I HLL U" "'SECURITY CLASSIFICATION OF THIS PAGE

PAGE

Is. REPORT SECURITY CLASSIFI TIVE MARKINGS

2.L SECURITY CLASSIFICATION. -5 8 rION/AVAILABILITY OF REPORT

1" 2b. DECLASSIFICATIONIDOWNGI Unlimited

4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S)

6a. NAME OF PERFORMING ORGANIZATION 5b. OFFICE SYMBOL 7. NAME OF MONITORING ORGANIZATION

State University of New York (Ofcplica le)

at Stony Brook Same as 8A

6c. ADDRESS (City. State and ZIP Code) 7b. ADDRESS (City. State and ZIP Code)

Department of PhysicsSUNYStony Brook, N. Y. 11794 Same as 8B

S.. NAME OF FUNDING/SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER

ORGANIZATION (If aiplicabZe)

ONR N00014-84-C-0261

Sc. ADDRESS (City. State and ZIP Code) 10. SOURCE OF FUNDING NOS.

800 N. Quincy Street PROGRAM PROJECT TASK WORK UNIT

Arlington, Virginia 22217 ELEMENT NO. NO. NO. NO.

11. TITLE (lInclude Security Clauificationl

Experimental Studies of Josephson Effect ... .

12. PERSONAL AUTHOR(S)

James Lukens13. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Yr.. Mo.. Day) 15. PAGE COUNT

Final FROM .B.L3i5_0 . TO 890930 900906 21 pages16. SUPPLEMENTARY NOTATION

reprints enclosed

17. COSATI CODES I. SUBJECT TERMS IContinue on rtvese If ltcUcry and identify by block number)

FIELD GROUP SUB. GR.

Josephson Devices, SQUID

19. ABSTRACT (Continue on rueers if neccuary and Identlfy by block number)

See attached summary

20, ISTRIBUTION/AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION

UNCLASSIFIED/UNLIMITED Q SAME AS RPT. 0 OTIC USERS 0 Unclassified

22s. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELC:PHONE NUMBER 22c, OFFICE SYMBOL

tlnclu de Area Code)

James Lukens, Project Director (516)632-8081

DD FORM 1473, 83 APR EDITION OF 1 JAN 73 IS OBSOLETE.90 09 SECURITY CLASSIFICATION OF THIS PAGE

Do 0: _

OFFICE OF NAVAL RESEARCH

FINAL TECHNICAL REPORT

for

15 March 1984 through 30 September 1989

for Accession For

NTIS GRA&IDIC TABUnannounced [

Contract N00014-84-C-0261 Justification

Task No. NR 604-004 By

Josephson Devices Distribution/

Availability Codes

lAvail and/orDist Special

James Lukens ".flj_

The Research Foundation ofState University of New York

P.O. Box 9Albany, New York 12201

Reproduction in whole, or in part, is permitted for any purpose of the United StatesGovernment.

* This document has been approved for public release and sale; its distribution isunlimited.

F

( Final report on ONR contract #N00014-84-C-0261

Summary of work completed under the contract

During the course of this contract, research has been conducted in three distinct

areas, all involving the Josephson effect:

A. Development of Josephson junction arrays as tunable sources of microwave

through submillimeter radiation,

B. Conformation of general relativistic effects, equivalent to the red shift, for

charged particles, .

C. Studies of the quantum mechanics of macroscopic variables, in particular

of the flux linking a SQUID........ : 2 "

The fundamental studies of Josephson oscillators A.), which had been supported

under previous ONR contracts, were concluded during the first part of the contract.

The present status of this work is discussed in detail in the reprint attached as reference

A. The success of this work has lead to very active ongoing research programs in my

laboratory, supported by several sources to develop technical applications of these

oscillators. In particular, techniques for coupling junctions distributed over many

wavelengths were developed and applied to the design of a broadly tunable submilli-

meter source generating 7pW of power at 400GHz. In addition, recent results on

frequency agile microwave sources demonstrate modulation of a 16GHz carrier over

3GHz at modulation frequencies in excess of 500MHz.

Our focus during the middle years of the contract was on B.), the study of effects

predicted by general relativity on charged particles moving through a gravitational field.

These experiments, which are described in the reprint in appendix B, produced the first

demonstration that the electrochemical potential difference between two

superconducting wires varies with gravitational potential (by about 1 part in 1016 per

meter in the earth's field). This effect is essentially the equivalent for charged particles

of the well known gravitational red shift for photons. Use was made of essentially ideal

Josephson effect "batteries" phase-locked to a microwave source to establish precisely

related EMFs at different heights. As part of this work techniques were developed to

measure potential differences of 10- 22V.

During the final phase of the contract, we began work to test predictions that

macroscopic variables, such as the flux through a SQUID loop, display quantum

mechanical properties such as tunneling and coherent superposition of the flux wave

functions belonging to different states. This work has been continued under a subse-

quent ONR grant mnd he:. produced the first evidence that quantum theory for

transitions between the fluxoid states in a degenerate two level SQUID is correct. The

results under this contract are presented in the reprint in appendix C. Here a novel

technique is demonstrated for using a two junction composite SQUID to make what is

essentially a "man made atom" with energy levels and lifetime which can be controlled

in situ by the experimenter.

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APPENDIX B

VOLUME58, NUMBER 12 PHYSICAL REVIEW LETTERS 23 MARCH 1987

Test for Relativistic Gravitational Effects on Charged Particles

A. K. Jain, J. E. Lukens, and J.-S. Tsai(a)Department of Physics, State University of New York, Stony Brook, New York 11794

(Received 14 November 1986)

Experimental results are presented which provide the first measurement of the effects of a gravitation-al field on charged particles, equivalent to the red shift for photons. Two Josephson-effect batteries(V"-300 pV) having a vertical separation -of 7.2 cm are connected in opposition by superconductingwires. A voltage difference of 2.35x 10- 21 V is maintained between these batteries by means of thegravitational red shift. The emf around this loop is, however, measured to be less than I X 10-22 V, con-sistent with the predicted invariance of the gravito-electrochemical potential along the wires.

PACS numbers: 04.80.+z, 74.50.+r

The strong equivalence principle, which states that lo- These gravitational effects can be 10 to 15 orders ofcally the effects of a uniform gravitational field are indis- magnitude greater 9 than the relativistic shift in potentialtinguishable from those due to an accelerating reference difference discussed above. However, they do not con-frame, is a fundamental postulate of general relativity, tribute to the net emf around a loop.A consequence of this equivalence, along with special In this Letter we report the results' ° of a null experi-relativity, is that the rates of identical clocks running in ment to compare the fractional change with height of thedifferent gravitational potentials will differ. This varia- electrochemical potential difference between two super-tion in clock rates implies the gravitational red shift of conducting wires to the fractional change in photon fre-photons, which was first observed by Pound and Rebka' quency over the same height, which is given by Av/v=X.and has, during the past two decades, been verified by a The schematic of the experiment is shown in Fig. 1. Thenumber of workers 2 to a present accuracy of better than "batteries" used to provide the highly precise referenceI part in 104. There have, however, previously been no emf's are Josephson junctions separated vertically by aexperimental tests of the corresponding effects for height z =7.2 cm and phase locked to a common exter-charged particles. nal microwave source. The relation between the fre-

One such effect occurs for the conduction electrons in quency v of the external radiation and the dc electro-a metal, such as a superconductor, in a gravitational field chemical potential difference across the junction, V-Ay,where, in equilibrium, the electrochemical potential pi is given by the Josephson equation aswill in general not be constant. Rather, a related quanti- 0)ty I! =p(l +X) called the gravito-electrochemical poten- V=nvo'(tial will be constant along the wire. 3-6 Here X=zg/c2 where the integer it is the order of the radiation-inducedand depends on the gravitational potential through the step in the junction I-V curve. Since the radiationheight z; g is the gravitational acceleration at the earth's reaching the upper junction is red shifted with respect tosurface and c the speed of light in a vacuum, givingX/z = 1.09 x 10 -i 6/m. For a resistanceless metal such asa superconductor, 7 should also be independent of posi-tion even with a dc current flowing. 6 Thus a circuit with CRYSTAL DATAtwo identical batteries (as measured by local observers) REFERENCE RECORDER I sQ UfDof emf V, separated in height by z and connected in op- 16,8 6Hzposition by superconducting wires, would have a net emf Phase lockedMicrowaveAV=XV around the zero-resistance loop leading to a iSourceloop current increasing linearly in time.4

Within a single wire a purely Newtonian gravitationalfield can produce a number of effects which shift the A ,-enuoio sQuIDelectrochemical potential. For instance, the gravitational MagnetomelerIforce on the electrons must be balanced by an electro- OCMO N ISstatic force giving rise to an electric field (the Schiff- soUC D I I 8:1 mlii GaussBarnhill field7). This latter effect is in fact contained in T--.5 Kthe potential fi if the electron rest mass is included, sinceX11c 2 = m gz. An electric field can also be produced be- FIG. 1. Schematic of the measurement circuit. The crossescause of the shift in the Fermi level induced by lattice denote the Josephson batteries. The darker lines indicate su-distortion due to the gravitational force on the ions.8 perconducting portions of the circuit.

@ 1987 The American Physical Society 1165

VOLUME 58, NUMBER 12 PHYSICAL REVIEW LETTERS 23 MARCH 1987

that reaching the lower junction, the potential differences radiation at the junction.' 3 At a constant gravitationalacross the upper and lower junctions, V, and V, respec- potential the time-averaged fluxoid quantization condi-tively, will differ slightly so that tion for the loop of inductance L as shown in Fig. 1 is

th~n '4

V. = V(l -). (2)

(4o/27r) (02- 01) +Lit + 'lo =mclo, (3)In the absence of additional relativistic effects on theCooper pairs, one would thus expect the net emf AV in where Ia is the flux applied to the loop and Ithe loop containing the junctions to beXV-2xl0- 2 V, =(A12 -AI1)/2. For simplicity, identical junctions andsince V"-300 pV and X--7x 10- 18 for this experiment, locking ranges for I and 2 have been assumed.Such a nonzero value for AV would lead, as discussed The emf which drives the changes in the loop currentbelow, to a time-varying flux through the loop. This It is, from Eq. (3),could be detected by use of a superconducting quantuminterference device (SQUID) magnetometer, as shown in 2 62 d - 0 + 4no --Fig. 1. The relativistic prediction that I=const in the 2;r d( 2 ;r dt

superconducting leads, however, implies that the poten- where Ao-toz-o 1. For the SQUID's used in thesetial difference V between the leads varies with height so measurements f't=2;rLJ/(Do .• I, thus the terms in 6 canthat V(z) V(O)(I -Xq). The symbol Xq, which in be neglected, givingtheory equals X, has been introduced to allow for the pos- dl _ d°ba (DOsibility of corrections to the predicted relativistic effects - L- - + -n Aw.on charged particles. The loop emf is thus AVV di d 2rx (X Xq) and is predicted to be zero on the basis of the The response of this phase-locked system to a small fre-strong equivalence principle. quency difference Aw, as described by Eq. (3), is the

The basic technique for the detection of extremely same as the response to an applied flux varying linearlysmall voltage differences between two Josephson junc- with time for the system biased below the junction criti-tions by monitoring of the flux change in a superconduct- cal current in the absence of radiation. For example, ifing loop was dcveloped 11 shortly after the discovery of (% is changed (or Ao; 0), a loop current will be in-the Josephson effect. The apparatus used for these men- duced. As long as I, < J, mi will remain constant and Itsurements is a refined version of that previously used by will increase (nearly) linearly with 4,, (or time) untilus to demonstrate the universality of Eq. (1) for different l,=J, then mi will change and It will decay in steps oftypes of Josephson junctions 12 to 2 parts in 1016. The approximately Oo/L as flux quanta enter the loop. In theJosephson junctions used here are lead-alloy tunnel junc- linear region, for I, <<J, the rate of change in the looptions with critical currents of I' 900 ,uA. The junctions current is related to Aw (for constant (Da) byare shunted with low-inductance 0.5-f1 resistors to en- dl,sure nonhysteretic operation and have I-V characteris- -L I--AV=- V (4)tics that are extremely close to those predicted for idealad 0)

resistively shunted junctions (RSJ model). The junc- with the junction voltage V=nvo0 .tions and superconducting leads connecting them are It is important to note from this discussion that, withinplaced over a niobium ground plane giving a total loop the constraint of fluxoid quantization, it is possible forinductance of L"-I nH, associated primarily with the the two junctions to have slightly different average volt-coupling inductor to the magnetometer. The microwave ages while in a fixed fluxoid state, phase locked to theradiation is coupled to the junctions by our placing each applied radiation. This result, contained in Eq. (4), hasjunction at the end of a capacitively grounded microstrip previously been demonstrated with a modified version ofwhich is in turn coupled to a coaxial cable through a the apparatus in which separate coaxial cables extendedsmall section of coplanar waveguide. In order to mini- from each junction to outside the cryostat.1 2 This per-mize any extraneous shifts in the relative phases of the mitted the creation of a small frequency difference byradiation reaching the two junctions, these two coaxial Doppler shifting of the radiation going to one of thecables are connected, just outside the sample cell and junctions and served to demonstrate that the sensitivityhalfway between the junctions, to a common cable lead- of the apparatus is indeed correctly predicted by Eq. (4).ing out of the cryostat. The sensitivity limit of the apparatus at a loop emf of

In general Josephson junctions will phase lock to ap- l XI0 - 22 V over a typical 10-h run implies that theplied radiation over a range of bias current Al, Jsin(8, change in the loop flux (D due to any extraneous sources-g/2), where the maximum current variation-or lock- must be less than 2x 10-4o/h. To provide shieldinging range-is dependent on microwave power. Here, against such small changes in the ambient flux (a, Mu-8=i---nco1: (wto=21rvi) is the difference between the metal shielding was used to reduce the ambient field tolinear component Oi of the phase of the Josephson oscil- about 1 mG. To stabilize the remaining flux the super-lations in the ith junction and the phase of the applied conducting loop was enclosed in a superconducting NbTi

1166

VOLUME 58, NUMBER 12 PHYSICAL REVIEW LETTERS 23 MARCH 1987

box which was placed inside a large lead can. The tem-perature of the box was regulated to ± 50 pK to prevent (a) V 30ojVthermally induced flux motion. In addition both junc--_ -_...._-_---tions were fabricated on sapphire substrates to minimizeany thermal gradients across the junction and prevent athermoelectric current from being induced in the loop. mDeliberate temperature shifts of 1 mK caused changes in0 of less than of I x 10-(Do even during temperaturetransients.

Additional factors which could cause All or A12 tochange and generate a loop current 11 are changes in the -microwave power or frequency or in the junction biascurrents Ii. Since there are common sources of mi- o.0 .o 4.0 6.0 8.0 10.0crowave and bias current for both junctions, the changes Time in hoursmentioned above, except for the gravitational red shift in FIG. 2. Curve a, (D vs time with the junctions phase lockedfrequency, will affect the junctions equally. For identical to lb.8-GHz radiation biased at 300 uV on the ninth-orderjunctions these extraneous effects would then produce step.' Curve b, (Fm vs time with the junctions biased on theequal changes in All and A12 causing no change in I,. zeroth step. Curve c, (,,(V'=300 uV)-4)m(V-0pV) vsEven with the small asymmetries present in the actual time. The dashed line shows the expected signal if i ratherapparatus, it has proven possible-as detailed below-to than #i remained constant along the superconducting wire. Thereduce the effects of variations in the bias current and solid lines in a and c are the respective best linear fits as de-

microwave sources to negligible levels. scribed in the text.Current from the common dc bias source was divided

between the junctions so that to first order a change inthe source current produced no change in I,. The drift in dependence of (Dm on time is highly reproducible fromthe current source of I part in 104 during a run corre- run to run. To check that the residual signal is due tosponded to an apparent voltage of about 10-23 V. The the rf SQUID, the measurement was repeated with thefrequency drift of the microwave source, which was junction bias current reduced so that the junctions werephase locked to a quartz-crystal oscillator, was I kHz/d. on their zeroth step (i.e., V=0 pV). The resulting vari-This resulted in a coherent drift in the voltages of the ation of cm with time, shown in Fig. 2, curve b, is essen-two Josephson batteries of about a picovolt over the tially identical to that in Fig. 2, curve a, indicating thatcourse of the measurement. However, the symmetric there is no measurable change in the flux 4) through theconfiguration, with equal microwave path lengths to the sample loop during the run. The maximum flux changetwo junctions, reduced the flux change through the loop through the sample loop as a result of relativistic effectsbecause of this coherent drift to a rate of 4.2x 10 5 (DO/ is just the difference between curves a and b, i.e.,d, corresponding to an apparent loop emf of less than (,,(V=300 uV)- m(V=0 uV) as shown in Fig. 2,10-24 V. The power output of the microwave source curve c.was regulated to about 2 parts in 103 by use of a leveling A linear-regression analysis of the data in Fig. 2, curveloop. However, the power reaching the junctions varied c, taking into account the presence of I/f noise in theby about 1.5% during the measurement because of a SQUID output, gives for the magnitude and error limitschange in the attenuation of the coaxial cable as the heli- (2a) of the loop voltage (3 ± 6) x 10-23 V. Nearly theum level in the cryostat dropped. This variation, which same limits are obtained from the data in Fig. 2, curve a,was the largest source of sample-related flux drift, pro- if the final 4 h of data at low helium level are discarded.duced an estimated change in (F of 5x 10 -4(o over 12 h In this case we obtain (I ± 8) x 10 -23 V.to give an apparent loop emf of 0.4x 10-22 V. We thus conclude that the net emf in a superconduct-

The loop flux was monitored for continuous 10-h ing loop containing two Josephson-effect batteries atperiods with use of a commercial rf SQUID magnetome- different gravitational potentials is not the difference inter located in the helium bath. The magnetometer out- the battery electrochemical potentials, AV=XV, butput (Dn, (referred to the sample loop) versus time is rather is indistinguishable from zero, consistent with theshown in Fig. 2, curve a, for the loop junctions phase relativistic prediction that #J rather than p is constantlocked on their ninth-order steps to 16.8-GHz radiation. along the superconducting wires. In particular we haveThis gives a junction voltage of "='300pV, and a tested the hypothesis that the potential difference be-difference between the two junction voltages, as a result tween the wires varies with height as V(z) =V(O)of the gravitational red shift, of 2.35x10 - 21 V. (, is x(l -Xq). Our results imply that X/IXq=-I ± 0.04, asessentially time independent except near the end of the compared with the predicted equality of X and Xqrun when the helium level is near the rf SQUID. This We would like to acknowledge useful conversations

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VOLUME58, NUMBER 12 PHYSICAL REVIEW LETTERS 23 MARCH 1987

with Jeeva Anandan as well as the assistance of Karen 1987), p. 224.Springer in sample fabrication. This work was supported 5R. M. Brady, Ph.D. thesis, University of Cambridge, 1982by the U. S. Office of Naval Research with additional (unpublished).support, for facilities used in this work, from the Nation- 6j. Anandan, Class. Quantum Gray. I, L51 (1984).al Science Foundation under Grant No. DMR-86-40195, 7L. I. Schiff and M. V. Barnhill, Phys. Rev. 151, 1067and the U. S. Air Force Office of Scientific Research. (1966).8A. J. Dessler et al., Phys. Rev. 168, 737 (1968).

9W. M. Fairbank, Physica (Amsterdam) 109B& 1l0B, 1404(1982).

10A. K. Jain, J. E. Lukens, and J.-S. Tsai, Bull. Am. Phys.(a)present address. Central Research Laboratory, NEC Soc. 31, 495 (1986), gives preliminary results of these mea-

Corporation, Kawasaki, Kanagawa 213, Japan. surements.IR. V. Pound and S. A. Rebka, Phys. Rev. Lett. 4, 337 11J. Clarke, Phys. Rev. Lett. 21, 1566 (1968).

(1960). 12J.-S. Tsai, A. K. Jain, and J. E. Lukens, Phys. Rev. Lett.2R. F. C. Vessot and M. W. Levine, Gen. Relativ. Gravita- 51, 316 (1983).tion 10, 181 (1979), C. 0. Alley, in Quantum Optics, Experi- 13See, for example, K. K. Likharev, Dynamks of Josephsonmental Gravity and Measurement Theory, edited by P. Mey- Junctions and Circuits (Gordon and Breach, Nc%& York,stre and M. 0. Scully (Plenum, New York, 1983), p. 363. 1986), Chap. 10.

3j. Anandan, Phys. Lett. 105A, 280 (1984). 14The full general-relativistic treatment of fluxoid quantiza-4For a discussion of general relativistic effects in supercon- tion may be found in Ref. 4. The treatment of gravitation asductors and references to earlier work, see J. Anandan, in New an additive effect to the nonrelativistic equations, presentedTechniques and Ideas in Quantum Measurement Theory, edit- here for simplicity, is equivalent in the limit of weak uniform

ed by D. Greenberger (N.Y. Academy of Science, New York, fields applicable to this experiment.

1168

APPENDIX CVOLUME63, NUMBER 16 PHYSICAL REVIEW LETTERS 16 OCTOBER 1989

Thermal Activation in a Two-Dimensional Potential

S. Han, J. Lapointe, and J. E. LukensDepartment of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794

(Received 13 July 1989)

Measurements have been made of the transition rates between fluxoid states of a system with twoJosephson junctions having two macroscopic degrees of freedom. The data, taken over a temperaturerange of 0.85 to 4.41 K, are in excellent agreement with the predictions for thermally activated transi-tions. No evidence is seen for recently reported apparent rate suppression below that predicted forthermal activation.

PACS numbers: 74.50.+r, 05.40.+j, 85.25.Cp

Josephson junctions have proven to be excellent sys- with a damping determined both by the junctions' resis-tems in which to study the dynamics, both classical and tance and the environment. Here, Do is the flux quan-quantum, of a macroscopic coordinate in a metastable tum, Y-L/I is the ratio of the loop inductances, andpotential. In general, careful experiments have yielded _=2rLIc/4o, where I, is the sum of the junctions' criti-excellent agreement with theory for systems with one de- cal currents. (Po "=I2ir/Oo]. Figure 2 shows a plot ofgree of freedom, i.e., a single, small junction. Recent re- this potential for f6 -4.14 and y -19.5 (the case for thesults by Sharifi, Gavilano, and Van Harlingen' 2 (SGV) junctions measured here) with 0, -0.6500 and 0 xdc - 0.on thermal activation from the zero-voltage state of a dc For these parameters the potential consists of two localSQUID (a system with two degrees of freedom) have, minima connected by a saddle point. For the values of Phowever, shown marked disagreement with classical and y here, there are at most two local minima withtheory for two-dimensional (2D) potentials and have U(0,0dC) increasing roughly parabolically at large 0been interpreted as showing a dramatic suppression of and Od. For values of (xdC > 0 the barrier amplitude isthe thermal activation rate from that predicted. In this reduced, finally going to zero for Zxdc- 0.405o (forLetter we report the results of thermal activation mea-surements on a similar two-dimensional system of 0.25Josephson junctions (a combined rf-dc SQUID) in which (a)the potential can be independently determined to veryhigh accuracy.

The system studied here, as shown in Fig. 1, consists " o.ooof two Josephson junctions in parallel in a low induc-tance (1) superconducting loop (the dc SQUID) which isincorporated as part of a larger inductance (L) super-conducting loop (the rf SQUID). The macroscopic -0.25 , ..............dynamical variables which describe this system are the -0.5 0.0 0.5 1.0 1.5

fluxes 0 (which can be monitored) and 0d& linking, re-

spectively, the rf and de SQUID's. The fluxes applied tothese loops, 0., and O.xdc, can be independently con- (b)trolled. The dynamics of this system can be visualized as 2 300motion in a 2D potential, which for symmetric dcSQUID having equal junctions is given by

100

-pcos2,dCcoSI, 1) - 02-. ' . 0.5 1.0 1.5

FIG. 2. The 2D potential U(4,O, ) of our sample SQUIDO'x with symmetric dc SQUID having identical junctions. The

_ E m mutual inductance between two loops is taken to be 1/2. The'--x.o Ipotential for fl-4.14, y"19.5, Ox-0.65o, and 0, -0 show-

M[geto ing (a) equipotential contours at the above value of externalfluxes and (b) U(0,0). U continues to increase monotonically

(roughly parabolically) for greater values of 0 and 4c thanFIG. I. Schematic of SQUID sample and coupling coils, those shown.

1712 © 1989 The American Physical Society

VOLUME63, NUMBER 16 PHYSICAL REVIEW LETTERS 16OCTOBER 1989

Ox- (o) so that no metastable states exist. Also for 5.4 'xdc > 0 the path linking the wells through the saddle (a)

point is no longer a straight line. The details of this po-tential along with the techniques used for the measure- L 3.0ment of its parameters will be presented in a forthcom- _Jing extended paper.

The longitudinal (transverse) small oscillation fre-quencies parallel (perpendicular) to the path from the 1.o0._.well through the saddle point are wjw (wtw) in the well 0.58 0.62 'o.066 0.70and wl, (wa) at the saddle point. The predicted rate of l/T (K")

thermally activated escape from such a two-dimensional 150 40 Nwell is given in the classical limit by Ben-Jacob et al. 3 to .be10

I2D -- expt-AU(,pd,)/kyT, (2) 0o 4

2r V50 V(mV)

where AU is the energy difference between the metasta-ble potential well and the saddle point and fl - floal. I

f ")iwwtw/tats and a, is a damping-dependent suppres- 2 3 4 5sion of the rate which can be much less than one in both T (K)high- and low-damping limits, i.e., for o) much greater or FIG. 3. Thermal activation rates between fluxoid states formuch less than RC. R and C are, respectively, the the symmetric potential (0, - -L (Do) shown as the inset in (a).junction's resistance and capacitance. For moderate to (a) Rate vs I/T for fixed barrier height with best linear fitheavy damping a-(I+G24)-G/2, where G-11 (line). (b) Barrier height vs T for fixed rate (r-I s-9. ForRCwt . In the extreme underdamped limit, a, << I is also points indicated by squares, data have been corrected for tern-predicted since perature dependence of critical current. Best fit by Eq. (2) as-

suming damping due to R. (solid line), or damping proportion-2 Irwa I (AU 2 al to exp[-AU/kuT] from quasiparticles (dashed line). Theat . .wRCi T inset of (b) shows the dc SQUID I-V curve for T-1.5 K,

tw R lw - 4 1ac -O indicating low subgap leakage.

Recent results for single-current biased junctions provideevidence for this extreme underdamped limit with adamping ,lccexp(-AU/kRT), where AU is the super- For the special case where 0,, is a half-integer fluxconducting energy gap, as expected for quasiparticles. 4-6 quantum, the potential is symmetric about 0-0., soHowever, the extension of these results to multidimen- that AU(Oxd) is the same for each well [see inset, Fig.sional potentials is still open to question. 7 3(a)], varying from a maximum of 143 K for Oxdc-0 to

All measurements were made in highly shielded cryo- 0 for 4)xdc > 0.40500, where the potential has only onestats (either 4He or dilution refrigerator). All signals minimum. Figure 3(a) shows the measured rate atwere inductively coupled to the sample, which was en- which the sample hops between two equal energy fluxoidclosed in a NbTi can, through leads having low- states versus temperature with constant AU (fixed Oxdc).temperature low-pass filters. For low-temperature mea- If fil is temperature independent in this range, one ex-surements a resistive shunt, heat sunk to the dilution re- pects from Eq. (2) that a linear relationship exists be-frigerator mixing chamber, was placed across the mag- tween ln(r) and I/Tsincenetometer input to filter any high-frequency signals gen-erated by the SQUID magnetometer. The junctions ln(F)-AU/kT+In(f/2z). (3)used were nominal I x I pm 2 Nb/AI2O3/Nb tunnel junc- As Fig. 3 shows, the observed dependence is in facttions with very low subgap leakage. The I-V curve of an linear, giving best-fit values of AU-32.2 ± 0.4 K andunshunted (without L) SQUID coprocessed with the ln(fl/27) -23.9 ± 0.3. These agree well with the valuessample SQUID is shown ;n the inset of Fig. 3(b). These of AU-33.5 ± 3 K and ln(fl/2r) -24.4 ±0.2 calculateddata give a single-junction differential resistance of from measured sample parameters, as discussed below.R -320 fl above the gap and a critical current of Here a, -I1 has been used; taking the damping to beI, -5.8 pA. From the dc SQUID resonance of this l/R, would give a, - 0.9-an insignificant correction.unshunted SQUID with Oxdc > 0, a capacitance of The greatest uncertainty in the calculated value of AUC-46± 6 fF was determined in agreement with the comes from the knowledge of L, which has been calculat-specific capacitance obtained by Lichtenberger et al. for ed from the loop geometry, giving L-230 pH ± 10%.similar junctions.3 The inductance I of the dc SQUID Measurements on single-junction SQUID's9 ,1 havewas determined from the sample as discussed below, shown such calculations to be reliable for the geometry

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VOLUME63, NUMBER 16 PHYSICAL REVIEW LETTERS 16 OCTOBER 1989

used here. Both y and the difference in the junctions' 0.80 3.00

critical currents, 8- (12-I )/I , can be determined from - -.

the equilibrium dependence of 0) on D. measured when4) xdc is a half-integer flux quantum so that the systemhas no metastable states. These data give y-19.5 and 0 .8 < 5% justifying the treatment of the junction parame- 0.75 -4.0ters as equal. P is determined by measuring the value of b

D.,: at which the system escapes from its metastable state

for Oxdc 0. Since this escape will occur slightly beforethe barrier is reduced to zero, a small, model-dependent OX Occorrection to P based on Eq. (2) is required. This correc- 0.700 . .5. 100 . . 2.0.0O0 .5 .0 15 2.00tion, at most, increases the calculated AU above by 5% T (K)over that which would be obtained by assuming the es-cape occurred for AU-0. FIG. 4. Mean (0x) (circles) and width a (squares) of the

cae occrrd3(a) is only thermal activation probability distribution (see text) vs T atThe transition rate as shown in Fig. 3(a)&-0.1220o. The solid and dashed lines are, respectively,measurable over a narrow range of temperature for fixted the predicted mean and width from Eq. (2) using parametersAU. To measure the temperature dependence of this rate given in text. Inset. A typical distribution used to calculateover a broad range of temperature, AU was adjusted at (Ox) and a at one temperature.each temperature to give approximately the same rate,10 < r < 50. These data are plotted as AU' vs T in Fig.3(b). Here AU'-AU+kBTln[r/I s-1 is the measured AU* =2.5AU as needed by SGV to explain their data inbarner normalized for the small variations in n() the thermal activation regime. The energies associatedamong the points. Again 4D.," -2(o and AU().:dc) was with the symmetric potential over the temperature rangeadjusted by varying (Ndc and calculated from the param- of the measurements are summarized in Table I. Foreters as above. comparison, we note that both for these data and for

The theoretical curve for thermal activation has been those of SGV, hw - kBT, whereas hot >>kBT withplotted using Eq. (3) taking the damping to be 1/Rn 1/o, 8. Even for our lowest-temperature data therewhich gives at 0.9. The uncertainty in the indepen- are at least ten energy levels in the wells, satisfying thedently measured values of AU ± 10% implies an accept- validity criteria for the classical [Eq. (2)] and semiclassi-able fit to the data in Fig. 3(b) for any temperature- call" rate calculations. The expected crossover tempera-independent damping less than 0.05 0l - 1if the moderate ture T, for tunneling-dominated transitions is T., -0.35damping expression for a, is used. The dashed line K in the zero-damping limit. Thus quantum correction11

in Fig. 3(b) is calculated assuming the damping q to Eq. (2) should be negligible for our data. Naor,ecexp(-AU/kBT) as expected for quasiparticles and Tesche, and Ketchen 12 have also reported excellentusing the extreme underdamped expression for a,. This agreement wit,. thermal activation theory for their mea-is not consistent with our data, however, we emphasize surements for transitions for T> 1.7 K in a highlythat this result may imply only that extrinsic damping damped, current-biased dc SQUID with wo/wot I.(e.g., losses from the SQUID loop) dominates. A more direct comparison with the results of SGV is

We conclude that the data presented in Fig. 3 are en- provided by the data in Fig. 4. Here, for Oxdc-0.1 2200,tirely consistent with thermal activation [Eq. (2)] and, in 0., is increased at a constant rate reducing the barrierparticular, do not admit an effective barrier height height until the SQUID undergoes a change of fluxoid

state by escaping from a metastable well. Measurementsof this type on a single-junction SQUID have been de-

TABLE I. Characterisic. energies (in units of kelvin) of the scribed in detail elsewhere. 9 10 These measurements areb.mm~raeI doubk-e-.ll potential (4A - 1 4k,) hi~h describes repeated approximately 104 times at a fixed temperaturethe system for data in Fig. 3. At each temperature OAd, has giving a histogram of transition probability versus 0,been adjusted to give the values of AU shown. (see Fig. 4, inset). The first and second moments of this

T-0.85 K T4.41 K distribution, (<)) and a, respectively, are predicted tovary with temperature' 3 as shown by the solid andAU 18 92 dashed lines in Fig. 4 assuming a potential and transitionhoiW. 1.85 3.15 rate given by Eqs. (1) and (2) and using the sample pa-h Oh. 14.6 14.8 rameters above.hwb 1.36 2.61 In calculating (4D,) vs T the critical applied flux 0, ath u 14.4 14.2 which AU-0 is used as a temperature-independentAU/hoi,, 9.7 29AU/hwOh,(u) 1.2 -6 fitting parameter, 0.,- 0 .88200, which is consistent______________-1.2_-6_ with P measured for Oid -0. As can be seen, the agree-

1714

VOLUME63, NUMBER 16 PHYSICAL REVIEW LETTERS 16OCTOBER 1989

ment with thermal activation predictions is excellent. S. Chakravarty and S. Kivelson. This work was support-This contrasts with similar measurements by SGV in ed by the U. S. Office of Naval Rcsearch.which they found for their system that d(0x)fdT wasabout a factor of 3 less than predicted. The width a alsovaries with temperature as expected, although it is about5% greater than predicted. This is consistent with the IF. Sharifi, J. L. Gavilano, and D. J. Van Harlingen, Phys.excess width expected from the Nyquit flux noise on the Rev. Lett. 61, 742 (1988).sample due to the shunt resistor across the magnetome- 2F. Sharifi, J. L. Gavilano, and D. J. Van Harlingen, IEEEter. The temperature of this shunt varied from 50 to 180 Trans. Mag. 25, 1174 (1989).mK as the sample temperature was increased. 3E. Ben-Jacob, D. J. Bergman, Y. Imry, B. J. Matkowsky,

Measurements have been made of thermally activated and Z. Schuss, J. Appl. Phys. 54, 6533 (1983); C. D. Tesche, J.transitions between fluxoid states of a SQUID system Low Temp. Phys. 44, 119 (1981).having two macroscopic degrees of freedom, in which the 4P. Silvestrini, S. Pagano, Roberto Cristiano, 0. Liengme,

energy potentials could be accurately determined and and K. E. Gray, Phys. Rev. Lett. 60, 844 (1988).5P Silvestrini, 0. Liengme, and K. E. Gray, Phys. Rev. B 37,where existing classical and semielassical theory should 1525 (1988).be valid. Excellent agreement has been found with the 6J. R. Kirtley, C. D. Tesche, W. J. Gallagher, A. W. Klein-prediction of thermal activation theory. In particular, no sasser, R. L. Sandstrom, S. I. Raider, and M. P. A. Fisher,hint was seen of the very large apparent rate suppression Phys. Rev. Lett. 61, 2372 (1988).recently reported in a similar system by SGV. In view of 7R. Landauer (private communication); S. Kivelson (privatethe similarities of the potentials for these two measure- communication).ments (in particular, the ratios of the longitudinal and 8A. W. Lichtenberger, C. P. McClay, R. J. Mattauch, M. J.transverse level spacing to kBT), our results should serve Feldman, S. K. Pan, and A. R. Kerr, IEEE Trans. Mag. 25,to focus on the key features of the system used by SGV 1247 (1989).responsible for their anomalous results. I or example, 9D. B. Schwartz, B. Sen, C. N. Archie, and J. E. Lukens,our results indicate that such anomalies cannot be a gen- Phys. Rev. Lett. 55, 1547 (1985).erl fresults indifa to-imesionoalo ils a n in t e - ID B Schwartz, Ph.D. thesis, State University of New Yorkeral feature of two-dimensional potentials even in the at Stony Brook, 1986 (unpublished).limit where hot> k 8 T so that level quantization of the I H. Grabert, P. Olschowski, and Ulrich Weiss, Phys. Rev. Btransverse mode is important. 36, 1931 (1987).

We would like to acknowledge the contributions of C. 12M. Naor, C. D. Tesche, and M. B. Ketchen, Appl. Phys.N. Archie, Aloke Jain, and B. Sen in the initial stages of Lett. 41, 202 (1982).this work as well as numerous helpful conversations with 13j. Kurkijgrvi, Phys. Rev. B 6, 832 (1972).

1715

Final report on ONR contract #N00014-84-C-0261

List of Publications

"Length Dependent Properties of SNS Microbridges," J.E. Sauvageau, R.H. Ono, A.K.

Jain, K. Li, and J.E. Lukens, IEEE Trans. on Mag., MAG-21, 854 (1985).

"Suspended Metal Mask Techniques in Josephson Junction Fabrication," R.H. Ono, J.E.Sauvageau, A.K. Jain, D.B. Schwartz, K.T. Springer, and J.E. Lukens, J. Vac. Sci. Tech.B, 3(l), (1985).

"The Decay of Metastable States in Systems with Highly Damped Josephson Junctions,"D.B. Schwartz, B. Sen, C.N. Archie, A.K. Jain, and J.E. Lukens, Proceedings of theThird IC SQUID Conference, Berlin (June 1985).

"Quantitative Study of the Effect of the Environment on Macroscopic QuantumTunneling," D.B. Schwartz, B. Sen, C.N. Archie, and J.E. Lukens, Phys. Rev. Lett. 55,1547-1550 (1985).

"Phase-locking in Distributed Arrays of Josephson Oscillators," J.E. Sauvageau, A.K.Jain, and J.E. Lukens, IEEE Trans. on Mag., MAG-23, No. 2, 1048-1050 (1987).

"Variation of the Electrochemical Potential Difference in a Gravitational Field," A.K.Jain, J.E. Lukens, and J.-S. Tsai, IEEE Trans. on Mag., MAG-23, No. 2, 450-453 (1987).

"Test for Relativistic Gravitational Effects on Charged Particles," A.K. Jain, J.E.Lukens, and J.-S. Tsai, Phys. Rev. Lett. 58, 1165-1168 (1987).

"Submillimeter wave generation using Josephson junction arrays," K. Wan, A.K. Jain,and J.E. Lukens, Appl. Phys. Lett. 54, 1805-1807 (1989).

"Thermal Activation in a Two-Dimensional Potential," S. Han, J. Lapointe, and J.E.Lukens, Phys. Rev. Lett. 63, 1712-1715 (1989).

"Josephson Arrays as High Frequency Sources," Chapter 4 in Modern SuperconductingDevices, S. Ruggiero and D. Rudman, Eds., Academic Press, Boston, 1990.

"Observation of Final State Effects on Macroscopic Quantum Transitions," J. Lapointe,S. Han, J. Lukens, to be published in Proceedings of LT-19.

"Observation of Incoherent Relaxation by Tunneling in a Macroscopic Two-StateSystem," S. Han. J. Lapointe, and J.E. Lukens, submitted to Physical Review Letters.

Final report on ONR contrac.,#NOOO14-84-C-O261

list of students with ONR support receiving Ph.D.s

Peter Kopietz

Joseph Sauvageau