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Physical Review A

Loschmidt echo of a two-level qubit coupled to nonuniform anisotropic XY chains in a transverse field

Ming Zhong, Peiqing Tong

Published Nov 11, 2011• 12 citations
Publisher

Abstract

The Loschmidt echo (LE) of a central two-level qubit coupled to nonuniform anisotropic $XY$ 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 $XY$ 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.

DOI
10.1103/physreva.84.052105

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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.
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 $\ensuremath{\lambda}$ in spin chain diverges along the critical line $\ensuremath{\lambda}=1$, which suggests an alternative measurement of the quantum criticality of the spin chain.
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.
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.
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.
We study the influence of disorder on the dynamical quantum phase transitions (DQPTs) in the transverse-field anisotropic XY chain after a sudden quench. The formulas to calculate the Loschmidt echo and Fisher zeros of the inhomogeneous XY chain are derived in real space. It is found that new DQPTs appear in the presence of weak disorder for the quenches across the Ising transition and two quantum phase transitions (Ising and anisotropic transitions). As the disorder strength increases, it's demonstrated that the Fisher zeros coalesce to regions on the imaginary axis in the thermodynamic limit. The DQPTs occur at the critical times corresponding to the boundaries of the Fisher zeros regions on the imaginary axis. However, for the quench across the anisotropic transition, the Fisher zeros regions appear even for the very weak disorder. Finally, for the quench within the same phase, the weak disorder is found having no effect on the DQPTs, but the DQPTs will be washed out in the presence of strong disorder.