To test whether the theory of macroscopic quantum tunneling (MQT) is applicable to systems with 2 degrees of freedom, we experimentally investigated the switching current distribution of a dc SQUID. Using sample parameters determined from measurements at T=4.2 K, we are able to make quantitative comparison to the theories from 8 mK to 4.2 K. The excellent agreement between the data and the MQT theory demonstrates that tunneling from the zero-voltage state of the dc SQUID is well described by the quantum mechanics

Han, Siyuan

Li, Shaoxiong

Qiu, Wei

Wang, Zhen

Yu, Yang

Zhang, Yu

KU ScholarWorks

VOLUME 89, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 2002Quantitative Study of Macroscopic Quantum Tunneling in a dc SQUID:A System with Two Degrees of FreedomShao-Xiong Li, Yang Yu, Yu Zhang, Wei Qiu, and Siyuan HanDepartment of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045Zhen WangKARC, Communication Research Laboratory, Ministry of Posts and Telecommunications,588-2 Iwaoka, Owaoka-cho, Nishi-ku, Kobe 651-24, Japan(Received 2 October 2001; published 7 August 2002)098301-1To test whether the theory of macroscopic quantum tunneling (MQT) is applicable to systems with 2degrees of freedom, we experimentally investigated the switching current distribution of a dc SQUID.Using sample parameters determined from measurements at T  4:2 K, we are able to make quantitativecomparison to the theories from 8 mK to 4.2 K. The excellent agreement between the data and the MQTtheory demonstrates that tunneling from the zero-voltage state of the dc SQUID is well described by thequantum mechanics.DOI: 10.1103/PhysRevLett.89.098301 PACS numbers: 85.25.Dq, 03.65.–w, 74.50.+rexceeded the theoretical prediction by more than 40%. (T  0:3). As one can see, in the ’ (longitudinal)Whether quantum mechanics is valid for macroscopicvariables is one of the most fascinating issues of funda-mental physics [1]. The experimental studies, especiallyquantitative tests of the theory for macroscopic variables inthe quantum regime, provide important insights to ourunderstanding of the physical world. In the past few years,devices based on the Josephson effect, such as theJosephson junction (JJ) and the superconducting quantuminterference device (SQUID), have been proven effectivesystems to perform this test. For instance, in a JJ, the phasedifference  across the junction is a macroscopic variableand the dynamics of the junction is identical to a particle’smotion in a one-dimensional (1D) washboard potential.Experiments in 1D systems, such as current biased JJsand rf SQUIDs, have yielded results in very good agree-ment with the theoretical predictions of macroscopic quan-tum tunneling (MQT) [2–6]. In contrast, experiments insystems with 2 degrees of freedom (2DF) have producedsignificant divergences. For example, using a dc SQUID,which contains two JJs and, hence, has two macroscopicdegrees of freedom, Sharifi, Gavilano, and Harlingen(SGH) reported the observation of anomalous suppressionof thermal activation (TA) from the two-dimensional (2D)potential well of a dc SQUID [7]. SGH suggested that theapparent suppression of TA rate may rise from an enhancedpotential barrier caused by interaction between the twomacroscopic degrees of freedom in the dc SQUID. Onthe other hand, experiments by Han, Lapointe, andLukens (HLL) [8] and Lefevre-Seguin, Turlot, Urbina,Esteve, and Devoret (LTUED) [9] showed that, in thethermal regime, the activation energy of the 2D SQUIDsagreed very well with the potential barrier. More impor-tantly, SGH’s result in the quantum regime significantlydisagreed with the theoretical prediction as the measuredwidth of switching current distribution-at temperature wellbelow the quantum-classical crossover temperature Tco0031-9007=02=89(9)=098301(4)$20.00 Notice that this disagreement could not be accounted forby measurement uncertainties and the barrier enhancementsuggested by SGH would actually make the matter muchworse. However, the HLL and LTUED’s experiments didnot address the quantum regime so that the experimentalevidence available thus far [7] seems to indicate that,despite its great success in describing 1D systems, MQTtheory might not be applicable to even the simplest 2Dsystems, such as dc SQUIDs with no flux bias. Further-more, the understanding of MQT in dc SQUIDs is crucialto quantum computing with SQUID qubits, in which dcSQUIDs, as the most sensitive flux detectors available [10],serve as the readout devices. In this Letter, we report theresults of a quantitative study of the escape rate of anunderdamped dc SQUID, a system with 2DF, in both theTA and quantum tunneling regimes. In contrast to SGH’sresult, our data are in excellent agreement with the MQTtheory, demonstrating that quantum mechanics also applieswell to macroscopic systems with 2DF.A dc SQUID consists of two JJs connected in parallel bya small superconducting loop of inductance L (Fig. 1 in-set). The critical current of each junction is I0 (assumingidentical JJs). The macroscopic variables of this system arephases, 1 and 2, across the two junctions, respectively.For a dc SQUID with current bias I and flux bias 	e f	0, where	0 is the magnetic flux quantum, the dynamicsof the system can be treated as a fictitious particle of massC, which is the sum of the two junctions’ capacitance,moving in a 2D potential [11] U’;’dc EJ cos’dc=2 cos’ x’ Tj2=4, where EJ "I0=e is the sum of two junctions’ Josephson couplingenergy, ’  1  2=2 is the average phase differenceacross the device, ’dc  2  1, j  ’dc  2f=T isthe normalized circulating current, x  I=2I0, and T 2LI0=	0. Figure 1 shows a 3D plot of the dc SQUIDpotential with zero flux bias (f  0) and a small T value2002 The American Physical Society 098301-1VOLUME 89, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 2002direction, the potential has a sequence of saddle points andlocal minima (wells) located at ’dc  0, while along the’dc (transverse) direction, the potential rises sharply asj’dcj increases. Hence, a particle initially trapped in apotential well would escape along the longitudinal direc-tion near the vicinity of ’dc  0.At high temperatures (T  Tco), escape from the 2Dpotential well is dominated by TA over the potential barrierwith the transition rate given by [11,12] 2at expUkBT; (1)where U is the minimum height of the potential barrier(i.e., through the saddle point), T is temperature, and at(0< at < 1) is a damping dependent factor [13]. The‘‘attempt frequency’’  is given by   !lw!tw=!ts,where !lw!tw is longitudinal (transverse) small oscilla-tion frequency in the well, and !ts is the transverse oscil-lation frequency at the saddle. Equation (1) is equivalent tothe TA rate of a 1D system having the attempt frequencyrenormalized by a factor of !tw=!ts. For dc SQUIDs withT  2, such as the one studied here and in [7], the ratio!tw=!ts is close to unity. Hence, the TA rate of dc SQUIDswith T  2 should closely follow that of a 1D system.At T  Tco, MQT becomes the dominant escapemechanism. The tunneling rate of dc SQUIDs with f  0is given by [14]:  f2D;c!02120p 7:2Uh!01=2exp7:2Uh!0 f2D;c1D; (2)where !0  2I0=CJ	01=21 x21=4 is the plasma fre-quency and CJ is the shunt capacitance of each junction.f2D;c is a dimensionless function of   22p1T 1 x1=2 and c  90 exp2Tco=T  5=4, where is a measure of interaction between the 2DF, c is theFIG. 1. The 2D potential for a dc SQUID with f  0, T 0:3, and x  Ib=2I0  0:25. The inset is a schematic of the dcSQUID.098301-2critical parameter for instanton splitting, and Tco h!0=2kB is the quantum-thermal crossover temperature.Equation (2) clearly shows that the tunneling rate of a dcSQUID with zero flux is closely related to that of a singleJosephson junction. In particular, for dc SQUIDs withT  2, such as our sample and that of SGH (T ’0:4), there is no instanton splitting and f2D ’ 1. Noticethat the disagreement between the result of [7] and theMQT theory is so significant that no reasonable adjustmentof sample parameters could reconcile the data with Eq. (2).The switching current distribution Px is related to theescape rate throughPx xdx=dtexp1dx=dtZ x0x0dx0: (3)To quantitatively compare experimental data with varioustheories, it is convenient to use the ‘‘escape temperature’’Tesc defined through [15]   =2 expU=kBTesc. Itis straightforward to show that, for TA Tesc ’ T, while forquantum tunneling Tesc ’ Tco. For dc SQUIDs biased atx & 1, Tesc depends linearly on 3=2 [16], where  isknown as the width of Px. Because in our experimentthe widths were extracted directly from the measured Px,which did not involve the use of any theoretical model andsample parameter, it was used in our data analysis andpresentations.The sample was a dc SQUID consisting of two nomi-nally identical NbN=AlN=NbN JJs each having a diameterof 2 m. The critical current of the dc SQUID is 2I0 35:8 A, which was determined by fitting Px at 4.2 K,where the system was in the thermal regime. Because ofthe large energy gap of NbN (2 ’ 5:4 meV), I0 remainedconstant below 4.2 K. Notice that in the quantum regime itis essential to have an independent measurement of thesample’s shunt capacitance for making a quantitative com-parison between the data and theoretical predictions. In ourexperiment, the capacitance of the dc SQUID, C  2CJ 380 30 fF, was determined from resonant activation(RA) measurement [15] at 4.2 K. The RA data also yieldeda quality factor, Q  RC!0  200, indicating that theeffect of damping on tunneling at T  Tco was negligible[17,18]. The inductance of the dc SQUID loop, estimatedfrom the modulation depth of critical current and loop geo-metry, is L  5 1 pH. Using these sample parameters,the crossover temperature is found to be Tco  0:29 K.We measured the switching current distribution of thedc SQUID using a time-of-flight technique similar to thatdescribed in [19]. Each measured distribution consisted of2 10420 000 escape events. Because heating and externalnoise could cause the width  to flatten out at low tem-peratures, which could be mistaken as the evidence forMQT, cautions must be taken to eliminate them. In ourexperiment, the dc SQUID was enclosed in a helium-filledcopper sample cell thermally anchored to the mixingchamber of a dilution refrigerator. EMI filters, cryogeniclow-pass filters and microcoax microwave filters [20],098301-2VOLUME 89, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 2002battery-powered low-noise preamplifiers, and shielded en-closure were used to protect the sample from externalnoise. Connections to the computer and ac-powered instru-ments were made via optically coupled isolation ampli-fiers. Extensive diagnostic tests were made using lowcritical current (Ic  1–10 A) junctions to ensure thatthe effects of noise from the environment and measurementcircuitry was negligible down to 8 mK. Distributions with< 15 nA have been observed using the same setup,demonstrating the effectiveness of the shielding. A mu-metal cylinder provided 60 dB attenuation to shield thesample from ambient magnetic field fluctuations. For im-proved stability, the data were taken with zero flux bias(f  0) and the temperature was regulated to within 1%and 0:3% of the set point at T below and above 0.6 K,respectively.Figure 2 shows a series of measured switching currentdistributions and their comparisons with the 2D TA andMQT theories. Notice that Eq. (3) is valid if the temporalescape probability at constant bias currents was Pesct 1 et, which was confirmed by the measured Pesctshown in Fig. 2(a). The data in Fig. 2 show that, in both thethermal (T > 2Tco) and quantum (T > Tco=2) regimes, thedistributions, including their shape, width, and position,agree very well with the theoretical predictions. Below0.15 K the data were essentially the same, as expectedfor a system with very low damping [17,18]. Tunnelingrates obtained from the switching current distributions areplotted as a function of barrier height in Fig. 3. Consideringthat all theoretical calculations were performed using theFIG. 2. The measured temperature dependence of the switch-ing current distributions. The symbols are the experimental dataand the lines are calculated from the theories of thermal activa-tion and MQT for systems with 2DF. Inset (a): The temporalescape probability at bias x  0:975 and T  10 mK showingthe expected exponential decay. Inset (b): Comparison of themeasured distribution (solid dots) at T  300 mK with thetheoretical prediction of quantum correction (solid line), T  0MQT (dashed line), and thermal activation (dash-dotted line).098301-3sample parameters determined from measurements at4.2 K, the agreement between the data and theories is quiteremarkable. Our data provide strong evidence supportingthe validity of MQT theory for systems with 2DF.However, as shown in Fig. 2(b), in the crossover regionthe measured distributions deviate significantly from thatof T  0 MQT theory (the dashed line). By taking intoaccount the effects of thermal enhancement to MQT [17],the agreement between the data and theory (the solid line)improves. However, the remaining disagreement cannot beentirely accounted by the thermally enhanced MQT theory,indicating the need for further studies in this temperaturerange, where neither TA nor MQT is the dominant escapemechanism.A critical comparison of the experimental data andvarious theories can be obtained by examining the T de-pendence of the distribution width [21]. In Fig. 4 and itsinset, the measured  vs T2=3 is compared with the MQTand TA theories using the sample parameters 2I0 35:8 A, C  380 fF, and R  1500 . At T > 0:6 K,the data show a linear dependence on T2=3, as expectedfrom Eq. (1) of TA theory (the solid line) without adjust-able parameters. The excellent quantitative agreement be-tween the data and the theory of TA clearly shows that theactivation energies were equal to the 2D potential barriers,in agreement with the results of HLL and LTUED. Be-low T  0:15 K, the data became T independent withinthe uncertainties of measurements. In the quantum li-mit, the measured dataT  0:15 K  40:6 1:6 nA isconsistent with the 41.2 nA calculated from the T  0MQT theory (the dashed line). Notice that in the thermalFIG. 3. Tunneling rates vs potential barrier at various tempera-tures. The symbols are data and the solid lines are theoreticalpredictions. The first three data sets from the left are T  10 mK(triangles), 150 mK (circles), and 300 mK (crosses). Inset: Themeasured (symbols) and predicted (lines) tunneling rates vsbarrier height for T  10 mK (triangles and solid line),150 mK (circles and dotted line), and 300 mK (crosses anddashed line).098301-3FIG. 4.  vs T2=3 of the data (symbols) and the predictions ofthermally enhanced quantum tunneling of Ref. [12] (solid line),thermal activation (dot-dashed line), and 2D MQT (dashed line)with arrow indicating Tco. The log-log plot magnifies the lowtemperature region, while the inset shows the linear dependenceof  on T2=3 in the thermal regime.VOLUME 89, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 2002limit, for samples with Q 1,  is almost independent ofC but quite sensitive to R. The situation is just the oppositein the quantum regime. For instance, using C  395 fF,we found  40:6 nAdataT  0:15 K, while increas-ing R from 1:5 k to 1:5 M results in a negligiblechange in the calculated width. Hence, the data at T <0:15 K again show that in the quantum regime the MQTtheory describes the behavior of a system of 2DF very well.Finally, we examine the crossover region. From theinterception of the straight lines extrapolated from thedata in quantum and thermal limits, we found Tco 0:29 0:01 K, which agrees very well with the theoreticalvalue of 0.290 K calculated from the sample parameters.However, for 0:15 K< T < 0:6 K, the measured dataTdeviates systematically from both the T-independent be-havior of the MQT theory and the simple  / T2=3 scalingbehavior of the TA theory (Fig. 4). Similar smooth cross-over behavior was observed previously in Josephson junc-tions by Martinis et al. [5]. Because in both the thermal andquantum limits the data agree very well with the theoreticalpredictions, the larger width in the crossover region couldnot be due to heating or external noise. One mechanismthat could result in a larger width is the thermal enhance-ment of MQT, which produces an improved, albeit stillunsatisfactory, agreement with the data for 0:15 K< T <Tco. For Tco < T < 0:6 K, the theory of quantum correc-tion [17] produced negligible improvement. In the cross-over region, the shape of Px also significantly deviatesfrom that calculated from Eq. (3), which could occur whentunneling from excited levels contributes significantly tothe escape process [19].In summary, we have measured the temperature depend-ence of the switching current distributions of a dc SQUID098301-4from well below (< 0:025Tco) to well above ( 14Tco) thequantum-classical crossover temperature. The experimen-tal ability to control and characterize the sample and thetheoretical capability to accurately model the system fa-cilitate a quantitative comparison between the theoriesand experiment. Our result in the thermal regime agreesvery well with the works of HLL and LTUED, supportingthe TA theory. More importantly, without the use of ad-justable parameters, the data in the quantum regime agreeexcellently with the MQT theory, demonstrating incon-trovertibly that the theory of MQT correctly describes thebehavior of dc SQUIDs—a macroscopic quantum systemwith 2DF. The result also assures that the MQT theorycan be applied to design the readout circuits made ofdc SQUIDs for flux based superconducting quantumlogic gates.We thank R. Alexander for technical support. This workwas supported in part by AFOSR (F49620-99-1-0205), thestate of Kansas (S99041), and the NSF (DMR-9876874).[1] Quantum Tunneling in Condensed Media, Modern Prob-lems in Condensed Matter Sciences Vol. 34, edited byY. Kagan and A. J. Leggett (North-Holland, Amsterdam,1992).[2] R. F. Voss and R. A. Webb, Phys. Rev. Lett. 47, 265(1981).[3] S. Washburn, R. A. Webb, R. F. Voss, and S. M. Faris,Phys. Rev. Lett. 54, 2712 (1985).[4] D. B. Schwartz, B. Sen, C. N. Archie, and J. E. Lukens,Phys. Rev. Lett. 55, 1547 (1985).[5] M. H. Devoret, J. M. Martinis, and J. Clarke, Phys. Rev.Lett. 55, 1908 (1985).[6] J. Clarke et al., Science 239, 992 (1988).[7] F. Sharifi, J. L. Gavilano, and D. J. V. Harlingen, Phys.Rev. Lett. 61, 742 (1988).[8] S. Han, J. Lapointe, and J. Lukens, Phys. Rev. Lett. 63,1712 (1989).[9] V. Lefevre-Seguin et al., Phys. Rev. B 46, 5507 (1992).[10] M. Mück, J. B. Kycia, and J. Clarke, Appl. Phys. Lett. 78,967 (2001).[11] C. D. Tesche, J. Low Temp. Phys. 44, 119 (1981).[12] E. Ben-Jacob et al., J. Appl. Phys. 54, 6533 (1983).[13] M Büttiker, E. P. Harris, and R. Landauer, Phys. Rev. B 28,1268 (1983).[14] B. I. Ivlev and Y. N. Ovchinnikov, Sov. Phys. JETP 66, 378(1987).[15] J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev. B35, 4682 (1987).[16] J. Kurkijärvi, Phys. Rev. B 6, 832 (1972).[17] H. Grabert, P. Olschowski, and U. Weiss, Phys. Rev. B 36,1931 (1987).[18] A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149,374 (1983).[19] S. Han et al., Science 293, 1457 (2001).[20] A. B. Zorin, Rev. Sci. Instrum. 66, 4296 (1995).[21] A. Garg, Phys. Rev. B 51, 15 592 (1995).098301-4

Quantitative study of macroscopic quantum tunneling in a dc SQUID: A system with two degrees of freedom

https://kuscholarworks.ku.edu/bitstream/1808/1619/1/e098301.pdf

Quantitative study of macroscopic quantum tunneling in a dc SQUID: A system with two degrees of freedom

Abstract

Similar works

Full text

Available Versions

KU ScholarWorks