Time-resolved statistics of nonclassical light in Josephson photonics

The interplay of the tunneling transfer of charges and the emission and absorption of light can be investigated in a setup, where a voltage-biased Josephson junction is connected in series with a microwave cavity. We focus here on the emission processes of photons and analyze the underlying time-dependent statistics using the second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ and the waiting-time distribution $w\left(\tau \right)$. Both observables highlight the crossover from a coherent light source to a single-photon source. Due to the nonlinearity of the Josephson junction, tunneling Cooper pairs can create a great variety of nonclassical states of light even at weak driving. Analytical results for the weak driving as well as the classical regime are complemented by a numerical treatment for the full nonlinear case. We also address the question of possible relations between $g^{\left(2\right)}\left(\tau \right)$ and $w\left(\tau \right)$ as well as the specific information which is provided by each of them.

Figure 1

(a) The second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ for weak driving $n_0=0.2$. The parameter $\kappa$ allows to tune the system from a harmonic oscillator $\left(\kappa =0\right)$ to a two-level system $\left(\kappa =2\right)$. (b) The second-order correlation function $g^{\left(2\right)}\left(\tau \right)$, and (c) the normalized waiting-time distribution $w\left(\tau \right)$ for the special cases of a harmonic oscillator (red lines) and a two-level system (blue lines). The two-level system is characterized by damped Rabi oscillations, the harmonic oscillator by a constant value [for $g^{\left(2\right)}\left(\tau \right)$], and an exponential decay with the decay rate $\gamma n_0$ [for $w\left(\tau \right)$]. The long-time behavior of $w\left(\tau \right)$ for the two-level system in the weak-driving regime is given by that of the harmonic oscillator. (d) The normalized waiting-time distribution $w\left(\tau \right)$ at $\kappa =1.5$ for various driving strengths $n_0$. In the strong-driving limit, it shows approximately the behavior of an exponential decay similar to the harmonic case, however, superimposed by oscillations.

Figure 2

The second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ in the numerical weak-driving/few-photon limit $n_0={(\sqrt{\kappa }{E}_J^{*}/\hslash \gamma )}^2\to 0$, where it approaches the analytical result given by Eq. (14), for various values of $\kappa$ (with $\kappa =0$ corresponding to the harmonic oscillator and $\kappa =2$ to a two-level system). The nonlinearity for finite $\kappa$ becomes apparent in suppressed (enhanced) transitions to the second-excited state for $\kappa <4(\kappa >4)$, resulting in $g^{\left(2\right)}\left(0\right)={(1-\kappa /2)}^2$. Compare to Fig. 1 for the effects of stronger driving (see main text).

Figure 3

The second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ for strong driving $n_0={18}^2$. At special values of $\kappa$ (dotted lines) certain transition matrix elements vanish, reducing the number of involved cavity states. This yields pronounced (damped) oscillations in $g^{\left(2\right)}\left(\tau \right)$ in a certain $\kappa$ interval around these values. Lines below $\kappa =2$ can be traced back to the first zero at $3.83$ of the Bessel function $J_1$, e.g., $T_{2,3}\propto 〈2|:J_1\left(\sqrt{4\kappa n}\right)/\sqrt{n}:|2〉=0$ for $\kappa =1.27$, the line at $\kappa =3.31$ to the second zero.

Figure 4

The second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ as a function of detuning $\mathrm{\Delta }$ at $\kappa =1$ for weak driving $n_0=0.2$. Oscillations with frequency $f\approx \mathrm{\Delta }/2\pi$ are observable with an amplitude, which increases with increasing detuning. Compare to the analytical result given by Eq. (14) for the effects occurring in the weak-driving limit $n_0={(\sqrt{\kappa }{E}_{\mathrm{J}}^{*}/\hslash \gamma )}^2\to 0$.

Figure 5

In case that renewal theory applies to the system, $g^{\left(2\right)}\left(\tau \right)$ and $w\left(\tau \right)$ provide identical information and the two Fano factors deduced from them are identical: $F^{\text{FCS}}=F^{\text{WTD}}$. The measure $|F^{\text{WTD}}/F^{\text{FCS}}-1|$ (a) highlights the deviations of the system from renewal theory, which vanish at $\kappa =0$ (harmonic oscillator), $\kappa =2$ (two-level system), and $n_0\ll 1$ (weak-driving regime, where nonlinearities are not crucial). The dotted lines refer to two cross sections (b) at weak and strong driving, revealing sub-Poissonian distributions of emitted photons.

Figure 6

The waiting-time distribution (solid line) $w\left(\tau \right)$ for the two-photon resonance in the special case of a parametric amplifier, $\kappa =0.001,n_0^{2\mathrm{ph}}=0.02\ll 1$, which approximately coincides with the analytical result given by Eq. (21). The short- and long-time decays are indicated by the analytical results (dashed lines), corresponding to the probability of detecting the first or second photon of an emitted pair.