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Study of the phase dynamics in moderately damped Josephson junctions and its applications

Suman Dhamala

Published Apr 1, 2021• 0 citations
OpenAlex

Abstract

The observation of macroscopic quantum tunneling (MQT) manifests the quantumness of macroscopic variables, such as the phase difference across a Josephson junction (JJ). Since then, MQT has become a hallmark experiment to show the quantum nature of JJ-based devices. It has also paved a way to test other macroscopic quantum effects and has led to engineering a variety of artificial atoms, i.e., qubits. However, in the progress of quantum circuits for different applications, it is crucial to understand the influence of thermal and quantum fluctuations, dissipation, and noise to various quantum devices. Thus, the role of the factors mentioned above and the junction parameters $E_{J}$ (Josephson coupling energy) and $E_{C}$ (charging energy) can be scrutinized by understanding the dynamics of the fictitious phase particle. Similarly, in the application perspective, Szilard engine based on SET device are too slow to convert heat into work and are not efficient in-terms of power performance. To overcome the drawbacks, we proposed a high-efficiency fast Szilard engine based on a flux logic device (FLD) and investigated its property numerically and optimize its performance. \\\\ In the first part of the dissertation, we studied the phase dynamics of moderately damped Josephson junctions. The time of flight technique is used to measure the switching current distributions (SCDs). Different dynamical states, thermal activation (TA), MQT, and phase diffusion (PD) are identified from SCDs through well-defined criteria. We observed unambiguous evidence of MQT when temperature is below $\sim0.5\text{-}0.8$ K. The importance of the tilt in the phase dynamics of the junctions, which has been overlooked in previous studies, is clarified. We found that the junction with larger $E_{J}$ but the same $\omega_{p0}$ requires greater tilt and lower temperature to enter MQT regime. \\ \\ In the second part of the dissertation, to overcome the limitation of the SET based Szilard engine, we used superconducting quantum interference device (SQUID) as a one-bit flux logic cell to realize a fast Szilard engine for optimal power. In the past decade, due to technological advancement, the study of energy fluctuations in nano-scale systems such as optical or electrical traps and single-electron tunneling (SET) devices has transformed Szilard's engine to experimental realizations. However, the heat engine realized are slow to convert heat into work limiting their performance in speed and power. The SQUID-based flux logic device (FLD) has a high characteristic frequency, $\omega_\text{p}$/2$\pi \sim$10$^{10}$ Hz, making it an excellent candidate for realizing a high-efficiency Szilard engine with unprecedented performance. We showed that, on average, the proposed heat engine can extract k$_\text{B}T$ln2 of heat per cycle from the thermal reservoir in the quasi-static (QS) limit. In addition, we showed that the FLD-based Szilard engine delivers maximum power when operated about 200 times faster than the threshold speed of quasi-static operation. Our result demonstrates that Szilard's engine's performance based on FLD exceeds other nano-device implementations by orders of magnitude. \\

Related papers

Macroscopic quantum phenomena in superconductors: study of phase dynamics and dissipation in moderately damped Josephson junctions

D. Massarotti

The topic of the PhD project is a comparative study of phase dynamics and macroscopic quantum phenomena in moderately damped NbN, YBaCuO grain boundary (GB) Josephson junctions (JJs) and hybrid devices. This type of research activity responds to the needs of better identifying phase dynamics in JJs in the moderately damped regime, which is going to be more and more common in hybrid nanostructures. Issues on a more detailed understanding of coherence, dissipation and noise in the various devices have a relevant role in the progress of quantum circuits. In the last few years, studies of phase dynamics and macroscopic quantum phenomena have been extended to junctions composed of materials other than the traditional low critical temperature superconductors (LTS) and to novel types of structures with unconventional barriers composed for instance of graphene sheets or of grain boundaries. Progress in engineering new materials into junctions and in understanding and controlling the physics of interfaces may offer novel solutions for junctions of superior quality and complementary functionalities, and therefore may lead in the long run to improve also specific qubit performances. For LTS JJs, once the barrier thickness and the critical current density (Jc) have been fixed, a reduction in its size unavoidably leads to a lowering of the critical current and determines a quite different phase dynamics re-normalized to the new scaling energy. Lower critical currents Ic result in lower Josephson energies EJ, and higher levels of dissipation are expected. The range of the energy dynamical parameters is significantly enlarged, and it is technologically easier to reproducibly realize nontrivial configurations. These devices are characterized by intermediate levels of dissipation (moderately damped regime) and by phase diffusion phenomena. The low Jc limit seems to be characteristic also of all futuristic nanohybrids devices incorporating nanowires, and the moderately damped regime (MDR) is intrinsically more common than it could be expected. Measurements of switching current distribution (SCD) in these last years have turned to be standard tools to investigate phase dynamics in unconventional and hybrid systems and nanostructures. High critical temperature superconductors (HTS) are an example of unconventional systems, because of the d-wave order parameter symmetry and of the presence of low-energy quasiparticles, which are expected to induce high level of dissipation and as a consequence to spoil macroscopic quantum coherence. A comparative study in systems so different and complementary is of great interest. During the PhD program, we have performed measurements of SCD, both in the thermal and quantum regime down to 20 mK, on moderately damped NbN/MgO/NbN JJs and on YBCO biepitaxial GB JJs. The NbN JJs exhibit a crossover from thermal activation (TA) to macroscopic quantum tunneling (MQT) regime at about 90 mK. After the MQT saturation, the width of the SCD follows the expected dependence in the TA regime. Deviations are evident in proximity and above the crossover temperature T* (about 1.6K) where we find effects of anti-correlation between the bath temperature and thermal fluctuations since the width of the histograms starts to collapse, and indications of a transition to a third regime called phase diffusion (PD) are evident, such as the change of the switching distributions symmetry. Experimental data have been analyzed through numerical fitting of the switching probability and Monte Carlo simulations, and we found a good agreement with theoretical expectations based on multiple re-trapping processes, which strongly depend on the junction damping. The combined analysis of these three regimes allowed us to extract the fundamental parameters, which completely characterize the dynamics of such moderately damped NbN JJs. In addition, clear signatures of the PD regime have been detected and numerical methods have been developed to estimate the dissipation level in the MDR. As a term of comparison, the studies on moderately damped systems have been also carried out on HTS junctions, which have different levels of intrinsic dissipation induced by low energy quasi-particles, and are for this reason of great impact as a reference. In such JJs we have clearly observed a direct transition from the MQT to the PD regime. The width of the SCDs is constant below 100mK, which is a clearly indication of a quantum activation regime. Above 100mK the negative temperature derivative of the width is consistent with a diffusive motion due to multiple escape and re-trapping in the potential wells. Thermal behavior of the width has been fitted through numerical simulations, thus providing the estimate of the damping factor. These conclusions of a PD regime also in YBaCuO JJs are supported by data in presence of an externally applied magnetic field, and are consistent with the numerical outcomes. Numerical simulations of the thermal behavior of the SCD width for different values of the quality factor ranging from 1 to 10 have been carried out. Such numerical simulations allow us to reconstruct a (Q, kBT/EJ) phase diagram summarizing the various activation regimes. The transition curve between the PD regime and the running state following thermal or quantum activation has been determined numerically by varying the damping factor Q as function of the ratio between the thermal energy and the Josephson energy. The result of the calculation is a universal curve. A phase diagram valid in a large range of dissipation conditions emerges as a functional guide to classify the switching behavior and to settle the fundamental junction parameters and energies in the MDR. It is therefore a reference for phase dynamics of novel types of junction and system for which the nature of the current induced transition from the superconducting to the normal state has not been completely clarified.

•Apr 2, 2013•0 citations

Measurements of Phase Dynamics in Planar Josephson Junctions and SQUIDs

D. Z. Haxell, Erik Cheah, Filip Křížek +8 more

We experimentally investigate the stochastic phase dynamics of planar Josephson junctions (JJs) and superconducting quantum interference devices (SQUIDs) defined in epitaxial InAs/Al heterostructures, and characterized by a large ratio of Josephson energy to charging energy. We observe a crossover from a regime of macroscopic quantum tunneling to one of phase diffusion as a function of temperature, where the transition temperature T^{*} is gate-tunable. The switching probability distributions are shown to be consistent with a small shunt capacitance and moderate damping, resulting in a switching current which is a small fraction of the critical current. Phase locking between two JJs leads to a difference in switching current between that of a JJ measured in isolation and that of the same JJ measured in an asymmetric SQUID loop. In the case of the loop, T^{*} is also tuned by a magnetic flux.

arXiv:2204.05619•Physical Review Letters•Feb 23, 2023•22 citations

Macroscopic quantum tunneling and retrapping processes in moderately damped YBaCuO Josephson junctions

D. Massarotti, Luigi Longobardi, Luca Galletti +3 more

The moderately damped regime in Josephson junctions (JJ) is quite common in devices characterized by low critical currents and, therefore, by low Josephson energies. Measurements of the switching current distribution (SCD) are a direct way of determining the phase dynamics in the nontrivial case of moderate damping, which is becoming more common with advances in nanopatterning superconductors and in materials science for fabricating hybrid systems. We report measurements of SCDs, in both the thermal and quantum regimes for moderately damped YBaCuO grain boundary biepitaxial JJs. A direct transition from a phase diffusion regime to macroscopic quantum tunnelling occurs at about 130 mK. The crossover to the quantum regime is determined by the magnetic field and the phase dynamics can be described by a fully consistent set of junction parameters obtained by numerical simulation.

Low Temperature Physics•Mar 1, 2013•9 citations

Coherence properties of superconducting flux qubits

Samuele Spilla

The research work discussed in this thesis deals with the study of superconducting Josephson qubits. Superconducting qubits are solid-state artificial atoms which are based on lithographically defined Josephson tunnel junctions properties. When sufficiently cooled, these superconducting devices exhibit quantized states of charge, flux or junction phase depending on their design parameters. This allows to observe coherent evolutions of their states. The results presented can be divided into two parts. In a first part we investigate operations of superconducting qubits based on the quantum coherence in superconducting quantum interference devices (SQUID). We explain experimental data which has been observed in a SQUID subjected to fast, large-amplitude modifications of its effective potential shape. The motivations for this work come from the fact that in the past few years there have been attempts to interpret the supposed quantum behavior of physical systems, such as Josephson devices, within a classical framework. Moreover, we analyze the possibility of generating GHZ states, namely maximally entangled states, in a quantum system made out of three Josephson qubits. In particular, we investigate the possible limitations of the GHZ state generation due to coupling to bosonic baths. In the second part of the thesis we address a particular cause of decoherence of flux qubits which has been disregarded until now: thermal gradients, which can arise due to accidental non equilibrium quasiparticle distributions. The reason for these detrimental effects is that heat currents flowing through Josephson tunnel junctions in response to a temperature gradient are periodic functions of the phase difference between the electrodes. The phase dependence of the heat current comes from Andreev reflection, namely an interplay between the quasiparticles which carry heat and the superconducting condensate which is sensitive to the superconducting phase difference. Generally speaking, the flux qubit states are characterized by different values of the phase difference through their Josephson junctions. Consequently, the phase-dependent thermal current through a device subject a temperature gradient is related to the phase-dependent qubit states. We study how the thermal currents change according to the state of the qubits hence yielding a measurement of the qubit state. This in turn leads to an impact of temperature gradient on the dynamics of the system. We show that flux qubits in the Delft qubit design can have limitations of the decoherence time to the order of microseconds as a result of this newly discovered source of decoherence. In contrast, the fluxonium qubit is found to be well protected due to its superinductance.

•Feb 16, 2015•0 citations

Weak coupling Josephson junction as a current probe: effect of dissipation on escape dynamics

Jani M. Kivioja, T E Nieminen, Julien Claudon +3 more

We have studied the temperature dependence of escape phenomena in various underdamped Josephson junctions (JJs). The junctions had different Josephson coupling energies E J which were relatively small, but larger than the charging energy E C . Upon increasing the temperature T , we first observe the usual cross-over between macroscopic quantum tunnelling and thermally activated (TA) behaviour at temperatures k B T ∼ ℏω p , where ω p is the plasma frequency of the junction. Increasing T further, the width of the switching current distribution has, counterintuitively, a non-monotonic temperature dependence. This can be explained by the novel cross-over from TA behaviour to underdamped phase diffusion. We show that this cross-over is expected to occur at temperatures such that k B T ∼ E J (1 − 4/π Q ) 3 /2, where Q is the quality factor of the junction at the plasma frequency, in agreement with experiment. Our findings can be compared with detailed model calculations which take into account dissipation and level quantization in a metastable well. Particular attention is paid to the sample with the smallest E J , which shows extensive phase diffusion even at the lowest temperatures. This sample consists of a dc-SQUID and a single JJ close to each other, such that the SQUID acts as a tunable inductive protection for the single junction from fluctuations of a dissipative environment. By varying the flux through the dc-SQUID, we present, for the first time, experimental evidence of the escape of a JJ from the phase diffusion regime to the free running state in a tunable environment. We also show that in the zero voltage state the losses mainly occur at frequencies near the plasma resonance.

New Journal of Physics•Aug 26, 2005•15 citations

Characterization of Moderately Damped Low Tc Josephsonjunctions through Measurements of Switching Current Distributions

D. Massarotti, Luigi Longobardi, G. Rotoli +6 more

We report on measurements of the switching current probability both in the thermal and quantum regime on moderately damped NbN/MgO/NbN Josephson junctions with very low critical current density (’ 2A/cm2). We observed three distinct regimes: below 90 mK the width of the switching distributions is constant within the experimental error, while from 90 mK to 1.6 K the width increases with temperature, this change of behavior signaling the standard transition from macroscopic quantum tunneling (MQT) to thermal activation (TA) regime respectively. Above 1.6 K we find effects of anti-correlation between the bath temperature and thermal fluctuations since the width of the histograms starts to collapse, and indications of a transition to a third regime called Phase Diffusion (PD) are evident, such as the change of the switching distributions symmetry. Experimental data have been analyzed through numerical fitting of the switching probability and Monte Carlo simulations, and we found a good agreement with theoretical expectations based on multiple re-trapping processes, which strongly depend on the junction damping. The parameters which completely characterize the dynamics of such moderately damped Josephson junctions have been also extracted.

Physics Procedia•Jan 1, 2012•1 citation
The observation of macroscopic quantum tunneling (MQT) manifests the quantumness of macroscopic variables, such as the phase difference across a Josephson junction (JJ). Since then, MQT has become a hallmark experiment to show the quantum nature of JJ-based devices. It has also paved a way to test other macroscopic quantum effects and has led to engineering a variety of artificial atoms, i.e., qubits. However, in the progress of quantum circuits for different applications, it is crucial to understand the influence of thermal and quantum fluctuations, dissipation, and noise to various quantum devices. Thus, the role of the factors mentioned above and the junction parameters EJE_{J}EJ​ (Josephson coupling energy) and ECE_{C}EC​ (charging energy) can be scrutinized by understanding the dynamics of the fictitious phase particle. Similarly, in the application perspective, Szilard engine based on SET device are too slow to convert heat into work and are not efficient in-terms of power performance. To overcome the drawbacks, we proposed a high-efficiency fast Szilard engine based on a flux logic device (FLD) and investigated its property numerically and optimize its performance. \\ In the first part of the dissertation, we studied the phase dynamics of moderately damped Josephson junctions. The time of flight technique is used to measure the switching current distributions (SCDs). Different dynamical states, thermal activation (TA), MQT, and phase diffusion (PD) are identified from SCDs through well-defined criteria. We observed unambiguous evidence of MQT when temperature is below ∼0.5-0.8\sim0.5\text{-}0.8∼0.5-0.8 K. The importance of the tilt in the phase dynamics of the junctions, which has been overlooked in previous studies, is clarified. We found that the junction with larger EJE_{J}EJ​ but the same ωp0\omega_{p0}ωp0​ requires greater tilt and lower temperature to enter MQT regime. \ \ In the second part of the dissertation, to overcome the limitation of the SET based Szilard engine, we used superconducting quantum interference device (SQUID) as a one-bit flux logic cell to realize a fast Szilard engine for optimal power. In the past decade, due to technological advancement, the study of energy fluctuations in nano-scale systems such as optical or electrical traps and single-electron tunneling (SET) devices has transformed Szilard's engine to experimental realizations. However, the heat engine realized are slow to convert heat into work limiting their performance in speed and power. The SQUID-based flux logic device (FLD) has a high characteristic frequency, ωp\omega_\text{p}ωp​/2π∼\pi \simπ∼1010^{10}10 Hz, making it an excellent candidate for realizing a high-efficiency Szilard engine with unprecedented performance. We showed that, on average, the proposed heat engine can extract kBT_\text{B}TB​Tln2 of heat per cycle from the thermal reservoir in the quasi-static (QS) limit. In addition, we showed that the FLD-based Szilard engine delivers maximum power when operated about 200 times faster than the threshold speed of quasi-static operation. Our result demonstrates that Szilard's engine's performance based on FLD exceeds other nano-device implementations by orders of magnitude. \
The topic of the PhD project is a comparative study of phase dynamics and macroscopic quantum phenomena in moderately damped NbN, YBaCuO grain boundary (GB) Josephson junctions (JJs) and hybrid devices. This type of research activity responds to the needs of better identifying phase dynamics in JJs in the moderately damped regime, which is going to be more and more common in hybrid nanostructures. Issues on a more detailed understanding of coherence, dissipation and noise in the various devices have a relevant role in the progress of quantum circuits. In the last few years, studies of phase dynamics and macroscopic quantum phenomena have been extended to junctions composed of materials other than the traditional low critical temperature superconductors (LTS) and to novel types of structures with unconventional barriers composed for instance of graphene sheets or of grain boundaries. Progress in engineering new materials into junctions and in understanding and controlling the physics of interfaces may offer novel solutions for junctions of superior quality and complementary functionalities, and therefore may lead in the long run to improve also specific qubit performances. For LTS JJs, once the barrier thickness and the critical current density (Jc) have been fixed, a reduction in its size unavoidably leads to a lowering of the critical current and determines a quite different phase dynamics re-normalized to the new scaling energy. Lower critical currents Ic result in lower Josephson energies EJ, and higher levels of dissipation are expected. The range of the energy dynamical parameters is significantly enlarged, and it is technologically easier to reproducibly realize nontrivial configurations. These devices are characterized by intermediate levels of dissipation (moderately damped regime) and by phase diffusion phenomena. The low Jc limit seems to be characteristic also of all futuristic nanohybrids devices incorporating nanowires, and the moderately damped regime (MDR) is intrinsically more common than it could be expected. Measurements of switching current distribution (SCD) in these last years have turned to be standard tools to investigate phase dynamics in unconventional and hybrid systems and nanostructures. High critical temperature superconductors (HTS) are an example of unconventional systems, because of the d-wave order parameter symmetry and of the presence of low-energy quasiparticles, which are expected to induce high level of dissipation and as a consequence to spoil macroscopic quantum coherence. A comparative study in systems so different and complementary is of great interest. During the PhD program, we have performed measurements of SCD, both in the thermal and quantum regime down to 20 mK, on moderately damped NbN/MgO/NbN JJs and on YBCO biepitaxial GB JJs. The NbN JJs exhibit a crossover from thermal activation (TA) to macroscopic quantum tunneling (MQT) regime at about 90 mK. After the MQT saturation, the width of the SCD follows the expected dependence in the TA regime. Deviations are evident in proximity and above the crossover temperature T* (about 1.6K) where we find effects of anti-correlation between the bath temperature and thermal fluctuations since the width of the histograms starts to collapse, and indications of a transition to a third regime called phase diffusion (PD) are evident, such as the change of the switching distributions symmetry. Experimental data have been analyzed through numerical fitting of the switching probability and Monte Carlo simulations, and we found a good agreement with theoretical expectations based on multiple re-trapping processes, which strongly depend on the junction damping. The combined analysis of these three regimes allowed us to extract the fundamental parameters, which completely characterize the dynamics of such moderately damped NbN JJs. In addition, clear signatures of the PD regime have been detected and numerical methods have been developed to estimate the dissipation level in the MDR. As a term of comparison, the studies on moderately damped systems have been also carried out on HTS junctions, which have different levels of intrinsic dissipation induced by low energy quasi-particles, and are for this reason of great impact as a reference. In such JJs we have clearly observed a direct transition from the MQT to the PD regime. The width of the SCDs is constant below 100mK, which is a clearly indication of a quantum activation regime. Above 100mK the negative temperature derivative of the width is consistent with a diffusive motion due to multiple escape and re-trapping in the potential wells. Thermal behavior of the width has been fitted through numerical simulations, thus providing the estimate of the damping factor. These conclusions of a PD regime also in YBaCuO JJs are supported by data in presence of an externally applied magnetic field, and are consistent with the numerical outcomes. Numerical simulations of the thermal behavior of the SCD width for different values of the quality factor ranging from 1 to 10 have been carried out. Such numerical simulations allow us to reconstruct a (Q, kBT/EJ) phase diagram summarizing the various activation regimes. The transition curve between the PD regime and the running state following thermal or quantum activation has been determined numerically by varying the damping factor Q as function of the ratio between the thermal energy and the Josephson energy. The result of the calculation is a universal curve. A phase diagram valid in a large range of dissipation conditions emerges as a functional guide to classify the switching behavior and to settle the fundamental junction parameters and energies in the MDR. It is therefore a reference for phase dynamics of novel types of junction and system for which the nature of the current induced transition from the superconducting to the normal state has not been completely clarified.
We experimentally investigate the stochastic phase dynamics of planar Josephson junctions (JJs) and superconducting quantum interference devices (SQUIDs) defined in epitaxial InAs/Al heterostructures, and characterized by a large ratio of Josephson energy to charging energy. We observe a crossover from a regime of macroscopic quantum tunneling to one of phase diffusion as a function of temperature, where the transition temperature T^{*} is gate-tunable. The switching probability distributions are shown to be consistent with a small shunt capacitance and moderate damping, resulting in a switching current which is a small fraction of the critical current. Phase locking between two JJs leads to a difference in switching current between that of a JJ measured in isolation and that of the same JJ measured in an asymmetric SQUID loop. In the case of the loop, T^{*} is also tuned by a magnetic flux.
The moderately damped regime in Josephson junctions (JJ) is quite common in devices characterized by low critical currents and, therefore, by low Josephson energies. Measurements of the switching current distribution (SCD) are a direct way of determining the phase dynamics in the nontrivial case of moderate damping, which is becoming more common with advances in nanopatterning superconductors and in materials science for fabricating hybrid systems. We report measurements of SCDs, in both the thermal and quantum regimes for moderately damped YBaCuO grain boundary biepitaxial JJs. A direct transition from a phase diffusion regime to macroscopic quantum tunnelling occurs at about 130 mK. The crossover to the quantum regime is determined by the magnetic field and the phase dynamics can be described by a fully consistent set of junction parameters obtained by numerical simulation.
The research work discussed in this thesis deals with the study of superconducting Josephson qubits. Superconducting qubits are solid-state artificial atoms which are based on lithographically defined Josephson tunnel junctions properties. When sufficiently cooled, these superconducting devices exhibit quantized states of charge, flux or junction phase depending on their design parameters. This allows to observe coherent evolutions of their states. The results presented can be divided into two parts. In a first part we investigate operations of superconducting qubits based on the quantum coherence in superconducting quantum interference devices (SQUID). We explain experimental data which has been observed in a SQUID subjected to fast, large-amplitude modifications of its effective potential shape. The motivations for this work come from the fact that in the past few years there have been attempts to interpret the supposed quantum behavior of physical systems, such as Josephson devices, within a classical framework. Moreover, we analyze the possibility of generating GHZ states, namely maximally entangled states, in a quantum system made out of three Josephson qubits. In particular, we investigate the possible limitations of the GHZ state generation due to coupling to bosonic baths. In the second part of the thesis we address a particular cause of decoherence of flux qubits which has been disregarded until now: thermal gradients, which can arise due to accidental non equilibrium quasiparticle distributions. The reason for these detrimental effects is that heat currents flowing through Josephson tunnel junctions in response to a temperature gradient are periodic functions of the phase difference between the electrodes. The phase dependence of the heat current comes from Andreev reflection, namely an interplay between the quasiparticles which carry heat and the superconducting condensate which is sensitive to the superconducting phase difference. Generally speaking, the flux qubit states are characterized by different values of the phase difference through their Josephson junctions. Consequently, the phase-dependent thermal current through a device subject a temperature gradient is related to the phase-dependent qubit states. We study how the thermal currents change according to the state of the qubits hence yielding a measurement of the qubit state. This in turn leads to an impact of temperature gradient on the dynamics of the system. We show that flux qubits in the Delft qubit design can have limitations of the decoherence time to the order of microseconds as a result of this newly discovered source of decoherence. In contrast, the fluxonium qubit is found to be well protected due to its superinductance.
We have studied the temperature dependence of escape phenomena in various underdamped Josephson junctions (JJs). The junctions had different Josephson coupling energies E J which were relatively small, but larger than the charging energy E C . Upon increasing the temperature T , we first observe the usual cross-over between macroscopic quantum tunnelling and thermally activated (TA) behaviour at temperatures k B T ∼ ℏω p , where ω p is the plasma frequency of the junction. Increasing T further, the width of the switching current distribution has, counterintuitively, a non-monotonic temperature dependence. This can be explained by the novel cross-over from TA behaviour to underdamped phase diffusion. We show that this cross-over is expected to occur at temperatures such that k B T ∼ E J (1 − 4/π Q ) 3 /2, where Q is the quality factor of the junction at the plasma frequency, in agreement with experiment. Our findings can be compared with detailed model calculations which take into account dissipation and level quantization in a metastable well. Particular attention is paid to the sample with the smallest E J , which shows extensive phase diffusion even at the lowest temperatures. This sample consists of a dc-SQUID and a single JJ close to each other, such that the SQUID acts as a tunable inductive protection for the single junction from fluctuations of a dissipative environment. By varying the flux through the dc-SQUID, we present, for the first time, experimental evidence of the escape of a JJ from the phase diffusion regime to the free running state in a tunable environment. We also show that in the zero voltage state the losses mainly occur at frequencies near the plasma resonance.
We report on measurements of the switching current probability both in the thermal and quantum regime on moderately damped NbN/MgO/NbN Josephson junctions with very low critical current density (’ 2A/cm2). We observed three distinct regimes: below 90 mK the width of the switching distributions is constant within the experimental error, while from 90 mK to 1.6 K the width increases with temperature, this change of behavior signaling the standard transition from macroscopic quantum tunneling (MQT) to thermal activation (TA) regime respectively. Above 1.6 K we find effects of anti-correlation between the bath temperature and thermal fluctuations since the width of the histograms starts to collapse, and indications of a transition to a third regime called Phase Diffusion (PD) are evident, such as the change of the switching distributions symmetry. Experimental data have been analyzed through numerical fitting of the switching probability and Monte Carlo simulations, and we found a good agreement with theoretical expectations based on multiple re-trapping processes, which strongly depend on the junction damping. The parameters which completely characterize the dynamics of such moderately damped Josephson junctions have been also extracted.