Page 1

Disorder effects on exciton-polariton condensates

G. Malpuech, D. Solnyshkov

Institut Pascal, Nanostructures and Nanophotonics group

Clermont Université, Université Blaise Pascal, CNRS, France

Summary

The impact of a random disorder potential on the dynamical properties of Bose Einstein

condensates is a very wide research field. In microcavities, these studies are even more crucial

than in the condensates of cold atoms, since random disorder is naturally present in the

semiconductor structures. In this chapter, we consider a stable condensate, defined by a chemical

potential, propagating in a random disorder potential, like a liquid flowing through a capillary.

We analyze the interplay between the kinetic energy, the localization energy, and the interaction

between particles in 1D and 2D polariton condensates. The finite life time of polaritons is taken

into account as well. In the first part, we remind the results of [G. Malpuech et al. Phys. Rev.

Lett. 98, 206402 (2007).] where we considered the case of a static condensate. In that case, the

condensate forms either a glassy insulating phase at low polariton density (strong localization),

or a superfluid phase above the percolation threshold. We also show the calculation of the first

order spatial coherence of the condensate versus the condensate density. In the second part, we

consider the case of a propagating non-interacting condensate which is always localized because

of Anderson localization. The localization length is calculated in the Born approximation. The

impact of the finite polariton life time is taken into account as well. In the last section we

consider the case of a propagating interacting condensate where the three regimes of strong

localization, Anderson localization, and superfluid behavior are accessible. The localization

length is calculated versus the system parameters. The localization length is strongly modified

with respect to the non-interacting case. It is infinite in the superfluid regime whereas it is

strongly reduced if the fluid flows with a supersonic velocity (Cerenkov regime).

I Introduction

In a normal fluid, the viscosity arises because of the elastic scattering of the particles

which compose it. This includes both the scattering on the external potential, for example, the

walls of the capillary, and the scattering of the particles on each other, if their velocities are

different. In contrast to that, for a condensate of weakly interacting bosons (a Bose-Einstein

condensate – BEC), which will be the main object studied in this chapter, single independent

particles are replaced by collective sonic-like excitations [1, 2]. As a result, such condensate

propagating with a velocity smaller than the speed of sound cannot dissipate its kinetic energy by

scattering on a disorder potential or on the non-condensed particles. This collective behavior

results in a vanishing mechanical viscosity, called superfluidity.

However, sometimes the potential fluctuations can be large enough to destroy the

superfluid behavior of a Bose condensate by provoking its complete localization. The question of

the interplay between kinetic energy, localization energy, and the interaction between particles

has been widely studied in solid state physics since the seminal work of Anderson [3] which

Page 2

described the localization of electrons in a disordered media. Some works have addressed these

questions for a gas of bosons in the eighties [4], and this activity took an enormous theoretical

and experimental expansion since the observation of the BEC of cold atoms [5]. Particularly

interesting to study is the simple case of a 1D weakly interacting Bose gas moving in a disorder

potential. Two different model frameworks are typically considered: discrete and continuous.

The discrete lattice models usually employ the Bose-Hubbard Hamiltonians by which Mott

insulator, Bose Glass, superfluid, or Anderson localized phases are described [6]. The continuous

models are usually employed for the description of a relatively weak and smooth potential,

where one cannot apply the tight-binding approximation. The theoretical modeling can be

performed in this case with the Gross-Pitaevskii equation [7,8 and refs. therein].

Exciton-polaritons are the quasi-particles formed of cavity photons strongly coupled with

quantum well excitons, which are expected to behave as weakly interacting bosons, at least at

relatively low densities. Despite their short life time, they can thermalize to a quasi-thermal

(Bose) distribution [9, 10, 11,12, 13] which can in principle allow the polariton gas to undergo a

Berezinskii-Kosterlitz-Thouless phase transition towards a superfluid state [14, 15, 16]. In CdTe

or GaN cavities, this superfluid behavior of the condensed phase was not observed because of

the presence of a strong in plane disorder which tends to localize the condensate, leading to the

formation of a glassy phase [17]. In cleaner GaAs-based samples, the generation of a superfluid

is in principle simpler and the observation of a renormalized linear dispersion above the

condensation threshold has been reported [18].

Another possible way for generating a polariton superfluid besides the BKT (equilibrium)

phase transition is to use the resonant excitation configuration as proposed in 2004 [19] and in

2008 for spinor polaritons [20]. The idea is to pump a polariton state with a laser, which should

be slightly blue-detuned from the bare polariton dispersion. If the blue shift induced by the inter-

particle interactions in the macro-occupied pumped state exactly compensates the detuning, the

laser and the polariton mode become resonant, and the dispersion of elementary excitations is

similar to the equilibrium case, and the pump state can propagate as a superfluid. This

configuration has been recently used [21, 22] to generate a high density flux of moving

polaritons and to study their elastic scattering on a large in-plane defect. A substantial decrease

of the flux dissipation by elastic scattering has been observed, but the expected singular character

of the superfluid formation under resonant pumping has not been evidenced. However, this type

of experiment is really opening a new research field. It reveals the enormous potential of the

polariton system to study quantum hydrodynamic effects when a moving quantum fluid hits a

large defect (typically larger than the fluid healing length). As predicted [23, 24], this

configuration has allowed the observation of oblique solitons [25] (2-dimensional stable solitons

[26]), whereas the accounting of the spin degree of freedom allowed to predict the formation of

oblique half solitons [27]. Another very promising configuration is given by the fabrication of

high quality 1D GaAs microwires [28]. In these samples the radiative particle life time can reach

30 ps, which is one order of magnitude longer than in other material systems. Under non

resonant excitation, the 1D character allows the formation of a high density non-equilibrium

condensate moving along the wire, spatially independent from the pumping region. Because of

the long life time, the propagation for large distances can take place without a substantial decay

of the particle density. This is, therefore, a quite ideal configuration, where the motion of a

Page 3

condensate in a random continuous potential can be studied versus the velocity and the density of

particles.

In this chapter we do not consider the specific case of a condensate hitting a single

potential barrier and we do not study the formation of topological defects, such as solitons.

Parametric instabilities are also not taken into account. We consider a stable condensate, defined

by a chemical potential, propagating in a random disorder potential, like a liquid flowing through

a capillary. This chapter is organized as follows. In the first section, we give an overview, and a

critical discussion of the literature devoted to the disorder effects on polaritons. In the second

section, we recall the main expected properties of a static Bose Einstein condensate placed in a

disorder potential, analyzing the interplay between localization and interaction effects. In the last

section, we finally consider a propagating condensate, first in the linear non-interacting limit,

mainly discussing Anderson localization of polaritons, and then taking into account the

interactions. We then discuss the critical condition required for the occurrence of superfluidity.

We finally address the question of the interplay between the kinetic energy, the localization

energy, the interaction between particles, and the lifetime. To summarize the different

possibilities for our interesting system, we plot a phase diagram.

II Historical overview

The role of the structural disorder on the linear optical properties of microcavities was

first evidenced by Resonant Rayleigh Scattering experiments. An exciton-polariton eigenstate is

which defines the polariton energy through the

dispersion relation ( ) E k . Because the polariton is a mixed state of two particles having different

masses, its dispersion is not parabolic. If the in-plane translational invariance is broken by the

normally characterized by a wave vector k

presence of disorder (acting on one or both polariton components), the polariton wave k

anymore a good eigenstate. Such wave, for example resonantly created by a laser, scatters

toward the “elastic ring” of isoenergetic polariton states as shown experimentally in [29]. This

perturbative description is correct if the disorder amplitude is small with respect to the kinetic

energy. If the kinetic energy is small compared with the disorder amplitude, the particles become

strongly localized which provokes a strong change of the particle dispersion, as discussed for

example by Agranovich and Litinskaia in 2001 [30]. In the hypothetic case where the decay

processes such as life time, or phonon scattering are negligible, this process leads to an iso-

distribution of particles on the elastic ring. If the coherence is sufficient, this process should lead

to a weak localization of the polariton waves called Anderson localization. So far this process

has not been yet neither observed, nor described in polaritonic systems. It is typically dominated

by the short radiative life time of particles which limits the spatial extension of a polariton wave

much more than localization effects. The other consequence of the effect of disorder is the

inhomogeneous broadening of the polariton line. As a result, the sum of the widths of the lower

and upper polariton modes is not constant versus the exciton-photon detuning, but shows a

minimum [31]. This result was interpreted in the 90’s as a “motional narrowing” effect which led

to some controversy [32]. Another important aspect relies on the type of material used to grow

the structure which strongly affects the amplitude of the disorder potential. GaAs based samples

is not

Page 4

show the best structural qualities with the inhomogeneous broadening of the polariton line which

can be as low as 0.1 meV. InGaAs QWs are a bit less good, with values of the order of 0.5 meV.

In CdTe-based structures the inhomogeneous broadening value is typically a few meVs. It is

typically 10 times larger in GaN based samples, and again about 5-10 times larger in organic

based structures. Disorder affects both the excitonic and photonic parts of the polariton modes,

but the typical correlation lengths for both are different.

After the study of linear properties of the microcavities, the non-linear optical response of

microcavities has been explored under resonant and non-resonant excitation. Under non-resonant

excitation, the goal of experimentalists was to achieve polariton lasing, first suggested by

Imamoglu in 1996 [33]. A non-resonant laser pulse creates high energy electron-hole pairs which

bind to form an incoherent exciton reservoir which in turn forms an exciton-polariton condensate

in the ground state. The polariton condensation is possible because of exciton-exciton and

exciton-phonon interaction. Because of the finite polariton life time and limited efficiency of the

relaxation processes, the polariton condensation is in principle an out-of-equilibrium process.

However, different regimes can be distinguished depending on the type of materials used, on the

exciton-photon detuning, polariton life time and on the size of the pumping spot. A

thermodynamic regime can be defined [11], corresponding to the achievement of a quasi-thermal

distribution function. In such a case, important features, such as the critical condensation density,

or the polarization of the polariton condensate can be extracted from thermodynamic

calculations, which often have the advantage of being analytical. On the other hand, another

regime, called kinetic, does exist as well, where the condensate features are fully governed by

the dynamics of the system. This feature and the existence of the two regimes in a given

structure with a possible transition between them have been demonstrated in all types of

semiconductor microcavities: CdTe [11], GaAs [34], GaN [13]. Technically, the first clear

evidence of the feasibility of the Imamoglu’s proposal has been published by Le Si Dang in

1998 [35]. However, the work, which is now mostly cited and recognized as being the one where

polariton condensation was observed, is the Nature paper of 2006 by Kasprzack et al. [11]. The

sample and the experiment performed were the very same as in 1998, but three new

measurements were added. First, the distribution function was measured and found to be close to

an equilibrium distribution function. Second, the spatial coherence was found to pass from 0 to

30 % at distances of about 5-10 µm. Third, the condensate was found to be linearly polarized

above threshold, which is another confirmation of the condensation taking place, because the

polarisation is the order parameter of such phase transition in a spinor system [36]. With these

new data, the observation of polariton condensation close to thermal equilibrium (“polariton

BEC”) was claimed. Since that time, there is a strong tendency to state that polariton

condensation is a non-equilibrium process and that the achievement of equilibrium (which was

one of the important results of [11]) is unimportant or impossible. If this is indeed true, it would

be probably fair to slightly rewrite the history. The build up of linear polarization, pinned along

crystallographic axis in the polariton laser regime was demonstrated before the Ref. [11], by the

group of Luis Vina [37]. Also, a “non-equilibrium” condensation had been reported earlier, in

two papers of 2005 by M. Richard et al. In Ref. [38], condensation took place in finite-k states

because of the use of a small pumping spot, as it was understood later. The coherence between

different k-states was evidenced directly. In Ref. [39], condensation was taking place in the

Page 5

ground state, but was stated as non-equilibrium, because of the use of a pulsed pumping laser.

The condensate was found to be spatially localized in several different spots linked with the

presence of an in-plane disorder potential. However, the angular width of the emission remained

narrow, well below the diffraction limit imposed by a single emitting spot. This evidenced that

these different spots were emitting in phase together, demonstrating the onset of spatial

coherence, one year before it was made by interferometric technique in the Nature of 2006. The

most convincing evidence of the build up of a spatially coherent condensate was given later, in

the Nature Physics of 2010 by Wertz et al [28], where a coherence degree larger than 80 % was

found for distances over 200 micrometers (50 times the De Broglie Wavelength). In this last

work however and similarly in the PRL of 2005 [38], the condensates generated are completely

out of equilibrium because of the use of a small pumping spot which limits the overlap between

the thermal exciton reservoir and the polariton condensate itself.

The conclusion one can draw from this brief historical overview is that if the achievement

of quasi-equilibrium is absolutely uninteresting, as suggested in many recent works, then

possibly other works than Ref. [11] could be cited as the first evidence of the polariton

condensation achievement, depending on the importance given to the achievement of spatial

coherence. One could interestingly notice that spontaneous symmetry breaking is often referred

to as the “smoking gun” of Bose condensation. From that point of view, the clearest evidence of

polariton condensation could be the J.J. Baumberg’s PRL of 2008 [40], where the build-up of a

condensate polarisation above threshold was observed, with a polarization direction varying

randomly from one experiment to another, and not pinned to any crystallographic axis.

Let us now go back to the main topic of this chapter, which is the effect of disorder on

polariton condensates. From this point of view, the experiments performed on CdTe-based

samples are really of strong interest, because of the relatively large disorder present in these

samples. In [39] already, the condensate was found to be strongly inhomogeneous in real space,

peaked around the in-plane fluctuations of the potential. The formation of vortices pinned to

these defects was already suggested, which, we remind, was later demonstrated clearly in the

Science [41]. In the Nature paper [11], evidences of localization were even stronger, with the

appearance of flat dispersion around k=0.

III Static condensate in a disorder potential

Let us first consider a static gas of bosons at 0 K in a random potential as discussed in [17]. A

realization of this potential is shown on the figure 1. The ground state of this system can be

found by solving the Gross-Pitaevskii equation by minimization of the free energy for a fixed

number of particles. In a non-interacting case, at thermal equilibrium all particles are strongly

localized in the deepest site, so-called Lifshits state. In practice, thermal activation or the out-of-

equilibrium system character allow the occupation of many localized states having possibly

different energies (Lifshits tail). Such case is shown on the figure 2a and 2b demonstrating the

spatial distribution of particles and their dispersion, which is quasi-parabolic near the ground

state, characterized by substantial inhomogeneous broadening. If one considers a weakly

interacting Bose Gas, it fills all localized states having their energy below the chemical potential