Noise switching at a dynamical critical point in a cavity-conductor hybrid

Coupling a mesoscopic conductor to a microwave cavity can lead to fascinating feedback effects which generate strong correlations between the dynamics of photons and charges. We explore the connection between cavity dynamics and charge transport in a model system consisting of a voltage-biased Josephson junction embedded in a high-$Q$ cavity, focusing on the behavior as the system is tuned through a dynamical critical point. On one side of the critical point the noise is strongly suppressed, signaling the existence of a regime of highly coherent transport, but on the other side it switches abruptly to a much larger value. Using a semiclassical approach we show that this behavior arises because of the strongly nonlinear cavity drive generated by the Cooper pairs. We also uncover an equivalence between charge and photonic current noise in the system which opens up a route to detecting the critical behavior through straightforward microwave measurements.

Figure 1

(a) Average current as a function of $E_J/E_J^B$ for $\delta =0$; dashed lines are from the semiclassical calculation discussed below. (b)–(d) Corresponding evolution of the Wigner function, for $E_J/E_J^B=0.5$, 1.0, and 1.5 with ${\mathrm{\Delta }}_0=0.2$. (e) Behavior of $\langle p\rangle$ as a function of ($\delta ,E_J$) for ${\mathrm{\Delta }}_0=0.1$; the system has a critical point at ($0,E_J^B$) and a line of first-order transitions along $\delta =0$ for $E_J>E_J^B$. Except where stated otherwise, these results are calculated numerically using the full master equation (2) [43].

Figure 2

Current Fano factor $F_{\text{CP}}$ as a function of $E_J/E_J^B$ for a range of ${\mathrm{\Delta }}_0$ values. The inset shows the critical scaling of ${\mathrm{\Delta }}_0^2\langle \delta I^2\rangle$. In each case the solid curves are numerical solutions of the master equation, the dashed curves are the predictions of the semiclassical calculation discussed below, and $\delta =0$.

Figure 3

Behavior of the (a) amplitude $A/A_0$, where $A_0=z_1/2{\mathrm{\Delta }}_0$, and (b) phase $\varphi$ of the classical stable (unstable) fixed points shown as black (red) lines. (c) Classical current function, $\mathcal{I}(\alpha ,{\alpha }^{*})/{\tilde{E}}_J$, overlaid by the locus of fixed points (black line). The maximum in $\mathcal{I}$ coincides with the critical point (red dot). In each case $\delta =0$.

Figure 4

(a) Evolution of the photonic Fano factor, $F_{\mathrm{ph}}\left(\omega \right)$, with $E_J/E_J^B$. Calculated numerically using the full master equation with ${\mathrm{\Delta }}_0=0.2, \delta =0$. (b) Comparison of the locations of the minima in $F_{\mathrm{ph}}\left(\omega \right)$ (red points) with the imaginary parts of the semiclassical eigenvalues ${\mathrm{\Gamma }}_{\pm }$ (black dashes).

Figure 5

Current Fano factor, $F_{\text{CP}}$, as a function of $E_J/E_J^B$ for large quantum fluctuations, ${\mathrm{\Delta }}_0=0.5$ (green), and closer to the semiclassical limit, ${\mathrm{\Delta }}_0=0.2$ (red). Results from the extended quantum master equation incorporating an explicit junction degree of freedom converge to the approach without fluctuations of the bias voltage (solid lines; cf. Fig. 2) for ${\gamma }_J/\gamma \to 0$. The impact of low-frequency voltage noise is seen most strongly when the system is close to the (classical) dynamical critical point, i.e., $E_J\approx E_J^B, {\mathrm{\Delta }}_0\ll 1$. Notably, the exact correspondence of the current Fano factor, $F_{\text{CP}}$, and the photonic Fano factor, $F_{\mathrm{ph}}(\omega =0)$ (symbols) accessed via $g^{\left(2\right)}$ [cf. Eq. (8)], persists for finite ${\gamma }_J$.