This thesis reports our recent progress in the study of quantum dissipation theories and their applications in chemical dynamics and nonlinear spectroscopies in condensed phases. This work is supported by National Science Fund for Distinguished Young Scholars\footnote{Control and Detection of Molecular Dynamics, No 29825502,Yijing YAN, 1999-2001} and National Natural Science foundation of China\footnote{Theory of Coherent Control of Molecular Dynamics and Bond-Selective Chemistry, No 29892161, Qingshi ZHU,1996-2000} and Research Grants Council of Hong Kong Government.\footnote{Many-body Interaction in Chemical Dynamics, HKUST680/96P} The thesis consists of two parts. In the first part, we develop the theory of dissipative dynamics in condensed phases, together with some applications in chemical dynamics problems. In the second part, the quantum dissipative dynamical equation is further combined with the spectroscopic formalism for nonlinear optical measurements in condensed phases. Together with the development of high efficient evaluation method, we carry out calculations of femtosecond nonlinear spectroscopies of a model molecule in condensed phases.
In the theoretical study of control of molecular dynamics by laser, the first problem is how to calculate the propagation of the system that couples to an external field. For low-dimensional problems, one can solve the \Sch equation for the time-dependent wavefunction of the complete system directly. Wavefunction-based methods scales exponentially with the number of degrees of freedom and hence rapidly becomes intractable even for medium-sized systems or for condensed phase dynamics. However, for a such large system, one is usually interested in a small primary part called as ``system'', of which the detailed dynamics is needed only. The remaining part acts as the surrounding or ``bath'' that is responsible for the irreversible dissipation processes in the system. We shall therefore use the reduced density operator to describe the dynamics of the primary system in the presence of dissipative bath. Part I of this thesis is to develop theories of quantum dissipation, together with some typical applications such as proton transfer in an organic molecule and quantum stochastic resonance in a spin system. This part includes the following four chapters.
In Chapter 1, we outline some theoretical and numerical tools used in the reduced description of quantum dynamics. We summarize some concepts and algebra in the Liouville space reduced density operator dynamics, and their relations, if any, to the more familiar Hilbert space wavefunction dynamics. Numerical algorithm for the Liouville space propagators used in this thesis will also be presented.
In Chapter 2, we use an algebraic approach to revisit and further bridge between two most commonly used quantum dissipation theories, the Bloch-Redfield (BR) theory and a class of Fokker-Planck (FP) equations. The nature of the common approximation scheme involving in these two theories is analyzed in detail. In principle the compact algebraic form of BR quantum dissipation equation is equivalent to the traditional BR theory. However, by introducing of the Brownian like dissipative mode, the new BR formulation avoids the tedious superoperator (tensor) algorithm in the conventional BR theory, and therefore, greatly enhance the computional efficiency. Proposed is also a generalized Fokker-Planck equation that preserves the general positivity of the reduced density operator, and meanwhile also satisfies the detailed balance relation up the second order moments in phase space. Both $T_1$-relaxation and pure-$T_2$ dephasing are considered, and their temperature dependences are shown to be very different. Finally, we present some numerical results of dynamics on B surface of I$_2$ in dissipative media.
In Chapter 3, a simple version of the FP equation developed earlier is implemented numerically to study the laser-driven intramolecular hydrogen transfer reaction dynamics in thioacetylacetone molecule. Different forms of external pulsed driving fields are exploited and their ability to compete with concurring relaxation processes is investigated. Energy relaxation and pure dephasing were shown to have rather different influences on the reaction yield.
In Chapter 4, we study a nonlinear phenomenon in the dynamics of two-level systems under the influence of both dissipation and periodic driving. The rate-matching condition for quantum stochastic resonance despite its different appearance is found to be physically the same as that in the classical case. Analyzed are also the Rabi-resonance and its implication to quantum stochastic resonance. We demonstrate that no matter how weak the driving field is, transport could involve about 70\% population in the vicinities of the third-harmonic as well as the fundamental-harmonic Rabi-resonance. Recovered is also an adiabatic passage condition in which the transport carries a nearly 100\% population in the low frequency and strong driving limit.
In Part II of the thesis, we make use of the quantum dissipation theory developed in the first part (more precisely the Chapter 2) to develope the nonlinear spectroscopic formulation in condensed phases. The new nonlinear spectroscopy formalism is more concision and numerically more efficient than the conventional one. We further propose an efficient numerical method for the evaluation of the key quantity --- field-dressed response function in the nonlinear spectroscopic signals. This part includes three chapters as follows.
In Chapter 5, we consider the transient probe absorption spectroscopies in condensed phases. We first extend the generalized quantum FP equation developed in Chapter 2 for single-surface molecular systems to the two-surface cases involved in the optical measurements. Included are the $T_1$-vibrational energy relaxation and the pure-$T_2$ dephasing of both electronic and nuclear degrees of freedom in two-surface molecular systems in nonlinear optical measurements. The resulting quantum dissipation theory is also of the dynamical semigroup form preserving the positivity of reduced density operator. It also satisfies the detailed-balance relation up to the second-order moments in phase-space at any temperatures. The quantum dissipation dynamics is further combined with spectroscopic formalisms for the efficient evaluation of the pump-probe absorption of two-surface molecules in condensed phases. Proposed are two approaches, polarization formalism and the field-dressed response function formalism, to the calculation of nonlinear spectroscopies in the weak probe detection regime. Numerical demonstrations are made for the integrated transient and the dispersed transient signals of a model molecule in dissipative media.
In Chapter 6, we investigate the theory of time-frequency resolved fluoresence spectropies in condensed phases. As we did in Chapter 5, we propose two approaches, polarization formalism and the field-dressed response function formalism for it. Numerical calculations of fluorescences, together with their comparison with transiant probe signals, are carried out for the same molecule in the above chapter.
In Chapter 7, we present in detail a highly efficient numerical method to evaluate the key quantity in our spectroscopic theory --- field-dressed response function. Our newly developed numerical method is based on a mixed Heisenberg/\Sch picture of the field-dressed response function, which reduced the two-dimensional time-grid problem almost to two one-dimensional time-grid problems.
Finally, we summarize the main contributions and results of the thesis. The ultimate goal of this research is to coherent control of dynamics in important and practice system in condensed phase or complex large molecular. The progress made in this thesis will be very useful for our future research.