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Vortex–antivortex pair dynamics in an exciton–polariton condensate

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2009 New J. Phys. 11 113048(http://iopscience.iop.org/1367-2630/11/11/113048)
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New Journal of Physics
The open–access journal for physics

Vortex–antivortex pair dynamics in an
exciton–polariton condensate
M D Fraser1,2,4 , G Roumpos3 and Y Yamamoto2,3
1
Institute for Nano Quantum Information Electronics, University of Tokyo,
4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
2
National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku,
Tokyo 101-8430, Japan
3
Edward L. Ginzton Laboratory, Stanford University,
Stanford, CA 94305-4085, USA
E-mail: mdfraser@nii.ac.jp
New Journal of Physics 11 (2009) 113048 (12pp)

Received 11 June 2009
Published 25 November 2009
Online at http://www.njp.org/
doi:10.1088/1367-2630/11/11/113048

The study of superfluid and Berezinskii–Kosterlitz–Thouless phases
in exciton–polaritons requires an understanding of vortex dynamics in adissipative unconfined condensate. In this paper we study the motion of dynamic
vortex–antivortex pairs and show that vortex pair stability defined as orderedmotion as opposed to rapid separation or recombination is the result of balance
between dissipative velocities in the condensate and interaction with thermal
polaritons. The addition of a trapping potential is further shown to considerablyenhance the lifetime of a single-vortex pair in this system. These investigations
have important consequences for interpretation of recent results and future
investigations of two-dimensional superfluid phases in polariton condensates.
Abstract.

4

Author to whom any correspondence should be addressed.

New Journal of Physics 11 (2009) 113048
1367-2630/09/113048+12$30.00© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2Contents

1. Introduction: two-dimensional polariton condensation
2. Dissipative Gross–Pitaevskii equation (GPE)
3. Dissipative vortex pair dynamics
3.1. Classification of vortex pair trajectories . . . . . . . . . . . . . . . . . . . . .
3.2. Contributions to vortex velocity . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Condensate velocity profiles . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Effect of trapping potential and interactions on vortex pair motion . . . . . . .
4. Conclusion
Acknowledgments
References

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1. Introduction: two-dimensional polariton condensation

Superfluid behavior is well known in liquid helium and atomic Bose–Einstein condensates(BEC) and extensive studies have been made of the properties of quantized vortices in thesesystems [1, 2]. A newer type of condensate receiving a lot of current attention; it occurs not in agas or liquid of ‘real’ particles, but instead occurs in quasi-particles in semiconductors known as
exciton–polaritons (hereafter referred to as polaritons) [3, 4]. While the condensation of bosonicexcitonic particles has been studied for many years, the strong coupling of excitons and photons
reduces the effective mass by such a degree that condensation occurs in GaAs at temperaturesof order 10 K [3] and is expected at room temperature in other materials [5]. However, thesmall effective mass comes at the expense of finite particle lifetime, implying that althoughpolaritons can reach equilibrium with the lattice [6], a steady-state condensate is formed inplace of a condensate in true thermal equilibrium. One of the advantages of a condensate
embedded within a semiconductor matrix is the ability to manipulate the internal properties anddimensionality through material changes and the use of heterostructures. Although not strictly
a necessary geometry, many of the recent observations of polariton condensation occur in twodimensional (2D) quantum wells embedded within microcavities in order to reach the strong
coupling regime.
Superfluidity is an expected property of this condensate-like system [7, 8], and though
not direct evidence of this, quantized vortices have already been observed [9, 10]. Quantizedvortices are topological excitations of phase coherent systems and exist commonly in responseto an applied rotational field, but in 2D spontaneous formation as a result of phase fluctuations
is also possible [11]. Spontaneous vortex excitation is known to destroy long-range order in a
reduced dimensional infinite system preventing superfluidity [12, 13] at nonzero temperature.
However, below a certain temperature, the thermal energy is not sufficient to generate distinct
single vortices, but only the lower energy vortex–antivortex pairs (or vortex pairs for short).
As the vortex-induced phase gradient is now largely localized between the vortex pair,superfluidity can be recovered. This 2D superfluid phase characterized by the presence ofbound vortex–antivortex pairs is known as the Berezinskii–Kosterlitz–Thouless (BKT) phase,occurring at temperatures below TBKT [14].
A Bose–Einstein condensate like transition can also be recovered in a 2D system providing
the condensate is confined to a finite area of a size comparable to the phonon de Broglie
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wavelength in the fluid, thereby excluding long-wavelength fluctuations through a discretedensity of states [15]. Although the transition from a normal (thermal) state to a BKT phase is
discontinuous at TBKT , the transition from BEC to BKT is a continuous function of condensate
size and temperature, both contributing to the vortex pair density in the system [16]. The
polariton condensate is thus an ideal system in which to study crossovers between various 2D
condensate and superfluid phases.
The BKT phase has previously been reported in superfluid 4 He films [17], atomic
hydrogen films [18] and dilute gas atomic BEC [19]. In each of these cases however, althoughconvincing evidence for the BKT phase is presented, the microscopic nature of the state (thevortex–antivortex pairs) has yet to be observed. The small characteristic size scale of vortices(healing length ξ ) to the condensate size, results in a population of vortex pairs and/or freevortices far in excess of one, making single-vortex pair observation difficult. The evidence
supporting the BKT phase in atomic BEC however, is the observation of a proliferation of freevortices with increased temperature believed to be due to pair breaking when the temperature
exceeds the pair binding energy.
Recently, however, direct evidence of a single vortex–antivortex pair has been reported in
an exciton–polariton condensate [10]. The observation is based mainly on phase dislocationsin interferometry experiments indicative of a single vortex–antivortex pair. Due to the low
polariton effective mass m ∼ 10−5 m 0e (m 0e being the free electron mass), the vortices are
significantly larger and for a typical condensate density and radius (L ∼ 10–15 µm) one vortex
pair can be comparable to the system size. Furthermore, unlike previous quantized vortex
observations in polariton condensates, due to the low disorder in this sample (Ṽd . 0.1 meV),this vortex pair is believed to be unpinned. This recent observation in particular suggests thenecessity of a proper understanding of vortex and vortex pair dynamics in this system.
In contrast to the superfluid He and atomic BEC systems which generally have aparticle number conservation over a dynamical timescale, the polariton condensate occurs
in quasi-equilibrium, the result of a steady-state process of continuous stimulated scatteringfrom a polariton reservoir (pump) and the finite polariton lifetime τpol ∼ ps [20]. While theproperties and dynamics of quantized vortices [21, 22] and vortex pairs [23] have received
considerable theoretical attention in superfluid helium and atomic condensates, given its steadystate dissipative nature, the polariton condensate phase is distinct and the dynamical natureof quantized vortices currently unknown. The theoretical investigation of vortex dynamics
presented here is directly relevant to the recent experimental observation of a single-vortex
pair and to further investigations of vortex nucleation and BEC–BKT crossover in a polariton
condensate.
2. Dissipative Gross–Pitaevskii equation (GPE)

Condensate dynamics are usually modeled using a form of the time-dependent GPE todescribe evolution of the condensate order parameter ψ(r, t) [24]. This technique has been
extended to describe non-condensate components including quantum and thermal depletionand finite temperature effects [25, 26]. In this work we apply a dissipative GPE previously
shown to contain the essential parameters necessary for simulation of polariton condensates[9, 20, 27]. This dissipative GPE is coupled to a thermal reservoir population n R (r, t) bystimulated scattering (R(n R (r, t))) and interactions (with coupling constant gR ). Our study
here specifically differs from other studies of vortex pair dynamics in that the non-condensate
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reservoir population is not uniquely determined by coupling with the condensate, but also by
the spatial pumping profile. In addition because of the steady-state nature, this unconfinedcondensate possesses a unique velocity profile.
We start by defining the usual single-particle Hamiltonianh̄ 2 ∇ 2
Ĥ 0 = −+ Vext (r),(1)
2mwhere m is the polariton effective mass and Vext (r) is any external confining or disorder potential
profile. The dissipative time-dependent GPE is then given as∂ψ(r, t)
ih̄ 
2
ih̄
= Ĥ 0 −γC − R(n R (r, t)) + gC |ψ(r, t)| + gR n R (r, t) ψ(r, t),(2)∂t2
where the loss and gain terms of −ih̄γC /2 and ih̄ R(n R (r, t))/2, respectively, describe theprocess of polariton decay as photons leaving the cavity at rate γC (or alternatively withlifetime τC ) and stimulated scattering into the condensate at a rate R(n R (r, t)) determined by
the reservoir population distribution n R (r, t). This reservoir population is described by a rateequation model∂n R (r, t)= Pl (r) − γR n R (r, t) − R(n R (r, t))|ψ(r, t)|2 ,∂t(3)

where Pl (r) is the laser excitation profile (in photons µm−2 ps−1 ) and γR is the reservoir
polariton loss rate.
Previous investigation of vortices with this model has not considered vortex pair dynamics,
and it is not presently clear how a vortex pair should behave in a dissipative polariton
condensate. Previous experimental observations of quantized vortices in polariton condensates
have been carried out in materials with a level of disorder potential sufficient to spatially pin
the vortex, prohibiting any vortex dynamics and allowing simple experimental observation intime-integrated measurements. However, the recent clear observation of a spontaneously formed
vortex–antivortex pair occurs in the GaAs-based system where the disorder is low enough for thevortex pair to remain unpinned and as such, dynamics and stability are critical to understanding
this experimental result and is the main motivation for this research.
3. Dissipative vortex pair dynamics

The key parameters in the model of equations (2) and (3) which are expected to influence vortex
dynamics in the absence of any trapping potential are the condensate density n C , reservoir
density n R , scattering rate R(n R (r, t)) and condensate polariton lifetime τC . It is difficult to
study the effect of any one of these parameter independently, as in this dynamic model changing
either the lifetime or scattering rate both alter the final steady-state condensate population.
As the condensate√ density critically determines the vortex size through the healing lengthparameter ξ = h̄/ 2mgC n C , both R(n R (r, t)) and τC are adjusted together in order to maintainan approximately constant steady-state condensate density. Equation (4) describes the steadystate relationship between these parameters (for a uniform system):R(n R (r))γR=.
n R (r)Pl /γC − n C
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5

sc
sc
sc
sc
sc

Figure 1. Vortex pair trajectory for (a) a conservative condensate in a squarewell (abrupt walls of 1 meV depth) circularly symmetric trapping potentialof radius r = 15 µm, and (b) dissipative condensates at various points in theparameter space depicting a range of paths between rapid separation and rapid
recombination. The corresponding scattering rate coefficient is indicated (inµm2 ps−1 ). The initial vortex pair positions are rv = ±4.5 µm along the y-axis
and the pumping spot radius is L = 15 µm.
The scattering rate R(n R (r)) is given a linear dependence on n R (r) with coefficient Rsc . The
study of dissipative vortex–antivortex pair dynamics here specifically addresses GaAs-basedpolariton condensates; however general arguments will be drawn relevant to all other material
systems. From a manually ascribed initial state, the condensate is allowed to evolve in time
until a steady-state distribution is achieved (constant energy and static particle distribution).
The vortex pair phase profiles are then artificially imprinted along the y-axis (at positions±dv /2 where dv is the vortex pair separation), with a mirror symmetry about the x-axis.
The dynamics of an unpinned vortex–antivortex pair in a spatially infinite conservative (nondissipative) superfluid is simple and well understood. Due to the interaction of the mutual phasegradients the vortex–antivortex pair undergoes a linear motion perpendicular to its dipole axis,
with direction determined by the dipole orientation [2] (here, along the x-axis). The vortex pair
will then travel at a velocity dependent on the vortex separation dv given by|κ||vp | =
(5),
2π dv
where κ = l(h/m)ẑ is the quantized circulation of each vortex with vector perpendicular to thex–y plane and l = ±1. In the absence of any dissipation, the vortex pair will maintain separationof dv . The introduction of a confining potential Vext (r) creates a boundary with which the vortex
pair interacts, perturbing its one-dimensional motion. The form of the perturbation depends on
the type of confining potential. In a conservative system, the vortex pair will track the boundary
to preserve the energy of the vortex pair periodically returning to its starting position. Figure 1(a)shows the numerical solution of the conservative GPE in a square well trap (abrupt walls) giving
an example of this type of motion. The trajectory in a harmonic trap is qualitatively similar, but
this example is closer to that of a top-hat pumping profile considered in the rest of this paper forthe dissipative condensate.
New Journal of Physics 11 (2009) 113048 (http://www.njp.org/)

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Figure 2. Map of n R (depicted as the circle/triangle marker size) with scattering

rate R(n R (r)) and condensate polariton lifetime τC . The condensate density isheld relatively constant in these simulations and corresponds to the different
lines labeled with n C , but does not have a significant effect on the vortex
trajectory. The vortices that leave the condensate are indicated as red triangles
and those that recombine as blue circles. The background in this plot comprises
the approximate steady-state analytical form of n R , with magnitude indicated by
the color bar in µm−2 .
3.1. Classification of vortex pair trajectories
The vortex pair dynamics in a dissipative system are found to differ considerably. The initially
imprinted pair with separation dv is found instead to choose from a continuum of different
trajectories based on the parameters chosen for the simulation, some examples of which are
shown in figure 1(b). These motions range from the vortex pair splitting and leaving thecondensate directly to recombining rapidly long before reaching the edge of the condensate.
In these simulations, a condensate density n C ≈ 570 µm−2 and a pumping spot of radius
L = 15 µm are used. This condensate size is chosen such that the vortex pair is not initiallyperturbed by the presence of a boundary and the motion largely independent of the condensate
size can be initially observed, from which general conclusions are drawn.
To gain an understanding of the effect of these parameters, a map of n R (size of circles) withvariation in R(n R (r, t))/n R (r, t) and τC is plotted in figure 2. The background map (saturated atsmall scattering rate) is the steady-state analytical contribution (equation (4)). In this parameter
space the vortex pair trajectories are divided into two groups according to whether the vortex
pair recombines in the center of the condensate before reaching the edge (depicted as blue
circles) and vortex pairs splitting and leaving the condensate (red triangles). A clear transition
between these two different groupings is apparent and observed to be largely dependent on
R(n R (r, t)).
If the initial vortex pair spacing dv is altered in this simulation, it does not affect this trend
other than to slightly shift the transition between the two classifications of vortex pair evolution.
This remains the case provided the vortex pairs are not initially overlapping slightly (dv . 2ξ ) or
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Figure 3. A map of the numerical vortex velocities as a function of the R(n R (r))

and τC parameter space. The recombining vortex pairs are again indicated as blue
circles, and the radially separating ones as red triangles.
the vortex is initially separated from the boundary by an amount of order |L − rv | . 2ξ . In longtimescale experimental scenarios, where a statistical ensemble of initial vortex pair separations
is expected, this then corresponds to the same trend but with a blurred cross-over transition
between the two classifications. The condensate population is also found to not impose mucheffect on this crossover (within experimentally reasonable ranges) as indicated in figure 2 where
the different lines indicate different condensate population density.
Attention is now turned to a map of vortex velocities constructed from this numerical dataand shown in figure 3. The√ vortex velocities are presented as a fraction of the condensate soundvelocity defined as c = gC n C /m. The distinction between the two classifications of vortex
dynamics is further clear in these data. For very low scattering rates (which corresponds to highreservoir population density), the vortex pair recombines rapidly. This vortex velocity drops
rapidly from the sound velocity as the scattering rate is increased, stopping when the cross-overto radially separating vortex pairs is achieved, remaining roughly constant thereafter at a fractionof the sound velocity. This analysis suggests two different mechanisms altering the vortex pairvelocity vector.
3.2. Contributions to vortex velocity
In a conservative condensate, the absence of any forces implies the vortex velocity vL
will coincide with that of the local superfluid flow vL = vs in correspondence with thefamiliar Magnus force (equation (6)) [2, 28]. In rotating trapped condensate for example, thevortex lattice will rotate at the same angular velocity as the condensate and thus appears
stationary in the rotating frame. In a dissipative system, i.e. through interaction with thermal
population and excitations at finite temperature, energy is transferred between these noncondensate populations and energetically unfavorable vortex states will decay by approaching
the condensate boundary decaying via excitations at the edge [31]:fM = mn C κ × (vL − vs ).
New Journal of Physics 11 (2009) 113048 (http://www.njp.org/)(6)

8
The superfluid velocity vs is defined as being the superfluid velocity far from the vortex
line. Thus, in the present vortex pair system, the superfluid velocity vs consists of velocity
contributions from each of the two vortices (v L1 and v L2 ) and the radially dissipating polaritons
due to repulsive interactions and a lack of confinement. Neglecting the condensate boundary, tocalculate the local velocity field at the vortex v1 (at rv1 with κv1 ) in the presence of vortex v2(at rv2 with κv2 ) and the radial velocity gradient (vC (r)) we can use
κv2 × (rv1 − rv2 )+ vC (r).
vs (rv1 ) =(7)
2π |rv1 − rv2 |2
The second source of forces on the vortex pair is the interaction of vortices with noncondensate population which creates a drag force dependent on the interaction energy withthis non-condensate population n R . This force is commonly broken up into longitudinal andtransverse components,


Bn R
κ
B 0nR
fD = −mn C
κ×× (vn − vs ) − mn C
κ × (vn − vs ),(8)2(n C + n R )|κ|2(n C + n R )where vn is the velocity of the normal component. The origin of these drag forces is usually
attributed to interaction with thermally excited non-condensate modes, where the perpendicularcomponent is commonly known as the Iordanskii force [29, 30]. The form of the friction
coefficients B and B 0 has been evaluated theoretically [28, 31] and experimentally [2], although
exact determination is very much dependent on the exact details of the system and thecontributions to the magnitude of the Iordanskii force are particularly complex. We find,however, that conclusions can be drawn relatively independent of the exact magnitude of thesecoefficients.
3.3. Condensate velocity profiles
As the condensate is unconfined, we expect there to be some radially dissipative condensate
velocity in the system amounting to a continual radial loss of particles due to particle repulsive
interactions, giving the form of vC (r). Figure 4(a) shows the numerically evaluated steady-state
unconfined condensate phase profile S(r). The condensate velocity is simply the gradient of thephase (vs = h̄/m∇ S(r)) and the radial velocity profile can be calculated as in figure 4(b). The
point at which the velocity increases discontinuously is the approximate condensate boundary.
Note that the velocities within the condensate possess a similar fraction of c as do thosedetermined for splitting pairs in figure 3.
In the absence of energy dissipation, we expect vortices in the condensate to move withthe same velocity vector as the local superfluid flow. The contributions to the magnitudeof the vector vC (r) are from the same contributions as for the condensate sound velocity, namely,
the condensate interaction energy gC |ψ(r)|2 and the effective mass m. The condensate radial
velocity also exhibits an inverse correspondence with the reservoir density n R (r) (induced by areduction in scattering coefficient Rsc ). The presence of this thermal reservoir with which thecondensate interacts introduces a drag force on the vortex pair. As the energy of a vortex pairis proportional to its separation (equation (9)), in the presence of finite gR n R , energy will betransferred from the vortex pair to the thermal reservoir, inducing a drag force on each vortex
towards the pair midpoint: dv
mn C |κ|2
Ev =
ln
.
(9)
2π
ξ
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Figure 4. (a) Steady-state unconfined condensate phase profile S(r) and (b) theradial cross-section of the velocity profile. The particle velocity within thecondensate increases with the scattering coefficient R(n R (r))/n R (r) in response
to a reduction in drag forces through decreasing reservoir density n R (r).
Based on this simple analysis, the occurrence of two distinct modes of vortex motion
differing from that of a conservative condensate appears to arise from contributions of two
separate sources, namely the dissipative velocity of an unconfined condensate and the draginduced by interaction with thermal particles. Allowing the two forces to cancel (fM + fD = 0),
the velocity vector v L1 of vortex v1 is given by
Bn R |κv1 |B 0 n R |κv1 |v L1 = vs +
κ̂v1 × (vn − vs ) −κ̂v1 × κ̂v1 × (vn − vs ) ,(10)2(n C + n R )2(n C + n R )where κ̂v1 = κv1 /|κv1 |. Making substitutions for vs relevant for this specific case (seeequation (7)) reveals the contributions of these two sources to v L1 . If we assume the two
vortices are initially at positions rv1 = (dv /2) ŷ and rv2 = −(dv /2) ŷ with circulation vectorsof κv1 = +(h/m)ẑ and κv2 = −(h/m)ẑ, respectively, yields the form of vs for v1,
vs (rv1 ) = +h̄x̂ + |vC (dv /2)| ŷ.
mdv

(11)

Figure 5 shows a schematic of various contributions to the vortex velocity with (a) showingvC (r) and (b) and (c) showing the radially diverging and recombining vortex pairs, respectively.
The first term in equation (10) corresponds to the contribution of dissipative superfluid flow
and the second term describing contributions of non-condensate interaction effects. While theexact trajectory depends on the accurate coefficients, it is clear that the first term contributes
an outward velocity in the + ŷ-direction with some curvature due to the +x̂ component as aresult of the vortex pair linear trajectory. Thus, on its own (i.e. when n R is small compared
to n C ), it describes a splitting and radially dissipating vortex pair as depicted in figure 5(b).
If we ignore the presence of the radial dissipative velocity (vC (r) = 0), the effect of the dragforces on v1 in the second term of equation (10) can be described, which are proportional to n R .
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Figure 5. Schematic illustration of the different components contributing to

the vortex velocity, where the large dotted circle indicates the approximate
condensate boundary and the small circles represent a sample of instantaneous
vortex positions with time increasing to the right of the schematic. (a) Shows
the radial dissipative velocity vC (r), the result of repulsive interactions and lackof confinement. When the velocity in (a) is dominant, the vortex vector in (b)results and the vortex pair separates. When drag forces dominate due to large n R ,the vortex pair recombines as in (c).
The perpendicular component labeled vd⊥ = (Bn R |κ|/2(n C + n R ))κ̂ × (vn − vs ) contains components in directions +x̂ and − ŷ while the longitudinal drag vdk = −(B 0 n R |κ|/2(n C +n R ))κ̂ ×[κ̂ ×(vn −vs )] contains vector components in directions −x̂ and − ŷ as illustrated in figure 5(c).
Thus, independent of the magnitude of the coefficients B and B 0 this set of equations generally
describes a trajectory directed inwards (− ŷ-direction) towards the mid-point of the vortex pair.
Clearly, if this analysis is performed again with the second vortex v2 the same results will beachieved with oppositely directed ŷ-vectors, demonstrating that this simple analytical model
agrees with our previous numerical results.
3.4. Effect of trapping potential and interactions on vortex pair motion
The application of a trapping potential will prevent radial particle escape by limiting any
radial superflow present in the system and its presence can be used to check the effect of thissuperflow on vortex pair motion. Furthermore, the onset of the BKT-phase occurs not onlywith temperature, but also with confinement size. In most current experiments the finite laser
pumping area is sufficient to restrict the condensate area (due to finite lifetime and diffusion
length). Experimental investigations of BKT transitions in a polariton condensate are likely tobenefit from the use of a trapping potential to confine the gas, particularly if the lifetime isincreased. Effective square well and harmonic profile traps for polaritons have been previously
experimentally demonstrated [7, 32, 33] allowing greater control over the condensate density
profile.
In figure 6 we show the numerical trajectories for (a) a diverging vortex pair and (b) fora recombining vortex pair. In figure 6(a), when a trapping potential is added, the vortex pairdoes not immediately split and leave the condensate, but recovers some of its circular motion
remnant of the conservative scenario. This implies that the velocity of the radial dissipative flow
as expected is the main contributor to perturbing the motion for this parameter range. In thiscase, the presence of a trapping potential extends the vortex pair lifetime by many times. In
figure 6(b) where the parameters dictate a rapidly recombining vortex pair, adding a trapping
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Figure 6. Vortex pair trajectories for parameters yielding (a) radially diverging

vortex pair and (b) recombining vortex pair. Within these plots the effects ofapplying a trapping potential and turning off the interaction with the reservoir
component are also displayed as separate trajectories.
potential has no obvious effect on the trajectory, implying influence of any radial condensate
velocity is negligible for these parameters.
Figure 6 also demonstrates the effect of turning off the interactions with the reservoir
polaritons (setting gR = 0). In figure 6(a) the effect of turning off interactions only is to reducedrag forces towards the condensate center and allows the vortex pair to leave the condensate
more directly, via a shorter path. However, in figure 6(b) for the recombining pair, turning off
reservoir interactions completely prevents the recombination of the vortex pair and the vortices
leave the condensate directly with the velocity vector comparable to vs .
4. Conclusion

We have studied the dynamics and stability of a single-vortex pair in a dissipative model ofa polariton condensate. While vortex pairs are essentially stable against recombination andradial dissipation in conservative condensates, in an unconfined dissipative condensate, thevortex pair either splits and leaves the condensate or recombines quickly a short distancefrom the nucleation location. It is found that the cross-over of these two behaviors is a resultof competition between a radially outward force due to the radially dissipating condensate
polaritons and the interaction of the vortices with non-condensate population which strongly
inhibits vortex motion and induces a drag force towards the pair mid-point as the vortex pairloses energy to the reservoir. We note that the long timescale vortex pair stability appropriate
to conservative (atomic) condensates can be recovered in this system through the applicationof a trapping potential and pumping the system such that the reservoir density is not excessive.
These observations thus have direct relevance to the interpretation of recent observations of a
dynamic vortex–antivortex pair in a polariton condensate and to the extension of these studies
to push the system into the BKT regime.

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Acknowledgments

This work is supported by the Special Coordination Funds for Promoting Science andTechnology, Navy/SPAWAR Grant N66001-09-1-2024, and MEXT.
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New Journal of Physics 11 (2009) 113048 (http://www.njp.org/)