In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho analisamos as soluÃÃes para a equaÃÃo de movimento para os osciladores de Lane-Emden, onde a massa à dada por m(t)=t^α, onde α>0. Os osciladores de Lane-Emden sÃo osciladores harmÃnicos amortecidos, onde o fator de amortecimento depende do tempo, γ(t)=α/t. Obtivemos as expressÃes analÃticas de q(t), dq(t)/dt, and p(t)=m(t)(dq(t)/dt) para α=2 e α=4. Discutimos as diferenÃas entre as expressÃes da hamiltoniana e da energia para sistemas dependentes do tempo. TambÃm, comparamos nossos resultados com aqueles do oscilador de Caldirola-Kanai. Usamos o mÃtodo dos invariantes quÃnticos e uma transformaÃÃo unitÃria para obter a funÃÃo de onda exata de SchrÃdinger, ψn (q,t), e calcular para n=0 a entropia conjunta (entropia de Leipnik) dependente do tempo e as informaÃÃes Fisher para posiÃÃo (Fq) e para o momento (Fp) para duas classes de osciladores harmÃnicos quÃnticos amortecidos. Observamos que a entropia de Leipnik nÃo varia no tempo para o oscilador Caldirola-Kanai, enquanto diminui e tende a um valor constante (ln(e/2)) para tempos assintÃticos para o oscilador de Lane-Emden. Isto à devido ao fato de que, para este Ãltimo, o fator de amortecimento diminui à medida que o tempo aumenta. Os resultados mostram que a dependÃncia do tempo da entropia de Leipnik à bastante complexa e nÃo obedece a uma tendÃncia geral de aumento monotonicamente com o tempo e que Fq aumenta enquanto Fp diminui com o aumento do tempo. AlÃm disso, FqFp aumenta e tende a um valor constante (4/ℏ^2 ) no limite em que t->∞. NÃs comparamos os resultados com os do bem conhecido oscilador de Caldirola-Kanai. / In this work we analyze the solutions of the equations of motions for two Lane-Emden-type Caldirola-Kanai oscillators. For these oscillators the mass varies as m(t)=t^α, where α>0.We obtain the analytical expression of q(t), dq(t)/dt, and p(t)=m(t)(dq(t)/dt) for α=2 and α=4. These are damped-like harmonic oscillators with a time-dependent damping factor given by γ(t)=α/t. We discuss the differences between the expressions for the hamiltonian and the mechanical energy for time-dependent systems. We also compared our results to those of the well-known Caldirola-Kanai oscillators. We use the quantum invariant method and a unitary transformation to obtain the exact SchrÃdinger wave function, ψn (q,t), and calculate for n=0 the time-dependent joint entropy (LeipnikÂs entropy) and the position (Fq) and momentum (Fp) Fisher information for two classes of quantum damped harmonic oscillators. We observe that the joint entropy does not vary in time for the Caldirola-Kanai oscillator, while it decreases and tends to a constant value (ln(e/2)) for asymptotic times for the Lane-Emden ones. This is due to the fact that for the latter, the damping factor decreases as time increases. The results show that the time dependence of the joint entropy is quite complex and does not obey a general trend of monotonously increase with time and that F_q increases while F_p decreases with increasing time. Also, FqFp increases and tends to a constant value (4/ℏ^2 ) in the limit t->∞.We compare the results with those of the well-known Caldirola-Kanai oscillator.

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.