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Sarkar P, Bhattacharjee JK. Nonlinear parametric oscillator: A tool for probing quantum fluctuations. Phys Rev E 2020; 102:052204. [PMID: 33327191 DOI: 10.1103/physreve.102.052204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 10/15/2020] [Indexed: 06/12/2023]
Abstract
Nanomechanical oscillators have, over the last few years, started probing regimes where quantum fluctuations are important. Here we consider a nonlinear parametric oscillator in the quantum domain. We show that in the classical subharmonic resonance zone, the quantum fluctuations are finite but greatly magnified depending on the strength of the nonlinear coupling. This should make such oscillators useful in probing quantum fluctuations.
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Affiliation(s)
- Prasun Sarkar
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700 032, India
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2
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Hasegawa T. Communication: Constant uncertainty molecular dynamics: A simple and efficient algorithm to incorporate quantum nature into a real-time molecular dynamics simulation. J Chem Phys 2016; 145:171101. [DOI: 10.1063/1.4966917] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Taisuke Hasegawa
- Department of Chemistry, Graduate School of Science, Kyoto University, Sakyoku, Kyoto 606-8502, Japan
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Choi S, Onofrio R, Sundaram B. Ehrenfest approach to open double-well dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042907. [PMID: 26565304 DOI: 10.1103/physreve.92.042907] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2015] [Indexed: 06/05/2023]
Abstract
We consider an Ehrenfest approximation for a particle in a double-well potential in the presence of an external environment schematized as a finite resource heat bath. This allows us to explore how the limitations in the applicability of Ehrenfest dynamics to nonlinear systems are modified in an open system setting. Within this framework, we have identified an environment-induced spontaneous symmetry breaking mechanism, and we argue that the Ehrenfest approximation becomes increasingly valid in the limit of strong coupling to the external reservoir, either in the form of an increasing number of oscillators or increasing temperature. The analysis also suggests a rather intuitive picture for the general phenomenon of quantum tunneling and its interplay with classical thermal activation processes, which may be of relevance in physical chemistry, ultracold atom physics, and fast-switching dynamics such as in superconducting digital electronics.
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Affiliation(s)
- Stephen Choi
- Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
| | - Roberto Onofrio
- Dipartimento di Fisica e Astronomia "Galileo Galilei," Università di Padova, Via Marzolo 8, Padova 35131, Italy
- Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA
| | - Bala Sundaram
- Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
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Sinha SS, Mondal D, Bag BC, Ray DS. Quantum diffusion in a fermionic bath. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:051125. [PMID: 21230455 DOI: 10.1103/physreve.82.051125] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2010] [Indexed: 05/30/2023]
Abstract
We propose a scheme for quantum brownian motion of a particle in a fermionic bath. Based on the spin coherent-state representation of the noise operators and a canonical thermal distribution of the associated c numbers, we derive a quantum analog of generalized Langevin equation for quantum-mechanical mean position of the particle subjected to an external force field. The approach allows us to map the quantum problem on a classical setting. The quantum dispersion around the mean can be estimated order by order by a set of quantum correction equations up to a desired degree of accuracy for a given nonlinear potential. We derive a quantum diffusion equation for free particle and show that quantization, in general, enhances the mean-square displacement. Increase in temperature leads to suppression of mean-square displacement. The method is based on canonical quantization procedure and may be used for understanding diffusive transport and thermally activated processes in a fermionic bath.
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Shit A, Chattopadhyay S, Banik SK, Chaudhuri JR. Generalized Einstein Relation in Tilted Periodic Potential: A Semiclassical Approach. J Phys Chem B 2010; 114:7854-63. [DOI: 10.1021/jp912069k] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Anindita Shit
- Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India
| | - Sudip Chattopadhyay
- Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India
| | - Suman Kumar Banik
- Department of Chemistry, Bose Institute, 93/1 A P C Road, Kolkata 700009, India
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Ghosh P, Chattopadhyay S, Chaudhuri JR. Stochastic resonance in a generalized quantum Kubo oscillator. J Phys Chem B 2010; 114:1368-79. [PMID: 20041674 DOI: 10.1021/jp909858c] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
We discuss stochastic resonance in a biased linear quantum system that is subject to multiplicative and additive noises. Starting from a microscopic system-reservoir Hamiltonian, we derive a c-number analogue of the generalized Langevin equation. The developed approach puts forth a quantum mechanical generalization of the "Kubo type" oscillator which is a linear system. Such a system is often used in the literature to study various phenomena in nonequilibrium systems via a particular interaction between system and the external noise. Our analytical results proposed here have the ability to reveal the role of external noise and vis-a-vis the mechanisms and detection of subtle underlying signatures of the stochastic resonance behavior in a linear system. In our development, we show that only when the external noise possesses a "finite correlation time" the quantum effect begins to appear. We observe that the quantum effect enhances the resonance in comparison to the classical one.
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Affiliation(s)
- Pradipta Ghosh
- Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India
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Barik D, Ray DS. Anharmonic quantum contribution to vibrational dephasing. J Chem Phys 2004; 121:1681-9. [PMID: 15260719 DOI: 10.1063/1.1766007] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Based on a quantum Langevin equation and its corresponding Hamiltonian within a c-number formalism we calculate the vibrational dephasing rate of a cubic oscillator. It is shown that leading order quantum correction due to anharmonicity of the potential makes a significant contribution to the rate and the frequency shift. We compare our theoretical estimates with those obtained from experiments for small diatomics N(2), O(2), and CO.
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Affiliation(s)
- Debashis Barik
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
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Banerjee D, Bag BC, Banik SK, Ray DS. Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects. J Chem Phys 2004; 120:8960-72. [PMID: 15267831 DOI: 10.1063/1.1711593] [Citation(s) in RCA: 53] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed [Phys. Rev. E 65, 021109 (2002); 66, 051106 (2002)] to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers' turnover and the quantum Arrhenius plot.
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Affiliation(s)
- Dhruba Banerjee
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
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Barik D, Bag BC, Ray DS. Numerical simulation of transmission coefficient using c-number Langevin equation. J Chem Phys 2003. [DOI: 10.1063/1.1628227] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Barik D, Banik SK, Ray DS. Quantum phase-space function formulation of reactive flux theory. J Chem Phys 2003. [DOI: 10.1063/1.1579473] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Banerjee D, Banik SK, Bag BC, Ray DS. Quantum Kramers equation for energy diffusion and barrier crossing dynamics in the low-friction regime. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:051105. [PMID: 12513465 DOI: 10.1103/physreve.66.051105] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2002] [Indexed: 05/24/2023]
Abstract
Based on a true phase space probability distribution function and an ensemble averaging procedure we have recently developed [Phys. Rev. E 65, 021109 (2002)] a non-Markovian quantum Kramers equation to derive the quantum rate coefficient for barrier crossing due to thermal activation and tunneling in the intermediate to strong friction regime. We complement and extend this approach to weak friction regime to derive quantum Kramers equation in energy space and the rate of decay from a metastable well. The theory is valid for arbitrary temperature and noise correlation. We show that depending on the nature of the potential there may be a net reduction of the total quantum rate below its corresponding classical value, which is in conformity with earlier observation. The method is independent of path integral approaches and takes care of quantum effects to all orders.
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Affiliation(s)
- Dhruba Banerjee
- Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India
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Hai W, Lee C, Chong G, Shi L. Chaotic probability density in two periodically driven and weakly coupled Bose-Einstein condensates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026202. [PMID: 12241262 DOI: 10.1103/physreve.66.026202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2001] [Indexed: 05/23/2023]
Abstract
Using the idea of the macroscopic quantum wave function and the definition of the classical chaos, we analytically reveal that the probability density of two periodically driven and weakly coupled Bose-Einstein condensates is deterministic but not predictable. Numerical calculation for the time evolutions of the chaotic probability density demonstrates the analytical result and exhibits the nonphysical implosions and ultimate unboundedness. A method for controlling the implosions and unboundedness is proposed through adjustment of the initial conditions that leads the probability density to periodically oscillate.
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Affiliation(s)
- Wenhua Hai
- Department of Physics, Hunan Normal University, Changsha 410081, China.
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Banik SK, Bag BC, Ray DS. Generalized quantum Fokker-Planck, diffusion, and Smoluchowski equations with true probability distribution functions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:051106. [PMID: 12059528 DOI: 10.1103/physreve.65.051106] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/17/2001] [Revised: 02/20/2002] [Indexed: 05/23/2023]
Abstract
Traditionally, quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasiprobability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using true probability distribution functions is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their coordinates and momenta, we derive a generalized quantum Langevin equation in c numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion, and Smoluchowski equations are the exact quantum analogs of their classical counterparts. The present work is independent of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (the Smoluchowski equation being considered in the overdamped limit).
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Affiliation(s)
- Suman Kumar Banik
- Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India
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Banerjee D, Bag BC, Banik SK, Ray DS. Approach to quantum Kramers' equation and barrier crossing dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:021109. [PMID: 11863505 DOI: 10.1103/physreve.65.021109] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2001] [Indexed: 05/23/2023]
Abstract
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.
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Affiliation(s)
- Dhruba Banerjee
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
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Sundaram B, Zaslavsky GM. Wave analysis of ray chaos in underwater acoustics. CHAOS (WOODBURY, N.Y.) 1999; 9:483-492. [PMID: 12779844 DOI: 10.1063/1.166421] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The dispersion of a wave packet in an acoustic medium is considered in the paraxial wave approximation, where the effective potential, due to variation of the speed of propagation, varies both with depth and propagation distance. The analysis of the resulting parabolic equation, similar to the Schrodinger equation, clearly demonstrates the role of ray chaos in enhancing the dispersion of the initial packet. However, wave coherence effects are also seen that suppress the effects of the ray chaos in a manner analogous to the effects of quantum chaos. (c) 1999 American Institute of Physics.
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Affiliation(s)
- Bala Sundaram
- Department of Mathematics, CSI-CUNY, Staten Island, New York 10314
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