped
atomic
condensate
left
panel
creates
density
hole
right
panel
larger
than
the
healing
length
The
friction
acting
on
this
hole
can
then
be
used
to
asess
the
Landau
critical
velocity
From
Ref
This
naturally
leads
us
to
the
topic
of
experimental
evidence
for
superfluid
behaviour
The
answer
however
will
not
be
clear
cut
one
as
we
have
seen
that
it
actually
depends
on
which
aspects
of
superfluidity
one
is
interested
in
such
as
NCRI
persistent
currents
quantized
vortices
or
reduced
friction
on
impurities
Historically
liquid
He
II
was
the
default
system
for
exploring
such
physics
however
new
systems
such
as
ultracold
gases
and
more
recently
microcavity
polaritons
the
subject
of
Chapter
have
also
entered
the
picture
In
the
case
of
liquid
helium
superfluidity
is
relatively
easy
to
detect
while
BEC
can
only
be
asessed
through
circumstantial
evidence
such
as
neutron
scattering
as
strong
interactions
oppose
any
large
density
changes
Ultracold
gases
on
the
other
hand
present
spectacular
BEC
onset
that
can
be
observed
in
the
density
profile
while
superfluidity
proves
somewhat
more
mysterious
Regarding
metastable
supercurrents
in
simply
connected
geometry
Chevy
and
coworkers
succeded
in
stirring
an
atomic
condensate
and
then
measuring
its
angular
momentum
using
the
precession
of
certain
collective
modes
Other
studies
were
also
shown
to
produce
metastable
single
vortex
state
and
even
vortex
lattices
that
persist
despite
not
being
the
ground
state
of
the
system
in
fact
it
can
be
shown
that
the
atomic
BEC
ground
state
is
characterised
by
the
absence
of
any
macroscopic
current
flow
Interferometric
measurements
are
commonly
employed
to
see
the
circulation
around
these
vortex
cores
However
the
lifetime
of
the
vortices
is
limited
by
how
long
it
takes
before
they
escape
the
magnetic
trap
To
circumvent
this
limitation
persistent
currents
in
multiply
connected
geometries
such
as
toroidal
traps
were
also
investigated
for
instance
in
Refs
and
As
far
as
the
Landau
criterion
is
concerned
an
experiment
similar
to
the
one
accelerating
charged
ions
through
liquid
helium
was
reported
in
Ref
The
authors
used
Raman
laser
to
transfer
fraction
of
the
condensed
atoms
into
different
hyperfine
state
As
result
those
atoms
no
longer
experienced
any
magnetic
trapping
but
only
the
effect
of
gravity
The
mobility
of
the
free
falling
atoms
was
then
computed
revealing
information
about
the
friction
acting
on
them
fairly
sharp
transition
with
sudden
increase
in
friction
was
observed
at
velocities
close
to
the
local
speed
of
sound
In
the
opposite
limit
of
macroscopic
"impurities
"large
compared
to
the
healing
length
Ref
details
the
use
of
blue
detuned
laser
beam
in
order
to
create
density
hole
which
is
then
moved
around
the
atomic
condensate
see
Fig
By
measuring
the
density
difference
immediately
ahead
and
behind
the
hole
one
can
deduce
the
frictional
force
acting
on
it
see
Appendix
This
experiment
resulted
in
critical
velocity
about
ten
times
smaller
than
the
speed
of
sound
presumably
due
to
vortex
creation
by
the
macroscopic
defect
Optical
lattices
Laser
trapping
An
electric
dipole
moment
placed
in
an
external
electric
field
aquires
an
energy
If
an
atom
is
placed
in
light
field
the
oscillating
electric
field
of
the
latter
induces
an
electric
dipole
moment
on
the
atom
with
the
free
space
permittivity
and
the
atomic
polarizability
Integrating
we
obtain
the
interaction
energy
of
the
atom
with
the
field
as
classical
model
of
the
atom
as
conducting
sphere
of
radius
yields
polarizability
of
In
practice
however
atomic
transitions
complicate
this
simple
picture
as
the
polarizability
depends
on
the
driving
frequency
of
the
light
field
In
the
framework
of
the
Drude
model
one
has
with
the
resonant
frequency
damping
time
and
the
static
polarizability
The
frequency
dependent
interaction
energy
thus
obtained
is
sometimes
called
the
ac
Stark
effect
If
the
light
frequency
is
smaller
than
that
of
the
atomic
resonance
the
induced
dipole
is
in
phase
with
the
electric
field
as
and
the
force
on
the
atom
will
point
in
the
direction
of
the
square
field
maxima
the
situation
is
of
course
reversed
if
instead
This
effect
was
first
used
in
Ref
to
trap
BEC
of
sodium
atoms
using
focused
infrared
laser
beam
the
dominant
transition
of
Na
is
the
familiar
yellow
emission
Interestingly
the
same
physics
can
be
used
to
create
an
optical
lattice
as
explained
below
Optical
lattice
Consider
plane
wave
laser
beam
propagating
along
the
axis
with
an
associated
electric
field
which
reflects
from
mirror
situated
at
The
interference
between
the
reflected
and
original
beam
produces
standing
wave
with
an
intensity
This
results
in
an
effective
periodic
lattice
for
the
atoms
with
lattice
spacing
where
Such
one
dimensional
lattice
was
first
demonstrated
in
Ref
By
tuning
the
strength
of
the
laser
field
one
can
adjust
the
lattice
depth
and
by
changing
the
laser
wavelength
one
can
modify
the
lattice
constant
Adding
more
laser
beams
can
produce
two
or
even
three
dimensional
lattices
in
variety
of
different
geometries
For
example
the
simplest
way
to
create
square
lattice
would
be
to
superimpose
second
lattice
in
the
direction
producing
an
optical
potential
of
depth
The
beam
along
should
be
orthogonally
polarized
with
respect
to
the
one
along
in
order
to
avoid
interference
effects
Band
structure
The
eigen
problem
of
particle
moving
in
periodic
potential
given
by
Eq
eq
square
potential
is
well
known
in
the
context
of
solid
state
band
theory
as
the
electrons
inside
solid
also
move
in
such
potential
produced
in
that
case
by
the
crystal
ions
The
energy
eigenstates
are
the
Bloch
waves
where
is
the
quasimomentum
restricted
to
the
first
Brillouin
zone
BZ
and
represents
the
band
index
The
functions
are
periodic
in
both
the
and
direction
with
period
To
see
how
these
bands
arise
one
can
expand
both
the
wavefunction
as
well
as
the
potential
in
Fourier
series
making
use
of
their
periodicity
Inserting
this
expansion
into
the
time
independent
Schr
odinger
equation
we
get
linear
system
of
equations
for
the
Fourier
coefficients
For
fixed
we
obtain
different
(Truncating
the
Fourier
expansion
to
finite
number
of
terms
results
in
bands
)eigenenergies
These
eigenenergies
form
bands
called
Bloch
bands
when
is
varied
inside
the
BZ
Each
band
as
certain
width
in
energy
unsurprisingly
called
the
bandwidth
and
consecutive
bands
are
separated
by
gaps
meaning
certain
energies
are
not
allowed
Furthermore
each
eigenenergy
has
an
associated
eigenfunction
completely
determined
by
the
set
of
Fourier
coefficients
with
index
The
solution
of
course
depends
on
the
value
of
and
we
can
distinguish
two
limiting
cases
depending
on
the
ratio
of
the
bandwidths
to
the
energy
gaps
In
the
weak
potential
limit
the
bandgaps
are
much
smaller
than
the
bandwidths
and
the
eigenenergies
have
pronounced
dependence
on
quasimomentum
The
opposite
limit
which
is
the
one
we
want
to
focus
on
is
the
regime
of
deep
lattices
also
called
the
tight
binding
TB
limit
The
band
dispersion
is
now
flatter
compared
to
the
previous
case
and
the
bandwidth
gap
ratio
is
much
smaller
Tight
binding
For
low
temperatures
in
the
TB
limit
only
the
lowest
energy
band
is
occupied
so
we
subsequently
drop
the
band
index
Since
the
potential
wells
are
so
deep
the
lowest
band
is
adequately
described
by
considering
localized
wavefunctions
at
each
well
at
the
minima
of
Eq
eq
square
potential
Suppose
now
that
function
exists
such
that
the
Bloch
waves
are
given
by
where
is
normalization
factor
and
are
the
well
positions
Eq
eq
wanier
represents
wave
whose
phase
and
amplitude
at
each
well
is
given
by
the
localized
function
called
the
Wannier
function
It
can
be
shown
that
Wannier
functions
centered
at
different
wells
are
orthogonal
to
each
other
and
form
complete
basis
The
Wannier
functions
are
thus
unitary
transformation
of
the
Bloch
functions
suitable
for
the
case
when
particle
dynamics
is
described
by
inter
well
tunneling
hopping
between
nearest
neighbour
wells
In
the
case
of
BEC
when
the
number
of
ultracold
atoms
per
lattice
well
is
small
the
"discrete
structure
"of
the
condensate
becomes
relevant
and
we
must
replace
the
GP
description
Eq
eq
TDGP
we
have
used
so
far
with
lattice
model
such
as
the
Bose
Hubbard
model
as
suggested
in
Ref
For
condensate
confined
in
the
optical
potential
Eq
eq
square
potential
with
an
additional
harmonic
trap
the
spinless
Bose
Hubbard
Hamiltonian
in
can
be
written
as
where
is
the
on
site
interaction
strength
and
is
the
energy
offset
due
to
the
harmonic
trap
Futhermore
denotes
the
number
operator
for
site
and
is
the
bosonic
creation
destruction
operator
for
that
site
Finally
the
kinetic
part
can
be
diagonalized
to
yield
the
Bloch
dispersion
where
the
tunneling
amplitude
is
an
exponentially
decreasing
function
of
The
characteristic
tunneling
energy
scale
is
given
by
the
bandwidth
which
in
this
case
is
Hamiltonian
Eq
eq
BH
ham
supports
zero
temperature
quantum
phase
transition
between
superfluid
and
Mott
insulating
phase
at
critical
value
of
the
ratio
This
transition
was
experimentally
observed
for
the
case
of
bosonic
BEC
in
optical
lattice
as
reported
in
Ref
Increasing
the
lattice
depth
the
authors
went
from
gapless
superfluid
regime
dominated
by
the
kinetic
term
Eq
eq
kinetic
part
to
gapped
Mott
insulator
state
where
the
atoms
are
localized
and
the
on
site
interaction
term
in
Eq
eq
BH
ham
is
dominant
Harper
Hofstadter
model
Landau
levels
In
Sec
we
introduced
the
quantized
Bloch
bands
characterising
particles
in
periodic
potential
similar
quantization
also
occurs
in
the
case
of
charged
particle
moving
perpendicular
to
constant
uniform
magnetic
field
and
the
allowed
energy
bands
are
called
Landau
levels
In
magnetic
field
the
relevant
energy
and
length
scales
for
particle
with
charge
and
mass
are
set
by
the
cyclotron
frequency
and
magnetic
length
respectively
In
this
Section
we
want
to
explore
the
fate
of
the
Bloch
bands
in
presence
of
magnetic
field
Alternatively
one
could
also
start
from
Landau
quantized
system
and
perturb
it
via
weak
periodic
potential
the
two
approaches
having
distinct
validity
ranges
Hofstadter
butterfly
One
of
the
simplest
TB
models
for
Bloch
electrons
in
uniform
magnetic
fields
is
the
Harper
Hofstadter
model
Its
relevant
dimensionless
parameter
is
the
ratio
of
flux
through
lattice
cell
to
one
Dirac
flux
quantum
(As
opposed
to
the
superconducting
flux
quantum
of
Sec
)Physically
reflects
(Not
to
be
confused
with
the
atomic
polarizability
also
denoted
by
)the
commensurability
of
the
lattice
period
and
the
magnetic
length
as
it
can
be
recast
in
the
form
The
resulting
single
particle
energy
spectrum
of
the
model
depends
on
the
rationality
of
If
for
and
co
prime
integers
the
Bloch
band
splits
into
distinct
magnetic
subbands
For
irrational
the
spectrum
consists
instead
of
an
infinite
number
of
energy
levels
forming
Cantor
set
The
union
of
all
allowed
energies
as
function
of
forms
self
similar
fractal
shown
in
Fig
and
known
as
Hofstadter
butterfly
In
the
continuum
limit
with
fixed
and
the
bands
group
into
clusters
that
can
be
identified
with
the
Landau
fan
predicted
in
the
absence
of
lattice
potential
Hofstadter's
butterfly
representing
the
internal
structure
of
the
lowest
Bloch
band
in
uniform
magnetic
field
For
rational
values
of
the
spectrum
has
magnetic
subbands
which
become
increasingly
narrow
as
gets
larger
Adapted
from
Ref
HH
experiments
Solid
state
experiments
in
the
Hofstadter
regime
are
notoriously
difficult
since
the
inter
atomic
spacing
of
crystal
lattice
is
few
AAngstr
oms
and
achieving
comparable
magnetic
length
would
require
field
of
about
Tesla
Signatures
of
the
Hofstadter
bands
have
instead
been
seen
in
artificial
superlattices
with
periodicities
of
tens
of
nanometers
In
the
context
of
neutral
ultracold
atoms
synthetic
gauge
potentials
paved
the
way
for
the
exploration
of
similar
physics
For
instance
rotating
harmonically
trapped
condensate
with
high
angular
velocity
comparable
to
the
radial
trap
frequency
allowed
BEC
experiments
to
enter
the
lowest
Landau
level
LLL
regime
Rotating
optical
lattices
have
also
been
realized
but
the
first
implementation
of
the
Harper
Hofstadter
HH
model
with
ultracold
atoms
employed
different
method
of
generating
the
artificial
magnetic
field
namely
laser
assisted
tunneling
based
on
linear
superlattice
potential
In
nutshell
tunneling
between
lattice
sites
required
atoms
to
absorb
and
emit
photons
produced
by
external
lasers
The
interaction
with
said
lasers
generated
net
phase
change
equivalent
to
the
flux
of
magnetic
field
The
total
phase
accumulated
by
an
atom
when
performing
loop
around
lattice
cell
plaquette
was
in
these
experiments
Aharonov
Bohm
phase
In
quantum
mechanics
the
phase
of
the
wavefunction
of
charged
particle
moving
between
positions
and
is
changed
by
the
presence
of
magnetic
field
by
where
the
integral
is
taken
along
an
oriented
path
from
point
""to
point
""Therefore
and
if
we
consider
smooth
closed
integration
contour
instead
of
two
separate
points
the
total
phase
change
or
Aharonov
Bohm
phase
along
is
where
is
the
magnetic
flux
through
smooth
surface
bounded
by
the
curve
with
positive
orientation
according
to
Stoke's
theorem
Peierls
phases
On
the
square
lattice
in
Fig
one
can
define
the
magnetic
phases
along
the
and
directions
in
similar
manner
as
square
lattice
pierced
by
homogeneous
static
magnetic
field
The
lattice
constant
is
and
the
curved
arrows
indicate
the
field
induced
hopping
phases
where
the
integrals
are
now
taken
along
the
links
joining
neighbouring
sites
Using
Eq
eq
AB
closed
the
flux
through
plaquette
can
be
related
to
the
total
phase
accumulated
on
contour
enclosing
the
plaquette
in
counterclockwise
sense
Note
that
is
gauge
independent
unlike
the
Peierls
phases
HH
Hamiltonian
To
obtain
the
HH
Hamiltonian
we
introduce
static
uniform
magnetic
field
into
the
single
band
Bose
Hubbard
model
Eq
eq
BH
ham
of
Sec
We
follow
Hofstadter's
original
treatment
and
neglect
on
site
interactions
(For
treatment
that
includes
interactions
in
the
Bogoliubov
approximation
see
Ref
)as
well
as
the
harmonic
trapping
The
kinetic
term
Eq
eq
kinetic
part
is
then
modified
according
to
Peierls
substitution
to
include
the
hopping
phases
These
phases
act
to
"frustrate
"the
hopping
such
that
for
noninteger
it
is
impossible
to
minimize
the
kinetic
energy
simultaneously
on
all
lattice
links
The
Hamiltonian
is
no
longer
invariant
under
translations
by
single
lattice
site
in
any
direction
because
fixing
gauge
for
the
vector
potential
reduces
the
physical
symmetry
of
the
lattice
In
fact
it
can
be
shown
that
translation
of
is
equivalent
to
gauge
transformation
suggesting
that
new
translation
operators
that
would
commute
with
Eq
eq
HH
hamiltonian
can
be
constructed
as
combination
of
translation
and
gauge
transformation
Magnetic
translation
operators
We
define
the
magnetic
translation
operators
MTOs
as
where
the
phases
are
to
be
determined
by
imposing
leading
to
While
the
specific
form
of
the
MTOs
depends
on
the
choice
of
gauge
their
commutator
is
gauge
independent
and
vanishes
only
if
is
an
integer
situation
equivalent
to
the
zero
field
case
In
order
to
proceed
we
point
out
that
Eq
eq
TxTy
has
transparent
physical
interpretation
If
we
act
on
the
single
particle
state
we
see
that
performing
counterclockwise
loop
around
unit
cell
results
in
the
accumulation
of
phase
proportional
to
the
flux
inside
the
cell
This
can
be
generalized
to
macrocell
containing
original
lattice
cells
to
deduce
that
For
the
commutator
vanishes
if
is
an
integer
The
smallest
possible
macrocell
is
given
by
and
is
called
the
magnetic
unit
cell
For
this
choice
of
unit
cell
form
complete
set
of
commuting
operators
so
we
can
find
simultaneous
eigenstates
labeled
by
the
quasimomentum
which
is
now
restricted
to
the
magnetic
Brillouin
zone
MBZ
HH
spectrum
Top
panels
Single
particle
energy
spectrum
left
of
the
HH
Hamiltonian
Eq
eq
HH
hamiltonian
for
and
dispersion
of
the
two
lowest
bands
middle
and
right
in
part
of
the
MBZ
Bottom
panel
Energy
gap
continuous
line
here
denotes
an
average
over
the
MBZ
and
bandwidth
of
dashed
line
for
The
dotted
vertical
line
marks
the
location
of
the
case
shown
above
To
obtain
the
spectrum
of
we
need
to
solve
its
associated
Schr
odinger
equation
We
expand
the
eigenstates
in
the
single
particle
basis
and
obtain
an
equation
for
the
coefficients
We
fix
the
Landau
gauge
for
simplicity
so
and
we
choose
the
magnetic
unit
cell
(In
the
symmetric
gauge
the
magnetic
unit
cell
is
larger
containing
sites
)for
which
the
commuting
MTOs
read
justifying
the
plane
wave
ansatz
(Here
is
defined
in
the
full
BZ
)subject
to
the
periodic
boundary
conditions
Substituting
this
ansatz
into
Eq
eq
HH
schrodinger
yields
the
Harper
equation
Calculating
the
single
particle
spectrum
of
Eq
eq
HH
hamiltonian
is
therefore
equivalent
to
solving
the
eigenvalue
equation
for
the
matrix
defined
as
(If
and
)For
integer
we
recover
the
lowest
Bloch
band
Eq
eq
tb
dispersion
while
for
this
band
splits
into
bands
with
dispersion
Spectrum
discussion
In
the
top
left
panel
of
Fig
we
show
an
example
of
the
spectrum
computed
by
diagonalizing
for
The
level
is
at
the
centre
of
the
middle
band
and
there
are
gaps
separating
the
bands
For
even
values
of
the
spectrum
is
symmetric
with
respect
to
and
the
two
central
bands
touch
at
zero
energy
Around
these
degeneracy
points
the
dispersion
is
linear
while
in
all
other
cases
it
is
quadratic
near
its
minimum
The
case
is
particularly
interesting
by
analogy
to
graphene
related
physics
as
there
are
two
inequivalent
"Dirac
points
"and
the
Hamiltonian
preserves
time
reversal
symmetry
Inspecting
the
middle
and
right
panels
we
note
that
despite
the
reduced
symmetry
of
the
dispersion
has
the
full
rotational
symmetry
of
the
underlying
square
lattice
One
can
formally
define
rotation
and
reflection
operators
which
together
with
the
MTOs
constitute
the
full
projective
symmetry
group
The
bottom
panel
of
Fig
shows
the
evolution
of
the
lowest
energy
gap
as
well
as
the
bandwidth
of
the
lowest
band
as
function
of
for
the
case
The
lowest
band
gets
flatter
as
the
continuum
LLL
eigenstates
are
the
solutions
of
the
lowest
band
of
the
HH
model
and
decreases
like
as
the
number
of
gaps
is
excluding
the
band
bottom
and
band
top
HH
topology
The
energy
bands
of
the
HH
model
are
topologically
nontrivial
as
characterized
by
their
non
zero
Chern
numbers
-topological
invariants
that
are
linked
to
the
quantization
of
the
Hall
conductivity
in
the
integer
quantum
Hall
QH
effect
In
fact
solutions
of
the
Harper
equation
Eq
eq
Harper
were
also
studied
in
connection
with
the
QH
effect
We
make
use
of
these
topological
properties
in
Chapter
where
we
consider
the
HH
model
in
the
presence
of
an
external
harmonic
trap
in
dissipative
system
Conservation
laws
for
the
GP
field
In
this
Appendix
we
use
classical
field
theory
to
derive
the
conservation
laws
associated
with
the
time
dependent
Gross
Pitaevskii
Eq
eq
TDGP
We
start
by
recasting
the
TDGP
using
Einstein's
summation
convention
with
the
notation
and
This
equation
can
be
deduced
by
extremalization
of
the
action
where
the
Lagrangian
density
is
given
by
For
compactness
let
(is
the
speed
of
light
but
it
is
just
constant
for
our
purposes
and
will
drop
out
from
all
final
results
)and
we
can
define
the
canonical
energy
momentum
tensor
following
the
usual
field
theory
prescription
where
the
indices
run
from
to
As
the
complex
scalar
field
and
satisfy
the
Euler
Lagrange
equations
resulting
in
Eq
eq
GP
field
and
its
complex
conjugate
we
have
from
which
energy
and
momentum
conservation
immediately
follow
(Conservation
of
angular
momentum
imposes
the
additional
requirement
that
)To
see
this
explicitly
we
split
the
temporal
and
spatial
parts
of
Eq
eq
conservation
laws
to
obtain
Direct
subsitution
into
Eq
eq
en
mom
tensor
shows
that
where
the
energy
density
is
given
by
while
with
the
energy
flux
density
Regarding
Eq
mom
cons
we
get
with
the
mass
current
density
compare
to
Eq
eq
bec
current
and
the
momentum
flux
density
tensor
Combining
all
the
above
one
can
rewrite
Eq
eq
conservation
laws
as
We
now
focus
on
Eq
eq
mom
cons
pi
which
shows
momentum
conservation
For
more
transparent
physical
interpretation
one
can
make
use
of
the
Madelung
transformation
Eq
eq
madelung
and
integrate
over
volume
obtaining
where
we
made
use
of
the
divergence
theorem
for
tensor
fields
in
order
to
express
the
volume
integral
as
an
integral
over
the
surface
which
encloses
Here
is
the
outward
pointing
unit
normal
to
at
each
point
and
we
also
introduced
the
total
momentum
of
the
field
with
the
velocity
Eq
eq
supervelocity
The
hydrodynamic
form
of
the
momentum
flux
density
then
reads
with
the
pressure
We
can
now
write
Eq
eq
gauss
in
vector
form
and
see
that
the
vector
between
square
brackets
is
the
amount
of
momentum
per
unit
time
and
area
"flowing
"through
surface
orthogonal
to
If
we
now
consider
stationary
regime
so
that
the
last
term
of
Eq
eq
vectorial
flow
drops
out
and
assume
the
potential
describes
some
fixed
obstacle
such
as
cylinder
inside
then
by
definition
the
"drag
"force
acting
on
surface
element
of
the
obstacle
is
the
momentum
flux
though
this
element
The
total
force
of
course
will
be
given
by
the
surface
integral
of
the
momentum
flux
or
equivalently
by
the
volume
integral
of
the
potential
gradient
in
Eq
eq
vectorial
flow
While
in
conservative
systems
this
drag
is
caused
by
pressure
gradients
across
the
obstacle
nonconservative
ones
such
as
the
resonantly
pumped
polariton
fluid
of
Chapter
also
have
viscous
contribution
to
the
momentum
flux
tensor
giving
rise
to
Navier
Stokes
type
physics
Microcavity
exciton
polaritons
In
Chapter
we
have
seen
among
other
things
that
interactions
are
essential
for
the
appearance
of
superfluidity
and
the
existence
of
BEC
is
not
sufficient
We
now
want
to
explore
similar
physics
in
the
context
of
nonlinear
optical
systems
as
their
nonlinear
polarization
is
able
to
mediate
appreciable
interactions
between
photons
One
way
to
achieve
the
strong
interactions
necessary
for
collective
fluid
like
behavior
of
the
many
photon
system
is
by
coupling
the
photons
with
matter
degrees
of
freedom
resulting
in
new
quasiparticles
called
polaritons
The
connection
between
BEC
and
spontaneous
coherence
effects
in
nonlinear
optical
systems
became
apparent
starting
with
the
experiments
on
exciton
polaritons
in
semiconductor
microcavities

However
the
steady
state
in
such
devices
typically
results
from
dynamical
balance
between
driving
and
dissipation
in
contrast
to
the
thermal
equilibrium
achieved
in
ultracold
atomic
gases
In
this
Chapter
we
review
the
physics
of
microcavity
exciton
polaritons
with
an
emphasis
on
their
hydrodynamics
properties
and
in
particular
the
new
effects
that
their
non
equilibrium
nature
introduces
Chapter
organization
This
Chapter
is
organized
as
follows
we
start
from
microscopic
description
of
semiconductor
microcavity
Sec
in
terms
of
electrons
holes
and
cavity
photons
Sec
before
switching
to
higher
level
weakly
interacting
Bose
model
for
polaritons
in
Sections
and
We
discuss
various
ways
of
pumping
the
system
as
well
as
the
important
roles
played
by
decay
Sec
and
disorder
Sec
leading
to
driven
dissipative
Gross
Pitaevskii
equation
at
the
mean
field
level
Sec
We
investigate
the
optical
limiter
and
bistable
regimes
of
resonantly
pumped
polariton
condensates
by
means
of
single
state
pump
only
ansatz
Sec
for
the
polaritonic
GP
equation
Finally
we
consider
three
state
solution
the
optical
parametric
oscillator
regime
Sec
obtained
by
parametric
scattering
of
pump
polaritons
into
the
signal
and
idler
states
In
Section
we
also
review
some
superfluid
related
phenomena
in
an
OPO
context
such
as
quantized
vortices
and
the
symmetry
breaking
resulting
in
Goldstone
mode
of
the
OPO
Bogoliubov
spectrum
Model
system
Consider
the
following
simplified
physical
system
at
zero
temperature
we
have
an
optical
planar
microcavity
containing
photons
confined
along
the
growth
direction
say
by
its
identical
metalic
mirrors
separated
by
distance
At
position
the
cavity
contains
quantum
well
QW
which
is
thin
semiconductor
layer
sandwitched
in
between
another
alloy
acting
as
barrier
The
setup
is
shown
in
Fig
for
an
in
depth
introduction
to
the
physics
of
semiconductor
microcavities
see
Ref
The
photons
create
electron
hole
hydrogenlike
bound
pairs
excitons
confined
by
the
barrier
inside
the
QW
Exactly
describing
the
system
of
electrons
holes
and
photons
taking
into
account
their
Coulomb
interaction
and
the
disorder
created
by
defects
inside
the
QW
as
well
as
imperfections
present
in
the
cavity
mirrors
is
daunting
task
that
would
provide
little
physical
insight
into
this
system
We
therefore
use
series
of
approximations
and
construct
an
effective
theory
that
is
more
physical
Pictorial
representation
of
microcavity
polaritons
as
electron
hole
pairs
excitons
inside
semiconductor
quantum
well
QW
coupled
with
the
photons
trapped
by
the
dielectric
Bragg
mirrors
of
the
cavity
Momentum
conservation
dictates
that
cavity
field
component
of
momentum
must
decay
into
external
radiation
of
frequency
emitted
at
an
angle
satisfying
From
Ref
Microscopic
description
We
start
by
writing
down
the
kinetic
energy
of
the
cavity
photons
where
we've
made
use
of
the
translational
symmetry
of
the
problem
in
the
plane
and
labeled
the
bosonic
creation
annihilation
operators
by
the
in
plane
momentum
These
operators
satisfy
the
standard
commutation
relations
The
confinement
along
the
direction
leads
to
the
quantization
of
the
photon
momentum
with
indexing
the
longitudinal
modes
Fig
depicts
the
photonic
field
corresponding
to
the
mode
as
standing
wave
representing
the
measurable
electric
field
inside
the
cavity
This
field
can
be
expressed
in
terms
of
photonic
operators
by
where
the
photon
mode
wavefunction
has
the
sinusoidal
shape
The
cavity
photon
dispersion
for
each
of
these
modes
is
therefore
with
the
speed
of
light
in
the
semiconductor
Expanding
the
square
root
for
one
gets
We
see
that
cavity
photons
as
opposed
to
free
space
photons
have
quadratic
dispersion
with
an
effective
mass
For
typical
cavities
used
in
experiments
is
of
the
order
of
free
electron
masses
see
Tab
We
now
turn
our
attention
to
the
semiconductor
QW
For
simplicity
consider
spin
polarized
direct
bandgap
semiconductor
such
as
GaAs
so
that
we
can
neglect
additional
spin
degrees
of
freedom
We
further
assume
that
the
dispersions
of
single
particle
states
in
the
conduction
and
valence
bands
have
the
quadratic
form
and
with
the
semiconductor
bandgap
and
and
the
effective
masses
for
electrons
and
heavy
holes
Excitons
have
typical
"extension
"see
Tab
for
typical
values
for
GaAs
based
microcavity
called
the
exciton
Bohr
radius
with
the
static
dielectric
constant
the
electron
charge
and
their
reduced
mass
Their
binding
energy
is
called
the
exciton
Rydberg
defined
as
see
Tab
We
further
define
and
as
the
operators
which
create
an
electron
in
the
empty
conduction
band
respectively
hole
in
the
filled
valence
band
and
which
obey
the
Fermi
anticommutation
rules
and
similar
for
We
can
therefore
write
the
electronic
Hamiltonian
in
terms
of
these
new
operators
as
Here
we
have
introduced
the
electron
and
hole
densities
and
The
matrix
element
of
the
Coulomb
potential
depends
on
cavity
quantization
area
but
this
dependence
drops
out
when
passing
from
discrete
sums
over
states
to
integrals
over
the
momenta
Finally
the
last
ingredient
of
our
microscopic
description
concerns
the
interaction
between
electrons
and
photons
Making
use
of
the
dipole
approximation
we
have
with
the
strength
written
in
the
dipole
gauge
where
is
the
inter
band
dipole
matrix
element
Effective
Hamiltonian
As
mentioned
in
Sec
we
are
interested
in
an
approximate
description
of
the
problem
where
we
can
consider
the
excitons
as
fundamental
quasiparticle
excitations
from
the
ground
state
of
the
semiconductor
and
treat
the
Coulomb
term
in
Eq
eq
coulomb
hamiltonian
as
an
effective
exciton
exciton
interaction
We
then
couple
the
resulting
excitons
to
light
and
obtain
the
so
called
weakly
interacting
boson
model
for
polaritons
The
name
stems
from
the
fact
that
at
moderate
electron
hole
densities
such
that
the
inter
exciton
distance
is
much
larger
than
one
can
assume
excitons
to
behave
esentially
as
composite
bosons
and
therefore
describe
them
using
the
Bose
creation
and
annihilation
operators
and
Making
an
Usui
transformation
and
truncating
the
interaction
terms
at
fourth
order
results
in
the
following
three
part
effective
Hamiltonian
where
we
have
defined
XX
XC
sat
sat
contains
the
kinetic
energy
of
the
excitons
and
cavity
photons
as
well
as
their
harmonic
coupling
ie
the
conversion
of
an
exciton
to
cavity
photon
at
the
Rabi
frequency
which
depends
on
the
overlap
between
the
wavefunctions
of
the
exciton
and
photon
through
the
oscillator
strength
surface
density
of
the
excitonic
transition
where
is
the
maximum
amplitude
of
the
cavity
photon
electric
field
at
one
of
its
antinodes
from
Eq
eq
electric
field
It
is
worth
noting
that
both
Eq
eq
dipolar
inter
and
Eq
eq
polariton
ham
make
use
of
the
rotating
wave
approximation
neglecting
antiresonant
terms
which
do
not
conserve
the
number
of
excitations
such
as
and
This
is
justified
so
long
as
which
is
normally
the
case
as
can
be
seen
in
Tab
which
reports
typical
parameters
of
GaAs
based
microcavity
with
embedded
QWs
Eq
eq
polariton
ham
can
be
diagonalized
by
means
of
unitary
transformation
of
the
form
The
normal
modes
of
are
called
upper
and
lower
polaritons
UP
and
LP
and
correspond
to
the
Bose
operators
and
with
eigenenergies
Dispersion
relation
for
the
two
polaritonic
branches
UP
LP
as
function
of
emission
angle
of
photons
outside
the
cavity
which
is
directly
related
to
the
momentum
of
polaritons
inside
the
cavity
The
exciton
and
cavity
photon
dispersions
are
depicted
with
dashed
red
lines
Due
to
their
light
mass
the
polaritons
have
very
sharp
dispersion
compared
to
excitons
Also
note
the
typical
anticrossing
form
characteristic
of
Hermitian
coupling
Parameters
eV
meV
and
meV
where
the
signs
refer
to
the
UP
and
LP
branches
and
the
Hopfield
coefficients
appearing
in
Eq
eq
hopfield
trans
are
LP
LP
The
two
branches
of
the
polaritonic
dispersion
relation
are
shown
in
Fig
together
with
the
dispersions
of
the
cavity
photons
and
excitons
One
can
therefore
think
of
polaritons
as
being
massive
photons
dressed
by
matter
excitations
Note
that
the
exciton
energy
can
be
approximated
as
with
the
exciton
mass
of
the
order
of
the
electron
mass
and
Since
as
we
have
already
noted
is
several
orders
of
magnitude
larger
than
the
cavity
photon
mass
one
can
in
practice
neglect
the
momentum
dependence
of
the
excitonic
dispersion
considering
it
as
flat
compared
to
the
cavity
photon
one
Furthermore
we
can
measure
energies
starting
from
and
denote
the
detuning
between
exciton
and
photon
bands
as
For
zero
detuning
the
splitting
between
the
two
polariton
branches
is
In
order
to
tune
in
experiments
one
tipically
builds
the
cavity
mirrors
with
wedge
quantifies
the
effective
repulsive
exciton
exciton
interaction
with
momentum
dependent
strength
that
can
be
calculated
from
the
Coulomb
exchange
term
in
the
Born
approximation
further
approximation
that
one
can
make
for
wave
vectors
much
smaller
than
is
to
reduce
to
contact
potential
as
shown
in
Ref
As
result
physically
represents
the
interaction
between
two
excitons
in
the
same
single
particle
momentum
eigenstate
Finally
the
last
term
of
Eq
eq
total
ham
can
be
interpreted
as
saturation
effect
due
to
the
Pauli
exclusion
principle
underlying
the
fermionic
character
of
the
excitons
The
exciton
saturation
density
generally
depends
on
the
specific
shape
of
the
internal
wavefunction
of
the
exciton
For
the
problems
we
will
be
treating
in
the
remainder
of
this
manuscript
it
can
be
considered
large
enough
to
justify
neglecting
this
anharmonic
contribution
to
the
total
Hamiltonian
Eq
eq
total
ham
see
Tab
Before
concluding
this
Section
one
should
mention
the
validity
limitations
of
the
weakly
interacting
boson
model
for
polaritons
reviewed
here
Its
derivation
implies
an
expansion
of
the
original
fermionic
operators
of
Eq
eq
coulomb
hamiltonian
in
powers
of
bosonic
operators
having
as
small
parameter
the
number
of
excitons
per
Bohr
radius
Truncating
this
series
at
fourth
order
results
in
an
upper
bound
on
the
exciton
density
of
around
At
higher
densities
an
electron
hole
plasma
like
state
is
formed
with
qualitatively
different
behaviour
We
see
that
this
model
is
particularly
useful
for
the
case
when
the
dominant
interaction
is
the
Coulomb
repulsion
between
the
excitons
Lower
polaritons
Assuming
the
Rabi
energy
to
dominate
over
the
kinetic
and
interaction
energies
we
can
neglect
the
upper
polaritons
and
project
Eq
eq
total
ham
on
the
basis
of
lower
polariton
states
only
Employing
the
approximations
discussed
above
we
obtain
the
lower
polariton
Hamiltonian
where
the
strength
of
the
repulsive
interaction
between
lower
polaritons
now
depends
on
in
plane
momentum
through
the
Hopfield
coefficients
(The
interaction
strength
can
usually
be
rescaled
to
by
simple
transformation
)This
effective
interaction
originating
from
the
binary
Coulomb
scattering
of
the
excitons
is
the
one
responsible
for
the
collective
behaviour
of
the
polaritons
and
in
particular
superfluid
hydrodynamics
an
aspect
which
we
will
explore
in
the
following
two
Chapters
Physically
the
quantities
and
are
the
excitonic
and
photonic
fractions
of
the
lower
polariton
mode
and
the
effective
mass
of
the
lower
polaritons
can
be
computed
from
the
curvature
of
their
dispersion
Eq
eq
polariton
dispersion
by
which
at
normal
incidence
reduces
to
if
we
neglect
the
flat
exciton
dispersion
in
the
limit
For
zero
detuning
we
see
that
and
the
light
and
matter
content
of
lower
polaritons
are
each
equal
to
As
long
as
the
interesting
physics
is
limited
to
small
compared
to
wavevectors
close
to
the
bottom
of
the
LP
branch
one
can
safely
approximate
the
LP
dispersion
as
being
parabolic
by
expanding
Eq
eq
polariton
dispersion
in
Taylor
series
around
and
truncating
at
second
order
We
obtain
the
expression
with
an
effective
mass
given
by
Eq
eq
LP
mass
In
practice
many
interesting
experimental
conditions
can
be
adequately
described
by
limiting
ourselves
to
the
LP
branch
with
quadratic
dispersion
Pumping
and
dissipation
We
have
so
far
assumed
that
the
mirrors
at
the
two
ends
of
the
optical
cavity
in
Fig
are
perfect
In
reality
of
course
this
is
not
the
case
and
photons
continuously
escape
from
the
cavity
In
other
words
decay
is
built
in
feature
of
this
system
and
rather
than
being
hindrance
it
is
actually
useful
as
it
allows
one
to
extract
information
about
the
system
By
looking
at
the
near
field
far
field
emission
from
the
cavity
one
can
measure
the
real
momentum
space
photon
density
The
polaritonic
spectrum
can
also
be
obtained
in
this
way
as
function
of
the
angle
of
emission
of
photon
out
of
the
cavity
as
shown
in
Fig
Since
the
system
is
lossy
one
needs
to
drive
it
by
means
of
an
incoming
laser
beam
called
the
pump
thus
replenishing
the
photons
and
achieving
stationary
state
It
is
important
to
note
however
that
this
state
should
not
be
confused
with
thermal
equilibrium
as
we
are
dealing
with
intrinsic
out
of
equilibrium
physics
here
Experiments
have
so
far
explored
two
main
ways
of
pumping
these
cavities
resonantly
coherent
and
non
resonantly
incoherent
We
will
not
concern
ourselves
with
the
latter
in
this
manuscript
but
we
point
the
interested
reader
to
the
excelent
review
in
Ref
and
references
therein
for
details
Resonant
pumping
is
done
by
means
of
continuous
wave
coherent
laser
beam
which
is
transmitted
through
one
of
the
mirrors
at
energies
and
momenta
close
to
the
LP
or
UP
The
coherence
of
the
beam
is
then
inherited
by
the
polariton
fluid
As
already
mentioned
conservation
of
in
plane
momentum
and
photon
frequency
dictates
that
laser
wave
coming
in
at
an
angle
and
frequency
resonantly
excites
microcavity
mode
with
wavevector
see
Fig
As
result
we
see
that
besides
balancing
the
losses
the
pump
can
also
be
used
to
tune
the
system
to
particular
desired
state
injecting
polaritons
of
given
momentum
and
energy
In
the
following
we
will
consider
that
the
pumping
is
done
quasi
resonantly
close
to
the
bottom
of
the
LP
branch
by
using
homogeneous
planewave
pump
of
amplitude
wavevector
frequency
Depending
on
the
pump
parameters
specifically
the
pumping
angle
which
in
turn
determines
one
can
distinguish
two
qualitatively
different
regimes
the
pump
only
state
where
the
only
stable
configuration
of
the
system
is
the
mode
at
and
the
regime
where
polaritons
at
undergo
stimulated
scattering
and
populate
two
other
modes
Indeed
if
the
pumping
angle
is
such
that
the
wavevector
it
excites
lies
above
the
inflection
point
of
the
LP
dispersion
the
system
can
enter
new
regime
called
optical
parametric
oscillator
see
Sec
for
details
about
OPO
To
quantitatively
account
for
the
driving
and
decay
one
can
use
the
input
output
formalism
that
was
extended
to
planar
microcavities
in
Ref
As
far
as
photonic
losses
are
concerned
we
must
first
note
that
apart
from
the
radiative
decay
channels
already
mentioned
there
is
also
nonradiative
contribution
due
to
photon
absorbtion
inside
the
cavity
The
total
cavity
photon
decay
rate
can
therefore
be
expressed
as
the
sum
where
the
radiative
decay
rate
is
for
normal
incidence
where
is
the
speed
of
light
in
vacuum
and
the
mirror
transmittivity
The
excitons
from
the
QW
are
of
course
also
subject
to
decay
processes
While
their
radiative
decay
can
only
take
place
via
one
of
the
cavity
modes
due
to
strong
light
matter
coupling
(In
general
one
is
in
the
strong
coupling
regime
when
is
greater
than
the
combined
losses
in
the
system
)we
must
include
nonradiative
recombination
processes
as
well
as
the
additional
dephasing
caused
by
interactions
with
carriers
inside
the
well
and
variations
in
well
thickness
All
this
results
in
an
effective
decay
rate
for
excitons
which
is
generally
weaker
than
the
photonic
one
quantified
by
Typical
values
of
both
rates
for
GaAs
based
microcavities
are
shown
in
Tab
Disorder
effects
It
is
noteworthy
to
remark
that
Hamiltonian
Eq
eq
total
ham
does
not
include
any
disorder
We
have
already
mentioned
that
there
are
two
distinct
disorder
classes
in
this
problem
One
is
the
excitonic
disorder
acting
on
length
scales
of
around
nanometers
which
tipically
affects
the
exciton
oscillator
strengths
but
not
the
spatial
polaritonic
density
This
type
of
disorder
stems
from
variations
in
the
QW
thickness
as
well
as
alloy
imperfections
and
is
typically
not
strong
enough
to
dissociate
excitons
Neglecting
excitonic
disorder
also
implies
that
each
localized
exciton
state
couples
to
precisely
one
extended
photon
state
The
resulting
polaritons
are
then
nothing
but
the
coherent
superposition
of
these
excitonic
and
photonic
states
Photonic
disorder
on
the
other
hand
acts
on
lengthscales
on
the
order
of
microns
see
Tab
and
therefore
can
disrupt
the
polariton
density
profile
Photonic
disorder
is
normally
caused
by
the
presence
of
imperfections
inherent
to
the
growth
process
of
the
cavity
mirrors
These
defects
as
well
as
any
layer
mismatch
act
as
scattering
centers
for
the
polaritons
and
can
be
included
in
the
Hamiltonian
as
an
additional
external
scalar
potential
The
simplest
example
is
that
of
spatially
localized
defect
of
strength
located
at
positon
which
can
be
approximated
by
the
potential
To
get
more
quantitative
idea
of
the
physics
involved
it
is
worth
describing
realistic
samples
typically
used
in
state
of
the
art
experiments
Instead
of
simple
metallic
mirrors
such
microcavities
are
usually
bound
by
series
of
up
to
around
distributed
Bragg
reflectors
DBRs
which
are
alternating
dielectric
layers
of
thickness
and
different
refractive
indices
achieving
total
reflectivity
of
over
Multiple
CdTe
or
GaAs
GaN
ZnO
QWs
few
nanometers
thick
are
placed
inside
the
cavity
of
length
at
the
antinodes
maxima
of
the
photonic
electric
field
in
order
to
maximize
the
coupling
between
excitons
and
photons
see
Eq
eq
omega
which
shows
maximum
coupling
for
When
distinct
QWs
are
placed
in
the
cavity
at
the
electric
field
maxima
the
light
matter
coupling
is
enhanced
by
factor
Furthermore
the
cavity
cutoff
frequency
is
normally
chosen
close
to
the
exciton
frequency
Typical
values
of
the
relevant
parameters
for
GaAs
based
microcavities
are
shown
in
Tab
Typical
parameters
of
GaAs
based
microcavity
with
embedded
QWs
Left
column
shows
properties
of
QW
while
the
right
one
refers
to
the
microcavity
For
these
values
the
ratio
between
saturation
and
Coulomb
interaction
is
small
From
Ref
Mean
field
description
As
it
turns
out
useful
approach
for
tackling
the
hydrodynamics
related
problems
that
we
will
encounter
in
the
following
Chapters
is
the
so
called
mean
field
approximation
In
the
context
of
conservative
weakly
interacting
Bose
gases
at
thermal
equilibrium
this
approach
leads
to
the
celebrated
Gross
Pitaevskii
equation
and
in
our
case
it
will
lead
to
its
driven
dissipative
counterpart
If
the
Rabi
energy
is
larger
than
the
pump
detuning
from
the
bottom
of
the
LP
branch
and
also
larger
than
the
decay
rates
and
we
can
start
from
the
lower
polariton
Hamiltonian
of
Sec
The
idea
is
that
one
replaces
the
lower
polariton
operators
in
Eq
eq
LP
ham
with
their
expectation
values
where
is
now
interpreted
as
classical
field
order
parameter
representing
lower
polariton
polarization
describing
the
state
with
momentum
label
The
mean
field
equation
for
is
then
obtained
from
the
Heisenberg
equation
of
motion
by
replacing
all
the
operators
with
their
corresponding
expectation
values
This
mean
field
treatment
is
of
course
exact
as
far
as
the
pumping
and
the
losses
are
concerned
as
well
as
for
any
two
operator
product
such
as
the
polariton
kinetic
energy
However
the
effective
lower
polariton
interactions
are
approximated
following
an
approach
similar
to
the
one
pioneered
by
Bogoliubov
in
the
context
of
atomic
BECs
namely
We
obtain
generalized
Gross
Pitaevskii
equation
GPE
in
momentum
space
under
coherent
driving
which
includes
losses
and
describes
the
time
evolution
of
LP
pt
Here
is
the
lower
polariton
loss
rate
is
the
Fourier
transform
of
the
external
potential
Eq
eq
defect
pot
and
the
effective
polariton
polariton
interaction
strength
is
given
by
Going
beyond
the
mean
field
approximation
to
include
quantum
and
thermal
fluctuations
of
the
quantum
field
is
not
an
easy
task
in
general
see
Ref
for
some
approaches
such
as
using
the
semiclassical
Wigner
representation
In
order
to
obtain
the
corresponding
form
of
Eq
eq
mom
GPE
in
real
space
we
introduce
the
Fourier
transform
of
the
classical
wavefunction
by
means
of
(Replacing
the
discrete
sums
over
states
by
continuous
integrals
is
justified
in
the
limit
of
large
system
)Knowing
that
Fourier
transforms
take
multiplication
to
convolution
we
immediately
get
the
real
space
equation
for
LP
Note
that
the
kinetic
energy
operator
has
particular
dependence
on
the
gradient
instead
of
the
usual
Laplacian
reflecting
the
non
parabolic
LP
dispersion
Furthermore
the
nonlocal
interaction
term
in
the
second
line
of
Eq
eq
convoluted
GPE
has
the
form
of
the
most
general
third
order
nonlinear
term
that
one
can
come
up
with
if
were
an
arbitrary
function
This
freedom
is
somewhat
restrained
in
practice
by
the
fact
that
is
related
to
Eq
eq
LP
inter
by
and
we
have
also
introduced
the
functions
which
are
the
Fourier
transforms
of
the
Hopfield
coefficients
These
coefficients
do
not
have
very
pronounced
momentum
dependence
as
in
fact
for
is
between
and
If
we
neglect
altogether
the
Hopfield
coefficients
in
Eq
eq
convoluted
GPE
and
consider
the
case
of
parabolic
lower
polariton
dispersion
we
finally
obtain
the
real
space
driven
dissipative
GPE
in
its
simplified
form
Here
we
have
also
neglected
the
momentum
dependence
of
the
polariton
loss
rate
justified
by
the
fact
that
at
large
momenta
the
diminishing
radiative
width
is
normally
compensated
by
non
radiative
broadening
To
make
stronger
connection
with
nonlinear
optics
we
note
that
similar
equation
holds
for
cavities
containing
nonlinear
medium
with
the
role
of
being
taken
by
the
susceptibility
of
the
medium
This
equation
allows
for
an
ab
initio
description
of
the
coherently
pumped
system
unlike
the
more
complicated
case
of
incoherent
pumping
where
one
generally
also
needs
phenomenological
description
of
the
various
relaxation
processes
Before
moving
on
we
note
that
one
can
also
describe
the
dynamics
of
polaritons
scattering
against
defect
using
the
two
component
GPE
for
the
coupled
exciton
and
cavity
fields
where
Eq
eq
component
GPE
is
usually
solved
numerically
as
we
shall
see
in
Chapter
Equation
of
state
Because
we
drive
the
system
with
the
coherent
plane
wave
continuous
pump
Eq
eq
pwpump
we
can
look
for
stationary
state
solutions
with
similar
oscillatory
behaviour
In
absence
of
an
external
potential
inserting
this
ansatz
into
Eq
eq
mom
GPE
we
obtain
the
equation
of
state
for
lower
polaritons
where
we
employ
the
simplified
notation
and
One
immediately
notices
that
unlike
the
case
of
the
atomic
mean
field
Eq
eq
atom
MF
the
phase
of
the
polariton
field
amplitude
is
set
by
the
pump
therefore
explicitly
breaking
the
gauge
symmetry
associated
with
global
rotations
of
the
condensate
phase
This
implies
that
unlike
the
atomic
BECs
no
Goldstone
branch
will
be
present
in
the
polaritonic
excitation
spectrum
Note
also
that
the
oscillation
frequency
of
the
condensate
wavefunction
is
fixed
by
the
pump
frequency
In
Chapter
we
will
compare
the
main
features
of
the
quasiparticle
excitation
spectrum
for
polaritons
to
equilibrium
systems
like
cold
atomic
gases
in
the
BEC
regime
Furthermore
locked
phase
clearly
prevents
the
formation
of
phase
dislocations
such
as
vortices
and
solitons
For
this
reason
it
has
been
suggested
and
experimentally
realised
that
the
defect
can
be
located
just
outside
the
pump
spot
leaving
the
condensate
phase
unbound
and
allowing
hydrodynamic
nucleation
of
vortices
vortex
antivortex
pairs
arrays
of
vortices
and
solitons
to
be
observed
when
the
fluid
collides
with
the
extended
defect
Similarly
nucleation
of
vortices
in
the
wake
of
the
obstacle
has
been
observed
in
pulsed
experiments
Pump
energy
blueshift
of
Eq
eq
pump
mf
sq
as
function
of
pump
laser
intensity
in
the
optical
limiter
left
panel
meV
and
bistable
right
panel
meV
regimes
The
arrows
in
the
right
panel
show
the
hysteretic
jumps
between
the
lower
and
upper
branches
at
different
intensities
while
the
dotted
middle
branch
is
single
mode
unstable
The
black
dot
marks
saddle
node
bifurcation
The
repulsive
polaritonic
interaction
is
responsible
for
blueshift
of
the
LP
dispersion
Introducing
the
pump
energy
blueshift
we
can
rewrite
Eq
eq
pump
mf
in
absolute
value
as
where
we
have
also
defined
the
pump
laser
intensity
One
can
now
distinguish
two
qualitatively
different
regimes
based
on
the
value
of
the
bare
pump
detuning
The
first
regime
called
optical
limiter
is
for
and
it
is
plotted
in
the
left
panel
of
Fig
We
see
that
the
pump
blueshift
increases
monotonically
with
increasing
pump
power
The
increase
is
sublinear
as
the
blueshift
moves
the
lower
polariton
dispersion
out
of
resonance
with
the
pump
The
second
regime
shown
in
the
right
panel
of
Fig
is
called
optical
bistability
and
is
no
longer
monotonic
The
blueshift
intensity
curve
has
characteric
shape
and
shows
hysteresis
increasing
the
pump
power
we
get
to
the
first
turning
point
filled
circle
in
the
figure
where
the
blueshift
suddenly
jumps
from
the
low
to
the
high
branch
while
the
reverse
process
of
decreasing
the
pump
power
shows
jump
at
second
turning
point
on
the
curve
Defining
an
interaction
renormalized
pump
detuning
we
see
that
this
quantity
is
positive
up
to
the
second
turning
point
on
the
curve
and
then
changes
sign
becoming
negative
on
the
higher
branch
The
middle
branch
of
the
curve
with
negative
slope
is
dynamically
unstable
while
the
two
turning
points
where
the
stable
and
unstable
solutions
meet
are
called
saddle
node
bifurcations
To
see
this
one
must
consider
small
perturbations
of
wave
vector
of
the
form
where
Substituting
this
into
Eq
eq
mom
GPE
and
linearising
with
respect
to
the
amplitudes
and
we
obtain
an
eigenvalue
problem
from
which
the
two
complex
eigenvalues
can
be
determined
in
straightforward
manner
Dynamical
stability
is
then
ensured
when
for
any
wave
vector
We
can
therefore
distinguish
two
main
types
of
instabilities
depending
on
the
value
of
for
the
degenerate
case
the
system
shows
Kerr
single
mode
or
pump
only
instability
which
only
involves
the
pump
wave
vector
while
the
general
case
of
implies
parametric
instability
signaling
the
onset
of
the
optical
parametric
oscillator
OPO
regime
This
regime
involves
two
separate
modes
apart
from
the
pump
namely
the
so
called
signal
and
idler
states
at
wavevectors
and
respectively
as
we
will
discuss
at
large
in
Sec
OPO
regime
The
driving
pumping
of
swing
by
person
using
the
standing
position
shown
as
an
albeit
contrived
example
of
degenerate
parametric
amplifier
From
Ref
To
understand
why
the
OPO
regime
arises
one
has
to
first
look
at
the
form
of
the
Heisenberg
equation
of
motion
Eq
eq
mom
GPE
This
equation
describes
the
coupled
dynamics
of
an
arbitrary
state
given
an
applied
optical
field
which
excites
polaritons
with
in
plane
wavevector
As
result
of
polariton
polariton
interactions
the
coherent
occupation
at
generic
wavevectors
and
can
scatter
into
and
process
one
can
represent
as
As
these
states
further
scatter
into
others
by
the
same
matter
wave
mixing
mechanism
we
get
cascade
of
so
called
satellite
states
with
diminishing
population
Out
of
the
multitude
of
possible
scattering
channels
we
limit
our
description
to
the
one
describing
the
process
which
is
precisely
the
parametric
scattering
of
two
pump
polaritons
into
the
signal
and
idler
states
that
we
just
introduced
in
Sec
As
it
turns
out
due
to
the
peculiar
shape
of
the
lower
polariton
dispersion
relation
Eq
eq
polariton
dispersion
this
scattering
process
can
occur
in
completely
resonant
way
with
conservation
of
both
in
plane
momentum
as
well
as
energy
expressed
by
the
phase
matching
conditions
and
we
have
introduced
here
the
signal
and
idler
frequencies
The
parametric
scattering
condition
that
has
to
be
satisfied
then
reads
We
must
further
emphasize
the
importance
of
the
unique
form
of
the
LP
dispersion
in
Eq
eq
conservations
as
it
allows
efficient
parametric
coupling
on
relatively
broad
range
of
pump
momenta
Parametric
scattering
would
not
be
possible
for
particles
with
quadratic
dispersion
in
the
atomic
case
for
instance
one
must
employ
the
use
of
an
optical
lattice
in
order
to
deform
the
dispersion
and
promote
parametric
amplification
While
this
may
all
seem
far
removed
from
everyday
experience
we
remind
the
reader
that
parametric
amplifiers
and
oscillators
are
commonly
found
in
mechanical
as
well
as
electronic
systems
For
instance
one
can
consider
person
on
swing
in
the
standing
position
as
shown
in
Fig
and
described
in
detail
in
Ref
In
this
particular
example
we
are
dealing
with
degenerate
parametric
amplifier
as
the
signal
and
idler
have
the
same
frequency
and
are
represented
by
the
swing
The
person
driving
the
swing
then
acts
as
the
pump
and
one
can
see
that
his
or
hers
up
down
motion
has
periodicity
which
is
twice
that
of
the
swing
therefore
in
this
case
schematic
depiction
of
the
polariton
parametric
scattering
process
with
the
signal
state
located
at
zero
momentum
and
the
pump
close
to
the
inflection
point
of
the
LP
dispersion
is
shown
in
Fig
It
is
clear
that
for
given
signal
wavevector
Eq
eq
conservations
then
uniquely
determines
the
pump
and
idler
states
Upper
UP
and
lower
LP
polariton
dispersions
are
shown
as
solid
black
lines
while
the
original
exciton
and
cavity
photon
are
represented
by
red
dashed
lines
as
function
of
the
laser
wavevector
The
red
circle
represents
the
pump
polaritons
being
parametrically
scattered
into
the
signal
upper
blue
triangle
and
idler
lower
green
triangle
states
The
scattering
process
depicted
by
the
curved
arrows
is
resonant
conserving
both
energy
and
momentum
Parameters
meV
and
The
fact
that
is
close
to
is
not
incidental
but
was
observed
in
multiple
experiments
performed
in
the
OPO
regime
as
well
as
numerical
simulations
of
the
full
GP
equation
An
intuitive
explanation
was
put
forward
based
on
the
interaction
induced
blueshift
of
the
LP
dispersion
We
now
turn
our
attention
to
the
mechanism
explaining
the
onset
of
the
parametric
scattering
instability
in
the
simple
case
of
the
optical
limiter
regime
shown
in
the
left
panel
of
Fig
As
already
mentioned
the
polariton
polariton
repulsive
interaction
causes
blueshift
of
the
LP
dispersion
proportional
to
the
pump
mode
population
This
shift
also
affects
the
signal
and
idler
frequencies
bringing
them
in
resonance
with
the
pump
frequency
The
pump
population
and
hence
the
shift
of
the
optical
limiter
is
monotonically
increasing
function
of
laser
intensity
quantified
by
Eq
eq
pump
mf
sq
As
result
there
is
certain
minimum
threshold
intensity
where
the
resonance
starts
and
the
pump
only
solution
becomes
unstable
towards
parametric
scattering
Further
increasing
the
laser
power
we
hit
an
upper
threshold
where
the
blueshift
becomes
too
large
and
the
resonance
is
lost
forbiding
parametric
oscillation
This
mechanism
defines
certain
window
of
pump
intensities
where
the
parametric
instability
can
arise
the
region
depicted
by
the
dashed
line
in
Fig
Inside
this
window
the
pump
only
solution
still
exists
but
it
is
unstable
towards
OPO
Signal
blue
upper
triangles
pump
red
circles
and
idler
green
lower
triangles
mean
field
energy
blueshifts
as
function
of
pump
laser
intensity
in
the
optical
limiter
regime
The
dashed
line
is
the
pump
blueshift
in
the
pump
only
solution
Eq
eq
pump
mf
sq
while
the
vertical
dotted
line
marks
the
pump
intensity
used
in
Fig
Parameters
are
meV
The
simplest
ansatz
one
can
make
for
describing
the
OPO
regime
above
threshold
is
to
consider
the
three
separate
states
signal
pump
and
idler
each
with
its
own
population
wavevector
and
frequency
in
the
form
where
energy
and
momentum
conservation
dictate
that
and
As
before
we
insert
this
ansatz
into
Eq
eq
mom
GPE
and
we
obtain
three
coupled
complex
nonlinear
equations
for
which
determine
the
dynamics
of
the
signal
pump
and
idler
It
is
important
to
note
that
these
equations
become
closed
only
if
we
neglect
the
multiple
scattering
processes
mentioned
at
the
start
of
this
Section
and
consider
only
the
conversion
of
two
pump
polaritons
into
signal
and
an
idler
polariton
We
will
not
give
the
full
form
of
these
OPO
equations
of
state
but
instead
refer
the
reader
to
Ref
where
complete
discussion
of
the
oscillation
threshold
and
ensuing
dynamics
is
given
Fig
shows
particular
solution
of
the
equations
for
the
optical
limiter
case
One
can
see
the
smooth
growth
of
the
signal
population
and
subsequent
depletion
of
the
pump
inside
the
OPO
dashed
region
It
is
of
course
never
implied
that
the
whole
OPO
window
is
dynamically
stable
and
in
fact
stability
analysis
needs
to
be
performed
by
adding
small
fluctuations
on
top
of
the
now
three
coupled
states
in
manner
similar
to
Eq
eq
pump
fluctuations
following
Ref
The
complete
treatment
will
be
presented
in
Chapter
Before
moving
on
it
is
worth
noting
that
limitation
of
the
ansatz
Eq
eq
state
ansatz
is
that
it
does
not
allow
solving
the
so
called
"selection
"problem
of
determining
the
value
of
above
threshold
As
result
this
value
has
to
be
considered
as
an
input
parameter
to
our
problem
Futhermore
similarly
to
the
pump
only
case
and
inherent
in
any
mean
field
approach
we
are
of
course
neglecting
any
quantum
fluctuations
of
the
three
macroscopically
occupied
modes
Last
but
not
least
we
are
also
neglecting
the
so
called
TE
TM
transverse
electric
transverse
magnetic
splitting
which
can
act
as
an
effective
spin
orbit
interaction
term
This
approximation
allows
one
to
work
with
single
spin
state
considering
the
circular
polarization
of
the
pump
to
be
fully
transmitted
to
the
signal
and
ider
as
well
We
now
turn
our
attention
to
separate
discussion
concerning
the
phases
of
the
three
OPO
states
As
before
the
phase
of
the
pump
is
fixed
by
the
outside
laser
and
now
the
sum
of
the
signal
and
idler
phases
is
given
by
the
matching
condition
Their
phase
difference
however
is
free
parameter
and
as
such
it
is
spontaneously
chosen
by
the
system
every
time
the
threshold
is
crossed
In
summary
the
OPO
equations
of
motion
for
the
three
states
posess
symmetry
corresponding
to
simultaneous
and
opposite
phase
rotation
of
the
signal
and
idler
that
is
This
intrinsic
symmetry
is
spontaneously
broken
above
threshold
random
value
of
the
phase
of
the
signal
and
hence
idler
being
selected
at
each
realization
of
the
experiment
We
can
thus
see
the
onset
of
OPO
as
phase
transition
in
out
of
equilibrium
conditions
signaling
the
spontaneous
macroscopic
occupation
of
the
signal
and
idler
states
Below
threshold
the
incoherent
signal
and
idler
emission
starts
being
stimulated
and
then
becomes
coherent
once
the
threshold
is
crossed
These
coherence
properties
were
studied
both
experimentally
as
well
as
theoretically
by
means
of
quantum
Monte
Carlo
methods
The
order
parameter
of
the
transition
is
the
matter
polarization
the
analog
of
magnetization
in
the
ferromagnetic
phase
transition
At
the
critical
point
noise
fluctuations
are
amplified
and
macroscopic
polarization
is
spontaneously
created
The
phase
transition
can
be
first
or
second
order
depending
on
the
pump
parameters
both
behaviors
have
been
observed
in
experiments
In
the
optical
limiter
case
detailed
above
both
pump
and
signal
are
continuous
functions
of
the
incident
laser
power
and
the
transition
is
second
order
as
the
signal
increases
smoothly
In
the
pattern
formation
language
one
clasifies
the
associated
bifurcation
as
being
of
the
supercritical
Hopf
type
If
one
considers
the
bistable
pump
regime
of
the
right
panel
of
Fig
however
things
get
more
complicated
as
the
bifurcation
can
now
be
of
the
subcritical
Hopf
type
and
hence
the
associated
OPO
initiates
discontinuously
It
is
relatively
well
known
that
in
an
infinite
uniform
two
dimensional
equilibrium
system
thermal
fluctuations
at
any
strictly
nonzero
temperature
are
strong
enough
to
destroy
the
fully
ordered
BEC
state
The
associated
phase
transition
in
that
case
is
instead
described
by
the
so
called
Berezinskii
Kosterlitz
Thouless
BKT
theory
and
is
driven
by
interactions
as
opposed
to
being
purely
statistical
phenomenon
As
the
polariton
system
is
out
of
equilibrium
and
in
practice
no
experimental
system
is
truly
infinite
anyway
the
exact
nature
of
the
OPO
transition
in
this
case
has
not
yet
been
fully
elucidated
although
it
seems
that
BKT
like
physics
has
been
both
predicted
and
observed
under
incoherent
pumping
The
breaking
of
the
symmetry
in
an
atomic
gas
below
the
BEC
critical
point
is
related
to
the
appearance
of
soft
Goldstone
mode
in
the
form
of
zero
sound
Similar
physics
arises
below
the
Curie
temperature
in
ferromagnet
where
the
magnon
excitations
spin
waves
are
the
corresponding
Goldstone
bosons
due
to
the
spontaneous
breaking
of
the
initial
rotational
symmetry
Since
the
resonantly
pumped
polariton
system
above
threshold
spontaneously
breaks
symmetry
Goldtone's
theorem
predicts
massless
mode
corresponding
in
this
case
to
spatial
twist
of
the
signal
idler
phases
Indeed
such
mode
has
been
identified
in
Ref
by
calculating
the
quasiparticle
excitation
spectrum
on
top
of
the
three
state
OPO
anzatz
Eq
eq
state
ansatz
The
Bogoliubov
spectrum
in
question
is
shown
in
Fig
and
the
Goldstone
mode
represented
by
the
solid
heavy
line
labeled
""Real
left
and
imaginary
right
parts
of
the
Bogoliubov
spectrum
of
collective
excitations
on
top
of
the
OPO
stationary
state
as
function
of
The
chosen
pump
intensity
is
marked
with
dotted
line
in
Fig
The
breaking
of
the
symmetry
corresponding
to
the
simultaneous
phase
rotation
of
the
signal
and
idler
is
manifest
in
the
presence
of
the
Goldstone
mode
Top
panels
Integrated
photon
emission
intensities
for
the
signal
red
dots
pump
blue
squares
and
idler
green
triangles
as
function
of
the
renormalized
pump
intensity
left
and
real
space
signal
emission
corresponding
to
positions
including
random
photonic
disorder
shown
as
contour
lines
right
Middle
panels
Photonic
component
of
the
OPO
spectrum
log
scale
as
function
of
energy
and
momentum
left
and
integrated
in
momentum
right
Note
that
the
emission
is
like
in
energy
Bottom
panels
Full
photonic
emission
left
and
filtered
real
space
emission
of
signal
pump
and
idler
last
panels
From
Ref
Note
that
both
the
real
and
imaginary
parts
of
the
frequency
of
this
gapless
mode
tend
to
zero
in
the
long
wavelength
limit
Close
to
this
point
the
real
part
has
non
zero
slope
due
to
the
finite
wavevector
of
the
injected
pump
polaritons
while
the
imaginary
part
has
parabolic
shape
This
means
that
as
opposed
to
the
BEC
case
localized
perturbation
in
the
OPO
regime
does
not
freely
propagate
as
sound
wave
Instead
one
should
observe
diffusive
like
behaviour
where
the
localized
signal
idler
phase
perturbation
decays
and
is
then
dragged
along
by
the
flow
of
the
pump
Another
important
difference
compared
to
the
BEC
spectrum
is
the
absence
of
singularity
of
the
Goldstone
branch
around
the
point
Further
insight
into
OPO
physics
for
realistic
experimental
conditions
can
be
gained
by
numerically
solving
the
full
component
GP
equation
eq
component
GPE
for
the
coupled
exciton
and
cavity
photon
fields
as
first
done
in
Ref
similar
approach
reviewed
in
detail
in
Ref
will
be
employed
in
Chapter
where
we
study
multicomponent
superfluidity
in
the
OPO
regime
Using
continuous
wave
pump
with
top
hat
profile
one
injects
polaritons
at
specific
wavevector
close
to
the
inflection
point
of
the
LP
dispersion
Above
certain
threshold
pump
power
random
numerical
noise
starts
getting
amplified
and
the
OPO
instability
sets
in
populating
the
three
states
After
the
system
reaches
steady
state
one
can
obtain
the
time
evolution
of
the
photon
and
exciton
fields
and
both
in
real
and
momentum
space
Since
experiments
measure
the
photonic
component
of
the
polariton
field
that
is
what
is
plotted
in
Fig
Filtering
in
narrow
cone
in
momentum
or
equivalently
in
energy
one
can
obtain
separately
the
emission
of
the
signal
pump
and
idler
states
while
the
energy
momentum
spectrum
can
be
calculated
as
the
discrete
Fourier
transform
of
the
total
photonic
wavefunction
recorded
at
equally
spaced
time
intervals
Inspecting
Fig
we
see
that
the
threshold
is
continuous
one
in
this
case
with
the
OPO
first
switching
on
in
narrow
region
in
the
center
of
the
system
and
then
extending
to
the
size
of
the
whole
pump
spot
before
finally
surviving
only
on
its
edge
when
pumping
is
increased
even
further
Interestingly
the
OPO
transition
is
signaled
by
the
appearance
of
moving
striped
pattern
in
the
polariton
density
see
bottom
left
panel
spontaneously
breaking
the
translational
symmetry
of
the
system
This
pattern
is
caused
by
the
interference
of
the
coherent
emission
of
the
three
states
and
is
similar
to
the
Rayleigh
nard
instability
in
the
field
of
fluid
dynamics
The
threshold
point
in
the
OPO
case
is
where
the
stimulated
scattering
rate
exceeds
the
losses
while
the
nard
cells
form
when
the
temperature
gradient
playing
the
role
of
pumping
exceeds
the
viscosity
dissipation
In
these
simulations
the
pump
and
hence
also
signal
and
idler
wavevectors
are
oriented
along
the
direction
so
the
fringes
appear
along
the
axis
They
are
however
usually
not
visible
in
experiments
due
to
the
time
integration
of
the
photonic
emission
in
most
experimental
setups
Seen
in
this
new
light
the
OPO
phase
transition
has
an
associated
Goldstone
mode
that
would
correspond
to
rigid
translation
of
the
spatial
stripe
pattern
The
photonic
component
of
the
OPO
spectrum
is
shown
in
the
middle
pannel
of
Fig
Integrating
the
spectrum
over
all
momenta
one
sees
that
as
result
of
being
in
the
steady
state
the
emission
is
like
in
energy
while
it
is
broadened
in
momentum
due
to
the
finite
size
of
the
pump
The
satellite
states
equally
spaced
in
energy
and
less
populated
are
also
visible
in
the
middle
left
panel
They
are
due
to
secondary
scattering
processes
from
the
signal
idler
into
the
pump
and
secondary
signal
secondary
idler
and
so
on
Finally
the
filtered
emission
from
the
signal
pump
and
idler
states
is
shown
in
the
lower
panels
of
Fig
Since
we
are
in
the
steady
state
the
density
profiles
are
time
independent
and
the
idler
emission
is
less
intense
due
to
the
smaller
radiative
decay
of
polaritons
at
large
momenta
One
important
difference
from
cold
atomic
gases
in
the
BEC
regime
is
the
presence
of
nonvanishing
currents
which
can
flow
across
the
microcavity
even
in
the
ground
state
in
polariton
system
These
currents
can
be
seen
in
the
second
panel
for
the
signal
state
and
are
due
the
complex
interaction
between
nonlinearities
dissipation
and
parametric
gain
Note
that
we
are
not
refering
here
to
the
uniform
flow
caused
by
the
signal
being
at
finite
momentum
as
in
fact
this
dominant
current
is
substracted
from
the
plot
OPO
superfluidity
Snapshots
of
the
time
evolution
of
the
phase
upper
row
and
the
density
bottom
row
of
an
OPO
polariton
condensate
triggered
by
pulsed
probe
carrying
vortex
From
Ref
While
still
on
the
topic
of
persistent
currents
we
can
make
the
connection
to
Sec
and
discuss
superfluid
related
phenomena
in
the
OPO
regime
prime
example
of
this
is
the
stability
of
quantized
vortices
reported
by
Sanvitto
and
coworkers
in
Ref
They
used
pulsed
narrow
probe
beam
in
Gauss
Laguerre
state
in
order
to
imprint
finite
angular
momentum
on
the
OPO
condensate
previously
created
by
means
of
wide
pump
spot
In
similar
fashion
to
the
rotating
BEC
experiments
performed
with
ultracold
gases
the
system
responded
by
forming
quantized
vortex
which
eventually
drifted
out
of
the
condensate
due
to
the
simply
connected
geometry
see
Fig
The
vortex
was
visible
in
both
the
signal
and
idler
state
and
its
lifetime
was
much
longer
than
the
average
polariton
lifetime
In
similar
direction
Ref
presented
theoretical
study
predicting
the
formation
of
spontaneous
quantized
vortices
in
the
OPO
regime
provided
the
pump
spot
is
spatially
narrow
These
vortices
were
shown
to
drift
to
their
stable
equilibrium
positions
caried
by
the
nontrivial
OPO
currents
discussed
in
the
previous
paragraph
Further
work
by
Tosi
et
al
investigated
the
appearance
and
dynamics
of
vortex
antivortex
pairs
created
by
pulsed
probe
much
narrower
than
the
pump
spot
It
was
shown
that
while
both
vortices
and
antivortices
form
along
the
edge
of
the
probe
laser
they
form
at
different
relative
positions
as
determined
by
the
steady
state
currents
We
will
deepen
our
investigation
of
OPO
superfluidity
in
Chapter
where
we
look
at
the
flow
of
the
signal
pump
and
idler
states
against
small
defect
Finally
departing
somewhat
from
the
OPO
regime
it
was
shown
in
Ref
that
despite
not
strictly
obeying
the
Landau
criterion
the
superfluid
density
of
incoherently
pumped
polaritons
need
not
vanish
However
due
to
their
nonequilibrium
nature
the
normal
fraction
would
remain
finite
even
at
zero
temperature
Last
but
not
least
the
same
publication
proposes
using
artificial
gauge
fields
generated
by
applying
real
magnetic
field
in
an
anti
Helmholtz
configuration
for
measuring
the
superfluid
and
normal
components
of
the
system
Applications
Drag
in
coherently
driven
polariton
fluid
In
conservative
quantum
liquid
flowing
past
small
defect
the
Landau
criterion
for
superfluidity
presented
in
Sec
links
the
onset
of
dissipation
at
critical
fluid
velocity
with
the
shape
of
the
fluid
collective
excitation
spectrum
In
particular
for
weakly
interacting
Bose
gases
the
dispersion
of
the
low
energy
excitation
modes
being
linear
implies
that
the
critical
velocity
for
superflow
coincides
with
the
speed
of
sound
Clearly
this
is
strictly
correct
only
for
vanishingly
small
perturbations
while
for
defect
with
finite
size
and
strength
the
critical
velocity
can
be
smaller
than
due
to
vortex
creation
by
the
macroscopic
defect
Experimental
images
of
the
real
top
row
and
momentum
bottom
row
space
polariton
density
extracted
from
the
near
far
field
light
emitted
from
the
cavity
The
different
columns
correspond
to
increasing
values
of
the
polariton
density
from
left
to
right
For
the
highest
density
polariton
superfluidity
is
apparent
as
suppression
of
the
real
space
modulation
panel
III
accompanied
by
the
collapse
of
the
Rayleigh
scattering
ring
panel
VI
From
Ref
However
even
for
perturbatively
weak
defects
in
out
of
equilibrium
systems
where
the
spectrum
of
excitations
is
complex
the
validity
of
the
Landau
criterion
has
to
be
questioned
In
the
particular
case
of
coherently
driven
polaritons
in
the
pump
only
configuration
it
has
been
predicted
and
later
observed
that
scattering
is
suppressed
at
either
strong
enough
pump
powers
or
small
enough
flow
velocities
see
Fig
Yet
on
closer
scrutiny
it
has
been
shown
that
despite
the
apparent
validity
of
the
Landau
criterion
the
system
always
experiences
residual
drag
force
even
in
the
limit
of
asymptotically
large
densities
or
small
velocities
This
result
has
been
proven
by
numerically
solving
the
Gross
Pitaevskii
equation
describing
the
resonantly
driven
polariton
system
in
presence
of
non
perturbative
extended
defect
Here
the
drag
force
exerted
by
the
defect
on
the
fluid
has
been
shown
to
display
smooth
crossover
from
the
subsonic
to
the
supersonic
regime
similar
to
what
it
has
been
found
in
the
case
of
non
resonantly
pumped
polaritons
In
this
Chapter
we
find
an
even
richer
phenomenology
for
the
dependence
of
the
drag
force
on
the
fluid
velocity
and
two
different
kinds
of
crossovers
from
the
sub
to
the
supercritical
regime
Furthermore
we
show
that
the
origin
of
the
residual
drag
force
which
in
agreement
with
Ref
lies
in
the
polariton
lifetime
only
can
be
demonstrated
even
within
linear
response
approximation
More
specifically
we
apply
the
linear
response
theory
described
in
Sec
for
the
case
of
an
equilibrium
condensate
and
extended
here
to
driven
dissipative
fluid
in
order
to
analytically
evaluate
the
drag
force
exerted
by
the
coherently
driven
polariton
fluid
in
the
pump
only
configuration
on
point
like
defect
To
simplify
the
formalism
we
restrict
our
analysis
to
the
case
of
resonant
pumping
close
to
the
bottom
of
the
lower
polariton
dispersion
where
the
dispersion
is
quadratic
Here
the
properties
of
the
collective
excitation
spectrum
have
been
shown
to
be
uniquely
determined
by
three
parameters
only
the
fluid
velocity
the
interaction
renormalised
pump
detuning
and
the
polariton
lifetime
In
particular
the
sign
of
the
detuning
determines
three
qualitatively
different
types
of
spectra
linear
for
diffusive
like
for
and
gapped
for
For
both
cases
of
linear
and
diffusive
spectra
we
find
qualitatively
similar
behaviour
of
the
drag
force
as
function
of
the
fluid
velocity
In
particular
the
drag
displays
crossover
from
subsonic
or
superfluid
regime
-characterised
by
the
absence
of
quasiparticle
excitations
-to
supersonic
regime
-where
Cherenkov
like
waves
similar
to
those
of
Sec
are
generated
by
the
defect
and
propagate
into
the
fluid
The
crossover
becomes
sharper
for
increasing
polariton
lifetimes
and
displays
the
typical
threshold
behaviour
for
with
critical
velocity
given
by
the
speed
of
sound
of
the
linear
regime
exactly
as
for
weakly
interacting
equilibrium
superfluids
in
the
case
of
perturbatively
weak
defects
This
behaviour
is
similar
to
the
one
predicted
for
polariton
superfluids
excited
non
resonantly
where
the
spectrum
in
that
case
is
diffusive
like
However
for
gapped
spectra
at
we
find
that
the
critical
velocity
governing
the
drag
crossover
exceeds
the
speed
of
sound
and
we
determine
an
analytical
expression
of
as
function
of
the
detuning
Furthermore
for
the
drag
has
threshold
like
behaviour
qualitatively
different
from
the
one
of
weakly
interacting
equilibrium
superfluids
with
the
drag
jumping
discontinuously
from
zero
to
finite
value
at
We
evaluate
the
drag
as
function
of
the
polariton
lifetime
and
find
for
all
three
cases
that
In
the
supercritical
regime
the
lifetime
tends
to
suppress
the
propagation
of
the
Cherenkov
waves
away
from
the
defect
and
therefore
to
suppress
the
drag
Instead
well
in
the
subcritical
regime
we
find
that
the
residual
drag
goes
linearly
to
zero
with
the
polariton
lifetime
in
agreement
to
what
was
found
in
Ref
by
making
use
of
non
perturbative
numerical
analysis
for
finite
size
defect
Similar
to
Ref
here
we
do
also
find
that
the
residual
drag
in
the
subcritical
regime
can
be
explained
in
terms
of
an
asymmetric
perturbation
induced
in
the
fluid
by
the
defect
in
the
direction
of
the
fluid
velocity
This
Chapter
is
structured
as
follows
In
Sec
we
briefly
review
the
linear
approximation
scheme
and
extend
it
to
the
case
of
resonantly
pumped
polariton
fluid
We
classify
the
three
types
of
collective
excitation
spectra
in
the
simplified
case
of
excitation
close
to
the
bottom
of
the
LP
dispersion
in
Sec
In
Sec
we
derive
the
drag
force
and
characterise
the
crossover
from
the
subsonic
to
the
supersonic
regime
in
the
three
cases
of
zero
positive
and
negative
detuning
In
this
section
we
also
evaluate
the
drag
as
function
of
the
polariton
lifetime
interpreting
therefore
the
results
of
Ref
Brief
conclusions
are
drawn
is
Sec
Linear
response
As
explained
in
Chapter
the
description
of
cavity
polaritons
resonantly
excited
by
an
external
laser
is
usually
formulated
in
terms
of
classical
non
linear
Schr
odinger
equation
or
Gross
Pitaevskii
equation
for
the
LP
field
see
Eq
eq
rspace
GPE
The
LP
dispersion
is
expressed
in
terms
of
the
photon
and
exciton
energies
the
photon
mass
and
the
Rabi
splitting
see
Eq
eq
polariton
dispersion
Because
polaritons
continuously
decay
at
rate
see
Sec
the
cavity
is
replenished
by
continuous
wave
resonant
pump
at
wavevector
we
will
later
assume
directed
along
the
direction
and
frequency
see
Eq
eq
pwpump
Note
that
as
discussed
in
Secs
-Eq
eq
basic
is
simplified
description
of
the
polariton
system
It
implies
that
the
interaction
nonlinearities
are
small
enough
not
to
mix
the
lower
and
upper
polariton
branches
and
that
we
pump
far
from
the
UP
dispersion
Moreover
starting
from
formulation
in
terms
of
coupled
exciton
and
photon
fields
the
polariton
lifetime
would
be
momentum
dependent
and
similarly
the
polariton
polariton
interaction
strength
is
not
contact
like
as
instead
assumed
in
Eq
eq
basic
However
as
shown
in
Appendix
these
simplifications
do
not
affect
our
results
qualitatively
rather
they
allow
us
to
write
them
in
terms
of
simpler
expressions
Furthermore
we
have
checked
that
whenever
the
system
is
excited
near
the
bottom
of
the
lower
polariton
dispersion
the
results
for
the
drag
force
reported
in
Sec
coincide
with
the
ones
obtained
using
the
photon
exciton
coupled
field
description
Eq
eq
component
GPE
introduced
in
Chapter
The
potential
in
Eq
eq
dispe
describes
defect
which
can
be
either
naturally
present
in
the
cavity
mirror
or
it
can
be
created
by
an
additional
laser
Later
on
we
will
assume
the
defect
to
be
point
like
and
weak
so
that
we
can
apply
the
linear
response
approximation
As
detailed
in
Sec
one
divides
the
response
of
the
LP
field
in
mean
field
component
corresponding
to
the
case
when
the
perturbing
potential
is
absent
and
fluctuation
part
reflecting
the
linear
response
of
the
system
to
the
perturbing
potential
see
Eq
eq
ansatz
atoms
By
substituting
eq
mfield
into
eq
basic
we
obtain
mean
field
equation
and
by
retaining
only
the
linear
terms
in
the
fluctuation
field
and
the
defect
potential
the
following
first
order
equation
in
the
equivalent
of
Eq
eq
GP
atoms
system
where
the
operator
is
given
by
compare
to
Eq
eq
ourL
for
the
atomic
case
with
We
solved
the
complex
cubic
mean
field
equation
eq
pump
mf
for
in
Sec
Here
we
want
to
study
the
response
of
the
system
to
the
presence
of
the
defect
and
how
different
behaviours
of
the
onset
of
dissipation
can
be
described
in
terms
of
the
different
excitation
spectra
one
can
get
for
polaritons
resonantly
pumped
close
to
the
bottom
of
the
LP
dispersion
Collective
excitation
spectra
for
the
subsonic
thick
solid
black
line
at
with
and
supersonic
dashed
red
line
at
regimes
and
for
an
interaction
renormalised
pump
detuning
and
Real
parts
of
the
spectra
are
plotted
in
the
left
panels
and
the
corresponding
imaginary
parts
in
the
right
panels
for
-note
that
in
our
description
the
spectrum
imaginary
parts
do
not
depend
on
the
fluid
velocity
Spectrum
of
collective
excitations
Momentum
space
response
top
row
arb
units
and
normalised
real
space
wavefunction
bottom
row
corresponding
to
the
nondiffusive
spectra
-in
Fig
System
parameters
and
is
left
column
middle
column
and
right
column
Momentum
space
response
top
row
arb
units
and
normalised
real
space
wavefunction
bottom
row
corresponding
to
the
diffusive
spectra
-in
Fig
Polariton
lifetime
and
left
column
middle
column
and
right
column
The
spectrum
of
the
collective
excitations
can
be
obtained
by
diagonalising
the
operator
in
the
momentum
space
representation
following
the
approach
of
Sec
see
Eq
eq
translated
where
The
description
of
the
spectrum
simplifies
in
the
case
when
the
pumping
is
close
to
the
bottom
of
the
LP
dispersion
that
can
be
approximated
as
parabolic
see
Eq
eq
parabolic
disp
where
is
the
fluid
velocity
and
is
the
LP
mass
of
Eq
eq
LP
mass
This
simplification
allows
one
to
describe
the
complex
spectrum
in
terms
of
three
parameters
only
namely
the
fluid
velocity
the
interaction
renormalised
pump
detuning
defined
in
Sec
and
the
LP
lifetime
One
observation
we
can
immediately
make
is
that
unlike
the
atomic
case
one
will
no
longer
have
sharp
corner
(Unless
of
course
)in
the
spectrum
at
as
that
feature
depended
on
the
equality
up
to
sign
between
the
diagonal
and
off
diagonal
elements
of
The
elementary
excitation
spectrum
of
coherently
driven
polaritons
reads
compare
this
to
the
spectrum
given
by
Eq
eq
boosted
bogoliubov
where
If
energies
are
measured
in
units
of
the
mean
field
energy
blue
shift
we
will
use
the
notation
and
then
the
fluid
velocity
is
measured
in
units
of
the
speed
of
sound
In
order
to
make
connection
with
the
current
experiments
note
that
for
blue
shifts
in
the
range
meV
typical
values
of
the
speed
of
sound
are
Similarly
for
common
values
of
the
LP
mass
the
range
in
momenta
in
Fig
comes
of
the
order
of
Hermitian
left
column
and
anti
Hermitian
right
column
coupling
between
two
damped
oscillators
of
energies
decay
rates
and
interaction
energy
Top
row
shows
the
coupling
matrices
while
middle
bottom
row
shows
the
evolution
of
the
real
complex
eigenvalues
with
increasing
From
Ref
The
spectrum
eq
spect
can
be
classified
according
to
the
sign
of
the
interaction
renormalised
pump
detuning
-see
Fig
For
panels
the
real
part
of
the
spectrum
lacks
the
sonic
behaviour
at
small
and
shows
gap
that
increases
with
while
the
imaginary
part
is
determined
by
the
polariton
lifetime
only
If
one
applies
the
Landau
criterion
for
the
real
part
of
the
spectrum
only
then
one
finds
critical
velocity
always
larger
than
the
speed
of
sound
for
If
the
fluid
velocity
is
subcritical
see
the
black
solid
lines
in
Fig
then
no
quasiparticles
can
be
excited
and
thus
for
infinitely
living
polaritons
the
fluid
would
experience
no
drag
when
scattering
against
the
defect
For
the
case
of
the
branch
touches
the
plane
in
one
point
see
left
column
of
Fig
resulting
in
localized
perturbation
around
the
defect
with
an
additional
stripe
pattern
of
wavevector
caused
by
the
interference
of
the
pump
with
the
momentum
of
the
scattered
state
For
supercritical
velocities
instead
see
the
red
dashed
lines
in
Fig
one
expects
dissipation
in
the
form
of
radiation
of
Cherenkov
like
waves
from
the
defect
into
the
fluid
In
the
supercritical
regime
the
set
of
wavevectors
for
which
form
closed
curve
in
space
with
no
singularity
of
the
derivative
see
middle
column
of
Fig
As
explained
in
the
geometrical
model
presented
in
Sec
this
means
the
radiation
can
be
emitted
in
all
possible
directions
around
the
defect
This
as
we
will
see
in
the
next
section
will
imply
that
the
drag
force
for
goes
abruptly
rather
than
continuously
from
zero
at
to
finite
value
at
The
spectrum
gap
closes
to
zero
in
the
resonant
situation
at
when
the
two
branches
touch
at
panels
of
Fig
Here
the
real
part
of
the
spectrum
displays
the
standard
sonic
dispersion
at
small
wavevectors
as
for
the
weakly
interacting
bosonic
gases
of
Chapter
with
slope
given
by
The
imaginary
part
as
in
the
previous
case
is
constant
and
equal
to
It
is
clear
therefore
that
in
this
case
when
one
recovers
the
equilibrium
results
valid
for
weakly
interacting
gases
where
the
critical
velocity
for
superfluidity
equals
the
speed
of
sound
and
the
drag
displays
threshold
like
behaviour
Here
in
the
supersonic
regime
the
closed
curve
has
instead
singularity
resulting
in
the
standard
Mach
cone
of
aperture
inside
which
radiation
from
the
defect
cannot
be
emitted
see
right
column
of
Fig
as
well
as
Fig
of
Sec
For
the
real
parts
of
the
two
Bogoliubov
branches
cross
and
stick
together
giving
rise
to
flat
regions
near
the
crossing
points
This
is
due
to
level
attraction
caused
by
the
anti
Hermitian
coupling
of
Eq
eq
opell
and
is
accompanied
by
splitting
of
the
imaginary
parts
as
illustrated
in
Fig
We
further
distinguish
two
separate
cases
and
In
the
first
case
panels
there
is
only
one
flat
region
in
momentum
space
and
the
intensity
of
the
Rayleigh
scattering
is
amplified
on
segment
situated
at
and
oriented
parallel
to
the
axis
as
shown
in
the
left
column
of
Fig
consequence
of
this
parametric
amplification
is
the
long
shadow
seen
in
the
real
space
image
As
the
imaginary
part
only
has
one
peak
corresponding
to
the
pump
this
is
the
precursor
of
the
Kerr
single
mode
instability
described
in
Sec
The
second
case
panels
has
different
momentum
space
topology
with
two
flat
regions
producing
two
distinct
peaks
in
the
imaginary
part
As
soon
as
these
become
positive
one
enters
the
regime
of
parametric
oscillation
see
Sec
with
the
signal
and
idler
momenta
situated
at
the
two
maxima
The
corresponding
far
field
image
plotted
in
the
middle
column
of
Fig
shows
two
pronounced
peaks
on
the
line
passing
through
the
interference
of
which
results
in
near
field
"zebra
like
"pattern
For
very
small
fluid
velocities
see
right
column
of
Fig
the
two
Bogoliubov
branches
cross
on
ring
of
wavevectors
giving
rise
to
cylindrical
wavefronts
We
note
that
spectra
-have
no
correspondence
in
equilibrium
systems
because
finite
polariton
lifetime
is
needed
in
order
to
insure
stability
Following
the
literature
we
refer
to
these
spectra
as
diffusive
like
We
also
note
that
for
these
spectra
even
if
considering
only
the
real
part
of
the
collective
excitation
spectrum
as
soon
as
the
fluid
is
in
motion
dissipation
in
the
form
of
waves
is
possible
However
we
will
see
that
similar
to
the
case
of
non
resonantly
pumped
polaritons
when
decreasing
and
accordingly
in
order
to
have
stable
solutions
this
situation
continuously
connects
to
the
case
where
threshold
like
behaviour
with
was
found
We
will
see
in
the
next
section
how
these
different
spectra
imply
only
two
qualitatively
different
types
of
crossover
of
the
drag
force
as
function
of
the
fluid
velocity
for
either
or
pump
detunings
Drag
force
as
function
of
the
fluid
velocity
for
different
values
of
the
pump
detuning
and
and
for
different
values
of
the
polariton
lifetime
-here
we
use
the
notation
and
Drag
force
The
steady
state
response
of
the
system
to
static
and
weak
defect
can
be
evaluated
starting
from
Eq
eq
linre
For
point
like
defect
this
can
be
written
in
momentum
space
as
while
the
other
component
can
be
obtained
by
complex
conjugation
and
by
substituting
The
drag
force
exerted
by
the
defect
on
the
fluid
is
given
by
the
expectation
value
of
the
operator
over
the
condensate
wavefunction
This
definition
is
justified
for
conservative
systems
in
Appendix
using
the
momentum
flux
tensor
In
the
steady
state
linear
response
regime
we
obtain
The
drag
is
clearly
oriented
along
the
fluid
velocity
If
then
the
integral
in
Eq
eq
dragf
is
finite
only
if
poles
exist
when
when
quasiparticles
can
be
excited
in
agreement
with
the
Landau
criterion
For
finite
polariton
lifetimes
however
it
is
clear
that
the
integral
will
always
be
different
from
zero
for
We
now
analyse
the
behaviour
of
the
drag
force
as
function
of
the
fluid
velocity
for
the
three
and
different
spectra
illustrated
in
the
previous
section
Drag
force
as
function
of
the
inverse
polariton
lifetime
in
the
subcritical
regime
and
supercritical
regime
In
both
cases
we
have
fixed
but
these
results
are
qualitatively
similar
for
any
other
value
of
the
pump
detuning
We
plot
in
the
right
panels
the
normalised
real
space
wavefunction
for
two
specific
values
of
and
For
the
linear
spectrum
at
in
the
equilibrium
limit
we
recover
for
the
drag
the
known
result
of
weakly
interacting
Bose
gases
in
two
dimensions
with
threshold
like
behaviour
at
critical
fluid
velocity
equal
to
the
speed
of
sound
This
limiting
result
is
plotted
as
bold
gray
line
in
the
panels
of
Fig
For
and
finite
lifetimes
we
find
smooth
crossover
from
the
subsonic
to
the
supersonic
regime
with
the
drag
being
closer
to
the
equilibrium
threshold
behaviour
for
decreasing
see
Fig
finite
lifetime
tends
to
increase
the
value
of
the
drag
in
the
subsonic
region
giving
place
to
residual
drag
force
similar
to
what
was
found
in
the
numerical
simulations
of
Ref
see
Fig
Instead
in
the
supersonic
region
the
finite
lifetime
tends
to
decrease
the
value
of
the
drag
In
the
case
of
diffusive
like
spectra
at
the
situation
is
qualitatively
very
similar
to
the
resonant
case
see
Fig
with
the
difference
that
now
in
order
to
have
stable
solutions
we
can
decrease
the
value
of
the
lifetime
only
by
decreasing
accordingly
also
the
value
of
the
pump
detuning
The
crossover
for
both
and
is
also
qualitatively
very
similar
to
the
case
of
non
resonantly
pumped
polaritons
where
the
spectrum
of
excitation
is
diffusive
like
In
Ref
similar
approach
was
used
to
analytically
derive
the
drag
for
nonresonantly
pumped
polaritons
in
finding
continuous
crossover
from
regime
dominated
by
viscous
drag
of
Stokes
type
to
one
dominated
by
wave
resistance
Futhermore
the
authors
proved
that
it
was
not
possible
to
separate
the
viscous
and
wave
resistance
components
of
the
drag
Shortly
after
the
publication
of
Ref
on
which
this
Chapter
is
based
Van
Regemortel
et
al
found
that
the
parametric
amplification
mechanism
described
in
Sec
can
increase
the
scattering
of
particles
in
the
flow
direction
for
low
condensate
speeds
leading
to
negative
drag
force
force
directed
opposite
to
the
flow
direction
To
see
how
this
comes
about
one
can
look
at
the
right
column
of
Fig
the
top
panel
shows
that
the
average
momentum
of
the
scattered
modes
is
in
the
positive
direction
This
in
turn
leads
to
pileup
of
the
fluid
density
behind
the
defect
as
can
be
seen
in
the
bottom
panel
Top
row
Photon
density
along
perturbed
by
circular
potential
of
radius
and
height
meV
centered
at
the
origin
in
the
subcritical
regime
The
pump
momentum
is
and
the
panels
from
left
to
right
correspond
to
decreasing
exciton
and
photon
decay
rates
and
meV
Bottom
row
Residual
drag
force
as
function
of
decay
rate
The
left
panel
compares
two
different
values
of
namely
black
solid
line
and
red
dashed
line
The
right
panel
shows
two
different
potential
heights
of
blue
dashed
line
and
meV
red
dashed
line
with
In
all
plots
the
pump
is
meV
blue
detuned
above
the
bare
LP
branch
Adapted
from
Ref
In
the
case
of
gapped
spectra
the
situation
is
however
qualitatively
different
see
Fig
For
infinitely
living
polaritons
the
drag
force
can
also
be
evaluated
analytically
and
its
expression
is
similar
to
Eq
eq
drag
but
with
critical
velocity
larger
than
the
speed
of
sound
the
expression
of
which
is
given
in
Eq
eq
criti
Therefore
now
the
drag
experiences
jump
for
rather
than
continuous
threshold
as
for
the
resonant
case
As
already
mentioned
in
the
previous
section
this
discontinuous
behaviour
of
the
drag
for
the
gapped
spectra
is
connected
to
the
fact
that
as
soon
as
quasiparticles
can
be
excited
by
the
defect
at
Cherenkov
like
waves
can
be
immediately
emitted
in
all
directions
rather
than
being
restricted
in
region
outside
the
Mach
cone
as
before
For
the
cone
was
gradually
closing
with
increasing
fluid
velocity
Both
the
increase
of
the
value
of
the
drag
in
the
subcritical
region
as
function
of
the
polariton
lifetime
and
the
decrease
in
the
supercritical
region
are
behaviours
common
to
all
the
types
of
spectra
We
plot
the
drag
force
as
function
of
in
Fig
for
two
values
of
the
fluid
velocity
and
specific
value
of
the
pump
detuning
though
we
have
checked
that
the
following
results
are
generic
For
we
find
that
the
residual
drag
is
finite
lifetime
effect
only
and
well
below
the
critical
velocity
the
drag
force
goes
linearly
to
zero
for
This
is
in
agreement
with
the
results
of
Ref
where
the
full
GP
equation
for
the
coupled
exciton
and
photon
fields
was
solved
numerically
for
finite
size
defect
see
the
bottom
row
of
Fig
for
the
numerical
drag
in
the
limit
of
asymptotically
large
densities
In
the
resonant
case
the
slope
of
the
drag
for
can
be
evaluated
analytically
starting
from
the
expression
eq
dragf
The
residual
drag
in
the
subsonic
regime
is
an
effect
of
the
broadening
of
the
quasi
particles
energies
Even
when
the
spectrum
real
part
does
not
allow
any
scattering
against
the
defect
for
the
broadening
produces
some
scattering
close
to
the
defect
This
results
in
perturbation
of
the
fluid
around
the
defect
asymmetric
in
the
direction
of
the
fluid
velocity
see
panel
of
Fig
similar
to
what
was
obtained
in
Ref
see
top
row
of
Fig
Instead
in
the
supersonic
regime
the
drag
force
is
weaker
in
the
non
equilibrium
case
with
respect
to
the
equilibrium
one
This
is
caused
by
the
finite
lifetime
tending
to
suppress
the
propagation
of
the
Cherenkov
waves
away
from
the
defect
as
shown
in
panel
of
Fig
Conclusions
and
discussion
To
conclude
we
have
analysed
the
linear
response
to
weak
defect
of
resonantly
pumped
polaritons
in
the
pump
only
state
and
we
have
been
able
to
determine
two
different
kinds
of
threshold
like
behaviours
for
the
drag
force
as
function
of
the
fluid
velocity
In
the
case
of
either
zero
or
positive
pump
detuning
one
can
continuously
connect
to
the
case
of
equilibrium
weakly
interacting
gases
of
Chapter
where
the
drag
displays
continuous
threshold
with
critical
velocity
equal
to
the
speed
of
sound
However
for
negative
pump
detuning
where
the
spectrum
of
excitations
is
gapped
the
drag
shows
discontinuity
with
critical
velocity
larger
than
the
speed
of
sound
In
this
sense
the
case
of
coherently
driven
microcavity
polaritons
in
the
pump
only
configuration
displays
richer
phenomenology
than
the
case
of
nonresonantly
pumped
polariton
superfluids
We
have
also
seen
that
the
absence
of
long
range
wake
does
not
imply
the
absence
of
dissipation
as
residual
drag
force
due
to
the
finite
polariton
lifetime
is
always
present
in
the
system
In
this
sense
one
can
say
that
we
are
not
dealing
with
superfluid
behaviour
in
strict
sense
GP
equation
for
the
LP
branch
If
one
starts
from
description
of
polaritons
in
terms
of
separate
exciton
and
cavity
photon
fields
rotation
into
the
LP
and
UP
basis
followed
by
neglecting
the
occupancy
of
the
upper
polariton
branch
as
explained
in
detail
in
Chapter
results
in
the
following
Gross
Pitaevskii
equation
compare
to
Eq
eq
mom
GPE
for
the
LP
field
in
momentum
space
LP
LP
LP
LP
LP
LP
LP
where
is
the
effective
LP
decay
rate
is
the
interaction
strength
and
where
In
these
expressions
the
coefficients
see
Eqs
eq
hopfield
and
eq
hopfield
are
the
Hopfield
coefficients
used
to
diagonalise
the
free
polariton
Hamiltonian
We
want
here
to
justify
the
simplified
description
done
in
Eq
eq
basic
If
we
follow
the
linear
response
expansion
as
in
eq
mfield
the
operator
in
momentum
space
analogous
to
eq
opell
reads
as
where
now
It
is
easy
to
show
that
the
eigenvalues
of
this
operator
coincide
with
our
approximated
expressions
eq
spect
in
the
limit
of
when
and
when
we
can
simply
rename
and
It
is
interesting
to
note
that
even
if
we
would
retain
the
linear
terms
in
in
the
expansion
of
this
would
result
in
renormalisation
of
the
fluid
velocity
in
the
expression
eq
spect
which
takes
into
account
the
blue
shift
of
the
LP
dispersion
due
to
the
interaction
Polariton
superfluidity
in
the
OPO
regime
Our
findings
This
Chapter
presents
joint
theoretical
and
experimental
study
of
an
OPO
configuration
see
Fig
where
wide
and
steady
state
condensate
hits
stationary
localized
defect
in
the
microcavity
Contrary
to
the
criterion
for
quantized
flow
metastability
for
which
the
signal
and
idler
display
simultaneous
locked
responses
we
find
that
their
scattering
properties
when
the
OPO
hits
static
defect
are
different
In
particular
we
investigate
the
scattering
properties
of
all
three
fluids
the
pump
the
signal
and
the
idler
in
both
real
and
momentum
space
We
find
that
the
modulations
generated
by
the
defect
in
each
fluid
are
not
only
determined
by
its
associated
Rayleigh
scattering
ring
but
each
component
displays
additional
rings
because
of
the
cross
talk
with
the
other
components
imposed
by
nonlinear
and
parametric
processes
We
single
out
three
factors
determining
which
one
of
these
rings
has
the
biggest
influence
on
each
fluid
response
the
coupling
strength
between
the
three
OPO
states
the
resonance
of
the
ring
with
the
blue
shifted
LP
dispersion
and
the
values
of
each
fluid
group
velocity
and
lifetime
together
establishing
how
far
each
modulation
can
propagate
from
the
defect
The
concurrence
of
these
effects
implies
that
the
idler
strongly
scatters
inheriting
the
same
modulations
as
the
pump
while
the
modulations
due
to
its
own
ring
can
propagate
only
very
close
to
the
defect
and
cannot
be
appreciated
However
the
modulations
in
the
signal
are
strongly
suppressed
and
not
at
all
visible
in
experiments
because
the
slope
of
the
polariton
dispersion
in
its
low
momentum
component
brings
all
Rayleigh
rings
coming
from
pump
and
idler
out
of
resonance
Pictorial
representation
of
the
optical
parametric
oscillator
OPO
state
inside
planar
microcavity
Cavity
modes
of
the
incoming
external
laser
black
arrow
couple
to
the
excitons
formed
by
electron
solid
spheres
-hole
hollow
spheres
pairs
inside
the
semiconductor
quantum
well
QW
creating
the
pump
red
polaritons
which
coherently
scatter
into
the
signal
blue
and
idler
green
states
All
three
types
of
polaritons
decay
into
external
radiation
emitted
through
the
top
distributed
Bragg
reflector
DBR
of
the
cavity
There
is
always
scattering
due
to
the
pump
Note
that
the
kinematic
conditions
for
OPO
are
incompatible
with
the
pump
and
idler
being
in
the
subsonic
regime
Thus
the
coupling
between
the
three
components
always
implies
some
degree
of
scattering
in
the
signal
In
practice
the
small
value
of
the
signal
momentum
strongly
suppresses
its
visible
modulations
as
confirmed
by
the
experimental
observations
Model
Exciton
photon
GPE
As
explained
in
Chapter
the
dynamics
of
polaritons
in
the
OPO
regime
and
their
hydrodynamic
properties
when
scattering
against
defect
can
be
described
via
classical
driven
dissipative
non
linear
Gross
Pitaevskii
equation
GPE
for
the
coupled
exciton
and
cavity
fields
The
dispersive
and
fields
decay
at
rate
and
are
coupled
by
the
Rabi
splitting
while
the
nonlinearity
is
regulated
by
the
exciton
coupling
strength
We
describe
the
defect
via
potential
acting
on
the
photonic
component
this
can
either
be
defect
in
the
cavity
mirror
or
localized
laser
field
In
the
conservative
homogeneous
and
linear
regime
the
eigenvalues
of
are
given
by
the
LP
and
UP
energies
Eq
eq
polariton
dispersion
The
cavity
is
driven
by
continuous
wave
laser
field
into
the
OPO
regime
Here
polaritons
are
continuously
injected
into
the
pump
state
with
frequency
and
momentum
and
above
pump
strength
threshold
they
undergo
coherent
stimulated
scattering
into
the
signal
and
idler
states
OPO
mean
field
blueshifts
and
fluctuation
Rayleigh
rings
in
the
linear
response
scheme
for
homogeneous
pumping
Left
panel
signal
blue
upper
triangles
pump
red
circles
and
idler
green
lower
triangles
mean
field
energy
blueshifts
in
units
of
vs
the
rescaled
pump
intensity
in
units
of
in
the
optical
limiter
regime
Parameters
are
meV
zero
cavity
exciton
detuning
meV
meV
and
The
shaded
area
is
stable
OPO
region
while
the
vertical
dashed
line
corresponds
to
the
pump
power
value
chosen
for
plotting
the
right
panel
Right
panel
blueshifted
LP
dispersion
eq
blues
with
superimposed
Rayleigh
curves
evaluated
within
the
linear
response
approximation
same
symbols
as
left
panel
The
two
rings
corresponding
to
the
signal
state
are
shrunk
to
zero
because
LP
GPE
in
momentum
space
As
first
step
it
is
useful
to
get
insight
into
the
system
behaviour
in
the
simple
case
of
homogeneous
pump
of
strength
numerical
study
of
the
coupled
equations
eq
gpequ
for
the
more
realistic
case
of
finite
size
top
hat
pump
profile
will
be
presented
in
Sec
To
further
simplify
our
analysis
we
assume
here
that
the
UP
dispersion
does
not
get
populated
by
parametric
scattering
processes
and
thus
by
means
of
the
Hopfield
coefficients
Eqs
eq
hopfield
and
eq
hopfield
we
project
the
GPE
eq
gpequ
onto
the
LP
component
where
As
explained
in
detail
in
Sec
we
obtain
Eq
eq
mom
GPE
LP
Here
is
the
effective
LP
decay
rate
the
interaction
strength
is
given
by
and
the
pumping
term
is
given
by
Note
that
the
dependence
on
the
exciton
exciton
interaction
strength
can
be
removed
by
rescaling
both
the
LP
field
and
the
pump
strength
something
we
will
do
later
on
to
all
effects
working
in
terms
of
energy
blueshifts
and
the
rescaled
pump
intensity
Linear
response
theory
Spectrum
of
collective
excitations
Cut
at
of
the
real
part
of
the
quasiparticle
energy
dispersion
plotted
versus
The
spectrum
is
evaluated
within
the
linear
approximation
scheme
and
the
parameters
are
the
same
ones
used
for
Fig
Linear
response
ansatz
As
already
explained
in
Sec
in
the
limit
where
the
homogeneously
pumped
system
is
only
weakly
perturbed
by
the
external
potential
we
apply
linear
response
analysis
The
LP
field
is
expanded
around
the
mean
field
terms
for
the
three
OPO
states
see
Eqs
eq
pump
fluctuations
and
eq
state
ansatz
where
Eq
eq
efflp
is
expanded
linearly
in
both
the
fluctuation
terms
and
as
well
as
the
defect
potential
At
zeroth
order
the
three
complex
uniform
mean
field
equations
can
be
solved
to
obtain
the
dependence
of
the
signal
pump
and
idler
energy
blueshifts
on
the
system
parameters
typical
behaviour
of
as
function
of
the
rescaled
pump
intensity
in
the
optical
limiter
regime
is
plotted
in
the
left
panel
of
Fig
At
first
order
one
obtains
six
coupled
equations
diagonal
in
momentum
space
for
the
six
component
vector
and
for
the
potential
part
To
understand
the
origin
of
the
factor
in
Eq
eq
fluct
it
is
sufficient
to
consider
one
component
only
and
start
from
Eq
eq
mfield
used
in
Sec
The
connection
becomes
clear
once
we
write
the
fluctuation
part
as
and
compare
it
to
Eq
eq
pump
fluctuations
of
Sec
Bogoliubov
matrix
In
eq
fluct
we
have
only
kept
the
terms
oscillating
at
the
frequencies
and
neglected
the
other
terms
in
the
expansion
which
are
oscillating
at
frequencies
far
from
the
LP
band
and
thus
with
negligible
amplitudes
In
the
particlelike
and
the
holelike
channels
the
Bogoliubov
matrix
determining
the
spectrum
of
excitations
can
be
written
as
where
the
three
OPO
states
components
are
mn
LP
mn
OPO
spectrum
In
absence
of
defect
potential
Eq
eq
fluct
is
the
eigenvalue
equation
for
the
spectrum
of
excitations
of
homogeneous
OPO
We
plot
in
Fig
typical
collective
dispersion
here
we
consider
the
same
system
parameters
as
the
ones
used
in
Fig
by
plotting
the
real
part
of
the
Bogoliubov
matrix
eigenvalues
as
function
of
cut
at
The
spectrum
has
six
branches
labeled
by
and
Even
though
these
degrees
of
freedom
are
mixed
together
at
large
momenta
one
recovers
the
LP
dispersions
shifted
by
the
three
states
energies
and
momenta
where
corresponds
to
the
particlelike
holelike
branch
The
OPO
solution
is
stable
shaded
area
in
Fig
as
far
as
Note
also
that
owing
to
the
symmetry
of
the
OPO
equations
of
motion
see
Sec
no
restoring
force
would
oppose
rotation
of
the
signal
idler
phases
meaning
that
the
generator
of
such
global
rotations
is
an
eigenvector
of
with
eigenvalue
Rayleigh
rings
The
shape
of
the
patterns
or
Cherenkov
like
waves
resulting
from
the
elastic
scattering
of
the
OPO
fluids
against
the
static
defect
can
be
determined
starting
from
the
spectrum
and
in
particular
evaluating
the
closed
curves
in
space
or
"Rayleigh
rings
"defined
by
the
condition
Even
if
they
do
not
appear
to
be
relevant
here
note
that
the
presence
of
non
vanishing
imaginary
part
of
the
excitation
spectrum
introduces
some
complications
Even
in
the
absence
of
any
Rayleigh
ring
the
drag
force
can
be
non
vanishing
as
we
saw
in
Chapter
for
the
pump
only
case
and
the
standard
Landau
criterion
may
fail
to
identify
critical
velocity
The
modulations
propagate
with
direction
orthogonal
to
each
curve
pattern
wavelength
given
by
the
corresponding
and
group
velocity
where
determines
the
distance
at
any
given
direction
over
which
the
perturbation
extends
away
from
the
defect
For
single
fluid
under
coherent
pump
the
qualitative
shape
of
the
modulation
pattern
generated
in
the
fluid
by
the
defect
is
mostly
determined
by
the
excitation
spectrum
Blueshifted
dispersion
and
rings
for
ks
For
OPO
the
spectrum
of
excitation
on
top
of
each
of
the
three
states
see
Fig
generates
six
identical
Rayleigh
rings
for
the
three
states
Note
that
the
Rayleigh
rings
can
be
found
by
finding
the
intersections
The
Rayleigh
rings
for
the
OPO
conditions
specified
in
Fig
are
clearly
visible
in
the
right
panels
of
Fig
where
we
plot
the
space
photoluminescence
filtered
at
the
energy
of
each
state
We
have
chosen
here
like
defect
potential
but
we
have
checked
that
our
results
do
not
depend
on
the
specific
shape
of
the
defect
potential
In
particular
we
have
also
considered
the
response
to
defects
with
smooth
Gaussian
like
profiles
whose
effect
is
only
to
partially
weaken
the
upstream
modulations
in
real
space
As
detailed
in
Sec
the
mean
field
OPO
ansatz
assumes
the
signal
momentum
to
be
an
input
parameter
rather
than
solving
the
full
"selection
problem
"This
is
in
contrast
to
both
the
full
numerical
solution
of
the
GP
Eq
eq
gpequ
Sec
as
well
as
experiments
Sec
where
the
signal
momentum
is
automatically
selected
typically
close
to
the
bottom
of
the
LP
band
by
parametric
scattering
processes
The
reason
why
parametric
scattering
processes
select
signal
with
momentum
close
to
zero
already
very
close
to
the
lower
pump
thresold
for
OPO
is
still
awaiting
an
explaination
linresp
Linear
response
of
the
three
OPO
states
to
static
defect
for
column
column
and
column
The
parameters
are
given
in
Fig
and
Column
Top
row
is
the
analogue
of
Fig
for
negative
signal
momentum
The
three
bottom
rows
show
the
rescaled
emission
in
real
left
panels
in
linear
scale
and
momentum
space
right
panels
in
logarithmic
scale
filtered
at
the
energies
of
the
signal
top
pump
middle
and
idler
bottom
Column
Same
as
column
for
positive
signal
momentum
Column
Vanishing
signal
momentum
the
corresponding
mean
field
blueshifts
and
Rayleigh
rings
are
plotted
in
Fig
The
inset
shows
the
Gaussian
filtered
see
main
text
signal
emission
In
particular
this
cannot
be
addressed
within
spatially
homogeneous
approximation
where
the
three
mean
field
solutions
for
pump
signal
and
idler
states
are
described
by
plane
waves
However
one
can
show
that
within
the
same
mean
field
approximation
scheme
when
increasing
the
pump
power
towards
the
upper
threshold
for
OPO
the
blue
shift
of
the
LP
polariton
dispersion
due
to
the
increasing
mean
field
polariton
density
causes
the
signal
momentum
to
converge
towards
zero
In
the
following
we
take
advantage
of
the
broad
range
of
possible
choices
for
and
thus
at
mean
field
level
and
analyze
three
distinct
cases
negative
vanishing
and
positive
signal
momenta
Vanishing
signal
momentum
For
the
OPO
conditions
considered
here
the
signal
momentum
is
at
and
thus
only
four
of
the
six
Rayleigh
rings
are
present
The
same
rings
are
also
plotted
in
the
right
panel
of
Fig
shifted
at
each
of
the
three
OPO
states
momentum
and
energies
It
is
important
to
note
that
even
though
the
three
OPO
states
have
locked
responses
because
they
display
the
same
spectrum
of
excitations
only
one
of
the
rings
is
the
most
resonant
at
with
the
interaction
blueshifted
LP
dispersion
where
are
the
mean
field
energy
blueshifts
measured
in
Fig
in
units
of
This
implies
that
the
most
visible
modulation
for
each
fluid
should
be
the
most
resonant
one
with
superimposed
weaker
modulations
coming
from
the
other
two
state
rings
Signal
for
ks
In
the
specific
case
of
Fig
the
signal
is
at
and
thus
produces
no
rings
in
momentum
space
The
other
four
rings
are
very
far
from
being
resonant
with
the
blueshifted
LP
dispersion
eq
blues
at
and
thus
the
signal
displays
only
an
extremely
weak
modulation
coming
from
the
next
closer
ring
which
is
the
one
associated
with
the
pump
state
We
estimate
that
the
signal
modulation
amplitudes
are
roughly
of
the
average
signal
intensity
and
about
factor
of
times
weaker
than
the
modulation
amplitudes
in
the
pump
fluid
To
show
that
the
signal
has
weak
modulations
coming
from
the
pump
we
apply
Gaussian
filter
to
the
real
space
images
see
the
inset
of
Fig
This
consists
of
the
following
procedure
The
original
data
for
the
real
space
profile
are
convoluted
with
Gaussian
kernel
obtaining
new
profile
where
short
wavelengths
features
are
smoothened
out
The
convoluted
image
is
then
subtracted
from
the
original
data
giving
and
effectively
filtering
out
all
long
wavelength
details
This
procedure
reveals
that
indeed
the
pump
imprints
its
modulations
also
into
the
signal
even
though
these
are
extremely
weak
thus
leaving
the
signal
basically
insensitive
to
the
presence
of
the
defect
Pump
and
idler
for
ks
Pump
and
idler
states
are
each
mostly
resonant
with
their
own
rings
at
and
at
respectively
Thus
one
should
then
observe
two
superimposed
modulations
in
both
pump
and
idler
filtered
emissions
the
stronger
one
for
each
being
the
most
resonant
one
However
the
modulations
associated
with
the
idler
only
propagate
very
close
to
the
defect
at
an
average
distance
before
getting
damped
and
thus
are
not
clearly
visible
For
the
OPO
conditions
considered
this
is
due
to
the
small
idler
group
velocity
as
the
dispersion
is
almost
excitonic
at
the
idler
energy
Conclusion
for
ks
We
can
conclude
that
for
the
typical
OPO
condition
with
signal
at
considered
in
Figs
and
the
signal
fluid
does
not
show
modulations
and
the
extremely
weak
scattering
inherited
from
the
pump
state
can
be
appreciated
only
after
Gaussian
filtering
procedure
of
the
image
In
contrast
the
idler
has
locked
response
to
the
one
of
the
pump
state
Finite
signal
momentum
Real
space
signal
pump
and
idler
OPO
one
dimensional
filtered
profiles
in
the
linear
response
approximation
OPO
filtered
emissions
along
the
direction
signal
at
panel
corresponds
to
Fig
signal
at
panel
corresponds
to
Fig
and
signal
at
panel
corresponds
to
Fig
In
each
panel
we
plot
the
filtered
profiles
of
signal
pump
and
idler
while
the
horizontal
gray
dashed
lines
represent
the
values
of
the
mean
field
emission
without
defect
Given
the
freedom
of
choice
for
the
signal
momentum
close
to
the
lower
OPO
threshold
we
consider
here
two
additional
cases
that
could
not
be
studied
neither
experimentally
nor
within
full
numerical
approach
but
that
instead
we
can
easily
analize
within
the
linear
response
theory
In
particular
we
have
left
fixed
the
pumping
conditions
and
meV
and
considered
two
opposite
situations
In
the
first
case
the
signal
has
finite
and
positive
momentum
and
thus
the
idler
is
at
low
momentum
The
results
are
shown
in
Fig
Here
we
see
that
all
six
Rayleigh
rings
are
clearly
visible
and
in
addition
as
the
idler
is
at
lower
momentum
compared
to
the
case
considered
in
Sec
and
thus
its
dispersion
steeper
the
idler
group
velocity
is
large
enough
to
appreciate
the
modulation
of
the
Rayleigh
ring
associated
to
this
state
As
result
each
of
the
three
filtered
OPO
emissions
exhibits
as
the
strongest
modulation
the
one
coming
from
its
own
Rayleigh
ring
including
the
signal
which
is
now
at
finite
and
large
momentum
In
this
case
the
OPO
response
of
each
filtered
state
profile
looks
completely
independent
from
the
other
as
if
we
were
pumping
each
state
independently
In
the
second
case
shown
in
Fig
the
signal
is
finite
and
negative
and
the
idler
is
now
at
very
large
momentum
where
its
dispersion
is
very
exciton
like
and
flat
and
thus
the
idler
has
very
small
group
velocity
and
its
own
modulations
are
visible
only
very
close
to
the
defect
For
this
case
we
can
appreciate
in
the
idler
filtered
profile
overlapped
modulations
from
all
the
three
state
Rayleigh
rings
note
that
because
the
signal
is
at
negative
momentum
its
modulations
have
an
opposite
direction
compared
to
the
ones
of
pump
and
idler
while
in
the
signal
we
can
mostly
see
the
signal
long
wavelength
modulations
and
only
very
weakly
the
pump
one
Real
space
profiles
of
three
uncoupled
signal
pump
and
idler
fluids
The
rescaled
profiles
are
obtained
by
setting
all
off
diagonal
couplings
in
Eq
eq
bogoliubov
opo
to
zero
resulting
in
three
uncoupled
signal
top
row
pump
middle
row
and
idler
bottom
row
fluids
The
three
columns
correspond
to
the
three
different
cases
analysed
within
the
linear
response
approximation
the
case
of
signal
at
left
column
corresponds
to
the
same
conditions
as
Fig
signal
at
middle
column
corresponds
to
Fig
and
signal
at
right
column
corresponds
to
Fig
Discussion
We
can
compare
the
different
modulation
strengths
of
the
three
OPO
profiles
by
looking
at
the
color
bars
plotted
next
to
the
profiles
In
order
to
better
compare
them
on
the
same
plot
we
show
in
Fig
the
one
dimensional
OPO
filtered
emissions
along
the
direction
rescaled
by
the
mean
field
solution
in
absence
of
the
defect
for
the
three
cases
analysed
above
While
for
the
OPO
conditions
with
signal
at
middle
panel
corresponding
to
Fig
the
imprinted
modulations
from
the
pump
are
hardly
visible
and
can
only
be
appreciated
after
Gaussian
filter
manipulation
both
OPO
cases
with
finite
momentum
signal
result
in
modulations
in
the
signal
with
an
amplitude
of
the
same
order
of
magnitude
of
both
pump
and
idler
fluids
In
Fig
we
instead
show
the
real
space
profiles
obtained
by
setting
all
off
diagonal
couplings
in
the
Bogoliubov
matrix
of
Eq
eq
bogoliubov
opo
to
zero
resulting
in
three
uncoupled
signal
top
row
pump
middle
row
and
idler
bottom
row
fluids
This
underlines
the
importance
of
the
coupling
between
the
three
fluids
in
the
three
different
OPO
regimes
we
analysed
within
the
linear
response
approximation
In
particular
for
the
OPO
conditions
such
that
the
signal
has
finite
and
positive
momentum
right
column
of
Fig
which
corresponds
to
the
conditions
shown
in
Fig
the
coupling
has
little
effect
and
the
three
fluids
respond
to
the
defect
in
practice
in
independent
way
each
Rayleigh
ring
influences
its
own
fluid
The
OPO
condition
with
finite
and
negative
momentum
which
corresponds
to
the
conditions
shown
in
Fig
are
shown
in
the
left
column
of
Fig
Here
we
can
see
that
the
coupling
between
the
three
fluids
plays
role
The
modulations
of
the
pump
can
be
appreciated
both
in
the
signal
profile
though
weakly
as
well
as
in
the
idler
profile
In
the
idler
fluid
its
own
modulations
can
only
propagate
very
close
to
the
defect
and
thus
the
only
really
appreciable
modulations
are
the
ones
inherited
from
the
pump
Finally
for
the
experimentally
relevant
case
of
an
OPO
with
signal
at
zero
momentum
middle
column
of
Fig
which
corresponds
to
the
conditions
shown
in
Fig
there
is
also
role
played
by
the
coupling
though
as
already
thoroughly
analysed
the
modulations
in
the
signal
inherited
from
the
pump
can
only
be
appreciated
after
Gaussian
filtering
manipulation
Finally
note
that
for
the
OPO
conditions
of
Sec
as
well
as
for
the
finite
momenta
cases
considered
in
Sec
the
subsonic
to
supersonic
crossover
of
the
pump
only
state
happens
at
pump
intensities
well
above
the
region
of
stability
of
OPO
-the
shaded
gray
regions
of
Fig
and
Fig
and
Thus
it
is
not
possible
to
study
case
where
the
pump
is
already
subsonic
and
at
the
same
time
promotes
stimulated
scattering
Numerical
analysis
OPO
spectrum
obtained
by
full
numerics
Left
panel
Photonic
component
of
the
OPO
spectrum
in
presence
of
point
like
defect
logarithmic
scale
as
function
of
the
rescaled
energy
versus
the
component
of
momentum
cut
at
for
top
hat
pump
see
text
for
the
space
profile
and
parameter
values
with
intensity
above
the
OPO
threshold
pump
wave
vector
in
the
direction
and
meV
The
symbols
indicate
the
signal
blue
upper
triangle
pump
red
circle
and
idler
green
lower
triangle
energies
as
well
as
the
two
momenta
on
each
state
Rayleigh
ring
at
Note
that
the
logarithmic
scale
results
in
fictitious
broadening
in
energy
of
the
spectrum
which
is
in
reality
like
see
right
panel
The
bare
LP
dispersion
including
its
broadening
due
to
finite
lifetime
is
plotted
as
shaded
grey
region
while
the
bare
UP
dispersion
as
black
dot
dashed
line
Right
panel
Momentum
integrated
spectrum
linear
scale
as
function
of
the
rescaled
energy
where
it
can
be
clearly
appreciated
that
the
emission
is
like
in
energy
The
results
obtained
within
the
linear
response
approximation
are
additionally
confirmed
by
an
exact
full
numerical
analysis
of
the
classical
driven
dissipative
non
linear
GPE
eq
gpequ
for
the
coupled
exciton
and
cavity
fields
in
the
case
of
finite
size
pump
Eq
eq
gpequ
is
solved
numerically
on
grid
of
points
and
separation
of
in
box
by
using
order
adaptive
step
Runge
Kutta
algorithm
Convergence
has
been
checked
both
with
respect
the
resolution
in
space
as
well
as
in
momentum
without
as
well
as
in
presence
of
the
defect
The
same
approach
has
been
already
used
in
the
literature
for
review
see
Refs
As
for
the
system
parameters
we
have
considered
LP
dispersion
at
zero
photon
exciton
detuning
dispersionless
excitonic
spectrum
and
quadratic
dispersion
for
photons
with
the
photon
mass
where
is
the
bare
electron
mass
The
LP
dispersion
Eq
eq
polariton
dispersion
is
characterised
by
Rabi
splitting
meV
Furthermore
the
exciton
and
cavity
decay
rates
are
fixed
to
meV
For
the
defect
we
choose
like
potential
where
its
location
is
fixed
at
one
of
the
points
of
the
grid
Note
that
in
finite
size
OPO
local
currents
lead
to
inhomogeneous
OPO
profiles
inside
the
pump
spot
despite
the
external
pump
having
top
hat
profile
with
completely
flat
inner
region
-as
shown
later
this
can
be
observed
in
the
filtered
OPO
profiles
evaluated
in
absence
of
defect
shown
as
dashed
lines
in
Fig
We
have
thus
chosen
the
defect
location
so
that
it
lies
in
the
smoothest
and
most
homogeneous
part
of
the
OPO
profiles
Also
we
have
checked
that
our
results
do
not
qualitatively
depend
on
the
strength
nor
on
the
sign
of
the
defect
potential
as
far
as
this
does
not
exceed
critical
value
above
which
it
destabilises
the
OPO
steady
state
regime
The
pump
has
smoothen
and
rotationally
symmetric
top
hat
profile
with
strength
meV
and
parameters
We
pump
at
in
the
direction
and
at
meV
roughly
meV
above
the
bare
LP
dispersion
By
increasing
the
pump
strength
we
find
the
threshold
above
which
OPO
switches
on
leading
to
two
conjugate
signal
and
idler
states
We
then
fix
the
pump
strength
just
above
this
threshold
where
we
find
steady
state
OPO
solution
which
is
stable
see
Ref
for
further
details
In
absence
of
the
defect
this
condition
corresponds
to
signal
state
at
and
meV
and
an
idler
at
and
meV
It
is
interesting
to
note
that
already
very
close
to
the
lower
pump
power
threshold
for
OPO
the
selected
signal
momentum
is
very
close
to
zero
This
contrasts
with
what
one
obtains
in
the
linear
approximation
scheme
where
instead
just
above
the
lower
OPO
threshold
there
exists
broad
interval
of
permitted
values
for
and
thus
-we
have
already
mentioned
this
"selection
problem
"for
parametric
scattering
in
Sec
We
evaluate
the
time
dependent
full
numerical
solution
of
eq
gpequ
until
steady
state
regime
is
reached
Here
both
its
Fourier
transform
to
momentum
and
energies
can
be
evaluated
numerically
We
plot
on
the
left
panel
of
Fig
cut
at
of
the
photonic
component
of
the
OPO
spectrum
in
presence
of
point
like
defect
as
function
of
the
rescaled
energy
versus
the
component
of
momentum
cut
at
In
the
right
panel
we
plot
instead
the
corresponding
momentum
integrated
spectrum
Here
we
can
clearly
see
that
the
presence
of
the
defect
does
not
modify
the
fact
that
the
OPO
emission
for
the
OPO
signal
blue
upper
triangle
pump
red
circle
and
idler
green
lower
triangle
states
has
completely
flat
dispersion
in
energy
thus
indicating
that
stable
steady
state
OPO
solution
has
been
reached
Note
that
in
the
spectrum
map
of
the
left
panel
of
Fig
the
logarithmic
scale
results
in
fictitious
broadening
in
energy
However
from
the
integrated
spectrum
plotted
in
linear
scale
in
the
right
panel
of
Fig
one
can
clearly
appreciate
that
this
emission
is
like
exactly
as
it
happens
for
the
homogeneous
OPO
case
Full
numerical
responses
to
static
defect
of
the
three
OPO
states
in
real
and
momentum
space
Filtered
OPO
emissions
signal
top
panels
pump
middle
and
idler
bottom
in
real
space
left
panels
in
linear
scale
and
momentum
space
right
panel
in
logarithmic
scale
obtained
by
full
numerical
evaluation
of
eq
gpequ
For
the
top
left
panel
of
the
signal
space
emission
Gaussian
filtering
is
applied
to
enhance
the
short
wavelength
modulations
of
this
state
revealing
that
the
modulations
corresponding
to
the
pump
state
are
also
imprinted
though
weakly
into
the
signal
The
symbols
indicate
the
pump
ring
diameter
extracted
from
fitting
the
upstream
modulations
and
resulting
in
density
wave
wavevector
coinciding
with
that
of
the
pump
Real
space
signal
pump
and
idler
OPO
one
dimensional
filtered
profiles
derived
from
finite
size
numerics
with
and
without
defect
Rescaled
OPO
filtered
emissions
along
the
direction
obtained
by
numerically
solving
the
GPE
Eq
eq
gpequ
of
Sec
While
the
dashed
lines
represent
the
filtered
emissions
of
signal
top
panel
pump
middle
and
idler
bottom
for
top
hat
pump
without
defect
the
solid
lines
are
the
same
OPO
conditions
but
now
for
defect
positioned
at
corresponding
to
the
vertical
dotted
lines
The
system
parameters
are
the
same
ones
as
those
of
Fig
Thus
the
effect
of
the
defect
is
to
induce
only
elastic
at
the
same
energy
scattering
now
the
three
OPO
states
emit
each
on
its
own
Rayleigh
ring
given
each
by
the
symbols
on
the
left
panel
of
Fig
which
represent
the
rings
at
cut
for
This
makes
it
rather
difficult
to
extract
the
separated
signal
pump
and
idler
profiles
by
filtering
in
momentum
as
done
previously
for
the
homogeneous
case
but
it
still
allows
to
filter
those
profiles
very
efficiently
in
energy
In
fact
because
the
emission
is
like
it
is
enough
to
fix
single
value
of
the
energy
to
the
one
of
the
three
states
thus
extracting
the
filtered
profiles
either
in
real
space
or
in
momentum
space
-we
have
however
checked
that
integrating
in
narrow
energy
window
around
does
not
quantitatively
change
the
results
The
results
of
the
above
described
filtering
are
shown
in
Fig
where
real
space
emissions
are
plotted
in
the
left
panels
while
the
ones
in
momentum
space
are
plotted
in
the
right
panels
We
observe
very
similar
phenomenology
to
that
one
obtained
in
the
linear
approximation
shown
in
Fig
The
signal
now
is
at
slightly
negative
values
of
momenta
thus
implying
very
small
Rayleigh
ring
associated
with
this
state
Thus
we
observe
that
only
the
modulations
associated
with
the
pump
are
the
ones
that
are
weakly
imprinted
in
the
signal
state
and
that
can
be
observed
by
means
of
Gaussian
filtering
inset
of
the
top
left
panel
We
have
fitted
the
upstream
wave
crests
and
obtained
the
same
modulation
wavevector
as
the
pump
one
blue
upper
triangles
Similar
to
the
linear
response
case
we
also
find
here
that
the
most
visible
perturbation
in
the
emission
filtered
at
the
idler
energy
is
the
one
due
to
the
pump
Rayleigh
ring
As
before
the
modulations
due
to
the
idler
Rayleigh
ring
cannot
propagate
far
from
the
defect
because
of
the
small
group
velocity
associated
with
this
state
In
Fig
we
instead
plot
the
corresponding
one
dimensional
profiles
in
the
direction
both
in
presence
solid
line
and
without
dashed
line
defect
Here
we
can
observe
that
even
if
the
pump
has
top
hat
flat
profile
as
also
commented
previously
in
absence
of
defect
the
finite
size
OPO
is
characterised
by
inhomogeneous
profiles
of
signal
pump
and
idler
because
of
localised
currents
Furthermore
we
observe
that
the
presence
of
defect
induces
strong
modulations
in
pump
and
idler
Notice
that
the
modulation
imprinted
by
the
pump
into
the
signal
is
hardly
visible
in
the
top
panel
of
Fig
without
the
Gaussian
filtering
Experiments
Experimental
OPO
spectrum
lower
panel
and
filtered
emissions
of
signal
top
pump
middle
and
idler
bottom
in
presence
of
structural
defect
The
six
panels
show
the
filtered
emission
profiles
in
real
space
Gaussian
filtering
to
enhance
the
short
wavelength
modulations
is
applied
in
the
right
column
The
extracted
wave
crests
from
the
idler
yellow
contours
in
the
bottom
panel
are
also
superimposed
on
the
pump
profile
middle
by
applying
phase
shift
We
now
turn
to
the
experimental
analysis
where
continuous
wave
laser
is
used
to
drive
high
quality
GaAs
microcavity
sample
into
the
OPO
regime
-details
on
the
sample
can
be
found
in
Refs
The
polariton
dispersion
is
characterised
by
Rabi
splitting
meV
and
the
exciton
energy
is
meV
while
the
cavity
exciton
detuning
is
slightly
negative
meV
The
pump
is
at
and
meV
and
at
pump
powers
times
above
threshold
an
OPO
appears
with
signal
at
small
wavevector
and
meV
and
idler
at
and
meV
The
defect
used
in
the
sample
is
localized
inhomogeneity
naturally
present
in
the
cavity
mirror
Note
that
the
exact
location
of
the
defect
can
be
extracted
from
the
emission
spectrum
and
is
indicated
with
dot
orange
symbol
in
the
profiles
of
Fig
To
filter
the
emission
at
the
three
states
energies
and
to
obtain
spatial
maps
for
the
three
OPO
states
one
uses
spectrometer
and
at
fixed
position
one
obtains
the
intensity
emission
as
function
of
energy
and
position
By
changing
one
can
build
the
full
emission
spectrum
as
function
of
energy
and
position
The
filtered
emission
for
each
OPO
state
is
obtained
from
the
integrals
with
meV
The
results
are
shown
in
Fig
for
respectively
the
signal
top
panel
pump
middle
and
the
idler
bottom
profiles
Energy
and
momentum
of
the
three
OPO
states
are
labeled
with
blue
upper
triangle
signal
red
circle
pump
and
green
lower
triangle
idler
while
the
localised
state
clearly
visible
just
below
the
bottom
of
the
LP
dispersion
is
indicated
with
the
symbol
The
bare
LP
dispersion
is
extracted
from
an
off
resonant
low
pump
power
measurement
as
well
as
the
emission
of
the
exciton
reservoir
and
that
of
the
UP
dispersion
each
in
different
scale
The
signal
profile
shows
no
appreciable
modulations
around
the
defect
locations
nor
could
any
be
observed
after
applying
Gaussian
filtering
procedure
to
the
image
In
contrast
in
agreement
with
the
theoretical
results
both
filtered
profiles
of
the
pump
and
idler
show
the
same
Cherenkov
like
pattern
We
extracted
the
wave
crests
from
the
idler
profile
yellow
contours
in
the
bottom
panel
and
superimposed
them
on
the
pump
profile
middle
panel
with
an
added
phase
shift
revealing
that
the
only
modulations
visible
in
the
idler
state
are
the
ones
coming
from
the
pump
state
Conclusions
To
conclude
we
have
presented
joint
theoretical
and
experimental
study
of
the
superfluid
properties
of
nonequilibrium
condensate
of
polaritons
in
the
so
called
optical
parametric
oscillator
configuration
by
studying
the
scattering
against
static
defect
We
have
found
that
while
the
signal
is
basically
free
from
modulations
the
pump
and
idler
lock
to
the
same
response
We
have
highlighted
the
role
of
the
coupling
between
the
OPO
components
due
to
nonlinear
and
parametric
processes
These
are
responsible
for
the
transfer
of
the
spatial
modulations
from
one
component
to
the
other
This
process
is
most
visible
in
the
clear
spatial
modulation
pattern
that
is
induced
by
the
nonsuperfluid
pump
onto
the
idler
while
the
same
modulations
are
only
extremely
weakly
transferred
into
the
signal
because
of
its
low
characteristic
wavevector
so
much
that
experimentally
cannot
be
resolved
The
main
features
of
the
real
and
momentum
space
emission
patterns
are
understood
in
terms
of
Rayleigh
scattering
rings
for
each
component
and
characteristic
propagation
length
from
the
defect
the
rings
are
then
transferred
to
the
other
components
by
nonlinear
and
parametric
processes
Much
interest
has
been
recently
devoted
to
aspects
related
to
algebraic
order
and
superfluid
response
in
drive
dissipative
polariton
condensates
Our
theoretical
and
experimental
results
further
stress
the
complexities
and
richness
involved
when
looking
for
superfluid
behavior
in
nonequilibrium
multicomponent
condensates
such
as
the
ones
obtained
in
the
OPO
regime
Landau
levels
in
driven
dissipative
cavity
arrays
In
Michael
Berry
showed
that
the
adiabatic
evolution
of
energy
eigenfunctions
with
respect
to
time
dependent
Hamiltonian
contains
phase
of
geometrical
origin
commonly
known
today
by
the
name
of
Berry's
phase
Although
Berry's
seminal
paper
gave
the
example
of
spinor's
evolution
under
slowly
changing
magnetic
field
geometric
phases
with
similar
origin
have
been
encountered
in
plethora
of
physical
contexts
ranging
from
hydrodynamics
to
quantum
field
theory
the
quantum
Hall
effect
and
topological
insulators
In
condensed
matter
context
the
eigenstates
of
electrons
in
periodic
lattice
can
be
labeled
by
band
index
and
the
crystal
momentum
The
Berry
phase
physics
arising
when
adiabatically
transitioning
between
neighbouring
wavevectors
was
appreciated
only
recently
This
phenomenon
can
explain
properties
such
as
electric
polarization
anomalous
Hall
conductivity
or
the
quantization
of
conductance
in
the
integer
quantum
Hall
effect
In
the
presence
of
synthetic
gauge
field
the
eigenstates
making
up
an
energy
band
can
have
nontrivial
geometrical
properties
as
encoded
in
the
Berry
connection
and
Berry
curvature

Understanding
the
geometry
of
eigenstates
in
band
is
of
great
importance
not
least
because
the
integral
of
the
Berry
curvature
over
the
Brillouin
zone
BZ
gives
the
first
Chern
number
the
topological
invariant
responsible
for
the
integer
quantum
Hall
effect
Consequently
there
has
been
much
work
in
recent
years
to
develop
new
techniques
with
which
to
probe
the
properties
of
energy
bands
in
photonics
and
ultracold
gases
For
example
the
Berry
curvature
can
be
measured
in
the
semiclassical
dynamics
of
wavepacket
in
an
optical
lattice


or
in
photon
transport
in
cavity
array
In
all
these
cases
the
physics
can
be
most
naturally
understood
by
recognising
that
the
Berry
curvature
acts
like
magnetic
field
in
momentum
space

The
analogy
between
Berry
curvature
and
magnetism
is
most
powerful
when
geometrical
energy
band
is
subjected
to
an
additional
weak
harmonic
potential
Then
in
the
effective
momentum
space
Hamiltonian
harmonic
potential
acts
like
the
kinetic
energy
of
particle
in
real
space
Just
as
the
physical
momentum
is
the
sum
of
the
canonical
momentum
and
the
magnetic
vector
potential
in
the
magnetic
Hamiltonian
so
the
physical
position
is
given
by
the
canonical
position
and
the
Berry
connection
of
band
in
the
effective
Hamiltonian


The
Berry
curvature
is
then
like
momentum
space
magnetic
field
For
certain
models
this
analogy
leads
to
clear
analytical
understanding
of
single
particle
dynamics

In
particular
we
will
focus
on
the
small
flux
limit
of
the
Harper
Hofstadter
Hamiltonian
which
is
model
that
has
recently
been
realized
in
multitude
of
experimental
configurations
ranging
from
ultracold
gases
solid
state
superlattices
and
silicon
photonics
to
classical
systems
such
as
coupled
pendula
and
oscillating
circuits
As
first
shown
in
Ref
the
eigenstates
of
this
model
in
the
presence
of
harmonic
trap
are
toroidal
Landau
levels
in
momentum
space
Not
only
would
an
observation
of
these
states
constitute
the
first
exploration
of
analogue
magnetic
states
in
momentum
space
but
also
the
first
experimental
study
of
magnetism
on
torus
Most
theoretical
works
on
momentum
space
Landau
levels
have
focused
on
conservative
dynamics
while
photonic
systems
naturally
include
driving
and
dissipation
In
this
Chapter
we
present
realistic
experimental
proposal
for
the
observation
of
these
states
in
driven
dissipative
lattice
of
cavities
such
as
the
array
of
coupled
silicon
ring
resonators
of
Ref
where
link
resonators
were
used
to
simulate
synthetic
gauge
field
for
photons
We
combine
this
set
up
with
harmonic
potential
introduced
for
example
by
spatial
modulation
of
the
resonator
size
We
demonstrate
numerically
that
the
main
features
of
momentum
space
Landau
levels
will
be
observable
spectroscopically
in
this
system
for
realistic
parameters
In
this
Chapter
we
also
emphasize
how
the
inherent
driving
and
dissipation
in
photonics
can
be
key
advantage
in
probing
properties
that
are
otherwise
inaccessible
Firstly
the
spectroscopic
measurements
discussed
here
are
sensitive
to
the
absolute
energy
of
state
From
this
we
show
how
to
extract
the
energy
shift
due
to
the
off
diagonal
matrix
elements
of
the
Berry
connection
relating
eigenstates
in
different
bands
and
Only
very
recently
has
the
first
measurement
of
such
effects
been
reported
in
ultracold
atomic
gases
and
the
approach
used
in
this
experiment
would
be
difficult
to
apply
to
photonics
set
up
The
scheme
we
present
may
therefore
be
useful
for
the
characterisation
of
energy
bands
in
topologically
nontrivial
photonic
systems
Secondly
since
the
photon
steady
state
depends
on
the
overlap
between
the
observable
spatial
amplitude
profile
of
the
drive
and
of
the
eigenstates
the
observables
will
depend
on
the
phase
of
the
eigenfunctions
and
thus
on
the
specific
synthetic
magnetic
gauge
that
is
implemented
in
given
experimental
realization
of
the
Harper
Hofstadter
Hamiltonian
using
synthetic
gauge
field
We
note
that
related
gauge
sensitivity
has
also
recently
been
of
much
interest
in
ultracold
gases
in
suitably
designed
time
of
flight
experiments

Experiments
on
synthetic
magnetic
fields
therefore
present
the
opportunity
of
straightforwardly
probing
gauge
dependent
physics
This
Chapter
is
organized
as
follows
in
Section
we
add
harmonic
trap
to
the
HH
model
introduced
in
Sec
We
explain
how
the
eigenstates
can
be
understood
as
momentum
space
Landau
levels
in
Section
before
we
discuss
the
breakdown
of
approximations
in
Section
focusing
on
the
energy
shift
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
Then
in
Section
we
add
driving
and
dissipation
to
the
model
In
Section
we
show
numerical
results
highlighting
gauge
dependent
effects
before
presenting
viable
proposal
for
photonics
based
experiment
in
Section
Finally
we
draw
conclusions
in
Section
The
trapped
Harper
Hofstadter
Model
In
this
Chapter
we
study
the
Harper
Hofstadter
Hamiltonian
in
the
presence
of
an
external
harmonic
trap
The
full
tight
binding
Hamiltonian
of
this
system
is
where
is
the
real
hopping
amplitude
and
are
the
creation
annihilation
operators
for
particle
on
square
lattice
at
site
The
harmonic
trap
is
of
strength
and
is
centered
at
position
which
in
general
need
not
coincide
with
lattice
site
Throughout
the
lattice
spacing
is
set
equal
to
one
As
explained
in
Sec
the
hopping
phases
are
the
Peierls
phases
gained
by
charged
particle
hopping
in
the
presence
of
perpendicular
magnetic
field
The
sum
of
the
phases
around
square
plaquette
of
the
lattice
is
therefore
equal
to
where
is
the
number
of
magnetic
flux
quanta
through
the
plaquette
with
For
neutral
particles
such
as
photons
these
phases
can
be
imposed
artificially
to
simulate
the
effects
of
magnetism
for
example
by
inserting
link
resonators
into
an
array
of
silicon
ring
resonators
as
mentioned
above
Although
the
sum
of
phases
around
plaquette
is
set
by
the
external
synthetic
flux
the
exact
form
of
the
hopping
phases
themselves
depends
on
the
choice
of
magnetic
gauge
In
the
Landau
gauge
for
example
such
that
only
the
hopping
amplitude
along
one
direction
is
modified
Conversely
in
the
symmetric
gauge
and
so
hopping
terms
along
both
and
are
affected
preserving
the
rotational
invariance
of
the
lattice
This
gauge
dependence
of
the
hopping
phases
is
reflected
in
the
spatial
profile
of
the
phase
of
an
eigenstate
of
In
photonics
experiment
this
phase
is
an
observable
quantity
as
the
intensity
response
of
system
to
given
external
driving
is
determined
by
the
overlap
of
the
spatial
amplitude
distribution
of
the
pump
with
the
eigenstates
Such
experiments
will
therefore
be
sensitive
to
the
synthetic
magnetic
gauge
as
we
discuss
in
Section
Toroidal
Landau
Levels
in
Momentum
Space
Having
introduced
the
full
Hamiltonian
in
Eq
eq
model
we
now
review
how
the
eigenstates
of
this
model
in
an
appropriate
limit
can
be
understood
as
toroidal
Landau
levels
in
momentum
space
Throughout
the
following
discussion
we
assume
that
the
trap
is
centered
at
the
origin
We
begin
from
the
eigenstates
of
the
Harper
Hofstadter
model
where
is
the
energy
dispersion
of
band
at
crystal
momentum
As
the
spatially
dependent
hopping
phases
in
break
translational
invariance
new
magnetic
translation
operators
must
be
introduced
to
define
larger
magnetic
unit
cell
containing
an
integer
number
of
magnetic
flux
quanta
Then
translational
symmetry
is
restored
and
Bloch's
theorem
can
be
applied
to
write
the
eigenstates
as
where
is
the
periodic
Bloch
function
and
is
the
number
of
lattice
sites
Thanks
to
the
new
larger
unit
cell
the
crystal
momentum
and
the
periodic
Bloch
functions
here
are
defined
in
the
smaller
magnetic
Brillouin
zone
MBZ
For
example
hereafter
we
take
the
number
of
flux
quanta
per
plaquette
to
be
of
the
form
where
is
an
integer
Then
the
magnetic
unit
cell
can
be
chosen
to
be
times
larger
than
the
original
unit
cell
while
the
MBZ
is
times
smaller
than
the
original
BZ
Adding
the
harmonic
trap
breaks
all
translational
symmetry
of
the
lattice
but
we
can
use
the
eigenstates
of
as
basis
in
which
to
expand
the
new
wave
function
Substituting
this
expansion
into
the
full
Schr
dinger
equation
it
can
be
shown
that
the
expansion
coefficients
satisfy
""""where
is
the
matrix
valued
Berry
connection
To
proceed
we
consider
the
harmonic
trap
to
be
sufficiently
weak
compared
to
the
bandgap
that
we
can
make
single
band
approximation
This
assumes
that
only
one
coefficient
is
non
negligible
and
that
the
external
trap
does
not
significantly
mix
different
energy
bands
Then
Eq
eq
first
reduces
to
where
we
have
introduced
the
effective
momentum
space
Hamiltonian
For
the
moment
we
focus
on
the
first
two
terms
we
discuss
the
role
of
the
last
term
which
comes
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
in
detail
in
the
next
subsection
As
can
be
seen
there
is
close
analogy
between
the
first
two
terms
in
the
momentum
space
Hamiltonian
and
that
of
charged
particle
in
an
electromagnetic
field
in
real
space
In
this
analogy
the
role
of
the
particle
mass
is
played
by
while
the
scalar
potential
is
replaced
by
the
energy
band
dispersion
and
the
magnetic
vector
potential
by
the
intra
band
Berry
connection
We
note
that
both
the
magnetic
vector
potential
and
the
Berry
connection
are
gauge
dependent
quantities
We
hereafter
refer
to
the
gauge
choice
for
the
Berry
connection
as
the
Berry
gauge
and
the
gauge
choice
for
real
space
magnetic
vector
potential
as
the
magnetic
gauge
From
the
Berry
connection
we
can
also
define
the
geometrical
Berry
curvature
which
acts
like
momentum
space
magnetic
field
The
topology
of
momentum
space
also
plays
crucial
role
here
as
the
MBZ
is
topologically
equivalent
to
torus
One
important
consequence
of
this
is
of
course
that
the
integral
of
Berry
curvature
over
the
whole
MBZ
is
quantised
in
units
of
the
first
Chern
number
In
the
above
analogy
with
magnetism
this
means
that
the
particle
is
confined
to
move
on
the
surface
of
torus
while
the
Chern
number
counts
the
number
of
magnetic
monopoles
contained
inside
The
above
analogy
with
magnetism
is
particularly
powerful
because
there
are
natural
limits
for
our
model
Eq
eq
model
in
which
the
eigenstates
of
Eq
eq
mag
and
hence
of
Eq
eq
dual
are
known
analytically

We
will
focus
on
the
flat
band
limit
in
which
the
bandwidth
is
much
smaller
than
the
trapping
energy
for
the
energy
bands
of
with
this
assumption
improves
as
decreases
or
as
increases
In
this
limit
we
can
firstly
approximate
so
that
the
first
term
is
analogous
to
the
kinetic
energy
of
particle
in
uniform
magnetic
field
Secondly
we
can
approximate
so
that
the
second
term
of
is
just
constant
energy
shift
Hence
the
corresponding
eigenstates
can
be
understood
as
toroidal
Landau
levels
in
momentum
space
We
note
that
the
opposite
limit
in
which
the
trapping
energy
is
small
compared
to
the
bandwidth
also
yields
very
interesting
physics
including
the
realisation
of
Harper
Hofstatder
model
in
momentum
space
As
shown
in
Ref
the
momentum
space
toroidal
Landau
levels
form
semi
infinite
ladders
of
states
where
we
have
introduced
the
Landau
level
quantum
number
and
where
can
be
recognised
as
the
analogue
of
the
cyclotron
frequency
Again
we
note
that
here
we
have
neglected
the
contribution
from
the
last
term
in
Eq
eq
dual
as
this
will
be
discussed
in
the
next
subsection
As
can
be
shown
from
the
Diophantine
equation
for
the
Hall
conductivity
for
odd
values
of
the
Chern
number
of
all
bands
except
the
middle
band
is
Then
as
the
Chern
number
is
related
to
the
uniform
Berry
curvature
as
where
is
the
MBZ
area
the
Berry
curvature
is
given
by
While
the
above
spectrum
does
not
directly
depend
on
the
toroidal
topology
of
momentum
space
the
topology
does
enter
into
the
eigenstate
degeneracy
which
is
equal
to
as
well
as
into
the
analytical
form
of
the
eigenstates
in
the
MBZ
For
example
for
the
bands
with
the
eigenstates
can
be
written
as
where
are
the
Hermite
polynomials
and
is
the
analogue
of
the
magnetic
length
Here
we
have
taken
the
Berry
gauge
to
have
Landau
form
parallel
to
the
unit
vector
in
momentum
space
We
have
also
assumed
that
the
MBZ
is
of
length
in
one
momentum
direction
and
in
the
other
direction
corresponding
to
magnetic
unit
cell
of
plaquettes
containing
one
flux
quantum
While
this
choice
of
MBZ
is
valid
in
any
magnetic
gauge
of
the
underlying
Harper
Hofstadter
model
it
is
particularly
natural
when
the
hopping
phases
in
are
in
the
Landau
magnetic
gauge
as
we
shall
discuss
further
below
The
Berry
connection
We
now
study
the
effects
of
the
last
term
in
the
momentum
space
Hamiltonian
Eq
eq
dual
which
comes
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
This
can
be
recognised
as
momentum
space
counterpart
of
the
real
space
geometrical
scalar
potential
previously
studied
in
atomic
systems
In
these
systems
the
scalar
potential
arises
from
real
space
Berry
connections
which
can
be
created
for
example
by
using
spatially
dependent
optical
fields
to
dress
the
atoms
In
the
flat
band
limit
we
have
checked
numerically
that
we
can
approximate
the
momentum
space
geometrical
scalar
potential
as
and
so
this
contributes
only
uniform
constant
energy
shift
This
term
was
not
considered
in
previous
works
as
these
works
focused
on
systems
such
as
ultracold
atomic
gases
where
the
absolute
energy
is
not
easily
experimentally
observable
As
we
will
discuss
in
the
next
section
spectroscopic
measurements
in
photonics
are
sensitive
to
the
energy
of
state
therefore
such
corrections
may
be
extracted
experimentally
We
note
that
an
analogous
effect
has
also
been
derived
for
the
effective
momentum
space
magnetic
Hamiltonian
for
trapped
particle
in
an
ideal
flat
band
and
predicted
for
the
frequency
spectrum
associated
with
excitonic
states
in
transition
metal
dichalcogenides
To
see
under
what
conditions
the
off
diagonal
elements
of
the
Berry
connection
are
relevant
we
compare
the
energy
obtained
from
an
exact
numerical
diagonalization
of
Eq
eq
model
with
the
analytical
eigenenergy
predicted
by
Eq
eq
ladders
We
focus
on
the
lowest
ladder
of
states
associated
with
band
in
Eq
eq
ladders
at
energies
which
are
below
the
onset
of
the
second
ladder
around
energy
This
allow
us
to
easily
identify
which
numerical
eigenvalue
should
correspond
to
which
Landau
level
quantum
number
We
introduce
two
dimensionless
parameters
and
to
quantify
deviations
between
the
numerics
and
analytics
The
former
represents
the
error
in
the
"zero
point
energy
and
is
defined
as
the
energy
of
the
lowest
numerical
state
relative
to
the
analytical
Landau
level
The
latter
is
the
level
spacing
error
which
we
define
as
the
difference
between
the
numerical
energy
spacing
between
two
neighbouring
states
and
the
analytical
spacing
between
states
with
and
quantum
numbers
Considering
first
the
"zero
point
energy
error
the
analytical
energy
of
the
Landau
level
is
given
from
Eq
eq
ladders
by
where
we
have
used
that
and
where
we
calculate
the
uniform
energy
shift
as
the
average
band
energy
over
the
MBZ
This
definition
generalises
our
flat
band
approximation
to
account
for
the
non
zero
bandwidth
of
the
lowest
band
We
then
define
the
dimensionless
parameter
as
This
dimensionless
error
is
plotted
with
dashed
line
as
function
of
for
in
the
top
panel
of
Fig
At
small
there
is
large
bandgap
between
the
lowest
two
Harper
Hofstadter
energy
bands
but
the
lowest
band
also
has
large
bandwidth
In
this
regime
the
single
band
approximation
is
reasonable
while
the
flat
band
approximation
breaks
down
leading
to
large
errors
This
limit
requires
different
analytical
approach
as
previously
presented
in
Ref
To
account
for
the
error
at
large
we
include
the
shift
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
Eq
eq
shift
We
calculate
this
shift
numerically
from
the
eigenstates
of
the
Harper
Hofstadter
model
and
we
incorporate
it
into
second
parameter
This
is
plotted
as
solid
line
in
the
top
panel
of
Fig
As
can
be
seen
this
shift
dramatically
reduces
the
error
in
the
zero
point
energy
at
large
We
also
calculate
this
shift
considering
only
the
effects
of
band
mixing
with
the
second
lowest
band
this
is
indistinguishable
on
this
scale
from
the
full
shift
This
can
be
understood
from
the
dependence
on
the
bandgaps
in
Eq
eq
shift
which
shows
that
the
contributions
of
high
energy
bands
are
suppressed
As
discussed
further
in
Section
it
would
be
possible
experimentally
to
extract
the
energy
of
the
lowest
state
this
could
constitute
the
first
direct
measurement
of
the
effects
of
the
off
diagonal
matrix
elements
of
the
Berry
connection
in
photonics
system
Top
panel
"Zero
point
energy
error
with
solid
curve
and
without
dashed
line
the
shift
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
Including
just
the
first
term
in
the
sum
Eq
eq
shift
gives
an
identical
curve
to
the
one
of
The
chosen
trap
strength
is
Bottom
panel
Level
spacing
error
for
the
same
trap
strength
considering
solid
curve
dashed
curve
dashed
dotted
curve
and
dotted
curve
We
turn
now
to
the
level
spacing
error
This
can
be
expressed
as
where
we
have
used
that
the
analytical
level
spacing
from
Eq
eq
ladders
is
simply
We
plot
the
level
spacing
error
in
the
bottom
panel
of
Fig
for
and
As
can
be
seen
here
there
is
large
variation
in
the
errors
at
small
due
to
the
large
bandwidth
On
the
other
hand
we
see
that
for
where
the
flat
band
approximation
improves
In
this
regime
the
level
spacing
error
is
much
smaller
than
the
zero
point
error
This
is
because
when
the
shift
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
Eq
eq
shift
is
approximately
uniform
over
the
MBZ
at
large
it
just
acts
as
uniform
energy
shift
on
all
the
states
in
ladder
with
band
index
Consequently
this
shift
drops
out
of
the
level
spacing
error
between
states
leaving
only
higher
order
band
mixing
terms
From
perturbation
theory
it
is
expected
that
mixing
with
other
bands
leads
to
negative
energy
shift
on
states
in
the
lowest
band
and
indeed
this
can
be
seen
in
both
and
in
the
small
negative
errors
found
at
large
Driving
and
dissipation
We
now
include
in
our
model
the
driving
and
dissipation
that
are
an
integral
part
of
the
proposed
photonics
experiment
We
assume
there
are
uniform
and
local
losses
characterized
by
loss
rate
and
that
the
pump
is
monochromatic
with
frequency
and
spatial
profile
Following
the
treatment
of
Ref
we
replace
the
bosonic
creation
and
annihilation
operators
with
their
expectation
values
as
can
be
justified
for
noninteracting
system
The
steady
state
evolution
of
the
photon
field
amplitude
in
cavity
then
follows
that
of
the
pump
as
Combining
Hamiltonian
evolution
with
pumping
and
losses
one
arrives
at
set
of
linear
coupled
equations
that
can
be
solved
numerically
for
the
steady
state
where
we
have
reintroduced
the
position
of
the
harmonic
trap
centre
although
unless
otherwise
specified
we
set
in
our
simulations
The
expectation
values
correspond
to
the
number
of
photons
at
site
whereas
the
intensity
spectrum
is
given
by
their
total
sum
as
function
of
pump
frequency
These
observables
can
be
directly
related
to
the
eigenstates
of
the
Hamiltonian
in
Eq
eq
model
Firstly
the
different
eigenmodes
of
driven
dissipative
system
will
appear
as
peaks
in
the
transmission
and
or
absorption
spectra
under
coherent
pump
The
resonance
peaks
will
be
broadened
by
the
decay
rate
while
the
area
of
the
peaks
will
depend
on
the
overlap
between
the
spatial
amplitude
profile
of
the
pump
and
the
underlying
eigenstate
of
at
that
energy
Secondly
when
the
pump
frequency
is
set
on
resonance
with
given
mode
the
intensity
profiles
in
both
real
and
momentum
space
reproduce
the
wave
function
of
that
mode
This
corresponds
respectively
to
measuring
the
near
field
and
far
field
spatial
emission
of
photons
from
the
cavity
array
We
note
that
the
far
field
emission
is
simply
the
Fourier
transform
of
the
real
space
wave
function
and
so
will
be
function
of
crystal
momentum
defined
in
the
full
BZ
To
reach
the
MBZ
further
processing
step
is
required
for
example
if
the
Harper
Hofstadter
Hamiltonian
is
in
the
Landau
gauge
and
if
we
choose
magnetic
unit
cell
of
plaquettes
along
the
appropriate
transformation
takes
particularly
simple
form
where
is
the
wave
function
coefficient
in
the
MBZ
while
is
that
in
the
original
BZ
In
this
expression
is
an
integer
while
is
the
magnetic
reciprocal
lattice
vector
where
the
factor
of
is
due
to
the
enlarged
magnetic
unit
cell
We
note
that
for
other
magnetic
gauges
or
for
other
choices
of
the
magnetic
unit
cell
this
transformation
will
in
general
be
more
complicated
In
this
sense
we
call
this
choice
of
magnetic
unit
cell
"natural
choice
when
the
Harper
Hofstadter
Hamiltonian
is
in
the
Landau
gauge
In
the
rest
of
the
Chapter
we
denote
the
momentum
in
the
original
BZ
as
and
that
in
the
MBZ
as
Results
and
discussion
Pumping
and
gauge
dependent
effects
Intensity
spectra
for
different
pumping
conditions
pumping
the
single
site
pumping
with
Gaussian
profile
centered
at
site
with
width
homogeneous
pumping
across
all
lattice
sites
and
pumping
with
random
phase
across
all
lattice
sites
These
results
were
obtained
by
numerical
solving
Eq
eq
linear
problem
for
the
steady
state
in
lattice
of
sites
with
and
Black
solid
curves
correspond
to
using
the
Landau
gauge
while
green
dashed
ones
to
the
symmetric
gauge
The
dotted
vertical
lines
with
labels
indicating
the
value
of
mark
the
states
which
were
selected
for
later
analysis
The
spectra
in
panels
and
are
identical
for
both
gauges
As
introduced
above
spectroscopic
measurements
can
be
used
in
driven
dissipative
photonics
experiment
to
study
the
trapped
Harper
Hofstadter
model
and
hence
toroidal
Landau
levels
in
momentum
space
In
this
section
we
focus
on
the
effects
of
the
pumping
exploring
how
different
pumping
schemes
excite
the
eigenstates
with
different
weights
We
find
that
such
spectroscopic
measurements
are
sensitive
to
the
underlying
synthetic
magnetic
gauge
chosen
in
given
implementation
of
the
Harper
Hofstadter
Hamiltonian
To
best
illustrate
these
gauge
dependent
effects
we
present
the
results
of
numerically
solving
Eq
eq
linear
problem
for
the
steady
state
in
large
lattice
of
sites
with
and
These
parameters
are
chosen
to
highlight
the
key
features
of
different
pumping
schemes
we
will
present
numerical
results
for
more
realistic
experimental
system
in
Section
The
numerical
code
was
written
in
Julia
and
is
available
online
as
supplemental
material
to
Ref
The
intensity
spectrum
of
the
steady
state
as
function
of
pump
frequency
is
shown
in
Fig
where
we
compare
results
for
both
the
Landau
and
symmetric
gauge
for
four
pumping
schemes
discussed
in
turn
below
For
simplicity
we
limit
ourselves
to
pump
frequencies
located
between
the
two
lowest
lying
Harper
Hofstadter
bands
of
the
untrapped
system
This
allows
us
to
focus
only
on
states
within
the
first
ladder
of
the
trapped
system
Eq
eq
ladders
with
At
higher
energies
the
clear
identification
of
states
is
more
difficult
as
more
than
one
ladder
of
toroidal
Landau
levels
can
overlap
as
shown
for
example
in
Section
Single
site
pumping
-The
first
and
simplest
case
that
we
consider
is
that
of
pumping
single
site
at
an
off
center
lattice
site
These
results
are
shown
in
Fig
panel
where
the
uniform
energy
spacing
of
the
toroidal
Landau
levels
can
be
clearly
observed
For
this
pumping
scheme
we
find
no
significant
differences
between
the
spectra
for
the
Landau
or
symmetric
gauge
This
is
to
be
expected
as
changing
the
gauge
is
equivalent
to
changing
the
relative
phase
between
different
sites
but
as
we
are
only
pumping
one
site
this
phase
difference
is
unimportant
Instead
for
both
magnetic
gauges
we
see
that
the
peak
height
is
very
small
for
low
energy
states
rising
to
maximum
as
energy
increases
before
decreasing
once
more
This
behaviour
can
be
understand
by
considering
the
form
of
the
real
space
wavefunctions
of
In
real
space
the
eigenstates
are
rings
of
finite
width
which
increase
in
radius
as
the
energy
increases
as
can
be
seen
in
Fig
Analytically
we
can
predict
how
the
ring
radius
scales
with
energy
by
remembering
that
the
term
in
Eq
eq
ladders
is
the
momentum
space
kinetic
energy
where
is
the
physical
position
operator
in
the
lowest
band
From
this
we
deduce
that
as
can
be
confirmed
numerically
Therefore
if
one
pumps
an
off
center
site
there
will
only
be
limited
range
of
rings
that
will
have
radii
that
will
overlap
with
the
pump
spot
and
so
be
excited
Here
we
have
set
the
pump
spot
to
be
at
position
and
from
the
above
scaling
the
toroidal
Landau
level
that
best
overlaps
with
this
pump
will
have
quantum
number
which
is
in
good
agreement
with
the
numerical
results
shown
in
Fig
Real
space
reconstruction
of
the
states
and
of
Fig
Gaussian
pumping
-We
now
consider
Gaussian
pump
as
the
next
logical
step
up
in
complexity
from
single
site
pump
This
has
the
form
and
we
choose
and
for
the
pump
centre
to
be
at
as
for
single
site
pumping
The
results
are
shown
in
Fig
The
main
effect
is
as
expected
that
more
states
become
visible
in
both
the
low
smaller
and
high
energy
larger
sections
of
the
spectrum
This
is
because
the
pump
has
greater
spatial
width
and
so
overlaps
with
larger
range
of
real
space
eigenstates
However
we
can
also
see
that
the
intensity
spectrum
now
depends
on
the
underlying
magnetic
gauge
as
the
phase
of
the
eigenstates
is
important
In
particular
more
high
energy
peaks
can
be
seen
for
the
Landau
gauge
than
for
the
symmetric
gauge
This
can
be
most
easily
understood
by
noting
that
in
momentum
space
the
symmetric
gauge
states
also
have
ring
like
structure
see
bottom
panel
of
Fig
where
the
ring
radius
increases
with
To
see
this
we
note
that
in
the
symmetric
gauge
the
real
space
wavefunctions
have
phase
which
winds
around
the
ring
as
where
is
the
polar
angle
around
the
ring
This
phase
winding
sets
the
radius
of
the
rings
in
momentum
space
as
scaling
that
can
be
confirmed
numerically
and
seen
in
Fig
bottom
panel
The
white
spot
close
to
the
edges
of
the
rings
in
these
figures
is
due
to
destructive
interference
with
the
pump
As
the
Fourier
transform
of
the
Gaussian
pump
is
again
Gaussian
it
follows
that
only
limited
range
of
low
energy
symmetric
gauge
momentum
space
states
will
have
good
overlap
with
the
pump
The
high
energy
portion
of
the
spectrum
is
therefore
washed
out
compared
to
its
Landau
gauge
counterpart
where
states
have
higher
amplitude
close
to
the
centre
of
the
BZ
and
so
better
overlap
with
the
pump
Momentum
space
reconstruction
of
the
eigenstates
Top
row
states
corresponding
to
and
in
Fig
using
the
Landau
gauge
Bottom
row
states
corresponding
to
and
in
Fig
using
the
symmetric
gauge
Before
continuing
we
also
note
that
for
sufficiently
large
values
of
the
symmetric
gauge
rings
in
momentum
space
will
increase
to
the
point
where
they
touch
the
BZ
boundaries
When
this
occurs
self
interference
patterns
appear
in
the
wave
function
as
shown
for
example
in
Fig
The
extra
ring
like
structures
appearing
for
in
Fig
are
due
to
the
close
proximity
of
states
pertaining
to
other
ladders
with
Note
that
in
order
to
excite
such
high
energy
states
we
have
used
pump
with
homogeneous
amplitude
and
random
onsite
phase
as
will
be
presented
in
the
fourth
pumping
case
below
Homogeneous
pumping
with
uniform
phase
-If
we
now
take
the
limit
of
very
wide
Gaussian
we
reach
homogeneous
pump
profile
extended
over
all
lattice
sites
The
results
for
this
pumping
scheme
are
shown
in
panel
of
Fig
Now
and
we
see
that
the
intensity
spectrum
is
strongly
magnetic
gauge
dependent
In
the
Landau
gauge
firstly
there
are
visible
peaks
for
only
half
of
the
states
This
can
be
understood
by
noting
that
homogeneous
pump
in
real
space
is
function
in
momentum
space
centered
in
the
middle
of
the
BZ
If
we
consider
the
Landau
gauge
eigenstates
in
the
full
BZ
as
shown
in
the
top
row
of
Fig
we
see
that
the
states
with
an
even
value
of
have
an
even
number
of
nodes
with
lobe
at
the
BZ
center
Conversely
the
states
with
odd
values
of
have
an
odd
number
of
nodes
including
one
at
the
BZ
center
These
feature
can
be
related
back
to
the
properties
of
the
Hermite
polynomials
in
the
analytical
eigenstates
in
the
MBZ
Eq
eq
chi
Consequently
only
states
with
even
values
of
have
good
overlap
with
the
pump
and
the
intensity
spectrum
contains
half
the
expected
peaks
now
separated
by
twice
the
toroidal
Landau
level
energy
spacing
In
the
symmetric
gauge
secondly
we
find
only
one
out
of
every
four
states
for
homogeneous
pumping
as
can
be
seen
in
the
inset
of
Fig
This
is
due
to
the
fact
that
on
square
lattice
the
angular
momentum
is
conserved
modulo
respecting
the
fold
rotational
symmetry
The
peak
intensity
gets
smaller
for
larger
because
of
the
diminishing
overlap
of
the
localized
central
pump
with
the
increasing
momentum
space
ring
discussed
above
Momentum
space
reconstruction
of
the
eigenstates
in
the
full
BZ
using
the
symmetric
gauge
and
homogeneous
pumping
with
random
on
site
phase
Top
row
states
corresponding
to
Bottom
row
states
corresponding
to
Parameters
are
the
same
as
in
Fig
Homogeneous
pumping
with
random
on
site
phase
-As
the
fourth
scheme
we
consider
pump
with
uniform
amplitude
over
the
lattice
but
random
site
dependent
phase
The
phases
are
chosen
from
random
uniform
distribution
and
have
values
in
the
interval
The
bottom
panel
of
Fig
was
obtained
by
averaging
over
distinct
realizations
of
these
random
phases
This
results
in
relatively
even
intensity
distribution
for
both
gauges
where
we
can
associate
peak
to
each
toroidal
Landau
level
in
this
energy
window
While
such
pumping
scheme
would
therefore
be
the
best
way
to
excite
all
the
eigenstates
and
to
fully
probe
the
momentum
space
physics
we
note
that
this
would
also
be
difficult
to
achieve
in
an
experiment
Momentum
space
reconstruction
of
the
eigenstates
in
the
full
BZ
using
the
Landau
top
row
and
symmetric
gauge
middle
and
bottom
row
and
spatially
homogeneous
pump
with
random
on
site
phase
We
have
considered
the
state
for
different
trap
positions
For
the
top
and
middle
rows
we
have
from
left
to
right
trap
in
the
center
and
whereas
for
the
bottom
row
we
chose
the
positions
and
Parameters
are
the
same
as
in
Fig
Top
row
Intensity
spectrum
for
lattice
of
sites
with
and
Second
row
Profile
of
the
state
in
real
space
left
momentum
space
center
and
the
population
over
bands
in
the
MBZ
right
for
the
conservative
system
Third
row
Reconstruction
of
the
mode
wavefunction
in
real
space
left
momentum
space
center
and
the
population
over
bands
in
the
MBZ
right
obtained
in
the
driven
dissipative
system
by
pumping
at
on
resonance
with
the
desired
mode
black
dotted
line
in
the
top
panel
The
like
pump
at
is
visible
as
dark
square
in
the
left
panel
Bottom
row
Slice
along
the
line
in
the
MBZ
solid
black
line
analytical
prediction
of
Eq
eq
chi
blue
dotted
line
and
population
over
bands
for
the
nondissipative
system
dashed
orange
line
Before
continuing
we
give
final
example
of
an
interesting
gauge
dependent
effect
that
could
be
studied
experimentally
in
this
system
Unlike
the
physics
discussed
above
this
is
not
directly
related
to
the
pumping
but
instead
to
the
behaviour
of
the
wave
function
under
change
in
the
centre
of
the
harmonic
trap
As
derived
in
Ref
moving
the
harmonic
trap
in
space
changes
the
boundary
conditions
on
the
wave
function
in
the
MBZ
We
note
that
although
this
derivation
was
made
explicitly
for
the
magnetic
Landau
gauge
in
the
MBZ
numerically
we
observe
here
that
this
physics
is
also
seen
in
the
full
BZ
in
both
gauges
As
shown
in
Fig
shift
in
the
harmonic
trap
centre
in
one
direction
shifts
the
observed
momentum
space
pattern
in
the
perpendicular
direction
This
behaviour
can
be
understood
as
realisation
of
Laughlin's
Gedankenexperiment
for
the
quantum
Hall
effect
but
now
in
momentum
space
As
we
observe
the
momentum
space
wave
function
returns
to
itself
after
the
harmonic
trap
has
been
moved
lattice
sites
for
the
magnetic
Landau
gauge
but
lattice
sites
for
the
magnetic
symmetric
gauge
reflecting
the
underlying
translational
symmetry
of
in
the
two
different
gauges
Results
for
realistic
experimental
parameters
We
now
present
numerical
results
for
system
parameters
within
current
experimental
reach
to
demonstrate
that
the
essential
characteristics
of
toroidal
Landau
levels
could
be
probed
experimentally
for
the
first
time
in
photonics
We
choose
small
lattice
of
only
sites
with
losses
of
this
loss
rate
is
in
the
same
range
as
those
present
in
the
experiment
of
Ref
Such
large
loss
rate
broadens
the
peaks
in
the
intensity
spectrum
making
closely
spaced
eigenenergies
harder
to
resolve
From
Eq
eq
ladders
we
see
that
the
level
spacing
is
given
by
and
so
we
can
increase
the
energy
spacing
by
applying
stronger
harmonic
potential
chosen
here
as
Increasing
the
strength
of
the
harmonic
trap
improves
our
flat
band
approximation
but
weakens
the
single
band
approximation
To
compensate
for
this
we
consider
larger
value
of
for
which
the
larger
band
gap
reduces
band
mixing
effects
As
in
the
experiment
of
Ref
we
work
in
the
Landau
gauge
for
the
Harper
Hofstadter
Hamiltonian
with
hopping
phases
given
by
In
order
to
model
the
experimental
pumping
scheme
where
light
was
injected
into
single
resonator
at
the
edge
of
the
system
via
an
external
integrated
waveguide
we
consider
localized
pump
on
single
site
situated
on
the
upper
border
of
the
system
at
The
corresponding
intensity
spectrum
is
shown
in
the
st
row
of
Fig
Apart
from
the
expected
broadening
due
to
larger
losses
the
peaks
observed
correspond
well
to
the
expected
eigenenergies
As
discussed
above
single
site
pumping
limits
the
number
of
visible
peaks
as
the
heights
of
the
peaks
at
low
energies
are
suppressed
due
to
the
poor
overlap
of
the
real
space
eigenstate
with
the
pump
position
However
as
this
pumping
scheme
is
closest
to
that
used
in
experiments
we
emphasise
that
even
in
this
case
enough
peaks
can
be
observed
to
extract
quantitative
measurements
of
the
toroidal
Landau
level
spacing
The
orange
vertical
dashed
lines
show
the
first
ladder
of
eigenstates
of
from
Eq
eq
model
We
also
note
that
here
for
frequencies
larger
than
we
also
start
to
see
states
from
the
second
ladder
see
Eq
eq
ladders
which
are
depicted
as
green
vertical
dash
dotted
lines
Their
proximity
to
the
first
ladder
states
means
they
cannot
be
easily
resolved
as
separate
peaks
in
the
dissipative
spectrum
Setting
the
pump
frequency
at
the
energy
indicated
by
the
black
dotted
line
we
plot
the
numerical
near
and
far
field
emission
in
the
left
and
center
panels
of
the
rd
row
of
Fig
This
corresponds
to
the
wave
function
in
real
space
and
in
the
full
BZ
respectively
By
applying
the
transformation
in
Eq
eq
trans
we
can
also
map
the
wave
function
in
the
full
BZ
to
the
population
over
bands
in
the
MBZ
as
shown
in
the
right
panel
of
the
rd
row
of
Fig
For
comparison
we
plot
these
quantities
for
the
corresponding
numerical
eigenstate
of
Eq
eq
model
in
the
second
row
of
Fig
for
which
there
is
no
pumping
and
dissipation
We
find
very
good
qualitative
agreement
between
the
numerical
results
in
the
MBZ
and
the
analytical
toroidal
Landau
level
Eq
eq
chi
with
as
expected
We
can
make
quantitative
comparison
with
this
analytical
eigenstate
by
taking
slice
along
the
dash
dotted
vertical
lines
in
the
right
column
of
rows
and
these
cuts
are
shown
in
the
bottom
panel
of
Fig
along
with
dotted
blue
curve
indicating
the
analytical
eigenstate
As
can
be
seen
there
is
excellent
agreement
between
the
numerics
without
driving
and
dissipation
and
the
analytical
result
We
have
checked
that
reducing
makes
this
fit
even
better
pointing
towards
band
mixing
effects
Introducing
pumping
and
dissipation
distorts
the
eigenstate
but
many
characteristic
features
are
still
clearly
observable
It
is
particularly
interesting
to
note
that
in
the
driven
dissipative
steady
state
in
real
space
shown
in
the
left
panel
of
the
rd
row
of
Fig
the
photon
distribution
breaks
the
rotational
symmetry
of
the
ring
eigenstate
While
this
can
be
physically
understood
as
decaying
cyclotron
orbit
with
an
inverse
lifetime
set
by
in
terms
of
eigenmodes
the
exponential
decay
and
more
generally
the
breaking
of
the
rotational
symmetry
results
from
the
interference
of
several
modes
which
overlap
in
frequency
due
to
the
relatively
large
value
of
In
the
same
way
that
real
space
Landau
levels
give
rise
to
real
space
cyclotron
orbits
under
the
effect
of
the
magnetic
field
the
observation
of
momentum
space
Landau
levels
can
provide
clear
evidence
of
cyclotron
orbit
in
momentum
space
under
the
effect
of
the
Berry
curvature
whose
effect
is
indeed
that
of
momentum
space
magnetic
field
Finally
we
briefly
summarize
how
one
can
practically
measure
the
contribution
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
see
Eq
eq
shift
from
the
intensity
spectrum
Starting
from
an
experimental
spectrum
one
first
needs
to
select
particular
peak
and
determine
its
label
by
comparing
the
MBZ
reconstruction
with
the
analytical
result
The
distance
between
two
neighbouring
peaks
gives
the
level
spacing
Finally
to
separate
the
shift
from
the
Harper
Hofstadter
ground
state
energy
in
Eq
eq
ladders
one
can
make
use
of
the
fact
that
the
former
depends
on
the
trap
strength
while
the
latter
does
not
Preparing
two
otherwise
identical
samples
with
different
trap
strengths
and
subtracting
the
ground
state
energy
will
then
allow
for
direct
measurement
of
the
contribution
from
the
off
diagonal
matrix
elements
of
the
Berry
connection
Conclusion
In
conclusion
we
have
shown
that
the
observation
of
toroidal
Landau
levels
in
momentum
space
is
within
experimental
reach
for
state
of
the
art
driven
dissipative
photonic
systems
Our
proposal
combines
the
recent
realisation
of
the
Harper
Hofstatder
model
in
an
array
of
silicon
based
coupled
ring
resonators
in
Ref
with
harmonic
potential
which
could
be
introduced
through
spatial
modulation
of
the
resonator
size
We
have
presented
numerical
results
to
show
that
even
for
very
small
lattices
in
the
presence
of
driving
and
strong
dissipation
key
characteristics
of
the
toroidal
Landau
levels
can
still
be
extracted
This
would
be
first
direct
investigation
of
analogue
magnetic
eigenstates
in
momentum
space
We
have
also
emphasised
that
the
proposed
photonics
experiment
would
be
able
to
highlight
momentum
space
analog
of
the
cyclotron
motion
as
well
as
to
measure
the
energy
shift
due
to
the
off
diagonal
matrix
elements
of
the
Berry
connection
which
as
these
are
inter
band
geometrical
properties
are
hard
to
access
by
other
means
We
have
also
discussed
how
the
spectroscopic
measurements
presented
here
are
sensitive
to
the
specific
synthetic
magnetic
gauge
implemented
in
an
experiment
Finally
an
interesting
outlook
would
be
to
include
the
effect
of
photon
photon
interactions
in
the
model
as
the
degenerate
ground
states
predicted
in
for
weakly
interacting
trapped
Harper
Hofstadter
model
may
lead
to
interesting
nonlinear
dynamical
features
In
the
longer
run
when
the
synthetic
gauge
field
is
combined
with
strong
interactions
one
can
hope
to
observe
the
hallmarks
of
fractional
quantum
Hall
physics
Conclusions
Conclusions
Summary
of
the
manuscript
The
main
theme
of
this
thesis
was
the
study
of
well
established
phenomena
namely
superfluidity
and
magnetism
in
novel
context
of
driven
dissipative
photonic
systems
which
are
subject
to
losses
that
need
to
be
compensated
by
continuous
pumping
After
introducing
the
basic
concepts
in
the
setting
of
ultracold
atomic
gases
in
thermal
equilibrium
we
looked
at
how
the
effects
of
driving
and
dissipation
changed
this
picture
In
particular
we
investigated
the
response
of
microcavity
exciton
polaritons
scattering
against
stationary
defect
with
an
eye
on
superfluid
like
effects
and
then
turned
to
the
momentum
space
magnetism
of
resonator
arrays
Polaritons
In
the
case
of
polaritons
we
looked
at
the
single
fluid
pump
only
regime
as
well
as
the
three
coupled
fluids
of
the
optical
parametric
oscillator
regime
created
by
the
parametric
scattering
of
the
pump
to
the
signal
and
idler
states
We
found
that
none
of
the
two
regimes
displayed
frictionless
flow
in
the
strict
sense
of
the
word
The
pump
only
fluid
showed
density
modulation
localized
close
to
the
defect
even
in
the
"superfluid
regime
"resulting
in
residual
drag
force
that
is
entirely
due
to
the
finite
polariton
lifetime
On
the
other
hand
we
showed
that
the
optical
parametric
oscillator
regime
in
the
optical
limiter
configuration
always
displays
propagating
density
modulations
in
the
pump
signal
and
idler
and
therefore
violates
the
definition
of
frictionless
flow
in
stronger
conceptual
way
Futhermore
we
determined
two
distinct
types
of
threshold
like
behaviour
of
the
drag
force
as
function
of
fluid
velocity
in
the
pump
only
case
one
of
which
has
no
direct
analog
in
equilibrium
weakly
interacting
atomic
gases
In
the
optical
parametric
oscillator
case
we
singled
out
three
factors
which
together
determine
the
amplitude
of
the
density
modulations
and
explained
why
for
typical
experimental
conditions
the
signal
state
may
appear
superfluid
Harper
Hofstadter
In
the
last
part
of
the
manuscript
we
have
discussed
how
momentum
space
Landau
levels
could
be
observed
experimentally
in
driven
dissipative
photonic
systems
by
breaking
time
reversal
invariance
by
means
of
synthetic
gauge
field
and
creating
topologically
nontrivial
energy
bands
We
have
presented
realistic
proposal
for
engineering
these
states
by
combining
harmonic
trap
with
an
artificial
magnetic
field
for
photons
in
two
dimensional
ring
resonator
array
An
observation
of
momentum
space
Landau
levels
would
be
the
first
realisation
of
magnetism
on
toroidal
surface
We
have
also
demonstrated
that
the
main
properties
of
momentum
space
Landau
levels
may
be
observed
spectroscopically
for
systems
with
realistic
experimental
parameters
Furthermore
spectroscopic
measurements
will
be
able
to
access
the
absolute
energy
of
eigenstates
measuring
novel
geometrical
features
of
energy
bands
such
as
contribution
from
the
inter
band
Berry
connection
The
pumping
and
losses
present
in
these
experiments
also
open
the
way
towards
studying
analogue
cyclotron
orbits
in
momentum
space
Outlook
In
the
future
more
in
depth
studies
of
polariton
superfluidity
especially
regarding
the
optical
parametric
oscillator
regime
are
needed
In
experiments
one
could
look
at
multiply
connected
geometries
and
on
the
theoretical
side
use
nonequilibrium
Keldysh
field
theory
which
was
already
successfully
employed
in
the
case
of
incoherently
pumped
systems
in
order
to
rigorously
compute
the
superfluid
and
normal
fractions
As
far
as
the
resonator
arrays
are
concerned
it
would
be
interesting
to
include
photon
photon
interaction
in
future
studies
paving
the
way
to
fractional
quantum
Hall
physics
in
driven
dissipative
setting