Non-Markovian dynamics of the mixed-state geometric phase of dissipative qubits

Wei Guo, Jian Ma, Xiaolei Yin, Wei Zhong, and Xiaoguang Wang

Phys. Rev. A 90, 062133 – Published 29 December 2014

Abstract

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 $π$ -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.

Received 27 November 2013
Revised 28 July 2014

DOI:https://doi.org/10.1103/PhysRevA.90.062133

Authors & Affiliations

Wei Guo¹, Jian Ma², Xiaolei Yin¹, Wei Zhong¹, and Xiaoguang Wang^1,*

¹Department of Physics, Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China
²Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

^*xgwang@zimp.zju.edu.cn

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Vol. 90, Iss. 6 — December 2014

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Figure 1
Plot of $Φ_{g}$ from Eq. (29) and $arg 〈 ψ (0) | ψ (T) 〉$ under the evolution of the Jaynes-Cummings model. Blue circles represents $Φ_{g}$ and red squares represents $arg 〈 Ψ (0) | Ψ (T) 〉$ . The initial angle is fixed to $π / 4$ with $W$ varying from zero to $1.5 ω_{0}$ and a phase jump also exists.
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Figure 2
Plot of $Φ_{g} / π$ from Eq. (29) as a function of coupling strength $W$ and initial angle $θ$ for (a) $λ = 5 ω_{0}$ , where the dynamics are Markovian and $Φ_{g}$ decays monotonically with $W$ and $θ$ , and (b) $λ = 0.05 ω_{0}$ , where the dynamics are in the non-Markovian regime and it is obvious that under the RWA, $Φ_{g}$ has a sudden change of $π$ in the vicinity of the nodal point, where $Φ_{g}$ is ill defined.
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Figure 3
Plots of $Φ_{g} / π$ from Eq. (29) and $\arg 〈 ψ (0) | ψ (T) 〉 / π$ from Eq. (30) under the RWA, with either initial angle $θ$ or coupling strength $W$ fixed; $λ = 0.05 ω_{0}$ in all plots. Red squares represent $Φ_{g}$ and blue crosses represent $arg 〈 Ψ (0) | Ψ (T) 〉$ . (a) The coupling strength $W / ω_{0} = 0.1$ and the initial angle $θ$ runs from 0 to $π$ . (b) The coupling strength $W / ω_{0} = 0.5$ and $θ$ varies. (c) The initial angle $θ$ is kept at $π / 3$ while the coupling strength $W / ω_{0}$ runs from 0 to $1.5$ , which is far beyond the strong-coupling regime. (d) The initial angle $θ$ is kept at $π / 10$ and $W / ω_{0}$ runs from 0 to 1.5. We observe that as long as $arg 〈 ψ (0) | ψ (T) 〉$ stays the same, $Φ_{g}$ shows no ill behavior, as shown in (a), and vice versa, which is verified in (a)–(c).
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Figure 4
Numerical results, given by the HEOM, of $Φ_{g} / π$ without the RWA as a function of coupling strength and initial angle: (a) $λ = 5 ω_{0}$ and the geometric phase under the RWA decrease monotonically, just like in the RWA case, and (b) $λ = 0.05 ω_{0}$ and the geometric phase is continuous in the whole parameter space, which is totally different from $Φ_{g}$ under the RWA.
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Figure 5
Numerical results of the geometric phase (in units of $π$ ) and the modulus of $〈 ψ (0) | ψ (T) 〉$ without the RWA versus the initial angle, with $W = 0.5 ω_{0}$ and $λ = 0.05 ω_{0}$ , i.e., within the strong-coupling regime and the non-Markovian regime. It is clear that the modulus never reaches 0, thus $Φ_{g}$ is continuous and well defined for all initial states.
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Figure 6
Comparison of the geometric phase with and without the RWA as a function of the initial angle $θ$ ; both Markovian and non-Markovian regimes are considered. Blue line represent the geometric phase acquired through the HEOM (without the RWA) and red squares are the geometric phase with the RWA. The parameters are as follows: (a) $λ = 5 ω_{0}$ , which lies in the Markovian regime, and the coupling strength $W = 0.05 ω_{0}$ , which is in the weak-coupling regime; (b) $λ = 5 ω_{0}$ , which lies in the Markovian regime, and $W = 0.5 ω_{0}$ , which is in the strong-coupling regime; (c) $λ = 0.05 ω_{0}$ , which lies in the non-Markovian regime, and $W = 0.05 ω_{0}$ , which is in the weak-coupling regime; and (d) $λ = 0.05 ω_{0}$ , which lies in the non-Markovian regime, and $W = 0.5 ω_{0}$ , which is in the strong-coupling regime.
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