Energy Relaxation in a 1-D Polariton Condensate

0.26 MB·English

by  M. Wouters

Energy Relaxation in one-dimensional polariton condensates
M. Wouters,1 T. C. H. Liew,1 and V. Savona1
1Institute of Theoretical Physics, Ecole Polytechnique F´ed´erale de Lausanne EPFL, CH-1015 Lausanne, Switzerland
(Dated: January 13, 2011)
Westudythekineticsofpolaritoncondensationaccountingforthecondensationprocessaswellas
theenergyrelaxationofcondensedpolaritonsduetotheirscatteringwithphononsandexcitons. By
assuming a Boltzmann kinetic description of the scattering process, we show that intra-condensate
1 relaxationcanbeaccountedforbyanadditionaltime-dependenttermintheGross-Pitaevskiiequa-
1 tion. As an example, we apply the formalism to the experimental results recently obtained in
0 polariton microwires [E. Wertz, et al., Nature Phys. 6, 860 (2010)1]. In the presence of a local
2 non-resonantopticalpump,adynamicbalancebetweenspatially dependentrelaxation andparticle
loss develops and excites a series of modes, roughly equally spaced in energy. Upon comparison,
n
excellent agreement is found with the experimentaldata.
a
J
PACSnumbers: 71.36.+c,03.75.Kk,05.70.Ln,42.65.-k
2
1

I. INTRODUCTION descriptionof the collective excitations that characterize
]
l a quantum fluid. To correctly describe these properties
l
a Researchers now have two decades of experience in at thermal equilibrium, symmetry-breaking theories are
h the study of semiconductor microcavities2 and exciton- instead necessary. These approaches model the conden-
s- polaritons, the mixed quasiparticles that arise from sate as a classical field, in terms of the Gross-Pitaevskii
e strong light-matter coupling. Inspired by the early pre- equation, possibly coupled to a quantum model for the
m diction ofBose-Einsteincondensationof excitons3–5, im- excitedstates,inanalogytothetheoryofatomicconden-
. pressive progress in cold-atom condensation6, and mo- sates24,26,29. Intuition suggests that we can use a mix of
t
a tivated by the prediction of ultra-low threshold polari- the two theories in which higher energy excitons are de-
m ton lasing7,8, many experiments have been dedicated to scribed by Boltzmann-like equations and condensed po-
- the realization of a Bose-Einstein condensate of polari- laritonsbyGross-Pitaevskii-likeequations. Suchamodel
d tons9–12. As mixed quasiparticles, polaritons are ex- wasconstructedbyWoutersandCarusotto30 to describe
n
pected to interact with each other (due to underlying polariton condensates, whilst similar approaches have
o
c exciton-exciton interactions) whilst maintaining a light been developed in atomic condensates31–33.
[ effective mass (due to the trait of cavity photons). As Still, in experiments condensation can take place into
bosons, polaritons should relax into the ground state of a state above the ground state34,35 and in planar mi-
3
the system, producing a transition to a phase coherent crocavitiescondensedpolaritonrelaxationwasevidenced
v
0 state capable of a laser-like emission without the need downa potentialgradient10. In periodic structures, con-
2 for population inversion13–16. This transitionis the non- densationwasobservedintobothin-phaseandantiphase
3 equilibrium analog of Bose-Einstein condensation and Blochstatesandthefurtherrelaxationintotheloweren-
5 can occur at high temperature due to the light effective ergy in-phase state demonstrated with increasing pump
8. polariton mass. Although research studies usually use intensity11. Particularlystunning relaxationeffects have
0 an optical excitation of polaritons, progress in electrical appearedinrecentexperimentsonpolaritonmicrowires1.
0 injection sets a direct path for practical applications in Here condensation was observed into a series of energy
1 theformofpolariton-basedlight-emittingdiodes17–19. A modesthatappearroughlyequallyspacedinenergy. Sur-
:
v growing trend towards studying highly confined struc- prisingly,theenergyspacingistoolargetobeattributed
i tures, in which polaritons are confined into zero or one to quantization of the center-of-mass motion (the mi-
X
dimensional structures14,20–23, opens the way to the re- crowire is too long). To describe these effects theoret-
r alisation of microstructured polariton devices. ically it is essential to have a model that accounts for
a
A theoretical description of the condensation of mi- both the energy relaxation of excitons into condensed
crocavity polaritons is a challenging24–27 task, especially polaritons and relaxation of polaritons within the con-
whenonewishestoaccountforspatialdependence. Since densedfractionofthe polaritongas(polaritonscanrelax
the polaritonlifetime is shorter than the relaxationtime theirenergybyscatteringwithphononsorhotexcitons).
(in today’s samples), the system always lies in a state While the first order process is included in most kinetic
out of thermal equilibrium, with no well defined chemi- theories of polariton condensation, the second has never
calpotentialortemperature. Forthisreasononerequires been theoretically addressed, to our knowledge.
a kinetic model that accountsfor the excitation, scatter- In this work we derive a generalized version of the
ing, relaxationand decay processes occurring in the sys- Gross-Pitaevskii equation (GPE), that describes intra-
tem. Semiclassical Boltzmann rate equations28 can be condensate relaxation induced by the coupling with a
used to describe the thermal relaxation processes that reservoir. The starting point of the derivation is the
lead to condensation, however they completely lack of a Boltzmann collision term for the condensate density. As
2
aresult,the GPEacquiresanadditionaltime-dependent assumptionwhenmodelingintra-condensatekinetics. In-
rate that models the relaxationprocess. As an example, deed, due to the high occupation of both initial and fi-
we apply the formalism to the recent experiment carried nalstatesinascatteringprocess,stimulatedscatteringis
out by Wertz et al. on polariton microwires.1. The in- largelydominant. Wethereforerestricttothestimulated
coherent optical pump is modeled using a classical rate relaxation processes. At high densities, relevant for po-
equation coupled to the generalized GPE. The optical laritoncondensates,theseprocessesaremainlypolariton-
pump itself introduces a spatially structured potential exciton scattering50 rather than polariton-phonon scat-
(an effect also shown in experiments using resonant ex- tering28. The in-scattering term at frequency ω reads
citation36). Only a full account of the energy relaxation
′
withinthe condensateandofthe spatialdynamicsofpo- dnω(x,t)= nω(x,t) r(ω,ω )nω′(x,t)dt, (3)
laritonsinteractingwiththispotentialcanreproducethe ω′
X
various modal patterns measured experimentally.
′
where r(ω,ω ) is the net scattering rate from the mode
Thepaperisorganizedasfollows. InSec.II,wederive
′
at frequency ω to the mode at ω. Due to particle num-
the term in the GPE that describes the relaxation. Sim-
ber conservation, the relaxation rate obeys the relation
ulations of relaxation in nonresonantly excited 1D wires
′ ′
r(ω,ω ) = r(ω ,ω). The first term in the expansion
are presented in Sec. III. Conclusions are presented in
−
of the rate as a function of the frequency difference is
Sec. IV.
therefore
r(ω,ω′)=κ (ω′ ω)+ (ω′ ω)3 , (4)
II. DESCRIPTION OF THE RELAXATION × − O −
where the relaxation constant κ has(cid:0)the dimens(cid:1)ion of an
The degeneracy of the Bose gas implies that a good inverse density. Here, we don’t use a microscopic model
account of the condensate kinetics is already given by a of the scattering process. Rather, we keep κ as a free
mean field description, formulatedin terms of a classical parameter that we adjust to fit the experimental data.
field ψ(x,t). This approach is common to most recent The resulting value is consistent with the scattering on
theoreticalworks37–44 that describe most existing exper- hot excitons as discussed by Porras,et al.50.
imentaldataobtainedabovethecondensationthreshold. Underthechangeofthedensity(3),the wavefunction
Moreadvancedquantumkinetictheoriesareonlyneeded varies as
to model the details of phase fluctuations45–47.
A kinetic model formulatedin terms of transitionsbe- nω(x,t)+dnω(x,t)
ψω(x,t)+dψω(x,t)= ψω(x,t),
tween states characterized by their frequencies, involves s nω(x,t)
a coarse graining of the time. This is analogous to the (5)
coarse graining of space used in the semiclassical Boltz- where we used the fact that a stimulated relaxationpro-
mann equation48 to describe the density as a function cess does not change the phase of ψ. Expanding Eq. (5)
of positionand momentumwhile fulfilling the constraint to first order in dnω, substituting Eqs. (3) and (4) and
imposed by the uncertainty principle. We will use the using the inverse transform (2), one obtains for the re-
following definition of the time dependent spectrum of laxation dynamics of the wave function
the bose field ψ(x,t):
∂ψ(x,t) κn¯(x,t) i∂
= µ¯(x,t) ψ(x,t). (6)
t
ψω(x,t)= 1 eiωt′ ψ(x,t′)dt′, (1) ∂t 2 (cid:20) − ∂t(cid:21)
T t−T
Z In the derivation, boundary terms in the partial integra-
tion that scale as 1/T were neglected, because we con-
where T is the coarse graining time step. When using
sider the limit of a large coarse graining time where 1/T
this formalism to model a steady-state spectrum, it is
important that ~/T is chosen smaller than the energy is smaller than any other relevant frequency scale. In
Eq. (6), n¯ and µ¯ are defined as
spacingofthefeaturestoberesolved. Then,thephysical
result should not be affected by the precise value of T. t
1
The inverse transform reads n¯(x,t)= ψ(x,t′)2dt′, (7)
T t−T | |
Z
ψ(x,t)= e−iωtψω(x,t). (2) 1 1 t ∗ ′ i∂ ′ ′
µ¯(x,t)= Re ψ (x,t) ψ(x,t)dt .(8)
Xω n¯(x,t) (cid:20)T Zt−T ∂t (cid:21)
Spontaneous scattering is a quantum feature that can- The first expression defines the density averaged over
not be described within a classical field model such the coarse graining interval T. Analogously, ~µ¯ can
as the GPE and has only been treated in a handful be suggestively read as an averaged chemical potential.
of theories46,47,49. Neglecting spontaneous scattering We stress however that no actual chemical potential can
mightlimitthe accuracyinmodeling the kinetics oflow- be defined for this non-equilibrium system, and Eq. (8)
occupationstates. Itishoweveraparticularlywellsuited draws its only justification from the formal derivation
3
presented above. The real part is taken in Eq. (8), be- ~GP(x) is an effective potential experienced by polari-
cause it is readily shownwith the Madelung transforma- tons, causedby interactions between polaritons with the
tion ψ = √neiθ, that the imaginary part of the integral reservoir and also the pump field, P(x), described with
in Eq. (8) scales as 1/T. Equation (6) gives the term a strengthG. The largepotential experiencedby polari-
that was sought to describe the stimulated relaxation of tons at the ends of the wire is accountedfor throughthe
a classicalBose field due to localscattering with phonon introduction of boundary conditions ψ(x= L)=0.
±
like particles and is the central result of this work. The dynamics of the reservoir field, representing opti-
The right hand side in Eq. (6) resembles the fre- cally injected hot excitons injected, is described by:
quency dependent amplification term that was intro-
∂N(x,t)
duced in Ref. 51 to describe an energy dependent gain = ΓR+βRR ψ(x,t)2 N(x,t)+P(x)
∂t − | |
mechanism due to scattering from a reservoir: ∂ψ/∂t =
(cid:16) (cid:17) (11)
(P/ΩK)[ΩK −i∂/∂t]ψ. Two main differences should be where ΓR is the decay rate of reservoir polaritons and
emphasized. First, here the relaxation term is propor-
the parameter β < 1 quantifies the backaction of the
tional to the polaritondensity n¯, whereas in Ref. 51, the condensate on the reservoir46. Such a parameter is nec-
relaxation is proportional to the gain from the reservoir
essary due to the simplified model of a single reservoir
P. The secondimportant difference is that the gaincut-
used for the exciton states; if β = 1, then unphysical
off frequency ΩK is replaced by the average frequency instabilities may develop in the solution of the coupled
µ¯. As a consequence, particle loss and gain balance each Eqs.(9)and(11)46. Thereasonisthatforβ =1thegain
other and there is no net gain in Eq. (6). The modes
saturationmechanismisoverestimated,giventhesimpli-
with frequency ω > µ¯ experience loss, where the ones
fied description of the reservoir in terms of Eq. (11). At
with ω < µ¯ are amplified. Energy absorption from the
the same time, the assumption β = 0 would make the
reservoir into the condensate is thus not included in our
condensate density diverge, because the gain saturation
model. The relaxation (6) is due to the interaction with
mechanism is needed to determine the steady state con-
an environment that is effectively at zero temperature.
densate density at a given pump intensity. Ultimately,
a microscopic description of the reservoir made with a
continuous spectrum of modes, would in principle make
III. RELAXATION IN 1D WIRES the introductionofthisphenomenologicalparameterun-
necessary. The choice β < 1 is therefore obliged, given
As an example of application, we consider a 1D mi- the reservoir model (11). Finally, note that the value
crowire1 under non-resonant optical excitation. In most of β makes no qualitative difference to the spectral and
situationsinwhichanincoherentopticalpumpispresent, spatial condensate pattern (implying only an overall de-
agooddescriptionofthecondensationkineticsisgivenby pendenceofthecondensatedensity)solongasitissmall
coupling the GPE to a kinetic equation for the reservoir enough to prevent instabilities.
population.30 To this purpose, we introduce a complex We have solved Eqs. (9-11) with parameters52 corre-
condensate field, ψ(x,t), coupled to a reservoir popula- spondingtothe experimentofWertz1 etal.,whoconsid-
tion, N(x,t). Neglecting the polarization degree of free- ered two cases: excitation at the wire center and excita-
dom, the evolution of condensed polaritons is given by tion near one edge. In both cases the time evolution is
the generalized GPE for a Bose gas with losses and gain calculated over a long (4000ps) period during which the
fromareservoir,supplementedwiththedissipationterm system reaches a steady state. Energy distributions are
(6): then calculated by Fourier transforming the fields.
i~∂ψ(x,t) = EˆLP + i~[RRN(x,t)−ΓC] A. Excitation in wire center
dt 2
(cid:18)
2
+V(x,t)+g ψ(x,t)
| | The energy distributions with excitation at the wire
~κn¯(x,t) ∂ center are shown in reciprocal (left) and real space
+ iµ¯(x,t)+ ψ(x,t), (9)
2 dt (right) in Fig. 1. The presence of the pump introduces
(cid:20) (cid:21)(cid:19)
excitons into the reservoir and also creates the spa-
where the kinetic energy eigenvalues of EˆLP are: tially varying potential, V(x). Together with a slight
blueshift caused by polariton-polaritoninteractions, this
ELP(k)= EC(k)+2 EX(k)−12 (EC(k)−EX(k))2+Ω2. pcoontednetnisaaltedeftoerrmmsindeusetthoetheenesrcgaytteartinwghoifchpaartipcloelsarfritoomn
q (10) the reservoir. The condensed particles are created with
Here EC(k) and EX(k) represent the cavity photon and zero in-plane wavevector, in the center of the pumping
exciton dispersion relations, respectively, and Ω is the spot, and are pushed away by the potential V(x) so
exciton-photon coupling constant. Polaritons scatter that they fill the wire. As the pump intensity increases,
from the reservoir into the condensate at a rate RR and a clear blueshift of the condensed polaritons is visible
radiatively decay at a rate ΓC. V(x,t) = ~RRN(x,t)+ and stimulation of the energy relaxation takes over.
4
Propagating polaritons form a spectral pattern with Energy Rate
several modes, roughly equally spaced, in agreement
with the experimental results1. V DE0
Κn
G
x DE
DE
0
FIG. 2. (color online) a) Illustration of quantized energy re-
laxationaspolaritonspropagatedownthepump-inducedpo-
tential, V(x). b) Sketch of the incoming (proportional to
κn¯(x,t))andoutgoingrates(Γ)forthelowerenergymodein
a relaxation step, as a function of the energy difference with
a higher energy mode.
To confirm the necessary role of energy relaxation in
the generation of a series of condensate modes at dis-
creteenergies,werepeatedourcalculationssettingκ=0.
Fig.3showsthatinthiscaseallpolaritonscondenseinto
a single energy mode determined mostly by V(x=0).
FIG. 1. (color online) For excitation in the wire center, the
energydistributionofcondensedpolaritonsinreal(right)and
momentum(left) spaceisshown forincreasingpumppowers.
Thewhitecurvesshowthebarepolaritondispersiongivenby
Eq. (10) in momentum space (left) and the effective polari-
ton potential, V(x), in real space (right). In each plot the
intensityisscaledtothemaximumintensityandplottedwith
a logarithmic color scale. The data were convoluted with a
small spectral linewidth (δ = 0.02meV) to account for the
finitespectral resolution in the experiment.
Theenergyrelaxationintroducesagainmechanismfor
thelowerenergypolaritonstatesandonecanimaginethe FIG. 3. (color online) Same as the bottom panels of Fig. 1
relaxation of polaritons as they travel down the pump- with κ=0.
inducedpotential,asillustratedinFig.2a. Therateofre-
laxationbetweenanytwomodesisanincreasingfunction
of their energy difference, ∆E, to lowestorder according
to Eq. (4), as illustrated by the blue (dark-grey) line in B. Excitation near wire edge
Fig. 2b. For the formation of the condensate to occur in
a lower energy mode, in the steady state, the gain must The case of excitation near the wire edge is shown in
exactlybalancethelossesinthe process. Inotherwords, Fig. 4. The same phenomenology as in the case of exci-
theincomingrateduetothisrelaxationmustbalancethe tation at the wire center can be identified; condensation
outgoing rate, which is mostly due to radiative decay, il- at a blueshifted energy and energy relaxation at higher
lustrated by the green (light-grey) line in Fig. 2b. The intensities. However, one can now observe the striking
balance occurs at the energy ∆E and relaxationis only appearance of harmonic modes due to a potential trap
0
expected to appear between modes separated at this en- formedbetweenthewireedgeandthepump-inducedpo-
ergy. The further outwardpropagationof polaritons can tential, V(x). This feature is in close agreement with
allow a cascade effect in which polaritons relax in units the experimental results1. The positioning of the optical
of∆E intoaseriesoflevels(Fig.2a). Astherateispro- pump near the wire edge introduces a strong asymme-
0
portional to κn¯(x,t) and to the energy step, the modes try between relaxation on the two sides of the pump. It
are roughly equally spaced in energy if we assume that is observed theoretically and experimentally that relax-
thedensityintheinitialmodeisapproximatelythesame ation is more effective when polaritons interact with the
forallrelaxationsteps. ThedatabyWertzetal. support wire edge.
this. We note that our model is limited in its quantitative
5
duces the experimental character.
IV. CONCLUSIONS
Wehavepresentedatheoreticalformalismthatmodels
both the condensation process, as polaritons relax from
high-energyhot-excitonstatesintothecondensedpolari-
tonfield,aswellasfurtherenergyrelaxationtakingplace
within the condensate itself as polaritons interact with
excitonsandphonons. Thetheorygoesbeyondthe stan-
dardGross-Pitaevskiitheories,whichdonotdescribethe
condensationprocessorenergyrelaxation,andtheBoltz-
mannrateequations,whichdonotallowadescriptionof
coherence,interference or the structure of modes in con-
fined systems.
We applied the formalism to the recent experiment of
Wertz1 et al. This experiment uncovered a number of
remarkableeffectsinthephysicsofnonequilibriumBose-
Einsteincondensationinconfinedsystems. Thepresence
of optically induced potentials changes the modal distri-
FIG. 4. (color online) Same as in Fig. 1 for excitation near bution in the system. Condensation initiates strongest
thewire edge. locally,atthe positionof the opticalpumping. However,
the dispersion of polaritons and their further relaxation
allows lower energy modes to host the condensate in a
description of the power dependence. Particularly near spatially dependent way. Our model reproduces well the
the wire edge, higher pump powers were required exper- experimental findings. In particular it explains the puz-
imentally to observe the same effects. Our energy re- zling feature ofdiscrete energymodes of the condensate,
laxation model assumes that the relaxation rate is pro- observed experimentally. The excitation of this mode
portionalto the condensed polariton population and the ladder is naturally attributed to the energy relaxation
energy difference in a relaxation process. In reality, we mechanism within the condensate. A selection rule in
expecttheratetoalsodependonthetotalexcitonpopu- the energy relaxationsteps is determined by the balance
lationandnottobe suchasimple functionofthe energy between gain and loss for the final energy mode of the
difference. The study of these issues is beyond the scope relaxation process. In the case of excitation at the wire
of our work; it would require further experimental char- edge,the narrowlaser-inducedtrapcreates a clear setof
acterization and a microscopic model of the polariton- harmonic modes.
reservoir kinetics. Although we also neglected exciton We aknowledge enlightening discussions with J. Keel-
diffusion, intrinsic reservoir dynamics and nonlinear loss ingandA.Kavokin. ThisworkwassupportedbyNCCR
mechanisms,webelievethatourmodelcapturesthemost QuantumPhotonics(NCCRQP),researchinstrumentof
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P Schwendimann,56, 7554 (1997). withFWHM=3µm[Ref.1].Thecavityphotondispersion
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