Generation and stability of diversiform nonlinear localized modes in exciton–polariton condensates

We propose a scheme to generate and stabilize one- and two-dimensional dark, bright, dark-like, bright-like solitons, and vortices with m = 1 and m = 2 in a nonresonantly incoherent pumped exciton–polariton condensate. A spatially modulating pumping is introduced, which can compensate (counteract) the loss (gain) originated from the nonlinear excitation of the stable homogeneous polariton. The numerical simulations show that the balance between the gain and loss in this scheme can support and stabilize various nonlinear modes, not just stable dark solitons which have been found in the previous studies. Our proposal may provide a way to generate, stabilize, and control nonlinear modes in the nonresonantly pumped exciton–polariton system.

The exciton-polaritons can be made to form condensates out of equilibrium, which are best understood as a steady-state balance between pumping and decay, rather than true thermal equilibrium [39]. The exciton-polariton condensates are described by a high-dimensional saturated nonlinear Schrödinger equation (SNLSE) with the Kerr nonlinear term and the gain and loss terms, and the formations and stability of nonlinear modes in this system are of particular interest. For the case of resonant coherent pumping, the bright [27,28] and dark solitons [26,29,34] were predicted and observed in experiments. However, for the nonresonant incoherent homogeneous pumping, dark solitons are unstable, and it can only evolve in a short time in one-dimensional [32] and two dimensional [33] cases. So the spatially periodic [23], ring-shaped [18,24,35] and Gaussian-shaped [22,30,31] pumping have been proposed to stabilize these localization states. However, these nonlinear localized states are still unstable, because the balance between the nonlinear gain and the constant loss is not realized.
In the previous studies on exciton-polariton condensations, the soliton is generated by the initial random noise, and formed by assuming a balance between gain and loss [30]. However, when we discuss the nonlinear excitations (such as solitons and vortices) on the basis of homogeneous condensates, it is difficult to realize the balance between the nonlinear gain and the invariable constant loss. The purpose of this paper is threefold. Firstly, based on the homogeneous steady state, we propose a spatially distribution pumping to obtain the balance between the nonlinear gain and the invariable loss. Secondly, we directly solve the nonlinear steady state, rather than obtaining the solutions by evolving an initial noise. Finally, after introducing the spatial distribution pumping, we can obtain stable dark soliton, bright soliton, bright-like, dark-like soliton, and vortices with m = 1 and m = 2 by our numerical method.
In this paper, to achieve the balance between gain and loss, we construct an incoherent pumping which consists of a homogeneous pumping and a Gaussian pumping. Then, various nonlinear stable states are found directly. The stability of nonlinear modes is proved by the linear stability analysis and the evolution. Thanks to the spatial distribution pumping and the method of finding solution, we not only find the dark soliton, bright ground soliton, bright-like soliton, and dark-like soliton in one-dimensional system, the bright soliton, bright-like soliton, dark-like soliton, and the vortices with m = 1 and 2 in two-dimensional system, but also prove their stability by the evolution and the linear stability analysis.
The article is arranged as follows. In section 2, we give an introduction of the model under study. In section 3, various soliton solutions, their properties, and stability are studied for the one-and two-dimensional systems, respectively. In the last section, we summarize the main results.

Model
Using a mean-field theory, we describe the dynamics of two dimensional exciton-polariton condensates by a dissipative Gross-Pitaevskii equation for the polariton field Ψ, coupled to the rate equation of the density of the excitonic reservoir n R where P u (r) is the exciton creation rate determined by the incoherent pumping profile, m * is the effective mass of lower polaritons, γ C and γ R are the polariton and exciton loss rates, R is the condensation rate, g C represents the nonlinear interaction between polaritons, g R is the interaction between polaritons and reservoir excitons. The incoherent pumping is constructed by P u (r) = P 0 + P 1 exp[−(x 2 + y 2 )/w 2 0 ], which consists of a cw field P 0 and Gaussian field, respectively.

Soliton solutions and properties
In order to discuss the soliton solutions of equations (2a) and (2b), their properties and stability, we first assume a plane-wave solution u = u 0 exp(iβs) and the constant reservoir density n = n 0 of equations (2a) and (2b), and consider a homogeneous pumping P(r) = σ 7 (i.e. σ 8 = 0 or P 1 = 0). When the pumping is weak, the reservoir density n 0 = σ 7 , and the condensate u 0 = 0. With the increase of the pumping, the balance between loss and gain is obtained at the pumping threshold value P th = σ 7 = σ 4 σ 3 . Above the threshold value, the steady homogenous condensate density is given by the condensate dimensionless energy is here, the reservoir density n 0 = σ 4 σ 3 .
Above the threshold pumping, we can consider the nonlinear excitation on the basis of homogeneous condensates. From the last term i (σ 3 n − σ 4 ) u of equation (2a), one can find that as u and n are the soliton profile being spatial dependence, the constant loss σ 4 can not be balanced by the nonlinear saturated gain term σ 3 n directly, here we assume n = n(r) = σ 7 1+σ 6 |u| 2 . Thus the spatial distribution pumping, i.e. Gaussian inhomogeneous pumping, is proposed to compensate the denominator of the nonlinear saturated gain term. Of course, the periodic-, Bessel-inhomogeneous pumping can also be used to realized the function.

The soliton solutions for one-dimensional system
We now present the soliton solutions of equation (5) for one-dimensional system, and check their stability by using the numerical simulations. Under the homogeneous pumping, the condensate is steady and homogenous, and the nonlinear modes are excited on a homogeneous background of the steady condensate wave function for the bright and dark solitons. Once the condensate wave function is the localization nonlinear mode with the homogeneous background, the nonlinear saturated gain term can not be balanced by the constant loss directly. An inhomogeneous pumping should be proposed to suppress part of the nonlinear saturated gain caused by the nonlinear excitation, meanwhile the homogeneous pumping suppresses the constant loss. The designed pumping can suppress the imaginary part of the exciton-polariton system availably, though an absolute balance of the gain and loss can not be realized.
Using the Newton conjugate gradient method [40] and the trial solution [41], we first find the nonlinear steady state of equation (5), then make the linear stability analysis by equation (7) and study the evolutions of the obtained steady states by numerically solving equations (2a) and (2b). The dark solitons and their stability with β = 0.1 are shown in figure 1. From the profile of dark soliton, since a hump contributes to the nonlinear saturated gain term in the denominator, the coefficients of inhomogeneous pumping σ 8 < 0 are taken to reduce the gain. Figures 1(a) and (b) illustrate the power and stability curves as a function of the intensity of inhomogeneous pumping σ 8 . From the stability curves, one can find the negative inhomogeneous pumping stabilizes the dark solitons. In figures 1 (c1)-(f1), the red solid line (blue dashed-dotted line) denotes the profile |ψ| (the phase φ) of the steady state by solving equation (5).
The stability is proved further by a numerical evolution of equations (2a) and (2b), and adding a random perturbations into the initial value of evolution, i.e., the initial value is taken as u(s = 0, ξ, η) = ψ(ξ, η)(1 + εf 1 ) and n(s = 0, ξ, η) = n (ξ, η)(1 + εf 2 ), where ε = 0.1 and f 1,2 are the random variables uniformly distributed in the interval [0, 1], and s = 100 denotes 54.5 ns. Comparing with the stability of the dark solitons obtained in [32], the evolving stability is improved, which results from the introduced inhomogeneous pumping and the calculation method. Here, the initial conditions for the evolutions are actually closer to the physical reality than ones in [32]. In figures 1(c2)-(f2), the projections The phase jump is the typical characteristics of dark soliton. From figures 1(e2)-(f2), it is seen that there are some oscillations of phase near ξ = 0 at certain time, but they disappear and a reverse of phase happen for a long evolution time, and the phase jump is clear. It should be noted that the dark soliton without the inhomogeneous pumping [32] or with the positive inhomogeneous pumping decays very quickly in a short time. For the negative inhomogeneous pumping, the dark solitons are stable when reaching the threshold value σ 8 = −0.2 as shown in figures 1(b) and (e1)-(f1).
When we find the stable state solution, we use the periodic boundary condition for bright soliton and the nonlinear modes with uniform background. For the dark soliton, the same boundary condition is also taken by taking u(ξ) = u(ξ) − u(ξ − ξ 0 ) as the solution, where u(ξ) and u(ξ − ξ 0 ) are both dark soliton solutions of equation (5) due to the translational invariance of equations (2a) and (2b). If ξ 0 is enough large, u(ξ) − u(ξ − ξ 0 ) is also the solution, thus, the periodic boundary condition is reasonable. And the finite boundary condition is used to obtain the evolution results. Where we take ξ 0 = 80 and u 0 is the value of the stable state in the neighborhood of ±ξ 0 . We also adopt the similar boundary condition for the two-dimensional system. Generally, it is difficult to obtain the dark and bright solitons in the same system, since the dark soliton results from the defocusing nonlinearity and the bright soliton results from the focusing nonlinearity. However, in the exciton-polariton condensate system, there exists the Kerr nonlinearity, saturated nonlinearity and the nonresonant pumping simultaneously, which can support bright solitons.
In figure 2, we show the bright soliton with the uniform background (the bright-like soliton) and the dip-type soliton (the dark-like soliton sharing a similar intensity profile of the dark soliton without the phase jump). Figures 2(a) and (b) show the power and stability curves as a function of the intensity of inhomogeneous pumping σ 8 . Figures 2(c1)-(e1) show the initial profiles by solving the steady state equation (5), and figures 2(c2)-(e2) corresponds to the evolutions. It is seen from the power curve in figure 2(a) that the bright-like soliton exists for σ 8 > 0, the dark-like soliton for σ 8 < 0, but no bright-like or dark-like soliton for σ 8 = 0 because the power is zero. From the stability curve in figure 2(b), one can find that the dark-like solitons are stable, whereas the bright-like soliton are unstable, but, whose profile can conserve even at the evolution time 109 ns from the numerical results in figure 2(c2). It is obvious that the negative (positive) inhomogeneous pumping can be used to generate and stabilize the dark-like (bright-like) soliton. From the phase of wave function denoted by the blue dashed-dotted line in figures 2(d2) and (e2), one can find that the profiles in figures 2(d1) and (e1) are indeed the dark-like solitons, which can be also called the gray-like soliton and the black-like soliton, respectively. In this subsection, we present the bright-like soliton and the dark-like soliton for one dimensional system, and the stability is due to the introduced inhomogeneous pumping.

The soliton solutions for two-dimensional system
It is well known that the solitons of the high-dimensional SNLSE are generally unstable except for a bright ground soliton. However, it is interesting to find stable high-dimensional solitons in the exciton-polariton model (2a) and (2b), because the SNLSE has the imaginary part and the Kerr nonlinearity. Except for the vortex, there has been no report on the high-dimensional stable spatial solitons. In this subsection, we study the interaction between the imaginary part and the Kerr-and saturated nonlinearities and find other stable high-dimensional spatial solitons.
The two dimensional bright-like soliton and dark-like soliton are shown in figure 3. Figures 3(a) and (b) show the power and stability curves as a function of the intensity of inhomogeneous pumping σ 8 , and figures 3(c1)-(e1) and (c2)-(e2) are the profiles and evolutions, respectively. The power curve plotted in figure 3(a) is different from the case in figure 2(a), but sharing a similar tendency. Once use the homogeneous incoherent pumping (σ 8 = 0), the bright-like soliton and the dark-like soliton can not be obtained. It is seen from the stability curve in figure 3(b) that the stability is strengthened with the increasing of |σ 8 | for σ 8 < 0, and all above these two-dimensional nonlinear modes are stable. Although the stability curve in figure 3(b) shows that the bright-like soliton are unstable, the numerical evolution in figure 3(c2) shows that the profile of the soliton is still conserved at evolution time 436 ns.
If a phase factor exp(imθ) (θ = arctan η ξ is the azimuthal coordinate) is imprinted onto the initial state, the vortex steady state can be found. Though there exists the vortex in the exciton-polariton system, it is still interesting to study how to enhance stability of vortex and find a stable high-charged vortex. For balancing the nonlinear gain and constant loss, we still search the vortex with the uniform background, rather than ones in [22].
In figure 4, we show the vortex with m = 1 and its stability. The power and stability curves as a function of the intensity of inhomogeneous pumping σ 8 are shown in figures 4(a) and (b), and figures 4(c1)-(f1) and (c2)-(f2) are the profiles and the evolution, respectively. From figure 4(b), we find the vortices with m = 1 become more stable with increasing |σ 8 | for σ 8 < 0, and unstable for σ 8 > 0. Compared figure 4(c1) with its evolution result shown in figure 4(c2), the vortex will disappear for a long evolution time when σ 8 = 0.2. Through the linear stability analysis in figure 4(b), one can find the vortex in figure 4(d1) is unstable for σ 8 = 0. Because the value Re(λ) is so small, the profile and phase are still remained after evolving 436 ns, which means that the vortex of m = 1 is stable even in the absence of the inhomogeneous pumping. By examining the initial phase with the evolution results, one can find the vortex rotates. For σ 8 = −3.0, the vortex has a core at the center, characterized the phase shown in the inset. The core disappears after a long evolution time, but the vortex can be considered to be stable as its profile and phase are still remained. Thus, for the case of m = 1, all the vortices with uniform background are stable so long as σ 8 0.
Furthermore, we investigate whether the above conclusions are suitable for the higher-charged vortices. In figure 5    σ 8 = 0 are unstable, which break down into two vortices, and are stable for σ 8 < −0.1. So, we see that the inhomogeneous pumping is very important for stabilizing the higher-charged vortices.

Some special soliton solutions
By taking another sets of parameters, σ 3 = 0.2 achieved by setting P 0 , β = 0.35, and the other parameters being same, we obtain one-and two-dimensional bright ground solitonas shown in figure 6. The dipole soliton and the vortex with the zero-background can also be found, but they are unstable. Though there is a wide existence interval for the bright ground soliton from the power curves in figure 6(a), very narrow interval of σ 8 can be found to stabilize these solitons, only in the neighborhood of the intersection point shown in figure 6(b). From these results, we know the Gaussian potential possesses the trapping function for these nonlinear modes, but its balance function is more important for the stability of soliton. From these evolution results, we know that it is easier to generate and stabilize the nonlinear modes with the uniform background than ones with the zero-background.
After some numerical simulations, we find all above results are independent on the width of Gaussian pumping and the function form, the similar nonlinear solution for equations (1a) and (1b) can be obtained by replacing the Gaussian pumping with other pumping, such as Bessel pumping, periodic pumping, and so on. The uniform part in the pumping is also very important to stabilize the nonlinear modes, because it can balance the constant loss.

Summary
In conclusion, we have proposed a scheme to generate and stabilize the nonlinear modes by introducing the incoherent pumping in the exciton-polariton condensate. The introduced pumping contains a homogeneous part that balances the constant loss, in addition to an inhomogeneous part that compensates the gain or loss caused by the denominator of nonlinear saturated gain terms. The bright, dark, bright-like, and dark-like solitons in one dimensional system, and the bright, bright-like, dark-like solitons and vortices with m = 1 and m = 2 in two-dimensional system are found. It is demonstrated that the stabilities of these nonlinear modes can be realized by engineering the inhomogeneous pumping. The realizations of the diversiform and stable nonlinear modes are resulted from the balance between the nonlinear saturated gain and the constant loss availably in the exciton-polariton system. The results presented here may be useful for understanding the physical properties of condensates out of equilibrium, and guiding experimental studying of condensate soliton, which may have potential applications in polariton condensates for information storage and processing or quantum simulators.