Discover Quantum Science

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

Mahgoub A. Salih, Mohamed. Y. Shirgawi

Published Jul 6, 2026• 0 citations

Abstract

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 $$[t_1,t_2]$$ . 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 $$t'<t_1$$ to post-window times $$t>t_2$$ with an amplitude proportional to $$e^{-\lambda _c\Delta t_b}M_\textrm{pre}$$ , where $$\Delta t_b=t_2-t_1$$ and $$M_\textrm{pre}$$ is a history-weighted pre-window survival functional. The switching distribution $$P(t)=-\dot{S}(t)$$ then contains a Markovian clustering peak at $$t_2$$ 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.

DOI: 10.1007/s44464-026-00029-5

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

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Discover Quantum Science

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

Mahgoub A. Salih, Mohamed. Y. Shirgawi

Published Jul 6, 2026• 0 citations

Publisher

Abstract

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

[t_1,t_2]

. 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'<t_1

to post-window times

t>t_2

with an amplitude proportional to

e^{-\lambda _c\Delta t_b}M_\textrm{pre}

, where

\Delta t_b=t_2-t_1

and

M_\textrm{pre}

is a history-weighted pre-window survival functional. The switching distribution

P(t)=-\dot{S}(t)

then contains a Markovian clustering peak at

t_2

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.

DOI: 10.1007/s44464-026-00029-5

Related papers

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

Mahgoub A. Salih, Mohamed. Y. Shirgawi

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 spectroscopyAbstract 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

Mar 26, 20260 citations

Resonant enhancement of MQT in Josephson junctions: the influence of coherent two-level systems

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We report a theoretical study of the macroscopic quantum tunneling (MQT) in small Josephson junctions containing randomly distributed two-level systems. We focus on the magnetic field dependent crossover temperature $T_{cr}$ between the thermal fluctuation and quantum regimes of switching from the superconducting (the zero-voltage) state to a resistive one. In the absence of two-levels systems the crossover temperature shows a smooth decrease with an applied magnetic field characterized by an external flux $Φ$ . Beyond that we predict a narrow peak in the dependence of $T_{cr}(Φ)$ occurring in the intermediate range of $Φ$ . The effect becomes more pronounced as the junction size increases. We explain this effect quantitatively by a strong resonant suppression of a 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 report a theoretical study of the macroscopic quantum tunneling (MQT) in small Josephson junctions containing randomly distributed two-level systems. We focus on the magnetic field dependent crossover temperature $T_{cr}$ between the thermal fluctuation and quantum regimes of switching from the superconducting (the zero-voltage) state to a resistive one. In the absence of two-levels systems the crossover temperature shows a smooth decrease with an applied magnetic field characterized by an external flux $Φ$. Beyond that we predict a narrow peak in the dependence of $T_{cr}(Φ)$ occurring in the intermediate range of $Φ$. The effect becomes more pronounced as the junction size increases. We explain this effect quantitatively by a strong resonant suppression of a potential barrier for the Josephson phase escape that is due to the coherent quantum Rabi oscillations in two-level systems present in the junction.

arXiv:1410.4388Oct 16, 20140 citations

Resonant enhancement of macroscopic quantum tunneling in Josephson junctions: Influence of coherent two-level systems

M. V. Fistul

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 ${T}_{cr}$ the thermal fluctuations of the Josephson phase induce such a switching, and as $T<{T}_{cr}$ 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 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 ${T}_{cr}$ the thermal fluctuations of the Josephson phase induce such a switching, and as $T<{T}_{cr}$ 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.

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Suppression of macroscopic quantum tunneling in a large Josephson junction coupled to a resonator

Brad R. Trees, Yaser H. Helal, Joshua Schiffrin +1 more

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 ${U}_{\mathit{int}}$ 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 $n=1∕2$ . We find that for any value of $n$ in the range $0<n<1$ the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for $n=1∕2$ , 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 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 ${U}_{\mathit{int}}$ 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 $n=1∕2$. We find that for any value of $n$ in the range $0<n<1$ the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for $n=1∕2$, 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.

Physical Review BDec 13, 20076 citations

Effect of temperature on the resonant macroscopic quantum tunneling in small Josephson junctions

B. Ruggiero, P. Silvestrini

The study of the quantum behaviour of Josephson junctions is interesting not only by itself, but also for the technological importance in electronics employing superconducting weak links. Recently great attention has been devoted to the effects of Resonant Macroscopic Quantum Tunneling (RMQT) between levels the same energy in neighbouring wells of the potential shape describing the junction. This can produce voltage spikes along the supercurrent branch of the I-V curves and peaks in the current switching distributions. Schmidt, Clealand, and Clarke have presented a detailed analysis of the resonant tunnel dynamics for the T=0 case. The present paper is based on a different theoretical approach, first proposed by Larkin and Ovchinnikov, which allows one to include explicitly the temperature. We study the phenomenon in its time dependence and this allows us to consider the lifetime of the voltage spikes as well as the effect of the RMQT on the quantum escape rate out of the V=0 state.>The study of the quantum behaviour of Josephson junctions is interesting not only by itself, but also for the technological importance in electronics employing superconducting weak links. Recently great attention has been devoted to the effects of Resonant Macroscopic Quantum Tunneling (RMQT) between levels the same energy in neighbouring wells of the potential shape describing the junction. This can produce voltage spikes along the supercurrent branch of the I-V curves and peaks in the current switching distributions. Schmidt, Clealand, and Clarke have presented a detailed analysis of the resonant tunnel dynamics for the T=0 case. The present paper is based on a different theoretical approach, first proposed by Larkin and Ovchinnikov, which allows one to include explicitly the temperature. We study the phenomenon in its time dependence and this allows us to consider the lifetime of the voltage spikes as well as the effect of the RMQT on the quantum escape rate out of the V=0 state.>

IEEE Transactions on Applied SuperconductivityJun 1, 19952 citations

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

[t_1,t_2]

t'<t_1

to post-window times

t>t_2

with an amplitude proportional to

e^{-\lambda _c\Delta t_b}M_\textrm{pre}

, where

\Delta t_b=t_2-t_1

and

M_\textrm{pre}

is a history-weighted pre-window survival functional. The switching distribution

P(t)=-\dot{S}(t)

then contains a Markovian clustering peak at

t_2

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

T_{cr}

between the thermal fluctuation and quantum regimes of switching from the superconducting (the zero-voltage) state to a resistive one. In the absence of two-levels systems the crossover temperature shows a smooth decrease with an applied magnetic field characterized by an external flux

Φ

. Beyond that we predict a narrow peak in the dependence of

T_{cr}(Φ)

occurring in the intermediate range of

Φ

. The effect becomes more pronounced as the junction size increases. We explain this effect quantitatively by a strong resonant suppression of a potential barrier for the Josephson phase escape that is due to the coherent quantum Rabi oscillations in two-level systems present in the junction.

{T}_{cr}

the thermal fluctuations of the Josephson phase induce such a switching, and as

T<{T}_{cr}

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.

{U}_{\mathit{int}}

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∕2

. We find that for any value of

n

in the range

0<n<1

the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for

n=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.