aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Decay in an asymmetric SQUID
J. P. Palomares-B´aez and B. Ivlev Instituto Potosino de Investigaci´on Cientifica y Tecnol´ogica,San Luis Potos´ı, San Luis Potos´ı 78231, Mexico Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ıSan Luis Potos´ı, San Luis Potos´ı 78000, Mexico
Quantum tunneling in an asymmetric (with strongly different capacitances) SQUID is studied.Since capacitances play a role of masses one phase, related to a large mass, becomes ”heavy” andremains always a constant in a tunneling process. Tunneling in an asymmetric SQUID becomesone-dimensional with a condition of optimization of tunneling probability with respect to a value ofthe ”heavy” phase. An unusual temperature dependence of the tunneling probability is obtained.It has a finite slope at zero temperature and a transition between thermally assisted tunneling andpure activation can be not smooth depending on current through a SQUID.
PACS numbers: 85.25.Dq, 74.50.+r
I. INTRODUCTION
A decay of a zero voltage state via quantum tunnel-ing of a phase across a potential barrier in Josephsonjunctions is possible.
Tunneling in a single Joseph-son junction is similar to a conventional one-dimensionalquantum mechanical process. In this case the tunnelingmechanism is described by theory of Wentzel, Kramers,and Brillouin (WKB). Tunneling occurs from a classi-cally allowed region which is a conventional potential wellwhere energy levels are quantized.
Quantum coherencebetween potential wells was demonstrated.
Besides single Josephson junctions, superconductingquantum interference devices (SQUID) are also a mat-ter of active investigation.
A SQUID consists of twojunctions and, therefore, represents a two-dimensionalsystem where macroscopic quantum tunneling is also pos-sible.Tunneling in multi-dimensional systems was describedin literature.
According to those theories, there is acertain underbarrier path where a wave function is local-ized and it decays along the path. However, a tunnelingscenario in two (many) dimensions can substantially dif-fer from that main-path mechanism. For example, in aSQUID a main path can split by two ones and evenby an infinite number of equivalent paths which interferproviding multi-path tunneling. See also .Capacitances of two junctions in a SQUID play a roleof masses. When one capacitance is large the massesbecome very different and one phase becomes ”heavy”.This provides an additional interest for study of an asym-metric SQUID. A behavior of an asymmetric SQUID af-ter tunneling was investigated. In this paper we alsofocus on strongly asymmetric SQUIDs, namely, on tun-neling process in them.During tunneling process a motion along the ”heavy”coordinate is weakly generated and the process of bar-rier crossing becomes almost one-dimensional when the”heavy” phase is fixed. This fixed value should be de-termined from a condition of maximum of a tunnelingprobability. That program has been performed in the paper. Thereare two unusual features of results.First, the tunneling probability, as a function of tem-perature, has a finite slope at low temperature. This con-trasts to a temperature dependence for a one-dimensionalbarrier where that slope is zero.Second, a transition at a finite temperature betweenthermally assisted tunneling and pure activation changesits character when current approaches the critical value.At those currents temperature dependence of tunnelingprobability exhibits a finite jump of slopes at the transi-tion temperature. When current is not too close to thecritical value the transition is smooth as for a one dimen-sional barrier.In Sec. III we apply to a SQUID a semiclassical for-malism of Hamilton-Jacobi. In Sec. IV the method ofclassical trajectories in imaginary time is used which ac-counts for an optimization of tunneling probability withrespect to a value of the ”heavy” phase. In Sec. VII itis argued that experimental observations of the proposedphenomena in SQUID is real.
II. FORMULATION OF THE PROBLEM
We consider a dc SQUID, consisting of two Joseph-son junctions with phases ϕ and ϕ , with no dissipationwhen the two junctions are inductively coupled. Criticalcurrents of the junctions are equal but capacitances, C and C , are different so that M = C C ≫ . (1)A classical behavior of phases corresponds to conserva-tion of the total energy E = E J ω "(cid:18) ∂ϕ ∂t (cid:19) + M (cid:18) ∂ϕ ∂t (cid:19) + E J (cid:20) − cos ϕ − cos ϕ − j ( ϕ + ϕ ) + 12 β ( ϕ − ϕ ) (cid:21) , (2) FIG. 1: (Color online) Curves of a constant energy V ( x, y ) = E at α = 0 .
9. Tunneling occurs along the dashed line, y = y ,where an underbarrier wave function is localized. where the dimensionless current j = I/ I c , the Joseph-son energy E J = ~ I c / e , the plasma frequency ω = p eI c / ~ C , and the coupling parameter β = 2 πLI c / Φ are introduced. Here I c and L are critical current and in-ductance of each individual junction. The magnetic fluxquantum is Φ = π ~ c/e . As follows from Eqs. (1) and(2), ϕ is a ”heavy” phase.Below we consider a large β and the total current I close to its critical value, (1 − j ) ≪
1. New variables areintroduced by the relations ϕ = π x − p − j ) + 3 xβ (3) ϕ = π y − p − j ) + 3 yβ . Below time is measured in the unit of t = √ βω r α α , (4)where the coupling parameter is α = 1 β p − j ) . (5)The energy (2) takes the form E = ~ Bt " (cid:18) ∂x∂t (cid:19) + M (cid:18) ∂y∂t (cid:19) + V ( x, y ) , (6)where B = B √ (cid:18) αα (cid:19) / , B = 9 E J ~ ωβ / . (7)The potential energy is V ( x, y ) = V ( x ) + V ( y ) − αxy α , (8)where V ( x ) = x − x . B in Eq. (6) is called semiclas-sical parameter. When B is large the phase dynamics ismainly classical. Below we consider that case, 1 ≪ B . y y −y y (a)(b)E(y) δ y yx,y |ψ( )| FIG. 2: (a) Form of δE ( y ). (b) Corresponding density distri-bution is Gaussian. It is plotted for some x under the barrier. III. DESCRIPTION OF TUNNELING
A classical dynamics of phases in a SQUID relates toEqs. (6) and (8). The effective particle moves in the clas-sically allowed region, in a vicinity of the point x = y = 0,which is restricted by the potential barrier. As known,the particle can tunnel through the barrier resulting inexperimentally observable phase jumps. Character oftunneling depends on coupling strength α between thetwo junctions. At α = 0 .
90 the curves of equal potential, V ( x, y ) = E , are shown in Fig. 1. An effective particletunnels from one classically allowed region (the potentialwell) to another (the outer region).To quantitatively study the problem of two-dimensional tunneling one should solve the Schr¨odingerequation with the exact potential (8). Since the potentialbarrier is almost classical one can apply a semiclassicalmethod when a wave function has the form ψ ∼ exp( iBσ ) , (9)where the classical action is ~ Bσ . σ satisfies the equationof Hamilton-Jacobi (cid:18) ∂σ∂x (cid:19) + 12 M (cid:18) ∂σ∂y (cid:19) + V ( x, y ) = E. (10)We define the energy E by the relation E = ~ Bt . (11)At a large M a solution of Eq. (10) can be written in theform σ = σ + σ where σ is small and σ is given by σ ( x, y ) √ i Z x dx r x − x − αx y α − E + δE ( x , y )+ √ M Z y dy q δE ( x, y ) − y + y , (12)where δE ( x, y ) is some function to be specified. It is easyto conclude that the correction σ is small (proportional y y −y y (a)(b)E(y) δ yx,y |ψ( )| y FIG. 3: (a) Form of δE ( y ). (b) Corresponding density distri-bution, plotted for some x under the barrier, is not Gaussian.See the text. to 1 / √ M ) when the derivative ∂δE/∂x is small (propor-tional to 1 /M ). So we consider below δE ( y ).The function δE ( y ) is determined by a state in the wellfrom which tunneling occurs.When the case of Fig. 2(a) is realized the last termin Eq. (12) provides a Gaussian distribution of densityaround the line y = y shown in Fig. 2(b) for some x under the barrier. This is analogous to a conventionalscenario of tunneling in two dimensions. In the case of δE ( y ) of Fig. 3(a) at y < y the densitydrops down under the barrier due to the second term inEq. (12) and at y < y due to the first one. Thereforethe underbarrier density in that case is also localized ina vicinity of the line y = y . It is shown in Fig. 3(b)for some x under the barrier. This is not a Gaussiandistribution but one of the type exp[ − c ( y − y ) / ]. At y < y the parameter c ∼ √ M is large.The situations in Figs. 2 and 3 relate to different typesof states in the potential well from which tunneling oc-curs. In the case of Fig. 2 a Gaussian distribution ofdensity holds also in the well because the last term inEq. (12) dominates. In the case of Fig. 3 the distribu-tion in the well is analogous to Fig. 3(b) when the partat y < y is horizontal and, therefore, the state is dis-tributed over a finite distance y in the well.In the both cases, Figs. 2 and 3, tunneling occurs alongthe certain line y = y and y should be determinedfrom the condition of maximum of a tunneling proba-bility. This corresponds to classical mechanics when aparticle does not move along a ”heavy” direction. Withan exponential accuracy a tunneling probability is thesame for the both types of states in the well. IV. TUNNELING PROBABILITY
Since tunneling occurs along the line y = y one canuse a WKB approach as in a one-dimensional case. The T B _ l n_1 Γ FIG. 4: (Color online) Γ is tunneling probability and tem-perature T is measured in the units of ~ ω/ √ β . The numbersmark values of the parameter (1 − j ) β . probability of tunneling with a fixed energy E isΓ( E ) ∼ exp[ − BA ( E )] , (13)where A ( E ) = √ Z dx p v ( x ) − E. (14)The one-dimensional potential v ( x ) is given by v ( x ) = x − x − αy α x + y − y . (15)The integration in Eq. (14) is restricted by the classicallyforbidden region where E < v ( x ).Tunneling probability at a fixed temperature accountsfor the Gibbs factor and is determined byΓ ∼ exp (cid:20) − BA ( E ) − E T (cid:21) , (16)with a subsequent optimization with respect to E . Tak-ing Eq. (11), one can write Eq. (16) in the formΓ ∼ exp ( − BA T ) , (17)where A T = A ( E ) + Eθ . (18)The parameter θ is connected with temperature θ = 2 T √ β ~ ω r α α . (19)Minimization of A T with respect to energy defines E by the equation 1 θ = 1 √ Z dx p v ( x ) − E . (20)The parameter y should be chosen to minimize A T . By T B _ l n_1 Γ FIG. 5: (Color online) Amplification of the lower set of curvesin Fig. 4. Left parts of the curves relate to a thermally assistedtunneling and right parts pertain to pure activation. introducing imaginary time t = iτ the action A T can bewritten in the form A T = Z /θ dτ " (cid:18) ∂x∂τ (cid:19) + v ( x ) , (21)where the classical trajectory under the barrier is deter-mined by Newton’s equation ∂ x∂τ = 2 x − x − αy α (22)with zero velocities at the terminal points, τ = 0 and τ = 1 /θ . According to Eq. (20), 1 /θ is the underbarriertime of motion between two terminal points. In terms oftrajectories, the condition of minimum A T with respectto y takes the form2 y − y = 2 αθ α Z /θ xdτ. (23)For a strongly asymmetric SQUID, a large M , tun-neling occurs along a straight line y = y shown inFig. 1. The action (21) depends on two parameters, α and T √ β/ ~ ω .The tunneling probability satisfies the relation1 B ln 1Γ = √ (cid:18) αα (cid:19) / A T (cid:16) α, T p β/ ~ ω (cid:17) . (24)A recipe of calculation of the action A T is the following.At fixed α , y , and θ one should find a solution of Eq. (22)with zero velocities, ∂x/∂τ = 0, at τ = 0 , /θ . Thatsolution has to be inserted into the relation (23) whichdefines y at fixed θ and α . The solution with the defined y should be substituted into Eq. (21) which produces A T (cid:0) α, T √ β/ ~ ω (cid:1) . We demonstrate in Sec. V how thisscheme works in the case of low temperatures. V. TUNNELING AT LOW TEMPERATURES
At low temperatures the energy E should be close tothe minimum of the potential v ( x ) providing a long un- derbarrier time 1 /θ . With the value of energy E = 1 + 2 α (1 + α ) y (25)the action takes the form A T = 4 √ − √ αy α + Eθ . (26)A minimization with respect to y of the action (26), ac-counting for (25), is equivalent to Eq. (23). The resultingaction, at low dimensionless temperature θ , is A T = 4 √ − α α θ. (27) VI. RESULTS
We performed a numerical solution of Eq. (22). The re-sults for the tunneling probability are presented in Figs. 4and 5 where temperature T is measured in the units of ~ ω/ √ β . Each curve in Figs. 4 and 5 consists of two parts.To the left of a dot each curve relates to above trajectorycalculations corresponding to thermally assisted tunnel-ing. To the right of a dot a curve is solely due to thermalactivation1 B ln 1Γ = ~ ωT √ β α ( (1 − α )(1 + 2 α ) , α < / , / < α (28)The activation energy is given by the saddle point V ( x s , y s ) which coinsides with the crossing point of twocurves in Fig. 1. The steepest descent in Fig. 1 goes alongthe direction x = y . The saddle point { x s , y s } is deter-mined by the conditions ∂V ( x, y ) /∂x = ∂V ( x, y ) /∂y = 0.At 1 / < α x s = y s = 23(1 + α ) (29)and at α < / x s , y s = 1 + 2 α ± √ − α α ) . (30)At α < / < (1 − j ) β in Figs. 4 and 5. At 1 / < α the transitionto the activation regime reminds type I phase transition.A derivative with respect to temperature jumps at thosepoints which can be observed in Fig. 5.Numerically calculated curves in Figs. 4 and 5 matchat low temperatures the analytical dependence followedfrom Eqs. (24) and (27). At low temperatures1 B ln 1Γ = 815 (cid:18) αα (cid:19) / (cid:20) − T √ β ~ ω α √ α (1 + 2 α ) √ α (cid:21) . (31) l n β _1 Γ _ B β _1 Γ _ B β _1 Γ _ B β _1 Γ _ B thermal + quantumthermal FIG. 6: (Color online) Tunneling probability Γ versus currentfor various values of the dimensionless temperature T √ β/ ~ ω .The dashed curve separates thermally assisted tunneling froma pure thermal activation. We note that the slope in the temperature dependence(31) is finite.The tunneling probability Γ as a function of the pa-rameter (1 − j ) β is plotted in Fig. 6 for different valuesof the dimensionless temperature T √ β/ ~ ω . This plotshows how Γ depends on current at a fixed temperature. VII. DISCUSSIONS
Quantum tunneling across a one-dimensional static po-tential barrier is described by WKB theory. Accord-ingly, in two dimensions the main contribution to tunel-ing probability comes from an extreme path linking twoclassically allowed regions. The path is a classical tra-jectory with real coordinates which can be parametrizedby imaginary time. The underbarrier trajectory is a so-lution of Newton’s equation in imaginary time. Underthe barrier the probability density reaches a maximumat each point of the trajectory along the orthogonal di-rection with respect to it. Along that direction the den-sity has a Gaussian distribution. Therefore around thetrajectory, which plays a role of a saddle point, quantumfluctuations are weak. The wave function, tracked alongthat trajectory under the barrier, exhibits an exponen-tial decay analogous to WKB behavior. This constitutesa conventional scenario of tunneling in multi-dimensionalcase which can be called main-path tunneling. In a symmetric (not very asymmetric) SQUID besidesthe conventional main-path tunneling also multi-pathtunneling is possible. In that case a density distributionunder a barrier is not as in Fig. 2(b), but of the type as inFig. 3(b). For a very asymmetric SQUID, considered inthis paper, the both mechanisms result in the same tun-neling exponents since the underbarrier channel shrinksto the line y = y due to the mass difference. We used a semiclassical approximation when there aremany levels in the well. This approach sometimes isnot appropriate in one-dimensional Josephson junctionswhere a barrier is weakly transparent but neverthelessthere is only a few levels in the well (say, five). In aSQUID based on two such junctions the number of lev-els can be roughly estimated as 5 ×
5. In our case ofa strongly asymmetric SQUID that number should bemultiplied by the large parameter √ M . Therefore theapproximation of a large number of levels in the well isreasonable.We propose two peculiarities of tunneling in an asym-metric SQUID which do not exist in a single junction andin a not very asymmetric SQUID.One of them is temperature dependence of tunnelingprobability at low temperature. According to Eq. (31),the curves in Fig. 5 have a finite slope at low temperature.In one dimension the slope is zero due to the exponentexp( − const/T ) instead of T in Eq. (31).The second peculiarity is an unusual transition be-tween thermally assisted tunneling and pure activationmarked by dots in Fig. 5. At 2 < (1 − j ) β the transitionis smooth but at (1 − j ) β < M ≃
35 and β ≃
15 correspond to reality in experiments with SQUIDsand fit the developed theory. It is more convenient inexperiments to obtain a set of curves as in Fig. 6 sinceusually measurements are run at a fixed temperature.A dependence on temperature, as in Fig. 5, also can beobtained. This would provide an experimental check ofthe predicted dependences on temperature and current
VIII. CONCLUSION
Quantum tunneling in an asymmetric (with stronglydifferent capacitances) SQUID is studied. Since capaci-tances play a role of masses one phase, related to a largemass, becomes ”heavy” and remains always a constant ina tunneling process. Tunneling in an asymmetric SQUIDbecomes one-dimensional with a condition of optimiza-tion of tunneling probability with respect to a value ofthe ”heavy” phase. An unusual temperature dependenceof the tunneling probability is obtained. It has a finiteslope at zero temperature and a transition between ther-mally assisted tunneling and pure activation can be notsmooth depending on current through a SQUID.
Acknowledgments
We thank G. Blatter, S. Butz, V. B. Geshkenbein, andA. V. Ustinov for valuable discussions. A. O. Caldeira and A. Leggett, Ann. Phys. (N.Y.) ,374 (1983). A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. , 1510 (1983) [Sov. Phys. JETP , 876 (1983)]. M. H. Devoret, J. M. Martinis, and J. Clarke, Phys. Rev.Lett. , 1908 (1985). M. H. Devoret, D. Esteve, C. Urbina, J. Martinis, A.Creland, and J. Clarke, in
Quantum Tunneling in Con-densed Media , edited by A. Leggett and Yu. Kagan (North-Holland, Amsterdam, 1992). L. D. Landau and E. M. Lifshitz,
Quantum Mechanics (Pergamon, New York, 1977). J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev.Lett. , 1543 (1985). J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev.B , 4682 (1987). A. Wallraf, T. Duty, A. Lukashenko, and A. V. Ustinov,Phys. Rev. Lett. , 037003 (2003). M. V. Fistul, A. Wallraf, and A. V. Ustinov, Phys. Rev. B , 060504 (2003). A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. ,1 (1987). Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature (Lon-don) , 786 (1999). J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J.E. Lukens, Nature (London) , 43 (2000). C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N.Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd,and J. E. Mooij, Science , 773 (2000). Y.-C. Chen, J. Low Temp. Phys. , 133 (1986). B. I. Ivlev and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. , 668 (1987) [Sov. Phys. JETP , 378 (1987)]. C. Morais Smith, B. Ivlev, and G. Blatter, Phys. Rev. B , 4033 (1994). S. X. Li, Y. Yu, Yu. Zhang, W. Qiu, S. Han, and Z. Wang,Phys. Rev. Lett. , 098301 (2002). F. Balestro, J. Claudon, J. P. Pekola, and O. Buisson,Phys. Rev. Lett. , 158301 (2003). M. G. Castellano, F. Chiarello, R. Leoni, F. Mattioli, G.Torrioli, P. Carelli, M. Cirillo, C. Casmelli, A. de Waard,G. Frossati, N. Grønbech-Jensen, and S. Poletto, Phys.Rev. Lett. , 177002 (2007). K. Mitra, F. W. Strauch, C. J. Lobb, J. R. Anderson, F.C. Wellstood, and E. Tiesinga, Phys. Rev. B , 214512(2008). A. U. Thomann, V. B. Geshkenbein, and G. Blatter, Phys.Rev. B , 184515 (2009). B. Ivlev, Phys. Rev. B, to be published (2010). C. G. Callan and S. Coleman, Phys. Rev. D , 1762(1977). S. Coleman, in
Aspects of Symmetry (Cambridge Univer-sity Press, Cambridge, 1985). A. Schmid, Ann. Phys. , 333 (1986). U. Eckern and A. Schmid, in
Quantum Tunneling in Con-densed Media , edited by A. Leggett and Yu. Kagan (North-Holland, Amsterdam, 1992). B. I. Ivlev and V. I. Melnikov, Phys. Rev. B , 6889(1987). B. Ivlev, arXiv:0903.5100. B. I. Ivlev, Phys. Rev. A , 052106 (2006). M. Marthaler and M. I. Dykman, Phys. Rev. A76