In a process called parametric down-conversion, a dc-biased Josephson junction coupled to two microwave resonators emits photon pairs when the Josephson frequency matches the sum of the two resonance frequencies. Recent experiments have shown that such a setup permits analyzing the correlation of the radiation. Motivated by these results, we study theoretically the full counting statistics of a non-degenerate parametric oscillator below the threshold of instability. Furthermore, we analyze the second-order coherences of the radiation and discuss thermal effects on the radiation statistics. We provide results for the driving strength at which the Cauchy-Schwarz inequality is most strongly violated. Additionally, we study the impact of asymmetry in the linewidth of the modes-a distinctive property of the non-degenerate resonance effect. In particular, we find that the radiation from the mode with the larger linewidth preferably takes the total detuning, while the other mode emits photons at its resonance frequency. arXiv:1903.11529v2 [cond-mat.mes-hall]

Bloch oscillations in a single Josephson junction in the phase-slip regime relate current to frequency. They can be measured by applying a periodic drive to a DC-biased, small Josephson junction. Phase-locking between the periodic drive and the Bloch oscillations then gives rise to steps at constant current in the I-V curves, also known as dual Shapiro steps. Unlike conventional Shapiro steps, a measurement of these dual Shapiro steps is impeded by the presence of a parasitic capacitance. This capacitance shunts the junction resulting in a suppression of the amplitude of the Bloch oscillations. This detrimental effect of the parasitic capacitance can be remedied by an on-chip superinductance. Additionally, we introduce a large off-chip resistance to provide the necessary dissipation. We investigate the resulting system by a set of analytical and numerical methods. In particular, we obtain an explicit analytical expression for the height of dual Shapiro steps as a function of the ratio of the parasitic capacitance to the superinductance. Using this result, we provide a quantitative estimate of the dual Shapiro step height. Our calculations reveal that even in the presence of a parasitic capacitance, it should be possible to observe Bloch oscillations with realistic experimental parameters.

At a bifurcation point, a small change of a parameter causes a qualitative change in the system. Quantum fluctuations wash out this abrupt transition and enable the emission of quantized energy, which we term photons, below the classical bifurcation threshold. Close to the bifurcation point, the resulting photon counting statistics is determined by the instability. We propose a generic method to derive a characteristic function of photon counting close to a bifurcation threshold that only depends on the dynamics and the type of bifurcation, based on the universality of the Martin-Siggia-Rose action. We provide explicit expressions for the cusp catastrophe without conservation laws. Moreover, we propose an experimental setup using driven Josephson junctions that exhibits both a fold and a pitchfork bifurcation behavior close to a cusp catastrophe.

No abstract

As a function of the driving strength, a degenerate parametric oscillator exhibits an instability at which spontaneous oscillations occur. Close to threshold, both the nonlinearity as well as fluctuations are vital to the accurate description of the dynamics. We study the statistics of the adiation that is emitted by the degenerate parametric oscillator at threshold. For a weak nonlinearity, we can employ a quasiclassical description. We identify a universal Liouvillian that captures the relevant long-time dynamics for large photon-numbers. We find that the cumulants obey a universal power-law scaling as a function of the nonlinearity. The Fano factor shows a maximum close, but not coinciding, with the threshold. Moreover, we predict a certain ratio of the first three cumulants to be independent of the microscopic details of the system and connect the results to experimental platforms.