Loschmidt echo with a nonequilibrium initial state: Early-time scaling and enhanced decoherence

Abstract

We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (E) which is subjected to a small and sudden quench at t = 0 so that its state at t = 0 + , remains the same as the ground state of the initial environmental Hamiltonian before the quench; this 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. Using a generic short-time scaling of the decay rate, we establish that in the early time limit, the rate of decay of the LE (or the overlap between two states generated from the initial state evolving through two channels ) close to the quantum critical point (QCP) of E is independent of the quenching. 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. In the limit of large quench amplitude the non-equilibrium initial condition is found to result in a drastic increase in decoherence at large times, even far away from a QCP. These generic results are verified analytically as well as numerically, choosing E to be a transverse Ising chain where the transverse field is suddenly quenched. The emergence of the classical world from the quantum world, namely decoherence, or the quantum-classical transition through reduction of a pure state to a mixed state has been a subject of perpetual interest to the physics community 1-4 . The concept of the LE has been proposed in connection to this quantum-classical transition in quantum chaos to describe the hypersensitivity of the time evolution of a system to the perturbation experienced by its surrounding 5-12 . The LE is defined as follows: if a quantum state |ψ evolves with two Hamiltonians H and H ′ , respectively, the LE is the measure of the overlap given byIn recent years, the temporal evolution of the LE has been studied in the vicinity of a QCP. In this context, the central spin model (CSM) where a central spin (CS) is coupled globally to all the spins of an environment (E) which is chosen to be a transverse Ising chain has been introduced 13 . The CS is assumed to be in a pure state initially while the spin chain is in the ground state. The interaction between the central spin and the environment effectively leads to two Hamiltonians which provide two channels of time evolution of the environmental ground state and lead to a decay in the LE. It has been reported that in the limit of weak coupling between the central spin and the environment, the LE shows a sharp decay close to the QCP of the spin chain and right at the QCP, it shows a collapse and revival as a function of time t. This collapse and revival of the LE can be taken to be an indicator of the proximity to a QCP. It can also be shown that the CS makes a transition to a mixed state when the LE vanishes.The CSM has been generalized to a more generalized e...

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