Cross-time memory kernels in macroscopic quantum tunneling non-Markovian signatures in Josephson junctions

We propose a population-level description of cross-time memory effects in macroscopic quantum tunneling of a current-biased Josephson junction subject to a dynamically suppressed interval [t₁,t₂] . Starting from the Caldeira–Leggett influence functional, we identify the controlled approximations under which bath-mediated correlations between instantons can be coarse-grained into a Volterra equation for the survival probability S ( t ). The resulting cross-time kernel couples pre-window times t'<t₁ to post-window times t>t₂ with an amplitude proportional to e-λ cΔ tbMₚre , where Δ tb=t₂-t₁ and Mₚre is a history-weighted pre-window survival functional. The switching distribution P(t)=-Ṡ(t) then contains a Markovian clustering peak at t₂ and a non-Markovian post-window tail. We clarify that the kernel is a controlled phenomenological reduction, not an exact microscopic identity, and we state explicit positivity, validity, and thermal-crossover conditions. Numerical diagnostics are reformulated in terms of S ( t ) and show how the full switching distribution changes with barrier duration and pre-barrier history. The analysis indicates that ordinary resistively shunted junctions have too short a memory time for nanosecond barriers, whereas engineered high- Q resonator environments provide a plausible route to testing the predicted peak-plus-tail signatures.