Three-photon process for producing a degenerate gas of metastable alkaline-earth-metal atoms

We present a method for creating a quantum degenerate gas of metastable alkaline-earth-metal atoms. This has yet to be achieved due to inelastic collisions that limit evaporative cooling in the metastable states. Quantum degenerate samples prepared in the ${}^1{S}_0$ ground state can be rapidly transferred to either the ${}^3{P}_2$ or ${}^3{P}_0$ state via a coherent three-photon process. Numerical integration of the density-matrix evolution for the fine structure of bosonic alkaline-earth-metal atoms shows that transfer efficiencies of $\simeq 90$% can be achieved with experimentally feasible laser parameters in both Sr and Yb. Importantly, the three-photon process can be set up such that it imparts no net momentum to the degenerate gas during the excitation, which will allow for studies of metastable samples outside the Lamb-Dicke regime. We discuss several experimental challenges to successfully realizing our scheme, including the minimization of differential ac Stark shifts between the four states connected by the three-photon transition.

Figure 1

Laser configuration for the three-photon excitation scheme. A Bose-Einstein condensate (BEC) sits in an optical dipole trap (ODT) in a region with a uniform magnetic field, $\vec{B}=B\widehat{z}$. Three lasers cross at the BEC and drive the transition to the metastable state. These lasers address the ${}^1{S}_0\to {}^3{P}_1,{}^3{P}_1\to {}^3{S}_1$, and ${}^3{S}_1\to {}^3{P}_{0\left(2\right)}$ transitions with Rabi frequencies of ${\mathrm{\Omega }}_{{}^1{S}_0{,}^3{P}_1},{\mathrm{\Omega }}_{{}^3{P}_1{,}^3{S}_1}$, and ${\mathrm{\Omega }}_{{}^3{S}_1{,}^3{P}_{0\left(2\right)}}$, respectively. The angles of incidence, ${\phi }_{{}^3{P}_1{,}^3{S}_1}$ and ${\phi }_{{}^1{S}_0{,}^3{P}_1}$, between the excitation laser beams and the $x$ axis can be chosen so as to eliminate the net momentum transfer to the condensate during the excitation process. The double-headed black arrows indicate the polarization of the optical fields, which decomposes to either $|\sigma =1\rangle +|\sigma =-1\rangle$ or $|\sigma =0\rangle$ in the circular basis, depending on whether the laser is polarized parallel or perpendicular to the magnetic field. The angle between the ODT polarization vector and the magnetic field, $\theta$, can be tuned to control dephasing due to differential ac Stark shifts between the four atomic states (see Sec. 3).

Figure 2

The Zeeman level structure of bosonic AE atoms relevant for the three-photon process. The colored double-headed arrows indicate dipole-allowed transitions for each optical field after projection of its polarization onto the quantization axis. These arrows are solid for the transitions closest to resonance and dotted for transitions that are far off resonance. The polarization projection depends on the angle, ${\alpha }_{\ell ,{\ell }^{\prime }}$, between the laser polarization vector and the magnetic field, $\vec{B}={\mathrm{\Omega }}_B\widehat{z}/{\mu }_B$ (see Fig. 1), where $\ell$ $\left({\ell }^{\prime }\right)$ is the term symbol of the lower (upper) state of the transition and we have taken $\hslash =1$. The excitation laser strength is $E_{\ell ,{\ell }^{\prime }}={\mathrm{\Omega }}_{\ell ,{\ell }^{\prime }}/\langle {\ell }^{\prime }\parallel \vec{d}\parallel \ell \rangle$ with $\langle {\ell }^{\prime }\parallel \vec{d}\parallel \ell \rangle$ the reduced dipole matrix element. The detuning, ${\mathrm{\Delta }}_{\ell }$, is referenced to the lowest-energy Zeeman state of $|\ell \rangle$. Here, we have depicted a three-photon transition to either ${|}^3{P}_2,-2\rangle$ or ${|}^3{P}_0,0\rangle$, but ${|}^3{P}_2,2\rangle$ can be reached by reversing the magnetic field or detuning each laser to the blue of the highest-energy Zeeman state that it addresses. Radiofrequency transitions from ${|}^3{P}_2,\pm 2\rangle$ can prepare other Zeeman states in ${}^3{P}_2$.

Figure 3

Results of the numerical integration of the optical Bloch equations for bosonic AE atoms given in Eq. (5). Solid curves show ground-state populations while dashed and dash-dotted curves show target-state populations $({|}^3{P}_0,0\rangle$ and ${|}^3{P}_2,-2\rangle$, respectively). The populations in other states are negligible and not shown. Each laser turns on instantaneously at time zero and is instantaneously extinguished at time $t_e$. All the parameters for these simulations can be found in Table 1.

Figure 4

The excitation fraction of the target state, ${|}^3{P}_2,\pm 2\rangle$ $\left({|}^3{P}_0,0\rangle \right)$, at $t_{\pi }$ as a function of the three-photon detuning is shown in the top (bottom) row. The left column shows the results for Sr and the right column shows the results for Yb. Except for the three-photon detuning, which is varied, the parameters for the simulations are identical to those used in Fig. 3 and reported in Table 1. The ${\text{sinc}}^2$ fits (solid lines) yield full widths at half maximum for the three-photon transitions to ${}^3{P}_2$ $\left({}^3{\mathrm{P}}_0\right)$ of 10.0 (7.2) kHz for Sr and of 17.6 (15.5) kHz for Yb. The amplitude mismatch between the wings of the fit and the simulation is likely caused by the convolution of power broadening and the transform limit.

Figure 5

The ac polarizabilities of ${|}^3{P}_0,0\rangle$ (solid orange), $|{}^{3}P_1,\pm 1\rangle$ (dotted blue), ${|}^3{P}_2,\pm 2\rangle$ (dash-dotted magenta), and ${|}^3{S}_1,\pm 1\rangle$ (dashed green) relative to $|{}^{1}S_0,0\rangle$ in atomic units. The top two panels show the polarizability of Sr for ODT light polarized parallel and perpendicular to the quantization axis, respectively. The bottom two panels are the corresponding plots for Yb. Every zero crossing indicates a magic wavelength. We indicate the magic wavelengths of specific interest here using black arrows. For ${|}^3{P}_2,\pm 2\rangle$ and $|{}^{3}P_1,\pm 1\rangle$, the laser polarization can tune the indicated magic wavelengths over a wide range (see Table 3). Note that the position of the ${|}^3{P}_0,0\rangle$ and ${|}^3{P}_2,\pm 2\rangle$ magic wavelengths flips with respect to the $|{}^{3}P_1,\pm 1\rangle$ magic wavelength in the near IR as the polarization angle changes from $0^{\circ }$ to ${90}^{\circ }$. This indicates the presence of a doubly magic wavelength at a particular polarization angle that equalizes the ac Stark shifts of three energy levels.