Multiple-output microwave single-photon source using superconducting circuits with longitudinal and transverse couplings

Single-photon devices at microwave frequencies are important for applications in quantum information processing and communication in the microwave regime. In this work we describe a proposal of a multioutput single-photon device. We consider two superconducting resonators coupled to a gap-tunable qubit via both its longitudinal and transverse degrees of freedom. Thus, this qubit-resonator coupling differs from the coupling in standard circuit quantum-electrodynamic systems described by the Jaynes-Cummings model. We demonstrate that an effective quadratic coupling between one of the normal modes and the qubit can be induced and this induced second-order nonlinearity is much larger than that for conventional Kerr-type systems exhibiting photon blockade. Assuming that a coupled normal mode is resonantly driven, we observe that the output fields from the resonators exhibit strong sub-Poissonian photon-number statistics and photon antibunching. Contrary to previous studies on resonant photon blockade, the first-excited state of our device is a pure single-photon Fock state rather than a polariton state, i.e., a highly hybridized qubit-photon state. In addition, it is found that the optical state truncation caused by the strong qubit-induced nonlinearity can lead to an entanglement between the two resonators, even in their steady state under the Markov approximation.

Figure 1

(a) Schematic circuit layout and (b) couplings and dissipative channels of our proposal. A gap-tunable qubit (e.g., a flux or charge qubit) couples with two superconducting (e.g., transmission-line) resonators with strengths $G_i$ for $i=1,2$. The eigenfrequencies for the $i\text{th}$ resonator and the qubit are ${\omega }_i$ and ${\omega }_q$, respectively. A capacitor $C$ is used to directly mediate the two resonators and results in a coupling strength $g$. On the left-hand side, two coherent microwave drives, with strengths ${\epsilon }_1$ and ${\epsilon }_2$, are applied to the resonators 1 and 2, respectively. On the right-hand side, the single-photon output microwave photons are collected from ports 1, 2, and 3. Ports 1 and 2 are semi-infinite transmission lines connected to resonators 1 and 2. This results in photon escape rates ${\kappa }_{1,1}$ and ${\kappa }_{2,2}$, respectively. Port 3 is the joint output transmission line from resonators 1 and 2, with photon escape rates ${\kappa }_{1,2}$ and ${\kappa }_{2,1}$. We assume that the input fields $b_{\text{in},i}$ for these three ports are all independent vacuum noises. Besides escaping into the transmission lines, the photons in the two resonators can also dissipate into the environment with intrinsic rates ${\kappa }_{\mathrm{in},\mathrm{i}}$. For the qubit, the decay (dephasing) rate is $\mathrm{\Gamma } \left({\mathrm{\Gamma }}_f\right)$.

Figure 2

Lowest energy levels for the Hamiltonian in Eq. (14). The supermode $A_{+}$ couples with the qubit with quadratic form, while the supermode $A_{-}$ decouples from the qubit. The frequency difference between these two supermodes is ${\mathrm{\Delta }}_2$. When ${\mathrm{\Delta }}_2=0$, these two modes are degenerate. The effective drives for the supermodes $A_{+}$ and $A_{-}$ are ${\epsilon }_{+}$ and ${\epsilon }_{-}$, respectively, as shown in Table 1.

Figure 3

Time evolutions of the probabilities for the system described by $H=H_s$ (shown with solid curves) and $H=H_{\text{eff}}$ (marked with symbols) in the (a) nondissipative and (b) dissipative cases. The initial state is $|0,0\rangle |g\rangle$. (a) Without considering any decay channels, the time evolution of the probabilities $P(0,0)$ and $P\left({\psi }_{1+}\right)$ exhibit the Rabi oscillations between the states $|0,0\rangle$ and $P\left({\psi }_{1+}\right)$. (b) Decay of the probabilities assuming the decoherent rates $\mathrm{\Gamma }={\mathrm{\Gamma }}_f/2={\gamma }_1={\gamma }_2=1$. We find that the sum of $P(0,0)$ and $P\left({\psi }_{1+}\right)$ is almost equal to 1 for all the evolution times. Thus, this sum can be considered as a fidelity measure of optical state truncation resulting in photon blockade. Here we consider that the two modes are degenerate, i.e., ${\mathrm{\Delta }}_2=0$.

Figure 4

(a) Two-resonator photon-number probabilities $P(n_1,n_2)$ and (b) photon-number probabilities $P^{\left(k\right)}$ of the resonator $k=1$ and 2 for the output steady state (i.e., for $t\to \infty$) for the same parameters as in Fig. 3. It can be found that the multiphoton states are hardly excited.

Figure 5

Single-resonator quasiprobability distributions $W^{\left(s\right)}$ with the parameter (a) $s=0$ (corresponding to the Wigner function), (b) $s=1/2$, and (c) $s=0.54$ for the steady-state solutions ${\rho }_{\mathrm{ss}}^{\left(1\right)}={\mathrm{Tr}}_2\left({\rho }_{\mathrm{ss}}\right)$ of the first resonator as a function of its canonical position $\text{Re}\left({\alpha }_1\right)$ and momentum $\text{Im}\left({\alpha }_1\right)$. The corresponding plots for the second resonator for ${\rho }_{\mathrm{ss}}^{\left(2\right)}={\mathrm{Tr}}_1\left({\rho }_{\mathrm{ss}}\right)$ are very similar to these and thus are not presented here. The other parameters used here are the same as those in Fig. 3. The negativity of the QPD shown in (c) clearly reveals the nonclassical character of the state generated via photon blockade. We note that the parameter $s=0.54$ was chosen to be slightly larger than the nonclassical depth $s_0=0.537$ of the state (or more precisely, of the corresponding perfectly truncated qubit state). Thus, the QPD shown in (c) is nonpositive, as indicated by the blue region.

Figure 6

Two-time second-order correlation functions $g_{21}\left(\tau \right)$ (red dashed curve), $g_{22}\left(\tau \right)$ (blue dotted curve), and $g_{23}\left(\tau \right)$ (black solid curve) as functions of the delay time $\tau$ for ports 1, 2, and 3. These three second-order correlation functions are much smaller than 1 and show dips at zero-time delay $\tau =0$. These dips reveal strong sub-Poissonian photon-number statistics, since $g_{2i}(\tau =0)\approx 0$, while the increase of $g_{2i}\left(\tau \right)$ with increasing $\tau$ from $\tau =0$ reveals photon antibunching. The parameters used here are the same as those in Fig. 3.

Figure 7

Average photon escape rate $N_1 \left(N_3\right)$ and the second-order correlation function $g_{21}\left(0\right) \left[g_{23}\left(0\right)\right]$ from port 1 (port 3) versus the drive detuning ${\mathrm{\Delta }}_{+}$ and the phase difference $\theta$ for (a)–(d) the degenerate case $({\mathrm{\Delta }}_2=0)$ and (e)–(h) the nondegenerate case $({\mathrm{\Delta }}_2=10)$. Experiments could adjust the coupling capacity $C$ between the two resonators, to change the frequency separation between the two supermodes. In the plot of the second-order correlation function $g_{2i}\left(0\right)$, the solid black closed loops in (a), (b), (e), and (f) correspond to ${log}_{10}\left[g_{2i}\left(0\right)\right]=-1$ and the points inside the loops indicate that the output fields exhibit a strong sub-Poissonian character. It can be found that, in both degenerate and nondegenerate cases, $g_{21}\left(0\right)$ and $g_{23}\left(0\right)$ can display dips around ${\mathrm{\Delta }}_{+}=0$ and $\theta =0$. The other parameters used here are the same as those in Fig. 3.

Figure 8

Logarithmic negativity $E_c$ versus the coupling ratio $\beta$. Here we set $\theta =0$ and ${\mathrm{\Delta }}_{+}=0$. Other parameters are the same as those of the nondegenerate case in Fig. 7. The vertical dashed line is at $\beta =1$.