PHYSICAL REVIEW A 86, 013636 (2012)

Dissipative solitons and vortices in polariton Bose-Einstein condensates
Elena A. Ostrovskaya,1 Jasur Abdullaev,1 Anton S. Desyatnikov,1 Michael D. Fraser,2 and Yuri 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, Tokyo, Japan(Received 7 May 2012; published 23 July 2012)
We examine spatial localization and dynamical stability of Bose-Einstein condensates of exciton polaritons
in microcavities under the condition of off-resonant spatially inhomogeneous optical pumping both with andwithout a harmonic trapping potential. We employ the open-dissipative Gross-Pitaevskii model for describing
an incoherently pumped polariton condensate coupled to an exciton reservoir, and reveal that spatial localizationof the steady-state condensate occurs due to the effective self-trapping created by the polariton ﬂows supportedby the spatially inhomogeneous pump, regardless of the presence of the external potential. A ground state of thepolariton condensate with repulsive interactions between the quasiparticles represents a dynamically stable bright
dissipative soliton. We also investigate the conditions for sustaining spatially localized structures with nonzero
angular momentum in the form of single-charge vortices.
DOI: 10.1103/PhysRevA.86.013636

PACS number(s): 03.75.Kk, 03.75.Lm, 71.35.Lk, 71.36.+c

I. INTRODUCTION

Since the ﬁrst observation of spontaneous exciton-polariton
condensation in a microcavity [1], signiﬁcant effort in thisvigorous ﬁeld has been directed towards understanding theproperties of superﬂuid ﬂow in the novel nonequilibriumpolariton superﬂuid [2–5]. Recently, considerable attention
was drawn to the controlled creation of spatially localized
collective excitations of the exciton-polariton Bose-Einsteincondensate (BEC). Most notably, moving bright solitons havebeen predicted [6] and observed in a pioneering proof-ofprinciple experiment [8]. The properties of polariton solitons,
superior to those of optical solitons in semiconductor cavity
lasers, namely, the picosecond response times and large
nonlinearities, suggest that polariton BEC can offer novel
functionalities for information-processing devices [7]. The
bright solitons observed in [8] require a resonant excitation
regime at nonzero in-plane momentum to make use of thedispersion properties of the lower-polariton branch that allow
the polaritons to have a negative effective mass. In this regime,strong repulsive interactions between the quasiparticles can bebalanced by the effective dispersion to achieve localization.
A similar mechanism for spatial localization via effective
mass management is well explored for atomic BECs in
optically induced band-gap structures [9]. In the absenceof the spectral gap, however, localization by means ofeffective mass management is only possible in one spatial
dimension. The question whether localization of polariton
BEC with repulsive interactions can occur in the regimeof spontaneous condensation at zero in-plane momentum(positive effective mass) and nonresonant excitation, remains
open.
In this paper we examine, both analytically and numerically,
formation and dynamical stability of a steady-state BEC
of exciton polaritons in microcavities in the presence of a
spatially localized (Gaussian) optical pump. By employing
the open-dissipative Gross-Pitaevskii model [10] describing
an incoherently pumped BEC coupled to the exciton reservoir
and successfully used in theoretical description of a numberof signiﬁcant experiments (see, e.g., [2,4]), we analyze the1050-2947/2012/86(1)/013636(7)mechanisms for creating and sustaining two-dimensional
spatially localized structures, such as dissipative solitons andvortices. In addition to the trap-free case, we analyze thestructure of the localized states with the addition of a harmonic
external potential that can be created, e.g., by engineered stressof the microcavity [11]. In the latter case, the localizationof the repulsive polariton BEC is due to both the harmonic
conﬁnement and the inhomogeneous pump, in contrast to thecase of a trapped atomic BEC with a positive scattering length
in thermal equilibrium.
We show, analytically and numerically, that, even in theabsence of external potentials localization of the steadystate polariton BEC occurs due to the effective self-trapping
created by the polariton ﬂows due to spatially inhomogeneous
off-resonant optical pumping. Spatial localization of thecondensate occurs due to the internal balance of superﬂuid
density ﬂows which is identical to that responsible forsupporting dissipative “antisolitons” in optical systems. Such
localization of continuously self-defocusing solitons occurs
despite the repulsive interactions between the quasiparticles,
and is analogous to optical gain-guiding effect [12,13] inits reliance on continuous pumping. Thus a ground stateof the polariton BEC represents a dissipative soliton whichis spatially localized and dynamically stable, regardless ofthe presence of the external potential. A harmonic trapping
potential dramatically modiﬁes the structure of the steady state
due to the competition with the spatially inhomogeneous pump(see Fig. 1).
A single vortex created in a localized steady-state BEC
by phase imprinting is similarly supported by the continuous
inhomogeneous pump as a spatially localized dissipative
vortex soliton. Akin to a vortex line in a trapped atomic BEC
with a positive scattering length, it is dynamically unstable[14]. In the absence of potential, a vortex line spirals out ofthe polariton condensate and the condensate restores to its
ground steady state. We show that the addition of fabricated(i.e., nonrotating) harmonic potential modiﬁes the threshold of
the optical pumping required to sustain a steady state with anangular momentum and leads to the possibility of a long-term
survival of the vortex.

013636-1©2012 American Physical Society

ELENA A. OSTROVSKAYA et al.
150

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PHYSICAL REVIEW A 86, 013636 (2012)
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100
0
10

0
100

0

y

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0
-10

-10

y

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-10

x

-10

FIG. 1. (Color online) Typical steady-state density of a harmonically trapped polariton condensate in the regime of (a) narrow and(b) wide pump (see text).
II. MODEL

We consider a spontaneously formed exciton-polariton
BEC (k|| = 0, lower-polariton branch) under the continuouswave nonresonant excitation. The model ﬁrst suggested in [10]consists of a mean-ﬁeld equation for the polariton condensate
wave function and a rate equation for the inhomogeneousdensity of the exciton reservoir. We write it here in the form
used in [4]:


h̄2 2
∂
h̄
= −
∇⊥ + V (r ,t) + i (RnR − γc ) ,
∂t
2m
2
(1)
∂nR
= −(γR + R||2 )nR (r ,t) + P (r ),∂tih̄

where V (r ,t) = U (r ) + gc ||2 + gR nR (r ,t). Here  is thecondensate wave function, nR is the exciton reservoir density,
P (r ) is the inhomogeneous optical pump, and U (r ) is the external potential. The critical parameters deﬁning the condensate
dynamics are the loss rates of the polaritons γc and reservoir
excitons γR , the stimulated scattering rate R, condensate
coupling to the reservoir gR , and the coefﬁcient gc quantifying
the nonlinear interaction of polaritons. In what follows, wewill consider a radially symmetric external trapping potential
U (r) = U02 r 2 , where U0 = 0 in a trap-free case.
The model can be rewritten in dimensionless form by using
the following characteristic scaling units of time, energy, andlength:
T = 1/γc , E = h̄γc ,L=h̄
mLP γc,(2)where mLP is the lower-polariton effective mass. Here, wechoose the parameters close to those of the experimental setupof [4], with mLP = 10−4 me , gc = 6 × 10−3 meV μm2 , gR =2gc , γc = 0.33 ps−1 , γr = 1.5γc , and R = 0.01 μm2 ps−1 . In
what follows we use and plot the dimensionless (normalized)
variables and parameters.
We recall that, in the spatially homogeneous case, one canestimate the threshold pumping power at which the condensate
appears in the system, as described in [10]. This threshold
is approximately determined by the values of parameters forwhich stimulated gain equals loss:
Pth ≈

γR γc
.
R

(3)

The ratio of the polariton nc to exciton reservoir nR densities
in the steady homogeneous state can then be deﬁned asncγc=(P̄ − 1),(4)nRγRwhere normalized pump is deﬁned as P̄ = P /Pth  1. In
general, even in a spatially inhomogeneous case, the typicaldynamical behavior of the model equations (1) displays rapidtransition to the steady state of the polariton BEC for P̄ > 1.
Below we consider the properties of the steady state
supported by an inhomogeneous (Gaussian) optical pump,
P (r) = P0 exp(−r 2 /σ 2 ), with or without an additional trapping potential. The spot size of the optical pump is considered
to be small compared to the size of the sample, so that the local
density approximation [15] is not applicable.
III. STEADY STATE OF THE CONDENSATE
A. Theory

For a general continuous-wave pump, the condensate wave
function in the steady state can be looked for in the following
general form:  = ψ(x,y) exp(−iμt), where μ is the energy(chemical potential) of the steady state. If both the externalpotential and the pump are radially symmetric, we can rewrite
the model system in polar coordinates (x,y) → (r,θ ) andreduce the problem to ﬁnding radially symmetric steadystates ψ(x,y) = ψ(r) exp(imθ ), where m is the phase winding
number [16] (topological charge of a vortex), and the groundstate corresponds to m = 0.
The steady-state wave function ψ(r) obeys the following
stationary equation:∇r2 ψ − 2V (r)ψ − i Rn0R − γc ψ + 2μψ = 0,(5)where ∇r2 = ∂ 2 /∂r 2 + (1/r)∂/∂r − m2 /r 2 , V (r) = gc |ψ|2 +gR n0R + U (r), and the steady-state reservoir density is foundfrom Eqs. (1) as
n0R =

P (r)
.
γR + R|ψ|2(6)

All physical parameters in Eq. (5) are dimensionless, and under
the adopted scaling (2), γc = 1. However, we formally retain
this parameter in subsequent formulas.
The stationary wave function can be further separated
into the real amplitude and phase: ψ(r) = (r) exp[iφ(r)].
Following the analysis suggested in [17] for the generalized
complex Ginzburg-Landau equation, we can obtain asymptotic
behavior of amplitude and phase of the condensate wave
function. In the absence of a trapping potential, U (r) = 0, therequirement of spatial localization, P (r) → 0, |ψ(r)| → 0,
imposed on both the pump and the condensate wave function
at large r leads to the following asymptotic behavior of theamplitude and the phase at r → ∞:(r) ∼ A exp(−p− r), φ(r) ∼ B + p+ r,(7)where p± = [(μstants. This asymptotic behavior shows that the condensate
decays slower than the pump ﬁeld, and therefore the spatial
proﬁle of the condensate will signiﬁcantly differ from that ofthe pump.

013636-22+ γc2 /4)1/2 ± μ]1/2 , and A,B are real con-

DISSIPATIVE SOLITONS AND VORTICES IN . . .

PHYSICAL REVIEW A 86, 013636 (2012)

Conversely, at r → 0, the asymptotic behavior is found tobe(r) ∼ Cr |m| exp(−q− r 2 ), φ(r) ∼ q+ r 2 ,

(8)

−1

where q− = (1/2)(1 + |m|) [V (0) − μ] and q+ = (1/4)(1 +|m|)−1 [Rn0R (0) − γc ], and for the ground state C = (0) ≡0 . For V (0) < μ, the second derivative of the amplitudechanges sign at r = 0, i.e., the condensate displays a central“dip” in its density proﬁle and acquires a ring shape, as seenin Fig. 1(a).
The presence of the harmonic trap U (r) = U02 r 2 dramatically modiﬁes the asymptotic behavior of the condensate wave
function for r → ∞ [18], so that(r) ∼ A exp −U02 r 2 + (μ − 1/2)ln(2U0 r) ,(9)φ(r) ∼ B + γc ln(2U0 r).
The asymptotic behavior of a trapped polariton BEC at r → 0
is described by Eq. (8).
For a spatially localized (e.g., Gaussian) pump, P (r),the spatial localization of the condensate in the absence ofan added trapping potential is counterintuitive. Indeed, theeffective potential V (r) formed by the repulsive nonlinearity(due to polariton scattering) and interaction with the exciton
reservoir is antitrapping. Therefore the balance of nonlinearityand dispersion responsible for nonlinear localization in conservative condensate systems cannot be achieved. Physically,
the existence of the spatially localized steady state of thepolariton condensate can be understood by examining theinternal ﬂows of the dissipative polariton superﬂuid, analogous
to the Pointing vector ﬂows in dissipative optical systems[17,19]. Indeed, by deﬁning the superﬂuid current density(ﬂux) in the standard way j = Im( ∗ ∇), we can recast
Eq. (5) in the form of coupled equations for the amplitudeof the condensate wave function  and radial component of
the stationary ﬂux J = jr = nc vr , where nc = 2 (r) is thecondensate density, and vr is the radial component of the ﬂow
velocity v = (dφ/dr)er + (m/r)eθ . The equations for thesevariables take the following form:
1 d(rJ ) − Rn0R − γc 2 = 0,
r dr
(10)
2
∇r  − 2[V (r) − μ] − VJ  = 0,
where VJ = J 2 −4 = (dφ/dr)2 . The ﬁrst equation of thesystem is the continuity equation for the stationary ﬂow withsource and sink. The ﬁrst term in this equation is simply thedivergence of the ﬂux, D = ∇ · j, which serves as a local
measure of gain (D > 0) or loss (D < 0). A steady state
exists if generation of the polariton superﬂuid via continuous
pumping is balanced by its dissipation. This condition can beformulated in terms of the ﬂux as follows [19]:∞D(r)r dr = 0.

(11)similar dynamical one-dimensional system with linear loss
and spatially localized gain and without additional externalpotential were found in [13]. The key feature of our system issimilar: for the given parameters of gain and loss, the chemical
potential, μ, is unique.
The second equation in (10) highlights the compositionof the effective potential supporting the condensate wave
function as its eigenstate with the corresponding eigenvalue
μ. It is clear that the radial ﬂux that exists due to the spatiallyinhomogeneous phase of the condensate φ(r) = const, forms
an attractive potential that can trap a spatially localized boundstate even in the absence of the external trapping U0 = 0.
In the linear limit (i.e., for very small condensates and/or
nonlinearity) this effective trapping potential due to the ﬂux isapproximately given by
VJ (r) ≈(12)

The effect of existence of the localized linear modes supported
purely by the localized gain is known as “gain guiding” inoptics [12].
B. Numerical results

The existence of the localized steady state of the dissipative
polariton BEC with the properties described in the previous
section can be demonstrated by the numerical simulation of
the time-dependent model equations (1). Figure 2 shows the
typical dynamical behavior and transition to the steady groundstate (m = 0) of the polariton component for intermediate
values of pumping power (P̄0 = 3) in the trap-free regime.
In addition, from the numerical simulations we can extract
and plot the following dynamical quantity:
μ(t) = −1 S [∇ 2 ||2 − 2|∇|2 − 4V ||2 ]d r
,
2 
4S || d r

(13)

where S is the numerical integration domain. In the steadystate limit this quantity coincides with the chemical potentialof the condensate μ(t) → μ. As can be seen from the
dynamical simulations for P̄ > 1 (Fig. 2), the steady-state
regime is reached quickly, and the chemical potential is welldeﬁned. Alternatively, the steady state is found by solving
the stationary equation (5) by the standard one-dimensional
relaxation method. The condensate wave function displays
characteristic inhomogeneous amplitude and phase proﬁleφ(r) (seen in Fig. 3) with the limiting behavior well describedby Eqs. (7) and (8).
40

5
4

30

Φ0 20

0

Remarkably, this condition can be satisﬁed regardless of thesign of the nonlinear interactions in the systems, i.e., forboth repulsive (as in the case of polaritons) and attractive
nonlinearities. The nonlinearity affects the energy (chemical
potential) of the steady state and its spatial extent according toEqs. (7) and (9). We note that nonlinear eigenstates for a very

γc2
(P̄0 − 1)2 r 2 .
4

μ

10

(a)
0
0

3
2
1

(b)

0
20

40

t

60

0

20

40

60

t

FIG. 2. Typical time evolution of (a) the condensate peak densityand (b) the chemical potential for P¯0 = 3, without a trapping potential.

013636-3

ELENA A. OSTROVSKAYA et al.
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PHYSICAL REVIEW A 86, 013636 (2012)
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(a )

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(b )

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(b)

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–

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0.1

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µ

(a)

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(c )

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J

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J

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(d )

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D
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D

0

-3

-5
0

10

20

r

30

40

-6

2
0
0

5

10

2

4

15

r

FIG. 3. (Color online) Radial proﬁles of the condensate groundstate in (a) trap-free and (b) harmonically trapped case. Bothamplitude, (r), and phase, φ(r), are shown in comparison to theGaussian pump proﬁle P (r)/Pth with the amplitude P̄0 = P (0)/Pth =6. Proﬁles of the radial ﬂux J and its divergence D (scaled up by thefactor of 10) for the ground state are shown for the (c) trap-free and(d) trapped cases, respectively. The harmonic potential of the depth
U02 = 0.1 is shown in (b) by the gray curve.

From Fig. 3(c) it is seen that the inhomogeneous phase of
the condensate wave function results in the nonzero ﬂux of thepolariton superﬂuid. Due to the spatially localized pump, thepolariton superﬂuid is continuously generated in the core areaof the steady state D > 0 and dissipated on its wings D < 0.
Thus the ground state of the strongly repulsive polariton
BEC is a spatially localized dissipative soliton. Its spatial
localization is owed to the balance of the superﬂuid currents,
and the internal structure of the currents is similar to that ofa continuously “self-defocusing” dissipative “antisoliton” of
the generalized complex Ginzburg-Landau equation describedin [19].
The presence of a harmonic trap modiﬁes the asymptotic
behavior of the condensate and phase according to Eq. (9),as shown in Fig. 3(b). Nevertheless, the structure of theinternal ﬂow within the ground state of the condensate remains
qualitatively the same, as seen in Fig. 3(d). In the trapped
regime the internal ﬂow of the polariton superﬂuid currents
lead to signiﬁcant distortions in the condensate density proﬁle,nc (r). In particular, a wide pump with P̄0
1 leads to awell-pronounced central peak [shown in Fig. 1(b) for σ 2 = 40
and P̄0 = 10] noted in numerical simulations of a similar
model of polariton BEC [20] and also hinted at in experimental observations of a trapped exciton-polariton condensate[11].
IV. TRAPPED VS TRAP-FREE REGIME

The ground m = 0 state of the polariton condensate canbe characterized by the dependence of chemical potential
μ on the parameters of the pump. In the absence of the

P0

6

8

10

FIG. 4. (Color online) Dependence μ(P̄0 ) for the ground statem = 0 without the harmonic potential (dashed line) and in thepresence of a harmonic trapping potential U (r) = U02 r 2 with U02 =0.1 (solid line). Vertical dashed and solid lines indicate the cut-off
values of P̄0 for the ground state given by the linear limit (14). Marked
points (a) and (b) correspond to the condensate proﬁles shown in
Fig. 3. The width of the pump spot is σ 2 = 30 in both cases.

trapping potential, this dependence is determined by theﬂux balance condition, and in the limit of low condensate
densities the dependence of μ on the pump intensity islinear μ ∼ P̄0 = P0 /Pth . In the presence of the harmonic
potential, this condition is modiﬁed by the trap, as seenin Fig. 4.
For a ﬁxed width of the pump, the cut-off value ofP̄0 corresponds to the linear limit |ψ| → 0, and deviates
quite signiﬁcantly from the value P̄0 = 1 determined by thethreshold behavior (3) in the homogeneous excitation case.
The linear limit is determined by the cutoff for the groundstate in the effective two-dimensional (2D) potential formed by
the combination of the external harmonic trap U (r), repulsive
potential due to the spatially localized pump VR (r) = gR n0R ,and the internal inhomogeneous ﬂux VJ (r) given by Eq. (12).
This cutoff can be estimated from the condition that theeffective potential in the linear limit is trapping (attractive)
rather than antitrapping (repulsive). Thus, the bound state
appears in the effective potential at the value of P̄0 given by
the positive root of the equation:
P̄02 − 2 1 +4gR 1γc R σ 2P̄0 + 1 +

4U0
γc2

= 0.

(14)

According to this formula, for the parameters in Fig. 4, theground state appears in the trap-free system at P̄0 ≈ 1.97.
The presence of the harmonic trap with U02 = 0.1 lowers
the threshold to P̄0 ≈ 1.61. The predicted tendency of theharmonic conﬁnement to lower the threshold of the steadystate formation compared to the trap-free case, agrees with
the numerically calculated cut-off values in Fig. 4. Above thethreshold, the harmonic trap leads, for the same parameters

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r

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(b)
(c)

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90

100

σ2

φ

-6

σ5

10

15

0

5

10

r

of the pump, to the formation of the steady-state BEC withstronger spatial localization and higher peak density, as can beseen from comparison of Figs. 3(a) and 3(b).
For a ﬁxed strength of the harmonic conﬁnement, thewidth of the pump spot determines different trapping regimes.
Intuitively, for an excitation spot much wider than the trap, thelimiting behavior should be described by a homogeneous pumpapproximation P (r) ≈ P0 [15,20]. In the opposite regime of anarrow excitation spot, the limiting behavior should be close
to that of a free condensate. Contrary to this intuitive picture,
the dependence of the chemical potential μ on the width ofthe excitation spot σ 2 calculated numerically shows dramaticdifferences for trapped and free condensates (Fig. 5). Most
notably, a trapped condensate displays well-pronounced bistability, whereby both narrow and wide pumps can support two
different steady states with the same energy. In addition, thecondensate density proﬁles are strongly spatially modulated(see Fig. 6). This is due to the presence of two competing
spatial scales. One of them, ∼ σ , is deﬁned by the pump, andthe other one, ∼ rTF , is deﬁned by the trapping potential and isgiven by the characteristic radius of the wave function in theThomas-Fermi limit. The latter is obtained by neglecting bothphase and density gradients in Eqs. (10), as the solution to theequation:
= μ.

(15)

In the regime of a wide pump, the condensate density canbe reasonably well approximated by the radially symmetricThomas-Fermi proﬁle with the radius rTF given by Eq. (15),

15

r
12

18

(c)

15

Φ

12

φ

ΦTF

6
3

6

0

3

-3

0

σ 5

-6

r

10

(f)

9

9

0

γc gR
r2
P̄0 exp − TF2
R
σ

D

0
-3

0

0

FIG. 5. (Color online) Dependence μ(σ 2 ) for the ground statem = 0 achieved at P̄0 = 5 without and with a harmonic trapping
potential U (r) = U02 r 2 with U02 = 0 (dashed), U02 = 0.01 (solid), andU02 = 0.01 (thick solid line). The vertical dotted line marks the cut-off
value of the pump width for the trap-free ground state given by thelinear limit expression (14) resolved with respect to σ 2 . Marked
points [(a)–(c)] correspond to the condensate proﬁles shown in
Fig. 6.

2+
U02 rTF

J

3

3

20

15

(e)

6

6

5.5
0

10

r
9

ΦTF

9

6

5

12

(b)

15

6.5

0

15

18

(a)

D

-3
-6

5σ

0

7

J

3

Φ TF

6

7.5

(d)

9

15

J
D

0

5

r

10

15

FIG. 6. (Color online) Radial shapes of the condensate groundstates in the harmonically trapped case for U02 = 0.1, P̄0 = 5 and thewidth of the pump spot σ 2 corresponding to the points marked inFig. 5. (a)–(c) Both amplitude, (r), and phase, φ(r), are shown incomparison to the condensate proﬁle obtained in the Thomas-Fermi
approximation, TF (gray). (d)–(f) Proﬁles of the radial ﬂux J and
its divergence D (scaled up by the factor of 10) for the ground states
shown in the panels (a)–(c).

as can be seen in Fig. 6(a). In this regime, the ground statein the form of a dissipative soliton becomes dynamicallyunstable with respect to azimuthal perturbations for P̄01,
as established in [20]. There it was shown that this instability
can lead to rotational symmetry breaking whereby multiple
vortex states enter the condensate.
2
For a very narrow pump, σ 2
rTF, the Thomas-Fermi
radius exactly coincides with the well-known expression for aconservative BEC with repulsive interparticle interactions in a0
= μ/U02 . In our numericalharmonic trapping potential: rTF
0
simulations we ﬁnd that rTF ≈ rTF for a wide range of values
σ < rTF . In this parameter regime the condensate density
proﬁle strongly deviates from the Thomas-Fermi proﬁle[Figs. 6(b) and 6(c)], as neither phase nor density gradients canbe neglected. The condensate experiences strong modulationsof density with a notable density dip along the line D = 0
separating the regions of loss and gain. The condensate peakdensity is also damped as the pump narrows down and the
dip at r = 0 appears for the values of σ corresponding to
the negative value of q− in (8). We note that the point (b)in Fig. 5 corresponds to q− = 0 for U02 = 0.1. Under the013636-5

ELENA A. OSTROVSKAYA et al.

PHYSICAL REVIEW A 86, 013636 (2012)

conditions of a strong (P̄01) narrow pump the steadystate is dynamically unstable and exhibits strong peak density
oscillations.
V. SINGLE VORTEX STATE

Spatially localized dissipative vortices are similarly foundas higher-order steady states of the system with m = 0[21], both in a trap-free and harmonically trapped polariton
BEC. Numerically, the system relaxes to a single steadystate vortex when a phase factor exp(imθ ) is imprinted
onto a condensate “seed” at the initial stages of evolution.
The dependence of the chemical potential μ on the pump
amplitude P̄0 is plotted in Fig. 7 for a vortex with m = 1.
As expected, the chemical potential and the cutoff for thisexcited state are higher than those of the ground state of
the trap-free condensate. In contrast, in a harmonic potential
the vortex and the ground state become nearly degenerate(see Fig. 7).
The global phase of the vortex is imposed not only by thevorticity but also by the nonzero ﬂux, and therefore depends
on both azimuthal (θ ) and radial (r) coordinates. Using theexpression (8) for the radial dependence of the phase near thevortex core, we ﬁnd that the lines of the constant phase follow
a spiral trajectory described by the equations
xs (θ ) = θ/q+ cos θ, ys (θ ) = θ/q+ sin θ.

16

a

2

U0 =0.1

14

2

U0 =0
12
10

μ
c

8
6
4
2
0

1

2

3

4

5

6

7

8

9

10

P0
20

45 8

20

250

10

35 4

10

200

y0

25 0

0

-10

15 -4

-10

-20

(a)

-20 -10 0

x

10 20

5 -8

(b)
-8

-4

0

x

4

150
100

-20
-20 -10 0
8

x

(c)
10

20

50
0

FIG. 7. (Color online) Top: Dependence μ(P̄0 ) for the vortex
state m = 1 without and with the harmonic trapping potential
U (r) = U02 r 2 for σ 2 = 30. The semitransparent line indicates thecorresponding dependence for a trapped m = 0 state from Fig. 4.
Bottom: (a) Condensate density and (b) phase for a dynamicallyunstable m = 1 stationary vortex state in a trap-free condensate atP̄0 = 8. (c) Metastable rotating vortex state supported by a harmonic
trap (U02 = 0.1) at P̄0 = 6.

This spiral phase structure is shown in Fig. 7(b) for thenumerically found localized vortex state in Fig. 7(a), without
the trapping potential. It is a signature of the polariton BEC
vortex observed in the experiments [2], and also appears asa characteristic feature of vortex states in other dissipative
systems [17,22,23].
The ground state (m = 0) of the polariton condensate is
dynamically stable in the absence of a trap. In contrast, we
ﬁnd that a single vortex created in a localized steady-state
BEC by phase imprinting is dynamically unstable and, in theabsence of potential, spirals out of the condensate, as predicted
for the case of trapped dissipative BEC of alkali atoms[14,24]. As can be seen from comparison of Figs. 4 and 7, theaddition of a stationary (i.e., nonrotating) harmonic potential
dramatically modiﬁes the threshold of the optical pumping
required to sustain a steady state of a polariton BEC with anangular momentum and leads to the possibility of a long-term
survival of the vortex in the form of an eccentric rotatingstate shown in Fig. 7(c). The detailed study of the stabilizing
effect of the trapping potential on the vortex dynamics andthe dynamics near the onset of rotational symmetry breaking[20] is beyond the scope of this study. However, we have
conﬁrmed that a wide pump (corresponding to small density
gradients in the harmonic trap) tends to stabilize the vortex
state.
VI. CONCLUSIONS

We have characterized the 2D stationary regime of thepolariton BEC in the framework of an open-dissipative GrossPitaevskii model coupled to an exciton reservoir and described,
both analytically and numerically, properties of the ground andexcited states (vortices) depending on the pump and trapping
parameters. We have shown that the spatial localization of
the condensate, even in the absence of a trapping potential,
is supported by the balance of the internal superﬂuid ﬂows
established by an inhomogeneous nonlinear gain due to opticalpump and linear loss due to the decay of the polaritons. The
ground state of the condensate can therefore be described asa continuously self-defocusing dissipative soliton. We havealso investigated localized dissipative vortex states and haveshown that they can display metastable behavior in nonrotating
trapping potentials.
Finally, we note that the open-dissipative model with theinhomogeneous pump used here and successfully employed
for theoretical description of experiments with microcavity
polaritons, has many similarities to optical systems withlocalized gain landscapes [13,22]. These and similar studies
suggest that the 2D stationary regime of polariton BEC can potentially display a rich variety of localized states which are notexhausted by radially symmetric conﬁgurations. In particular,
the experiments with polariton BEC excited by the elliptical
optical pump have demonstrated pattern formation due to theappearance of standing waves [25]. Furthermore, the recent
experiment with polariton BECs created by two pump spots[26] demonstrates novel possibilities arising from interactionof two dissipative solitons with variable separation. The clear
understanding of the structure of phase gradients underpinningthe existence of steady-state polariton condensates presented
in our work enables us to gain immediate insight into the013636-6

DISSIPATIVE SOLITONS AND VORTICES IN . . .

PHYSICAL REVIEW A 86, 013636 (2012)

properties and outcomes of such interactions. The possibility
to create a rich variety of localized states by varying the spatial
properties of the off-resonant optical pump are currently underinvestigation and will be reported elsewhere.

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ACKNOWLEDGMENT

This work was supported by the Australian ResearchCouncil (ARC).

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