Victor Mukherjee, Shraddha Sharma, Amit Dutta
We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment though initial state is not chosen to be the ground state of the environment. Rather, we assume that the spin chain is subjected to a small and sudden quench at $t=0$ so that the state of the environment at $t=0^+$ remains the same as the ground state of the initial Hamiltonian before the quench, which leads to a non-equilibrium situation. This state now evolves with two Hamiltonians, the final Hamiltonian following the quench and its modified version which incorporates an additional term arising due to the coupling of the central spin to the environment. The overlap between the states evolved through different channels starting from the same initial state are calculated as a function of $t$. Our study shows that the rate of the decay of the LE close to the quantum critical point of the environmental spin chain in the early time limit is independent of the quenching; this is established by proposing a generic short-time scaling of the decay rate. We do also study the temporal evolution of the LE and establish the presence of a crossover to a situation where the quenching becomes irrelevant. These generic scaling proposals are verified analytically as well as numerically, choosing the spin chain to be a transverse Ising chain where the transverse field is suddenly quenched. In that case, we also observe the absence of perfect collapse and revival for the LE and a faster decay of the amplitude of the LE when one starts from a non-equilibrium initial state generated following a sudden quench.
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http://arxiv.org/abs/1203.0737
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