# Precision measurement of atom-dimer interaction in a uniform planar Bose gas

Chloé Maury, Brice Bakkali-Hassani, Guillaume Chauveau, Franco Rabec, Sylvain Nascimbène, Jean Dalibard, Jérôme Beugnon

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Chloé Maury, Brice Bakkali-Hassani, Guillaume Chauveau, Franco Rabec, Sylvain Nascimbène, et al. Precision measurement of atom-dimer interaction in a uniform planar Bose gas. Physical Review Research, 2023, 5 (1), pp.L012020. 10.1103/PhysRevResearch.5.L012020. hal-04022876

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Distributed under a Creative Commons Attribution 4.0 International License
Precision measurement of atom-dimer interaction in a uniform planar Bose gas

C. Maury, B. Bakkali-Hassani, G. Chauveau, F. Rabec, S. Nascimbene, J. Dalibard, and J. Beugnon

Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université,
11 Place Marcelin Berthelot, 75005 Paris, France

<figure><img src="image_2.png" /></figure>

(Received 14 November 2022; accepted 12 January 2023; published 15 February 2023)

Cold quantum gases, when acted upon by electromagnetic fields, can give rise to samples where isolated atoms coexist with dimers or trimers, which raises the question of the interactions between these various constituents. Here we perform microwave photoassociation in a degenerate gas of $^{87}$Rb atoms to create weakly bound dimers in their electronic ground level. We determine the complete energy diagram of one hyperfine manifold of the least-bound level, which we accurately reproduce with a simple model. Then, from the density-induced shift of the photoassociation line, we measure the atom-dimer scattering length for two weakly bound states of the molecular potential.

DOI: 10.1103/PhysRevResearch.5.L012020

*Introduction.* Cold atomic gases constitute a unique platform to study many-body phenomena. They offer the possibility to relate macroscopic observables, e.g., the equation of state of the fluid, to microscopic quantities such as two- and three-body interactions, as illustrated by the introduction of the “contact parameter” [1]. More generally, with a suitable control of these few-body interactions, quantum gases can host simultaneously atoms and molecules, opening intriguing connections with quantum chemistry [2–7].

The coexistence of atoms and molecular dimers in a quantum gas raises the question of whether a universal description of their interactions exists. In a spin-1/2 Fermi gas close to the unitary limit, the knowledge of the atom-atom scattering length *a* is sufficient to predict the scattering length *a*<sub>ad</sub> characterizing the interaction between an atom and a weakly bound dimer [3,8–17], as well as *a*<sub>dd</sub>, the scattering length for dimer-dimer interaction [18]. In a Bose gas in the vicinity of a scattering resonance, the search for a universal relation between *a* and *a*<sub>ad</sub> is more subtle due to the Efimov effect [19], i.e., the existence of a large number of three-body bound states when *a* increases, which requires the introduction of the so-called “three-body parameter” [20–24]. Outside a resonance, *a* is comparable to the range of the potential and the existence of a van der Waals universality relating *a*<sub>ad</sub> and *a* for weakly bound dimers remains an open question [25–27].

Experimentally, most studies of atom-dimer interactions in quantum gases concentrated so far on inelastic scattering [28–38] and atom-exchange reactions [39–42]. Elastic collisions have been studied in a Fermi mixture, in which several partial waves contributed to the scattering process [43] and, more recently, in the context of sympathetic cooling of a

molecular gas [44]. In this Letter, we study elastic interactions between atoms and s-wave dimers in a rubidium Bose gas. The dimers are prepared either in the least or second-to-least bound vibrational (n = −1 and n = −2) level using microwave photoassociation. Due to the low temperature of the gas, atom-dimer collisions only occur in the s-wave channel, and can be parametrized by the scattering length *a*<sub>ad</sub>. We first present our measurement of the full Zeeman diagram of the relevant hyperfine manifold for the least-bound level. This allows us to identify the dimer state which is best suited in terms of weak sensitivity to magnetic perturbations. Then, we present a precise spectroscopic measurement of the scattering length *a*<sub>ad</sub>, using a uniform atomic gas to minimize inhomogeneous broadening of the signal.

*Experimental methods.* We use uniform planar Bose gases of $^{87}$Rb atoms at low temperature ($k_B T/h \lesssim 400$ Hz) prepared in the strongly degenerate regime, as described in Ref. [45]. The confinement along the vertical direction *z* is created by an optical accordion lattice operating at the wavelength $\lambda = 532$ nm. We load the atoms from a three-dimensional (3D) Bose-Einstein condensate around a single node of this optical standing wave in a large lattice spacing configuration ($\approx 10$ µm) and then compress this gas to a short lattice spacing ($\approx 2$ µm) so as to reach a quasi-two-dimensional (quasi-2D) regime. The resulting approximate harmonic potential of frequency $\omega_z/2\pi \approx 3.7$ kHz along the vertical direction is strong enough to reach the regime $k_B T$, $\mu_a \ll \hbar\omega_z$, where $\mu_a$ is the chemical potential of the gas. This ensures the 2D character of the fluid at the thermodynamic level. However, the thickness of the cloud along *z*, $\ell_z = \sqrt{\hbar/m\omega_z} \approx 180$ nm, remains much larger than the scattering length *a* $\approx 5$ nm, so that collisions keep a 3D character [46].

The in-plane confinement is created by shaping the profile of a repulsive optical dipole trap with the same wavelength $\lambda$. A digital micromirror device allows us to design arbitrary-shaped box potentials. In this work, we mostly use a disk-shaped box potential of radius 20 µm [Fig. 1(a)]. However, for some specific sequences requiring a mixture of Zeeman sublevels, we use instead an array of microtraps, as shown in

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<figure><img src="image_3.png" /><figcaption>FIG. 1. (a),(b) Clouds used for microwave spectroscopy with either (a) a single component state $|f=1, m_f=0, \pm 1\rangle$ or (b) a binary mixture of these hyperfine states (bar length: 10 µm). (c) Relevant levels for a pair of atoms occupying the $f=1$ or $f=2$ hyperfine sublevels of the electronic ground state of $^{87}$Rb. The dissociation limits of molecular state manifolds, represented as dashed lines, are separated by $h\nu_{hf} \approx h \times 6.8$ GHz. The molecular potentials are represented by thick continuous lines. The graph is limited to relatively large interatomic distances where van der Waals interaction dominates, hence the superposition of singlet and triplet potentials at the scale of the figure. We also indicate the values of the total spin angular momentum *F* of the dimer. Here, we focus on the least-bound vibrational levels *n* = -1 and *n* = -2 of the ($f=1; f=2$) subspace, with zero orbital angular momentum. These levels are located ~25 and 642 MHz below the dissociation energy, respectively. The dimers are produced by microwave photoassociation of two atoms either both in *f* = 1 (M₁ and M₂ lines) or both in *f* = 2 (M'₁ line).</figcaption></figure>

Fig. 1(b). The 2D density $n_a$ of the cloud is tuned from about 20 to 100 µm⁻² by removing a controlled fraction of the atoms from the large density configuration [47].

In most experimental sequences, a magnetic field $B_0$ in the range 0.7–2 G, aligned along the *z* axis, defines the quantization axis [48]. For photoassociation spectroscopy, we use a microwave field of frequency $\approx 6-7$ GHz and of amplitude $B_{mw} \approx 30$ mG, which has nonzero components in all $\pi$, $\sigma_\pm$ polarizations. We detect the formation of dimers by losses in the gas and we identify for each target state the resonance frequency at which these losses are maximal.

**Atomic and molecular levels.** For the $^{87}$Rb atom, the electronic ground level is split by the hyperfine interaction into two sublevels of spin angular momentum $f=1$ and $f=2$. It results from the coupling $\hat{H}_{hf} \propto \hat{s} \cdot \hat{i}$ between the electron spin $s=1/2$ and the nuclear spin $i=3/2$, with $\hat{f} = \hat{s} + \hat{i}$. The two sublevels are separated by an energy difference $h\nu_{hf} \approx h \times 6.8$ GHz. Atoms can be prepared in any of the

<figure><img src="image_4.png" /><figcaption>FIG. 2. Energy diagram of the ($f=1; f=2$) subspace. The colors of the experimental points encode the composition of the initial atomic state. The set of solid lines is the result of the simple model described in the text, with two adjustable parameters *U* and $E_0$. The state $|\Psi_0^{(n=-1)}\rangle$ used for measuring the atom-dimer scattering length is highlighted with a red solid line. The numbers on the right give, for each state, the quantum number $M_F$ associated with the *z* component of the spin angular momentum $\hat{F}$.</figcaption></figure>

eight states $|f, m_f\rangle$, where $m_f$ is associated to the projection along *z* of $\hat{f}$. In addition, any binary mixture of these atomic states can be prepared using a series of suitable coherent transfer pulses between these states.

The relevant energy levels for a pair of two atoms labeled A and B are sketched in Fig. 1(c). The unbound states with zero asymptotic kinetic energy are represented as dashed lines. Three subspaces ($f=1; f=1$), ($f=1; f=2$), and ($f=2; f=2$) separated by $h\nu_{hf}$ can be identified. For each subspace, we indicate the possible values of the total spin (nuclear + electron) angular momentum *F* (with $\hat{F} = \hat{f}_A + \hat{f}_B$), taking into account the required exchange symmetry for two bosonic atoms. Molecular states corresponding to bound dimers are sketched with solid horizontal lines. We have represented for each subspace the two least-bound vibrational levels $n = -1$ and $n = -2$ with zero orbital angular momentum and binding energies $\approx -25$ and $-600$ MHz, respectively.

**Microwave photoassociation spectroscopy.** In a first experiment, we photoassociate pairs of atoms in $|f=1, m_f=0\rangle \otimes |f=1, m_f=0\rangle$ into weakly bound dimers in the ($f=1, f=2$) subspace, on which we now focus. We target the least-bound ($n=-1$) vibrational level with zero orbital angular momentum and a binding energy $\sim -h \times 25$ MHz [49]. We show in Fig. 2 the measurement of the position of three lines (blue circles) corresponding to the absorption of a photon with a $\pi$ or $\sigma_\pm$ polarization as a function of the applied external magnetic field $B_0$. The rest of the diagram is obtained by preparing the initial atomic gas in other pure hyperfine states $|f=1, m_f = \pm 1\rangle$ or in binary mixtures of $|f=1, m_f=0, \pm 1\rangle$. Experiments with mixtures require a modification of the experimental protocol: as the two species have different magnetic moments, residual magnetic field gradients lead to spatial separation of the two components and prevent the dimer formation. We circumvent this problem by
loading atoms in an array of microtraps whose size (about 5 µm) is small enough to prevent phase separation [Fig. 1(b)].

*Energy diagram.* The energy diagram of Fig. 2 was previously computed using coupled channel calculations [50] and quantum defect theory [51], and it has already been partially measured [52]. We provide below a simple discussion to understand its main features and predict quite accurately the positions of the molecular lines. In all that follows, the dominant term is the hyperfine Hamiltonian $\hat{H}_{hf} \propto \hat{s}_A \cdot \hat{i}_A + \hat{s}_B \cdot \hat{i}_B$, which gives rise to the three subspaces $(f, f')$ of Fig. 1. The binding energies of the dimer states and the Zeeman shifts are small compared to the splitting $h\nu_{hf}$ between these subspaces, which allows for a perturbative analysis.

We address first the $B_0 = 0$ case and we consider the total electron spin of the system $\hat{S} = \hat{s}_A + \hat{s}_B$, where $\hat{s}_i$ designates the electron spin operator of atom $i$. Collisions between two $^{87}$Rb atoms involve two channels associated to singlet $S = 0$ and triplet $S = 1$ potentials, with slightly different scattering lengths. The wave functions of the weakly bound dimers are localized at a large interparticle distance, of the order of a few nanometers. At such a distance, van der Waals interactions, which are spin independent and thus identical for the three subspaces $(f, f')$, dominate and the two potentials are very close to each other. We can thus treat the difference between them by a perturbative approach and model it by the operator

$$ \hat{H}_{\text{st}} = U \hat{s}_A \cdot \hat{s}_B = \frac{U}{2} (\hat{S}^2 - 3/2), \quad (1) $$

where the adjustable parameter $U$ is small compared to the binding energy of the dimer and to $h\nu_{hf}$.

Since $\hat{H}_{st}$ commutes with $\hat{F}^2$, it only couples states with the same value of $F$. More precisely, the restriction of $\hat{H}_{st}$ to the subspace $(f = 1, f = 2)$ splits the $F = 2$ manifold from the $F = 1, 3$ manifolds with the energy shifts $\Delta E_{F=1} = \Delta E_{F=3} = -2\Delta E_{F=2} = U/4$ [53]. Note that due to the large size of a weakly bound dimer, additional magnetic couplings $\propto \hat{s}_A \cdot \hat{i}_B + \hat{s}_B \cdot \hat{i}_A$ play a negligible role [54].

The presence of a magnetic field, described by the Zeeman Hamiltonian $\hat{H}_Z$ (with $\hat{H}_Z \approx 2\mu_B B_0 \hat{S}_z$ if nuclear magnetism is neglected), mixes all $F$ states. However, rotation invariance around the $z$ axis ensures that the projection along $z$ of the spin angular momentum $\hat{F}$ commutes with the Hamiltonian. Since we focus on weakly bound dimers with zero orbital angular momentum, we can label all relevant states by their spin composition and we use the quantum number $M_F$ associated to $\hat{F}_z$ to label the dimer states in Fig. 2.

More quantitatively, the diagonalization of the 36 × 36 matrix of the spin Hamiltonian $\hat{H}_{hf} + \hat{H}_{st} + \hat{H}_Z$ provides the energy shift of the dimer states as a function of magnetic field. It leads, for the least-bound level of the subspace $(f = 1; f = 2)$ studied here, to the 15 continuous lines shown in Fig. 2. These lines are obtained by adjusting two parameters: the coupling [55] $U$ and the energy $E_0 \equiv E_{F=1}(B=0)$. We find $U = h \times 2.875(5)$ MHz and $E_0 = h \times -24.985(1)$ MHz. This model provides an excellent agreement with all 15 lines in the explored range of magnetic fields. The distance [56] between the best fitted energy diagram and our measurements is 3.7 kHz, which is slightly larger than the uncertainty on the measured molecular line positions ($\lesssim 1$ kHz). The

remaining deviations remind us that the simple model described by Eq. (1) is not exact. A similar study could be performed to measure the Zeeman diagram of weakly bound states in the $(f = 1; f = 1)$ and $(f = 2; f = 2)$ subspaces. However in our case, the lifetime of the atomic sample prepared in $|f = 2\rangle$ is limited by hyperfine changing two-body collisions. It is comparable to the required excitation times which makes these experiments challenging.

*The dimer state* $|\Psi_0\rangle$. In Fig. 2, we highlighted one particular dimer state, labeled $|\Psi_0^{(n=-1)}\rangle$, and that plays a central role in the following. It is an eigenstate of the restriction of the Hamiltonian $\hat{H}_{hf} + \hat{H}_{st} + \hat{H}_Z$ to the $(f = 1, f = 2)$ subspace and its energy is independent of the magnetic field as long as the coupling to the two other subspaces $(f = 1, f = 1)$ and $(f = 2, f = 2)$ induced by $\hat{H}_{st} + \hat{H}_Z$ can be neglected. Its spin component is $(\sqrt{3}|F = 3, M_F = 0\rangle - \sqrt{2}|F = 1, M_F = 0\rangle)/\sqrt{5}$ and it is a pure electronic spin triplet [53]. When photoassociated from two atoms prepared in a state with a null magnetic moment, it provides us with a “clock” free-bound transition, up to second-order $B$-dependent shifts $\sim (\mu_B B)^2/(h\nu_{hf})$.

We have also measured the position of the state with the same spin component in the $n = -2$ level of the same $(f = 1; f = 2)$ subspace. We label it $|\Psi_0^{(n=-2)}\rangle$ and we found the value $h \times -642.219(1)$ MHz for its binding energy at zero magnetic field.

*Atom-dimer interaction.* We now move to the main result of this Letter regarding the measurement of the elastic interaction between a dimer and the atomic bath from which it is photoassociated. We have studied the interaction of dimers in state $|\Psi_0^{(n)}\rangle$ for $n = -1, -2$ with a bath of atoms in $|f = 1; m_f = 0\rangle$ and the interaction of $|\Psi_0^{(n=-1)}\rangle$ with a bath of atoms in $|f = 2; m_f = 0\rangle$. These three cases correspond to the three microwave association lines, respectively $M_1, M_2$, and $M'_1$, shown in Fig. 1(c).

We first detail the study of the $M_1$ line, associating atom pairs in $|f = 1, m_f = 0\rangle \otimes |f = 1, m_f = 0\rangle$ into the dimer state $|\Psi_0^{(n=-1)}\rangle$. We detect the formation of dimers by losses in the gas as a function of the microwave frequency $v$. A typical signal on the $M_1$ transition is reported in the upper right inset of Fig. 3. The reference frequency $v_0$ corresponds to the resonance frequency extrapolated in the zero-density limit [57]. For this reported spectrum, the peak frequency $v_m$ is displaced with respect to $v_0$ by $\approx 670$ Hz and the measured full width at half maximum is $\approx 1$ kHz. We attribute it to the finite lifetime of the dimers, which can decay through two channels. First, dipolar relaxation within the dimer can produce a pair of atoms by releasing the energy $h\nu_{hf}$ [58]. Second, two-body inelastic collisions between atoms and molecules can play a significant role, as observed for other alkali molecules [28–30,52].

To study the interaction between the produced dimers and the atom gas, we determine the variation of the peak frequency $\Delta v \equiv v_m - v_0$ with the 2D atomic density $n_a$. For each density, we adjust the duration of the excitation time to keep the relative depletion at resonance $\delta n_a/n_a$ at a given value. We show the results obtained for three different values of $\delta n_a/n_a$ in Fig. 3. We observe a shift of the resonant frequency which goes up to 800 Hz at our maximum density $n_a \sim$
<figure><img src="image_5.png" /><figcaption>FIG. 3. Frequency shift of the M₁ line as a function of the 2D density of the gas. The three different symbols correspond to a measured relative depletion signal of 20% (square), 14% (diamond), and 8% (circle). All data are adjusted by a common linear fit. The left-hand inset shows the result of a similar measurement for the second-to-least bound level (M₂ line in Fig. 1). The right-hand inset shows a typical microwave photoassociation signal for nₐ = 95 µm⁻². The variation of the width of the photoassociation signal with density is reported in [53].</figcaption></figure>

100 atoms/µm². All data collapse on a single curve, which confirms that we operate in the weak excitation regime. We fit a linear function to the data and obtain Δν/nₐ = -7.3(3) Hz/µm² [59].

In order to interpret this shift within a mean-field approach, we introduce the interaction parameter g<sub>ad</sub> = 2πa<sub>ad</sub>ħ²/m<sub>r</sub>, where m<sub>r</sub> = 2m<sub>a</sub>/3 is the reduced mass of the atom-dimer system. We assume that all interactions occur in the s-wave regime, because of the very low relative momenta between the unbound atoms and the dimer. The photoassociation process must bring to the sample (i) the energy in the zero-density limit ħv₀, (ii) the interaction energy between the dimer and the atom bath, and (iii) the energy -2μ<sub>a</sub>, since two atoms are removed from the bath. Denoting ρ<sub>a</sub>(z) the 3D density profile of the atom bath and f<sub>d</sub>(z) the distribution function of the dimer [with the normalization ∫dz ρ<sub>a</sub>(z) = n<sub>a</sub> and ∫dz f<sub>d</sub>(z) = 1], we find using the mean-field value of μ<sub>a</sub> in the low-temperature limit [53]

$$h\Delta\nu = g_{ad} \int dz \rho_a(z) f_d(z) - 2\mu_a = \left( \frac{\sqrt{3} a_{ad}}{2a_1} - 2 \right) \mu_a, \quad (2)$$

where a<sub>f</sub> (with f = 1, 2) denotes the s-wave scattering length for the collision between two atoms in state |f, m<sub>f</sub> = 0⟩. In all cases the dimer density is low enough so that dimer-dimer interactions can be safely neglected.

The atomic 2D density n<sub>a</sub>, or equivalently the chemical potential μ<sub>a</sub>, are inferred via Ramsey spectroscopy following the method used in Refs. [60,61], which probes the difference of mean-field interaction energies experienced by the atoms in two different internal states. We measure the density-dependent component Δν' of the microwave frequency that allows a complete transfer, after the two Ramsey pulses, of the gas from the |f = 1, m<sub>f</sub> = 0⟩ to the |f = 2, m<sub>f</sub> = 0⟩ atomic

state. It reads (see [60,61])

$$h\Delta\nu' = \frac{1}{2} \frac{a_2 - a_1}{a_1} \mu_a \quad (3)$$

and it depends only on the knowledge of the aforementioned atomic scattering lengths a<sub>f</sub> of states |f, m<sub>f</sub> = 0⟩. Such a measurement of this frequency shift thus provides a direct calibration of the chemical potential of the gas. Preparing the sample in the same experimental conditions as for the measurement of the frequency shift of the dimer line, we obtain Δν'/nₐ = -0.52(2) Hz/µm². Combining Eq. (2) with Eq. (3) we obtain the atom-dimer scattering length:

$$a_{ad} = \frac{4}{\sqrt{3}}a_1 + \frac{1}{\sqrt{3}}\frac{\Delta\nu}{\Delta\nu'}(a_2 - a_1), \quad (4)$$

an expression which is immune to systematic errors in the calibration of the density nₐ. Using the known values of a₁ = 100.9a₀ and a₂ - a₁ = -6a₀ with a₀ the Bohr radius [62], we obtain

$$a_{ad}^{(n=-1)}[\text{bath in } f=1] = 184(2)a_0, \quad (5)$$

where the quoted error takes into account only the uncertainties on Δν and Δν' [63]. This value is notably different from the result of the impulse approximation a<sub>ad</sub><sup>impulse</sup> = 4(a₁ + a₁₂)/3 ≈ 261a₀, where a₁₂ is the interspecies scattering length describing the interaction between an f = 1 and an f = 2 atom. This approximation consists in summing independently the scattering amplitudes of an atom of the bath with each atom of the dimer [64].

Using the M₂ line we have also determined the atom-dimer scattering length of the |Ψ<sub>0</sub><sup>(n=-2)</sup>⟩ with the same bath. We obtain

$$a_{ad}^{(n=-2)}[\text{bath in } f=1] = 21(7)a_0 \quad (6)$$

for the atom-dimer scattering length (see inset of Fig. 3). The large difference between a<sub>ad</sub><sup>(n=-1)</sup> and a<sub>ad</sub><sup>(n=-2)</sup> for an identical bath and identical spin states shows the key role of the dimer radial wave function in the scattering process. We note that a related work was performed with two-photon photoassociation of <sup>87</sup>Rb atoms for a state in the n = -2, (f = 1; f = 1) subspace and trapped in a harmonic potential [65]. An atom-dimer scattering length of -180(150)a₀ was reported, where the large uncertainty could be attributed to the difficulty of accurately modeling the experimental signal in an inhomogeneous cloud.

Lastly, using the M'₁ line we measured the interaction of the dimer state |Ψ<sub>0</sub><sup>(n=-1)</sup>⟩ with a bath of atoms initially all in state |f = 2, m<sub>f</sub> = 0⟩. The fitted slope is now 8.1(9) Hz/µm², leading to

$$a_{ad}^{(n=-1)}[\text{bath in } f=2] = 165(7)a_0, \quad (7)$$

a value close to the result of an atom bath in state |f = 1, m<sub>f</sub> = 0⟩. The similarity between these two results, combined with the small difference between the two bath scattering lengths a₁ and a₂, is compatible with the existence of a “van der Waals universality”, which may allow one to link a and a<sub>ad</sub> for the least-bound dimers.

**Conclusions.** We have presented a precise measurement of the scattering length a<sub>ad</sub> characterizing the interaction be-
tween atoms and weakly bound dimers in a degenerate Bose gas. This result provides a first step in the search for a possible van der Waals universality for this problem. Our method can be straightforwardly adapted to other alkali-metal bosonic species. Some of them provide easily accessible Feshbach resonances, making it possible to study the emergence of Efimov physics on $a_{ad}$. In addition, our precise determination of the whole energy diagram of a weakly bound dimer manifold

paves the way to the implementation of microwave Feshbach resonances [66,67], using for instance strong microwave fields directly generated on atom chips [68].

*Acknowledgments.* This work is supported by ERC (Grant Agreement No. 863880) and ANR (Grant No. ANR-18-CE30-0010). We thank O. Dulieu, F. Chevy, C. Greene, J. D'Incao, S. Kokkelmans, and D. Papoular for fruitful discussions.

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