Quench Dynamics and Stability of Dark Solitons in Exciton–Polariton Condensates

Abstract

Exciton–polariton condensates (EPCs) have emerged as a paradigmatic platform for investigating nonequilibrium quantum many-body phenomena, particularly due to their intrinsic open-dissipative nature and strong nonlinear interactions governed by the interplay between stimulated scattering and reservoir-mediated damping. Recent advances in Feshbach resonance engineering now enable precise tuning of interaction strengths, opening new avenues to explore exotic nonlinear excitations in these driven-dissipative systems. In this work, we systematically investigate the quench dynamics and stability of dark solitons in repulsive one-dimensional EPCs under sudden parameter variations in both nonlinear interaction strength g and pump intensity P. Through a Hamiltonian variational approach that incorporates reservoir damping effects, we derive reduced equations of motion for soliton velocity evolution that exhibit remarkable qualitative agreement with direct numerical simulations of the underlying open-dissipative Gross–Pitaevskii equation. Our results reveal three distinct dynamical regimes: (i) stable soliton propagation at intermediate pump powers, (ii) velocity-dependent soliton breakup above critical pumping thresholds, and (iii) parametric excitation of soliton trains under simultaneous interaction quenches. These findings establish a quantitative framework for understanding soliton dynamics in nonresonantly pumped EPCs, with implications for quantum fluid dynamics and nonequilibrium Bose–Einstein condensates.

1. Introduction

Exciton–polariton condensates (EPCs) in semiconductor microcavities represent a highly promising platform for investigating fundamental quantum many-body problems and nonequilibrium phenomena. Owing to their inherent nonequilibrium character, polariton condensates have garnered significant attention from the physics community over the past two decades. As hybrid quasiparticles, exciton–polaritons exhibit properties of composite bosons, arising from the strong coupling between quantum-well excitons and cavity photons. The photonic component endows polaritons with a low effective mass and a readily excitable nature, while the excitonic component mediates interactions between polaritons [1,2]. Benefiting from these synergistic attributes inherited from their constituent components, polaritons can more readily form macroscopic coherent quantum states—namely, exciton–polariton condensates—without requiring the ultralow temperatures necessary for conventional atomic condensates. To date, multiple experimental groups have achieved exciton–polariton condensation at elevated temperatures [3,4,5,6,7] and even at room temperature [7,8]. Collectively, the experimental realization of exciton–polariton condensates has inaugurated a new frontier of research, where the study of nonlinear excitations and nonequilibrium dynamics remains a persistent and captivating focus.

Soliton formation in atomic Bose–Einstein condensates (BECs) is classically understood to arise from the balance between dispersion and nonlinear interactions [9,10]. Unlike conservative systems (e.g., atomic BECs), where solitons can persist indefinitely, exciton–polariton solitons exhibit finite lifetimes due to the rapid radiative decay inherent to their hybrid quasiparticle nature. Consequently, sustaining a stationary soliton state requires continuous pumping to compensate for polariton losses. These nonlinear excitations display heightened instability in higher-dimensional systems—specifically, in two or three dimensions, solitons readily decay into vortex states. Such dynamics have made them a focal point of extensive research. Notable experimental and theoretical studies include reports of solitons [11,12,13,14,15,16,17,18] and vortices [19,20,21,22] in microcavity polariton condensates. In particular, Refs. [23,24] have specifically examined dark solitons in polariton condensates, analyzing their behavior under resonant and nonresonant pumping configurations, respectively.

Distinct nonlinear excitations such as bright and dark solitons emerge from competing interaction regimes in quantum many-body systems. While attractive interactions (characterized by negative s-wave scattering lengths $a<0$) stabilize bright solitons [25], repulsive interactions promote dark soliton formation. Feshbach resonances provide an unparalleled platform for investigating these nonlinear phenomena through precise manipulation of interaction strengths via magnetic field tuning of scattering lengths [26,27]. Recent advances demonstrate that quenching interaction strengths using Feshbach resonances can induce dynamical phase transitions, as evidenced by dark soliton splitting in atomic Bose–Einstein condensates when crossing critical interaction thresholds [28]. This approach has further revealed rich nonlinear dynamics in systems ranging from ultracold atoms [29] to exciton–polariton condensates [30], where parametric amplification and soliton trains emerge under controlled interaction quenches [31]. The interplay between Feshbach-tuned interactions and nonequilibrium dynamics continues to unlock new regimes in quantum fluid behavior.

Soliton splitting represents a fundamental nonlinear phenomenon that manifests across diverse physical systems, including optical fibers [32,33] and BECs [34,35]. In photonic systems, higher-order dispersion or nonlinear effects can trigger soliton splitting. This phenomenon is typically accompanied by the formation of quantum shock waves, characterized by spectral broadening and wave-breaking effects. Quantum shock waves have emerged as a distinct nonlinear phenomenon attracting significant research attention. One formation mechanism involves quantum coherence between the background and wavepackets [36]. These shock waves exhibit characteristic abrupt changes in physical properties across the wavefront [37]. Recent theoretical research demonstrates that controlled generation of shock waves in polariton condensates can be achieved through modulation of external potentials and incoherent pumping [38]. Shock waves have not only been observed in BECs [39,40,41,42] but were found in optics [43,44,45], water waves [46,47], and even Rydberg atomic gases [48,49]. Despite these amazing results, the physics of quantum dynamics and nonlinear excitations using resonance techniques is still not completely understood. So far, there have not been studies on the quench dynamics and stability of dark solitons in polariton condensates.

In this work, we are motivated to launch a systematic investigation of the quench dynamics and stability of dark solitons in repulsive, one-dimensional EPCs subjected to the sudden quenches of both the nonlinear interaction parameter and the laser pump strength. Specifically, when the interaction strength undergoes an abrupt change, the initial soliton splits and generates a pair of faster asymmetric solitons. To this end, by adopting a dissipative Gross–Pitaevskii description of EPCs, we investigate the static and dynamical properties of dark soliton. First, the equation of motion for the center of mass of the dark soliton’s center is derived analytically by using the Hamiltonian approach. The resulting equation captures how the combination of the quench and the open-dissipative character can affect the properties of the dark soliton. Further numerical solutions are designed to study the dynamics of the dark soliton by sudden quenches of both the nonlinear interaction parameter and the laser pump strength, which are in good agreement with the analytical results.

The present paper is structured as follows. In Section 2, we introduce the theoretical framework and derive the effective Gross–Pitaevskii equation under the fast reservoir limit. Next, in Section 3, we investigate the quench dynamics of dark solitons within exciton–polariton condensates, presenting results from both analytical treatments and direct numerical simulations. Finally, in Section 4, we summarize our key findings.

2. The Theoretical Model

In the present work, we investigate the dynamics of dark solitons in polariton condensates using diverse quench engineering. To this end, we consider an incoherently far off-resonantly pumped exciton–polariton condensate in a one-dimensional (1D) setting. In the mean-field approximation, the behavior of the polariton condensate is governed by the generalized open-dissipative Gross–Pitaevskii (GP) equation, with the condensate wave function $\psi \left(x,t\right)$ coupled to the hot exciton reservoir density $n_R\left(x,t\right)$ [50,51,52] so that

$$\begin{array}{ccc}\hfill i\hslash {\displaystyle \frac{\partial \psi }{\partial t}}& =& \left[-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}+g_C{\left|\psi \right|}^2+g_R{n}_R+{\displaystyle \frac{i\hslash }{2}}\left(R{n}_R-{\gamma }_C\right)\right]\psi ,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial n_R}{\partial t}}& =& P-\left({\gamma }_R+R{\left|\psi \right|}^2\right)n_R.\hfill \end{array}$$

(2)

Before proceeding, we remark that the detailed derivation of Equations (1) and (2) can be found in the review work of Ref. [52].

In Equations (1) and (2), m is the effect mass of the polariton, $g_C$ is proportional to the polariton–polariton interaction strength, $g_R$ is proportional to the interaction strength between the hot exciton reservoir and polariton condensate, the condensate polaritons are continuously replenished from the hot exciton reservoir at a rate R, ${\gamma }_C$ and ${\gamma }_R$ characterize the loss rate of the polariton and hot excitons, respectively, and P represents of the strength of the laser pump. Note that the parameters of $g_C$, $g_R$, and R in Equations (1) and (2) have been rescaled into the one-dimensional case by the width d of the nanowire thickness as ($g_C\to g_C/\sqrt{2\pi d}$, $g_R\to g_R/\sqrt{2\pi d}$, and $R\to R/\sqrt{2\pi d}$). We remark that the model system described by Equations (1) and (2) corresponds to an exciton–polariton Bose–Einstein condensate under nonresonant pumping, realized in a wire-shaped microcavity analogous to that implemented in Ref. [53]. This geometry confines polaritons to a quasi-one-dimensional (1D) channel.

In relevant experiments, the values of the parameters can be set as follows: $m=5\times {10}^{-5}{m}_e$, $g_C=0.475\mathsf{\mu }\mathrm{eV}·\mathsf{\mu }\mathrm{m}$, $g_R=0.95\mathsf{\mu }\mathrm{eV}·\mathsf{\mu }\mathrm{m}$, $R=2.24\times {10}^{-4}\mathsf{\mu }{\mathrm{mps}}^{-1}$,${\gamma }_R=1.5\times {\gamma }_C=0.25{\mathrm{ps}}^{-1}$, and $P=1.1{\mathrm{ps}}^{-1}\mathsf{\mu }{\mathrm{m}}^{-2}$. We study the quench dynamics in a repulsive one-dimensional exciton–polariton condensate, where both the nonlinearity parameter and the laser pump parameter undergo a sudden quench.

For subsequent analysis, all variables need to be dimensionless in Equations (1) and (2). In the dimensionless process, we set $x^{\prime }=x/r_h$ and $t^{\prime }=t/{\tau }_0$, where ${\tau }_0=r_h/c_s=m{r}_h^2/\hslash$, $c_s=\hslash /m/r_h$ is the local sound velocity, and $r_h$ is the healing length in the condensate. As such, Equations (1) and (2) can be rewritten into

$$\begin{array}{ccc}\hfill i{\displaystyle \frac{\partial \psi }{\partial t}}& =& \left[-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}+g{\left|\psi \right|}^2+{\overline{g}}_R{\overline{n}}_R+{\displaystyle \frac{i}{2}}\left[\overline{R}{\overline{n}}_R-{\overline{\gamma }}_C\right]\right]\psi ,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\overline{n}}_R}{\partial t}}& =& \overline{P}-\left({\overline{\gamma }}_R+\overline{R}{\left|\psi \right|}^2\right){\overline{n}}_R,\hfill \end{array}$$

(4)

with $m{r}_h^2{g}_C/{\hslash }^2\equiv g$, $m{r}_h^2{g}_R/{\hslash }^2\equiv {\overline{g}}_R$, $m{r}_h^{2}R/\hslash \equiv \overline{R}$, $m{r}_h^2{\gamma }_C/\hslash \equiv {\overline{\gamma }}_C$, $m{r}_h^2{\gamma }_R/\hslash \equiv {\overline{\gamma }}_R$, $n_R\equiv {\overline{n}}_R$, and $t^{\prime }\to t$ and $x^{\prime }\to x$ set in the above equations. Furthermore we consider $\psi \left(x,t\right)={\psi }^{\prime }\left(x,t\right)exp\left[-ig{n}_{0}t\right]$. The resulting dimensionless equations take the form

$$\begin{array}{ccc}\hfill i{\displaystyle \frac{\partial }{\partial t}}{\psi }^{\prime }& =& \left[-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}+g\left({\left|{\psi }^{\prime }\right|}^2-n_0\right)+{\overline{g}}_R{\overline{n}}_R+{\displaystyle \frac{i}{2}}\left(\overline{R}{\overline{n}}_R-{\overline{\gamma }}_C\right)\right]{\psi }^{\prime },\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial t}}{\overline{n}}_R& =& \overline{P}-\left({\overline{\gamma }}_R+\overline{R}{\left|{\psi }^{\prime }\right|}^2\right){\overline{n}}_R.\hfill \end{array}$$

(6)

Then, we restrict our analysis to the fast reservoir limit, characterized by the condition ${\overline{\gamma }}_C\ll {\overline{\gamma }}_R$. Expanding the reservoir dynamics to leading order, the hot exciton reservoir density $n_R$ satisfies

$${\overline{n}}_R={\displaystyle \frac{\overline{P}}{{\overline{\gamma }}_R}}-{\displaystyle \frac{\overline{P}\overline{R}}{{\overline{\gamma }}_R^2}}{\left|{\psi }^{\prime }\right|}^2.$$

(7)

Under the fast reservoir limit, the dynamics of the condensate results from Equations (5) and (6) reads

$$\begin{array}{c}\hfill i{\displaystyle \frac{\partial }{\partial t}}{\psi }^{\prime }+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}{\psi }^{\prime }-g\left({\left|{\psi }^{\prime }\right|}^2-n_0\right){\psi }^{\prime }=P\left({\psi }^{\prime }\right),\end{array}$$

(8)

where $P\left({\psi }^{\prime }\right)$ represents the nonequilibrium nature, i.e., the intrinsic kinds of gain and loss of the polariton condensate, which is given by

$$\begin{array}{ccc}\hfill P\left({\psi }^{\prime }\right)& =& \left[{\overline{g}}_R{\overline{n}}_R+{\displaystyle \frac{i}{2}}\left(\overline{R}{\overline{n}}_R-{\overline{\gamma }}_C\right)\right]{\psi }^{\prime },\hfill \\ & =& {\overline{g}}_R\left({\displaystyle \frac{\overline{P}}{{\overline{\gamma }}_R}}-{\displaystyle \frac{\overline{P}\overline{R}}{{\overline{\gamma }}_R^2}}{\left|{\psi }^{\prime }\right|}^2\right){\psi }^{\prime }+{\displaystyle \frac{i}{2}}\left[\overline{R}\left({\displaystyle \frac{\overline{P}}{{\overline{\gamma }}_R}}-{\displaystyle \frac{\overline{P}\overline{R}}{{\overline{\gamma }}_R^2}}{\left|{\psi }^{\prime }\right|}^2\right)-{\overline{\gamma }}_C\right]{\psi }^{\prime }.\hfill \end{array}$$

(9)

In what follows, we plan to investigate the quench dynamics and stability of dark solitons in repulsive one-dimensional exciton–polariton condensates undergoing a sudden quench simultaneously in the nonlinearity and pump parameters based on both analytically and numerically solving Equations (8) and (9). The emphasis and value of this work is to investigate how the relationship between the dissipation and pumping can affect the quench dynamics of the nonlinear excitations in a nonequilibrium condensate.

3. The Hamiltonian Approach and Dynamics of the Dark Soliton Induced by the Quenched Interaction

The goal of Section 3 is to derive the equation of motion for the center of mass of the dark soliton by solving Equation (8) subjected to time-dependent perturbations provided by Equation (9) in the limit of a fast reservoir.

As a first step, we initially consider the simple case where $P\left({\psi }^{\prime }\right)=0$ in Equation (8), i.e., in the absence of the open-dissipative. Meanwhile, we limit the case of $g>0$ corresponding to the repulsive interaction. There exists an exact dark soliton solution to Equation (8), taking the form of

$${\psi }^{\prime }=\sqrt{n_0}\left[i{\displaystyle \frac{u}{c_s}}+\sqrt{1-{\displaystyle \frac{u^2}{c_s^2}}}tanh\left[\sqrt{1-{\displaystyle \frac{u^2}{c_s^2}}}{\displaystyle \frac{x-ut}{\sqrt{2}{r}_h}}\right]\right],$$

(10)

with $n_0$ being the background density and $r_h$ being the healing length or coherence length. We recall that $c_s$ accounts for the speed of sound while u denotes the velocity of the dark soliton itself. Now, we recall the fundamental properties of dark solitons. A soliton is often referred to as a “black” soliton when the velocity is zero. However, for a moving (“gray”) soliton, the minimum value of density $n_{\min }$ increases in proportion to the square of the soliton’s velocity, i.e., $n_{\min }=u^2$.

Next, we consider the open-dissipative properties as perturbations to the condensate and study the finite-amplitude collective excitations in a homogeneous condensate as captured by $P\left({\psi }^{\prime }\right)\ne 0$. In other words, our aim is to investigate the quench dynamics of dark solitons in condensates with nonequilibrium properties. To the end, we plan to use the Hamiltonian approach [2,12,16,54,55] to solve Equation (8). At the heart of the Hamiltonian approach of quantum dynamics for dark solitons is the assumption that the parameters of the dark solitons supported by Equation (8) become slow functions of time in the presence of perturbation in Equation (9), i.e., $u\to u\left(t\right)$, while the functional form of the dark soliton in Equation (10) remains unchanged. As such, the time-dependent variation in the soliton’s parameters can be analytically derived by the time evolution of the dark soliton’s energy as follows,

$${\displaystyle \frac{dE}{dt}}=\int dx\left(P^{*}\left({\psi }^{\prime }\right){\displaystyle \frac{d}{dt}}{\psi }^{\prime }+P\left({\psi }^{\prime }\right){\displaystyle \frac{d}{dt}}{\psi }^{\prime *}\right).$$

(11)

In Equation (11), the excitation energy of the dark soliton can be calculated by

$$\begin{array}{cc}\hfill E& ={\displaystyle \frac{1}{2}}\int dx\left({\left|{\displaystyle \frac{\partial {\psi }^{\prime }}{\partial x}}\right|}^2+g{\left(n_0-{\left|{\psi }^{\prime }\right|}^2\right)}^2\right),\hfill \\ \hfill & ={\displaystyle \frac{\sqrt{2}{n}_0{\left(c_s^2-u^2\right)}^{3/2}\left(1+2g{r}_h^2{n}_0\right)}{3c_s^3{r}_h}}.\hfill \end{array}$$

(12)

Before proceeding, we establish methodological distinctions between our Hamiltonian approach and related analytical frameworks in Refs. [50,51]: (i) While Ref. [50] investigates linear excitations and their spectral properties through Bogoliubov’s theory within the linear response formalism, our work focuses on quench dynamics of nonlinear excitations—specifically dark solitons—exhibiting non-perturbative temporal evolution. (ii) Both our study and Ref. [51] address dark soliton dynamics but employ distinct approaches: Carretero-González et al. in Ref. [51] utilize reductive perturbation theory to derive a Korteweg–de Vries (KdV) equation with linear damping, valid under weak reservoir coupling where reservoir excitations remain infinitesimal. Conversely, we operate in the fast reservoir limit by directly solving the microscopic Equation (8), capturing nonlinear soliton–reservoir interactions without perturbative approximations.

At the end of the analytical calculation, by inserting Equations (9) and (12) into Equation (11), one can smoothly derive the equation of motion for the velocity of the dark soliton, which takes the form

$$\begin{array}{cc}\hfill {\displaystyle \frac{du}{dt}}& ={\displaystyle \frac{r_h}{3\sqrt{2}\left(1+2g{r}_h^2{n}_0\right)c_s{\overline{\gamma }}_R^2}}\hfill \\ \hfill & \times \left\{6c_s^{2}u{\overline{\gamma }}_R\left(\overline{P}\overline{R}-{\overline{\gamma }}_C{\overline{\gamma }}_R\right)-2u{n}_0\overline{P}{\overline{R}}^2\left(2u^2+c_s^2\right)\right\}.\hfill \end{array}$$

(13)

We find that, for a certain situation where ${\overline{\gamma }}_C\ll {\overline{\gamma }}_R$, the trajectory of the dark soliton over long-term evolution indeed conforms to the form of Equation (13). According to this formula, we understand that the quench dynamics of the dark solitons is solely dependent on the quench interaction strength and the open-dissipative nature of the reservoir system.

Now, we are ready to investigate the quench dynamics and stability of dark solitons in repulsive one-dimensional exciton–polariton condensates undergoing a sudden quench simultaneously in the nonlinearity and pump parameters based on both analytically and numerically solving Equations (8) and (9) by designing two scenarios. The first scenario involves exciton–polariton condensates undergoing a sudden quench of the nonlinearity parameter; i.e., the strength of the interaction changes from $g_1$ to $g_2$. On the other hand, the second scenario involves exciton–polariton condensates undergoing a sudden quench of the laser pump parameter; i.e., the strength of the pump changes from ${\overline{P}}_1$ to ${\overline{P}}_2$. This can be achieved by either varying the coupling constant g of the system or changing the laser pump P in the experiment. The above expression or above result can also be illustrated in diagrammatic form, as shown in Figure 1 and Figure 2.

We revisit the scenario of non-quenching dynamics under nonresonant pumping, where the interaction strength remains consistent before and after the non-quenching process—consistent with the setup described in Ref. [56] (as illustrated in Figure 1a). As depicted, in an open-dissipative system, dark solitons eventually decay after persisting for a finite duration, with their lifetime modulated by dissipation parameters. The dynamical trajectory and lifetime of such dark solitons can be obtained via real-time evolution of the Gross–Pitaevskii equation.

Figure 1b–d present the time evolution of the minimum condensate density associated with the dark soliton, corresponding to $g_2={\displaystyle \frac{1}{2}}{g}_1$, $g_2=2g_1$, and $g_2=4g_1$, respectively. Key observations from Figure 1 are summarized as follows: When the post-quench interaction strength is halved relative to the initial value (Figure 1b), shock waves emerge during dark soliton evolution. For a post-quench strength doubled relative to the initial value (Figure 1c), the soliton splits into two distinct solitons, accompanied by the formation of a shock wave. In contrast, when the post-quench strength is quadrupled (Figure 1d), the soliton splits during evolution, producing a pair of asymmetric solitons without any shock wave formation. Soliton splitting occurs due to changes in the interaction strength, which alter the dynamical phase of the soliton. This causes the initial soliton to split into two faster-moving solitons, while the original soliton becomes narrower.

Figure 2a illustrates the quench dynamics of a 1D dark soliton in a conventional Bose–Einstein condensate (BEC) via quenching the interaction strength, i.e., in the absence of open-dissipative effects. In this conservative system, the velocity of the dark soliton remains constant during propagation. This figure validates the correctness of our scheme by demonstrating the reproducibility of dark soliton quench dynamics in conservative regimes. By comparing with Figure 1b, we observe that while a shock wave still emerges for $g_2={\displaystyle \frac{1}{2}}{g}_1$, the presence of dissipation reduces the lifetime of both the soliton and the shock wave. Notably, Figure 2d shows that numerical solutions of Equations (5) and (6) (plotted as red circles) agree well with those obtained from solving Equation (8) (plotted as a blue curve) within the allowable error range. Figure 2b,c present the quench dynamics of a 1D dark soliton under laser pumping intensity quenching, corresponding to ${\overline{P}}_2=27$ and ${\overline{P}}_2=30$, respectively. The insets in (a), (b), and (c) display the density profiles at $t=20$. These results reveal that shock waves persist during dark soliton propagation even under laser pumping intensity quenching.

4. Discussion and Conclusions

In summary, we have investigated the quench dynamics of dark solitons in a polariton condensate under nonresonant pumping. The dynamics are studied through independent quenches of the interaction strength and the pump laser intensity. Using a Hamiltonian approach, we have derived analytical expressions governing the dark soliton’s equation of motion for its velocity. During the evolution, phenomena such as splitting of the initial soliton, generation of asymmetric solitons, and formation of shock waves may occur under specific scenarios. Our numerical solutions of the modified open-dissipative Gross–Pitaevskii equation confirm these analytical findings.