Scattering of Two Photons from Two Distant Qubits: Exact Solution

We consider the inelastic scattering of two photons from two qubits separated by an arbitrary distance $R$ and coupled to a one-dimensional transmission line. We present an exact, analytical solution to the problem, and use it to explore a particular configuration of qubits that is transparent to single-photon scattering, thus highlighting non-Markovian effects of inelastic two-photon scattering: strong two-photon interference and momentum dependent photon (anti)bunching. This latter effect can be seen as an inelastic generalization of the Hong-Ou-Mandel effect.

Figure 1

Top: schematic of the scattering setup considered in this Letter. Two counterpropagating photons at equal momenta are scattered from an antisymmetrically detuned configuration of qubits separated by a distance $R$. Bottom: cartoon of a possible experimental realization of two flux qubits coupled capacitively to the waveguide and two circulators directing the incoming (blue) photons to the qubits and scattered (red) photons to the detectors.

Figure 2

Diagrammatic representation of the closed set of equations for the two-photon part $T^{(2)}$ of the scattering matrix, full two-photon vertex $W=W^{(i)}+W^{(r)}$, full two-qubit Green’s function $G(E)$, irreducible two-photon vertex $W^{(i)}=W_{s^{\prime }s}^{(i)}(E)a_{s^{\prime }}^{†}{a}_s$, effective single-photon vertex $V={\overline{V}}_s^{†}(E)a_s+V_{s^{\prime }}(E)a_{s^{\prime }}^{†}$, and qubit self-energy $\mathrm{\Sigma }(E)$. Small black circles depict bare vertices with incoming (outgoing) wavy lines corresponding to the absorption (emission) of a photon; the undirected photon lines include both processes, $v=v_s^{†}{a}_s+v_{s^{\prime }}{a}_{s^{\prime }}^{†}$. Horizontal lines depict the evolution of the qubit states.

Figure 3

(a) Probability density as a function of the difference in the outgoing momenta ${\mathrm{\Delta }}^{\prime }$ for the scattering of two counterpropagating photons (LR) at equal momenta ($\mathrm{\Delta }=0$) to the same (LL) or opposite (LR) directions, and (b) to the same direction for several values of the ratio ${\mathrm{\Gamma }}_2/{\mathrm{\Gamma }}_1$.

Figure 4

Second order correlation function for two photons scattered to the same (autocorrelation) or opposite directions (cross-correlation). Initial state consists of two counterpropagating photons at equal momenta. All parameters are the same as in Fig. 3. Inset shows the structure of the first antibunching dip, together with the same dip for $\mathrm{\Gamma }R=1$, $\mathrm{\Gamma }\tau \in [0,2]$ (dashed).