Diagrammatic approach to multiphoton scattering

We present a method to systematically study multiphoton transmission in one-dimensional systems comprised of correlated quantum emitters coupled to input and output waveguides. Within the Green's function approach of the scattering matrix $(S$ matrix), we develop a diagrammatic technique to analytically obtain the system's scattering amplitudes while at the same time visualize all the possible absorption and emission processes. Our method helps to reduce the significant effort in finding the general response of a many-body bosonic system, particularly the nonlinear response embedded in the Green's functions. We demonstrate our proposal through physically relevant examples involving scattering of multiphoton states from two-level emitters as well as from arrays of correlated Kerr nonlinear resonators in the Bose-Hubbard model.

Figure 1

Two photonic waveguides coupled to a many-body system with Hamiltonian, $H_{\text{sys}}$ that is described by the operators $a_1,\cdots ,a_N$. The left and right waveguides are bilinearly coupled to $a_1$ and $a_N$, respectively.

Figure 2

Diagram for two-point Green's function.

Figure 3

Examples of all possible diagrams for (a) four-point Green's function and (b) six-point Green's function. Note that some of the diagrams drawn here are simply the disconnected products of the lower-order diagrams. Generally, one can construct all diagrams for higher-order Green's functions by building upon the lower-order diagrams and drawing purely connected ones.

Figure 4

Scattering photons from a two-level system.

Figure 5

Only possible scattering diagram of the $2m\text{−}\mathrm{point}$ Green's functions for a two-level quantum emitter being probed by $m$ photons.

Figure 6

Schematic diagram of two collocated (ignore the distance in the sketch) two-level atoms coupled to a waveguide.

Figure 7

Two collocated atoms. Plots of $|G(p_1,p_2;k_1,k_2){|}^2$ where (i) ${\omega }_d^2>{\gamma }_c^2/4$ with ${\omega }_d=1,2$, and 3 [figures (a)–(c)] and (ii) ${\omega }_d^2<{\gamma }_c^2/4$ with ${\omega }_d=0,0.05$, and 0.1 [figures (d)–(f)]. For all the plots, ${\omega }_c=12,{\gamma }_c=0.25$, and the total incoming momentum, $E_i=2{\omega }_c+3{\gamma }_c$. We define $\mathrm{\Delta }k\equiv k_1-k_2$ and $\mathrm{\Delta }p\equiv p_1-p_2$. In figures (a)–(c), under (i), the spectra exhibit eight peaks with the same width, while in figures (d)–(f), under (ii), the spectra exhibit only four peaks with the width becoming smaller as the detuning ${\omega }_d$ decreases except at zero detuning, where the width is much wider.

Figure 8

Sketch of a one-dimensional nonlinear cavity array coupled to input and output waveguides.

Figure 9

Bose-Hubbard model with $N=2$. Plots of $|G_{RR;LL}{|}^2$ (incoming photon channel labels $LL$ are dropped) of Bose-Hubbard model with $N=2$. $U=0,0.1,4,10,20$, and 200, respectively, above. $\mathrm{\Delta }k\equiv k_1-k_2$ and $\mathrm{\Delta }p\equiv p_1-p_2$. ${\omega }_0=100,\gamma =0.25$, and the total incoming momentum, $k_1+k_2=2{\omega }_0+\frac{U}{2}-\sqrt{4J^2+\frac{U^2}{4}}$, all in units of $J$.

Figure 10

Positions of the peaks on the $\mathrm{\Delta }k-\mathrm{\Delta }p$ plane as determined by Eq. (18). The different colors denote different paths: red represents paths that go through $|{\epsilon }_{-}^{\left(1\right)}\rangle$ twice, blue represents paths that go through both $|{\epsilon }_{\pm }^{\left(1\right)}\rangle$, and green represents paths that go through $|{\epsilon }_{+}^{\left(1\right)}\rangle$ twice.

Figure 11

Bose-Hubbard model. Plots of $|G_{RR;LL}{|}^2$ (incoming photon channel labels $LL$ are dropped) of the Bose-Hubbard model with the open boundary condition. $N=2,5,10$ and $U=4,10,200$ are shown. $\mathrm{\Delta }k\equiv k_1-k_2$ and $\mathrm{\Delta }p\equiv p_1-p_2$. ${\omega }_0=100,\gamma =0.25$, and the total incoming momentum, $k_1+k_2=\text{highest}\text{doubly}\text{excited}\text{eigenvalue}$, all in units of $J$. The schematic diagram at the top right corner depicts the scaling of the level spacings.

Figure 12

Bose-Hubbard model with $N=5$. Plots of $|G_{RR;LL}{|}^2$ (incoming photon channel labels $LL$ are dropped) of the Bose-Hubbard model with the open boundary condition. $N=5$ and $U=0.1,4,10$, and 200, respectively. $\mathrm{\Delta }k\equiv k_1-k_2$ and $\mathrm{\Delta }p\equiv p_1-p_2$. ${\omega }_0=100,\gamma =0.25$, and the total incoming momentum, $k_1+k_2=\text{highest}\text{doubly}\text{excited}\text{eigenvalue}$, all in units of $J$.

Figure 13

Cluster decomposition structure of the one-, two-, and three-photon $S$ matrix. The $n$-photon $S$ matrix is made up of $2m\text{−}\mathrm{point}$ connected Green's functions, for $m⩽n$. The $\delta$ function in the one-photon $S$ matrix is present only if the input from the corresponding waveguide is nonzero.