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arXiv:1402.0758 · Europhysics Letters (EPL)

Loschmidt echo and dynamical fidelity in periodically driven quantum systems

Shraddha Sharma, Angelo Russomanno, Giuseppe E. Santoro, Amit Dutta

Published Jun 1, 2014• 63 citations
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Abstract

We study the dynamical fidelity $\mathcal{F} (t)$ and the Loschmidt echo $\mathcal{L} (t)$, following a periodic driving of the transverse magnetic field of a quantum Ising chain (back and forth across the quantum critical point) by calculating the overlap between the initial ground state and the state reached after $n$ periods $τ$. We show that $\log{\mathcal{F}}(nτ)/L$ (the logarithm of the fidelity per-site) reaches a steady value in the asymptotic limit $n\to \infty$, and we derive an exact analytical expression for this quantity. Remarkably, the steady state value of $[\log{\mathcal{F}}(nτ\to \infty)]/L$ shows memory of non-trivial phase information which is instead hidden in the case of thermodynamic quantities; this conclusion, moreover, is not restricted to 1-dimensional models.

DOI
10.1209/0295-5075/106/67003

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We study the dynamical fidelity F(t)\mathcal{F} (t)F(t) and the Loschmidt echo L(t)\mathcal{L} (t)L(t), following a periodic driving of the transverse magnetic field of a quantum Ising chain (back and forth across the quantum critical point) by calculating the overlap between the initial ground state and the state reached after nnn periods τττ. We show that log⁡F(nτ)/L\log{\mathcal{F}}(nτ)/LlogF(nτ)/L (the logarithm of the fidelity per-site) reaches a steady value in the asymptotic limit n→∞n\to \inftyn→∞, and we derive an exact analytical expression for this quantity. Remarkably, the steady state value of [log⁡F(nτ→∞)]/L[\log{\mathcal{F}}(nτ\to \infty)]/L[logF(nτ→∞)]/L shows memory of non-trivial phase information which is instead hidden in the case of thermodynamic quantities; this conclusion, moreover, is not restricted to 1-dimensional models.
Analytical expressions are derived for the fidelity and for the quantum Loschmidt echo in the broken phase of the Lipkin-Meshkov-Glick (LMG) model for large particle number NNN. A detailed comparison is given between this model and some other models for the decaying behaviors of the two quantities in the neighborhood of critical points. The two quantities show qualitatively different decaying behaviors on the two sides of the critical point. Specifically, in the broken phase they behave in a way similar to those in models such as the one-dimensional Ising chain in a transverse field. In contrast, in the symmetric phase, the similarity is to those in models such as the Dicke model.
We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise from the level crossings of this matrix. Our exact results are expressed in terms of “excited-state” thermodynamic Bethe ansatz equations, whose solutions involve non-trivial Riemann surfaces. By evaluating our formulas, we provide explicit numerical results for the quench from the Néel state, and we determine the first few non-analytic points.
A quantum phase transition is generally thought to imprint distinctive characteristics on the nonequilibrium dynamics of a closed quantum system. Specifically, the Loschmidt echo after a sudden quench to a quantum critical point-measuring the time dependence of the overlap between initial and time-evolved states-is expected to exhibit an accelerated relaxation followed by periodic revivals. We here introduce a new exactly solvable model, the extended Su-Schrieffer-Heeger model, the Loschmidt echo of which provides a counterexample. A parallell analysis of the quench dynamics of the three-site spin-interacting XY model allows us to pinpoint the conditions under which a periodic Loschmidt revival actually appears.
Abstract We drive the one-dimensional quantum Ising chain in the transverse field from the paramagnetic phase to the critical point and study its free evolution there. We analyze excitation of such a system at the critical point and dynamics of its transverse magnetization and Loschmidt echo during free evolution. We discuss how the system size and quench-induced scaling relations from the Kibble–Zurek theory of non-equilibrium phase transitions are encoded in quasi-periodic time evolution of the transverse magnetization and Loschmidt echo.
We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically time-varying transverse field of frequency , we arrive at the characteristic cumulant generating function G ( u ), which is then used to calculate the work distribution function P ( W ). By applying the Floquet theory we show that, in the infinite time limit, P ( W ) converges, starting from the initial ground state, towards an asymptotic steady state value whose small- W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h 0 , the initial transverse field, , and the equilibrium quantum critical point , which we find to generate a sequence of non-equilibrium critical points , with l integer. When , we find a ‘universal’ edge singularity in P ( W ) at a threshold value of which is entirely determined by . The form of that singularity—Dirac delta derivative or square root—depends on h 0 being or not at a non-equilibrium critical point h * l . On the contrary, when , G ( u ) decays as a power-law for large u , leading to different types of edge singularity at . Generalizing our calculations to the case in which we initialize the system in a finite temperature density matrix, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.
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.