Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1208.1701

A quantum effect in the classical limit: nonequilibrium tunneling in the Duffing oscillator

Shu-Hao Yeh, Dong-Bang Tsai, Che-Wei Huang, Md. Manirul Ali, Alec Maassen van den Brink

Published Aug 8, 2012• 0 citations
arXivHTMLPDF

Abstract

For suitable parameters, the classical Duffing oscillator has a known bistability in its stationary states, with low- and high-amplitude branches. As expected from the analogy with a particle in a double-well potential, transitions between these states become possible either at finite temperature, or in the quantum regime due to tunneling. In this analogy, besides local stability, one can also discuss global stability by comparing the two potential minima. For the Duffing oscillator, the stationary states emerge dynamically so that a priori, a potential-minimum criterion for them does not exist. However, global stability is still relevant, and definable as the state containing the majority population for long times, low temperature, and close to the classical limit. Further, the crossover point is the parameter value at which global stability abruptly changes from one state to the other. For the double-well model, the crossover point is defined by potential-minimum degeneracy. Given that this analogy is so effective in other respects, it is thus striking that for the Duffing oscillator, the crossover point turns out to be non-unique. Rather, none of the three aforementioned limits commute with each other, and the limiting behaviour depends on the order in which they are taken. More generally, as both $\hbar\To0$ and $T\To0$, the ratio $\hbarω_0/k_\mathrm{B}T$ continues to be a key parameter and can have any nonnegative value. This points to an apparent conceptual difference between equilibrium and nonequilibrium tunneling. We present numerical evidence by studying the pertinent quantum master equation in the photon-number basis. Independent verification and some further understanding are obtained using a semi-analytical approach in the coherent-state representation.

DOI
10.48550/arxiv.1208.1701

Related papers

Unraveling the switching dynamics in a quantum double-well potential

Qile David Su, Rodrigo G. Cortiñas, Jayameenakshi Venkatraman +1 more

The spontaneous switching of a quantum particle between the wells of a double-well potential is a phenomenon of general interest to physics and chemistry. It was broadly believed that the switching rate decreases steadily as a function of the size of the energy barrier. This view was challenged by a recent experiment on a driven superconducting Kerr nonlinear oscillator (often called the Kerr-cat qubit or the Kerr parametric oscillator), whose energy barrier can be continuously increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a steplike decrease termed the ``staircase.'' The view challenged by the experimental staircase demands a deep review of our understanding of the role of quantum effects in double wells. In this work, we use a Lindbladian model of dissipation to derive a semianalytical formula for the switching rate, resolving a continuous transition between tunneling-dominated dynamics and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat part and the steep part of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: the dephasing of interwell superpositions freezes tunneling via the quantum Zeno effect, and the decay from one excited state to another limits the time during which tunneling can be attempted. This physical understanding allows us to pinpoint the critical drive amplitude that separates the flat and steep part of each step using an equation involving the rates of the two dissipative processes and the tunnel splitting. In addition, analyzing the transition matrix elements in the formula shows that in the regime of a few ($\ensuremath{\lesssim}10$) states in the well and under moderate to low temperatures, highly excited states are populated predominantly via cascaded and direct thermal heating rather than quantum heating. At very low temperatures, we find that the perturbation induced by the non-Hermitian Hamiltonian part of the Lindbladian model becomes increasingly important and facilitates a form of quantum heating that has not been identified before. We numerically map the activation mechanism as a function of drive amplitude, damping rate, and temperature. Our theory deepens the understanding of switching dynamics between metastable quantum states, highlights the importance of a general interplay between tunneling and dissipation, and identifies a novel quantum regime in activated transitions.

arXiv:2501.00209•Physical Review A•Sep 2, 2025•4 citations

Quantum tunnelling and thermally driven transitions in a double-well potential at finite temperature

Robson Christie, Jessica K. Eastman

Abstract We explore dissipative quantum tunnelling, a phenomenon central to various physical and chemical processes, utilizing a model based on a double-well potential. This paper aims to bridge gaps in understanding the crossover from thermal activation to quantum tunnelling, a domain still shrouded in mystery despite extensive research. By numerically investigating a model derived from Caldeira–Leggett’s work on quantum Brownian motion, examining both Lindblad and stochastic Schrödinger dynamics, we offer new insights into the transition states in the crossover region. Contrary to a common belief that temperature strongly dampens all quantum effects, our findings reveal that under certain conditions temperature instead alters the nature of tunnelling from a deterministic and periodic process to a stochastic yet still very quantum phenomenon. This underscores the profound influence of quantum effects on transition rates and the critical role of temperature in modulating tunnelling behaviours. Additionally, we introduce a new model for quantum Brownian motion that takes Lindblad form and is formulated as a modification of the widely known model found in Breuer and Petruccione. In our approach, we remove the zero-temperature singularity resulting in a better description of low-temperature quantum Brownian motion near a potential minima. Despite these advancements, we recognize persistent challenges in accurately simulating the dynamics at extremely low temperatures for arbitrary potentials, particularly those that cannot be closely approximated locally by a quadratic function.

arXiv:2312.12101•Journal of Physics A Mathematical and Theoretical•May 14, 2024•4 citations

Quantum behavior of the Duffing oscillator at the dissipative phase transition

Qiming Chen, Michael Fischer, Yuki Nojiri +8 more

The non-deterministic behavior of the Duffing oscillator is classically attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective which predicts a single unique steady state. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and experimentally reconcile the classical and quantum descriptions as indicated by the Liouvillian spectral theory. We demonstrate that the two classically regarded steady states are in fact quantum metastable states. They have a remarkably long lifetime but must eventually relax into the single unique steady state allowed by quantum mechanics. By engineering their lifetime, we observe a first-order dissipative phase transition and reveal the two distinct phases by quantum state tomography. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition and form an essential step towards understanding the intriguing phenomena in driven-dissipative systems.

arXiv:2206.06338•Nature Communications•May 20, 2023•43 citations

Quantum dynamics in a cubic potential in the semi-classical limit: Smearing of the homoclinic bifurcation

Akshay Pal, Jayanta K. Bhattacharjee

We consider the dynamics of a particle inside the metastable well of a cubic potential. In the classical picture the particle can oscillate inside the well when its total energy is less than a critical value Ec at which point a homoclinic bifurcation occurs. The quantum particle is always supposed to tunnel out. We set up a semi classical dynamics (which cannot capture tunneling effects) of the quantum particle which can be analyzed in terms of fixed points and stabilities and thus yields quantitative results. We find that the quantum fluctuations smear out the homoclinic bifurcation and for energies greater than a new critical value, substantially less than Ec, the particle always escapes from the well. This implies that even for an initial wave packet centered at the minimum of the well, there is a critical value of the width for which the particle tunnels out even in this semi-classical limit.

Physics Open•Oct 16, 2020•2 citations

Crossover from Classical to Quantum Behavior in Duffing Oscillator through “Pseudo-Lyapunov Exponent”

Yukihiro Ota, Ichiro Ohba

We discuss the quantum–classical correspondence in a specific dissipative chaotic system, Duffing oscillator. We quantize it on the basis of quantum state diffusion (QSD) that is a certain formulation for quantum open systems and a useful tool for analyzing nonlinear problems and their classical limit. We define a “pseudo-Lyapunov exponent” and investigate the behavior of this quantity varying Planck constant effectively. We show that in a dissipative system there exists some critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we show that the role of dissipation changes for quantized Duffing oscillator since the effect of dissipation is important for the occurrence of chaos in classical mechanics but suppresses it in quantum region.

Journal of the Physical Society of Japan•Jan 1, 2003•0 citations

Coherent oscillations and incoherent tunnelling in one - dimensional asymmetric double - well potential

Viktor A. Benderskii, Efim I Kats

For a model 1d asymmetric double-well potential we calculated so-called survival probability (i.e. the probability for a particle initially localised in one well to remain there). We use a semiclassical (WKB) solution of Schroedinger equation. It is shown that behaviour essentially depends on transition probability, and on dimensionless parameter which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunnelling. For the potential describing a finite motion (double-well) one has always a regular behaviour. For the small value of the parameter there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for the large value of the parameter no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunnelling levels grows proportionally to the value of this parameter (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of the parameter one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behaviour.

•Jan 1, 2002•0 citations
For suitable parameters, the classical Duffing oscillator has a known bistability in its stationary states, with low- and high-amplitude branches. As expected from the analogy with a particle in a double-well potential, transitions between these states become possible either at finite temperature, or in the quantum regime due to tunneling. In this analogy, besides local stability, one can also discuss global stability by comparing the two potential minima. For the Duffing oscillator, the stationary states emerge dynamically so that a priori, a potential-minimum criterion for them does not exist. However, global stability is still relevant, and definable as the state containing the majority population for long times, low temperature, and close to the classical limit. Further, the crossover point is the parameter value at which global stability abruptly changes from one state to the other. For the double-well model, the crossover point is defined by potential-minimum degeneracy. Given that this analogy is so effective in other respects, it is thus striking that for the Duffing oscillator, the crossover point turns out to be non-unique. Rather, none of the three aforementioned limits commute with each other, and the limiting behaviour depends on the order in which they are taken. More generally, as both ℏ\To0\hbar\To0ℏ\To0 and T\To0T\To0T\To0, the ratio ℏω0/kBT\hbarω_0/k_\mathrm{B}Tℏω0​/kB​T continues to be a key parameter and can have any nonnegative value. This points to an apparent conceptual difference between equilibrium and nonequilibrium tunneling. We present numerical evidence by studying the pertinent quantum master equation in the photon-number basis. Independent verification and some further understanding are obtained using a semi-analytical approach in the coherent-state representation.
The spontaneous switching of a quantum particle between the wells of a double-well potential is a phenomenon of general interest to physics and chemistry. It was broadly believed that the switching rate decreases steadily as a function of the size of the energy barrier. This view was challenged by a recent experiment on a driven superconducting Kerr nonlinear oscillator (often called the Kerr-cat qubit or the Kerr parametric oscillator), whose energy barrier can be continuously increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a steplike decrease termed the ``staircase.'' The view challenged by the experimental staircase demands a deep review of our understanding of the role of quantum effects in double wells. In this work, we use a Lindbladian model of dissipation to derive a semianalytical formula for the switching rate, resolving a continuous transition between tunneling-dominated dynamics and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat part and the steep part of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: the dephasing of interwell superpositions freezes tunneling via the quantum Zeno effect, and the decay from one excited state to another limits the time during which tunneling can be attempted. This physical understanding allows us to pinpoint the critical drive amplitude that separates the flat and steep part of each step using an equation involving the rates of the two dissipative processes and the tunnel splitting. In addition, analyzing the transition matrix elements in the formula shows that in the regime of a few ($\ensuremath{\lesssim}10$) states in the well and under moderate to low temperatures, highly excited states are populated predominantly via cascaded and direct thermal heating rather than quantum heating. At very low temperatures, we find that the perturbation induced by the non-Hermitian Hamiltonian part of the Lindbladian model becomes increasingly important and facilitates a form of quantum heating that has not been identified before. We numerically map the activation mechanism as a function of drive amplitude, damping rate, and temperature. Our theory deepens the understanding of switching dynamics between metastable quantum states, highlights the importance of a general interplay between tunneling and dissipation, and identifies a novel quantum regime in activated transitions.
Abstract We explore dissipative quantum tunnelling, a phenomenon central to various physical and chemical processes, utilizing a model based on a double-well potential. This paper aims to bridge gaps in understanding the crossover from thermal activation to quantum tunnelling, a domain still shrouded in mystery despite extensive research. By numerically investigating a model derived from Caldeira–Leggett’s work on quantum Brownian motion, examining both Lindblad and stochastic Schrödinger dynamics, we offer new insights into the transition states in the crossover region. Contrary to a common belief that temperature strongly dampens all quantum effects, our findings reveal that under certain conditions temperature instead alters the nature of tunnelling from a deterministic and periodic process to a stochastic yet still very quantum phenomenon. This underscores the profound influence of quantum effects on transition rates and the critical role of temperature in modulating tunnelling behaviours. Additionally, we introduce a new model for quantum Brownian motion that takes Lindblad form and is formulated as a modification of the widely known model found in Breuer and Petruccione. In our approach, we remove the zero-temperature singularity resulting in a better description of low-temperature quantum Brownian motion near a potential minima. Despite these advancements, we recognize persistent challenges in accurately simulating the dynamics at extremely low temperatures for arbitrary potentials, particularly those that cannot be closely approximated locally by a quadratic function.
The non-deterministic behavior of the Duffing oscillator is classically attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective which predicts a single unique steady state. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and experimentally reconcile the classical and quantum descriptions as indicated by the Liouvillian spectral theory. We demonstrate that the two classically regarded steady states are in fact quantum metastable states. They have a remarkably long lifetime but must eventually relax into the single unique steady state allowed by quantum mechanics. By engineering their lifetime, we observe a first-order dissipative phase transition and reveal the two distinct phases by quantum state tomography. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition and form an essential step towards understanding the intriguing phenomena in driven-dissipative systems.
We consider the dynamics of a particle inside the metastable well of a cubic potential. In the classical picture the particle can oscillate inside the well when its total energy is less than a critical value Ec at which point a homoclinic bifurcation occurs. The quantum particle is always supposed to tunnel out. We set up a semi classical dynamics (which cannot capture tunneling effects) of the quantum particle which can be analyzed in terms of fixed points and stabilities and thus yields quantitative results. We find that the quantum fluctuations smear out the homoclinic bifurcation and for energies greater than a new critical value, substantially less than Ec, the particle always escapes from the well. This implies that even for an initial wave packet centered at the minimum of the well, there is a critical value of the width for which the particle tunnels out even in this semi-classical limit.
We discuss the quantum–classical correspondence in a specific dissipative chaotic system, Duffing oscillator. We quantize it on the basis of quantum state diffusion (QSD) that is a certain formulation for quantum open systems and a useful tool for analyzing nonlinear problems and their classical limit. We define a “pseudo-Lyapunov exponent” and investigate the behavior of this quantity varying Planck constant effectively. We show that in a dissipative system there exists some critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we show that the role of dissipation changes for quantized Duffing oscillator since the effect of dissipation is important for the occurrence of chaos in classical mechanics but suppresses it in quantum region.
For a model 1d asymmetric double-well potential we calculated so-called survival probability (i.e. the probability for a particle initially localised in one well to remain there). We use a semiclassical (WKB) solution of Schroedinger equation. It is shown that behaviour essentially depends on transition probability, and on dimensionless parameter which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunnelling. For the potential describing a finite motion (double-well) one has always a regular behaviour. For the small value of the parameter there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for the large value of the parameter no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunnelling levels grows proportionally to the value of this parameter (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of the parameter one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behaviour.