Single-photon nonlinearities using arrays of cold polar molecules

We model single-photon nonlinearities resulting from the dipole-dipole interactions of cold polar molecules. We propose utilizing “dark state polaritons” to effectively couple photon and molecular states; through this framework, coherent control of the nonlinearity can be expressed and potentially used in an optical quantum computation architecture. Due to the dipole-dipole interaction the photons pick up a measurable nonlinear phase even in the single-photon regime. A manifold of protected symmetric eigenstates is used as basis. Depending on the implementation, major sources of decoherence result from nonsymmetric interactions and phonon dispersion. We discuss the strength of the nonlinearity per photon and the feasibility of this system.

Figure 1

Level scheme for utilizing slow-light polaritons. $|e⟩$ and $|a⟩$ can be coupled via a two-photon transition, or $|e⟩$ can be a mixed-parity state.

Figure 2

(Color online) (a) Nonlinear phase as a function of time for a 1D array with $N=36$ molecules (solid line) and $N=81$ (dashed line). Here $t_{\pi }\left(N=36\right)=\hslash \pi /\left(2{\chi }_{\text{eff}}\right)$ is the expected phase gate time for the $N=36$ system calculated by projecting the dipole Hamiltonian [Eq. (5)] onto the Dicke manifold. It is obvious from this graph that the projected $\Theta =\pi$ phase time ($t=t_{\pi }$, solid black line for $N=36$, broken for $N=81$) deviates strongly from the exact one $\left(t\approx 3.5t_{\pi }\right)$. The main reason for this deviation is the importance of transition processes out of the Dicke manifold as confirmed in panel (b) where we plot the fidelity to stay in the state $|2⟩$.

Figure 3

(Color online) $\xi /\kappa$ for varying $E\left[B/{\mu }_0\right]$ in SrO. (a) $|g_i⟩=|J_{\text{SrO}},M_{J,\text{SrO}}⟩={|0,0⟩}_i$ and $|e_i⟩={|1,0⟩}_i$. (b) $|g_i⟩={|1,0⟩}_i$ and $|e_i⟩={|2,0⟩}_i$. $B$ is the rotational constant of the molecule, and ${\mu }_0$ is the maximum ground-state dipole moment, in our case assumed to be ${\mu }_0=\left|{\mathit{\mu }}_{eg}\right|$.

Figure 4

(Color online) (a) Nonlinear phase as a function of time for a 1D array with $N=36$ molecules (solid line) and $N=81$ (dashed line) in the presence of an external dc field, $\xi /\kappa =0.05$. The latter is used to implement the MPM. Here $t_{\pi }\left(N=36\right)=\hslash \pi /\left(2{\tilde{\chi }}_{\text{eff}}\right)$ is the expected phase gate time from our perturbative analysis for the $N=36$ system (indicated by a solid grid line). The corresponding time for the $N=81$ system is indicated by the dashed vertical line. For the two cases the actual time at which the phase gate is accomplished, $\left|\text{cos}\left(\Theta /2\right)\right|=0$ is close to the calculated $t_{\pi }$ indicating the validity of the perturbative analysis specially for $N=36$. The deviations can be accounted for by higher order corrections in perturbation theory. The fidelity of remaining in the $|2⟩$ state is shown in panel b. The latter decreases as either the ratio $\xi /\kappa$ or $N$ increases, consistently with Fermi's golden rule predictions.

Figure 5

(Color online) (a) Nonlinear phase as a function of time for a 2D array with $N=36$ molecules (solid line) and $N=81$ (dashed line). The solid and dashed vertical lines indicated the expected phase gate time according to our perturbative analysis. The ratio of $\xi /\kappa =1$ is outside the perturbative regime however the dynamics exhibits almost the behavior expected from an ideal ${\widehat{J}}_z^2$ evolution. The fidelity to remain in the $|2⟩$ state is shown in panel b. Note that in 2D the fidelity improves as $N$ is increased.

Figure 6

(Color online)$1-F_1\left(t\right)$, where $1-F_1\left(t\right)/{\left(\xi +4B_0\right)}^2/\sqrt{\beta }$ is the probability of decay of a single excitation for a 1D system with $N=36$ (solid), $N=81$ (dashed) and $\xi$, $B_0$ and $\beta$ are system dependant parameters. The solid and dashed vertical lines are at the time where the perturbative treatment predicts the implementation of the phase gate for the $N=36$ and 81 systems, respectively.

Figure 7

(Color online) $1-F_1\left(t\right)$, where $1-F_1\left(t\right)/{\left(\xi +4B_0\right)}^2/\sqrt{\beta }$ is the probability of decay of a single excitation for a 2D system with $N={20}^2$ (solid), $N={81}^2$ (dashed) and $\xi$, $B_0$, and $\beta$ are system dependant parameters. In this case we had to go to larger system sizes to show the decrease in fidelity with increasing $N$. The top (bottom) axis scales are for $N={81}^2\left({20}^2\right)$.

Figure 8

(Color online) Phonon excitation spectrum in 1D. $a$ is the separation between the molecules. The units of $\omega \left(q\right)$ are $U_{dd}/\sqrt{\beta }$.

Figure 9

(Color online) Phonon excitation spectrum in 2D for the two different branches of the phonon spectrum. $q_x$ and $q_y$ label the 2D quasimomenta. The units of ${\omega }_{1,2}\left(q_x,q_y\right)$ are $U_{dd}/\sqrt{\beta }$.