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Cross-Time Memory Kernelsin Macroscopic Quantum TunnelingNon-Markovian Signatures in Josephson Junctions

Mahgoub A. Salih, Mohamed. Y. Shirgawi

Published Mar 26, 2026• 0 citations
Publisher

Abstract

Abstract When a Josephson junction is coupled to an environment with memory time $\lambda_c^{-1}$ exceeding a dynamically engineered suppression window $[t_1,t_2]$, bath-mediated correlations link pre-window and post-window tunneling events in a way that no local-in-time escape rate can reproduce. Using the Caldeira--Leggett influence-functional framework, we derive a Volterra integro-differential survival equation containing, alongside the usual Markovian and local-memory terms, an explicit \emph{cross-time} kernel $K_{\text{cross}}(t,t')$ that couples times $tt_2$ while contributions from within the suppressed interval remain exponentially small. The resulting switching-time distribution displays two separable signatures: a clustering peak at $t_2$ with jump ratio $\varepsilon^{-1}$ ($\varepsilon \ll 1$ being the in-window rate suppression), and a post-window memory tail whose decay constant equals the bath correlation rate $\lambda_c$ and whose \emph{amplitude} is set by the pre-window history functional $M_{\text{pre}}$. The amplitude also scales as $e^{-\lambda_c \Delta t_b}$ with $\Delta t_b = t_2 - t_1$, providing a direct spectroscopic handle on the bath memory time. Standard resistively shunted junctions, with $\lambda_c^{-1} \sim 10\,\text{ps}$, render the cross-time signal unobservable for nanosecond barriers; coupling the junction to a high-$Q$ microwave resonator ($Q \sim 10^5$, $\omega_r/2\pi \sim 5\,\text{GHz}$) extends the effective memory time to microseconds, bringing both signatures within reach of current superconducting circuit technology. \bigskip\noindent\textbf{Keywords:} Non-Markovian dynamics; Cross-time memory kernels; Macroscopic quantum tunneling; Josephson junctions; Switching-time distribution; Caldeira--Leggett model; Volterra integro-differential equations; Bath correlation spectroscopy

DOI
10.21203/rs.3.rs-9103716/v1

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Abstract When a Josephson junction is coupled to an environment with memory time λc−1\lambda_c^{-1}λc−1​ exceeding a dynamically engineered suppression window [t1,t2][t_1,t_2][t1​,t2​], bath-mediated correlations link pre-window and post-window tunneling events in a way that no local-in-time escape rate can reproduce. Using the Caldeira--Leggett influence-functional framework, we derive a Volterra integro-differential survival equation containing, alongside the usual Markovian and local-memory terms, an explicit \emph{cross-time} kernel Kcross(t,t′)K_{\text{cross}}(t,t')Kcross​(t,t′) that couples times tt2tt_2tt2​ while contributions from within the suppressed interval remain exponentially small. The resulting switching-time distribution displays two separable signatures: a clustering peak at t2t_2t2​ with jump ratio ε−1\varepsilon^{-1}ε−1 (ε≪1\varepsilon \ll 1ε≪1 being the in-window rate suppression), and a post-window memory tail whose decay constant equals the bath correlation rate λc\lambda_cλc​ and whose \emph{amplitude} is set by the pre-window history functional MpreM_{\text{pre}}Mpre​. The amplitude also scales as e−λcΔtbe^{-\lambda_c \Delta t_b}e−λc​Δtb​ with Δtb=t2−t1\Delta t_b = t_2 - t_1Δtb​=t2​−t1​, providing a direct spectroscopic handle on the bath memory time. Standard resistively shunted junctions, with λc−1∼10 ps\lambda_c^{-1} \sim 10\,\text{ps}λc−1​∼10ps, render the cross-time signal unobservable for nanosecond barriers; coupling the junction to a high-QQQ microwave resonator (Q∼105Q \sim 10^5Q∼105, ωr/2π∼5 GHz\omega_r/2\pi \sim 5\,\text{GHz}ωr​/2π∼5GHz) extends the effective memory time to microseconds, bringing both signatures within reach of current superconducting circuit technology. \bigskip\noindent\textbf{Keywords:} Non-Markovian dynamics; Cross-time memory kernels; Macroscopic quantum tunneling; Josephson junctions; Switching-time distribution; Caldeira--Leggett model; Volterra integro-differential equations; Bath correlation spectroscopy
Abstract 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][t_1,t_2][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′<t1t'<t_1t′<t1​ to post-window times t>t2t>t_2t>t2​ with an amplitude proportional to e−λcΔtbMpree^{-\lambda _c\Delta t_b}M_\textrm{pre}e−λc​Δtb​Mpre​ , where Δtb=t2−t1\Delta t_b=t_2-t_1Δtb​=t2​−t1​ and MpreM_\textrm{pre}Mpre​ is a history-weighted pre-window survival functional. The switching distribution P(t)=−S˙(t)P(t)=-\dot{S}(t)P(t)=−S˙(t) then contains a Markovian clustering peak at t2t_2t2​ 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.
Building upon our prior work [1], we present a unified stochastic drift model (SdM) for superconducting charge qubits based on memory multi-fractional Brownian motion (mmFBM). The classical sector employs a time-dependent Hurst exponent H(t) and adaptive memory kernel K(t,s), capturing non-stationary 1/f^beta noise and long-range temporal correlations inaccessible to conventional models. The quantum extension is formulated via a time-dependent Caldeira--Leggett environment with spectral density J(omega;t) = eta(t) omega_c^{1-s(t)} omega^{s(t)} exp(-omega/omega_c), where s(t) = 2H(t)-1, consistently reproducing beta(t) = 2H(t)-1. Four central results emerge: (1) relaxation and noise amplitudes act independently on energy decay; (2) time-varying H(t) matches experimental 1/f spectra more accurately than any constant exponent; (3) adaptive kernel dynamics preserve correlations without artificial damping; and (4) simulations predict coherence times (T1 ~ 5.00 x 10^6 ns, T2 ~ 4.18 x 10^5 ns) consistent with theory when charge noise dominates. The qubit exhibits stretched-exponential Ramsey and echo decay, non-Markovian dephasing, and a temperature-driven quantum-to-classical crossover. We derive the effective time-local Lindblad master equation, establish the classical mmFBM limit at high temperatures, and provide experimentally testable scaling relations. The non-exponential decay patterns reveal fundamental limitations of Markovian approaches, and the framework guides the design of noise-resilient qubit architectures.
High-resolution (sub-microsecond to nanosecond) time-resolved measurements of the dynamics of superconducting electronic devices, such as Josephson tunnel junctions and SQUIDs, are indispensable for fundamental physics research, such as quantum mechanics of macroscopic variables. We describe the development of a time-resolved measurement technique that enables direct measurements, in the time domain, of the temporal evolution of Josephson junctions with a nanosecond resolution. The technique was applied to study escape dynamics of Josephson junctions, as macroscopic quantum systems, in the quantum tunnelling regime. The measured probability of junctions remaining in the initial metastable state—the survival probability—as a function of time decayed exponentially, agrees very well with the theoretical prediction of the incoherent tunnelling process.
We employ a functional integral technique to calculate the macroscopic quantum tunneling rate at zero temperature of a current-biased Josephson junction weakly coupled to a resonator. We allow for the effects of environmental dissipation on both the junction and the resonator via baths of harmonic oscillators, and we consider both cases of weak and strong junction damping. The resonator can be, in principle, any quantum harmonic oscillator that couples to the junction bilinearly in either the coordinate or the velocity. The Josephson phase difference $\ensuremath{\phi}$ is the tunneling variable in this system, and the low-lying bound states in the junction's potential energy well are weakly metastable due to tunneling through a finite-width energy barrier. There has been some interest in using such biased junctions as qubits with the resonator providing multiqubit coupling. The main result of this work is that coupling to the resonator has a suppressive effect on the junction's tunneling: the stronger the coupling strength between the junction and resonator, the greater the reduction of the tunneling rate. Including damping to the junction also suppresses tunneling, a well-known result, but damping of the resonator actually reduces the magnitude of suppression; i.e., damping the resonator partially counteracts the suppression provided directly by the junction-resonator coupling. Details of the junction-resonator coupling yield interesting variations on this theme that may be useful for qubit design. For example, the coupling Uint{U}_{\mathit{int}}Uint​ between a current-biased junction and an AlN dilatational resonator should depend on the angular frequency of the resonator ${\ensuremath{\omega}}_{R}$ in a power-law fashion [A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)], specifically ${U}_{\mathit{int}}\ensuremath{\propto}{({\ensuremath{\omega}}_{R})}^{n}$, where n=1∕2n=1∕2n=1∕2. We find that for any value of nnn in the range 0<n<10<n<10<n<1 the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for n=1∕2n=1∕2n=1∕2, the tunneling rate is maximally suppressed for ${\ensuremath{\omega}}_{R}∕{\ensuremath{\omega}}_{J}\ensuremath{\approx}1$, where ${\ensuremath{\omega}}_{J}$ is the bias current-dependent plasma frequency of the junction.
We report a theoretical study of the macroscopic quantum tunneling (MQT) in small Josephson junctions containing randomly distributed two-level systems. We focus on a Josephson phase escape for switching from the superconducting (the zero-voltage) state to a resistive one. Above the crossover temperature Tcr{T}_{cr}Tcr​ the thermal fluctuations of the Josephson phase induce such a switching, and as T<TcrT<{T}_{cr}T<Tcr​ the regime of the MQT occurs. In the absence of two-level systems (TLSs) a magnetic field applied parallel to the junction plane results in a smooth reduction of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$, where $\mathrm{\ensuremath{\Phi}}$ is an applied magnetic flux. As the TLSs are present in Josephson junctions we obtain a resonant enhancement of the MQT. This phenomenon manifests itself by a narrow peak in the dependence of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$ occurring in the intermediate range of $\mathrm{\ensuremath{\Phi}}$, i.e., $0<\mathrm{\ensuremath{\Phi}}<{\ensuremath{\phi}}_{0}$ (${\ensuremath{\phi}}_{0}$ is the magnetic flux quantum). We explain this effect quantitatively by a strong resonant suppression of the potential barrier for the Josephson phase escape that is due to the coherent quantum Rabi oscillations in two-level systems present in the junction.
We develop a complete theoretical framework for open quantum systems coupled to scale-invariant environments. We show that such environments are universally described by unparticle baths characterized by a single scaling dimension dUd_{\mathcal{U}}dU​. This work provides the proof of the uniqueness theorem, the formalism of the resulting non-Markovian dynamics, and applications to several physical systems. From the uniqueness theorem, we derive the non-Markovian memory kernels, the exact noise kernel including vacuum and thermal contributions, and a fractional generalization of the Caldeira-Leggett master equation for arbitrary dUd_{\mathcal{U}}dU​. The scaling dimension governs a rich phase structure, including a thermalization transition at dU=3/2d_{\mathcal{U}}=3/2dU​=3/2, the Ohmic boundary at dU=2d_{\mathcal{U}}=2dU​=2, and a decoherence transition at dU=5/2d_{\mathcal{U}}=5/2dU​=5/2 in the thermal regime, beyond which long-time quantum coherence is protected. Three realizations are studied. For the quantum Ising model at criticality, coupling to the energy operator in (1+1)(1+1)(1+1) dimensions gives dU=3/2d_{\mathcal{U}}=3/2dU​=3/2, producing 1/f1/f1/f noise, while the (2+1)(2+1)(2+1)D case yields dU≈1.413d_{\mathcal{U}}\approx1.413dU​≈1.413 from the conformal bootstrap. In inflationary cosmology, massless scalar and graviton baths in de Sitter spacetime give dU=2d_{\mathcal{U}}=2dU​=2, predicting linear decoherence growth consistent with the quantum-to-classical transition. For high-energy astrophysical neutrinos, the decoherence rate Γdecoh∝B(E,TU)L5−2dUΓ_{\mathrm{decoh}}\propto \mathcal{B}(E,T_{\mathcal{U}})L^{5-2d_{\mathcal{U}}}Γdecoh​∝B(E,TU​)L5−2dU​ provides an observable signature of the scaling dimension. We also compare the framework with Caldeira-Leggett and Lindblad approaches, analyze the validity regimes, and discuss experimental implications for trapped-ion simulators, neutrino telescopes, and superconducting qubits.