Polariton ring condensates and sunflower ripples in an expanding quantum liquid

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by  Gabriel Christmann

submitted to PRL’11
Polariton ring condensates and sunflower ripples in an expanding quantum liquid
Gabriel Christmann,1,∗ Guilherme Tosi,1 Natalia G. Berloff,2 Panos Tsotsis,3 Peter S.
Eldridge,4 Zacharias Hatzopoulos,4 Pavlos G. Savvidis,3,4 and Jeremy J. Baumberg1
1NanoPhotonics Centre, Cavendish Laboratory, University of Cambridge,
J. J. Thomson Avenue, Cambridge CB3 0HE, England, UK.
2Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge, CB3 0WA, UK.
3Department of Materials Science and Technology,
2 University of Crete, P.O. Box 2208, 71003 Heraklion, Greece.
1 4FORTH-IESL, PO Box 1527, 71110 Heraklion, Crete, Greece.
0 (Dated: January 11, 2012)
2
Opticallypumpinghighqualitysemiconductormicrocavitiesallowsforthespontaneousformation
n of polariton condensates, which can propagate over distances of many microns. Tightly focussed
a
pumpspotsherearefoundtoproduceexpandingincoherentbottleneckpolaritonswhichcoherently
J
amplifytheballisticpolaritonsandleadtotheformationofunusualringcondensates. Thisquantum

0 liquid is found to form a remarkable sunflower-like spatial ripple pattern which arises from self
1 interference with Rayleigh-scattered coherent polariton waves in the Cˇerenkov regime.

]
PACSnumbers: 71.36.+c,42.65.Yj,73.63.Hs,78.67.De
l
l
a
h
Condensing solid state bosonic quasiparticles brings coherent. We show that spatial ripples appearing as in-
-
s prospects for macroscopic quantum-coherent integrated terferometric sunflowers arise from elastic scattering in
e
m devices. Although still cryogenic, advances in coher- the superfluid.
ent atom matter-wave optics [1, 2] shows what might be Themicrocavitiesconsistofa5λ/2AlGaAscavitycon-
.
t possible with room temperature condensation, as sought taining four sets of three GaAs quantum wells placed at
a
m for years with superconductors. In the last decade it the antinodes of the cavity electric field, surrounded by
has become apparent that polaritons formed from cou- two AlGaAs/AlAs Bragg mirrors of 35 (bottom) and 32
-
d pling semiconductor excitons with cavity photons pro- (top) pairs. The cavity quality factor is measured to
n ducebosonicquasiparticlesthatcancondense[3,4],even exceed Q>16000, with cavity photon lifetime τ =9ps.
c
o
at room temperature [6], with superfluid transport and Strong coupling is obtained with a Rabi splitting be-
c
[ a number of unusual vortex states now discussed [7–9]. tween lower and upper polariton energies of 9 meV. All
Polariton condensation requires high-Q optical cavi- data presented here use a negative cavity detuning of
1
ties, large light-matter Rabi coupling exceeding thermal -5meV, although other negative detunings give similar
v
energies(¯hΩ>k T),andefficientrelaxationintobosonic results. Excitation is provided by a single-mode narrow-
4 B
1 quasiparticles from the high-energy injected fermionic linewidth CW laser focused to a 1µm diameter spot and
1 carriers. Successful condensation in microcavities made tunedtothefirstspectraldipatenergiesabovethehigh-
2 of III-V arsenides [3], II-VI tellurides [4], III-V nitrides reflectivity mirror stopband at 750nm. To prevent un-
.
1 [6, 10] and anthracene [11] have relied on natural spa- wanted sample heating the pump laser is chopped at
0 tial traps produced by the disordered energy landscape 100Hz with an on:off ratio of 1:30. Both real and k-
2
toconfinepolaritonssufficientlytoinitiatecondensation, space images of the photoluminescence (filtering out the
1
as well as artificially created ones. This is largely due to pumplaser)canbeimaged,eitherontheentranceslitsof
:
v theexistenceofa‘bottleneck’[12]intherelaxationpath- aspectrometer/CCDcombination,ordirectlyonaCCD.
i
X waywherepolaritonscollect, thusformingahigh-energy We observe that the micron-scale polariton reservoir
reservoir from which they cannot relax further. Conden- resulting from the tight pump spot efficiently feeds a
r
a sation requires an additional localized concentration of macroscopic quantum state in the form of a spatial ring
the bosons. However such disorder-induced localization which appears above the threshold pump power for con-
severely hampers development of chip-based condensate densation (Fig.1a). While spatial images below thresh-
circuits. old show maximum incoherent emission from the cen-
Herewedemonstrateanefficientschemetopumpsuch tral pumped spot (Fig.1b), above threshold the coherent
non-equilibrium quantum liquids, that produces unusual condensate emission develops in a ring surrounding the
coherent spatial ring states. These macroscopic conden- pump 10µm across (Fig.1c) independent of precise po-
sate rings experience spatially-localized stimulated scat- sition on the semiconductor wafer, which becomes more
tering as the bosons expand ballistically. Despite their pronounced at high powers (Fig.1d). The entire conden-
outwards acceleration the quantum liquid remains phase sate is spatially coherent (as observed also with inter-
2
FIG. 1: (a) Integrated polariton emission vs pump power, insets are real space emission images. Black lines are guide to the
eyeabove/belowthresholdat9mW.Greyregioncorrespondstopolaritoncondensate. (b-d)Realspaceimages(loggreyscale)
ofemissionforthepowerscorrespondingtopointsmarked1-3in(a)atI =0.3,1.1,and3.0I . (e-g)Correspondingemission
p th
intensities for 1-3 on energy vs position maps (log scale). Horizontal dashed line is the non-blueshifted k=0 energy.
a Near the pump spots, above threshold emission is sup-
pump
ℏ𝜔 ℏ𝜔(𝑘,𝑟) pressed around the condensate energy and only at larger
𝑟 distances do polariton states below the condensate be-
𝑐
come available.
ring
Δ0 𝜔𝑐 condensate The condensate is created at in-plane k=0 at the bot-
𝑟 tomoftheblue-shiftedpolaritondispersioninthemiddle
b
of the pump spot, and gains momentum as it accelerates
E (eV)
1.542 5µm 10µm 15µm outwards (Fig.2a). This is confirmed by resolving the
emission from the dispersion relation E(k) at each posi-
1.540
tion (Fig.2b). These observations form the 2D counter-
1.538
0µm part to that observed in 1D polariton wires[13], leading
-2 0 2 -2 0 2 -2 0 2 -2 0 2 to the formation of condensates with circular symmetry
𝑘𝑥(𝜇𝑚−1) 𝑘𝑥(𝜇𝑚−1) 𝑘𝑥(𝜇𝑚−1) 𝑘𝑥(𝜇𝑚−1) extended over >100µm2.
The ring condensation mechanism is revealed by the
FIG.2: (a)Schematicshowingdispersionrelationsatincreas-
ingradialposition,togetherwith(b)correspondingmeasured weak uncondensed hot background polariton lumines-
emission from dispersion relations at each position. cence in the same spectra (Figs.1e-g) above the conden-
sate energy. The Λ-shaped emission observed originates
5meV above the k=0 polariton and drops in energy with
ferometry [4], not shown). The smooth emission profile distance from the centre (dashed) until it meets the ring
observed below threshold acquires a peculiar ‘sunflower’ condensate exactly at its intensity maximum at r = r .
c
patternofinterferenceripplesabovethreshold,whichare Suprisingly, Λ-shaped emission is also observed below
always observed (Fig.1c,d) and stable for many minutes. threshold. These hot polaritons possess a large momen-
The origin of these ring states lies in the spatially- tumandcomefromthehighmomentumpolaritonreser-
dependent energy shifts ∆(r), which create an optically- voirclosetothebottleneck. Despiteballisticallyexpand-
controlled potential landscape. Polaritons are highly ing from the pump spot, they decelarate, thus dropping
nonlinear quasiparticles, and experience strong mutual down the polariton dispersion (see sequence in Fig.2a).
repulsion resulting in energy blue shifts with increasing Whilst condensed polaritons cannot inelastically scatter,
density [4, 5]. This effect can be clearly seen in the blue the incoherent polaritons decelerate in a systematic way.
shifts of the condensate energy ω = ∆(0) = ∆ with Eventually this incoherent reservoir meets the energy of
c 0
increasing laser power (Fig.1e-g) which show the spec- thecondensateandamplifiesitstrongly,leadingtoaring
tra emitted at different distances from the pump spot. shaped condensate.
3
1.544 scattering events are suppressed. Below threshold, inco-
a ) b
).b Vem1.542 houertewnatrdpsowlairtihtoansdebcuaiyldleinnggtuhpofa2t.1kµ=mr=. 0Hoawreevaecrctehleeryateexd-
rA( ytisn ( ygrenE111...555433086 ( 0tip) puanleardtieisontncimeinunloaatanemydposlncifiaectstatetarioitnenga(ittshretchysuiasnrcseelontwho.terABetoisstehnroceolsanhrdogleedn,psoethdpe)-
etn 𝑟𝑐 1 2 4 610 2 4 61002 ringimmediatelyformsatrc=5.6µm(Fig.3c)andthein-
i no 𝐼 )m 6Pump power (mWc ) tseunbssietqyutehnetrebailnlicsrteiacspesrobpyagaalmtioonstotfetnh-efocldon(dFeings.3aate).isTihne-
𝑝
issim ( su 5 cfurertahseedr u(tpot4h.4eµdmis)pdeurseiotno tghiveinbgluela-srhgiefrtsowuthwicahrdmvoevleocωic-
E pump id ties. As the excitation power increases the ring shrinks
a
0 5 10 15 20 r g 4 in diameter linearly with the blue shift (Fig.3c), and the
n 2 3 4 ring intensity increases linearly (as Fig.1a).
Radial distance (m) iR Blue shift (meV)
This shrinking ring tracks the position at which the
1.544
d e Λ reservoir intersects with the increasingly blue-shifted
)V condensate emerging from the pumped spot (Fig.3d).
e 1.543
m These extracted energies of the incoherent reservoir po-
( y 1.542 laritonsatincreasingradialdistance(Λ-shapedinFig.1e-
g g) change little in shape with power. They are rigidly
re 𝐼𝑝 blueshifted, but less than the condensate mode itself
n 1.541
E 3 4 (Fig.3b). Hence the reservoir polaritons are not injected
10 10
Intensity (arb) at a fixed energy above the polariton dispersion, but ex-
perience thermally-induced energy changes at the pump
FIG. 3: (a) Averaged radial intensity for increasing excita- spot(believedtobeduetolocalheatingattheverycen-
tion powers (black lines), with excitation profile (red line). treofthepumpspotwhichred-shiftstheexcitonlocally).
(b)Energyofthebottomofthelowerpolaritonbranchatthe A second feature is the constant deceleration rate of the
excitation spot (circles) and energy of the Λ tip (squares) vs expanding reservoir polariton cloud (note the lack of po-
excitation power. (c) Distance of the ring condensate from
laritonemissionfrominsidetheΛlineinFigs.1e-gimplies
excitation spot vs power. Lines are guide to the eye. (d,e)
that this expansion is not diffusive but ballistic). Since
Energy position and intensity of the incoherent reservoir po-
¯hk˙ = −F this implies a constant opposing force F on
laritonsvs distanceforincreasingpowers. Whitecirclesmark
condensate energies (different dataset). the polaritons (as in Bloch oscillators [15]). This force
seems to arise from interactions of the incoherent reser-
voir polaritons with excitons near the pump spot. Such
The ring condensate cannot be described as indepen- interactions cannot occur with the coherent polaritons
dentlyacceleratingpolaritonsineachradialdirectionbe- becausethecondensate,beingasinglewavefunction,can-
cause of the varying k (r) = K[ω − ∆(r)] set by the notdeceleratesimultaneouslyallaroundtheringthrough
c c
position dependent blueshifts. Instead the condensate single local scattering events (as these cannot conserve
wavefunction is a superposition of different k-states at energy and momentum). The preservation of incoherent
different radial positions, polaritons throughout this decelaration process (Fig.3e)
supports this scenario.
ψ(r,t)=ei{K[ωc−∆(r)].r−ωct}g(r)e−t/τc(r) (1) The characteristic sunflower self-interference ripples
(Fig.4a) clearly demonstrate that a component of the
where τ is the polariton decay time, and the dispersion
c (cid:112) quantum liquid propagates in non-radial (spiral) direc-
relation K(ω)=k ≈ 2m∗ω/¯h for small k. As the con-
tions. The patterns are persistent and regular, and can
densateexpands,itsvelocitychangessothatthedistance
be decomposed by local 2D Fourier transforms within
ρ is reached after a time τ,
different regions of the image. This analysis reveals
(cid:90) ρ dK(cid:12)(cid:12) that at every point the ∆k corresponding to the fringe
τ = dω(cid:12)(cid:12) dr (2) periods and directions (formed by interference between
0 ωc theoutwards-movingcondensateandaccompanyingscat-
Theamplitudeg(r)isstronglyamplifiedatthecritical tered waves) lies on a circle touching the origin and of
radiusr (Fig.3a)whenstimulatedscatteringconvertsin- radius k = K(ω ) (Fig.4b). This is exactly the pat-
c c c
coherent polaritons from the reservoir into the coherent tern expected for isotropic Rayleigh scattering of the
expanding condensate of Eq.(1). This unusual conden- outwards-travellingcondensateandisexperimentallyob-
sate thus has a wavevector that varies with radial posi- served (Fig.4c) [16].
tion, andis amacroscopic object sothat individual local Confirmationofthisanalysisisprovidedfirstlybysim-
4
(cid:113)
a d (cid:96)0 = ¯h2/2mκ = 1.95µm for m = 10−4me. The ran-
dom potential therefore creates obstacles of size on the
order of the healing length for the polariton flow. It is
well-known [20] that linear sound waves are emitted as
a result of supersonic flow around such small impurities
whereas for larger obstacles the emitted waves are not
e linearbutaredispersiveshocksconsistingofobliquesoli-
tons. Such Cherenkov emission of sound waves is clearly
10µm seen in Fig.4e. Observation of this pattern implies the
breakdown of superfluidity in the polariton condensate.
A simple estimate shows that away from the pump spot
) b c the flow velocity, u = h¯∇φ/m where φ is the phase of
1
− ψ, becomes much greater than the local sound veloc-
𝑚 Δ𝑘 ity cs = (cid:112)U0ρ/m where the density ρ = |ψ|2. The
𝜇
( Mach number, M, which is the ratio of the flow ve-
𝑦
𝑘 locity to the local speed of sound, is of the order of
Δ (cid:112)µ/U ρ, where µ is the chemical potential of the sys-
0
tem. Neglecting the quantum pressure term it follows
Δ𝑘 (𝜇𝑚−1) 𝑘 (𝜇𝑚−1) from (3) that µ = U0(γ −κ)/(ηU0 +Γ), ρ(0) = µ/U0
𝑥 𝑥 (cid:112)
and ρ < ρ(0)exp[−2κ(r − R)/¯h µ/m]/(r − R) away
from the pumping spot. It follows that M (cid:29) 1 as soon
FIG. 4: (a) Real space image of the emission (log scale), to- (cid:110) (cid:112) (cid:111)
as exp 2κ(r−R)/¯h µ/m (r − R) (cid:29) 1 which takes
getherwith(b)Fouriertransformand(c)k-spaceimagespa-
tial filtered within the rectangle marked. (d) Simulated in- place within a couple of healing lengths away from the
terference of the condensate with Rayleigh scattered waves. boundary of the pump spot. Comparing to experiments
(e) Simulation of the spatial polariton density using the cGL of Amo and cowokers [9] where the superfluid is at sub-
equation (see text).
sonic flow velocities, here the flow velocity is supersonic
which places the system in the Cˇerenkov regime where
scattering is allowed.
ulations that interfere a circular wave (the condensate)
with weaker (Rayleigh-scattered) plane waves in random In conclusion, expanding ring condensates are thus
directions and with random phases (Fig.4d). The result- clearly non-equilibrium quantum liquids, which can be
ingimagefromthissimplemodelreproducesimpressively efficiently excited and locally amplified using incoherent
well the experimental observations, with 6% of the total reservoirs. However they also suffer Rayleigh scattering,
energyintheRayleighcomponent. Amoreaccuratethe- which leads to condensate fragmentation once the ran-
oretical description of the system starts from the mean dompotentialexceedsacertainamplitude(thusdemand-
field equation [17–19] in the form of the cubic Ginsburg- ing high quality microcavity samples). Our methods fi-
Landau (cGL) equation, nally allow the generation of 2D condensates in arbitary
locations, which can be spatially manipulated for use in
i¯h∂∂ψt =(cid:2)−¯h22m∇2+U0|ψ|2+Vext(r)+iP(r,ψ)−iκ(cid:3)ψ cfoorheerleecnttriscoallildy-sptuamtepdeedviccoensd,eannsdatoepcehnipusp. the possibility
(3) We acknowledge EPSRC grants EP/G060649/1,
EP/F011393, EU CLERMONT4 PITNGA-2009235114
where U0 is the strength of the interaction, and the de- and INDEX PITNGA-2011-289968 and Spanish MEC
tails of pumping are included in P(r,ψ) = γ −ηi¯h∂t − (MAT2008-01555). G.T. acknowledges FPI scholarship
Γ|ψ|2 (inside the pump spot only). We take the po- support from the Spanish MICINN.
lariton decay rate κ = 0.1meV, radius of pump spot
R=10µm, and consider pumping at 3 times threshold so
γ = 3κ, with Γ/U = 0.3 where Γ is the rate of non-
0
linear dissipation leading to gain saturation [19]. The
dimensionless parameter η=0.1 describes the energy re- ∗ Electronic address: [email protected]
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[3] H. Deng et al. Science 298, 199-202 (2002).
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ext [4] J. Kasprzak et al. Nature 443 409-414 (2006).
modelled by a random distribution of Fourier modes of
[5] G. Tosi et al., Nat. Phys. (2011).
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