Quantum Systems Connected by a Time-Dependent Canonical Transformation

We study both classical and quantum relation between two Hamiltoniansystems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other istime-dependent Hamiltonian system. The quantum unitary operatorrelevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.     

Quantum particles and classical particles

The relation between wave mechanics and classical mechanics is reviewed, and it is stressed that the latter cannot be regarded as the limit of the former as 0. The motion of a classical particle (or ensemble of particles) is described by means of a Schrödinger-like equation that was found previously. A system of a quantum particle and a classical particle is investigated (1) for an interaction that leads to stationary states with discrete energies and (2) for an interaction that enables the classical particle to act as a measuring instrument for determining a physical variable of the quantum particle.     

On Algebraic Integrability of the Deformed Elliptic Calogero-Moser Problem

Abstract

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the ? ⁴ type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms — the (Rashba)–Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spectrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.     

Quantitative conditions for time evolution in terms of the von Neumann equation

The adiabatic theorem describes the time evolution of the pure state and gives an adiabatic approximate solution to the Schr ¨odinger equation by choosing a single eigenstate of the Hamiltonian as the initial state. In quantum systems, states are divided into pure states(unite vectors) and mixed states(density matrices, i.e., positive operators with trace one). Accordingly, mixed states have their own corresponding time evolution, which is described by the von Neumann equation. In this paper, we discuss the quantitative conditions for the time evolution of mixed states in terms of the von Neumann equation. First, we introduce the definitions for uniformly slowly evolving and δ-uniformly slowly evolving with respect to mixed states, then we present a necessary and sufficient condition for the Hamiltonian of the system to be uniformly slowly evolving and we obtain some upper bounds for the adiabatic approximate error. Lastly, we illustrate our results in an example.     

A Necessary Condition for Quantum Adiabaticity Applied to the Adiabatic Grover Search

Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. Recently a practicable necessary condition well suited for many-body systems has been proved. Here we tailor this condition for estimating run times of adiabatic quantum algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that the thus obtained lower bound on the run time of this algorithm reproduces \( \sqrt{N} \) scaling (with N being the number of database entries) of the explicitly known optimum run time. This is in contrast to the poor performance of the known sufficient adiabatic conditions, which guarantee adiabaticity only for a run time on the order of O(N), which does not constitute any speedup over the classical database search. This observation highlights the merits of the new adiabatic condition and its potential relevance to adiabatic quantum computing.     

Star products and geometric algebra

The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficients of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.     

Statistische Bewegungsgesetze abgeschlossener Systeme und Entropie-Vermehrung

The most general dynamical laws describing the evolution of isolated systems are discussed. These may be described by linear transformations which in classical physics apply to probability-distributions in quantum physics to density operators. Entropy does not decrease if and only if the equipartition is invariant under the dynamical transformation. This invariance follows in a natural way for isolated systems from the interpretation of entropy as lack of information. If entropy is conserved for quantum systems the dynamical transformation becomes a unitary transformation generated by a Hamiltonian whereas for classical systems a generalized form ofLiouville's equation may be derived.     

Nonadiabatic dynamics of two strongly coupled nanomechanical resonator modes

The Landau-Zener transition is a fundamental concept for dynamical quantum systems and has been studied in numerous fields of physics. Here, we present a classical mechanical model system exhibiting analogous behavior using two inversely tunable, strongly coupled modes of the same nanomechanical beam resonator. In the adiabatic limit, the anticrossing between the two modes is observed and the coupling strength extracted. Sweeping an initialized mode across the coupling region allows mapping of the progression from diabatic to adiabatic transitions as a function of the sweep rate.     

Measure synchronization in hybrid quantum-classical systems

Measure synchronization in hybrid quantum-classical systems is investigated in this paper. The dynamics of the classical subsystem is described by the Hamiltonian equations, while the dynamics of the quantum subsystem is governed by the Schrödinger equation. By increasing the coupling strength in between the quantum and classical subsystems, we reveal the existence of measure synchronization in coupled quantum-classical dynamics under energy conservation for the hybrid systems.     

Nonlinear description of Yang-Mills cosmology: cosmic inflation and the accompanying Hannay's angle

Hannay's angle is a classical analogue of the "geometrical phase factor" found by Berry in his research on the quantum adiabatic theorem. This classical analogue is defined if closed curves of constant action variables return to the same curves in phase space after an adaibatic evolution. Adiabatic evolution of Yang-Mills cosmology, which is described by a time-dependent quartic oscillator, is investigated. Phase properties of the Yang-Mills fields are analyzed and the corresponding Hannay's angle is derived from a rigorous evaluation. The obtained Hannay's angle shift is represented in terms of several observable parameters associated with such an angle shift. The time evolution of Hannay's angle in Yang-Mills cosmology is examined from an illustration plotted on the basis of numerical evaluation,and its physical nature is addressed. Hannay's angle, together with its quantum counterpart Berry's phase, plays a pivotal role in conceptual understanding of several cosmological problems and indeed can be used as a supplementary probe for cosmic inflation.     

Quantum-Classical Correspondence for Adiabatic Shortcut in Two- and Three-Level Atoms

The methods of quickly achieving the adiabatic effect through a non-adiabatic process has recently drawn widely attention both in quantum and classical regime. In this work ,we study the classical adiabatic shortcut for two- and three-Level atoms by transforming the quantum version into classical one via quantum-classical corresponding theory. The results shows that, the additional couplings between the oscillators can be used to speed up the adiabatic evolution of coupled oscillators. Furthermore, we find that the quantum-classical correspondence theory still holds for the couter-adiabatic driving Hamiltonian for the TQD. This means that, we can obtain the counter-adiabatic driving Hamiltonian for a classical system by averaging over its quantum correspondence in a quantum system. This provides a feasible way to study the classical adiabatic shortcut and the simulation for the quantum adiabatic shortcut in a classical system.

     

The Exact Solution,Berry's Phase and Coherent State for the Driven Generalized Time-Dependent Harmonic Oscillator

The Lewis'invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for thirr system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adiabatic limit. The relation between the exact Lewis'phase and the corresponding classical phase angle L'discusrred.     

Factorization of wave functions of the quantum integrable particle systems

The relation between the characteristics of the equilibrium configurations of the classical Calogero-Moser integrable systems and properties of the ground state of their quantum analogs is found. It is shown that under the condition of factorization of the wave function of these systems the coordinates of classical particles at equilibrium are zeroes of the polynomial solutions of the second-order linear differential equation. It turns out that, under these conditions, the dependence of classical and quantum minimal energies on the parameters of the interaction potential is the same.     

Classical chaos with Bose-Einstein condensates in tilted optical lattices

A widely accepted definition of "quantum chaos" is "the behavior of a quantum system whose classical limit is chaotic." The dynamics of quantum-chaotic systems is nevertheless very different from that of their classical counterparts. A fundamental reason for that is the linearity of Schr?dinger equation. In this paper, we study the quantum dynamics of an ultracold quantum degenerate gas in a tilted optical lattice and show that it displays features very close to classical chaos. We show that its phase space is organized according to the Kolmogorov-Arnold-Moser theorem.     

Adiabatic condition for nonlinear systems

The adiabatic approximation is an important concept in quantum mechanics. In linear systems, the adiabatic condition is derived with the help of the instantaneous eigenvalues and eigenstates of the Hamiltonian, a procedure that breaks down in the presence of nonlinearity. Using an explicit example relevant to photoassociation of atoms into diatomic molecules, we demonstrate that the proper way to derive the adiabatic condition for nonlinear mean-field (or classical) systems is through a linearization procedure, using which an analytic adiabatic condition is obtained for the nonlinear model under study.     

Geometry of the Parameter Space of a Quantum System: Classical Point of View

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.     

The performance characteristics of an irreversible quantum Otto harmonic refrigeration cycle

In this paper, an irreversible quantum Otto refrigeration cycle working with harmonic systems is established. Base on Heisenberg quantum master equation, the equations of motion for the set of harmonic systems thermodynamic observables are derived. The simulated diagrams of the quantum Otto refrigeration cycle are plotted. The relationship between average power of friction, cooling rate, power input, and the time of adiabatic process is analyzed by using numerical calculation. Moreover, the influence of the heat conductance and the time of iso-frequency process on the performance of the cycle is discussed.     

Correlations in classical and quantum systems far from thermal equilibrium

We consider classical systems described by a Fokker-Planck equation or a generalized Fokker-Planck equation and quantum systems described by a density matrix equation or by a generalized Fokker-Planck equation using the principle of quantum classical correspondence. We split the corresponding operators of the equation of motion into a part which refers to the proper system and another one which describes the coupling of the proper system to the external world (reservoirs). We demonstrate that by use of conservation laws, referring to the proper systems, exact relations hold for certain moments, valid for all temperatures and coupling constants of the reservoirs. Using the concepts of a previous paper we describe then a perturbation theoretical approach which allows in a simple manner to determine a number of important correlation functions (moments of the total system). The time dependent case is briefly discussed. The applicability and usefulness of the present procedure is demonstrated by the example of the single-mode laser yielding e.g. expressions for the atom-field correlation.