Rydberg optical Feshbach resonances in cold gases

We propose a scheme to efficiently tune the scattering length of two colliding ground-state atoms by off-resonantly coupling the scattering state to an excited Rydberg molecular state using laser light. For the $s$-wave scattering of two colliding ${}^{87}\mathrm{Rb}$ atoms, we demonstrate that the effective optical length and pole strength of this Rydberg optical Feshbach resonance can be tuned over several orders of magnitude, while incoherent processes and losses are minimized. Given the ubiquity of Rydberg molecular states, this technique should be generally applicable to homonuclear atomic pairs as well as to atomic mixtures with $s$-wave (or even $p$-wave) scattering.

Figure 1

(a) Sketch of a Rydberg optical Feshbach resonance: Two ground-state atoms colliding with relative energy $\epsilon$ are off-resonantly coupled by laser light (vertical arrow) to an eigenstate of an excited Rydberg molecular potential with spontaneous emission rate ${\gamma }_{\mathrm{m}}$ (wiggling arrow). Here, $V_L\left(R\right)$ and $V_U\left(R\right)$ are the ground and Rydberg Born-Oppenheimer potentials, respectively [32]. $\mathrm{\Omega }$ is the laser Rabi frequency, $\delta$ is the detuning from molecular resonance and $\mathrm{\Delta }$ is the detuning from the asymptotic atomic Rydberg state energy, and $R_C$ is the Condon point. (b) Lowest-lying Rydberg molecular Born-Oppenheimer potential [32] (continuous black line) of the $\mathrm{Rb}(27S_{1/2}$)$\mathrm{Rb}(5S_{1/2},F=1$) $M_K=1/2$ dimer, together with the computed eigenstates of the potential (see text).

Figure 2

Parameters of the harmonic approximation of the outermost minimum of the Rydberg molecular state plotted in Fig. 1. $\omega \left(n\right)$ is the frequency of the fitted harmonic potential, $|E_{\text{min}}\left(n\right)|$ is the corresponding depth of this potential, and $R_C$ is the principal quantum number dependent location of this minimum used in the Rydberg OFR [32].

Figure 3

Analytical prediction Eq. (5) (solid line) and coupled channel results (dashed line) of the real (green thin lines) and imaginary (blue tick lines) parts of the scattering length $\alpha \left(\epsilon \right)$ vs the detuning $\delta /{\gamma }_m$ from the lowest-lying molecular states of the dimers (a) $\mathrm{Rb}(27S_{1/2}$)$\mathrm{Rb}(5S_{1/2},F=1$) and (b) $\mathrm{Rb}(40S_{1/2}$)$\mathrm{Rb}(5S_{1/2},F=1$) and $\mathrm{\Omega }/\left(2\pi \right)=0.5$ and $0.2\mathrm{MHz}$, respectively. The two ground-state Rb atoms collide with relative energy $\epsilon =1\mu \text{K}·k_B$, with $k_B$ being the Boltzmann constant. The analytical (coupled channel) value of the optical length ${\ell }_{\mathrm{opt}}\left(\epsilon \right)$ is ${\ell }_{\mathrm{opt}}\left(\epsilon \right)\simeq 4900a_0$ ($5270a_0$) and $11500a_0$ ($13400a_0$) for panels (a) and (b), respectively. The linewidth of the molecular state is ${\gamma }_m/\left(2\pi \right)\simeq 144$ and 54 kHz in panels (a) and (b), respectively.

Figure 4

Comparison between the thermal average of the scattering length ${\overline{\alpha }}^T$ (dashed lines) [see definition in Eq. (7)] and the energy-dependent scattering length $\alpha \left(\epsilon \right)$ (solid lines) for the molecular state of $\mathrm{Rb}(27S_{1/2}$)$\mathrm{Rb}(5S_{1/2}$). The real and imaginary parts of the thermal average and the energy-dependent scattering lengths are indicated by thin-green and thick-blue lines, respectively. The results are obtained with the single resonance approximation (5), and calculated for the Rydberg excitation $\mathrm{Rb}(27S_{1/2}$), temperatures $T=1$ and $5\mu \text{K}$ (dashed lines), and energies $\epsilon =2\pi \times {10}^{-12}{k}_B\hslash$ and $2\pi \times 5\times {10}^{-12}{k}_B\hslash \mathrm{MHz}$ (solid lines). Insets: Details of the real and imaginary parts of the scattering lengths around $\delta =0$. The thermally averaged scattering length is smaller than the single energy scattering length and also shows an asymmetric dependence on $\delta$ (in contrast with the symmetrical behavior of the single energy curves).

Figure 5

For $T=1\mu \text{K}$, (a) thermally averaged pole strength $s_{\mathrm{res}}^T$ and (b) thermally averaged optical length ${\ell }_{\mathrm{opt}}^T$ vs the Rydberg excitation $n$. See text and Table 1.