Nonreciprocal atomic scattering: A saturable, quantum Yagi-Uda antenna

Recent theoretical studies of a pair of atoms in a one-dimensional waveguide find that the system responds asymmetrically to incident fields from opposing directions at low powers. Since there are no explicit time-reversal symmetry-breaking elements in the device, this has caused some debate. Here we show that the asymmetry arises from the formation of a quasidark state of the two atoms, which saturates at extremely low power. In this case the nonlinear saturability explicitly breaks the assumptions of the Lorentz reciprocity theorem. Moreover, we show that the statistics of the output field from the driven system can be explained by a very simple stochastic mirror model and that at steady state, the two atoms and the local field are driven to an entangled, tripartite $\left|W\right.〉$ state. Because of this, we argue that the device is better understood as a saturable Yagi-Uda antenna, a distributed system of differentially tuned dipoles that couples asymmetrically to external fields.

Figure 1

(a) Schematic illustration of two atoms in a waveguide, driven by a field incident from the left with amplitude $〈a_{\mathrm{in}}〉=\alpha$, or from the right with $〈b_{\mathrm{in}}〉=\beta$. (b) Hybridized model, in which an incident field scatters strongly (i.e., reflects) off the $\left|G\right.〉=\left|gg\right.〉\leftrightarrow \left|B\right.〉\sim \left|eg\right.〉-\left|ge\right.〉$ transition. For a field incident from the left, interference between direct and indirect excitation channels couple weakly to the dark state $\left|D\right.〉\sim \left|eg\right.〉+\left|ge\right.〉$. Driving from the right does not couple to $\left|D\right.〉$. (c) Reflectance $\mathcal{R}$ and phase shift $\phi$ versus dimensionless detuning, $\delta =\delta {\omega }_1/\gamma$, for a field reflected off a single atom.

Figure 2

Correlation functions, $g^{\left(1\right)}\left(\tau \right)$ and $g^{\left(2\right)}\left(\tau \right)$, for reflection and transmission, under $\alpha$ driving. We can calculate these correlation functions using the full four-dimensional Hilbert space, the adiabatically eliminated two-dimensional space, and the “flapping-mirror” toy model. The differences between these calculation methods are $\sim {(\delta /\alpha )}^2={10}^{-6}$, so are not visible on this scale.

Figure 3

Transmittance versus incident power, $|{\alpha }_{\text{in}}{|}^2/\gamma$ or $|{\beta }_{\text{in}}{|}^2/\gamma$. The dark state becomes saturated for ${\left|\alpha \right|}^2\gtrsim {\delta }^2$, so that ${\mathcal{T}}_{\alpha }$ has a dark-state saturation plateau for ${\delta }^2\lesssim {\left|\alpha \right|}^2/\gamma \lesssim 1$ wherein ${\mathcal{T}}_{\alpha }\approx 2/3=p_D$. ${\mathcal{T}}_{\beta }$ is independent of $\delta$. The bright state also becomes saturated when ${\left|\alpha \right|}^2,{\left|\beta \right|}^2\gtrsim \gamma$, so that ${\mathcal{T}}_{\alpha }$ and ${\mathcal{T}}_{\beta }$ both asymptote to unity when ${\left|\alpha \right|}^2,{\left|\beta \right|}^2\gtrsim \gamma$. Solid curves are calculated using the full four-dimensional Hilbert space. Dashed curves are calculated for the two-dimensional slow subspace (adiabatic elimination) which are valid for ${\left|\alpha \right|}^2,{\left|\beta \right|}^2\ll \gamma$, and are plotted with a small offset for visibility. Also shown is the “diode efficiency” $\mathcal{E}$ (dotted) for $\delta ={10}^{-3}$, as defined in Ref. [2].

Figure 4

Radiated flux from the dark state, $\left|D\right.〉$, moving left, ${\mathcal{A}}_{\text{out}}$, and right, ${\mathcal{B}}_{\text{out}}$, showing strong asymmetry in the emission profile, analogous to a Yagi-Uda directional transmitter. The main panel shows flux over long time scales $\gamma t\sim {\delta }^{-2}\gg 1$; inset shows flux over short time scales, $\gamma t\sim 1$.

Figure 5

Top: Contour plot of the “diode efficiency” $\mathcal{E}$ as a function of interatom path length $\delta \varphi$, and the detuning of one atom $\delta {\omega }_1$, assuming $\delta {\omega }_2=0$. Bottom: Line cut through contour plot along dashed line. The peak efficiency, $\mathcal{E}=2/3$, occurs along the line $\delta \varphi =-\delta {\omega }_1\equiv \delta$. Allowing $\delta {\omega }_2$ to vary does not yield a higher value for $\mathcal{E}$.