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This paper proposes a way of generating polariton solitons (PSs) in a semiconductor microcavity using Stark effect as the trigger mechanism. A Stark pulse performing as the writing beam is used to excite non-resonant fluctuations of polariton, which finally evolves into bright PSs. It is found that a branch of PS solutions versus pump parameters could be found through optimizing parameters of the Stark pulse, and polarization of the generated PS is dependent on the writing beam.
© 2015 Optical Society of America
1. Introduction
Polaritons in semiconductor microcavities are composite half-light half-matter quasi-particles arising from the strong coupling between quantum-well excitons and cavity photons [1–3 ]. Cavity polaritons exhibit some important advantages, for example a small effective mass compared with atoms and a short lifetime (optical property) as well as the fast response and a low power under the strengthened polariton-polariton interaction (excitonic property) [4–7 ]. The unique properties let cavity polaritons be promising for various applications, including Bose-Einstein condensation, superfluidity, vortices [2,7–9 ], etc. Moreover, recent realizations of optical bistability [4,10,11 ] and polariton lasers [11–14 ] promote the implementation of polaritonic devices. These findings, especially the observations of condensation and bistability phenomenon, lay the foundations for the realization of solitons.
Studies of spatial solitons in a strong-coupling regime initialized the investigation of various polariton solitons (PSs) in microcavities, such as dark, bright, and spin-dependent vectorial PSs [3–6,15–18 ]. Compared with those occurred in the weak-coupling regime [18–21 ], the response time and the pumping intensity required to operate PSs in the strong-coupling regime can be substantially reduced, because the nonlinear effect and optical response speed of semiconductor microcavities have been improved by 2 ~3 orders in magnitude.
Usually, PSs can be excited in semiconductor microcavities by defect potential (i.e., type I) or a pulsed writing beam (i.e., type II) [3–6,15–18 ], under the balanced dispersion and nonlinear scattering of the waveform. For type I excitation, an existing defect potential in a microcavity could not be adjusted to fit the requirements of forming PSs when the operation condition changes [22,23 ]. For type II, coherent writing beams were usually used. The extraction of the polariton properties from interference experiments is relative difficult [24]. Besides, temporary fluctuation of the polariton is accompanied by accumulation of exciton reservoirs, preventing fast manipulation and characterization of the polariton condensate [24,25 ].
Recently, studies have demonstrated that Stark effect can induce fast dynamics and switch of polariton state without excitation of additional exciton [24,25 ]. The underlying principle is to control the energy of the polariton branches using the Stark shift (i.e., blueshift of both lower and upper polariton energies) due to a writing beam which is far red-detuned from the excitonic line. Different from the methods using coherent writing beam, the far red-detuned writing beam ensures a much faster control of the dynamics of the system [24,25 ]. Inspired by these works, this paper proposed the idea and carried out theoretical studies of PS formation through Stark effect. A Stark pulse performing as the writing beam is used to excite temporal fluctuation of the polariton field, which evolves to bright PS due to balance between the dispersion and nonlinear effect. Compared with the way of using coherent writing beams, the dynamic mechanism caused by Stark effect is density-independent and might provide ways of faster manipulation of PS. This is promising feature favorable for application in ultrafast optical information processing.
2. Theoretical model
Coupled equations of the photon and exciton can be expressed as [5,6,26,27 ]
wherein, subscripts 1 and 2 correspond to the right circularly polarized (RCP) and left circularly polarized (LCP) modes, respectively. ξ 1,2, φ 1,2 and E in1, in2 denote the photonic, excitonic and the pump field respectively. The time, t, the transverse coordinates (i.e., r), and the wave number of pump field, kin, are normalized to 1/Ω, r 0, and 1/r 0, respectively. , where c is the light velocity in vacuum, n is the refractive index of the quantum well, and Ω is the Rabi frequency. Parameter α denotes the cross exciton-exciton interaction constant, and γp (γe) is the photon (exciton) decay rate normalized to Ω.In our case, a quasi-one dimensional lateral potential is considered (seeing Eq. (3). This lateral confinement can be achieved through mirror patterning of the microcavity [6,28-30 ]. Besides, wave vector of the coherent pump and the exciting field is only nonzero in the x direction, so as to facilitate formation of one dimensional PS.
Transformations that and are used to simplify analysis of PS solutions, and Eqs. (1) and (2) are simplified to
In Eqs. (4) and (5) , we suppose resonance frequencies of the exciton (i.e., ) and the cavity (i.e., ) of the free running microcavity are equal, and taking them as reference, i,e, . Thus, .When Stark effect takes place, ωX 0 changes to [25], so in Eq. (5) should be replaced by δ + δs 1, s 2, and
where ωp and Es denote the frequency and intensity of the Stark field normalized to Ω respectively. In the simulation, a GaAs/AlAs microcavity with InGaAs/GaAs quantum wells as the gain region is considered. The values of typical parameters are α = −0.1, γp = 0.1, γe = 0.1, δ = 0.05, kin = 1.7, ωp = −5, n = 3.5, ω 0 = 0, and Ω = 10 meV [4,17,18 ]. The Stark pulse Fs () is used to add temporal fluctuations to Es (The pulse parameters Is, As, and Bs are optimized constants [17]).3. Numerical results
Firstly, we consider the case when the RCP and LCP modes are equally pumped, i.e., Ein 1 = Ein 2 = Ein. The homogeneous solutions of polariton along the x-direction (i.e., solutions of ψ 1,2 for Fin 1 = Fin 2 = Fs = 0) is analyzed before searching their soliton solutions. In this case, ψ 1 = ψ 2 = ψ and E 1 = E 2 = E. In the following discussion, excitonic part ψ is use to represent the polariton. For the photonic part E, similar results can be obtained, so the results are not repeated here. Figure 1(a) shows the value of |ψ| as a function of Es. An anticlockwise bistable loop of |ψ| is observed when Es changes increasingly and then decreasingly between 0 and 1.5. In Fig. 1(a), three values of Ein ( = 0.2, 0.19, and 0.18) are considered. The bistable region becomes wider and moves to smaller value of Es when Ein increases. Similarly, Fig. 1(b) shows the value of |ψ| as a function of Ein, for three values of Es. It is observed that the bistable region becomes narrower and moves to smaller values of Ein with Es increasing.
Fig. 1 Bistable curves of PS. (a) Maximum value of |ψ(0, y)| versus Stark pump strength, Es. (b) Maximum value of |ψ(0, y)| versus pump strength, Ein.
As proved in former studies, PS solutions could be found within the bistable region. Result of Fig. 1 indicates that bistable region exists for |ψ| versus either Es or Ein. Thus, PS branch can be found respectively through variation of Es and Ein. For the case of PS excitation through Stark effect, the arriving of Stark pulse causes blue-shift of the polariton branch, and induces dynamic fluctuations of the polariton field. If parameters of the Stark pulse are properly selected, the fluctuations may evolve into a soliton owning to balance between the dispersion and nonlinear effect. The stability of the PS is determined by their sensitive to the parameters of the Stark pulse, i.e., stable/unstable PS solution is insensitive/sensitive to the excitation parameters. Figure 2 shows the PS solutions searched along the bistable curves of |ψ| versus Es and Ein. The asterisks denote the peak values of stable PSs, and the pluses denote the unstable ones. The output profiles of |ψ| in the space and time domain for fixed values of Es and Ein are given in Fig. 3 , which reflect the process of PS formation. It is observed that the profiles of |ψ| become unchanged after a short time interval of oscillations from t = 0 (i.e., after the arrival of the Stark pulse).
Fig. 2 Solutions of PS. (a) Maxima of the excitonic components of stable (unstable) PSs as a function of stark pumping strength, Es. (b) Maxima of the excitonic components of stable (unstable) PSs as a function of pumping strength, Ein. In (a) Ein = 0.2, in (b) Es = 0.
Fig. 3 Soliton waveforms. (a), (b), (c), (d) correspond to point P1, P2, P1’, P2’ of Fig. 2. In (a) and (b) Es = 1.437, Ein = 0.2, in (c) and (d) Es = 0, Ein = 0.2602.
In the above discussions, the pump field and the stark pulse are assumed to be equally injected to the RCP and LCP modes. Figure 4 shows the case when the right and left circular polarization of the Stark pulses are unequally injected. The ratio of Stark pulses in the two polarization directions is defined as η = max(Fs 2)/max(Fs 1). Similarly, the ratio of right/left circular polarization of the PS is defined as ρ = max(|ψ 2|)/max(|ψ 1|) (ρ = min(|ψ 2|)/max(|ψ 1|)) when the left solitons are bright (dark).
Fig. 4 Soliton waveforms and values of ρ when RCP and LCP Stark pulses are unequally injected. (a) Values of ρ versus η. (b) and (c) are waveforms of solitons excited by Stark field when η = 0.2; (d) and (e) are waveforms of solitons excited by Stark field when η = 0.5; In the simulation, Es = 1.445, Ein = 0.2.
Similar to the case of using coherent writing pulse [27], the polarization of the generated PS is dependent on the Stark pulse. In Fig. 4(b) and 4(c), when η = 0.2, the RCP Stark pulse can trigger bright PS in the same polarization direction, while in the LCP direction, the Stark pulse is not strong enough to trigger a bright PS. Due to cross interaction between the two polarization modes, a dark PS in the LCP direction is excited passively [17,27 ]. When η > 0.4, in either the RCP or the LCP direction, bright PS with nearly the same peak amplitude can be triggered, seeing Fig. 4(d) and 4(e). As indicated by Fig. 4(a), ρ ≈0.3 when η < 0.4 and ρ switches to about 1 when η > 0.4.
When RCP and LCP modes are unequally pumped, i.e., Ein 1 ≠ Ein 2, the bistable regions in the two polarized direction are not overlapped. This means that a fixed value of Es might be within the bistable region of only one polarization mode, and the Stark pulse can only excite PS with the same polarization. Typical bistable curves and profiles of Stark pulse induced PS is given Fig. 5(a) and 5(b) respectively. In Fig. 5(a), for Ein 2 = 0.9Ein 1, the point of Es = 1.31 is within/without the bistable region of the RCP/LCP mode. Correspondingly, a bright PS in the RCP direction is excited by the Stark pulse. In the LCP direction, a dark soliton appears passively owning to cross interaction of the two polarization modes.
Fig. 5 Bistable and soliton waveform, for Ein 2 = 0.9Ein 1, Ein 1 = 0.2. (a) bistable curves; (b) PS waveforms when Es = 1.31.
4. Conclusion
In conclusion, we have analyzed the formation of polariton solitons through Stark effect in semiconductor microcavities. It is found that both the Stark and the pump field can induce bistability of polariton. Thus, PS branch versus intensity of either the Stark or the pump filed was observed by excitation of an additional Stark pulse. Different from former method that uses coherent writing pulse to excite PS, the Stark pulse causes dynamical fluctuations without accumulation of exciton reservoirs, which make it possible to manipulate PS easier and faster than that using coherent writing beam. Hence, the proposed method provides a novel way of trigger ultrafast PS, which has great potentials to advance future all optical information processing.
Acknowledgments
The authors gratefully acknowledge the supports of NSFC (61106045), the Open Research Fund of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences under Grant (SKLST201302), Open Research Fund of State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yatsen Unversity (OEMT-2015-KF-04), the PCSIRT (IRT1218), and the 111 Project (B14039).
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