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

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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

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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

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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

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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

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values µ. The condensate forms independent lakes separated by potential barriers (figure 2c) and

forms a glassy phase.

This glassy phase of a Bose condensate can be called a Bose glass, but this term is

usually applied to the glassy phase formed due to excessive inter-particle interactions, which

inhibit the hopping between different sites in lattice models, which is never the case for

polaritons for reasonable parameters. Our glassy phase, therefore, has to be called Anderson

glass, and the distinction between the two glasses has become clear since Ref. [6], where the

transitions between all four possible phases (Anderson glass, Superfluid, Bose glass, Mott

Insulator) were studied.

Even if the system is not in thermodynamic equilibrium and several independent

condensates are formed in the potential minima, depending on the particle density the

neighboring sites having non-connected non-interacting ground states can start to overlap to

finally synchronize, as shown in [42]. However, in 2D the randomness of the potential forbids

the creation of a conduction band for the particles until the classical percolation threshold is

reached. The spectrum of elementary excitations of the condensate is shown on the figure 2d. It

shows the flat dispersion observed experimentally. The superfluid fraction, calculated using the

twisted boundary condition technique, is close to zero. This picture corresponds to the one

experimentally observed. If one increases the density further, the classical percolation threshold

is reached and a fraction of the fluid becomes superfluid as demonstrated by the linear dispersion

of the elementary excitations shown on the figure 2f.

Figure 1. A typical disorder potential characterized by rms fluctuation V0=0.5 meV and

correlation length lc=2 µm.

Page 7

Fig. 2. Spatial images (top panel) and quasiparticle spectra (bottom panel) for a realistic disorder

potential. The figures shown correspond to densities 0, 6x1010 and 2x1012 cm. The red lines are

only guides to the eye, showing parabolic, flat and linear-type dispersions. Colormap of the panel

(b) is the same as (c).

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10

0

10

1

Temperature (K)

10

2

10

3

10

6

10

8

10

10

10

12

Polariton density (cm−2)

Bose Glass

Polariton diode

LED

VCSEL

Superfluid

Fig. 3. Phase diagram for exciton-polaritons in a disordered microcavity in the thermodynamic

limit. The Bose glass phase can also be called Anderson glass. Superfluidity is only possible

above certain density, depending on the disorder rms fluctuations.

Page 9

05 10 1520 2530

0

0.2

0.4

0.6

0.8

1

x (µm)

g1

Figure 4. Coherence degree for 3 different condensate densities. In the linear case (almost zero

density) the coherence length is given approximately by the de Broglie wavelength. The

condensed part is assumed to be 100% coherent, but its density varies from point to point due to

the presence of the disorder potential.

A phase diagram from Ref. [17] summarizing these results is shown in figure 3. It demonstrates

that exciton-polaritons in a disordered microcavity can present several different phases, and the

superfluidity can be recovered only above certain density threshold, linked with the rms

amplitude of the disorder. Since the polariton density is limited by the possible loss of the strong

coupling, only relatively good material systems with low disorder, such as GaAs, can be

expected to show signs of superfluidity.

In the figure 4, we show a calculation of the first order coherence for a finite size 1D system,

assuming a thermal uncondensed fraction of particles with a homogeneous spatial distribution

(high kinetic energy of uncondensed particles provides their weak sensitivity to the disorder).

The coherence degree fluctuates, showing maximum values at the potential minima, because the

density of the condensate is higher in these minima. This is again in good agreement with the last

work of the Deveaud’s group on the very same CdTe sample [43]. The density and coherence

fluctuations of the condensate associated with the presence of the disorder potential is measured,

and related to the formation of a Bose glass phase (which, we remind, should rather be called

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Anderson glass). For some puzzling reasons, the authors insist that this effect is the result of the

strong non-equilibrium nature of their system whereas it is obviously not the case.

Similar works, considering 1D condensates in disordered systems have then been

published by the group of Vincenzo Savona [7,8]. They reached the same conclusion that the

insulating to superfluid phase transition occurs at the percolation threshold. However, one should

notice that in an infinite 1D disorder potential with a Gaussian distribution, there is no

percolation threshold. This transition therefore only exists either in a finite system, or in a system

with a well defined higher bound for the potential fluctuations.

One should point out that, in 2007, an alternative explanation to the absence of

superfluidity in polariton condensates was proposed both by the Cambridge group of P.

Littlewood [44] and the Trento group of I. Carusotto [45] who attributed the formation of a

diffusive (flat) dispersion to the non-equilibrium nature of the polariton codensate. It is indeed

well known that in the resonant pumping case, the external laser driving the system strongly

affects the dispersion of the excitations of the driven mode which can be diffusive, parabolic, or

even linear for some precise value of the pumping intensity [19, 20]. In [45], a simple reservoir-

condensate model was proposed which gave essentially the same result as the resonant pumping

scheme, being very similar to the latter from the mathematical point of view. Depending on the

decay time constant of the reservoir, the dispersion of the elementary excitations of the

condensate can be parabolic, linear, and, when using an unphysically short decay time for the

reservoir (5 times shorter than for the condensate), diffusive. This result is mathematically

interesting, but it is clearly completely unrelated to the observed experimental result. The latter

statement is also supported by several experimental studies. In [18], a GaAs structure less

affected by disorder than CdTe sample has been studied. The dispersion of elementary

excitations was claimed to pass from parabolic to linear above threshold. The data could be

judged not so convincing, but it is however clear that the dispersion was not diffusive in that

case. Finally, tons of experimental works demonstrating the strong disorder effect in CdTe

cavities came. One can mention the observation of the condensation in a single potential trap of a

few microns by D. Sanvitto et al. [46], or the observation of a Josephson-like dynamics between

two localized states where the condensation is taking place [47]. Also, observation of vortices

[48] and half vortices [41] was made possible because of the pinning of the flux by the disorder

potential.

IV Localization and superfluidity of a moving condensate

In this second section we discuss a bit more subtle problem of a moving fluid, interacting

or not, flowing against a defect or through a disorder potential. This has been a very active field

of research since the seminal work of Anderson [3]. It is by far beyond our capacities to give a

complete review on this topic and we are going to concentrate on a few specific results.

a) Anderson localization

Brownian motion of classical particles in a random potential leads to diffusive dynamics. A

particle freely propagates for a typical distance (mean free path, a distance between two

impurities) and is scattered in a random direction. The width of the particle distribution around

the initial position increases as

probability distribution therefore infinitely broadens in time. In his initial work, Anderson

Dt , where D is the diffusion coefficient and t the time. The

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considered a 3D lattice model. The energies of the sites are random. They are connected by some

matrix element of hopping between the sites. Anderson shown that in such a model, the wave

function of a single quantum particle does not always spread with time but sometimes remains

exponentially localized on the initial site. The transparent explanation given by Anderson is that

the electron cannot hop to the neighboring site if the energy mismatch exceeds the hopping

matrix element. In other words, the diffusion coefficient is zero. In 3D, the existence of a

mobility edge has been demonstrated. However, in 1D and 2D any weak random potential is

expected to localize a quantum particle, whatever its kinetic energy, unless some extra terms are

added to the Hamiltonian, such as the spin-orbit coupling or the interactions. Localization has

been evidenced for light waves[49], microwaves [50], sound waves [51], and matter waves [52].

Anderson localization is strongly associated with the concept of phase coherence as we are going

to explain below, which is not evident from the initial interpretation given by Anderson.

Let us consider a particle moving in a random 1D potential. This particle hits a defect and

is either transmitted or reflected with certain probabilities. In a classical scheme, the reflected

fraction will follow a classical random walk, but at a later time it is expected to collide with the

same barrier again and participate in the transmission. For a quantum particle, this random walk

will lead to destructive interferences, and the reflected part does not give any contribution to the

transmission. The wave is therefore attenuated at each barrier, which leads to an exponential

decay of the wave function. The effect is really subtle, because the role of interference is to

cancel all multi-reflection channels for the transmission, which play a crucial role in the classical

diffusive motion.

If one consider 1D systems, a standard approach used [53,54] to calculate the localization

length is the so-called “phase formalism” [55], where the Schrödinger equation is solved in a

Born approximation using a localized wave function Ansatz. Here, for the sake of demonstration,

we choose to use a slightly different approach.

We consider a propagating plane wave with a wave vector k0. The bare dispersion of the

particles

(

0

E k

is assumed to be quasi-parabolic. The wave is propagating in a continuous

( )

)

disorder potential

V r with a Gaussian random distribution of energies with a root mean

square fluctuation

0

V and a correlation length

c L . The action of this potential on the propagating

particles is treated as a perturbation. It does not provoke any change in the dispersion of the

particles, but simply induces some elastic (Rayleigh) scattering processes which are assumed to

provoke a decay of the number of the propagating particles (scattering to the other k states). This

assumption is based on the fact that, in the Anderson picture, the backscattered part of the wave

will never interfere again constructively with the propagating wave.

From the Fermi golden rule, the decay rate of the propagating particles reads

12

π

τ

where

''kk kk

MV

ψψ

−

=

,

'

kk

=

, and

k

V is the Fourier transform of the potential. The final

result essentially depends on the density of final states, which depends itself on the

dimensionality of the problem:

()

2

if

f

MEE

δ

=−

(1)

Page 12

()

/2

2

2

00

2

1

()

dd

c

d

d

L

x V DE E k

τ

==

(2)

where d is the dimensionality, 0 k is the wave vector,

the density of states at the energy of the propagating wave.

From the equation (2) one can extract the inverse of the propagation length as

2

x the exciton fraction.

()

0

()

d

DEE k

=

is

()()

/2

2

2

000

2

0

0

2

1

τ

()

dd

c

dd

dd

mL

k

m

k

kxV DE E k

v

γ

τ

====

(3)

In 1D and 2D, and for parabolic dispersions, this formula reads:

2

2

2

10

4 2

k

0

2

c

D

m L

xV

γ=

(4)

2 2

2

2

20

4

0

2

c

D

m L

k

xV

γ=

(5)

The formula (4) is similar to formula (13) of [53], derived from the phase formalism,

π.

except that our numerical prefactor 2 is replacing 2

In a system with a finite lifetime, a supplementary decay source should be considered and

the total decay rate can be written as:

()()

(

)

2

/2

2

2

000

2

0

0

1

2

()

dd

c

dd

R

mx

mL

k

kxV DEE k

k

γ

τ

−

==+

(6)

where

R

τ is the photon life time in the cavity. The scattering of excitons by phonons is

neglected, but it could be easily introduced, that would bring additional temperature dependence

into the formula. The figure 5 shows, in blue and red for 1D and 2D respectively, the localization

lengths of polaritons for a rms potential fluctuation of 0.25 meV and a correlation length of

1 micron. The Rabi splitting is taken 20 meV and the exciton-photon detuning is zero. The cavity

photon life time is taken 30 ps. The black line shows the localization length induced by the life

time. For these parameters, Anderson localization of photons can be evidenced both in 2D and

1D, despite the considerable effect of the life time. The 1D and 2D cases do not appear

extremely different. Particles are more localized in 1D than in 2D for small energies. The

situation is opposite for higher energies. The exact position of the crossing of the two curves

depends on the coherence length of the disorder.

Page 13

0 0.51 1.52

0

20

40

60

80

100

120

140

Energy (meV)

Localisation lentgh (µm)

2D

1D

Lifetime

Figure 5: Localization length

1

d γ− of the propagating polariton condensate (from formula 6). The

black curve shows the localization length in a disorder-free structure (life time only).

b) Superfluidity

A superfluid is a fluid moving without mechanical viscosity. As first understood by

Landau [1], this extraordinary property can arise in a fluid if the dispersion of the elementary

excitations of the fluid is linear. In that case, the fluid can flow without friction as long as its

velocity remains below the speed of sound

sc given by the slope of this linear dispersion. In

1947, Bogoliubov [2] established the link between superfluidity and condensation by showing

that the dispersion of the excitations of weakly interacting bosons follows, at low k, the linear

shape proposed by Landau. Physically, the suppression of viscosity is associated with the

suppression of elastic scattering processes provoked by the roughness of the surface along which

the fluid in propagating.

Below we will demonstrate the Bogoliubov linearization procedure for a condensate of

weakly interacting bosons propagating with a certain wave vector. Let us consider the Gross-

Pitaevskii equation :

( , ) r t

iT

t

∂

where T is the kinetic energy operator, namely (

(

0

0

e

ψ

is a solution of the equation (7) with :

2

( , ) r t( , ) r t( , ) r t

ψ∂

ψα ψ

)

ψ

=+

(7)

2/ 2m

−Δ

for a parabolic dispersion.

A plane wave

)

i k rt

μ−

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2

000

()

k

μω α ψ=+

(8)

where

00

()k

ω

is the kinetic energy.

We then consider a small perturbation of the condensate wave function:

(

, r t

ψ

The non linear term of (1) reads:

2

( , )( , ) r tr t

ψψψ≈

which means that the non-linear term couples δψ and

Therefore, the solutions of the equation 7 can be written as

)

0

ψ= δψ+

.

22

2

0000

22*

ψψδψψ δψ++

*

δψ

.

()

()

()

0

0

2

()

*

0

1

ik k rt

i krt

i k rt

eAeB e

ω−

ω−

μ−

ψψ

=

−−

++

(9)

Inserting (9) into (7), neglecting high order terms yields the linearized system of equations

below :

(

(

)( )

k

(

)

()

()

)

2

2

000

2

*2

0000

2

22

AAB

BkkBA

μωωα ψαψ

μ ω−ωα ψαψ

+=++

=−++

(10)

The solutions of the system (10) are given by:

()

2

2

1212

012

22

ωα ψ

±

−Δ −ΔΔ −Δ

=±+Δ −Δ

(11)

where

(

(

)()

)

( )

k

10000

2000

2kk

ω

k

k

ω

ω

ωΔ =

Δ =

−

−

−

For a parabolic dispersion for

0

~kk

()

2

0

0

0

k

m

kk

m

α ψ

ω±

=±−

(12)

(

)

(

)

(

)

2

2

00000

2

22

22

000000

()(2)

()(2)

kkk

A

kkk

ωωωα ψ

ωωωα ψα ψ

±

±

±

−+−+

≡

−+−++

Page 15

(

)

(

)

(

)

2

2

0000

2

22

22

00000

() ( )k

() ( )

k

k

B

k

ωωω α ψ

−

ωωω α ψ

−

α ψ

±

±

±

+−

≡

+−+

(13)

The figure 5 shows the dispersions of elementary excitations of a polariton condensate

propagating with a wavevector

10 m-1. Panel (a) illustrates the Čerenkov regime, where the

elastic scattering towards an excited state with the same energy as the condensate is possible

because the renormalization of the dispersion due to the interactions is not strong enough. Panel

(b) corresponds to the superfluid case, where the dispersion is so strongly renormalized that the

elastic scattering towards any excited state is no more possible, and therefore, due to the Landau

criterion, the viscosity of the propagating polariton fluid drops to zero.

6

Page 16

Figure 6. Bare polariton dispersion (black dotted line) and the dispersions of elementary

excitations (black and blue/light grey solid lines) in the Čerenkov (a) and superfluid (b) regimes.

Elastic scattering towards excited states is indicated by a red arrow.

The solution (12) tells us that the excitations of the condensate are created by a kind of

parametric process symmetric with respect to the energy and the wave vector of the condensate.

However, when the final state is far from the initial dispersion (far means that the energy

difference becomes comparable or larger than μ ), the amplitude of the corresponding state

vanishes, which means that this state does not exist and that the excitation is just a plane wave

(free particle), essentially unperturbed by the presence of the condensate. The condensate itself is

expected to be stable against parametric processes if ω± are real values, which is always the case

for a parabolic dispersion in 1D.

OPO-like processes are very well known in the polariton community because of the non-

parabolicity of the polariton branch which makes them resonant. They have therefore been

strongly studied since the year 2000 and the pioneering works of Savvidis [56], Stevenson [57],

and the strong theoretical contribution of Ciuti [58, 59]. After the claiming by the Yamamoto’s

group of the observation of this linear Bogolubov dispersion [18], some doubts came from the

fact that they did not observe the negative part of the dispersion. Recently the Lausanne group

attacked this question, by trying to observe this “ghost branch” by heterodyne detection [60].

One should notice that off resonant emission coming from “ghost branches” has already been

evidenced [59] under resonant excitation. In fact, even if these new branches (eigen values)

come from OPO-like processes, such processes are not balanced. The figure 7 below shows the

expected emission spectra for the same parameters as for figure 6. The dispersion is indeed

perturbed, but the “ghost branch” is not easily visible.

The second comment is that superfluidity is fully linked with inter-particle interactions. A

non-interacting gas moving with any velocity in a weak random potential gets exponentially

localized because of Anderson localization. If interactions are considered, the same gas can show

a superfluid behavior which means that it does not even “feel” the presence of the disorder

during the propagation.

Page 17

Figure 7: Emission from the elementary excitations of a polariton condensate, calculated

using formula 11 and 13. The condensate has an in-plane wave vector 1.5 106 m-1. The

2

0

α ψ

is 0.2 meV. All states are assumed to emit light proportionally to the

coefficients calculated with the formula 13, including a linewidth of 0.1 meV. The “Ghost

Branch” can be guessed to the right of the main dispersion.

c) Phase diagram

In what follows, we consider a 1D condensate moving in a random disorder potential.

This condensate is characterized by a fixed energy, which means that we neglect the possible

occurrence of parametric instabilities, and the possible formation of topological defects, such as

solitons. This is a strong approximation. In [61] for example, such situation is considered

numerically and the condensate is found “unstable” in a wide range of parameters, which means

that the propagating flux cannot be characterized by a single frequency. Therefore, in what

follows, we assume that the only possible dissipative process for the propagating condensate are

backscattering processes, which could also be called “emission of Čerenkov waves”. The

qualitative picture we propose is summarized on the phase diagram shown on the figure 3. As

said in the previous section, at negligible interaction energy, bosons in a 1 D system are known

to be always localized in a random potential, whatever their kinetic energy. Two different

localization regimes can however be distinguished. If the kinetic energy

interaction energy

0

ω is much smaller than

the amplitude of the random potential

0

V , the condensate is classically localized in some

independent localised states, referred to as Liftshits states in [53]. The localisation length is

typically given by the correlation length Lc of the disorder potential. If

still localised but the physical mechanism is more subtle, relying on Anderson localization, as

00

V

ω >

, the bosons are

Page 18

discussed in the previous sections. A propagating plane wave is backscattered by the disorder

with some finite scattering time proportional to the density of states at the propagating energy

3

0

1

2

0

2

c

mL V

D

k

τ=

, where 0 k is the wave vector of the wave, m - the mass and

c L - the correlation

length of the disorder potential. The localization length is given by

0

1D

k

m

τ

and is no more

directly proportional to the coherence length of the disorder potential. The smooth crossover

between the two regimes takes place approximately when

00

V

ω =

.

This picture is dramatically modified by the interactions. Their first role is to bleach the

disorder potential, whose effective height is given by

2

00

V

α ψ−

. This reduces the kinetic

energy at which the transition between strong and Anderson localization takes place to

2

000

V

ω α ψ−=

. Second, the interactions provide a finite superfluid velocity

2

0

sv

m

α ψ

=

. A

condensate propagating with a velocity smaller than

sv shows no viscosity because of the

suppression of the elastic backscattering – the mechanism, which is also responsible for the

occurrence of the Anderson localization. However, if the system is strongly localized, it cannot

be called superfluid; because there is no motion of the fluid in this case. Therefore, superfluidity

can take place only if the kinetic energy of the condensate is large enough for the particles not to

be strongly localized, but small enough to have the condensate velocity below the speed of sound

in the media, which is expressed by the condition

22

0000

/2

V

α ψ−ω α ψ<<

. If

2

00

/2

ω α ψ>

, a superfluid enters the so-called Čerenkov regime. We show that in this regime,

the backscattered amplitude is enhanced with respect to the non-interacting case, providing a

reduced localization length.

An important remark regards the difference of this model with the one describing

disordered lattices. In the lattices, the coupling between regularly spaced localized states can

result in the formation of a conduction band allowing superfluidity. In the case of a continuous

random potential, the delocalization threshold corresponds to the percolation of the wave

function, and the formation of a finite conduction band below this threshold is not expected.

Quantitatively, one can calculate the localization length using the formula (6) in which

the renormalized density of states calculated from the dispersion (11) is used. This approach is

perturbative. The hypothesis made is that the inter-particle interaction is the main process

leading to a well defined renormalized dispersion for the elementary excitations of the

condensate, which is stable against the parametric processes. Second, the disorder is also a

perturbation, which can only provoke scattering processes within this dispersion. In fact, there is

a wide range of parameters [61], were both mechanisms should be taken into account

simultaneously, leading to the instability of the condensate, something which we neglect here.

Page 19

Figure 8: Phase diagram of condensed polaritons in the presence of disorder.

10-5 m0. Everything below the red line is localised. Above the red and blue: Delocalised or

Čerenkov. Above the red and below the blue: Superfluid.

For a parabolic dispersion (a valid approximation for polaritons with relatively small

wavevectors), one can obtain a simple analytical expression for the density of excited states at

the energy of the propagating condensate:

0

V = 1 meV, m = 5

()

10

2

0

0

1

()

D

DE E k

k

µ

km

==

−

if

2 2

0 k

m

µ

>

(14)

()

10

()0

D

DE E k

==

if

2 2

0 k

m

µ

<

(15)

Since the density of available states for the subsonic case is zero,

1D

γ

drops to zero as

well, and the condensate can propagate infinitely far without any scattering. For supersonic

waves 1D

γ

is given by :

()

22

2

4 2

k

2

0

2

100

4 2

k

0

2

,()

c

D

m L V

−

k µxmµ

mµ

γ=Θ−

(16)

where Θ is the Heaviside function

Page 20

The denominator describes the enhancement of the scattering by disorder in the Čerenkov

regime. One should note that

1D

γ

goes to infinity close to the transition to a superfluid state

which corresponds to a fully localized system. The hypothesis we made to derive the formula is

that the disorder is acting as a weak perturbation on the propagating wave. This hypothesis

breaks down close to the transition point between the superfluid and Čerenkov regimes.

In a system with a finite lifetime, a supplementary decay source should be considered and

the total decay rate can be written as:

()

(

)

22

22

0

2

10

4 2

k

0

0

2

,1

c

D

R

m L V

−

m

k

k µxx

mµ

γ

τ

=+−

(17)

Figure 9 shows a plot of

1

1D

γ− which is the propagation length for interaction energy of

1 meV. In the low energy range, the kinetic energy is at least twice smaller than the interaction

energy and polaritons are superfluid. Their propagation is only limited by their lifetime. When

the speed of sound is reached (kinetic energy is one half of the interaction energy), the density of

states available for elastic scattering processes passes from 0 to infinity and the condensate

becomes very strongly localized. First, we can expect that this abrupt transition can be smoothed

in a real system because, of finite linewidth, finite size effects and so on. Second, it is not clear at

all if our approximation (stability of the condensate against parametric processes) remains valid

in this case. The dashed line shows a plot of the decay in the non-interacting case (same as for

figure 5) for comparison. The extra localization induced by the renormalization of the dispersion

in the Čerenkov regime is clearly evidenced. It is interesting to notice, that for infinite life time

particles our approach gives at the transition point a localization length passing from infinity to

zero.

a)

b)

Figure 9. a) Propagation length of a 1D polariton condensate as a function of its energy in

the presence of disorder

c L = 1µm,

0

V =0.25 meV. Solid line:

2

0

α ψ

= 1 meV, dashed line, no

interaction (same case as for figure 5)

b) Density profile of a 1D polariton condensate propagating in the presence of disorder.

The transition from superfluid to Čerenkov regime is visible as a cusp.

Page 21

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 [28] and even more [62]

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 [65] for instance) can

take place.

References

[1] L.D. Landau, The theory of superfluidity of helium II, J. Phys. USSR 5, 71 (1941).

[2] N.N. Bogoliubov, On the theory of superfluidity, J. Phys. USSR, 11, 23 (1947); N.N.

Bogoliubov, Lectures on Quantum Statistics, Vol 1 Quantum Statistics, Gordon and Breach

Science Publisher, New York, London, Paris, (1970).

[3] P.W. Anderson, Phys. REv. 109, 1492, (1958).

[4] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S.Fisher, Phys. Rev. B 40, 546 (1989).

[5] M.N. Anderson et al., Science 269,198 (1995). ;K.B. Davis et al. Phys. Rev. Lett. 75, 3669

(1995).

[6] R.T. Scalettar, G.G. Batrouni, and G.T. Zimanyi, Phys. Rev. Lett. 66, 3144 (1991).

[7] L. Fontanesi, M. Wouters, and V. Savona Phys. Rev. A 81, 053603 (2010).

Page 22

[8] L. Fontanesi, M. Wouters, V. Savona, Phys. Rev. A 83, 033626 (2011).

[9] G. Malpuech, A. Kavokin, A. Di Carlo, and J. J. Baumberg, Phys. Rev. B 65, 153310 (2002).

[10] D. Porras, C. Ciuti, J. J. Baumberg, and C. Tejedor, Phys. Rev. B 66, 085304 (2002).

[11] J. Kasprzak et al., Nature 443, 409, (2006).

[12] J. Kazprzak, D.D. Solnyshkov, R. André, Le Si Dang, G. Malpuech, Phys. Rev. Lett. 101,

146404, (2008).

[13] Jacques Levrat, Raphaël Butté, Eric Feltin, Jean-François Carlin, Nicolas Grandjean,

Dmitry Solnyshkov, and Guillaume Malpuech, Phys. Rev. B 81, 125305 (2010).

[14] G. Malpuech, Y.G. Rubo, F. P. Laussy, P. Bigenwald and A. Kavokin, Semicond. Sci.&

Technol. 18, Special issue on microcavities, edited by J.J. Baumberg and L. Vina, S 395, (2003).

[15] Jonathan Keeling, Phys. Rev. B 74, 155325 (2006).

[16] G. Malpuech and D. Solnyshkov, arXiv:0911.0807 (2009).

[17] G. Malpuech, D. Solnyshkov, H. Ouerdane, M. Glazov, I. Shelykh, Phys. Rev. Lett. 98,

206402 (2007).

[18] S. Utsunomiya et al. Nature Physics, 4, 700, (2008).

[19] I. Carusotto and C. Ciuti, Phys. Rev. Lett. 93, 166401 (2004).

[20] D. D. Solnyshkov, I. A. Shelykh, N. A. Gippius, A. V. Kavokin, and G. Malpuech, Phys.

Rev. B 77, 045314 (2008).

[21] A. Amo et al., Nature, 457, 291, (2009).

[22] A. Amo et al., Nature Physics,5, 805, (2010).

[23] S. Pigeon et al. Phys. Rev. B, 83, 144513, (2011).

[24] G. A. El et al., Phys. Rev. Lett. 97, 180405 (2006).

[25] A. Amo et al., Science, 332, 1167, (2011).

[26] A. M. Kamchatnov et al., Phys. Rev. Lett., 100, 160402 (2008).

[27] H. Flayac, D. Solnyshkov, G. Malpuech, Phys. Rev. B 83, 193305, (2011).

[28] E. Wertz et al., Nature Physics 6, 860 (2010).

[29] R. Houdré, C. Weisbuch, R. P. Stanley, U. Oesterle, and M. Ilegems Phys. Rev. B 61,

R13333 (2000).

[30] M. Litinskaia, G.C. La Rocca, and V.M. Agranovich Phys. Rev. B 64, 165316 (2001).

[31] D. M. Whittaker, P. Kinsler, T. A. Fisher, M. S. Skolnick, A. Armitage, A. M. Afshar, M.

D. Sturge, and J. S. Roberts Phys. Rev. Lett. 77, 4792 (1996).

[32] For more details, see, for ex. chapter 3 of “Cavity Polaritons”, A. Kavokin, G. Malpuech,

Elevier (2003).

[33] A. Imamoglu, J.R. Ram, Phys. Lett. A 214, 193, (1996).

[34] E. Wertz et al. Appl. Phys. Lett. 95, 051108 (2009).

[35] Le Si Dang et al. Phys. Rev. Lett. 81 3920, (1998).

[36] I. A. Shelykh et al. Phys. Rev. Lett. , 97, 066402, (2006).

[37] L. Klopotowski et al. Solid State Com., 139, 511 (2006).

[38] M. Richard et al., Phys. Rev. Lett. 94 187401 (2005).

[39] M. Richard et al. Phys. Rev. B 72, 201301 (2005).

[40] J.J Baumberg et al., Phys. Rev. Lett. 101, 136409, (2008).

[41] K. G. Lagoudakis et al., Science, 326, 974 (2009).

[42] M. Wouters, Phys. Rev. B 77, 121302 (2008).

[43] F. Manni et al., Phys. Rev. Lett. 106, 176401 (2011).

[44] F. M. Marchetti, J. Keeling, M. H. Szymańska, and P. B. Littlewood, Phys. Rev. Lett. 96,

066405 (2006)

[45] M. Wouters and I. Carusotto Phys. Rev. Lett. 99, 140402 (2007).

[46] D. Sanvitto, A. Amo, L. Viña, R. André, D. Solnyshkov, and G. Malpuech, Phys. Rev.B 80,

045301, (2009).

[47] K. G. Lagoudakis, B. Pietka, M. Wouters, R. André, and B. Deveaud-Plédran Phys. Rev.

Lett. 105, 120403 (2010).

Page 23

[48] K. G. Lagoudakis et al., Nature Physics, 4, 706, (2008).

[49] Wiersma, D. S., Bartolini, P., Lagendijk, A. & Righini, R. Nature 390, 671–673 (1997); 4.

Scheffold, F., Lenke, R., Tweer, R. & Maret, G. Nature 398, 206–270 (1999).; Storzer, M.,

Gross, P., Aegerter, C. M. & Maret, G. Phys. Rev. Lett. 96, 063904 (2006); Schwartz, T., Bartal,

G., Fishman, S.&Segev,M.. Nature 446, 52–55 (2007). Lahini, Y. et al., Phys. Rev. Lett. 100,

013906 (2008).

[50] Dalichaouch, R., Armstrong, J. P., Schultz, S., Platzman, P. M. & McCall, S. L. Nature 354,

53–55 (1991); Chabanov, A. A., Stoytchev, M. & Genack, A. Z. Nature 404, 850–853 (2000).

[51] Weaver, R. L. WaveMotion 12, 129–142 (1990).

[52] J. Billy et al., Nature 453, 891, (2008); G. Roati et al., Nature 453, 895, (2008).

[53] L. Sanchez Palencia et al. Phys. Rev. Lett. 98, 210401, (2007).

[54] G. Madugno, Rep. Prog. Phys. 73 (2010) 102401 and ref. Therein

[55] I. M. Lifshits et al., Introduction to the Theory of Disordered Systems (Wiley, New York,

1988).

[56] P. G. Savvidis et al. Phys. Rev. Lett. 84, 1547, (2000).

[57] R. M. Stevenson et al., Phys. Rev. Lett. 85, 3680, (2000).

[58] C. Ciuti et al, Phys. Rev. B 62 R4825 (2000) ; C. Ciuti et al., Phys. Rev. B 63, 041303(R)

(2001).

[59] P.G. Savvidis et al., Phys. Rev. B 63, 041303(R) (2001).

[60] Verena Kohnle et al. Phys. Rev. Lett. 106, 255302, (2011).

[61] T. Paul, M. Albert, P. Schlagheck, P. Leboeuf, and N. Pavloff Phys. Rev. A 80, 033615

(2009); M. Albert, T. Paul, N. Pavloff, and P. Leboeuf Phys. Rev. A 82, 011602 (2010).

[62] J. Bloch, private communication.

[63] C.W. Lai et al, Nature 450, 529 (2007).

[64] E. A. Cerda-Mendez, D. N. Krizhanovskii, M. Wouters, R. Bradley, K. Biermann, K. Guda,

R. Hey, P. V. Santos, D. Sarkar, and M. S. Skolnick, Phys. Rev. Lett. 105 116402 (2010).

[65]. Flayac, D. Solnyskov, G. Malpuech, Phys. Rev. B. 83, 045412 (2011).