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Abstract
We review recent progress in open quantum system approach to the description of quarkonium in the quark-gluon plasma. A particular emphasis is put on the Lindblad equations for quarkonium and its numerical simulations.
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Lee TY, Lu CY, Chou CC. Moving Boundary Truncated Grid Method: Application to the Time Evolution of Distribution Functions in Phase Space. J Phys Chem A 2021; 125:476-491. [PMID: 33372780 DOI: 10.1021/acs.jpca.0c09525] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The moving boundary truncated grid (TG) method, previously developed to integrate the time-dependent Schrödinger equation and the imaginary time Schrödinger equation, is extended to the time evolution of distribution functions in phase space. A variable number of phase space grid points in the Eulerian representation are used to integrate the equation of motion for the distribution function, and the boundaries of the TG are adaptively determined as the distribution function evolves in time. Appropriate grid points are activated and deactivated for propagation of the distribution function, and no advance information concerning the dynamics in phase space is required. The TG method is used to integrate the equations of motion for phase space distribution functions, including the Klein-Kramers, Wigner-Moyal, and modified Caldeira-Leggett equations. Even though the initial distribution function is nonnegative, the solutions to the Wigner-Moyal and modified Caldeira-Leggett equations may develop negative basins in phase space originating from interference effects. Trajectory-based methods for propagation of the distribution function do not permit the formation of negative regions. However, the TG method can correctly capture the negative basins. Comparisons between the computational results obtained from the full grid and TG calculations demonstrate that the TG method not only significantly reduces the computational effort but also permits accurate propagation of various distribution functions in phase space.
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Affiliation(s)
- Tsung-Yen Lee
- Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Chun-Yaung Lu
- Texas Advanced Computing Center, The University of Texas at Austin, Austin, Texas 78758, United States
| | - Chia-Chun Chou
- Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan
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3
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Zhdanov DV, Bondar DI, Seideman T. No Thermalization without Correlations. PHYSICAL REVIEW LETTERS 2017; 119:170402. [PMID: 29219440 DOI: 10.1103/physrevlett.119.170402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2016] [Indexed: 06/07/2023]
Abstract
The proof of the long-standing conjecture is presented that Markovian quantum master equations are at odds with quantum thermodynamics under conventional assumptions of fluctuation-dissipation theorems (implying a translation invariant dissipation). Specifically, except for identified systems, persistent system-bath correlations of at least one kind, spatial or temporal, are obligatory for thermalization. A systematic procedure is proposed to construct translation invariant bath models producing steady states that well approximate thermal states. A quantum optical scheme for the laboratory assessment of the developed procedure is outlined.
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Jang S. Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition. J Chem Phys 2016; 144:214102. [PMID: 27276940 DOI: 10.1063/1.4952477] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/30/2023] Open
Abstract
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.
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Affiliation(s)
- Seogjoo Jang
- Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, New York 11367, USA and Ph.D. Programs in Chemistry and Physics, and Initiative for Theoretical Sciences, Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016, USA
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5
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Bondar DI, Cabrera R, Campos A, Mukamel S, Rabitz HA. Wigner-Lindblad Equations for Quantum Friction. J Phys Chem Lett 2016; 7:1632-1637. [PMID: 27078510 DOI: 10.1021/acs.jpclett.6b00498] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Dissipative forces are ubiquitous and thus constitute an essential part of realistic physical theories. However, quantization of dissipation has remained an open challenge for nearly a century. We construct a quantum counterpart of classical friction, a velocity-dependent force acting against the direction of motion. In particular, a translationary invariant Lindblad equation is derived satisfying the appropriate dynamical relations for the coordinate and momentum (i.e., the Ehrenfest equations). Numerical simulations establish that the model approximately equilibrates. These findings significantly advance a long search for a universally valid Lindblad model of quantum friction and open opportunities for exploring novel dissipation phenomena.
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Affiliation(s)
- Denys I Bondar
- Princeton University , Princeton, New Jersey 08544, United States
| | - Renan Cabrera
- Princeton University , Princeton, New Jersey 08544, United States
| | - Andre Campos
- Princeton University , Princeton, New Jersey 08544, United States
| | - Shaul Mukamel
- University of California , Irvine, California 92697, United States
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6
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Ferialdi L. Exact Closed Master Equation for Gaussian Non-Markovian Dynamics. PHYSICAL REVIEW LETTERS 2016; 116:120402. [PMID: 27058061 DOI: 10.1103/physrevlett.116.120402] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2015] [Indexed: 06/05/2023]
Abstract
Non-Markovian master equations describe general open quantum systems when no approximation is made. We provide the exact closed master equation for the class of Gaussian, completely positive, trace preserving, non-Markovian dynamics. This very general result allows us to investigate a vast variety of physical systems. We show that the master equation for non-Markovian quantum Brownian motion is a particular case of our general result. Furthermore, we derive the master equation unraveled by a non-Markovian, dissipative stochastic Schrödinger equation, paving the way for the analysis of dissipative non-Markovian collapse models.
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Affiliation(s)
- L Ferialdi
- Mathematisches Institut, Ludwig-Maximilians Universität, Theresienstraße 39, 80333 Munich, Germany
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7
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Smirne A, Bassi A. Dissipative Continuous Spontaneous Localization (CSL) model. Sci Rep 2015; 5:12518. [PMID: 26243034 PMCID: PMC4525142 DOI: 10.1038/srep12518] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2015] [Accepted: 06/22/2015] [Indexed: 11/09/2022] Open
Abstract
Collapse models explain the absence of quantum superpositions at the macroscopic scale, while giving practically the same predictions as quantum mechanics for microscopic systems. The Continuous Spontaneous Localization (CSL) model is the most refined and studied among collapse models. A well-known problem of this model, and of similar ones, is the steady and unlimited increase of the energy induced by the collapse noise. Here we present the dissipative version of the CSL model, which guarantees a finite energy during the entire system's evolution, thus making a crucial step toward a realistic energy-conserving collapse model. This is achieved by introducing a non-linear stochastic modification of the Schrödinger equation, which represents the action of a dissipative finite-temperature collapse noise. The possibility to introduce dissipation within collapse models in a consistent way will have relevant impact on the experimental investigations of the CSL model, and therefore also on the testability of the quantum superposition principle.
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Affiliation(s)
- Andrea Smirne
- 1] Dipartimento di Fisica, Università degli Studi di Trieste, Strada Costiera 11, I-34151 Trieste, Italy [2] Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
| | - Angelo Bassi
- 1] Dipartimento di Fisica, Università degli Studi di Trieste, Strada Costiera 11, I-34151 Trieste, Italy [2] Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
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8
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Tameshtit A. Zero-point energies, the uncertainty principle, and positivity of the quantum Brownian density operator. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:042103. [PMID: 22680520 DOI: 10.1103/physreve.85.042103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2011] [Indexed: 06/01/2023]
Abstract
High-temperature and white-noise approximations are frequently invoked when deriving the quantum Brownian equation for an oscillator. Even if this white-noise approximation is avoided, it is shown that if the zero-point energies of the environment are neglected, as they often are, the resultant equation will violate not only the basic tenet of quantum mechanics that requires the density operator to be positive, but also the uncertainty principle. When the zero-point energies are included, asymptotic results describing the evolution of the oscillator are obtained that preserve positivity and, therefore, the uncertainty principle.
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Kumar J, Sinha S, Sreeram PA. Dissipative dynamics of a harmonic oscillator: a nonperturbative approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:031130. [PMID: 19905085 DOI: 10.1103/physreve.80.031130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2009] [Indexed: 05/28/2023]
Abstract
Starting from a microscopic theory, we derive a master equation for a harmonic oscillator coupled to a bath of noninteracting oscillators. We follow a nonperturbative approach, proposed earlier by us for the free Brownian particle. The diffusion constants are calculated analytically and the positivity of the master equation is shown to hold above a critical temperature. We compare the long time behavior of the average kinetic and potential energies with known thermodynamic results. In the limit of vanishing oscillator frequency of the system, we recover the results of the free Brownian particle.
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Affiliation(s)
- Jishad Kumar
- Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur 741252, India
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11
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Sinha S, Sreeram PA. Nonperturbative approach to quantum Brownian motion. Phys Rev E 2009; 79:051111. [PMID: 19518420 DOI: 10.1103/physreve.79.051111] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2008] [Revised: 03/05/2009] [Indexed: 11/07/2022]
Abstract
Starting from the Caldeira-Leggett model, we derive the equation describing the quantum Brownian motion, which has been originally proposed by Dekker purely from phenomenological basis containing extra anomalous diffusion terms. This nonperturbative approach yields explicit analytical expressions for the temperature dependence of the diffusion constants. At high temperatures, additional momentum diffusion terms are suppressed and classical Langevin equation can be recovered and at the same time positivity of the density matrix is satisfied. At low temperatures, the diffusion constants have a finite positive value. However, below a certain critical temperature, the master equation does not satisfy the positivity condition as proposed by Dekker.
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Affiliation(s)
- Subhasis Sinha
- Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, 741252, India
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12
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Grossmann F, Koch W. A finite-difference implementation of the Caldeira–Leggett master equation. J Chem Phys 2009; 130:034105. [DOI: 10.1063/1.3059006] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
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13
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Breuer HP, Vacchini B. Three-dimensional Monte Carlo simulations of the quantum linear Boltzmann equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:036706. [PMID: 17930360 DOI: 10.1103/physreve.76.036706] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2007] [Indexed: 05/25/2023]
Abstract
Recently the general form of a translation-covariant quantum Boltzmann equation has been derived, which describes the dynamics of a tracer particle in a quantum gas. We develop a stochastic wave function algorithm that enables full three-dimensional Monte Carlo simulations of this equation. The simulation method is used to study the approach to equilibrium for various scattering cross sections and to determine dynamical deviations from Gaussian statistics through an investigation of higher-order cumulants. Moreover, we examine the loss of coherence of superpositions of momentum eigenstates and determine the corresponding decoherence time scales to quantify the transition from quantum to classical behavior of the state of the test particle.
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Affiliation(s)
- Heinz-Peter Breuer
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany.
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14
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Adler SL. Normalization of collisional decoherence: squaring the delta function, and an independent cross-check. ACTA ACUST UNITED AC 2006. [DOI: 10.1088/0305-4470/39/45/015] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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15
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Hornberger K. Master equation for a quantum particle in a gas. PHYSICAL REVIEW LETTERS 2006; 97:060601. [PMID: 17026153 DOI: 10.1103/physrevlett.97.060601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2006] [Indexed: 05/12/2023]
Abstract
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
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Affiliation(s)
- Klaus Hornberger
- Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 Munich, Germany. www.klaus-hornberger.de
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16
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Barnett SM, Jeffers J, Cresser JD. From measurements to quantum friction. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2006; 18:S401-S410. [PMID: 21690742 DOI: 10.1088/0953-8984/18/16/s02] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
We present a quantum theory of friction in which interactions with the surrounding medium are described by generalized measurements of the particle's position and momentum. The theory predicts intrinsically quantum contributions to the particle's steady-state energy and to the associated diffusion in position. We discuss the physical significance of these and demonstrate their significance in ensuring a well behaved theory.
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Affiliation(s)
- Stephen M Barnett
- Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK
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17
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Vacchini B. Theory of decoherence due to scattering events and Lévy processes. PHYSICAL REVIEW LETTERS 2005; 95:230402. [PMID: 16384279 DOI: 10.1103/physrevlett.95.230402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2005] [Indexed: 05/05/2023]
Abstract
A general connection between the characteristic function of a Lévy process and loss of coherence of the statistical operator describing the center of mass degrees of freedom of a quantum system interacting through momentum transfer events with an environment is established. The relationship with microphysical models and recent experiments is considered, focusing on the recently observed transition between a dynamics described by a compound Poisson process and a Gaussian process.
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Affiliation(s)
- Bassano Vacchini
- Dipartimento di Fisica dell'Università di Milano and INFN, Sezione di Milano, Via Celoria 16, I-20133, Milan, Italy
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18
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Petruccione F, Vacchini B. Quantum description of Einstein's Brownian motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:046134. [PMID: 15903753 DOI: 10.1103/physreve.71.046134] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2004] [Indexed: 05/02/2023]
Abstract
A fully quantum treatment of Einstein's Brownian motion is given, stressing in particular the role played by the two original requirements of translational invariance and connection between dynamics of the Brownian particle and atomic nature of the medium. The former leads to a clearcut relationship with a generator of translation-covariant quantum-dynamical semi-groups recently characterized by Holevo, the latter to a formulation of the fluctuation-dissipation theorem in terms of the dynamic structure factor, a two-point correlation function introduced in seminal work by Van Hove, directly related to density fluctuations in the medium and therefore to its atomistic, discrete nature. A microphysical expression for the generally temperature dependent friction coefficient is given in terms of the dynamic structure factor and of the interaction potential describing the single collisions. A comparison with the Caldeira-Leggett model is drawn, especially in view of the requirement of translational invariance, further characterizing general structures of reduced dynamics arising in the presence of symmetry under translations.
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Affiliation(s)
- Francesco Petruccione
- School of Pure and Applied Physics, Howard College, University of KwaZulu-Natal, Durban, 4041, South Africa.
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21
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López JL. Nonlinear Ginzburg-Landau-type approach to quantum dissipation. Phys Rev E 2004; 69:026110. [PMID: 14995523 DOI: 10.1103/physreve.69.026110] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2003] [Revised: 11/04/2003] [Indexed: 11/07/2022]
Abstract
We formally derive two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium. These models mainly consist of a quantum, dissipative O(Planck 3) hydrodynamic/O(Planck 4) stochastic correction to the frictional (Caldeira-Leggett-)Schrödinger equation. The main ingredient lies in the use of the hydrodynamic/stochastic fluid model approach associated with the quantum Fokker-Planck equation and the identification of the associated pressure field. Then, Madelung transformations set the problem in the Schrödinger picture of dissipative quantum mechanics. We also describe the stationary dynamics associated with both systems.
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Affiliation(s)
- José L López
- Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.
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22
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Kimura G, Yuasa K, Imafuku K. Bifurcation phenomenon in spin relaxation. PHYSICAL REVIEW LETTERS 2002; 89:140403. [PMID: 12366033 DOI: 10.1103/physrevlett.89.140403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/03/2001] [Indexed: 05/23/2023]
Abstract
Spin relaxation in a strong-coupling regime (with respect to the spin system) is investigated in detail based on the spin-boson model in a stochastic limit. We find a bifurcation phenomenon in temperature dependence of relaxation constants, which is never observed in the weak-coupling regime. We also discuss inequalities among the relaxation constants in our model and show the well-known relation 2gamma(T)>or=gamma(L), for example, for a wider parameter region than before.
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Affiliation(s)
- Gen Kimura
- Department of Physics, Waseda University, Tokyo 169-8555, Japan.
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23
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Vacchini B. Non-Abelian linear Boltzmann equation and quantum correction to Kramers and Smoluchowski equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:027107. [PMID: 12241323 DOI: 10.1103/physreve.66.027107] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2002] [Indexed: 11/06/2022]
Abstract
A quantum linear Boltzmann equation, constructed in terms of the operator-valued dynamic structure factor of the macroscopic system the test particle is interacting with, is proposed. Due to this operator structure it is a non-Abelian linear Boltzmann equation and when expressed through the Wigner function it allows for a direct comparison with the classical one. Considering a Brownian particle, the corresponding Fokker-Planck equation is obtained in a most direct way taking the limit of small energy and momentum transfer. A typical quantum correction to the Kramers equation thus appears, describing diffusion in position and further implying a correction to Einstein's diffusion coefficient in the high temperature and friction limit in which the Smoluchowski equation emerges.
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Affiliation(s)
- Bassano Vacchini
- Dipartimento di Fisica dell'Università di Milano and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy
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24
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Jang S, Cao J, Silbey RJ. Fourth-order quantum master equation and its Markovian bath limit. J Chem Phys 2002. [DOI: 10.1063/1.1445105] [Citation(s) in RCA: 80] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
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25
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Maddox JB, Bittner ER. Quantum dissipation in unbounded systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:026143. [PMID: 11863623 DOI: 10.1103/physreve.65.026143] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2001] [Indexed: 05/23/2023]
Abstract
In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie--Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wave function of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to nonclassical equations of motion. Until very recently these schemes have been limited to conservative systems. In this paper, we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme that incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy, we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.
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Affiliation(s)
- Jeremy B Maddox
- Department of Chemistry, University of Houston, Houston, Texas 77204, USA.
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26
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27
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Vacchini B. Test particle in a quantum gas. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:066115. [PMID: 11415181 DOI: 10.1103/physreve.63.066115] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2000] [Indexed: 05/23/2023]
Abstract
A master equation with a Lindblad structure is derived, which describes the interaction of a test particle with a macroscopic system and is expressed in terms of the operator valued dynamic structure factor of the system. In the case of a free Fermi or Bose gas the result is evaluated in the Brownian limit, thus obtaining a single generator master equation for the description of quantum Brownian motion in which the correction due to quantum statistics is explicitly calculated. The friction coefficients for Boltzmann and Bose or Fermi statistics are compared.
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Affiliation(s)
- B Vacchini
- Dipartimento di Fisica dell'Università di Milano, Sezione di Milano, Via Celoria 16, I-20133, Milan, Italy.
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