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arXiv:1106.5654 · Physical Review A

Decoherence in an exactly solvable qubit model with initial qubit-environment correlations

V. G. Morozov, Steven Mathey, GERD RÖPKE

Published Feb 1, 2012• 79 citations
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Abstract

We study a model of dephasing (decoherence) in a two-state quantum system (qubit) coupled to a bath of harmonic oscillators. An exact analytic solution for the reduced dynamics of a two-state system in this model has been obtained previously for factorizing initial states of the combined system. We show that the model admits exact solutions for a large class of correlated initial states which are typical in the theory of quantum measurements. We derive exact expressions for the off-diagonal elements of the qubit density matrix, which hold for an arbitrary strength of coupling between the qubit and the bath. The influence of initial correlations on decoherence is considered for different bath spectral densities. Time behavior of the qubit entropy in the decoherence process is discussed.

DOI
10.1103/physreva.85.022101

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We study a model of dephasing (decoherence) in a two-state quantum system (qubit) coupled to a bath of harmonic oscillators. An exact analytic solution for the reduced dynamics of a two-state system in this model has been obtained previously for factorizing initial states of the combined system. We show that the model admits exact solutions for a large class of correlated initial states which are typical in the theory of quantum measurements. We derive exact expressions for the off-diagonal elements of the qubit density matrix, which hold for an arbitrary strength of coupling between the qubit and the bath. The influence of initial correlations on decoherence is considered for different bath spectral densities. Time behavior of the qubit entropy in the decoherence process is discussed.
We study the bath dynamics in the dephasing model of a two-state quantum system (qubit) coupled to an environment of harmonic oscillators.This model was shown [Morozov et al., Phys.Rev. A, 2012 85, 022101] to admit the analytic solution for the qubit and environment dynamics.Using this solution, we derive the exact expression for the bath reduced density matrix in the presence of initial qubit-environment correlations.We obtain the non-equilibrium phonon distribution function and discuss in detail the time behavior of the bath energy.It is shown that only the inclusion of dynamic correlations between the qubit and the bath ensures the proper time behavior of the quantity which may be interpreted as the "environment energy".
The influence of the initial preparation on dephasing in open quantum dynamics is studied using an exactly solvable model of a two-level system (qubit) interacting with a bosonic bath. It is found that, for some classes of nonselective preparation measurements, qubit-bath correlations lead to a significant enhancement of coherence in the qubit at the initial stage of evolution. The time behavior of the qubit purity and entropy in the regime of enhancement of coherence is considered for different temperatures and coupling strengths.
We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit) models.The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators.The entropy production for this model in the regimes of "complete" and "incomplete" decoherence is discussed.As another example, we consider the damped Jaynes-Cummings model describing a spontaneous decay of a two-level system into the field vacuum.It is shown that, for all strengths of coupling, the open system passes through the mixed state with the maximum information entropy.
Based on the exact dynamics of a two-qubit system and environment, we investigate system-environment (SE) quantum and classical correlations. The coupling is chosen to represent a dephasing channel for one of the qubits and the environment is a proper thermal bath. First we discuss the general issue of dilation for qubit phase damping. Based on the usual thermal bath of harmonic oscillators, we derive criteria of separability and entanglement between an initial XXX state and the environment. Applying these criteria to initial Werner states, we find that entanglement between the system and environment is built up in time for temperatures below a certain critical temperature Tcrit{T}_{\mathrm{crit}}Tcrit​. On the other hand, the total state remains separable during those short times that are relevant for decoherence and loss of entanglement in the two-qubit state. Close to Tcrit{T}_{\mathrm{crit}}Tcrit​ the SE correlations oscillate between separable and entangled. Even though these oscillations are also observed in the entanglement between the two qubits, no simple relation between the loss of entanglement in the two-qubit system and the build-up of entanglement between the system and environment is found.
It is not always justified to model the environment of a realistic quantum system as a collection of harmonic oscillators or two-level systems. To explore interesting physics associated with a composite environment, here we study the dynamics of a qubit coupled to a second two-level system, which is in turn coupled to a harmonic-oscillator bath. That is, the qubit of interest is in the presence of a composite environment consisting of the second two-level system and a conventional harmonic-oscillator bath. We investigate the issue of pointer states of decoherence for the qubit for different coupling strengths between the qubit and such a composite environment. It is shown that for weak-, intermediate-, or strong-coupling strengths, the qubit's reduced density matrix has a coupling-strength-dependent diagonal representation, thus yielding coupling-strength-dependent computational pointer states. Interestingly, the associated decoherence rate can decrease as the coupling strength increases.
The thermodynamics of a qubit undergoing dephasing due to the coupling with the external environment is discussed. First of all, we assume the dynamics of the system to be described by a master equation in Lindblad form. In this framework, we review a standard formulation of the first and second law of thermodynamics that has been known in literature for a long time. After that, we explicitly model the environment with a set of quantum harmonic oscillators choosing the interaction such that the global dynamics of system and bath is analytically solvable and the Lindblad master equation is recovered in the weak-coupling limit. In this generalized setting, we can show that the correlations between system and bath play a fundamental role in the heat exchange. Moreover, the internal entropy production of the qubit is proven to be positive for arbitrary coupling strength.