Photon control and coherent interactions via lattice dark states in atomic arrays

Ordered atomic arrays with subwavelength spacing have emerged as an efficient and versatile light-matter interface, where collective interactions give rise to sets of super- and subradiant lattice states. Here, we demonstrate that highly subradiant states, so-called lattice dark states, can be individually addressed and manipulated by applying a spatial modulation of the atomic detuning. More specifically, we show that lattice dark states can be used to store and retrieve single photons with near-unit efficiency, as well as to control the temporal, frequency, and spatial degrees of freedom of the emitted electromagnetic field. Furthermore, we demonstrate how to engineer arbitrary coherent interactions between multiple dark states and thereby manipulate information stored in the lattice. These results pave the way towards quantum optics and information processing using atomic arrays.

Figure 1

Checkerboard detuning pattern. (a) The checkerboard lattice is a non-Bravais lattice with two atoms per unit cell, such that neighboring atoms have detunings of opposite sign relative to the natural transition frequency ${\omega }_0$. (b) Momentum-space representation of a square lattice (black square) and the checkerboard lattice that emerges at finite detuning (blue square). The detuning couples the momentum states $\mathbf{k}$ and $\mathbf{k}+\mathit{\pi }/d$ (dashed lines) with strength $\mathrm{\Delta }$ and creates a three-level system such that both momentum states decay to the ground state with their respective collective decay rates. (c) Band structure for the nondetuned lattice with spacing $d=0.2{\lambda }_0$ and circular in-plane polarization. The radiating states are located around the origin of the Brillouin zone $\mathrm{\Gamma }$ and zero decay is observed outside the light cone. (d) Upon introducing a finite detuning, a band gap opens. The momentum states around $\mathrm{\Gamma }$ mix with those close to the corner $M$ and the original dark band acquires a nonzero decay rate.

Figure 2

Experimental setup and efficiency of photon retrieval. (a) Setup to store a photon in the two-dimensional atomic array. The incoming photon is split by a beam splitter and impinges the lattice from both sides. Conversely, the retrieved or outgoing photon is equally emitted on both sides of the array and needs to be recombined. (b) Retrieval error of a single photon in Gaussian detection modes of waist $\rho$ for lattices with spacing $d=0.3{\lambda }_0$ and different total number of atoms, $N$. In all figures, we consider the atoms to be circularly polarized, $\wp =(1,1j,0)/\sqrt{2}$. The solid lines represent the error if the Gaussian photon is retrieved immediately after being stored ($t_s=0$). The dashed lines show the error if a finite amount of time $t_s{\gamma }_0=50$ goes by before retrieving the photon. (c) Retrieval error as a function of storage time for waists $\rho /d=5$ (blue) and $\rho /d=7$ (orange), as well as for the optimal waist ${\rho }_{\mathrm{opt}}$ for any given storage time $t_s$ (green). A $21\times 21$ lattice is considered. (d) Curvature of the band around the Brillouin zone corner $M$ as a function of lattice spacing along the directions shown in the inset. The band curvature determines the rate at which the different momentum components dephase. Lattices with larger curvatures and stored excitations with larger momentum width dephase faster.

Figure 3

Temporal shape of the emitted photon. (a) Optimized temporal detuning sequences $\mathrm{\Delta }\left(t\right)$ for an infinite lattice and emitted photons of the following shapes: Blackman window (blue), Tukey window (red), triangular window (green), and sinusoidal window (orange). Lower plot: Population emitted in a Gaussian mode of waist $\rho =1.2{\lambda }_0$ per unit of time $dn/dt$ for the different detuning sequences and for a $21\times 21$ lattice with lattice spacing $d=0.2{\lambda }_0$. (b) Detuning sequences optimized for a finite lattice and resulting temporal shape of the emitted photon for Blackman windows of different durations. Again, a $21\times 21$ lattice with lattice spacing $d=0.2{\lambda }_0$ is used. The inset shows the spectrum of the emitted electric field. The resulting photon has only one central frequency and its width increases as its temporal duration decreases.

Figure 4

Frequency profile of the emitted photon. Frequency profile of the emitted electric field for different detuning strengths and for spacings (a) $d=0.3{\lambda }_0$ and (b) $d=0.2{\lambda }_0$. As the detuning is increased, the central frequency peaks splits into two. The insets show the band structure of the lattice with zero detuning. The sign of $J_{\mathrm{\Gamma }}-J_M$ determines whether the most prominent peak is found at frequencies larger or lower than the natural transition frequency ${\omega }_0$. The same color scale is used for both insets.

Figure 5

Direction of the emitted photon. (a) The photon is initially stored at the $X$ point by coupling it to the origin of the Brillouin zone $\mathrm{\Gamma }$ (green dots). Applying spatial detunings with a periodicity of three atoms along the $x$ direction, the $X$ point is coupled to the two red dots with momentum $\mathbf{k}=(\pm \pi /3d,0)$. In real space, the photon to be stored impinges the array in the perpendicular direction (green arrows), whereas the excitation is retrieved in a superposition of the four red arrows which form an angle $sin\theta \approx {\lambda }_0/6d$ with the perpendicular axis. We compute the magnitude of the electric field along the directions ${\mathit{\kappa }}_{\theta }=2\pi /{\lambda }_0(sin\theta ,0,\pm cos\theta )$ contained in the $xz$ plane for two different detunings: ${\mathrm{\Delta }}_A/{\gamma }_0=e^{i{\mathbf{Q}}_{+}{\mathbf{r}}_j}/2+e^{i{\mathbf{Q}}_{-}{\mathbf{r}}_j}/2$ (blue) and ${\mathrm{\Delta }}_B/{\gamma }_0=-i{e}^{i{\mathbf{Q}}_{+}{\mathbf{r}}_j}/2+i{e}^{i{\mathbf{Q}}_{-}{\mathbf{r}}_j}/2$ (orange), where ${\mathbf{Q}}_{\pm }=(\pm 2\pi /3d,0)$. (b) Same for detuning patterns with a periodicity of four atoms. The $X$ point is in this case coupled to the origin $\mathrm{\Gamma }$ and the states $\mathbf{k}=(\pm \pi /2d,0)$ and the photon is emitted in a superposition of six directions, two perpendicular to the array and four which form an angle $sin\theta \approx {\lambda }_0/4d$ with the normal axis. The most general detuning pattern is ${\mathrm{\Delta }}_j=\beta e^{i{\tilde{\mathbf{Q}}}_{+}{\mathbf{r}}_j}+{\beta }^{*}{e}^{i{\tilde{\mathbf{Q}}}_{-}{\mathbf{r}}_j}+\delta e^{i{\mathbf{Q}}_{\pi }{\mathbf{r}}_j}$, where ${\tilde{\mathbf{Q}}}_{\pm }=(\pm \pi /2d,0)$ and ${\mathbf{Q}}_{\pi }=(\pi /d,0)$. Three detuning patterns are considered: pattern C with ${\beta }_C=0.75{\gamma }_0$ and ${\delta }_C=0$, pattern D with ${\beta }_D=0.75e^{i\pi /4}{\gamma }_0$ and ${\delta }_D=0$, and pattern E with ${\beta }_E=0.5e^{i\pi /4}{\gamma }_0$ and ${\delta }_E=0.7{\gamma }_0$. In all cases, $41\times 41$ lattices with spacing $d=0.3{\lambda }_0$ are used and the stored Gaussian photon has a waist of $\rho =12d$.

Figure 6

Rabi flopping of two momentum dark states. (a) Quality factor of the coherent oscillations, limited by decoherence effects arising from a finite momentum width of the dark state stored at the $X$ point of the Brillouin zone. The quality factor is plotted versus momentum in reciprocal space (lower $x$ axis) or, analogously, versus the beam waist such that $99.8\%$ of the excitation lies within the corresponding momentum (upper axis). The inset shows how the detuning profile ${\mathrm{\Delta }}_R=\mathrm{\Delta }{(-1)}^{n_y}$ couples momentum states $X$ (blue) and $M$ (orange). (b) Rabi oscillations between momentum states with spread $\rho =1.8{\lambda }_0$ centered around $X$ (blue trace) and $M$ (orange trace) for $\mathrm{\Delta }=0.3{\gamma }_0$ and $\mathrm{\Delta }=5{\gamma }_0$ in a $21\times 21$ lattice of spacing $d=0.3{\lambda }_0$.

Figure 7

Engineering coherent interactions. Cyclic population transfer between (a) three and (b) four lattice dark states and (c) alternative four-atom dynamics with partially suppressed population at states $\mathbf{k}=(\pi ,\pm \pi /2)$ (orange and red). The different detuning patterns have the following Fourier components: (a) ${\mathbf{Q}}_{\pm }=(\pm 2\pi /3d,0)$ and (c) ${\tilde{\mathbf{Q}}}_{\pm }=(\pm \pi /2d,0)$ with amplitudes $\pm 5i$; (b) ${\tilde{\mathbf{Q}}}_{\pm }$ with amplitudes $5\sqrt{2}{e}^{\pm i\pi /4}$ and ${\mathbf{Q}}_{\pi }=(\pi /d,0)$ with amplitude 5. High-quality coherent interactions between multiple dark states require large arrays and narrow excitation distributions in reciprocal space. Here, we consider a $61\times 61$ lattice with spacing $d=0.4{\lambda }_0$ in (a) and $d=0.3{\lambda }_0$ in (b) and (c), and incident photons with waists $\rho =16d$.

Figure 8

Frequency modulation of emitted photon. Frequency profile of the emitted electromagnetic field from a $21\times 21$ lattice with spacing $d=0.3{\lambda }_0$ and an initial Gaussian state of waist $\rho =6d$. (a) A checkerboard detuning pattern with strength $\mathrm{\Delta }=0.75{\gamma }_0$ is applied and the frequency of the homogeneous detuning is $\mathrm{\Omega }=2\pi {\gamma }_0$. (b) The parameters are $\mathrm{\Delta }=3{\gamma }_0$ and $\mathrm{\Omega }=6\pi {\gamma }_0$. The three traces correspond to different ratios of the relevant parameter $\delta /\mathrm{\Omega }$.

Figure 9

Effect of defects. (a) Relative decrease in efficiency $(\eta -{\eta }_{\mathrm{def}})/\eta$ as a function of ${\sum }_{d\in \mathrm{defects}}|E_d{|}^2/{\sum }_{l\in \mathrm{lattice}}{\left|E_l\right|}^2$. A linear relation with proportionality constant $\alpha =1.19$ is found between both quantities (black dashed line). The different colors represent results for systems with different sets of holes: orange has a hole at ($2d,0$); grey has three holes at ($2d,0$), ($-d,-d$), and ($6d,3d$); red corresponds to a set of seven holes at (0,0), ($2d,0$), ($-2d,2d$), ($-3d,d$), ($5d,-d$), ($4d,4d$), and ($8d,8d$); cyan corresponds to a set of nine holes, the seven in red plus ($-d,d$) and ($0,-3d$). The origin of the lattice is considered to be at (0,0). (b) Infidelity versus beam waist $\rho$ for different single-atom defects. The blue dots correspond to a defect at position ($8d,8d$), the green dots to a defect at ($4d,3d$), and the orange to a defect at ($2d,0$). The black dashed line is the result for the corresponding finite lattice without defects. In all cases, a lattice with $d=0.3\lambda$ and $21\times 21$ atoms is considered.