Efficient state transfer in an ultracold dense gas of heteronuclear molecules

Abstract

Polar molecules have bright prospects for novel quantum gases with long-range and anisotropic interactions¹, and could find uses in quantum information science² and in precision measurements^3,4,5. However, high-density clouds of ultracold polar molecules have so far not been produced. Here, we report a key step towards this goal. We start from an ultracold dense gas of loosely bound ⁴⁰K⁸⁷Rb Feshbach molecules^6,7 with typical binding energies of a few hundred kilohertz, and coherently transfer these molecules in a single transfer step into a vibrational level of the ground-state molecular potential bound by more than 10 GHz. Starting with a single initial state prepared with Feshbach association⁸, we achieve a transfer efficiency of 84%. Given favourable Franck–Condon factors^9,10, the presented technique can be extended to access much more deeply bound vibrational levels and those exhibiting a significant dipole moment.

Main

Strategies for the realization of an ultracold dense gas of polar molecules have generally followed two different approaches. The first is to directly cool ground-state polar molecules by means of buffer gas cooling¹¹, Stark deceleration^12,13 or velocity filtering¹⁴. However, direct cooling strategies have typically been restricted to the millikelvin temperature range. Photoassociation schemes¹⁵, on the other hand, have the advantage of using high-density ultracold atomic clouds as a promising starting point. The combination of photoassociation and pump–dump schemes has recently resulted in the formation of ground-state polar molecules⁹. However, the realization of an ultracold dense gas of polar molecules has so far not been reported. Photoassociation schemes rely on spontaneous emission to populate bound molecular levels in the ground-state molecular potential^9,16. The spontaneous processes result in a large spread of population among several continuum and bound molecular states. Two-colour photoassociation schemes suffer from the small wavefunction overlap between the continuum states of colliding atoms and the localized molecular states, resulting in the necessity of large optical power, which then opens up other loss channels^17,18. This wavefunction overlap can be significantly enhanced by starting from a weakly bound molecule as compared with the continuum state of two colliding atoms. Weakly bound molecules in well-defined near-dissociation quantum states can be efficiently created using magnetic-field-tunable Feshbach resonances in ultracold atomic gases⁸. The resulting ensembles of Feshbach molecules constitute an excellent launching stage for the application of coherent optical transfer schemes to produce molecules in deeply bound vibrational levels. The combination of heteronuclear Feshbach molecule creation^6,7,19,20 with coherent de-excitation schemes²¹ might therefore result in the creation of an ultracold dense gas of polar molecules.

In the homonuclear Rb₂ system, Feshbach molecules have been used as a starting point for coherent two-photon transfer into a state bound by ∼h×500 MHz (ref. 22). In addition, radiofrequency manipulation schemes have been used in the Rb₂ system, achieving binding energies of >h×3 GHz (ref. 23). These results represent a powerful demonstration of the degree of control available in ultracold gases. Here, we report on coherent transfer of heteronuclear Feshbach molecules into a more deeply bound vibrational level of the ground-state molecular potential (see Table 1). We thus demonstrate a key step towards the realization of a quantum degenerate ultracold gas of polar molecules. Using two phase-coherent laser fields to couple the initial Feshbach state |i〉 and the target state |t〉 to a common electronically excited state |e〉 (Fig. 1a), stimulated Raman adiabatic passage²⁴ (STIRAP) is used to coherently transfer the population of |i〉 to |t〉. Through this transfer process, we produce a dense ultracold cloud of heteronuclear molecules with a binding energy of >h×10 GHz. An essential prerequisite for the successful demonstration of STIRAP in this system has been careful one-photon spectroscopic investigation of the so-far unknown ⁴⁰K⁸⁷Rb excited-state molecular potential below the 4S_1/2+5P_1/2 threshold and precise two-photon spectroscopy of more deeply bound vibrational levels in the ground-state molecular potential of ⁴⁰K⁸⁷Rb.

Table 1 Basic properties of KRb molecules in different vibrational levels of the ground-state molecular potential. Comparison of the binding energy, internuclear distance (‘size’) and dipole moment of the Feshbach molecules, the current target state K|9/2,−9/2〉+Rb|1,1〉 v=−3 and the absolute ground state X ¹Σ⁺(v=−99) in the singlet potential³⁰. The current target state has 92.5% triplet character and is therefore denoted by a³Σ⁺(v=−3). v is the vibrational quantum number as counted from the threshold, with the least bound level labelled as v=0.

Figure 1: Schematic diagram of the STIRAP scheme.

a, Schematic diagram of molecular potentials and molecular vibrational levels involved in the coherent two-photon transfer scheme. The phase-coherent lasers 1 and 2 couple the initial state |i〉 and the target state |t〉 to the intermediate state |e〉 in the excited-state molecular potential connecting to the 4S_1/2+5P_1/2 threshold. Ω₁ and Ω₂ denote the corresponding Rabi frequencies. δ and Δ are the two-photon and single-photon detuning, respectively. γ_x represents the decay rate of state x. b, Calculated molecular energy structure of the ⁴⁰K⁸⁷Rb molecule in the ground-state molecular potential below the K|9/2,−9/2〉 Rb|1,1〉 atomic threshold. All levels have total spin-projection quantum number M_F=−7/2. The initial Feshbach state |i〉 and the target state |t〉 are highlighted by the red bold lines. The states observed in dark-resonance spectroscopy (see the text) are plotted with dashed blue lines. c,d, Comparison of single-photon excitation rates for the |i〉→|e〉 transition, starting from Feshbach molecules (c) as compared with photoassociation rates from free atoms (d). The excitation rates are determined by measuring loss of population in state |i〉 from the trap in each case. For this set of data, |e〉 is a level in the 2(1) v=−9 hyperfine manifold. In d, the Rb (K) atom number is plotted with the red filled (open blue) circles. The red solid and blue dashed lines are simultaneous fits to the decaying Rb and K atom numbers. We assume that the loss rate (Ṅ_atom) is the same for both components at any time. These data demonstrate that the strength of the |i〉→|e〉 transition (the first leg of the STIRAP process) is markedly enhanced by starting from Feshbach molecules. With an initial molecule density of n_molecules≈5×10¹¹ cm⁻³ and initial atomic densities of n_atoms≈10¹³ cm⁻³, we observe an enhancement in the initial loss rate of Ṅ_molecules/Ṅ_atoms=(180±60) at a magnetic field close to the Feshbach resonance (B=546.88 G) (refs 28, 29).

Starting from a near-quantum-degenerate Bose–Fermi mixture of ⁸⁷Rb and ⁴⁰K atoms confined in a single-beam optical dipole trap, we use magnetic-field ramps across a Feshbach resonance at 546.7 G to create 2×10⁴ heteronuclear Feshbach molecules with a density of ≈5×10¹¹ cm⁻³ (see the Methods section). At a magnetic field of B=545.88 G, the Feshbach molecules have a binding energy of h×(270±50) kHz and can be directly, and selectively, imaged by high-field resonant absorption imaging (see the Methods section). The Feshbach molecules are in a superposition of the open channel K|F=9/2,m_F=−9/2〉+Rb|F=1,m_F=1〉 and the adiabatically connecting closed channel K|7/2,−7/2〉+Rb|1,0〉 (refs 6, 7). Here, F denotes the total atomic spin and m_F is the spin projection along the magnetic-field direction.

The Feshbach molecules are manipulated with light derived from a phase-coherent Raman laser system. A Ti:sapphire laser is offset-locked to a temperature-stabilized Fabry–Perot cavity, resulting in a linewidth of <20 kHz and absolute long-term frequency stability better than 2 MHz. The second laser beam is derived from an external cavity diode laser phase-locked to the Ti:sapphire laser. The two beams are π-polarized with respect to the magnetic field and propagate collinearly when irradiated onto the molecular ensemble.

In the first step, we carry out one-photon, bound–bound spectroscopy of the electronically excited ⁴⁰K⁸⁷Rb^* 2(1) and 2(0^+/−) molecular potentials for energies within h×400 GHz below the 4S_1/2+5P_1/2 threshold. Starting from the Feshbach molecules at B=545.88 G, laser 1 is used to couple the molecules to vibrational levels in the excited-state molecular potentials. These electronically excited molecules subsequently decay into a multitude of molecular and atomic states where they are invisible to our detection method. Using loss spectra, we identify vibrational series of different excited-state molecular potentials, which will be reported elsewhere. After an extensive search, we have chosen the 2(0⁺) v=−14 state as an intermediate coupling state |e〉. Here, v denotes the vibrational quantum number as counted from the 4S_1/2+5P_1/2 threshold. This state is located 213 GHz below the 4S_1/2+5P_1/2 threshold. Comparing one-photon loss rates starting from Feshbach molecules with photoassociation rates of free atoms, we observe an enhancement in the excitation rate by (180±60) (Fig. 1c,d).

An essential prerequisite for STIRAP is precise knowledge of possible target states. Whereas a theoretical model of the electronic ground-state potential has recently been published by Pashov et al. ²⁵, the ground-state level structure of ⁴⁰K⁸⁷Rb in the near-threshold regime has never been probed experimentally. Using two-photon dark-resonance spectroscopy²⁶, we probe the binding energy of vibrational levels in the K|9/2,−9/2〉+Rb|1,1〉 and K|7/2,−7/2〉+Rb|1,0〉 channels at B=545.88 G. We probe levels with binding energies less than h×10.5 GHz, which is accessible to our phase-locked laser system. Figure 2 shows a typical dark-resonance spectrum when scanning the frequency difference (δ) of the two phase-coherent laser fields in the vicinity of the frequency splitting between the two molecular ground-state levels. The presence of the near-resonant laser 2 dresses the transition |t〉→|e〉 (Fig. 1), leading to destructive interference for the absorption probability of laser 1 on the probe transition |i〉→|e〉 for δ=0. We observe one-photon loss of Feshbach molecules with a width of ∼6 MHz and a narrow dark-resonance feature in the vicinity of δ=0. At B=(545.88±0.05) G, we measure a binding energy of h×3.1504(10) GHz and h×10.49238(15) GHz for the K|9/2,−9/2〉+Rb|1,1〉 v=−2 and −3 states, respectively. For K|7/2,−7/2〉+Rb|1,0〉 v=−3, we find h×7.31452(15) GHz. The measured binding energies deviate by less than 0.1% from the theoretical prediction based on the potentials by Pashov et al. ²⁵.

Figure 2: Dark resonance in the molecular system.

In this case, the target state |t〉 is the K|7/2,−7/2〉+Rb|1,0〉 v=−3 state and the intermediate state |e〉 is a state in the 2(1) v=−12 hyperfine manifold. We shine a 10 μs square pulse of Raman light onto the Feshbach molecules and the remaining fraction of Feshbach molecules is detected. For δ=0, the loss of Feshbach molecules is strongly suppressed owing to the emergence of a dark state. For this particular set of data, Δ≈−2π×0.8 MHz. The inset shows the central dark-resonance feature.

With possible target states and coupling states precisely determined, we use STIRAP to transfer the molecules coherently into the 10.49238 GHz bound vibrational level of the ground-state molecular potential. STIRAP is known to be a robust coherent transfer scheme in a three-level system. When two-photon resonance (δ=0) is maintained, the molecules are transferred between |i〉 and |t〉 with negligible population in the lossy excited state |e〉 throughout the process. Figure 3a shows the counterintuitive STIRAP pulse sequence used in the experiment. In the first step, laser 2 is turned on, coupling the target state |t〉 to the intermediate state |e〉. While the intensity of laser 2 is ramped down from I₂^max to 0 within τ_p=20 μs, the intensity of laser 1 is ramped up to its maximum value I₁^max, thereby adiabatically transferring the population from the Feshbach state into the deeply bound target state |t〉. Reversing the pulse sequence reverses the process and the transfer occurs from the deeply bound level |t〉 to the initial state |i〉, as shown for t_pulse>0 in Fig. 3a.

Figure 3: Time evolution of the coherent two-photon transfer.

a, Counterintuitive STIRAP pulse sequence. b, Measured population (black circles) in the initial Feshbach state |i〉 during the STIRAP pulse sequence. To measure the population in the Feshbach state |i〉 at various times t_pulse, the phase coherent light fields 1 and 2 are switched off simultaneously and the Feshbach molecules are subsequently imaged by resonant absorption imaging (see the Methods section). Starting with population of 1 in |i〉, the Feshbach molecules are coherently transferred to the target state |t〉 by the first pulse sequence (t=−τ_p to t=0). In the deeply bound target state, the molecules are invisible to the detection light. Reversing the pulse sequence, the molecules are transferred back to the initial state |i〉 (t=0 to t=τ_p). The solid line is a fit to the experimental data on the basis of equation (1). With γ_e=2π×5.7 MHz and γ_i=2π×97 Hz, we obtain Ω₂^max/Ω₁^max=1.7 and γ_t=2π×3 kHz. Note that the thus-determined lifetime of the target state of τ_t=1/(2πγ_t)≈330 μs is shorter than the measured lifetime of 850 μs (Fig. 5) in the absence of light fields. The extra loss is presumably due to residual population of the lossy excited state during the STIRAP process. The dashed line is the corresponding calculated population of the target-state population during the process.

Figure 3b shows the time dependence of the measured population of the initial Feshbach state |i〉 during the pulse sequence. Molecules in the Feshbach state can be detected by direct high-field resonant absorption imaging, whereas molecular population in the target state is invisible to the light. During the pulse sequence, we observe the hiding of the molecules in the more deeply bound vibrational level and the transfer back into the initial state after reversal of the pulse sequence. We observe an efficiency of the double STIRAP sequence of 71%, corresponding to an efficiency of 84% for a single pulse. The transfer process can be described by an open three-level system in the rotating-wave approximation:

Here, |〈Ψ_x(t)|Ψ_x(t)〉|² and γ_x are the population and decay rate of state x, respectively (Fig. 1). The time dependence of the transfer is determined by the exact pulse shape and the ratio of the maximum Rabi frequencies Ω₁^max and Ω₂^max. Given the pulse shape shown in Fig. 1a, we can extract the Rabi frequency ratio used in the experiment. Given I₂^max/I₁^max=0.27, we obtain Ω₂^max/Ω₁^max=(1.7±0.1), corresponding to a ratio of the effective transition dipole moments of (3.3±0.3).

The absolute coupling strengths can be extracted from an analysis of the STIRAP efficiency as a function of the two-photon detuning δ, as shown in Fig. 4. We have measured the line shape both for a single STIRAP process (|i〉 → |t〉) and for a double STIRAP (|i〉 → |t〉 → |i〉). A comparison between a simulation based on equation (1) with Ω₂/Ω₁=1.7±0.1 and Δ=−2π×(50±5) MHz and the experimental single STIRAP line-shape data yields Ω₁^max=2π×(3.6±0.5) MHz and Ω₂^max=2π×(6.1±0.9) MHz at I₁^max=(3.7±1.5) W cm⁻² and I₂^max(1.0±0.4) W cm⁻². This translates into effective transition dipole moments of d₁^eff=(0.050±0.015) e a₀ and d₂^eff=(0.17±0.04) e a₀.

Figure 4: STIRAP line shape.

a, STIRAP line shape for Δ=−2π×50 MHz, Ω₁=2π×3.6 MHz and Ω₂/Ω₁=1.7 both for single STIRAP (transfer from |i〉 to |t〉) and double STIRAP (transfer from |i〉 to |t〉 and back). The asymmetry in the line shape is due to the finite single-photon detuning Δ (see Fig. 1). b, Difference between the double and single STIRAP signal as plotted in a. The difference corresponds to the fraction of Feshbach molecules transferred into the deeply bound state and back at a given δ.

Finally, we have measured the lifetime of the deeply bound molecules in the target state. Starting from the molecules in state |i〉, we apply a single STIRAP pulse to transfer the molecules into the target state |t〉. The deeply bound molecules are held in the optical dipole trap for varying times t_hold. A second STIRAP pulse then transfers the molecules back to the initial state, where they can be detected by high-field resonant absorption imaging. Figure 5 shows the decay of the deeply bound molecules. We observe a lifetime of the molecules in |t〉 of (850±200) μs as compared with a lifetime of the initial state of (10.3±0.3) ms. We observe that the lifetime of both Feshbach molecules and deeply bound molecules decreases with increasing atom density. This suggests that atom–molecule collisions limit the molecule lifetime.

Figure 5: Lifetime of the v=−3 molecules.

We observe a lifetime of 850 μs. The inset shows the decay of the initial Feshbach molecules, which have a lifetime of 10.3 ms.

In summary, we have demonstrated coherent optical transfer of heteronuclear Feshbach molecules into a 10 GHz bound vibrational level of the ground-state molecular potential. Our experiments show that the binding energy of the molecules can be significantly enhanced in a single STIRAP step. Given promising Franck–Condon factors through a number of intermediate states^9,10, the presented techniques can be extended to binding energies ≫h×10 GHz by referencing both lasers to a frequency comb bridging almost arbitrary frequency gaps²⁷. The production of a polar molecular quantum gas by means of coherent optical transfer of heteronuclear Feshbach molecules might hence come within experimental reach.

Methods

Feshbach molecule formation

After evaporation of ⁴⁰K and ⁸⁷Rb in an Ioffe–Pritchard magnetic trap, 5×10^6 87Rb and 1×10^6 40K atoms are loaded into a single-beam optical dipole trap operating at 1,064 nm. At this stage, the temperature of the atoms is about 3 μK. In the optical trap, radiofrequency adiabatic rapid passage is used to prepare a ⁴⁰K|9/2,−9/2〉 and ⁸⁷Rb|1,1〉 spin mixture. It is in this combination of Zeeman states that the Feshbach resonance at a magnetic field of 546.7 G occurs between K and Rb. After ramping the magnetic field to 555 G, we further evaporate the mixture in the optical dipole trap to a temperature of T/T_c≈1. Here, T_c is the critical temperature for the onset of Bose–Einstein condensation of the Rb cloud. Starting from the near-degenerate mixture of 1×10^5 40K atoms and 3×10^5 87Rb atoms, we finally use a magnetic-field ramp across the 546.7 G Feshbach resonance to couple the open scattering channel to the bound molecular state, thereby associating about 2×10⁴ heteronuclear Feshbach molecules. The molecules have a temperature of 300 nK and a density of ≈5×10¹¹ cm⁻³. To suppress inelastic collisions of the molecules with remaining unbound atoms, we remove about 95% of the left-over atoms.

Direct resonant absorption imaging of heteronuclear Feshbach molecules

The weakly bound heteronuclear Feshbach molecules (binding energy of 270 kHz) are detected by direct absorption imaging. The imaging uses light resonant with a cycling transition starting from the ⁴⁰K|9/2,−9/2〉 state. For small binding energies of the heteronuclear molecules, the atomic cycling transition of ⁴⁰K is detuned by only a few megahertz from the molecular transition. Light resonant with the atomic transition will therefore dissociate the molecules and then scatter on the resulting K atoms. To distinguish the absorption signal of weakly bound Feshbach molecules from unbound ⁴⁰K atoms, we flip the spin of residual ⁴⁰K|9/2,−9/2〉 atoms with a radiofrequency π-pulse on the atomic |9/2,−9/2〉→|9/2,−7/2〉 transition before imaging. Note that direct imaging of molecules at high magnetic field is favourable for heteronuclear Feshbach molecules. In the heteronuclear case, both the ground and the excited electronic potentials share a 1/R⁶ dependence. The difference in energy between these two states therefore varies more slowly with the internuclear separation than in the homonuclear case, where the excited-state long-range potential varies as 1/R³.