Radiation linewidth of a long Josephson junction in the flux-flow regime.

Theoretical model for the radiation linewidth in a multi-fluxon state of a long Josephson junction is presented. Starting from the perturbed sine-Gordon model with the temperature dependent noise term, we develop a collective coordinate approach which allows to calculate the finite radiation linewidth due to excitation of the internal degrees of freedom in the moving fluxon chain. At low fluxon density, the radiation linewidth is expected to be substantially larger than that of a lumped Josephson oscillator. With increasing the fluxon density, a crossover to a much smaller linewidth corresponding to the lumped oscillator limit is predicted. 74.50.+r, 03.40.Kf, 74.80.Dm Typeset using REVTEX Permanent address: Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow district, 142432, Russia Permanent address: Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel 1 A long Josephson junction is an example of a distributed nonlinear oscillator which time-dependent response is associated with its intrinsic spatial dynamics. An externally applied magnetic field H penetrates into the junction in form of Josephson fluxons (solitons), each of them carrying one magnetic flux quantum Φ0. As schematically shown in Fig. 1, fluxons move across the junction under the influence of the bias current generating an electromagnetic radiation at the junction boundary. The frequency f of the radiation is given by the Josephson relation f = V/Φ0, where V is a dc voltage induced by the fluxon motion. For a lumped (short) Josephson junction, the linewidth δf of the emitted radiation is determined by thermal fluctuations of current passing through the junction. Assuming a Nyquist noise spectrum, for the current-biased lumped Josephson tunnel junction the full linewidth at half power is given by expression [1,2] δf ≡ ∆fT · f = 4πkBT Φ20 R D RS , (1) where kB is Boltzmann’s constant, T is the temperature. The linewidth depends on the differential resistance RD = dV/dI at the junction bias point and the static resistance RS = V/I, where I is the bias current flowing through the junction. There are two main regimes which characterize the fluxon motion in long junctions. First, a shuttle-like resonant fluxon motion gives rise to zero-field steps (ZFSs) in the dc current-voltage (I−V ) characteristics of the junction. In this regime fluxons and antifluxons undergo reflections from the junction boundaries and the radiation frequency is determined by the junction length L and the fluxon velocity v as fZFS = v/2L . Second, in the high magnetic field, the so-called flux-flow regime occurs and is manifested by a flux-flow step (FFS) on the I−V curve. In this regime fluxons are created at one boundary of the junction and annihilate at the other boundary. The radiation frequency f = v/dfl is determined by the spacing between moving fluxons dfl. In general, for both ZFS and FFS regimes, the radiation linewidth δf should be related to thermal fluctuations of the fluxon velocity v. Joergensen et al. [3] obtained a striking general result that, in spite of a different nature of 2 the phase slippage in small and long junctions, the linewidth of the resonant single-fluxon radiation in a long junction is given by the same Eq. (1), except for a missing factor of 4 due to the modified Josephson relation f = V/2Φ0 at the ZFS. Experiment [3] showed a reasonable agreement with that theory, though some excess linewidth broadening has been seen. For the multi-fluxon state which forms FFS it can be argued that internal degrees of freedom in the moving fluxon chain may yield a significant contribution in δf . Here, in contrast to the resonant single-fluxon case, local variations of the fluxon spacing dfl change the radiation frequency. Recent FFS radiation linewidth measurements by Koshelets et al. [4] showed the scaling of δfFF as predicted by Eq. (1) but with an effective temperature Teff being by factor of 8 larger than the physical temperature of their experiment. Since there was no theory for the radiation linewidth δf in the flux-flow regime, the reason for the excess noise was not resolved. In this paper we present a theoretical model for the radiation linewidth in the fluxflow regime for long Josephson junctions. We start from the perturbed sine-Gordon model with the spatially and temporarily dependent noise current. Using the collective coordinate approach we calculate the finite radiation linewidth due to the internal degrees of freedom in the moving fluxon chain. The obtained analytical result is evaluated for the relevant parameter range of experimentally studied flux-flow oscillators. First, we introduce main characteristic quantities for the system. Mean frequency of Josephson oscillations for a long junction in the flux-flow regime is 〈f〉 = v 〈H〉Λ Φ0 ≡ β 〈H〉 , (2) where v is the velocity of the fluxon chain, 〈H〉 is the average magnetic field, Λ = 2λL + t is the effective magnetic thickness of the junction (λL is the London penetration depth, and t is the insulator thickness). The quantities essential for the linewidth problem are the mean square deviation of the frequency f