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Journal of Physics A Mathematical and Theoretical

Geometric phase of a qubit in dephasing environments

Jerzy Dajka, Marcin Mierzejewski, J. Łuczka

Published Dec 12, 2007• 21 citations
Publisher

Abstract

The geometric phase of a qubit coupled dephasingly to ohmic or non-ohmic bosonic environment exhibits a strong dependence on both an initial state of the system and spectral properties of the environment. Analysing the exact reduced dynamics we show that the geometric phase can be more easily maintained for the subohmic and ohmic environments than for the superohmic environment. We show the existence of the initial states for which the geometric phase is robust against coupling to the environment.

DOI
10.1088/1751-8113/41/1/012001

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The geometric phase of a qubit coupled dephasingly to ohmic or non-ohmic bosonic environment exhibits a strong dependence on both an initial state of the system and spectral properties of the environment. Analysing the exact reduced dynamics we show that the geometric phase can be more easily maintained for the subohmic and ohmic environments than for the superohmic environment. We show the existence of the initial states for which the geometric phase is robust against coupling to the environment.
We investigated the geometric phase for interaction between superconducting two-qubit system in dephased environment. The Pancharatnam phase and the Berry phase are studied. Numerical results are discussed. By considering the differently initial conditions, we find that the time-dependent Pancharatnam phase keeps the initial entangling message. On the other hand, the transition of Pancharatnam phase is dependent of the phase change in the superconducting two-qubit coupling system. Our results may be helpful to implement the time-dependent geometric quantum computation.
We find the geometric phase of a two-level system undergoing pure dephasing via interaction with an arbitrary environment, taking into account the effect of the initial system-environment correlations. We use our formalism to calculate the geometric phase for the two-level system in the presence of both harmonic-oscillator and spin environments, and we consider the initial state of the two-level system to be prepared by a projective measurement or a unitary operation. The geometric phase is evaluated for a variety of parameters such as the system-environment coupling strength to show that the initial correlations can affect the geometric phase very significantly even for weak and moderate system-environment coupling strengths. Moreover, the correction to the geometric phase due to the system-environment coupling generally becomes smaller (and can even be zero) if initial system-environment correlations are taken into account, thus implying that the system-environment correlations can increase the robustness of the geometric phase.
The evolution of quantum entanglement of a pair of non-interacting qubits is studied for the case where one of them is non-dissipatively but dephasingly coupled to the environment. The reduced non-Markovian dynamics of the qubits is exact for an arbitrary strength of coupling to the environment and the arbitrary frequency spectrum of environment fluctuations. While for the subohmic and ohmic environments the entanglement diminishes, for the superohmic zero-temperature environment it survives for a long time.
We explore the dynamics of spin-boson model for the Ohmic bath by employing the master equation approach and obtain an explicit expression of reduced density matrix. We also calculate the geometric phases of the spin-boson model by making use of the analytical results and discuss how the dissipative bosonic environment affects geometric phases. Furthermore, we investigate the dynamics of quantum discord and entanglement of two qubits each locally interacting with its own independent bosonic environments. It is found that the decay properties of quantum discord and entanglement are sensitive to the choice of initial state's parameter and coupling strength between system and bath.
Based on an exactly-solvable spin-boson model, the geometric phase (GP) of a system of two qubits collectively coupled to a quantum bath is investigated. In this scenario, the decomposition of GP into the sum of GP of the subsystem should be forbidden. We demonstrate that the environment-induced modifications of the GP usually include not only the pure decoherence effect, but also the information about the environment evolution. We also show the dependence of the GP upon the entangled initial state and extend it to the case of multi-qubits system.
We theoretically study the quantum speed limits (QSLs) of a qubit system coupled to a thermal dephasing environment with an Ohmic-like spectral density. Based on the geometric QSLs time bound, which is derived by employing the trace distance to quantify the geodesic between two distinguishable states in dynamical evolution, we study the influences of the temperature and spectral density of the environment on the QSLs time of the dephasing qubit. We also investigate the interplay between the QSLs time, the environmental temperature, and the spectral density of the environment. It has been demonstrated that the QSLs time closely depends on the transition frequency and the dynamical behavior (e.g., coherence trapping) of the dephasing qubit. For a fixed Ohmicity parameter of the environmental spectral density, the increase in environmental temperature can enhance the QSLs time bound. In addition, when the environmental temperature remains constant, the increase in the Ohmicity parameter initially leads to a reduction in the QSLs time bound, which is then followed by an increase in the time bound of QSLs. Our results can help to better understand the QSLs in the dynamics of open quantum systems and have potential application in the modulation of QSLs time in the dephasing qubit by engineering the spectral density of the environment.