Dissipative Magnetic Soliton in a Spinor Polariton Bose–Einstein Condensate

Magnetic soliton is an intriguing nonlinear topological excitation that carries magnetic charges while featuring a constant total density. So far, it has only been studied in the ultracold atomic gases with the framework of the equilibrium physics, where its stable existence crucially relies on a nearly spin-isotropic, antiferromagnetic, interaction. Here, we demonstrate that magnetic soliton can appear as the exact solutions of dissipative Gross–Pitaevskii equations in a linearly polarized spinor polariton condensate with the framework of the non-equilibrium physics, even though polariton interactions are strongly spin anisotropic. This is possibly due to a dissipation-enabled mechanism, where spin excitation decouples from other excitation channels as a result of gain-and-loss balance. Such unconventional magnetic soliton transcends constraints of equilibrium counterpart and provides a novel kind of spin-polarized polariton soliton for potential application in opto-spintronics.


I INTRODUCTION
Spinor polariton condensate in semiconductor microcavities [1][2][3][4] provides a unique out-ofequilibrium platform for exploring exotic nonlinear excitations with spin textures, which may even transcend usual restrictions of equilibrium systems. Formed from strong couplings between excitons and photons, polaritons possess peculiar spin properties: the J z ±1 (spin-up or spin-down) spin projections of the total angular momentum of excitons along the growth axis of the structure directly correspond to the right-and left-circularly polarized photons absorbed or emitted by the cavity, respectively [1]. Therefore, the properties of a spinor polariton fluid (e.g., density and phase distributions) can be probed from the properties of the emitted light [5]. In addition, the polariton-polariton interaction features a strong spin anisotropy [6][7][8], with a repulsive interaction between same spins ( g > 0) and a weaker, attractive, interaction between opposite spins ( g 12 < 0). Furthermore, a polariton condensate is intrinsically open dissipative, distinguishing it fundamentally from its atomic counterpart [9]. Recently, half-soliton [10,11] and half-vortices [12,13] behaving like magnetic monopoles have been experimentally observed in spinor polariton condensates under coherent pumping. There, the key prerequisite for such excitation is the spinanisotropic antiferromagnetic interaction, while dissipation only occurs as a perturbation. Instead, below, we present a new kind of polariton soliton that carries magnetic charges-dissipative magnetic soliton (DMS). In particular, whereas magnetic soliton cannot occur in equilibrium condensates with strongly spinanisotropic (antiferromagnetic) interactions, it can nevertheless appear in non-equilibrium spinor polariton condensates harnessing dissipation as essential resources.
Magnetic soliton [14][15][16] is a localized nonlinear topological excitation, which exhibits a density dip in one component and a hump in the other, but featuring a constant total density. It is a fundamentally important entity in the nonlinear context, as it provides an exceptional example of exact vector soliton solution that can exist outside the paradigmatic Manakov limit (g g 12 ); within this limit, a multicomponent nonlinear system is integrable [17][18][19]. It also attracts considerable interests in the condensed matter, offering interesting perspectives as regards many-body phenomenon of solitonic matter [20]. So far, magnetic soliton has only been realized in a spinor Bose-Einstein condensate (BEC) with nearly-isotropic spin interactions of antiferromagnetic type [14,21,22], 0 < g − g 12 ≪ g. This requirement is essential because it makes the density depletion-inevitably induced alongside spin excitation-strongly suppressed by a high energy cost, thus ensuring the characteristic constant density background of magnetic soliton. Beyond this regime, a stable magnetic soliton cannot occur in an atomic superfluid.
In this work, we theoretically show that a stable magnetic soliton can be formed in a linearly polarized polariton condensate under non-resonant excitations with a spatially homogeneous pump, even though g − g 12 > g. It is an exact soliton solution to the multicomponent driven-dissipative Gross-Pitaevskii (GP) equation, preserving its energy over infinitely long times-so coined as DMS. It stems from a dissipation-enabled mechanism rather than an energetic mechanism (cf. Figure 1): the spinpolarization excitation, originally coupled to other dissipative excitations in a multicomponent quantum fluid, becomes decoupled conditionally on the local balance of gain and loss, thus allowing non-decaying localized spin texture far from the spin-isotropic Manakov limit. We remark that DMS exists for a time-independent and spatially uniform pump, which affords an appealing advantage in view of potential application [23,24]: while polariton soliton has been well known to promise applications in opto-spintronics, present schemes for the generation and stabilization of solitons usually rely on complex engineering of the space-time profile of the pump [25][26][27][28][29][30], which requires optical isolation that has hitherto been challenging to integrate at acceptable performance levels and introduce redundant and power-hungry electronic components.
The structure of the paper is as follows. In Section II, we present our theoretical model of dissipative Gross-Pitaevskii equations, based on which we solve for the novel magnetic solitons that carry magnetic charges while featuring a constant total density. In Section IV, we present a comprehensive study of the physical mechanism of the magnetic solitons with the help of the dynamic structure factors. Finally, we conclude with a summary in Section V, and all the detailed mathematical derivations are outlined in Section A

II DISSIPATIVE GROSS-PITAEVSKII EQUATIONS AT QUASI-1D
Motivated by Ref. [31], we consider a spinor polariton BEC formed under a homogeneous incoherent pumping in a wireshaped microcavity that bounds the polaritons to a quasi-1D channel in the following geometry: In the x-direction, the polariton BEC is homogeneous; in the y-direction, the wire size d is sufficiently small compared to the wire length, thus providing a strong lateral quantum confinement. Moreover, the incoherent pump is also restricted to a small transverse size comparable to d. When Z 2 /(md 2 ) ≫ gn 0 , where m is the effective mass of polaritons and n 0 is the 1D polarion density, the polarion motion in the y direction can be seen as frozen. In this case, the order parameter for the polariton BEC at quasi-1D can be effectively described by a complex vector [32][33][34][35][36], ψ(x, t) ≡ [ψ 1 (x, t), ψ 2 (x, t)] T , in the circular basis. Here, ψ 1 and ψ 2 are the spin-up and spin-down wavefunctions, and we denote the density in each component by n 1 and n 2 , respectively. The system dynamics is governed by driven-dissipative GP equations coupled to a rate equation for the reservoir density n R [37][38][39][40][41]: iZ

FIGURE 1 | Schematics of DMS formation in an open-dissipative
polariton BEC under incoherent pumping, which is coupled to a reservoir via its density n R . On top of the steady state, there exist density (blue patch) and spin-polarization (red patch) excitation channels, which are usually coupled. However, once the pump balances the loss, the two channel, i.e., density excitation and spin-polarization excitation decouples (white curve). In such, the spin-polarization excitation is immune of the reservoir and a non-decaying DMS in spin-density excitation channel can occur.
Frontiers in Physics | www.frontiersin.org December 2021 | Volume 9 | Article 805841 Here, interactions between polaritons are typically g 12 < 0, g > 0, and |g 12 | < g. The interaction between the condensate and reservoir is modeled by constant g R . Condensed polaritons decay at a rate c C but are replenished from the reservoir at a rate R. This process is captured by D s iZ(Rn R − c C )/2. Reservoir polariton decays at a rate c R and is driven by an off-resonant continuouswave pump, which is spatially homogeneous. Note that, here, we have assumed that the reservoir lacks spin selectivity due to infinite fast spin relaxation [37].
The steady-state solutions of Eqs. 1-3 are given by ψ 0 where ψ 0 1(2) and n 0 R denote the steady-state condensate wavefunction of each component and reservoir density, respectively. As shown, the steady-state polariton BEC has a uniform density determined by n 0 P/c C − c R /R and is linearly polarized with a stochastic polarization direction in the absence of pinning [1]. Note that Eqs. 1-3 in the limit of fast reservoir [3,42] are of immediate relevance in the context of the complex Ginzburg Landau equations [23,24]. In the following, we choose system parameters where such steady state is within the modulation stable regime [42][43][44][45] (this can be further seen in Section IV).

III DISSIPATIVE MAGNETIC SOLITON
On top of the steady state, two kinds of excitations can occur: density excitation and spin-polarization excitation. These excitations are, in general, correlated with each other and with the reservoir, so that fluctuations in one channel can induce that in another and are dissipative. As shown below, the central result of this work is that under the condition the spin-polarization excitation decouples from other dissipative channels, such that it can support a new kind of nonlinear excitation against the steady-state background in situations not allowed in the equilibrium case. We look for an analytical solution satisfying Eqs. 1-3, which describes a moving soliton with velocity υ. For simplicity, hereafter, we will denote η x − υt. To describe populations in each component, we rewrite n 1 n 0 (1 + δn 1 )/2 and n 2 n 0 (1 − δn 2 )/2 in terms of the total density n 0 and the dimensionless variables δn 1 (2) . We, moreover, define a linear polarization angle φ r and global phase φ g . The order parameter can then be generically written as with μ R g R c C /R. We consider general boundary conditions: lim η→±∞ δn 1(2) (η) 0 and lim η→±∞ z η φ r(g) (η) 0. Our goal next is to determine n 0 , δn 1(2) , φ r , and φ g .
Exact solutions for Eqs. 1-3 can be found under condition (5). The detailed calculations can be found in Appendix A. The results are: φ r arctan Here, U υ/ n 0 (g − g 12 )/2m is a dimensionless velocity and ξ s Z/ 2mn 0 (g − g 12 ) denotes the spin healing length.
A typical space-time profile of the above soliton solution is illustrated in Figure 2A for g 12 − 0.1g. The density distribution n 1 (2) in each component and φ r and φ g at a chosen time are shown in Figure 2B. We see that, unlike half-solitons, the vector soliton here is characterized by a density notch in one component and a hump in the other, whereas n 1 + n 2 ≡ n 0 is constant, i.e., it is magnetic soliton (see Figure 2A and top panel of Figure 2B). The linear polarization angle φ r and the global phase φ g vary simultaneously in space (see bottom panel of Figure 2B): φ r always jumps by π across the soliton, lim η→+∞ φ r − lim η→−∞ φ r π, regardless of soliton velocity. In contrast, the phase jump of φ g is velocity dependent, with the maximum shift − π only for stationary case.
To verify the above analytical solution, we have numerically solved Eqs. 1-3 starting from an initial order parameter given by Eqs. 6-9 for t 0 along with n R (0) c C /R. Comparisons of numerical and analytical solutions show perfect agreement; see Figure 2B for t/τ 15. We have numerically verified the stability of our solution by time evolving an initial order parameter where n 1 (0) − n 2 (0) is perturbed from Eq. 7 while keeping n 0 n 1 (0) + n 2 (0) fixed.
The polarization texture of polariton magnetic soliton in Eqs. 6-9 can be characterized by standard Stokes parameters [33,46,47], S(η) (S x , S y , S z ), with S x (η) 2R(ψ * 1 ψ 2 )/n 0 , S y (η) 2I(ψ * 1 ψ 2 )/n 0 , and S z (η) (|ψ 1 | 2 − |ψ 2 | 2 )/n 0 . Here, R and I denote the real and imaginary part, respectively. For the stationary case U 0, S(η) is entirely in the (S x , S z ) plane and presents an ingoing divergent spin texture whose direction defines the magnetic charge (see top panel of Figure 2C): The degree of circularization S z reaches unity at the center, while the linear polarization S x flips its direction crossing the soliton core due to the π jump in φ r . In comparison, a moving soliton (see bottom panel of Figure 2C) has a broadened localization width, l w ξ s / 1 − U 2 √ , and its polarization becomes strongly elliptical in the (S y , S z ) plane near the core, with a decreased circular polarization (i.e., magnetization) given by S z (η) 1 − U 2 √ . However, the linear polarization still flips across the soliton, independent of U. To see whether the polariton magnetic soliton in an opendissipative spinor condensate decays with time, we calculated its energy E as [41,48,49].
Here, the second term corresponds to the spin-spin interaction associated with S z , and the third term is the energy associated with the density depletion. Once the gain balances loss, as formulated by Eq. 5, we derive straightforwardly (see Eq. B2 in Appendix)  Such non-decaying polariton magnetic soliton, therefore, belongs to dissipative solitons [50][51][52].
Such dissipative magnetic soliton (DMS) is quite unconventional, as its creation cannot be understood along the line of the well known example in the equilibrium context. In Bose condensed atomic gas, the key prerequisite for creating magnetic soliton is an antiferromagnetic interaction satisfying g − g 12 ≪ g, i.e., close to the spin-isotropic Manakov limit (g g 12 ). This condition, as can be seen from Eq. 10, creates a large energy separation between the density and spin-polarization excitations: The density depletion from n 0 near the soliton core requires much more energy than that associated with S z , making the former energetically suppressed and thus ensuring a constant total density that characterizes magnetic soliton. However, such scenario fails here because polaritons feature g − g 12 > g.

IV DISSIPATION-ENABLED FORMATION MECHANISM
To understand this unconventional phenomenon, the fact that Eqs. 6-9 are exact solutions offer a "sweet point." We see that the balance of gain and loss [Eq. 5] is the key for fixing the background density at n 0 P/c C − c R /R. Simultaneously, this gives rise to a closed real equation for the magnetization [dS z (y)/dy] 2 + S 4 z (y) − (1 − U 2 )S 2 z (y) 0 with y η/ξ s ; that is, the spin polarization excitation is decoupled from other excitation channels. We emphasize that such conditionally coherent dynamics has a fundamentally different origin from that in a purely conservative system such as atomic BEC. As such, magnetic soliton in the former case can occur far from the Manakov limit, in contrast to the latter where it is only possible when the deviation from g g 12 is small breaking slightly system integrability.
The above dissipation-enabled decoupling of excitations is at the heart of DMS formation, which also manifests itself in the linear excitation regimes, e.g., in the excitation spectrum and linear response function. Briefly, to describe a spinor polariton BEC linearly perturbed from the steady state, we substitute Eq. 6 into Eqs. 1-3 and follow the standard Bogoliubov-de Gennes (BdG) approach (see details in Appendix C). The eigen-energy Zω q of excitations solves the equation being the free-particle energy. Two decoupled equations follow: the quadratic equation immediately yields Zω q ±Zω S for the energy of the spin-polaridzation excitation, whereas the cubic equation reflects the coupled linear excitations in the reservoir and density channel of polariton BEC. Importantly, we see ω s is purely real (Figure 3A), regardless of whether the reservoir is fast or slow compared to the polariton BEC. This feature of the linear spin polarization excitation contrasts to the linear density excitation that generically exhibits a complex energy Zω D and eventually damps out. The latter is most transparent in the fast reservoir limit c R /c C ≫ 1. There, an adiabatic elimination of the reservoir gives Zω D − iΓ/2 ± Zω 0 , with Zω 0 ε 0 q [ε 0 q + (g + g 12 )n 0 − 2g R ZΓ/R] − Γ 2 /4 and Γ n 0 n 0 R R 2 Z/(c R + n 0 R). Note that ω D is purely imaginary for |q| ≤ q c due to polariton losses, with q c m( α 2 + Γ 4 /4 √ − α 2 )/Z 2 . Here, we introduced parameter α P/P th − 1 and the threshold value P th c R c C /R. In Figures  3A,B, we show the complex spectrum of linear density excitation for c R /c C ≫ 1 where the analytical results agree with numerical solutions of BdG equations. Note that the damping spectrum in Figure 3B shows that the considered steady state is indeed modulationally stable.
To further visualize the decoupling of excitations as a result of the balance between gain and loss, we analyze the linear response of the system. Considering an external density perturbation described by λe i(qx−ωt) + H. c with λ ≪ 1 is acted on the exciton-polariton BEC, we calculate the density static structure factor S D (q) and the spin-density static structure factor S S (q) [53]. For simplicity of analytical derivation, we assume fast reservoir limit and obtain (see the details in Appendix C) and we also find S S (q) 0. Figure 3C shows lim q→∞ S D (q) → 1, meaning the response of a polariton BEC to a density perturbation is exhausted by the density excitation, without collateral generations of excitations in other excitation sectors. If the system is instead subjected to a spin-dependent perturbation λσ z e i(qx−ωt) + H. c, we find S S (q) Zq 2 /(2mω S ), which approaches unity for q → ∞ and S D (q) 0 (see Figure 3D). This further verifies that a perturbation in the spin polarization sector only induces spin excitations.

V CONCLUDING DISCUSSIONS
Summarizing, we theoretically show that a new kind of soliton DMS can be created in a spinor polariton condensate. The value and significance of our work are twofold. First, DMS has no atomic counterpart and relies crucially on the open-dissipative property of the system, in contrast to solitons discussed in Refs. [25-30, 37, 54, 55] and half-solitons in Refs. [10,11]. Second, DMS provides a rare example of exact solutions to the dissipative GP equations at quasi-1D. In the future, it is interesting to explore concrete proposals for the experimental observation of the predicted phenomenon within feasible facilities and to study the unique quantum many-body physics associated with a collection of DMSs with same (opposite) magnetic charges. Furthermore, in our present Frontiers in Physics | www.frontiersin.org December 2021 | Volume 9 | Article 805841 theoretical illustration, the condition of Eq. 5 reduces to D s 0, but the concept of dissipation-enabled decoupled excitations applies for generic cases where D s ψ s 0 rather than D s 0 holds. Thus, it is also interesting to explore in a broader context other new kinds of dissipative solitons that can arise from excitation decoupling.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
ZL and YH have developed and supervised the research projects with the help of W-ML. CJ and RW have done the detailed calculations. All the authors contribute to writing the manuscript.
FUNDING BEC with n 0 1 n 0 2 n 0 /2, where n 0 P/c C − c R /R and n 0 R c C /R. We further have μ T 1 2 (g + g 12 )n 0 + g R n 0 R . For linear excitations, we follow the standard procedures of Bogoliubov decomposition and write [40]. and It's convenient to rewrite the excited components in Eq. (C1) in terms of u d u 1q + u 2q and v d v 1q + v 2q , and u s u 1q − u 2q and v s v 1q − v 2q , which are then subsequently substituted into Eqs. A1-A3. Retaining only the first-order terms of the fluctuations, we obtain the Bogoliubov-de Gennes (BdG) equation as