PHYSICAL REVIEW B VOLUME 49, NUMBER 6 1 FEBRUARY 1994-II Macroscopic quantum tunneling in a dc SQUID: Instanton splitting C. Morais Smith Theoretische Physik, Eidgenossische Technische Hochschule —Honggerberg, CH-8093 Zu'rich, Switzerland and Universidade Estadual Paulista Julio de Mesquita, UNESP, 17100 Bauru, Brazil B. Ivlev Theoretische Physik, Eidgenossische Technische Hochschule —Honggerberg, CH-8093 Zu'rich, Switzerland and L. D. Landau Institute for Theoretical Physics, 11?9940Moscow, Russia G. Blatter Theoretische Physik, Eidgenossische Technische Hochschule —Honggerberg, CH-8093 Zu'rich, Switzerland (Received 24 May 1993; revised manuscript received 22 October 1993) The theory of macroscopic quantum tunneling is applied to a current-biased dc SQUID which consti- tutes a system of two interacting quantum degrees of freedom coupled to the environment. The decay probability is obtained in the exponential approximation for the overdamped case. Close to the critical driving force of the system, the decay of the metastable state is determined by a unique instanton solu- tion describing the symmetric decay of the phases in each of the two Josephson juctions. Upon reducing the external driving force a new regime is reached where the instanton splits. The doubling of the decay channels reduces the decreasing of the decay rate in the quantum regime. A current-temperature phase diagram is constructed based on the Landau theory of phase transitions. Depending on the external pa- rameters the system develops either a first- or a second-order transition to the split-instanton regime. I. INTRODUCTION During the last decade the subject of macroscopic quantum phenomena has attracted a great deal of in- terest. Macroscopic systems are inherently coupled to their environment which makes it often necessary to take dissipation into account. ' Some scientifically interest- ing as well as technologically important examples exhibit- ing this kind of phenomena are superconducting quan- tum interference devices (SQUID) and current-biased Josephson junctions (JJ). In the former, the macroscopic variable is the flux trapped in the ring, while in a JJ, the phase of the junction is the variable which can be subject to quantum effects. In both examples, the decay proba- bility is determined in the exponential approximation by the action evaluated for the (classical) extremal trajectory in an imaginary time formulation (instanton). In this work we are interested in the quantum dynam- ics of a system which is equivalent to an interacting two- particle problem, i.e., the decay of a metastable state in a dc SQUID. The device contains two JJ which makes it a particularly interesting model system as it goes beyond a single degree of freedom. Under well-defined conditions, it develops an instability where the action becomes ex- tremal along two close-lying trajectories. Such a splitting of the instanton makes the two phases decay asymmetri- cally and strongly influences the value of the decay rate. A deeper understanding of the SQUID behavior is of great technological importance since these devices are used, for instance, as magnetic-flux detectors, allowing the construction of very sensitive magnetometers. The macroscopic quantum tunneling (MQT) behavior of a dc SQUID with a very small ring inductance or with currents flowing in the two junctions nearly in phase has been considered before by Chen. ' In both cases, only the one-instanton solution arises. Ivlev and Ovchinnikov" have calculated the instanton splitting in a dc SQUID for the underdamped case. Here, we concentrate on the effect of dissipation and study the decay process in the overdamped limit. The standard method to treat the effect of damping on MQT is the theory introduced by Caldeira and Leggett (CL).' In their formalism the dissipation is treated by coupling the particle to a thermal bosonic bath. The CL theory assumes that each degree of freedom of the envi- ronment is only slightly perturbed by the coupling with the particle such that the bath can be represented by an ensemble of harmonic oscillators. Notice that this weak- coupling hypothesis does not exclude the possibility of strong dissipation, since the environment is composed of an infinite number of degrees of freedom. In the CL model, the tunneling rate for a dissipative macroscopic quantum variable is computed at zero temperature. The formalism has been generalized to finite temperatures by Larkin and Ovchinnikov in the limit of strong dissipa- tion, and by Grabert, Weiss, and Hanggi ' in the general case. For a recent review see Refs. 12 and 13. In the present work we consider the decay mode of a metastable state in a dc SQUID consisting of two identi- cal JJ arranged symmetrically in a superconducting loop and connected to the outside world by two leads (Fig. 1). The device is current biased near to criticality, i.e., the current I;, i = 1,2, in each branch is chosen to be slightly smaller than the critical current I, . The critical current in a JJ is nothing but the maximum current which can be transported by Cooper pairs. When the current exceeds 0163-1829/94/49(6)/4033(10)/$06. 00 49 4033 1994 The American Physical Society 4034 C. MORAIS SMITH, B. IVLEV, AND G. BLATTER FIG. 1. dc SQUID consisting of two identical Josephson junctions arranged symmetrically in a superconducting loop. The phase differences y& and y2 in each JJ are the macroscopic coordinates decaying out of a metastable state. In order to ob- serve quantum effects, the system is current biased near to criti- cality, i.e., the currents I& and I2 in each branch are slightly smaller than the critical current I, . The effective shape of the potential is modi6ed by changing the total current I=I, +I2. transition depends on the bias current and on the temper- ature. For small values of the external current and close to the crossover temperature To to the classical regime, the system develops a first-order phase transition, while for temperatures close to zero and higher bias current, the transition is continuous. The outline of this paper is as follows: In Sec. II we derive and discuss the action describing our system. In Sec. III we calculate the saddle points with the corre- sponding instantons. The decay rate in the exponential approximation is computed in Sec. IV and the phase dia- gram for the four different current-temperature regimes (classical single saddle, classical double saddle, quantum single instanton, and quantum split instanton) is dis- cussed in detail. The conclusions are presented in Sec. V. II. EUCLIDEAN ACTION The probability for the decay of a metastable state at finite temperature T is related to the imaginary part of the free energy F 2r= —ImF . this maximum value, a finite voltage appears across the junction, indicating that the additional current has to be carried by quasiparticles. The system under consideration develops a very in- teresting behavior: upon changing external parameters such as the bias current I=I&+I2 or the temperature T, several physically different situations can be realized. At high temperatures and currents, the system is in the clas- sical regime and decays through a unique saddle by thermal activation. Upon reducing the current a new phase arises at I=I„,where the saddle splits into two lower ones and the decay probability of the thermal pro- cess increases. Instead, if the current is kept constant and the temperature is reduced, the system enters the quantum regime and different regions can be reached: For very high currents, the coupling between the two JJ is strong and both phases decay symmetrically (single- instanton region). For lower currents, the coupling be- comes weaker and the decay ceases to be symmetric. In this case, two close-lying decay channels become avail- able, i.e., the instanton splits into two. The bifurcation of the trajectory reduces the action and consequently, the decay rate determined by the split-instanton solution is larger than the rate obtained for the single instanton in the same range of temperature and current. The phenomenon of instanton splitting occurs at currents larger than I„,i.e., in the region where the classical re- gime is still governed by a single saddle. For smaller currents I &I„ the quantum regime naturally involves two instanton trajectories. Identifying the action with a thermodynamic potentia1, we can reformulate the prob- lem in terms of the Landau theory of phase transitions. On this basis, the transition between the one-instanton and the two-instanton phases can be either continuous (a second-order phase transition), or it can be discontinuous (a first-order phase transition). The order of the phase I =fe (2) where Sz is the Euclidean (imaginary time) action evalu- ated along the instanton trajectory and f is a prefactor, which is obtained by considering fiuctuations around this trajectory. The instanton path is a saddle point of the ac- tion, i.e., a path which extremizes the action. The Eu- clidean action of a system consisting of two identical cou- pled Josephson junctions in the presence of a bias current 1s S@=S.+S, +Sd, where 2 C'o c 2m 2 2 '2 ~%a+ a1 ar +EJ[( I cosy, )+(—1 —cosy2)] I (V i+@2)2I, (3a) is the action of two independent JJ. Here and in the fol- lowing we have set the Boltzmann constant equal to uni- ty, k~ = I. The first term is the kinetic energy, where the capacitance C of the junction plays the role of the "mass" in the system; No=he/2e is the Aux quantum; tp& and tp2 are the phase differences across the two junctions which are periodic functions of the imaginary time ~ with a period 4!T. The second term is the potential energy of the junctions, FJ=(4c/2')I„where I, is the critical current of an individual junction. The last term In the semiclassical approximation, it can be shown that the imaginary part of F is dominated by the classical tra- jectory in the inverted potential (bounce or instanton) and the above expression takes the form 49 MACROSCOPIC QUANTUM TUNNELING IN A dc SQUID: . . . 4035 represents the driving force in the system, which can be externally controlled via a change of the bias current I. The interaction between the two JJ is given by 2 A/T E S;=J dr (q', —q'z) (3b) o where L is the total inductance of the loop. The expres- sion (3b) corresponds to the classical energy of a loop with total inductance L and enclosing a flux 4,E=4 /2L. The value of 4 is determined from the condition that the macroscopic phase of the wave func- tion must be single valued around the loop. In a dc SQUID, the gauge-invariant phase differences y, and y2 across the junctions determine the flux 4=(4o/2n)(y& —tpz). The present system is intermedi- I ate between a zero- and a one-dimensional problem as realized in a rf SQUID (d =0) (Refs. 1,2) or a tunneling vortex line (d =1) (Refs. 14,15}, for example. For the vortex problem, the macroscopic variable is given by the displacement field u (x,r), where x denotes the coordi- nate along the string. The interaction energy Ez(qr, qr2—) /2LI, in the dc SQUID then maps to the elastic energy fdx e&(B„u) /2 in the vortex problem with c. & denoting the line tension of the string. The last term in the action, Sd, is the dissipative contri- bution due to the coupling to the environment. Here we consider the resistively shunted junction (RSJ} model such that we can use the Caldeira-Leggett model in order to describe the coupling between the phases y;, i =1,2, of the JJ and the heat bath; (3c) with g denoting the phenomenological friction coeacient. The action (3) describes a situation with a vanishing external magnetic fiux threading the loop of the SQUID. If the external flux is nonzero but an integral number of flux quanta, the action can still be written in the form (3). For arbitrary external fiux, the SQUID acquires an effective asymmetry. However, our results are general and the splitting of the instanton occurs in any case. The oscillatory cosine term and the driving term pro- portional to I/I, [both in (3a)] combined with the in- teraction potential in (3b) give rise to the tilted wash- board potential, which is shown in Fig. 2(a). The minima are arranged along the diagonal y&=y2 and an illustra- tion is given in Fig. 2(b), where the equipotential lines are plotted. Experimentally, macroscopic quantum tunnel- ing is studied at currents close to criticality, i.e., 2I, —I ((I„when the effective potential barrier becomes small. In this limit, the potential energy can be approxi- mated by a cubic form. Introducing the dimensionless variables, f i+0'2 ~ 2 0'& 0'2+—+—and q= 3E 8 3 3E the Euclidean action can be rewritten in the form (neglecting a constant term), 2 2 SE =g dz — +— + V(p, q) G Bp G Bq 2 Bz 2 Bz 0 T Bp Bp . z zl dz& ln sin 7T p Z 7T Z ] 2 T Bq ~ Bq dz& ln sin &To az — az Z Z) where the effective potential is V(p, q)= p — —,'p +q (1+a)——,'pq (4) FIG. 2. (a) Tilted washboard potential acting on the phase differences y& and y& in the presence of a bias current I. The shape of the potential results from a periodic cosine term (po- tential energy of the junctions) combined with the bias driving term [ ~ I(qr, +yz) ] and with the interaction term [ ~ (y, yz)'/L]. (b) Eq—uipotential lines plotted for the tilted washboard potential given in (a). The minima are located along the diagonal pl=f2. Depending on the external parameter I, the phases y& and y2 decay from the metastable state following the diagonal (single-instanton phase) or following two close- lying trajectories with q&0 (split-instanton phase). MORAIS SMITH, B. IVLEVV, AND G. BLATTER Here we haveave introduced =8n CEJ T2/cI262 the dimensionle por ant driving pparameter, J/2~g, as well he as the with E=+(2I I—I )—/I, and P= LI, . 0 The parameter g is suis supposed to be 1 h obl em in the semicl o al- semic assical approxi- pA ~ 1 ~ 1 ~ ~ ~ I lg pl ~ I II 1' ~ ~ ~g, i ~ ( I ~ I ~ ~ ~ ~I, mass mation, g/A)&1 a 49 h or er that quantum t note that c. is a um tunnelin can im- sid erin b ~ g g ias currents I e er, since we are h ar t at the imit o ef. 10) is di o small inducta i erent from th in this work e limit of sm t ay in theex rp ins a factor c and h and c pla di an ence, the fact the 1 sua ized b i e latterr can be e imit of small in qs. (3a) and (3 ); in phases, while h E) th term lidh pi s, in cont st to e. ana yze more carefulleu yt eeffecti e values of F e poten- t. For o. (1 e avior for di lute minim small currentsF n s, thereisan b 1 siderations expect that f a ~ rom these stati t'nt a, a unique ccur. It turns out that s ins s ould p 0 ~here th d a. e exact sha e e ynamics become pe o the critical 1 in ec. III. ' ' a me a, (T) will be ob- III. SADD LK POINTS AND INSTANTTON SPLITTING I neext step, we look fo 'g = pp site limit wh ++1o 1 th f e inetic terms E g ct pattve ones (overdam edn e dissi a at t ey are m tim g p 1 imes, ' ns o motion foror imaginar y FIG. 3. Detailed view of of th P ff t'1 d' ss variables esse in term oosin th mass) and r «I th e potential can to its critical va e potential is controlled bP o s minimum t (, ) gp ,q, ), a maximumat p&, so a& the o »q2), d two s ddlt, ) at {p&, ew saddle point. merge into 2, which becom, w ic becomes 2 +—p —p+ —q'= ' 2 dz, cot 0 Z Z 1 —2(1+a)q+ 3pq = dz, cot 0 7T Z$ Z Z 1 (5b) Equation (5b) can be tr q (z) =qo(z) =0. trivially satisfied h case, the present r 1 J h e o reedom ( son junction i. p interacts with at ai a thermal bath at a 49 MACROSCOPIC QUANTUM TUNNELING IN A dc SQUID: 4037 finite temperature T. The solution of the remaining equa- tion (instanton) has been obtained by Larkin and Ovchin- nikov. It is given by 4 T 1 po(z) =— (6) 1 —cosz +1 —( T/To ) where [po(z), qo(z}]. On the other hand, for a &a„ the opera- tor Q also acquires a negative eigenvalue, 2(a —a, ) &0, and a "dangerous" mode is developed. As a conse- quence, the single instanton present for a) a, will split into two instantons at a & a, . In terms of the operators (10), the equations of motion (5a) and (5b) can be written in the equivalent form, REJ Tc = E(I) 2778 pp —3 2+ 3@2 Qq=3pq (1 la} (1 lb) is the crossover temperature separating the classical from the quantum regime. Inserting po, qo back into (4) we obtain the action S0=4nv]e (I) 1 ——1 T Q Next, we investigate the possibility that the dynamical equations (Sa) and (Sb) may develop a second solution which further reduces the action (4). In particular, we look for a solution p, q which bifurcates away from the solution pQ, qQ at some critical value of the dynamical pa- rameter a. Defining p (z) =po(z)+p(z) and q(z)=qo(z)+q(z), we can rewrite the action in the com- pact form, S=Sc+—f dz[qQq+pPp] Qq=o. (12) Solving Eq. (12) we thus immediately obtain the critical parameter a, where the spectrum of Q goes to zero, Q(a, }q,=0. Here q, denotes the eigenfunction of Q be- longing to the lowest eigenvalue 2(a —a, }, i.e., Qq, =2(a —a, )q, . Expanding q(z) in a Fourier series, q(z) = g C„e'"', (13) and remembering that' Our task now is to determine the corrections p and q for a & a, . To lowest order, we can set p =0 and q is the solution of the (linear) eigenvalue problem, Gz p 6fz pq ('9) cot Z Z 1 oo =2 g sin[m (z —z, }], 2 (14) where we have introduced the two operators we obtain the following equation for the Fourier coefficients C„, 5SP = =2—3po(z) 5P Pp~ vp TQ 00 ~n~C„+(I+a) C„=2 g C e (15) + f dzi cot 7T TQ IT a Bz i (10a) where TQ+ T b—=—ln and 5S Q = =P+2a . 5q pp, pp (lob) Making use of the instanton technique developed by Callan and Coleman, ' we know that the operator P does not only have a zero eigenvalue (translation mode), but also a negative one (unstable mode) which is responsible for the imaginary part in the energy and hence, for the finite decay rate of the metastable state. From Eq. (1) we immediately see that the value of the parameter a plays a critical role in this problem. For large a, the spectrum of Q is positive definite and all the q modes are stable. Upon decreasing a (e.g. , via reduction of I) the lowest eigenval- ue of Q can become negative and an unstable q mode is developed. The value of a where this instability occurs for the first time is defined as the critical value a, . Hence, adding 2a, to the operator P shifts the negative eigenvalue to zero. For a) a„only P has a negative ei- genvalue and the action is extremal for the trajectory Using the ansatz C„=Be "~" ( A + /n[) (17) are fulfilled. Since Eq. (15}is linear in C„, the overall am- plitude B cannot be obtained at this stage, but has to be determined by taking higher-order terms in Eq. (11) into account. Equation (18) defines the critical parameter a, (T) below which the lowest eigenvalue of Q becomes negative, a, =—+Q —, ' —(T/To) 1 (19) Note that with the above calculation we have simul- taneously determined the eigenfunction q & (z) correspond- (A, B constants) it is straightforward to verify that we ob- tain a solution if the conditions aTQ and —a +a+I- T 4038 C. MORAIS SMITH, B. IVLEV, AND G. BLATTER 49 ing to the negative eigenvalue 2(a —a, ) of g in the region a &a„as well as the lowest order approximation q' '(z) to the split-instanton solution q(z)%0 appearing below a„ q(z)=q' '(z)=Bq, (z) tion (4) results in a lower action than that obtained with qo(z) =0. Using the Landau theory of phase transitions, ' we can treat the action as the thermodynamic potential of the system and we can determine the temperature-current phase diagram in terms of the dimensionless variables, a, Tob/n— / ' + ( ( inz Tn = —oo (20) e» (2I, I )/3—8—: T and i, =— = = . (21) AEJ 4 4I, The condition (18) relates the current I with the tem- perature T and as such determines the critical tempera- ture T, (a} below which the two instantons (po, +Bq, ) occur. As in the classical regime, we can verify that this bifurcation appears in the region a &a„ i.e., in the low- current regime. In this region, the symmetric solution [qo(z) =0] is still valid. However, it can be easily verified that the substitution of the solutions +Bq, (z) into the ac- l.O ep "s "cs The phase diagram comprises four phases (see Fig. 4). Let us consider initially the large currents region, c &1. At high temperatures, 8&t9o, and very large currents (very small i.) the system behaves classically and the decay process occurs through a single saddle. Crossing the boundary Oo=v'i (remember that To=heEz/2m') we enter the quantum regime which is characterized by the single-instanton solution. Decreasing the current J for a fixed temperature T, or decreasing the temperature for a fixed current, we finally cross the critical line H, =i.+&i,—1 [see (18)] and reach the split-instanton phase. Instead, if the system is kept in the high- temperature regime and the current is reduced, a new classical phase arises at i=1 where the saddle splits into two (see also Fig. 3). Upon decreasing the temperature for a fixed value of current in the region c & 1, the system enters the quantum regime characterized by the double- instanton solution. The crossover line 6o separating the classical two-saddle regime from the quantum two- instanton one, can be determined from the curvature of the potential at the saddle: from the equation of motion written in terms of Matsubara frequencies co„=2m.nT/A we obtain 0.38 l.0 l.6 2~Io 8 V(p', q') c&1, gp'2 FIG. 4. The temperature-current phase diagram in terms of the dimensionless variables 8=(qmP) T/AEJ and i =P (2 I /I, )/4 det—ermining temperature T and driving current I, respectively. At large currents (c&1) the classical single-saddle regime at high temperatures is separated by the crossover line 80(c)=&c from the quantum single-instanton re- gime. This behavior is the same as for a system consisting of a single degree of freedom. For the present system consisting of two interacting degrees of freedom, several new critical lines arise in the temperature-current phase diagram. At high tem- peratures, the classical single-saddle regime is separated by the crossover line c=1 from the classical double-saddle regime; in the low-current region (c) 1) the double-saddle classical regime is separated from the two-instanton quantum one by the line Oo(c) ( 1+&4c—3)/2; and in the large-current region (c & 1 ) a new critical line 8,(c)=(c+&c—1)' appears in the quantum regime, separating the single-instanton phase from the split- instanton one. At low temperatures, the transition through the critical line 0, (c) is smooth, analogous to a second-order phase transition. At temperatures near to the crossover temperature Oo(c), the splitting of the instanton is abrupt, as in a first-order phase transition. 0 is the tricritical point where the second- and first-order phase transitions merge. The decay rates along the lines c; =const (i =1, 2, 3) and 0, =const (j =1, 2) are illus- trated in Figs. 5 and 6, respectively. where (i3 V/i}p' ) ~, is the second derivative of the poten- tial along the steepest descent of the saddle (p' and q' are the two orthogonal main directions at the saddle). The crossover temperature Ho(i, ) is then given by Ho(i, ) = —,'(1+v 4i, —3) . (22) IV. TUNNELING RATES AND PHASE DIAGRAM In Sec. III we have calculated the correction q' '(z} to qo(z) =0, corresponding to the trajectory minimizing the action in the region a (a, . Substituting Eq. (20) into Eq. (9), remembering that in the present approximation P =0, and making use of Qq, =2(a —a, )q„we obtain the quad- ratic form: Therefore, the point 8(i, =l)=1 is a crossing point be- tween single and/or double and classical and/or quantum regimes. Although the two-instanton phase appears also for c, & 1, we are interested in the splitting of the instan- ton, which occurs only in the region ~ & 1 ~ In the follow- ing we will restrict our analysis to this region. The quan- tum tunneling rate for the two cases a & a, and o,' & a, will be discussed in Sec. IV. 49 MACROSCOPIC QUANTUM TUNNELING IN A dc SQUID: 4039 S=So+g(a —a, )c, , where c&=B f dzq&(z), (23) In a final step we have to determine the amplitude B. From (25a) we infer that p(z) =B R(z), where R (z) is a function obeying the following inhomogeneous integral equation: S=So+g(a —a, )cf +g5c &, (24) where 5 is a coefficient to be determined by considering higher-order corrections in the expansion of the action. In this case, the two eigenvalue problems for P and Q be- come coupled and we have to solve the inhomogeneous equations, and So is given by (8). At this point, we have been able to solve the problem in the two regions a & a, and a )e, . However, one has to remember that the constant B in Eq. (20) is still un- known. Moreover, in the critical regime the eigenvalue 2(a —a, ) of the dangerous mode is small and it is no longer sufficient to consider only quadratic deviations from the classical trajectory. Both problems are solved by taking higher-order corrections into account, which is done in the following. From the Landau theory of phase transitions, with c & playing the role of the order parame- ter, we know that near the critical point where the coefficient of the quadratic term vanishes, the expansion of the free energy must contain also a fourth-order term. Hence, the action should take the form [2—3po(z}]R(z)+ f dz& cotT ~ BR OTTO 7T Z } Using (26) we can easily verify that Z Z] =—', q&(z) . (30) B = (a —a, },2Q 3U (31) where v = f" dz q & (z)R (z}. Substituting now (31) into (29), we obtain the expression for the parameter 5, 3U 4u (32) (1+y lnl )R„—2y g R e I — I —'S (33) It remains to determine the function R(z). Note that taking a=0 in Eq. (Sb), we obtain the homogeneous equation corresponding to (30). Following the same steps as we have done to find q & (z), we obtain an equation for the Fourier coefficients R„, Pp = —',q where y = T/To and (25a) Q(a, )q = —2(a —a, )q+3pq, (25b) S„—g Ck C„ k= —oo where the higher-order term ~p in (25a) has been neglected [see (1 la)]. The solution of the homogeneous equation for Q(a, ) is q' '. As a drops below its critical value a„ the solution q =q' ' is slightly distorted and we make the ansatz q =q' '+q"'. Substituting this ansatz into (25b), neglecting q'" on the right-hand side, multi- plying the entire equation by q' ', and integrating we ob- tain 3 zp zq& z =2a —a, dzq& z, 26 which is a useful expression relating the unknown func- tion p(z) to the well-known dangerous mode q&(z). In- serting (25a), (25b), and (26) into (9) and neglecting the term proportional to P, we eventually arrive at the fol- lowing expression for the action, S=SO+ —(a —a, )B'f" dz q', (z) . (27) Comparison with Eq. (24) minimized with respect to c&, g(a —a, )' S=So— (28} a, —a5= 2uB (29) allows us to express the important parameter 6 via the norm u = f dz q f(z) of the eigenfunction q& and the amplitude B, '+/n —k/ . . r y e —b [ Ikl+ l n —kl & '+ k[ k= —oo (34} Tf drq~—(r) e ', (35) 4 where R (x) is a continuous function corresponding to the discrete Fourier coefficients R„. In this case, the value of 5 turns out to be positive and thus characterizes a second-order phase transition. It follows that a tempera- ture T* should exist where 5(T*)=0. T then corre sponds to a tricritical point into which the curves for the first- and the second-order phase transitions merge. Below T*, 5 is positive and the transition is smooth if we vary a from a &a, to a & a„corresponding to a second- This equation can be solved analytically in two limits, namely, T=TO and T=o. Near To, we have r =1 and a, =l [see (19)]. Considering only the three leading terms in the series (33}and (34) we obtain after some alge- bra 5= —(9/32m. ), which characterizes a first-order phase transition since 5 (0. On the other hand, close to the absolute zero of the temperature, T=0, y ~ n ~ can be treated as a continuous variable x. In this limit b =y [see (16)] and the equation to be solved takes the form (1+~x ~ )R (x)—2f dyR (y)e 4040 C. MORAIS SMITH, B. IVLEV, AND G. BLATTER 49 order phase transition. On the other hand, above T*, the transition is abrupt, 6&0, and it corresponds to a first- order phase transition. From the result obtained for the underdamped case, " where the phase diagram has the same features as pointed out here, we expect that the tri- critical temperature T* must be very close to the cross- over temperature Tc. Therefore, we can expand 5(y) in a Taylor series around the value y = 1, 5(y) =5(1)+ (y —1),B6 ay (36) cc e —E(I)/T I ) (37) where T is the temperature and E (I) is the activation en- ergy, which is determined by the extremum of the free energy in (3) as a function of (p, q). In the high-current region (i &1) the activation energy E, (I)=4EJE (I)/3, whereas in the low-current region (i) 1) the saddle splits and the activation energy becomes Ed(I)=EJE (I)(2 +3a —a')/3. In the region i & 1, as T drops below the crossover temperature To(I), the system enters the quan- tum regime and the decay rate takes the form and calculate the value of y for which 5(y) =0. Perform- ing some tedious but simple calculations we obtain that the tricritical temperature T*=0.97 Tp. We are now in a position to determine the behavior of the decay rate as a function of temperature and current in the classical and quantum regimes. As usual, the classi- cal decay rate is given by the Arrhenius law, rithm of the decay rate I, 3P A' o. = — 1nI 16~g (40) 2 3/2 g Op —3L 0 o, =3i—9 —a (i, 8), (41a) (41b) (41c) where a (L, 6)=3P fig(a a, ) /64m—r15 Not.e that a is a function of c, and a, and 5 are functions of both c and 8. Let us analyze the result (41) in detail (Figs. 4 —6). To begin with, we keep the current (i, ) fixed and discuss the functional dependence o(8), see Fig. 5. For a large 5.0 O and using the variables defined in (21), 8 (related to the temperature T) and i (related to the difference between the bias current I and the critical current 2I, in the sys- tem) one can rewrite the decay rates in the simple form (remember i & 1), So(I, T)/RIp~e (38) —g[(a —a )el+bc] ]/Ar, e ''"f" dc e —[S —g(a —a ) /46]/A'-e C (39) where we have made use of (28) in the last expression. Comparison of the results (38) and (39) shows that I, & I p and thus the instanton splitting leads to a relative increase in the decay rate. For negative 5, a term propor- tional to c, has to be considered in the action, i.e., the term proportional to p should be retained in Eq. (9) in order to provide a complete description. Defining the new dimensionless variable for the loga- where Sc(I, T) =4m.7)e (I)[1—T /3TO ] is the action eval- uated for the single-instanton solution. At T= Tp the quantum and the classical results merge. For systems consisting of only one degree of freedom, the relaxation in the quantum regime is entirely described by Eq. (38). The saturation of the decay rate at low temperatures as obtained in (37) and (38) is in agreement with experimen- tal observations on a rf SQUID. ' ' In our model con- taining two coupled degrees of freedom, the quantum re- gime is described by Eq. (38) only down to the tempera- ture T, (I). Below T, (I) the instanton splits as a conse- quence of the development of a new unstable mode. For 5 & 0, the decay rate can be obtained by singling out the integration over the dangerous mode, I.O ] e(l ) 8 ( 2) 8(&) 0 s 0 0.5 c ] =0.20 (.o 8 FIG. 5. Logarithm of the decay rate vs temperature in terms of the dimensionless variables o = —(3P I/16m q)lnI' and 0=(rivrP) T/AEJ for three different regions within the phenome- nological temperature-current phase diagram (see Fig. 4). For a large current (~1=0.20) and at high temperatures, the system decays following Eq. (41a), o., ~ 1/0 (dotted line). Upon reduc- ing the temperature, the quantum regime is reached at 0,"'=0.45 and the decay rate becomes oo~ const —9' [see (41b)]. The behavior is the same as for a system consisting of a single degree of freedom because the coupling term between the two phases in the SQUID is relevant and forces them to decay symmetrically. If we fix the bias current to a lower value (~2=0.64) the system decays by thermal activation at high tern- peratures, enters the quantum regime described by Eq. (41b) at 0o '=0.80, and then reaches a new phase in the quantum regime at 0', '=0.66, where the instanton splits and the relative value of the decay rate increases [dashed line, see (41c)]. The transition at 0,' ' is smooth, like in a second-order phase transition. Re- ducing further the bias current (F3=0.96), the system behaves in the same manner (0o '=0.98 and 0', '=0.97) except for an abrupt change in the action at 0,' '„a behavior corresponding to a first-order phase transition. 49 MACROSCOPIC QUANTUM TUNNELING IN A dc SQUID: . . . 2.0 l.0 0 08 I I0 current (s&=0.20, remember that large currents corre- spond to small e) and at high temperatures, the system decays following (41a), o, ~ 1/8. Upon reducing the tem- perature, the system enters the quantum regime at 8o"=0.45 and relaxes according to (41b), eo-const —8 . At large bias current, the coupling between the two phases in the SQUID is very effective and they decay symmetrically. Reducing the bias current (cz=0.64), the system decays by thermal activation at high tempera- tures, enters the quantum regime described by the single instanton at 00 '=0.8, and then reaches a new phase in the quantum regime at 8,' '=0.66 (see also Fig. 4), where the instanton splits and the decay rate I, is larger than the value I 0 obtained for the single-instanton solution [see Eq. (41c)]. The transition at 8,'2' is smooth, like in the second-order phase transition. If the bias current is further lowered (~3=0.96), the system behaves in the same manner except for an abrupt change in the action (corresponding to a first-order transition) as the split- instanton region is entered. The behavior of the system at fixed temperature 8 and with varying current c is illus- trated in Fig. 6. The second-order transition to the split- instanton region is probably difficult to observe experi- mentally. However, the discontinuity in the decay rate occurring for c3=0.96 in Fig. 5 and for 02=0.95 in Fig. 6 is a signal of the onset of instanton splitting. V. CONCLUSIONS In the present work, the quantum decay of a current- carrying state in a dc SQUID has been studied in the overdamped limit. The analysis of this system compris- ing two coupled macroscopic quantum degrees of free- dom has shown that a much richer behavior is developed than in a system with only a single degree of freedom. FIG. 6. Logarithm of the decay rate vs current in terms of the dimensionless variables o = —(3P'A/16m')lnl' and c=P (2 I/I, )—/4 for fixed values of temperature (8& =0.5, 82=0.95). For large currents (small c), the system is in the thermal regime described by Eq. (41a), 0., ~ c (dotted line). At the value co"=0.25(co '=0.90), the single-instanton regime is reached [see Eq. (41b), o 0 ~ 3i—const] and at 2 "=0.53( i',~' =0.93), the instanton splits [dashed line, see Eq. (41c)]. The splitting can be smooth (case 1) or sharp (case 2). Beyond the crossover temperature 8o(a), which exists also in the latter case and indicates the appearance of a non- trivial trajectory in imaginary time, an additional critical temperature 8, (a) arises in this system marking a qualita- tive change in the nature of the tunneling process. Considering the problem in terms of the Landau theory of phase transitions, a phase diagram relating the temper- ature T and the bias current I has been constructed. Four distinct phases have been found, depending on the values of the external parameters (temperature and bias current) which influence the shape of the potential. At high temperatures, the decay out of the metastable state is due to thermal activation and the system behaves clas- sically. In this regime the decay rate is given by the usual Arrhenius law. Depending on the value of the external current, two different situations can be realized in the classical limit. At high temperatures 8(~)&8o(a) and large currents c &1, the system escapes through a single saddle, whereas at lower currents (s & 1) the saddle splits into two. For c (1, upon decreasing the temperature, in the region 8(c) & 8o(c), the system decays due to quantum tunneling through the barrier. The quantum regime is characterized by two different regions. At large enough bias current I (small s) and high enough temperature T, i.e., in the region 8, (~) & 8(a) & 8o(~), the phases y& and q&2 in the two Josephson junctions are strongly correlated and they decay symmetrically. This behavior is the same as for a system composed of only a single degree of free- dom coupled to the environment. Its dynamics is de- scribed by the single-instanton solution and has been dis- cussed before by Larkin and Ovchinnikov for the case of strong dissipation. Below the critical line 8, (a), the in- stanton splits and two degenerate extremal trajectories appear. In this regime the two phases are only weakly correlated and they decay in an asymmetric way. We have calculated analytically the split-instanton solution corresponding to the quantum behavior in the region 8(s) &8,(s). Surprisingly, the solution for the over- damped case has a very simple form, in contrast to the el- liptic functions obtained for the underdamped case." The splitting of the instanton reduces significantly the de- creasing of the tunneling rate in the quantum regime. Focusing our attention on this new transition line 8, (c), we have observed that near the critical regime, the fluc- tuations of the mode corresponding to a transition be- tween the two trajectories become large, requiring to go beyond the quadratic approximation in the action (the fourth-order term is needed in this case). At low temper- atures, the transition through the critical region is smooth, analogous to a second-order phase transition. Instead, at temperatures near to the crossover tempera- ture 8o(i) separating the quantum regime from the classi- cal one, the splitting of the instanton is abrupt, as in a first order-phase transition. Above and close to the tri- critical point 0*, where the second- and first-order phase transitions merge, the dangerous mode fluctuates so strongly that sixth-order corrections have to be included in the action. It would be interesting to observe experimentally the variation of the decay rate as a function of the tempera- ture T and the bias current I in the region corresponding C. MORAIS SMITH, B. IVLEV, AND G. BLATTER to the instanton splitting. In Ref. 20, the transition rate from the zero-voltage metastable minimum in the wash- board potential of a dc SQUID has been measured as a function of applied Aux and temperature. However, the devices used in this study are characterized by very small inductances (P« 1) and in this limit, the splitting cannot be observed. 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