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Spontaneous emission (SE) from a two-level atom in a photonic crystal (PC) with anisotropic oneband model is investigated using the fractional calculus. Analytically solving the kinetic equation in terms of the fractional exponential function, the dynamical discrepancy of SE between the anisotropic and isotropic systems is discussed on the basis of different photon density of states (DOS) and the existence of incoherent diffusion field that becomes even more clearly as the atomic transition frequency lies close to the band edge. With the same atom-field coupling strength and detuning in the forbidden gap, the photon-atom bound states in the isotropic system turn into the unbound ones in the anisotropic system that is consistent with the experimental observation in P hys. Rev. Lett. 96, 243902 (2006). Dynamics along different wavevectors with various curvatures of dispersion is also addressed with the changes of the photon DOS and the appearance of the diffusion fields.

Modulation instability (MI) in a coupled resonator optical waveguide (CROW) and photonic-crystal waveguide (PCW) with nonlinear Kerr media was studied by using the tight-binding theory. By considering the coupling between the defects, we obtained a discrete nonlinear evolution equation and termed it the extended discrete nonlinear Schrödinger (EDNLS) equation. By solving this equation for CROWs and PCWs, we obtained the MI region and the MI gains, G(p,q), for different wavevectors of the incident plane wave (p) and perturbation (q) analytically. In CROWs, the MI region, in which solitons can be formed, can only occur for pa being located either before or after pi/2, where a is the separation of the cavities. The location of the MI region is determined by the number of the separation rods between defects and the sign of the Kerr coefficient. However, in the PCWs, pa in the MI region can exceed the pi/2. For those wavevectors close to pi/2, the MI profile, G(q), can possess two gain maxima at fixed pa. It is quite different from the results of the nonlinear CROWs and optical fibers. By numerically solving the EDNLS equation using the 4th order Runge-Kutta method to observe exponential growth of small perturbation in the MI region, we found it is consistent with our analytic solutions.

We suggest a better mathematical method, fractional calculus, for studying the behavior of the atom-field interaction in photonic crystals. By studying the spontaneous emission of an atom in a photonic crystal with a one-band isotropic model, we found that the long-time inducing memory of the spontaneous emission is a fractional phenomenon. This behavior could be well described by fractional calculus. The results show no steady photon-atom bound state for the atomic resonant transition frequency lying in the proximity of the allowed band edge which was encountered in a previous study (Woldeyohannes and John 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R43). The correctness of this result is validated by the 'cut-off smoothing' density of photon states (DOS) with fractional calculus. By obtaining a rigorous solution without the multiple-valued problem for the system, we show that the method of fractional calculus has a logically concise property.

We study non-equilibrium microcavity-polariton condensates (MPCs) in a harmonic potential trap theoretically. We calculate and analyze the steady state, collective-excitation modes and instability of MPCs. Within excitation modes, there exist Nambu-Goldstone modes that can reveal the pattern of the spontaneous symmetry breaking of MPCs. Bifurcation of the stable and unstable modes is identified in terms of the pumping power and spot size. The unstable mechanism associated with the inward supercurrent flow is characterized by the existence of a supersonic region within the condensate.PACS number: 05.30. Jp, 03.75.Kk, 47.20.Ky, 71.36.+c In past years, there have been intensive searches for a new Bose-Einstein condensate in solids.Researchers found such a candidate called the microcavity-polariton condensate (MPC), which has been created from the interaction of cavity photons and confined excitons in the strong-coupling regime [1,2]. Growing research activities in this MPC can be attributed to the system being intrinsically out-ofequilibrium determined by the dynamical balance between interactions, trapping potentials, pumping and decay [3]. Rich phenomena from non-equilibrium many-body physics are accessible in this system. Many phase-transition signatures from inhomogeneous MPCs, such as spectral and spatial narrowing and first-order coherence, were studied by Balili et al. [8]. There still exists superfluidity even though the MPC involves a non-equilibrium dissipative character [4-6]. Due to the continuous pumping and disorders of MPCs, the instability of rotationally symmetric states and vortices appear spontaneously without stirring or rotating MPCs [6, 7]. Non-equilibrium MPCs also show the spontaneously rotational symmetry breaking and are unstable towards the formation of vortices and spontaneous array of vortices without any rotational drive [9-11]. Although the spontaneous symmetry breaking followed by non-equilibrium MPCs has been demonstrated, the properties of spontaneous-symmetry-breaking states still need to be understood and are worthily studied. Non-equilibrium MPCs have been theoretically investigated by several groups, for instance, Keeling et al. studied the slow dynamics of MPCs by eliminating the reservoir dynamics [9, 10], whileWouters and Carusotto couple the dynamics of polaritons from condensate to the reservoir with ratediffusion equation in homogeneous systems [12]. Both of them concluded the same excitation behavior of diffusive Goldstone modes at low momentum, which is recognized as a unique feature coming from the driven-dissipative systems [13]. In this Letter, we shall apply the complex Gross-Pitaveskii equation (cGPE) to study MPCs with pumping, and decay dynamics in a harmonic potential trap [9,10].We study collective excitations and instability of MPCs, and show that the existence of Nambu-Goldstone modes within collective-excitation modes can reveal the pattern of the spontaneous