Dark soliton of polariton condensates under nonresonant PT-symmetric pumping
DDark soliton of polariton condensates under nonresonant PT - symmetric pumping Chunyu Jia Zhaoxin Liang ∗ Department of Physics, Zhejiang Normal University, Jinhua, 321004, China (Dated: February 25, 2020)A quantum system in complex potentials obeying parity-time ( PT ) symmetry could exhibit allreal spectra, starting out in non-Hermitian quantum mechanics. The key physics behind a PT -symmetric system consists of the balanced gain and loss of the complex potential. In this work,we plan to include the nonequilibrium nature, i.e. the intrinsic kinds of gain and loss of a system,to a PT - symmetric many-body quantum system, with the emphasis on the combined effects ofnon-Hermitian due to nonequilibrium nature and PT symmetry in determining the properties of asystem. In this end, we investigate the static and dynamical properties of a dark soliton of a polaritonBose-Einstein condensate under the PT - symmetric non-resonant pumping by solving the driven-dissipative GrossPitaevskii equation both analytically and numerically. We derive the equation ofmotion for the center of mass of the dark solitons center analytically with the help of the Hamiltonianapproach. The resulting equation captures how the combination of the open-dissipative characterand PT - symmetry affects the properties of dark soliton, i.e. the soliton relaxes by blending withthe background at a finite time. Further numerical solutions are in excellent agreement with theanalytical results. At present, there are significant interests and ongoingefforts in investigating parity-time ( PT )- symmetric non-Hermitian quantum mechanics [1–5] in different contextsof physical systems. The motivation behind this researchline is twofold. First, the introduction of PT - symme-try has gone conceptually beyond the Hermitian quan-tum mechanics. Traditionally, it was well believed thatany physical phenomena in quantum systems must bedescribed by Hermitian Hamiltonians with real eigenval-ues. However, non-Hermitian systems with the PT - sym-metic complex potential can have real spectra, staringout in non-Hermitian quantum mechanics [2, 6]. Second,the physical systems with the PT - symmetry can pro-vide some enlightening applications, such as single-mode PT lasers [7, 8] and unidirectional reflectionless PT -symmetric meta material at optical frequencies [9]. Upto now, extension of PT symmetry to other branches ofphysics is a hot topic, such as mechanical oscillators [10],optical waveguides [11], and optomechanical systems [12].Along this research line, there are timely efforts of ex-tending the studies of PT - symmetry to non-equilibriumquantum systems with emphasis on the combined effectsof PT - symmetry and the intrinsic non-equilibrium na-ture on a quantum system. In this end, Bose-Einsteincondensates (BECs) of polaritons in quantum well semi-conductor microcavities [13–16] has opened up intrigu-ing possibilities to explore PT - symmetric non-Hermitianquantum mechanics beyond thermal equilibrium. On theone hand, as polaritons undergo rapid radiative decay,their population in the condensate is maintained by per-sistent optical pumping. Hence a polariton condensateis inherently in non-equilibrium with an open-dissipativecharacter. Its mean-field physics can be well captured bya GrossPitaevskii equation (GPE) with gain and loss [17–27], where the nonlinearity results from the strongly andrepulsively interacting excitons. On the other hand, there are ongoing experimental investigations on non-Hermitian PT - symmetric systems in the context of po-lariton condensates [28, 29]. For example, the non-trivialtopology of eigenmodes and unusual transport proper-ties in the vicinity of exceptional points are currentlyunder investigation [29]. Moreover, the direct observa-tion of non-Hermitian degeneracies in a chaotic exciton-polariton billiard has been done in Ref. [28].In this work, we propose and investigate an alternativeroute to study non-equilibrium PT - symmetric many-body quantum system in the context of a polariton con-densate with PT - symmetric non-resonant pumping. Inmore details, we focus on the dynamics of a dark soli-ton in a polariton codensate under the nonresonant spa-tial dependent pumping with PT - symmetry by solvingopen-dissipative GP equation coupled to a rate equationfor a density with a combination of analytical and numer-ical approaches. In this end, we first use the Hamiltonianapproach to derive the equation of motion for the solitonparameters analytically, i.e. the velocity of the dark soli-ton. Then we compare these analytical results with thenumerical solutions for the trajectory of dark solitons bysolving the GP equation. We find a remarkable agree-ment between the two as it’s expected. Our work wouldopen a new perspective of including the nonequilibriumnature, i.e. the intrinsic kinds of gain and loss of a sys-tem, to a PT - symmetric many-body quantum system.We are interested in the dynamics of the polariton con-densate formed under uniform non-resonant pumping ina wire-shaped microcavity, as motivated by Ref. [30]. Atthe mean field level, the condensate can be well describedby the driven-dissipative GPE characterized by the time-dependent condensate order parameter of ψ ( x, t ), cou-pled to a rate equation for a density, n R ( x, t ), of an un-condensed reservoir of high-energy near-excitonic polari- a r X i v : . [ c ond - m a t . o t h e r] F e b FIG. 1. The schematic profiles of the real and imaginary partsof the non-resonant PT symmetry pumping based on Eq. (3). tons as follows [17–24] i (cid:126) ∂ψ ( x, t ) ∂t = (cid:104) − (cid:126) m eff ∇ + g C | ψ | + g R n R ( x, t )+ i (cid:126) Rn R ( x, t ) − γ C ) (cid:105) ψ ( x, t ) , (1) ∂n R ( x, t ) ∂t = P ( x ) − (cid:16) γ R + R | ψ | (cid:17) n R ( x, t ) . (2)In Eqs. (1) and (2), ψ ( x, t ) and n R ( x, t ) represent the or-der parameter of the polariton condensate and the reser-voir density respectively. Here, m eff = 10 − m e labels theeffective mass of polaritons with m e being the free elec-tron mass. The g C and g R characterize the strength ofnonlinear interaction of polaritons and the interactionstrength between reservoir and polariton respectively.The condensed polaritons with a finite lifetime γ − C arecontinuously replenished from reservoir polaritons at arate R . High energy exciton-like polaritons are injectedinto the reservoir by laser pump P ( x ) and relax at thereservoir loss rate γ R .In this work, we consider that the PT - symmetricpumping of P ( x ) [13–16] in Eq. (2) consist of a con-stant part P and the spatial dependent part as P ( x ) = P + P PT ( x ). For the pumping rate P to be PT symmet-ric, P PT must be successive action of the parity (P) andtime-reversal (T) operator: PT { P PT ( x ) } = P ∗PT ( − x ) = P PT ( x ). In more details, we are interested in P PT ( x ) inthe following form P PT = P R + iP I = V sech (cid:18) U ξx h (cid:19) + iV sech (cid:18) U ξx h (cid:19) tanh (cid:18) U ξx h (cid:19) . (3)with V and V being the amplitudes of the real andimaginary part of nonresonant PT - symmetric pump-ing and ξ = x − v s t , x h being the healing length, and U = (cid:112) − v s . As shown in Fig. 1, we plot the real partand imaginary part of P PT , corresponding the even andodd functions respectively. Note that the imaginary part represents the gain-loss effects due to the PT - symmetricpumping. We remark that the reservoir density n R of Eq.(2) is supposed to be complex under the PT - symmetricpump, which will challenge the physical explanation of n R . Our strategy is to obtain an effective equation of ψ (see Eq. (14) below) by replacing the n R in Eq. (1) inthe fast reservoir limit.The emphasis and value of this work is to include thenonequilibrium nature characterized the intrinsic kinds ofgain and loss of a system, to a PT - symmetric many-bodyquantum system, with the emphasis on the combined ef-fects of non-Hermitian due to both nonequilibrium na-ture and PT symmetry in determining the properties ofa system. It’s clear now that the intrinsic kinds of gainand loss of a polariton condensate are described by theparameters of R and γ C in Eq. (1); in contrast, the imag-inary part of PT - symmetric pumping in Eq. (3) cap-tures the key physics of PT symmetry inducing gain-losseffects of the polariton condensate. Hence, the competi-tion of non-Hermitian due to both nonequilibrium natureand PT symmetry in determining the properties of a sys-tem can be determined by the rich interplay among fourparameters of R , γ C in Eq. (1) and V , V in Eq. (3).In what follows, we focus on the static and dynamicalproperties of a dark soliton of a polariton condensate un-der the PT - symmetric non-resonant pumping by solvinga mean-field driven-dissipative GP description both ana-lytically and numerically.For convenience of later analysis, we proceed to recastEqs. (1) and (2) into a dimensionless form by rescalingspace time in the units of healing length r h = (cid:126) / ( mc s )and τ = r h /c s . Here c s = ( g C n ∗ C /m ) / is a local soundvelocity in the condensate and n ∗ C is a characteristic valueof the condensate density. For a cw background, it isconvenient to choose n ∗ C = n C = ( P − P th ) /γ C with P th = γ R γ C /R . The dimensionless equations for thenormalized condensate wave function ¯ ψ = ψ/ (cid:112) n ∗ C ,andreservoir density ¯ n R = n R /n ∗ C take the form i ∂ ¯ ψ∂t = (cid:20) − ∂ ∂x + (cid:12)(cid:12) ¯ ψ (cid:12)(cid:12) + ¯ g R ¯ n R + i (cid:0) ¯ R ¯ n R − ¯ γ C (cid:1)(cid:21) ¯ ψ, (4) ∂ ¯ n R ∂t = ¯ P + ¯ P PT ( x ) − (cid:16) ¯ γ R + ¯ R (cid:12)(cid:12) ¯ ψ (cid:12)(cid:12) (cid:17) ¯ n R , (5)In above, the dimensionless parameters are defined asfollows: ¯ g R = g R g C , ¯ R = (cid:126) Rg C , ¯ γ C = (cid:126) γ C g C n ∗ C , ¯ γ R = (cid:126) γ R g C n ∗ C , and¯ P ( r ) = ¯ P + ¯ P PT ( x ) = (cid:126) P ( x,t ) g C n ∗ C .The goal of this work is to investigate the dynamicsof the dark soliton of the polariton condensate under in-coherent PT -symmetric pumping. Generally speaking,a dark soliton is referred to as the finite amplitude col-lective excitation of a homogeneous condensate. Inspiredby this physical picture of a dark soliton, we considerperturbations of the condensate wave function and thereservoir density in the following general form as Refs.[18, 24] ¯ ψ ( x, t ) = ψ ( x, t ) exp[ − i (1 + ¯ g R ¯ γ C / ¯ R ) t ] , (6)¯ n R ( x, t ) = ¯ g R ¯ γ C / ¯ R + m R ( x, t ) . (7)By plugging Eqs. (6) and (7) into Eqs. (4) and (5), wecan obtain that the perturbations ψ ( x, t ) and m R ( x, t )are governed by the dynamical equations i ∂ψ∂t = (cid:20) − ∇ + | ψ | − g R m R + i Rm R (cid:21) ψ, (8) ∂m R ∂t = ¯ P − (cid:16) ¯ γ R + ¯ R | ψ | (cid:17) m R + ¯ γ C (cid:16) − | ψ | (cid:17) . (9)Blow, we plan to present a detailed analysis on the dy-namics of the dark soliton by solving Eqs. (8) and (9)analytically and numerically.Before investigating the effects of PT - symmetricpumping on the soliton, we first briefly review someimportant features of a dark soliton in the nonlinearSchr¨odinger equation, corresponding to m R ( x, t ) = 0 inEq. (8), i.e. in the absence of the open-dissipative and PT - symmetric pumping effects. In such, there exists anexact dark-soliton solution, which takes the form ψ s ( ξ = x − υ s t ) = (cid:112) − υ s tanh( (cid:112) − υ s ξ ) + iυ s , (10)with υ s being the velocity of the traveling dark soliton.In particular, for a moving soliton, the minimum valueof density n minC = | ψ s ( ξ = 0) | increases in proportion tothe square of the soliton velocity n minC = υ s .Next, we proceed to include both the open-dissipativeeffects and PT - symmetric pumping as captured by m R ( x, t ) (cid:54) = 0. In such a case, Equation (10) is not theexact solution of Eq. (8) any more. Directly followingRefs. [18, 24, 31], we plan to adopt the Hamiltonianapproach for the perturbation theory of soliton. At theheart of the Hamiltonian approach of quantum dynamicsfor a dark soliton is based on that, in presence of pertur-bation, the dark soliton’s velocity become slow functionsof time, but the functional form of the dark soliton re-mains unchanged, i.e. v s → v s ( t ) in Eq. (10). Then, thetime evolution of the parameter of v s can be routinelydetermined by calculating the time variation of the darksoliton’s energy dEdt = υ s (cid:90) + ∞−∞ dξ (cid:18) F (cid:0) m R , ψ s (cid:1) ∂ψ ∗ s ∂ξ + F ∗ (cid:0) m R , ψ s (cid:1) ∂ψ s ∂ξ (cid:19) , (11)with F ( m R , ψ ) = (cid:0) ¯ g R + i ¯ R (cid:1) m R ψ and the energy of thedark soliton being given by E = 12 (cid:90) + ∞−∞ dξ (cid:34)(cid:12)(cid:12)(cid:12)(cid:12) ∂ψ s ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) − | ψ s | (cid:17) (cid:35) = 43 (cid:0) − υ s ( t ) (cid:1) . (12)With the knowledge of the dark soliton’s energy of Eq.(12), we can easily calculate the time variation of the soliton’s energy, i.e. the left hand of Eq. (11). However,the calculations of the right-hand side of Eq. (11) requirethe information of the reservoir density of m R .In this work, we limit ourselves to the experiment-related fast reservoir limit, i.e. in the parameter regimeof ¯ γ C (cid:28) ¯ γ R . In such, directly following Refs. [18, 24], wecan obtain the density of reservoir as follows m R ( ξ ) = ¯ γ C (cid:16) − | ψ | (cid:17) + ¯ P PT ( ξ )¯ γ R , (13)with ¯ P PT being the dimensionless form PT - symmetricpumping of Eq. (3). Substituting Eq. (13) into Eq. (8),we can obtain i ∂ψ∂t = − ∇ ψ + (cid:18) − ¯ g R ¯ γ C ¯ γ R (cid:19) (cid:16) | ψ | − (cid:17) ψ + ¯ g R ¯ γ R ¯ P PT ψ + i R ¯ γ C ¯ γ R (1 − | ψ | ) ψ + i P PT ¯ γ R ψ. (14)Equation (14) is the key result of this Letter. In de-tail, the first term in the second line of Eq. (14) labelsthe PT - symmetric potential, giving the PT - symmetry-induced non-Hermitian; the second term in the secondline of Eq. (14) represents the non-Hermitian due to theintrinsic non-equilibrium nature; the last term in the sec-ond line of Eq. (14) can only exist in the case of both ¯ P PT and ¯ γ R being non-zero. In such, Equation (14) can cap-ture the combined effects of non-Hermitian due to bothnonequilibrium nature and PT symmetry in determiningthe properties of a system by including the nonequilib-rium nature characterized the intrinsic kinds of gain andloss of a system, to a PT - symmetric many-body quan-tum system.Finally, by plugging Eqs. (12) and (13) into Eq. (11),one can readily arrive at the equation of motion for thedark soliton’s velocity as follows dυ s dt = F eff , (15)with F eff representing an effective force given by F eff = 13 ¯ R ¯ γ C ¯ γ R υ s (cid:0) − υ s (cid:1) + 13 ¯ RP ¯ γ R υ s + π
32 ¯ RP ¯ γ R (cid:112) − υ s . (16)Based on Eq. (15), we can regard the dynamics of thedark soliton as the motion of a classical particle of mass m eff = 1 subjected to an external force F eff . As a firstcheck of validity of Eq. (15), we consider the case of van-ishing PT - symmetric pumping, i.e. P = P = 0. Insuch a case, Equation (15) can exactly cover the corre-sponding results in Ref. [18].Based on Eq. (15), the effects of PT -symmetric pump-ing in the non-equilibrium scenario on the dark solitoncan be explained as follows: (i) The first term in Eq.(16) comes from the effects due to the non-equilibriumnature; the second and third terms originate from the (a) (b) (c)(d) (e) (f)(g) (h) (i)(j) (k) (l) FIG. 2. Dynamics of a 1D dark soliton with the initial ve-locity υ s = 0 .
35 in the case of weak pumping. Shown arecontour plots of n C ( x, t ) and the real part of m R ( x, t )and thedependence n minC ( t ) computed using Eqs. (15) and (16). Pa-rameters are ¯ g R = 2, ¯ γ C = 3, and ¯ γ R = 15, ¯ R = 1 .
5, and(a)-(c) P = 0, P = 0; (d)-(f) P = 0 . P = 0 .
3; (g)-(i) P = 0 . P = 0 .
6; (j)-(l) P = 0 . P = 0 . real and imaginary part of PT -symmetric pumping. (ii)The last two terms in Eq. (16) contains two competitiveparts: while the action of the first part leads to accelera-tion of the dark soliton, the second part can be rewrittenas π
32 ¯ RP ¯ γ R (1 − υ s ) in the limit of slow velocity. Then theimaginary part of PT - symmetric pumping affects thedark soliton by giving a background fixed force of π
32 ¯ RP ¯ γ R and slowing down the motion of the dark soliton withthe force of − π
64 ¯ RP ¯ γ R υ s . With the knowledge of v s bysolving Eq. (15), we can proceed to obtain determinesdarkness of a dark soliton through the simple relation n min C ( t ) = v ( t ).Above, we have developed the analytical physical pic-ture of the dark soliton in a polariton condensate underthe nonresonant PT - symmetric pumping. Below we areready investigate how the combined effects of the non-equilibrium nature and PT - symmetry on a dark soliton(see Figs. 2) by numerically solving Eqs. (8) and (9)with the initial condition of Eq. (10).As a benchmark for later analysis, we recall the case ofthe vanishing PT - symmetric pumping ( P = P = 0), corresponding to the scenario of Ref. [18]. As shown inFigs. 2 (a)-(b), the non-equilibrium nature of the modelsystem is to blend the soliton with the background ata finite time. Meanwhile, trajectory and lifetime pre-dicted by Eq. (15) (see Fig. 2 (c)) are in excellent agree-ment with direct numerical simulations. We begin withdiscussing the effect of PT - symmetric pumping char-acterized by the parameters of P and P on the darksoliton of a polariton condensate with the fixed nonequi-librium nature related parameters. For this purpose,we devise the following scenario: we fix the real part P of PT - symmetric pumping and investigate the roleof the imaginary part P on the dark soliton. In Figs.2, we show the evolution of the condensate density dis-tribution n C ( x, t ) = | ψ ( x, t ) | and associated perturba-tions of the polariton reservoir density m R ( x, t ) in thefollowing figure (left and middle columns, respectively).The right column in figure shows the time dependence ofthe minimum value of the condensate density associatedwith the dark soliton. The red line shows the darkness n minC ( t ) = υ s ( t ) calculated analytically using Eq. (15),and shows an excellent agreement with numerics. Asshown in the left column of Fig. 2, although the lifetimeof the dark soltion becomes to be shorter with the in-crease of the P , the corresponding propagation time forthe dark soliton reaches t ∼ ps, which is much longerthan the condensate and reservoir relaxation times. Notethat with the further increase of P , it’s supposed thatthere will exist a quantum phase transition from the PT -symmetry phase to PT - symmetry broken phase [32, 33].The dark soliton should be destroyed in the parameterregime of PT - symmetry broken phase, which is beyondthe scope of this work.In summary, we have investigated the dynamics of darksolitons appearing in a polariton condensates under non-resonant PT - symmetric pumping. In particularly, wehave derived analytical expression of the equation of mo-tion of the velocity of the dark soliton. Within the frame-work of Hamiltonian approach, our analytical results cap-ture the essential physics as to how the combined effectsof the open-dissipative and PT Symmetry affects the life-time of the dark solitons. We also solve the modifiedopen-dissipative GP equation numerically. The numeri-cal results find agreement with the analytically ones. Wealso demonstrate that the dark solitons can exist in along time which being influenced by the PT - symmetricparameters. Acknowledgement—
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