Overlapping of nonlinear resonances and the problem of quantum chaos

Ugulava, A.; Chotorlishvili, L.; Nickoladze, K.

doi:10.1103/physreve.68.026216

Cited by 17 publications

(29 citation statements)

References 8 publications

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“…If ψ(ϕ) ≡ G(ϕ) means either ce n (ϕ) or se n (ϕ), then G(ϕ) and G(π − ϕ) satisfy the same equation (9) and the same boundary conditions (11). Therefore these functions differ from each other only by the constants.…”

Section: B Periodical Solution Of Mathieu -Schrodinger Equationsmentioning

confidence: 99%

“…So this part may be considered as an attempt to study quantum chaos in the essentially quantum domain. This study is based on our previous papers [9,10,11,12]. In addition, in the present paper, we will discuss in details possible experimental realizations and applications of the theoretical concepts in the organic chemistry and polyatomic organic molecules.…”

Section: Paper Is Organized As Followsmentioning

confidence: 99%

“…Note that functions (23) correspond also to the well-known asymptotic (l → 0) forms of the Mathieu functions [9]:…”

Section: The Physical Problems Are Reduced To Quantum Pendulummentioning

confidence: 99%

“…there is one eigenfunction for each of four boundary conditions and m equal to the number of zeroes in the interval 0 < ϕ < π/2. The functions (13) represent a complete system of eigenfunctions of the equation (9).…”

Section: B Periodical Solution Of Mathieu -Schrodinger Equationsmentioning

confidence: 99%

“…Eigenfunctions ψ n can be recorded with the help of the Mathieu functions [9,10]: even ce n (ϕ, l) and odd se n (ϕ, l). Appropriate eigenvalues are usually designated by a n (l) and b(l).…”

Section: B Periodical Solution Of Mathieu -Schrodinger Equationsmentioning

confidence: 99%

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Quantum chaos and its kinetic stage of evolution

Chotorlishvili

Ugulava

2010

Physica D: Nonlinear Phenomena

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Usually reason of irreversibility in open quantum-mechanical system is interaction with a thermal bath, consisting form infinite number of degrees of freedom. Irreversibility in the system appears due to the averaging over all possible realizations of the environment states. But, in case of open quantum-mechanical system with few degrees of freedom situation is much more complicated.Should one still expect irreversibility, if external perturbation is just an adiabatic force without any random features? Problem is not clear yet. This is main question we address in this review paper. We prove that key point in the formation of irreversibility in chaotic quantum-mechanical systems with few degrees of freedom, is the complicated structure of energy spectrum. We shall consider quantum mechanical-system with parametrically dependent energy spectrum. In particular, we study energy spectrum of the Mathieu-Schrodinger equation. Structure of the spectrum is quite non-trivial, consists from the domains of non-degenerated and degenerated stats, separated from each other by branch points. Due to the modulation of the parameter, system will perform transitions from one domain to other one. For determination of eigenstates for each domain and transition probabilities between them, we utilize methods of abstract algebra. We shall show that peculiarity of parametrical dependence of energy terms, leads to the formation of mixed state and to the irreversibility, even for small number of levels involved into the process. This last statement is important. Meaning is that, we are going to investigate quantum chaos in essentially quantum domain.In the second part of the paper, we will introduce concept of random quantum phase approximation. Then along with the methods of random matrix theory, we will use this assumption, for derivation of muster equation in the formal and mathematically strict way.

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Section: B Periodical Solution Of Mathieu -Schrodinger Equationsmentioning

confidence: 99%

Section: Paper Is Organized As Followsmentioning

confidence: 99%

“…Note that functions (23) correspond also to the well-known asymptotic (l → 0) forms of the Mathieu functions [9]:…”

Section: The Physical Problems Are Reduced To Quantum Pendulummentioning

confidence: 99%

Section: B Periodical Solution Of Mathieu -Schrodinger Equationsmentioning

confidence: 99%

Section: B Periodical Solution Of Mathieu -Schrodinger Equationsmentioning

confidence: 99%

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Quantum chaos and its kinetic stage of evolution

Chotorlishvili

Ugulava

2010

Physica D: Nonlinear Phenomena

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Non-reversible evolution of quantum chaotic system. Kinetic description

Chotorlishvili

Skrinnikov

2008

Physics Letters A

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It is well known that the appearance of non-reversibility in classical chaotic systems is connected with a local instability of phase trajectories relatively to a small change of initial conditions and parameters of the system. Classical chaotic systems reveal an exponential sensitivity to these changes. This leads to an exponential growth of the initial error with time, and as the result after the statistical averaging over this error, the dynamics of the system becomes non-reversible.In spite of this, the question about the origin of non-reversibility in quantum case remains actual. The point is that the classical notion of instability of phase trajectories loses its sense during quantum consideration.The current work is dedicated to the clarification of the origin of non-reversibility in quantum chaotic systems.For this purpose we study a non-stationary dynamics of the chaotic quantum system. By analogy with classical chaos, we consider an influence of a small unavoidable error of the parameter of the system on the non-reversibility of the dynamics.It is shown in the paper that due to the peculiarity of chaotic quantum systems, the statistical averaging over the small unavoidable error leads to the non-reversible transition from the pure state into the mixed one.The second part of the paper is dedicated to the kinematic description of the chaotic quantummechanical system. Using the formalism of superoperators, a muster kinematic equation for chaotic quantum system was obtained from Liouville equation under a strict mathematical consideration.

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