Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker-Planck equation and a quantum master equation from a system-bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.
The nonlinear optical response of liquids subjected to a series of N femtosecond laser pulses is calculated using a multimode harmonic model for nuclear motions, with nonlinear coupling to the radiation field through the coordinate dependence of the electronic polarizability. Using electronically off-resonant optical fields, this multidimensional spectroscopy is shown to provide direct information regarding the homogeneous or the inhomogeneous nature of the spectral density obtained from optical birefringence measurements. Complementary information can be obtained using infrared pulses where the multiple time correlation functions of the nuclear dipole moment (rather than the electronic polarizability) are being probed.
Reduced equations of motion for a two-level system strongly coupled to a harmonic oscillators bath are constructed by extending the hierarchy of equations introduced by Tanimura and Kubo [J. Phys. Soc. Jpn. 58 (1989) 101]. The set of equations treats the bath in a nonperturbative manner and is applicable to a low-temperature system with taking into account the correlation time of noise. By numerically calculating linear absorption spectra for different temperatures, we demonstrate that the present theory is not afflicted with the dynamical positivity problem that occurs at low temperatures without the rotating wave approximation. Remarkable changes are found in the spectra when the temperature is lower than the resonant energy of the two-level system. KEYWORDS: quantum dissipative system, Gaussian-Markovian noise, nonperturbative treatment, low temperature, hierarchy of equations DOI: 10.1143/JPSJ.74.3131Quantum systems in a dissipative environment have been a subject of great interest for many years.1,2) There are several approaches to deal with such systems, 3-7) but the most commonly used approach is based on the reduced equations of motion, which are obtained by tracing over the heat-bath degrees of freedom under the rotating wave approximation (RWA) or high-temperature approximation. The perturbative approximation together with the factorization condition is also commonly employed. These approximations, however, limit the applicability of the equations.8) Due to the advent of experimental technology, we can now test a system under ultimate conditions, i.e., at very low temperatures, in a very short time scale with extreme accuracy. It is therefore crucial to establish a reliable theory that can accurately treat the effects of dissipation under such extreme conditions. In this letter, we show that we can remove the limitations by extending the hierarchy treatments in the equations of motion developed by Tanimura and Kubo.
9)We consider a spin-Boson system modeled by a two-level system (TLS) coupled to a harmonic oscillators bath. The total Hamiltonian is expressed as 1)
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