Universal Quantum Fluctuations of a Cavity Mode Driven by a Josephson Junction

We analyze the quantum dynamics of a superconducting cavity coupled to a voltage-biased Josephson junction. The cavity is strongly excited at resonances where the voltage energy lost by a Cooper pair traversing the circuit is a multiple of the cavity photon energy. We find that the resonances are accompanied by substantial squeezing of the quantum fluctuations of the cavity over a broad range of parameters and are able to identify regimes where the fluctuations in the system take on universal values.

Figure 1

(a) Schematic diagram and (b) effective circuit model of the system. It consists of a SQUID containing two Josephson junctions (JJs) in series with a microwave cavity, modeled as a set of $LC$ oscillators, across which a voltage $V$ is applied. The SQUID acts as a single effective junction whose Josephson energy can be tuned by application of a flux, $\Phi$. Coupling of the cavity to a transmission line [shown in (a)] leads to dissipation. High-$Q$ modes have voltage nodes at the point where the bias voltage line joins the center conductor, one such mode shape is indicated in (a). Definitions of the effective circuit parameters are given in 26.

Figure 2

Average energy $⟨n⟩$ calculated numerically for the full quantum problem (full curves) compared with stable fixed point amplitudes (dashed curves) for the $p=1$, 2 resonances. The stable fixed point changes from $A_{\mathrm{I}}$ to $A_{\mathrm{II}}$ at $E_J=0.405$ (0.666) for $p=1(2)$ and the threshold where $A=0$ becomes unstable for $p=2$ is $E_J^c=0.278$. The inset is a magnified part of the $p=2$ curves. Adopting units where $\hslash {\omega }_0=1$, we take $\hslash \gamma ={10}^{-3}$, ${\Delta }_0=0.06$, ${\delta }^{(1)}={\delta }^{(2)}=0$, values which are used throughout.

Figure 3

Fluctuations in the energy $F$ (lower full curve) and quadrature $\Delta X_{\varphi =0}$ (upper full curve) calculated numerically as a function of $E_J$ for $p=1$. Analytic results for the fixed points $A_{\mathrm{I}}^{(1)}$ (for $E_J<0.405$) and $A_{\mathrm{II}}^{(1)}$ (for $E_J\ge 0.405$) are shown as dashed curves. The horizontal line is the value of $F$ given by Eq. (14).

Figure 4

Fluctuations at the $p=2$ resonance. The main plot shows $F$, numerics (full curve) are compared with analytic results when $A_0>0$: the dashed curve is for the $A_{\mathrm{I}}$ fixed points and the horizontal line, given by Eq. (14), describes the $A_{\mathrm{II}}$ fixed points. Threshold is $E_J^c=0.278$ and the bifurcation between $A_{\mathrm{I}}$ and $A_{\mathrm{II}}$ fixed points is at $E_J=0.666$. The inset compares analytic (dashed curves) and numerical (full curves) calculations of $\Delta X_{\varphi =\pm \pi /4}$ below threshold.