Efficient on-chip source of microwave photon pairs in superconducting circuit QED

We describe a scheme for the efficient generation of microwave photon pairs by parametric down-conversion in a superconducting transmission line resonator coupled to a Cooper-pair box serving as an artificial atom. By properly tuning the first three levels with respect to the cavity modes, the down-conversion probability may reach the percentage level at good fidelity. We show this by numerically simulating the dissipative quantum dynamics of the coupled cavity-box system and discussing the effects of dephasing and relaxation in the solid state environment. The setup analyzed here might form the basis for a future on-chip source of entangled microwave photons, e.g., using Franson’s idea of energy-time entanglement.

Figure 1

(Color online) Schematic setup for the proposed parametric down-conversion (PDC) experiment in superconducting circuit cavity electrodynamics, with a Cooper-pair box (CPB) interacting with the three lowest modes of a transmission line resonator, whose voltage distributions are shown.

Figure 2

(Color online) (a) Excitation energies (transition frequencies) of the CPB, measured with respect to the ground state energy ${ϵ}_0\equiv 0$ as a function of gate charge $N_G$. The energies are plotted in units of $E_C$ for $E_J∕E_C=3$. At particular gate values (away from the “qubit” regime near $N_G=1∕2$), the CPB transition frequencies are related in an integer ratio, $({ϵ}_2-{ϵ}_1)∕{ϵ}_1=1:1$ or $2:1$, respectively. This enables us to match them with the cavity modes, giving rise to particularly efficient degenerate or nondegenerate PDC ($2\omega \mapsto \omega +\omega$ or $3\omega \mapsto \omega +2\omega$, respectively). (b) Simplified transition scheme for the nondegenerate PDC process considered in the text, with a detuning $\Delta$ of the intermediate state $\mid 1,2\omega ⟩$: ${ϵ}_1=\omega +\Delta$. In this plot, only the energy contribution of the CPB is indicated in terms of the vertical position of the levels. The incoming and outgoing photons are displayed as well. (c) Alternative illustration of the same scheme: Here, the level position indicates the full energy of the $\mathrm{CPB}+\text{cavity}$ system, and the coupling strengths governing the individual transitions are shown (see main text).

Figure 3

(Color online) Parametric down-conversion probability $P_{\mathrm{PDC}}=P\left(\mid \omega ,2\omega ⟩\right){\kappa }^2∕{\mid \alpha \mid }^2$ as a function of the microwave input frequency ${\omega }_{\mathrm{in}}$ and the detuning of the intermediate state. The dashed line indicates the analytical resonance condition (see main text), the full line denotes the location of minimal $Q_1$ (high fidelity of PDC), and the dotted line is the cross section shown in Fig. 4.

Figure 4

(Color online) Top: The parametric down-conversion probability $P_{\mathrm{PDC}}$ as a function of input frequency along the cross section indicated in Fig. 3, and the probabilities $P_{\omega }=fP\left(\mid \omega ⟩\right)$ and $P_{2\omega }=fP\left(\mid 2\omega ⟩\right)$ for one or two photons in the cavity, rescaled in the same manner, with $f={\kappa }^2∕{\mid \alpha \mid }^2$ (ideally $P_{\mathrm{PDC}}=P_{\omega }=P_{2\omega }$). The probability of an excited qubit, $P_2=fP\left(\mid 2⟩\right)$, is also shown. Bottom: The nonideality parameters $Q_1=\gamma P_{\mid 1,2\omega ⟩}∕\left(\kappa P_{\mid \omega ,2\omega ⟩}\right)$ and $Q_2=P_{\mid \omega ⟩}∕P_{\mid \omega ,2\omega ⟩}-1$, reaching a minimum at the middle peak (thin line: for half the dephasing rate of ${\gamma }_{\phi }∕\omega =2\times {10}^{-4}$).

Figure 5

(Color online) (a) Parametric down-conversion probability $P_{\mathrm{PDC}}$ and (b) the nonideality parameter $Q_2$ as a function of the detunings between the Cooper-pair box energy levels ${ϵ}_{1,2}$ and the cavity modes (controlled by $E_J$ and $N_G$). At each point, the microwave input frequency ${\omega }_{\mathrm{in}}$ has been chosen to minimize the parameter $Q_1$ (see Fig. 4). The dotted line in (a) indicates the fixed value of ${ϵ}_2$ for which Fig. 3 has been plotted (where $P_{\mathrm{PDC}}$ was plotted as a function of $\Delta$ and ${\omega }_{\mathrm{in}}$). The values of $P_{\mathrm{PDC}}$ plotted here correspond to those obtained along the full yellow line in Fig. 3, which indicated the locus of minimum $Q_1$ (high quality of emitted photon pairs).

Figure 6

(Color online) Schematic of the setup needed to generate and test energy-time entanglement according to Franson. A time delay $\Delta t$ between the long and short arms of the Mach-Zehnder interferometers is needed, exceeding the decay time ${\kappa }^{-1}$ of the cavity producing the photon pairs.