Echo State Condition at the Critical Point

Recurrent networks with transfer functions that fulfill the Lipschitz continuity with K=1 may be echo state networks if certain limitations on the recurrent connectivity are applied. It has been shown that it is sufficient if the largest singular value of the recurrent connectivity is smaller than 1. The main achievement of this paper is a proof under which conditions the network is an echo state network even if the largest singular value is one. It turns out that in this critical case the exact shape of the transfer function plays a decisive role in determining whether the network still fulfills the echo state condition. In addition, several examples with one neuron networks are outlined to illustrate effects of critical connectivity. Moreover, within the manuscript a mathematical definition for a critical echo state network is suggested

Loschmidt echo with a non-equilibrium initial state: early time scaling and enhanced decoherence

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.Comment: 5 pages, 6 figures; New results, figures and references added, title change

Universal nonequilibrium signatures of Majorana zero modes in quench dynamics

The quantum evolution after a metallic lead is suddenly connected to an electron system contains information about the excitation spectrum of the combined system. We exploit this type of "quantum quench" to probe the presence of Majorana fermions at the ends of a topological superconducting wire. We obtain an algebraically decaying overlap (Loschmidt echo) ${\cal L}(t)=| < \psi(0) | \psi(t) > |^2\sim t^{-\alpha}$ for large times after the quench, with a universal critical exponent $\alpha$=1/4 that is found to be remarkably robust against details of the setup, such as interactions in the normal lead, the existence of additional lead channels or the presence of bound levels between the lead and the superconductor. As in recent quantum dot experiments, this exponent could be measured by optical absorption, offering a new signature of Majorana zero modes that is distinct from interferometry and tunneling spectroscopy.Comment: 9 pages + appendices, 4 figures. v3: published versio

One-half of the Kibble-Zurek quench followed by free evolution

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.Comment: 19 pages, version accepted for publicatio

Disordered Kitaev chain with long-range pairing: Loschmidt echo revivals and dynamical phase transitions

We explore the dynamics of long-range Kitaev chain by varying pairing interaction exponent, $\alpha$. It is well known that distinctive characteristics on the nonequilibrium dynamics of a closed quantum system are closely related to the equilibrium phase transitions. Specifically, the return probability of the system to its initial state (Loschmidt echo), in the finite size system, is expected to exhibit very nice periodicity after a sudden quench to a quantum critical point. Where the periodicity of the revivals scales inversely with the maximum of the group velocity. We show that, contrary to expectations, the periodicity of the return probability breaks for a sudden quench to the non-trivial quantum critical point. Further, We find that, the periodicity of return probability scales inversely with the group velocity at the gap closing point for a quench to the trivial critical point of truly long-range pairing case, $\alpha < 1$. In addition, analyzing the effect of averaging quenched disorder shows that the revivals in the short range pairing cases are more robust against disorder than that of the long rang pairing case. We also study the effect of disorder on the non-analyticities of rate function of the return probability which introduced as a witness of the dynamical phase transition. We exhibit that, the non-analyticities in the rate function of return probability are washed out in the presence of strong disorders.Comment: 13+ pages, 8 figures, new results adde

Testing quantum adiabaticity with quench echo

Adiabaticity of quantum evolution is important in many settings. One example is the adiabatic quantum computation. Nevertheless, up to now, there is no effective method to test the adiabaticity of the evolution when the eigenenergies of the driven Hamiltonian are not known. We propose a simple method to check adiabaticity of a quantum process for an arbitrary quantum system. We further propose a operational method for finding a uniformly adiabatic quench scheme based on Kibble-Zurek mechanism for the case when the initial and the final Hamiltonians are given. This method should help in implementing adiabatic quantum computation.Comment: This is a new version. Some typos in the New Journal of Physics version have been correcte