High-resolution molecular spectroscopy for producing ultracold absolute-ground-state ${}^{23}\mathrm{Na}{}^{87}\mathrm{Rb}$ molecules

We report a detailed molecular spectroscopy study of the lowest excited electronic states of ${}^{23}\mathrm{Na}{}^{87}\mathrm{Rb}$ for producing ultracold ${}^{23}\mathrm{Na}{}^{87}\mathrm{Rb}$ molecules in the electronic, rovibrational, and hyperfine ground state. Starting from weakly bound Feshbach molecules, a series of vibrational levels of the $A{}^1{\mathrm{\Sigma }}^{+}\text{−}b{}^3\mathrm{\Pi }$ coupled excited states is investigated. After resolving, modeling, and interpreting the hyperfine structure of several lines, we successfully identify a long-lived level resulting from the accidental hyperfine coupling between the $0^{+}$ and $0^{-}$ components of the $b{}^3\mathrm{\Pi }$ state, satisfying all the requirements for the population transfer toward the lowest rovibrational level of the $X{}^1{\mathrm{\Sigma }}^{+}$ state. Using two-photon spectroscopy, its binding energy is measured to be 4977.308(3) ${\mathrm{cm}}^{-1}$. We calibrate all the transition strengths carefully and also demonstrate Raman transfer of Feshbach molecules to the absolute ground state.

Figure 1

Two-photon path used in this work to create ground-state ${}^{23}\mathrm{Na}{}^{87}\mathrm{Rb}$ molecules and the relevant ${}^{23}\mathrm{Na}{}^{87}\mathrm{Rb}$ potential energy curves including spin-orbit coupling involved in the proposed three-level system. The $|1\rangle$ state is the Feshbach state (for which the squared radial probability density is drawn) with a dominant $a{}^3{\mathrm{\Sigma }}^{+}$ character and is coupled through the pump laser $L_1$ to the $|2\rangle$ level belonging to the strongly mixed $A{}^1{\mathrm{\Sigma }}^{+}\text{−}b{}^3\mathrm{\Pi }$ system. The dump laser $L_2$ couples $|2\rangle$ to the absolute NaRb ground-state level $|3\rangle$. The dashed area represents the energy range explored in the experiment.

Figure 2

Computed absolute values of the TDMs for the pump and dump transitions. The energy range represented by the dashed area in Fig. 1 and explored in the experiment is featured as vertical dashed lines. In the model (see the text) the $|1\rangle$ state is the uppermost vibrational level $v_a=21$ with $J_a=0$ and $|3\rangle \equiv |v_X=0,J_X=0\rangle$. Open red squares (green diamonds) denote the TDMs for the pump transition driven by $L_1$ (lower horizontal energy scale) from level $|1\rangle$ toward $A\text{−}{b}_{0^{+}}$ ($A\text{−}{b}_1$). Open black circles denote the TDMs for the dump transition driven by $L_2$ (upper horizontal energy scale) from the $A\text{−}{b}_{0^{+}}$ component of level $|2\rangle$ to $|3\rangle$.

Figure 3

One-photon loss spectroscopy of Feshbach molecules for searching the intermediate level for STIRAP. Vibrational levels from $v^{\prime }=53$ to $v^{\prime }=67$ of the mixed $A\text{−}{b}_{0^{+}}\text{−}{b}_1$ system are observed. Note that the experimental signal for $v^{\prime }=54,56$ was too weak to be observed.

Figure 4

(a) High-resolution spectrum recorded at 335.6 G of the $|v^{\prime }=60,J^{\prime }=1\rangle$ level (with dominant $b_1$ character) exhibiting a large and fully resolved hyperfine structure. The data shown by black squares are obtained with vertical polarized light ($\pi$ transitions), while the red circles are obtained with horizontal polarization ($\sigma$ transitions). Being a piecewise spectrum, the intensities of the lines are arbitrarily normalized from one piece to the other. (b) Comparison between the experimentally extracted TDMs (lower panel) with those yielded by the perturbative model (upper panel). Lines are represented as vertical bars as follows. The upper panel is for $\sigma$ transitions (with $M_F^{\prime }=3$ in red and and $M_F^{\prime }=1$ in green) and $\pi$ transitions (with $M_F^{\prime }=2$ in blue) and the lower panel is for $\sigma$ transitions in orange and $\pi$ transitions in black. (c) Same as (b) for the $|v^{\prime }=60,J^{\prime }=2\rangle$ level with zero energy corresponding to an $L_1$ frequency of 244.534 386 THz.

Figure 5

High-resolution spectrum recorded at 335.6 G of the $|v^{\prime }=59,J^{\prime }=1\rangle$ level (with dominant $A\text{−}{b}_{0^{+}}$ character) with partially resolved hyperfine structure. Black squares denote vertically (v) polarized light ($\pi$ transitions) and red circles horizontally (h) polarized light ($\sigma$ transitions). The vertical bars indicate line positions from the theoretical model with the same color convention as in Fig. 4.

Figure 6

(a) High-resolution spectrum (recorded at 335.6 G) showing 21 resolved lines for the hyperfine structure of the $|v^{\prime }=55,J^{\prime }=1\rangle$ level. Black squares denote vertical polarized light ($\pi$ transitions) and red circles horizontal polarization ($\sigma$ transitions). The vertical bars indicate line positions from the theoretical model with the same color convention as in Fig. 4. The red open circle marks the hyperfine level used in STIRAP. (b) Blowup of the central group of lines. The bars indicate the position of the calculated levels for $\sigma$ transitions ($M_F^{\prime }=3$ in red and $M_F^{\prime }=1$ in green) and $\pi$ transitions ($M_F^{\prime }=2$ in blue).

Figure 7

Computed values of the energy spacings $\mathrm{\Delta }$ between the vibrational levels of $0^{+},v_{0^{+}}^{\prime },J^{\prime }=1$ with a dominant $b$ character and the corresponding $0^{-},v_{0^{-}}^{\prime },J^{\prime }=0$ levels. Calculations are done with an ab initio $R$-dependent SO coupling [22] (black circles) and with an $R$-independent (atomic) one. The $v^{\prime }=55$ level is circled as the one that is used for STIRAP.

Figure 8

Computed (upper panel) and observed (lower panel) spectra for the $|v^{\prime }=55,J=1\rangle$ level. The zero energy corresponds to the level energy position without Zeeman and/or hfs. The upper panel is for $\sigma$ transitions (with $M_F^{\prime }=3$ in red and and $M_F^{\prime }=1$ in green) and $\pi$ transitions (with $M_F^{\prime }=2$ in blue) and the lower panel is for $\sigma$ transitions (orange) and $\pi$ transitions (black). The experimental intensities are calibrated following Sec. 3d.

Figure 9

Pump transition strength calibration from the Feshbach state $|1\rangle$ to the selected hyperfine level $|2\rangle$ in Fig. 6, based on the measurement of the variations of the population of Feshbach molecules when the $L_1$ laser is on with a power of 0.456 mW and a waist of about 45 $\mu \mathrm{m}$. (a) Line shape obtained with an $L_1$ pulse length of 10 $\mu \mathrm{s}$. (b) Time evolution of Feshbach molecules with $L_1$ on resonance. The red curves result from the simultaneous fit of the two data sets to extract the TDM and the excited-state lifetime.

Figure 10

Two-photon spectroscopy of the $|v^{″}=0,J^{″}=0\rangle$ and $|v^{″}=0,J^{″}=2\rangle$ ground-state levels via the $|v^{\prime }=55,J^{\prime }=1\rangle$ excited intermediate level. The scheme indicates that $L_1$ is kept on resonance, while the frequency of $L_2$ is expressed as the detuning ${\mathrm{\Delta }}_2$ with respect to the resonance. A much larger dump transition Rabi frequency (induced by a laser power as high as 27 699.9 mW/${\mathrm{cm}}^2$) than the pump one is used to obtain this spectrum, so the hyperfine structure is not resolved. The red curves are Gaussian fits for extracting the line centers.

Figure 11

Two-photon spectrum of the $X{}^1{\mathrm{\Sigma }}^{+}$, $|v^{″}=0,J^{″}=0\rangle$ level with resoled hyperfine structure. The Rabi frequency of the vertically polarized pump ($v_p$) laser $L_1$ is about $2\pi \times 0.096$ MHz. Two $M_F^{″}=+2$ levels are observed with a vertically polarized dump ($v_d$, or $\pi$-polarized) laser $L_2$ with an intensity of 5.1 mW/${\mathrm{cm}}^2$; one $M_F=+3$ and three $M_F=+1$ hyperfine levels are observed with an $L_2$ horizontally polarized dump ($h_d$) laser $L_2$ with an intensity of 51.0 mW/${\mathrm{cm}}^2$. The solid curves are Gaussian fits of the data for extracting the line centers and the vertical dashed lines depict the theoretical predictions (Table 2).

Figure 12

Two-photon dark-state spectroscopy for calibrating the dump transition strength. The $L_2$ frequency is kept on resonance and the $L_1$ frequency (and thus the detuning ${\mathrm{\Delta }}_1$) is scanned. The red solid curve is the result of the fit of the signal using Eq. (3) with a pump Rabi frequency of $2\pi \times 0.169$ MHz and an excited lifetime of $\tau =238$ ns. The dump Rabi frequency extracted from the fit is $2\pi \times 0.225$ MHz, which corresponds to a normalized Rabi frequency of ${\overline{\mathrm{\Omega }}}_2=2\pi \times 23.2\mathrm{kHz}\times \sqrt{\mathrm{I}/(\mathrm{mW}/{\mathrm{cm}}^2)}$ and a TDM of 0.021 a.u.