Antibunched $N$-photon bundles emitted by a Josephson photonic device

We propose an experimentally feasible scheme for generating antibunched $N$-photon bundles by a dc voltage-biased Josephson junction in series with a superconducting microwave resonator and a charge qubit. Each resonant tunneling Cooper pair leads to the excitation of the charge qubit and resonator with $N$ photons simultaneously via the ac Josephson effect. Meanwhile, the charge qubit with strong anharmonicity is utilized to regulate the tunneling behavior of the Cooper pairs; that is, the presence of $N$ photons in the resonator prevents the next tunnel event. So the resonator contains only $N$ photons or none, and all other possibilities are greatly suppressed. Combined with the system's dissipation, the resonator can emit its energy in antibunched bundles of $N$ strongly correlated photons with high purity and an in situ tunable emission rate. Such a nonclassical source could be useful in applications in the field of quantum information science.

Figure 1

Schematic of the proposed setup. A dc-SQUID composed of two identical Josephson junctions is coupled to both an $LC$ resonator and a charge qubit via the biased voltage $V$.

Figure 2

(a) Bare energy structure of the qubit and resonator, where the red arrow indicates a super-Rabi oscillation between $|0,g\rangle$ and $|N,e\rangle$ states. (b) Dynamics of the state populations $P_{0g}$ and $P_{2e}$ for the two-photon resonance; that is, the red solid and green dash-dotted curves are simulated with the original Hamiltonian $H_{\mathrm{I}}$, while the black dashed and blue dotted curves are achieved with the effective Hamiltonian $H_{\mathrm{eff}}$. The parameters are chosen to be $\omega /2\pi =7$ GHz, $\delta /2\pi =5$ GHz, $E_J/2\pi =0.7$ GHz, and $\lambda =0.2$.

Figure 3

A tiny fraction of one quantum trajectory showing a complete period of two-photon emission, where the red triangles denote the successive one-photon emissions and the blue one indicates the flip of the qubit. The damping rates are $\kappa /2\pi =0.1$ GHz and $\gamma =\kappa /20$. The other parameters are chosen to be the same as those in Fig. 2.

Figure 4

(a) Zero-delay $n\mathrm{th}$-order photon correlation functions $g^{\left(n\right)}$ versus $\mathrm{\Delta }/\omega$, where $\mathrm{\Delta }={\omega }_J-\delta$. (b) The standard second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ and the generalized second-order correlation function $g_2^{\left(2\right)}\left(\tau \right)$ versus $\kappa \tau$ for the two-photon resonance. The parameters are chosen to be the same as those in Fig. 3.

Figure 5

Purity and rate of the two-photon emission. (a) ${\pi }_2$ and (b) $S_2$ versus the imperfect tuning $\mathrm{\Delta }V$ and the qubit's decay rate $\gamma$ for a fixed cavity decay rate $\kappa /2\pi =0.1$ GHz. The other parameters are chosen to be the same as those in Fig. 2.

Figure 6

A series of emission events extracted from the Monte Carlo simulations over 40 quantum trajectories for the two-photon resonance. The single-, two-, and four-photon emissions are denoted by green stars, red circles, and the blue diamond, respectively. The parameters are chosen to be the same as those in Fig. 3.

Figure 7

The different $N$-photon coupling rates $|g_0^N|$ versus the cavity's zero-point fluctuation $\lambda$, where we choose the parameter $E_J/2\pi =0.7$ GHz.

Figure 8

(a) and (b) Dynamics of the state populations $P_{0g}$ and $P_{Ne}$ for the three- and four-photon resonances without regard to dissipation; that is, the red solid and green dash-dotted curves are simulated with the effective Hamiltonian $H_{\mathrm{eff}}$, while the black dashed and blue dotted curves are achieved with the original Hamiltonian $H_{\mathrm{I}}$. (c) and (d) Purities and rates of the three- and four-photon emissions. The parameters are chosen to be the same as those in Fig. 5.