1. Introduction

Cavity polaritons, which exhibit neither exciton nor photon characteristics, are half-matter half-light quasi-particles arising from the strong coupling between excitons and photons [1,2]. The nonlinear effect of polaritons in quantum well semiconductor microcavities is an interesting area of recent research. Related topics including polariton bistability [3, 4], parametric wave-mixing [5, 6], vortices and Bose-Einstein condensation [7, 8] have been investigated extensively. Furthermore, studies of spatial solitons in a strong-coupling regime have initialized the investigation of polariton solitons in microcavities [9–11]. This is because, when compared with the weak-coupling regime [12–14], the nonlinear effect and optical response speed of semiconductor microcavities can be significantly improved by 2 to 3 orders of magnitude in the strong-coupling regime. Hence, the response time and the required pumping intensity to operate polariton solitons can be efficiently reduced. In addition, it is expected that both lifetime and relaxation kinetics of polaritons can be easily modified by changing the photon-exciton detuning, which can be used to design a specific dispersion relation for soliton formation.

Recently, dark and bright polariton solitons of semiconductor microcavities have been reported [10, 15] and verified experimentally [2, 9]. However, these studies only considered the scalar polariton solitons, and the polarization of polariton fields was ignored. It is known that the polariton fields in microcavities are vectorial polarization dependence which governed by the underlying spin-dependent polariton-polariton scattering processes. Two possible spins, where are + 1 and –1 correspond to the right circular polarization (RCP) and left circular polarization (LCP) respectively, are supported inside microcavities [16–18]. This polarization dependence of polaritons adds the complexity of their nonlinearities and provides more flexible ways to form solitons. Hence, it is of great importance to study the polariton solitons with polarization dependence. In this paper, vectorial polariton solitons is studied numerically in a semiconductor microcavity. A modified 4th-order Runge-Kutta method is used to find the soliton solutions. Based on this method, various combinations of polariton solitons in the two polarization directions are detected and analyzed for both vertical and oblique pumping.

2. Theoretical model

The polariton fields are usually described by two coupled equations for their excitonic and photonic components. We start our discussions with the polarization-resolved dimensionless model for the intracavity photon field, E, and for the exciton field, ψ [10, 15–19]

For zero-pumping condition, if we neglect the nonlinear effect and assume E _1,2 = E _{1 s ,2 s}exp(ikx), ψ _1,2 = ψ _{1 s ,2 s}exp(ikx) and ω ₀ = 0, the dispersion relationship of polaritons can be obtained through the eigenvalues of coupled Eqs. (1) and (2). This paper only refer to the lower polariton branch, E_LP, and it can be expressed as [10, 15]

In order to find the bright soliton solutions for a value of E_in, we first guess the initial values of excitons, ψ _1,2(0, x), and photons, E _1,2(0, x). This can be done by assuming that the analytical expressions of ψ _1,2(0, x), and E _1,2(0, x) to be ψ _1,2(0, x) = ψ _{1 s ,2 s}[1 + A _1,2exp(−B _1,2 x ²)] and E _1,2(0, x) = E _{1 s ,2 s}[1 + C _1,2exp(−D _1,2 x ²)] respectively, where ψ _{1 s ,2 s} and E _{1 s ,2 s} are the homogenous solutions in the lower bistable branch, and A _1,2, B _1,2, C _1,2, and D _1,2 are real constants. The differential terms ∂_xψ _1,2(0, x) and ${\partial }_x^2{\psi }_{1,2}(0,x)$ can be deduced from central difference method. Hence, ψ _1,2(Δt, x) and E _1,2(Δt, x) can be obtained for the next time step by 4th-order Runge-Kutta method (Δt denotes the time step). Repeating the above steps, we can obtain the values of ψ _1,2(mΔt, x) and E _1,2(mΔt, x) at any time step where m is an integer. As unstable solitons are sensitive to the initial conditions, ψ _1,2(0, x) and E _1,2(0, x) should be carefully selected, i.e., different values of A _1,2 are tested until the profiles of ψ _1,2 and E _1,2 remain unchanged with the increase of t. On the other hand, the soliton solutions are stable if they are insensitive to the initial conditions. Using similar method, dark solitons can also be excited.

In the following simulations, we concentrate our investigation on the bistable regime and the excitonic component is taken as an example to discuss polariton solitons. Typical parameters of a GaAs/AlAs microcavity with InGaAs/GaAs quantum wells as the gain region are used to model cavity polaritons [10, 15–17]. Hence, the values of the mentioned parameters are assumed to be: η = –0.1, γ_p = 0.1, γ_e = 0.1, E_{in 1} = E_in, and E_{in 2} = 0.8E_in. δ, k_in and E_in are variables to be specified latter.

3. Pumping at normal incidence

In this section, we consider the case of normal incidence, i.e., k_in = 0. Figure 1 shows the homogonous solutions of the excitonic components of polaritons in the two polarization directions. The solid (dotted) curves mark the stable (instable) states. We can see that the two polarization modes show a distinct ‘S’ shape as a function of pump strength, E_in. Because the pumping into the two polarization modes is asymmetric, the hysteresises of the two modes are not overlapped. A vertical-broken line separates Fig. 1 into two regions. In region I (II), the RCP (LCP) mode exhibits a dominate hysteresis. Its counterpart, the LCP (RCP) mode, also shows a reverse (similar) hysteresis due to nonlinear interaction between the two modes.

 

Fig. 1 The homogenous solutions of the excitonic components of the polaritons as a function of pumping strength, E_in, for k_in = 0 and δ = −0.7.

Download Full Size | PDF

Next, the solutions of vectorial solitons are searched along the bistable curves of the homogenous solutions, and the results are given in Fig. 2 . The squares (dots) denote the peak (dip) values of bright (dark) solitons, and the stable solitons are also marked by solid lines. Figures 2(a) and (c) correspond to region I. In Fig. 2(a), a branch of bright and a branch of dark solitons are found to bifurcate from the homogonous solutions of the RCP mode. The entire branch of bright solitons is unstable, and the branch of dark solitons is separated into a stable and an unstable part by a fold bifurcation point. Furthermore, due to nonlinear interaction, the bright (dark) solitons of RCP mode also arouse small-amplitude dark (bright) solitons in the LCP mode, see Fig. 2(c). Hence, vectorial bright-dark and dark-bright solitons are found in region I. In region II, bright-bright and dark-dark combinations of solitons are found, see Fig. 2(b) and (d).

 

Fig. 2 Maxima (minima) of the excitonic components of bright (dark) polariton solitons as a function of pumping strength, E_in, for k_in = 0 and δ = −0.7. The solid-dotted curves represent the background amplitude, and the squares (dots) denote the peak (dip) values of the bright (dark) solitons.

Download Full Size | PDF

Figure 3 gives an example for the excitation of polariton solitons through the proposed method. Figure 3(a) and (b) correspond to points P1 and P1’ marked in Fig. 2 respectively (i.e., vectorial bright-dark solitons). For clarity, the dark solitons of LCP mode are plot using negative values of |ψ ₂|. We can see that the profiles of ψ ₁ and ψ ₂ become unchanged after a short time interval of oscillations from t = 0. It is also noted that the solitons are stationary, i.e., the peak position of solitons in the x direction does not change with time.

 

Fig. 3 The excitonic components of polaritons in time and real space domains. (a) and (b) correspond to, respectively, points P1 and P1’ marked in Fig. 2. For clarity, the dark soliton is plot using negative value of |ψ ₂|.

Download Full Size | PDF

Figure 4 shows the profiles of other types of vectorial solitons, i.e., dark-bright, bright-bright, and dark-dark solitons that correspond to points P2 – P2’, P3 – P3′ and P4 – P4’ respectively marked in Fig. 2. These entangled states of vectorial solitons induce additional flexibilities and are thought to provide a promising way for information processing.

 

Fig. 4 Profiles of the excitonic components of polaritons. (a) – (f) correspond to, respectively, points P2, P3, P4, P2’, P3′, and P4’ marked in Fig. 2.

Download Full Size | PDF

4. Pumping at oblique incidence

This section considers the case at oblique incidence. We choose k_in = 1.7, δ = –0.03, and the microcavity maintains within the bistable regime. In this case, the bistable curves and the maxima (minima) of bright (dark) solitons are given in Fig. 5 . In Fig. 5(a), we find a branch of bright solitons for the RCP mode. Different from the case of normal incidence, a part of the bright solitons becomes stable. In the branch of solitons, the solutions are stable (instable) above (below) the fold bifurcation point, which is in accordance with the result of [15, 19]. Corresponding to the bright solitons of RCP mode, a branch of dark solitons of LCP mode is found in Fig. 5(c). Thus, vectorial bright-dark solitons are formed. In region II, vectorial bright-bright solitons are found, see Fig. 5(b) and (d).

 

Fig. 5 Maxima (minima) of the excitonic components of bright (dark) polariton soliton as a function of pump strength, E_in, for k_in = 1.7 and δ = −0.03. Other descriptions are the same as those of Fig. 2.

Download Full Size | PDF

Figure 6 repeats the plot of Fig. 3 except that the pumping is oblique, i.e., k_in = 1.7. In this case, the profiles of the solitons become asymmetric and the solitons is nonstationary. From Fig. 6, the velocity of the solitons is deduced to be 0.31. It is close to the first order dispersion of the polariton, ∂_kE_LP, which equals 0.3 for k_in = 1.7.

 

Fig. 6 The excitonic components of polaritons in time and real space domains. (a) and (b) correspond to, respectively, points P1 and P1’ marked in Fig. 5. For clarity, the dark soliton is plot using negative value of |ψ ₂|.

Download Full Size | PDF

From the above results, we can see that the vectorial solitons are the combinations of a strong and a weak solitons in the two polarization directions. In region I (II), the strong and weak solitons are of the same (reverse) type. However, these results are obtained from a relative small value of |η|. In some cases (i.e., the active region has multiple quantum wells), the value of |η| may be even larger. Strong nonlinear interactions can greatly change the entangled solitons in the two polarization directions. In Fig. 7 , η = –0.5 is considered and the results correspond to region II. Different from the case for η = –0.1 (i.e., vectorial bright-bright solitons are found in region II), the combination of solitons is dark-bright, and the amplitudes of solitons are large for both polarizations. We also find that all the soliton solutions are stable.

 

Fig. 7 Maxima (minima) of the excitonic components of bright (dark) polariton solitons as a function of pump strength, E_in, for k_in = 1.7, δ = −0.03, and η = −0.5. Other descriptions are the same as those of Fig. 2.

Download Full Size | PDF

5. Conclusion

In conclusion, we have analyzed the formation of vectorial polariton solitons in semiconductor microcavities. In the simulation, RCP and LCP polaritons are in different bistable regimes due to an asymmetric pumping. For normal pumping, bright and dark solitons can bifurcate from homogonous solutions of one polarization mode which shows a dominant bistability. In the counterpart polarization mode, reverse (in region I) or same (in region II) types of solitons with small amplitudes are aroused due to nonlinear interaction. Thus, vectorial bright-dark, dark-bright, bright-bright, and dark-dark solitons are found. For oblique pumping, bright-dark and bright-bright solitons are found in regions I and II respectively. Moreover, there is a fold bifurcation point separating the soliton branchs into a stable and an instable part. The strength of nonlinear interaction between the two plarization modes also plays an important role in the formation of solitons, i.e., when the two polarization modes have strong interactions, they have reverse types of solitons in region II, and the amplitudes of solitons for both modes are large.