Observation of bound state self-interaction in a nano-eV atom collider


Quantum mechanical scattering resonances for colliding particles occur when a continuum scattering state couples to a discrete bound state between them. The coupling also causes the bound state to interact with itself via the continuum and leads to a shift in the bound state energy, but, lacking knowledge of the bare bound state energy, measuring this self-energy via the resonance position has remained elusive. Here, we report on the direct observation of self-interaction by using a nano-eV atom collider to track the position of a magnetically-tunable Feshbach resonance through a parameter space spanned by energy and magnetic field. Our system of potassium and rubidium atoms displays a strongly non-monotonic resonance trajectory with an exceptionally large self-interaction energy arising from an interplay between the Feshbach bound state and a different, virtual bound state at a fixed energy near threshold.


Scattering resonances are an important feature of quantum mechanics and arise whenever asymptotically free particles are coupled to an unstable bound state of the system. While the underlying mechanism of the quasi-bound formation depends on the system, its energy and lifetime can generally be determined by measurements of the position of the scattering resonance and its width. In nearly all systems, coupling between the asymptotically free states and the bound state affects the observable energy of the quasi-bound state. In the non-relativistic theory, a pair of asymptotically free particles, A and B, colliding at a kinetic energy E have both background and resonant contributions to their scattering amplitude f(E). Figure 1a represents diagrammatically the scattering of particles A and B at a resonance. The background scattering amplitude, fbg(E), is added to the resonant scattering amplitude, fres(E), which is an infinite sum of contributions from different order processes that start and end with asymptotically free particles A and B. When only the leading order, zero-loop term is present then EAB(E) = E0 is the bound state energy in the absence of coupling. However, if we include all loop orders then the quasi-bound state’s energy becomes1

$$E_{{\mathrm {AB}}}(E) = E_0 + \underbrace {\left\langle {{\mathrm {AB}}} \right|\hat W\hat G_{{\mathrm{bg}}}(E)\hat W^\dagger |{\mathrm {AB}}\rangle }_{\delta E(E)}.$$
Fig. 1

Feynman diagrams for resonant collisions. a The total scattering amplitude f(E) for asymptotically free particles A and B is the sum of the background contribution fbg and the resonant contribution fres. The lowest order contribution to fres has A and B forming the quasi-bound state AB which then decays back to the individual particles. The next order adds a self-energy correction to the bound state energy. AB decays into particles A and B which reform into the quasi-bound state AB. b Analogous self-energy correction for the mass of the Z0 boson in the elastic scattering of electrons and positrons

The interpretation of the energy shift δE is that the bare bound state |AB〉 dissociates into free particles by the action of the coupling operator \(\hat W^\dagger\), the individual particles propagate according to the background Green’s operator \(\hat G_{{\mathrm{bg}}}(E)\), after which they associate back into the bound state by \(\hat W\). This process is similar to self-energy effects that alter the mass of force-carrying particles, such as the Z0 boson (Fig. 1b) in relativistic electron–positron interactions2. In all cases, when the bare energy of the intermediate state is fixed we can only measure the re-normalised, dressed bound state energy EAB: it is impossible to observe the actual self-energy correction.

In atomic systems one can often change the energy of a molecular bound state using a magnetic field which leads to magnetic Feshbach resonances in the collisions of atoms3,4,5,6. Specifically, through the application of a magnetic field B, the energy of the bare (uncoupled) bound state varies according to E0 = δμ(B − Bc), where δμ is the difference in magnetic moments between the free atoms and the molecular state AB, and Bc is the field at which the bound state energy is equal to the background channel’s threshold energy. This tunability has made Feshbach resonances pivotal in modern atomic physics, as one can tailor the interactions for ultracold and quantum degenerate gases7, but it also means that, with few exceptions8,9,10,11,12, the study of Feshbach resonances in ultracold gases has invariably been conducted on trapped samples, where the atomic collision energy is defined by the sample temperature. As temperatures are sufficiently low for collisions to remain in the threshold regime, such experiments cannot provide insight into self-energy effects. Instead, one must probe the resonance over a range of both collision energy and magnetic field.

In this study, we use the tunability of a magnetic Feshbach resonance in atomic collisions to directly observe the effect of self-energy corrections to the quasi-bound state energy. We detect these corrections as a non-monotonic and non-linear change of the resonance energy with respect to changes in the magnetic field as expected for an atomic system with strong coupling and a virtual bound state near threshold13. We find that the resonance position is well-described by a coupled-channels model.


Scattering amplitude of an s-wave Feshbach resonance

The background and resonant part of the elastic scattering process represented by Fig. 1 combine to give rise to a scattering amplitude expressed as14,15

$$\begin{array}{*{20}{l}} {f(E,\Omega )} \hfill & = \hfill & {\underbrace {\frac{{\left( {{\mathrm{e}}^{2{\mathrm{i}}\delta _{{\mathrm{bg}}}(E)} - 1} \right)}}{{2{\mathrm{i}}p/\hbar }} + f_{\ell > 0}(E,\Omega )}_{f_{{\mathrm{bg}}}(E,\Omega )}} \hfill \\ {} \hfill & {} \hfill & { + \underbrace {\left( { - \frac{{e^{2{\mathrm{i}}\delta _{{\mathrm{bg}}}(E)}}}{{2p/\hbar }}\frac{{{\mathrm{\Gamma }}(E)}}{{E - E_{{\mathrm{AB}}}(E) + {\mathrm{i\Gamma }}(E)/2}}} \right)}_{f_{{\mathrm{res}}}(E)},} \hfill \end{array}$$

where \(p = \sqrt {2mE}\) is the magnitude of the relative momentum of the two particles with reduced mass m, δbg(E) is the s-wave background scattering phase shift, Γ(E) is an energy-dependent resonance width, and EAB(E) is the energy of the quasi-bound state AB. Higher angular momentum scattering amplitudes are included in the direction-dependent term \(f_{\ell > 0}(E,\Omega )\) with Ω = (θ, ϕ) the direction in spherical polar coordinates. Substituting E0 = δμ(B − Bc) into Eq. (1), and the result into Eq. (2), the resonant scattering amplitude becomes

$$f_{{\mathrm{res}}}(E,B) = \frac{{{\mathrm{e}}^{2{\mathrm{i}}\delta _{{\mathrm{bg}}}(E)}}}{{2p/\hbar }}\frac{{{\mathrm{\Gamma }}_B(E)}}{{B - B_{{\mathrm{res}}}(E) - {\mathrm{i\Gamma }}_B(E)/2}},$$

where ΓB(E) = Γ(E)/δμ, and

$$B_{{\mathrm{res}}}(E) = B_{\mathrm {c}} + \left[ {E - \delta E(E)} \right]/\delta \mu ,$$

is the resonant magnetic field. By measuring Bres(E) at sufficiently high energies, we can infer both δμ and Bc, since \( \delta E(E) {E \rightarrow \infty\atop \longrightarrow} 0\)16, and hence determine the self-energy of the quasi-bound state.

System under study

We explore the collision energy and magnetic field dependence of an s-wave Feshbach resonance in 40K + 87Rb using an optical collider12,17,18. Specifically, we address a resonance that occurs between 87Rb atoms in the |F = 1, mF = 1〉 state and 40K atoms in the \(\left| {\frac{9}{2}, - \frac{9}{2}} \right\rangle\) state near 546 G (104 G = 1 T)19,20,21,22. As these states are the absolute ground states of 40K and 87Rb all collisions are elastic, and the cross section at a collision energy E near the resonance can be written as1,6

$$\begin{array}{*{20}{l}} {\sigma (E,B)} \hfill & = \hfill & {{\displaystyle\int} \left| {f(E,\Omega )} \right|^2d\Omega } \hfill \\ {} \hfill & = \hfill & {\frac{{4{\mathrm{\pi}}\hbar ^2}}{{mE}}{\mathrm{sin}}^2\left[ {\delta _{{\mathrm{bg}}}(E) + {\mathrm{tan}}^{ - 1}\left( {\frac{{{\mathrm{\Gamma }}_B(E)/2}}{{B - B_{{\mathrm{res}}}(E)}}} \right)} \right]} \hfill \\ {} \hfill & {} \hfill & { + \sigma _{\ell > 0}(E),} \hfill \end{array}$$

where the first term is a Beutler–Fano profile1,5,23 as a function of B for a given energy E, and \(\sigma _{\ell > 0}(E)\) accounts for the non-zero contribution to the cross section from higher \(\left( {\ell\, > \, 0} \right)\) partial waves which become important in the 40K + 87Rb system when E/k > 100 μK, where k is the Boltzmann constant. The magnetic field dependence of the cross section is experimentally determined through measurements of the fraction of atoms scattered in a collision of two cold atomic clouds

$$S(E,B) = \frac{{\alpha (E)\sigma (E,B)}}{{1 + \alpha (E)\sigma (E,B)}},$$

where α(E) is a scaling factor determined by the peak densities of the clouds and their transverse spatial overlap at the time of collision (see Supplementary Note 1). By measuring S(E,B) as a function of B for fixed E, we determine δbg(E), ΓB(E), and Bres(E).

Experimental procedure

We begin our experiment by loading an ultracold mixture of 40K atoms in the \(\left| {\frac{9}{2},\frac{9}{2}} \right\rangle\) state and 87Rb atoms in the |2, 2〉 state into a double-well far off-resonant optical dipole trap. The double well is formed by the intersection of a horizontal laser beam with two vertical laser beams generated by rapidly switching between two frequency pairs that drive a two-axis acousto-optic deflector18,24. The initial well separation is 80 μm, or approximately twice the vertical laser beam waist. We transfer Rb atoms to the |1,1〉 state and separate the double well in the presence of a strong magnetic field gradient which preferentially pushes Rb atoms into one well and K atoms into the other. The two wells are moved to final positions zK = 2.0 mm for K and zRb = −(mK/mRb)zK = −0.92 mm for Rb, where mi are the masses of 40K and 87Rb, respectively. We prepare the internal states of each species of atoms using microwave and radio-frequency transitions between hyperfine states at a magnetic field of ~9 G, and we purge the wells of unwanted states and species using resonant light pulses. At the end of the state preparation, we have 3.0(3) × 105 atoms of each species in their respective wells at temperatures of 1.1(1) μK for K and 0.8(1) μK for Rb, as determined by time-of-flight absorption images. Throughout this work, numbers in parentheses correspond to the one standard deviation uncertainty combining statistical and systematic contributions. With the atoms in the desired internal quantum states, we use a dedicated, water-cooled coil pair in the Helmholtz configuration to create a stable and homogeneous magnetic field around the 546 G Feshbach resonance. The current in these coils is regulated to a fractional stability of <10−5, and the magnetic field is calibrated using Rabi spectroscopy between the 87Rb |2, 0〉 and |1, 0〉 states to an accuracy of 5 mG.

Figure 2 shows the operation (see also Supplementary Movie 1) of our optical collider. From their initial positions in Fig. 2a, the atomic clouds of Rb and K are accelerated towards each other along the horizontal guide beam by steering the vertical beams of the crossed traps as illustrated in Fig. 2b. The traps are moved at velocities vRb and vK such that vRb/vK = −mK/mRb. This keeps the centre-of-mass (COM) of the collisional partners at rest in the laboratory frame and ensures that expanding spheres of scattered particles are centred around a common point. Just before the collision we turn off all laser beams (Fig. 2c), so that the atoms collide in free space (Fig. 2d). After the collision, we switch the magnetic field off (see Methods) and wait until the K cloud has moved 350 μm from the collision point. At this time a halo of scattered particles will have formed (Fig. 2e), and we record the distribution of K atoms using absorption imaging perpendicular to the collision axis as in Fig. 2g. We then wait until the Rb atoms have also moved 350 μm from the collision point (Fig. 2f) and acquire an image of the Rb atoms (Fig. 2h); a frame-transfer CCD allows us to acquire images of both species of atoms in rapid succession. We determine the collision energy by acquiring two pairs of images with a time-of-flight difference of 5 ms, measuring the distance travelled by the unscattered atoms in that time, and then calculating the kinetic energy. The uncertainty in the determination of the collision energy from both systematic and statistical sources is 2% of the value.

Fig. 2

Optical collider procedure. a 40K (blue, right) and 87Rb (red, left) atoms, prepared in the desired internal quantum states, are held in two crossed optical dipole traps separated by ~3 mm; COM indicates the centre-of-mass for pairs of K and Rb atoms. b The two traps are accelerated towards each other, keeping mRbvRb = −mKvK, so that pairs of of K and Rb atoms have zero total momentum on average, and their COM is at rest in the laboratory frame. c When the wells are separated by ≈60 μm, the optical traps are switched off. d The two clouds collide in free space. e, f The K and Rb collision halos expand at different rates. We image the K halo at the time represented in e, and then wait to image Rb until its halo has expanded to the equivalent size (f). g, h 3D collision halos of K (g) and Rb (h) are projected onto a 2D plane by the process of absorption imaging. We integrate the projected images in the vertical direction to obtain line densities. Line densities are fitted to extract the scattered fraction


Using our optical collider, we measure the scattered fraction of atoms S from absorption images (see Methods) as a function of magnetic field with the 40K + 87Rb system prepared in states \(\left| {\frac{9}{2}, - \frac{9}{2}} \right\rangle\) (40K) and |1,1〉 (87Rb) for collision energies E/k = 10 to 300 μK. Figure 3a presents example data acquired at collision energy E/k = 52(1) μK. While this energy, equivalent to 4.5 neV, is far below that of conventional particle colliders, it remains two orders of magnitudes higher than the typical energy scales for studies of Feshbach resonances in trapped ultracold atomic gases as set by the sample temperature. The fraction S has a pronounced Beutler–Fano lineshape associated with an interspecies Feshbach resonance centred on a magnetic field of ~546 G. In Fig. 3b, c we present post-collision absorption images of 40K and 87Rb for the field values where σ (and hence S) attains its maximum (Fig. 3b) and minimum (Fig. 3c). At the minimum, no discernible scattering halo is visible as a result of destructive quantum interference, where the background phase shift δbg(E) of Eq. (5) is cancelled by the resonant contribution; only outgoing clouds of unscattered particles are visible in the absorption images. For the maximum, where the interference is constructive, as well as away from resonance, a halo of isotropically scattered particles emerges. The 40K image in Fig. 3b also reveals an additional feature located at the position of the outgoing 87Rb cloud caused by multiple scattering (see also Supplementary Fig. 1 which shows the result of a numerical simulation elucidating the effect of multiple scattering).

Fig. 3

Imaging and analysis of 40K + 87Rb collisions. a Measured scattered fraction (circles) as a function of magnetic field for E/k = 52(1) μK; error bars indicate one standard deviation uncertainties. Solid line is a fit to Eqs. (5) and (6). b, c Absorption images and density profiles of K (b) and Rb (c) acquired at the magnetic fields indicated by the horizontal arrows. Light blue and light red coloured areas in density profiles indicate unscattered atoms and dark red and dark blue areas indicate scattered atoms. The double-headed arrow in b indicates an additional 40K feature (see text and Supplementary Fig. 1)

Data analysis

We fit Eqs. (5) and (6) to our data allowing α, \(\sigma _{\ell > 0}\), δbg, ΓB, and Bres to vary as functions of energy. Figure 3a shows the fitted curve for the E/k = 52(1) μK case, which is typical across our entire energy range. Figure 4a–c show our fitted values for Bres, ΓB, and δbg as functions of energy along with predictions from a coupled-channels model based on published 40K + 87Rb interaction potentials25 that have been adjusted to match our measured resonance positions (see Methods).

Fig. 4

Parameters describing the 40K + 87Rb Feshbach resonance. ac Resonance position Bres, width ΓB, and background phase shift δbg as a function of collision energy E for the resonance near 546 G. Solid lines are predictions from a coupled-channels model. Circles are measurements using the collider, and error bars are the one standard deviation uncertainty obtained from the fits to Eqs. (5) and (6). The dashed grey line in a indicates the resonance position in the absence of self-energy effects extrapolated from the asymptotic behaviour of the fitted model for Bres(E)


A particularly striking feature of our measurements is that the resonance position Bres is not a monotonic function of E (Fig. 4a). From zero energy, Bres(E) decreases to a minimum value near E/k = 75 μK before approaching a linear asymptote. This curvature is solely the result of the self-interaction energy δE(E) due to higher-order resonant scattering processes, such as those shown in Fig. 1a. Without these processes, the resonance position would follow the linear curve Bc + E/δμ in Fig. 4a. The gap between the two curves, therefore, is a direct measure of the self-energy δE(E)/δμ. The marked non-monotonic behaviour, as opposed to only being non-linear, is due to the negative background scattering length which is associated with a virtual bound state just below threshold in the energetically open channel26,27,28. The effect of the virtual bound state on the open channel scattering13,29 can also be seen in Fig. 4b,c as similarly non-monotonic behaviours of the background phase shift and resonance width, respectively. In fact, the initial rise of δbg for E/k < 50 μK in Fig. 4c corresponds to the Wigner threshold regime30 and is a direct measure of the background scattering length1, \(a_{{\mathrm{bg}}} = - \hbar\, \mathop{\lim}\limits_{E \to 0}\,{\mathrm{tan}}\delta _{{\mathrm{bg}}}(E)/p\). Here, we calculate abg = −185.80(35) a0 from our theoretical model, which is negligibly different from the value calculated with the unmodified potentials (−185.65a0) and is consistent with previous estimates22,31. This non-monotonic behaviour is qualitatively different to systems with a positive background scattering length, where both the background phase shift and resonance width scale as E1/2 for energies up to the van der Waals energy (EvdW/k ≈ 600 μK for 40K + 87Rb). While measurements of molecular-binding energies also probe self-energies22,31, we note that this is the first direct observation of such self-interaction effects in Feshbach resonances for continuum states as previous experiments worked with resonances with widths that were too small to generate appreciable curvature10,12.

Although we have focused on one particular resonance, our technique is more general and can be applied to both isolated and overlapping Feshbach resonances. By performing our measurements at fixed collision energy, we isolate the functional dependence of both the background and resonant scattering phases and eliminate the need for a priori knowledge about how these parameters vary with energy. Elastically dominated resonances can be described by a simple Beutler–Fano function, and even resonances with significant inelastic scattering have analytic descriptions as functions of magnetic field at fixed energy32. The ability to study resonances with significant inelastic widths is a marked improvement over molecular-dissociation experiments, where a stable molecular state is a required starting point8,9. Moreover, by measuring the number of scattered atoms we avoid the need for an external interferometer to measure the scattering phase10. Indeed, the number of scattered atoms can be regarded as the output of an internal interferometer, where the path length for one arm at a given energy is fixed and yields the background phase, and the path length of the other, resonant, arm can be varied with an external field: in this case, a magnetic field. Given the method’s generality and the relatively high energies that we can access with our collider, we expect that this work can be extended to study Feshbach resonances in higher partial waves33,34, where the well-defined collision axis can unambiguously determine the orbital angular momentum without observing resonance multiplets35. Furthermore, we can investigate the interference between Feshbach and shape resonances9 and the effect of shape resonances on the self-energy, of which this study with an s-wave virtual state is a special case. By associating pairs of atoms into molecules, we can investigate atom–molecule and molecule–molecule collisions where there is a dearth of accurate theoretical predictions36,37. Finally, by changing the collisional opacity through tuning of the elastic cross section via a Feshbach resonance, we can study multiple scattering effects in jet production in unitary Fermi gases38.


Counting of scattered atoms

We count atoms by acquiring absorption images of both the rubidium and potassium clouds and converting those to density plots. The frequency of our probe laser system is not sufficiently tunable to image clouds at magnetic fields of >50 G, so we must turn off the Feshbach magnetic field prior to image acquisition. After the two atom clouds collide, we wait until the initial atom clouds are separated by 160 μm before we switch off the Feshbach magnetic field, which ensures that the magnetic field is present during the entire collision. We then wait until each cloud has travelled 350 μm from the collision point before we acquire an absorption image of the atoms; due to the differences in the speed of the Rb and K clouds, we take these images at two different times. Depending on the collision energy, the delay of K image acquisition with respect to turning off the Feshbach field coil varies between 0.5  and 4 ms. The large electrical currents (>100 A) flowing in the coil that need to be switched off, combined with metal components surrounding the atoms, means that eddy currents, and hence stray magnetic fields, are present during imaging. As a result, we image in an unknown magnetic field, and while we optimise the probe laser frequency for each collision energy, the absolute number of atoms that we extract from our absorption images is not directly comparable between experiments conducted at different energies. Therefore, we calculate the fraction of atoms scattered by the collision and compare that to theoretical predictions instead of the total number of scattered atoms, as it eliminates a free parameter from our fitted model.

Determination of scattered fraction

We measure the scattered fraction by integrating the absorption images along an axis perpendicular to the collision axis z to produce line density distributions ni(z), where i can be either K or Rb. Each ni(z) can be expressed as the sum of three terms \(n_i(z) = N_i^{\mathrm {u}}P_i^{\mathrm {u}}(z) + N_i^{\mathrm {s}}P_i^{\mathrm {s}}(z) + N_i^{\mathrm {m}}P_i^{\mathrm {m}}(z)\), where each probability density \(P_i^x(z)\) is normalised to unity. The distribution \(P_i^{\mathrm {u}}(z)\) represents the distribution of unscattered atoms and is a Gaussian distribution. \(P_i^{\mathrm {s}}(z)\) describes scattered atoms whose distribution is proportional to the differential cross-section. In computing this, we neglect the initial spatial distribution and the distribution of total momentum, both of which only lead to negligible blurring of the final pattern. Finally, \(P_i^{\mathrm {m}}(z)\) accounts for multiply scattered atoms whose distribution cannot be described using the differential cross-section. These multiply scattered atoms are mostly K atoms due to the higher density of the Rb cloud, and the potassium atoms have a Gaussian distribution centered at the position of the Rb cloud as shown in Supplementary Fig. 1. We fit ni(z) to our integrated densities to extract the number of unscattered \(N_i^{\mathrm {u}}\), scattered \(N_i^{\mathrm {s}}\), and multiply scattered \(N_i^{\mathrm {m}}\) atoms. The scattered fraction S is then

$$S = \frac{{\mathop {\sum}\limits_i (N_i^{\mathrm {s}} + N_i^{\mathrm {m}})}}{{\mathop {\sum}\limits_i (N_i^{\mathrm {u}} + N_i^{\mathrm {s}} + N_i^{\mathrm {m}})}}.$$

Resonance parameters from theoretical modelling

We integrate coupled-channels equations based on the Hamiltonian described in Pashov et al.25 which includes the internal structure of the atoms in the presence of a magnetic field, the Born–Oppenheimer potentials for the X1Σ+ and a3Σ+ molecular states, the centrifugal barrier for partial wave \(\ell\), and the magnetic dipole–dipole interaction. The Hamiltonian conserves the total spin projection and parity, and the weak nature of the dipole–dipole interaction means that we need only consider partial waves \(\ell = 0\) and \(\ell = 2\) in our calculations. Parameters are extracted by a fit of the calculated elastic cross-section to Eq. (5). We modify the long-range dispersion forces Cnrn for n = 6, 8, and 10 by multiplying each term by retardation correction factors fn(r) obtained from the and K and Rb values \(f_n^{\mathrm{K}}(r)\) and \(f_n^{{\mathrm{Rb}}}(r)\) from ref. 39 as

$$f_n(r) = \frac{1}{2}\left( {\frac{{f_n^{\mathrm{K}}(\beta r)}}{{f_n^{\mathrm{K}}(\beta R_{{\mathrm{outer}}})}} + \frac{{f_n^{{\mathrm{Rb}}}(\beta r)}}{{f_n^{{\mathrm{Rb}}}(\beta R_{{\mathrm{outer}}})}}} \right).$$

At r = Router, the correction factors are unity which ensures continuity of the potentials. For large r, fn(r) r−1. We perform a least-squares fit to our data with β as a free parameter, and we find β = 1.000(65). If we instead fit to only the measurement of ref. 22, calculated at an energy of E/k = 1 nK, then β = 0.960(19). These modifications have little effect on the inferred resonance width or background phase shift. For more details, see ref. 40.

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.


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This work was supported by the Marsden Fund of New Zealand (Contract No. UOO1121).

Author information

N.K. conceived the project. R.T. performed experiments with support from A.B.D. R.T. implemented the magnetic field servo with assistance from M.C. R.T. analysed the data and conducted coupled channels calculations. E.T. provided theory. R.T. and N.K. prepared the manuscript with input and comments from all authors. N.K. supervised the project.

Correspondence to Niels Kjærgaard.

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