Statistics of localized phase slips in tunable width planar point contacts

Baumans X.D., Cerbu D., Adami O., Zharinov V.S., Verellen N., Papari G., Scheerder J.E., Zhang G., Moshchalkov V.V., Silhanek A.V., Van de Vondel J.

Superconducting nanowires currently attract great interest due to their application in single-photon detectors and quantum-computing circuits. In this context, it is of fundamental importance to understand the detrimental fluctuations of the superconducting order parameter as the wire width shrinks. In this paper, we use controlled electromigration to narrow down aluminium nanoconstrictions. We demonstrate that a transition from thermally assisted phase slips to quantum phase slips takes place when the cross section becomes less than ∼150 nm2. In the regime dominated by quantum phase slips the nanowire loses its capacity to carry current without dissipation, even at the lowest possible temperature. We also show that the constrictions exhibit a negative magnetoresistance at low-magnetic fields, which can be attributed to the suppression of superconductivity in the contact leads. These findings reveal perspectives of the proposed fabrication method for exploring various fascinating superconducting phenomena in atomic-size contacts. Nanostructured superconductors allow dissipationless electrical transport to be exploited in technologically relevant devices. Here, the authors follow how detrimental fluctuations of the superconducting order parameter evolve in Al atomic contacts as their width is controlled by electromigration.

Massarotti D., Stornaiuolo D., Lucignano P., Galletti L., Born D., Rotoli G., Lombardi F., Longobardi L., Tagliacozzo A., Tafuri F.

We have identified anomalous behavior of the escape rate out of the zero-voltage state in Josephson junctions with a high critical current density Jc. For this study we have employed YBa2Cu3O7-x grain boundary junctions, which span a wide range of Jc and have appropriate electrodynamical parameters. Such high Jc junctions, when hysteretic, do not switch from the superconducting to the normal state following the expected stochastic Josephson distribution, despite having standard Josephson properties such as a Fraunhofer magnetic field pattern. The switching current distributions (SCDs) are consistent with nonequilibrium dynamics taking place on a local rather than a global scale. This means that macroscopic quantum phenomena seem to be practically unattainable for high Jc junctions. We argue that SCDs are an accurate means to measure nonequilibrium effects. This transition from global to local dynamics is of relevance for all kinds of weak links, including the emergent family of nanohybrid Josephson junctions. Therefore caution should be applied in the use of such junctions in, for instance, the search for Majorana fermions.

Massarotti D., Pal A., Rotoli G., Longobardi L., Blamire M.G., Tafuri F.

The interfacial coupling of two materials with different ordered phases, such as a superconductor (S) and a ferromagnet (F), is driving new fundamental physics and innovative applications. For example, the creation of spin-filter Josephson junctions and the demonstration of triplet supercurrents have suggested the potential of a dissipationless version of spintronics based on unconventional superconductivity. Here we demonstrate evidence for active quantum applications of S-F-S junctions, through the observation of macroscopic quantum tunnelling in Josephson junctions with GdN ferromagnetic insulator barriers. We show a clear transition from thermal to quantum regime at a crossover temperature of about 100 mK at zero magnetic field in junctions, which present clear signatures of unconventional superconductivity. Following previous demonstration of passive S-F-S phase shifters in a phase qubit, our result paves the way to the active use of spin filter Josephson systems in quantum hybrid circuits. Spin triplet superconductivity may benefit spintronics, providing dissipation-free spin-polarized currents. Here, the authors demonstrate macroscopic quantum tunnelling in spin filter Josephson junctions containing a ferromagnetic insulator barrier of GdN, evidencing unconventional superconductivity below 100 mK.

Murphy A., Weinberg P., Aref T., Coskun U.C., Vakaryuk V., Levchenko A., Bezryadin A.

We perform measurements of phase-slip-induced switching current events on different types of superconducting weak links and systematically study statistical properties of the switching current distributions. We employ two types of devices in which a weak link is formed either by a superconducting nanowire or by a graphene flake subject to proximity effect. We demonstrate that independently of the nature of the weak link, higher moments of the distribution take universal values. In particular, the third moment (skewness) of the distribution is close to $\ensuremath{-}1$ both in thermal and quantum regimes. The fourth moment (kurtosis) also takes a universal value close to 5. The discovered universality of skewness and kurtosis is confirmed by an analytical model. Our numerical analysis shows that introduction of extraneous noise into the system leads to significant deviations from the universal values. We suggest using the discovered universality of higher moments as a robust tool for checking against undesirable effects on noise in various types of measurements.

Bezryadin A.

Aref T., Levchenko A., Vakaryuk V., Bezryadin A.

We measure quantum and thermal phase-slip rates using the standard deviation of the switching current in superconducting nanowires. Our rigorous quantitative analysis provides firm evidence for the presence of quantum phase slips (QPSs) in homogeneous nanowires at high bias currents. We observe that as temperature is lowered, thermal fluctuations freeze at a characteristic crossover temperature ${T}_{q}$, below which the dispersion of the switching current saturates to a constant value, indicating the presence of QPSs. The scaling of the crossover temperature ${T}_{q}$ with the critical temperature ${T}_{c}$ is linear, ${T}_{q}\ensuremath{\propto}{T}_{c}$, which is consistent with the theory of macroscopic quantum tunneling. We can convert the wires from the initial amorphous phase to a single-crystal phase, in situ, by applying calibrated voltage pulses. This technique allows us to probe directly the effects of the wire resistance, critical temperature, and morphology on thermal and quantum phase slips.

Li P., Wu P.M., Bomze Y., Borzenets I.V., Finkelstein G., Chang A.M.

An aluminum nanowire switches from superconducting to normal as the current is increased in an upsweep. The switching current (I(s)) averaged over upsweeps approximately follows the depairing critical current (I(c)) but falls below it. Fluctuations in I(s) exhibit three distinct regions of behaviors and are nonmonotonic in temperature: saturation well below the critical temperature T(c), an increase as T(2/3) at intermediate temperatures, and a rapid decrease close to T(c). Heat dissipation analysis indicates that a single phase slip is able to trigger switching at low and intermediate temperatures, whereby the T(2/3) dependence arises from the thermal activation of a phase slip, while saturation at low temperatures provides striking evidence that the phase slips by macroscopic quantum tunneling.

Aref T., Bezryadin A.

We present a method for in situ tuning of the critical current (or switching current) and critical temperature of a superconducting MoGe nanowire using high bias voltage pulses. Our main finding is that as the pulse voltage is increased, the nanowire demonstrates a reduction, a minimum and then an enhancement of the switching current and critical temperature. Using controlled pulsing, the switching current of a superconducting nanowire can be set exactly to a desired value. These results correlate with in situ transmission electron microscope imaging where an initially amorphous nanowire transforms into a single crystal nanowire by high bias voltage pulses. We compare our transport measurements to a thermally activated model of Little's phase slips in nanowires.

Hübler F., Lemyre J.C., Beckmann D., v. Löhneysen H.

We explore charge imbalance in mesoscopic normal-metal/superconductor multiterminal structures at very low temperatures. The investigated samples, fabricated by e-beam lithography and shadow evaporation, consist of a superconducting aluminum bar with several copper wires forming tunnel contacts at different distances from each other. We have measured in detail the local and nonlocal conductance of these structures as a function of the applied bias voltage $V$, the applied magnetic field $B$, the temperature $T$, and the contact distance $d$. From these data the charge-imbalance relaxation length ${\ensuremath{\lambda}}_{{Q}^{\ensuremath{\ast}}}$ is derived. The bias-resolved measurements show a transition from dominant elastic scattering close to the energy gap to an inelastic two-stage relaxation at higher bias. We observe a strong suppression of charge imbalance with magnetic field, which can be directly linked to the pair-breaking parameter. In contrast, practically no temperature dependence of the charge-imbalance signal was observed below 0.5 K. These results are relevant for the investigation of other nonlocal effects such as crossed Andreev reflection and spin diffusion.

Bezryadin A., Goldbart P.M.

The application of single molecules as templates for nanodevices is a promising direction for nanotechnology. We use suspended deoxyribonucleic acid molecules or single-walled carbon nanotubes as templates for fabricating superconducting devices and then study these devices at cryogenic temperatures. Because the resulting nanowires are extremely thin, comparable in diameter to the templating molecule itself, their electronic state is highly susceptible to thermal fluctuations. The most important family of these fluctuations are the collective ones, which take the form of Little's phase slips or ruptures of the many-electron organization. These phase slips break the quantum coherence of the superconducting condensate and render the wire slightly resistive (i.e., not fully superconducting), even at temperatures substantially lower than the critical temperature of the superconducting transition. At low temperatures, for which the thermal fluctuations are weak, we observe the effects of quantum fluctuations, which lead to the phenomenon of macroscopic quantum tunneling. The modern fabrication method of molecular templating, reviewed here, can be readily implemented to synthesize nanowires from other materials, such as normal metals, ferromagnetic alloys, and semiconductors.

Pekker D., Shah N., Sahu M., Bezryadin A., Goldbart P.M.

Superconducting nanowires fabricated via carbon-nanotube-templating can be used to realize and study quasi-one-dimensional superconductors. However, measurement of the linear resistance of these nanowires have been inconclusive in determining the low-temperature behavior of phase-slip fluctuations, both quantal and thermal. Thus, we are motivated to study the nonlinear current-voltage characteristics in current-biased nanowires and the stochastic dynamics of superconductive-resistive switching, as a way of probing phase-slip events. In particular, we address the question: Can a single phase-slip event occurring somewhere along the wire--during which the order-parameter fluctuates to zero--induce switching, via the local heating it causes? We explore this and related issues by constructing a stochastic model for the time-evolution of the temperature in a nanowire whose ends are maintained at a fixed temperature. We derive the corresponding master equation as tool for evaluating and analyzing the mean switching time at a given value of current. The model indicates that although, in general, several phase-slip events are necessary to induce switching via a thermal runaway, there is indeed a regime of temperatures and currents in which a single event is sufficient. We carry out a detailed comparison of the results of the model with experimental measurements of the distribution of switching currents, and provide an explanation for the counter-intuitive broadening of the distribution width that is observed upon lowering the temperature. Moreover, we identify a regime in which the experiments are probing individual phase-slip events, and thus offer a way for exploring the physics of nanoscale quantum tunneling of the superconducting order parameter.

Sahu M., Bae M., Rogachev A., Pekker D., Wei T., Shah N., Goldbart P.M., Bezryadin A.

Phase slips are topological fluctuations that carry the superconducting order-parameter field between distinct current-carrying states. Owing to these phase slips, superconducting nanowires acquire electrical resistance. In such wires, it is well known that at higher temperatures phase slips occur through the process of thermal barrier-crossing by the order-parameter field. At low temperatures, the general expectation is that phase slips should proceed through quantum tunnelling events, which are known as quantum phase slips. However, resistive measurements have produced evidence both for and against the occurrence of quantum phase slips. Here, we report evidence for the observation of individual quantum phase-slip events in homogeneous ultranarrow wires at high bias currents. We accomplish this through measurements of the distribution of switching currents for which the width exhibits a rather counter-intuitive, monotonic increase with decreasing temperature. Importantly, measurements show that in nanowires with larger critical currents, quantum fluctuations dominate thermal fluctuations up to higher temperatures. Measurements of the distribution of stochastic switching currents in homogeneous, ultra-narrow superconducting nanowires provide strong evidence that the low-temperature current-switching in such systems occurs through quantum phase slips—topological quantum fluctuations of the superconducting order parameter via which tunnelling occurs between current-carrying states.

Shah N., Pekker D., Goldbart P.M.

We study the stochastic dynamics of superconductive-resistive switching in hysteretic current-biased superconducting nanowires undergoing phase-slip fluctuations. We evaluate the mean switching time using the master-equation formalism, and hence obtain the distribution of switching currents. We find that as the temperature is reduced this distribution initially broadens; only at lower temperatures does it show the narrowing with cooling naively expected for phase slips that are thermally activated. We also find that although several phase-slip events are generally necessary to induce switching, there is an experimentally accessible regime of temperatures and currents for which just one single phase-slip event is sufficient to induce switching, via the local heating it causes.

Altomare F., Chang A.M., Melloch M.R., Hong Y., Tu C.W.

Quantum phase slips have received much attention due to their relevance to superfluids in reduced dimensions and to models of cosmic string production in the early universe. Their establishment in one-dimensional superconductors has remained controversial. Here we study the nonlinear current-voltage characteristics and linear resistance in long superconducting Al wires with lateral dimensions $\ensuremath{\sim}5\text{ }\text{ }\mathrm{nm}$. We find that, in a magnetic field and at temperatures well below the superconducting transition, the observed behaviors can be described by the nonclassical, macroscopic quantum tunneling of phase slips, and are inconsistent with the thermal activation of phase slips.

Strachan D.R., Smith D.E., Fischbein M.D., Johnston D.E., Guiton B.S., Drndić M., Bonnell D.A., Johnson A.T.

Electromigrated nanogaps have shown great promise for use in molecular scale electronics. We have fabricated nanogaps on free-standing transparent SiN(x) membranes which permit the use of transmission electron microscopy (TEM) to image the gaps. The electrodes are formed by extending a recently developed controlled electromigration procedure and yield a nanogap with approximately 5 nm separation clear of any apparent debris. The gaps are stable, on the order of hours as measured by TEM, but over time (months) relax to about 20 nm separation determined by the surface energy of the Au electrodes. A major benefit of electromigrated nanogaps on SiN(x) membranes is that the junction pinches in away from residual metal left from the Au deposition which could act as a parasitic conductance path. This work has implications to the design of clean metallic electrodes for use in nanoscale devices where the precise geometry of the electrode is important.

Domínguez D., Berger J.

Vedin R., Lidmar J.

We study switching current distributions in superconducting nanostrips using theoretical models and numerical simulations. Switching current distributions are commonly measured in experiments and may provide a window into the microscopic switching mechanisms. As the current through a superconducting strip is increased from zero it will at some point switch to the normal dissipative state. Due to thermal and/or quantum fluctuations the switching current will be random and follow a certain distribution depending on sweep rate, temperature, material properties, wire length, and width. By analyzing the resulting distribution it is possible to infer the transition rate for a switch, which can be related to the free-energy barrier separating the metastable superconducting state and the normal one. We study different switching scenarios and show using simulations how data taken for different sweep rates can be combined to obtain the switching rate over a wider interval of currents. In doing this, it is necessary to account for a time delay between the initiation of the switching event and its detection. Published by the American Physical Society 2025

Uhl K., Hackenbeck D., Koelle D., Kleiner R., Bothner D.

Nulens L., Chaves D.A., Harb O.J., Scheerder J.E., Lejeune N., Brahim K., Raes B., Silhanek A.V., Van Bael M.J., Van de Vondel J.

Massarotti D.

The novel opportunities offered by nanotechnologies and materials science have enlarged the physical conditions of occurrence of the Josephson effect. Semiconducting, ferromagnetic, topological-insulator, and graphene barriers lead to unconventional and anomalous aspects of the Josephson coupling, which might be useful to respond to some issues on key problems of solid-state physics. However, the complexity of the layout and the competing physical processes occurring in the junctions pose novel questions on the interpretation of their phenomenology. Measurements of critical current fluctuations have turned to be imaging tools of different dissipation processes, whose origin may depend on the type of the barrier, the nature of the interface, the geometry and dimensionality of the device. In this chapter, we classify some significant behaviors of unconventional junctions in terms of the temperature properties of the critical current fluctuations, thus providing accurate arguments to distinguish and understand physical processes occurring in quite different dynamical regimes. These notions are universal and apply to all kinds of junctions.

Nulens L., Dausy H., Wyszyński M.J., Raes B., Van Bael M.J., Milošević M.V., Van de Vondel J.

We fabricated an asymmetric nanoscale SQUID consisting of one nanobridge weak link and one Dayem bridge weak link. The current phase relation of these particular weak links is characterized by multivaluedness and linearity. While the latter is responsible for a particular magnetic field dependence of the critical current (so-called vorticity diamonds), the former enables the possibility of different vorticity states (phase winding numbers) existing at one magnetic field value. In experiments the observed critical current value is stochastic in nature, does not necessarily coincide with the current associated with the lowest energy state and critically depends on the measurement conditions. In this paper, we unravel the origin of the observed metastability as a result of the phase dynamics happening during the freezing process and while sweeping the current. Moreover, we employ special measurement protocols to prepare the desired vorticity state and identify the (hidden) phase slip dynamics ruling the detected state of these nanodevices. In order to gain insights into the dynamics of the condensate and, more specifically the hidden phase slips, we performed time-dependent Ginzburg-Landau simulations.

Harrabi K., Mekki A., Bahlouli H., Mathieu P.

• The work presented in this paper is a study of dissipative state in NbTi thin film on a sapphire substrate. In response to an electric step current exceeding the critical current I c , a voltage appears after a certain delay time td. Near the transition temperature T c , the type of dissipation was identified as phase slip center. And the delay of nucleation is analyzed according to a Time-Dependent Ginzburg-Landau system of equations, leaving a unique time parameter for fitting: the thermal cooling time of the film on its substrate. Although different, this situation bears similarities with photon detection using a Superconducting Single Photon Detector (SSPD), which implies the formation of a dissipative on the site of absorption; absorption; therefore, its repetition rate performance is limited by the film cooling time (reset time). Moreover, estimation of the temperatures attained in the core of the phase slip center from the blackbody radiation theory of the acoustic phonons revealed that they were larger than to be larger than the substrate temperature and remained below the transition temperature T c . • Hereby, it is understood that the above-mentioned manuscript has not been published, nor accepted for publication, elsewhere, or under editorial review for publication elsewhere. We have investigated the dynamics of dissipative states in NbTi superconducting thin films using electrical step pulse excitations on a sapphire substrate close to T c . Excitation of a superconducting wire with a current pulse larger than the depairing critical current, I c , gives rise to a non-equilibrium superconducting state. We identified two dissipative states, the hotspot (HS) and the phase slip center (PSC). Both are signaled by a voltage that emerges after a certain delay time t d . We focused on the phase slip phenomenon in the vicinity of T c and measured the dependence of the delay times on the value of the applied current pulse; these were fitted with the time-dependent Ginzburg-Landau (TDGL) theory due to Tinkham. The film thermal relaxation times were deduced for the phase slip center close to T c for different samples. In addition, the temperatures at the center of the PSCs were estimated from the blackbody radiation model, and the results are consistent with the PSC formalism.

Dausy H., Nulens L., Raes B., Van Bael M.J., Van de Vondel J.

In this work, we study the current-phase relation ($\mathrm{C}\mathrm{\ensuremath{\Phi}}\mathrm{R}$) of lithographically fabricated molybdenum germanium (${\mathrm{Mo}}_{79}{\mathrm{Ge}}_{21}$) nanobridges that is intimately linked with the nanobridge's kinetic inductance. We do this by imbedding the nanobridges in a superconducting quantum interference device (SQUID). We observe that, for temperatures far below ${T}_{c}$, the $\mathrm{C}\mathrm{\ensuremath{\Phi}}\mathrm{R}$ is linear, as long as the condensate is not weakened by the presence of a supercurrent. We demonstrate lithographic control over the nanobridge kinetic inductance, which scales with the nanobridge aspect ratio. This allows the ${I}_{c}(B)$ characteristic of the SQUID to be tuned. The SQUID properties that can be controlled in this way include the SQUID's sensitivity and the positions of the critical-current maxima. These observations can be of use for the design and operation of future superconducting devices, such as magnetic memories or flux qubits.

Zgirski M., Foltyn M., Savin A., Norowski K.

We study correlated switchings of a superconducting nanobridge probed with a train of current pulses. For pulses with a low repetition rate each pulse transits the superconducting bridge to the normal state with probability $P$ independent of the outcomes in the preceding pulses. We show that with the reduction of the time interval between pulses long-range correlation between pulses occurs: stochastic switching in a single pulse raises the temperature of the bridge and affects the outcome of the probing for the next pulses. As a result, an artificial intricate stochastic process with an adjustable strength of correlation is produced. We identify the regime where apparent switching probability exhibits the thermal hysteresis with discontinuity at a critical current amplitude of the probing pulse. This engineered stochastic process can be viewed as an artificial phase transition and provides an interesting framework for studying correlated systems. The process resembles the familiar transition from the superconducting to normal state in the current-bias nanowire, proceeding through a phase slip avalanche. Due to its extreme sensitivity to the control parameter, i.e., electric current, temperature, or magnetic field, it offers the opportunity for ultrasensitive detection.

Harrabi K., Mekki A., Bahlouli H., Ladan F.R.

We have studied the characteristics of different metastable states in NbTi thin film deposited on sapphire substrate in a region very close to the transition temperature T $$_{c}$$ , which was estimated to be about 7.6 K in our sample. Localized dissipative zones are induced (phase-slip centers (PSC) and hot spots (HS)) when a current pulse larger than the depairing critical ( $$I_{c}$$ ) current is sent through the filament. These resistive zones appear after a delay time at zero voltage (transient superconductivity) that depends on the thermal cooling time of the material. A time-dependent Ginzburg–Landau (TDGL) theory developed by M. Tinkham allows to extract the gap relaxation time from the measured time delays at a temperature very close to T $$_{c}$$ . Furthermore, it appears that, well below T $$_{c}$$ , this relaxation time is dominated by the thermal equilibration time of the film on its substrate. In addition, the niobium-based material showed a clear evidence that PSC can be considered as precursors for hot spots.

Collienne S., Raes B., Keijers W., Linek J., Koelle D., Kleiner R., Kramer R.B., Van de Vondel J., Silhanek A.V.

In this work, we show that targeted and controlled modifications of the Josephson-junction properties of a bridge-type $\mathrm{Nb}$ nanoSQUID can be achieved by an electroannealing process allowing us to tune and tailor the response of a single device. The electroannealing consists in substantial Joule heating produced by large current densities followed by a rapid temperature quench. We report on a highly nontrivial evolution of the material properties when performing subsequent electroannealing steps. As the current density is increased, an initial stage characterized by a modest improvement of the superconducting critical temperature and normal-state conductivity of the bridges, is observed. This is followed by a rapid deterioration of the junction properties, i.e., decrease of critical temperature and conductivity. Strikingly, further electroannealing leads to a noteworthy recovery before irreversible damage is produced. Within the electroannealing regime where this remarkable resurrection of the superconducting properties are observed, the nanoSQUID can be operated in nonhysteretic mode in the whole temperature range and without compromising the critical temperature of the device. The proposed postprocessing is particularly appealing in view of its simplicity and robustness.

Puglia C., De Simoni G., Giazotto F.

The investigation of the switching current probability distribution of a Josephson junction is a conventional tool to gain information on the phase slips dynamics as a function of the temperature. Here we adopt this well-established technique to probe the impact of an external static electric field on the occurrence of phase slips in gated all-metallic titanium (Ti) Josephson weak links. We show, in a temperature range between 20 mK and 420 mK, that the evolution of the phase slips dynamics as a function of the electrostatic field starkly differs from that observed as a function of the temperature. This fact demonstrates, on the one hand, that the electric field suppression of the critical current is not simply related to a conventional thermal-like quasiparticle overheating in the weak-link region. On the other hand, our results may open the way to operate an electrostatic-driven manipulation of phase slips in metallic Josephson nanojunctions, which can be pivotal for the control of decoherence in superconducting nanostructures.

Parlato L., Salvoni D., Ejrnaes M., Massarotti D., Caruso R., Satariano R., Tafuri F., Yang X.Y., You L., Wang Z., Pepe G.P., Cristiano R.

We report on measurements of the switching current distributions on two-dimensional NbN superconducting nanostrip single-photon detectors (SNSPD), 5 nm thick and 80 nm wide, in an interval of temperatures from 6 K down to 0.3 K and compare the data with those obtained for similar NbTiN nanostrips. The standard deviations of the switching distributions show an extended region at high temperatures where multiple phase slip switching events occur. This is probably related to a decreasing critical current and an increasing electron and phonon heat capacities. In this temperature region, the width of the switching distribution, and therefore the dark count rate, is considerably reduced down to values below those observed at the lowest temperature. Finally, we also quantify the energy scale of the fluctuation phenomena. The proposed experimental approach may result in a powerful tool for the diagnostic of SNSPD operation mode.

Harrabi K.

We report on the study of the transient voltage response of superconducting NbTi strips to an over-critical current pulse ($${I}> {I}_\mathrm{c}$$, where $${I}_\mathrm{c}$$ is the pair-breaking current). In this experiment, a localized normal spot appeared for a current amplitude larger than the critical current. The induced metastable superconducting state was identified as either a hotspot or phase-slip center. These two dissipative modes share the feature of the voltage response occurring after a delay time $${t}_\mathrm{d}$$, a solution of the time-dependent Ginzburg–Landau (TDGL) theory developed by M. Tinkham. The gap relaxation time was subsequently deduced from fitting the experimental data with the TDGL theory. An agreement was found by choosing an effective gap relaxation time $$\tau _{\Delta }= 4.75 \, \hbox {ns}$$ for a thickness of 50 nm. Assuming the proportionality to sample thickness, this indicates a thermal relaxation time of 96 ps/nm for a NbTi film sputtered at room temperature on polished crystalline $$\hbox {Al}_{{2}} \hbox {O}_{{3}}$$. If we assume that the electron and the phonon specific heats have the same ratio as for pure Nb, then it results in a phonon heat escape time $$\tau _\mathrm{es}/{d} = 32 \, \hbox {ps/nm}$$.

Massarotti D., Tafuri F.

We will review concepts, theory and experimental results on whether and in which conditions a quantum system, governed by a single macroscopic degree of freedom interacting with its environment, can tunnel out of a metastable state. The macroscopic quantum tunneling (MQT) experiments discussed in this chapter demonstrate that $$\varphi $$ is indeed a quantum variable. Differently from the tunneling of a microscopic entity, coupling to the environment plays a major role in the macroscopic analog, and can be so strong that the motion in the classically accessible region is highly damped.