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Nonmonotonicity in the QuantumClassical Transition: Chaos Induced by Quantum Effects Arie Kapulkin1 and Arjendu K. Pattanayak2 1128 Rockwood Crescent, Thornhill, Ontario L4J 7W1 Canada 2Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057 (Received 16 February 2007; published 11 August 2008) The classicalquantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative @ is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the doublewell Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective @ is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantumclassical transition in this case is nonmonotonic in @. DOI: 10.1103/PhysRevLett.101.074101 PACS numbers: 05.45.Mt, 03.65.Sq Open nonlinear quantum systems are critical in understanding the foundations of quantum behavior, particularly the transition from quantum to classical mechanics. For example, it has been argued that quantum systems decohere rapidly when the classical counterpart is chaotic, with the decoherence rate determined by the classical Lyapunov exponents of the system [1]. This applies to entanglement and fidelity as well [2–4], since decoherence amounts to entanglement with the environment. A powerful way of studying open quantum systems is the quantum state diffusion (QSD) approach [5]. This enables the resolution of the paradox that in the absence of a QSDlike formulation, classical chaos cannot be recovered from quantum mechanics, indicating that the @ ! 0 limit is singular. Brun et al. [6] studied the convergence towards classical trajectories for a chaotic system with quantum Poincare´ sections of the quantities hx^i and hp^ i. They showed that the classical chaotic attractor is recovered when the system parameters were such that @ was small relative to the system’s characteristic action. As the relative @ increased, the attractor disappeared gradually, suggesting a persistence of chaos into the quantum region, consistent with later, more quantitative analyses [7,8]. Related work [9] studied a quantum system that is being continuously weakly measured, which leads to similar equations as those for QSD [10]. This also showed that chaos is recovered in the classical limit, and that it persists, albeit reduced, substantially into the quantum regime. Another related study [11] of coupled Duffing oscillators, showed that quantum effects, specifically entanglement, persist in a quantum system even when the system is classical enough to be chaotic. The prevailing paradigm is that chaos is classical, and is suppressed quantum mechanically. Do quantum effects always decrease chaos, however? A closed Hamiltonian quantum system studied within a Gaussian wave packet (WP) approximation [12] manifested chaos absent classically. This has been understood to be an artifact of the approximation, since the full quantum system is not chaotic. Followup work with an open system [13] also manifested quantum chaos, but it is not clear if this was not due to the approximations made. However, contrary to the prevailing paradigm, the classicaltoquantum transition [14] for the kicked rotor was shown to be nonmonotonic in the degree of diffusion, which is related to the degree of chaos in the problem. In this Letter we show evidence of chaos being induced by quantum effects. Specifically, in a QSD system with a nonchaotic classical limit, as we increase the relative @, chaos emerges, due to explicitly quantum effects (tunneling and zeropoint energy) and as @ is increased further, the chaos disappears. Although being reported for the first time, this intriguing result is arguably relatively common. Moreover, it shows that the quantumclassical transition for nonlinear systems is in general not monotonic in @. The QSD evolution equation for a realization j i of the system interacting with a Markovian environment is jd i i @ ^H j idt X j h ^Lyj i ^L j 1 2 ^Lyj ^L j 1 2 h ^Lyj ih ^L ji j idt X j ^L j hLji j id j; (1) where ^H is the Hamiltonian and the Lindblad operators ^L j model coupling to an external environment. The density matrix is recovered as the ensemble mean M over different realizations as ^ Mj ih j [5]. The d j are independent normalized complex differential random variables satisfying M d j 0; M d jd j0 0; M d jd j0 jj0dt. Consider specifically the classical driven dissipative Duffing oscillator x 2 x_ x3 x g cos t ; (2) for a particle of unit mass in a doublewell potential, with dissipation , and driving amplitude g and frequency . Chaotic behavior obtains for certain ranges of , g, [15]. Chaos is found through Poincare´ maps [obtained by recording (x, p) at time intervals of 2 = ]showing a strange attractor, or the behavior of the time series x t , or through a positive Lyapunov exponent. To quantize this problem [6,7] choose ^H and ^L for Eq. (1) as ^H ^H D ^H R ^H ex, PRL 101, 074101 (2008) PHYSICAL REVIEW LETTERS week ending 15 AUGUST 2008 00319007=08=101(7)=074101(4) 0741011 © 2008 The American Physical Society
Object Description
Collection Title  Scholarly Publications by Carleton Faculty and Staff 
Journal Title  Physical Review Letters 
Article Title  Nonmonotonicity in the quantumclassical transition: chaos induced by quantum effects 
Article Author 
Pattanayak, Arjendu Kapulkin, Arie 
Carleton Author 
Pattanayak, Arjendu 
Department  Physics 
Field  Science and Mathematics 
Year  2008 
Volume  101 
Publisher  American Physical Society 
File Name  038_PattanayakArjendu_NonmonotonicityInTheQuantumClassicalTransition.pdf; 038_PattanayakArjendu_NonmonotonicityInTheQuantumClassicalTransition.pdf 
Rights Management  This document is authorized for selfarchiving and distribution online by the author(s) and is free for use by researchers. 
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Contributing Organization  Carleton College 
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Language  English 
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Article Title  Page 1 
FullText  Nonmonotonicity in the QuantumClassical Transition: Chaos Induced by Quantum Effects Arie Kapulkin1 and Arjendu K. Pattanayak2 1128 Rockwood Crescent, Thornhill, Ontario L4J 7W1 Canada 2Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057 (Received 16 February 2007; published 11 August 2008) The classicalquantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative @ is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the doublewell Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective @ is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantumclassical transition in this case is nonmonotonic in @. DOI: 10.1103/PhysRevLett.101.074101 PACS numbers: 05.45.Mt, 03.65.Sq Open nonlinear quantum systems are critical in understanding the foundations of quantum behavior, particularly the transition from quantum to classical mechanics. For example, it has been argued that quantum systems decohere rapidly when the classical counterpart is chaotic, with the decoherence rate determined by the classical Lyapunov exponents of the system [1]. This applies to entanglement and fidelity as well [2–4], since decoherence amounts to entanglement with the environment. A powerful way of studying open quantum systems is the quantum state diffusion (QSD) approach [5]. This enables the resolution of the paradox that in the absence of a QSDlike formulation, classical chaos cannot be recovered from quantum mechanics, indicating that the @ ! 0 limit is singular. Brun et al. [6] studied the convergence towards classical trajectories for a chaotic system with quantum Poincare´ sections of the quantities hx^i and hp^ i. They showed that the classical chaotic attractor is recovered when the system parameters were such that @ was small relative to the system’s characteristic action. As the relative @ increased, the attractor disappeared gradually, suggesting a persistence of chaos into the quantum region, consistent with later, more quantitative analyses [7,8]. Related work [9] studied a quantum system that is being continuously weakly measured, which leads to similar equations as those for QSD [10]. This also showed that chaos is recovered in the classical limit, and that it persists, albeit reduced, substantially into the quantum regime. Another related study [11] of coupled Duffing oscillators, showed that quantum effects, specifically entanglement, persist in a quantum system even when the system is classical enough to be chaotic. The prevailing paradigm is that chaos is classical, and is suppressed quantum mechanically. Do quantum effects always decrease chaos, however? A closed Hamiltonian quantum system studied within a Gaussian wave packet (WP) approximation [12] manifested chaos absent classically. This has been understood to be an artifact of the approximation, since the full quantum system is not chaotic. Followup work with an open system [13] also manifested quantum chaos, but it is not clear if this was not due to the approximations made. However, contrary to the prevailing paradigm, the classicaltoquantum transition [14] for the kicked rotor was shown to be nonmonotonic in the degree of diffusion, which is related to the degree of chaos in the problem. In this Letter we show evidence of chaos being induced by quantum effects. Specifically, in a QSD system with a nonchaotic classical limit, as we increase the relative @, chaos emerges, due to explicitly quantum effects (tunneling and zeropoint energy) and as @ is increased further, the chaos disappears. Although being reported for the first time, this intriguing result is arguably relatively common. Moreover, it shows that the quantumclassical transition for nonlinear systems is in general not monotonic in @. The QSD evolution equation for a realization j i of the system interacting with a Markovian environment is jd i i @ ^H j idt X j h ^Lyj i ^L j 1 2 ^Lyj ^L j 1 2 h ^Lyj ih ^L ji j idt X j ^L j hLji j id j; (1) where ^H is the Hamiltonian and the Lindblad operators ^L j model coupling to an external environment. The density matrix is recovered as the ensemble mean M over different realizations as ^ Mj ih j [5]. The d j are independent normalized complex differential random variables satisfying M d j 0; M d jd j0 0; M d jd j0 jj0dt. Consider specifically the classical driven dissipative Duffing oscillator x 2 x_ x3 x g cos t ; (2) for a particle of unit mass in a doublewell potential, with dissipation , and driving amplitude g and frequency . Chaotic behavior obtains for certain ranges of , g, [15]. Chaos is found through Poincare´ maps [obtained by recording (x, p) at time intervals of 2 = ]showing a strange attractor, or the behavior of the time series x t , or through a positive Lyapunov exponent. To quantize this problem [6,7] choose ^H and ^L for Eq. (1) as ^H ^H D ^H R ^H ex, PRL 101, 074101 (2008) PHYSICAL REVIEW LETTERS week ending 15 AUGUST 2008 00319007=08=101(7)=074101(4) 0741011 © 2008 The American Physical Society 