Quantitative indicator for semiquantum chaos (original) (raw)
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arXiv preprint chao-dyn/9309004, 1993
We consider a system in which a classical oscillator is interacting with a purely quantum mechanical oscillator, described by the Lagrangian L = 1 2 ˙x2 + 1 2 ˙A2 − 1 2 (m2 + e2A2)x2 , where A is a classical variable and x is a quantum operator. With 〈x(t)〉 = 0, the relevant variable for the quantum oscillator is 〈x(t)x(t)〉 = G(t). The classical Hamiltonian dynamics governing the variables A(t), ΠA(t), G(t) and ΠG(t) is chaotic so that the results of making measurements on the quantum system at later times are sensitive to initial conditions. This system arises as the zero ...
Sensitivity to initial conditions in quantum dynamics: an analytical semiclassical expansion
Physics Letters A, 2004
We construct a class of systems for which quantum dynamics can be expanded around a mean field approximation with essentially classical content. The modulus of the quantum overlap of mean field states naturally introduces a classical distance between classical phase points. Using this fact we analytically show that the time rate of change (trc) of two neighbouring classical trajectories is directly proportional to the trc of quantum correlations. Coherence loss and nonlocality effects appear as corrections to mean field dynamics and we show that they can be given in terms of classical trajectories and generalized actions. This result is a first step in the connection between quantum and classically chaotic dynamics in the same sense of semiclassical expansions for the density of states. We apply the results to the nonintegrable (classically chaotic) version of the N-atom Jaynes-Cummings model. * Permanent Address.
Statistical mechanics of quantum-classical systems
The Journal of Chemical Physics, 2001
The statistical mechanics of systems whose evolution is governed by mixed quantum-classical dynamics is investigated. The algebraic properties of the quantum-classical time evolution of operators and of the density matrix are examined and compared to those of full quantum mechanics. The equilibrium density matrix that appears in this formulation is stationary under the dynamics and a method for its calculation is presented. The response of a quantum-classical system to an external force which is applied from the distant past when the system is in equilibrium is determined. The structure of the resulting equilibrium time correlation function is examined and the quantum-classical limits of equivalent quantum time correlation functions are derived. The results provide a framework for the computation of equilibrium time correlation functions for mixed quantum-classical systems.
Quantum and Classical Correlations in Gaussian Open Quantum Systems∗
2014
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable quantum correlations (quantum entanglement and quantum discord) for a system consisting of two noninteracting bosonic modes embedded in a thermal environment. We solve the Kossakowski-Lindblad master equation for the time evolution of the considered system and describe the entanglement and discord in terms of the covariance matrix for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. We study the time evolution of logarithmic negativity, which characterizes the degree of entanglement, and show that in the case of an entangled initial squeezed thermal state, entanglement suppression takes place for all temperatures of the environment, including zero temperature. We analyze the time evolution of the Gaussian quantum discord, which i...
The low-density limit of quantum systems
The present paper concludes a research programme developed, in the past three years, by various authors in Several papers. The basic idea of this programme is to expiain the irreversible behaviour of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system interacling with a reservoir and, in the first attempts to deal with this problem, only limits of observables of the system and deduced master equations were considered. In our approach we study the limits of quantities related to the whole compound syrtem. As a corollary we obtain an explanation of the physical origins of the quantum Brownian motion. In the present paper we study state at invene temperature p and fugacity r, through an interaction of the scattering type,
On the Discrimination Between Classical and Quantum States
Foundations of Physics, 2011
With the purpose of introducing a useful tool for researches concerning foundations of quantum mechanics and applications to quantum technologies, here This work has been supported by SanPaolo Foundation, Quantum Candela EU Project.
An outline of quantum probability
1 INDEX Introduction (1a.) Foundations of quantum theory (1b.) Quantum probability and the paradoxes of quantum theory (1c.) Von Neumann' s measurement theory (1d.) Contemporary measurement theory (1e.) Open systems and quantum noise (1f.) Stochastic calculus (1g.) Laws of large numbers and central limit theorems (1h.) Conditioning PART I: ALGEBRAIC PROBABILITY THEORY (2.) Algebraic probability spaces (3.) Algebraic random variables (4.) Stochastic Processes (5.) The local algebras of a stochastic process (6.) Independence (7.) Example: quantum spin systems (8.) A combinatorial lemma (9.) The Boson law of large numbers for independent random variables (10.) The central limit theorem for product maps (11.) Boson and Fermion Gaussian maps (12.) The quantum commutation relations as GNS representations (13.) The quantum commutation relations (14.) De Finetti' s theorem (15.) Conditioning: expected subalgebras (16.) Conditional amplitudes on B(H o ) (17.) Transition expectations and Markovian operators (18.) Markov chains, stationarity, ergodicity (19.) Conditional density amplitudes, potentials and invariant weights (20.) Multiplicative functionals and the discrete Feynman integral (21.) Quantum Markov chains and high temperature superconductivity models (22.) Kümmerer's Markov chains (23.) The algebraic states of Fannes, Nachtergaele and Slegers (24.) 1-dependence and the Ibragimov-Linnik conjecture (25.) 1-dependent quantum Markov chains 2 (26.) Commuting conditional density amplitudes (27.) Diagonalizable states (28.) A nonlinear chain of harmonic oscillators (29.) Generalized random walks (30.) The diffusion limit of the coherent chain (31.) Cecchini' s Markov chains PART II : STOCHASTIC CALCULUS (32.) Simple stochastic integrals (33.) Semimartingales and integrators (34.) Forward derivatives (35.) The o(dt)-notation (36.) Stochastic differential equations (37.) Meyer brackets and Ito tables (38.) The weak Itô formula (39.) The unitarity conditions (40.) The Boson Lévy theorem PART III : CONDITIONING (41.) The standard space of a von Neumann algebra (42.) The ϕ-conditional expectation
Quantum Statistical Mechanics for Nonextensive Systems: Prediction for Possible Experimental Tests
Physical Review Letters, 1998
The traditional basis of description of many-particle systems in terms of Green functions is here generalized to the case when the system is nonextensive, by incorporating the Tsallis form of the density matrix indexed by a nonextensive parameter q. This is accomplished by expressing the many-particle q Green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical Green function which now depends on this parameter. We study one-and two-particle Green functions in detail. From the one-particle Green function, we deduce some experimentally observable quantities such as the one-particle momentum distribution function and the one-particle energy distribution function. Special forms of the twoparticle Green functions are related to physical dynamical structure factors, some of which are studied here. We deduce different forms of sum rules in the q formalism. A diagrammatic representation of the q Green functions similar to the traditional ones follows because the equations of motion for both of these are formally similar. Approximation schemes for one-particle q Green functions such as Hartree and Hartree-Fock schemes are given as examples. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the q averages. ͓S1063-651X͑99͒10201-0͔
Nonclassical correlation in a multipartite quantum system: Two measures and evaluation
Physical Review A, 2008
There is a commonly recognized paradigm in which a multipartite quantum system described by a density matrix having no product eigenbasis is considered to possess nonclassical correlation. Supporting this paradigm, we define two entropic measures of nonclassical correlation of a multipartite quantum system. One is defined as the minimum uncertainty about a joint system after we collect outcomes of particular local measurements. The other is defined by taking the maximum over all local systems about the minimum distance between a genuine set and a mimic set of eigenvalues of a reduced density matrix of a local system. The latter measure is based on an artificial game to create mimic eigenvalues of a reduced density matrix of a local system from eigenvalues of a density matrix of a global system. Numerical computation of these measures for several examples is performed.