The influence of magnetic vortices motion on the inverse ac Josephson effect in asymmetric arrays

We report on the influence a preferential magnetic vortices motion has on the magnitude of the inverse ac Josephson effect (the appearance of dc current Shapiro steps) and the coherent operation of asymmetrical parallel arrays of YBaCuO Josephson junctions (JJ) irradiated with microwave (MW) radiation in the presence of an applied magnetic field B. The preferential direction of motion of the Josephson vortices is due to the asymmetry-induced ratchet effect and has a dramatic impact: for a particular positive dc bias current I when the flux-flow is robust multiple pronounced Shapiro-steps are observed consistent with a coherent operation of the array. This suggests an efficient emission/detection of MW in related applications. In contrast, when we reverse the direction of I, the flux-flow is reduced and the Shapiro-steps are strongly suppressed due to a highly incoherent operation that suggests an inefficient emission/detection of MW. Remarkably, by changing B slightly, the situation is reversed: Shapiro steps are now suppressed for a positive I, while well pronounced for a reverse current -I. Our results suggest that a preferential vortex-flow has a very significant impact on the coherent MW operation of superconducting devices consisting of either multiple JJs or a single long JJ asymmetrically biased. This is particular relevant in the case of flux-flow oscillators for sub-terahertz integrated-receivers, flux-driven Josephson (travelling-wave) parametric amplifiers, or on-chip superconducting MW generators which usually operate at bias currents in the Shapiro step region.

2 oscillators for sub-terahertz integrated-receivers, flux-driven Josephson (travelling-wave) parametric amplifiers, or on-chip superconducting MW generators which usually operate at bias currents in the Shapiro step region.
Achieving coherent microwave (MW) operation of Josephson junctions (JJ)-based devices operating in the presence of an applied magnetic field B is essential in many applications such as MW generators/detectors 1-2 , Josephson flux-flow-oscillators (FFO) currently used as sub-terahertz integrated-receivers in radio-astronomical research or atmospheric science projects 3,4 or flux-driven Josephson (travelling-wave) parametric amplifiers [5][6][7][8][9][10][11] used in MW generation/detection [6][7][8] or read-out of a flux-qubit 9 . In all these applications, such JJ-based devices are dc current biased and usually operate in an environment where they are simultaneously exposed to both a (remnant) magnetic field B field and microwave (MW) radiation. The exposure to MW can be due to an external source or due to internally generated radiation via the ac Josephson effect. Depending on applications the presence of either B or MW can be essential for, or detrimental to, their operation. Such JJbased devices are inherently asymmetrical to some degree (the asymmetry can be structural or current bias) and therefore the Lorentz force induces a preferential Josephson vortex flow or intermittent vortex motion. On the other hand, an applied MW induces Shapiro steps on the dc current-voltage characteristics. From this perspective, understanding how the Lorentz forcedriven asymmetric flux motion impacts on either 1) the height of MW induced Shapiro resonances or 2) on the array's coherent operation is essential. In the first case it will help minimizing the unwanted interference of Shapiro resonances on the performance of high frequency operation of superconducting devices and their noise spectral density. In the second case, it would be beneficial to our understanding of reaching coherent operation for superconducting MW generators/receivers and FFOs. Here, we address both these cases as explained in the following. The applied magnetic field B penetrates such devices as an ensemble of magnetic flux quanta , known as Josephson vortices whose dynamics are very sensitive to applied direct I or/and alternating currents Iac (and voltages) originating from MW exposure. These devices being inherently asymmetrical to some degree, operate in a non-zero voltage state where I produces a Lorentz force that induces a preferential Josephson vortex flow. Thus, the lattice of vortices is moving with a speed (proportional to the measured dc voltage V) whose magnitude depends on the value of I: positive or negative. This phenomenon is called a B-field induced magnetic Josephson vortex ratchet effect [12][13][14][15][16][17][18][19][20] . Josephson ratchets based on very large asymmetric JJ-arrays have showed remarkable features such as an ability to amplify the self-induced electromagnetic radiation 19 . Another fundamental phenomenon in JJ-based superconducting devices is the so-called inverse ac Josephson effect, i.e., the appearance of resonant dc Shapiro steps 21 on the dc current-voltage characteristics (IVC) in the presence of an applied MW radiation. The Shapiro steps appear at multiple voltages of the ac Josephson effect relation f=nV/0, with n=1, 2,3,… Surprisingly, the influence of a preferential flux-flow on the strength of the inverse ac Josephson effect, i.e., on the magnitude the Shapiro steps in asymmetrical arrays or on their coherent operation have never been investigated. This is important to fully understand the physics behind the response, coherent operation, and The JJ-arrays were fabricated by depositing high-quality epitaxial, 100 nm thick c-axis     Fig.3c). Interestingly, this behaviour is qualitatively similar to that observed for single Josephson junctions [21].
To However, there is a significant difference in that we consider not only the applied current and the external magnetic field, but also take into account the field gradient trapped in neighbouring holes proportional to the electrical current in the junction connecting these holes. The field gradient and self-induction was ignored in 34 , which is key for our analysis of asymmetric 4a. In contrast (see Fig. 1b in supplementary material) for a negative bias current of I=-0.67 there is a low degree of coherence among the array with no junctions performing in-phase oscillations, leading to a relatively suppressed first Shapiro step in Fig. 4a. Interestingly, there is a strong correlation between the degree of dynamic coherence in the oscillations within the 10 JJ-array and the corresponding static flux configuration in the 9 holes at the I values corresponding to the Shapiro steps. Thus, for values of I around 1.05 when the first Shapiro step is well pronounced (see Fig.4a), the B field configuration is well structured with significant differences in the values of neighbouring loops for most loops in the array (see Fig. 1c in the supplementary material). Alternatively, we may say that in this case the JJ-array is significantly polarized, with strong circulating supercurrents around most loops to account for the rather large differences in the B values. This is in high contrast to the case when I take values around -0.67 and the first Shapiro step is strongly suppressed: in this case the B field configuration is

Supplementary Material
See the supplementary material for the details related to JJ-array fabrication and numerical simulations.
Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request.