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1 Powered by TCPDF ( This is an electronic rerint of the original article. This rerint may differ from the original in agination and tyograhic detail. Kivioja, J. M.; Nieminen, T. E.; Claudon, J.; Buisson, O.; Hekking, F. W J; Pekola, J.P. Observation of Transition from Escae Dynamics to Underdamed Phase Diffusion in a Josehson Junction Published in: Physical Review Letters DOI: /PhysRevLett Published: 22/06/2005 Document Version Publisher's PDF, also known as Version of record Please cite the original version: Kivioja, J. M., Nieminen, T. E., Claudon, J., Buisson, O., Hekking, F. W. J., & Pekola, J. P. (2005). Observation of Transition from Escae Dynamics to Underdamed Phase Diffusion in a Josehson Junction. Physical Review Letters, 94(24), 1-4. [247002]. DOI: /PhysRevLett This material is rotected by coyright and other intellectual roerty rights, and dulication or sale of all or art of any of the reository collections is not ermitted, excet that material may be dulicated by you for your research use or educational uroses in electronic or rint form. You must obtain ermission for any other use. Electronic or rint coies may not be offered, whether for sale or otherwise to anyone who is not an authorised user.

2 Observation of Transition from Escae Dynamics to Underdamed Phase Diffusion in a Josehson Junction J. M. Kivioja, 1 T. E. Nieminen, 1 J. Claudon, 2 O. Buisson, 2 F. W. J. Hekking, 3 and J. P. Pekola 1 1 Low Temerature Laboratory, Helsinki University of Technology, POB 3500, FIN HUT, Finland 2 Centre de Recherches sur les Très Basses Temératures, laboratoire associé à l Université Joseh Fourier, CNRS, BP 166, Grenoble-cedex 9, France 3 Laboratoire de Physique et Modélisation des Milieux Condensés, CNRS and Université Joseh Fourier, BP 166, Grenoble-cedex 9, France (Received 14 January 2005; ublished 22 June 2005) We have investigated the dynamics of underdamed Josehson junctions with a relatively small Josehson couling E J. In addition to the usual crossover between macroscoic quantum tunneling and thermally activated (TA) behavior we observe, for the first time, the transition from TA behavior to underdamed hase diffusion. Above the crossover temerature, the threshold for switching into the finite-voltage state becomes extremely shar. We roose a (T, E J ) hase diagram with various regimes and show that for a roer descrition of it, dissiation and level quantization in a metastable well are crucial. DOI: /PhysRevLett PACS numbers: r, Dq, A hysteretic Josehson junction (JJ) switching from its metastable zero-voltage state into a stable finite-voltage state has recently been used as a readout device for suerconducting quantum bit systems in many exeriments [1]. Switching measurements also enable one to erform conventional large bandwidth current measurements, and recently there have been roosals to use hysteretic JJs as ammeters for studying henomena like non-gaussian noise [2]. Often it may be advantageous to reduce the critical current I c of the detecting junction in order to increase the measurement sensitivity. Yet the hysics governing switching henomena of small I c junctions ultimately differs from those with larger I c [3]. How far can one reduce I c while still maintaining the useful features of the detector? Desite a considerable amount of work on junctions with moderate I c [4], no clear icture based on a systematic study exists as yet of temerature-deendent escae henomena in such junctions. In this Letter we resent such a study. The dynamical variable of a JJ is the hase difference of the suerconducting wave function in the two electrodes. Within the resistively and caacitively shunted junction (RCSJ) model, the dynamics of is governed by the Josehson energy E J hi c =2e, the charging energy E C e 2 =2C J (C J is the junction caacitance), and a shunt resistance R resonsible for dissiation. For later use we also define the bare lasma frequency! 8E J E C = h and the quality factor Q! RC. As we will detail below, the behavior of the junction at a given temerature T can be classified according to the hase diagram of Fig. 1(a). The overdamed case Q<1 with E J * E C was studied in detail by Vion et al. [3], who uncovered the existence of a hase diffusion regime at finite T with the aearance of a small voltage, rior to switching to a voltage on the order of twice the suerconducting ga [5]. As far as the underdamed case Q>1 is concerned, most exeriments were done on junctions with relatively high I c. Deending on T such junctions escae out of the metastable zerovoltage state either via macroscoic quantum tunneling (MQT) or thermal activation (TA) rocesses, switching directly to the finite-voltage state. In this Letter, we focus on the regime Q>1 with relatively small I c, such that E J k B T E C. We show, theoretically and exerimentally, that a regime exists where escae does not lead to a finite-voltage state, but rather to underdamed hase diffusion (UPD) [the shaded region in Fig. 1(a)]. According to the RCSJ model, the dynamics of a JJ biased with a current I is that of a article (mass m h 2 = 8E C ) whose osition is. It moves in a tilted cosine otential U E J cos I=I c under a viscous force. The bias current renormalizes the lasma frequency, such that! m 1 d 2 U=d 2 8E J E C q 0 = h with q I=I c. The cosine otential has wells where the article can be localized; then has constant average value, and the average voltage across the junction is zero. For nonzero I the quantum levels in the otential well are metastable and the article can escae from a given well either via TA over, or MQT through the barrier. For large junctions TA I; T! 2 e U=kBT for the h d 2e 2 Rdt TA escae rate and MQT I 12! q 6 2 U= h! e 36=5 U= h! for the MQT rate [6]. In the cubic aroximation the barrier height is U 2 3 E Jq 3 0. Below the crossover temerature T 0 h! =2 k B the dominant escae mechanism is MQT. Aroximating the total escae rate as I; T TA I; T MQT I, the escae robability in the time interval 0 t can be written as P I; T 1 e R I;T dt 0. If dissiation is weak, uon escae from the well the article moves down the otential and hase is running freely, hence the voltage reaches a finite value, =05=94(24)=247002(4)$ The American Physical Society

3 (a) E J /hω (b) U( ϕ ) I MQT kt B 0 escae hω CJ ω Ic Γ MQT R J TA escae escaing into running state T <30mK D 175 mk underdamed hase diffusion overdamed dynamics k B T/hω Γ TA Rs ϕ RUNNING STATE about 2. However, if dissiation is strong enough, there is a finite robability that, uon escae from the well, the article is relocalized in the next well instead of running down the otential: the hase then diffusively moves from one metastable well to another, see Fig. 1(b). In this UPD regime the average voltage across the junction is much smaller than 2 [3]. As it was ointed out over a decade ago [7], hase diffusion can occur even in a hysteretic junction due to the deendence of dissiation on frequency! [4]. Our exeriment corresonds to the simlified equivalent circuit with frequency deendent dissiation as resented in the inset of Fig. 1(b). After switching to the running state, the dominant art of dissiation comes from small!, governed by R! 0, tyically given by the large junction subga resistance, on the order of 1M. In the hase diffusion regime the hase mainly oscillates in a well at the lasma frequency and thus the dissiation is characterized by R!, which is much smaller, tyically on the order of vacuum imedance Z 0 377, since [Fig. 1(b)] acts as a short. Here we will consider junctions that are underdamed even at!, in contrast to [3]. UPD 335 mk 200 na 130 na I c=60na Q>1 R/R q Q<1 FIG. 1 (color). (a) The various oeration regimes of a Josehson junction with low E J. For details, see text. The UPD region corresonds to R s 500, C J 100 ff,and 100 s. (b) The dynamics in the uer well and schematic behavior uon escae. Inset: equivalent circuit of the junction with frequency deendent dissiation. The dissiated energy between neighboring otential maxima can be aroximated by E D 8E J =Q and if the article has energy less than E D above the next barrier to, it simly diffuses to the next well. The maximum ossible dissiated ower due to hase diffusion can be written as 1 2eV 2 h E D, where V is the average voltage across the junction. By equating this with the alied bias ower I m V,we find the maximum ossible hase diffusion current I m 4I c = Q, which is identical in form to the well-known retraing current formula, but now the value of Q is that at lasma frequency!. For I<I m, there is nonzero robability that the hase relocalizes after escae. The gray area in Fig. 1(a) resents the UPD regime, where escae does not necessarily lead to the transition into a running state. The condition TA I m ;T D 1= determines the searatrix E D J T D between the TA and UPD regions in Fig. 1(a) with current ulses of length : E D J 3 2 k BT D 1 I m =I c 3=2 ln! =2 : (1) Similarly, for T<T 0, the searatrix between MQT and UPD is found from MQT I m 1= ; E D J is indeendent of T, and given by Eq. (1) with T 0 relacing T D. We resent exerimental data of two samles, a dc- SQUID and a single JJ. They were fabricated using standard electron beam lithograhy and aluminum metallization in a UHV evaorator. The AlOx tunnel barriers were formed by basic room temerature oxidation of Al. The dc- SQUID consists of two wide suerconducting lanes connected by two short suerconducting lines with tunnel junctions in the middle forming the dc-squid loo of area m 2 (see the inset in Fig. 2). The loo inductance was determined [8] to be around 100 H, small as comared to the calculated Josehson inductance (L 0 2 I c 400 H er junction). The dc-squid thus behaves almost like a single JJ, whose I c can be tuned. The other measured samle was a single junction between long inductive biasing lines. The normal state resistances of the dc-squid and the single JJ were 1:3 k and 0:41 k yielding for I c 199 and 630 na, resectively. Assuming a secific value of 50 ff= m 2, the caacitances of the samles were estimated to be 100 and 130 ff, resectively. Both measured samles had strongly hysteretic I-V characteristics with retraing currents well below 1 na. The exerimental setu is resented in the inset in Fig. 2. Switching robabilities have been measured by alying a set of traezoidal current ulses through the samle and by measuring the number of resulting voltage ulses. At the samle stage we used low ass RC filters (surface mount comonents near the samle). In the measurements on a single junction we used surface mount caacitors ( 680 F), but in the dc-squid measurements we had filters in series with resistors, with 5nFcaacitance to ground. The resistors were R s 500 and 680 in the measurement on a dc-squid and on a single junction,

4 R s L s L s R s I 10 µ m V P= # V ulses # I ulses P R s L s L s R s Φ=0.41Φ 0 Φ=0.28Φ 0 Φ=0 T=325 mk I (na) FIG. 2 (color). Cumulative histograms of the dc-squid at different temeratures and at different magnetic fields. Curves are shifted for clarity and the vertical sacing between ticks corresonds to unity escae robability. Solid lines are from simulations described in the text; dotted blue lines corresond to P I; T. Inset: scanning electron micrograh of the measured dc-squid and the exerimental circuit. resectively. Bonding wires with inductance of order nh connect the samle to filters. In Fig. 2 we resent the measured cumulative switching histograms (oen circles) of the dc-squid at different fluxes and temeratures with 200 s. At the lowest temeratures, the histograms can be well fitted by the MQT model, giving I c 200, 128, and 55 na, for = 0 0, 0.28, and 0.41, resectively. For 0 we also lot the escae robability P I; T defined earlier (dotted blue lines). In Fig. 3 we show the measured histogram osition I 50% P I 50% 0:5 and the width I I 90% I 10% for both samles. The dc-squid measurements were done both at negative and ositive values of flux in order to ensure that the external flux had not changed. The osition in Fig. 3(a) is normalized to the corresonding value of I c at zero temerature. At low T all the measured data are consistent with MQT results. On increasing T the arameters are constant u to the estimated crossover temerature T 0.ForT>T 0 the width is increasing and the osition is moving down as the TA model redicts. The qualitative agreement is good for most of the results u to the temerature T D. At T D, I starts to decrease abrutly. Moreover, the osition I 50% saturates at the same value 0:35I c. For the single JJ, the saturation occurs at 0:3I c. The dc-squid data measured at 0:41 0 show no FIG. 3 (color). (a) The ositions (I 50% ) and (b) the widths ( I) of the histograms. Black solid and dotted lines are results (with known junction arameters) of standard TA and MQT models, resectively, ignoring dissiation. Blue, red, and orange solid lines are the arameters of the simulated histograms based on the LO model discussed in the text. trace of thermal activation at any temerature, and underdamed hase diffusion revails down to temeratures below T 0 50 mk where MQT dominates. If we assume a realistic shunt imedance R! 500 (the value of the surface mount resistors), we obtain Q 4 at I c 200 na and C J 100 ff, which yields I m 0:35I c, like in the exeriment. In the diagram of Fig. 1 we also resent I c of the dc-squid at fluxes 0, 0:28 0, and 0:41 0 by horizontal dashed lines. It can be seen that the intersections of the dashed lines and the boundary of the UPD regime are very close to the exerimental values of T D. The saturation of the histograms and their reentrant steeness is thus a manifestation of the crossover from TA escae into UPD due to dissiation. In the case of a single JJ we obtain I m 0:3I c and T D 650 mk, yielding Q 4:4. This corresonds to R! 230, somewhat smaller than the anticiated value R! 680. Figure 2 shows that the standard TA and MQT models cannot account for our observations. Excet for the data at

5 the lowest temeratures, the measured histograms deviate from P I; T. Dissiation alone cannot exlain the difference. The basic TA model yields a width I / T 2=3 [6] and it can be seen in Figs. 2 and 3 that the dc-squid has weaker temerature deendence even well below T D.In the dc-squid there are just few energy levels in the well and thus the assumtions of continuous energy sectrum are not valid [6]. Within the semiclassical model of Larkin and Ovchinnikov (LO) [9], the total escae robability is calculated using P LO 1 k k, where k is the robability of finding the article in a state k after the current ulse of length. The kinetic equation can be written as d k dt j kj j jk k k k. We take into account transitions jk between neighboring levels and tunneling out, k. The relaxation rates between levels j and j 1 are well aroximated by j 1;j j! =4Q. Detailed balance yields j;j 1 e E j E j 1 j 1;j. The ositions of the levels and the escae rates are calculated using the results of Ref. [9]. The final state k is calculated numerically using R 1 0 e A t 0 dt, where A is the transition matrix. I is set to zero in the beginning, and 0 is Boltzmann distributed. The effect of the relocalizing dissiation must be taken into account also in the quantized energy level model. Using again E D 8E J =Q, and the fact that the energy difference between the two successive maxima is 2 E J I=I c, the level energy E must satisfy U E<E J 2 I=I c 8=Q (2) to allow switching into the free running state. If (2) is violated, the corresonding tunneling rate is set to zero [10]. Note that (2) yields the condition I>I m if we set U E, but in Eq. (2) we assume that after tunneling the starting oint is not at the otential maximum. The solid red lines in Fig. 2 resent results of simulations with quantized energy levels and dissiation for the data of the dc-squid at zero and 0:28 0 fluxes. At 0:41 0 I c is so small that the escae robability is large even at zero bias (excet at the lowest temeratures). This means that the hase is moving constantly rather than infrequently escaing from a localized state, and thus our model does not work anymore. The only fitting arameter was Q, and the fitted values were in a very reasonable range. At 0 we find Q 6 at the lowest temerature, and it decreases with increasing temerature u to 4 at 325 mk. At 0:28 0 Q 3 to 4, again decreasing with temerature. In Fig. 3 we resent I 50% and I of the simulated histograms. We resent also the simulated arameters for single junction (Q 4) [5]. In the measurements on a single junction, the number of energy levels is large ( 20) and the their searation is smaller than their width. Our model assumes well-searated levels and thus the agreement with a simle TA model is better for a single junction at T<T D, esecially in Fig. 3(a), whereas reentrant steeness of the histograms could not be exlained with the basic TA model. The agreement between simulation and measurements on a dc-squid is excellent. The osition and the width of the measured histograms coincide and, in articular, at higher temeratures the simulated histograms for both samles start to get steeer again. At higher temeratures the uer energy levels, whose escae rate is significant with smaller otential tilting angles, are oulated as well. The histograms thus eak at smaller currents and the condition (2) is not necessarily fulfilled anymore. What remains in the measured (and simulated) histograms is the escae from the states at the tail of the Boltzmann distribution above the dissiation barrier. In summary, our measurements demonstrate for the first time how a Josehson junction can transit from the macroscoic quantum tunneling through thermal activation into underdamed hase diffusion under the variation of temerature and magnetic flux. A model including a hase diffusion barrier and level quantization yields a quantitative fit of the observations. The Academy of Finland, EU IST-FET-SQUBIT2, and the French ACI, IPMC, and IUF rograms are acknowledged for financial suort. We thank J. Männik, A. O. Niskanen, T. T. Heikkilä, and M. A. Paalanen for helful discussions. Note added. Since the submission of this manuscrit, similar exerimental results were reorted in [11]. Their interretation differs from ours in details. [1] D. Vion et al., Science 296, 886 (2002); I. Chiorescu et al., Nature (London) 431, 159 (2004); S. Saito et al., Phys. Rev. Lett. 93, (2004); J. Claudon et al., ibid. 93, (2004). [2] J. Tobiska and Yu. V. Nazarov, Phys. Rev. Lett. 93, (2004); J. P. Pekola, ibid. 93, (2004). [3] D. Vion et al., Phys. Rev. Lett. 77, 3435 (1996). [4] A. Franz et al., Phys. Rev. B 69, (2004); Y. Koval, M. V. Fistul, and A. V. Ustinov, Phys. Rev. Lett. 93, (2004); P. A. Warburton et al., J. Al. Phys. 95, 4941 (2004); Joachim Sjöstrand et al., cond-mat/ [5] In this Letter, we ignore the T deendence of. [6] U. Weiss, Quantum Dissiative Systems (World Scientific, Singaore, 1999), 2nd ed.. [7] J. M. Martinis and R. L. Kautz, Phys. Rev. Lett. 63, 1507 (1989); R. L. Kautz and J. M. Martinis, Phys. Rev. B 42, 9903 (1990). [8] H. S. J. van der Zant, D. Berman, and T. P. Orlando, Phys. Rev. B 49, (1994). [9] A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eks. Teor. Fiz. 91, 318 (1986); [Sov. Phys. JETP 64, 185 (1987)]. A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eks. Teor. Fiz. 87, 1842 (1984); [Sov. Phys. JETP 60, 1060 (1984)]. [10] With the low Q values in the exeriment, the hase relaxes in the next well in a time! s, far shorter than the tyical time interval between hase diffusion events (10;...; 100 s for bias currents used here). [11] V. M. Krasnov et al., cond-mat/ ; J. Männik et al., cond-mat/