week ending
26 APRIL 2013

PHYSICAL REVIEW LETTERS

PRL 110, 170407 (2013)

Self-Localization of Polariton Condensates in Periodic Potentials
E. A. Ostrovskaya,1 J. Abdullaev,1 M. D. Fraser,2 A. S. Desyatnikov,1 and Yu. S. Kivshar1
1

Nonlinear Physics Centre, Research School of Physics and Engineering,
The Australian National University, Canberra ACT 0200, Australia
2
National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan(Received 10 October 2012; published 26 April 2013)
We predict the existence of novel spatially localized states of exciton-polariton Bose-Einsteincondensates in semiconductor microcavities with fabricated periodic in-plane potentials. Our theoryshows that, under the conditions of continuous nonresonant pumping, localization is observed for a wide
range of optical pump parameters due to effective potentials self-induced by the polariton flows in thespatially periodic system. We reveal that the self-localization of exciton-polaritons in the lattice may occur
both in the gaps and bands of the single-particle linear spectrum, and is dominated by the effects of gain
and dissipation rather than the structured potential, in sharp contrast to the conservative condensates ofultracold alkali atoms.
DOI: 10.1103/PhysRevLett.110.170407

PACS numbers: 05.30.Jp, 03.75.Kk, 03.75.Lm, 71.36.+cIntroduction.—Ultracold atomic gases loaded into periodic ‘‘lattice’’ potentials provide a fascinating tool for
exploring fundamental quantum many-body physics, aswell as offering an opportunity to build quantum simulatorsand scalable quantum information processing systems [1].
In recent years, polariton condensates created in semiconductor microcavities [2] have emerged as an attractive
alternative to the atomic Bose-Einstein condensates(BECs), owing to the relatively high condensation temperatures, direct momentum and real-space imaging via
the cavity photoluminescence, and possibility to manipulate condensates using both optical pump and structure of
the microcavity [3]. Considerable effort has been directed
towards achieving the polariton condensation in latticepotentials and observing macroscopic population of thesingle-particle band-gap spectrum. So far, the majority ofthe experiments with exciton-polaritons in periodic potentials have been performed in a pulsed excitation regime,
which has allowed us to populate and image rapidly decaying high momentum states. The staggered ‘‘-state’’higher-order Bloch states have been demonstrated in a
one-dimensional (1D) lattice created by metal deposition
onto the microcavity [4]. Similarly, condensation into thesecond band has been achieved in a two-dimensional (2D)square lattice [5], and the Brillouin zone structure of the
kagome lattice has been mapped out [6]. Recently, condensation into an energy state corresponding to a spectral
gap has been demonstrated in a 1D lattice created bymodulation of quantum wires in a microcavity [7].
Condensation in tunable acoustically created lattices has
also been investigated [8,9].
One of the fundamental effects due to the band-gap
structure is localization of the condensate in real space.
In kagome lattices [6] this could be achieved via dispersionless dynamics in the flat spectral band. Alternatively,
spatial localization in spectral gaps of the band-gap
0031-9007=13=110(17)=170407(5)structure can be achieved through the balance of repulsive
nonlinearity and modified effective dispersion due to thenegative effective mass. This mechanism has been exploredin the context of gap solitons in photonic [10] and matterwave [11] band-gap structures. Scarce theoretical descriptions of exciton-polaritons in periodic potentials so far
either attributed localization phenomena to intrinsic structural disorder [12] or disregarded the driven-dissipative
nature of the polariton condensate arising due to finite
lifetime of quasiparticles and the presence of the opticalpump [13,14]. This distinct nature of polariton condensates
becomes especially important in the continuous-wave (cw)nonresonant excitation regime [2], which creates condensates with spontaneously established spatial coherence
and long coherence times. The gain-assisted localization
and related dynamics plays a critical role for such condensates [15,16], in contrast to atomic BECs in thermal
equilibrium.
In this Letter, we identify and analyze the mechanism
for localization of a dissipative polariton condensate in afabricated periodic potential under incoherent cw excitation. We show that the driven-dissipative nature of thepolariton condensate gives rise to effective potentials
local to the condensate, namely a defect-type potential
introduced by the pump-induced polariton reservoir and apotential self-induced by the strongly modulated flow ofpolaritons away from the excitation spot. Trapping of thepolariton condensate in these effective potentials, which
supersede the external lattice potential, is responsible forits spatial localization.
Model.—We assume that a near-equilibrium polariton
condensate is formed via cw incoherent excitation with aspatially inhomogeneous optical pump. Above condensation threshold, it can be described by the mean field opendissipative model for a polariton condensate coupled to an
incoherent polariton reservoir density [17],170407-1

Ó 2013 American Physical Society

PRL 110, 170407 (2013)



r2
@
i
¼  ? þ VNL þ ðRnR  c Þ þ VL ;
@t
2
2
@nR
¼ ðR þ Rjj2 ÞnR ðr~; tÞ þ Pð~rÞ:@tiweek ending
26 APRIL 2013

PHYSICAL REVIEW LETTERS(1)

Here  is the condensate wave function, VL ð~rÞ is a periodic
potential experienced by the polaritons, and VNL ¼
gc jj2 þ gR nR is an effective nonlinear potential depending both on the condensate density and the reservoir density, nR , which is determined by the pump Pð~rÞ. Parameters
defining the condensate dynamics are the relaxation ratesof the condensed polaritons c and reservoir polaritons R ,the stimulated scattering rate R, and the polariton-polariton
interaction strengths gR , gc . Equation (1) is written in thedimensionless form by using the characteristic scaling
units of time 1=c , energy @c , and length ½@=ðmLP c Þ1=2 ,where mLP is the lower-polariton effective mass [18].
Under this scaling c ¼ 1; however, we formally retain
this parameter hereafter. The scaling units are calculated using the parameters close to those of the experimental setup of [19], with mLP ¼ 103 me , gc ¼ 6 
103 meV m2 , gR ¼ 2gc , c ¼ 0:02 ps1 , R ¼ 1:5c ,and R ¼ 0:01 m2 ps1 .
1D localization.—First, we consider the simplest modelof a quasi-one-dimensional polariton condensate (e.g.,
confined in a microwire [20]) subject to a periodic potential
approximated by a harmonic function VL ðxÞ¼V0 sin2 ðkl xÞ,
with the lattice constant a ¼ =kl . Periodicity of thepotential imposes a band-gap structure onto the lower
polariton (LP) dispersion branch [4], and we can consider
population of this band-gap structure by polaritons depending on the pump parameters. To this end, we look for thesteady state solutions ðxÞ ¼ c ðxÞ expðiEtÞ of Eq. (1)governed by the complex Ginzburg-Landau equation. We
assume the Gaussian pump profile PðxÞ ¼ P0 expð2 x2 Þ
and relate its peak intensity to the homogeneous thresholdintensity Pth ¼ R c =R [17].
As has been established both experimentally (see, e.g.,[15,20]) and theoretically [16,18,21], nonresonant cw
pump leads to spatial localization of polariton BEC even
without a trapping potential. This effect, resulting in formation of dissipative solitons [18,22] is due to the balanceof gain and loss in the system, and is dramatically different
from formation of bright solitons in the coherent fields
due to conventional balance of nonlinearity and dispersion(diffraction). Below, we demonstrate how spatial localization of the condensate due to the balance between gain andloss is affected by a periodic potential.
Narrow excitation.—A spatially narrow pump PðxÞ cantarget a single lattice site and therefore produce an on-siteor off-site excitation. The peak of the off-site excitation
spot can be arbitrarily offset from the minimum of thelattice potential. As can be seen from Eq. (1), in the caseof the near-threshold pump intensity, P0 =Pth  1, nonlinear effects can be neglected. In this case, assuming astraightforward balance of gain and loss, Eq. (1) would
describe a condensate in the periodic potential VL ðxÞwith the addition of a linear defect potential induced by
the pump, gR n0R  ðgR =R ÞPðxÞ, where n0R ¼ P=ðR þ
Rj c j2 Þ is the steady state reservoir density. The energyof the lowest-order defect modes is plotted as a functionof pump intensity in Fig. 1, together with the band-gap
spectrum imposed by the periodic potential.
Taking this spectrum as a frame of reference, we
examine the energy and spatial structure of the localized
steady state depending on the pump intensity. The respective dependence is plotted in Fig. 1 and shows that, fornear-threshold on-site excitation, the energy of the localized state is indeed very close to that of a symmetriclinear defect mode. As the ratio P0 =Pth grows, so does theenergy of the localized state until it crosses into the bandof the linear band-gap spectrum. This is in contrast to
dissipative photonic lattice solitons [23] and defect states[24], which have been previously found to exist exclusively in the spectral gaps. An additional feature of thepolariton steady states is transition, with growing pumpintensity, from narrow states localized in a vicinity of asingle potential well [Fig. 2(a)] to broader states withexponentially decaying and spatially modulated tails[Figs. 2(b) and 2(c)].
To understand the spatial structure of the localizedstates, we perform the Madelung transformation c ðxÞ ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃðxÞ exp½iðxÞ, and write down the steady statepﬃﬃﬃﬃ
equations for the condensate density  ¼  and phasegradient (flow velocity) v ¼ x :

7

(c)
6

(d)

(b)
5

E 4

(a)

3

2

1

1

2

3

4

5

P0/PthFIG. 1 (color online). Energy of the steady-state polariton
condensate obtained with cw on-site (closed circles) or =4
off-site (open circles) narrow Gaussian pump with the width
2 ¼ 0:6. Shaded (open) areas show bands (gaps) of thesingle-particle Bloch-wave spectrum in the lattice potential
VL ðxÞðV0 ¼ 5:0; kl ¼ 1:5Þ. Dashed lines are linear defectstates in the combined effective potential VL ðxÞ þ gR PðxÞ=R(see text).

170407-2

5

−5

−5

−10
−10

−5

0

x

5

5

10

−10
−20

−10

0

x

10

0

−5

−5

−10
−20

−10

0

x

10

−10
20 −20

8

8

4

4
−8

−4

20

(d)

0

12

16

0

4

8

10

20

(d)

−π/a

0



v2
 þ 2E ¼ 0;
xx  2 VL þ gc 2 þ gR n0R þ2ð2 vÞx  ðRn0R  c Þ2 ¼ 0:

(2)Given the rapid decay of the pump at large x, asymptotic
behavior of the condensate near-linear density away fromthe pump spot is determined by the combination of thelattice potential VL ðxÞ and a self-induced effective potential due to the phase gradient, v2 ðxÞ=2. These two potentials for the narrow and wide states (a) and (b,c) in
Figs. 1 and 2 are shown in Figs. 3(a)–3(c). The strong
contribution of the Bloch states from the edges of theBrillouin zone into the composition of the localized state[Fig. 3(d)] leads to steep spatial gradients of the phase,ðxÞ. As a result, the flow of the condensate out of theexcitation spot induces a periodic, tight-binding KronigPenney-like potential [Fig. 3(a)], and the band-gap spectrum due to the linear lattice potential becomes of little
relevance to the energy eigenstates. The minima of theeffective potential are offset from the minima of the latticepotential, which can cause the condensate density peaks
to coincide with the maxima of the lattice potential [seeFig. 2(a)]. The tunnelling of the condensate out of theexcitation spot is strongly inhibited by the tight-binding
effective potential, until the modulations of the flow aresmoothed out by the stronger pump. As demonstrated in
Figs. 2(b) and 2(c), the strongly pumped condensate density extends over many wells of the periodic potential. In
contrast to the lattice potential, the self-induced potential

0

x

x

FIG. 2 (color online). Spatial profiles of the condensate densitylnððxÞÞ and pump intensity lnðPðxÞÞ in the lattice potential
VL ðxÞ with V0 ¼ 5:0 and kl ¼ 1:5, corresponding to the peak
pump intensity P0 =Pth ¼ 1:27 [point (a) in Fig. 1], P0 =Pth ¼
2:53 [point (b)], P0 =Pth ¼ 5:06 [point (c)], and P0 =Pth ¼ 4:30
[point (d)]. Panel (a) also shows the lattice potential lnðVL Þ (notethe difference in x-scale).

−4

0

(c)

8

−8

−4

x

12

0

0

−8

x

4

−10

0

4

8

v2/2
VL

(b)

16

12

0

20

5

(c)

ln(P), ln(ρ)

VL , v 2/2

0

(a)

16

VL , v 2/2

ln(P), ln(ρ)

0

20

20

ln(P)
ln(ρ)

(b)

ln(VL)

intensity (arb. units)

5(a)week ending
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PHYSICAL REVIEW LETTERS

PRL 110, 170407 (2013)

k

4

8

π/a

FIG. 3 (color online). Spatial profiles of the lattice potential
VL ðxÞðV0 ¼ 5:0; kl ¼ 1:5Þ and the effective linear potential dueto the phase gradient v2 =2 corresponding to the peak on-sitepump intensities (a) P0 =Pth ¼ 1:27, (b) P0 =Pth ¼ 5:06, and (c) anoncentered pump (=4 off site) with P0 =Pth ¼ 4:30. Panel(d) shows the k-space spectrum of the steady state [Fig. 2(a)]corresponding to the effective potential in (a).

can become physically meaningless far on the tails of thecondensate [shaded areas in Fig. 3(a)], where the density
is negligible and any dynamical fluctuations can destroy
regular phase structure.
Although the narrow gap states [Fig. 2(a)] resemble well
localized gap solitons of a conservative BEC [11], they
have no connection to the Bloch states at the band edgeof the first Brillouin zone in the lattice potential, rather
branching off the linear defect states. Their spatial extent isdetermined purely by the phase gradients established dueto the balance of gain and loss in the system. The energy ofthe localized state can lie in either a spectral gap or band
imposed by the lattice [25] since it is supported by theeffective potentials created by the polariton flow and therole of the band-gap structure is diminished. Localizationof in-band, as well as gap states, is strongly affected by thelattice potential modulating the flow, as illustrated in the
Supplemental Material [25].
In the off-site excitation regime, the localized states canalso be formed both within gaps and bands of the linear
band-gap spectrum of the potential VL ðxÞ [Fig. 1], and thethreshold for the formation of these states is higher thanthat of the on-site state. Remarkably, the asymmetry ofthe defect created by the offset pump translates to theasymmetry of both the effective potential generated by
the polariton flow and the condensate density [Figs. 2(d)and 3(c)]. Such asymmetric localized states are supportedby the interface between two different effective potentials,
and therefore resemble ‘‘mixed-gap’’ interface states of an
atomic BEC in anharmonic lattices [26].

170407-3

20

P
ρ

(a)

12

P, ρ, VL

VL

y 0

8

VL

3

P
ρ
VL

50

20

φ

6

60

40

(b)

16

VL, v 2/2

P, ρ, VL , φ

9 (a)week ending
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PHYSICAL REVIEW LETTERS

PRL 110, 170407 (2013)

40

(b)

30
20

4

−20
10

−20

−10

0

10

0

20

−20

−10

10

(c)

P
ρ

P, ρ, φ

8
6

φ

V0

4
2
0
−60 −40 −20

10

−40
-40
40

20

x

intensity (arb. units)

x

0

0

20

40

60

1.0

0

20

0
0

40

π

(c)

(d)

10

20

30

40

50

60

3

(d)

2

20

0.8

1

0.6

y

0

0

0.4

0

−π/a

−20

0

−40
-40

π/a

FIG. 4 (color online). (a),(c) Condensate density ðxÞ, phaseðxÞ, and lattice potential VL ðxÞðV0 ¼ 5:0; kl ¼ 1:5Þ corresponding to the peak on-site pump intensity P0 =Pth ¼ 1:27
and widths 2 ¼ 6  104 and 2 ¼ 6  105 , respectively.
Dashed curve in (c) is the analytical approximation of ðxÞ
[Eq. (3)]. (b) Lattice potential VL and the self-induced potential,
v2 =2, corresponding to (a). (d) Spectrum of the state (c) within
the first Brillouin zone of the lattice potential.

Wide excitation.—By fixing the pump intensity andextending its width significantly beyond the single latticewell, we find a variety of spatially extended nearly energydegenerate localized states. One example of such a gap
state and its supporting effective potential is presented in
Figs. 4(a) and 4(b). These states strongly resemble truncated nonlinear Bloch states in a conservative BEC [27].
However, the localization in this case is due to the effective
potential created by the polariton flow out of the finite gain
region.
A very wide, locally homogeneous, near-threshold pumpgenerates extended states that locally resemble a ground
Bloch state [Fig. 4(c)] of the lattice potential VL ðxÞ. In thisregime, the asymptotic approximation of the phase profile
at x ! 0, derived in [18] for the case of untrapped polariton
condensate becomes valid for most of the excitation region[Fig. 4(c), dashed line], and the effective potential due tothe polariton flow becomes harmonic,(3)

The density profile is a typical ground state supported by
the combination of the periodic and harmonic effective
potential [Fig. 4(c)], and its k-space spectrum [Fig. 4(d)]displays localization around k ¼ 0 characteristic of theground state in the lattice [4]. Thus, in this regime
the dissipative condensate in a periodic potential mimics
the behavior of a conservative condensate in the combined
periodic and harmonic potentials.

−2
-20

0

x

k


2
2c P0v ðxÞ  1 x2 :
4 Pth

0

−1

0.2

x

2

-20

φ

0

20

40

-π

−3
0

10

20

30

40

50

60

x

FIG. 5 (color online). Spatial profiles and cross-sections ofthe (a),(b) condensate density and (c),(d) phase, corresponding
to a narrow in-gap state in a ‘‘square’’ lattice VL ðx; yÞðV0 ¼
2:0; kl ¼ =4Þ.

2D localization.—Similarly to the 1D case, we find that
excitation spots of varying width can create narrow andwide localized steady states of the polariton condensate
both in the gaps and bands of the single-particle band-gap
spectrum corresponding to a 2D ‘‘square’’ lattice potential
VL ¼ V0 ½sin2 ðkl xÞ þ sin2 ðkl yÞ. Examples of a narrowstate density and its phase are shown in Fig. 5. Although
the detailed description of such states is beyond the scopeof this work, we note that the phase structure displays sharpdomain walls associated with strong phase gradients, similar to those in 1D case, which leads to the tight spatial
localization of the condensate.
Conclusions.—We have demonstrated that, under theconditions of spatially inhomogeneous nonresonant incoherent cw excitation, a polariton condensate can bespatially localized in a periodic in-plane potential owing
to the polariton flows established due to the balance
between gain and loss. Periodic potential strongly modulates the velocity of the polariton flow out of the pumping region, and the polaritons become self-trapped in the
resulting effective potential. This novel mechanism of
localization ensures that, depending on the pump intensity, condensate can be formed and localized both in gaps
and bands of the single-particle band-gap spectrum of theperiodic potential. Localization of the dissipative condensate is therefore irrelevant to the requirement of thenegative effective mass which is essential for the gap
soliton formation in conservative (atomic) BECs withrepulsive interactions [11].
The predicted features of the real-space localization ofa dissipative polariton condensate subject to a periodic
potential are expected to have strong consequences forany polariton devices incorporating modulated potentials,
and should be investigated further.

170407-4

PRL 110, 170407 (2013)

PHYSICAL REVIEW LETTERS

This work was supported by the Australian ResearchCouncil. Discussions with D. Tanese, G. Malpuech, S.
Höfling, and A. Kavokin are gratefully acknowledged.

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