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arXiv:1001.3454 · Physical Review A

Non-Markovian effect on the geometric phase of a dissipative qubit

Juanjuan Chen, Jun‐Hong An, Qingjun Tong, Hong‐Gang Luo, C. H. Oh

Published Feb 24, 2010• 38 citations
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Abstract

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.

DOI
10.1103/physreva.81.022120

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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.
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.
Geometric phases are investigated for a Josephson qubit interacting with the Markovian and non-Markovian environments at arbitrary temperature according to a kinematic approach. In order to obtain the condition allowing experimental observation of geometric phase, we also estimate the decoherence times. The result shows that the Aharonov-Anandan phase may be different from the Berry phase under the environment perturbations. By comparing with the numerical result of the geometric phase as a function of the initial angle, we find that at low temperature, the effect of Markovian environment is larger than the non-Markovian one on the geometric phase. An approach is proposed to understand the accumulative effects of the geometric phase and its robustness to high-frequency noise, which may be helpful for geometric quantum information processing.
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 investigate the geometric phase of a two-level atom (qubit) coupled to a bosonic reservoir with Lorentzian spectral density and find that for the non-Markovian dynamics in which the rotating-wave approximation (RWA) is performed, the geometric phase has a $\ensuremath{\pi}$-phase jump at the nodal point. However, the exact result without the RWA given by the hierarchical equations of motion method shows that there is no such phase jump or nodal structure in the geometric phase. Thus our results demonstrate that the counterrotating terms significantly contribute to the geometric phase in the multimode Hamiltonian under certain circumstances.