Ehrenfest Time Research Articles

Layered chaos in mean-field and quantum many-body dynamics

We investigate the dimension of the phase-space attractor of a quantum chaotic many-body ratchet in the mean-field limit. Specifically, we explore a driven Bose-Einstein condensate in three distinct dynamical regimes---Rabi oscillations, chaos, and self-trapping regimes---and for each of them we calculate the correlation dimension. For the ground state of the ratchet formed by a system of field-free noninteracting particles, we find four distinct pockets of chaotic dynamics throughout these regimes. We show that a measurement of local density in each of the dynamical regimes has an attractor characterized by a higher fractal dimension, ${D}_{R}=2.59\ifmmode\pm\else\textpm\fi{}0.01$, ${D}_{C}=3.93\ifmmode\pm\else\textpm\fi{}0.04$, and ${D}_{S}=3.05\ifmmode\pm\else\textpm\fi{}0.05$, compared to the global measure of current, ${D}_{R}=2.07\ifmmode\pm\else\textpm\fi{}0.02$, ${D}_{C}=2.96\ifmmode\pm\else\textpm\fi{}0.05$, and ${D}_{S}=2.30\ifmmode\pm\else\textpm\fi{}0.02$. The deviation between local and global measurements of the attractor's dimension corresponds to an increase towards higher condensate depletion, which remains constant for long time scales in both Rabi and chaotic regimes. The depletion is found to scale polynomially with particle number $N$, namely, as ${N}^{\ensuremath{\beta}}$ with ${\ensuremath{\beta}}_{R}=0.51\ifmmode\pm\else\textpm\fi{}0.004$ and ${\ensuremath{\beta}}_{C}=0.18\ifmmode\pm\else\textpm\fi{}0.004$ for the two regimes. Thus, we find a strong deviation from the mean-field results, especially in the chaotic regime of the quantum ratchet. The ratchet also reveals quantum revivals in the Rabi and self-trapping regimes but not in the chaotic regime, with revival times scaling linearly in particle number for Rabi dynamics. Based on the obtained results, we outline pathways for the identification and characterization of emergent phenomena in driven many-body systems. This includes the identification of many-body localization from the many-body measures of the system, the influence of entanglement on the rate of the convergence to the mean-field limit, and the establishment of a polynomial scaling of the Ehrenfest time at which the mean-field description fails to describe the dynamics of the system.

Quantum revival patterns from classical phase-space trajectories

A general semiclassical method in phase space based on the final value representation of the Wigner function is considered that bypasses caustics and the need to root-search for classical trajectories. We demonstrate its potential by applying the method to the Kerr Hamiltonian, for which the exact quantum evolution is punctuated by a sequence of intricate revival patterns. The structure of such revival patterns, lying far beyond the Ehrenfest time, is semiclassically reproduced and revealed as a consequence of constructive and destructive interferences of classical trajectories.

Quantum-classical correspondence in integrable systems

We find that the quantum-classical correspondence in integrable systems is characterized by two time scales. One is the Ehrenfest time below which the system is classical; the other is the quantum revival time beyond which the system is fully quantum. In between, the quantum system can be well approximated by classical ensemble distribution in phase space. These results can be summarized in a diagram which we call Ehrenfest diagram. We derive an analytical expression for Ehrenfest time, which is proportional to ħ1/2. According to our formula, the Ehrenfest time for the solar-earth system is about 1026 times of the age of the solar system. We also find an analytical expression for the quantum revival time, which is proportional to ħ-1. Both time scales involve ω(I), the classical frequency as a function of classical action. Our results are numerically illustrated with two simple integrable models. In addition, we show that similar results exist for Bose gases, where 1/N serves as an effective Planck constant.

Operator spreading in quantum maps

Operators in ergodic spin-chains are found to grow according to hydrodynamical equations of motion. The study of such operator spreading has aided our understanding of many-body quantum chaos in spin-chains. Here we initiate the study of "operator spreading" in quantum maps on a torus, systems which do not have a tensor-product Hilbert space or a notion of spatial locality. Using the perturbed Arnold cat map as an example, we analytically compare and contrast the evolutions of functions on classical phase space and quantum operator evolutions, and identify distinct timescales that characterize the dynamics of operators in quantum chaotic maps. Until an Ehrenfest time, the quantum system exhibits classical chaos, i.e. it mimics the behavior of the corresponding classical system. After an operator scrambling time, the operator looks "random" in the initial basis, a characteristic feature of quantum chaos. These timescales can be related to the quasi-energy spectrum of the unitary via the spectral form factor. Furthermore, we show examples of "emergent classicality" in quantum problems far away from the classical limit. Finally, we study operator evolution in non-chaotic and mixed quantum maps using the Chirikov standard map as an example.

Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices.

The out-of-time-ordered correlator (OTOC) is a measure of quantum chaos that is being vigorously investigated. Analytically accessible simple models that have long been studied in other contexts could provide insights into such measures. This paper investigates the OTOC in the quantum bakers map which is the quantum version of a simple and exactly solvable model of deterministic chaos that caricatures the action of kneading dough. Exact solutions based on the semiquantum approximation are derived that tracks very well the correlators until the Ehrenfest time. The growth occurs, surprisingly, at the exponential rate of the classical Lyapunov exponent which is half of that expected semiclassically. This exponential growth is modulated by slowly changing coefficients. Beyond this time, saturation occurs at a value close to that of random matrices. Using projectors for observables naturally leads to truncations of the unitary time-t propagator and the growth of their singular values is shown to be intimately related to the growth of the out-of-time-ordered correlators.

Scrambling and entanglement spreading in long-range spin chains

We study scrambling in connection to multipartite entanglement dynamics in regular and chaotic long-range spin chains, characterized by a well defined semi-classical limit. For regular dynamics, scrambling and entanglement dynamics are found to be very different: up to the Ehrenfest time they rise side by side departing only afterwards. Entanglement saturates and becomes extensively multipartite, while scrambling continues its growth up to the recurrence time. Remarkably, the exponential behaviour of scrambling emerges not only in the chaotic case, but also in the regular one, when the dynamics occurs at a dynamical critical point.

Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators.

Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τ_{E} in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N-particle systems. We first show how the growth of OTOCs up to τ_{E}=(1/λ)logN is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τ_{E}, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. Tothis end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit ℏ→0 for fixed N, including quantum-chaotic single- and few-particle systems.

Quantum walk on a toral phase space

A quantum walk on a toral phase space involving translations in position and its conjugate momentum is studied in the simple context of a coined walker in discrete time. The resultant walk, with a family of coins parametrized by an angle is such that its spectrum is exactly solvable with eigenangles for odd parity lattices being equally spaced, a feature that is remarkably independent of the coin. The eigenvectors are naturally specified in terms using the q-Pochhammer symbol, but can also be written in terms of elementary functions, and their entanglement can be analytically found. While the phase-space walker shares many features in common with the well-studied case of a coined walker in discrete time and space, such as ballistic growth of the walker position, it also presents novel features such as exact periodicity, and formation of cat-states in phase space. The participation ratio (PR), a measure of delocalization in walker space, is studied in the context of both kinds of quantum walks; while the classical PR increases as there is a time interval during which the quantum walks display a power-law growth ∼t0.825. Studying the evolution of coherent states in phase space under the walk enables us to identify an Ehrenfest time after which the coin-walker entanglement saturates.

Complex saddle trajectories for multidimensional quantum wave packet and coherent state propagation: Application to a many-body system.

A practical search technique for finding the complex saddle points used in wave packet and coherent state propagation is developed which works for a large class of Hamiltonian dynamical systems with many degrees of freedom. The method can be applied to problems in atomic, molecular, and optical physics and other domains. A Bose-Hubbard model is used to illustrate the application to a many-body system where discrete symmetries play an important and fascinating role. For multidimensional wave packet propagation, locating the necessary saddles involves the seemingly insurmountable difficulty of solving a boundary value problem in a high-dimensional complex space, followed by determining whether each particular saddle found actually contributes. In principle, this must be done for each propagation time considered. The method derived here identifies a real search space of minimal dimension, which leads to a complete set of contributing saddles up to intermediate times much longer than the Ehrenfest timescale for the system. The analysis also gives a powerful tool for rapidly identifying the various dynamical regimes of the system.

Post-Ehrenfest many-body quantum interferences in ultracold atoms far out of equilibrium

Far out-of-equilibrium many-body quantum dynamics in isolated systems necessarily generate interferences beyond an Ehrenfest time scale, where quantum and classical expectation values diverge. Of great recent interest is the role these interferences play in the spreading of quantum information across the many degrees of freedom, i.e.~scrambling. Ultracold atomic gases provide a promising setting to explore these phenomena. Theoretically speaking, the heavily-relied-upon truncated Wigner approximation leaves out these interferences. We develop a semiclassical theory which bridges classical and quantum concepts in many-body bosonic systems and properly incorporates such missing quantum effects. For mesoscopically populated Bose-Hubbard systems, it is shown that this theory captures post-Ehrenfest quantum interference phenomena very accurately, and contains relevant phase information to perform many-body spectroscopy with high precision.

A unified time scale for quantum chaotic regimes

We present a generalised time scale for quantum chaos dynamics, motivated by nonextensive statistical mechanics. It recovers, as particular cases, the relaxation (Heisenberg) and the random (Ehrenfest) time scales. Moreover, we show that the generalised time scale can also be obtained from a nonextensive version of the Kolmogorov–Sinai entropy by considering the graininess of quantum phase space and a generalised uncorrelation between subsets of the phase space. Lyapunov and regular regimes for the fidelity decay are obtained as a consequence of a nonextensive generalisation of the mth point correlation function for a uniformly distributed perturbation in the classical limit.

Fractal Weyl laws and wave decay for general trapping

We prove a Weyl upper bound on the number of scattering resonances in strips for manifolds with Euclidean infinite ends. In contrast with previous results, we do not make any strong structural assumptions on the geodesic flow on the trapped set (such as hyperbolicity) and instead use propagation statements up to the Ehrenfest time. By a similar method we prove a decay statement with high probability for linear waves with random initial data. The latter statement is related heuristically to the Weyl upper bound. For geodesic flows with positive escape rate, we obtain a power improvement over the trivial Weyl bound and exponential decay up to twice the Ehrenfest time.

Global Normal Form and Asymptotic Spectral Gap for Open Partially Expanding Maps

We consider a $${\mathbb{R}}$$ -extension of one dimensional uniformly expanding open dynamical systems and prove a new explicit estimate for the asymptotic spectral gap. To get these results, we use a new application of a “global normal form” for the dynamical system, a “semiclassical expression beyond the Ehrenfest time” that expresses the transfer operator at large time as a sum over rank one operators (each is associated to one orbit). In this paper we establish the validity of the so-called “diagonal approximation” up to twice the local Ehrenfest time.

Out-of-time-order correlators in quantum mechanics

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.

Counting statistics of chaotic resonances at optical frequencies: Theory and experiments.

A deformed dielectric microcavity is used as an experimental platform for the analysis of the statistics of chaotic resonances, in the perspective of testing fractal Weyl laws at optical frequencies. In order to surmount the difficulties that arise from reading strongly overlapping spectra, we exploit the mixed nature of the phase space at hand, and only count the high-Q whispering-gallery modes (WGMs) directly. That enables us to draw statistical information on the more lossy chaotic resonances, coupled to the high-Q regular modes via dynamical tunneling. Three different models [classical, Random-Matrix-Theory (RMT) based, semiclassical] to interpret the experimental data are discussed. On the basis of least-squares analysis, theoretical estimates of Ehrenfest time, and independent measurements, we find that a semiclassically modified RMT-based expression best describes the experiment in all its realizations, particularly when the resonator is coupled to visible light, while RMT alone still works quite well in the infrared. In this work we reexamine and substantially extend the results of a short paper published earlier [L. Wang et al., Phys. Rev. E 93, 040201(R) (2016)2470-004510.1103/PhysRevE.93.040201].

Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions

The semiclassical long-time limit of free evolution of quantum wave packets on the torus is under consideration. Despite the simplicity of this system, there are still open questions concerning the detailed description of the evolution on time scales beyond the Ehrenfest time. One of the approaches is based on the limiting Wigner or Husimi distributions of time-evolved wave packets as the Planck constant tends to zero and time tends to infinity. We derive explicit expressions for semiclassical measures corresponding to all time scales and the corresponding stages of evolution: classical-like motion, spreading of the wave packet, and its revivals.

About the Concept of Quantum Chaos

The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work on the subject is, even at present, a superposition of several approaches expressed in different mathematical formalisms and weakly linked to each other. The purpose of this paper is to supply a unified framework for describing quantum chaos using the quantum ergodic hierarchy. Using the factorization property of this framework, we characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time. We also outline a generalization of the quantum mixing level of the kicked rotator in the context of the impulsive differential equations.

Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System.

It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator C(t) for the classical and quantum kicked rotor-a textbook driven chaotic system-and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K→0, while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time t_{E}: transitioning from a time-independent value of t^{-1}lnC(t) at t<t_{E} to its monotonic decrease with time at t>t_{E}. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996)PRBMDO0163-182910.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.124101] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.

Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger’s equation with a potential which is the sum of a periodic function and a general smooth function. We identify two dimensionless parameters: (re-scaled) Planck’s constant and the ratio of the lattice spacing to the scale of variation of the external potential. We consider the special case where both parameters are equal and denote this parameter ϵ. In the limit ϵ↓0, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to “Ehrenfest time” t∼ln1/ϵ. To leading order, the center of mass and average quasi-momentum of these solutions evolve along trajectories generated by the classical Hamiltonian given by the sum of the Bloch band energy and the external potential. We then derive all corrections to the evolution of these observables proportional to ϵ. The corrections depend on the gauge-invariant Berry curvature of the Bloch band and a coupling to the evolution of the wave-packet envelope, which satisfies Schrödinger’s equation with a time-dependent harmonic oscillator Hamiltonian. This infinite dimensional coupled “particle-field” system may be derived from an “extended” ϵ-dependent Hamiltonian. It is known that such coupling of observables (discrete particle-like degrees of freedom) to the wave-envelope (continuum field-like degrees of freedom) can have a significant impact on the overall dynamics.

Strong scarring of logarithmic quasimodes

We consider a semiclassical (pseudo)differential operator on a compact surface (M,g), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit γ at some energy E0. For any ϵ>0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ϵh|logh| in the semiclassical limit, but which still exhibit a strong scar on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.