Disorder effects on exciton-polariton condensates
G. Malpuech, D. Solnyshkov
Institut Pascal, Nanostructures and Nanophotonics group
Clermont Université, Université Blaise Pascal, CNRS, France
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).
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  which
described the localization of electrons in a disordered media. Some works have addressed these
questions for a gas of bosons in the eighties , and this activity took an enormous theoretical
and experimental expansion since the observation of the BEC of cold atoms . 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 . 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 . 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 .
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  and in
2008 for spinor polaritons . 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  (2-dimensional stable solitons
), whereas the accounting of the spin degree of freedom allowed to predict the formation of
oblique half solitons . Another very promising configuration is given by the fabrication of
high quality 1D GaAs microwires . 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
condensate in a random continuous potential can be studied versus the velocity and the density of
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 . 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 . 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 . This result was interpreted in the 90’s as a “motional narrowing” effect which led
to some controversy . 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
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 . 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 , 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 , GaAs , GaN . Technically, the first clear
evidence of the feasibility of the Imamoglu’s proposal has been published by Le Si Dang in
1998 . 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. . 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 . 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 ) 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. , by the
group of Luis Vina . Also, a “non-equilibrium” condensation had been reported earlier, in
two papers of 2005 by M. Richard et al. In Ref. , 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. , condensation was taking place in the
We show the results of a hydrodynamic simulation of the propagation of a polariton
condensate with a wavevector in a disordered system taking into account the finite particle
lifetime and the backscattering induced by the disorder, both included in the equation (17). The
condensate is injected (presumably by non-resonant pumping) at the point x=0 µm and
propagates to the right. The potential U(x) felt by the condensate at the point x is composed of
the disorder potential and the interaction energy linked with the local density. Figure 9b shows
the steady-state situation, obtained when the interaction energy significantly exceeds the kinetic
one at the injection point, and the condensate is therefore initially superfluid. Since the density is
decreasing with the coordinate even in the superfluid regime because of the finite lifetime, the
condensate propagates down a potential slope (created by the gradient of the density) and is
therefore accelerated, until it reaches the (local) speed of sound. At this point, the condensate
becomes supersonic and backscattering starts to play a major role, strongly reducing the
localization length. When the potential energy is completely transferred into kinetic energy, the
condensate starts to move faster and therefore the backscattering is reduced again. The
localization length increases almost back to the value induced by the finite lifetime.
The transition between the superfluid and supersonic phases is a horizon at which strong
instabilities may develop, but their study will be a subject of a separate work.
V Conclusion and perspectives.
The main result of this chapter is to calculate the localization length of an interacting
polariton condensate, moving in a weak random potential. To our knowledge this type of
calculation was never published before for this type of system. We consider the transition from a
superfluid to an Anderson localized condensate. The localization length is found to be reduced in
the Čerenkov regime. Till 2009, polariton lifetime was typically below 10 ps, and the
propagation length of particles was typically limited by this factor. Another case realized is the
one of large disorder were the condensate is classically localized. The fabrication of new
structures, of high quality, and showing polariton life time of 30 ps  and even more 
should allow to study in details localization and superfluidity phenomena. Another interesting
direction relates to the realization of periodically modulated samples [63, 64]. In such type of
structures, a wide variety of new phenomena (such as Bloch oscillations  for instance) can
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