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

We have studied the temperature dependence of escape phenomena in various underdamped Josephson junctions (JJs). The junctions had different Josephson coupling energies EJ which were relatively small, but larger than the charging energy EC. Upon increasing the temperature T, we first observe the usual cross-over between macroscopic quantum tunnelling and thermally activated (TA) behaviour at temperatures kBT ∼ ℏωp, where ωp 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 kBT ∼ EJ(1 − 4/πQ)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 EJ, 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.


Introduction
In quantum computing one needs, besides a quantum bit, also a way to measure the quantum state of the system. Recently, in many superconducting quantum bit experiments, the hysteretic Josephson junction (JJ) escaping from its zero voltage state has successfully been used in the detection of the quantum state [1,2]. In these applications, a hysteretic JJ is also providing inductive protection against phase fluctuations. Escape measurements also enable one to perform conventional large bandwidth current measurements with extensive statistical averaging, and recently there have been proposals to use escape measurements as classical ammeters for studying phenomena like shot-noise [3]- [5]. For many purposes, it may be advantageous to reduce the critical current I c of the detecting junction in order to increase the measurement sensitivity. Yet the physics governing escape phenomena of small I c junctions, i.e. junctions where the thermal energy k B T is of the order of the Josephson coupling energy E J =¯h 2e I c , ultimately differs from those with larger I c , and this sets a limit on how far one can reduce I c still maintaining the useful features of the detector [6]- [8]. In the present work, we have investigated experimentally, (a) (b) (c) Figure 1. In the RCSJ model, the real JJ (a) is modelled by an ideal junction, which has capacitance and dissipative shunting impedance in parallel (b). Normally the dissipative environment is strongly frequency-dependent and the model presented in (c) is more realistic.
in this limit, the phase dynamics in underdamped JJs and dc-SQUIDs. Studied samples were either in the weak Josephson coupling regime with small I c or in the intermediate regime between 'standard' strong coupling junctions (E J k B T ) and the junctions with weak coupling. We will see that, in this intermediate regime, the usual escape from a single metastable state and the underdamped phase diffusion (UPD) both play a role.

Resistively and capacitively shunted junction (RCSJ) model
JJ is a weak link (e.g. tunnel junction) between two superconducting electrodes. It can be characterized by its coupling energy E J = 0 2π I c , charging energy E C = e 2 /2C J and dissipation R(ω) = 1/Re {Y(ω)}. Here C J is the capacitance of the junction and 0 = h/2e the flux quantum. The dynamics of a JJ can be described by the RSCJ model presented in figure 1(b). In this model the junction capacitance and dissipative resistor are in parallel with an ideal JJ. This leads to the model where a fictive phase particle of mass m =h 2 /8E C , whose position is given by the difference of the superconducting phase of the junction, ϕ, resides in a tilted cosine potential U(ϕ) = −E J (cos ϕ + I/I c ϕ) schematically presented in figure 2. The motion of ϕ is also affected by viscous drag [9] h 2e Here Q(ω) = ω p R(ω)C J is the quality factor of the junction and is the plasma frequency, i.e. the angular frequency of small oscillations around the metastable minimum of the potential and q 0 ∼ = √ 2(1 − I/I c ) at currents close to I c (ω 0 p is the plasma frequency at zero bias). When the biasing current is close to the critical value, the potential is well approximated by the cubic form U(q) = 3 U (q/q 0 ) 2 (1 − 2 3 q/q 0 ), where q = ϕ/2 − π 4 + 1 2 q 0 , and U = 2 3 E J q 3 0 is the height of the potential barrier. The dynamics of the JJ can be either overdamped (Q < 1) or underdamped (Q > 1). In the underdamped dynamics, the I-V characteristic of the junction is hysteretic: with increasing current, the voltage will jump abruptly from zero to V ≈ 2 BCS /e at I = I sw < I c . Here BCS is the superconducting gap. In contrast, in the case of Q < 1 the I-V characteristic is non-hysteretic and the voltage increases continuously. Dynamics of a hysteretic JJ. The top frame presents an example of a measured I-V characteristic. In the superconducting state (S-state) the phase mainly oscillates in the well and the average voltage is close to zero. The phase particle can escape from the well either by thermal activation (TA) or macroscopic quantum tunnelling (MQT) and the system switches to the free running state (N-state). In the N-state, the voltage across the junction is approximately twice the superconducting gap (2 Al ≈ 360 µV). The escaping is a stochastic process, and by sweeping the current repeatedly, one can measure the distribution of the switching currents (inset in the I-V characteristics figure). In this work we used a pulse technique, which yields the integral of this distribution directly. Figure 2 presents schematically the dynamics of a hysteretic JJ. The upper inset shows an example of a measured I-V characteristic and in the main frame we present the corresponding dynamics of the phase particle in the tilted cosine potential. There are two distinguishable states of the system: the superconducting S-state and the high voltage N-state. In the first one the phase has constant average value and the voltage across the junction is close to zero. With increasing current the phase particle will 'escape' from the metastable S-state and switch to the second state, where the phase is running freely. In this N-state the voltage is about twice the superconducting gap (2 BCS ≈ 360 µV for aluminum). Escape is fully a stochastic phenomenon and the value of the switching current changes from current sweep to another. With decreasing current, the dissipation is slowing down the phase particle and at I = I r it will be relocalized again to the S-state. This retrapping current I r differs from that of escaping and it strongly depends on dissipation.

5
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT The phase particle can escape from a metastable well in the S-state either via TA over or MQT through the barrier. For strong coupling junctions (hω p U), we can assume that there is a continuum of levels within a metastable potential well, which leads to the TA rate where U is the height of the potential barrier [10] and a t is the dissipation-dependent prefactor, which is of the order of unity with typical experimental parameters [11]. The tunnelling rate out of the bottom of the potential well can be calculated using standard WKB approximations leading to [10] where [12,13]. The dissipation is described by α = 1/2Q. The total escape rate can be approximated by TOT (I ) MQT (I ) + TA (I ) and the total escape probability in the time interval 0 t τ can be written as P = 1 − e − τ 0 TOT [I(t)] dt . The two conventional methods to investigate escape dynamics are as follows: (i) by ramping biasing current through the junction and by measuring the distribution of the switching currents or (ii) by measuring the escape rate directly at different values of the bias current. Escape rates per unit time can be determined via the escape probability in a measurement with a set of current pulses with fixed amplitude and duration, determining the statistical probability of junction to switch into a high-voltage free running state. Usually, one can measure the switching probability as a function of current pulse amplitude P(I), which yields cumulative histograms of switching currents (see figure 8).

The effects due to frequency-dependent Q
In a typical experimental setup, the junction is also capacitively shunted by the stray capacitance of, e.g., the leads, besides by the junction capacitance. A more realistic equivalent circuit of the junction is thus that presented in figure 1(c) instead of the simple RCSJ-model of figure 1(b). At low frequencies, the dissipation is mostly determined by the junction subgap resistance, which is usually of the order of 1 M , but at high frequencies, e.g., at ω p , the impedance is typically small because C s is short and R s is small. In the S-state, the phase mainly oscillates in a well at plasma frequency and it may transit from one well to another in a time which is of the order of the inverse plasma frequency. The dissipation is thus characterized by R(ω p ) in this case. Without a specially designed environmental circuit, this high-frequency dissipation is usually of the order of vacuum impedance Z 0 = √ µ 0 / 0 ≈ 377 . Yet after transition into the N-state, a dominant part of dissipation takes place at low frequencies. Thus the junction can have very large Q at low frequencies with very small retrapping current, yet in the supercurrent branch the junction can be overdamped [14,15].

dc-SQUID
A dc-SQUID consists of a superconducting loop and two weak links. In the limit of small loop inductance L loop (β L ≡ 2πL loop I c 0 1 [9]), the potential energy of the dc-SQUID can be written as where γ is the phase across the dc-SQUID, the magnetic flux through the loop and δ determines the asymmetry of the dc-SQUID (E J1,J2 = E J (1 ± δ)). The supercurrent (I sc = 2ē h ∂U J /∂γ) can in this case be written as Hereafter, we will consider δ = 0. Then we can write I = I c sinγ, where I c = 2I 0 c | cos(π / 0 ) |. The potential energy of the dc-SQUID has also an inductive contribution due to the inductance of the dc-SQUID loop. However, if the Josephson inductance L J = 0 /2πI c is much larger than the loop inductance (β L 1), this term will not affect the dynamics and we can neglect it [16]. In an ideal case, the dc-SQUID is a tunable single junction with maximum critical current twice that of one junction in the loop (δ = 0). Both our dc-SQUIDs are in the β L 1 limit.

Current threshold detection using JJs
Schematic examples of circuits used in a current threshold detection are presented in figures 3(a) and (b). In these setups, the goal is to measure the current induced by a generic current source. This current is called I x . The thick blue lines indicate the superconducting parts of the circuits, and the principal difference between circuits shown in figures 3(a) and (b) is that the circuit in figure 3(a) can be used for measuring currents induced by externally biased circuits in contrast  to figure 3(b), where the measured circuit is purely inductively shunted and it can be used for measuring a persistent supercurrent. The circuit presented in figure 3(a) can be used in a conventional current measurement and it has also been proposed to be used, e.g., in measurements of shot-noise [3]- [5]. In these measurements the JJ is providing the way to perform large bandwidth current measurements with extensive statistical averaging. The scheme presented in figure 3(b) is similar to what has been used, e.g., in the 'quantronium' experiment [1,17,18], but the circuit has also been proposed for measurements of quantum errors in Cooper pair pumping [19].
In both examples, figures 3(a) and (b), the measured current I x runs in parallel with biasing current I bias through the measuring junction (or a dc-SQUID as in figure 3). A direct measurement can be obtained by measuring the corresponding change in escape probability for a constant I bias . From this change I x can be extracted. However, the working point of the detector is not constant in this measurement, which can induce an error. A second measurement method is to regulate I bias in order to keep the escape probability equal in the absence and in the presence of the signal to be measured. The current through the measuring junction is then equal in the two cases and loading errors, e.g. the change of the Josephson inductance and superconducting phase, can be neglected. I x is then directly given by the variation of I bias .
The sensitivity of the current threshold detection with constant bias current pulses can be defined as S = dP/dI (the derivative of the cumulative histogram). In current measurements, the resolution is δI = δP/S. With TA and MQT models, the maximum sensitivity is reached with the current pulse amplitude, which corresponds approximately to the 70% escape probability. In figure 4, we present the maximum sensitivity as a function of the critical current and capacitance,

8
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Table 1. The parameters of the measured samples. Both the critical current I c (Ambegaokar-Baratoff value calculated from the normal state resistance [9]) and the sample capacitance C J (calculated based on the junction area and 50 fF µm −2 ) are given for the whole tunable circuit in case of dc-SQUID samples. calculated by using TA and MQT models. We have not taken into account the small effects of dissipation. A single JJ with fixed oxidation parameters is covering only a line on the (I c , C)-plane in figure 4 with an almost constant sensitivity. For typical oxidation parameters of Al-AlOx-Al junctions this is presented as a red region. We have used the values 1 k µm −2 and 50 fF µm −2 . By decreasing the junction area, both critical current and junction capacitance decrease. However, the sensitivity (dP/dI) max remains fixed. Therefore it is beneficial to use a dc-SQUID configuration instead, where one can tune the critical current and capacitance independently and thus increase sensitivity remarkably. The number of switching events is binomially distributed, whereby we find that the standard deviation of the measured escape probability is where P is the measured escape probability and N tot is the total number of current pulses on the measurement. The maximum of dP occurs at P = 0.5 and it vanishes at P = 0 and at P = 1. The current resolution is thus not the best at the current value where the maximum sensitivity is reached, but at a probability which is slightly higher instead (P 0.8).

Experimental details and measured samples
The parameters of the measured samples are presented in table 1. All the measured samples were made out of aluminum and they were fabricated using standard electron beam lithography and aluminum metallization in a UHV evaporator. The AlOx tunnel barriers were formed by basic room temperature oxidation. We report here on measurements of three different samples: two with intermediate coupling energy (E J > k B T ) and one with small coupling (E J k B T ). Sample SQ1 has a conventional dc-SQUID geometry, which consists of two wide superconducting planes connected with two short superconducting lines with tunnel junctions in the middle forming the dc-SQUID loop of area 20 × 39 µm 2 (see figure 5). The purpose of wide planes was to reduce loop inductance, In the measurements on a sample SQ1, we have R S = 500 and C S ≈ 5 nF. In the case of sample SQ1, we also had a 20 k resistor in parallel with voltage amplifier, to speed up the retrapping. and the measured value was around 120 pH, which was small as compared to the calculated Josephson inductance (L J = 0 /2πI c = 3.2 nH per junction). The dc-SQUID sample can thus indeed be considered as a single JJ, whose I c can be tuned. The loop inductance was estimated from the measured resonant voltage determined by C J and loop inductance [20]. Sample JJ1, also with intermediate Josephson coupling, was a single junction between long inductive biasing lines.
The schematic of the sample with low coupling energy, SQJJ, is presented in figure 10. The sample consists of a single tunnel junction, a dc-SQUID and a long inductive line connected together in the middle. We assumed that the dc-SQUID is providing the tunable environment for the single junction. The distance between a dc-SQUID and a single junction was approximately 100 µm and a long inductive line was connected in between for separate biasing. The length of the line was around 2.5 mm leading to an inductance of approximately lµ 0 ≈ 3 nH.
The schematic of the experimental setup is presented in figure 5. Measurements were done in dilution refrigerators with minimum temperatures around 20-30 mK. The refrigerators were equipped with strongly filtered lines (Thermocoax ® and π-filters). Switching probabilities were measured by applying a set of short trapezoidal current pulses through the sample and by measuring the number of resulting voltage pulses. The statistical switching probability is thus simply P = number of V pulses /number of I pulses . Current pulses were created by applying voltage pulses either from the PC data acquisition card or from Agilent 33220A function generator through a large (100 k -10 M ) resistor. Voltage across the sample was amplified and recorded by using the same data acquisition card. The normal delay time between current pulses was 500 µs, and it was measured to be long enough for cooling the sample after the dissipative switching event. For reducing flux noise we have used both superconducting lead and lowtemperature µ-metal shields in 4 K helium bath.
At the sample stage we used low pass RC-filters (surface mount components near the sample) as presented in figure 5. We use 4-wire configuration in all measurements, and thus there were two RC-filters connected to each electrode of the samples. In the measurements on samples JJ1 and SQJJ we used surface mount capacitors (C s = 680 pF), but in the measurements on sample Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT SQ1 we had π-filters in series with resistors, with C s ∼ 5nF capacitance to ground. The resistors were R s = 500 in the measurements on SQ1 and 680 in the measurement on samples JJ1 and SQJJ. The sample was connected to filters by ultrasonic bonding. The inductance of the bonding wires is of the order of nH, but the accurate value is not known. The agreement between theory and experiments (described below) is good by assuming that the inductive reactance is negligible at frequencies close to the plasma frequency (f p = ω p /2π =1-10 GHz with current bias, see table 1).

JJs with intermediate coupling
With decreasing critical current, the simple picture based on TA and MQT to explain the switching from the zero-voltage S-state to the free running N-state fails when E J is comparable tohω p or k B T [21]. The first condition is due to the small number of quantized energy levels (QELs) inside the well in this limit and thus the continuum approximation is no longer valid. The latter one is explained by a change in the escape dynamics due to dissipation effects. Typically, for a 500 environmental impedance, these effects occur for E J 10k B T which defines the condition of intermediate coupling.

Phase-diffusion regime
In the S-state, the voltage across the junction is not necessarily exactly zero, because phase can have 2π-slips, which causes a small average voltage across the junction. This phase diffusion is a dissipative process and therefore it can be harmful in applications where the junction is used, e.g., as a quantum-state detector.
When critical current is decreasing, dissipation is starting to play a more important role. The quality factor which is proportional to √ E J is decreasing with decreasing E J . On the other hand, the escape rate is significant in the range of currents where the barrier height is comparable to the thermal energy. For strong coupling junctions, E J k B T , this is the case when the bias current is only slightly below the critical current, but for small junctions this rate can be large even without tilting the cosine potential at all (I/I c = 0). At small critical currents, the successive barrier tops are thus close in energy after escaping. If escape occurs at small enough currents, there is a non-zero probability that it will be followed by immediate relocalization to the next minimum due to dissipation. In this process the phase is diffusing from one well to another even though the junction can be underdamped and thus hysteretic. The maximum biasing current when this diffusion occurs can be given as [21] where the value of Q corresponds to damping at plasma frequency ω p . Below I m the phase can be localized again and the voltage rise per one escape event is negligibly small and we are not able to count these events in a switching probability measurement, nor can we measure the average voltage due to these rare 2π phase-slip events.
The behaviour of the junction at different values of E J and T is described by the phase diagram in figure 6. The white area describes the region where switching to the N-state after escaping is certain and thus the measurements of the escape dynamics are possible. The shaded area in figure 6 presents the phase diffusion regime, where escaping does not necessarily lead The different operation regimes of a small JJ [21]. The thick black line shows the cross-over temperature between TA and MQT regimes. The white area shows the regime where the conventional escape into a resistive state dominates (with current pulses of ∼100 µs length and with ∼500 shunting impedance). Inset: the cosine potential. The dynamics inside a well (upper well) and the schematic dynamics after leaving the upper well.
to a transition into the running state with 100 µs current pulses and with 500 environmental impedance (C J = 100 fF). This area is here divided into two different regimes: underdamped (Q > 1) and overdamped (Q < 1) phase diffusion regimes. In contrast to UPD, where the phase has to localize in the succeeding minimum, in overdamped phase diffusion regime the phase can slide down the potential with almost constant velocity.

Energy level quantization
For low lying energy levels, the potential is close to the harmonic one and the level energies are E n ≈hω p (n + 1 2 ). The number of QELs inside the potential well can be approximated by N = √ E J /2E c (1 − I/I c ) 5/8 . For example, in a 1 µm 2 Al-AlOx-Al junction with usual oxidation parameters they are of the order of 10 levels at zero bias and this number is decreasing strongly with increasing bias current. In this case, the basic continuum approximations no longer hold and more adequate models should be used. We take into account level quantization using the model of Larkin and Ovchinnikov [22,23]. The scheme of the model is presented in figure 7. This semi-classical model takes into account the influence of the shape of the potential.
The total escape probability is calculated using P esc (τ) = 1 − k ρ k (τ), where ρ k (τ) is the probability of finding the particle in level k after a current pulse of length τ. The kinetic equation of the phase particle can be written as dρ k /dt = j (γ kj ρ j − γ jk ρ k ) − k ρ k . Due to the nearly parabolic shape of the potential, the model takes into account only the transitions γ kj between Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Figure 7. The dynamics inside a well. The model takes into account the anharmonicity of the potential. The model of the dynamics considers the transitions between the nearest-neighbouring levels and the escape out of these levels.
the nearest-neighbouring levels and the tunnelling out, k . The relaxation rate between levels j and (j − 1) are well approximated by equation γ j−1,j = jω p /4Q. We also assume detailed balance: γ j,j−1 = e −β(E j −E j−1 ) γ j−1,j . The positions and escape rates are calculated using the results in [22].
For calculating the full dynamics properly, non-localized states above (but close to) the barrier top must be included in the model as well. Actually, also the states just below the barrier (E > U − 0.4hω p ) must be calculated in a similar manner. The expressions for calculating the shape of these broad levels is given in [22].
The final state ρ ≡ [ρ 1 ρ 2 · · · ρ k ] is calculated by numerically integrating the equation is the tridiagonal transition matrix including all the transition elements. The current bias is set to zero in the beginning, and the initial state ρ(0) is Boltzmann distributed. The possibility that the phase particle can relocalize in the succeeding well after tunnelling must be taken into account in the model as well. The dissipated energy in transition from one well to another can be approximated by E d ≈ 8E J /Q and by using that and the fact that the energy difference between the two successive maxima is −2πE J I/I c , we can write that the level energy E must fulfil the condition to secure switching into the N-state. If condition (7) is not fulfilled, tunnelling rate at that energy level is set to zero in the model. This means that in the next well the thermal distribution is reached immediately. With the low Q values in the experiment, the phase relaxes in the next well in a time ∼ω −1 p ∼ 100 ps, which is far shorter than the typical time interval between phase diffusion events with studied current bias values (10-100 µs), and the assumptions of zero tunnelling rate and immediate recovery to the Boltzmann distribution in the next well are thus valid. Note that equation (7) gives the same threshold as equation (5) for the special case U = E, but in equation (7) we have taken into account the fact that, after tunnelling, the energy of the phase particle is not necessarily that at the potential maximum.

Results
In figure 8 we present the measured cumulative switching histograms (open circles) of sample SQ1 at different fluxes and temperatures. If we assume that the two junctions in the dc-SQUID are identical, we can infer that the corresponding critical currents at the lowest temperature are In the diagram of figure 6, we also present the critical current of SQ1 at fluxes 0, ±0.28 0 and ±0.41 0 by horizontal dashed lines. The phase diagram was calculated assuming a realistic shunting impedance R(ω p ) 500 (the value of the surface mount resistors). It can be seen that the intersections of the dashed lines and the boundary of the phase diffusion regime are very close to the experimentally determined temperatures T D . The cross-over from thermal escape into the UPD regime is thus causing the re-entrant steepness of the histograms.
At the temperature T D , the position of the sample SQ1 histograms is saturating at about the same value I 0.35I c at all magnetic fluxes (JJ1 at I 0.3I c ). If we assume R(ω p ) 500 , we obtain Q ≈ 4 (I c = 200 nA and C J = 100 fF), which yields indeed I m 0.35I c through equation (5) like in the experiment. In the case of a single junction R(ω p ) 681 gives Q ≈ 13 yielding I m ≈ 0.1I c instead of the measured 0.3I c . The latter current would rather correspond to R(ω p ) 230 giving Q ≈ 4 and an estimated T D = 700 mK which is consistent with the measurements.
In section 2.4 we discussed the sensitivity of the JJ threshold detection characterized by the steepness of the cumulative histograms. In the UPD regime, however, the re-entrant steepness is due to missed escape events and relocalization, and it might be harmful in some detector applications, where dissipation is to be avoided.  Figure 8 shows that the plain TA-MQT models cannot account for our observations. Except for the data at the lowest temperatures, the width and the position of the measured histograms deviate from the simulated ones (dotted line). Dissipation alone cannot explain the difference. The basic TA model yields I ∝ T 2/3 [10], and it can be seen in figures 8 and 9 that the dc-SQUID has weaker temperature dependence even well below T D . In these samples there are just few energy levels in the well and thus the assumptions of continuous energy spectrum are not valid here [10]. We present results of QEL-model simulations for data at zero and ±0.28 0 fluxes. At ±0.41 0 , I c is so small that the escape probability is large even at zero bias (except at the lowest temperatures). This means that the phase is running constantly rather than infrequently escaping from one well to another, and thus our model does not work anymore. The only fitting parameter was the quality factor Q, and the fitted values (presented in the inset of figure 8) were in a very reasonable range. At = 0 we find Q ≈ 6 at the lowest temperature, and it decreases with increasing temperature up to 4 at 325 mK. At = ±0.28 0 , Q ≈ 3-4, again decreasing with temperature. The agreement between simulation and measurements is excellent. The position and the width of the measured histograms coincide and, in particular, at higher temperatures the simulated histograms also start to get steeper again. At higher temperatures, the upper energy levels, whose escape rate is significant with smaller potential tilting angles, are populated as well. The histograms thus peak at smaller currents and the condition of equation (7) is not necessarily fulfilled anymore. Dissipation is screening part of the events and the histogram gets distorted. What remains in the measured (and simulated) histograms is the escape from the levels at the tail of the Boltzmann distribution above the dissipation barrier. The measured samples can thus be tuned from a pure escape regime deep into phase diffusion by varying the temperature and flux. Figure 10 shows a schematic representation of the sample SQJJ with tunable environment and micrographs of the junctions in this sample. When the coupling energy E J is of the order of k B T , the escape rate is significant even without bias current. This means that the junction is dissipative also with zero bias. Figure 6 demonstrated that when I c < 60 nA, the system is in the phase diffusion regime even at the lowest temperatures with 500 environmental impedance. The measured weak coupling junctions are thus not obeying simple escape dynamics from a metastable potential well into a free running state, but the escape occurs over a dissipation barrier instead. The measured histograms of both the dc-SQUID and the single junction of sample SQJJ (data not shown here) indeed show qualitatively similar behaviour to that presented in figures 9(a) and (b) at = ±0.41 0 .

Zero-bias resistance
Predominantly unshunted hysteretic junctions can be overdamped due to frequency-dependent environmental impedance (R(ω p ) ∼ Z 0 ≈ 377 ). This can be the case especially on the samples with small E J , which usually also means small ω p like in the sample SQJJ. Small coupling energy (E J ∼ k B T ) yields also to large escape rate and further to large phase diffusion rate. If the rate is large enough, the average voltage across the junction starts to be measurable (but still 2 /e).
In SQJJ, the escape rate of the single junction is large, i.e. the phase diffusion is strong, leading to a small measurable voltage ( 2 /e) across the junction. Therefore the I-V characteristics present a finite slope even at zero current. Ingold et al have shown that for overdamped junctions this zero-bias resistance can be given as [9,24] where I 0 is the modified Bessel function and R a shunting impedance. The zero-bias resistances R 0 due to phase diffusion of both the single junction and the dc-SQUID are plotted at different temperatures by using lock-in technique in figure 11. They are fitted by the predicted resistance given by equation (8) using R = 270 as a fit parameter. The theory closely follows the measurement over two decades in resistance. The measurement noise level sets the minimum measurable resistance to somewhere around 10 , which corresponds to the flat background in figures 11(a) and (b). R 0 of the dc-SQUID was also measured at different magnetic fields ( figure 11(a)).
We observe that R 0 can be tuned by changing the magnetic flux inside the dc-SQUID. The zero-bias resistance is increasing strongly with decreasing critical current as equation (8) predicts. With critical currents of the order of 10 nA, R 0 is larger than 10 even at the lowest temperatures.

Tunable environment
Escape from the phase diffusion branch to the running state is a complicated process and it does not have similar simple analytical expressions as, e.g., the basic TA has. Yet, this thermally activated process is strongly environment-dependent as our Monte Carlo simulations, as clearly demonstrated by [14,15]. Turlot et al [25] have measured how thermally activated escape rate over a tilted cosine potential barrier varies with changing impedance of the frequency-dependent environment. It was shown in their work that if 1/Re(Y(ω)) does not vary too rapidly close to the plasma frequency, the dominant part of the dissipation happens in the vicinity of ω p [25,26].

Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
With frequency-dependent environment, the TA rate TA thus approximately varies like 1/Re(Y(ω p )) with changing impedance (equation (1)).
In this work, we measured the dynamics of escape from the phase diffusion regime to the free running state by varying the environmental impedance. The escaping rate follows a similar Arrhenius law as the standard TA, but now escape occurs over a dissipation barrier U s − U i instead (s corresponds to a saddle point and i to the initial state) [6]. The prefactor a d is similarly a dissipation-dependent parameter as a t for TA over cosine potential barrier (equation (1)). We thus assume that the escape rate PD TA from the phase diffusion regime varies also like 1/Re(Y(ω p )). In our sample, the environment presented by the dc-SQUID on the single junction, and vice versa, are strongly inductive. The Josephson inductance of the dc-SQUID at zero field is of the same order of magnitude (L J ≈ 4.5 nH) as the inductance of the long line, but it increases strongly with increasing magnetic field. We consider the dc-SQUID here as a tunable environmental inductance of the single junction and measured how changing the Josephson inductance of the dc-SQUID is affecting escape dynamics of the single junction.
The sample is approximated by an equivalent circuit presented in figure 10(b). In this model the dc-SQUID is approximated by a parallel combination of Josephson inductance and junction capacitance similarly as Warburton et al [27] successfully did with high-T c intrinsic JJs. We have not taken into account the fact that close to = 0.5 0 , the dc-SQUID is also in the dissipative phase diffusion regime, which might affect the escape dynamics. The parallel resistance of the dc-SQUID is assumed to be large. Notice that this does not mean that the dissipation that the dc-SQUID can see at high frequencies is small. The capacitance C s = 10 pF is assumed to be the sum of capacitances of the surface mount capacitors and of the stray capacitances of the line. The high frequency dissipation is modelled with resistance R s , which is most probably close to the value of surface mount resistors (∼340 ) or to the value of vacuum impedance (Z 0 ≈ 377 ). Figure 12(a) presents the calculated real part of the admittance of the model environment of the single junction which corresponds to the dc-SQUID with the inductive line ( figure 10). In calculations, we used the value 3 nH for the inductance of the long line and the parameters of the dc-SQUID are given in table 1. The impedance is modified by changing the flux through the SQUID. The environmental admittance of the single junction at the attempt frequency ω p is presented in figure 12(b).
We measured the escape probability of the single junction as a function of flux through the dc-SQUID. For measurement, the current bias was fed through the single junction line (electrode A in figure 10) to the long inductive line (C) and the voltage was measured between these electrodes. At each temperature, we first measured the amplitude of the current pulse, which corresponds to approximately 70% switching probability at zero flux. We measured escape probability of the single junction as a function of flux at this fixed current amplitude. Results are presented in figure 12(c), and the inset of figure 12(c) shows I 70% at zero flux at each measuring temperature. Figure 12(c) shows that the measured escape probability indeed follows the general behaviour of the real part of the modelled environmental admittance at plasma frequency of the single junction plotted in figure 12(b) as was predicted above. We do not have a quantitative model for the flux dependence of the escape rate, but it is obvious that the dc-SQUID acts as a (c) Measured switching probability of the single junction of sample SQJJ at constant current bias as a function of applied magnetic flux at several temperatures between 25 and 450 mK. Curves are shifted for clarity and the vertical spacing between ticks corresponds to an escape probability of unity. Inset: the current, which corresponds to 70% escape probability at various temperatures, as measured on the single junction of sample SQJJ.
tunable environment for the single junction such that it can decouple the external noise at a certain value of flux. At = 0, the plasma frequency of the dc-SQUID is larger than that of the single junction due to different thickness of the oxide barriers (see table 1). In this case, Y(ω p ) of the single junction presented by the dc-SQUID is large. With increasing flux, the plasma frequency of the dc-SQUID reduces. Around = ±0.3 0 it coincides with that of a single junction and environmental admittance is reduced and thus the escape rate is significantly suppressed. The behaviour is qualitatively the same at all temperatures, but at higher temperatures the escape rate is strongly enhanced close to = ±0.5 0 . One possible explanation for this is that the phase diffusion in the dc-SQUID is playing a role which is not taken into account in our model. The flux dependence of the escape rate in the single junction suggests that dissipation in the zero voltage state takes place mainly at the plasma frequency. Furthermore, the dominant part of the dissipation is, in our case, physically happening far away from the junction and the quasi-particle resistance is not playing an important role.

Concluding remarks
Our measurements confirm that dissipation plays a more important role in the phase dynamics of JJs when the critical current decreases. The parameters of the measured samples were in the intermediate range, where the simple escape from a single metastable state, and phase diffusion both play a role. The sample with the smallest critical current consists of a single JJ and a dc-SQUID close to each other, the latter of which acts as a tunable inductive protection for the single junction. We were able to study the effects of dissipation on escape dynamics by varying the temperature, flux and environment. The measurements with tunable environment show that escape from the phase diffusion regime depends strongly on the environment. In summary, our observations show that in order to use junctions with E J ∼ k B T for current detection purposes, one should pay particular attention to the environmental circuit. For low Q values, these junctions will be in the UPD regime, which might be harmful in threshold current measurements. This regime is also dissipative, which is undesirable in some applications, e.g., in the detection of a quantum state of a superconducting quantum bit. Our measurements with tunable environment confirm that the inductive leads can provide a way to further decrease the critical current without losing the beneficial properties in threshold current measurements.