We analyze the influence of noise on magnetic properties of a superconducting loop which contains three

Josephson junctions. This circuit is a classical analog of a persistent current (flux) qubit. A loop supercurrent

induced by external magnetic field at the presence of thermal fluctuations is calculated. In order to

get connection with experiment we calculate the impedance of the low-frequency tank circuit which is inductively

coupled with a loop of interest. We compare obtained results with the results in quantum mode —

when the three junction loop exhibits quantum tunneling of the magnetic flux. We demonstrate that the

tank-loop impedance in the classical and quantum modes have different temperature dependence and can be

easily distinguished experimentally

Il’ichev, E.

Omelyanchouk, A.N.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Fizika Nizkikh Temperatur, 2008, v. 34, No. 6, p. 520–524Magnetic flux noise in the three Josephson junctionssuperconducting ringE. Il’ichev1 and A.N. Omelyanchouk21Institute of Photonic Technology, 9 Albert-Einstein Str., 07745, Jena, Germany2B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine47 Lenin Ave., Kharkov 61103, UkraineE-mail: omelyanchouk@ilt.kharkov.uaReceived January 11, 2008We analyze the influence of noise on magnetic properties of a superconducting loop which contains threeJosephson junctions. This circuit is a classical analog of a persistent current (flux) qubit. A loop super-current induced by external magnetic field at the presence of thermal fluctuations is calculated. In order toget connection with experiment we calculate the impedance of the low-frequency tank circuit which is in-ductively coupled with a loop of interest. We compare obtained results with the results in quantum mode —when the three junction loop exhibits quantum tunneling of the magnetic flux. We demonstrate that thetank-loop impedance in the classical and quantum modes have different temperature dependence and can beeasily distinguished experimentally.PACS: 85.25.Am Superconducting device characterization, design, and modeling;85.25.Hv Superconducting logic elements and memory devices; microelectronic circuits.Keywords: Josephson junction, SQUID, flux qubit, thermal fluctuations.1. IntroductionMagnetic flux quantization in superconductors isused, in particular, for realization of very sensitive mag-netometers. One of them is so-called radio-frequency (rf)SQUID [1]. The sensor of rf SQUIDs is a single junctioninterferometer — a Josephson junction which is incorpo-rated in a superconducting ring with a sufficiently smallinductance L. When an external flux e is applied to aninterferometer loop, the circulating supercurrent I is in-duced and a flux  i is admitted into the ring: i e IL  . (1)The phase difference across a Josephson junction equalsto a normalized magnetic flux in the interferometer loop:   2 20i n , (2)where 0 is the flux quantum, and n is an integer. Sincethe Josephson current is related to the phase difference:I I c sin, (3)where I c is the critical current Eq. (1) can be rewritten:    e sin (4)where e e 2 0 / is the normalized external flux andthe constant  2 0LI c / (5)is the normalized inductance of the interferometer.From Eq. (4) it is clearly seen that the magnetic pro-perties of a single junction interferometer are defined byparameter . If   1 the  e ( ) dependence is unique (seeFig. 1) and corresponding mode of SQUID operation isso-called nonhysteretic. If  	 1 the  e ( ) dependence ismultivalued (see Fig. 1) and corresponding mode ofSQUID operation is hysteretic.A rf SQUID basically consists of a sensor (usually asingle junction interferometer) inductively coupled to aradio-frequency-biased tank circuit. The flux  e appliedto the sensor changes the effective inductance (or/and theeffective resistance) of the tank-sensor arrangement.Thus, a flux change can be detected as changes in phase(or/and amplitude) of the voltage across the tank circuit.© E. Il’ichev and A.N. Omelyanchouk, 2008The classical mode for the single junction interferome-ter as well as for corresponding rf SQUID in the presenceof fluctuations have been investigated in detail theoreti-cally as well as experimentally [2–7]. On the other handthe quantum mode for this device is difficult to realize.Since an interferometer should be hysteretic, it requiresthe finite LI c product, and therefore, the finite couplingwith environment. In order to avoid this problem, a sub-stitution of the geometrical inductance to the Josephsonone has been proposed. Indeed, if the number of Jo-sephson junctions in the loop m 	 2 and for suitable junc-tions parameters, a double degenerated state exists at anygeometrical inductance L. One of the simplest realizationshere is a three-junction interferometer, which is calledalso a persistent current (or flux) qubit [8]. Such qubitwas fabricated by several teams and quantum regime wasconvincingly demonstrated.If external magnetic flux is    2 n (6)the hysteretic interferometer exhibits double-degeneratedenergy states, see Fig. 2. These states correspond to thedifferent directions of the interferometer current. If tem-perature is low enough and for suitable parameters ofJosephson junctions the magnetic flux can tunnel be-tween the two potential minima. Below we will call thesystems under consideration quantum if in their dynamicsthere is quantum tunneling. If, for some reasons, thequantum tunneling is suppressed, we will call these sys-tems classical ones.It is clear that the presence of the quantum tunnelingensures the absence of the hysteresis at the e ( ) depend-ence. On the other hand jumps between two energy min-ima can be originated by the fluctuations and the hyster-esis will be washed out. Therefore for both casesconsidered above the mode of the rf SQUID operationwill be nonhysteretic. It arises a natural question: by ana-lyzing a SQUID output signal is it possible to distinguishbetween quantum mode (interferometer with «quantumleak») and classical mode (interferometer in presence ofthe fluctuations)? We address this paper to that question.2. Classical mode of a flux qubit in the presenceof fluctuationsThe studied system presents the superconducting cir-cuit (ring) with three Josephson junctions, see Fig. 3. Weconsider the case of small self inductance of the ringL  0, therefore  e i . The phases across each junc-tion in the qubit loop  i satisfy to   1 2 3   e .In the frame of RSJ model for Josephson junctions thecurrent through each junction is:ICeddt eRddtI I t ii iii ci i i     2 21 2 322   sin ( ), , ,(7)Magnetic flux noise in the three Josephson junctions superconducting ringFizika Nizkikh Temperatur, 2008, v. 34, No. 6 5210 >1<1eFig. 1.  ( )e for rf SQUID in nonhysteretic and hystereticmodes.a–6 –4 –2 0 2 4 6–404–1.5 –1.0 –0.5 0 0.5 1.0 1.5–0.7–0.6–0.5–0.4qU(,=0)bquantumthermoactivated21Fig. 2. Contour plot of the potential U ( , )  (13),   0.8 (a).Bistable potential profile along line 1–2 in Fig. 2,a at  e ,  0.8 (b).1 23JJ JeFig. 3. The scheme of persistent current (flux) qubit.where  is the Plank constant, e is the electron charge, C iand Ri are the junctions’ capacitances and resistances, re-spectively. We restrict ourself to a practical case, whentwo junctions in the loop are identical: C C C1 2  ,I I Ic c c1 2  , R R R1 2  and the third junction hasslightly smaller critical current (with the same criticalcurrent density) I Ic c3  , 0 5 1.   and thereforeC C3  , R R3  /. The presence of the «white noise»is given by the fluctuation currents I ti ( ) with correlator  	     I t I t kT R t ti i i( ) ( ) / ( )2 a n d m e a n v a l u e 	 I ti ( ) 0.In dimensionless units:R ceRI 2 / , R t  (8)and for negligible capacitance, Eq. (7) can be rewritten:I I d dc/ / sin, , ,      1 2 1 2 1 2        d d3 3 3/ sin ( ) . (9)The correlators of  i are:  	         i i iD( ) ( ) ( )2 , (10)where D kT EJ / , E I eJ c  / 2 .By introducing the phases  and 2 1 2    , 2 1 2    ,and taking into account that   3 2 e , Eq. (7) can bepresented in the form:d d     / cos sin / ( )   1 2 2 1 ,( ) / sin cos sin ( )1 2 2            d d e     3 2 11 2( ) / ( ) .These equations can be reduced to:ddU    ( ) , (11)ddU    11 2( )( ) . (12)whereU e( , ) cos cos cos ( )         122 (13)is the effective potential and the random forces are:  	         ( ) ( ) ( )D ,  	       ( ) ( )( )( )31 2 2D .The Langevin equations (11), (12) describe the ran-dom motion of the «particle» with coordinates (, ) in theperiodic potential (13), which is a set of bistable cells(eight-shaped contours in Fig. 2,a). We have numericallyintegrated these stochastic equations by Ito’s method (seee.g. [9]) for different values of parameter  and thestrength of the fluctuations D. The typical traces of  ( )and  ( ) are shown in Fig. 4. They correspond to randommotion in bistable potential, Fig. 2. The arrows indicatethe switching from one unit cell in Fig. 2,a to another.With knowledge of  ( ) and  ( ) the average circulatingcurrent in the ring is obtained as:I Ie c( ) ( )    		sin   . (14)The averaging  		 includes for each value offlux e the average over time of traces (   ( ), ( )) and theaverage over set of 50 traces.The calculated in such way the current-flux curves fordifferent values of D and different values of parameter are presented in Fig. 5 and Fig. 6.From Eqs. (11)–(13) the Fokker-Planck equation fordistribution function P( , )  can be reconstructed:522 Fizika Nizkikh Temperatur, 2008, v. 34, No. 6E. Il’ichev and A.N. Omelyanchouk2000 4000 6000 8000t2000 4000 6000 8000t0 10000–20240 10000–4–202Fig. 4. The random motion of phases  and  in bistable poten-tial (Fig. 2,b) for  e  ,   08. and D  01. .0 0.1 0.2 0.3 0.4 0.50.10.20.30.40.50.60.7I/I c123e/2Fig. 5. The dependences I e( ) for   08. and D  05. (1),D  025. (2), D  01. (3).P UPDPUP    211 2221231 2 222DP( ) . (15)The Fokker-Planck equation (15) admits the stationarypotential solution (see [9]) in the special case  1, i.e.when all three junctions are identical. For 1the analyt-ical solution reads:PDUNexp ( , )2 , (16)N d dDU    exp ( , )2. (17)Since the potential U is 2 periodical function of vari-ables  and  the average current in the ring is:IId dDUdc 020202022     sin exp( ) ( , )   dDUexp2( , ). (18)In Fig. 7 we compare the numerical results (circles)and the I e( ) obtained from the analitical formula (18)(solid line) for the case 1and D  0 2. . This comparisonwas used as an additional calibration of our numericalprocedure, which is working at arbitrary values of .3. The probing of qubit’s state in classical and quan-tum modesAs we wrote in introduction we probe a qubit withmaking use of a tank circuit by the impedance measure-ment technique [10]. It was convincingly shown [10,11]that the observable phase difference  ( )e between tankcurrent I rf and tank voltage Vrf both in classical andquantum modes reads:tan ( ) eedId , (19)where  is const, which characterizes the inductive cou-pling of the qubit with the tank circuit. By using the re-sults of Section 2 we calculated output signal  ( )e forclassical noise affected mode. For different levels ofnoise D (all of those correspond to nonhysteretic regime)the dependencies  ( )e are shown in Fig. 8.In the quantum noise free mode (see Appendix A) thenonhysteretic behavior is achieved by the tunneling be-tween two wells. The phase shift  ( )e in this case is de-scribed by Eqs. (19), (A.1). It is presented in Fig. 8 for thesame as in the classical case value of the qubit-tank cou-pling constant  and experimentally realized qubit’s pa-rameters I p0 200 GHz,   1.5 GHz.Comparing classical and quantum modes (Figs. 8,9)we have found that in quantum mode the dip on the  ( )edependence remains constant at wide temperature rangekT  . This reflects the fact that tunneling splitting does not depend on the temperature. The depth of the dipis changed with temperature — excitations to the upperlevel depress the value of the average qubit’s current. Forclassical mode the situation is rather different. First of allfor reasonable set of qubit parameters it is impossible toget such narrow and profound dip similar to obtained inMagnetic flux noise in the three Josephson junctions superconducting ringFizika Nizkikh Temperatur, 2008, v. 34, No. 6 5230 0.1 0.2 0.3 0.4 0.50.10.20.30.40.50.60.70.80.91.0132I/I ce/2Fig. 6. The dependences I e( ) for D  01. and   05. (1),   08.(2),  1(3).0 0.1 0.2 0.3 0.4 0.50.5I/I ce/2Fig. 7. The comparision of numerical (circles) and analitical(solid line) calculations. D  02. ,  1.quantum mode. Moreover, the temperature dependence of ( )e dip demonstrates that its width strongly depends onT . Therefore by analyzing the temperature dependence of ( )e one can easily distinguish between quantum andclassical modes.In conclusion we analyzed the temperature depend-ence of imaginary part of impedance for three-junctionloop-tank circuit arrangement in quantum and classicalmodes. We argued that impedance for these modes havequite different temperature dependencies and, therefore,can be easily distinguished experimentally.Appendix AQuantum mode of a flux qubitSince the tunnel splitting in flux qubit is much smallerthan the difference between upper energy levels, qubitsare effectively two-level quantum systems. In the twolevel approximation a flux qubit can be described by thepseudo-spin HamiltonianH t x z( )    ! , (A.1)where  x ,  z are the Pauli matrices;  is the tunnelingamplitude. The qubit bias is given by !  I fp e0 , whereI p is the magnitude of the qubit persistent current andf e e  / /0 1 2. The stationary energy levels can beeasily found from the Hamiltonian (A.1):E" " ( )! ! 2 2 , (A.2)and the average value of the qubit current at temperatureT is:IIkTp( )!!!!2 22 2tanh . (A.3)Note, that the dependence (A.2) is valid within the nar-row interval ofe near e  0 2/ , where the potentialU(13) is bistable.1. A. Barone and G. Paterno, Physics and Applications of theJosephson Effect, Wiley, New York (1982).2. V.A. Khlus and I.O. Kulik, Sov. Tech. Phys. Lett. 20, 283(1975).3. L.D. Jackel, R.A. Buhrman, and W.W. Webb, Phys. Rev.B10, 2782 (1974).4. E. Il’ichev, V. Zakosarenko, V. Schultze, H.-G. Meyer,H.E. Hoenig, V.N. Glyantsev, and A. Golubov, Appl. Phys.Lett. 72, 731 (1998).5. R. de Bruyn Ouboter and A.N. Omelyanchouk, PhysicaB216, 37 (1995).6. E. Il’ichev, V. Zakosarenko, R.P.J. IJsselsteijn, and V. Schult-ze, JLTP 106, 503 (1997).7. E. Il’ichev, V. Zakosarenko, V. Schultze, H.E. Hoenig,H.-G. Meyer, K.O. Subke, H. Burkhard, and M. Schilling,Appl. Phys. Lett. 76, 100 (2000).8. J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van derWal, and S. Lloyd, Science 285, 1036 (1999).9. C.W. Gardiner, Handbook of Stochastic Methods, 2nd ed.,Shpringer, Berlin (1990).10. E. Il’ichev, A.Yu. Smirnov, M. Grajcar, A. Izmalkov, D. Born,N. Oukhanski, Th. Wagner, H.-G. Meyer, W. Krech, andA. Zagoskin, Fiz. Nizk. Temp. 30, 823 (2004) [Low Temp.Phys. 30, 620 (2004)].11. S.N. Shevchenko, Eur. Phys. J. B61, 187 (2008).524 Fizika Nizkikh Temperatur, 2008, v. 34, No. 6E. Il’ichev and A.N. Omelyanchouk0.4 0.44 0.48 0.52 0.56 0.6–0.05–0.04–0.03–0.02–0.0100.01123e/2Fig. 8. The phase shift  ( )e in classical mode.   08. andD  03. (1), D  02. (2), D  01. (3).0.4 0.5 0.6–0.3–0.2–0.1012e/2Fig. 9. The phase shift  ( )e in quantum mode. T / .  01 (1),T /  1(2).

Magnetic flux noise in the three Josephson junctions superconducting ring

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Magnetic flux noise in the three Josephson junctions superconducting ring

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)