Josephson photonics with a two-mode superconducting circuit

We analyze the quantum dynamics of two electromagnetic oscillators coupled in series to a voltage-biased Josephson junction. When the applied voltage leads to a Josephson frequency across the junction which matches the sum of the two mode frequencies, tunneling Cooper pairs excite photons in both modes simultaneously leading to far-from-equilibrium states. These states display highly nonclassical features including strong antibunching, violation of Cauchy-Schwartz inequalities, and number squeezing. We obtain approximate analytic results for both the regimes of low and high photon occupancies which are supported by a full numerical treatment. The impact of asymmetries between the two modes is explored, revealing a pronounced enhancement of number squeezing when the modes are damped at different rates.

Figure 1

Effective circuit model of the system. It consists of a Josephson junction (JJ) in series with two $LC$ oscillators, across which a voltage $V$ is applied. The two $LC$ oscillators are assumed to have different angular frequencies ${\omega }_a={\left(L_a{C}_a\right)}^{-1/2}\ne {\omega }_b={\left(L_b{C}_b\right)}^{-1/2}$.

Figure 2

Autocorrelations $g_{aa\left(bb\right)}^{\left(2\right)}\left(0\right)$ (left) and cross-correlations $g_{ab}^{\left(2\right)}\left(0\right)$ (right) of the two modes vary with the strength of zero-point fluctuations ${\Delta }_{a\left(b\right)}$ in the two oscillators. For weak driving, $E_J=0.2E_J^c$, the autocorrelations (symbols) are given by (12) (lines) when ${\Delta }_a$, or simultaneously ${\Delta }_a$ and ${\Delta }_b$ are tuned. The reduced cross-correlations $g_{ab}^{\left(2\right)}\left(0\right)-1/\left(2n\right)$ (lines) obey the general relation (11) with the mean of the autocorrelations $[g_{aa}^{\left(2\right)}\left(0\right)+g_{bb}^{\left(2\right)}\left(0\right)]/2$ depicted as symbols for the case of symmetric damping $r=1$.

Figure 3

(a) Average occupation, $\langle n\rangle =n_a=n_b$, (b) Fano factor, $F=F_a=F_b$, as a function of $E_J/E_J^c$ for symmetric oscillators. The full curves are the linearized results and the other curves are for $\Delta =0.1$ (dashed curves), $\Delta =0.3$ (dotted curves), and $\Delta =0.6$ (dash-dotted curves).

Figure 4

Comparison of the oscillator occupation numbers (a) and Fano factor (b) obtained from numerical solution of the quantum master equation for $\Delta =0.1$, 0.3, and 0.6 with corresponding semiclassical calculations over the range $E_J^c<E_J<E_J^{c2}=2.5E_J^c$. In (a) both the semiclassical oscillator energy, $A_0^2$, and occupation number, $n_a$, are scaled by $4{\Delta }^2$.

Figure 5

Steady-state occupations $n_a$ and $n_b$ (full lines) compared with the classical values of $A_0^2$ and $B_0^2$ at the stable fixed points (dashed lines) for (a) $\Delta =0.3$, (b) $\Delta =0.6$. Results are shown for $r=1,1/2$, and $r=1/3$ in each case. Note that the semiclassical amplitudes are zero for $E_J<E_J^c$. The above-threshold bifurcation occurs at $E_J^{c2}/E_J^c=2.5$, 4.0, and 8.7 for $r=1,1/2$, and $r=1/3$, respectively.

Figure 6

Steady-state Fano factors of the modes (a) calculated semiclassically (small-$\Delta$ limit) and calculated numerically using the master equation for (b) $\Delta =0.3$ and (c) $\Delta =0.6$. In each case results are shown for $r=1$ (full lines), $r=1/2$ (dashed lines), and $r=1/3$ (dotted lines). For $r=1/2$ and $r=1/3$ the upper curves are for oscillator $B$ and the lower ones for oscillator $A$. The bifurcation occurs at $E_J^{c2}/E_J^c=2.5$, 4.0, and 8.7 for $r=1,1/2$, and $r=1/3$, respectively. Note that the semiclassical results in (a) are for $E_J^c<E_J<E_J^{c2}$ while (b) and (c) cover a broader range of $E_J$ values.

Figure 7

Wigner function of oscillator A for $r=1/3,E_J/E_J^c=6$, and (a) $\Delta =0.3$, (b) $\Delta =0.6$. Negative regions are apparent in both cases, though more strongly in (b). The Fano factors associated with the states are $F_a=0.19$ (a) and $F_a=0.22$ (b).

Figure 8

Steady-state occupation numbers $n$ (a) and Fano factors $F$ (b) of the oscillators for ${\Delta }_a=0.4,{\Delta }_b=0.2$ with $r=1$ and $1/2$. In (a) the semiclassical predictions are shown as a dotted line and the numerical results as a full line in each case.