Singlet Pathway to the Ground State of Ultracold Polar Molecules

TL;DR

The paper presents a new two-photon pathway to produce ultracold polar molecules in their ground state using only singlet-to-singlet optical transitions, simplifying the process by avoiding complex intermediate state searches.

Contribution

It introduces a novel singlet pathway that bypasses the need for hyperfine mixing, enabling efficient transfer to the ground state in bi-alkali molecules.

Findings

Successfully demonstrated the pathway with ${}^{6} extrm{Li}{}^{40} extrm{K}$ molecules.

Achieved large, balanced Rabi frequencies for both transitions.

Method applicable to other molecular species.

Abstract

Starting from weakly bound Feshbach molecules, we demonstrate a two-photon pathway to the dipolar ground state of bi-alkali molecules that involves only singlet-to-singlet optical transitions. This pathway eliminates the search for a suitable intermediate state with sufficient singlet-triplet mixing and the exploration of its hyperfine structure, as is typical for pathways starting from triplet dominated Feshbach molecules. By selecting a Feshbach state with a stretched singlet hyperfine component and controlling the polarization of the excitation laser, we assure coupling to only a single hyperfine component of the $\textrm{A}^{1}\Sigma^{+}$ excited potential, even if the hyperfine structure is not resolved. Similarly, we address a stretched hyperfine component of the $\textrm{X}^{1}\Sigma^{+}$ rovibrational ground state, and therefore an ideal three level system is established. We demonstrate this pathway with ${}^{6}\textrm{Li}{}^{40}\textrm{K}$ molecules. By exploring deeply bound states of the $\textrm{A}^{1}\Sigma^{+}$ potential, we are able to obtain large and balanced Rabi frequencies for both transitions. This method can be applied to other molecular species.

Tables

Table 1. Table 1: Measured resonant frequencies and normalized Rabi frequencies of transitions from X 1 ​ Σ + ​ | v ′′ = 47 ⟩ superscript X 1 superscript Σ ket superscript 𝑣 ′′ 47 \textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=47} to deeply bound A 1 ​ Σ + ​ | v ′ = 23 − 29 ⟩ superscript A 1 superscript Σ ket superscript 𝑣 ′ 23 29 \textrm{A}^{1}\Sigma^{+}\ket{v^{\prime}=23-29} excited states.

${\text{X}}^1{\Sigma }^{+}|v^{\prime \prime }=47⟩\to {\text{A}}^1{\Sigma }^{+}|v^{\prime }⟩$
$v^{\prime }$	$f$ (THz)	${\overline{\Omega }}_P$ (kHz/$\sqrt{\mathrm{mW}/{\mathrm{cm}}^2}$)
23	267.842232(1)	2$\pi \times 2.3\left(2\right)$
24	271.383075(1)	2$\pi \times 1.6\left(2\right)$
25	274.898734(2)	2$\pi \times 5.1\left(2\right)$
26	278.388059(2)	2$\pi \times 0.7\left(1\right)$
27	281.846301(1)	2$\pi \times 8.8\left(3\right)$
28	285.281804(1)	2$\pi \times 3.0\left(3\right)$
29	288.688289(1)	2$\pi \times 3.1\left(3\right)$

Table 2. Table 2: Measured resonance frequencies and normalized Rabi freqencies for the coupling (a) between A 1 ​ Σ + ​ | v ′ = 29 ⟩ superscript A 1 superscript Σ ket superscript 𝑣 ′ 29 \textrm{A}^{1}\Sigma^{+}\ket{v^{\prime}=29} and low lying X 1 ​ Σ + ​ | v ′′ = 2 − 3 ⟩ superscript X 1 superscript Σ ket superscript 𝑣 ′′ 2 3 \textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=2-3} ; (b) between A 1 ​ Σ + ​ | v ′ = 23 ⟩ superscript A 1 superscript Σ ket superscript 𝑣 ′ 23 \textrm{A}^{1}\Sigma^{+}\ket{v^{\prime}=23} and X 1 ​ Σ + ​ | v ′′ = 0 − 2 ⟩ superscript X 1 superscript Σ ket superscript 𝑣 ′′ 0 2 \textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=0-2} using π 𝜋 \pi polarization as initially the hyperfine coupling of excited states is not considered.

(a)                    ${\text{A}}^1{\Sigma }^{+}|v^{\prime }=29⟩\to {\text{X}}^1{\Sigma }^{+}|v^{\prime \prime }⟩$
$v^{\prime \prime }$	$f$ (THz)	${\overline{\Omega }}_S$ (kHz/$\sqrt{\mathrm{mW}/{\mathrm{cm}}^2}$)
2	458.389566(5)	2$\pi \times 43\left(8\right)$
3	451.871767(10)	2$\pi \times 152\left(52\right)$
(b)                    ${\text{A}}^1{\Sigma }^{+}|v^{\prime }=23⟩\to {\text{X}}^1{\Sigma }^{+}|v^{\prime \prime }⟩$
$v^{\prime \prime }$	$f$ (THz)	${\overline{\Omega }}_S$ (kHz/$\sqrt{\mathrm{mW}/{\mathrm{cm}}^2}$)
0	450.841975(2)	2$\pi \times 12\left(2\right)$
1	444.148838(5)	2$\pi \times 84\left(18\right)$
2	437.543516(8)	2$\pi \times 256\left(59\right)$

Full text

The rich physics of ultracold dipolar quantum gases Trefzger et al. (2011); Baranov et al. (2012) based on the long-range and anisotropic interaction has moved into the focus of many experimental research activities. Studies on Rydberg atoms Schauss et al. (2015); Labuhn et al. (2016) and recent results with strongly magnetic atoms Böttcher et al. (2019); Chomaz et al. (2019) represent two important realizations of dipolar systems. A third approach is using polar molecules, which combine strong dipolar interactions and long lifetimes when trapped in deep optical lattices or low dimensional traps. Therefore, they are prominent candidates for research in the areas of quantum simulation Micheli et al. (2006); Gadway and Yan (2016) and quantum information DeMille (2002); Yelin et al. (2006); André et al. (2006). Moreover, polar molecules are an important platform for metrology Kozlov and Labzowsky (1995); Hudson et al. (2006); Andreev et al. (2018) and are a testbed for ultracold chemistry Carr et al. (2009); Bell and Softley (2009); Quéméner and Julienne (2012); Jin and Ye (2012). For the difficult task of producing trapped ultracold molecular samples several methods were explored Wynar et al. (2000); Bethlem et al. (2000); Kraft et al. (2006); Shuman et al. (2010), and over the recent years much progress was made with laser cooling of molecules from a buffer gas source Anderegg et al. (2018). However, the highest numbers and densities and lowest temperatures for ultracold molecular samples were produced by association of ultracold atoms via magnetically tunable Feshbach resonances Köhler et al. (2006). To access large dipole moments, such weakly bound molecules need to be transferred into their rovibrational ground state by stimulated Raman adiabatic passage (STIRAP) Vitanov et al. (2017). The ground state was achieved for several heteronuclear bi-alkali combinations Ni et al. (2008); Takekoshi et al. (2014); Molony et al. (2014); Park et al. (2015a); Guo et al. (2016). This bottom-up approach is the only one that recently reached the remarkable milestone of quantum degeneracy for the case of KRb De Marco et al. (2019). To obtain molecules with higher ground state dipole moments other bi-alkali combinations with larger mass ratios are being investigated. Obtaining efficient ground state transfer requires as a cornerstone studying the molecular structure and identifying a suitable electronically excited state. Most experiments so far follow the strategy that was pioneered in KRb Ni et al. (2008) by using an intermediate excited state of mixed singlet and triplet character. This facilitates a coherent transfer from a Feshbach molecule state with triplet character to the singlet ground state. To gain full control over the molecular state, spectroscopically resolved hyperfine components of the excited state need to be identified, as coupling to unresolved hyperfine levels will lead to a low STIRAP efficiency Vitanov and Stenholm (1999). Identifying such states typically requires extensive spectroscopic surveys Kotochigova et al. (2009); Debatin et al. (2011); Zabawa et al. (2011); Park et al. (2015b); Guo et al. (2017); Rvachov et al. (2018) to find states where the triplet admixture leads to large hyperfine splitting, while the singlet admixture is strong enough to address the ground state. Moreover, even if the hyperfine structure is resolved, off-resonant coupling to other hyperfine components of the intermediate state can lead to an undesired superposition of several hyperfine components of the ground state Liu et al. (2019).

In this letter, we present a pathway to the ground state that avoids the use of perturbed potentials by only using singlet-to-singlet transitions for both pump and Stokes couplings as illustrated in Fig. 1. For addressing the $\textrm{X}^{1}\Sigma^{+}$ rovibrational ground state of the molecules, the $\textrm{A}^{1}\Sigma^{+}$ potential is explored to search for a suitable intermediate state. Here, we demonstrate that by making use of deeply bound vibrational states in this potential, we are able to achieve sufficient Franck-Condon factors (FCF) to the ground state with moderate laser powers. The use of deeply bound vibrational states has the advantage that states of mixed singlet-triplet character are less frequent, due to the larger level spacing. To allow for the excitation to the unperturbed $\textrm{A}^{1}\Sigma^{+}$ potential, it is necessary to employ a Feshbach resonance with significant admixture from the singlet ground state potential. For the $\textrm{A}^{1}\Sigma^{+}$ potential, its hyperfine structure is typically not resolved due to the absence of spin and internal magnetic field. We then demonstrate how to address a sole hyperfine component of the ground state, even if the hyperfine structure of the excited state is not resolvable. Our pathway is facilitated by selecting the Feshbach resonance, such that the singlet admixture to the Feshbach molecular state consists of only one hyperfine component. This singlet component corresponds to a fully stretched state of the nuclear spin projections. Starting from the stretched state and applying circularly polarized spectroscopy light ensures that only one ground state hyperfine component can be addressed. In the following, this scheme is applied to ${}^{6}\textrm{Li}{}^{40}\textrm{K}$ molecules. We describe the selection of a suitable Feshbach state and present our spectroscopic results for several deeply bound vibrational states of the $\textrm{A}^{1}\Sigma^{+}$ potential as well as our measurements of the ground state by two-photon spectroscopy.

Our experiments start from a quantum degenerate mixture of $10^{5}$ ${}^{6}\textrm{Li}$ and $8\times 10^{4}$ ${}^{40}\textrm{K}$ atoms in a magnetic trap, which is sympathetically cooled via evaporative cooling of ${}^{87}\textrm{Rb}$ Taglieber et al. (2008). After expunging ${}^{87}\textrm{Rb}$ from the magnetic trap, the Fermi-Fermi mixture is transferred into a crossed optical dipole trap. This is followed by preparation of suitable atomic hyperfine states and magneto-association of typically $10^{4}$ Feshbach molecules Voigt et al. (2009). The molecules are detected by absorption imaging in the presence of the magnetic field after Feshbach molecular association, where atoms and weakly bound molecules can be simultaneously detected. We reduce the background due to remaining unbound ${}^{6}\textrm{Li}$ atoms by transferring them to a different Zeeman level, which is off-resonant to the imaging light.

For the molecular association in previous work we employed a Feshbach resonance at $15.54\,$mT Voigt et al. (2009). To select an initial molecular state for the spectroscopy that contains a sole singlet admixture, we use a different Feshbach resonance at $21.56\,$mT. This is achieved by preparing the lithium and potassium atoms in the $\ket{F_{\mathrm{Li}}=\frac{1}{2},m_{F,\mathrm{Li}}=-\frac{1}{2}}$ and $\ket{F_{\mathrm{K}}=\frac{9}{2},m_{F,\mathrm{K}}=-\frac{9}{2}}$ hyperfine states respectively, where $F$ is the hyperfine quantum number, and $m_{F}$ is the projection of hyperfine angular momentum on the quantization axis. The sum of the projection quantum numbers is $M=-5$, which is conserved during molecule formation. As most Feshbach resonances of the ${}^{6}\textrm{Li}-{}^{40}\textrm{K}$ mixture are narrow Wille et al. (2008), the closed molecular channel should be considered. In the molecular basis $\ket{S,m_{S},m_{I,\mathrm{Li}},m_{I,\mathrm{K}}}$, the total projection quantum number then equals to $M=m_{S}+m_{I,\mathrm{Li}}+m_{I,\mathrm{K}}=-5$. Hence, only one spin singlet and three spin triplet states contribute to the closed channel: $\ket{0,0,-1,-4}$, $\ket{1,0,-1,-4}$, $\ket{1,-1,0,-4}$, and $\ket{1,-1,-1,-3}$. Since the nuclear spin quantum numbers are $I_{\mathrm{Li}}=1$ and $I_{\mathrm{K}}=4$, the only contributing spin singlet state is fully stretched. The singlet and triplet states are mixed by the hyperfine coupling, which can be well described by the Asymptotic Bound State Model Tiecke et al. (2010). Based on this, we estimate the admixture of the spin singlet state to the molecular eigenstate at the Feshbach resonance to be $52$% sup . This significant singlet character of the Feshbach state is an excellent starting point for the spectroscopy of $\textrm{A}^{1}\Sigma^{+}$.

To search for a suitable intermediate state for locating the ground state, we study the excitation from the initial Feshbach state $\textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=47,J^{\prime\prime}=0}$ to the rotational excited states in $\textrm{A}^{1}\Sigma^{+}$ potential $\textrm{A}^{1}\Sigma^{+}\ket{v^{\prime},J^{\prime}=1}$ as shown in Fig. 1, where $v$ is the vibrational quantum number. Since only ${}^{1}\Sigma$ states are involved and the $J^{\prime\prime}=0\rightarrow J^{\prime}=0$ transition is forbidden for the total angular momentum $J$ of the molecules, the quantum number $N$ for molecular rotation is not conserved. Hence, starting from a Feshbach state with $N^{\prime\prime}=0$, we can only probe excited states of $\textrm{A}^{1}\Sigma^{+}$ with $N^{\prime}=1$. As a consequence, the rotational angular momentum projection $m_{N}$ needs to be considered. In the extended uncoupled molecular basis $\ket{S,m_{S},m_{I,\mathrm{Li}},m_{I,\mathrm{K}},N,m_{N}}$ the following three states are possible $\ket{0,0,-1,-4,1,0}$, $\ket{0,0,0,-4,1,-1}$, $\ket{0,0,-1,-3,1,-1}$, if we drive $\pi$ transitions and the total projection $M=m_{I,\mathrm{K}}+m_{I,\mathrm{Li}}+m_{N}=-5$ is conserved. These states can be mixed due to hyperfine coupling originating from nuclear-spin rotation, nuclear quadrupole, and the nuclear-spin dipole interactions. In the $\textrm{A}^{1}\Sigma^{+}$ potential the hyperfine coupling constants are expected to be small sup and the hyperfine structure is not resolved. Therefore, the excitation can couple to different hyperfine components. To achieve excitation to the sole singlet hyperfine component $\ket{0,0,-1,-4,1,-1}$ in the $\textrm{A}^{1}\Sigma^{+}$ potential, we apply $\sigma^{-}$-polarized light. As stretched states do not couple, this argument is also valid, if the hyperfine coupled basis is considered. Since our Feshbach state is in $\ket{F^{\prime\prime}=5,m_{F}^{\prime\prime}=-5}$, the addressable excited hyperfine states are $F^{\prime}=4,5,6$. With $\sigma^{-}$ polarization we can address only the $\ket{F^{\prime}=6,m_{F}^{\prime}=-6}$ stretched state.

A suitable intermediate vibrational state of the $\textrm{A}^{1}\Sigma^{+}$ potential needs to have a good FCF with the Feshbach molecular state as well as the vibrational ground state of the $\textrm{X}^{1}\Sigma^{+}$ potential. On the one hand, the overlap with the ground state is favorable, if deeply bound vibrational states at long wavelengths are used, as shown in Fig. 2(b). On the other hand, at long wavelengths the good overlap with the singlet component of the closed channel dominated Feshbach molecular state ceases, as inferred from Fig. 2(a). In the experiment we cover a wavelength range of $1038\,$nm - $1120\,$nm, where good FCFs for both Pump and Stokes transitions are expected. It should be noted that the variation of the transition strength throughout this range is determined by the vibrational wave-function overlap. This is because for the $\textrm{X}^{1}\Sigma^{+}\rightarrow\textrm{A}^{1}\Sigma^{+}$ transition the transition dipole matrix element (TDM) does not vary significantly with the internuclear distance, as we find from our ab-initio calculation sup .

We perform one-photon spectroscopy for the $\textrm{A}^{1}\Sigma^{+}$ vibrational levels by switching off the optical dipole trap and applying laser pulses during time-of-flight (TOF) before imaging. We use a gain-chip laser diode, which is stabilized by a tunable frequency offset lock to a high-finesse optical resonator. The laser frequency is determined with $1\,$MHz resolution using a beat setup with an optical frequency comb system. Our identification of the transition lines is facilitated by previous polarization labeling spectroscopy in a heat pipe Grochola et al. (2012). By mass scaling of the Dunham coefficients determined by this work to the ${}^{6}\textrm{Li}{}^{40}\textrm{K}$ isotopologue sup , we are able to address individual vibrational states. Here, the large level separation occurring for deeply bound states leads to the unambiguous assignment of the vibrational level index. The locations of seven deeply bound excited states are measured with highly resolving spectroscopy, and given in Tab. 1. As an example, the spectrum for the transition to $\textrm{A}^{1}\Sigma^{+}\ket{v^{\prime}=23}$ is shown in Fig. 3(a). The laser intensity is reduced to avoid line broadening. From simultaneous curve fitting to the observed spectrum, Fig. 3(b), and the exponential decay of the molecular number with the irradiation time, Fig. 3(c), the transition strength is inferred from the normalized Rabi frequency $\bar{\Omega}_{\mathrm{p}}$. Consistent with the prediction of the FCFs shown in Fig. 2(a), we do not observe a reduction of the transition strength of this state as compared with the strengths measured at lower wavelengths as summarized in Tab. 1.

An important feature of all observed lines is the absence of hyperfine-structure for the measured linewidth of $5\,$MHz, if the excitation is driven by $\sigma^{-}$ light. This is expected as the stretched hyperfine state is the only available final state. An observable hyperfine structure is expected to occur for spin-triplet states. In this case, the atomic hyperfine splitting for potassium atoms of a few tens of MHz can be used as an estimate for the molecular case. Since no hyperfine structure is observed over the $160\,$MHz span of the measurement shown in Fig. 3(a), it can be excluded that the observed vibrational states are perturbed by a significant contribution of the $\textrm{b}^{3}\Pi$ states via spin-orbit coupling sup . Further, in similar measurements no hyperfine structure is resolved, if the excitation is driven with $\pi$-polarized light, which allows to address several hyperfine states. This is consistent with estimates of the dominant quadrupole hyperfine interaction for the $\textrm{A}^{1}\Sigma^{+}$ potential based on ab-initio calculations sup .

To address the low lying vibrational states of the $\textrm{X}^{1}\Sigma^{+}$ ground state, we make use of the stretched hyperfine state $\ket{0,0,-1,-4,1,-1}$ of the $\textrm{A}^{1}\Sigma^{+}$ potential as an intermediate state with projection quantum number $M^{\prime}=-6$. For the rovibrational ground state with $N^{\prime\prime}=0$, only the stretched hyperfine component with $M^{\prime\prime}=-5$ can couple to the excited state. In this case $\sigma^{-}$-polarized Stokes light needs to be applied. Consequently, off-resonant coupling to other hyperfine states is prevented for both transitions.

We perform two-photon spectroscopy during TOF to determine the energy difference between the rovibrational ground state and the intermediate state. For the spectroscopy, a pump pulse duration of $18\,\mathrm{\mu s}$ is used and the Stokes pulse is switched on and off $2\,\mathrm{\mu s}$ earlier and respectively later than the pump pulse.

The search for the ground state levels is assisted by the $\textrm{X}^{1}\Sigma^{+}$ potential curve of Tiemann et al. (2009). We first identify the transition frequencies to $\textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=3}$ by scanning the Stokes laser detuning, while keeping the pump laser on resonance with respect to the transition to $\textrm{A}^{1}\Sigma^{+}\ket{v^{\prime}=29}$. The Stokes light is derived from a low power diode laser system. As shown in Fig. 2(b), lower vibrational states have significantly smaller FCFs. Therefore, it is impossible to address the $\textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=0}$ with the same intermediate state, unless laser power of several Watts is available.

To facilitate the search via other intermediate vibrational levels we make use of a tunable dye laser system (Coherent 699). The DCM dye in use covers a wavelength range of $640-{}690\,$nm with a power up to $300\,$mW applied to the molecules. The frequency of the dye laser is controlled by our previously described interferometric laser frequency stabilization Brachmann et al. (2012), which allows continuous in-lock scanning of the frequency over many gigahertz in steps of $1\,$MHz. A more deeply bound intermediate state $\textrm{A}^{1}\Sigma^{+}\ket{v^{\prime}=23}$ is utilized to access the low lying vibrational states including the $\textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=0}$ with sufficient FCFs. All measured transition frequencies are summarized in Tab. 2(b). The predicted frequencies deviate from the measured values by approximately $3\,$GHz, which is within the uncertainty of $6\,$GHz given in Tiemann et al. (2009).

The results of two-photon spectroscopy for measuring the ground state energy for the $\textrm{X}^{1}\Sigma^{+}\ket{v^{\prime\prime}=0,N^{\prime\prime}=0}$ state are shown in Fig. 4. In Fig. 4(a), the Feshbach molecular signal is restored due to the Autler-Townes effect, if the Stokes laser is tuned to resonance. If the polarization of the Stokes laser is changed to $\sigma^{+}$, no coupling is expected and therefore no line is observed. Fig. 4(b) shows the Autler-Townes splitting (ATS) by keeping the Stokes laser on resonance and scanning the pump laser frequency. To confirm the correct assignment of the ground state, we address the rotationally exited $\ket{v^{\prime\prime}=0,N^{\prime\prime}=2}$ state, as shown in Fig. 4(c). From the separation of the two resonance signals a rotational constant of $\mathrm{B_{0}}=h\times 8.742(3)\,$GHz is inferred.

The transition strengths $\bar{\Omega}_{\mathrm{S}}$ to the $\textrm{X}^{1}\Sigma^{+}$ vibrational states are obtained from the measured widths of the spectra as shown in Fig. 4(a) and the applied laser intensities. The results are shown in Tab. 2. The scaling of the transition strengths between different lines is in good agreement with the ratio of the respective FCFs shown in Fig. 2(b). The strong coupling to the $v^{\prime\prime}=0$ ground state can be inferred from the observed ATS, which shows a Stokes Rabi frequency $\Omega_{\text{S}}$ around $2\pi\times 8\,$MHz. $\Omega_{\text{S}}$ of $500\,$kHz can be achieved, even if we switch to a low power diode laser. As the measured Rabi frequency for the pump excitation is of the same order, we have established favorable conditions for a fast STIRAP transfer.

While our pathway method was demonstrated here for the case of ${}^{6}\textrm{Li}{}^{40}\textrm{K}$, it can be applied to other bi-alkali molecules, where a Feshbach state can be found that contains singlet admixture, like LiCs or KCs. Other molecular species that are currently studied, like LiYb, RbSr, and CsYb, have only spin-doublet ground and excited electronic states. We believe that the method of using a single stretched hyperfine state can be extended to such molecules.

To conclude, we demonstrated a pathway to access the rovibrational ground state of ultracold bi-alkali molecules using only singlet states. We found that the $\textrm{A}^{1}\Sigma^{+}$ potential offers good transition strengths to the Feshbach and ground states with moderate laser powers. Starting from a Feshbach state with a stretched hyperfine component of the singlet admixture allows addressing a pure hyperfine component of the ground state, even if the hyperfine structure is not resolved. An ideal three-level system is established and is robust against off-resonant coupling to other hyperfine components. This pathway has the advantage that it does not require a singlet-triplet mixing with fully resolved hyperfine structure in the excited state manifold. Our method is applicable to other types of molecules and represents a simplified and efficient pathway to their ground state.