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Journal of Yangzhou University

DECOHERENCE OF QUANTUM BIT IN THE WEAK-COUPLING LIMIT

Yan Chen

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

The decoherence of the quantum bit is studied in the approximation of weak coupling in the Liouville space. The dynamic equation of N quantum bits is given in the weakly coupling limits and under the non-Markovian approximation. The rate of decoherence is given in the approximation of different time order. Based on the above results, the general form of criterion for decoherence-free states is proposed. Also a new method for quantum correction is proposed from the non-Markovian dynamic equation in the weak coupling limit.

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The thesis is devoted to the study of the non-Markovian dynamical process in several different physical systems. Specifically, we investigate the non-Markovian entangle- ment dynamics of two quantum bits (qubits) in a thermal squeezed bath, the non-Markovian dynamics of a nanomechanical resonator (NMR) measured by a quantum point contact (QPC) detector and, the non-Markovian evolution of two-time correlation functions (CF’s) of a two-level atom coupled to a thermal bosonic bath. First, in the investigation of the non-Markovian entanglement dynamics of two qubits in a common squeezed bath, we see a remarkable difference between the non-Markovian entanglement dynamics and its Markovian counterpart. We show that a non-Markovian decoherence-free state is also decoherence free in the Markovian regime, but all the Markovian decoherence-free states are not necessarily decoherence free in the non-Markovian domain. We extend our calculation from a squeezed vacuum bath to a squeezed thermal bath, where we see the effect of finite bath temperatures on the entanglement dynamics. Second, we also investigate the dynamics of a NMR subject to a measurement by a low-transparency QPC or tunnel junction in the non-Markovian domain. We derive the non-Markovian number-resolved (conditional) and unconditional master equations valid to second order in the tunneling Hamiltonian without making the rotating-wave approximation and the Markovian approximation, generally made for systems in quantum optics. We find considerable difference in dynamics between the non-Markovian cases and its Markovian counterparts. We also calculate the time-dependent transport current through the QPC which contains information about the measured NMR system. We find an extra transient current term proportional to the expectation value of the symmetrized product of the position andmomentum operators of the NMR. This extra term, with a coefficient coming from the combination of the imaginary parts of the QPC reservoir correlation functions, was generally ignored in the studies of the same problem in the literature. But we find that it has a substantial contribution to the total transient current in the Non- Markovian case and differs qualitatively and quantitatively from its Markovian counterpart. Thus it may serve as a witness or signature of non-Markovian features for the coupled NMR-QPC system. Finally, we use the quantum master equation approach to derive, valid to second order in the system-environment interaction Hamiltonian, non-Markovian evolution equations of two-time CF’s of system operators at finite environment temperatures with any initial separable system-environment states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equations are valid for both Her- mitian and non-Hermitian system coupling operator cases. We then give conditions on which the derived evolution equations reduced to the case of quantum regression theorem (QRT)in the weak system-environment coupling case, and apply the derived evolution equations to a problem of a two-level system (atom) coupled to a bosonic environment (electromagnetic fields) with $L = L^(＋)$.

臺灣大學物理研究所學位論文•Jan 1, 2010•0 citations
The decoherence of the quantum bit is studied in the approximation of weak coupling in the Liouville space. The dynamic equation of N quantum bits is given in the weakly coupling limits and under the non-Markovian approximation. The rate of decoherence is given in the approximation of different time order. Based on the above results, the general form of criterion for decoherence-free states is proposed. Also a new method for quantum correction is proposed from the non-Markovian dynamic equation in the weak coupling limit.
The decoherence of two initially entangled qubits in non-Markovian reservoirs has been investigated exactly without Born-Markovian approximation or rotating-wave approximation (RWA). The nonperturbative quantum master equation is derived and its exact solution is obtained. The decoherence behavior of two qubits, initially entangled in Bell states, has been investigated in the weak coupling regime and strong coupling regime. The results show that the counter-rotating wave terms have obvious influence on the decoherence behavior in the non-Markovian regime.
We develop a dynamical non-Markovian description of quantum computing in the weak-coupling limit, in the lowest-order approximation. We show that the long-range memory of the quantum reservoir (such as the 1/t4{1/t}^{4}1/t4 one exhibited by electromagnetic vacuum) produces a strong interrelation between the structure of noise and the quantum algorithm, implying nonlocal attacks of noise. This shows that the implicit assumption of quantum error correction theory---independence of noise and self-dynamics---fails in long time regimes. We also use our approach to present pure decoherence and decoherence accompanied by dissipation in terms of the spectral density of the reservoir. The so-called dynamical decoupling method is discussed in this context. Finally, we propose a minimal decoherence model, in which the only source of decoherence is vacuum. We optimize the fidelity of quantum-information processing under the trade-off between the speed of the gate and the strength of decoherence.
We study the properties of a refined weak-coupling limit that preserves complete positivity in order to describe non-Markovian dynamics in the spin-boson model. With this tool, we show the system presents a rich non-Markovian phenomenology. This implies a dynamical difference between entanglement and coherence: the latter undergoes revivals, whereas the former not, despite the induced dynamics being fully incoherent. In addition, the evolution presents ``quasieternal'' non-Markovianity, becoming nondivisible at any time period where the system evolves qualitatively. Furthermore, the method allows for an exact derivation of a master equation that accounts for a reversible energy exchange between system and environment. Specifically, this is obtained in the form of a time-dependent Lamb shift term.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
We study non-Markovian decoherence phenomena by employing projection-operator formalism when a quantum system (a quantum bit or a register of quantum bits) is coupled to a reservoir. By projecting out the degree of freedom of the reservoir, we derive a non-Markovian master equation for the system, which is reduced to a Lindblad master equation in Markovian limit, and obtain the operator sum representation for the time evolution. It is found that the system is decohered slower in the non-Markovian reservoir than the Markovian because the quantum information of the system is memorized in the non-Markovian reservoir. We discuss the potential importance of non-Markovian reservoirs for quantum-information processing.
The thesis is devoted to the study of the non-Markovian dynamical process in several different physical systems. Specifically, we investigate the non-Markovian entangle- ment dynamics of two quantum bits (qubits) in a thermal squeezed bath, the non-Markovian dynamics of a nanomechanical resonator (NMR) measured by a quantum point contact (QPC) detector and, the non-Markovian evolution of two-time correlation functions (CF’s) of a two-level atom coupled to a thermal bosonic bath. First, in the investigation of the non-Markovian entanglement dynamics of two qubits in a common squeezed bath, we see a remarkable difference between the non-Markovian entanglement dynamics and its Markovian counterpart. We show that a non-Markovian decoherence-free state is also decoherence free in the Markovian regime, but all the Markovian decoherence-free states are not necessarily decoherence free in the non-Markovian domain. We extend our calculation from a squeezed vacuum bath to a squeezed thermal bath, where we see the effect of finite bath temperatures on the entanglement dynamics. Second, we also investigate the dynamics of a NMR subject to a measurement by a low-transparency QPC or tunnel junction in the non-Markovian domain. We derive the non-Markovian number-resolved (conditional) and unconditional master equations valid to second order in the tunneling Hamiltonian without making the rotating-wave approximation and the Markovian approximation, generally made for systems in quantum optics. We find considerable difference in dynamics between the non-Markovian cases and its Markovian counterparts. We also calculate the time-dependent transport current through the QPC which contains information about the measured NMR system. We find an extra transient current term proportional to the expectation value of the symmetrized product of the position andmomentum operators of the NMR. This extra term, with a coefficient coming from the combination of the imaginary parts of the QPC reservoir correlation functions, was generally ignored in the studies of the same problem in the literature. But we find that it has a substantial contribution to the total transient current in the Non- Markovian case and differs qualitatively and quantitatively from its Markovian counterpart. Thus it may serve as a witness or signature of non-Markovian features for the coupled NMR-QPC system. Finally, we use the quantum master equation approach to derive, valid to second order in the system-environment interaction Hamiltonian, non-Markovian evolution equations of two-time CF’s of system operators at finite environment temperatures with any initial separable system-environment states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equations are valid for both Her- mitian and non-Hermitian system coupling operator cases. We then give conditions on which the derived evolution equations reduced to the case of quantum regression theorem (QRT)in the weak system-environment coupling case, and apply the derived evolution equations to a problem of a two-level system (atom) coupled to a bosonic environment (electromagnetic fields) with L=L(＋)L = L^(＋)L=L(＋).