Intrinsic Decoherence Mechanisms in the Microcavity Polariton Condensate
A.P.D. Love,1D.N. Krizhanovskii,1D.M. Whittaker,1R. Bouchekioua,1D. Sanvitto,2S. Al Rizeiqi,1R. Bradley,1
M.S. Skolnick,1P.R. Eastham,3R. Andre ´,4and Le Si Dang4
1Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
2Departamento de Fisica de Materiales, Universidad Autonoma de Madrid, 28049 Madrid, Spain
3Department of Physics, Imperial College, London SW7 2AZ, United Kingdom
4Institut Ne ´el, CNRS and Universite ´ J. Fourier, 38042 Grenoble France
(Received 17 March 2008; published 7 August 2008)
The fundamental mechanisms which control the phase coherence of the polariton Bose-Einstein
condensate (BEC) are determined. It is shown that the combination of number fluctuations and
interactions leads to decoherence with a characteristic Gaussian decay of the first-order correlation
function. This line shape, and the long decay times (?150 ps) of both first- and second-order correlation
functions, are explained quantitatively by a quantum-optical model which takes into account interactions,
fluctuations, and gain and loss in the system. Interaction limited coherence times of this type have been
predicted for atomic BECs, but are yet to be observed experimentally.
DOI: 10.1103/PhysRevLett.101.067404 PACS numbers: 78.67.?n, 42.50.Ar, 78.45.+h, 78.55.?m
The recent observations of polariton condensation in
semiconductor microcavities have provided a new, solid
state system for the study of Bose-Einstein Condensate
(BEC) phenomena [1,2]. The Bose-condensed state exhib-
its characteristic properties including massive occupation
of the ground state [1–3] and long range spatial coherence
, which due to thevery small mass of polaritons (10?5of
the free electron mass) occur at high critical temperatures
of 20–30 K . Like atomic BECs, the polariton conden-
sate is a mesoscopic system containing a few hundred
particles, so number fluctuations are important. The fun-
damental mechanisms limiting the coherence times of such
mesoscopic condensates have been discussed in the case of
atomic BECs , but have yet to be observed experimen-
In this work, we demonstrate that the phase coherence
time of such systems is intrinsically limited by the com-
bined effect of number fluctuations and interactions in the
coherent state. Since the polaritons interact with each
other,the numberfluctuations lead tofluctuationsinenergy
which cause the coherence to be lost, giving a character-
istic Gaussian decay of the first-order correlation function
?g?1????? which we observe experimentally. These effects
have not been observed previously for polariton conden-
sates, because of dominating pump laser noise. Our mea-
surements, using a noise-free diode laser, reveal very long
coherence times up to ?150 ps, up to 30 times longer than
those deduced previously [1,2], making it realistic to study
other condensate phenomena, such as Josephson oscilla-
tions, in polariton systems.
It is notable that the polariton condensate differs from an
ideal BEC in that it is not in equilibrium. Polaritons decay,
as externally emitted photons, with a lifetime ?1:5 ps, so
the steady-state population has to be maintained by a pump
laser. This continual replacement of particles is a unique
aspect of the polariton condensate, and has to be accounted
for in any theoretical treatment of the system. Here, we
present a model which shows that the decoherence mecha-
nism discussed above, which is essentially a property of an
equilibrium condensate, can be observed despite the short
lifetime. This is possible because the effective time for the
replacement of polaritons is lengthened by stimulated scat-
tering terms in the gain, which dominate above threshold.
The lengthening is demonstrated directly by experimental
measurements of the second-order g?2?coherence function,
which show a decay time of ?150 ps. Using only experi-
mentally derived parameters as input, our quantum-optical
model explains simultaneously the time dependence of
both g?1?and g?2?functions, which we show have very
different physical origin.
The sample employed here is a piece from the CdTe-
based microcavity wafer used in Ref.  where BEC with
extended spatial coherence was reported. Figure 1 shows
spectra from the bottom of the lowerpolariton (LP)branch,
excited nonresonantly with a semiconductor diode laser
free from intensity fluctuations on a 0.1–1 ns time scale,
below and above the threshold for condensation. Above
threshold the spectrum breaks up into a series of lines with
?0:2 meV separation, which are more than 30 times nar-
rower than that below threshold. The marked narrowing
and the accompanying superlinear increase of the intensity
with excitation power [Fig. 1 (inset)] are characteristic of
the formation of a macroscopically occupied polariton
condensate. We note that the resolved spectral structure
and the narrow linewidths (long coherence times) are not
observable under multimode laser excitation [1,2], where
as we argue later, laser intensity fluctuations obscure the
quantum properties of the polariton BEC.
To understand the origin of the coexisting condensates
revealed by the high spectral resolution, spectrally and
PRL 101, 067404 (2008)
8 AUGUST 2008
© 2008 The American Physical Society
spatially resolved images were obtained below [Fig. 2(a)]
and above[Figs.2(b)–2(d)] threshold. Belowthreshold the
contrast, above threshold each mode has a distinct spatial
pattern, with intensity maxima separated by ?2–3 ?m.
The modes nevertheless show strong spatial overlap.
Also the intensity of the modes decreases going to higher
energy in the spectrum. We attribute the coexistence of the
overlapping condensed states to weak spatial fluctuations
(?0:2–0:5 meV) in the cavity mode energy [1,7] which
result in localized polariton modes at different energies .
Above threshold condensation into each localized mode
occurs, and the modes become spectrally resolved due to
the accompanying strong narrowing.
Having achieved spectral resolution of localized states,
we investigate the time dependence of the phase coherence
of the strongest individual condensate mode using a Mach-
Zehnder interferometer. Figure 3(a) shows the decay of the
first-order correlation function g?1???? for a single mode
above threshold. It is found to have a Gaussian form with a
coherence time ??1?
of this phase coherence time is plotted as a function of the
intensity of the emission, which is proportional to the
number of particles in the condensed state . The coher-
ence time first grows rapidly with increasing intensity,
from ?1:5 ps around threshold, until it saturates to a value
of ?120–150 ps , at a pump power approximately 2
times threshold. The higher energy modes are found to
exhibit coherence times close to that of the most intense
mode within experimental error.
More information about the character of the condensate
was obtained from measurements of the second-order (in-
c of ?120 ps. In Fig. 3(b), the variation
FIG. 2 (color online).
ages (?2 ?m resolution) of the polariton emission for P ?
3 mW (a) and P ? 30 mW (b)–(d). The detection energies E1
to E3 for images (b) to (d) are given on the images and
correspond to detection at the energies of the individual, resolved
modes of Fig. 1.
(a)–(d) Spectrally resolved, spatial im-
0 50 100 150
-0.5 0.0 0.5
correlation function g(2)
Delay time, τ (ps)
correlation function g(1)
Intensity (arb. units)
Coherence time, τc
Delay time, τ (ns)
-0.5 0.0 0.5
versus delay time for a single condensate mode above threshold.
(b) Phase coherence time ??1?
of the condensate mode versus
emission intensity showing a strong increase at threshold fol-
lowed by saturation at powers approximately a factor of 2 above
threshold. (c)–(d) Measured g?2?second-order correlation func-
tion versus delay time at powers P ? 6 and 30 mW, below (c)
and a factor of 2 above (d) threshold, respectively. The full lines
are fits with values for g?2??0? and the coherence time, ??2?
and 1.5 ps below threshold and 1.1 and 100 ps above.
(a) Variation of g?1?first-order correlation function
c of 2
Photon Energy (eV)
Intensity (arb. units)
bottom of the LP branch for detection angle ? ? ?0?at
excitation powers P ? 1 mW and 30 mW, respectively, below
and above the threshold (Pth? 10–15 mW) for condensation.
Below threshold, the linewidth is broad (FWHM ? 1:5 meV),
determined by the polariton lifetime (?1:5 ps). By contrast,
above threshold the emission consists of a number of very
narrow lines with energy separations of ?0:2 meV and linewidth
of 0.06 meV (limited by the spectrometer resolution: the true
linewidths (decay times) are obtained from the first-order corre-
lation measurements). The inset shows the strongly nonlinear
variation of the intensity of mode E1with power.
Spectra corresponding to polariton emission from the
PRL 101, 067404 (2008)
8 AUGUST 2008
tensity) correlation function g?2???? using a Hanbury
Brown–Twiss (HBT) setup . The g?2?results, which
provide essential parameters for the modelling of the co-
herence below, are presented in Figs. 3(c) and 3(d) for two
pump powers, below and a factor 2 above threshold. In
both cases photon bunching is observed with the measured
However, the decay times (??2?
(detector resolution limited) below and ?100 ps above
threshold, respectively. Using an expression relating
account the detector efficiency, the resolution time of the
detectors (?40 ps) and ??2?
pected value of g?2??0? ? 2 for a thermal state below
threshold, we calibrated the efficiency of the detectors.
The true value of g?2??0? above threshold is then found to
be ?1:1 ? 0:015, slightly above 1, the prediction for a
coherent state. This value is consistent with values in
Ref. , but it should be noted that the present studies
eliminate any fluctuations from multimode excitation.
We now discuss our model for the first and second-order
coherence treating initially an isolated, equilibrium BEC,
then introducing the nonequilibrium character. Consider an
isolated state, with a Gaussian probability distribution for
the number of polaritons P?n? expected, for example, for a
coherent state. This is characterized by its mean ? n and
variance (fluctuations in particle number) ?2. The state
evolves under a nonlinear Hamiltonian H ? ?a?aa?a,
describing the polariton-polariton interaction. To find
g?1????, we evaluate ha????a?0?i, which corresponds to
removing a polariton at time t ? 0 and putting it back at
t ? ?. The phase change is ? times the difference in energy
of states with n and n ? 1 polaritons, that is ???n2? ?n ?
1?2? ? 2??n. Averaging over the probability function
meas???? having a maximum at ? ? 0 of ?1:04.
c ) are different: ?40 ps
meas??? to the true value of g?2??0?, which takes into
, and assuming the ex-
dnP?n?exp?2i??n? ? exp??2?2?2?2?
which has the Gaussian form observed experimentally in
We obtain the variance ?2from the second-order corre-
lation function, g?2??0? ? 1 ? ??2? ? n?=? n2. Using the
measured g?2??0? of 1.1, and estimating ? n ? 500 from the
emission intensity, we obtain ?2? 25500. As shown be-
low, ? is estimated to be 2 ? 10?5ps?1from the blueshift
(0.5 meV) from below to above threshold. This gives a
decay time ?c? 220 ps, close to the experimental value of
?150 ps. An equilibrium model including number-
fluctuations and interactions thus explains both the
g?1???? line shape and the quantitative values of the decay
The coherent mode is, of course, not isolated, as it is not
in equilibrium; polaritons are lost by external emission at a
rate determined by the cavity linewidth ?, and are replaced
from the reservoir of particles in other modes . This
disrupts the Hamiltonian evolution on a time scale ?r,
which depends on the loss rate, but is generally slower
than ??1, because the stimulated component of the scat-
tering into the mode exactly replaces the particle which is
lost. This time scale is obtained most directly from the
decay of g?2????, that is ??2?
down which allows us to see the equilibrium condensate
physics, despite the short cavity lifetime.
To make these considerations more quantitative, we
have solved a generalization of a model for atom lasers
with interactions . We treat the case where the occupa-
tion ofthe mode is less than or comparable to the saturation
value ns. The population dynamics are given by a master
equation of the form
c ? 100 ps. It is this slowing
n ? ns
?n ? 1?
?n ? 1? ? ns
? ???n ? 1?Pn?1? nPn
where ncis a measure of the pumping strength. Above threshold, the steady-state Pnhas a Gaussian form with mean
? n ? nc? nsand variance nc. Solving this model, we find approximate expressions for the correlation functions
? n2exp??? ???;
g?2???? ? 1 ?ns
jg?1????j ? exp??2?2nc?2?exp????=2? n?
? exp??4?2nc?=? ??exp????=4? n?
?? ?? ? 1?;
?? ?? ? 1?;
where the decay rate ? ? ? ? n?=ncis much slower than that
of the bare cavity mode.
It is the slowed decay of g?2?, with decay rate ? ? ?
? n?=?? n ? ns?, which determines that we are in the early
time regime ? ?? ? 1 for g?1?as given by Eq. (4a). In this
regime, the first factor in g?1?is identical to the Gaussian
expression for an equilibrium condensate of Eq. (1), since
the variance ?2? nc. The second factor, exp????=2? n?,
corresponding to a Schawlow-Townes phase diffusion, is
much slower. From the measured g?2??0? of 1.1, we obtain
ns? 25000 which predicts ? ??1to be ?100 ps when ? n ?
500, in very good agreement with the experimental decay
PRL 101, 067404 (2008)
8 AUGUST 2008
of g?2?(100 ps). The deduced value of ? ? means that we are
in the regime ? ?? ? 1 long enough to see almost the entire
Gaussian decay of g?1?with the decay time of 220 ps,
shown above to be in quantitative agreement with experi-
ment. Using experimentally determined input parameters
and our modeling, we thus explain quantitatively the decay
times of both g?1?and g?2?, corresponding to very different
In addition to the exchange of particles above between
the condensate and the reservoir, there is a direct interac-
tion through the nonlinear Hamiltonian, which means that
noise in the reservoir will cause energy fluctuations and
decoherence. This effect explains the short coherence
times andlargelinewidthsobtained previouslyusingmulti-
mode laser excitation. For such a laser, we measure inten-
sity fluctuations of ?15–20% on a nanosecond time scale;
comparable variations in the reservoir population will be
expected. The interaction with the reservoir population
causes a blueshift of the mode energy to ?0:5 meV above
the low power peak [Fig. 1]. The fluctuations will thus be
20% of this, ?0:1 meV, which translates into a coherence
time ?20 ps, consistent with the experimental results
under multimode excitation (?6 ps).
Even with a noise-free excitation laser, there will be
thermal fluctuations in the reservoir population, which
can cause decoherence. If the mean reservoir population
is Nr, the blueshift of the mode due to the interactions is
?4?Nr. Estimating Nr? 104using the pump power, we
deduce ? ? 2 ? 10?5ps?1. Assuming the reservoir modes
are independent, the variance in the population will be
?Nr, so the arguments leading to Eq. (1) give a coherence
time of ?8Nr?2??1=2? 180 ps. We conclude that decoher-
ence due to the thermal fluctuations is unlikely to obscure
the intrinsic effects of interactions between the polaritons
in the mode.
In conclusion, we show that study of the polariton con-
densed state in semiconductor microcavities reveals new
physics of interacting BECs. We demonstrate the roles of
interactions and fluctuations in determining the phase co-
herence and the lengthening of the second-order correla-
tion decay by stimulation. A quantum-optical model is
presented which explains both these properties using a
treatment of a driven interacting condensate. These results
show that the polariton condensate exhibits properties
characteristic of an equilibrium BEC, even though it is
exchanging particles with its environment. The true coher-
ence properties of the polariton condensate reported here
were obscured in previous work by fluctuations in excita-
tion intensity of the multimode laser sources employed.
The work was supported by EU projects Clermont 2
RTN-CT-2003-503677 and Stimscat 517769, and EPSRC
Grants No. GR/S09838/01 and No. GR/S76076/01. D.N.
Krizhanovskii thanks the EPSRC for financial support (EP/
E051448). We thank K. Burnett for a very helpful
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 D.N. Krizhanovskii et al., Phys. Rev. B 75, 233307
 D.M. Whittaker et al., Phys. Status Solidi (a) 164, 13
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 L.K. Thomsen and H.M. Wiseman, Phys. Rev. A 65,
063607 (2002) and references therein.
 The coherence of atom BECs is measured in atom-laser
experiments, where the coherence is limited by the life-
time of atoms in the trap (few ms) and not by the intrinsic
interactions. See, for example, M. Kohl, T.W. Hansch, and
T. Esslinger, Phys. Rev. Lett. 87, 160404 (2001).
 D. Sanvitto et al., Phys. Rev. B 73, 241308 (2006);
M. Richard et al., Phys. Rev. B 72, 201301 (2005).
 Coexistenceof macroscopically
states has been reported for the resonantly pumped optical
Krizhanovskii et al., Phys. Rev. Lett. 97, 097402 (2006).
 The g?1???d? function is measured as a function of excita-
tion power at a constant delay ?dof 50–100 ps. The co-
herence time is deduced from ?c? ?d
assuming Gaussian decay of g?1???d?.
 This coherence time corresponds to a spectral linewidth of
?10 ?eV. This is the true linewidth of the spectra of
Fig. 1 which are resolution limited to ?60 ?eV.
 R. Loudon, The Quantum Theory of Light (Oxford
University Press, Oxford, 2000).
 J. Kasprzak et al., Phys. Rev. Lett. 100, 067402 (2008).
 The reservoir consists of all the occupied modes in the
system, predominantly exciton states at high wave vector.
PRL 101, 067404 (2008)
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8 AUGUST 2008