The population decay due to a small opening in an otherwise closed cavity supporting chaotic classical dynamics displays a quantum correction on top of the classical exponential form, a pure manifestation of quantum coherence that acquires a universal form and can be explained by path interference. Being coherent, such enhancement is prone to decoherence effects due to the coupling of the system to an external environment. We study this interplay between incoherent and coherent quantum corrections to decay by evaluating, within a Caldeira-Leggett scenario, off-diagonal contributions to the decoherence functional coming from pairs of correlated classical paths in the time regime where dissipative effects are neglected and decoherence does not affect the classical dynamics, but quantum interference must be accounted for. We find that the competing effects of interference and decoherence lead to a universal nonmonotonic form for the survival probability depending only on the coupling strength and macroscopic parameters of the cavity.
We demonstrate that in Hamiltonian many-body systems at equilibrium, any kind of time dependent correlation function c(t) can always be expanded in a series of (complex) exponential functions of time when its Laplace transform C̃(z) has single poles. The characteristic frequencies can be identified as the eigenfrequencies of the correlation. This is done without introducing the concepts of fluctuating forces and memory functions, due to Mori and Zwanzig and extensively used in the literature in the last decades. Our method is based on a different projection technique in the Hilbert space S of the system and shows that appropriate approximations of the exponential series are related to the contraction of S to a finite, usually small, number of dimensions. The time dependence of correlation functions is always described in detail by a multiple-exponential functionality also at long times. This result is therefore also valid for correlation functions of many-body Hamiltonian systems for which a power-law dependence, observed in restricted time ranges and predicted to be the asymptotic one, can be considered at most as a useful approximate modeling of long-time behavior.
We study the time evolution of the reduced density matrix for the Ohmic spin boson model out of an uncorrelated but otherwise arbitrary initial state. We consider arbitrary bias $\ensuremath{\epsilon}$ and tunneling $\mathrm{\ensuremath{\Delta}}$ at zero temperature for a weak coupling $\ensuremath{\alpha}$ to the bosonic bath. Using the real-time renormalization group method, we present a consistent weak-coupling expansion one order beyond the Bloch-Redfield approximation within a renormalized perturbation theory with analytical results covering the whole crossover regime from small times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\ll}1$ to large times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\gg}1$, where $\mathrm{\ensuremath{\Omega}}=\sqrt{{\ensuremath{\epsilon}}^{2}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}^{2}}$ denotes the Rabi frequency in terms of the renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$. In addition, for exponentially small or large times, we perform a nonperturbative resummation of all logarithmic terms. We show that standard Born approximation schemes calculating the effective Liouvillian of the kinetic equation up to first order in $\ensuremath{\alpha}$ are not sufficient to account for various important corrections one order beyond the Bloch-Redfield solution. (1) The resummation of all secular terms $\ensuremath{\sim}{(\mathrm{\ensuremath{\Gamma}}t)}^{n}$ is necessary to obtain the correct exponential decay of all terms of the time evolution with decay rate $\mathrm{\ensuremath{\Gamma}}$ or $\mathrm{\ensuremath{\Gamma}}/2$, together with the correct pre-exponential functions. (2) The resummation of all logarithmic terms at high and low energies leads to a renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$ and to pre-exponential functions of logarithmic and power-law form. (3) The fact that two eigenvalues of
L(E) are close to each other by $O(\mathrm{\ensuremath{\Gamma}})$ requires degenerate perturbation theory for times $\mathrm{\ensuremath{\Gamma}}t\ensuremath{\sim}O(1)$, where certain terms of the Liouvillian in $O({\ensuremath{\alpha}}^{2})$ are needed to calculate the stationary state and the time evolution of the nonoscillating purely decaying modes up to $O(\ensuremath{\alpha})$. In contrast to the zero-bias case, we find two further interesting results for the time dynamics of the oscillating modes. (4) The terms of the pre-exponential functions with a strong time dependence show a leading long-time tail $\ensuremath{\sim}\ensuremath{\alpha}/(\mathrm{\ensuremath{\Omega}}t)$, besides other subleading terms $\ensuremath{\sim}\ensuremath{\alpha}/{(\mathrm{\ensuremath{\Omega}}t)}^{2}$ well-known from the zero-bias case. (5) The terms of the pre-exponential functions with a weak (logarithmic) time dependence vary according to a power law ${(\frac{1}{\mathrm{\ensuremath{\Omega}}t})}^{2\ensuremath{\alpha}\frac{{\ensuremath{\epsilon}}^{2}}{{\mathrm{\ensuremath{\Omega}}}^{2}}}$ for exponentially large times. The power-law exponent depends on the bias and has to be contrasted to the one at exponentially small times where it crosses over to the bias-independent result $2\ensuremath{\alpha}$. We discuss that the complexity to calculate one order beyond Bloch-Redfield approximation is rather generic and applies also to other models of dissipative quantum mechanics.