Precision molecular spectroscopy for ground state transfer of molecular quantum gases
One possibility for the creation of ultracold, high-phase-space-density quantum gases of molecules in the rovibrational ground state relies on first associating weakly-bound molecules from quantum-degenerate atomic gases on a Feshbach resonance and then transfering the molecules via several steps of coherent two-photon stimulated Raman adiabatic passage (STIRAP) into the rovibronic ground state. Here, in ultracold samples of Cs Feshbach molecules produced out of ultracold samples of Cs atoms, we observe several optical transitions to deeply bound rovibrational levels of the excited molecular potentials with high resolution. At least one of these transitions, although rather weak, allows efficient STIRAP transfer into the deeply bound vibrational level of the singlet X ground state potential, as recently demonstrated . From this level, the rovibrational ground state level can be reached with one more transfer step. In total, our results show that coherent ground state transfer for Cs is possible using a maximum of two successive two-photon processes or one single four-photon STIRAP process.
Ultracold and dense molecular samples in specific deeply bound rovibrational levels are of high interest for fundamental studies in physics and chemistry. They are expected to find applications in high resolution spectroscopy and fundamental tests [2, 3], few-body collisional physics [4, 5], ultracold chemistry , quantum processing , and in the field of dipolar quantum gases and dipolar Bose-Einstein condensation [8, 9]. Ideally, full control over the molecular wave function is desired, i.e. full (quantum) control over the internal and external degrees of freedom of the molecules. High phase space densities are needed for molecular quantum gas studies. For many of the envisaged studies and applications, initial preparation of the molecular sample in the rovibronic ground state, i.e. the lowest energy level of the electronic ground state, is desired. Only in this state one can expect sufficient collisional stability.
But how is it possible to produce dense samples of ultracold molecules in the rovibrational ground state? Laser cooling of atoms, which has lead to the production of quantum degenerate atomic Bose and Fermi gases , can so far not be adapted to the case of molecular systems as suitable cycling transitions are not available. Versatile non-optical cooling and slowing techniques such as buffer gas cooling and Zeeman slowing in combination with molecule trapping [11, 12, 13] have been developed, but high molecular densities and in particular high phase space densities are yet to be reached. An alternative route to producing ultracold molecular samples is given by first producing ultracold atomic samples and then associating molecules out of the atomic sample. While this technique is so far limited to the production of selected species of dimer molecules, it has the advantage that ultra-low temperatures and high particle densities are easily inherited from the atomic precursor sample. There are essentially two association techniques, photoassociation  and magnetically induced Feshbach association [15, 16]. In photoassociation experiments [17, 18, 19, 20], ultracold samples of deeply bound molecules have been created. Additional techniques such as vibrational cooling  should allow selective pumping into the rovibrational ground state and open up the prospect for high molecular phase space densities. In Feshbach association experiments [21, 22], high-density samples of weakly bound molecules are produced. For dimer molecules composed of Fermions, collisional stability of the highly excited molecules is assured as a result of a Pauli blocking effect, and molecular Bose-Einstein condensation could be achieved in the limit of extremely weak binding .
Here, we are interested in combining the techniques of Feshbach association and coherent molecular state transfer to produce quantum gases of molecules in the rovibrational ground state of the lowest electronic state. As usual, and are the vibrational and rotational quantum numbers, respectively. The molecules, produced on a Feshbach resonance and hence initially very loosely bound, are to be transferred in a few successive steps of coherent two-photon laser transfer to the rovibrational ground state, acquiring more and more binding energy in each step. The general idea is sketched in Fig.1A for the case of Cs. Each two-photon step involves an excited state level. Population transfer into this level needs to be avoided to prevent loss due to spontaneous emission. One possibility is to use the technique of stimulated Raman adiabatic passage (STIRAP) , which is very robust and largely insensitive to laser intensity fluctuations. The scheme has several advantages. First, production of Feshbach molecules out of a quantum degenerate atomic sample can be very efficient . Second, the optical transition rate on the first transition starting from the Feshbach molecules is greatly enhanced in comparison to the free atom case. Further, the scheme is fully coherent, not relying on spontaneous processes, allowing high state selectivity, and involving only a comparatively small number of intermediate levels. A ground state binding energy of typically 0.5 eV for an alkali dimer can be removed essentially without heating the molecular sample, as the differential photon recoil using pairwise co-propagating laser beams driving the two-photon transitions is very small. If losses and off-resonant excitations can be avoided, the scheme essentially preserves phase space density and coherence of the initial particle wave function, allowing the molecular sample to inherit the high initial phase space density from the atomic precursor sample.
Certainly, several challenges have to be met: Going from weakly bound Feshbach to tightly bound ground state molecules corresponds to a large reduction in internuclear distance. Consequently, the radial wave function overlap between successive levels is small, and a compromise has to be found between the number of transitions and the minimum tolerable wave function overlap. To keep the complexity of the scheme low, one or at most two two-photon transitions are desirable. Accordingly, suitable intermediate levels have to be identified that allow a balanced division of wave function overlap, as given by the Franck-Condon factors, between the different transitions. For example, for a four-photon transition scheme with Cs as shown in Fig.1A the Franck-Condon factors are all on the order of . We emphasize that the identification of the first excited level and hence of the first transition starting from the Feshbach molecules is of crucial importance. Detailed calculations determining the wave function overlap are generally missing, and estimates on the Franck-Condon factors using hypothetical last bound states of either the singlet or triplet potentials of an alkali dimer molecule do not necessarily reflect the transition dipole moments adequately. In addition, for electronic molecular states or energy regions where spectroscopic data is missing, the precise energy of the excited state levels above the atomic threshold is known only with a large uncertainty which can approach the vibrational spacing of a few nanometers. Hence, considerable time has to be spent on searching for weak transitions starting from the initial Feshbach molecules.
In a pioneering experiment, Winkler et al.  demonstrated that the STIRAP technique can efficiently be implemented with quantum gases of weakly bound Feshbach molecules. In this work, the transferred molecules, in this case Rb, were still weakly bound with a binding energy of less than of the binding energy of the rovibronic ground state, and the intermediate excited state level was close to the excited-atom asymptote. Here, we observe several optical transitions starting from a weakly bound Feshbach level to deeply bound rovibrational levels of the mixed excited Ab molecular potentials of the Cs molecule in a wavelength range from 1118 to 1134 nm, far to the red of the atomic D and D transitions. The Cs molecular potentials are shown in Fig.1A. We observe the levels as loss from an ultracold sample of Cs Feshbach molecules as shown in Fig.1B. We observe two progressions, one that we attribute to the Ab potentials and one that we associate to the triplet potential. From the loss measurements, we determine the transition strengths and find that the stronger transitions should be suitable for STIRAP to an intermediate, deeply bound rovibrational level of the singlet X potential with . Recently, we could implement STIRAP into . For the case of the dimer molecule KRb, Ni et al.  could demonstrate quantum gas transfer all the way into the rovibrational ground state of the singlet X molecular potential. Here, the transfer could be achieved in only a single step as a result of the favorable run of the excited state potentials, which is generally the case for heteronuclear molecules composed of alkali atoms . Also recently, transfer to the rovibrational ground state of the lowest triplet state a of Rb could be achieved .
2 Preparation of a sample of weakly bound Feshbach molecules
We produce ultracold samples of molecules on two different Feshbach resonances, one near 1.98 mT and one near 4.79 mT . In both cases, essentially following the procedure detailed in Ref., we first produce an ultracold sample of typically Cs atoms in the lowest hyperfine sublevel in a crossed optical dipole trap. As usual, is the atomic angular momentum quantum number, and its projection on the magnetic field axis. The trapping light at 1064.5 nm is derived from a single-frequency, highly-stable Nd:YAG laser. The offset magnetic field value for evaporative cooling is 2.1 mT. We support optical trapping by magnetic levitation with a magnetic field gradient of 3.1 mT/cm. We then produce weakly bound Feshbach molecules out of the atomic sample . We produce a sample every 8 s, i.e. our spectroscopic measurements are performed at a rate of one data point every 8 s. In order to be able to search for optical transitions over large frequency ranges it is advantageous to work with the shortest possible sample preparation times. For this reason we stop evaporative cooling slightly before the onset of Bose-Einstein condensation (BEC), which also makes sample preparation somewhat less critical. The temperature of the initial atomic sample is then typically about 100 nK. At higher temperatures and hence lower phase space densities the molecule production efficiency is reduced, so that there is a trade off between ease of operation and molecule number. We note that for our ground state transfer experiments reported in Ref. we produce a pure atomic BEC at the expense of longer sample preparation times.
The spectrum of weakly-bound Feshbach levels near the two-free-atom asymptote is shown in Fig.2 . For molecule production at the Feshbach resonance at 4.79 mT, we first ramp the magnetic field from the BEC production value to 4.9 mT, about 0.1 mT above the Feshbach resonance. We produce the molecular sample on a downward sweep at a typical sweep rate of 0.025 mT/ms. The resulting ultracold sample contains up to 11000 molecules, immersed in the bath of the remaining ultracold atoms. The resonance at 4.79 mT is a -wave resonance , and hence the molecules are initially of -wave character, i.e. , where is the quantum number associated with the mechanical rotation of the nuclei. However, there is a weakly bound -wave Feshbach state () belonging to the open scattering channel right below threshold. This state couples quite strongly to the initial -wave state, resulting in an avoided state crossing (as shown in the inset to Fig.2), on which the molecules are transferred to the -wave state upon lowering the magnetic field [30, 1]. Upon further lowering the magnetic field to less than 2.0 mT, the molecules acquire more and more character of a closed channel -wave state on a second, very broad avoided crossing. Here, we perform spectroscopy in this transition range from open channel to closed channel -wave character. At a magnetic field value of 2.0 mT, the binding energy of the molecules is near MHz with respect to the two-atom asymptote, where is Planck’s constant.
For molecule production at the Feshbach resonance at 1.98 mT, we simply ramp the magnetic field down from the initial BEC production value. Again, we produce an ultracold molecular sample with about 11000 molecules. The molecules in have -wave character, i.e. . When we lower the magnetic field to 1.6 mT, the binding energy of the molecules is also near MHz with respect to the two-atom asymptote.
For spectroscopy, we release the molecules from the trap after magnetic field ramping is completed and perform all subsequent experiments in free flight without any other light fields on except for the spectroscopy laser.
For molecule detection in both cases, we reverse the magnetic field ramps . The -wave molecules are dissociated on the -wave Feshbach resonance at 1.98 mT, and the -wave molecules are dissociated on the -wave Feshbach resonance at 4.79 mT. Prior to the reverse magnetic field ramp, we apply a magnetic field gradient of 3.1 mT/cm for about 5 ms to separate the molecular from the atomic sample in a Stern-Gerlach-type experiment. Finally, we detect atoms by standard absorption imaging. The minimum number of molecules that we can detect is on the order of 200 molecules.
We perform optical spectroscopy on Feshbach molecules in the wavelength region around 1125 nm. Based on symmetry considerations, there are two sets of electronically excited states that we address in the spectroscopic measurements presented here, namely the Ab coupled state system and the electronically excited states. We first discuss transitions to the coupled state system. Transitions to the latter state are discussed in Sec. 3.2.
3.1 Transitions to the Ab coupled electronically excited states
We are primarily interested in transitions from Feshbach levels to rovibrational levels of the Ab electronically excited states. In the heavy alkali dimers, most notably in Cs, the A state and the b state are strongly coupled by resonant spin-orbit interaction [32, 33], yielding the coupled states in Hund’s case (c) notation. The singlet component of the states allows us to efficiently couple to deeply bound X state levels, specifically to the level of the ground state potential, as has recently been shown in a coherent transfer experiment . We have chosen to do spectroscopy in the wavelength range of 1118 nm to 1134 nm above the 6S+6S dissociation threshold of the Cs dimer. This corresponds to a detuning of roughly 2300 cm from the cesium D line and to an energy range of approximately 12572 cm to 12450 cm above the rovibronic ground state X . This region was chosen in order to give a balanced distribution of transition dipole moments in a 4-photon transfer scheme to the rovibronic ground state. In addition, the wavelengths of the four lasers used in the transfer experiments were chosen such that they lie within the energy range covered by the infrared fiber-based frequency comb that we use as a frequency reference in the state transfer experiments.
The transitions of interest here lie outside the energy regions for which Fourier transform spectroscopic data was obtained at Laboratoire Aimé Cotton from transitions to the X state . The vibrational progression of the states is highly perturbed by the resonant spin-orbit coupling and exhibits an irregular vibrational spacing. Molecular structure calculations are complicated by the spin-orbit coupling and calculated term values are highly sensitive to the coupling. Prior to the experiments discussed here the absolute energies of the vibrational levels of the Ab excited state levels were poorly known in the region of interest from 1118 nm to 1134 nm. We therefore perform a broad range search by irradiating the weakly-bound Feshbach molecules at a fixed wavelength for a certain irradiation time of up to ms and by recording the number of remaining molecules as a function of laser frequency. In one run of the experiment one particular laser frequency is queried. We thus take data points at the repetition rate of our experiment, which is given by the sample preparation time of 8 seconds. Based on the available laser intensity from and an estimate of the dipole transition moments for the strongest expected lines, we chose a frequency step size of about 100 MHz to 150 MHz for initial line searching. We obtain the laser light at 1118 nm - 1134 nm from a grating-stabilized external cavity diode laser. For coarse frequency scanning, the laser is free running and tuned via a piezoelectric element on the grating of the laser. For more precise measurements, we lock the laser to a narrow-band optical resonator that can be tuned via a piezoelectric element. Fig.3 A shows a typical loss spectrum starting from Feshbach state for excitation near 1126 nm, measured at a magnetic field of 1.98 mT. In this particular case we find three resonances, which we associate with the rotational splitting of the excited state level, , where is the rotational quantum number. Based on molecular structure calculations we identify this level as the 225th one of the progression with an uncertainty of about two in the absolute numbering. We zoom in on these three transitions in Fig.3 B, C, and D and record loss resonances at reduced laser intensity in order to avoid saturation of the lines. For these measurements, the laser is locked to the narrow-band optical resonator and the resonator in turn is stabilized to the optical frequency comb to assure reproducibility and long term frequency stability. As one can expect, the loss is strongest on the transition to the level, and it is weakest on the transition to . All lines have an excited state spontaneous decay rate of around MHz, in agreement with the typical expected lifetimes of excited molecular levels. The transition to shown in Fig.3 D is of special interest to the current work. It has been used as intermediate excited state level for coherent transfer to X in our recent experiments .
By fitting a two level model that takes into account decay from the upper level to a series of such measurements obtained with different laser intensities, we determine the transition strength as given by the normalized Rabi frequency. As the Feshbach molecules scatter photons and spontaneously decay to other molecular levels, the number of Feshbach molecules decays as a function of laser detuning according to , where is the molecule number without laser irradiation and is the irradiation time. From the fit we obtain the Rabi frequency on resonance and the excited state spontaneous decay rate . We determine the normalized Rabi frequency to kHz for , where is the laser intensity. This value is sufficient to perform STIRAP given the available laser power . The corresponding transition strengths for and are kHz and kHz , respectively. The absolute values of these transition strengths bear an estimated uncertainty of 20 % because the laser beam parameters for the spectroscopy laser are not well determined.
We also record the time dependence of the molecular loss on some of the stronger lines. For this, we step the laser irradiation time from to s, while laser is kept on resonance. The result is shown in Fig.4 A for the transition at 1126.173 nm for two different values of the excitation laser intensity.
We note that the transition strength for a particular line starting from Feshbach level strongly depends on the value of the magnetic field, as evidenced in Fig.4 B. Loss resonances for the transition at 1126.173 nm at 1.9 mT and 2.2 mT are shown. For ground state transfer , we choose a magnetic field of around 1.9 mT, which is somewhat below the magnetic field region where state is strongly curved, but above the avoided state crossing with state , as seen in Fig.2. The pronounced bending of is the result of a strong avoided crossing between two s-wave Feshbach levels . For magnetic field values beyond 3.0 mT the level can be associated to the asymptote, where , is the atomic angular momentum quantum number of the -th atom, respectively. Below 2.0 mT the level can be associated to the asymptote. It is hence of closed channel character and much more deeply bound with respect to its potential asymptote, effectively by twice the atomic hyperfine splitting, improving the radial wave function overlap with the excited state levels. This increases the transition strength. Trivially, the resonance frequency is shifted as the binding energy is reduced for larger magnetic field values. Coupling to the excited state level is reduced from kHz to kHz when the magnetic field is changed from 1.9 mT to 2.2 mT.
As will be discussed in Sec.4 it is advantageous to be able to choose different Feshbach states as a starting state for ground state transfer experiments. Therefore, we probe transitions from Feshbach level to Ab levels. Fig.5 shows loss resonances to the same excited state levels as shown in Fig.3, only that now the initial Feshbach level is instead of . In this case, the transition to is the strongest, while the transition to is very weak, but can be detected. A comparison of the transition strengths from to the excited state level , giving kHz versus to giving kHz shows that level could also be potentially used as a starting level for coherent population transfer to deeply bound levels of the ground state but requires longer STIRAP times in order to assure sufficient adiabaticity . The excited state level in turn couples to in the ground state, as in previous work .
In addition to the transition near 1126 nm we find a series of other excited state levels that we assign to the Ab coupled state system. These are listed in Table 1. The assignment to either the Ab system or to the electronically excited state discussed below is primarily based on the spacing between neighboring vibrational levels and in addition on the pattern of loss resonances associated with each particular vibrational level. Resonant spin-orbit coupling in the case of the states leads to an irregular vibrational spacing. In contrast, the state is not perturbed by spin-orbit interaction and therefore has a regular vibrational progression. The levels near 1126 nm and near 1123 nm have been used to detect dark resonances with deeply bound levels of the X state . The ability to couple to these essentially purely singlet ground state levels unambiguously assigns the corresponding excited state levels to the system. The data given in Table 1 does not represent a fully exhaustive study of the Ab coupled states in the wavelength range of interest. In fact, for the most part we observe those levels of the system that have a dominant A state contribution, as determined from molecular structure calculations.
3.2 Transitions to the electronically excited state
The Feshbach levels that serve as starting levels for the spectroscopy are of mixed X and a character. In the wavelength range explored here, excitation to the electronically excited triplet state is possible from the a component of the Feshbach molecules. In fact, for a heavy molecule as Cs, the state is better described by the two separate electronic states and , denoted by the Hund’s case notation. The has been previously studied by Fourier transform spectroscopy . This state is not of prime interest for the present work as transitions from this state down to the X ground state are expected to be strongly suppressed, but would be important for STIRAP transfer into the rovibrational ground state level of the shallow triplet a potential . Certainly, it is important to be able to distinguish rovibrational levels belonging to the state from the ones belonging to the system, because otherwise time would be wasted in searching for ground state dark resonances that are very weak or even do not exist. Fig.6 A shows a typical loss spectrum for one of the lines that we detected near 1127.37 nm. Due to hyperfine splitting, levels of triplet character exhibit a much richer substructure than the levels used for ground state transfer. Several components can be identified as a result of rotational and excited state hyperfine splitting. Zoomed-in regions are shown in Fig.6 B, C, D, and E. We have observed a regularly spaced series of optical transitions which we attribute to the excited state as listed in Table 1. The levels are well reproduced by molecular structure calculations using the Dunham coefficients from Ref.. The vibrational numbering used here is the same as in that work. However, it relies on the absolute energy position of the potential, T, which was not determined precisely in Ref. .
We have performed optical spectroscopy starting from weakly bound Cs Feshbach molecules into deeply bound rovibrational levels of the mixed excited state system and the excited triplet state. At least one of the observed transitions, namely the one at 1126.173 nm starting from the Feshbach level at an offset magnetic field value of 1.9 mT to the excited level of the system, is strong enough to allow efficient STIRAP transfer into deeply bound rovibrational levels of the singlet X ground state potential. The use of this transition for STIRAP has recently been demonstrated in Ref.. In that work, the deeply bound rovibrational level of the X ground state potential was populated in the molecular quantum gas regime with 80% efficiency. The rovibrational ground state of the X ground state potential can thus be reached from the atomic threshold with a maximum of two two-photon STIRAP transfers. Dark resonances connecting to have recently been observed , and two-step STIRAP into has recently been implemented . For future experiments, the use of Feshbach level as the initial state might be advantageous. Level can be more easily populated, as the Feshbach resonance connected to this level is at a low magnetic field value of 1.98 mT , where the atomic background scattering length has a moderate value of 155 a, where a is Bohr’s radius. The use of this resonance avoids excitation of collective motion of the atomic BEC as a result of a large mean field interaction near the Feshbach resonance at 4.79 mT , where the atomic background scattering length is about 935 a. The transition starting from level appears to be strong enough to allow STIRAP, this time via the excited state level of the system. An attractive strategy for the production of a BEC of ground state molecules relies on the addition of a three-dimensional optical lattice. Starting from the atomic BEC, pairs of atoms at individual lattice sites can be produced in a superfluid-to-Mott-insulator transition  with high efficiencies of up to 50% . These pairs can then be very efficiently associated on a Feshbach resonance  and subsequently transferred to the rovibronic ground state with STIRAP. The lattice has the advantage of shielding the molecules against inelastic collisions during the association process and subsequent state transfer. In particular, it should allow long STIRAP pulse durations, allowing us to resolve the weak hyperfine structure of ground state molecules . As proposed by Jaksch et al. , dynamical melting of the lattice should ideally result in the formation of a BEC of molecules in the rovibronic ground state in a Mott-insulator-to-superfluid-type transition.
We are indebted to R. Grimm for generous support and we thank T. Bergeman, H. Salami, J. Hutson, J. Aldegunde, and E. Tiemann for valuable discussions. We gratefully acknowledge funding by the Austrian Ministry of Science and Research (BMWF) and the Austrian Science Fund (FWF) in form of a START prize grant and by the European Science Foundation (ESF) in the framework of the EuroQUAM collective research project QuDipMol. R.H. acknowledges support by the European Union in form of a Marie-Curie International Incoming Fellowship (IIF).
|WL [nm]||Energy above||Assignment|
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