a b

In situ we first study whether intensity interferometry can be used to characterize the distributions of non-interacting Feshbach molecules. We perform an HBT experiment in which we measure the fluctuations in the density distribution of individual clouds of molecules on a grid of detectors after a time of flight (TOF) 30 . Particles originating from sources A and B can arrive at detectors 1 and 2 in two different ways (A→1, B→2 and A→2, B→1) (Fig. 1). If the two paths are indistinguishable, then their quantum mechanical amplitudes interfere, as reflected in the two-point density correlation function. The correlation relates to the Fourier transform of the source density distribution in the case of a distribution of sources with no phase coherence. As a result, if the positions of the molecules are initially discretized by an optical lattice, the interference pattern exhibits constructive interference peaks whose separation and width are inversely related to the lattice spacing and in situ cloud size, respectively 13,22 .

There are two requirements for measuring a high-visibility HBT interference pattern: particle shot-noise-limited detection and a detector spatial resolution better than the interference peak width. To fulfil these conditions, we develop a molecular quantum gas microscope [31][32][33] to simultaneously prepare and detect over one hundred 23 Na 87 Rb molecules by dissociating them into atoms and using the atoms as tags for the molecule positions.

We form the ultracold bosonic molecules from degenerate gases of Na and Rb atoms by magnetoassociation 34 . To start, dual Na and Rb Bose-Einstein condensates (BECs) are prepared in a single layer of a vertical lattice. Fine adjustment of atom numbers is accomplished using a bichromatic dimple trap, allowing the reproducible production of clouds of several hundred atoms per species. Both species are prepared in their |F = 1, m F = 1〉 states, where F and m F are the total-angular-momentum quantum number and its projection along the quantization axis, respectively. Next, we make the mixture miscible by nulling the interspecies scattering length using an s-wave Feshbach resonance at 347.6 G (ref. 35 ). The atoms are then loaded into a two-dimensional (2D) in-plane optical lattice with spacing a = 752 nm. The lattice depth is increased to 36E r,Na (where E r,Na = h 2 /8m Na a 2 is the recoil energy for Na, m Na is the atomic mass and h is the Planck constant), crossing the superfluid-to-Mott-insulator transition and freezing tunnelling for both species. We associate the atoms into molecules by adiabatically ramping the magnetic field across the resonance at a rate of 2.5 G ms -1 , changing the scattering length from attractive to repulsive. For lattice sites containing one atom of each species, the theoretical conversion efficiency is close to unity (Supplementary Section III). We selectively remove the remaining free Na and Rb atoms by transferring them to the |2, 2〉 hyperfine state and applying resonant light pulses on the optical cycling transitions. The transfer and removal is repeated four times for each species to achieve high efficiency.

To realize a quantum gas microscope for molecules, we employ a three-step process to extract site-resolved positions (Fig. 2a). We first ramp the magnetic field back to the attractive side of the Feshbach resonance, increasing the Franck-Condon overlap between the bound and free states of atoms on a site. We note that a confinement-induced weakly bound state exists here due to the lattice 36 ; therefore, an adiabatic ramp across the resonance does not break apart the molecules. The confinement-induced molecules are then dissociated by addressing Na with another series of microwave transfers and optical removal pulses. The remaining Rb atoms serve as markers for the Feshbach molecule locations. Light-assisted collisions during the optical removal pulses are estimated to be negligible (Supplementary Section VI). After ramping the magnetic field to zero, the in-plane lattice depth is increased to 6,000E r,Rb and the atoms are loaded into a light-sheet potential. Optical molasses beams both cool the Rb atoms and scatter photons, which are collected with a high-numerical-aperture objective and imaged onto a camera. The full-width at half-maximum of the point spread function is 1,010(15) nm, which is sufficient for single-site resolution using a reconstruction algorithm 33 . Figure 2b shows an example image of an in situ cloud of 103 molecules. A radially averaged density profile, averaged over 30 experimental repetitions, is shown in Fig. 2c. We measure a central molecule filling of 0.15(1), compared with the Rb parity-projected central filling of 0.73(2) (Supplementary Section III). The molecule filling fraction can be increased in future work by improving the overlap between the atomic clouds, lowering their temperature and increasing the vertical confinement to obtain stronger interactions at the superfluid-to-Mott-insulator transition.

A high molecule detection fidelity is essential for obtaining high-contrast HBT interference as well as for future applications of molecule microscopy. Possible limitations on the molecule detection fidelity include imperfect atom tagging of sites that had molecules as well as site-to-site hopping and loss of the Rb-tag atoms during fluorescence imaging. We first measure the false-negative molecule detection rate due to imperfect tagging. After performing the tagging protocol consisting of dissociating the molecules and optically removing Na, we ramp the field to the repulsive side of the Feshbach resonance and remove the Rb-tag atoms. The false-negative rate of 1.2(1)% is obtained by repeating the tagging protocol before fluorescence imaging to identify failures from the first attempt. During our 0.5 s imaging exposure time, we measure site-to-site hopping of 0.1(1)% and 1.7(3)% loss. The latter is consistent with the measured atom lifetime limited by background gas collisions. We additionally measure a false-positive rate of 0.34(5)% due to Rb atoms that did not associate into molecules and were not removed before imaging (Supplementary Section V).

We further confirm that the detected particles are molecules by measuring their binding energy as a function of magnetic field using dissociation spectroscopy (Fig. 3). Driving the |1, 1〉→|2, 2〉 transition in Na yields the molecular dissociation spectrum, showing a sharp onset at a microwave frequency shifted by binding energy E b from the atomic transition. Molecules that are not dissociated are not detected in fluorescence imaging. We find close agreement between our measured binding energies and a coupled-channel calculation using the BOUND package 37 based on the parameters mentioned elsewhere 38 . Confinement effects are ignored since they are smaller than the resolution of the measurement.

Having established our molecule detection procedure, we now observe the HBT effect by measuring the density-density correlation function g (2) (d) after a TOF (Fig. 4). The correlation function is defined as

where n(x) is the number operator at detection position x, and d is the displacement between the detection positions. Maximal particle bunching is indicated by g (2) = 2, whereas for uncorrelated particles, g (2) = 1. For a Gaussian source cloud of half-width s (at e -1/2 of the maximum), the half-width of the interference peaks is given by δ = ℏt/ms (at e -1 of the maximum), where m is the particle mass, t is the TOF and ℏ is the reduced Planck constant. Although particle shot-noise-limited imaging in a 2D plane can theoretically achieve peak correlation amplitudes of g (2) = 2, the amplitude reduces if the width of the peaks is narrower than the detector size (in our case, one lattice site). Therefore, we carefully choose our source cloud size and use the largest possible TOF, given the constraints from the size of the lattice beams, to maximize the HBT amplitude.

A larger signal-to-noise ratio of the interference pattern can be achieved for higher in situ lattice filling fractions. Since we achieve higher fillings with atoms than with molecules, we first benchmark the interferometry protocol with Rb atoms. We prepare a gas with 189(20) Rb atoms frozen in a 2D optical lattice (depth, 66E r,Rb ), with peak filling of 0.86(2) and an average source size s = 7(1) sites. We abruptly turn off the 2D lattice to initiate a 9.4(1) ms TOF in the vertical lattice. The vertical lattice confinement is set to ω Rb = 2π × (3, 4, 1,000) Hz, providing negligible radial confinement. The TOF satisfies the far-field detection condition t ≫ 2π/ω r , where ω r = 2π × 14 kHz is the on-site radial trap frequency of the 2D lattice for Rb (ref. 39 ). Following the TOF, we turn on the 2D lattice to pin the distribution for imaging.

The observed atomic HBT correlations are shown in Fig. 4c. We observe a high-contrast interference pattern with average correlation peak amplitudes of 1.80 (12) and an average background value of 0.999 (6). The measured peak separation and width from the one-dimensional (1D) cut (Fig. 4d) is 75.9(4) and 2. For each field, we drive the |1, 1〉→|2, 2〉 transition in Na to break apart the molecules (upper-left inset) and measure the resulting dissociation spectrum. We extract the bound-free transition frequency from the onset of the spectrum. We also measure the free atomic transition frequency at each field. The difference between the atomic and molecular transition frequencies gives the binding energy (blue circles). The black dashed line is the predicted binding energy from a coupled-channel calculation. The lower-right inset shows an example molecular dissociation spectrum at 343.4 G (red circles) fit to an asymmetric Gaussian (solid black line). The error bars in the inset are the s.e.m. The error on the binding energies is taken as the half-width of the shaded region of the spectrum.

of the peak width 40 . The symmetric pattern verifies that the 2D lattice axes are orthogonal and the lattice spacings along both axes are identical to better than 0.5%. This implies that g (2) (d) is invariant for d reflected across the x = 0, y = 0, x = y and x = -y symmetry axes, justifying the averaging of the weaker molecular correlations across the lattice symmetries to reduce noise.

We repeat the HBT interferometry with 56(13) molecules and a mean source size s ≈ 17 sites, which is expected to produce interference peaks whose widths are of the order of the lattice spacing. The protocol is the same as that used for atoms, with molecules released from the 2D lattice at a magnetic field of 335.1 G (with binding energy E b /h ≈ 20 MHz). Figure 4e,f shows the observed molecular correlations averaged across the lattice symmetries. Since the TOF is the same as that used in the Rb-atom correlation measurement, the smaller correlation peak spacing for the molecules is a direct result of their increased mass. The measured spacing is 60.0(5) sites, consistent with the theoretical expectation of 60.3(6) sites. Although the correlation peaks are narrower than for Rb (<1 site), the peak amplitude remains large at 1.58 (13). The average baseline is 1.04(1), with the deviation from unity caused by correlations at all distances due to shot-to-shot molecule number fluctuations. The interference contrast is sensitive to the preparation of the molecules in the same internal state and the same motional state of the vertical lattice, since these quantum numbers can provide which-path information during the free expansion; therefore, the measured contrast of 0.54 (13) indicates a high degree of indistinguishability of the molecules (Supplementary Section IX).

To conclude, we have demonstrated site-resolved measurements of density correlation functions in a non-interacting molecular quantum gas after TOF expansion. The observation of the HBT effect with molecules paves the way towards realizing other quantum optical phenomena with molecules of increasing complexity 41 . By transferring the Feshbach molecules to the rovibrational ground state 1,42 , which has been already demonstrated for NaRb (ref. 43 ), a molecular lattice gas with strong dipolar interactions can be prepared. The interplay of quantum statistics and interactions can give rise to interesting signatures in HBT measurements 20,21 . In addition, many correlated quantum states predicted to be realizable with polar molecules also exhibit real-space density correlations that can be directly measured with a molecule microscope, including Wigner crystals 44 and Mott solids with rational lattice fillings 45 . Finally, by encoding a pseudospin using a rotational degree of freedom, the microscope can be used to study spin correlations in quantum magnets 29 . Fluorescence (a.u.) g ( 2) g ( 2) 〈g ( 2) (d)〉

Letters

NATURe PHysics

Methods

Here we describe the formation of ultracold bosonic 23 Na 87 Rb molecules by magnetoassociation from degenerate gases of Na and Rb atoms. NaRb molecules have been produced in bulk mixtures elsewhere 34 . We start by creating dual Na/Rb BECs with typically 2 × 10 5 atoms of each species in the |F = 1, m F = -1〉 state, using Na as a sympathetic coolant for Rb during forced evaporative cooling in a quadrupole magnetic trap followed by evaporation in a crossed 1,064 nm optical dipole trap. To prepare a 2D system, we load the condensates into a 1,064 nm light sheet with tight confinement along the vertical direction (ω Na = 2π × (24, 117, 1,900) Hz, ω Rb = 0.86ω Na ). We then transfer both atom species to the |F = 1, m F = 1〉 state, the entrance channel for the relevant Feshbach resonance. To further increase the vertical confinement, the atoms are loaded into a single layer of a 3.8-μm-spacing vertical lattice created by two 1,064 nm beams intersecting at 16° (ω Na = 2π × (14, 20, 4,500) Hz). For our optical lattice approach of molecule formation, we need the central density of the clouds to be of the order of one atom per site for each species. Given our trap parameters, this requires the ability to reproducibly generate small condensates of the order of one hundred atoms. We achieve this by performing a second stage of evaporative cooling in a tightly focused bichromatic dimple trap, which allows for independent adjustment of the atom number in each species (Supplementary Section II).

At a near-zero magnetic field, Na and Rb BECs are immiscible. To increase the overlap of the spatial distributions before magnetoassociation, we tune the interspecies scattering length using an s-wave Feshbach resonance at 347.6 G to bring the clouds into the miscible regime 35 . We quickly ramp the field above resonance to 415.9 G and then slowly decrease the field to the zero crossing of the interspecies scattering length at 351.9 G in 20 ms. We subsequently load the 2D mixture into an in-plane square lattice with spacing a = 752 nm created by the fourfold interference of a single 1,064 nm beam (105 μm waist) in a bowtie configuration 46 . We freeze tunnelling for both species by ramping the in-plane lattice depth to 36E r,Na , where E r,Na = h 2 /8m Na a 2 is the recoil energy for Na. Next, we ramp the magnetic field below resonance at a rate of 2.5 G ms -1 to form Feshbach molecules. For lattice sites containing one atom of each species, the theoretical conversion efficiency is very close to unity (Supplementary Section III). We selectively remove the remaining free atoms by transferring them to |2, 2〉 with a microwave Landau-Zener sweep at a field of ~346.6 G (molecular binding energy E b /h ≈ 0.7 MHz) and applying resonant light on the |2, 2〉 to |3, 3〉 optical cycling transition. The transfer and removal is repeated four times for each species to achieve high efficiency, as described earlier.

For imaging, the magnetic field is brought to 0 G, the in-plane lattice depth is increased to 6,000E r,Rb and the Rb atoms are loaded back into the light sheet at a depth of 140 μK. Optical-molasses cooling light scatters photons into a 0.5-numerical-aperture objective, with ~10 4 photons collected on the camera per atom per second.