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arXiv:1810.00260 · Physical Review A

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

Marc Andrew Valdez, Gavriil Shchedrin, Fernando Sols, Lincoln D. Carr

Published Jun 17, 2019• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.99.063609

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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, DR=2.59\ifmmode±\else\textpm\fi0.01{D}_{R}=2.59\ifmmode\pm\else\textpm\fi{}0.01DR​=2.59\ifmmode±\else\textpm\fi0.01, DC=3.93\ifmmode±\else\textpm\fi0.04{D}_{C}=3.93\ifmmode\pm\else\textpm\fi{}0.04DC​=3.93\ifmmode±\else\textpm\fi0.04, and DS=3.05\ifmmode±\else\textpm\fi0.05{D}_{S}=3.05\ifmmode\pm\else\textpm\fi{}0.05DS​=3.05\ifmmode±\else\textpm\fi0.05, compared to the global measure of current, DR=2.07\ifmmode±\else\textpm\fi0.02{D}_{R}=2.07\ifmmode\pm\else\textpm\fi{}0.02DR​=2.07\ifmmode±\else\textpm\fi0.02, DC=2.96\ifmmode±\else\textpm\fi0.05{D}_{C}=2.96\ifmmode\pm\else\textpm\fi{}0.05DC​=2.96\ifmmode±\else\textpm\fi0.05, and DS=2.30\ifmmode±\else\textpm\fi0.02{D}_{S}=2.30\ifmmode\pm\else\textpm\fi{}0.02DS​=2.30\ifmmode±\else\textpm\fi0.02. 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 NNN, namely, as N\ensuremathβ{N}^{\ensuremath{\beta}}N\ensuremathβ with \ensuremathβR=0.51\ifmmode±\else\textpm\fi0.004{\ensuremath{\beta}}_{R}=0.51\ifmmode\pm\else\textpm\fi{}0.004\ensuremathβR​=0.51\ifmmode±\else\textpm\fi0.004 and \ensuremathβC=0.18\ifmmode±\else\textpm\fi0.004{\ensuremath{\beta}}_{C}=0.18\ifmmode\pm\else\textpm\fi{}0.004\ensuremathβC​=0.18\ifmmode±\else\textpm\fi0.004 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.
We uncover signatures of quantum chaos in the many-body dynamics of a Bose-Einstein condensate-based quantum ratchet in a toroidal trap. We propose measures including entanglement, condensate depletion, and spreading over a fixed basis in many-body Hilbert space, which quantitatively identify the region in which quantum chaotic many-body dynamics occurs, where random matrix theory is limited or inaccessible. With these tools, we show that many-body quantum chaos is neither highly entangled nor delocalized in the Hilbert space, contrary to conventionally expected signatures of quantum chaos.
The mean-field limit of a bosonic quantum many-body system is described by (mostly) nonlinear equations of motion which may exhibit chaos very much in the spirit of classical particle chaos, i.e., by an exponential separation of trajectories in Hilbert space with a rate given by a positive Lyapunov exponent $\ensuremath{\lambda}$. The question now is whether $\ensuremath{\lambda}$ imprints itself onto measurable observables of the underlying quantum many-body system even at finite particle numbers. Using a Bose-Einstein condensate expanding in a shallow potential landscape as a paradigmatic example for a bosonic quantum many-body system, we show that the system loses its coherence at an exponentially fast rate. Furthermore, we show that the rate is given by the Lyapunov exponent associated with the chaotic mean-field dynamics. Finally, we demonstrate that this chaos-induced loss of coherence imprints itself onto the visibility of interference fringes in the total density after time of flight, thus, opening the possibility to measure $\ensuremath{\lambda}$ and with it the interplay between chaos and nonequilibrium quantum matter in a real experiment.
Within the past five years Albert Einstein's concept of a dilute atomic Bose Condensate has been realized in many experimental laboratories. Temperatures in the nano-Kelvin regime have been achieved using magnetic and optical trapping of laser and evaporatively cooled atoms. At such temperatures the relative de Broglie wavelengths of the gaseous trapped atoms can become long compared to their mean spacing, and through a process of bosonic amplification a "quantum phase transition" takes place involving 10 3 to 10 11 atoms, most of which end up in identical single particle quantum states, whose length scales are determined by the external trap. Rb, Na, Li and atomic H have been trapped at variable densities of the order of 10 13 /cm 3 , and in traps of varying geometries. Of these all but Li have an effective repulsive atomic pair interaction, but utilization of molecular Feshbach resonances allows the interactions of other species to be tuned over wide ranges of strengths, including control of the sign of the effective atomic pair interactions. Fully quantum and macroscopic systems at such low densities are a theorist's dream: simple Hartree type mean field theory provides a startlingly accurate description of density profiles, low energy excitation frequencies, and such a description, commonly called the Non-linear Schrödinger Equation (NLSE) or the Gross-Pitaevskii (GP) Equation, will be explored here. The NLSE appropriate for attractive atomic interactions is known to lead to chaotically unstable dynamics, and eventual implosion of the condensate should the local number density exceed a critical value. In this work we illustrate this type of chaotic collapse for attractive condensates, and then explore the types of chaotic dynamics of solitons and vortices, which are the signatures of dynamical non-linearity in the repulsive case. Finally the implications of the symmetry-breaking associated with phase rigidity are explored in model simulations of repulsive condensates: condensates with repulsive atomic interactions break into phase domains when subject to weak shocks and, perhaps surprisingly, break into chaotic "laser-speckle" type patterns as the shock level increases. The fully quantum mechanical NLSE thus displays a full range of chaotic types of motion: from particle-like chaotic collisions of solitons and vortices to fully developed time dependent wave chaos.
We investigate both classical and quantum chaotic dynamics of a two-mode Bose-Einstein condensate driven by a continuously periodic field. We calculate the quantities, including quantum Fisher information (QFI), fidelity, and linear entropy, to characterize the quantum chaos and compare these quantities with the maximum Lyapunov exponent (MLE) of the corresponding classical system. The behaviors of the quantum quantities and the classical quantity show a strong correspondence regardless of the classical regular and chaotic regions in the parameter space. A quantitative link between the MLE and the short-time dynamics of the QFI is established. Particularly, we find that our continuously driven model is significantly superior to other models in accelerating entanglement generation through chaos, which is evidenced by the shorter time required for the QFI to reach the Heisenberg limit. We briefly discuss the underlying mechanism of this finding and its application in saving time costs and improving measurement accuracy as well.
Abstract We study the dissipative Bose-Hubbard model on a small ring of sites in the presence of a chiral drive and explore its long-time dynamical structure using the mean-field equations and by simulating the quantum master equation. Remarkably, for large enough drivings, we find that the system admits, in a wide range of parameters, a chaotic attractor at the mean-field level, which manifests as a complex Wigner function on the quantum level. The latter is shown to have the largest weight around the approximate region of phase space occupied by the chaotic attractor. We demonstrate that this behavior could be revealed via measurement of various bosonic correlation functions. In particular, we employ open system methods to calculate the out-of-time-ordered correlator, whose exponential growth signifies a positive quantum Lyapunov exponent in our system. This can open a pathway to the study of chaotic dynamics in interacting systems of photons.
We explore the role of atomic correlations in a harmonically trapped Bose-Einstein condensate coupled to a dissipative cavity where both the atoms and the cavity are blue-detuned from the external pumping laser. Using a genuine beyond-mean-field many-body approach, we extract density distributions and many-body correlations to unveil a pathway to chaos at large pump power through a hierarchical self-organization of the atoms where the atoms transition from a single-well optical lattice to a double-well optical lattice. Correlated states of the atoms emerge and are characterized by local superfluid correlations in phases which are globally superfluid or Mott insulating. Local superfluid-Mott transitions are precluded by a dynamical instability to chaos which occurs via quasiperiodic attractors. Our results explain the mechanism behind the dynamical instabilities observed in experiments.