Spin Rings in Semiconductor Microcavities I. A. Shelykh, 1,2 T. C. H. Liew, 1,3 and A.V. Kavokin 3,4 1 ICCMP, Universidade de Brasilia, 70904-970 Brasilia DF, Brazil 2 St. Petersburg State Polytechnical University, 195251, St. Petersburg, Russia 3 School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, United Kingdom 4 Physics Faculty, University of Rome II, 1, via della Ricerca Scientifica, 00133, Roma, Italy (Received 28 August 2007; revised manuscript received 21 January 2008; published 18 March 2008) New effects of self-organization and polarization pattern formation in semiconductor microcavities, operating in the nonlinear regime, are predicted and theoretically analyzed. We show that a spatially inhomogeneous elliptically polarized optical cw pump leads to the formation of a strongly circularly polarized ring in real space. This effect is due to the polarization multistability of cavity polaritons which was recently predicted. The possible switching between different stable configurations allows the realization of a localized spin memory element, suitable for an optical data storage device. DOI: 10.1103/PhysRevLett.100.116401 PACS numbers: 71.36.+c, 42.55.Sa, 42.65.Pc Introduction.—Self-organization and pattern formation are among the most interesting phenomena in various non- linear systems in physics, chemistry, and biology. In quan- tum physics nonlinearity arises from many-particle interactions, which being treated within the framework of the mean-field approximation result in the Hartree-Fock equations for interacting fermions and the Gross-Pitaevskii (GP) equation for interacting bosons. The latter is widely used for the description of the dynamics of atomic Bose- Einstein condensates (BECs) [1]. Mathematically, the GP equation is equivalent to the nonlinear Schro ¨dinger equa- tion of classical nonlinear optics. It describes a variety of intriguing phenomena in nonlinear media, such as vortex formation [2], self-focusing, and soliton propagation [3]. Recently, examples of self-organization were reported in the system of interacting 2D excitons, where the formation of ring patterns in the emission distribution was experi- mentally observed in the nonlinear regime [4]. This phe- nomenon was initially attributed to the superfluid phase transition. More recent models, however, identified the crucial role of the separation of classical electron and hole plasmas with emission from the sharp circular bound- ary between these two regions [5]. Because of strong dephasing this phenomenon can be described by classical diffusion equations for electrons and holes, rather than by a quantum equation of the GP type for a spatially coherent excitonic BEC. Cavity polaritons seem to be more appropriate candi- dates for the observation of quantum nonlinear phe- nomena. Being combinations of the cavity photon and 2D exciton, they have extremely small effective mass (about 10 4 –10 5 of the free electron mass) and, at the same time, they interact efficiently with one another. Polariton-polariton interactions lead to various nonlinear effects in microcavities, including parametric scattering [6] and bistability [7]. Because of the long decoherence time [8] and the fact that in the low density limit they behave as weakly interacting bosons [9], the dynamics of the polar- iton system can be described by the GP equation [10,11]. Being treated coherently, polariton-polariton interactions result in the suppression of Rayleigh scattering [10] and ring pattern formation [11] in polariton systems. Both effects are due to the renormalization of the dispersion of elementary excitations and an associated superfluid tran- sition in the system. An important peculiarity of cavity polaritons is related to their spin degree of freedom [12]. Polaritons have two possible spin projections on the structure growth axis, 1, corresponding to the right (  ) and left (  ) circular polarizations of emitted photons. In the case of nonzero in-plane wave vector these two components are mixed by TE-TM splitting [13]. A further mixing of the linear polar- izations appears due to the spin dependent polariton- polariton interaction [14], which affects the superfluid properties of the system [11,15] and leads to remarkable nonlinear effects in polariton spin relaxation, such as self- induced Larmor precession and inversion of the linear polarization during the scattering act [12,16]. Recently, the scalar semiclassical approach based on the Gross-Pitaevskii equation was extended to account for the two polarization states of resonantly pumped cavity po- laritons [17]. It was shown that the nonlinear, polariza- tion dependent polariton-polariton interactions result in a multistability of the driven polariton mode. In this Letter we analyze how the polarization multi- stability and hysteresis can lead to polarization pattern formation in realistic semiconductor microcavities. Qualitatively, the polarization multistability and hys- teresis can be understood as follows. Let us consider a quantum microcavity resonantly driven at k  0 by a cw laser beam with circular polarization degree:  pump c  P   P  P   P  ; (1) where P  and P  represent the intensities of the   and   components of the pump, respectively. The polariton wave function   satisfies the driven spinor Gross- Pitaevskii equation [17], which in the stationary regime PRL 100, 116401 (2008) PHYSICAL REVIEW LETTERS week ending 21 MARCH 2008 0031-9007= 08=100(11)=116401(4) 116401-1 © 2008 The American Physical Society