Nonperturbative Resummation of Divergent Time-Local Generators

Abstract

Time-local generators of open quantum systems are generically divergent at long times, even though the reduced dynamics remains regular. We construct, by analytic continuation, nonperturbative dynamical maps consistent with these generators. For the weak-coupling unbiased spin--boson model, this construction yields an explicit dynamical map that nonperturbatively resums the TCL generator and exposes how the divergences signal the approach to a singular time at which the reduced dynamics becomes noninvertible. The reconstructed map is validated against TEMPO simulations at short times and the exactly solvable rotating-wave model at all times. In the full spin--boson model, the same continuum mechanism produces both an early-time anisotropy, with a measurable phase shift that provides a signature of the environmental correlation and the pointer direction, and a late-time singularity at which the reduced dynamics becomes noninvertible. By contrast, in the rotating-wave model the map approaches this point without reaching it and remains invertible at all times. These results establish a nonperturbative framework for reconstructing reduced dynamics from divergent time-local generators, diagnosing the onset of noninvertibility, and identifying experimentally accessible early-time signatures of environment-induced anisotropy.

Figures (7)

• Figure 1: Finite-time loss of invertibility of the dynamical map induced by system--environment interactions. Markovian relaxation, nonsecular coherence mixing, and the Khalfin tail jointly drive the smallest singular value toward zero. The repeated crack-like features reflect successive destructive interference between the exponential contribution and the bath correlation function, progressively suppressing coherence. At the final crack, a singular value vanishes, marking noninvertibility. Beyond this point, resurrection of the initial superposition from the reduced dynamics is no longer possible.
• Figure 2: Representative long-time dynamics of the coherence amplitude $|f(t)|$. (a,b) Log--linear and log--log plots showing the crossover from exponential (Markovian) to power-law (non-Markovian) decay. Near the phase lock-in time $t_P$, interference produces oscillations. Black: Ohmic bath ($s=1$, $\lambda^2=0.025$); red: sub-Ohmic bath ($s=1/3$, $\lambda^2=0.01$). In both cases $\omega_c=4$, $\Delta=1$.
• Figure 3: Coherence magnitude (a) and phase (b) near $t_P$ in the RWA model. Constructive interference produces bursts, while destructive interference leads to near-extinction. The phase undergoes a rapid rotation approaching a $\pi$ slip, consistent with the map remaining invertible. Red dotted: exact solution; black solid: two-component model. Parameters: $s=1/3$, $\lambda^2=0.01$, $\omega_c=4$, $\Delta=1$.
• Figure 4: Numerical validation of the continuation procedure. (a) Early-time generator element $L_{11,11}(t)$: resummed TCL (blue) agrees with the exact result (red). (b) Near $t_P$, destructive interference suppresses $f(t)$ and produces a sharp generator spike. The generator reconstructed from the disentangled map (black) reproduces the exact result (red) before, at, and after the spike. Parameters: $s=1/3$, $\lambda^2=0.01$, $\omega_c=4$, $\Delta=1$, $T=0$.
• Figure 5: Real part of the nonsecular transfer amplitude $\Phi_{21,12}(t)$. The disentangled map (red), obtained by numerically solving the reconstruction equation \ref{['Eq:Cdot']}, is compared with TEMPO simulations (black dashed) and the Bloch--Redfield prediction (blue). The disentangled result agrees closely with TEMPO and shows that the anisotropy is encoded as a phase shift of oscillations along the pointer direction. Bloch--Redfield, which retains only the pole contribution, misses the branch-cut part of the anisotropy and instead predicts a rotated oscillation axis with no phase shift. The figure therefore provides an early-time, experimentally accessible signature of the continuum contribution that is absent in pole-only treatments. Parameters: $\Delta=1$, $\omega_c=4$, $s=1$, $\lambda^2=0.025$, $\theta=-0.625$, $T=0$.