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arXiv:quant-ph/0208176 · Journal of Modern Optics

Damped quantum interference using stochastic calculus

David Cienfuegos Salgado, J. L. Sánchez‐Gómez

Published Jan 1, 2003• 2 citations
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Abstract

It is shown how the phase-damping master equation, either in Markovian and non-Markovian regimes, can be obtained as an averaged random unitary evolution. This, apart from offering a common mathematical set-up for both regimes, enables us to solve this equation in a straightforward manner just by solving the Schrödinger equation and taking the stochastic expectation value of its solutions after an adequate modification. Using the linear entropy as a figure of merit (basically the loss of quantum coherence) four distinct kinds of environment are suggested.

DOI
10.1080/09500340308234546

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It is shown how the phase-damping master equation, either in Markovian and non-Markovian regimes, can be obtained as an averaged random unitary evolution. This, apart from offering a common mathematical set-up for both regimes, enables us to solve this equation in a straightforward manner just by solving the Schrödinger equation and taking the stochastic expectation value of its solutions after an adequate modification. Using the linear entropy as a figure of merit (basically the loss of quantum coherence) four distinct kinds of environment are suggested.
It is shown how any Lindbladian evolution with selfadjoint Lindblad operators, either Markovian or nonMarkovian, can be understood as an averaged random unitary evolution. Both mathematical and physical consequences are analyzed. First a simple and fast method to solve this kind of master equations is suggested and particularly illustrated with the phase-damped master equation for the multiphoton resonant Jaynes-Cummings model in the rotating-wave approximation. A generalization to some intrinsic decoherence models present in the literature is included. Under the same philosophy a proposal to generalize the Jaynes-Cummings model is suggested whose predictions are in accordance with experimental results in cavity QED and in ion traps. A comparison with stochastic dynamical collapse models is also included.
We suggest a novel proposal to express decoherence in open quantum systems by jointly employing spectral and stochastic methods. This proposal, which basically perturbs the unitary evolution operator in a random fashion, allows us to embrace both markovian and nonmarkovian situations with little extra effort. We argue that it can be very suitable to deal with models where an approximation neglecting some degrees of freedom is undertaken. Mathematical simplicity is also obtained both to solve some master equations and to arrive at experimentally measured decoherence functions.
A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equation derived by the non-conventional treatment of the damping theory.
Suppose a single qubit is repeatedly prepared and evolved under imperfectly-controlled conditions. A drunk model represents uncontrolled interactions on each experimental trial as random or stochastic terms in the qubits Hamiltonian operator. Time evolution of states is generated by a stochastic differential equation whose sample paths evolve according to the Schrödinger equation. For models with Gaussian white noise which is independent of the qubits state, the expectation value of the solution obeys a master equation which is identical to the high-temperature limit of the Bloch equation. Drunk models predict that experimental data can appear consistent with decoherence even if qubit states evolve by unitary transformations. Examples are shown in which reversible evolution appears to cause irreversible information loss. This paradox is resolved by distinguishing between the true state of a system and the estimated state inferred from an experimental dataset.
We study the non-Markovian quantum interference phenomenon of a multi-state atomic system coupled to a bosonic dissipative environment by using the exact master equations derived in this paper. Two examples involving four-level systems with or without external control fields are investigated. Our results show that non-Markovian master equations are capable of exhibiting quantum interference phenomena in a temporal domain that has not fully explored before. In particular, we show that the environmental memory is instrumental in the onset of non-Markovian quantum interference pattern in different time scales.