Spin-boson dynamics beyond conventional perturbation theories

A generalized approximation scheme is proposed to describe the dynamics of the spin-boson problem. Being nonperturbative in the coupling strength nor in the tunneling frequency, it gives reliable results over a wide regime of temperatures and coupling strength to the thermal environment for a large class of bath spectral densities. We use a path-integral approach and start from the exact solution for the two-level system population difference in the form of a generalized master equation (GME). Then, we approximate interblip and blip-sojourn interactions up to linear order, while retaining all intrablip correlations to find the kernels entering the GME in analytical form. Our approximation scheme, which we call weakly interacting blip approximation, fully agrees with conventional perturbative approximations in the tunneling matrix element (noninteracting-blip approximation) or in the system-bath coupling strength in the proper parameter regime.

Figure 1

(Color online) Generic path with $2n=6$ transitions at flip times $t_1,t_2,\dots ,t_{2n}$. The system is in an off-diagonal state (blip) of the reduced density matrix (RDM) in the time intervals $t_{2j}-t_{2j-1}$ (blue online) and in a diagonal state (sojourn) during the intervals $t_{2j+1}-t_{2j}$. Due to the initial preparation, the initial sojourn obeys the constraint ${\eta }_0=+1$.

Figure 2

(Color online) Bath-induced nonlocal in time correlations among tunneling transitions. The interactions $S_{2j,2j-1}$, $R_{2j,2j-1}$, and $Y_{j,j-1}$ (intradipole and blip-preceding-sojourn interactions) which appear in the influence phase ${\Phi }_{\mathrm{intra},\mathrm{bps}}$ [cf. Eq. (12)] are symbolized by the wiggled lines (blue and magenta online, respectively). The double-dashed lines denote the interdipole interactions ${\Lambda }_{j,k}$, while the bold-dotted lines are the remaining blip-sojourn interactions $X_{j,k}$ contained in the influence phase ${\Phi }_{\mathrm{inter}}$ [cf. Eq. (16)].

Figure 3

(Color online) Generic path and bath-induced correlations retained (a) in the NIBA and (b) in the extended NIBA. In both approximations, the interblip and blip-sojourn correlations ${\Lambda }_{j,k}$ and $X_{j,k}$, respectively, which appear in the influence phase ${\Phi }_{\mathrm{inter}}$ [see Eq. (12)], are neglected. Within the extended NIBA, the blip-preceding-sojourn interaction $Y_{j,j-1}$ (magenta online), which contributes to the influence phase ${\Phi }_{\mathrm{bps}}$ [see Eq. (14)] and is being neglected in the NIBA, is retained. The first sojourn is treated differently, according to the initial preparation (here, we choose the preparation of class B).

Figure 4

(Color online) Irreducible kernel $K^{\left(s\right)}\left(t\right)$ (a) in the NIBA and (b) in the extended-NIBA (blue online). The extended-NIBA kernel is obtained from the NIBA one by replacing the function $R\left(t\right)$ with $\tilde{R}\left(t\right)\equiv R\left(t\right)-t\dot{R}\left(t\right)$. Hence, the blip becomes “dressed.”

Figure 5

(Color online) A comparison between the standard NIBA and the extended NIBA for an Ohmic bath $(s=1)$ is shown. The parameters are $T=0.1$, $\alpha =0.1$, and $\epsilon =1$ (in units of the tunneling frequency $\Delta$). One can see that in the chosen intermediate parameter regime, the extended NIBA coincides with the conventional one at short times. However, it predicts a different asymptote from the NIBA one (see inset). Hence, at moderate damping and temperatures, the blip-preceding-sojourn correlations retained in the extended NIBA become important.

Figure 6

(Color online) Generic paths and bath-induced correlations retained in the WIBA. The linearized influence functional ${\Phi }_{\mathrm{inter}}^{\left(n\right)}$ yields (a) one single blip-blip correlation or (b) blip-sojourn interaction for each path.

Figure 7

(Color online) Irreducible symmetric WIBA kernel. Notice that the lowest order reproduces the extended-NIBA case [see Fig. 4b]. The inner bubble represents the contribution coming from the sum of all orders in ${\Delta }^2$, which coincides with the symmetric part of the solution of the master equation for $P\left(t\right)$ in the extended-NIBA case.

Figure 8

(Color online) Time evolution of $P\left(t\right)$ for an Ohmic symmetric two-state system. The WIBA (full lines) coincides with the NIBA (dashed lines) and the extended NIBA (dotted lines) over the whole range of parameters. Here, $\alpha =0.1$ and $T=0.1$ (in units of $\Delta$) have been chosen. For such a choice, the WCA (dot-dashed lines) slightly deviates from all other predictions.

Figure 9

(Color online) Time evolution of $P\left(t\right)$ for Ohmic damping and finite bias. The chosen parameters are $\alpha =0.01$, $T=0.1$, and $\epsilon =1$ (in units of $\Delta$). The WIBA (full lines) coincides with the WCA (dot-dashed lines) for low temperatures and small coupling, while the NIBA (dashed lines) and the extended NIBA (dotted lines) predict an unphysical asymptote, for an asymmetric system.

Figure 10

(Color online) Time evolution of $P\left(t\right)$ for $\alpha =0.1$ and $T=0.1$. The WIBA (full lines) exhibits more coherence than the QUAPI (lines with bullets), with its predictions being closer to the QUAPI than the WCA (dot-dashed lines). The extended NIBA (dotted lines) is also getting closer to the QUAPI predictions, the NIBA (dashed lines) still predicting a strong localization in the left well.

Figure 11

(Color online) Time evolution of $P\left(t\right)$ for $\alpha =0.25$ and $T=0.01$. The WCA (dot-dashed lines) completely fails in describing the short-time dynamics and also predicts a wrong asymptote. The WIBA (full lines) as well as the extended NIBA (dotted lines) very well agree with the QUAPI predictions (lines with bullets), despite some spurious oscillations still contained in the theory. The disagreement with the NIBA is striking, already at this intermediate damping, although the NIBA correctly describes the short-time dynamics.

Figure 12

(Color online) Time evolution of $P\left(t\right)$ for $\alpha =0.25$ and $T=0.1$. In this regime the extended NIBA (dotted lines) as well as the WIBA (full lines) are almost indistinguishable from QUAPI (lines with bullets), whereas both WCA (dot-dashed lines) and NIBA (dashed lines) fail in describing the dynamics, the latter being, however, able to reproduce the short-time dynamics.

Figure 13

(Color online) Time evolution of $P\left(t\right)$ for ${\delta }_s=0.01$ and ${\omega }_c=200$ ($T=0.1$ and $\epsilon =1$). Here, one sees no difference between WCA (dot-dashed lines) and WIBA (full lines), showing that the chosen parameters lie in an effective weak-coupling regime.

Figure 14

(Color online) Time evolution of $P\left(t\right)$ for ${\delta }_s=0.01$ and ${\omega }_c=1$ ($T=0.1$ and $\epsilon =1$). Here, we notice discrepancies between WIBA (full lines) and QUAPI (full lines with bullets), since the extended-NIBA predictions (dotted lines) also account for too many oscillations. Nevertheless, it gives much better results than NIBA (dashed lines), predicting unphysical values for $P\left(t\right)$. Finally, the WCA (dot-dashed lines) is so far the best model in this regime, although it lies apart as well from the QUAPI predictions, since the perturbative approach for such parameters begins to fail.