Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincaré recurrences for a rotation map: only three distinct revival times can occur, with specified weights. A link is thus established between quantum revivals and recurrences in a coarse-grained discrete-time dynamical system.
Wave packets in a system governed by a Hamiltonian with a generic nonlinear spectrum typically exhibit both full and fractional revivals. It is shown that the latter can be eliminated by inducing suitable geometric phases in the states, by varying the parameters in the Hamiltonian cyclically with a period T. Further, with the introduction of this natural time step T, the occurrence of near revivals can be mapped onto that of Poincaré recurrences in an irrational rotation map of the circle. The distinctive recurrence time statistics of the latter can thus serve as a clear signature of the dynam…
Recurrence in the classical random walk is well known and described by the Pólya number. For quantum walks, recurrence is similarly understood in terms of the probability of a localized quantum walker to return to its origin. Under certain circumstances the quantum walker may also return to an arbitrary initial quantum state in a finite number of steps. Quantum state revivals in quantum walks on circles using coin operators which are constant in time and uniform across the path have been described before but only incompletely. In this paper we find the general conditions for which full-quantum…
Recurrence of a random walk is described by the Polya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full-revival of its quantum state. Localization for two dimensional quantum walks is known to exist in the sense of non-vanishing probability distribution in the asymptotic limit. We show on the example of the 2-D Grover walk that one can exploit the effect of localization to construct stationary solutions. Moreover, we find full-revivals of a quantum state with a period of two steps. We prove that there cannot be longer cycles for…
We study holographic models related to global quantum quenches in finite size systems. The holographic set up describes naturally a CFT, which we consider on a circle and a sphere. The enhanced symmetry of the conformal group on the circle motivates us to compare the evolution in both cases. Depending on the initial conditions, the dual geometry exhibits oscillations that we holographically interpret as revivals of the initial field theory state. On the sphere, this only happens when the energy density created by the quench is small compared to the system size. However on the circle considerab…
Quantum carpets- in position and momentum space- woven by the self-interference of de Broglie wave of an atom or an electron, trapped in an infinitely deep potential well, are explained. The recurrence of self-similar structures in designs of these carpets mimics the phenomena of quantum revivals and fractional revivals. We identify fractional revivals of various order by means of these space-time and momentum-time interference patterns.
We investigate recurrence phenomena in coupled two degrees of freedom systems. It is shown that an initial well localized wave packet displays recurrences even in the presence of coupling in these systems. We discuss the interdependence of these time scales namely, classical period and quantum revival time, and explain significance of initial conditions.
Within the framework of simple perturbation theory, recurrence time of quantum fidelity is related to the period of the classical motion. This indicates the possibility of recurrence in near integrable systems. We have studied such possibility in detail with the kicked rotor as an example. In accordance with the correspondence principle, recurrence is observed when the underlying classical dynamics is well approximated by the harmonic oscillator. Quantum revivals of fidelity is noted in the interior of resonances, while classical-quantum correspondence of fidelity is seen to be very short for …
Analytical eigensolutions for Morse oscillators are used to investigate quantum resonance and revivals and show how Morse anharmonicity affects revival times. A minimum semi-classical Morse revival time Tmin-rev found by Heller is related to a complete quantum revival time Trev using a quantum deviation parameter that in turn relates Trev to the maximum quantum beat period Tmax-beat. Also, number theory of Farey and Thales-circle geometry of Ford is shown to elegantly analyze and display fractional revivals. Such quantum dynamical analysis may have applications for spectroscopy or quantum info…
In this work we reconsider the problem discussed by Caldeira and Leggett (CL), and generalize their results for a quantum oscillator coupled bilinearly to a reservoir with dense discrete spectrum of harmonic oscillators. We show that for such systems dynamic evolution of any state of the CL Hamiltonian consists of recurrence cycles with partial revivals of the initial state amplitude. This revival or Loschmidt echo appears in each cycle due to reverse (from the reservoir) transitions (not existing in the limit of a macroscopic bath). Width and number of the Loschmidt echo components increase w…
Quantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum Pólya number can be seen.
A systematic study of fractional revival at two sites in $XX$ quantum spin chains is presented and analytic models with this phenomenon are exhibited. The generic models have two essential parameters and a revival time that does not depend on the length of the chain. They are obtained by combining two basic ways of realizing fractional revival in a spin chain each bringing one parameter. The first proceeds through isospectral deformations of spin chains with perfect state transfer. The second arises from the recurrence coefficients of the para-Krawtchouk polynomials with a bi-lattice orthogona…
Increasing fidelity is the ultimate challenge of quantum information technology. In addition to decoherence and dissipation, fidelity is affected by internal imperfections such as impurities in the system. Here we show that the quality of quantum revival, i.e., periodic recurrence in the time evolution, can be restored almost completely by coupling the distorted system to an external field obtained from quantum optimal control theory. We demonstrate the procedure with wave-packet calculations in both one- and two-dimensional quantum wells, and analyze the required physical characteristics of t…
Quantum mechanical objects or nanoobjects have been proposed as bits for information storage. While time-averaged properties of magnetic, quantum-mechanical particles have been extensively studied experimentally and theoretically, experimental investigations of the real time evolution of magnetization in the quantum regime were not possible until recent developments in pump-probe techniques. Here we investigate the quantum dynamics of effective spin systems by means of analytical and numerical treatments. Particular attention is paid to the quantum revival time and its relation to the magnetiz…
Cyclotomic Structure and Exact Recurrences in Floquet Many-Body Quantum Dynamics
Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文学系）；Department of Physics and Computer Science, Wilfrid Laurier University（威尔弗里德·劳里埃大学物理与计算机科学系）；Perimeter Institute for Theoretical Physics（基础理论物理研究所）
We examine the dynamics of subsystems of bipartite and tripartite quantum systems with nonlinear Hamiltonians. We consider two models which capture the generic features of open quantum systems: a three-level atom interacting with a single-mode radiation field, and a three-level atom interacting with two field modes which do not directly interact with each other. The entanglement of specific initially unentangled states of the atom-field system is examined through the time-varying subsystem von Neumann entropy (SVNE). The counterparts of near-revivals and fractional revivals of the initial stat…
We study a special kind of semiclassical limit of quantum dynamics on a circle and in a box (infinite potential well with hard walls) as the Planck constant tends to zero and time tends to infinity. The results give detailed information about all stages of evolution of quantum wave packets: semiclassical motion, collapses, revivals, as well as intermediate stages. In particular, we rigorously justify the fact that the spatial distribution of a wave packet is most of the time close to uniform. This fact was previously known only from numerical calculations. We apply the obtained results to a pr…
We review the definition of hypergeometric coherent states, discussing some representative examples. Then we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of $k$-th powers of nonlinear $f$-oscillator annihilation operators, with $f$ of hypergeometric type. These "$k$-hypercats" can be written as an equally weighted superposition of hypergeometric coherent states $|z_l\rangle, l=0,1,\dots,k-1$, with $z_l=z e^{2πi l/k}$ a $k$-th root of $z^k$, and they interpolate between number and coherent states. This fact motiva…
We formulate continuous time quantum walks (CTQW) in a discrete quantum mechanical phase space. We define and calculate the Wigner function (WF) and its marginal distributions for CTQWs on circles of arbitrary length $N$. The WF of the CTQW shows characteristic features in phase space. Revivals of the probability distributions found for continuous and for discrete quantum carpets do manifest themselves as characteristic patterns in phase space.
We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. [1] showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle [2], expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al. [3].
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard（哈佛大学工程与应用科学学院）；Department of Physics, Harvard University（哈佛大学物理系）；Department of Chemistry and Chemical Biology, Harvard University（哈佛大学化学与化学生物学系）；Harvard College, Harvard University（哈佛学院）；School of Physics, Peking University（北京大学物理学院）；TU Dresden, Institute of Theoretical Physics（德累斯顿工业大学理论物理研究所）；Computational Physics Laboratory, Tampere University（坦佩雷大学计算物理学实验室）
