B. Georgeot
Quantum algorithms are built enabling to find Poincaré recurrence times and periodic orbits of classical dynamical systems. It is shown that exponential gain compared to classical algorithms can be reached for a restricted class of systems. Quadratic gain can be achieved for a larger set of dynamical systems. The simplest cases can be implemented with small number of qubits.
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We study numerically the properties of entanglement of two interacting, or noninteracting, particles evolving in a regime of quantum chaos in the quantum Chirikov standard map. Such pairs can be viewed as interacting, on noninteracting, Einstein-Podolsky-Rosen pairs in a regime of quantum chaos. The analysis is done with such tools as the Loschmidt echo of entanglement and the Poincaré recurrences of entanglement in presence of absorption. The obtained results show unusual features of the entropy of entanglement and the spectrum of Schmidt decomposition with their dependence on interactions at…
Poincare recurrence theorem states that any finite system will come arbitrary close to its initial state after some very long but finite time. At the statistical level, this by itself does not represent a paradox, but apparently violates the second law of thermodynamics, which may lead to some confusing conclusions for macroscopic systems. However, this statement does not take gravity into account. If two particles with a given center of mass energy come at the distance shorter than the Schwarzschild diameter apart, according to classical gravity they will form a black hole. In the classical c…
In this paper by using geometric techniques, we provide upper bounds for the Poincaré recurrence time of a quantum mixed state with discrete spectrum of energies. In the case of discrete but finite spectrum we obtain two type of upper bounds; one of them depends on the uncertainty in the energy, and the other depends only on the (finite) number of states. In the case of discrete but non-finite spectrum we obtain in the same way two upper bounds defining the number of relevant states according to an statistical measurement. These bounds correspond to two different situations in the quantum recu…
We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouville's theorem is then proposed. We proceed to study Poincare recurrence in C*-algebras by mimicking the measure theoretic setting. The results are interpreted as recurrence in quantum mechanics, similar to Poincare recurrence in classical mechanics.
A blueprint for experiments exploring the Poincaré quantum recurrence theorem
本文研究弱耦合环境中的$N$量子比特系统，解析与数值刻画复活概率及指数标度的回归时间，并提出利用超导多量子比特平台实验检验庞加莱量子回归。
The evolution of the ground state wave function of a zero-temperature Bose-Einstein condensate (BEC) is well described by the Hamiltonian Gross-Pitaevskii (GP) equation. Using a set of appropriately interleaved unitary collision-streaming operators, a quantum lattice gas algorithm is devised which on taking moments recovers the Gross-Pitaevskii (GP) equation in diffusion ordering (time scales as square of length). Unexpectedly, there is a class of initial conditions in which their Poincare recurrence is extremely short. Further it is shown that the Poincare recurrence time scales with diffusio…
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.
We investigate the quantum recurrence phenomena in periodically driven systems. We calculate the classical period and the quantum recurrence time and develop their interdependence. We further predict the behavior of the recurrence phenomena for the power law potentials.
Quantum systems in specific regimes display recurrences at the period of the periodic orbits of the corresponding classical system. We investigate the excited hydrogen atom in a magnetic field -- a prototypical system of 'quantum chaos' -- from the point of view of the de Broglie Bohm (BB) interpretation of quantum mechanics. The trajectories predicted by BB theory are computed and contrasted with the time evolution of the wavefunction, which shows pronounced features at times matching the period of closed orbits of the classical hydrogen in a magnetic field problem. Individual BB trajectories…
Quantum vortex structures and energy cascades are examined for two dimensional quantum turbulence (2D QT) at zero temperature. A special unitary evolution algorithm, the quantum lattice gas (QLG) algorithm, is employed to simulate the Bose-Einstein condensate (BEC) governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a short Poincaré recurrence time. It is demonstrated that such short recurrence times are destroyed as the nonlinear interaction is strengthened. The similar loss of Poincaré recurrence is also reported in 3D QT [1] Energy…
The recurrence phenomena of an initially well localized wave packet are studied in periodically driven power-law potentials. For our general study we divide the potentials in two kinds, namely tightly binding and loosely binding potentials. In the presence of an external periodically modulating force, these potentials may exhibit classical and quantum chaos. The dynamics of a quantum wave packet in the modulated potentials displays recurrences at various time scales. We develop general analytical relations for these times and discuss their parametric dependence.
Recurrence behavior of wave packets in coupled higher dimensional systems and periodically driven systems is analyzed, which takes place in the realm of higher coupling/modulation strength. We analyze the wave packet dynamics close to nonlinear resonances developed in the systems and provide the analytical understanding of recurrence times.We apply these analytical results to investigate the recurrence times of matter waves in optical lattice in the presence of external periodic forcing. The obtained analytical results can experimentally be observed using currently available experimental setup…
Periodically-driven systems are ubiquitous in science and technology. In quantum dynamics, even a small number of periodically-driven spins leads to complicated dynamics. Hence, it is of interest to understand what constraints such dynamics must satisfy. We derive a set of constraints for each number of cycles. For pure initial states, the observable being constrained is the recurrence probability. We use our constraints for detecting undesired coupling to unaccounted environments and drifts in the driving parameters. To illustrate the relevance of these results for modern quantum systems we d…
We present numerical study of a model of quantum walk in periodic potential on the line. We take the simple view that different potentials affect differently the way the coin state of the walker is changed. For simplicity and definiteness, we assume the walker's coin state is unaffected at sites without potential, and is rotated in an unbiased way according to Hadamard matrix at sites with potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks were studied numerically. It is found that of the six …
Recurrence Theorem for Open Quantum Systems
本文基于非厄米描述建立开放量子系统回归定理，阐明PT与赝厄米对称性的保护作用，并提出分析NH系统信息动力学的新方法。
We study the phase space of periodically modulated gravitational cavity by means of quantum recurrence phenomena. We report that the quantum recurrences serve as a tool to connect phase space of the driven system with spectrum in quantum domain. With the help of quantum recurrences we investigate the quasi-energy spectrum of the system for a certain fixed modulation strength. In addition, we study transition of spectrum from discrete to continuum as a function of modulation strength.
The dynamics of vortex solitons in a BEC superfluid is studied. A quantum lattice-gas algorithm (localization-based quantum computation) is employed to examine the dynamical behavior of vortex soliton solutions of the Gross-Pitaevskii equation (phi^4 interaction nonlinear Schroedinger equation). Quantum turbulence is studied in large grid numerical simulations: Kolmogorov spectrum associated with a Richardson energy cascade occurs on large flow scales. At intermediate scales a k^{-6} power law emerges, in a classical-quantum transition from vortex filament reconnections to Kelvin wave-acoustic…
Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence
本文研究量子零和博弈中的矩阵乘法权重更新与量子复制者动力学，证明系统具有量子信息论守恒定律和Poincaré回归，并推广经典博弈结果。
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