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arXiv:1012.1051 · Physical Review A

Precision of electromagnetic control of a quantum system

Ching-Kit Chan, Lu Jeu Sham

Published Sep 22, 2011• 10 citations
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Abstract

Coherent control of a quantum system is limited both by the decoherence due to environment and the quantum nature of the control agent. The high fidelity of control demanded by fault-tolerant quantum computation and the intrinsic interest in nonclassical effects from the interplay between control and dissipation are motivations for a detailed study of the interaction dynamics between the quantum system and the macroscopic environment and control agent. We present a detailed time-evolution study of a two-level system interacting with a laser pulse and the electromagnetic vacuum in the multimode Jaynes-Cummings model. A diagrammatic formalism allows easy identification of coherent dynamics and relaxation of the two-level system. We demonstrate a computational method of dynamics with precise error bounds for fast operations versus slow decoherence, spanning the Markovian and non-Markovian regimes. Comparison against an exact model solution of our results with existing approximations of the master equation shows the lack of accuracy in the latter.

DOI
10.1103/physreva.84.032116

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Coherent control of a quantum system is limited both by the decoherence due to environment and the quantum nature of the control agent. The high fidelity of control demanded by fault-tolerant quantum computation and the intrinsic interest in nonclassical effects from the interplay between control and dissipation are motivations for a detailed study of the interaction dynamics between the quantum system and the macroscopic environment and control agent. We present a detailed time-evolution study of a two-level system interacting with a laser pulse and the electromagnetic vacuum in the multimode Jaynes-Cummings model. A diagrammatic formalism allows easy identification of coherent dynamics and relaxation of the two-level system. We demonstrate a computational method of dynamics with precise error bounds for fast operations versus slow decoherence, spanning the Markovian and non-Markovian regimes. Comparison against an exact model solution of our results with existing approximations of the master equation shows the lack of accuracy in the latter.
A precise time-dependent control of a quantum system relies on an accurate account of the quantum interference among the system, the control, and the environment. A diagrammatic technique has been recently developed to precisely calculate this quantum correlation for a fast multimode coherent photon control against slow relaxation, valid for both Markovian and non-Markovian systems. We review this formalism in comparison with the existing approximate theories and extend it to cases with controls by photon state other than the coherent state.
Decoherence of the dispersive Jaynes–Cummings model which is placed in a lossy cavity is studied by means of the time-convolutionless non-Markovian quantum master equation. The effect of the interaction between a two-level atom and photon field on the damping operator of the quantum master equation is made clear by calculating the decay of coherence and entanglement of the relevant system. It is shown that although the interaction effect on the damping operator is small when the cavity loss has the Markovian property, it becomes significantly large when the cavity loss is subject to the non-Markovian process. Furthermore, the quantum channel of the two-level atom derived by the dispersive Jaynes–Cummings model is investigated. The decay of entanglement of the two two-level atoms is examined to show the importance of the interaction effect on the damping operator.
We perform a first-principles derivation of the general master equation to study the non-Markovian dynamics of a two-level atom (2LA) interacting with an electromagnetic field (EMF). We use the influence functional method, which can incorporate the full back reaction of the field on the atom, while adopting Grassmannian variables for the 2LA and the coherent-state representation for the EMF. We find exact master equations for the cases of a free quantum field and a cavity field in the vacuum. In response to the search for mechanisms to preserve maximal coherence in quantum computations in ion trap prototypes, we apply these equations to analyze the decoherence of a 2LA in an EMF, and find that decoherence time is close to relaxation time. This is at variance with the claims by authors who studied the same system but used a different coupling model. We explain the source of difference and argue that, contrary to common belief, the EMF, when resonantly coupled to an atom, does not decohere it as efficiently as a bath does on a quantum Brownian particle. The master equations for non-Markovian dynamics derived here are expected to be useful for exploring new regimes of 2LAEMF interaction, which is becoming physically important experimentally.
In quantum computation, it is of paramount importance to locate the parameter space where maximal coherence can be preserved in the qubit system. In recent years environment-induced decoherence based the quantum Brownian motion (QBM) models have been applied to two level systems (2LS) interacting with an electromagnetic field, leading to the general belief that 2LS are easily decohered. In a recent paper C. Anastopoulos and B. L. Hu [Phys. Rev. A62, (2000) 033821] derived a new exact non-Markovian master equation at zero temperature, from which they showed that this belief is actually misplaced. For a two-level atom (2LA)- electromagnetic field (EMF) system the decoherence time is rather long, comparable to the relaxation time. Theoretically this is because the dominant interaction is the hatsigmapm\\hat \\sigma_{\\pm}hatsigmapm​ type of coupling between the two levels (what constitutes the qubit) and the field, not the hatsigmaz\\hat \\sigma_zhatsigmaz​ type, which shows the QBM behavior. Depending on the coupling the field can act as a resonator (in an atom cavity) or as a bath (in QBM) and produce very different decoherent behavior in the system. This is not new to Cavity QED experimentalists: the 2LA-EMF system maintaining its coherence in sufficiently long duration is the reason why they can manipulate them so well to show interesting quantum coherence effects.
We examine the dynamics of bipartite quantum entanglement between a two-level atom and the electromagnetic field. We begin with the Jaynes-Cummings model with a single field mode and examine in detail the exact time evolution of entanglement, including cases where the atomic state is initially mixed and the atomic transition is detuned from resonance. We then explore the effects of other nearby modes by calculating the exact time evolution of entanglement in more complex systems with two, three, and five field modes. For these cases we can obtain exact solutions which include the strong-coupling regime. Finally, we consider the entanglement of a two-level atom with the infinite collection of modes present in the intracavity field of a Fabre-Perot cavity. In contrast to the usual treatment of atom-field interactions with a continuum of modes using the Born-Markov approximation, our treatment in all cases describes the full non-Markovian dynamics of the atomic subsystem under general couplings. Only when an analytic expression for the infinite mode case is desired do we need to make a weak coupling assumption which at late times approximates Markovian dynamics.
We study the non-Markovian dynamics of multilevel quantum systems coupled with a bosonic dissipative environment. Based on the known exact quantum-state diffusion (QSD) equations, we propose a systematic approach to derive exact time-convolutionless master equations for multilevel quantum systems. Through a combination of analytical and numerical approaches, we extract the non-Markovian dynamics of quantum interference in different time scales. Also, we demonstrate the evolution of quantum interference in a four-level system controlled by an external electromagnetic field. Our findings are extended to few-body quantum networks, with a universal formalism established.