Long-lived entanglement of molecules in magic-wavelength optical tweezers

Experimental apparatus

In our experimental apparatus^61,62, we produce ultracold ⁸⁷Rb¹³³Cs (hereafter RbCs) molecules trapped in one-dimensional arrays of optical tweezers at wavelength 1,065.512 nm (hereafter 1,066 nm). The molecules are trapped inside an ultrahigh vacuum glass cell, with the tweezers formed by focusing light through a high numerical aperture objective lens placed before this cell. The molecules are formed by associating Rb and Cs atoms as described in ref. ⁴¹.

Magic-wavelength tweezers

For the work presented here, we have added a set of tweezers at a magic wavelength of 1,145.31 nm. This light is in the vicinity of a weakly allowed electronic transition^11,44 and eliminates the differential a.c. Stark shift hΔα_a.c. (ref. ⁶³) for the rotational transition |↓⟩ → |↑⟩. We determine the magic wavelength by measuring hΔα_a.c. with a Ramsey procedure and setting the frequency of the traps so that hΔα_a.c. is eliminated (Hepworth, T. R. et al., manuscript in preparation). In front of the objective lens, the polarization of the tweezers is parallel to the quantization axis set by the external magnetic field. The array of tweezers is created with an acousto-optic modulator (AOM) placed before the objective lens (Extended Data Fig. 1a). By applying multiple radio-frequency (RF) tones to the AOM, we form multiple diffracted beams to generate the tweezers. We dynamically switch and move the tweezers by changing the RF tones applied to the AOM to manipulate the trapped molecules mid-routine. By imaging Cs atoms trapped in the magic tweezers, we calibrate the change in tweezer position (at the focal plane) with the change in RF frequency applied to the AOM as 397(7) nm MHz⁻¹.

We perform parametric heating measurements⁶⁴ of Cs atoms trapped in the magic tweezers to characterize their 1/e² beam waists. To do this, we modulate the intensity of the traps and measure a loss feature that occurs when the modulation frequency is twice that of the trap frequency. We assume the light in the focal plane is well described by a Gaussian beam and take the polarizability of the Cs atoms to be \(918(3)\times 4{\rm{\pi }}{\varepsilon }_{0}{a}_{0}^{3}\) (ref. ⁶⁵) to obtain the 1/e² waist 1.76(4) μm.

For efficient transfer of molecules between different tweezer arrays, the tweezers must be well overlapped. We overlap the tweezers in the radial directions by imaging Cs atoms in both sets of tweezers and moving the magic tweezers until the positions of the atoms overlap. This enables us to overlap the centre of the tweezers to sub-micrometre accuracy. This method is much less sensitive to the overlap in the direction of tweezer-light propagation. We coarsely overlap the arrays in this direction by moving a lens in the expansion telescope of the 1,145 nm light so that atoms in both arrays are in focus on our imaging camera. We expect that there could be an alignment error of up to a few micrometres in this direction.

To transfer molecules between the two arrays, we start with the tweezers overlapped. We ramp up the power of the magic tweezers and then ramp down the power of the 1,066 nm array. During this step, the separation between neighbouring tweezers is approximately 4 μm. After this transfer, we switch off excess tweezers to discard the excess molecules. At the end of the experiment, we transfer the molecules back to the 1,066 nm array before disassociating them and reimaging their constituent atoms. During this process, we map the internal state of the molecule onto atomic position for multistate readout⁴¹.

To tune the dipolar interaction strength between molecules, we tune the separation of the molecules by chirping the frequency of the RF tones that generate their tweezers. For all the experiments presented in the main text, we move a pair of molecules symmetrically around their mean position to minimize the chance that a molecule is preferentially heated during the movement process.

Magic-frequency stabilization

In previous work trapping RbCs molecules in magic-wavelength traps¹², the single-molecule coherence time was limited by the frequency stability of the trapping laser. The laser was stabilized to a cavity of finesse approximately 400, and a frequency stability (standard deviation) of 0.76 MHz was achieved.

For this work, we reference an external-cavity diode laser (ECDL; Toptica DL pro) at 1,145 nm to an ultralow expansion cavity (Stable Laser Systems) with a finesse of about 3.7 × 10⁴. We stabilize this laser with a fast feedback loop (Toptica FALC pro) and achieve a linewidth of around 5 kHz. To allow for future scaling to larger arrays, we source the trapping light from a vertical-external-cavity surface-emitting laser (Vexlum VALO), which provides up to 4 W of optical power. We stabilize the beat note between this laser and the ECDL. Feedback to the laser frequency is achieved using a piezo-electric actuator mounted on a mirror in the laser cavity. With stabilization, the standard deviation of the beat-note signal is 80(20) kHz. Therefore, we expect the frequency of the trapping light to be stable to within 80(20) kHz.

Tweezer-intensity noise

In Fig. 1c, we show the measured single-molecule coherence times \({T}_{2}^{* }\) as a function of the detuning Δ_magic of the tweezers from the magic frequency f_magic. We model the effect of intensity noise in our experiment to understand the behaviour of \({T}_{2}^{* }\) with Δ_magic and briefly discuss that model here.

We determine f_magic and the sensitivity of the molecules to Δ_magic with a Ramsey procedure (Hepworth, T. R. et al., manuscript in preparation). The differential a.c. Stark shift hΔα_a.c. is proportional to the power P of each tweezer and Δ_magic. The scaling constant k = 923(3) mHz MHz⁻¹ mW⁻¹ relates these such that Δα_a.c. = kΔ_magicP. The power in each tweezer is measured before the objective lens; we estimate that the transmission from this location to the science cell is 0.48(1).

To model the intensity noise, we assume that there is Gaussian noise on P such that it is sampled from a Gaussian distribution with mean ⟨P⟩ and standard deviation σ_P. For the measurement in Fig. 1, ⟨P⟩ = 0.36 mW. This noise is mapped to Δα_a.c. with standard deviation σ_α = kΔ_magicσ_P. Therefore, the Ramsey contrast \(C(T)=\exp [-{(2{\rm{\pi }}{\sigma }_{\alpha }T)}^{2}/2]\equiv \exp [-{(T/{T}_{2}^{* })}^{2}]\) (ref. ⁶⁶). Hence, \({T}_{2}^{* }=1/(\sqrt{2}{\rm{\pi }}{\sigma }_{\alpha })\) and the solid line in Fig. 1c shows the predicted behaviour when σ_P/⟨P⟩ = 0.7%.

Achieving magic trapping conditions for multiple tweezers

For the experiment in Fig. 1, we prepare single molecules in pairs of tweezers separated by 8.6(2) μm. They are generated using a frequency difference of Δf = 21.7 MHz between the two RF tones applied to the AOM and the power per tweezer is actively stabilized to ⟨P⟩ = 0.36 mW. Therefore, we expect Δα_a.c. would be different by δ = k⟨P⟩Δf = 7.2 Hz. The data in Fig. 1a are fitted with a damped sinusoidal function with frequency ν. For the tweezer that is closer to f_magic (blue, filled points), we fit ν = 999.26(2) Hz, and for the tweezer that is further detuned (red, empty points), we fit ν = 992.49(1) Hz. This is a frequency difference of 6.77(3) Hz, approximately 6% smaller than expected.

For the experiments in Figs. 2 and 3, the tweezer separation is 2.78(5) μm. Each tweezer has a time-averaged power of ⟨P⟩ ~ 0.3 mW and is generated by RF tones with a frequency difference of Δf = 7.011 MHz. This difference in detuning from the magic frequency results in a difference in transition frequency between the two molecules of δ = k⟨P⟩Δf ~ 2 Hz.

To engineer the regime δ ≪ J, we minimize δ by minimizing Δf while maintaining the same tweezer separation. To do this, we modulate the tweezer intensities in antiphase at a frequency of 500 kHz with a duty cycle of 0.35. Simultaneously, we modulate the frequency of an RF tone applied to a compensation AOM so that, ideally, the light forming the two tweezers has an identical frequency. A schematic of the modulation scheme is shown in Extended Data Fig. 1b. The 500 kHz modulation frequency is far above any parametric resonances and we do not observe any change in the molecule-loss rate due to the modulation. We do not actively stabilize the tweezer intensity when operating in this regime. We have verified that this modulation does not affect single-molecule coherence by repeating measurements such as those in Fig. 2c (top) with and without this modulation. We attribute the non-zero value of δ reported in the main text to the non-zero decay time of tones in the amplifier that drives this compensation AOM.

In future, we plan to scale to larger molecule arrays with methods that will not require this compensation AOM. For example, by using a spatial light modulator to form the magic tweezers, as we do for the 1,066 nm tweezers⁴¹, all tweezers will have the same frequency. Alternatively, a pair of crossed acousto-optic deflectors could be used to create arrays of magic-wavelength tweezers with a constant frequency across the array⁶⁷. Moreover, we note that all sites in a magic-wavelength (one-dimensional) optical lattice would have the same frequency.

Microwave excitation

In our experiment, we prepare RbCs molecules in the absolute internal ground state |↓⟩ = |N = 0, M_N = 0, m_Rb = 3/2, m_Cs = 7/2⟩. Here, N is the rotational quantum number, M_N is its projection, m_Rb is the projection of the nuclear spin of Rb and m_Cs is the projection of the nuclear spin of Cs. We couple this state to the excited rotational state |↑⟩ = |N = 1, M_N = 1, m_Rb = 3/2, m_Cs = 7/2⟩. Both of these states are stretched with maximum projections of angular momentum. In our experiment, the quantization axis is set by the externally applied magnetic field (about 181.7 G), which stays approximately constant for all science stages of the experiment.

The transition |↓⟩ → |↑⟩ is magnetically insensitive. The dominant contribution to the Zeeman shifts of the states |↓⟩ and |↑⟩ is associated with the projection of the nuclear spins. However, as these are both stretched states with the same m_Rb and m_Cs, their nuclear-spin Zeeman shifts are equal. The rotational Zeeman effect is very small^68,69, leading to a differential Zeeman shift of about 5 Hz G⁻¹ × h. In our experiment, we stabilize the magnetic field to the approximately 10 mG level so that the differential shift does not vary from shot to shot. We expect that this magnetic field noise will limit single-molecule coherence times to the approximately 10 s level.

We drive the molecular transition |↓⟩ → |↑⟩ with microwaves radiated from a dipole Wi-Fi antenna mounted approximately 10 cm from the vacuum chamber. The frequency of the transition in free space (or in a perfectly magic tweezer) is 980.38559837(4) MHz (Hepworth, T. R. et al., manuscript in preparation). The resultant microwaves are not well polarized, so it would be possible to drive transitions to other rotational states. For this reason, we use Rabi frequencies ≲10 kHz such that off-resonant excitation to other states is negligible⁴¹ and each molecule can be considered a two-level system. For kilohertz-scale Rabi frequencies, we drive the antenna with an Agilent E4400B source and typically input a microwave power of about 0 dBm to the antenna. We vary the phase of this source when measuring the parity Π presented in Fig. 3. For hertz-scale Rabi frequencies, we use an Anritsu MG369xC source set to about −15 dBm with a further 55 dB of attenuation. We amplitude modulate this source with an arbitrary function generator (Tektronix AFG3022C) when using the Hann pulse for direct microwave entanglement. The sources are combined before the antenna with an RF switch (Minicircuits ZFSWA2R-63DR+) and are referenced to the same 10 MHz GPS signal to ideally maintain a constant, but arbitrary, relative phase. We attribute the observed dφ/dt for the state |Ψ⁺⟩ (Fig. 3d, bottom, green data points) to a slight phase drift between these microwave sources.

Microwave pulse sequences

We probe single-molecule coherences and generate entanglement with spin exchange using the Ramsey pulse sequences shown in Figs. 1a and 3b, respectively. In both sequences, we apply two π/2 pulses on the single-molecule transition |↓⟩ → |↑⟩ with a hold time T between them. Both pulses have the same phase which we use to define the \(\widehat{x}\)-axis of the Bloch sphere. Neither of these sequences includes any rephasing pulses. The microwaves drive the transition with Rabi frequency Ω = 5.0(1) kHz.

For the measurement in Fig. 1, the first pulse prepares each molecule in the state \((| \downarrow \rangle +{\rm{i}}| \uparrow \rangle )/\sqrt{2}\). The non-zero microwave detuning (Δ ≈ 1 kHz) causes the phase to accumulate between |↓⟩ and |↑⟩ during the hold, and the second pulse projects this onto the states |↓⟩ and |↑⟩. The populations of these states oscillate as a function of T with frequency ν = Δ − Δα_a.c.. For the entangling sequence of Fig. 3b, the microwaves are resonant with the transition and the pulse sequence prepares the pair state |Φ(T)⟩ (equation (1)).

When entangling molecules with direct microwave excitation, as shown in Fig. 3c, we set the Hann pulse such that the peak Rabi frequency Ω₀ = 2.245 Hz. For the π/2 pulses used to read out the parity of the entangled state |Ψ⁺⟩, we use square pulses that drive the transition |↓⟩ → |↑⟩ with Rabi frequency Ω = 882(3) Hz.

Experimental statistics

To obtain statistics, we repeat each experimental sequence many times. Data points in figures show the average state populations from these repeats and error bars show the 1σ binomial confidence intervals, calculated using the Jeffreys prior^70,71,72 and are indicative of the number of repeats used to obtain each data point. Most data presented here are obtained by postselecting to ignore experimental runs in which molecule formation was unsuccessful or molecules were not detected in the states |↓⟩ or |↑⟩. We perform this postselection by using optical tweezers to map these cases onto distinct spatial configurations of atoms following the methods reported in ref. ⁴¹. Briefly, Rb atoms may be transferred into three distinct tweezer arrays: one flagging molecule-formation errors, one to detect |↑⟩ and one to detect |↓⟩. The Cs atom remains in the original tweezer array. At the end of each experimental sequence, we capture atomic fluorescence images to determine the atom locations. These are then used for shot-to-shot postselection, with a successful shot requiring recovery of both the Cs atom and the Rb atom in either the |↑⟩ or |↓⟩ array. For the data presented in Fig. 3a, in which we measure molecular lifetimes, we postselect to remove only detectable molecule-formation errors.

With postselection, we can obtain statistics for single- and two-molecule cases in a single set of experimental runs using the same sequence. For example, for each value of Δ in Fig. 2b, we repeat the experiment about 400 times. In 25% of runs, we successfully form and detect exactly one molecule in either the state |↓⟩ or |↑⟩. Therefore, each data point in the top panel represents about 100 samples of the binomial distribution, and the error bars are calculated accordingly. Likewise, in 7% of runs, we successfully form and detect exactly two molecules, and each data point in the bottom panel reflects about 30 samples.

Simulations of the two-molecule system

To simulate the dynamics of the two-molecule system, we use the Python package QuTiP⁷³ and model its time evolution with different microwave pulses and hold times.

The Hamiltonian that describes a pair of molecules interacting by the dipolar spin-exchange interaction in the presence of microwave coupling between |↓⟩ and |↑⟩ with a Rabi frequency Ω is³²

Here, \({H}_{{\rm{mol}}}^{(i)}=\frac{1}{2}h\varOmega ({\sigma }_{i}^{+}+{\sigma }_{i}^{-})-h{\varDelta }_{i}{| \uparrow \rangle }_{i}{\langle \uparrow | }_{i}\) is the single-particle Hamiltonian of molecule i and \({{\mathcal{I}}}^{(i)}\) is its identity operator. \({H}_{{\rm{int}}}=\frac{1}{2}hJ({\sigma }_{1}^{+}{\sigma }_{2}^{-}+{\sigma }_{2}^{+}{\sigma }_{1}^{-})\) is the interaction Hamiltonian, \({\sigma }_{i}^{+}\equiv {| \uparrow \rangle }_{i}{\langle \downarrow | }_{i}\) is the raising operator for molecule i and \({\sigma }_{i}^{-}\equiv {| \downarrow \rangle }_{i}{\langle \uparrow | }_{i}\) is the lowering operator for molecule i. hJ is the interaction strength and Δ_i is the microwave detuning from the transition \({| \downarrow \rangle }_{i}\to {| \uparrow \rangle }_{i}\). We allow for the fact that there may be a small difference \(\delta \equiv \varDelta {\alpha }_{{\rm{a.c.}}}^{(2)}-\varDelta {\alpha }_{{\rm{a.c.}}}^{(1)}={\varDelta }_{1}-{\varDelta }_{2}\) in the differential a.c. Stark shifts of the molecules as they are in different traps. We generally denote pair states as \(| ab\rangle \equiv {| a\rangle }_{1}\otimes {| b\rangle }_{2}\).

Our molecules are predominantly, but not exclusively, formed in the three-dimensional motional ground state⁴¹. This causes shot-to-shot noise in J as the separation averaged over the molecular wavefunctions varies. We incorporate this in our model with a Monte Carlo method: the dynamics are averaged over 200 iterations for which we assume that shot-to-shot noise in J is such that, in each experimental iteration, J is sampled from a Gaussian distribution with mean ⟨J⟩ and standard deviation σ_J.

Eigenstates in the absence of microwaves

Equation (2) gives the Hamiltonian H that describes our system of two interacting molecules. In the absence of microwave radiation, H simplifies to

in the basis {|↓↓⟩, |↓↑⟩, |↑↓⟩, |↑↑⟩}. The eigenstates of H₀ are |↓↓⟩, |↑↑⟩,

where N_± are normalization constants.

In the main text, we consider the limit of strong interactions (that is, ∣J∣/∣δ∣ → ∞), where \(| {\widetilde{\varPsi }}^{\pm }\rangle \to | {\varPsi }^{\pm }\rangle \equiv (| \downarrow \uparrow \rangle \pm | \uparrow \downarrow \rangle )\sqrt{2}\). However, the non-zero value of δ in our experiment gives rise to eigenstates that are slightly asymmetric. The eigenstates of our system, taking J = 5.20 Hz and δ = 220 mHz, are \(| {\widetilde{\varPsi }}^{+}\rangle =0.722| \downarrow \uparrow \rangle +0.692| \uparrow \downarrow \rangle \) and \(| {\widetilde{\varPsi }}^{-}\rangle =0.692| \downarrow \uparrow \rangle -0.722| \uparrow \downarrow \rangle \), in which the coefficients are given to three significant figures.

In Fig. 2b (bottom), we show microwave spectroscopy in which we drive the transition |↓↓⟩ → |Ψ⁺⟩. The asymmetry in the probability amplitudes |↓↑⟩ and |↑↓⟩ in |Ψ⁺⟩ is the reason why we measure a slightly higher population in the state |↓↑⟩ than in the state |↑↓⟩. This has only a slight effect on the achieved entanglement fidelity, the dominant limitations of which are the non-zero value of σ_J and leakage errors caused by Raman scattering from the tweezer light.

Design of direct-entanglement pulse

For the demonstration of the two-molecule microwave gate shown in Fig. 3c, we use a pulse with a simple shape. We choose the parameters of this pulse using our Monte Carlo model with the parameters fitted from the data in Fig. 2.

First, we model and optimize the pulse assuming that there is no noise in J. We take J to be equal to the measured value of ⟨J⟩ (5.20 Hz) and consider three simple pulse shapes: a square pulse (Ω(t) = Ω₀ for 0 < t < τ, 0 otherwise), a Hann pulse (Ω(t) = Ω₀ sin²(πt/τ)) and a Blackman–Harris pulse (Ω(t) = Ω₀ [a₀ − a₁cos(2πt/τ) + a₂cos(4πt/τ) − a₃cos(6πt/τ)] for a₀ = 0.35875, a₁ = 0.48829, a₂ = 0.14128 and a₃ = 0.01168). Here, Ω(t) is the Rabi frequency that we drive the single-molecule transition |↓⟩ → |↑⟩, Ω₀ is the peak Rabi frequency and τ is the pulse duration. For each pulse shape, we vary Ω₀ and calculate P_↓↑ + P_↑↓ as a function of τ and the microwave detuning Δ (Extended Data Fig. 3, inset). P_↓↑ + P_↑↓ is a good proxy for the fidelity of the entangling gate because pairs that are not entangled preferentially occupy the states |↓↓⟩  and |↑↑⟩. This gives an optimum value of τ and Δ for each value of Ω₀, with an associated maximum (P_↓↑ + P_↑↓)_max. We show the behaviour of (P_↓↑ + P_↑↓)_max on τ in Extended Data Fig. 3 (top); a longer pulse duration generally allows higher fidelity entanglement because a smaller Rabi frequency can be used to minimize off-resonant excitation to |↑↑⟩.

We now consider fluctuations in J. With the optimum pulse parameters obtained above, we use our Monte Carlo model to recalculate (P_↓↑ + P_↑↓)_max when σ_J = 1 Hz (Extended Data Fig. 3, bottom). The effect of σ_J is to favour larger Rabi frequencies (that is, smaller τ), which spectrally broaden the excitation feature. We expect that, out of the pulse shapes considered, a Hann pulse will achieve the highest (P_↓↑ + P_↑↓)_max. The corresponding pulse parameters are τ = 328 ms, Δ = 3.069 Hz and Ω₀ = 2.245 Hz; we use these for the experiments presented in Fig. 3.

Expected interaction strength

Here we consider the strength of the spin-exchange interaction between the molecular pair states |↓↑⟩ and |↑↓⟩. First, we consider the case in which the molecules can be treated as point particles with zero temperature. Then, we estimate the effect that the non-zero temperature and wavefunction size have on this interaction strength.

Point-particle and zero-temperature case

The strength of the dipole–dipole interaction between the states |↓↑⟩ and |↑↓⟩ is³²

Here, r_i is the position vector of molecule i and θ is the angle between the quantization axis and the intermolecular vector. ε₀ is the vacuum permittivity. \({d}_{\downarrow \uparrow }\equiv \langle \uparrow | {\widehat{d}}_{1}| \downarrow \rangle \) is the relevant matrix element for the dipole operator \({\widehat{d}}_{1}\) that corresponds to the σ⁺ transition that we use. At zero electric field, \({d}_{\downarrow \uparrow }=d/\sqrt{3}\), where d = 1.225(11) D is the RbCs molecule-frame electric dipole moment⁷⁴.

For all experiments, the intermolecular axis is parallel to the quantization axis (that is, θ = 0). We apply no external electric fields and assume that the stray electric field is negligible. For the experiment presented in Fig. 1, we prepare molecules at a separation |r₁ − r₂| = 8.6(2) μm. Therefore, if the molecules were point particles pinned to the centre of their respective optical tweezer, we would expect J = 0.24(1) Hz. For this reason, we limit the interrogation time for this measurement to T ≲ 2 s so that these interactions are insignificant. Likewise, for the experiments presented in Figs. 2 and 3, |r₁ − r₂| = 2.78(5) μm giving J = 7.0(4) Hz. In both cases, the uncertainty in J reflects the uncertainty in the molecular separation.

Effect of motional excitation

We fit the microwave spectroscopy shown in Fig. 2 with a Monte Carlo model, in which J is sampled from a normal distribution for every iteration of the experiment. Using this model, we extract the mean ⟨J⟩ = 5.20(5) Hz and standard deviation σ_J = 1.0(1) Hz.

We expect that motional excitation of the molecules causes the reduction in ⟨J⟩ from the expected value and is the dominant contribution to σ_J. To estimate the magnitude of this effect, we numerically calculate the matrix elements

where \(| {{\bf{n}}}_{i}\rangle \equiv | {n}_{x}^{i},{n}_{y}^{i},{n}_{z}^{i}\rangle \) is the three-dimensional wavefunction for molecule i, labelled by the number of motional quanta in each of the three directions. Here, we define the x-axis as the quantization axis, the y-axis as the other radial axis of the tweezers and the z-axis as the direction of tweezer-light propagation, as shown in Extended Data Fig. 1a. We assume that the trapping potential is harmonic and the three axes are separable such that

where \({H}_{{n}_{r}}\) are the Hermite polynomials and the index r runs over the three separable axes. \({\beta }_{r}=2{\rm{\pi }}\sqrt{m{\nu }_{r}/h}\) and ν_r are the confinement length and trap frequency along the r-axis, respectively, and the normalization constant \(C({n}_{r})=1/\sqrt{({2}^{{n}_{r}}{n}_{r}!{\beta }_{r}{{\rm{\pi }}}^{1/2})}\).

Extended Data Fig. 2 shows calculations of selected values of \(\bar{J}\). In general, \(\bar{J}\) is a six-dimensional matrix; we show the three slices of this matrix in which the motional quanta of the molecules along one axis is varied, whereas there is no motional excitation along the other axes. For this calculation, the separation between the most likely positions of the molecules is 2.78 μm along the x-axis. The molecules are trapped in tweezers of waist 1.76 μm and intensity 4.5 kW cm⁻². We neglect the effect of the tweezer confining the first molecule on the second molecule (and vice versa) and assume that fluctuations in the relative positions of the tweezers are negligible as they are formed from a common source⁶⁰. We take the polarizability of the molecules at the magic wavelength to be \(720\times 4{\rm{\pi }}{\varepsilon }_{0}{a}_{0}^{3}\) (ref. ¹¹) such that the trap frequencies are ν_x = ν_y = 3.0 kHz and ν_z = 0.4 kHz.

We estimate that 58(6)% of molecules formed in the 1,066 nm array occupy the three-dimensional motional ground state^41,75. Furthermore, we expect that most of the motionally excited molecules have just one motional quantum. Therefore, the most likely scenario is that, when a pair of molecules is formed, one occupies the motional ground state and the other has one motional quantum. Assuming negligible heating as the molecules are transferred to the magic tweezers, the relevant matrix element \(\bar{J}\approx 5.5(3)\,{\rm{Hz}}\). This is approximately equal to our measured value of ⟨J⟩, and the stochastic occupancy of the motional states will give rise to σ_J.

In future, we expect that moving to more confining traps (for example, by trapping the molecules in an optical lattice) will allow smaller separations and reduce the wavefunction spread, leading to an increase in ⟨J⟩ and a reduction in σ_J. We note that σ_J could also be reduced by increasing the fraction of molecules that occupy the three-dimensional motional ground state by reducing atomic heating before association^62,75.