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Acta Physica Sinica

Geometric phase evolution of atom under non-Markovian environment

Liming Zheng, Songhao Liu, Wang Fa-Qiang, (1)华南师范大学信息光电子科技学院，广州 510006; (2)暨南大学信息科学技术学院，广州 510632

Published Jan 1, 2009• 2 citations
Publisher

Abstract

The geometric phase of two-level atom, under the non-Markovian effect of environment, is investigated from the microscopic Hamiltonian. The results show that the geometric phase of atom strongly coupled with cavity field is larger than that of atom weakly coupled with cavity field, and the difference between them becomes larger as the loss of environment increases. In the case of relatively small environment loss, the geometric phase of atom shows continuous or discontinuous evolution with time depending on different initial atom states, and the discontinuity range increases when the environment loss increases. In a word, the geometric phase exhibits complicated characteristics under non-Markovian environment.

DOI
10.7498/aps.58.2430

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The geometric phase of two-level atom, under the non-Markovian effect of environment, is investigated from the microscopic Hamiltonian. The results show that the geometric phase of atom strongly coupled with cavity field is larger than that of atom weakly coupled with cavity field, and the difference between them becomes larger as the loss of environment increases. In the case of relatively small environment loss, the geometric phase of atom shows continuous or discontinuous evolution with time depending on different initial atom states, and the discontinuity range increases when the environment loss increases. In a word, the geometric phase exhibits complicated characteristics under non-Markovian environment.
We studied the geometric phase of a two-level atom coupled to an environment with Lorentzian spectral density. The non-Markovian effect on the geometric phase is explored analytically and numerically. In the weak coupling limit, the lowest order correction to the geometric phase is derived analytically and the general case is calculated numerically. It was found that the correction to the geometric phase is significantly large if the spectral width is small, and in this case the non-Markovian dynamics has a significant impact on the geometric phase. When the spectral width increases, the correction to the geometric phase becomes negligible, which shows the robustness of the geometric phase to the environmental white noises. The result is significant to the quantum information processing based on the geometric phase.
The geometric phase of a two-level atom non-resonantly coupled to a non-Markovian dissipative environment is investigated. Compared to an earlier work [Chen J. J. et al. Phys. Rev. A 81 (2010) 022120 ] in which the non-Markovian effect has a serious correction on geometric phase, we find that the geometric phase can be stabilized by detuning in non-Markovian dissipative decoherence. Moreover, the geometric phase approaches the unitary geometric phase with the increase of detuning for any initial polar angle, which shows that the geometric phase is not only resilient to the Markovian noise but is also resilient to the non-Markovian noise when a large detuning between the qubit and environment is considered. Our results may be helpful for geometric quantum computation.
Dynamics of an open system is vividly influenced by the structure of environments. This paper studies in detail the dynamics of a two-level atom in the presence of an overall environment composed of two hierarchies. The first hierarchy is just a single lossy cavity while the second hierarchy consists of a number of other lossy cavities. The atom is coupled directly to the first hierarchy but indirectly to the second one via the couplings between the two hierarchies. We show that even when the coupling between the atom and the first hierarchy is weak the atom's dynamics can become non-Markovian if the number of cavities in the second hierarchy or/and the coupling between the two hierarchies are large enough. We also analyze the case when the coupling between the atom and the first hierarchy is strong and show that the non-Markovian dynamics exhibits different patterns depending on both the number of cavities in the second hierarchy and the coupling between the two hierarchies.
The geometric phases of a two-level atom interacting with non-Markovian environments are calculated and the non-Markovian effects on the geometric phases are discussed in this paper. Three kinds of methods that describe the non-Markovian process, projection superoperator technique, memory kernel master equation and post-Markovian master equation, are used in the discussions. The results show that when the dissipation rate is large, the non-Markovian effects change the geometric phase more strikingly than the small one.
We theoretically study the geometric effect of quantum dynamical evolution in the presence of a nonequilibrium noisy environment. We derive the expression of the time-dependent geometric phase in terms of the dynamical evolution and the overlap between the time evolved state and initial state. It is shown that the frequency shift induced by the environmental nonequilibrium feature plays a crucial role in the geometric phase and evolution path of the quantum dynamics. The nonequilibrium feature of the environment makes the length of the evolution path become longer and reduces the dynamical decoherence and non-Markovian behavior in the quantum dynamics.
Geometric phases and information flows of a two-level system coupled to its environment are calculated and analyzed. The information flow is defined as a cumulant of changes in trace distance between two quantum states, which is similar to the measure for non-Markovianity given by Breuer. We obtain an analytic relation between the geometric phase and the information flow for pure initial states, and a numerical result for mixed initial states. The geometric phase behaves differently depending on whether there are information flows back to the two-level system from its environment.