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Josephson Vortex Bloch Oscillations: Single Pair Tunneling Effect

We consider the Josephson vortex motion in a long one--dimensional Josephson junction in a thin film. We show that this Josephson vortex is similar to a mesoscopic capacitor. We demonstrate that a single Cooper pair tunneling results in nonlinear Bloch--type oscillations of a Josephson vortex in a current-biased Josephson junction. We find the frequency and the amplitude of this motion.

Single electron and Cooper pair tunneling remain a fundamental problem in normal metal and superconductivity physics. Under detail experimental and theoretical study is electron transport through tunnel junctions in mesoscopic systems [1,2]. A particular attention has been given also to the classical and quantum dynamics of Josephson vortices in Josephson junctions and arrays of Josephson junctions. Very fast classical motion of Josephson vortices with velocities approaching the Swihart velocity was treated experimentally in annular Josephson tunnel junctions [3,4]. The ballistic motion of vortices in twodimensional arrays of Josephson junctions is under thorough theoretical and experimental study [5,6]. It was shown that the spin waves excitation results in an intrinsic friction when a vortex is moving in an array of Josephson junctions [7,8,9,10,6]. Vortices tunneling is treated theoretically and experimentally in two-dimensional arrays of Josephson junctions [11,12,6].
In this Letter we consider the effect of a quantum single Cooper pair tunneling on the classical motion of a Josephson vortex in a long Josephson junction.
We point out that a Josephson vortex in a thin film is similar to a mesoscopic capacitor. We show that at sufficiently low temperatures it results in nonlinear Bloch-type oscillations in a current-biased Josephson junction. We find the amplitude and frequency of this motion.
To develop the analogy between a Josephson vortex and a mesoscopic capacitor let us consider a superconducting film with a Josephson junction along the x axis (see Fig. 1). The motion of a Josephson vortex along this Josephson junction induces voltage V (x, t). The value of the voltage drop across the Josephson junction is equal to [13] V =h 2e where ϕ(x, t) is the phase difference. This voltage results in a certain charge Q moving simultaneously with the Josephson vortex. As ϕ(x, t) = ϕ(x − vt) and where d is the film thickness, C is the specific capacity of the Josephson junction, where j is the current density. We treat here the case of zero dissipation, i.e., the case of an ideal Josephson junction and calculate the current density taking into account the superconducting, tunneling and displacement currents. We neglect the resistive (leakage and quasi-particle tunneling) currents as they can be minimized by the Josephson junction design and by lowering the temperature.
An additional friction force arise as a result of Josephson vortex Cherenkov radiation of small amplitude electromagnetic waves propagating along the tunnel junction [14]. The contribution of this radiation friction force becomes important only when the Josephson vortex velocity is approaching the Swihart velocity.
Note, that the dissipation due to Josephson vortex Cherenkov radiation in a long Josephson junction is similar to the dissipation due to spin waves excitation in Josephson junction arrays [7,8,9,10,6].
In the thin film limit the density of the superconducting current and magnetic field in the superconductor decrease with the characteristic space scale [15] To emphasize on the mesoscopic effects we consider the case of a small size Josephson vortex, i.e., we assume that l ≪ λ eff . Then the relation between the superconducting current density and the phase difference becomes nonlocal and the continuity equation leads to a nonlinear integro-differential equation determining the value of ϕ(x, t) [16,17]. In particular, in the case of d ≪ λ and l ≪ λ eff this equation reads [18] sin ϕ + 1 where is the plasma frequency, Φ 0 is the flux quantum, and j c is the Josephson critical current density.
Let us first consider a stationary Josephson vortex. In this case v = 0 and the space distribution of the phase difference ϕ(x) is given by the exact solution of Eq. (4) [16,17] ϕ(x) = 2 arctan x l J + π, If the velocity v of a Josephson vortex is small, i.e., v ≪ ω p l J the solution of Eq. (4) is given by Eq. (7), where instead of x we substitute x − vt. In particular, it follows from Eq. (7) that the size of a Josephson vortex moving with a velocity v ≪ ω p l J is of the order of l J .
The motion of a Josephson vortex induces an electrical field localized inside the Josephson junction. Using Eq. (1) we find that the energy of this electrical field is equal to It follows from Eq. (8) that E v ∝ v 2 . Combining Eqs. (8) and (2) we find that at the same time E v ∝ Q 2 . We can thus introduce the Josephson vortex mass M and effective capacity C eff presenting E v as where and Let us now consider the motion of a Josephson vortex along a Josephson junction. We treat the case when the driving force is the Lorentz force F L resulting from a bias current across the Josephson junction. Then the equation of motion where j is the bias current density. Using Eq. (2) we rewrite Eq. (13) as It follows from Eqs. (12) and (13) where T is the temperature and k B is the Boltzmann constant. Note, that the inequality given by Eq. (15)  Josephson vortex is continuing to increase with the rate given by a v the process repeats itself periodically. It follows from Eqs. (12) and (13) that the frequency ω and the amplitude l m of this periodic Josephson vortex motion are given by The dependence of the Josephson vortex velocity on time is shown in Fig. 2.
To estimate the values of ω and l m we substitute in Eq. (17) the expression for l J given by Eq. (6) and the specific capacity C in the form where ǫ is the dielectric constant and d 0 is the distance between the banks of the Josephson junction. It results in the formulae Using the data λ ≈ 3 × 10 −5 cm, d ≈ 10 −6 cm, d 0 ≈ 10 −8 cm, and ǫ ≈ 5, we find an estimation for the values of ω and l m in the form ω ≈ 2 × 10 15 j j c s −1 , and l m ≈ 1.5 × 10 −9 j c j cm.
It follows from Eqs. (19) and (20) that if j/j c ≈ 10 −8 then ω ≈ 2 × 10 7 s −1 and l m ≈ 0.15 cm. Note, that for the same data as above and the critical current density j c ≈ 10 5 A/cm 2 we estimate l J ≈ 1.5 × 10 −4 cm, C eff ≈ 7.5 × 10 −3 cm and the characteristic temperature To summarize, we show that a Josephson vortex in a thin film is similar to a mesoscopic capacitor. A single Cooper pair tunneling results in Josephson vortex nonlinear Bloch-type oscillations in a current-biased Josephson junction. We find the frequency and the amplitude of this motion.
We are grateful to Y. Gefen, D.E. Khmelnitskii and M. Tinkham for useful discussions. This work was supported in part by the Foundation Raschi.