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arXiv:0804.0241 · Physical Review A

Decoherence of Bell states by local interactions with a dynamic spin environment

Cecilia Cormick, Juan Pablo Paz

Published Jul 29, 2008• 26 citations
arXivHTMLPDF

Abstract

We study the evolution of a system of two qubits, each of which interacts locally with a spin chain with nontrivial internal Hamiltonian. We present an exact solution to this problem and analyze the dependence of decoherence on the distance between the interaction sites. In the strong coupling regime we find that decoherence increases with increasing distance. In the weak coupling regime the dependence of decoherence with distance is not generic (i.e., it varies according to the initial state). Decoherence becomes independent of distance when the latter is over a saturation length $l$. Numerical results for the Ising chain suggest that the saturation scale is related to the correlation length $\ensuremath{\xi}$. For strong coupling we display evidence of the existence of non-Markovian effects (such as environment-induced interactions between the qubits). As a consequence the system can undergo a quasiperiodic sequence of ``sudden deaths and revivals'' of entanglement, with a time scale related to the distance between qubits.

DOI
10.1103/physreva.78.012357

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We study the evolution of a system of two qubits, each of which interacts locally with a spin chain with nontrivial internal Hamiltonian. We present an exact solution to this problem and analyze the dependence of decoherence on the distance between the interaction sites. In the strong coupling regime we find that decoherence increases with increasing distance. In the weak coupling regime the dependence of decoherence with distance is not generic (i.e., it varies according to the initial state). Decoherence becomes independent of distance when the latter is over a saturation length lll. Numerical results for the Ising chain suggest that the saturation scale is related to the correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ. For strong coupling we display evidence of the existence of non-Markovian effects (such as environment-induced interactions between the qubits). As a consequence the system can undergo a quasiperiodic sequence of ``sudden deaths and revivals'' of entanglement, with a time scale related to the distance between qubits.
We study the dynamics of disentanglement of two qubits initially prepared in a Bell state and coupled at different sites to an Ising spin chain in a transverse field (ITF) playing the role of a dynamic spin environment. The initial state of the whole system is prepared in a tensor product state ${\ensuremath{\rho}}_{\mathrm{Bell}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{\mathrm{chain}}$, where the state of the chain is taken to be given by the ground state $|G({\ensuremath{\lambda}}_{i})\ensuremath{\rangle}$ of the ITF Hamiltonian ${H}_{E}({\ensuremath{\lambda}}_{i})$ with an initial field ${\ensuremath{\lambda}}_{i}$. At time t=0+t={0}^{+}t=0+, the strength of the transverse field is suddenly quenched to a new value, ${\ensuremath{\lambda}}_{f}$, and the whole system (chain +++ qubits) undergoes a unitary dynamics generated by the total Hamiltonian ${H}_{\mathrm{Tot}}={H}_{E}({\ensuremath{\lambda}}_{f})+{H}_{I}$, where HI{H}_{I}HI​ describes a local interaction between the qubits and the spin chain. The resulting dynamics leads to a disentanglement of the qubits, which is described through Wootters' concurrence, due to their interaction with the nonequilibrium environment. The concurrence is related to the Loschmidt echo, which in turn is expressed in terms of the time-dependent covariance matrix associated with the ITF. This permits a precise numerical and analytical analysis of the disentanglement dynamics of the qubits as a function of their distance, bath properties, and quench amplitude. In particular, we emphasize the special role played by a critical initial environment.
This paper analyzes the decoherence induced on a single qubit by the interaction with a spin chain with nontrivial internal dynamics (XYXYXY-type interactions). The aim of the paper is to study the existence and properties of the so-called universal regime, in which the decoherence time scale becomes independent of the strength of the coupling with the environment. It is shown that, although such a regime does exist, as previously established by Cucchietti et al. [Phys. Rev. A 75, 032337 (2007)], it is not a clear signature of a quantum phase transition in the environment. In fact, this kind of universality also exists in the absence of quantum phase transitions. A universal regime can be related to the existence of an energy scale separation between the Hamiltonian of the environment and the one characterizing the system-environment interaction. The results presented also indicate that in the strong-coupling regime the quantum phase transition does not produce an enhancement of decoherence (as opposed to what happens in the weak-coupling regime).
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We investigate the reduced dynamics of two qubits coupled to an interacting quantum spin bath modeled by a XXZ spin chain. By using the method of time-dependent density matrix renormalization group (t-DMRG), it is shown that when the spin bath is in an ordering state, the reduced dynamics of qubits strongly depends on the range of the spin chain that qubits coupled to. Only for ordering spin baths, both decoherence and entanglement show qualitatively different behaviors when the coupling range changes. In particular, In particular, we find that in the parametic regime where there is no ordering in the bath, the coupling range is irrelevant. On the other hand, for ordering spin baths, when the coupling range increases from local coupling to infinite range coupling, the reduced dynamics of qubits exhibits a transition in which initially entangled states no longer suffers from the entanglement sudden death when the coupling range is large.
We theoretically study the non-Markovian disentanglement dynamics of a two-qubit system coupled to nonequilibrium environments with nonstationary and non-Markovian random telegraph noise statistical properties. The reduced density matrix of the two-qubit system can be expressed as the Kraus representation in terms of the tensor products of the single qubit Kraus operators. We derive the relation between the entanglement and nonlocality of the two-qubit system which are both closely associated with the decoherence function. We identify the threshold values of the decoherence function to ensure the existences of the concurrence and nonlocal quantum correlations for an arbitrary evolution time when the two-qubit system is initially prepared in the composite Bell states and the Werner states, respectively. It is shown that the environmental nonequilibrium feature can suppress the disentanglement dynamics and reduce the entanglement revivals in non-Markovian dynamics regime. In addition, the environmental nonequilibrium feature can enhance the nonlocality of the two-qubit system. Moreover, the entanglement sudden death and rebirth phenomena and the transition between quantum and classical nonlocalities closely depend on the parameters of the initial states and the environmental parameters in nonequilibrium environments.
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