Photon scattering from a system of multilevel quantum emitters. II. Application to emitters coupled to a one-dimensional waveguide

In a preceding paper we introduced a formalism to study the scattering of low-intensity fields from a system of multilevel emitters embedded in a three-dimensional ($3\mathrm{D}$) dielectric medium. Here we show how this photon-scattering relation can be used to analyze the scattering of single photons and weak coherent states from any generic multilevel quantum emitter coupled to a one-dimensional ($1\mathrm{D}$) waveguide. The reduction of the photon-scattering relation to $1\mathrm{D}$ waveguides provides a direct solution of the scattering problem involving low-intensity fields in the waveguide QED regime. To show how our formalism works, we consider examples of multilevel emitters and evaluate the transmitted and reflected field amplitude. Furthermore, we extend our study to include the dynamical response of the emitters for scattering of a weak coherent photon pulse. As our photon-scattering relation is based on the Heisenberg picture, it is quite useful for problems involving photodetection in the waveguide architecture. We show this by considering a specific problem of state generation by photodetection in a multilevel emitter, where our formalism exhibits its full potential. Since the considered emitters are generic, the $1\mathrm{D}$ results apply to a plethora of physical systems such as atoms, ions, quantum dots, superconducting qubits, and nitrogen-vacancy centers coupled to a $1\mathrm{D}$ waveguide or transmission line.

Figure 1

Schematic of photon scattering from a generic system of emitters coupled to a waveguide. The emitters can be either a simple two-level system with a decay $\mathrm{\Gamma }$ or have multiple levels. These can be separated into two subspaces: an excited-state manifold $M_e$ and a ground-state manifold $M_g$. The couplings between the two manifolds ${\widehat{\mathcal{V}}}_{+}\left({\widehat{\mathcal{V}}}_{-}\right)$ are assumed to be perturbative while the excited states experience decay modeled by the Lindblad operators ${\widehat{\mathcal{L}}}_k$. The couplings within the excited-state and ground-state manifolds are shown by the wiggly and straight arrow-headed lines, respectively. The $1\mathrm{D}$ waveguide supports both forward and backward propagating modes of an input photon represented by the operators $a_f$ and $a_b$, respectively. Furthermore, the symbols $r$ and $t$ represent the reflection and transmission coefficients satisfying the relation ${\left|r\right|}^2+{\left|t\right|}^2=1$. Photons scattered from the emitters can decay to outside modes and into the waveguide with decay rates of ${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}_{1\text{D}}$, respectively.

Figure 2

Schematic diagram of the energy level structure of emitters with (a) single optical transition, (b) two optical transitions in $\mathsf{V}$ configuration. Here $|0\rangle$ is the ground state and $|i=1,2\rangle$ the excited states of the emitter. The linewidth of the excited states is given by $\mathrm{\Gamma }$'s, and the $\delta$'s are detuning of the transition with respect to the frequency of the incoming photon. The coupling strength of the transitions to the waveguide mode is given by $\mathcal{A}$'s.

Figure 3

Transmitted intensity ${\left|\mathcal{T}\right|}^2=|\langle {\widehat{a}}_{\text{out},\text{R}}^{†}{\widehat{a}}_{\text{out},\text{R}}\rangle /\langle {\widehat{a}}_{\text{in}}^{†}{\widehat{a}}_{\text{in}}\rangle |$ for a single (a) two-level emitter and (b) three-level emitter in the $\mathsf{V}$ configuration coupled to a 1D waveguide. For (a) we consider the parameters $\delta ={\omega }_{11}-{\omega }_{00}-\omega$ and different values of $\beta$ while for (b) we consider ${\delta }_1=-\delta , {\delta }_2=\mathrm{\Gamma }-\delta , \beta =0.99$, and coupling $\mathrm{\Omega }=2\mathrm{\Gamma }$ or 0, and we plot the results for $\mathrm{\Delta }\varphi =0$ and $\mathrm{\Delta }\varphi =\pi$.

Figure 4

Two emitters in a waveguide (top) with individual level structures (center), and combined level structure in the single-excitation limit (bottom).

Figure 5

Schematic of light scattering from two generic emitters located at the position ${\text{z}}_A$ and ${\text{z}}_B$ in a double-sided waveguide with a right-going input photon pulse. Here ${\mathcal{T}}_i$ and ${\mathcal{R}}_i$ signify the single-emitter transmitted and reflected amplitudes, respectively. Amplitude for transmitted and reflected light for scattering involving two emitters are on the other hand given by ${\mathcal{T}}_{ij}$ and ${\mathcal{R}}_{ij}$, respectively. The wiggly lines signify field-mediated interactions between the emitters in terms of the non-Hermitian Hamiltonian ${\tilde{\mathcal{H}}}_{\text{nh}}$ as discussed in the text. The wiggly circles with arrows inside symbolize the scattering event.

Figure 6

Transmission ${\left|\mathcal{T}\right|}^2=|\langle {\widehat{a}}_{\text{out},R}^{†}{\widehat{a}}_{\text{out},R}\rangle /\langle {\widehat{a}}_{\text{in}}^{†}{\widehat{a}}_{\text{in}}\rangle |$ for two two-level emitters coupled to a 1D waveguide. The parameters used for the plots are (a) $\beta =1$ and comparing four different values of the phase distance $k\mathrm{\Delta }z$, (b) transmission as a function of the phase distance $k\mathrm{\Delta }z$ for ${\delta }_1={\delta }_2=\delta =0.1\mathrm{\Gamma }$ and $0.3\mathrm{\Gamma }, \beta =0.99$.

Figure 7

Transmission ${\left|\mathcal{T}\right|}^2=|\langle {\widehat{a}}_{\text{out},R}^{†}{\widehat{a}}_{\text{out},R}\rangle /\langle {\widehat{a}}_{\text{in}}^{†}{\widehat{a}}_{\text{in}}\rangle |$ from a two-emitter system. Here we consider a combination of a two-level emitter and a three-level emitter in the $\mathsf{V}$ configuration coupled to a 1D waveguide. The parameters used for the plots are as follows: for (a) ${\delta }_A=-3\mathrm{\Gamma }-\delta , {\delta }_B=-2\mathrm{\Gamma }-\delta , \beta =1, k\mathrm{\Delta }z=2\pi$, and $\mathrm{\Omega }=5$, while for (b) ${\delta }_1=4\mathrm{\Gamma }-\delta , {\delta }_2=-\delta , {\delta }_3=6\mathrm{\Gamma }-\delta , {\mathrm{\Gamma }}_{1,1\text{D}}=0.1\mathrm{\Gamma }, {\mathrm{\Gamma }}_{2,1\text{D}}=\mathrm{\Gamma }, {\mathrm{\Gamma }}_{3,1\text{D}}=3\mathrm{\Gamma }, k\mathrm{\Delta }z=1$, and $\mathrm{\Omega }=2$.

Figure 8

Three-level emitter with a $\mathrm{\Lambda }$-type level structure consisting of two ground states ($|0\rangle , |1\rangle$) and one excited state ($|2\rangle$).

Figure 9

(a) Fidelity of the antisymmetric superposition state $|{\mathrm{\Psi }}^{-}\rangle$ as a function of the detection time $t_c$ normalized with the pulse duration $T$. We plot here for ${\mathrm{\Gamma }}_{0,1\text{D}}={\mathrm{\Gamma }}_{1,1\text{D}}, \delta =0, {\varphi }_z=0, \beta =1, {\omega }_{01}=5\frac{2\pi }{T}$, and an average number of photons $\overline{n}=0.8$. Resolving the detection time determines the phase of the generated state. The detection time has an arbitrary offset determined by the spatial position of the detectors. (b) Fidelity of superposition-state generation as a function of the $\beta$ factor for different values of $\overline{n}$, the average photon number in the coherent pulse.

Figure 10

Contour for evaluating the principal-value integral.