Tunable bandwidth and nonlinearities in an atom-photon interface with subradiant states

Optical nonlinearities at the single-photon level are key features to build efficient photon-photon gates and to implement quantum networks. Such optical nonlinearities can be obtained using an ideal two-level system such as a single atom coupled to an optical cavity. While efficient, such atom-photon interface however presents a fixed bandwidth, determined by the spontaneous emission time and thus the spectral width of the cavity-enhanced two-level transition, preventing an efficient transmission to bandwidth-mismatched atomic systems in a single quantum network. In the present work, we propose a tunable atom-photon interface making use of the direct dipole-dipole coupling of two slightly different atomic systems. We show that, when weakly coupled to a cavity mode and directly coupled through dipole-dipole interaction, the subradiant mode of two slightly detuned atomic systems is optically addressable and presents a widely tunable bandwidth and single-photon nonlinearity.

Figure 1

cw laser of power $P_{\mathrm{laser}}$ pumps the cavity mode and we measure the reflected power $P_{\mathrm{reflected}}$. Both QDs interact equally with the cavity mode at a rate $g$. The varying parameters are the frequency detuning of the QDs ${\mathrm{\Delta }}_{12}=\frac{{\omega }_1-{\omega }_2}{2}$ and the direct dipole-dipole coupling with a rate ${\mathrm{\Omega }}_{12}$. $|e_i\rangle$ and $|g_i\rangle$ are the excited and ground state of the $i\mathrm{th}$ QD, respectively. The spontaneous decay rates $\gamma$ into the environment can change depending on the direct coupling.

Figure 2

Energy diagram for the QDs. $|ee\rangle$ and $|gg\rangle$ represent the states with both QDs excited or in the ground state. The other states are entangled states and are described in the main text. The dotted arrows represent spontaneous decays and the double arrows depict coherent coupling of levels through the cavity mode. (a) ${\mathrm{\Omega }}_{\mathbf{12}}=\mathbf{0}$, ${\mathrm{\Delta }}_{\mathbf{12}}=\mathbf{0}$; only the symmetric state $|+\rangle$ couples to the cavity by absorbing or emitting a photon with a rate $\sqrt{2}g$. The antisymmetric state $|-\rangle$ is a dark state and doesn't interact with the cavity. The spontaneous decay is the same for all states and is equal to the single QD decay rate $\gamma$. (b) ${\mathrm{\Omega }}_{\mathbf{12}}>\mathbf{0}$, ${\mathrm{\Delta }}_{\mathbf{12}}=\mathbf{0}$; QDs are dipole-dipole coupled. The entangled eigenstates are still completely symmetric and antisymmetric but they are now energy split by $\pm {\mathrm{\Omega }}_{12}$ and the spontaneous decay rates are modified by $\pm {\gamma }_{12}$, respectively. (c) ${\mathrm{\Omega }}_{\mathbf{12}}=\mathbf{0}$, ${\mathrm{\Delta }}_{\mathbf{12}}>\mathbf{0}$; QDs are detuned; the new entangled eigenstates $|{-}^{\prime }\rangle$ and $|{+}^{\prime }\rangle$ are a superposition of the previous totally symmetric and antisymmetric states. $|{-}^{\prime }\rangle$ now couples to the cavity mode. There is no dipole-dipole coupling so the spontaneous decay rates are not modified and are still equal to $\gamma$ for all the states. (d) ${\mathrm{\Omega }}_{\mathbf{12}}>\mathbf{0}$, ${\mathrm{\Delta }}_{\mathbf{12}}>\mathbf{0}$. Detuned and dipole-dipole coupled QDs: the eigenstates are energy split by $\sqrt{{\mathrm{\Omega }}_{12}^2+{\mathrm{\Delta }}_{12}^2}$; $|{+}^{″}\rangle$ and $|{-}^{″}\rangle$ are both coupled to the cavity and the spontaneous decay rates are modified.

Figure 3

Modulus of the decomposition coefficients $A$ (lower curve) and $B$ (upper curve) for the entangled states $|{+}^{\prime }\rangle$ and $|{-}^{\prime }\rangle$ of Eq. (15) as a function of detuning between the two QDs. Parameters are $\{g,\gamma ,\kappa \}=\{20,0.6,200\}$ $\mu \mathrm{eV}$. The vertical line indicates detuning ${\mathrm{\Delta }}_{12}={\mathrm{\Gamma }}_0$.

Figure 4

Modulus of the decomposition coefficients $\mu$ (lower curves) and $\nu$ (upper curves) for the entangled states $|{+}^{″}\rangle$ and $|{-}^{″}\rangle$ of Eq. (17). Dashed lines show the analytical results of Eq. (18). Parameters are the same as Fig. 3 but now ${\mathrm{\Omega }}_{12}=31\mu \mathrm{eV}$ corresponding to a distance of $d=10\text{nm}$.

Figure 5

Reflectivity of the cavity containing two QDs. Panel (a) represents the QDs in resonance and without dipole-dipole coupling. Panels (b) and (c) are the cases with dipole-dipole coupling and detuning, respectively. (d) The two QDs are both detuned and dipole-dipole coupled. (e) Same as (d) but with smaller detuning. The three rows show the same spectra but for increasing incident laser power of (i) 1 pW, (ii) 1 nW, and (iii) 10 nW. Parameters are $\left\{g,\kappa ,\gamma \right\}=\left\{20,200,0.6\right\}\mu \mathrm{eV}$.

Figure 6

Reflectivity vs incident laser power. In dotted black for one quantum dot, in dashed blue for the superradiant state $|{+}^{″}\rangle$ of two dipole-dipole coupled, detuned quantum dots with ${\mathrm{\Delta }}_{12}=20\mu \mathrm{eV}$, in solid red the subradiant state $|{-}^{″}\rangle$ with the same detuning, and in dashed-dotted yellow the same state $|{-}^{″}\rangle$ but for a smaller detuning of ${\mathrm{\Delta }}_{12}=10\mu \mathrm{eV}$. The parameters are $\left\{g,\kappa ,\gamma \right\}=\left\{20,200,0.6\right\}\mu \mathrm{eV}$. The pump laser and the cavity mode are each time tuned to the frequency of the state in consideration.

Figure 7

Color map of the reflectivity of the subradiant state $|{-}^{″}\rangle$ as a function of the incident laser power and of the QDs detuning ${\mathrm{\Delta }}_{12}=\frac{{\omega }_1-{\omega }_2}{2}$. Dipole-dipole interaction is kept the same with ${\mathrm{\Omega }}_{12}=31\mu \mathrm{eV}$. The laser and cavity frequency are set to ${\omega }_L={\omega }_c=\frac{{\omega }_1+{\omega }_2}{2}-\sqrt{{\mathrm{\Delta }}_{12}^2+{\mathrm{\Omega }}_{12}^2}$, in resonance with $|{-}^{″}\rangle$. The subradiant state saturates at lower incident power for smaller detuning since its antisymmetric component is more pronounced. For very small detuning the state is completely antisymmetric and dark and doesn't couple to cavity. Inset: ${\mu }^2$ the coefficient controlling the critical photon number for the subradiant state as a function of detuning. Parameters as in Fig. 6.

Figure 8

Normalized second-order correlation function $g_2\left(\tau \right)$ as a function of delay between two detections. In dotted black for one quantum dot; in dashed blue for the superradiant state $|{+}^{″}\rangle$ with ${\mathrm{\Delta }}_{12}=20\mu \mathrm{eV}$. The subradiant state $|{-}^{″}\rangle$ has been shown at two different detunings ${\mathrm{\Delta }}_{12}=20\mu \mathrm{eV}$ in solid red and ${\mathrm{\Delta }}_{12}=10\mu \mathrm{eV}$ in dashed-dotted yellow. The cavity and laser frequency are matched to the respective state and $P_{\mathrm{laser}}=10\text{pW}$. Parameters are the same as in Fig. 6.