Ehrenfest Time Research Articles

Quantum instability and Ehrenfest time for an inverted harmonic oscillator

We use out-of-time order correlators (OTOCs) to investigate the quantum instability and Ehrenfest time for an inverted harmonic oscillator (IHO). For initial states located in the stable manifolds of the IHO we find that the corresponding OTOC exhibits identical evolutionary characteristics to the saddle point before the Ehrenfest time. For initial states located in the unstable manifolds, the OTOCs still grow exponentially but the time to maintain exponential growth is related to the center position of its wave packet in phase space. Moreover, we use the Husimi Q function to visualize the quantum wave packets during exponential growth of the OTOCs. Our results show that quantum instability exists at arbitrary orbits in the IHO system, and the Ehrenfest time in the IHO system depends not only on the photon number of the initial system but also on the central positions of the initial states in phase space.

Lyapunov exponent as a signature of dissipative many-body quantum chaos

A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large-q limit of a q-body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of q≥4 based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment. Published by the American Physical Society 2024

Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results of Burq, Gérard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss Lt,xqc-estimates on intervals of length log⁡λ⋅λ−1 for initial data whose frequencies are comparable to λ, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

Particle Trajectories for Quantum Maps

We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by the propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is “less chaotic.” In addition, we present numerical simulations of these effects. In proving this result, we provide a characterization of a type of semi-classical defect measure we call uniform defect measures. We also prove derivative estimates of a function composed with a flow on the torus.

The Van Vleck Formula on Ehrenfest Time Scales and Stationary Phase Asymptotics for Frequency-Dependent Phases

The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on Ehrenfest time scales, i.e. \(\hbar \)-dependent time intervals which most commonly take the form \(|t| \le c|\log \hbar |\). Our derivation is based on an approximation to the integral kernel often called the Herman–Kluk approximation, which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by Robert (Rev Math Phys 22(10):1123-1145, 2010) , this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman–Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.

Breakdown of quantum-classical correspondence and dynamical generation of entanglement

The {\it exchange} interaction arising from the particle indistinguishability is of central importance to physics of many-particle quantum systems. Here we study analytically the dynamical generation of quantum entanglement induced by this interaction in an isolated system, namely, an ideal Fermi gas confined in a chaotic cavity, which evolves unitarily from a non-Gaussian pure state. We find that the breakdown of the quantum-classical correspondence of particle motion, via dramatically changing the spatial structure of many-body wavefunction, leads to profound changes of the entanglement structure. Furthermore, for a class of initial states, such change leads to the approach to thermal equilibrium everywhere in the cavity, with the well-known Ehrenfest time in quantum chaos as the thermalization time. Specifically, the quantum expectation values of various correlation functions at different spatial scales are all determined by the Fermi-Dirac distribution. In addition, by using the reduced density matrix (RDM) and the entanglement entropy (EE) as local probes, we find that the gas inside a subsystem is at equilibrium with that outside, and its thermal entropy is the EE, even though the whole system is in a pure state. As a by-product of this work, we provide an analytical solution supporting an important conjecture on thermalization, made and numerically studied by Garrison and Grover in: Phys. Rev. X \textbf{8}, 021026 (2018), and strengthen its statement.

Chaotic Einstein–Podolsky–Rosen pairs, measurements and time reversal

We consider a situation when evolution of an entangled Einstein–Podolsky–Rosen (EPR) pair takes place in a regime of quantum chaos being chaotic in the classical limit. This situation is studied on an example of chaotic pair dynamics described by the quantum Chirikov standard map. The time evolution is reversible even if a presence of small errors breaks time reversal of classical dynamics due to exponential growth of errors induced by exponential chaos instability. However, the quantum evolution remains reversible since a quantum dynamics instability exists only on a logarithmically short Ehrenfest time scale. We show that due to EPR pair entanglement a measurement of one particle at the moment of time reversal breaks exact time reversal of another particle which demonstrates only an approximat time reversibility. This result is interpreted in the framework of the Schmidt decomposition and Feynman path integral formulation of quantum mechanics. The time reversal in this system has already been realized with cold atoms in kicked optical lattices in absence of entanglement and measurements. On the basis of the obtained results, we argue that the experimental investigations of time reversal of chaotic EPR pairs are within reach of present cold atom capabilities.

Control of eigenfunctions on surfaces of variable curvature

We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C 1 + C^{1+} regularity of the stable/unstable foliations.

Continuum reset dynamics as a pathway to Newtonian classical limit of Quantum Mechanics

The Quantum–Classical transition problem is investigated in a nonlinear oscillator model context. The main issue addressed here is: how quantum mechanics can reproduce Newtonian dynamics for a nonlinear oscillator. The used model is the Gamma Oscillator, and it was solved, in terms of series and semi-classically, in all orders. The Ehrenfest time scale was numerically determined and shown that it decreases as the classical action increases in the small action interval and tends to a smaller and constant value the classical action increases. The Newtonian regime is reaching if a continuum monitoring is considered, i.e., continuum reset dynamics, no matter how strong nonlinearity is. The numerical calculations did not show a correlation between Ehrenfest time and the complexity of the dynamics observed in the quantum phase space and measured by Shannon Entropy.

Ehrenfest Time at the Transition from Integrable Motion to Chaotic MotionSupported by the National Key R&D Program of China (Grants Nos. 2017YFA0303302 and 2018YFA0305602), the National Natural Science Foundation of China (Grant No. 11921005), and the Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01).

Ehrenfest time depends differently on the Planck constant in integrable motion and chaotic motion. We study how its dependence on the Planck constant changes when there is a continuous transition from regular motion to chaotic motion. We find that the dependence is a weighted compromise between its two distinct dependences in regular and chaotic motions. The study is carried out with the system of periodically driven anharmonic oscillator. As the system is quite typical, the result may apply generally.

Chaos and order in librating quantum planar elastic pendulum

We investigate, numerically and analytically, the quantum mechanics of planar elastic pendulum in libration. The classical phase space of the system consists of both regular Kolmogorov–Arnold–Moser trajectories and chaotic trajectories. We systematically study the former by using the canonical perturbation theory along with the Einstein–Brillouin–Keller quantization rule and the latter using out-of-time-ordered-correlator among other tools and techniques, like the Husimi distribution and energy level spacing distribution. We find that the deviations in the energy level statistics, due to the presence of nearly degenerate levels, disappear by considering the states with same symmetry. We also propose an eight-point-out-of-time-ordered-correlator whose growth rate between the exponential decay time (around which nonlinearity sets in) and the Ehrenfest time (after which quantum effect washes out the classical growth) can be used to relate the quantum correlator to the second largest Lyapunov exponent of the corresponding classical system.

Chaotic dynamics of a non-Hermitian kicked particle

We investigate both the classical and quantum dynamics of a kicked particle with symmetry. In chaotic situation, the mean energy of the real parts of momentum linearly increases with time, and that of the imaginary momentum exponentially increases. There exists a breakdown time for chaotic diffusion, which is obtained both analytically and numerically. The quantum diffusion of this non-Hermitian system follows the classically chaotic diffusion of Hermitian case during the Ehrenfest time, after which it is completely suppressed. Interestingly, the Ehrenfest time decreases with the decrease of effective Planck constant or the increase of the strength of the non-Hermitian kicking potential. The exponential growth of the quantum out-of-time-order correlators (OTOC) during the initially short time interval characterizes the feature of the exponential diffusion of imaginary trajectories. The long time behavior of OTOC reflects the dynamical localization of quantum diffusion. The dynamical behavior of inverse participation ratio can quantify the symmetry breaking, for which the rule of the phase transition points is numerically obtained.

Short-periodic-orbit method for excited chaotic eigenfunctions.

An alternative method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wave functions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wave functions is only of the order of the ratio t_{H}/t_{E}, with t_{H} the Heisenberg time and t_{E} the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynamics.

Quantum echo dynamics in the Sherrington-Kirkpatrick model

Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a well-defined chaotic semi-classical (large-N) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins N. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on N and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.

Flat trace statistics of the transfer operator of a random partially expanding map

We consider the skew-product of an expanding map E on the circle with an almost surely random perturbation τ = τ0 + δτ of a deterministic function τ0:. The associated transfer operator can be decomposed with respect to frequency in the y variable into a family of operators acting on functions on the circle:. We show that the flat traces of behave as normal distributions in the semi classical limit n, ξ → ∞ up to the Ehrenfest time n ⩽ ck log ξ.

Scrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the Ehrenfest timescale

Out-of-time-order correlators (OTOC), vigorously being explored as a measure of quantum chaos and information scrambling, is studied here in the natural and simplest multi-particle context of bipartite systems. We show that two strongly chaotic and weakly interacting subsystems display two distinct phases in the growth of OTOC.The first is dominated by intra-subsystem scrambling, when an exponential growth with a positive Lyapunov exponent is observed till the Ehrenfest time. This phase is essentially independent of the interaction, while the second phase is an interaction dominated exponential approach to saturation that is universal and described by a random matrix model. This simple random matrix model of weakly interacting strongly chaotic bipartite systems, previously employed for studying entanglement and spectral transitions, is approximately analytically solvable for its OTOC. The example of two coupled kicked rotors is used to demonstrate the two phases, and the extent to which the random matrix model is applicable. That the two phases correspond to delocalization in the subsystems followed by inter-subsystem mixing is seen via the participation ratio in phase-space. We also point out that the second, universal, phase alone exists when the observables are in a sense already scrambled. Thus, while the post-Ehrenfest time OTOC growth is in general not well-understood, the case of strongly chaotic and weakly coupled systems presents an, perhaps important, exception.

Quantum chaos associated with an emergent ergosurface in the transition layer between type-I and type-II Weyl semimetals

We present emergent ergosurfaces (ES) in a transition layer between type-I and type-II Weyl semimetals (WSMs). The Hawking temperature defined by the surface gravity at the acoustic event horizon which coincides with the ES when the tangent velocity $v_{\parallel}$ is small is in a measurable interval. On the type-II WSM side, i.e., inside the {ES when $v_{\parallel}$ is large}, the motion of the quasiparticles may be chaotic {after} a critical surface as they are governed by an effective inverted oscillator potential induced by the mismatch between the type-I and type-II Weyl nodes. In a relevant lattice model, we calculate out of time ordered correlators (OTOCs). We find that the OTOCs are fast scrambling with a quantum Lyapunov exponent in high temperature and the scrambling is saturated after the Ehrenfest time. This confirms the quantum chaotic behavior.

Out-of-Time-Order Correlators in Bipartite Nonintegrable Systems

Out-of-time-order correlators (OTOC) being explored as a measure of quantum chaos, is studied here in a coupled bipartite system. Each of the subsystems can be chaotic or regular and lead to very different OTOC growths both before and after the scrambling or Ehrenfest time. We present preliminary results on weakly coupled subsystems which have very different Lyapunov exponents. We also review the case when both the subsystems are strongly chaotic when a random matrix model can be pressed into service to derive an exponential relaxation to saturation.

Reversible Quantum Information Spreading in Many-Body Systems near Criticality.

Quantum chaotic interacting N-particle systems are assumed to show fast and irreversible spreading of quantum information on short (Ehrenfest) time scales ∼logN. Here, we show that, near criticality, certain many-body systems exhibit fast initial scrambling, followed subsequently by oscillatory behavior between reentrant localization and delocalization of information in Hilbert space. We consider both integrable and nonintegrable quantum critical bosonic systems with attractive contact interaction that exhibit locally unstable dynamics in the corresponding many-body phase space of the large-N limit. Semiclassical quantization of the latter accounts for many-body correlations in excellent agreement with simulations. Most notably, it predicts an asymptotically constant local level spacing ℏ/τ, again given by τ∼logN. This unique timescale governs the long-time behavior of out-of-time-order correlators that feature quasiperiodic recurrences indicating reversibility.

A semiclassical theory of phase-space dynamics of interacting bosons

We study the phase-space representation of dynamics of bosons in the semiclassical regime where the occupation number of the modes is large. To this end, we employ the van Vleck-Gutzwiller propagator to obtain an approximation for the Green’s function of the Wigner distribution. The semiclassical analysis incorporates interference of classical paths and reduces to the truncated Wigner approximation (TWA) when the interference is ignored. Furthermore, we identify the Ehrenfest time after which the TWA fails. As a case study, we consider a single-mode quantum nonlinear oscillator, which displays collapse and revival of observables. We analytically show that the interference of classical paths leads to revivals, an effect that is not reproduced by the TWA or a perturbative analysis.