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arXiv:1203.0737 · Physical Review B

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

Victor Mukherjee, Shraddha Sharma, Amit Dutta

Published Jul 30, 2012• 47 citations
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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 nonequilibrium 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 close to the quantum critical point (QCP) of $E$ is independent of the quenching. We 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 nonequilibrium 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.

DOI
10.1103/physrevb.86.020301

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We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (EEE) which is subjected to a small and sudden quench at t=0t=0t=0, so that its state at t=0+t={0}^{+}t=0+ remains the same as the ground state of the initial environmental Hamiltonian before the quench; this leads to a nonequilibrium 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 close to the quantum critical point (QCP) of EEE is independent of the quenching. We 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 nonequilibrium 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 EEE to be a transverse Ising chain where the transverse field is suddenly quenched.
Abstract We study the non-equilibrium evolution of concurrence of a Bell pair constituted of two qubits, through the measurement of Loschmidt echo (LE) under the scope of generalized central spin model. The qubits are locally coupled to a one dimensional disordered Ising spin chain. We first show that in equilibrium situation the derivative of LE is able to detect the extent of Griffiths phase that appeared in presence of disordered transverse field only. While in the non-equilibrium situation, the spin chain requires a temporal window to realize the effect of disorder. We show that within this timescale, LE falls off exponentially and this decay is maximally controlled by the initial states and coupling strength. Our detail investigation suggests that there actually exist three types of exponential decay, a Gaussian decay in ultra short time scale followed by two exponential decay in the intermediate time with two different decay exponents. The effect of the disorder starts appearing in the late time power law fall of LE where the power law exponent is strongly dependent on disorder strength and the final state but almost independent of initial states and coupling strength. This feature allows us to indicate the presence of Griffiths phase. To be precise, continuously varying critical exponent and wide distribution of relaxation time imprint their effect in LE in the late time limit where the power law fall is absent for quenching to a Griffiths phase. Here, LE vanishes following the fast exponential fall. Interestingly, for off-critical quenching LE attains a higher saturation value for increasing disorder strength, otherwise vanishes for a clean spin chain, referring to the fact that disorder prohibits the rapid decay of entanglement in long time limit. Moreover, we show that disorder is also able to destroy the light cone like nature of post quench quasi-particles as LE does not sense the singular time scales appearing for clean spin chain with qubits coupled at symmetric positions.
In this paper, we study the Loschmidt Echo (LE) of a two-dimensional Kitaev model residing on a honeycomb lattice which is chosen to be an environment that is coupled globally to a central spin. The decay of the LE is highly influenced by the quantum criticality of the environmental spin model, e.g., it shows a sharp dip close to the anisotropic quantum critical point (AQCP) of its phase diagram. The early time decay and the collapse and revival as a function of time at AQCP also exhibit interesting scaling behavior with the system size, which is verified numerically. It has also been observed that the LE stays vanishingly small throughout the gapless phase of the model. The above study has also been extended to the 1D Kitaev model, i.e. when one of the interaction terms vanishes.
We study the short-time Loschmidt dynamics after a small sudden quench near a quantum critical point. We show that the initial quadratic growth of the Loschmidt rate function is governed by the variance of the quench operator per system size. For a local quench operator, this variance density is exactly equal to the spatial sum of the equal-time connected two-point correlation function in the initial ground state. This relation connects the early-time Loschmidt response to static critical correlations. Assuming power-law correlations at criticality, we classify the finite-size scaling of the short-time coefficient by the scaling dimension of the quench operator. In one dimension, the coefficient is finite, logarithmically enhanced, or algebraically enhanced with system size. We illustrate this operator dependence in the transverse-field Ising chain, where transverse-field and longitudinal-field quenches couple to different critical operators. We also discuss the first correction beyond the quadratic regime using the fourth cumulant of the post-quench Hamiltonian.
We study the response to sudden local perturbations of highly excited quantum Ising spin chains. The key quantity encoding this response is the overlap between time-dependent wave functions, which we write as a transient Loschmidt Echo. Its asymptotics at long time differences contain crucial information about the structure of the highly excited nonequilibrium environment induced by the quench. We compute the echo perturbatively for a weak local quench but for arbitrarily large global quench, using a cumulant expansion. Our perturbative results suggest that the echo decays exponentially, rather than power law as in the low-energy orthogonality catastrophe, a further example of quench-induced decoherence already found in the case of quenched Luttinger liquids. The emerging decoherence scale is set by the strength of the local potential and the bulk excitation energy.
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.
A critically enhanced decay of the Loschmidt echo is characteristic of sudden quench dynamics near a quantum phase transition. Here, we demonstrate that the decay and revival of the Loschmidt echo follows power-law scaling in the system size and the distance from a critical point with equilibrium critical exponents. We reveal such dynamical scaling laws by analyzing relevant perturbations to the Loschmidt echo cast in a scaling invariant form. We confirm the validity and the generality of the predicted dynamical scaling laws with a diverse range of critical models such as Ising spin models with a short and long range interaction, a finite-component system phase transition, and a topological phase transition. Moreover, using the integrability of systems in the thermodynamic limit, we derive such scaling laws analytically from a microscopic analysis. Our finding promotes the Loschmidt echo to a quantitative non-equilibrium probe of criticality and allows for quantitative predictions on the role of criticality on various physical scenarios where the Loschmidt echo is central to describing non-equilibrium dynamics.