![Fig. 1: Experimental coherence properties of the polariton condensate. (a) First-order coherence function g(1)(τ) above threshold. (b) Dependence of the coherence time on pumping intensity. (c,d) Experimental intensity-intensity correlation function g(2)(τ) below (c) and a factor of two above (d) threshold. Correcting for experimental factors gives g(2)(0) = 1.1 for the emission above threshold. Reproduced from ref. [4].](/figures/fig-1-experimental-coherence-properties-of-the-polariton-38qvzguc.webp)
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July 2009
EPL, 87 (2009) 27002 www.epljournal.org
doi: 10.1209/0295-5075/87/27002
Coherence properties of the microcavity polariton condensate
D. M. Whittaker
1
and P. R. Eastham
2(a)
1
Department of Physics and Astronomy, University of Sheffield - Sheffield S3 7RH, UK, EU
2
Department of Physics, Imperial College London - London SW7 2AZ, UK, EU
received on 2 July 2009; accepted by M. C. Payne on 2 July 2009
published online 21 July 2009
PACS 71.36.+c – Polaritons (including photon-phonon and photon-magnon interactions)
PACS 78.20.Bh – Theory, models, and numerical simulation
PACS 42.50.Lc – Quantum fluctuations, quantum noise, and quantum jumps
Abstract – A theoretical model is presented which explains the dominant decoherence process
in a microcavity p ol arit on condensate. The mechanism which is invoked is the effect of self-phase
mo dul ati on, whereby interactions transform polariton number fluctuations into random energy
variations. The model shows that the phase coherence decay, g
(1)
(τ ), has a Kubo form, whi ch can
b e Gaussian or exponential, depending on whether the number fluctuations are slow or fast. This
fluctuation rate also determines the decay time of the intensity correlation functi on, g
(2)
(τ ), so it
can be directly determined experimentally. The model explains recent exp erim ental measurements
of a relatively fast Gaussian decay for g
(1)
(τ ), but also predicts a regime, further above threshold,
where the decay is much slower.
Copyright
c
EPLA, 2009
Introduction. – Microcavity polaritons are quasi-
particles arising from the strong coupling between exci-
tons and photons confined in planar cavity structures.
The observation of coherent emission f rom a CdTe micro-
cavity [1] has demonstrated that polaritons can form a new
type of quantum condensate. As in other quantum conden-
sates, such as atomic gases or superconductors, a key prop-
erty is the existence of an ord er parameter, the local phase,
which is correlated over large times and distances. The
polariton condensate presents interesting theoretical chal-
lenges since, unlike these other systems, it is mesoscopic,
typically consisting of a few hundred particles, and out
of equilibrium, with pumping required to maintain the
population against emission losses. In mesoscopic systems
order parameters fluctuate [2,3], so the phase correlations
decay. In this letter, we present a theory which shows that
the source of the fluctuations is variations in the number
of cond ensed particles, combined with polariton-polariton
interactions: it is this dynamics that is responsible for
the decoherence. Our theory shows that, under appro-
priate pumping conditions, existing microcavity struc-
tures should display much longer coherence times than
currently measured, opening up opportunities for experi-
ments manipu latin g the quantum state of the system.
The coherence of the polariton condensate can be
quantified by the decay of the first-order coherence func-
tion, g
(1)
(τ) ∝a
†
(0)a(τ), whose Fourier transform is
(a)
E-mail: p.eastham@imperial.ac.uk
the emission spectrum. For the polariton condensate this
function is directly revealed by coherence measurements
on the optical emission [1,4,5]. In a condensate we expect
long phase coherence times and thus a spectrally narr ow
emission above threshold. Recent experimental results [4]
(see fig. 1) show th at the decay time of g
(1)
(τ)is∼150 ps.
This is much longer than was originally believed [1,5], but
short compared to a laser or atomic gas. Furthermore,
the decay has a distinctive Gaussian form, and the
decay time is approximately constant above threshold.
The experiments also determine the intensity-intensity
correlation function (second-order coherence fun ction),
g
(2)
(τ), which reveals significant number flu ctuation s
(g
(2)
(0) > 1), decaying with a timescale ∼ 100 ps.
In ref. [4], along with the experiments, we quoted the
semiclassical results, eqs. (7) and (9) of this letter, which
we showed to be compatible with the measured values
of the coherence times. The discussion was limited to
the case of slow number fluctuations, whose presence is
directly evident in the experimental data. Here we show
that this regime is achieved due to critical slowing down in
the thr eshold region. At higher powers, where the critical
slowing down disappears and fluctuations become faster,
we predict that the phase coherence times will become
significantly longer. This is shown to be a manifestation
of the Kubo stochastic line-shape theory [6,7], in the
motional narrowing limit. We also present numerical
results for the threshold region, where mesoscopic effects
are important and the semiclassical results break down.
27002-p1
D. M. Wh ittaker and P. R. Eastham
Fig. 1: Experimental coherence properties of the p olari ton
condensate. (a) First-order coherence function g
(1)
(τ )above
threshold. (b) Dependence of the coherence time on pumping
intensity. (c,d) Experimental intensity-intensity correlation
function g
(2)
(τ ) below (c) and a factor of two above (d)
threshold. Correcting for experimental factors gives g
(2)
(0) =
1.1 for the emission above threshold. Reproduced from ref. [4].
This leads to a plateau in the variation of coherence time
with condensate occupation (fig. 2b), in agreement with
experiments (fig. 1b).
Rather than attempting a detailed microscopic model,
our treatment is based on general considerations of inter-
acting, open condensates. We argue that the observed
slowing-down of number fluctuations near to threshold
means that the pumping term that drives the system must
be saturable, that is the pumping decreases as the popula-
tion of the condensate grows. This, along with polariton-
polariton interactions, and fluctuations due to pumping,
is required in order to obtain agreement with the experi-
ments. However, one or more of these features are missing
from previous theories of condensate coherence [3,8–13].
Although a detailed description of th e incoherent pump-
ing process is beyond the scope of the present work, our
results should provide a useful guide to the development
of microscopic theories [8,10–14].
In the remainder of this paper, we briefly review the
semiclassical treatment in the limit of slow number fluctu-
ations [4]. We next explain how the Kubo stochastic line-
shape theory [6,7] can be applied to give a general form for
g
(1)
(τ), accounting for both interactions and number fl uc-
tuations. This expression reduces to the observed Gaussian
decay provided the number fluctuations are slow. We then
develop a solution to a quantum-mechanical model of
a pumped condensate, which has the same Kubo form
for g
(1)
(τ) in the semiclassical limit. Close to threshold
this soluti on gives a slow decay for number fluctuations,
γτ
g
(1)
(τ)
10
0
10
-1
(a)
50 500 <n>=1000
Mean Population <n>
Decay Time (γ
-1
)
g
(2)
g
(1)
(b)
mean field
threshold n=0
0 200 400 600
05001000
0
100
200
Fig. 2: (a) The decay of g
(1)
(τ ) for populations n = 50, 500
and 1000. On this logarithmic plot, a simple exponential is
linear, while a Gaussian is quadratic. (b) Decay times for
g
(1)
(τ )andg
(2)
(τ ), as a function of population, obtained from
the numerical solutions of eqs. (4) and (5) (solid li nes). The
non-linearity is κ =4× 10
−5
γ,andn
s
=2.5 × 10
4
, as derived
from the experimental results. The dashed lines are from the
analytic expressions, eqs. (7) and (9). The marked points
indicate the values of n used in (a).
and hence explains the observed line-shape as well as
the observed decay time. However, it also predicts that
further above threshold number fluctuations will be much
faster, and significantly longer coherence times should be
obtained.
Static limit. – In our discussions we neglect spatial
effects, which is justified in the CdTe system where
the emission spot is strongly localised by disorder. We
thus model the condensate mode as a single anharmonic
oscillator, with Hamiltonian
H = a
†
aω
0
+ κ (a
†
a)
2
, (1)
where ω
0
is the oscillator frequency, and κ the strength of
the polariton-polariton interaction.
For a condensate of interacting particles, the interac-
tions translate number fluctu ation s in the condensate into
random changes to its energy, and so the coherence is lost.
A similar effect, commonly termed “self-phase modula-
tion”, was originally observed for laser beams propagat-
ing in a non-linear Kerr medium [15]. If we assume that
the condensate has a Gaussian probability distribution for
the number of polaritons, with variance σ
2
, it is straight-
forward to obtain g
(1)
(τ) when the number fluctuations
are sufficiently slow [4]. It has a Gaussian form
|g
(1)
(τ)|=exp(−2κ
2
σ
2
τ
2
)=exp(−τ
2
/τ
2
c
). (2)
27002-p2
Coherence p roperties of the microcavity polariton condensate
As detailed in ref. [4] we can obtain, directly from
the measured data, values of σ
2
∼25000 and κ ∼2 ×
10
−5
ps
−1
, giving a decay time τ
c
∼200 ps, in reasonable
agreement with the experiments. The value for κ is consis-
tent with theoretical estimates [16,17] of the interaction in
a mode of linear size ∼5 µm.
The picture described above is essentially static; it
assumes that the time scale on which the number of
polaritons changes, τ
r
, is much longer than the coherence
time τ
c
, so the only relevant time evolution is caused by the
action of the Hamiltonian. The obvious problem with this
description is that the coherence time, τ
c
, is much longer
than the polariton lifetime, τ
0
∼2 ps, due to emission from
the cavity. This suggests that the microcavity system may
well not be in the quasi-static regime, and we need to
consider the processes by which the number fluctuations
occur in more detail.
Kubo approach. – The effect of intro d ucin g a time
scale τ
r
for fluctuations can be understood using the
Kubo stochastic line-shape theory [6], which describes the
decay of g
(1)
(τ) for emission associated with a transition
whose energy varies randomly in time. In our case, the
random vari ations in energy are a consequence of the
numb er fluctuations, with time scale τ
r
determined by
the pumping. When the random energies have a Gaussian
distribution, the Kubo theory gives
|g
(1)
(τ)|=exp
−
2τ
2
r
τ
2
c
(e
−τ/τ
r
+ τ/τ
r
−1)
. (3)
For τ
r
slow compared to the scale set by the variance
of the random energy distribution (τ
c
above), we are in
the static regime and a Gaussian decay with lifetime τ
c
is predicted. However, in the opposite regime, τ
r
≪τ
c
,
motional narrowing occurs, and the decay becomes a
simple exponential, with lifetime τ
2
c
/2τ
r
, much longer than
τ
c
. If we naively take τ
r
= τ
0
∼2 ps, we would clearly be
in the motional narrowing regime, giving an exponential
function, with a very slow decay ∼10 ns. The measurement
of the g
(1)
(τ) decay thus shows that τ
r
is, in fact,
much longer than τ
0
, and must be comparable to, or
greater than, τ
c
. This prediction is very well confirmed by
the measurement of the decay of g
(2)
(τ), which directly
determines the time scale for number fluctuations; the
experimental decay time is ∼100 ps, similar to τ
c
for the
same condensate population.
The explanation for the slow decay of g
(2)
(τ) comes
from laser physics, where it is well known that number
fluctuations are slowed close to the threshold. This can
be explained using a simple classical model where the
pumping pr ovides a gain which is saturable, that is, has
a dependence on the mode population. Above threshold,
the linear part of the gain term exceeds the loss, so
the population grows, and a non-linear saturation term
is required to obtain a finite steady state. However, the
response to small fluctuations, which is what the intensity
correlation experiment measures, depends only on the net
linear gai n. Thus near threshold, where the linear gain and
loss are closely matched, fluctuations in the system relax
very slowly. Haken [18] shows the close analogy of this
behaviour to the critical slowing down of fluctuations in
the vicinity of an equ ili bri um p hase transition.
Quantum model. – To put these considerations on
a more formal fo ot ing, we have developed a quantum
model of the polariton condensate which can be solved
analytically for g
(1)
(τ)andg
(2)
(τ). This model is a
generalisation of one studied by Thomsen and Wiseman [9]
in the context of atom lasers, extended to cover the full
range of mode occupancies; their model only applies to
the ‘far above threshold’ limit, where the gain is fully
saturated and slowing of number fluctuations no longer
occurs. The coherent mode is treated as an anharmonic
oscillator with a Kerr non-linearity, eq. (1). This mode
is coupled to a r eservoir, using the master equation
formalism for the density matrix ρ. Reservoir losses are
offset by a standard laser-like saturable pump term [19].
We thus obtain equations for the population distribution ,
P
n
= ρ
n,n
, and the coherence, u
n
=
√
nρ
n−1,n
e
−iω
0
t
:
˙
P
n
= γn
c
n
n + n
s
P
n−1
−
(n +1)
(n +1)+n
s
P
n
+γ[(n +1)P
n+1
−nP
n
], (4)
and
˙u
n
= γn
c
n
n + n
s
−
1
2
u
n−1
−
n +
1
2
n + n
s
+
1
2
u
n
+γ
nu
n+1
−(n −
1
2
)u
n
+2iκnu
n
. (5)
Here,
1
2
γ =1/τ
0
is the cavity decay rate, and n
c
and n
s
are
parameters describing the pump process
1
. n
c
characterises
the st rength of the pumping, while n
s
provides the
saturation: for n n
s
the gain decreases. Physically, this
corresponds to the depletion of the pump r eservoir by
the processes which populate the condensate. Far above
threshold, where the mean occupation n≫n
s
, this
model becomes identical to ref. [9]. Pumping terms like
this are required to give a finite condensate population,
and hence shoul d be derivable from microscopic kinetic
theories. Similar gain saturation effects appear in some
recent mean-field theories [16,20,21].
The steady-state solution of eq. (4) is
P
S
n
∝
n
n
c
(n + n
s
)!
≈exp
−
(n −
n)
2
2n
c
, (6)
where the Gaussian form is valid when
n = n
c
−n
s
≫1.
The variance of the population is σ
2
= n
c
, so that number
fluctuations in the threshold region,
n ≪n
s
, are super-
Poissonian. It is convenient to divide th e threshold region
1
In the terminology of ref. [19], γ = C, n
s
= A/B,andn
c
=
A
2
/BC. We have neglected some terms O(1/n
s
), since n
s
∼ 10
4
for
our system.
27002-p3
D. M. Wh ittaker and P. R. Eastham
into two: for
n 3
√
n
s
∼500 the Gaussian form is valid
for all n, because the non-physical n<0 states are not
significantly occupied, while for smaller values of
n these
states have to be explicitly excluded. In the former case,
the mean population n=
n.
Equations (4) and (5) are easily solved numerically for
populations of a few hu nd red particles. Figure 2b shows
the decay times for the correlation functions obtained
from these solutions, plotted as a function of the mean
population. As the population increases, the g
(2)
decay
time rises rapidly to ∼100 ps, then decreases. The g
(1)
time also rises quickly, then flattens for a while before
increasing again. Populations up to ∼500 correspond to
the experimental regime of fig. 1b, and the observed
g
(1)
behaviour has a very similar form. In fig. 2a the
actual decay of g
(1)
(τ) is plotted, for three population
values. For n= 50, corresponding to th e initial rise in
coherence time, the form is a simple exponential. In the
flat region, with n= 500 the decay starts off Gaussian,
before becoming exponential at longer delays; this is
the near static behaviour of the self-ph ase modulation
regime. The final rise occurs when the g
(2)
time shortens
and motional narrowing sets in. This is evident in the
decay curve for n= 1000, which star ts off with the same
Gaussian as n= 500, b ut much sooner becomes a slower
exponential.
We now turn to deriving semiclassical analytic solutions
to eqs. (4) and (5), which provide the dashed lines on
the figure. These solutions are valid in the regime where
n 3
√
n
s
∼ 500 and the Gau ssian in eq. (6) applies
without truncating the n<0 part. Equation (4) for the
population is independent of the non-linearity, so we can
quote standard laser theory results [19] for the intensity
correlation function:
g
(2)
(τ)=1+
n
s
n
2
exp
−
n
n
c
γτ
=1+
n
s
n
2
exp (−
γτ ), (7)
where
γ = nγ/n
c
is the slowed decay r ate. This result fits
the single experimental data point fairly well; using the
experimental measurements g
(2)
(0) = 1.1andn = 500 we
obtain n
s
= 25000 and
γ = γ/50. This gives a decay time
of ∼50 ps, in reasonable agreement with the measured
100 ps.
The first-order correlation function, g
(1)
(τ) ∝
n
u
n
(τ)e
iω
0
τ
, is obtained by solving eq. (5). The
solution is required with an initial condition u
n
(0) = nP
S
n
,
which is a similar Gaussian function to P
S
n
, but with
mean
n + n
c
/n. To obtain th e correlation function, we
follow the approach of Gar din er and Zoller [22]. Using a
Kramers-Moyal exp ansion, the difference operators are
converted into differentials, leading to a Fokker-Planck
equation for u, which we now write as u(n, t), with n a
continuous variable. To deal with the appearance of n
in the denominator of the pumping terms, we linearise
around the mean value, writing n =
n + n
c
/n + n
′
.Thus
we obtain
∂u
∂t
=2i
κ +
iγ
4n
c
n
′
u −
γ
2n
u
+
γ
n
n
c
∂
∂n
′
n
′
u +
1
2
(2n
c
)
∂u
∂n
′
, (8)
where we have omitted constant non-linear contributions
to the oscillator energy, which can be absorbed in a
renormalised ω
0
. This equation is solved in the Fourier
domain [22] to give
|g
(1)
(τ)|=exp
−
4n
c
κ
2
γ
2
(e
−
γτ
+
γτ −1)
×exp
n
c
4n
2
(e
−
γτ
−
γτ −1)
. (9)
The first factor is just the Kubo expression, eq. (3), with
τ
2
c
=1/2κ
2
σ
2
and σ
2
= n
c
, as before, and τ
r
=1/γ. This
constitutes the main result of our treatment: we obtain
Kubo type behaviour, with the fluctuation time τ
r
given
by the decay time of g
(2)
(τ). The second term corresponds
to the Schawlow-Townes decay, enh anced in the threshold
region due to the finite amplitude fluctuations. It is
generally much slower than the first, in the regime where
the expression i s valid.
Discussion. – Figure 2 shows very clearly the impor-
tance of the fluctuation time scale τ
r
∝1/
n (for large
n) on the coherence time. When n is increased τ
c
only
changes very slightly, in fact decreasing as n
c
= n
s
+
n
grows, b ut τ
r
shortens rapidly. This pushes the system
into the motional narrowing regime, where the lifetime
τ
2
c
/2τ
r
is proportional to
n. Note that we have to be care-
ful treating this as a prediction of a linear relationship
between coherence time and emission intensity for high
powers; we have simply kept n
s
constant and increased n
c
.
In the textbook laser model [19], n
s
is indeed independent
of pump power P ,andn
c
∝P . Though pumping of the
the polariton system is considerably more complex than
this, it is likely that n
c
increases more rapidly with P than
n
s
does. Thus the fluctuation decay time should decrease
with stronger pumping, potentially taking the condensate
into the motional narrowing regime. For this to be observ-
able it would, of course, be necessary that other mecha-
nisms should not take over and restrict the coherence as
the number fluctuation effect is suppressed.
One surprising feature of the experiments is that very
significant slowing down is still occurring at pump powers
of twice the threshold value P
th
. This is inconsistent
with the simplest assumption, n
c
∝P , since then ¯γ/γ =
(1 −P
th
/P ) is close to one at these powers. A full under-
standing of this requires microscopic models of the pump-
ing, but it may be exp lain ed by assuming that there are
a limited number of reservoir states which can provide
gain for the condensate, so the gain parameter n
c
must
become independent of P at high pumping. In a system
where the maximum achievable gain only just exceeds the
27002-p4