October 2007 EPL, 80 (2007) 17009 www.epljournal.org doi: 10.1209/0295-5075/80/17009
Why macroscopic quantum tunnelling in Josephson junctions differs from tunnelling of a quantum particle
A. O. Sboychakov1,2, Sergey Savel’ev1,3, A. L. Rakhmanov1,2,3 and Franco Nori1,4
1 Frontier Research System, The Institute of Physical and Chemical Research (RIKEN) - Wako-shi, Saitama, 351-0198, Japan 2 Institute for Theoretical and Applied Electrodynamics Russian Academy of Sciences - 125412 Moscow, Russia 3 Department of Physics, Loughborough University - Loughborough LE11 3TU, UK 4 MCTP, CSCS, Department of Physics, University of Michigan - Ann Arbor, MI 48109-1040, USA
received 14 June 2007; accepted in final form 17 August 2007 published online 17 September 2007
PACS 74.50.+r – Tunneling phenomena; point contacts, weak links, Josephson effects
Abstract – We show that the macroscopic quantum tunnelling of a fluxon in a Josephson junction cannot be described, even qualitatively, as the tunnelling of a quantum particle in a potential U(ϕ), where the phase difference ϕ plays the role of particle position, if the length of the junction d exceeds a fluxon length. We calculate the probability per unit time of tunnelling (orescaperate),Γ,whichhasaformΓ=A exp(−B). In contrast to particles, where the B is proportional to d, our field-theory predicts a different behavior of B for either usual, 0–π,or stacks of Josephson junctions, giving rise to a renormalization of Γ by many orders of magnitude.
Copyright c EPLA, 2007
Macroscopic quantum tunnelling (MQT) is one of the we emphasize that the problem we are considering is few manifestations of quantum effects in macroscopic far more general and is also applicable to MQT in long systems [1]. MQT was experimentally observed [1] in Josephson junctions (see, e.g., [6]). Moreover, MQT in Josephson junctions (JJs) in 1980s and has been stud- long Josephson junctions is of relevance for the so-called ied for different Josephson systems. General interest on phase qubits which might provide basic elements for future MQT is now fuelled not only by its fundamental inter- quantum information processing devices (see ref. [2] for est but as a readout mechanism for phase qubits [2]. a review). These phase qubits are current-biased JJs A new surge of interest on MQT occurred after the driven by microwaves, and exhibiting MQT. Controlling recent discovery of MQT in high-temperature layered the motion of Josephson vortices is also of interest for the superconductors [3,4]. The observed giant enhancement design of novel types of THz and ratchet devices [7]. of MQT could be attributed to the spatial structure of Using a field-theory approach, here we show that the the tunnelling fluxon [5]. This is in contrast to the stan- fluxon tunnels through the formation of single nucleus dard approach, where MQT is associated with a quantum followed by a classical motion through the contact. For particle tunnelling through an effective potential barrier. a long junction this results in a huge renormalization of The latter approach is correct only for very short junc- the tunnelling escape rate Γ, with respect to the “particle” tions, when one can ignore the spatial dependence of the approximation. The tunnelling probability differs for the phase difference ϕ across the junction. A naive guess would usual and 0–π junctions: for usual junctions, log Γ(d)is be that the spatial dependence of ϕ should suppress the proportional to d for d λJ and shows a maximum at MQT because of an increase of the potential energy U(ϕ) d ∼ λJ ; while log Γ(d) for 0–π junctions increases slower 2 from (∇ϕ) . This is in analogy to the positive energy of a with d for d λJ and does not exhibit a maximum within domain wall in a ferromagnet. However, further analysis the studied range of parameters. We also demonstrate that shows that the total potential energy can decrease simi- the strong renormalization of Γ for stacks of intrinsic JJs larly to the decrease of energy for a ferromagnet having occurs even for very short junctions of about 1 µm, in several domains. agreement with very recent experiments [4]. This offers Our work was motivated by the MQT recently observed potentially useful flexibility when designing readouts for in high-temperature layered superconductors. However, phase qubits [2].
17009-p1 A. O. Sboychakov et al.
General model. – Consider a scalar field ϕ in a finite These functions are orthogonal and normalized V S ψ ψ δ ψ volume with surface . The field action is given by V dx n m = nm. The solution can be expanded (c =1) in the basis ψn as ψ(τ,x)= n cn(τ)ψn(x). Multiplying t eq. (4) by ψn and performing space integration, we get S(t)= dt L(t)+ε0 dSϕ(t, x) f(x) , the system of equations for cn(τ) −∞ S 1 (3) 1 (4) 2 c¨n − µncn = − Unmkcmck − Unmklcmckcl (6) 1 ∂ϕ 1 2 2 6 L(t)=ε0 dx − (∇ϕ) − U(ϕ) , (1) mk mkl V 2 ∂t 2 with initial conditionsc ˙n(0) = cn(∞) = 0. Here, dot means imaginary time derivative, and where L is the Lagrangian and ε0 is an energy scale. Hereafter we ignore dissipation, which was shown [8] to (i) i i Un ... k = − dx ∂ U(ϕ0)/∂ϕ0 ψn...ψk. (7) be negligible. The equality δS/δϕ(t, x) = 0 determines V the classical equation of motion for the field ϕ(t, x). Substituting the series expansion for ψ into eq. (3) and The surface term in eq. (1) corresponds to the external using eq. (6) we express B in the form force acting on the field. It leads to the boundary condi- ∞ (3) (4) ε0 U U tion ∂ϕ/∂n|x∈S = f(x). We assume that the equation B τ nmk c c c nmkl c c c c ... . = d n m k + n m k l+ δS/δϕ = 0 has at least two static solutions corresponding 0 nmk 6 nmkl 12 to two energy minima. One of them, ϕ−(x), corresponds (8) ϕ to a global minimum, whereas the second one, 0(x), is We enumerate the eigenvalues of Lˆ in ascending order a local energy minimum. We calculate the probability Γ (µ0 <µ1 <µ2 <...). All µn’s are positive since ϕ0(x) (per unit time) of transition from the local minimum to corresponds to an energy minimum. With increasing the global one. In the semiclassical limit this probability external force f(x), the state ϕ0(x) becomes unstable ω B/π −B can be written as [9] Γ = 0 30 exp( ), where and B = 0. At this point, several lowest µn’s are zero. ω ϕ 0 is the oscillation frequency of the field near 0(x), At first, we assume that only the value µ0 vanishes. In B iS i∞ / S = cl( ) ,and cl is the action eq. (1) calculated this case, µ0 µn for n>0 when f(x)isclosetothe along classical trajectories. Introducing the imaginary critical value. According to eq. (8), the condition B → 0 t iτ time = , it was obtained [9] requires cn → 0. Thus, we can write a series expansion (0) (1) ∞ c τ µ c µ2c ... 2 n( )= 0 n + 0 n + . Substituting this expansion B − τ L τ ε Sϕ τ, f . (0) = d ( )+ 0 d ( x) (x) (2) cn n> 0 S into eq. (6), we find that =0for 0. In this case, the characteristic frequency ω0 in the expression for Γ is ω0 ≈ ϕ τ, √ (0) (3) The field ( x) in eq. (2) satisfies the imaginary- µ0. We introduce new variables c0 (τ)=3α(ζ)/U000, ∂2/∂τ 2 ∇2 ϕ − time classical equation of motion ( + ) ζ = ω0τ. In the main approximation with respect to µ0, ∂U(ϕ)/∂ϕ = 0 with the initial conditions ∂ϕ/∂τ|τ=0 =0, the system (6) with its initial conditions reduces to ϕ(∞, x)=ϕ0(x). d2α 3 dα(0) The quantum tunnelling of a macroscopic system occurs − α = − α2,α(∞)= =0. (9) dζ2 2 dζ with an appreciable probability only if the barrier is low. √ 1 α/ α − α ζ Then, we can use a perturbation approach and seek a After integration we get α(ζ)d 1 = . Substi- solution of the form ϕ(τ,x)=ϕ0(x)+ψ(τ,x), with |ψ| c(0) |ϕ | U ϕ ψ tuting 0 into eq. (8) and performing time integration 0 . Expanding the potential ( )inaseriesof and with the help of eqs. (9), we derive the expression for B integrating by parts, we obtain in the main approximation in µ0 ∞ 2 2ε0 1 ∂ (0) 5/2 (3) 2 B τ ψ Lˆ − ψ B =24ε0µ0 5(U000) . (10) = d dx 2 0 V 2 ∂τ (1) 3 4 The equations for cn at n>0 follow from eqs. (6) and 1 ∂ U(ϕ0) 3 1 ∂ U(ϕ0) 4 ψ ψ ... , the series expansion of cn(µ0), + ∂ϕ3 + ∂ϕ4 + (3) 6 0 24 0 2 (1) (1) 1 (3) (0) c − µnc − U c . 2 2 2 ¨n n = n00 0 (11) where Lˆ = −∇ + ∂ U(ϕ0)/∂ϕ0. The perturbation ψ(τ,x) 2 The solution of this equation, with the appropri- satisfies the equation of motion (1) ate initial conditions, can be written as cn (τ)= 2 3 4 √ ∂ 1 ∂ U(ϕ0) 2 1 ∂ U(ϕ0) 3 (3) (3) 2 − Lˆ ψ ψ ψ ... , [9U /(U ) ]νn( µ0τ), where ∂τ2 = ∂ϕ3 + ∂ϕ4 + (4) n00 000 2 0 6 0 −λ ζ ∞ n e 2 −λnζ νn(ζ)= dζ α (ζ )e with zero boundary and initial conditions. 2µ0λn 0 We consider the system of eigenfunctions ψn of the ζ ∞ 1 λ (ζ−ζ) 2 −λ ζ operator Lˆ corresponding to the eigenvalues µn + dζ e n dζ α (ζ + ζ )e n 2µ0 0 0 Lψˆ n = µnψn ,∂ψn/∂n|x∈S =0. (5) (12)
17009-p2 Why macroscopic quantum tunnelling etc.
(1) (1) and λn = µn/µ0 1. Writing c0 in the form c0 (τ)= (3) 2 √ [9/(U000) ]γ( µ0τ), we obtain, from eqs. (6), for γ(ζ): 2γ 0.2 d − γ − αγ − α υ(3)ν − 1υ(4)α3 , ζ2 = 3 3 n n 0 (13) d n=0 2
(3) (3) 2 (3) (4) (4) (3) where υn =(Un00) /U000,υ0 = U000/U000. The initial conditions for eq. (13) are evidently γ(∞)=dγ(0)/dζ =0. Substituting the obtained formulas into eq. (8) we derive the expression for the tunnelling exponent 0.1 µ ∞ B ∼ B(0) 45 0 ζα2 = 1+ (3) d 16U 0 000 z y
(3) 1 (4) 2 × 3γ +3 υn νn + υ0 α . (14) -d/2 0 d/2 x n=0 2 0.0 02468d/λ To compute B, we should first solve the eigenvalue J problem (5), and solve eqs. (9), (12), and (13). Similarly, higher-order corrections to B could be obtained. The Fig. 1: (Color online) The logarithm B of the escape rate approach described above can be generalized for the case Γ vs. d at I/Ic(d)=0.99 calculated in zeroth-order (dashed when several, m, eigenvalues are small at the critical blue curve) and first-order (solid red curve) approximation. The dashed green line corresponds to eq. (16), i.e.,the external force. In this case, we should solve the system “particle” approximation. The schematics of the JJ is shown in m c(0) n m of nonlinear equations for n ( ), and then find the inset. The dash-dotted curve is calculated without taking corrections to cn using a perturbation approach. The into account the second small parameter. existence of m small eigenvalues indicates that there exists m channels for tunnelling. The case of two small ϕ ϕ d/ λ I2/ parameters µ0 and µ1 will be essential for the case ˜0 = arccos cos 0( 2 J )+ 8 is found from the considered below: the quantum tunnelling of a fluxon. boundary condition. The solution of this equation exists if the current I is less than the critical value Ic(d). If Tunnelling in Josephson junctions. – Here we d 4λJ , the current density is approximately constant in apply our approach for the study of a fluxon tunnelling in the JJ and the function Ic(d) increases linearly with d; a JJ. The geometry of the problem is shown in the inset of if d λJ , the current flows near the junction edges and d fig. 1. Two superconducting bars overlap a length in the Ic(d) reaches the saturation value 4i0λJ . x l y -direction. The length of the JJ in the -direction is of First, we compute ϕ0(x) and the set of eigenvalues and d y 2 2 the order of or larger than . The current per unit length, eigenfunctions of the operator Lˆ = −d /dx +cosϕ0(x). J z , flows through the junction in the -direction. The prop- Then, we numerically solve the system (6) by the method erties of a JJ are described in terms of the gauge-invariant described above, and, finally, calculate B in the main and ϕ phase difference , which obeys the sine-Gordon equation. first-order approximations. We find that the first-order l d ϕ It can be shown that at , the stationary solution 0, correction to B does not exceed a few percent for any d, corresponding to an energy minimum, does not depend if I/Ic > 0.97. Thus, eq. (10) is applied for the calculation y y ψ on , and the -dependence of n contributes only to the of B with a good accuracy if the current I is close to B higher-order corrections to . In the geometry considered the critical value Ic. The dependence of B on the junction ϕ here we can write the sine-Gordon equation for as length d at I/Ic(d)=0.99 is shown in fig. 1. At small d,the d ϕ¨ − ϕxx +sinϕ =0, (15) tunnelling exponent linearly increases with , according to a well-known result obtained using an analogy with the where the time t and coordinate x are normalized quantum tunnelling of a particle [4,10]: by the Josephson frequency ωJ and length λJ .The i x i ld − j2 5/4 current density ( ) is given by the Josephson relation B Φ0 0 24(1 ) . QM = 2 (16) i(x)=i0sin ϕ(x), where i0 is the critical current density. 2πcωJ 5j We define the dimensionless current j = J/di0, potential U(ϕ)=U0 − cos ϕ,andε0 =Φ0i0λJ l/(2πcωJ ), where Φ0 This formula follows directly from eq. (10). Naturally, if d λ ϕ x ≈ j jx2/ is the flux quantum. The boundary conditions for eq. (15) J ,weget 0( ) arcsin( )+ 2. Neglecting the x ϕ0 x are dϕ/dx = ±I/2, x = ±d/2λJ , where I = jd/λJ . -dependence of cos ( ), from eq. (5) we obtain ϕ x √ The stationary solution 0( ) of eq. (15) corre- 2 µ0 ≈ 1 − j ,ψ0 ≈ 1/ d. (17) sponding√ to an energy minimum√ can be written as ϕ0(x) ϕ/ ϕ0 − ϕ |x| ϕ˜0 d cos ˜ cos = 2 , where the value Substituting these values in eq. (10) we obtain eq. (16).
17009-p3 A. O. Sboychakov et al.
Fig. 2: (Color online) The shape ψ(0,x)ofthenucleusofa Fig. 3: (Color online) Tunnelling exponent B vs. d for d λ d λ tunnelling fluxon at =2 J (green solid curve) and =8 J 0–π junction in the main approximation; I/Ic(d)=0.99. In d −2 (red and blue dashed curves). At large , two different functions the inset: B vs. d/λc at j =0.99, γs/λc =10 (red solid line) ψ(0,x) correspond to two channels of tunnelling: nucleation of for a stack of JJs. The slope of the asymptote of B(d)at a vortex (on the right sample edge) or an antivortex (on the d dmin (green dot-dashed line) is twice smaller than the I/I d . µ left); c( )=099. The dependence of 0 (red solid curve) slope of B(d)atd A pronounced departure of B(d) from the linear law (16) The distribution of ϕ is non-uniform in the 0–π junction arises at d 4λJ . The tunnelling exponent starts to even at I = 0 [11] and a 0–π JJ cannot be described within decrease with the growth of d and tends to a constant when the “particle approximation” for any d. The dependence d →∞. In this case, the eigenvalue µ1 becomes close to µ0 of the tunnelling exponent on d for a 0–π junction at (see the inset in fig. 2) and one should take into account I/Ic(d)=0.99 is shown in fig. 3. The analysis of the the existence of two tunnelling channels. The field ψ can (0) (0) eigenvalue problem shows that within the considered be written as ψ(τ,x)=c0 (τ)ψ0(x)+c1 (τ)ψ1(x). For range of d, there exists only one small parameter µ0 and d c(0) τ d λ µ ≈ µ small , 1 ( ) = 0, whereas for J we have 1 0 the function ψ(0,x) corresponds to the antivortex nucleus (0) (0) and c1 (τ) ≈±c0 (τ). The function ψ(0,x) is shown in at the left edge of the junction. fig. 2 for small and large d. At small d, ψ(0,x) ≈ const Tunnelling in stacks of junctions. – In layered and the analogy between field and particle tunnelling is superconductors described as a stack of JJs of height D,a valid. For larger d, ψ(0,x) is nonzero only near the right fluxon can tunnel through a certain contact located at D1. (vortex nucleation) or left (antivortex nucleation) edges This process can be described by the non-local Lagrangian of the junction, depending on the choice of the sign of (0) c1 (0). Thus, for d λJ the MQT has two stages. First, d d 2 ε0 ∂ ψ the vortex (or antivortex) nucleus tunnels through the L x ψ K x x x = d ( ; )∂x2 d right (or left) edge of the junction. Then, this vortex (or 2 0 0 antivortex) moves classically through the junction. The ∂ψ 2 ψ3 tunnelling probability starts to increase with the growth − − − j2 ψ2 , ∂τ 1 + (18) of d, due to the opening of the second tunnelling channel. 3 In addition to the usual JJ (0 junction), we calculate the probability of tunnelling for 0–π junctions. Such a which can be obtained following the approach in ref. [5]. ε i Dλ / eω λ junction naturally arises at the boundary of d and s-wave Here 0 = 0 c (4 J ), c is the out-of-plane penetra- superconductors [11] or can be fabricated using a ferro- tion depth, magnetic layer between s-wave superconductors. We K(x; x)=(γs/d) consider the usual current-phase relation to be valid at ∞ −d/2 17009-p4 Why macroscopic quantum tunnelling etc. χn =cosh(νnD1)cosh[νn(D−D1)]/ sinh(νnD), (20) REFERENCES [1] Takagi S., Macroscopic Quantum Tunneling (Cambridge 2 2 −1/2 νn = kn(1 − j ) +1, (21) University Press, Cambridge) 2002. [2] You J. Q. and Nori F., Phys. Today, 58, No. 11 (2005) 42. k πnλ /d γ s n = c , anisotropy , and interlayer distance . Note, [3] Inomata K., Sato S., Nakajima K., Tanaka A., that there are two different characteristic sizes in the Takano Y., Wang H. B., Nagao M., Hatano H. system. The current changes on the scale of λc, whereas and Kawabata S., Phys. Rev. Lett., 95 (2005) 107005; a characteristic size of the fluxon is γs λc. Following Lombardi F., Tafuri F., Barone A., Rotoli G., the numerical procedure described above and taking into Delsing P. and Claeson T., Phys. Rev. Lett., 94 (2005) account that the fluxon can tunnel through any junction, 087003; Bauch T. et al., Science, 311 (2006) 57. we calculate numerically the dependence of Γ on the size [4] Jin X. Y., Lisenfeld J., Koval Y., Lukashenko A., ¨ d of the junction. We also estimate the minimal width of Ustinov A. V. and Muller P., Phys. Rev. Lett., 96 the junctions, (2006) 177003. [5] Savel’ev S., Rakhmanov A. L. and Nori F., Phys. Rev. 98 2 1/4 Lett., (2007) 077002; 269901. dmin =2πγs/5(1 − j ) , (22) [6] Larkin A. I. and Lee P. A., Phys. Rev. B, 17 (1978) 1596; Kato T. and Imada M., J. Phys. Soc. Jpn., when the “particle” approximation becomes invalid and 65 (1996) 2963; Shnirman A., Ben-Jacob E. and we should take into account two tunnelling channels. The Malomed B., Phys. Rev. B, 56 (1997) 14677; Wallraff dependence of the tunnelling exponent B on d is shown A. et al., Nature (London), 425 (2003) 155; Li H., Shen S.-Q., Liang J.-Q. and Liu W.-M., Phys. Rev. B, 72 in the inset of fig. 3 for D1 = D/2. The function B(d) d d 17009-p5