Phase Diffusion in Low-E_J Josephson Junctions at Milli-Kelvin Temperatures

Abstract

Josephson junctions (JJs) with Josephson energy $E_J\lesssim 1$ K are widely employed as non-linear elements in superconducting circuits for quantum computing operating at milli-Kelvin temperatures. In the qubits with small charging energy $E_C$ ( $E_J/E_C\gg 1$ ), such as the transmon, the incoherent phase slips (IPS) might become the dominant source of dissipation with decreasing $E_J$. In this work, a systematic study of the IPS in low-$E_J$ JJs at milli-Kelvin temperatures is reported. Strong suppression of the critical (switching) current and a very rapid growth of the zero-bias resistance due to the IPS are observed with decreasing $E_J$ below 1 K. With further improvement of coherence of superconducting qubits, the observed IPS-induced dissipation might limit the performance of qubits based on low-$E_J$ junctions. These results point the way to future improvements of such qubits.

1. Introduction

Josephson junctions (JJs) with the Josephson energy $0.1$ K$<E_J<1$ K have been recently employed as non-linear elements of superconducting qubits (see, e.g., [1,2,3,4]). Though $E_J$ of these junctions remains much greater than the physical temperature of qubits (∼20 ÷ 50 mK), a non-zero rate of thermally-activated phase slips in these junctions might soon limit the coherence of superconducting qubits. Indeed, with the qubit coherence time exceeding 1 ms [5], even rare dissipative events might become significant. Thus, the study of incoherent phase slips, induced by either equilibrium (thermal) or non-equilibrium noise, might help better understand the limitations of the low-$E_J$ JJs as elements of quantum circuits operating at mK temperatures.

In the past, phase slips in JJs [6] and associated phase diffusion [7,8,9,10] attracted a great deal of experimental and theoretical attention. This effort was mainly aimed at better understanding of a crossover from the classical Josephson behavior (well-defined phase difference, strong quantum fluctuations of charge) to the Coulomb-blockade regime (localized charges, strong quantum fluctuations of phase) (see, e.g., [11,12,13,14] and references therein). The crossover is observed in ultra-small JJs with the Josephson energy $E_J$ of the same order of magnitude as the Coulomb energy $E_C={\left(2e\right)}^2/\left(2C_J\right)$ ($C_J$ is the effective JJ capacitance) provided the junctions are included in a circuit with the impedance Z greatly exceeding the quantum resistance $R_Q=h/{\left(2e\right)}^2\approx 6.5\mathrm{k}\Omega$. The rate of the coherent phase slip processes (the so-called quantum phase slips, or QPS) exponentially increases with decreasing the ratio $E_J/E_C$ [15].

The qubit dephasing by QPS [16] has been considered for the charge-sensitive transmon qubits [17] and heavy fluxonium qubits [1,18,19]. However, energy relaxation due to the incoherent phase slips in the ubiquitous elements of the multi-qubit circuits, the charge-insensitive transmons, has never been addressed. To fill this gap, in the current paper the dissipative processes in the low-$E_J$ Josephson junctions are systematically studied in the regime $E_J\gtrsim T\gg E_C$, which is relevant for operation of superconducting qubits shunted with a large external capacitance [13,14,15]. In this regime, quantum fluctuations of charge are strongly enhanced and the QPS are suppressed. To explore the dynamics of low-$E_J$ junctions at mK temperatures, we designed JJs with $E_J$ = 0.1–1 K and $E_C<10$ mK, and studied the IPS in these JJs with low-frequency transport measurements. The JJs in this work have been implemented as SQUIDs to allow in-situ tuning of $E_J$ by the magnetic field.

The paper is organized as follows. In Section 2, the known facts about the phase diffusion induced by incoherent phase slips in underdamped JJs are briefly reviewed. The sample design and experimental techniques are discussed in Section 3. The measurements of current-voltage characteristics (IVC) of low-$E_J$ devices are presented in Section 4. In Section 5, our results and the data reported by other groups are discussed with the focus on implications of the dissipation induced by incoherent phase slips for the operation of qubits that employ low-$E_J$ Josephson junctions. The conclusions are presented in Section 6.

2. Phase Diffusion in Underdamped Junctions

At $T=0$, the critical current $I_C^{AB}$ of a “classical” JJ ($E_J\gg E_C$) is provided by the Ambegaokar–Baratoff relation [6]

$$I_C^{AB}(T=0)=\frac{2e}{\hslash }{E}_J=\frac{\pi \Delta \left(0\right)}{2e{R}_N},$$

(1)

where $\Delta$ is the superconducting energy gap and $R_N$ is the normal-state resistance of a JJ. This relation has been derived by neglecting phase fluctuations. In the absence of non-equilibrium noise and charging effects, the voltage drop across a JJ is expected to be zero at $I<I_C^{AB}(T=0)$. The quantum phase fluctuations, which become strong at $E_J\lesssim E_C$, result in the so-called coherent quantum phase slips (CPS) in one-dimensional JJ chains (see [20,21,22] and references therein). The junction capacitance C plays the role of the effective mass of a fictitious particle that tunnels between the minima of the “washboard” potential $U\left(\phi \right)=-E_{J}cos\phi -\frac{\hslash I}{2e}\phi$[6]. Reduction of C and, thus, increase of $E_C$, facilitates tunneling and promotes CPS. The CPS shift the system energy levels and renormalize the effective Josephson coupling $E_J^{*}\sim E_J^2/E_C$. The CPS also result in energy dissipation in one-dimensional superconducting wires [23,24] as well as in the external circuits connected to the Josephson junctions.

In the regime $E_J\gg E_C$, on the other hand, the incoherent classical phase slips (IPS) induced by either non-zero temperature or non-equilibrium noise are expected to dominate. The IPS correspond to the over-the-barrier activation in the washboard potential [6]. In thermal equilibrium, the IPS rate depends exponentially on the temperature: ${\nu }_{IPS}\propto {\omega }_p{e}^{-\Delta U/k_{B}T}$[6]. Here, ${\omega }_p=\frac{1}{\hslash }\sqrt{2E_J{E}_C}$ is the plasma frequency which plays the role of the attempt rate, $\Delta U$ is the height of the potential barrier, which is close to $2E_J$ at currents $I\ll I_C^{AB}$.

The IPS process is analogous to a single flux quantum ${\Phi }_0$ crossing a JJ (the process is dual to the transfer of a single Cooper pair through the JJ [25]). Each phase slip generates a voltage drop $V\left(t\right)$ across the JJ, such that $\int V\left(t\right)dt={\Phi }_0$ and, in the presence of a current I, releases an energy $I{\Phi }_0$. Thus, the zero-voltage state can be destroyed by the energy dissipation due to the time-dependent phase fluctuations. At zero tilt of the “washboard” potential $U\left(\phi \right)$, the phase slips with different signs of the phase change occur with the same probability and, as a result, the average voltage across the junction is zero. However, when the junction is biased with a non-zero current I, the tilt of the washboard potential breaks the symmetry and a non-zero average voltage proportional to the phase slip rate is generated across the junction.

The dynamics of JJs depends on all sources of dissipation, such as IPS, thermally excited quasiparticles, etc. The low-dissipative (underdamped) regime, observed at $T\ll \Delta$ and in a high-impedance environment, is relevant to the operation of superconducting qubits. Typically, dissipation is highly frequency-dependent: it is strongly suppressed at low frequencies $\omega \ll {\omega }_p$ and, potentially, significantly enhanced at frequencies approaching ${\omega }_p$. This frequency-dependent dissipation leads to the phenomenon of underdamped phase diffusion [7,8,26]. Characteristic signatures of this regime are the absence of the zero-voltage superconducting state and the existence of a low-voltage ($V\ll \frac{2\Delta }{e}$) IVC branch, which extends up to $I_{SW}\ll I_C^{AB}$. The IVC is hysteric at currents $I<I_{SW}$: the low-V branch observed with increasing the current from 0 to $I_{SW}$ coexists with a high-voltage ($V\ge \frac{2\Delta }{e}$) branch observed with decreasing the current from $I>I_{SW}$ to zero. At high voltages $V>\frac{2\Delta }{e}$, the main dissipation mechanism is the Cooper pair breaking and generation of non-equilibrium quasiparticles. In the low-voltage state $V<\frac{2\Delta }{e}$, the energy gained by a system in the process of the over-the-barrier activation is dissipated mostly due to the Josephson radiation [27].

The theory of the DC transport in underdamped Josephson junctions in the regime $E_C\ll T\le E_J<\Delta$ in presence of a stochastic noise has been developed by Ivanchenko and Zilberman [28] (the IZ theory, see Appendix C). The IZ theory predicts that $I_{SW}\propto E_J^2$ at small $E_J$[29], in contrast to the dependence $I_C^{AB}\propto E_J$ for the regime $E_C,T\ll E_J$ (Equation (1)).

More recent analysis of the effect of non-zero temperature in the underdamped junctions was provided by Kivioja et al. [30]. By considering the quality factor at the plasma frequency, $Q\left({\omega }_p\right)$, and the energy dissipated between adjacent potential maxima $\Delta E_D\approx 8E_J/$$Q\left({\omega }_p\right)$, Kivioja et al. showed that the maximum possible power dissipated due to phase diffusion before switching to a state with $V\approx 2\Delta /e$ can be expressed as

$$\frac{2\pi V}{{\Phi }_0}\times \frac{\Delta E_D}{2\pi }=V\times I_{SW},$$

(2)

where $I_{SW}=4I_C/$$\pi Q$ is the maximum possible current carried by underdamped junctions in the phase diffusion (UPD) regime. At $I<I_{SW}$, there is a non-zero probability for a fictitious particle to be retrapped after escape from a local minimum of the potential $U\left(\phi \right)$. As a result, instead of a run-away state with $V=2\Delta /e$, the IVC demonstrates a non-zero slope at $I<I_{SW}$ due to the phase diffusion. The value of $R_0$, therefore, provides valuable information regarding the nature of damping in the junction circuits.

3. Experimental Techniques

All the samples studied in this work have been implemented as SQUIDs, in order to be able to in-situ tune $E_J$ by changing the magnetic flux $\Phi$ in the SQUID loop [6]:

$$E_J=2E_{J0}cos\left(\pi \frac{\Phi }{{\Phi }_0}\right),$$

(3)

Figure 1 schematically shows the design of a chain of SQUIDs formed by small ($0.2\times 0.2$ $\mathsf{\mu }$m${}^2$) JJs. The area of the SQUID loop, $A_{SQUID}$, varied between 6 $\mathsf{\mu }$m${}^2$ and 50 $\mathsf{\mu }$m${}^2$. The chains of SQUIDs had additional contact pads (shown in yellow in Figure 1) to provide access to individual SQUIDs or pairs of SQUIDs within a chain.

In order to reduce the rate of quantum phase slips ${\nu }_{QPS}\propto exp\left(-\sqrt{\frac{2E_J}{E_C}}\right)$[6], either the low-transparency JJs junctions with a relatively large in-plane area $A_{JJ}$ ($>1$ $\mathsf{\mu }$m${}^2$), or smaller junctions shunted with external capacitors (the design details are provided in Appendix A) have been used. In both cases, the charging energy $E_C$ was reduced below $\approx 10$ mK, and this allowed us to maintain a large ratio $E_J/E_C$ for all studied JJs.

The amplitude of variations of $E_J$ with the external magnetic field depends on scattering of parameters of individual JJs that form a nominally symmetric SQUID. This scattering did not exceed $10\%$ for the JJs with the normal-state resistance $R_N\approx$ 1 k$\Omega$ and $A_{JJ}$ = 0.02 $\mathsf{\mu }$m${}^2$. However, fabrication of the low-transparency JJs with $R_N\approx$ 100 k$\Omega$ and $A_{JJ}$ = 4 $\mathsf{\mu }$m${}^2$ (the nominal critical current density $I_C^{AB}/A_{JJ}\approx 5\times {10}^{-4}A/$cm${}^2$), which required very long oxidation times and high partial pressure of $O_2$, resulted in a larger ($\approx 30\%$) scattering of the $R_N$ values (Appendix A). This scattering was one of the reasons for different dependence $I_{SW}\left(B\right)$ observed for the nominally identical SQUID chains (see below). The parameters of representative samples are listed in

Table 1. Parameters of single Josephson junctions in SQUID chains. $R_N$ and $A_{JJ}$ are the normal-state resistance and the junction area, respectively. The Josephson energy $E_J=\pi \hslash \Delta /\left({\left(2e\right)}^2{R}_N\right)$ has been calculated using $R_N$ and $T_C=1.3$ K. The charging energy $E_C$, where C is the shunting capacitance, did not exceed 10 mK for all samples. The critical current $I_C^{AB}$ was calculated using Equation (1).

Sample	${\mathit{R}}_{\mathit{N}}$ (k$\mathbf{\Omega }$)	${\mathit{E}}_{\mathit{J}}$ (K)	${\mathit{A}}_{\mathit{JJ}}$ ($\mathsf{\mu }$m${}^2$)	${\mathit{I}}_{\mathit{C}}^{\mathit{AB}}$	${\mathit{I}}_{\mathit{SW}}$ (nA)
1	2.4	2.9	1.9	130	48
2	2.9	2.4	3.74	107	68
3	9.4	0.76	0.04	33	9
4	15.8	0.45	0.04	20	0.3
5	16.6	0.43	0.04	19	0.1
6	175	0.04	0.04	1.8	0.003

(the total number of tested samples exceeded 50 [31]).

4. Current-Voltage Characteristics of Low-${\mathit{E}}_{\mathit{J}}$ Junctions

The results below focus on data obtained at $T<200$ mK—in this temperature range one can neglect transport of the thermally-excited quasiparticles in $Al$-based superconducting circuits. Typical IVC measured at $T_{base}=25$ mK for the samples with $E_J\approx 1$ K and $E_J\ll 1$ K are shown in Figure 2. Several characteristic features of the IVC are addressed below.

4.1. The Switching Current $I_{SW}$ and the Zero-Bias Resistance $R_0$

Figure 2 shows how the switching current $I_{SW}$ and the zero-bias resistance $R_0$ measured at small DC voltages $V\ll 2\Delta /e$ and currents $I\ll I_{SW}$ are determined. Note that the zero-bias resistance per junction is twice as large as the zero-bias resistance of a SQUID. For the JJs with $E_J=0.76$ K (Figure 2a) a non-zero $R_0$ could not be detected within the accuracy of measurements conducted in this work ($\approx {10}^2\sim {10}^3\Omega$, depending on the magnitude of $I_{SW}$). This is the behavior expected in the “classical” regime $E_J\gg T,E_C$. At currents $I>I_{SW}$, the voltage across the chain approaches the value $N\times 2\Delta /e$, where N is the number of SQUIDs in the chain and $2\Delta \approx e\times 0.4$ mV is the sum of superconducting energy gaps in the electrodes that form a junction. For the chains with $E_J\ll 1$ K, the switching current is several orders of magnitude smaller than the Ambegaokar–Baratoff critical current (Figure 2c). With the magnetic field B approaching the value ${\Phi }_0/\left(2A_{SQUID}\right)$, the switching current vanishes and $R_0$ increases by orders of magnitude (red curves in Figure 2a,c). The IVC at $\Phi ={\Phi }_0/2$ resemble that observed in the Coulomb-blockade regime. Note that for most of the studied samples in this regime $E_C$ is close to the base temperature, so the Coulomb blockade is partially suppressed by thermal effects. The resistance $R_0(\Phi ={\Phi }_0/2)$ for the samples with $E_J\ll 1$ K is limited by the input resistance of the preamplifier (a few $G\Omega$).

The evolution of the IVC measured at different temperatures for $\Phi ={\Phi }_0/2$ is shown in Figure 3a. The $R_0\left(T\right)$ drop observed with an increase of temperature at $T>0.2$ K (Figure 3b) is due to an increasing concentration of thermally excited quasiparticles in $Al$ electrodes: the JJ becomes “shunted” by the quasiparticle current. The dependence $R_0\left(T\right)$ at $T>0.25$ K can be approximated by the Arrhenius dependence $R_0(\Phi ={\Phi }_0/2,T)\propto exp(\delta /k_{B}T)$ with $\delta \approx 2.1$ K. The activation energy $\delta$ is close to the superconducting energy gap $\Delta \approx 2.3$ K in Al electrodes with $T_C\approx 1.3$ K.

4.2. The IVC Hysteresis

For all studied samples, strong hysteresis of the IVC is observed at $\Phi =n{\Phi }_0$ where n is integer. The hysteresis is a signature of the underdamped junctions with the McCumber parameter $\beta \gg 1$ [32]. Observation of the hysteresis is also an indication that the noise currents $I_N$ in the measuring setup are significantly smaller than the switching current even for the samples with $I_{SW}$ in the sub-pico-A range (in the opposite limit, $I_N>I_{SW}$, the hysteresis vanishes, see Appendix C and Ref. [33]).

4.3. The $I_{SW}\left(B\right)$ Dependences

Even in the “classical” regime $E_J\gg E_C,T$, several unexpected results are obtained. Firstly, the dependence of $I_{SW}(\Phi )$ for some samples significantly deviated from the dependence

$$I_{SW}(\Phi )=I_{SW}(\Phi =0)\times \mid cos\left(\pi \frac{\Phi }{{\Phi }_0}\right)\mid ,$$

(4)

(see Figure 4c). These deviations can be at least partially explained by a relatively large scattering of parameters of individual JJs and non-uniformity of the local magnetic field in the SQUID loops due to the magnetic field focusing. Observation of a steeper drop of $I_{SW}$ with $\Phi \to 0.5{\Phi }_0$ than that predicted by Equation (3) can be attributed to violation of the condition $E_J(\Phi )\gg E_C$ and crossover to the Coulomb blockade regime.

Secondly, sub-gap ($V_{subgap}<2\Delta /e$) voltage steps on the IVC are observed (Figure 4a), which significantly reduced the accuracy of extraction of $I_{SW}$ and $R_0$ at the values of $\Phi$ close to ${\Phi }_0/2$. A possible reason for appearance of sub-gap steps might be the Fiske resonances due to the microwave resonant modes of the circuit [34]. Identifying the circuit elements that would be responsible for the corresponding resonance frequencies at $f_{res}=(75\mathsf{\mu }$eV$)/$$h\approx 18$ GHz (this frequency corresponds to a wavelength ∼2.5 mm for the electromagnetic wave propagating along the interface between a silicon substrate and vacuum) requires further investigation.

5. Discussion

There are several potential sources of dissipation in Josephson circuits at $T\ll \Delta$, such as non-equilibrium quasiparticles or two-level systems in the circuit environment (see, e.g., [35] and references therein). However, we are unaware of a mechanism other than the IPS that would explain the observed strong dependence of dissipation on the ratio $E_J/T$. Below, we focus on the IPS as the dominant dissipation mechanism in this work.

5.1. The Switching Currents $I_{SW}$

The results of $I_{SW}$ measurements for samples with different $E_J$ are summarized in Figure 5. Here, the data for four samples are plotted at different values of external magnetic flux threading the SQUIDs loop. The effective $E_J$ for these devices was calculated using Equation (2). For comparison, in Figure 5 the $I_{SW}$ data reported in literature for the JJs with large values of $E_J/E_C$ are also plotted. Scattering of $I_{SW}$ values at a given $E_J$ can be attributed, at least partially, to different noise levels in different experimental setups, and to a relatively wide range of $E_J/E_C$ values (see Table 2)—proliferation of QPS with decreasing $E_J/E_C$ can strongly reduce $I_{SW}$. Despite the data scattering, the general trend is clear: with decreasing $E_J$, the switching current decreases significantly faster than the linear dependence $I_{SW}\left(E_J\right)$ predicted by the Ambegaokar–Baratoff relationship (Equation (1)). At $E_J=$ 0.1 K, $I_{SW}$ is already two orders of magnitude smaller than $I_C^{AB}(T=0)$.

Qualitatively, the data are in agreement with theory of the DC transport in underdamped Josephson junctions with $E_C\ll E_J$ developed by Ivanchenko and Zilberman [28]. In the regime $E_C\ll T\le E_J<\Delta$, dissipation in the JJ circuits is associated with IPS induced by either an equilibrium noise generated by thermal excitation or a non-equilibrium noise. According to the IZ theory, the equation for the phase $\phi$ across a classical Josephson junction ($E_C\ll E_J$) can be written as

$$\frac{2e}{\hslash }(I_b+I_n)=\frac{1}{R}\frac{\partial \phi }{\partial t}+\frac{2e}{\hslash }{I}_{C}sin\phi ,$$

(5)

where $I_b$ is the bias current, $\langle I_n\left(0\right)I_n\left(\tau \right)\rangle \ge \frac{2k_{B}T}{R}\delta \left(\tau \right)$ is the delta-correlated Johnson–Nyquist noise across the resistance R connected in parallel with the junction. The superconducting part of the current as a function of bias voltage $V_B$ can be found by solving the corresponding Fokker–Planck equation (see Appendix D). With decreasing $E_J$, the theory predicts a quadratic drop of the maximum superconducting current that the junction can sustain (i.e., the switching current $I_{SW}$) [29]. This maximum value of the switching current is realized at a non-zero voltage $V_n$, which depends only on the voltage noise amplitude, so the zero-bias resistance in the IPS regime is expected to scale as $E_J^{-2}$.

The dependence $I_{SW}\left(E_J\right)$ predicted by the IZ theory in presence of additional Gaussian noise with amplitude of $V_{noise}=24$ $\mathsf{\mu }$V is plotted in Figure 5. This noise corresponds to the Johnson–Nyquist noise $\delta V_t=\sqrt{4k_{B}TR\Delta f}$ generated at $T=50$ mK by two 100 k$\Omega$ resistors connected in series with the device (Figure A2 in Appendix B). These chip resistors, designed for microwave applications, had a very small imaginary part of their impedance. The bandwidth was estimated as $\Delta f\approx {\omega }_p/2\pi$, where ${\omega }_p/2\pi \approx 1$ GHz is the plasma frequency of the shunted JJs. Though the theory (the solid curve in Figure 5) is in qualitative agreement with the experimental data, most of the experimental $I_{SW}$ values are an order-of-magnitude smaller than $I_{SW}$ predicted by the IZ theory. A possible explanation for this discrepancy might be more complex phase dynamics in the devices with a very high IPS rate, outside of the limits of applicability of the IZ theory. Another possibility is the exponentially strong sensitivity of the IPS rate to the noise level in different setups and the physical temperature of samples, the parameters that are not easy to control in most experiments.

5.2. The Zero-Bias Resistance $R_0$

Figure 6 shows $R_0$ as a function of $E_J$ measured in this work and by other groups for $Al/Al{O}_X/Al$ junctions. To simplify the Figure, the data in this work are plotted only for the sample with the lowest values of $R_0$ (sample 5); $R_0$ for other samples are approximately in line with the data from literature shown in Figure 6. The zero-bias resistance, being unmeasurably low at $E_J>1$ K, rapidly increases at $E_J<1$ K, and becomes much greater than the normal-state resistance $R_N$ at $E_J\le 0.1$ K. Instead of a well-defined “superconductor-to-insulator” transition at a certain value of $E_J/E_C$, a broad crossover between these two limiting regimes is observed. Note that different JJ samples (single junctions and arrays) demonstrate similar values of $R_0$, though their charging energies could vary over a wide range.

Figure 5 and Figure 6 show that the findings are in good agreement with the literature data on the highest values of $I_{SW}$ and lowest values of $R_0$ measured for low-$E_J$ junctions. Despite large scattering of the data in Figure 5 and Figure 6, a very rapid drop of $I_{SW}$ and increase of $R_0$ has been observed in most of the experiments as soon as $E_J$ becomes significantly less than 1 K. Figure 5 shows that for typical experimental conditions, the crossover between the “classical” behavior $I_C\propto E_J$ to the behavior controlled by the phase diffusion occurs at $E_J\approx 1$ K. Note that the literature data in Figure 5 and Figure 6 correspond to samples with different values of the ratio $E_J/E_C$. Given strong scattering of $I_{SW}$ and $R_0$, the effect of QPS is hard to detect. For the same reason, it is unclear if the impedance of the environment plays any significant role in these experiments; similar values of $I_{SW}$ could be observed for single JJ in a highly-resistive environment ($>100$ k$\Omega$ as in [36] and the setup in this work), single JJ in a low-impedance environment [37], and chains of SQUIDs frustrated by the magnetic field [31,35].

The observations are in line with an expected strong dependence of the IPS rate on the sample parameters in the regime $E_C\ll T\le E_J\ll \Delta$. At $E_J\gg T$, one can estimate the rate of the thermally-generated IPS as $\Gamma ={\omega }_{p}exp(-2E_J/k_{B}T)$, where ${\omega }_p$ is the plasma frequency (or an attempt rate) and $exp(-2E_J/k_{B}T)$ is the probability of the over-the-barrier excitation. For example, at $E_J=0.25$ K and ${\omega }_p/2\pi =1.32$ GHz, the rate decreases from $3\times {10}^5$ s${}^{-1}$ to $0.1$ s${}^{-1}$ if the physical temperature decreases from 50 mK to 20 mK. This might also explain why the experimental results are so sensitive to the noise level in the experimental setup.

Table 2. The literature data on $I_{SW}$ and $R_0$, ranked by $E_J$.

Reference	${\mathit{E}}_{\mathit{J}}$ (K)	${\mathit{E}}_{\mathit{J}}/{\mathit{E}}_{\mathit{C}}$	${\mathit{I}}_{\mathit{C}}^{\mathit{AB}}$ (nA)	${\mathit{I}}_{\mathit{SW}}$ (nA)	${\mathit{R}}_0$ (k$\mathbf{\Omega }$)
Watanabe 2003 [10] sample C	5.7	8	240	40	0.6
Kivioja 2005 [30]	5.2	500	220	145
Schmidlin 2013 [33], Figure 5.2	2.5	50	106	25	0.13
Shimada 2016 [29], SQUID at $\Phi /{\Phi }_0=0.375$	1.1	14	47	1.2	0.11
Weissl 2015 [36], SQUID at $\Phi /{\Phi }_0=0.26$	0.95	10	38	0.35	0.14
Watanabe 2003 [10] sample G	0.76	1	32		14
Jäck 2015 [38] Figure 4.6	0.54			1.5	13
Senkpiel 2020 [37]	0.47			0.3	33
Senkpiel 2020 [37]	0.23			0.07	143
Yeh 2012 [39]	0.18	1.3	6.5	0.35	31
Jäck 2017 [40]	0.17		7..5	0.05	400
Murani 2020 [14]	0.12		5	0.07
Senkpiel 2020 [37]	0.09			0.012	830
Kuzmin 1991 [41]	0.05	$\ll 1$		0.014	8000

Table 2. The literature data on $I_{SW}$ and $R_0$, ranked by $E_J$.

Reference	${\mathit{E}}_{\mathit{J}}$ (K)	${\mathit{E}}_{\mathit{J}}/{\mathit{E}}_{\mathit{C}}$	${\mathit{I}}_{\mathit{C}}^{\mathit{AB}}$ (nA)	${\mathit{I}}_{\mathit{SW}}$ (nA)	${\mathit{R}}_0$ (k$\mathbf{\Omega }$)
Watanabe 2003 [10] sample C	5.7	8	240	40	0.6
Kivioja 2005 [30]	5.2	500	220	145
Schmidlin 2013 [33], Figure 5.2	2.5	50	106	25	0.13
Shimada 2016 [29], SQUID at $\Phi /{\Phi }_0=0.375$	1.1	14	47	1.2	0.11
Weissl 2015 [36], SQUID at $\Phi /{\Phi }_0=0.26$	0.95	10	38	0.35	0.14
Watanabe 2003 [10] sample G	0.76	1	32		14
Jäck 2015 [38] Figure 4.6	0.54			1.5	13
Senkpiel 2020 [37]	0.47			0.3	33
Senkpiel 2020 [37]	0.23			0.07	143
Yeh 2012 [39]	0.18	1.3	6.5	0.35	31
Jäck 2017 [40]	0.17		7..5	0.05	400
Murani 2020 [14]	0.12		5	0.07
Senkpiel 2020 [37]	0.09			0.012	830
Kuzmin 1991 [41]	0.05	$\ll 1$		0.014	8000

6. Conclusions and Outlook

Phase slips in JJs have been actively studied over the last three decades in different types of Josephson circuits (single JJs, JJ arrays, etc.) over wide ranges of $E_J$ and $E_C$. In this work, we focus on the incoherent phase slips, which are expected to become the dominant source of dissipation at sufficiently low temperatures $T\ll \Delta$, where the concentration of quasiparticles becomes negligibly low.

It is observed that in all studied devices with $E_J<1$ K, the switching current $I_{SW}$ is significantly suppressed with respect to $I_C^{AB}$. At the same time, a very rapid growth of $R_0$ with decreasing Josephson coupling below $E_J\approx 1$ K is observed. Large scattering of the data might reflect a steep dependence of the rate of incoherent phase slips on the physical temperature and non-equilibrium noise in different experimental setups. Our observations are consistent with similar data that have been previously reported in the literature.

The observed enhanced dissipation in Josephson circuits with $E_J<1$ K might impose limitations on the further progress of superconducting qubits based on low-$E_J$ junctions. This important issue requires further theoretical and experimental studies. An especially important direction would be measurements of the coherence time in the qubits with systematically varied Josephson energy over the range $E_J$ = 0.1–1 K. One of the signatures of IPS-induced decoherence might be an observation of a steep temperature dependence of the coherence time at $T<100$ mK [42].