Multiple emitters in a waveguide: Nonreciprocity and correlated photons at perfect elastic transmission

We investigate interference and correlation effects when several detuned emitters are placed along a one-dimensional photonic waveguide. Such a setup allows multiple interactions between the photons and the strongly coupled emitters, and underlies proposed devices for quantum information processing. We show, first, that a pair of detuned two-level systems (2LS) separated by a half wavelength mimic a driven $\mathrm{\Lambda }$-type three-level system (3LS) in both the single- and two-photon sectors. There is an interference-induced transparency peak at which the fluorescence is quenched, leaving the transmitted photons completely uncorrelated. Slightly away from this separation, we find that the inelastic scattering (fluorescence) is large, leading to nonlinear effects such as nonreciprocity (rectification). We connect this nonreciprocity to inelastic scattering caused by driving a dark pole and so derive a condition for maximum rectification. Finally, by placing a true 3LS midway between the two 2LS, we show that elastic scattering produces only transmission, but inelastic scattering nevertheless occurs (the fluorescence is not quenched) causing substantial photon correlations.

Figure 1

Schematic for the “2-3-2” structure: a pair of 2LS, separated by distance $L$ and detuned by $\delta$, coupled to a 1D waveguide, with an additional driven $\mathrm{\Lambda }$-type 3LS placed in the middle. In the rotating frame, ${\omega }_s={\omega }_e-\mathrm{\Delta }$ where $\mathrm{\Delta }$ is the detuning of the classical beam. Throughout this paper, we assume that two identical photons with momentum $k$ are incident.

Figure 2

Schematic of the level structure of two detuned 2LS with $k_{0}L=2n\pi$ in the symmetric-antisymmetric basis. The effective $\mathrm{\Lambda }$-type 3LS is formed from the three states $|gg\rangle$, $|S\rangle$, and $|A\rangle$. In the case of two incident photons at the resonant frequency ${\omega }_0$, the doubly excited state $|ee\rangle$ is not populated because of interference similar to that causing EIT.

Figure 3

Pair of detuned 2LS: single-photon transmission spectrum $T\left(k\right)$ (left column) and two-photon inelastic flux $F\left(k\right)$ (right column) as a function of incident momentum $k$ for a separation of ${\lambda }_0/2$ (so $k_{0}L=\pi$). The detuning $\delta /\mathrm{\Gamma }$ is set to 0 (top row), 0.35 (middle row, EIT regime), and 1.5 (bottom row). In the right column, red dotted curves represent transmitted flux $F_{\text{R}}$, and brown solid curves represent total flux $F=F_{\text{R}}+F_{\text{L}}$. Note that (i) $F=0$ at $k={\omega }_0$, (ii) $T=0$ at $k={\omega }_0\pm \delta /2$, and (iii) $T\left({\omega }_0\right)=1$ when $\delta \ne 0$. The inset in panel (d) magnifies the peaks around ${\omega }_0=100\mathrm{\Gamma }$.

Figure 4

The two-photon correlation function $g_2\left(t\right)$ in the transmission channel for (a)–(c) a pair of detuned 2LS and (d) a 2-3-2 structure. The detunings are (a) $\delta /\mathrm{\Gamma }=0$, (b) $\delta /\mathrm{\Gamma }=0.35$, (c) $\delta /\mathrm{\Gamma }=1.5$, and (d) $\delta /\mathrm{\Gamma }=0.35$, in which $\mathrm{\Omega }/\mathrm{\Gamma }=3$ as well. The incident frequency $k$ is chosen for each case so that $T=50\%$ and $k$ is on the red-detuned side of the ${\omega }_0=100\mathrm{\Gamma }$ resonance. [The resonance itself $k={\omega }_0$ is not used because for case (a) $g_2\left(t\right)$ is ill defined because in the elastic channel all photons are reflected, $T\left(k\right)=0$, while for cases (b) and (c), $g_2\left(t\right)=1$ (gray line).] The characteristic time delay $\tau$ for each case is labeled. For panel (a), we use the fact that $\tau =2/\mathrm{\Gamma }$ for an array of identical 2LS [46]. The separation between the two 2LS in all cases is $L={\lambda }_0/2$.

Figure 5

Nonreciprocal effects for a pair of 2LS as a function of incident frequency $k$. (a) Transmitted inelastic flux $F\left(k\right)$ for several qubit detunings $\delta$. The peak position is at ${\omega }_0-{\delta }_{\text{opt}}/2\approx 100.031\mathrm{\Gamma }$. The inset magnifies the peaks. (b) Difference of transmitted flux due to left ($\to$) and right driving ($\leftarrow$) for two $L$ and two $\delta$. $(k_{0}L,\delta /\mathrm{\Gamma })=(0.98\pi ,-0.06)$ (solid green), $(0.98\pi ,-0.12)$ (dashed yellow), $(1.02\pi ,+0.06)$ (dotted-dashed black), and $(1.02\pi ,+0.12)$ (dotted magenta). (c) Unnormalized rectification factor given by the difference in transmitted intensity. The color scheme and parameters are the same as in (b). ${\omega }_0/\mathrm{\Gamma }=100$ in all panels.

Figure 6

For the 2-3-2 structure: single-photon transmission spectrum $T\left(k\right)$ (left column) and two-photon inelastic flux $F\left(k\right)$ (right column) as a function of incident momentum $k$ for 2LS-3LS-2LS separated by ${\lambda }_0/4$ (so $k_{0}L=\pi$). The detuning is $\delta /\mathrm{\Gamma }=0.35$, and the Rabi frequency of the classical beam $\mathrm{\Omega }/\mathrm{\Gamma }$ is 0 (top row) and 3 (bottom row). In the right column, red dotted curves represent transmitted flux $F_R$, and brown solid curves represent total flux $F=F_R+F_L$. Note that (i) $F\ne 0$ at $k={\omega }_0$, (ii) $T=0$ at both $k={\omega }_0\pm \delta /2$ and ${\omega }_0\pm \mathrm{\Omega }/2$, and (iii) $T\left({\omega }_0\right)=1$ when $\mathrm{\Omega }\ne 0$. The inset in panel (d) magnifies the peaks around ${\omega }_0=100\mathrm{\Gamma }$.

Figure 7

For the 2-3-2 structure excited at resonance, (a) the two-photon normalized power spectrum $S\left(\omega \right)/F$ as a function of output frequency $\omega$, and (b) the two-photon correlation function $g_2\left(t\right)$ in the transmission channel. In (a), both the transmitted fluorescence $S_R/F$ (red dotted line) and the total fluorescence $S_R/F+S_L/F$ (brown solid line) are shown. In panel (b), the uncorrelated value $g_2=1$ is labeled by a gray horizontal line, and the characteristic time delay $\tau$ is also labeled (see main text). The inset magnifies $\mathrm{\Gamma }t\in [0,20]$. Parameters used are $k={\omega }_0=100\mathrm{\Gamma }$, $L={\lambda }_0/2$, $\delta /\mathrm{\Gamma }=0.35$, and $\mathrm{\Omega }/\mathrm{\Gamma }=3$ such that $T=100\%$.

Figure 8

Nonreciprocal transmitted inelastic flux $F$ and intensity $X$ as a function of incident frequency $k$ for the 2-3-2 structure. (a) Difference of transmitted flux due to left ($\to$) and right driving ($\leftarrow$). (b) Unnormalized rectification factor given by the difference in transmitted intensity. Note that the value at $k=100\mathrm{\Gamma }$ in both panels is zero. Parameters used are the same as those used in Fig. 6.

Figure 9

Assessing the lossless and Markovian approximations: the single-photon transmission $T\left(k\right)$ as a function of incident frequency $k$ for a pair of detuned 2LS with $\delta /\mathrm{\Gamma }=0.35$. Loss: the difference between the exact lossless (orange dashed, $L={\lambda }_0/2$) and lossy (purple dotted) curves shows that modest loss has a significant effect on the EIT-like resonance: the peak transmission (at $k_0=100\mathrm{\Gamma }$) decreases from $100\%$ to $T\approx 56.9\%$ for ${\mathrm{\Gamma }}^{\prime }/\mathrm{\Gamma }=0.02$. Markovianity: For $L={\lambda }_0/2$ (and lossless), the Markovian result (green solid) is nearly indistinguishable from the exact result. In contrast, for $L=20{\lambda }_0$, the exact result (magenta dotted-dashed) is significantly different from the Markovian curve (which is the same for the two cases).