Quantum dynamics of a Josephson junction driven cavity mode system in the presence of voltage bias noise

We give a semiclassical analysis of the average photon number as well as photon number variance (Fano factor $F)$ for a Josephson junction (JJ) embedded microwave cavity system, where the JJ is subject to a fluctuating (i.e., noisy) bias voltage with finite dc average. Through the ac Josephson effect, the dc voltage bias drives the effectively nonlinear microwave cavity mode into an amplitude squeezed state $\left(F<1\right)$, as has been established previously [Armour et al., Phys. Rev. Lett. 111, 247001 (2013)], but bias noise acts to degrade this squeezing. We find that the sensitivity of the Fano factor to bias voltage noise depends qualitatively on which stable fixed point regime the system is in for the corresponding classical nonlinear steady-state dynamics. Furthermore, we show that the impact of voltage bias noise is most significant when the cavity is excited to states with large average photon number.

Figure 1

Schematic diagram of a $\lambda /2$ microwave cavity-embedded JJ device showing dc bias circuitry (see the main text for details). The center conductor of the cavity is weakly coupled via a capacitor to a probe transmission line at its right-hand end. Note that the upper ground plane of the cavity has been omitted for clarity.

Figure 2

Model lumped element cavity mode JJ circuit with the various phase coordinate definitions.

Figure 3

Average photon number $\langle N\rangle$ versus $E_J$ (normalized by $E_{J\mathrm{crit}})$ for different bias noise strengths $\kappa =0$ (solid line), $\kappa =0.01$ (dashed line), and $\kappa =0.05$ (dotted line). The curves are terminated in the neighborhood of the bifurcation points where the semiclassical validity criterion $\langle \delta b^{†}\delta b\rangle /B^2\le 0.1$ is violated.

Figure 4

Fano factor $F$ versus $E_J$ (normalized by $E_{J\mathrm{crit}})$ for different bias noise strengths $\kappa =0$ (solid line), $\kappa =0.001$ (dashed line), and $\kappa =0.002$ (dotted line). The inset shows a zoom-in of the Fano factor versus $E_J$ for a small interval in the type-I stable fixed-point regime.

Figure 5

Fano factor $F$ versus bias noise strength $\kappa$ for different $E_J$ (normalized by $E_{J\mathrm{crit}}^{\left(0\right)})$ with ${\mathrm{\Delta }}_0=0.06$. (a) Type-$\mathrm{I}$ regime for $E_J=0.30$ (solid line) and $E_J=0.60$ (dashed line). (b) Type-$\mathrm{II}$ regime for $E_J=1.50$ (solid line) and $E_J=2.00$ (dashed line).

Figure 6

Fano factor $F$ versus bias noise strength $\kappa$ for different $E_J$ (normalized by $E_{J\mathrm{crit}}^{\left(0\right)})$ with ${\mathrm{\Delta }}_0=0.12$. (a) Type-$\mathrm{I}$ regime for $E_J=0.30$ (solid line) and $E_J=0.60$ (dashed line). (b) Type-$\mathrm{II}$ regime for $E_J=1.50$ (solid line) and $E_J=2.00$ (dashed line).