Fast-responding property of electromagnetically induced transparency in Rydberg atoms

We investigate the transient optical response property of an electromagnetically induced transparency (EIT) in a cold Rydberg atomic gas. We show that both the transient behavior and the steady-state EIT spectrum of the system depend strongly on Rydberg interaction. Especially, the response speed of the Rydberg-EIT can be five times faster (and even higher) than the conventional EIT without the Rydberg interaction. For comparison, two different theoretical approaches (i.e., two-atom model and many-atom model) are considered, revealing that Rydberg blockade effect plays a significant role for increasing the response speed of the Rydberg-EIT. The fast-responding Rydberg-EIT by using the strong, tunable Rydberg interaction uncovered here is not only helpful for enhancing the understanding of the many-body dynamics of Rydberg atoms but also useful for practical applications in quantum information processing by using Rydberg atoms.

Figure 1

(a) Level configuration and excitation scheme of the two-atom model, which consists of two identical atoms, $A$ and $B$, with three internal states ${|1\rangle }_l, {|2\rangle }_l$, and ${|3\rangle }_l$ (Rydberg state), interacting via van der Waals (Rydberg) interaction. ${\mathrm{\Gamma }}_{12}$ (${\mathrm{\Gamma }}_{23}$): decay rate from ${|2\rangle }_l$ to ${|1\rangle }_l$ (from ${|3\rangle }_l$ to ${|2\rangle }_l$); ${\mathrm{\Delta }}_2$ (${\mathrm{\Delta }}_3$): one- (two-) photon detuning; ${\mathrm{\Omega }}_p$ (${\mathrm{\Omega }}_c$): half Rabi frequency of the probe (control) field coupling to the transition ${|1\rangle }_l\leftrightarrow {|2\rangle }_l$ (${|2\rangle }_l\leftrightarrow {|3\rangle }_l$) ($l=A,B$). (b) Time sequence for the probe (red dashed line) and the control (blue solid line) fields. (c) Schematic of Rydberg blockade in the many-atom model. The Rydberg interaction between atoms blocks the excitation of the atoms within blockade spheres (i.e., the ones with the boundary marked by the orange dashed lines). In each blocked sphere only one Rydberg atom (small yellow sphere) is excited, and excitations of other atoms (small blue spheres) to their Rydberg states are suppressed.

Figure 2

Transient response behavior of the Rydberg-EIT as a function of time $t$. (a) Normalized absorption of the two-atom model for ${\mathrm{\Omega }}_p=0.2{\mathrm{\Gamma }}_{12}$, characterized by $\mathrm{Im}\left({\rho }_{21}\right) [{\rho }_{21}\left(t\right)\equiv {\rho }_{21}^A\left(t\right)]$. The green dashed-dotted line is for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$ with the Rydberg interaction $[V_{AB}=1\mathrm{GHz}$ ($r_{AB}=3.10\mu \mathrm{m}$)]; the brown dotted line is for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$ with no Rydberg interaction ($V_{AB}=0$). (b) $\mathrm{Im}\left({\rho }_{21}\right)$ of the many-atom model for ${\mathrm{\Omega }}_p=0.05{\mathrm{\Gamma }}_{12}$ as a function of $t$. The green dashed-dotted line is for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$ with a high atomic density ($N_a=1.2\times {10}^{10}{\mathrm{cm}}^{-3}$) and hence significant Rydberg interaction; the brown dotted line is for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$ with a low atomic density ($N_a=1\times {10}^8{\mathrm{cm}}^{-3}$) and hence negligible Rydberg interaction. Results for a large control field, i.e., ${\mathrm{\Omega }}_c=2\pi \times 8\mathrm{MHz}$, are also shown. In both panels, blue solid lines and red dashed lines are for the case with significant Rydberg interaction and the case with negligible Rydberg interaction, respectively.

Figure 3

Transient response behavior of the Rydberg-EIT as a function of the probe-field detuning $\mathrm{\Delta }(\equiv {\mathrm{\Delta }}_2={\mathrm{\Delta }}_3)$. (a) Normalized absorption spectrum $\mathrm{Im}\left({\rho }_{21}\right)$ for $t=0$, where ${\mathrm{\Omega }}_c=0$. (b) $\mathrm{Im}\left({\rho }_{21}\right)$ at $t=0.14\mu \mathrm{s}$ for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$. (c) $\mathrm{Im}\left({\rho }_{21}\right)$ at $t=1.2\mu \mathrm{s}$ for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$. Both panels (b) and (c) are obtained from the two-atom model with ${\mathrm{\Omega }}_p=0.2{\mathrm{\Gamma }}_{12}$, where the blue solid line is the EIT spectrum with the Rydberg interaction $V_{AB}=1\mathrm{GHz}$ ($r_{AB}=3.10\mu \mathrm{m}$), while the red dashed line is the EIT spectrum without the Rydberg interaction ($V_{AB}=0$). Panels (d) and (e) are, respectively, the same with (a) and (b), but obtained by the many-atom model with ${\mathrm{\Omega }}_p=0.03{\mathrm{\Gamma }}_{12}$, where the blue solid line is the EIT spectrum with a high atomic density $N_a=1.2\times {10}^{10}{\mathrm{cm}}^{-3}$ (significant Rydberg interaction), and the red dashed line is the EIT spectrum with a low atomic density $N_a=1\times {10}^8{\mathrm{cm}}^{-3}$ (negligible Rydberg interaction).

Figure 4

The response time $T_R$ of the Rydberg-EIT for as a function of $r_{AB}$, obtained with the two-atom model for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$ (blue solid line) and ${\mathrm{\Omega }}_c=2\pi \times 8\mathrm{MHz}$ (red dashed line), with ${\mathrm{\Omega }}_p=0.3{\mathrm{\Gamma }}_{12}$.

Figure 5

Response time $T_R$ of the EIT as a function of the probe-field Rabi frequency ${\mathrm{\Omega }}_p$. Line 1 (line 2): $T_R$ for the EIT with the Rydberg interaction obtained in the many-atom (two-atom) model. Line 3 (line 4): $T_R$ for the EIT with no Rydberg interaction obtained in the many-atom (two-atom) model. Three blue points along the decreasing direction of line 1 indicate the tendency of the response time of the EIT with the Rydberg interaction as ${\mathrm{\Omega }}_p$ is increased, which means that $T_R$ will be decreased further for larger probe field.

Figure 6

Transient response behavior of the Rydberg-EIT as a function of the probe-field detuning $\mathrm{\Delta }(\equiv {\mathrm{\Delta }}_2={\mathrm{\Delta }}_3)$. (a) Normalized dispersion spectrum $\mathrm{Re}\left({\rho }_{21}\right)$ for $t=0$, where ${\mathrm{\Omega }}_c=0$. (b) $\mathrm{Re}\left({\rho }_{21}\right)$ at $t=0.14\mu \mathrm{s}$ for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$. (c) $\mathrm{Re}\left({\rho }_{21}\right)$ at $t=1.2\mu \mathrm{s}$ for ${\mathrm{\Omega }}_c=2\pi \times 4\mathrm{MHz}$. Both panels (b) and (c) are obtained from the two-atom model with ${\mathrm{\Omega }}_p=0.2{\mathrm{\Gamma }}_{12}$, where the blue solid line is the EIT spectrum with the Rydberg interaction $V_{AB}=1\mathrm{GHz}$ ($r_{AB}=3.10\mu \mathrm{m}$), and the red dashed line is the EIT spectrum without the Rydberg interaction ($V_{AB}=0$). Panels (d) and (e) are respectively the same with (a) and (b), but obtained by the many-atom model with ${\mathrm{\Omega }}_p=0.03{\mathrm{\Gamma }}_{12}$, where the blue solid line is the EIT spectrum with a high atomic density $N_a=1.2\times {10}^{10}{\mathrm{cm}}^{-3}$ (significant Rydberg interaction), and the red dashed line is the EIT spectrum with a low atomic density $N_a=1\times {10}^8{\mathrm{cm}}^{-3}$ (negligible Rydberg interaction)

Figure 7

Definition of the response time of a transient response process, described by a response function $f\left(\tau \right), \tau$ is dimensionless time. (a) Example: $f\left(\tau \right)=e^{-\tau }sin(3\tau +0.4\pi )+0.001$. The blue rectangle indicates that the variation of $f\left(\tau \right)$ reaches within the range $2{\mathrm{\Delta }}_{\mathrm{err}}$ around the steady-state value $f\left(\infty \right)=0.001$. The blue point denotes the response time. (b) Amplification of the blue rectangle shown in (a). The region with green color is the permitted relative error range for defining response time, marked by the upper boundary $f\left(\infty \right)+\left|f\left(\infty \right)\right|{\mathrm{\Delta }}_{\mathrm{err}}$ and the lower boundary $f\left(\infty \right)-\left|f\left(\infty \right)\right|{\mathrm{\Delta }}_{\mathrm{err}}$, with ${\mathrm{\Delta }}_{\mathrm{err}}=0.05$. The blue point is the response time of the transient response process.