Dissipative quantum mechanics beyond the Bloch-Redfield approximation: A consistent weak-coupling expansion of the Ohmic spin boson model at arbitrary bias

Abstract

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.