A pathway to ultracold bosonic ²³Na³⁹K ground state molecules

Quantum gases of ultracold polar molecules offer unprecedented novel opportunities for the investigation of dipolar collisions, quantum chemical processes and quantum many-body systems [1–5]. The new handle in comparison to atomic systems is given by the electric dipole moment of heteronuclear diatomic ground state molecules. The most successful approach to the creation of ultracold ensembles of polar ground state molecules is based on the association of two chemically different ultracold atomic alkali species. This process starts with magneto-association to weakly bound Feshbach molecules and continues with the coherent transfer of the weakly bound molecules to the rovibrational ground state of polar molecules using a STImulated Raman Adiabatic Passage (STIRAP) [6]. For the transfer from the initially magneto-associated molecular state $| i\rangle$ to the final molecular ground state $| f\rangle$, the STIRAP transfer consists of an adiabatic change of the dressed state composition by involving two coherent laser beams referred to as Pump and Stokes laser. The Pump laser couples $| i\rangle$ to a third excited state $| e\rangle$, the Stokes laser $| f\rangle$ to $| e\rangle$, resulting in a typical Λ-level scheme; see figure 1. During the time evolution the intermediate state $| e\rangle$ is not populated and therefore does not contribute to incoherent molecule losses due to spontaneous decay. In the case of bi-alkali molecules, the coupling between $| i\rangle$/$| f\rangle$ and $| e\rangle$ is governed by the Franck–Condon overlap, the singlet–triplet fraction and the hyperfine composition of the states. For bi-alkali heteronuclear molecules the ground state $| f\rangle$ always belongs to the X¹Σ⁺ potential while a weakly bound dimer state $| i\rangle$ exists mainly in the a³Σ⁺ potential; see figure 1(a). To act as an efficient coupling bridge, the choice of $| e\rangle$ is crucial. It needs to be chosen to couple well to the initial triplet as well as to the final singlet state. This can be achieved through the careful choice of mixed states in the ³Π/¹Σ⁺ or ³Σ⁺/¹Π potentials; see figure 1(a). Up to now, the production of trapped ground state molecules, either fermionic, such as ⁴⁰K⁸⁷Rb [7], ²³Na⁴⁰K [8] and ⁶Li²³Na [9], or bosonic, such as ⁸⁷Rb¹³³Cs [10, 11] and ²³Na⁸⁷Rb [12] has been reported. NaRb [12] and RbCs [10, 11] ground state molecules have been prepared by using coupled states in the A¹Σ⁺ and b³Π potentials. KRb [7] and fermionic ²³Na⁴⁰K [8] ground state molecules have been created using states in the c³Σ⁺ and B¹Π potentials. Additionally, ground state molecules for fermionic ²³Na⁴⁰K have been created using excited states in energetically higher D¹Π and d³Π potentials [13]; not shown in figure 1(a).

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In the case of ²³Na³⁹K, suitable transitions for the creation of ground state molecules have so far only been investigated experimentally in hot beam experiments accessing rotational states with N ≥ 6 [15]. Moreover, the work [15] does not include hyperfine structure and offset magnetic fields. In addition, previous theory work [16] describes a complete spectrum involving spin–orbit, Coriolis and spin–rotation interactions as well as Franck–Condon factors identifying promising singlet–triplet mixed states. However, the hyperfine structure still remains untreated although it is crucial for a successful STIRAP transfer [13]. In this paper, we present hyperfine resolved spectroscopic investigations of the ²³Na³⁹K molecule at bias magnetic fields for strongly mixed states known from [15, 16].

Using one-photon association spectroscopy, we first locate and characterize the strongly mixed $| {B}^{1}{\rm{\Pi }},v=8\rangle$ and $| {c}^{3}{{\rm{\Sigma }}}^{+},v=30\rangle$ states serving as a bridge between the a³Σ⁺ and X¹Σ⁺ potentials. A large Zeeman splitting at 150 G enables us to perform hyperfine resolved optical spectroscopy of the excited molecular state manifold. We present a simple model to determine the singlet–triplet admixtures of the excited states. Furthermore, we perform two-photon Autler–Townes spectroscopy [17] to locate the rovibronic ground state, determine the rotational constants and extract effective dipole transition matrix elements from our measurements. Thus, we present and fully characterize a pathway for the creation of rovibronic (v = 0, N = 0) ²³Na³⁹K ground state molecules for the first time.

In the following, we give an outline of the experimental setup and procedures; see section 2. In section 3.1 we present the model fit for the coupled excited states, and in section 3.2 we detail the spectroscopic one-photon measurements. This is followed by the two-photon ground state spectroscopy which includes the determination of the rotational constant and the transition dipole element of the Stokes transition in section 4.

The spectroscopic studies presented in this work are based on the preparation of an ultracold mixture of ²³Na+³⁹K. A description of the experimental setup can be found in [16, 18, 19]. Details of the experimental procedure can be found in [20, 21]. In the following, we briefly summarize the main experimental steps. First, ²³Na and ³⁹K atoms are loaded into a two-color magneto-optical trap, followed by simultaneous molasses cooling of both species. The atoms are optically pumped to the F = 1 manifold and loaded into an optically plugged magnetic quadrupole trap, where forced microwave evaporative cooling of ²³Na is performed. ³⁹K atoms are sympathetically cooled by ²³Na. After both atomic species are transferred into a 1064 nm crossed-beam optical dipole trap (cODT), a homogeneous magnetic field of 150 G is applied to ensure favorable scattering properties in the mixture [20]. The atoms are prepared in the $| f=1,{m}_{f}=-1{\rangle }_{\mathrm{Na}}+| f=1,{m}_{f}=-1{\rangle }_{{\rm{K}}}$ states. Then, forced optical evaporation is performed on both species by lowering the intensity of both beams of the cODT. The forced evaporation process is stopped when the atomic sample has a temperature of about 1 μK and a corresponding phase space density ≤0.1 for both species. To decrease the differential gravitational sag and assure a significant overlap of the two clouds, the cODT intensity is ramped up again. For the spectroscopy presented in this work, the magnetic field is either set to 130 G or left at 150 G yielding values at which the inter and intra species scattering properties allow for a long lifetime of the sample by minimizing three-body losses. This allows for a sufficiently long interaction time during photoassociation experiments, as required by the expected weak coupling between the atom pair at the scattering threshold and the electronically excited state molecules. The laser light for the Pump and the Stokes transitions at 816 nm and 573 nm respectively is generated by diode laser systems. The 816 nm light is generated by a commercial external-cavity diode laser (ECDL). The generation of the 573 nm light starts from a commercial ECDL operating at 1146 nm. The light is amplified by a tapered amplifier and subsequently frequency doubled in a self-built resonant bow-tie doubling cavity. Both lasers are locked to a commercial ultra-low expansion (ULE) glass cavity by using the Pound–Drever–Hall technique [22]. A sideband locking scheme involving widely tunable electro-optical modulators ensure the tunability of both locked lasers within the free spectral range of the ULE cavity which is 1.499 GHz. The finesse of the ULE cavity is 24 900 and 37 400 for 816 nm and 1146 nm light, respectively. The laser system setup is similar to the one described in [23].

Optical fibers spatially filter the light and ensure high quality beam properties. The polarization is set by a half-wave plate for each wavelength. The foci of the two beams on the atoms are adjusted to 1/e²-beam waists of 35 μm for the Pump laser and 40 μm for the Stokes laser. The maximum laser power is 25 mW for each laser. With the Pump beam at full power, the Stokes beam is reduced to ≤10 mW to avoid a depletion of ²³Na atoms in the trap center, originating from a strong repulsive dipole force. The beams are geometrically superimposed and orientated perpendicularly to the applied magnetic field.

3.1. Local model for the excited state manifold

In our experiment, we are specifically interested in a detailed understanding of the previously located [15] strongly mixed $| {B}^{1}{\rm{\Pi }},v=8\rangle$ and $| {c}^{3}{{\rm{\Sigma }}}^{+},v=30\rangle$ states. Hyperfine splitting and spin–orbit coupling for these two states can be modeled by a simple two-manifold coupled system, when neglecting contributions from other vibrational and rotational levels, from nucleus–nucleus interaction and nucleus–rotation coupling. This allows us to treat the two state manifolds separately in the following spectrally local model.

The excited state can be described by Hund's case (a) and that allows to consider the total angular momentum J = 1 as resulting from the coupling of the rotational angular momentum $\vec{N}$ and the spin $\vec{S}$ [24]. ²³Na and ³⁹K nuclear spins are ${i}_{\mathrm{Na}}={i}_{{\rm{K}}}=3/2$ leading to 3 × 4 × 4 = 48 different states in the singlet and triplet channel.

In absence of coupling the Hamiltonians for the investigated states, in the basis of $| B/c,J,{m}_{J},{i}_{\mathrm{Na}},{m}_{{i}_{\mathrm{Na}}},{i}_{{\rm{K}}},{m}_{{i}_{{\rm{K}}}}\rangle$, reads as [25]

$$\begin{eqnarray}&&{H}^{B}={\mu }_{0}{g}_{J}^{B}\vec{J}\cdot \vec{B}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{H}^{c}={\rm{\Delta }}+\vec{J}\cdot ({{A}}_{{\rm{K}}}{\vec{i}}_{{\rm{K}}}+{{A}}_{\mathrm{Na}}{\vec{i}}_{\mathrm{Na}})+{\mu }_{0}{g}_{J}^{c}\vec{J}\cdot \vec{B},\end{eqnarray} \tag{ 2 }$$

where A_K and A_Na are the hyperfine constants, μ₀ the Bohr magneton and $\vec{B}$ the applied magnetic field. The energy Δ corresponds to the unperturbed energy difference between the two states and we neglect the hyperfine term for the singlet state. The ²³Na hyperfine coupling A_Na is almost two orders of magnitude larger than A_K and it was already resolved in previous spectroscopic measurements on cold molecular beams [15]. The hyperfine splitting of ³⁹K is comparable or even smaller than the molecular state linewidth and remains mainly unresolved. At low field this allows to consider ${F}_{1}=| \vec{J}+{\vec{i}}_{\mathrm{Na}}| =(1/2,3/2,5/2)$ as a good quantum number, as visible in figure 2, with three groups of states at low magnetic field. Additionally, one can consider the total angular momentum $\vec{F}=\vec{J}+{\vec{i}}_{\mathrm{Na}}+{\vec{i}}_{{\rm{K}}}={\vec{F}}_{1}+{\vec{i}}_{{\rm{K}}}$, where the projection on the quantization axes m_F is the only preserved quantity even at large magnetic fields. The Zeeman term of equation (2) neglects nuclear contributions and can be expanded as μ₀m_J g_J B, where m_J is the projection of $\vec{J}$ on the magnetic field axis and g_J the Landé factor. In absence of coupling between singlet and triplet, one has g^B_J = g_N = g_L/(J(J + 1)) = 1/2 and ${g}_{J}^{c}={g}_{s}/2\approx 1$ [25, 26], where g_L and g_S are the known electron orbital and spin g-factors equal to 1 and 2.0023, respectively. Spin–orbit interaction couples singlet and triplet with strength ξ_Bc with selection rules ΔJ = 0, ${\rm{\Delta }}{m}_{{i}_{\mathrm{Na}}}=0$ and ${\rm{\Delta }}{m}_{{i}_{{\rm{K}}}}=0$. The coupling also already incorporates the vibrational wavefunction overlap of the two states. Hence, the problem reduces to solving the following (48 + 48) × (48 + 48) matrix

$$\begin{eqnarray}\left(\begin{array}{cc}{H}^{B} & {\xi }_{{Bc}}\\ {\xi }_{{Bc}} & {H}^{c}\end{array}\right).\end{eqnarray} \tag{ 3 }$$

Note that the coupling ξ_Bc shows its effect in two ways:

• As the Zeeman term remains significantly smaller than both the unperturbed singlet–triplet energy difference Δ and of the coupling strength ξ_Bc, one finds for strong fields compared to the hyperfine splitting (not shown in the later figure 3) the effective Landé factors ${g}_{J,{\rm{eff}}}^{B,c}$
  $$\begin{eqnarray}&&{g}_{J,{\rm{eff}}}^{B,c}=\displaystyle \frac{1}{2}\left({g}_{J}^{c}+{g}_{J}^{B}\mp {\rm{\Delta }}\displaystyle \frac{{g}_{J}^{c}-{g}_{J}^{B}}{\sqrt{4{\xi }_{{Bc}}^{2}+{{\rm{\Delta }}}^{2}}}\right)\end{eqnarray} \tag{ 4 }$$
  $$\begin{eqnarray}&&=\,{g}_{J}^{B,c}\pm \displaystyle \frac{1}{4}\left(1-\displaystyle \frac{{\rm{\Delta }}}{\widetilde{{\rm{\Delta }}}}\right),\end{eqnarray} \tag{ 5 }$$
  where $\widetilde{{\rm{\Delta }}}=\sqrt{4{\xi }_{{Bc}}^{2}+{{\rm{\Delta }}}^{2}}$ is the energy difference of the hyperfine centers at B = 0 in the the coupled case.
• The acquired triplet character of the singlet state will lead to a large hyperfine splitting, while the triplet one is decreased.

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3.2. One-photon spectroscopy

Using the located excited state as a bridge, we demonstrate two-photon spectroscopy of the rovibrational ground state $| {X}^{1}{\rm{\Sigma }},v=0,N=0\rangle ;$ as sketched in figure 1. The energy difference between ground and excited state is well known from spectroscopic data [15] and can be predicted within a precision of $\pm 0.08\,{{\rm{cm}}}^{-1}$. To exactly determine the energy of the rovibrational ground state, we fixed the Pump laser frequency on the transition with strongest losses from figure 2(d) within the singlet states which is marked with a red dashed curve. The state originates from the combination of two basis states which are labeled by $| B/c,1,-1,3/2,-1/2,3/2,-1/2\rangle$ and $| B/c,1,0,3/2,-1/2,3/2,-3/2\rangle$ where the quantum numbers are as described in section 3.1.

When the Stokes laser is on resonance, it is inducing a Stark shift on the excited state and a revival of the atom number is expected as sketched in figure 3(a). Figure 3(b) shows this protection when the detuning ${\delta }_{\mathrm{Stokes}}$ is scanned with a full width half maximum for the $| {X}^{1}{{\rm{\Sigma }}}^{+},v=0,N=0\rangle$ of about 405(35) MHz for a laser power of 5 mW. The observed peak of the transition energy is at 17 452.826(2) cm⁻¹.

Furthermore, to unambiguously identify the ground state, we performed an atom-loss scan also in the range of frequencies where the second rotationally excited state with N = 2 is expected; see figure 3(b). The observed energy difference of ${{\rm{\Delta }}}_{{N}=0\to {N}=2}=h\times 17.3(3)\,{\rm{GHz}}$ allows us to deduct the rotational constant to be B_v=0 = h × 2.89(5) GHz by using the relation ${{\rm{\Delta }}}_{{N}=0\to {N}=2}={B}_{0}\times 2(2+1)$. This value agrees with the one observed by using microwave spectroscopy [30]. The full width of half maximum for the protection is 133(27)MHz. This is smaller than for the N = 0 state, indicating a weaker coupling to the excited states, as expected from theoretical considerations.

To directly determine the coupling strength between the ground and the excited state, we fixed the Stokes laser frequency close to resonance and scanned the Pump laser frequency detuning ${\delta }_{\mathrm{Pump}}$. This scan reveals the well-known Autler–Townes splitting [17], the magnitude of which is proportional to the dipole matrix element of the transition. Figure 3(c) shows the measured remaining atom number with the typical double-loss feature. The asymmetric shape of the splitting originates from a residual detuning to the exact transition frequency of the Stokes laser which is determined to be ${\delta }_{\mathrm{Stokes}}=2\pi \times -22\,{\rm{MHz}}$. We derive the Rabi frequency ${{\rm{\Omega }}}_{\mathrm{Stokes}}=2\pi \times 23.5\,{\rm{MHz}}$ for an applied laser power of 5 mW, corresponding to a normalized Rabi frequency of ${\widetilde{{\rm{\Omega }}}}_{\mathrm{Stokes}}=2\pi \times 65.2\,{\rm{kHz}}\times \sqrt{I/({\rm{mW}}\,{{\rm{cm}}}^{-2})}$ and an effective transition dipole moment of 0.170 D. Both values, ${\delta }_{\mathrm{Stokes}}$ and ${{\rm{\Omega }}}_{\mathrm{Stokes}}$ are derived from a three-level master equation modeling the line shape shown in figure 3(c).

Within this work, we have characterized a two-photon scheme for the coupling of the diatomic scattering threshold in the a³Σ⁺ potential to the rovibrational ground state $| {X}^{1}{{\rm{\Sigma }}}^{+},v=0,N=0\rangle$. Using photoassociation spectroscopy, we have observed and characterized the excited state hyperfine manifolds of the coupled $| {B}^{1}{\rm{\Pi }},v=8\rangle$ and $| {c}^{3}{{\rm{\Sigma }}}^{+},v=30\rangle$ states in the bosonic ²³Na³⁹K molecules at 130 and 150 G, starting from an ultracold atomic quantum gas mixture in the states $| f=1,{m}_{f}=-1{\rangle }_{\mathrm{Na}}+| f=1,{m}_{f}=-1{\rangle }_{{\rm{K}}}$. By applying a spectral local model fit to the measurements, we have extracted the admixture of these states to be 26 %/74 %. Due to the strong singlet–triplet mixing, this part of the spectrum serves as an ideal bridge from the triplet atomic scattering threshold to the singlet rovibronic ground state molecules. Making use of this bridge, we have identified the rovibrational ground state and the second rotationally excited state in two-photon spectroscopy. From an Autler–Townes measurement we have extracted the Rabi-coupling between the excited and the ground state. Our work results in a fully characterized scheme for the conversion of ²³Na³⁹K Feshbach molecules to rovibrational ground state polar molecules and will allow for the efficient creation of ultracold ensembles of chemically stable bosonic ²³Na³⁹K ground state molecules.

We thank Eberhard Tiemann for enlightening discussions and suggestions. We gratefully acknowledge financial support from the European Research Council through ERC Starting Grant POLAR and from the Deutsche Forschungsgemeinschaft (DFG) through CRC 1227 (DQ-mat), project A03 and FOR2247, project E5. KKV and PG thank the Deutsche Forschungsgemeinschaft for financial support through Research Training Group (RTG) 1991.