Manipulation of a stable dark soliton train in polariton condensate

Yan Jiang; Gang Wang; Xuemei Sun; Shouhua Feng; Yan Xue

doi:10.1364/OE.27.010185

1. Introduction

Exciton polaritons [1] in semiconductor microcavity are a breed of bosonic optelectronic excitations mixed by cavity photons and intracavity excitons in the strong-coupling regime. These quasiparticles possess both a photonic and excitonic nature. While the fast escape of photons out of the microcavity makes polaritons intrinsically open-dissipative, their excitonic component provides strong repulsive interactions. With the optical pump to replenish the particles against the radiative loss, microcavity polaritons constitute an archetypal system to study the effect of quantum fluids, in which remarkable effects such as superfluidity [2], bistability [3], Bogoliubov excitation spectrum [4], and the hydrodynamic nucleation of quantized vortices [5–7] or dark solitons [8,9] have been observed.

Dark soliton, a fundamental nonlinear collective excitation, is a self-localized and shape-preserving stationary solution in a one-dimensional (1D) quantum degenerated fluids with positive mass and repulsive interactions. They can be thought of natural modes of nonlinear wave equations and have been observed in a wide variety of systems, such as atomic condensates [10,11], optical fibers [12], superconductors [13,14] and magnetic films [15]. In polariton system, due to the intrinsic dissipative nature, solitons are known to be qualitatively different from the ones present in Hamiltonian systems. They may exist with an unstable state and be evolved in a complicated and often chaotic manner [16, 17]. Several proposals are presented to improve the stability of the dark soliton in some particular cases: under the nonresonant excitation, the dark soliton trains with stability are firstly theoretically demonstrated in the spinor system with the help of potential step [18], and three years later, the controls of the stable dark soliton on demand [19] is reported in the scalar system. Under the coherent resonant excitation, scalar dark soliton trains [20] are also demonstrated with the pulsed excitation [21] based on the mechanism of quantum interference. Detailed studies about the dark soliton (trains) with stability are still required with the consideration of underlying application in photonics and quantum information.

The purpose of this paper is to, by exploiting the quantum interference in spatial domain, investigate the generation and stability of the sink-type dark soliton trains in a nonresonant polariton condensate excited with the cw homogeneous field. The nonresonant excitation with cw-homogeneous pump is crucial because of the fact in this context: the free evolution of the condensate phase in contrast to the resonant injection scheme that would imprint the phase. With the help of potential trap to mimic the finite-size effect, the polariton condensate is spatially separated into two parts to induce the quantum interference in spatial domain, which introduce the generation of a pair of sink-type dark solitons. The domain of the material parameters to observe the sink-type soliton train is also given, which is in accordance with the restriction of repulsive interaction for the generation of polariton dark soliton [22]. Further, the number of solitons in the train is controllable on demand in the domain of parameters such as potential width and amplitude. With the increase of pump power, the stationary sink-type dark soliton train could achieve enough energy to periodically oscillate and collide and finally enters into the final state of anti-dark soliton. Finally, with the consideration of disturbance from noise, these nonlinear excitations are approved to have infinite lifetime and remain spatially localized even for the periodically oscillating and colliding state, which don’t depend on the parameters of the condensate. The stability is further investigated with the analysis of Bogliubov method.

2. Model

In the analysis below, we consider the system of a wire-shaped microcavity modeled in Fig. 1, where the polaritons are bounded to a quasi-1D channel as implemented in Ref. [23]. A metallic mesas is deposited on top of the sample to create the potential trap V(x) for polariton particles with a tunable amplitude by applying an electric field [24]. The polariton condensate is assumed to be nonresonantly populated with a far detuned cw-homogeneous pump P over the whole sample. The polariton wave function ψ(x) evolves along the open-dissipative Gross-Pitaevskii equation (ODGPE), coupled to the rate equation of exciton reservoir density n_R(x):

\begin{array}{l} d ψ (x) & = [\frac{ℏ}{2 m^{*}} \frac{\partial^{2} ψ (x)}{\partial x^{2}} (i + D_{0} n_{R} (x)) + \frac{1}{2} \underset{R^{'}}{\underset{︸}{(R^{1 D} n_{R} (x) - γ_{C})}} ψ (x) - \frac{i}{ℏ} (g_{C}^{1 D} {| ψ (x) |}^{2} ψ (x) \\ + g_{R}^{1 D} n_{R} (x) ψ (x) + V (x) ψ (x))] d t + d W \\ \frac{\partial n_{R} (x)}{\partial t} & = D_{1} \frac{\partial^{2} n_{R} (x)}{\partial x^{2}} + P - (γ_{R} + R^{1 D} {| ψ (x) |}^{2}) n_{R} (x) \end{array}

where the real coefficients D₀ and D₁ represent respectively the energy relaxation in the condensate and the spatial diffusion rate of reservoir polaritons. m^* is the effective polariton mass, R^1D presents the incoming rate of polaritons from the exciton reservoir, γ_i(i = C, R) is the decay rate of polaritons and exciton reservoir,

g_{i}^{1 D}

(i = C, R) depicts the interaction coefficient between polaritons and exciton reservoir. dW is the quantum fluctuation noise given by a complex stochastic term in the truncated Wigner approximation (TWA)

\begin{array}{l} < d W (x) d W (x^{'}) > = 0 \\ < d W (x) d W^{*} (x^{'}) > = \frac{d t}{2 d x} (R^{1 D} n_{R} + γ_{C}) δ_{x, x^{'}} \end{array}

the potential trap V(x) described in this work is given by

V (x) = {\begin{array}{l} V_{0} & | x | \leq a \\ 0 & | x | > a \end{array}

here a is set to mimic the finite-size effect so that the condensate is spatially separated into two parts to induce the rich dynamics of counter-propagating polariton waves.

Fig. 1 Sketch of a potential sample consisting in a quasi-1D semiconductor microcavity [distributed Bragg reflectors (DBR)]. A metallic mesas is embedded and deposited in the central area with a tunable potential amplitude by applying an electric field.

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3. Generation and modulation of dark soliton train

In the scalar system with potential trap V(x) described in previous section, polariton condensates are spatially divided into two parts with the nonresonant excitation of one cw homogeneous field above the threshold P_th = γ_Cγ_R/R^1D. The quantum interference [21] between two neighbouring polariton condensates plays a keys role for the dynamics of the scalar system. Fig. 2(a) shows the dynamical evolution of the scalar system with parameters V₀ = 0.2mev, a = 3μm and P = 1.2P_th. Under the effect of homogeneous nonresonant pump, polariton particles are stimulated toward the lowest energy state and form the condensate in momentum space with a chemical potential $μ ≃ g_{C}^{1 D} {| ψ |}^{4} + g_{R}^{1 D} n_{R} {| ψ |}^{2}$ . In spatial space, however, rich nonlinear phenomena such as a pair of sink-type dark solitons appear. Two parts of polariton condensates could be understood as two counterpropagating nonlinear waves with dissipation, which heal from the potential step and collide at the center of potential trap to form the quantum interference with dissipation. While the constructive quantum interference occurs accompanied with the balance between the kinetic energy $\frac{ℏ^{2}}{2 m^{*}} ψ^{*} \nabla^{2} ψ$ and the nonlinear energy $g_{C}^{1 D} {| ψ |}^{4}$ , the phenomena of a pair of sink-type dark solitons appear. It has been predicted numerically [17] and analytically [22] that dark solitons in nonresonant polariton condensate relax by blending with the background at a finite time. With the help of potential trap, however, the blending is avoided and the stationary case is observed in our model with the consideration of both quantum and classical noise. Here t = 10⁴ps is the cutoff time of the simulation.

Fig. 2 (a) dynamical evolution of the system towards the generation of a pair of sink-type dark solitons, (b) |ψ| and phase of a pair of sink-type dark solitons as a function of spatial coordinate at t = 10⁴ps, (c) phase diagram depicting parameters for which sink-type dark soliton trains are achieved. Parameters are: (a) and (b) $γ_{R} = 2 γ_{C} = \frac{1}{15} {ps}^{- 1}$ , P = 1.2P_th; (c) $γ_{C} = \frac{1}{30} {ps}^{- 1}$ , α = γ_C/γ_R. other parameters are: V₀=−0.2mev, a = 3μm, m^* = 5 × 10⁻⁵m_e, R^1D = 2.24 × 10⁻⁴μm ps⁻¹, $g_{R}^{1 D} = 2 g_{C}^{1 D} = 0.95 μ e V μ m$

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Figure 2(c) presents a phase diagram of domain for the existence of sink-type dark soliton trains in the parameter space α = γ_C/γ_R and P. The shaded area, corresponding to parameters existing sink-type dark soliton trains, is limited by the maximum value of α given by the condition $(g_{R}^{1 D} γ_{C}) / (g_{C}^{1 D} γ_{R}) < 1$ . The limitation is originated from the nature for the creation of dark soliton: repulsion between particles. As demonstrated in Ref. [22], the character of nonlinearity is effectively switched from repulsive to attractive at $(g_{R}^{1 D} γ_{C}) / (g_{C}^{1 D} γ_{R}) > 1$ . The limitation of α is also investigated for the fixed parameter of γ_R = 100ps, not shown in the paper.

As discussed above, the generation of the sink-type dark soliton train is induced by the quantum interference originated from two counter propagating polariton waves. It has been pointed out that [21], under the excitation of two optical pulse, constructive and destructive quantum interference change periodically with the product of chemical potential of the condensate and distance between two pump fields. As shown in Fig. 3, the dependence of the quantum interference on the width of the potential trap is investigated with a fixed chemical potential by setting P = 1.2P_th. Here we set a < 10μm to be sure the finite-size effect so that polariton condensates are spatially separated into two parts. By modifying the width from a = 2μm to a = 9μm, the number of the solitons varies accordingly. The destructive quantum interference with odd number of sink-type solitons in the train is observed only with the condition of 1μm< a < 3μm, as shown in Fig. 3(a). With the further increase of a, the constructive quantum interference with even number of sink-type solitons in the train can always be observed and the only difference is the variation of the soliton number. The phenomena is a little different from what is expected for the periodical changes of quantum interference as discussed in Ref. [21]. The case is supposed to be relevant to the model of potential trap. Only the quantum interference with wavefunction in accordance with that of the bound states in the potential trap could be survived in the dynamical evolution of the system. Additionally, we would like to point out that, for sink-type soliton trains as shown in Fig. 3, all polariton particles condensate in an agglomeration region in momentum space with the same chemical potential. However, for the parameters where the sink-type soliton train is observed but easy to jump to another case with a small modification in parameters, a few polariton particles appear in the vicinity of the agglomeration region accompanied with a higher chemical potential, which is not shown in the paper.

Fig. 3 the phenomena of sink-type dark soliton trains as a function of spatial coordinate for different potential width while t = 10⁴ps. (a) a=2μm, (b) a=5μm, (c) a=8μm, (d) a=9μm. other parameters are the same as those in Fig. 2(a).

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Further, with a fixed width a of potential trap, the effect of pump power P on the sink-type dark soliton train is investigated as shown in Fig. 4. Similar as the case talked before, due to the action of bound states in potential trap, the type of quantum interference won’t be changed periodically with pump power P and depends on the depth of potential trap. The shallower the potential trap, the less number the bound states. The parity of wavefunction corresponding to the bound state represents the type of quantum interference possible to survive in the dynamical evolution. For the shallow trap such as V₀ = −0.2mev and a = 3μm (corresponding to the case of Fig. 2(a)), there is only one bound state with wavefunction of even parity. This predicts that only a pair of sink-type dark solitons could be observed in this case. With the increase of pump power up to P = 2.0P_th, as shown in Fig. 4(a), the sink-type soliton train obtain enough energy to move and brings on the generation of periodically oscillating and colliding sink-type dark soliton pairs. With the further increase of pump power up to P = 2.2P_th, the polariton particles achieve much more energy to escape the bound of potential trap and condensate in momentum space with higher energy of k ≠ 0. It is conductive to the generation of the anti-dark state as shown in Fig. 4(b).

Fig. 4 dynamical evolution of the system towards the generation of (a) a pair of periodically oscillating and colliding dark solitons as well as (b) anti dark soliton. Parameters are (a) P = 2.0P_th, (b) P = 2.2P_th. Other parameters are the same as those in Fig. 2(a).

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It is important to note that, besides two factors of pump power and potential width, there is another factor of V₀ noticeable in our system to modulate the dark soliton train. The larger value of V₀, the more number of bound states to observe corresponding dark soliton trains. With a fixed pump power P = 1.6P_th and a given potential width a = 3μm, Fig. 5 shows the variation of the sink-type dark soliton trains for different value of V₀. The number of solitons in the train appears from two to three and then to four with the increase of V₀ from 0.2mev up to 0.7mev. With the consideration that the phase of condensate changes with the amplitude of potential trap, it is supposed that the choice of the bound state for the possible quantum interference in this model is relative to the flux flow velocity while the polariton particle heals from the potential step, which is determined by the variation of the phase of condensate.

Fig. 5 dark soliton trains vs the amplitude of potential trap. (a) V₀ = 0.2mev, (b) V₀ = 0.7mev, (c) V₀ = 1.8mev. P = 1.6P_th and other parameters are the same as those in Fig. 2(a).

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4. Stability of the dark soliton train

In the previous simulation, the random low-intensity white noise is seeded both in the initial condition and during the evolution. The fact of the solution staying for a long time indicates that this is a steady state in a nonequilibrium system. Next, we would like to further check the stability of the solution within the Bogoliubov-de Geneess approximation. The small fluctuations around the stationary state can be decomposed into their Fourier components:

\begin{array}{l} ψ (x, t) = e^{- i μ t} ψ_{0} (x) [1 + \sum_{k} (u_{k} e^{i k x - i λ_{k} t} + v_{k}^{*} e^{- i k x + i λ_{k}^{*} t})], \\ n_{R} (x, t) = n_{R}^{0} (x) (1 + \sum_{k} (w_{k} e^{i k x - i λ_{k} t} + w_{k}^{*} e^{- i k x + i λ_{k}^{*} t})) \end{array}

where λ_k is the mode frequency of the wavenumber k, and u_k, v_k, w_k are small fluctuations. ψ₀(x) and

n_{R}^{0} (x)

are stationary solutions of the system for the dark soliton train, with μ the corresponding chemical potential. The imaginary part of λ_k is equal to the exponential growth rate of an unstable mode. Substituting Eq. (4) into the Eq. (1) and keeping the linear terms only, the eigenvalue problem λ_k(u_k, v_k, w_k)^T = A(u_k, v_k, w_k)^T of the elementary excitations can be derived, with the operator matrix A dependent on the stationary solution ψ₀(x).

Typical examples of the elementary excitation spectra are shown in Figs. 6(a)–(d), respectively corresponding to the sink-type dark soliton trains as shown in Figs. 3(a) and 3(b). These two cases represent the every possible scenario of dispersion of the sink-type dark soliton trains observed in our model. It is noted in Fig. 2(c) that the sink-type dark soliton trains in our model can be observed only within the parameter domain α = γ_C/γ_R ≤ 1/2 to maintain the repulsive property between polariton particles [22]. Here we would like to note that this condition is under the adiabatic condition of γ_C ≪ γ_R for polariton condensate. Therefore it is not surprised to observe the similar dispersion profiles (as shown in Figs. 6(a) and 6(b)) for the nonlinear excitation of dark soliton trains as those of the polariton condensate under the adiabatic condition [25, 26]: a Goldstone mode (indicated as + in the figure) whose real part is equal to zero and appears dispersionless at low k, accompanied with an imaginary part starting from zero in a quadratic way. Additionally, there is another kind of dispersion as shown in Figs. 6(b) and 6(d): although with the same real part as that in Fig. 3(a), there is a little difference in the imaginary part. It still starts from zero but not in a quadratic way and oscillation appears. Anyway, in all these cases, the imaginary part of λ_k no bigger than 0 depicts the stability of the sink-type dark soliton train created in our model, which is in accordance with the information provided by the numerical simulation with the consideration of noise.

Fig. 6 real and imaginary part of the excitation spectrum of the dark soliton train. (a) and (b) corresponding to the case of Figs. 3(a), (c) and (d) corresponding to the case of Fig. 3(b).

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5. Conclusion

In conclusion, we found that, with the action of potential trap to prepare the spatially separated binary condensate, the nonlinear excitation of sink-type dark soliton train could be generated and stabilized in a 1D nonresonant pumped polariton condensate. The mechanism is relative to the quantum interference originated from two counter propagating polariton waves. We proceed to explore the on demand control of the soliton number in the train with the help of potential parameters. The parameter domain to generate the sink-type dark soliton train is also presented, which renders them available for experimental observations. We’d like to note that the sink-type solution with stability [27] has also been discused recently with the help of spatially localized pumping where the heating effect is the considered. The present results on the generation and control of the sink-type dark soliton train could also have implication in matter-wave interferometry.

Funding

National Natural Science Foundation of China (NSFC) (11374125); China Postdoctoral Science Foundation (2015M581391, 2015M570266).

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Manipulation of a stable dark soliton train in polariton condensate

Abstract

1. Introduction

2. Model

3. Generation and modulation of dark soliton train

4. Stability of the dark soliton train

5. Conclusion

Funding

References

Cited By

Figures (6)

Equations (4)

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