We discuss quantum random walk of two photons using linear optical elements. We analyze the quantum random walk using photons in a variety of quantum states including entangled states. We find that for photons initially in separable Fock states, the final state is entangled. For polarization entangled photons produced by type II downconverter, we calculate the joint probability of detecting two photons at a given site. We show the remarkable dependence of the two photon detection probability on the quantum nature of the state. In order to understand the quantum random walk, we present exact analytical results for small number of steps like five. We present in details numerical results for a number of cases and supplement the numerical results with asymptotic analytical results.

We investigate the power-dependent photoluminescence spectra from a strongly coupled quantum dot-cavity system using a quantum master equation technique that accounts for incoherent pumping, pure dephasing, and fermion or boson statistics. Analytical spectra at the one-photon correlation level and the numerically exact multi-photon spectra for fermions are presented. We compare to recent experiments on a quantum dot-micropiller cavity system and show that an excellent fit to the data can be obtained by varying only the incoherent pump rates in direct correspondence with the experiments. Our theory and experiments together show a clear and systematic way of studying stimulated-emission induced broadening and anharmonic cavity-QED. Introduction.-Single quantum dot (QD) -cavity systems facilitate the realization of solid state qubits (quantum bits) and have applications for producing single photons [1,2,3] and entangled photons [4,5]. Rich in physics and potential applications, the coupled QDcavity has been inspiring theoretical and experimental groups to probe deeper into the underlying physics of both weak and strong coupling regimes of semiconductor cavity-QED (quantum electrodynamics). Key signatures of cavity-QED include the Purcell effect and vacuum Rabi oscillations. Although a well known phenomenon in atomic cavity optics [6], vacuum Rabi splitting in a semiconductor cavity was only realized a few years ago [7,8,9]. Inspired by the recent surge of related experiments, many researchers have been working hard to develop new theoretical tools to understand the semiconductor cavity-QED systems. For example, the persistent excitation of the cavity mode for large excitoncavity detunings was measured [10,11], and qualitatively explained by extended theoretical approaches that account for coupling between the leaky cavity mode and the exciton, and by showing that the main contribution to the emitted spectrum comes from the cavity-mode emission [12,13,14,15,16]. These formalisms assume an initially excited exciton or an initially excited leaky cavity mode, and they are valid for low pump powers. However, an interesting question that has been posed recently, e.g., see Refs. [17,18,19], is what is the role of an incoherent pump on the photoluminescence (PL) spectra, where the pump can excite the exciton or cavity mode? To experimentally investigate the pump-dependent spectra, two recent experiments have been respectively reported by Münch et al. [20] for a QD-micropillar system, and by Laucht et al.[21] for a QD-photonic crystal system; these measurements show the pump-induced crossover from strong to weak coupling.In this work, we present a master equation (ME) the-

We present a formal theory of single quantum-dot coupling to a planar photonic crystal that supports quasi-degenerate cavity modes, and use this theory to describe, and optimize, entangledphoton-pair generation via the biexciton-exciton cascade. In the generated photon pairs, either both photons are spontaneously emitted from the dot, or one photon is emitted from the biexciton spontaneously and the other is emitted via the leaky-cavity mode. In the strong-coupling regime, the generated photon pairs can be maximally entangled, in qualitative agreement with the simple dressed-state predictions of Johne et al. [Phys. Rev. Lett. vol. 100, 240404 (2008)]. We derive useful and physically-intuitive analytical formulas for the spectrum of the emitted photon pairs in the presence of exciton and biexciton broadening, which is necessary to connect to experiments, and demonstrate the clear failure of using a dressed-state approach. We also present a method for calculating and optimizing the entanglement between the emitted photons, which can account for post-sample spectral filtering. Pronounced entanglement values of greater than 80% are demonstrated using experimentally achievable parameters, even without spectral filtering.

Using resonant interaction between atoms and the field in a high quality cavity, we show how to generate a superposition of many mesoscopic states of the field. We study the quasi-probability distributions and demonstrate the nonclassicality of the superposition in terms of the zeroes of the Q-function as well as the negativity of the Wigner function. We discuss the decoherence of the generated superposition state. We propose homodyne techniques of the type developed by Auffeves et al [Phys. Rev. Lett. 91, 230405 (2003)] to monitor the superposition of many mesoscopic states.

Using resonant interaction between atoms and the field in a high quality cavity, we show how to realize quantum random walks as proposed by Aharonov et al [Phys. Rev. A {\bf48}, 1687 (1993)]. The atoms are driven strongly by a classical field. Under conditions of strong driving we could realize an effective interaction of the form $ iS^{x}(a-a^{\dag})$ in terms of the spin operator associated with the two level atom and the field operators. This effective interaction generates displacement in the field's wavefunction depending on the state of the two level atom. Measurements of the state of the two level atom would then generate effective state of the field. Using a homodyne technique, the state of the quantum random walker can be monitored.Comment: 6-page 4-fig. submitted Phy. Rev