Polarization-anisotropy induced spatial anisotropy of polariton amplification in planar semiconductor microcavities
Based on a microscopic many-particle theory we investigate the amplification of polaritons in planar semiconductor microcavities. We study a spatially perfectly isotropic microcavity system and excitation geometry. For this system, our analysis shows that a pump-induced vectorial polarization anisotropy can lead to a spatial anisotropy in the stimulated amplification of polaritons. This effect is brought about by the interplay of the longitudinal-transverse cavity mode splitting (TE-TM splitting) and the spin-dependence of the polariton-polariton scattering processes.
pacs:71.35.-y, 71.36.+c, 42.65.Sf, 42.65.-k
Introduction – In the past decade the parametric amplification of polaritons in planar semiconductor microcavities has been intensively investigated in theory and experiment, see, e.g., Refs. Savvidis et al., 2000; Huang et al., 2000; Ciuti et al., 2000; Stevenson et al., 2000; Saba et al., 2001; Whittaker, 2001 or the reviews given in Refs. Ciuti et al., 2003; Baumberg and Lagoudakis, 2005; Keeling et al., 2007. Whereas the polariton amplification was initially reported in a co-circularly polarized pump-probe setupSavvidis et al. (2000), for other configurations it can show a rich vectorial polarization dependence.Lagoudakis et al. (2002); Kavokin et al. (2003); Eastham and Whittaker (2003); Renucci et al. (2005); Kavokin et al. (2005a); Dasbach et al. (2005); Krizhanovskii et al. (2006); Sanvitto et al. (2006); Klopotowski et al. (2006); Schumacher et al. (2007a); Martin et al. (2007) This vectorial polarization dependence is governed by the underlying spin-dependent polariton-polariton scattering processes, longitudinal-transverse cavity mode splitting (TE-TM splitting), and possible structural anisotropy of the investigated systems.
In the most intensively studied configuration, a pump pulse induces a coherent polariton density close to the inflection point of the lower polariton branch (LPB), the so-called “magic angle”. In this configuration phase-matched resonant pairwise scattering of pump-induced polaritons into probe and four-wave mixing (FWM) directions is allowed which makes the polariton amplification most efficient. However, for oblique pump excitation the intrinsic cylindrical symmetry of the planar microcavity system is destroyed by the optical fields.
In this paper, we study an excitation geometry where the incoming pump does not necessarily spoil the symmetry of the system. As the simplest such configuration, we investigate the polariton amplification for excitation with a pump pulse, which is incident along the system’s symmetry axis, i.e., perpendicular to the plane of the embedded quantum well along the axis. In this excitation configuration the pump-excited polaritons carry zero in-plane momentum . As illustrated in Fig. 1, for pump frequency above the LPB, FWM processes triggered by fluctuations in the cavity photon field can give rise to pairwise off-axis scattering of pump-excited polaritons into two polaritons with finite and opposing in-plane momentum and . For a pump-induced exciton density above the stimulated amplification threshold these scattering processes can lead to strong off-axis () signals as has been demonstrated in Ref. Romanelli et al., 2007 for a slightly different excitation geometry. For the perfectly isotropic system studied in this work, spontaneous (fluctuation-induced) off-axis pattern formation is expected which is invariant under rotation about the propagation direction of the pump.
Based on a microscopic many-particle theory we discuss that even if the incoming optical pump field does not destroy the spatial isotropy of the system, a specific choice of its vectorial polarization state can lead to a pronounced azimuthal angular dependence of the spontaneous signals in the off-axis directions. We discuss the interplay of the longitudinal-transverse cavity mode splitting (TE-TM splitting) and the spin-dependent polariton-polariton scattering processes that leads to this spatial anisotropy, solely induced by a vectorial polarization anisotropy.
Microscopic theory – Our theoretical analysis is based on the microscopic many-particle theory presented in Ref. Schumacher et al., 2007a. It includes spin and frequency dependent polariton-polariton scatteringKwong et al. (2001a, b); Savasta et al. (2003); Schumacher et al. (2007b) and a TE-TM cavity mode splitting in the coupled cavity-confined field and exciton dynamics. Calculations are done for the same GaAs based microcavity system studied in Ref. Schumacher et al., 2007a but for a slightly smaller intrinsic dephasing for the excitonic polarization amplitude and the fields in the cavity modes, . Within this framework and for steady-state monochromatic pump excitation with frequency , the linearized coupled dynamics for field and excitonic polarization components with in-plane momenta and can be cast into the following simple form:
The vector groups together the dynamic variables for field components (in the TE and TM cavity modes, respectively) and excitonic polarization components (longitudinal or transversal to , respectively). Analyzing the linear stability of the pump-driven polariton dynamics against off-axis fluctuations, no incoming fields in the and directions are explicitly considered. The matrix is a -dependent but time independent matrix where all the system and pump parameters enter. The components of carry only the time dependence relative to the time dependence that is imposed by the pump field. This additional time dependence is determined by the eigenvalues of . The two-exciton continuum is treated in Markov approximation as in Ref. Schumacher et al., 2007a but non-Markovian contributions from excitation of the bound biexciton state are included in Eq. (1). This Markov approximation should be particularly well fulfilled for the present study since the dominant and thus relevant modes oscillate with or close to the pump frequency (cf. Fig. 1). We neglect the influence of excitonic phase-space filling nonlinearities which have been found to be insignificant for the stimulated polariton amplification.Schumacher et al. (2007a) To concentrate on the issues under investigation, we do not consider possible complications of our discussion from bistabilityWouters and Carusotto (2007) or multistabilityGippius et al. (2007) for the steady-state pump-induced polarization.
Within this framework, the eigenvalues of contain comprehensive information about temporal growth or decay (real part of ) and frequency (imaginary part of ) of the possible off-axis modes as long as the system dynamics is linear in the off-axis contributions. Information about the vectorial polarization state of these modes is contained in the respective eigenvectors. We investigate the system dynamics for a total (including both vectorial polarization states) pump-induced exciton density of , which is close to the amplification threshold, and pump frequency tuned below the bare exciton resonance. The data presented in the following quantitatively depend on the choice of excitation, cavity and other material parameters. However, the qualitative nature of the results is robust. We diagonalize for different in-plane momenta , and analyze the real-part of the eigenvalue with the largest real part. This gives us the growth rate of the initially fastest growing and therefore “most unstable” mode, which will eventually dominate the system dynamics for sufficiently long growth periods.
Results and Discussion – For excitation with a circularly polarized pump, Fig. 2(a) shows the real-part of the eigenvalue of with the largest real part, i.e., the maximum growth (for ) or minimum decay (for ) rate for each . (In the following loosely referred to as ‘growth’.) The largest growth rate is found on the ‘elastic circle’ where pairs of the off-resonantly excited pump polaritons can resonantly be scattered on the pump-renormalized polariton dispersion (cf. Fig. 1). No angular dependence of the displayed growth rates, and thus no anisotropy of the stimulated scattering is found. Note that for the chosen pump density the stimulated amplification threshold is not reached (all ) in Fig. 2(a), all modes are exponentially decaying and no significant off-axis scattering of light initiated by fluctuations is expected.
This situation changes for excitation with a linearly polarized pump as shown in Fig. 2(b). By choosing a particular orientation of the vectorial polarization of the pump, the spin-dependent polariton-polariton scattering together with the TE-TM cavity mode splitting induces a spatial anisotropy for the growth of off-axis signals over time. It is worthwhile to emphasize again that this anisotropy comes about without any anisotropy in the cavity or the quantum well. The stimulated scattering intrinsically depends on the orientation of to the pump’s linear polarization vector: Owing to the spin-dependence of the underlying polariton-polariton scattering processes, polaritons scattered off-axis are preferably polarized perpendicular to the pump’s vectorial polarization state.Schumacher et al. (2007a) This preference leads to different scattering rates into the TE or TM polarized modes, depending on the orientation of the off-axis momenta and of the scattered polaritons relative to the pump’s polarization vector. For the TE mode, the electric field vector is perpendicular to the plane of incidence, thus also perpendicular to . For the TM mode, the component of the electric field vector in the quantum well plane is parallel to . Therefore, for the results shown in Fig. 2(b), for parallel to the pump’s polarization vector, scattering in the TE mode is preferred. For perpendicular to the pump’s polarization vector, scattering into the TM mode is preferred. A similar anisotropy has been observed earlier but in the linear optical regime, in the ballistic propagation of cavity polariton wave packetsLangbein et al. (2007) and in Ref. Kavokin et al., 2005b where an optical spin Hall effect was proposed. In Refs. Kavokin et al., 2005b; Langbein et al., 2007 the TE-TM cavity mode splitting has led to a four-fold rotation symmetry in the results. In our case the spin-dependent many-particle interactions reduce the rotation invariance further to a two-fold rotation symmetry. As shown in Fig. 3(a), a four-fold rotation symmetry in the polariton amplification can only be regained if the polariton-polariton scattering is treated on the mean-field Hartree-Fock level, neglecting excitonic correlations in the scattering processes. For scattering with either parallel or perpendicular to the pump’s polarization vector, in Eq. (1) is block-diagonal. Consequently, in these directions the polaritons are scattered off-axis either into the TE or into the TM mode. For all other orientations, the eigenvectors of contain contributions from both TE and TM modes. Note that without correlations in Fig. 3(a) again the amplification threshold is not reached for the chosen density.
Finally, Fig. 3(b) shows the growth rate including correlations but for zero TE-TM cavity mode splitting. Circular symmetry is recovered and the amplification threshold is reached. This indicates that minimization or proper engineering of the TE-TM cavity-mode splitting – at least close to – can be used to rebuild the isotropy of the stimulated amplification mechanism. This way possible benefits of excitation with a linearly polarized pump, such as a lower threshold density or polarization-selective separation of the off-axis signals from the pump, respectively, may be accessible without inducing an undesired anisotropy in the system dynamics.
Strictly speaking, the linear stability analysis discussed above only reveals information about the initial growth of off-axis fluctuations as long as the system dynamics is linear in the resulting off-axis signals. To avoid a clear deviation from this situation in an experimental attempt to verify the discussed results, the pump frequency should be chosen such that the off-axis scattering of pump polaritons does not lead to a significant buildup of polariton density at the “magic angle”. This precaution is in analogy to Ref. Romanelli et al., 2007 and prevents that polariton scattering at the “magic angle” would eventually take over the nonlinear system dynamics with increasing intensity of the off-axis signals over time.
As an alternative to the steady-state experiment, all assumptions entering the linear stability analysis can be well satisfied in a typical pump-probe experiment: The pump pulse needs to be long enough, so that the steady-state limit is imitated in a good approximation. The incoming probe pulse must be weak so that even after amplification it does not significantly influence the polarization dynamics in the pump direction. The incoming probe pulse should also be short enough so that it does not considerably drive the polarization dynamics, but merely acts as a seed for the off-axis signal in the probe direction. With the pump pulse in normal incidence, the probe pulse is used to map out the amplification in the plane.
Remarks & Conclusions – We have shown that in a perfectly isotropic microcavity system a vectorial polarization anisotropy of the pump can induce a spatial anisotropy for the stimulated amplification of polaritons. This anisotropy is brought about by the combined influence of a TE-TM cavity mode splitting and the spin-dependent polariton-polariton scattering processes. Besides its general interest, this effect may have an influence on, e.g., the generation of correlated photon pairs in the stimulated amplification regime.Romanelli et al. (2007) Angular dependencies as discussed in this work are also expected to play a role in other systemsDiederichs et al. (2006); Schumacher et al. (2007c) as long as they are not overshadowed by structural imperfections.
Acknowledgments – The author gratefully acknowledges various fruitful discussions with Nai H. Kwong and Rolf Binder throughout this project. He also acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, project No. SCHU 1980/3-1). This work has further been supported by ONR, DARPA, and JSOP.
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