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arXiv:quant-ph/0610218 · Physical Review A

Loschmidt echo and Berry phase of a quantum system coupled to an XY spin chain: Proximity to a quantum phase transition

Zi-Gang Yuan, Ping Zhang, Shu‐Shen Li

Published Jan 5, 2007• 92 citations
arXivHTMLPDF

Abstract

We study the Loschmidt echo (LE) of a coupled system consisting of a central spin and its surrounding environment described by a general $XY$ spin-chain model. The quantum dynamics of the LE is shown to be remarkably influenced by the quantum criticality of the spin chain. In particular, the decaying behavior of the LE is found to be controlled by the anisotropy parameter of the spin chain. Furthermore, we show that due to the coupling to the spin chain, the ground-state Berry phase for the central spin becomes nonanalytical and its derivative with respect to the magnetic parameter $\ensuremath{\lambda}$ in spin chain diverges along the critical line $\ensuremath{\lambda}=1$, which suggests an alternative measurement of the quantum criticality of the spin chain.

DOI
10.1103/physreva.75.012102

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We study the Loschmidt echo (LE) of a coupled system consisting of a central spin and its surrounding environment described by a general XYXYXY spin-chain model. The quantum dynamics of the LE is shown to be remarkably influenced by the quantum criticality of the spin chain. In particular, the decaying behavior of the LE is found to be controlled by the anisotropy parameter of the spin chain. Furthermore, we show that due to the coupling to the spin chain, the ground-state Berry phase for the central spin becomes nonanalytical and its derivative with respect to the magnetic parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ in spin chain diverges along the critical line \ensuremathλ=1\ensuremath{\lambda}=1\ensuremathλ=1, which suggests an alternative measurement of the quantum criticality of the spin chain.
By introducing the Dzyaloshinsky-Moriya (DM) interaction, the Loschmidt Echo ( LE ) of a quantum system consisting of a central spin and its surrounding environment characterized by an XY spin chain was investigated analytically and numerically. At the critical points of the magnetic field, the LE presents an obvious decay. The decay amplitude can be tuned by the DM interaction. In some specific intervals the DM interaction can remarkably delay the decay of the LE . On the other hand, the DM interaction can change the effects of the anisotropy parameter on the LE.
By introducing the Dzyaloshinsky–Moriya (DM) interaction, the Loschmidt Echo (LE) of a quantum system consisting of a central spin and its surrounding environment characterized by an XY spin chain was investigated. The analytical expression of the LE was obtained. The effects of magnetic field, DM interaction, spin chain size, and anisotropy parameter on the evolution behavior of the LE were discussed numerically. At the critical points of magnetic field, the LE presents an obvious decay following with local oscillation. The decaying and oscillation behavior of the LE can be tuned by the intensity of the DM interaction. With increasing the spin chain size, the decaying amplitude is increased and the following oscillation behavior vanishes for the spin chain with larger size. The effect of the anisotropy parameter on the LE presents different properties at both sides of the knee point. The effect of the DM interaction on the decay of the LE is also related to the value of anisotropy parameter.
We investigate the Loschmidt echo of a quantum system where a central spin is symmetrically coupled to a compass spin-chain. It is found that the Loschmidt echo and its statistical distribution are able to detect the quantum criticality of the compass spin-chain at low temperature. By applying a spin flip operation to reverse the central spin's orientation, we find the controlled spin echo and its statistical distribution can effectively reveal quantum criticality at very high temperature. Moreover, we show that the geometric phase of a central spin can be used to indicate the quantum phase transition of a compass spin-chain and obeys scaling behavior in the vicinity of a quantum phase transition.
For a central system uniformly coupled to a XY spin-1/2 bath in a transverse field, we explicitly calculate the Loschmidt echo(LE) to characterize decoherence quantitatively. We find that the anisotropy parameter γγγ affects decoherence of the central system sensitively when γ∈[0,1]γ\in [0,1]γ∈[0,1]% , in particular, the LE becomes unit without varying with time when $γ=0 $, implying that environment-induced decoherence vanishes . Some other cases in which the LE is unit are discussed also.
The Loschmidt echo (LE) of a central two-level qubit coupled to nonuniform anisotropic XYXYXY chains in a transverse field is studied. A general formula for LE is derived, which we use to discuss the influence of the criticality of the environment on LE. It is found that for the periodic XYXYXY chain the behaviors of LE in the vicinity of the critical points are similar to those of the uniform case. It is different for the disordered transverse Ising chains. For the aperiodic chains, if the surrounding systems are bounded chains, the behaviors of LE are similar to those of the uniform case, while if the surrounding systems are unbounded chains, they are similar to those of the disordered case.