Quantum dissipation theory

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

As the foundation of quantum statistical mechanics, the theory of open quantum systems has remained an active topic of research since about the middle of the last century [1-40]. Its development has involved scientists working in fields as diversified as nuclear magnetic resonance, quantum optics and nonlinear spectroscopy, solid-state physics, material science, chemical physics, biophysics, and quantum information. The key quantity in quantum dissipation theory (QDT) is the reduced system density operator, defined formally as the partial trace of the total composite density operator over the stochastic surroundings (bath) degrees of freedom. 

NON-MARKOVIAN QUANTUM DISSIPATION THEORY then assumes [cf. Eq. (4.1)]... 

Baer R and Kosloff R 1997 Quantum dissipative dynamics of adsorbates near metal surfaces a surrogate Hamiltonian theory applied to hydrogen on nickel J. Chem. Rhys. 106 8862... 

Voth, G. A., Chandler, D., MiUer, W. H. (1989). Rigorous formulation of quantum transition state theory and its dynamical corrections. J. Chem. Phys. 91,7749-7760. Weiss, U. (1993). Quantum Dissipative Systems, World Scientific, Singapore. 

ABSTRACT We present a dynamical scheme for biological systems. We use methods and techniques of quantum field theory since our analysis is at a microscopic molecular level. Davydov solitons on biomolecular chains and coherent electric dipole waves are described as collective dynamical modes. Electric polarization waves predicted by Frohlich are identified with the Goldstone massless modes of the theory with spontaneous breakdown of the dipole-rotational symmetry. Self-organization, dissipativity, and stability of biological systems appear as observable manifestations of the microscopic quantum dynamics. 

Decoherence is an essential concept appearing in a system in which a quantum subsystem contacts classical subsystem(s) in one way or another. As is widely recognized, the SET cannot describe this dynamics since there is no mechanism in it to switch off the electronic coherence along the nuclear path. The decoherence problem is critically important not only in our nonadiabatic dynamics but in other contemporary science such as spin-Boson dynamics in quantum computation theory and more extensively a quantum theory in open (dissipative) systems [147]. The decoherence problem is also critical to chaos induced by nonadiabatic djmamics [136, 137,182, 453, 454]. Therefore, in the rest of this section, we pay deeper attention to the aspect of the effect of electronic state decoherence strongly coupled with the relevant nuclear motion. A review about the notion of decoherence related to quantum mechanical measmement theory is found in the papers by Rossky et al. [53]. 

The key quantity in quantum dissipative dynamics is the reduced system density operator, ps(t) = trBPT(0> Ihe bath-subspace trace over the total composite density operator. It is worth mentioning here that the harmonic bath described above assumes rather Gaussian statistics for thermal bath influence. Realistic anharmonic environments usually do obey Gaussian statistics in the thermodynamic mean field limit. For general treatment of nonperturbative and non-Markovian quantum dissipation systems, HEOM formalism has now emerged as a standard theory. It is discussed in the next section. 

A major focus of the theory of quantum dissipation in a spin-bath is the conspicuous thermal behavior of the reservoir. Our analysis clearly shows that, at temperatures close to zero, a spin-bath behaves almost in the same way as a bosonic bath, implying a universality in the nature of bath as TWO. At higher temperatures (below saturation temperature), the system-bath coupling tends to diminish, which is reflected in the emergence of coherence in the dynamics, and the behavior of a spin-bath differs significantly from that of a bosonic bath. In what follows, we consider two specific examples to illustrate these aspects. 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

MSN.52.1. Prigogine, Quantum theory of Dissipative Systems, Nobel Symposium 5, S. Claesson, ed.. Interscience, New York, 1967, pp. 99-129. 

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