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We study some discrete symmetries of unbiased (Hadamard) and biased quantum walk on a line, which are shown to hold even when the quantum walker is subjected to environmental effects. The noise models considered in order to account for these effects are the phase flip, bit flip, and generalized amplitude damping channels. The numerical solutions are obtained by evolving the density matrix, but the persistence of the symmetries in the presence of noise is proved using the quantum trajectories approach. We also briefly extend these studies to quantum walk on a cycle. These investigations can be relevant to the implementation of quantum walks in various known physical systems. We discuss the implementation in the case of NMR quantum information processor and ultracold atoms. © 2007 The American Physical Society.

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Physical Review A - Atomic, Molecular, and Optical Physics

Symmetries and noise in quantum walk

The Wigner transform of the master equation describing the reduced dynamics of the system, of a harmonic oscillator coupled to an oscillator bath, was obtained by Karrlein and Grabert. It was shown that for some special correlated initial conditions the master equation reduces, in the classical limit, to the corresponding classical Fokker-Planck equation obtained by Adelman. However, for separable initial conditions the Adelman equations were not recovered. We resolve this problem by showing that, for separable initial conditions, the classical Langevin equations are somewhat different from the one considered by Adelman. We obtain the corresponding Fokker-Planck equation and show that they exactly match the classical limit of the evolution of the Wigner function obtained from the master equation for separable initial conditions. We also discuss why thermal initial conditions correspond to Adelman's solution. © 2006 The American Physical Society.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Classical limit of master equation for a harmonic oscillator coupled to an oscillator bath with separable initial conditions

The (in)finite dimensional symplectic group of homogeneous canonical transformations is represented on the bosonic Fock space by the action of the group on the ultracoherent vectors, which are generalizations of the coherent states. © World Scientific Publishing Company.

Infinite Dimensional Analysis, Quantum Probability and Related Topics

Ultracoherence and canonical transformations

Canonical transformations are defined and discussed along with the coherent and the ultracoherent vectors. It is shown that the single-mode and the n-mode squeezing operators are elements of the group of canonical transformations. An application of canonical transformations is made, in the context of open quantum systems, by studying the effect of squeezing of the bath on the decoherence properties of the system. Two cases are analysed. In the first case, the bath consists of a massless bosonic field with the bath reference states being the squeezed vacuum states and squeezed thermal states while in the second case a system consisting of a harmonic oscillator interacting with a bath of harmonic oscillators is analysed with the bath being initially in a squeezed thermal state. © 2005 IOP Publishing Ltd.

Journal of Physics A: Mathematical and General

Applications of canonical transformations

The functional integral formalism is used to provide a unified treatment of the effect of squeezing of the bath and frequency modulation of the system-bath coupling on the decoherence and dissipation properties of an open quantum system. The system chosen is a standard one consisting of a particle in a harmonic oscillator potential interacting with a bath of harmonic oscillators by a coupling of the position-position kind. Using an Ohmic bath and the high-temperature limit, the coefficients of the master equation describing dissipation and decoherence are obtained and analyzed. © 2004 Elsevier B.V. All rights reserved.

Physica A: Statistical Mechanics and its Applications

Decoherence and dissipation of an open quantum system with a squeezed and frequency-modulated heat bath

The dynamics of an open quantum system exhibiting the quantum Brownian motion is analyzed when the coupling between the system and its environment is nonlinear, and the system and the reservoir are initially correlated. For couplings quadratic in the environment variables, the influence functional for the system is obtained perturbatively up to second order in the coupling constant, and then the propagator is explicitly evaluated when the particle is under the influence of a harmonic potential and an additional anharmonic potential, the so-called washboard potential. As an application of the propagator, the master equation and the Wigner equation are obtained for the quantum Brownian particle moving in a harmonic potential for the generalized correlated initial condition, and then for the specific case of the simplified “thermal” initial condition. The system is shown to obey the corresponding fluctuation-dissipation theorem. © 2003 The American Physical Society.

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

General quantum Brownian motion with initially correlated and nonlinearly coupled environment

We consider the general problem of a single two-level atom interacting with a multimode radiation field (without the rotating-wave approximation), and additionally take the field to be coupled to a thermal reservoir. Using the method of bosonization of the spin operators in the Hamiltonian, and working in the Bargmann representation for all the boson operators, we obtain the propagator for the composite system using the techniques of functional integration, under a reasonable approximation scheme. The propagator is explicitly evaluated for a simplified version of the system with one spin and a dynamically coupled single-mode field. The results are also checked on the known problem of quantum Brownian motion.

Propagator for a spin-Bose system with the Bose field coupled to a reservoir of harmonic oscillators

We present the quantum theory of a Stern-Gerlach system in contact with a linearly dissipative environment at an arbitrary temperature. The dynamics of the reduced density matrix of the system is calculated using the path integral technique, under a general condition when the system and its environment may not be decoupled initially. We analyze the behavior of the density matrix in the long-time limit and compute the time scales of decay of the elements off-diagonal in the coordinate and the momentum space, for the cases of high temperature, zero temperature, and intermediate temperature baths. © 2000 The American Physical Society.

Quantum theory of a Stern-Gerlach system in contact with a linearly dissipative environment

In this study, an attempt was made to solve the dynamics of the Stern-Gerlach spin in contact with a linearly dissipative environment at an arbitrary temperature. All the previous results in the high-temperature limit were recovered. It was observed that except for the extreme quantum case of zero temperature, the system shows the expected classical diffusive behavior over long times.