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Why Macroscopic Quantum Tunnelling in Josephson Junctions Differs from Tunnelling of a Quantum Particle

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Why Macroscopic Quantum Tunnelling in Josephson Junctions Differs from Tunnelling of a Quantum Particle 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.
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