Coherent Photon Manipulation in Interacting Atomic Ensembles

Coupling photons to Rydberg excitations in a cold atomic gas yields unprecedentedly large optical nonlinearities at the level of individual light quanta. Here, the basic mechanism exploits the strong interactions between Rydberg atoms to block the formation of nearby dark-state polaritons. However, the dissipation associated with this mechanism ultimately limits the performance of many practical applications. In this work, we propose a new approach to strong photon interactions via a largely coherent mechanism at drastically suppressed photon losses. Rather than a polariton blockade, it is based on an interaction-induced conversion between distinct types of dark-state polaritons with different propagation characteristics. We outline a specific implementation of this approach and show that it permits us to turn a single photon into an effective mirror with a robust and continuously tunable reflection phase. We describe potential applications, including a detailed discussion of achievable operational fidelities.

Popular Summary

Photons, the fundamental constituents of light, do not interact in a vacuum and are, accordingly, free to pass through one another unimpeded. This distinguishing property of light is pivotal to the success of optical-communication technologies that enable long-distance transmission of information. However, many other emerging technologies would instead benefit from having interacting photons, whereby one photon can strongly influence the propagation or state of others. One way to make photons interact with one another is to convert them into so-called polaritons, which are compositions of light and excitations in a material that may feature strong interactions such that the photons effectively acquire the interactions of the medium. Here, we present a new theoretical mechanism for engineering controlled photon-photon interactions.

We devise a setup in which the presence of one polariton can cause another to be converted between two different “flavors” that have drastically different propagation characteristics. We show how this all-optical switching mechanism can cause photons to effectively behave like reflective quantum mirrors. In such a situation, the photons bounce off one another like billiard balls. We propose that an experimental realization of our setup might be possible with ${}^{87}\mathrm{Rb}$ atoms in a Rydberg state (i.e., one electron held in a very high-energy orbital), with minimal decoherence.

The behavior that we have hypothesized opens the door for future experiments on processing quantum information with photons.

Figure 1

Illustration of the basic principle of nonlinear polariton switching. (a) A photon (target) propagates initially as one type of dark-state polariton (type A, gray sphere), but is subsequently converted to another kind of dark-state polariton (type B, blue sphere) upon interacting with a second (gate) polariton. (b) This induces a change in the dispersion relation that governs the propagation of light and thereby mediates an effective photon interaction at greatly suppressed losses.

Figure 2

(a) Schematics of the considered coupling scheme. Classical fields with indicated Rabi frequencies $\mathrm{\Omega }$ and ${\mathrm{\Omega }}_S$ establish EIT conditions for the counterpropagating photonic modes ${\widehat{\mathcal{E}}}_{\to }$ and ${\widehat{\mathcal{E}}}_{\leftarrow }$. The Rydberg state $|s⟩$ is subject to a spatially dependent level shift $V(z)$ upon interacting with the stored gate excitation as illustrated in (b). This shift modifies the underlying EIT conditions for ${\widehat{\mathcal{E}}}_{\to }$ and ${\widehat{\mathcal{E}}}_{\leftarrow }$, rather than perturbing them.

Figure 3

(a) Real part of the polariton spectrum in the absence of interactions, indicating the emergence of two slow-light dark-state polaritons. These are separately governed by the linear dispersion relations ${\omega }_{\to }(k)$ and ${\omega }_{\leftarrow }(k)$ as given by Eqs. (6) and (7), respectively. Here, ${\mathrm{\Omega }}_S/G=1$, $\mathrm{\Omega }/G=0.5$, and $\gamma /\mathrm{\Omega }=1$. (b) Imaginary part of the polariton spectrum for strong interactions, i.e., under conditions of a complete Rydberg blockade. In this case, one finds a single stationary-light dark-state polariton described by a quadratic dispersion relation ${\omega }_{\leftrightarrow }(k)$, Eq. (9). Here, $G/\mathrm{\Omega }=1$ and $\gamma /\mathrm{\Omega }=1$. For each bright-state branch there are two solutions with identical $\mathrm{Im}[\omega (k)]$. The blue to red color coding indicates the relative fraction of ${\widehat{\mathcal{E}}}_{\to }$ and ${\widehat{\mathcal{E}}}_{\leftarrow }$ comprising the underlying state of each polariton branch, while the gray scale indicates the overall atomic fraction.

Figure 4

Transmission coefficient in the absence of a stored gate excitation for various indicated values of ${\mathrm{\Omega }}_S/\mathrm{\Omega }$. The total optical depth is $2d=50$, while $\gamma /\mathrm{\Omega }=0.5$ and $G/\mathrm{\Omega }=0.1$.

Figure 5

(a) Illustration of target photon propagation in the presence of a stored gate excitation, where $R_1(\omega )$ and $T_1(\omega )$ are the reflection and transmission coefficients, respectively. (b) The resonant values of $R_1(\omega =0)$ and $T_1(\omega =0)$ are shown as a function of $d_b$ in red and blue, respectively. The inset shows the spatial dependence of the photonic amplitudes ${\widehat{\mathcal{E}}}_{\to }(z)$ and ${\widehat{\mathcal{E}}}_{\leftarrow }(z)$ within the medium for $d_b=5$, where the gray shaded region indicates the extent of the blockade volume established by the stored gate excitation. (c)–(e) The transmission and reflection spectra $T_1(\omega )$ and $R_1(\omega )$ are shown in blue and red, respectively, for various indicated values of $d_b$, where the stored excitation is located at the center of the medium. The total optical depth is $2d=50$, and the remaining parameters are $\gamma /\mathrm{\Omega }=0.5$, $\mathrm{\Omega }/G=0.1$, ${\mathrm{\Omega }}_S/G=0.5$. The transmission coefficient $T_0(\omega )$ in the absence of interactions is indicated by the gray shaded region for reference.

Figure 6

(a) Loss coefficient $A$ as a function of $d_b$ (solid line) compared with the conventional polariton blockade mechanism (dashed line). (b) Initial density matrix, ${\rho }_0(x,y)=\sin (x\pi /L)\sin (y\pi /L)$, of the stored spin wave for $L=5z_b$. (c)–(f) Final density matrix of the stored spin wave corresponding to the present coherent polariton-switching mechanism for various indicated values of $d_b$. Panels (g)–(j) show the final density matrix for the dissipative polariton blockade, which is shown to cause much stronger decoherence.

Figure 7

(a) Overall switch fidelity, taking into account both the target field switching as well as the gate photon storage and retrieval efficiencies. The gray lines show the fidelity of a classical switch realized for a dissipative polariton blockade nonlinearity (dashed) and the coherent polariton switching nonlinearity (solid). The red lines show the corresponding fidelities for a quantum transistor. (b) Fidelity of a two-photon phase gate realized with the proposed photon router (solid line) compared with the corresponding fidelity of a $\pi$-phase gate based on the dispersive phase shift obtained from the polariton blockade mechanism (dashed line). The black dot indicates the critical value of $d_b\sim 6$ below which a dispersive phase shift of $\pi$ is fundamentally impossible with the polariton blockade mechanism under EIT conditions.

Figure 8

Reflection efficiency, according to Eq. (22), for different indicated durations of a Gaussian incoming target pulse ${\mathcal{E}}_0$. The total optical depth is fixed at $2d=50$, $\gamma /2\pi =3.05\mathrm{MHz}$, and the classical Rabi frequencies are $\mathrm{\Omega }/2\pi =5\mathrm{MHz}$ and ${\mathrm{\Omega }}_S/2\pi =20\mathrm{MHz}$. The solid lines include a finite Rydberg-state linewidth ${\gamma }_S/2\pi =300\mathrm{kHz}$ [92]. The gate excitation is located at the center of the medium and generates an interaction region with $z_b=\sqrt[6]{C_6\gamma /{\mathrm{\Omega }}_S^2}\sim 8.7\mu \mathrm{m}$, as obtained for the $100S_{1/2}$ state of ${}^{87}\mathrm{Rb}$ atoms [93]. The dashed line indicates the ideal limit of a vanishing photon bandwidth and ${\gamma }_S=0$, corresponding to Eq. (20).