Facilitation-Induced Transparency and Single-Photon Switch with Dual-Channel Rydberg Interactions

We investigate facilitation-induced transparency (FIT) enabled by strong and long-range Rydberg atom interactions between two spatially separated optical channels. In this setting, the resonant two-photon excitation of Rydberg states in a target channel is conditioned by a single Rydberg excitation in a control channel. Through the contactless coupling enabled by the Rydberg interaction, the optical transparency of the target channel can be actively manipulated by steering the optical detuning in the control channel. By adopting a dressed-state picture, we identify two different interference pathways, in which one corresponds to Rydberg blockade and an emergent one results from facilitation. We show that the FIT is originated from the Rydberg interaction and the quantum interference effect between the two pathways, which is different from conventional electromagnetically induced transparency realized by single-body laser-atom coupling. We find that the FIT in such a dual-channel setting is rather robust, insensitive to changes of systemic parameters, and can be generalized to multichannel settings. Moreover, we demonstrate that such a FIT permits the realization of controllable single-photon switches, which also paves a route to detect Rydberg facilitation by using optical absorption spectra. Our study contributes to current efforts in probing correlated many-body dynamics and developing single-photon quantum devices based on Rydberg atom ensembles.

Figure 1

Scheme of dual-channel Rydberg atomic gas for realizing FIT. Two atomic ensembles, i.e., target channel $A$ and control channel $B$, are prepared in parallel and separated by distance $d$. (a) A weak probe field ${\hat{\mathcal{E}}}_p$ propagates in the channel $A$, whose dynamics is facilitated by the Rydberg excitation in the channel $B$. Long-range interchannel Rydberg interaction permits a contactless control for the two channels and hence the propagation of the probe photon. (b) Level diagram and excitation scheme. Atoms in the channel $A$ have three quantum states (i.e., $|1⟩$, $|2⟩$, $|3⟩$; $|3⟩$ is a Rydberg state) forming a Rydberg EIT, where resonant transitions $|1⟩\leftrightarrow |2⟩$ and $|2⟩\leftrightarrow |3⟩$ are, respectively, provided by weak probe field ${\hat{\mathcal{E}}}_p$ and strong coupling field (Rabi frequency $\mathrm{\Omega }$). In the channel $B$, a control field with Rabi frequency ${\mathrm{\Omega }}_c$ drives the transition between the ground state $|1⟩$ to the Rydberg state $|3⟩$ with detuning ${\mathrm{\Delta }}_c$. $V_{AB}$ is the van der Waals interaction between the atoms in the two channels.

Figure 2

Optical coherence and interchannel correlation in the two-atom system. (a) Level diagram and excitation scheme between the Rydberg blockade state $|1_A{3}_B⟩$ (occurring at ${\mathrm{\Delta }}_c=0$) to the facilitated state $|3_A{3}_B⟩$ (occurring at ${\mathrm{\Delta }}_c=-V_{AB}$) of the two-atom model. For more details, see the text. (b) FIT transparency window induced by the interchannel interaction $V_{AB}$, as seen in the absorption spectrum Im(${\rho }_{21}^A$) as a function of ${\mathrm{\Delta }}_c$. $\mathrm{\Delta }E$ is the energy separation between centers of the two peaks of the FIT spectrum. (c1) Im(${\rho }_{21}^A$) as a function of ${\mathrm{\Delta }}_c$ and $V_{AB}$. Im(${\rho }_{21}^A$) exhibits a single peak for smaller $V_{AB}$; it however splits into two peaks for larger $V_{AB}$. The dip between the two peaks becomes wider and deeper as $V_{AB}$ increases, similar to the behavior of EIT in three-level systems. Corresponding Rydberg excitations are shown in (b) (dashed red line for ${\rho }_{33}^A$; dot-dashed blue line for ${\rho }_{33}^B$), and also in (c2), which shows that the channel $A$ is fully blockaded independent of $V_{AB}$ (solid blue line) for ${\mathrm{\Delta }}_c=0$ but ${\rho }_{33}^A$ is facilitated around ${\mathrm{\Delta }}_c=-V_{AB}$ (dashed red line). (d) Analytical result (dotted red line) given by Eq. (6), which captures well with the one by solving the master equation (ME) numerically (solid black line). The quantum correlation $O_{AB}$ (dot-dashed blue line) is significant for the FIT effect. The parameters used are $\mathrm{\Omega }={\mathrm{\Omega }}_c=5{\mathrm{\Gamma }}_{12}^A$, ${\mathrm{\Omega }}_p=0.5{\mathrm{\Gamma }}_{12}^A$, ${\mathrm{\Gamma }}_{23}^A={10}^{-3}{\mathrm{\Gamma }}_{12}^A$, and ${\mathrm{\Gamma }}_{13}^B={\mathrm{\Gamma }}_{12}^A$; in (b),(d), $V_{AB}=15{\mathrm{\Gamma }}_{12}^A$.

Figure 3

Dressed-state picture and the scaling of the FIT peak. (a) Interaction-dressed subspace spanned by the basis $\{|2_A⟩,|3_A⟩\}\otimes \{|1_B⟩,|3_B⟩\}$, indicated by the dashed gray box. (b) Dressed eigenvalues ${ϵ}_j$ as functions of ${\mathrm{\Delta }}_c$ for $V_{AB}=15{\mathrm{\Gamma }}_{12}^A$; corresponding eigenstates $\{|{\psi }_j⟩;j=1,4\}$ (dashed gray lines) and $\{|{\psi }_j⟩;j=2,3\}$ (solid blue lines) are also shown. ${\mathrm{\Omega }}_p$ can resonantly drive the transition $|1_A⟩\to |{\mathrm{\Psi }}_F⟩$ ($|1_A⟩\to |{\mathrm{\Psi }}_E⟩$) at ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_c^{-}$ (${\mathrm{\Delta }}_c={\mathrm{\Delta }}_c^{+}$), where $|{\mathrm{\Psi }}_E⟩$ ($|{\mathrm{\Psi }}_F⟩$) is the blockade (facilitation) state (see Appendix pp3). (c) Energy separation $\mathrm{\Delta }E$ [described also in Fig. 2] as a function $V_{AB}$ for ${\mathrm{\Omega }}_c=3{\mathrm{\Gamma }}_{12}^A$ (yellow circles) and ${\mathrm{\Omega }}_c=5{\mathrm{\Gamma }}_{12}^A$ (green crosses) with fixed $\mathrm{\Omega }=3{\mathrm{\Gamma }}_{12}^A$ and ${\mathrm{\Gamma }}_{13}^B=0.1{\mathrm{\Gamma }}_{12}^A$, obtained by numerically solving the master equation, Eq. (3). Triangles: the result for the channel $B$ is driven by a three-level scheme (3LS). Inset: details of $\mathrm{\Delta }E$ in the different parameter regimes for small $V_{AB}$. Dashed red line: the line $\mathrm{\Delta }E=V_{AB}$, which matches well with the numerical data for larger $V_{AB}$. (d) Fidelities ${\mathcal{F}}_B$ (dashed red line) and ${\mathcal{F}}_F$ (dash-dotted blue line) as functions of ${\mathrm{\Delta }}_c$. Maximal fidelities overlap with the doublet peaks of the absorption spectrum $\mathrm{Im}({\rho }_{21}^A)$ (solid black line). (e) Mandel $Q$ factor as a function of ${\mathrm{\Delta }}_c$ for different $V_{AB}$, which is enhanced drastically at ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_c^{\pm }$ for large $V_{AB}$. (f) Suggested model with multiple target channels, in which the control channel (solid red circle) locates at the center of a ring (with radius $R_{\mathrm{fac}}$) where the target channels (solid blue circles) locate. Such a setting can facilitates the excitation of $N$-target channels (solid blue circles) within a shell (gray region). Shown in the box are lines of Im(${\rho }_{21}$) for $N=2,3,4$, indicating that FIT is supported for such a system.

Figure 4

Optical susceptibility. (a) Im(${\chi }_p$) as a function of ${\mathrm{\Delta }}_c$ (for $V_{AB}=15{\mathrm{\Gamma }}_{12}^A$) when the Rabi frequency of the probe field takes values of $0.01{\mathrm{\Gamma }}_{12}^A,0.1{\mathrm{\Gamma }}_{12}^A,0.5{\mathrm{\Gamma }}_{12}^A,$ and ${\mathrm{\Gamma }}_{12}^A$, respectively. (b1) Im(${\chi }_p$) as a function of ${\mathrm{\Omega }}_p$ and $V_{AB}$ for ${\mathrm{\Delta }}_c=-V_{AB}$ [corresponding to the left peak of (a)]. (b2) The same as (b1) but for small values of ${\mathrm{\Omega }}_p$. (c1),(c2) The same as (b1) and (b2) but with ${\mathrm{\Delta }}_c=0$ [corresponding to the right peak of (a)]. The result shows that when decreasing ${\mathrm{\Omega }}_p$, Im(${\chi }_p$) saturates for ${\mathrm{\Omega }}_p<0.1{\mathrm{\Gamma }}_{12}^A$, such that ${\chi }_p$ behaves as a linear susceptibility, independent of ${\mathrm{\Omega }}_p$.

Figure 5

Facilitation enabled single-photon switch. (a) The control channel $B$ is driven with the excitation localized at $z=0$ (a1). In the vicinity of $z=0$, ${\rho }_{33}^A$ is blockaded for ${\mathrm{\Delta }}_c=0$ or facilitated for ${\mathrm{\Delta }}_c=-V_{AB}$ (${\mathrm{\Delta }}_c=-V_{AB}/2$ is an intermediate case). They both result in a strong photon scattering, as can be seen from the decreasing relative intensity ${\mathcal{I}}_s$ (a2). Shown in (a3) and (a4) are ${\rho }_{33}^A$ and ${\rho }_{33}^B$ as functions of $z$ for ${\mathrm{\Delta }}_c=0$ (red dashed line), $-V_{AB}/2$ (green dash-dotted line), and $-V_{AB}$ (blue solid line), respectively. (b) When the channel $B$ is dynamically driven (b1), one can change the control detuning ${\mathrm{\Delta }}_c$ from a far-detuned value to $-V_{AB}$ (or $0$) to launch facilitated (blockade) single-photon switch, where the relative intensity of the probe field quickly decreases to zero (b2). Shown in (b3) and (b4) are ${\rho }_{33}^A$ and ${\rho }_{33}^B$ as functions of $z$ for ${\mathrm{\Delta }}_{cF}=0$ (red dashed line), and $-V_{AB}$ (blue solid line), respectively. In the simulation, system parameters $\mathrm{\Omega }={\mathrm{\Omega }}_c=3{\mathrm{\Gamma }}_{12}^A$, $V_{AB}=15{\mathrm{\Gamma }}_{12}^A$, and ${\mathcal{N}}_A=3\times {10}^{12}{\mathrm{cm}}^{-3}$ are used. See text for more details.

Figure 6

FIT with delocalized control-atom excitation. (a) Schematic of the delocalized control-atom excitation in the channel $B$. The delocalization modifies the atom-atom interaction form from $V_{AB}=-C_6/d^6$ to $V_{AB}=-C_6/|\mathbf{d}+{\mathbf{r}}_B{|}^6$, here $d=|\mathbf{d}|$ is the center-of-mass separation between the $A$ and $B$ channels and $r_B=|{\mathbf{r}}_B|$ is the relative variation of the control excitation. (b) Photon transmission $T$ as a function of ${\mathrm{\Delta }}_c$ for different $d$, but with fixed $V_{AB}(r_B=0)=15{\mathrm{\Gamma }}_{12}^A$, when the photon propagates to distance $L=20\mu \mathrm{m}$. Solid black line: the result of the ideal FIT (i.e., the case for localized control-atom excitation). (c) is similar to Fig. 5. To launch facilitated (blockade) switch, here ${\mathrm{\Delta }}_c$ is changed from the far-detuned value to $-V_{AB}$ (or $0$). (c1)–(c3) show the relative intensity of the probe field $I_s$ and the Rydberg population ${\rho }_{33}^{A(B)}$ in the facilitated regime; (d1)–(d3) are similar to (c1)–(c3) but for the blockade regime. Blue curves in these figures are results of single trajectory simulation in the delocalized regime. Their mean values are calculated by casting 300 trajectories for different initial delocalized distribution (see green-dotted line). The coherence of the control atom is modified during the operation of the photon switch, manifested by the real Re(${\rho }_{31}^B$) of the atomic coherence [see (c4) and (d4)]. System parameters used in the simulation are $\mathrm{\Omega }={\mathrm{\Omega }}_c=3{\mathrm{\Gamma }}_{12}^A$ and ${\mathcal{N}}_A=3\times {10}^{12}{\mathrm{cm}}^{-3}$.

Figure 7

Two-photon excitation scheme for the control channel $B$. (a) The channel $A$ is the same as that of Fig. 1. The control channel $B$ is however driven by a two-photon process, where the transition $|1⟩\leftrightarrow |2⟩$ ($|2⟩\leftrightarrow |3⟩$) is coupled by the control field of Rabi frequency ${\mathrm{\Omega }}_{c1}$ (${\mathrm{\Omega }}_{c2}$), with detuning $\mathrm{\Delta }$ (${\mathrm{\Delta }}_c$). (b) Level diagram of the system.

Figure 8

Atomic coherence and Rydberg excitations in the dual-channel model when the control channel $B$ is driven by a two-photon process. (a) FIT effect emerges in the absorption spectrum of the probe field, manifested by the imaginary part Im(${\rho }_{21}^A$) of the atomic coherence ${\rho }_{21}^A$ as a function of ${\mathrm{\Delta }}_c$ (solid black line). Corresponding Rydberg excitations ${\rho }_{33}^A$ (dot-dashed blue line) and ${\rho }_{33}^B$ (dashed red line) are also plotted, where the channel $A$ is blockaded at ${\mathrm{\Delta }}_c=0$, but facilitated around ${\mathrm{\Delta }}_c=-V_{AB}$. (b) As $V_{AB}$ increases, the FIT transparency window becomes wider and deeper. Two peaks are visible even for weak interaction strength (e.g., $V_{AB}=5{\mathrm{\Gamma }}_{12}^A$). System parameters used are ${\mathrm{\Omega }}_{c1}={\mathrm{\Omega }}_p=0.8{\mathrm{\Gamma }}_{12}^A$, ${\mathrm{\Omega }}_{c2}=\mathrm{\Omega }=4{\mathrm{\Gamma }}_{12}^A$, and $\mathrm{\Delta }=10{\mathrm{\Gamma }}_{12}^A$.

Figure 9

Three-atom model. (a1) FIT spectrum Im(${\rho }_{21}$) for the target channel as functions of ${\mathrm{\Delta }}_c$ and $V_{TT}$, which exhibits double peaks even for a strong $V_{TT}$, with $V_{CT}=15{\mathrm{\Gamma }}_{12}^A$. The inset shows the geometry of the system, in which the control channel (red circle) locates at the center of a ring (with radius $R_{\mathrm{fac}}$), and the target channels (two blue circles) locate on the shell. (a2) The corresponding contour map as functions of ${\mathrm{\Delta }}_c$ and $V_{TT}$ is sketched with $\mathrm{Im}[{\rho }_{21}-{\rho }_{12}(V_{TT}=0)]$. It is clear to see that the results reach saturation as $V_{TT}$ increases. (b1),(b2) are similar to (a1),(a2) but with $V_{CT}=10{\mathrm{\Gamma }}_{12}^A$.