Ultrafast Excitation Exchange in a Maxwell Fish-Eye Lens

The strong coupling of quantum emitters to a cavity mode has been of paramount importance in the development of quantum optics. Recently, also the strong coupling to more than a single mode of an electromagnetic resonator has drawn considerable interest. We investigate how this multimode strong coupling regime can be harnessed to coherently control quantum systems. Specifically, we demonstrate that a Maxwell fish-eye lens can be used to implement a pulsed excitation exchange between two distant quantum emitters. This periodic exchange is mediated by single-photon pulses and can be extended to a photon-exchange between two atomic ensembles, for which the coupling strength is enhanced collectively.

Figure 1

(a) 2D Maxwell fish-eye lens with two opposing quantum emitters. Within the lens, all rays emerging from one atom form circular arcs due to the radial refractive index gradient and converge at the second atom [23]. All light paths that connect the two atoms with one reflection at the bounding mirror have the same optical length. (b) Illustration of relevant coordinates and parameters in the lens. (c) Radial refractive index gradient within the lens [cf. Eq. (2)].

Figure 2

Atom and multimode dynamics in a Maxwell fish-eye lens. The excited state populations of the two emitters are shown as a function of time. The insets display the intensity expectation value of the quantized electric field $⟨{\mathit{E}}^{-}(\mathit{r}){\mathit{E}}^{+}(\mathit{r})⟩$ for five different times, as indicated by the links to the curves. Note that peaks of exceeding height were clipped in the insets. A video illustrating the dynamics is provided in the Supplemental Material [50].

Figure 3

Coherent excitation exchange in the Maxwell fish-eye lens. (a) Structured in analogy to Fig. 2. All parameters but the cutoff frequency ${\omega }_{\mathrm{c}}=0.1{\omega }_{\mathrm{a}}$ are left unchanged as compared to Fig. 2. Because of the narrower range of modes participating in the dynamics, a high exchange efficiency is achieved. (b) Illustration of the spectral engineering (${\omega }_{\mathrm{c}}=0.1{\omega }_{\mathrm{a}}$) in comparison with the cutoff used in Fig. 2. A video illustrating the dynamics is provided in the Supplemental Material [50].

Figure 4

Collective excitation exchange between two atomic ensembles in a Maxwell fish-eye lens. (a) Spatial distribution of the emitters in the lens. The two ensembles opposing each other consist of $N=100$ atoms each. The atoms are normal distributed with a standard deviation of $\sigma =0.02R$ around the two centers at $r_{\mathrm{a}}=0.6R$ at opposite positions in the cavity. (b) Collective excitation probabilities of the ensembles $|C_{\text{left}(\text{right})}{|}^2={\sum }_i|c_{i,\text{left}(\text{right})}{|}^2$ as a function of time. Initially, the excitation is distributed among the left ensemble as a symmetric Dicke state. The individual coupling strength of each atom of the ensemble is $g=0.05{\omega }_{\mathrm{a}}$, while the other parameters are analogous to Fig. 3.

Figure 5

Population of the three-excitation subspace $P(|3⟩)$ (blue) when counterrotating terms are taken into account. The dynamics of the atomic populations (gray) is shown in the background for comparison to Fig. 3. All parameters are identical to Fig. 3.