Bright sink-type localized states in exciton-polariton condensates

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by  Michał Kulczykowski

Bright sink-type localized states in exciton-polariton condensates
Michal Kulczykowski, Nataliya Bobrovska, and Michal Matuszewski
Institute of Physics Polish Academy of Sciences, Al. Lotniko´w 32/46, 02-668 Warsaw, Poland
The family of one-dimensional localized solutions to dissipative nonlinear equations includes a
variety of objects such as sources, sinks, shocks (kinks), and pulses. These states are in general
accompanied by nontrivial density currents, which are not necessarily related to the movement of
5 theobject itself. Weinvestigatetheexistenceandphysicalpropertiesofsink-typesolutionsinnon-
1 resonantly pumped exciton-polariton condensates modeled by an open-dissipative Gross-Pitaevskii
0 equation. Whilesinkspossessdensityprofilessimilartobrightsolitons,theyarequalitativelydiffer-
2 ent objects as they exist in the case of repulsive interactions and represent a heteroclinic solution.
We show that sinks can be created in realistic systems with appropriately designed pumping pro-
n
files. We also consider the possibility of creating sinks in a two-dimensional configuration with a
u
ring-shaped pumpingprofile.
J

1 PACSnumbers: 71.36.+c,03.75.Lm,42.65.Tg,78.67.-n
1

] I. INTRODUCTION cluding sources, sinks, shocks and pulses21–25. In gen-
s
a eral,theyexhibitnontrivialinternaldensitycurrentsand
g Strong coupling of semiconductor excitons to micro- may undergo complicated, sometimes even chaotic dy-
t- cavity photons results in the appearance of spectral res- namics15,26.
n
onances associated with mixed quantum quasiparticles In this paper, we demonstrate the existence and
a
u called exciton-polaritons1. These particles exhibit ex- stability of bright self-localized solutions of the open-
q tremely light effective masses, few orders of magnitude dissipative polariton model. These solutions are classi-
. smallerthanthemassofelectron,whichallowsfortheob- fiedassinks(anti-darksolitons)22,25,the namereflecting
t
a servationofphysicalphenomenarelatedtoBose-Einstein that their structure corresponds to terminating lines of
m
condensation already at room temperatures2–9. At the incoming density currents, with a local increase of loss.
- same time, polaritons exhibit strong exciton-mediated While sink-type solutions possessdensity profiles similar
d
interparticle interactions and picosecond lifetime due to tobrightsolitons,theyarequalitativelydifferentobjects.
n
their photonic component. They are actively studied In contrast to bright solitons, they exist in the case of
o
c both from the point of view of fundamental interest6–8 repulsive interactions and represent a heteroclinic solu-
[ and potential applications10,11. tionconnectingtwocounterpropagatingplanewaves. We

Inrecentyears,greatattentionhasbeendevotedtothe demonstrate the dynamics of sink formation and their
2
v study of nonlinear self-localized states of superfluid po- stability in a realistic model with appropriately chosen
1 laritons, such as dark and bright solitons12–17. Solitons pumpingprofile. Weinvestigatesystematicallytheprop-
9 are nonlinear wavepackets which preserve their shape ertiesofsinksandprovideanapproximateanalyticalfor-
7 thanks to the balance between dispersion and nonlin- mula for their shape. In the two-dimensional case, we
7 earity18. They have been applied to long-distance op- show that sink creation is hindered by the spontaneous
0
tical fiber communication19 as well as description of nu- proliferationofvortices,whichdestroythe supercurrents
.
1 merous physical systems. Polariton solitons have been necessary for the existence of a symmetric sink solution.
0 demonstratedbothinthecasesofresonant12,13 andnon-
5 resonant pumping14–17. To date, no bright states were
1
showntoexistinthenonresonantcasewithhomogeneous
: II. MODEL
v pumping.
i
X Polariton superfluids are inherently nonequilibrium
systems in which the balance between pumping and loss We consider a polariton condensate in the one-
ar is anessentialfactor2,9,20. In many of the previousstud- dimensional (1D) setting, e.g. trapped in a microwire27.
ies,this aspectwastreatedasanunwantedcomplication We model the system with the generalized open-
of the theory. Standard models, such as the conserva- dissipative Gross-Pitaevskii equation for the condensate
tive Gross-Pitaevskii equation, were frequently used to wavefunctionψ(x,t) coupledtothe rateequationforthe
describe solitons. However, it is well known that self- polariton reservoir density, nR(x,t)15,20,28
localized solutions in dissipative systems have qualita-
tively different properties than their conservative coun- ∂ψ ~2D∂2ψ
i~ = +g1D ψ 2ψ+g1Dn ψ
terparts. In the case of repulsive interactions, only one ∂t −2m∗ ∂x2 C | | R R
type of one-dimensional solution exists in the conserva- ~
+i R1Dn γ ψ, (1)
tive theory – dark or bright solitons, depending on the 2 R− C
siliygnooffqtuhaelietffateicvteivlye mdiaffsesr.enInt tlhoceadliizsesdipasttiavteescaesxei,sats,faimn-- ∂nR =P(x(cid:0)) (γR+R1(cid:1)D ψ 2)nR
∂t − | |
2
whereP(x)istheexcitoncreationratedeterminedbythe
∗
pumping profile, m is the effective massof lowerpolari-
tons, γ and γ are the polaritonand exciton loss rates,
C R
and (R1D,g1D) = (R2D,g2D)/√2πd2 are the rates of
i i
stimulated scattering into the condensate and the inter-
action coefficients, rescaled in the one-dimensional case.
Here, we assumed a Gaussian transverse profile of ψ 2
| |
and n of width d. In the case of a one-dimensional
R
microwire27, the profile width d is of the order of the
microwire thickness. We also introduced D = 1 iA
−
with A being a small constant accounting for the energy
relaxation in the condensate27–30.
To obtain a system of dimensionless evolution equa-
tions, it is possible to rescale time, space, wavefunction
amplitude and material coefficients according to t = τt,
x = ξx, ψ = (ξβ)−1/2ψ, nR = (ξβ)−1nR, R1D = FIG. 1: Density patterns created with counterpropagating
(ξβ/τ)R, (g1D,g1D) = (~ξβ/τ)(g ,g ), (γ ,γ ) =e waves. (a) Interference pattern in thelinear regime. (b) Sta-
C R C R C R
τ−1(γ ,eγ ),P(x)=(1/ξβeτ)P(x),whereξ =e ~τ/2m∗, tionarysink-typelocalized patterninamodelwithnonlinear
C R
dissipation.
while τeand β are arbitrary scalieng eparameters. We
p
rewrieteteheaboveequationinethedimensionlessform(we
omit the tildes for convenience)
Ginzburg-Landau equation22,23 (CGLE)
∂ψ ∂2 i
i∂t = −D∂x2 +gC|ψ|2+gRnR+ 2(RnR−γC) ψ, ∂ψ = iD ∂2 +iC ψ 2 iB ψ. (3)
(cid:20) (cid:21) ∂t ∂x2 | | −
∂nR =P(x) γ +Rψ 2 n . (2) (cid:20) (cid:21)
R R
∂t − | | This equation can be obtained from the system of equa-
(cid:0) (cid:1) tions (2) in the limit of fast relaxation time of the reser-
In the above transformations the norms of both fields voir,which is “slaved”by the slowerψ dynamics, and in
Nψ = |ψ|2dx and NR = nRdx are multiplied by the the linearized approximation |ψ|2 ≈ n0 ≡ ImB/ImC =
factor of β. (P P )/γ (n is the dynamical equilibrium density
th C 0
R R −
when the loss and gain are balanced). Here, we as-
sumed a homogeneous pumping P(x) = const > P
th
andintroducedthethresholdpowerP =γ γ /R. The
th R C
III. SINK-TYPE SOLUTIONS CGLE parameters are B = iγ /2 (g + iR/2)(1 +
C R
Rn /γ )P/γ and C = (g +iR/2−)PR/γ2 g with
0 A A R A − C
γ =Rn +γ . It is clear that the existence and stabil-
A. Description of sinks solutions A 0 R
ity of the homogeneous steady state with n >0 (which
0
in general can be also a plane wave solution) requires
Thestructureofsinksolutionscanbeunderstoodmost ImB > 0 and ImC > 0. Under these conditions, per-
easilyasaresultofinteractionoftwocounterpropagating turbations of the steady state with density n exponen-
0
nonlinear waves. In the linear case, two waves emitted tially decay,whichis the reasonforthe abovementioned
by distant sources give rise to a standard interference smoothening. Any areasofdensityhigher thann corre-
0
pattern, as shown in Fig. 1(a). In the case of a dissi- spondtonetloss,andthosewithdensitylowerthann to
0
pative model with nonlinear gain or loss coefficients, the net gain in Eq. (3). Nevertheless, nontrivial (non-plane
interference pattern can be replaced by a localized den- wave) stationary solutions can still exist15,22,23.
sity peak, sometimes exhibiting oscillating features, as Onecanformulateanothernecessaryconditionforsta-
depicted inFig. 1(b). Sink is aheteroclinic solutioncon- bility of sinks based on the modulational (or Benjamin-
necting two plane waves emitted by the sources at each Feir)stabilityofthe incomingplanewaves22. Inthe case
side. The two waves collide at the sink position, where of CGLE with real D > 0, this is assured by the condi-
the incoming density currents are dissipated22. tion ReC < 0, which in terms of Eq. (2) translates into
The sink density pattern is a result of the ability of P/P >(γ g /γ g ), as shown recently in17. We note
th C R R C
the dissipative medium to smoothen out density “dips” thatthis isanecessaryconditionforstability,andanac-
and “peaks”, which are present in the standard interfer- tualdomainofstabilityofplanewavesmaybesmaller28.
ence pattern. When one of the incoming waves reaches It may seem natural to treat sinks as dissipative ana-
theareaoccupiedbytheother,the resultinginterference logues of bright solitons. These states are, however,
leads to decay of waves. Let us consider one of the sim- qualitatively different from each other. Bright solitons
plest dissipative nonlinear wave equations, the complex exist in the conservative limit of the CGLE (3) with
3
10 16 intensity
(a)
14 P(x)=P e−(x/wb)α (P P )e−(x/ws)β (4)
8 max − max− 0
12
as shown in Fig. 2 (dashed lines), where we used the
6 10 smoothness parametersα=100 andβ =80. The profile
2
(x)| 8 P(x) oexfhthibeitSsujpuemrp-GsaoufsPsi(axn)taetrmxs=in±w(4b).andx=±ws,typical
| 4
6
The sink is created in the central area with pump-
4 ing intensity P = P0. The side areas with P = Pmax
2
are the sources of polariton waves. This flow is obtained
2
thankstotherepulsivepolariton-polaritonandreservoir-
0 0 polariton interactions g , g >0, which create an effec-
C R
-30 -20 -10 0 10 20 30
tivepotentialhillintheareaswithhighpumpingdensity
x
P = P . In result of interaction of the two nonlinear
max
14 16 wavespropagating towardsthe center, depending on the
(b) parameters of the system, interference or localized sink
12 14
patterns can appear, as shown in Fig. 2(a) and 2(b), re-
10 12 spectively. Note that at the interface between the high-
10 and low-pumping areas at x = ws in 2(b), oscillating
2 8 time-dependent states form,whic±h do not, however,pre-
(x)| 8 P(x) clude the formation of sinks. We obtained also different,
| 6
6 more complicated nonlinear patterns for other values of
4 parameters, especially in the case when the two sources
4
were relatively close to each other. These patterns did
2 2 not have a localized character such as the one shown in
Fig. 2(b). A smallvalue of relaxationcoefficient A=0.1
0 0
-150 -100 -50 0 50 100 150 wasinsomecasesnecessarytoattenuatehigh-momentum
x modes in simulations and obtain physically relevant so-
lutions.
FIG.2: Patternscreatedbypolaritonwavesemittedbyhigh- Figure 3(a) shows the dynamics of the sink creation
intensitysourcesonthetwosideswithpumpingprofilesasin process. Initially, we assumed zero density of excitons
(4)(dashedlines). Panel(a)presentsastationaryinterference and a small white noise in the polariton field, but we
patternobtainedbyintegration of (2)intimeforparameters checkedthatthefinalstationarystateispracticallyinde-
A = 0.1, R = 0.96, gC = 0.63, gR = 1.91, γC = 0.9, γR = pendent ofthe formof the initial condition. First,in the
0.6 at low pumping powers. (b) With a stronger pumping
areabetweenthesourcesacondensateiscreatedwithap-
P0 a nonlinear sink-type solution is created. Corresponding
parameters in physical units are: time unit τ = γ−1 = 3ps, proximatelyzeromomentum. Thewavefrontsgenerated
C
length unit ξ = 1.9µm, g = 3.9µeVµm2, R = 9×10−3µm2 by the sourcesgraduallymovetowardsthe center,where
ps−1 for d=2µm, m∗ =5×10−5me, and β =0.003. theycollidecreatingthestablesinkstructure. Thewaves
are“stopped”by the sink due to the nonlinearcharacter
of the gain and dissipation.
It is important to note that the sinks are in general
not completely stationary, as in Fig. 3(a), but may be
Im(C,B,D) = 0, which is the celebrated Nonlinear
putinmotionbytheimbalanceofmomentaofthewaves
Schr¨odinger equation31 (or Gross-Pitaevskii equation in
emittedbythetwosources,seeFig.3(b). Inthiscasethe
the context of degenerate bosons32). Sinks require non-
sink moves with a constant velocity proportional to the
zero incoming currents for their existence22, and exist
mismatchbetween the two wavevectors22. If the balance
in the case of a stable background, Re(CD) < 0, while
is restored after some period of time, the sink stops at
bright solitons exist only in the self-focusing (modula-
the new position. The sink may be also stopped after
tionally unstable) case with CD >0.
reachingtheweakersource,asdemonstratedinFig.3(b).
IV. DOMAIN OF EXISTENCE AND
B. Sinks in the exciton-polariton model PROPERTIES OF SINKS
Tocreatesink solutionsinthe modeldescribedby(2), Inthissectionwedescribeasystematicinvestigationof
one has to provide sources of counterpropagating waves stationary sink solutions of the exciton-polariton model
as described above. We consider the following pumping with homogeneous pumping, P(x) = const. We sub-
profilecreatedbyapumpingbeamwithspatiallyvarying stitute ψ(x,t) = φ(x)e−iµt and n (x,t) = n (x) into
R R
4
1.3
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
P
FIG.4: Phasediagramdepictingparametersforwhichasink
solution was obtained with (5) (gray area). Parameters are
R=0.76,gC =0.38,gR =0.76,γC =0.75,γR =1.
1.4 12
(a)
1.2 11
1 xmin 10
FIG. 3: (a) Evolution towards a stationary sink state from a 0.8 ||(cid:2)(cid:2)mmainx((xx))||k22 89 min
small initial noise. Parameters are as in Fig. 2(b). (b) The 0.6 (cid:1)min x
case of asymmetric pumping profile, with momentum mis- 7
match between the waves from the two sources. The sink 0.4
6
movestowardstheweakersource,whereitisstoppedbutnot
destroyed. 0.2 5
0 4
0.98 1 1.02 1.04 1.06 1.08 1.1 1.12
Eqs. (2) to obtain a single ordinary differential equation (cid:1)
1.4 22
for the profile of the stationary state
(b)
1.2 20
d2φ P
dx2 =−µφ+gC|φ|2φ+gRγ +Rφ2φ+ 1 18
R
| |
i RP i 0.8 16
+ 2γ +Rφ2φ− 2γCφ (5) min
R x
| | 0.6 14
We complement the above equation with boundary xmin
cmoentdriytiopnosin.tAwtitxho=ut0lo,swshoifchgewneerachliotys,etthoebfierstthdeesryivma-- 00..24 ||(cid:0)(cid:0)mmainx((xx))||22 1102
k
tive is equal to zero, dφ/dx = 0. At x = + the so-
lution tends to the plane wave with the norm∞equal to 0 8
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
φ2 = n = (P P )/γ (in practice we impose this
| | 0 − th C P
condition on the last point of the computational mesh).
We solve the boundary problem with the shooting FIG. 5: Panels show the values of |ψmin(x)|2, |ψmax(x)|2, x1
method using the Newton minimization algorithm. We andthewavevectork (seeFig. 1for descriptions). Onpanel
keep ddφx|x=0 =0andchangethe value ofφ(0)while solv- (a)wekeepaconstant valueofP =1.45butvarythechemi-
ing (5) with the Runge-Kutta algorithm on a certain in- calpotentialµ. Thevalueofxminquicklydecreasesindicating
terval0<x<x . TheNewtonmethodisthenusedto thatathigherµthesinkismoredenselyundulating. Thede-
max
findasolutionthatsatisfiesboundarycondition φ2 =n pendence of |ψmin(x)|2 and |ψmax(x)|2 shows that the sink
0
at x = x within a given tolerance. We the|n|extend “height”isincreased at higherµ. Inpanel(b)wechangethe
max
value of P while µ is chosen slightly higher than the lower
the interval boundary x slightly and repeat our pro-
cedure. This method pmroavxed to be an effective way to tdhernecsehoofld|ψµmminin(xf)o|r2,w|ψhmicahx(ax)s|i2nkonsoPluitnidonicaotcecsutrhs.atTsihnekdheepigehnt-
obtain a localized state on a large x domain.
is approximately constant while the background density n0
grows with P.
5
Figure4presentsaphasediagramshowingthedomain 1.44
ofexistenceofsink-typesolutionsintheparameterspace (a) numerical
1.42
of P and µ with values of other parameters fixed. The analytical
shaded area, corresponding to parameters for which the 2)| 1.4
x
algorithm converged to a sink solution, is limited from ψ( 1.38
|
below by the natural minimum given by the chemical
1.36
potential of the steady state
1.34
Pg -20 -10 0 10 20
µ =g n2+ R . (6)
min C 0 (γ +Rn2) x
R 0
1.8
The range of µ for which the sink solution exists turned 1.7 (b) numerical
out to be the largest for moderate pumping intensities 1.6 analytical
P ≈1.5Pth. Althoughtherangeofµappearstobesmall, 2x)| 1.5
it corresponds in fact to a broad range of wavenumbers ( 1.4
of incoming waves, from approximately zero to around | 1.3
6, as shown in Fig. 5(a) with a dash-dotted line. In 1.2
this figure, the vertical line corresponds to the minimal 1.1
-30 -20 -10 0 10 20 30
value of µ, given by Eq. (6). The dependence between
the chemical potential and the wavenumber of incom- x
ing waves can be calculated by taking the x
→ ±∞
limit away from the sink, for which Eq. (5) reduces to FIG. 6: Comparison of the numerical sink solution from the
µ=k2+g n2+Pg /(γ +Rn2) under the conditionof shootingmethodwiththeapproximateanalyticalsolutionfor
C 0 R R 0
balanced gain and loss. It is then clear that the neces- two sets of parameters. Parameters are R = 0.76, g = 0.38,
saryconditionfortheexistenceofsinksisthatthekinetic gR =2g, γC =0.75, γR =1, P =2.0, µ=1.2683 for (a) and
energy is much smaller than the nonlinear energy of the P =1.85, µ=1.2173 for (b).
wave.
The dashed and dotted lines in Figure 5(a) depict the
minimum and maximum density of the sink profile, see
densateapplyingtheMadelungtransformationforψ(x,t)
Fig. 1. With increasing µ, which corresponds to increas-
ing k vectorsof incoming waves,the sinks become larger ψ(x,t)=a(x)ei(ϕ(x)−µt),
and more highly modulated, while in the µ=µ limit (7)
min n (x,t)=n (x),
they transform smoothly to the flat homogeneous state R R
φ(x,t) = n0e−iµmint. The position of the first minimum where a(x) is the amplitude, ϕ(x) is the phase and µ is
ofthedensityxmin(solidline)decreaseswiththeincrease the chemical potential of the condensate. Neglecting the
of µ, which is related to the increasingly dense interfer- spatial derivatives of a(x), Eq. (2) can be rewritten as
ence pattern of the tails of the sink.
Figure5(b) showsthe dependence ofthe same proper- d 2
ties of the sink as described above, but with increasing iµ = i ga2(x)+gRnR(x)+ ϕ(x) +
dx
pumping power P. Here µ is kept at a slightly increased " (cid:18) (cid:19) #
level with respect to µ . This corresponds to a con- 1 d2
min (Rn (x) γ ) ϕ(x) (8)
stant small value of k. In this case, the increase of P − 2 R − C − dx2
(cid:20) (cid:21)
leads to the increase of average sink density, but with-
outalmostanychangeofthedifference φ 2 φ 2. The real part of Eq.(8), which can be interpreted as the
max min
On the other hand, the value of x s|hows|a−s|trong|ly continuity equation, gives
min
nonmonotonouscharacter,decreasingforsmallP andin-
2 d2 γ
creasing again for large P. This shows that in the case n (x)= ϕ(x) + C (9)
R R dx2 R
of small k the position of first minimum is not simply (cid:18) (cid:19)
related to the incoming wavevectork.
Ontheotherhand,solvingtheequationforreservoirden-
sity (2) in the steady state we get
V. ANALYTICAL SOLUTION P
n (x)= . (10)
R a2(x)R+γ
R
Ifthe “height”ofthe sinkis relativelysmallcompared
We expand this formula into Taylor series of degree two
to the steadystate density, ψ(x,t) n0 n0,andthe around a2(x)=n =(P/γ ) (γ /R)
|| |− |≪ 0 C R
amplitude ψ(x,t) is slowlyvaryinginspace,anapprox- −
imateanaly|ticalso|lutioncanbe found33. Werewritethe γ a2(x)Rγ 2PR+γ γ
C C C R
steady-state solution in a homogeneously pumped con- nR(x)=− P−R2 (11)
(cid:0) (cid:1)
6
Comparing Eqs. (9) and (11) we obtain
2PR d2 ϕ(x) PRγ +γ γ
a2(x)= dx2 − C C R (12)
− (cid:16) R(cid:17)γ2
C
From the imaginary part of Eq. (8), using (9) and (12)
we get
g d2
µ = 2PR ϕ(x) PRγ +γ2γ +
−Rγ2 dx2 − C C R
C (cid:20) (cid:18) (cid:19) (cid:21)
2 d2 γ d 2
C
+ g ϕ(x) + + ϕ(x) . (13)
R R dx2 R dx
(cid:20) (cid:18) (cid:19) (cid:21) (cid:18) (cid:19)
With the definition ξ(x) = d ϕ(x) we obtain the first
dx
order differential equation
g gP d
2 R ξ(x)+ξ2(x)+ (14)
R − γ2 dx
(cid:18) C(cid:19)
P γ g γ
R R C
+ g + µ =0 (15)
γ − R R −
(cid:20) (cid:18) C (cid:19) (cid:21)
With the solution
√δ 1√δ
ξ(x)= tan , (16)
Rγ 2 α
C !
where δ =R(PRgγ Rµγ gγ2γ +g γ3) andα=
C− C− C R R C
gPR g γ2. Calculating the amplitude with (12) we
− R C
finally obtain
1 δP 1γ √δ
a2(x) = 1+tan2 C x +
−RγC2 " α ( 2 α !)
PRγ +γ2γ (17)
− C C R
Comparison between the(cid:3)analytical and numerical so- FIG. 7: Snapshot of a two-dimensional solution of (19) after
a long time of evolution t = 1600, generated by a pumping
lution obtained with the shooting method is shown in
profile in the shape of a ring. Quantum vortices are clearly
Fig. 6. In general, very good agreement is obtained for
visible in the density (top) and phase (bottom) of ψ(x,y,t).
small k, when the sink profile is flat and broad, with Sinksolutionsin2Dareabsentduetoproliferationofvortices.
no oscillating tails. The tails obviously cannot be repro- Parameters are A = 0.1, R = 0.96, gC = 0.63, gR = 1.91,
duced by the approximate solution (17), which is visible γC =1, γR =0.6withP0 =6,Pmax =10,r1 =155, r2 =220.
in the (b) panel of Fig. 6.
VI. TWO-DIMENSIONAL CASE where g , γ , R, and P(x,y) are dimensionless pa-
R,C R,C
rameters obtained from physical ones in an analogous
way as in the 1D case.
Additionally, we performed a series of simulations to
investigate whether sink creation is possible in the two- One of possible choices of the pumping profile corre-
dimensional(2D)versionofthe exciton-polaritonmodel, sponds to a constant P = P0 pumping intensity on a
described by the equations circle of radius r1, surrounded by a ring of P = Pmax
with inner and outer radius r and r , respectively. In
1 2
∂ψ ∂2 ∂2 Figure 7 we show a typicalstate obtained with this kind
i = D + +g ψ 2+g n +
∂t − ∂x2 ∂y2 C| | R R of pumping profile and an evolution from a small ini-
(cid:20) (cid:18) (cid:19)
tial noise. In general, after a certain time of evolution,
i
+ (Rn γ ) ψ, (18) smallerorlagernumberofvorticesarespontaneouslycre-
R C
2 −
(cid:21) ated,andoftenastationarystatecouldneverbereached
∂nR =P(x,y) γ +Rψ 2 n , (19) even with a very long integration time. Vortices may
R R
∂t − | | appear spontaneously during condensation in a process
(cid:0) (cid:1)
7
analogoustotheKibble-Zurekmechanism34–36 aswellas ton model. In contrast to bright solitons of conserva-
due to the emergence of supercurrents in an inhomoge- tive models, sinks exist in the case of repulsive interac-
neous system7,37. Despite using various combinations of tionsandarecreatedinacollisionofcounter-propagating
system parameters, as well as asymmetric ring pumping waves. We studied the dynamics of sink formation in a
profiles,wewere notableto obtainanystable structures realistic one-dimensionalpolaritonmodel with appropri-
that would resemble one-dimensional sinks of the previ- ately chosen pumping profile. We studied the domain of
ous sections. existence of sinks in parameter space and their physical
We note that similar pumping profiles were used in properties. An approximate analytical formula for the
several experiments38,39 where multi-lobe or vortex pat- sink shape, valid in the case where sinks do not possess
ternswereobserved. However,theexperimentalpatterns oscillatingtails,was determined. Inthe two-dimensional
wereinmostcasesregular,whichsuggeststhatthey cor- ring-shaped confguration, sink solutions were not found
respond to the linear regime as in Fig. 2(a). In the due to the spontaneous appearance of vortices.
caseofstrongnonlinearinteractions,regularvortexchain
patterns could be observed with the resonant pumping
scheme38. Depending on the system parameters, these
could be destroyed by spontaneously nucleating vortices
createdthroughahydrodynamicinstability,whichiscon-
sistent with our simulations.
Acknowledgments
VII. CONCLUSIONS
We demonstratedthe existence and stability of a fam- We acknowledge support from the National Science
ily ofbrightsink solutionsofthe open-dissipativepolari- Center grant DEC-2011/01/D/ST3/00482.
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