Bose-Einstein Condensation of Long-Lifetime Polaritons in Thermal Equilibrium

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by  Yongbao Sun

Bose-Einstein Condensation of Long-Lifetime Polaritons in Thermal Equilibrium
Yongbao Sun,1∗ Patrick Wen,1 Yoseob Yoon,1 Gangqiang Liu,2 Mark Steger,2
Loren N. Pfeiffer,3 Ken West,3 David W. Snoke,2∗ and Keith A. Nelson,1
1Department of Chemistry and Center for Excitonics, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, USA
2Department of Physics, University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15218, USA
3Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
6
1 Theexperimentalrealization ofBose-Einsteincondensation(BEC)withatomsandquasiparticles
0 hastriggeredwideexplorationofmacroscopicquantumeffects. Microcavitypolaritonsareofpartic-
2 ularinterestbecausequantumphenomenasuchasBECandsuperfluiditycanbeobservedatelevated
c temperatures. However, polariton lifetimes are typically too short to permit thermal equilibration.
e Thishasledtodebateaboutwhetherpolaritoncondensationisintrinsicallyanonequilibriumeffect.
D Here we report the first unambiguous observation of BEC of optically trapped polaritons in ther-
malequilibriuminahigh-Qmicrocavity,evidencedbyequilibriumBose-Einstein distributionsover

5 broad ranges of polariton densities and bath temperatures. With thermal equilibrium established,
weverify that polariton condensation is a phasetransition with a well defineddensity-temperature
] phase diagram. The measured phase boundary agrees well with the predictions of basic quantum
s
a gas theory.
g
-
t
n The realization of exciton-polariton condensation in sist over whether polariton condensates can be called
a semiconductor microcavities from liquid helium temper- Bose-Einsteincondensates [27–29], in partrelatedto the
u
ature [1, 2] all the way up to room temperature [3–5] question of whether polariton condensation is intrinsi-
q
presents great opportunities both for fundamental stud- cally a nonequilibrium effect. It is thus of fundamental
.
t ies of many-body physics and for all-optical devices on importanceto investigatewhetherpolaritoncondensates
a
m the technology side. Polaritons in a semiconductor mi- canreachthermalequilibrium. Ofcourse,strictlyspeak-
crocavity are admixtures of the confined light modes of ing,BECcannotoccurinanidealinfinite2Dsystem,but
-
d the cavity and excitonic transitions, typically those of it has been shown [30, 31] that a 2D Bose gas in a large
n
excitons in semiconductor quantum wells (QWs) placed but finite trap has the same threshold behavior as a 3D
o
at the antinodes of the cavity. Quantum effects such as Bose gas in a finite trap of the same type. We can thus
c
[ condensation [1–5], superfluidity [6], and quantized vor- talk of an equilibrium BEC in 2D and 3D finite trapped

tices [7–11] have been reported. The dual light-matter systems using the same language.
3
nature permits flexible control of polaritons and their
v
1 condensates, facilitating applications in quantum simu- Trapping polaritons in a high-Q microcavity
8 lation. It is also straightforwardto measure the spectral
5 functions, A(k,ω), of polaritons, which can provide in- Themainchallengeinreachingfullthermalizationinpo-
2 lariton systems is to achieve a very long polariton life-
sights into the dynamics of many-body interactions in
0 time, longer than their thermalization time. The ther-
polariton systems. For cold atoms, the equilibrium oc-
1. cupationnumberscanbemeasured[12],butthespectral malizationtimeforthepolaritongaswasestimatedtobe
0 at least 40 ps for polaritons that are mostly exciton-like
function is not readily accessible. Observations of non-
6 [16], and can be even longer for more photon-like polari-
Hermitian physics [13] and phase frustration [14] have
1
tons which are less interactive. However, most samples
: shown that polaritons are an important complement to
v used in previous experiments have polariton lifetimes on
atomic condensates.
i the order of a few picoseconds. This suggests that an
X
However,inmostpreviousexperiments,the lifetimeof improvementofthecavityQbyatleastanorderofmag-
r
a the polaritonsinmicrocavitieshasbeen30psorless[15] nitude is needed, which is not trivial for GaAs fabrica-
due to leakage of the microcavity. Thus, although there tion technology. We have succeeded at this by growing
have been claims to partial thermalization of polaritons a GaAs-based high-Q microcavity structure by molecu-
[16, 17], no previous work has unambiguously shown a larbeamepitaxy. Themainchangefromsamplesusedin
condensationinthermalequilibrium,leadingtothecom- previousexperiments[2]wastodoublethenumberofthe
mondescriptionofpolaritoncondensatesas“nonequilib- quarter-wavelengthlayersinthedistributedBraggreflec-
rium condensates” [18–20]. The theory of nonequilib- tors (DBRs) that make up the mirrors of the cavity; the
rium condensation is still an active field [21–24]. Al- detailedrecipeandthedifficultiesinvolvedinfabricating
though polariton experiments and theory have shown alonglifetimesamplearedescribedintheSupplementary
that a great number of canonical features of condensa- Information. The new microcavity structure has a Q of
tion persist in nonequilibrium, e.g., superfluid behavior ∼320,000 and a cavity photon lifetime of ∼135 ps. This
[22, 23], some aspects may not [25, 26], anddebates per- correspondstoapolaritonlifetimeof270psatresonance,
2
which has been confirmed by the long-range (millimeter is slightly wider than the laser profile, because excitons
scale) propagation of polaritons created through either propagateupto10µm. Thepotentiallandscapeisnearly
resonant [32] or non-resonant[33] optical excitation. flatintheregionfrom−11µmto11µmindicatedbythe
Due to the light effective mass and inefficient scatter- white circle in Fig. 1a and the horizontallines in Fig. 1b
ingwithphonons,polaritonscanpropagateoverlongdis- and1c. PL wascollectedfromonly this regionfor deter-
tances, up to millimeters when they are mostly photonic minationofthepolaritonenergydistributionasdiscussed
[32, 33]. In order to guide them towardequilibrium with below. The nearly flat potential profile corresponds to a
aspecifiedlocationandgeometry,wemadeaspatialtrap. constant density of states in 2D. Additionally, a nearly
We createdanannularopticaltrapto localize polaritons homogeneous distribution was established in the field of
under non-resonant excitation. This method has been view,asevidencedbylittlechangeintheenergy-resolved
used previously in several experiments to confine polari- emission intensities in Fig. 1c (see Fig. S6 in Ref. [36]
tons [34–38]. The excitation pattern on the microcavity foradirectmeasurementofspatialprofilesundersimilar
is an annulus with a diameter of 38 µm, as shown in conditions).
Fig. 1a. In Fig. 1b, we show the normalized light inten- Polaritons are generated in the pump region and
sityplotforthe x=0sliceofthe ringpatterninFig.1a. stream away in all directions. However, only polari-
The light intensity in the center is nearly negligible; as tons that propagate into the center of the trap can meet
discussed above, it gives rise to a nearly flat potential and interact, leading to the accumulation of the densi-
for the polaritons in the center of the ring, in which the ties high enough for condensation. A near-field image of
variationin energyis much lessthan k T. We also plot- the sample was projectedonto a spatialfilter ata recon-
B
structed real-space plane of the sample surface to select
(a) (b) (c) only the PL frominside the trap(within the white circle
1 in Fig. 1a), and a far-field image of the PL that passed
20 through the spatial filter was projected onto an imag-
0.8
ing spectrometer, giving the intensity of the PL, I, as
m) 0.6 a function of both the in-plane wavevector component,
µ 0
y ( k||, and the corresponding energy, E(k||). The disper-
0.4 sion E(k ) is given in the supplementary information.
||
−20 Finally, I[E(k )] was converted into the number of po-
||
0.2
laritons,N[E(k )](hereaftersimplyN(E)),bytheuseof
||
−20 0 20 1 0.5 0 1604 1602 1600 1598 one single efficiency factor throughout the experiments.
x (µm) I(x=0) Energy (meV)
Detailed information about how to determine the effi-
FIG.1. (coloronline). (a)Reflectionoftheexcitationbeam ciency factor can be found in Ref. [36]. Crucially, the
from the sample surface. The white circle indicates the re- same efficiency factor was used for all the distributions
gionofthesamplethatisobservedinphotoluminescence(PL) sothat the absolute occupationnumbers ofdifferent dis-
imaging measurements after spatial filtering. (b)Normalized
tributions could be compared.
excitation light intensity along the x = 0 line through the
center of the excitation ring pattern shown in (a). (c) Spec-
Varying the polariton gas from nonequilibriumto
trallyresolvedPLalongx=0. ThePLwithinthesolidwhite
equilibrium
lines is collected and imaged onto the spectrometer CCD in
thefar-field geometry forthepolariton distribution measure- Tosee the effectofinteractionsonthermalization, N(E)
ments. The dashed white line indicates the photon energy was measured at two different cavity detunings, δ = −5
gradient deduced from thelow-density spectrum. meV and δ = 0 meV, for a series of pump powers. The
detuning δ is the energy difference between the cavity
ted the intensity of the photoluminescence (PL) emitted resonance and exciton energy at k = 0. Changing the
||
by lower polaritons as a function of both the PL energy detuningchangestheunderlyingexcitonicfractionofthe
and the sample position for the case of moderate pump polaritons, which governs the strength of their interac-
power below the condensation threshold in Fig. 1c. The tions. Positive values of detunings indicate polaritons
white dashed line indicates the emissionenergies at very are more exciton-like, while negative values of detunings
low pump powers; the slope of this line arises from the givepolaritonswhicharemostlyphoton-like. Hereδ =0
wedgeofthe cavitythickness,whichcausesa gradientin meV and δ = −5 meV correspond to excitonic fractions
the cavity photon energy. At the pump region, there is of 50% and 30%, respectively. This indicates that the
a blue shift of the polariton energy due to their interac- polaritons with δ =−5 meV have interactions which are
tions with each other as well as from repulsive interac- weaker by a factor of 3 than those at δ = 0 meV, and
tions between polaritons and excitons and free carriers. less well thermalization is expected.
AsseeninFig.1c,the barrieris notconstantaroundthe The measured distributions N(E) at both detunings
ring, varying by about 1 meV from one side to the other andvariouspumppowersareshowninFig.2. Thepump
duetoinhomogeneityinthepumpintensity. Thebarrier powers are reported in terms of the threshold power,
3
102 102 but the rest of the distribution does not fit the Bose-
(a) (b) Einstein functional form in Eq. (1), indicating that the
δ = −5 meV δ = 0 meV polaritons are not in thermal equilibrium. This behav-
101 101 ior is similar to that seen in many other experiments
with short-lifetime polaritons, e.g. Refs. [1, 17], and is
consistent with a nonequilibrium polariton condensate.
Thenonequilibriumdistributionhasbeenreproducedby
N(E)100 N(E)100 numerical solution to the quantum Boltzmann equation
[39]. Despite the long cavity lifetime, the photon-like
polaritons with weak interactions do not reach thermal
10−1 10−1 equilibrium.
In contrast, N(E) at δ = 0 is well described by
N (E) for all pump powers up to P = 1.1P . At
BE BE
pumppowerswellbelowP , N(E)is welldescribedby
10−2 10−2 BE
0 0.5 1 1.5 0 0.5 1 1.5 a single exponential function, i.e., a Maxwell-Boltzmann
E(k )−E (meV) E(k )−E (meV)
|| 0 || 0 distribution. Between P = 0.9P and P = 1.1P ,
BE BE
an upturn in the distribution at E = 0 meV is ob-
FIG. 2. (color online). Energy distributions of polaritons in
thecenterof thetrap at (a)δ =−5meV(b)δ=0 meVat a served, indicating that N(E) deviates from Maxwell-
bathtemperatureofTbath=12.5Katdifferentpumppowers Boltzmann statistics and must be described by a Bose-
(see supplementary information for values). The solid curves Einsteindistributionwiththereducedchemicalpotential
are best fits to the equilibrium Bose-Einstein distribution in |µ/k T| < 1. The fit values of T and µ used in Fig. 2b
B
Eq. (1). The fitted values of T and µ are shown in Fig. 3. are shown in Fig. 3 as the blue symbols. As seen in this
The power values from low to high are 0.12, 0.24, 0.45, 0.71,
figure, when the density is increased, T decreases from
0.93, 1.07, 1.10, 1.12 and 1.14 times of the threshold values
around20Ktoalowestvalueof13.9±0.2Kandµ/k T
PBE, which are 382 mW and 443 mW for detunings δ = 5 B
smoothly goes from −2.93±0.16 to −0.28±0.01. For
meV and δ = 0 meV, respectively.
pump powers greater than 1.1P , a condensate in the
BE
ground state appears. In the weakly-interacting limit,
P , defined below. The sample was immersed in a he- the condensatepeak shouldbe delta-functionlike,which
BE
lium bath that was kept at a temperature T = 12.5 is broadened in the presence of finite-size fluctuations
bath
K for both detuning positions. The measured values of [40].The high-energy tail of the top two curves has the
N(E) were fit to a Bose-Einstein distribution, given by same absolute value, indicating that the population in
the excited states saturates when there is a condensate,
1
N (E)= , (1) consistentwithaBose-Einsteincondensationphasetran-
BE e(E−µ)/kBT −1
sition for bosons in thermal equilibrium.
where T and µ are the temperature and chemicalpoten-
tial of the polaritons, respectively, and k is the Boltz- (a) (b)
B
mann constant. The ground state (k = 0) of the lower 25 0
||
polaritonshiftstohigherenergyasthedensityincreases, −0.5
due to the repulsive interpolariton interactions. We de- 20 K) −1
fined the ground state energy in each case as E = 0 so K) 12.5 K (fit−1.5 12.5 K
thatµ=0correspondstotheconditionforBose-Einstein T (fit15 22.5 K µ/kTfitB −2 22.5 K
condensation. The best fits of the data to N (E) were
BE −2.5
determinedusingT andµasfreeparametersinnonlinear
−3
least-squares regressions, and are shown as solid curves 10
0 200 400 600 800 1000 200 400 600 800
in Fig. 2. Ntot Ntot
When the polariton states are negatively detuned and
FIG. 3. (color online). (a) Effective temperatures of po-
haveveryweakinteractions,thefitstotheBose-Einstein
laritons for bath temperatures T =12.5 K (blue points) and
distribution are poor. As seen in Fig. 2(a), for the case
T =22.5 K (red points) at different pump powers, extracted
ofδ =−5meV,atlowdensitythedistributionhasarea-
by fitting the energy distributions (shown in Fig. 2b for the
sonable fit to a Maxwell-Boltzmann distribution (which 12.5 K case) to the equilibrium Bose-Einstein model. The
correspondstoasingleexponential,i.e.,astraightlineon dashed lines indicate the helium bath temperatures. (b) Re-
asemi-logplot),butasthepolaritondensityisincreased, duced chemical potential α = µ/kBT for bath temperatures
the distribution is no longer fully thermal. The hump at T = 12.5 K (blue points) and T = 22.5 K (red points) at
different pump powers .
E =0.5 meV is a manifestation of the bottleneck effect,
as was also observed in Ref. [17]. As the density is in-
creased further, a peak occurs which is condensate-like, The upturn in the shape of N(E) in the low en-
4
ergystatesunambiguouslydistinguishesN(E)asaBose- We choose the threshold as the onset of Bose amplifica-
Einstein distribution rather than a Maxwell-Boltzmann tion, i.e., N(k =0)=1.
||
distribution. Previous reports using short-lifetime sam- Based on this methodology, N were determined for
BE
ples [16, 17] showed fits of N(E) but did not show this a series of T values ranging from 10.0 K to 25.0 K.
bath
behavior; although a condensate peak appeared in some ThefitvaluesofT attheonsetofBose-Einsteinstatistics
cases, there was not a clear density-dependent evolution are plotted in Fig. 4a, showing the general trend of T
BE
froma thermalMaxwell-Boltzmanndistributiontoa de- slightlyhigherthanT ,asdiscussedearlierinthetext.
bath
generateBose-Einstein,condenseddistribution. Further- Therelativedeviationishighestatlowbathtemperature,
more, short-lifetime samples thermalized only when the when the heat capacity of the sample is lowest, allowing
microcavitywaspositivelydetuned[16]andthepolariton thelocalsampletemperaturetorisemoreduetothelaser
characteristics were mostly exciton-like so that the mo- heating.
tion of the polaritons was severely restricted (see Sup- ThephasediagramofBose-Einsteintransition,i.e.,the
plementary Information for a detailed discussion). In relation of N to T is shown in Fig. 4b. The black
BE BE
contrast, the long lifetime polaritons seen here at zero lineisthebestfitofalinearproportionality. Withinthe
detuning follow Bose-Einstein statistics throughout the uncertainty,thedataareconsistentwithalinearincrease
phase transition and propagate to fill the trap in spatial in threshold T with N , consistent with the expected
BE
equilibrium. phaseboundaryofaweaklyinteractingbosongasintwo
We emphasize that not only the curvature of the fits dimensions implied by the relation (2). This line can be
in Fig. 2 but also the absolute verticalscale of the fits is viewed as a phase boundary: above the line, the gas is
constrainedby the value of µ. We do not have a free pa- quantum-degenerate, and below it, the gas is classical.
rameter to change the overallintensity scaling factor for
(a) 30 (b) 1000
eachcurve. Thedatapointsgivetheabsoluteoccupation
numbers as indicated by the vertical scale in addition to 25
800
the relative occupation numbers at different pump pow-
20
teirosn. iWs ihnecnretahseedvatlouweaordf µzeirno,tthheisBionscer-eEaisnesstethinedaibsstorilbuute- T (K)BE15 NBE 460000
value of NBE(E). Thus, the fits are tightly constrained 10
by the requirement that we fit not only the shape of the 200
5
distributionbutalsothe relativeheightsofallthe curves
0 0
with only two parameters, T and µ. This constraint is 0 10 20 30 0 10 20 30
Bath temperature (K) T (K)
reflectedintheverysmallrelativeuncertaintiesinthefit BE
values of µ shown in Fig. 3b.
FIG. 4. (color online). (a) The critical temperature as
a function of lattice temperature for δ = 0 meV. The solid
Phase diagram of polariton Bose condensation
black line indicates TBE = Tbath. (b) Phase diagram of the
The bath temperature was also varied in the range of
transitiontoadegenerateBosegas. Thesolidblacklineshows
10.0–25.0 K. Good thermalization has been achieved thebest fit of a linear relation NBE ∝TBE.
across this range. In Fig. 3, we plot the fitted values
of T and µ/kBT for different pump powers. As can be It has been a longstanding assumption that the Bose
seen, when the bath temperature is low, the fit values condensation effects seen in polariton systems are a di-
of T at low densities are much higher than Tbath and rect result of the quantum nature of the system when
at higher densities they settle to temperatures slightly r ∼ λ , but up to now it has not been possible to di-
s T
above Tbath, while for a bath temperature of T = 22.5 rectlytestthis. By usinghigh-qualitymicrocavitieswith
K, the fitted temperatures stay pinned to the bath tem- lifetimes over an order of magnitude longer than those
perature,withinthe uncertainty. Thechemicalpotential ofprevioussamples,polaritonswithinatwo-dimensional
increases smoothly toward zero in each case as the den- flat optical trap are seen to unambiguously show ther-
sity is increased. mal Bose-Einstein statistics. This clearly distinguishes
Now that we have a well defined temperature rang- polariton condensation from the conventional lasing ef-
ing over which thermal equilibrium is established, it is fect in semiconductor materials.
straightforward to determine the phase diagram of po- Now that we have samples in which true equilibrium
lariton Bose-Einstein condensation. To determine the can be established, more new experiments are possible
phase diagram, i.e., to check the scaling law in Eq. (2), to test theoretical predictions of interacting Bose gases
we want to plot the total number of polaritons N as a whichhavebeenelusiveincoldatomexperiments. Addi-
function of the fit value of T at the threshold. tionally,studies canbe conductedofthe excitationspec-
trum of the interacting Bose gas, and of the crossover
r ∼n−1/2 ∼λ ∼ ~2 . (2) from 2D to 1D equilibrium which can be controlled
s T
smkBT by spatial shaping of the excitation light to make tai-
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7
Supplementary Information for “Bose-Einstein (a)
Condensation of Long-Lifetime Polaritons in
Thermal Equilibrium” DBRs
QWs
Detailed sample structure The structure of the
sample used in the experiments described in the main
text is shown in Fig. S1 and Fig. S2a: three sets of four
7-nm GaAs quantum wells (QWs) are embedded at the x 104
threeantinodesina3λ/2microcavity,withthefrontand (b) 3
back distributed Bragg reflectors (DBRs), composed of u.)
a.2
e (
y nc
a
ect1
k θ fl
|| z Re
x
0
700 710 720 730 740 750 760 770 780 790 800
Wavelength (nm)
FIG.S2. (coloronline)Schematicillustrationofmicrocavity
structureusedinthiswork. Thedarkandlight gray alterna-
tive layers indicate the distributed Bragg reflectors (DBRs)
that are used to confinethelight in thecavity. Setsof quan-
tum wells (QWs) are shown as the black lines. The wedged
cavity allows different cavity resonance frequencies to be se-
FIG. S1. (color online) 3D illustration of microcavity struc- lected. (b) Calculated white light reflectance of the sample
ture used in this work. The red lines indicate the intensity structure. The black arrow indicates the second reflection
distribution oftheconfinedoptical field. Theredconeshows minimumabovestopbandtowhichthelaserwavelengthwas
thepumplaser beam. Thedark and light gray alternate lay- tunedto match.
ersarethedistributedBraggreflectors(DBRs)thatareused
toconfinethelightinthecavity,shown astheorangeregion.
Quantumwells (QWs) are shown as theorange layers. around780nmarisefromlowerandupper polaritonres-
onances. In the measurements,we tuned the laser wave-
32 and 40 pairs, respectively, of Al Ga As/AlAs λ/4 length to the second reflection minimum above the stop
0.2 0.8
layers. Thismicrocavitystructureisidenticaltosemicon- band, as indicated by the black arrow.
ductor microcavity structures with short polariton life-
Background on exciton-polaritonsin semiconduc-
times [2] except that the number of layers in the DBRs
tor microcavities. Exciton-polaritons are formed in
is doubled. This leads to an significant increase in the
semiconductormicrocavitiesthroughstrongcouplingbe-
lifetime of the trapped cavity field to about 135 ps, im-
tweenopticalmodesofthemicrocavityandexcitontran-
plying a polariton lifetime at δ =0 of 270 ps, which has
sitions of a material embedded inside the microcavity
been verified by transport measurements [32].
[42–45]. For the case of a single microcavity mode and a
In Fig. S1, the cavity regionis indicated as the orange
singleexcitontransition,twopolaritonmodes,theupper
section, and red lines show a plot of the intensity distri-
and lower polaritons, are formed with energies E (k )
butionofthe confinedlightmodes inthe cavity. We also LP ||
and E (k ) given by:
show the ring-shaped optical excitation beam as the red UP ||
cone on the right side of the structure. The microcav-
1
ity structure is wedged along one direction as shown in ELP/UP(k||)= 2 EX(k||)+EC(k||)∓ Ω2+δ2(k||)
Fig.S2a,whichallowstuningofphotonresonanceacross h q (i3)
the exciton resonance, thus allowing control of the exci-
ton fraction in the polariton state. where k is the wave vector in the plane perpendicular
||
Although the concept of doubling the number of DBR to the microcavity confinement direction, E (k ) is the
X ||
layers is simple, the fabrication is not trivial, because it energy of the exciton transition, E (k ) is the energy
C ||
requiresmuchlongerfabricationtimes,approximately30 of the cavity mode, δ(k ) is the detuning energy defined
||
hoursofmolecular-beamepitaxy(MBE),withtightcon- as δ(k ) = E (k )−E (k ), and Ω is the strength of
|| C || X ||
trol during the entire growth process. If the growth pro- radiative coupling between the exciton and cavity field,
cess is not well controlled, inhomogeneities in the lower also known as full Rabi splitting energy. The confine-
levels will be amplified in higher levels. ment of light gives the cavity mode a parabolic disper-
Fig. S2b plots the simulated spectrum of white light sionintheplaneperpendiculartothe confinementdirec-
reflection of a microcavity structure. The stop band for tion: E ≃E (0)+~2k2/2m ,wherem istheeffective
C C || C C
opticaltransmissionspansfrom740nmto820nm. Peaks mass of the cavity field. This effective mass is typically
8
10−4 times lighter than the vacuum electron mass, and
(a)
about 10−3 times less than an exciton in a GaAs quan- 1620
tum well structure, so that E (k ) is essentially con- V)
X || me1610
stant with k||. The energies EX(k||),EC(k||),ELP(k||), y (
and EUP(k||) are given in Fig. S3 for three different val- erg1600
ues of δ(k|| =0). The energies were calculated using (3) En
1590
and parameters matching the sample structure used in
the experiments: Ω = 10.84 meV and E (0) = 1604.6 0
X (1b5).5 16 16.5 17 17.5 18 18.5
meV. x0.2
n
o
1625 cti0.4
(a) (b) (c) c fra0.6
1620 ni
o
cit0.8
1615 Ex
1
meV) meV)
Energy (1610 Energy ( 15.5 16Positi1o6n. 5on th1e 7samp1le7 .(5mm)18 18.5
1605
FIG. S4. (color online) (a) The calculated upper polariton
1600 (blue line) and lower polariton (red line) ground state (k =
||
0) energies at different positions of the sample. The dashed
1595 line indicates the exciton energies, and the dotted line shows
−10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10
k||(µm) k||(µm) k||(µm) thecavityenergies. (b)Excitonicfractionsofupperpolaritons
(blueline) and lower polaritons (red line) at differentsample
FIG.S3. (coloronline)Dispersioncurvesofpolaritonmodes positions.
at threerepresentativecavitydetunings(a) δ=−5meV,(b)
δ = 0 meV, and (c) δ = 5 meV. The dotted line shows the
confinedcavitymode,andthedashedlineshowsthebareex-
The characteristics of the polariton modes are deter-
citon mode. The blue and red solid lines indicate the upper
polariton (UP) and lower polariton (LP) branches, respec- mined by the coefficients, which depend on δ(k||). The
tively,arisingfromthestrongcouplingbetweencavitymodes lower polariton is more photon-like and the upper po-
and exciton modes. Oursample parameters were used in the lariton is more exciton-like for δ(k ) < 0, and the lower
||
calculations. polariton is more exciton-like and the upper polariton
is more photon-like when δ(k ) > 0. Due to the wedge
||
Thelengthofthe cavityincreasesmonotonicallyalong in the cavity thickness, we can easily tune the excitonic
one direction of the QW plane so that the energy of the fraction |X(k )|2 of lower polaritons by moving the ex-
||
cavity mode can be tuned relative to the exciton reso- citation spot to different sample positions, as shown in
nance energy, as shown in Fig. S4, allowing us to exper- Fig. S4b, where we plot |X(k = 0)|2 at different po-
||
imentally tune δ(k = 0). The energies of all modes in sitions on the sample. As seen in Fig. S3 and Fig. S4,
||
Fig.S3areplottedasafunctionofk , the in-planewave theenergiesandshapesofthepolaritondispersioncurves
||
vector. depend strongly on δ: positive detuning results in lower
The polariton modes are linear superpositions of the polaritons that are more exciton-like, with a heavier ef-
exciton and microcavity photon modes. The lower po- fective mass and stronger interactions with phonons and
lariton and upper polariton operators, Pˆ and Qˆ , re- other carriers, while negative detuning results in lower
k|| k||
spectively, can be written in terms of exciton and cavity polaritonsthataremorephoton-like,withasmallereffec-
operators, aˆ andˆb : tivepolaritonmassandweakerinteractionswithphonons
k|| k||
and other carriers.
Pˆ =X(k )aˆ +C(k )ˆb (4)
k|| || k|| || k||k|| Review of earlier thermalization data. In order
Qˆ =−C(k )aˆ +X(k )ˆb . (5)
k|| || k|| || k|| to justify the realization of Bose-Einstein condensation,
macroscopicoccupationofasinglequantumstate,which
Thecoefficients,X(k )andC(k ),arecalledtheexciton
|| || translates to the macroscopic quantum coherence, to-
and cavity Hopfield coefficients [46] and are given by
gether with an equilibrium Bose-Einstein distribution of
particles, which is crucial in deriving the temperature
1 δ(k )
|X(k )|2 = 1+ || (6) of the system and the phase diagram of the condensa-
|| 2 δ2(k||)+Ω2! tion, need to be verified [47–49]. Various systems in-
cluding a simple laser as well as short-lifetime microcav-
1 p δ(k )
|C(k )|2 = 1− || . (7) ity polaritons have shown the spontaneous emergence of
|| 2 δ2(k||k||)+Ω2! macroscopic quantum coherence, however, a lacking of
p
9
equilibrium distribution across the condensation thresh- veryshortmeanfreepath,whichimpliesaverylowdiffu-
olddisqualifiesthesesystemsfrombeingnamedasBose- sion constant. Since the polaritons were generated with
Einstein condensation. In the past decade, the polariton a single laser spot, this led to an inhomogeneous spatial
condensates have been widely described as a nonequi- distribution. The momentum-space data were collected
librium Bose-Einstein condensation. This leads to not byintegratingoverthe entirespatialprofile. Inthis type
onlyasimplenomenclatureproblem,butalsoafailureof of experiment with short-lifetime polaritons and a single
applying to the polariton condensates the well accepted Gaussianlaser excitation spot, the potential-energy pro-
knowledgeintheatomiccondensates,notmentionednew file felt by the polaritons is strongly renormalized, lead-
difficulties in disentangling the effect between the many- ing to both self-trapping of the condensate in a small,
body renormalization and nonequilibrium in polariton quasi-harmonic potential inside the laser spot [50], and
condensates. Itisthereforecrucialtoinvestigatewhether free streaming away from the spot [51]. In general, it
polaritons and their condensates can thermalize to form has been shown that integration of a spatially inhomo-
an equilibrium distribution. geneous distribution can lead to misleading fits to a
Two earlier works have addressed in depth the ther- Bose-Einstein distribution [52], although reasonable
malization of polaritons in short-lifetime samples. In assumptions for density and temperature variations can
Ref. [17], the polariton condensate in CdTe-based sam- give fits with an average temperature and chemical po-
pleswasstudiedoverawiderangeoflatticetemperatures tential [53]. Later work by the same group [54] with the
and polariton densities. In the reported data, the distri- sameshort-lifetimestructureusedalargelaserspotwith
butions N(E) were of two types. At high temperature a flat intensity profile and resonant detuning that gave
at low density, the distributions fit a classical Maxwell- longermean free path to reduce the spatial inhomogene-
Boltzmann distribution. At low temperature and high ity. Inthisconfiguration,thespatialcoherenceproperties
density,thedistributionshadapeakatE =0,butcould were measured. The power law for the spatial coherence
not be fit by an equilibrium Bose-Einstein distribution wasfoundtodepend cruciallyonthe nonequilibriumna-
with a well-defined temperature and chemical potential. ture of the polariton gas [54, 55].
For example, in the curve for T = 5.3 K in Fig. 1a of These early attempts in addressingthe thermalization
Ref. [17], the best fit to an equilibrium Bose-Einstein behavior of polaritons and their condensates serve an
distribution missed the data by about a factor of importantstepinunderstandingpolaritoncondensation,
10 at low energy, leading to the difficulty in extracting however, it clearly does not resolve the debate, as terms
the chemical potential, and also the temperature, at the such as “instrinsic nonequilibrium character” and “far
onset of condensation. There was no comparison of the from equilibrium” have still been widely used in the lit-
data at different densities, so that the meaningfulness eratures in the past 2 years [18–20, 26, 56].
of the fit values of chemical potential (in particular, the
Energy distributions at different temperatures.
prediction of the absolute number of particles from the
In Fig. S5, we show the energy distributions at three
chemical potential) could not be verified. The interac-
representative bath temperatures, i.e., T = 10.0 K,
tion strength of the polaritons was varied by changing bath
T = 17.5 K and T = 25.0 K, at the same cavity
the detuning, and it was shown that the distributions bath bath
detuning as that of Fig. 2 in the main text, i.e., δ = 0
became much further from equilibrium as the interac-
meV, for a series of pump powers. As seen in these fig-
tion strength decreased. The authorsthus conclude that
ures, the distributions at all temperatures fit well to the
polariton condensation is driven by kinetics and cannot
ideal Bose-Einstein function in Eq. (1) in the main text,
be termed as Bose-Einstein condensation (which is ther-
uptoaground-stateoccupationnumberN(0)∼2–3. At
modynamic phase transition) given the nonequilibrium
higher densities, the distribution deviates from the ideal
particle distributions.
Bose-Einstein distribution. This is not surprising, since
In Ref. [16], the interaction strength was also varied
many-body effects play an important role when there is
by changing the detuning, and it was shown that for
a largegroundstate occupation. The decreasein the en-
weak interactions, the distribution N(E) was far from
ergyrangeoftheobserveddataasweincreasetheexcita-
equilibrium, while for strongly excitonic detuning (6
tion power is a result of spectral narrowing, particularly
to 9 meV), the distribution became more thermalized.
whencondensationforms,asshowninFig.2inthemain
Reasonable fits to a Bose-Einstein equilibrium distribu-
text.
tion were reported, but the system as a whole was still
strongly in nonequilibrium. An intense pulsed laser was Goodness of fit. Following Ref. [36], we calculated po-
used, which gave a rapidly changing temperature and lariton number in the field of view by integrating the
density of the polaritons on time scales of tens of pi- fitted distribution
coseconds. It could be argued, however, that on short
gmS Emax
time scales of 1-2 picoseconds, the gas could be consid- N = N(E)dE (8)
fit 2π~2
ered to be in quasi-equilibrium. More significantly, the Z0
strongly excitonic nature of the polaritons gave them a where g = 2 is the spin degeneracy factor of polaritons,
10
(a) (b) (c)
102 102 102
P = 0.12 P
BE
101 P = 0.20 P BE 101 101
P = 0.30 P
BE
P = 0.40 P P = 0.08 P P = 0.07 P
BE BE BE
P = 0.55 P P = 0.19 P P = 0.16 P
E)100 BE E)100 BE E)100 BE
N( P = 0.75 P N( P = 0.39 P N( P = 0.31 P
BE BE BE
P = 0.88 P P = 0.63 P P = 0.67 P
BE BE BE
P = 0.98 P P = 0.98 P P = 1.00 P
10−1 BE 10−1 BE 10−1 BE
P = 1.04 P P = 1.12 P P = 1.14 P
BE BE BE
P = 1.05 P P = 1.15 P P = 1.19 P
BE BE BE
P = 1.07 P P = 1.17 P P = 1.20 P
10−2 BE 10−2 BE 10−2 BE
0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5
E(k )−E (meV) E(k )−E (meV) E(k )−E (meV)
|| 0 || 0 || 0
FIG. S5. Energy distributions of polaritons (dots) and the best fits to equilibrium Bose-Einstein model (solid lines) at a bath
temperature of T = 10.0 K (a), T = 17.5K (b) and T = 25.0 K(c), with corresponding threshold pump powers PBE = 435
mW,PBE =474mW,andPBE =557 mW,respectively. Thecavitydetuningisδ=0meV,thesameasthatinFig.2binthe
main text
mistheeffectivemassofpolaritons,S isareaofthefield 1.2
ofview,andE istheupperboundoftheenergylimit
max
collected by the objective lens. We also computed the 1.15
polariton number in the field of view by directly sum-
ming up the CCD count per second multiplied by the nt 1.1
Ni
appropriate conversionfactor, namely /fit
N 1.05
N τ
i i
N = (9)
int
ξMi 1
i
X
where N ,τ and M are the CCD counts per second,
i i i
0.95
the lifetime of the polariton state, and the density of 15 20 25
T (K)
k states at pixel i. ξ accounts for the overall optical BE
efficiency of the collection setup, including the loss in
the dispersive grating and the quantum efficiency of the FIG. S6. (color online) Goodness of fit at different bath
CCD. We define the goodness of fit as temperatures.
Nc
fit
gof = (10)
Nc
int
data to a Bose-Einstein distribution, and then integrat-
where the superscript c indicates the critical point of
ing the fit curve using Eq. (2) in the main text. This
Bosedegeneracydeterminedfrommaintext. Thisquan-
method allows us to account for the high-energy tail of
tity can be used as a quantitative measure of the overall
theparticledistributionwhichextendsoutsideofourde-
quality of Bose-Einstein fitting. Fig. S6 plots the gof at
tectedrange,due tothe factthathigh-energystates cor-
the criticalthresholdµ/k T =−ln2for the datashown
B respondto highk ,whichgive lightemissionoutside the
k
in Fig. 5b of the main text. As can be seen, this fac-
numerical aperture of our collection lens. Fig. S6 shows
tor varies within 5% for most of the temperatures. This
twoalternatewaysofdeducingthephasediagram,which
confirmsthehigh-qualityfits,andthusconfirmsfullther-
rely less on the theoretical fit.
malization of polariton gas in the range of studied tem-
Fig.S7(a)plotsthetotalnumberofparticlescomputed
peratures. The slight increase in the gof might come
fromsumming the CCD counts per secondas in Eq.(9),
from the temperature-dependent emission rate of polari-
withoutusinganyfit. Theblacklineisthebestlinearfit
tons, which was not taken into account in the current
y ∝ x. As can be seen, this plot deviates from linearity.
data analysis routine.
This deviationis a result of the change in the fractionof
Replot of phase diagram. Fig. 5b in the main text polaritons in the field of view at different temperatures.
shows the phase diagram deduced by fitting the N(E) As the temperature of the polaritons increases, the en-