Quantum Loschmidt Echo Research Articles

Singularities in the Loschmidt echo of quenched topological superconductors

We study the Loschmidt echo in the quenched two-dimensional $p$-wave topological superconductor. We find that if this superconductor is quenched out of the critical point separating its topological and nontopological phases into either of the two gapful phases, its Loschmidt echo features singularities occurring periodically in time where the second derivative of the Loschmidt echo over time diverges logarithmically. Conversely, we give arguments towards $s$-wave superconductors not having singularities in their Loschmidt echo regardless of the quench. We also demonstrate that the conventional mean-field theory calculates classical echo instead of its quantum counterpart, and show how it should be modified to capture the full quantum Loschmidt echo.

The quantum Loschmidt echo on flat tori

The quantum Loschmidt echo is a measurement of the sensitivity of a quantum system to perturbations of the Hamiltonian. In the case of the standard two-torus, we derive some explicit formulae for this quantity in the transition regime where it is expected to decay in the semiclassical limit. The expression involves both a two-microlocal defect measure of the initial data and the form of the perturbation. As an application, we exhibit a non-concentration criterium on the sequence of initial data under which one does not observe a macroscopic decay of the quantum Loschmidt echo. We also apply our results to several examples of physically relevant initial data such as coherent states and plane waves.

Quantum-to-classical transition in the work distribution for chaotic systems.

The work distribution is a fundamental quantity in nonequilibrium thermodynamics mainly due to its connection with fluctuation theorems. Here, we develop a semiclassical approximation to the work distribution for a quench process in chaotic systems that provides a link between the quantum and classical work distributions. The approach is based on the dephasing representation of the quantum Loschmidt echo and on the quantum ergodic conjecture, which states that the Wigner function of a typical eigenstate of a classically chaotic Hamiltonian is equidistributed on the energy shell. Using numerical simulations, we show that our semiclassical approximation accurately describes the quantum distribution as the temperature is increased.

Long-Time Dynamics of the Perturbed Schrödinger Equation on Negatively Curved Surfaces

We consider perturbations of the semiclassical Schrodinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.

Decay of quantum Loschmidt echo and fidelity in the broken phase of the Lipkin-Meshkov-Glick model

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 $N$. 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.

Complexity and instability of quantum motion near a quantum phase transition.

We show that the number of harmonics of the Wigner function, recently proposed as a measure of quantum complexity, can also be used to characterize quantum phase transitions. The nonanalytic behavior of this quantity in the neighborhood of a quantum phase transition is illustrated by means of the Dicke model and is compared to two well-known measures of the (in)stability of quantum motion: the quantum Loschmidt echo and fidelity.

Semiclassical approach to the quantum Loschmidt echo in deep quantum regions: From validity to breakdown

Semiclassical results are usually expected to be valid in the semiclassical regime. An interesting question is, in models in which appropriate effective Planck constants can be introduced, to what extent will a semiclassical prediction stay valid when the effective Planck constant is increased? In this paper, we numerically study this problem, focusing on semiclassical predictions for the decay of the quantum Loschmidt echo in deep quantum regions. Our numerical simulations, carried out in the chaotic regime in the sawtooth model and in the kicked rotator model and also in the critical region of a one-dimensional Ising chain in transverse field, show that the semiclassical predictions may work even in deep quantum regions, particularly for perturbation strength in the so-called Fermi-Golden-Rule regime.

Initial-value representation for the Loschmidt echo

We obtain an initial-value representation for the quantum Loschmidt echo from the semiclassical theory of Wigner function evolution, together with the classical first-order perturbation theory. In the limit of small actions, the amplitude of each trajectory reduces to unity, just as in the dephasing representation introduced by Vaníček [Phys. Rev. E 70, 055201(R) (2004)], but these trajectories are generated here by the mean Hamiltonian for both the forward and the backward motion. This slight change of action may substantially alter the phase. The amplitude correction depends on the second derivative of the action. This improved dephasing approximation is verified to work even for quadratic Hamiltonians, for which the semiclassical evolution is exact, thus extending the range of application beyond its original scope in quantum chaos.

Relatively-long time decay of Loschmidt echo of a Bose-Einstein condensate in a double-well potential

We study the long-time decay of quantum Loschmidt echo (LE) of a Bose-Einstein condensate (BEC) in a double-well potential. In the tunneling and self-trapping phases of the BEC, the LE has exponential and Gaussian decays, respectively, for relatively-long times. In the crossover region, the LE behaves differently from both the tunneling and the self-trapping phases. These results indicate that relatively-long time decay of the LE is suitable for characterizing the dynamical phase transition of the BEC.

Decay of Loschmidt echo in a Bose-Einstein condensate at a dynamical phase transition

We study the quantum Loschmidt echo (LE) in a Bose-Einstein condensate (BEC) in a double-well potential. The BEC may undergo a dynamical phase transition between two phases: a tunneling phase and a self-trapping phase. For sufficiently weak perturbation, the LE has Gaussian decay in both phases. While, for relatively strong perturbation, the LE has a Gaussian decay in the self-trapping phase and has a stretched exponential decay in the tunneling phase. This qualitative difference in the decaying law of the LE in the two phases provides a characterization of the dynamical phase transition of the BEC. The semiclassical theory is used to explain the numerically observed behaviors of the LE decay.

Loschmidt echo near a dynamical phase transition in a Bose-Einstein condensate

We show that the quantum Loschmidt echo can be employed to characterize the dynamical phase transition, from a tunnelling phase to a self-trapping phase, of a Bose–Einstein condensate in a double-well potential. The echo is found to have a relatively fast decay in the transition region, with a Gaussian decay in the self-trapping phase and a stretched exponential decay in the tunnelling phase.

Sensitivity of quantum motion to perturbation in a triangle map

We study quantum Loschmidt echo, or fidelity, in the triangle map whose classical counterpart has linear instability and weak chaos. Numerically, three regimes of fidelity decay have been found with respect to the perturbation strength epsilon. In the regime of weak perturbation, the fidelity decays as exp(-c epsilon(2)t(gamma)) with gamma approximately 1.7. In the regime of strong perturbation, the fidelity is approximately a function of epsilont(2.5), which is predicted for the classical fidelity [G. Casati, Phys. Rev. Lett. 94, 114101 (2005)], and decays slower than power-law decay for long times. In an intermediate regime, the fidelity has approximately an exponential decay exp(-c' epsilont).

Crossover of quantum Loschmidt echo from golden-rule decay to perturbation-independent decay.

We study the crossover of the quantum Loschmidt echo (or fidelity) from the golden-rule regime to the perturbation-independent exponential decay regime by using the kicked top model. It is shown that the deviation of the perturbation-independent decay of the averaged fidelity from the Lyapunov decay results from quantum fluctuations in individual fidelity, which are caused by the coherence in the initial coherent states. With an averaging procedure suppressing the quantum fluctuations effectively, the perturbation-independent decay is found to be close to the Lyapunov decay. We also show that the Fourier transform of the fidelity is determined directly by the initial state and the eigenstates of the Floquet operators of the two classically chaotic systems concerned. The absolute value part and the phase part of the Fourier transform of the fidelity are found to be divided into several correlated parts, which is a manifestation of the coherence of the initial coherent state. In the whole crossover region, some important properties of the fidelity, such as the exponent of its exponential decay and the short initial time within which the fidelity almost does not change, are found to be closely related to the properties of the central part of its Fourier transform.

Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo.

The overlap of two wave packets evolving in time with slightly different Hamiltonians decays exponentially approximate to e(-gammat), for perturbation strengths U greater than the level spacing Delta. We present numerical evidence for a dynamical system that the decay rate gamma is given by the smallest of the Lyapunov exponent lambda of the classical chaotic dynamics and the level broadening U(2)/Delta that follows from the golden rule of quantum mechanics. This implies the range of validity U > the square root of [lambdaDelta] for the perturbation-strength independent decay rate discovered by Jalabert and Pastawski [Phys. Rev. Lett. 86, 2490 (2001)].