Semiclassical dynamics of open quantum systems: Comparing the finite with the infinite perspective

Goletz, Christoph-Marian; Koch, Werner; Großmann, Frank

doi:10.1016/j.chemphys.2010.06.019

Cited by 20 publications

(19 citation statements)

References 49 publications

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“…In this point, however, we adopt a more recent development in statistical mechanics, in that we keep the number of oscillators large,

, but finite [ 8 , 9 , 10 , 11 , 12 ], so that the dynamics of the total system can be treated in the framework of the time-reversal invariant Hamiltonian mechanics of closed systems. It has been demonstrated for classical as well as for quantum systems [ 13 , 14 ], and is corroborated by the present work, that despite its time-reversal symmetry, this approach reproduces irreversible behaviour on all relevant timescales. Poincaré recurrences, which prevent true irreversibility in systems with a finite number of freedoms, occur only on timescales that diverge geometrically with N [ 15 ].…”

Section: Introductionsupporting

confidence: 87%

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Dittrich

Martı́nez

2020

Entropy

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We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N=1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol’d-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

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“…In this point, however, we adopt a more recent development in statistical mechanics, in that we keep the number of oscillators large,

Section: Introductionsupporting

confidence: 87%

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Dittrich

Martı́nez

2020

Entropy

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“…for the frequencies and we here choose the second parameter f co > f , such that extremely high frequencies which would not exchange energy with the system (not shown) are not considered. Other frequency distributions have been used in [34] as well as in [50], while the present one has been found favorable also in multi configuration time-dependent Hartree (MCTDH) calculations [51]. Now one could choose the coupling strength between system and environment according to a specific (continuous) spectral distribution, which is usually taken as Ohmic or sub-or super-Ohmic.…”

Section: B Numerical Resultsmentioning

confidence: 94%

Quantum approach to the thermalization of the toppling pencil interacting with a finite bath

Choudhury¹,

Großmann²

2021

Preprint

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We investigate the longstanding problem of thermalization of quantum systems coupled to an environment by focusing on a bistable quartic oscillator interacting with a finite number of harmonic oscillators. In order to overcome the exponential wall that one usually encounters in grid based approaches to solve the time-dependent Schrödinger equation of the extended system, methods based on the time-dependent variational principle are best suited. Here we will apply the method of coupled coherent states [D. V. Shalashilin and M. S. Child, J. Chem. Phys. 113, 10028 (2000)].By investigating the dynamics of an initial wavefunction on top of the barrier of the double well, it will be shown that only a handful of oscillators with suitably chosen frequencies, starting in their ground states, is enough to drive the bistable system close to its uncoupled ground state. The long-time average of the double-well energy is found to be a monotonously decaying function of the number of environmental oscillators in the parameter range that was numerically accessible.

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“…This price appears acceptable, though, as long as a faithful description of the processes of interest is required only over a correspondingly large, but finite time scale, as is the case, for example, in computational molecular physics and in quantum optics. Numerical experiments simulating decoherence with heat baths of finite Hilbert space dimension [ 79 , 80 , 81 ] provide convincing evidence that even with a surprisingly low number of bath modes, N of the order of 10, most relevant features of decoherence can be satisfactorily reproduced; see Figure 16 . This suggests to restrict the dimension of the meter sector of the Hilbert space underlying the Hamiltonian ( 58 ) accordingly to a finite number N ,

Like this, the Hamiltonian can be considered as a model of, e.g., a two-level atom in a high- Q microwave cavity [ 76 ].…”

Section: Quantum Measurement and Quantum Randomness In A Unitary Smentioning

confidence: 96%

“…Experience with similar models comprising finite baths [ 79 , 81 ], suggests the following scenario: For small values

, the time evolution comprises only a few, but typically incommensurate, frequencies and should appear quasi-periodic. Already for moderate numbers, say

, the unitary model will exhibit a similar behavior as has been observed for standard models of quantum optics and solid-state physics, known as “collapses and revivals” [ 76 ].…”

Section: Quantum Measurement and Quantum Randomness In A Unitary Smentioning

confidence: 99%

Quantum Chaos and Quantum Randomness—Paradigms of Entropy Production on the Smallest Scales

Dittrich

2019

Entropy

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Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro-to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: Bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase space, reminiscent of Gibbs' paradox in Statistical Mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos, and how the unitary time evolution in a closed system directly implies the "quantum death?? of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, that is, produce entropy without discernible source, could it have infiltrated from the macroscopic meter? This speculation is worked out for the case of spin measurement.

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Semiclassical dynamics of open quantum systems: Comparing the finite with the infinite perspective

Cited by 20 publications

References 49 publications

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Quantum approach to the thermalization of the toppling pencil interacting with a finite bath

Quantum Chaos and Quantum Randomness—Paradigms of Entropy Production on the Smallest Scales

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