Quantum optics in Maxwell's fish eye lens with single atoms and photons

We investigate the quantum optical properties of Maxwell's two-dimensional fish eye lens at the single-photon and single-atom level. We show that such a system mediates effectively infinite-range dipole-dipole interactions between atomic qubits, which can be used to entangle multiple pairs of distant qubits. We find that the rate of the photon exchange between two atoms, which are detuned from the cavity resonances, is well described by a model where the photon is focused to a diffraction-limited area during absorption. We consider the effect of losses on the system and study the fidelity of the entangling operation via dipole-dipole interaction. We derive our results analytically using perturbation theory and the Born-Markov approximation and then confirm their validity by numerical simulations. We also discuss how the two-dimensional Maxwell's fish eye lens could be realized experimentally using transformational plasmon optics.

Figure 1

Light rays propagating within the infinite 2D fish eye lens trace out perfect circles (dashed red lines). If a mirror of radius $R_0$ is introduced (black dotted circle), the trajectories remain closed (solid red lines). All light rays emerging from an arbitrary point within the lens (green dot) refocus at the antipodal point (blue star). The color code and the inset show the spatial variation of the refractive index as a function of the radius, where we assume that $n_0=1$ in Eq. (1). For $r>R_0$ the refractive index of the fish eye dips below 1.

Figure 2

(a) Schematic depiction of the two dipoles embedded in the fish eye cavity, which is surrounded by mirrors on all sides. (b) Spectrum of the cavity $({\omega }_l=\sqrt{l(l+1)}c/\left(R_0{n}_0\right), l=1,2,3\cdots )$ in the absence ($\kappa =0$) and the presence ($\kappa \ne 0$) of losses. The atomic resonant frequency ${\omega }_0$ is tuned between two resonances of the cavity. (c) Strength of the dipole-dipole interaction $\delta \omega ({\mathbf{r}}_1,{\mathbf{r}}_2)/{\mathrm{\Gamma }}_0$ between two atoms for four different lens radii: (i) $R_0=4.93\lambda$, (ii) $R_0=8.11\lambda$, (iii) $R_0=11.3\lambda$, and (iv) $R_0=14.48\lambda$, assuming lens thickness $b=\lambda /10$ and ${\mathrm{\Gamma }}_0=d_z^2{\omega }_0^3/\left(3\pi {\varepsilon }_0\hslash c^3\right)$. The lens radii are chosen such that the transition frequency of the atoms ${\omega }_0=2\pi c/\lambda$ lies halfway between the resonances of the lens ($l=1,2,3\cdots$). In particular, we chose the order parameters (i) $\nu =30.5$, (ii) $\nu =50.5$, (iii) $\nu =70.5$, and (iv) $\nu =90.5$, where $\nu =\frac{1}{2}[\sqrt{16{\pi }^2{(R_0{n}_0/\lambda )}^2+1}+1]$ and $n_0=1$. The atom on the left is positioned exactly $\lambda$ away from the mirror, whereas the position of the second atom is sweeped. The strength of the interaction peaks $\lambda$ away from the opposite mirror surface with a height that is independent of the radius of the lens and the interatomic distance. (d) Enlarged view of the dipole-dipole interaction near the antipodal point, showing that the width of the peak is approximately $\lambda /2$.

Figure 3

Schematic depiction of a realistic scheme for the photon transfer between the two atoms. The first atom emits the photon, while the second atom fully absorbs it. By applying classical time-dependent control pulses ${\mathrm{\Omega }}_1\left(t\right)$ and ${\mathrm{\Omega }}_2\left(t\right)$, the transfer can be initiated and the photon can be captured in the metastable state of the second atom, from which the photon can be read out using fluorescence techniques.

Figure 4

Excitation probability of two atoms within the cavity as a function of time. Initially, atom 1 is excited and atom 2 is in its ground state. As the system evolves, the two atoms repeatedly exchange a photon via the dipole-dipole interaction. The photon gradually decays from the cavity modes, leaving the atoms in their ground states. A fully entangled state with maximal fidelity is formed at $t=\pi /\left(4\delta \omega \right)$ (see arrow). The plot was obtained for $R_0=3.34\lambda$ with a cavity loss rate of $\alpha =\kappa /{\omega }_0=5\times {10}^{-4}$, assuming that the two atoms are located at two antipodal points within the lens such that $|{\mathbf{r}}_1|=|{\mathbf{r}}_2|=0.27R_0$ and ${\varphi }_1={\varphi }_2+\pi$.

Figure 5

Error $(1-F)$ of the entangling operation between two qubits located at two antipodal points within the lens ($|{\mathbf{r}}_1|=|{\mathbf{r}}_2|=0.27R_0$ and ${\varphi }_1={\varphi }_2+\pi$). (a) Error of the entangling operation as a function of the cavity loss rate $\alpha =\kappa /{\omega }_0$ for four different lens sizes $R_0\in \{1.75,3.34,8.11,14.5\}\lambda$, where the $R_0/\lambda$ ratio was chosen such that $\text{Re}\left(\nu \right)=\frac{1}{2}[\sqrt{16{\pi }^2{(R_0/\lambda )}^2+1}+1]\in \{10.5,20.5,50.5,90.5\}$. The error increases as the losses and lens radii increase. (b) Error of the entangling operation for a fixed loss rate $\alpha =5\times {10}^{-4}$ as a function of the detuning $\mathrm{\Delta }\nu =\text{Re}\left(\nu \right)-{\nu }_{\text{center}}$, where ${\nu }_{\text{center}}\in \{10.5,20.5,50.5,90.5\}$. Error is plotted for the same four lens radii as in (a). The error increases with radius and as the frequency approaches one of the resonances. Numerical results are shown with large dots. Good agreement is obtained between analytic and numerical data, confirming the validity of the Born-Markov analysis.

Figure 6

Maximum fidelity of the entanglement operation as a function of the lens radius. The fidelity was evaluated at discrete values of $R_0/\lambda$ that correspond to tuning the atomic frequency halfway between two resonances, i.e., $\text{Re}\left[\nu \right]=m+0.5$, where $m$ is an integer. The red line marked with circles was obtained from the exact analytical expression in Eq. (41), whereas the blue line marked with squares was obtained from the approximate expression in Eq. (42). Good agreement is obtained between the two curves. The loss rate was assumed to be $\alpha =\kappa /{\omega }_0=5\times {10}^{-4}$.

Figure 7

Physical realization of the fish eye lens using transformation plasmon optics. (a) Effective refractive index $n\left(d\right)=\text{Re}\left\{\tilde{n}\left(d\right)\right\}$ created as a function of the height of the dielectric $d$ deposited on the silver surface. The inset shows the material losses $\chi \left(d\right)=\text{Im}\left\{\tilde{n}\left(d\right)\right\}$ as a function of the dielectric height $d$. (b) Schematic depiction of the plasmonic fish eye lens. The two emitters are embedded in the dielectric. The height of the dielectric varies across the lens, which creates the effective refractive index distribution of the fish eye lens. The lens is surrounded by mirrors from all sides (the front part of the mirror has been removed to show the interior).