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An Example of the Decoherence Approach to Quantum Dissipative Chaos

Todd A. Brun

Published Aug 24, 1995• 4 citations
OpenAlex

Abstract

Quantum chaos---the study of quantized nonintegrable Hamiltonian systems---is an extremely well-developed and sophisticated field. By contrast, very little work has been done in looking at quantum versions of systems which classically exhibit {\\it dissipative} chaos. Using the decoherence formalism of Gell-Mann and Hartle, I find a quantum mechanical analog of one such system, the forced damped Duffing oscillator. I demonstrate the classical limit of the system, and discuss its decoherent histories. I show that using decoherent histories, one can define not only the quantum map of an entire density operator, but can find an analog to the Poincar\\'e map of the individual trajectory. Finally, I argue the usefulness of this model as an example of quantum dissipative chaos, as well as of a practical application of the decoherence formalism to an interesting problem.

Related papers

Quantum Dissipative Chaos

Todd A. Brun

Using the decoherence formalism of Gell-Mann and Hartle, a quantum system is found which is the equivalent of the classical chaotic Duffing oscillator. The similarities and the differences from the classical oscillator are examined; in particular, a new concept of quantum maps is introduced, and alterations in the classical strange attractor due to the presence of scale- dependent quantum effects are studied. Classical quantities such as the Lyapunov exponents and fractal dimension are examined, and quantum analogs are suggested. These results are generalized into a framework for quantum dissipative chaos, and there is a brief discussion of other work in this area.

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In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.

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Quantum chaos in open systems: a quantum state diffusion analysis

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Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wavepackets in the neighbourhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics.

arXiv:quant-ph/9509015•Journal of Physics A Mathematical and General•May 7, 1996•83 citations

Quantum Chaos in Open Systems: A Quantum State Diffusion Analysis

Todd A. Brun, Ian Colin Percival, Rüdiger Schack

Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wave packets in the neighborhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics. QMW Theory Preprint QMW-PH-95-33 1 1

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Quantization of Open System Based on Quantum State Diffusion II

Ichiro Ohba, Yukihiro Ota

We apply the QSD to the quantum version of Duffing oscillator and discuss the QCC and dissipative quantum chaos in this system. We calculate the constant phase map and the pseudo-Lyapunov exponent, varying Planck constant effectively and show there is a certain clear critical stage in which a crossover from classical to quantum behavior occurs. Furthermore, we discuss the effect of dissipation in the dissipative quantum chaos by means of a degrees of localization introduced in the QSD formalism[5]. Unfortunately, the pseudo-Lyapunov exponent is not a quantity characterized a dissipative system. Therefore, it will be necessary to discuss the connection between the classical trajectory (strange attractor) and the quantum observable which characterize a dissipative quantum system.

物性研究•Aug 20, 2004•0 citations

Decoherent Histories and Quantum Maps

Artur Scherer

Discrete maps play an important role in the investigation \nof dynamical features of complex classical systems, especially \nwithin the theory of chaos. Similarly, quantum maps have proven \nto be a very useful mathematical tool within the study of complex \nquantum dynamical systems. Using the decoherent histories formulation \nof quantum mechanics we consider a particular framework for studying \nquantum maps which is motivated by the method of classical symbolic \ndynamics. Symbolic dynamics is known to be a very powerful method \nspecifically invented for the purpose of representing classical dynamical \nsystems by a discrete model that is suitable for information theoretic studies. \nOur framework uses the decoherent histories formalism which, \nsimilarly to classical symbolic dynamics, allows one to introduce information \ntheoretic quantities with respect to system dynamics. \nOur research within this framework can be viewed as a contribution \ntowards the development of a general theory of "quantum \nsymbolic dynamics". \n\nWe start by considering a special but very important example \nfor a quantum map: the quantum baker's map, invented for \nthe theoretical investigation of quantum chaos. \nHere we use the decoherent histories formalism to examine \nthe coarse-grained evolution of this map \nwith regard to the question of how classical predictability of \nthe evolution depends on the character of coarse-graining. \n%is affected by the choice of coarse-graining. \nWe demonstrate that hierarchical coarse-grainings display \ninteresting features with respect to this question. \n\nWe proceed with the investigation of decoherence properties \nof arbitrary unitary quantum maps. A number of interesting results \nis obtained within the framework of arbitrarily long histories \nconstructed from a fixed projective partition of a finite \ndimensional Hilbert space. In particular, we derive simple \nnecessary decoherence conditions, which employ only a single \niteration of a given unitary quantum map. Furthermore, \na surprising result is obtained with regard to the \nfundamental question of how the choice of the \ninitial state affects decoherence of histories. \nWithin the considered framework we show that if decoherence \nis established for arbitrary history lengths and all initial \nstates from the smallest natural set of states that \ncan be associated with the framework, then we get decoherence \nof such histories for arbitrary initial states. \nFinally, we make first steps towards proving analogous\nresults for approximate decoherence and suggest \nvarious questions for future research.\n

•Jan 1, 2005•0 citations
Quantum chaos---the study of quantized nonintegrable Hamiltonian systems---is an extremely well-developed and sophisticated field. By contrast, very little work has been done in looking at quantum versions of systems which classically exhibit {\it dissipative} chaos. Using the decoherence formalism of Gell-Mann and Hartle, I find a quantum mechanical analog of one such system, the forced damped Duffing oscillator. I demonstrate the classical limit of the system, and discuss its decoherent histories. I show that using decoherent histories, one can define not only the quantum map of an entire density operator, but can find an analog to the Poincar\'e map of the individual trajectory. Finally, I argue the usefulness of this model as an example of quantum dissipative chaos, as well as of a practical application of the decoherence formalism to an interesting problem.
Using the decoherence formalism of Gell-Mann and Hartle, a quantum system is found which is the equivalent of the classical chaotic Duffing oscillator. The similarities and the differences from the classical oscillator are examined; in particular, a new concept of quantum maps is introduced, and alterations in the classical strange attractor due to the presence of scale- dependent quantum effects are studied. Classical quantities such as the Lyapunov exponents and fractal dimension are examined, and quantum analogs are suggested. These results are generalized into a framework for quantum dissipative chaos, and there is a brief discussion of other work in this area.
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wavepackets in the neighbourhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics.
Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wave packets in the neighborhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics. QMW Theory Preprint QMW-PH-95-33 1 1
We apply the QSD to the quantum version of Duffing oscillator and discuss the QCC and dissipative quantum chaos in this system. We calculate the constant phase map and the pseudo-Lyapunov exponent, varying Planck constant effectively and show there is a certain clear critical stage in which a crossover from classical to quantum behavior occurs. Furthermore, we discuss the effect of dissipation in the dissipative quantum chaos by means of a degrees of localization introduced in the QSD formalism[5]. Unfortunately, the pseudo-Lyapunov exponent is not a quantity characterized a dissipative system. Therefore, it will be necessary to discuss the connection between the classical trajectory (strange attractor) and the quantum observable which characterize a dissipative quantum system.
Discrete maps play an important role in the investigation \nof dynamical features of complex classical systems, especially \nwithin the theory of chaos. Similarly, quantum maps have proven \nto be a very useful mathematical tool within the study of complex \nquantum dynamical systems. Using the decoherent histories formulation \nof quantum mechanics we consider a particular framework for studying \nquantum maps which is motivated by the method of classical symbolic \ndynamics. Symbolic dynamics is known to be a very powerful method \nspecifically invented for the purpose of representing classical dynamical \nsystems by a discrete model that is suitable for information theoretic studies. \nOur framework uses the decoherent histories formalism which, \nsimilarly to classical symbolic dynamics, allows one to introduce information \ntheoretic quantities with respect to system dynamics. \nOur research within this framework can be viewed as a contribution \ntowards the development of a general theory of "quantum \nsymbolic dynamics". \n\nWe start by considering a special but very important example \nfor a quantum map: the quantum baker's map, invented for \nthe theoretical investigation of quantum chaos. \nHere we use the decoherent histories formalism to examine \nthe coarse-grained evolution of this map \nwith regard to the question of how classical predictability of \nthe evolution depends on the character of coarse-graining. \n%is affected by the choice of coarse-graining. \nWe demonstrate that hierarchical coarse-grainings display \ninteresting features with respect to this question. \n\nWe proceed with the investigation of decoherence properties \nof arbitrary unitary quantum maps. A number of interesting results \nis obtained within the framework of arbitrarily long histories \nconstructed from a fixed projective partition of a finite \ndimensional Hilbert space. In particular, we derive simple \nnecessary decoherence conditions, which employ only a single \niteration of a given unitary quantum map. Furthermore, \na surprising result is obtained with regard to the \nfundamental question of how the choice of the \ninitial state affects decoherence of histories. \nWithin the considered framework we show that if decoherence \nis established for arbitrary history lengths and all initial \nstates from the smallest natural set of states that \ncan be associated with the framework, then we get decoherence \nof such histories for arbitrary initial states. \nFinally, we make first steps towards proving analogous\nresults for approximate decoherence and suggest \nvarious questions for future research.\n