## Introduction

Resonances lie at the heart of quantum mechanical scattering1, 2. They generally result from the occurrence of a metastable state in the encounter between colliding particles and can lead to striking modifications of the scattering cross section. Such metastable states may be established by barriers in the potential landscape of interactions between particles leading to so-called shape resonances. Alternatively, they may take the guise of a quasi-bound molecular state in a given degree of freedom that is coupled to other degrees of freedom into which it can decay. In this scenario, Feshbach resonances arise when the energy of an interacting atom pair in an incident, energetically open channel approaches that of the quasi-bound state. The ability to harness such Feshbach resonance-mediated interactions has been pivotal for ultracold atomic physics for the past two decades. For example, magnetic Feshbach resonances3 offer a way of tuning both the strength and sign of interactions in ultracold gases and have opened up the opportunity of studying many important fundamental phenomena. These include the creation of bright matter solitons4, the investigation of the crossover from Bardeen–Cooper–Schrieffer pairing to a Bose–Einstein condensate for strongly interacting Fermi gases5,6,7,8,9,10,11,12, as well as the coherent formation of ultracold molecules13, 14.

A Feshbach resonance as a rule modifies the amplitude of elastic scattering where atoms leave in the internal quantum states of the incident channel. Generally, however, the quasi-bound state responsible for the resonance may couple to additional energetically open channels connected to separated atoms pairs in states different to the incoming channel. This describes an inelastic collision. In addition, sufficiently far above the threshold of the incoming channel, the energy of the bound state and thus the resonance position relative to the threshold depend linearly on an applied magnetic field B with a slope defining the differential magnetic moment δμ of the channels. This magnetic field dependence opens up the possibility of tuning the resonance location to the threshold (E res(B) → 0) for atoms in the incident channel and the threshold regime just above it.

Away from threshold, resonant s-wave scattering at relative collision energy E = ħ 2 k 2/(2μ) is typically described by the Breit–Wigner formula with an energy-independent width1. Here, μ = m/2 is the reduced mass of atoms with mass m, and ħk is the relative momentum with wavenumber k and the reduced Planck’s constant ħ. Near threshold, (E → 0), the coupling between the bound state of the closed channel to the incident collision channel is characterized by a width that depends on the relative collision energy as Γ(E) $$\sqrt E$$. In contrast, the partial widths $$\Gamma _j^{{\rm{inel}}}$$ corresponding to the decay from the bound state to inelastic channels j are independent of E (as well as B). The s-wave elastic scattering amplitude at magnetic field B is13, 15, 16

$$f(E,B) = - \frac{{{a_{{\rm{bg}}}}}}{{1 + ik{a_{{\rm{bg}}}}}} - {{\rm{e}}^{2i{\delta _{{\rm{bg}}}}}}\frac{{{a_{{\rm{bg}}}} \gamma }}{{E - {E_{{\rm{res}}}} + i{\Gamma _{{\rm{tot}}}}/2}},$$
(1)

where Γtot = Γ(E) + Γinel, δ bg = −arctan abg k is the background scattering phase shift with scattering length a bg, and the reduced width γ = Γ(E)/2ka bg is independent of both E and B. An essential insight is that for E → 0 the regime $$\Gamma (E) \, < \, {\Gamma ^{{\rm{inel}}}} = \mathop {\sum}\nolimits_j {\Gamma _j^{{\rm{inel}}}}$$ will be reached such that Γtot ≈ Γinel. The resonance position depends on B as E res = δμ(B − B 0), where B 0 is the field for which the resonance at threshold occurs. It is worth noting that the total (elastic plus inelastic) cross section

$${\sigma _{{\rm{tot}}}}(E,B) = \frac{{4\pi }}{k}{\rm{Im[}}f{\rm{(}}E,B{\rm{)]}}\hskip7pc$$
(2a)
$$\hskip2pc\begin{array}{l} = 4\pi a_{{\rm{bg}}}^2\left[ {1 + \frac{{2\left( {E - {E_{{\rm{res}}}}} \right) \gamma + {\gamma ^2}}}{{{{\left( {E - {E_{{\rm{res}}}}} \right)}^2} + {{\left( {{\Gamma _{{\rm{tot}}}}{\rm{/}}2} \right)}^2}}}} \right]\\ + \frac{{2\pi }}{k}\frac{{{a_{{\rm{bg}}}}\gamma {\Gamma ^{{\rm{inel}}}}}}{{{{\left( {E - {E_{{\rm{res}}}}} \right)}^2} + {{\left( {{\Gamma _{{\rm{tot}}}}{\rm{/}}2} \right)}^2}}},\\ \end{array}$$
(2b)

is consistent with the optical theorem15. The first and second terms of Eq. (2b) correspond to the elastic and inelastic contributions, respectively; in obtaining this expression from Eq. (1) we have assumed $$\left| {{a_{{\rm{bg}}}}k} \right| \ll 1$$ so that $${{\rm{e}}^{2i{\delta _{{\rm{bg}}}}}} \approx 1 + 2i{a_{{\rm{bg}}}}k$$.

In the realm of ultracold atomic gases Feshbach resonances have, since their inception in these systems17, predominantly been studied by measuring the loss rate of a trapped, stationary sample as a function of magnetic field. Here, the temperature of the sample would define a characteristic collision energy. Varying the temperature can hence provide some insight into to the energy dependence of the resonance18,19,20, but by its nature this method is associated with thermal broadening. Measurements in the energy domain have also been achieved by dissociating Feshbach molecules through a fast and non-adiabatic magnetic field ramp taking the molecular bound state considerably above threshold21,22,23. Here, decay into free atom pairs ensues in what can be described as a half-collision24. Obviously, this method relies on the ability to associate atoms into molecules in the first place, which is not generally applicable.

Recently, both experimental and theoretical collider-type approaches that consider energy as a tuning parameter for Feshbach resonances have emerged25,26,27,28. For example, Gensemer et al.27 explored resonant scattering behavior with respect to the relative collision energy through use of an atomic fountain that launched ultracold clouds of 133Cs atoms to collide in free space at a fixed magnetic bias field. In this way, resonances that would overlap at threshold could be resolved. A collider-like configuration was also employed to observe scattering from a Feshbach resonance at a finite energy by Genkina et al. by splitting a trapped cloud of 40K into two momentum components28.

In this study, we report on the use of an optical collider29, 30 based on steerable optical tweezers to explore the E and B-dependent scattering of 87Rb atoms for a Feshbach resonance. In particular, our approach can determine the differential magnetic moment δμ of the resonant quasi-bound level when this strongly couples to outgoing inelastic channels. This setting usually rules out methods relying on associating atoms into molecules31, 32. In our experiment, we measure the magnetic field dependence of the total number of particles lost from the two colliding clouds for a range of non-zero energies. We observe a shift of the magnetic Feshbach resonance, but thermal broadening and the k −1 fall-off of the inelastic cross section, defined in Eq. (2b), makes the determination of the shifted resonance position exceedingly difficult. However, by instead measuring the center-of-mass positions of the outgoing clouds of unscattered atoms we achieve a dispersively shaped signal, which through its zero-crossing accurately defines the resonant magnetic field at collision energies even far above threshold.

## Results

### System under study

For our demonstration, we consider a resonance between the |F = 2, m F = 0〉 ≡ |2, 0〉, and |1, 1〉 spin states of 87Rb located33, 34 at approximately B 0 = 0.940(7) mT (9.040 G), which is predicted33, 35 to have comparable elastic and inelastic widths of γ/δμ = 1.1 mG and Γinel/δμ = 2.4 mG respectively, and a residual magnetic moment of δμ = 1.998μ B, where μ B is the Bohr magneton. Figure 1a shows the relevant interaction potentials for this resonance. The s-wave bound state, which is supported by the |1, −1〉 + |2, 2〉 potential, is strongly coupled to the inelastic d-wave channels |1, 1〉 + |2, −1〉 and |1, 1〉 + |2, −2〉. The excess energy associated with an inelastic event, as measured in units of the Boltzmann constant k B, is several hundred microkelvin and is gained as kinetic energy as the atoms fly away from the interaction region. Figure 1b presents the expected E and B dependence of the total cross section σ tot according to Eq. (2b) for the domain to be covered in our experiments. The Feshbach resonance gives rise to a peak centered on B res = B 0 + E/δμ, which shifts linearly with the energy, while its height decreases due to the k −1 dependence of the inelastic part of σ tot.

### Apparatus and experimental procedure

Our experimental setup has been previously described36. Briefly, an ultracold ensemble of ≈3 × 106 87Rb atoms in the |2, 2〉 hyperfine ground-state, at a temperature of T≈1 μK, is loaded into a far-detuned optical crossed-beam trap. Both the horizontal and vertical beams, making up the dipole trap, are derived from a 1064 nm fiber laser and have waists of ≈80 and ≈40 μm, respectively, at the point of intersection. An acousto-optic deflector (AOD) provides position control for the vertical beam so that the crossing position can be moved along the horizontal guide beam. By toggling the frequency input of the AOD the single cloud of atoms can be split into two, and one of the samples (the projectile) is moved 1 mm along the horizontal guide. Here, we evaporatively cool both samples to below 1 μK by lowering the power of the horizontal trapping beam and initiate a sequence (see “Methods”) that prepares the projectile cloud in the |2, 0〉 state and the stationary cloud (the target) in the |1, 1〉 state. Using the AOD the projectile cloud is accelerated toward the target cloud to collide at a nominal relative energy, E nom, in the presence of a homogeneous magnetic field B. At a separation of approximately 80 μm, both vertical beams are switched off, so that the collision occurs while the clouds are confined in the horizontal beam. This procedure gives rise to a 1–3 ms time window prior to the clouds colliding during which atoms evolve ballistically. Approximately 10 ms after the two clouds separate we transfer the atoms in the |1, 1〉 state to the |2, 2〉 state using microwave adiabatic passage. Atoms that have not been scattered out of the horizontal guide beam are then detected using standard time-of-flight absorption imaging, where the sample is exposed to a probe laser beam resonant with the 87Rb F = 2 → F′ = 3 transition of the D2 line. As remarked previously, atoms involved in inelastic collision events gain several hundred microkelvin in kinetic energy, which far exceeds the depth of the horizontal guide and thus leads to their ejection. In the absence of resonant loss, the imaged projectile clouds contain ≈5 × 105 atoms at a temperature of ≈900 nK, while the target clouds contain ≈9 × 105 at a temperature of ≈600 nK.

### Collisional loss from a Feshbach resonance

Figure 2 shows loss spectroscopy data for the |1, 1〉 + |2, 0〉 → |1, −1〉 + |2, 2〉 Feshbach resonance as a function of B performed for five collision energies, from E nom/k B = 1.7 μK to 12.0 μK as determined from cloud positions in time-of-flight images. Two distinct loss features emerge from this data. First, as expected from Fig. 1b, we observe a significant peak in atom loss, which shifts with the relative collision energy of the two clouds. This loss adheres to the dashed gray line in the figure showing the resonance position predicted by coupled-channels calculations. With increasing energy the feature widens and becomes less distinct. This widening is a result of our acceleration scheme rather than the more fundamental energy dependence of Γ(E) at threshold. The spread in energies for the collider depends on the initial cloud temperature and is given by $$\delta E = \sqrt {2{E_{{\rm{nom}}}}{k_{\rm{B}}}T}$$ (see “Methods”). The gray shaded area in Fig. 2 illustrates δE associated with a cloud temperature of T = 600 nK, characteristic for our experiments. A second non-trivial loss feature occurs around the threshold resonant field B 0, and follows the vertical dash-dotted line in Fig. 2. We attribute this loss to second-order scattering, where a first elastic scattering event transfers (close to) all of the kinetic energy of a projectile atom to a target atom. Such an event adds a |1, 1〉 atom traveling along with the |2, 0〉 projectile cloud. It also adds a |2, 0〉 atom within the stationary |1, 1〉 target cloud. Hence, from elastic scattering, secondary |2, 0〉 + |1, 1〉 threshold collisions (E ≈ 0) within both target and projectile clouds ensue, and for the magnetic field B 0 a resonant inelastic loss will be encountered. Of course, s-wave elastic collisions populate an isotropic halo in momentum space so that secondary scattering occurs for a range of energies and not only at threshold. However, the subset of particles elastically scattered into the projectile and target modes cause the most pronounced loss, since they follow these high-density clouds axially while being radially confined by the horizontal trapping laser beam. Numerical modeling of our collider experiment at E nom/k B = 12.0 μK using the direct simulation Monte Carlo method30, 37 corroborates our interpretation of the secondary peak.

In conjunction with the inherent energy-dependent resolution of the optical collider, the reduction in the cross section σ tot with increasing energy, shown in Fig. 1b, makes a more quantitative comparison between the shifted peak position to theory increasingly challenging. In an attempt to more accurately pinpoint the shifting peak, we next focus our attention to how scattering via a narrow resonance modifies the spatial distributions of the target and projectile clouds.

### Spatial distortions of colliding clouds

As mentioned above, atoms in the two clouds are allowed to expand freely along the collision axis, $$\hat {\bf{z}}$$, for a period of time before they begin to overlap. They remain confined in the transverse radial directions. A one-dimensional, axial model of phase-space distributions in position z and momentum p z along $$\hat {\bf{z}}$$ then suffices to capture the physics at play. Due to the thermal spread $$\delta {p_z} = \sqrt {2m{k_{\rm{B}}}T}$$ the initial (Gaussian) phase-space distribution ρ p(z, p z , τ) of the target cloud stretches with time τ along $$\hat {\bf{z}}$$ as shown in Fig. 3a, b during the ballistic expansion38, 39. This introduces correlations between the position and momentum of a particle. Those at the front of the target cloud (relative to incident projectile atoms) have mainly negative momentum, while those at the back have positive momentum.

An incident test particle with momentum p z  = p p interacts resonantly with a particle in the expanding target cloud with momentum p z  = p t, if the difference in their momentum is p p − p t = [4mE res(B)]1/2 ≡ Δp res(B). The resonant interaction removes atoms from a horizontal strip through ρ t(z, p z , τ), with a location that depends on the magnetic field through E res(B)—in particular, this strip is centered on p z  = 0 when $$B = {B_0} + p_{\rm{p}}^2{\rm{/}}(4m\delta \mu ) \equiv {B_{{\rm{res}}}}$$ as illustrated in Fig. 3d. The loss of particles from the target cloud is imprinted onto its marginal spatial distribution, $${n_{\rm{t}}}(z,\tau ) = {\int} {{\rm{d}}{p_z}{\rho _{\rm{t}}}\left( {z,{p_z},\tau } \right)}$$. For B = B + > B res (Fig. 3c) interactions involving target atoms with a negative momentum p z  < 0, and thus z < 0 are predominantly lost due to the correlations introduced by the ballistic expansion. In contrast, B = B  < B res (Fig. 3e) leads to a predominant loss of target atoms with p z  > 0 and z > 0. Altogether, this means that the distribution n t(z, τ) shifts in the direction of $$+ \hat {\bf{z}}$$ or $$- \hat {\bf{z}}$$, depending on whether B > B res or B < B res, respectively.

We can extend these considerations to a thermal ensemble of projectile particles with mean momentum $${\overline p _{\rm{p}}}$$, and momentum spread δp z (Fig. 4a). Defining now B res in terms of $${\overline p _{\rm{p}}}$$, Fig. 4b–d illustrate the resulting interactions during transit of the projectile cloud for fields B = B + > B res, B = B res, and B = B  < B res, respectively. As a result of their encounter, the centers of the projectile and target clouds shift oppositely. For BB res the target and projectile shift in directions $$\pm \hat {\bf{z}}$$ and $$\mp \hat {\bf{z}}$$, respectively, and by symmetry the shift is zero for B = B res (Fig. 4c). The shift also becomes zero well away from the resonance.

### Differential magnetic moment from spatially dependent loss

From the above analysis, we expect to encounter a spatial imprint of the Feshbach resonance on our clouds. We hence turn to an investigation of the center-of-mass positions Δz CM of each of the two clouds after the collision, as an alternative to monitoring the total loss in atoms. Figure 5 presents examples of the magnetic field dependence of Δz CM as extracted from post-collision absorption images (see “Methods”) for both target and projectile for three values of E nom. We see that as B is scanned across the resonance, Δz CM(B) carries out an oscillation about zero and we take the zero crossing to define the resonance position B res. The oscillations observed in the target (Fig. 5a–c) and projectile (Fig. 5d–f) clouds are in antiphase as a result of the complementarity in loss (see Fig. 4b, d). Since the projectile cloud contains fewer atoms than the target cloud its center-of-mass performs a larger excursion. We stress that at the resonant field B = B res, no center-of-mass shift is expected in either cloud (Δz CM = 0) irrespective of the number of atoms in each cloud. This independence makes the determination of B res less sensitive to initial atom number fluctuations as compared to a simple loss measurement. From Fig. 5 a clear upward shift in B res with increasing E nom can be seen in both target and projectile clouds. Figure 6 displays the inferred shifts (see “Methods”) from Δz CM(B) for our entire data set as well as B res predicted from coupled channels calculations. The measured B res position agrees well with the predicted linear dependence of the theory, where the rate of the shift is given by δμ.

## Discussion

We have investigated the collision of two individually prepared clouds of ultracold atoms about a narrow Feshbach resonance with a significant inelastic component. Previous experiments with colliding clouds successfully analyzed the halos of particles that elastically scattered from incident samples to infer properties about their interaction23, 24, 28, 40,41,42,43,44. While a collisional halo also ensues from the process considered in this study45, its resonant enhancement is difficult to attain due to thermal broadening and the existence of three dominant exit channels. Rather than observing the scattered particles we therefore focused on the atoms remaining in the outgoing clouds following the collision. In this respect, the scheme put forward here is closely related to conventional Feshbach loss spectroscopy, but with the important advantage of adding the collision energy as a tuning knob. While the total number of remaining atoms demonstrated an energy dependence of the resonance position, we found that B res(E) could be inferred with significantly improved accuracy from the shift in center-of-mass positions of the clouds. This shift is relatively immune to fluctuations in initial atom numbers and is introduced by an imprint of the Feshbach resonance on the spatial atomic distribution. The occurrence of a spatial imprint hinges, crucially, on a position-momentum correlation introduced by ballistic expansion of the clouds prior to collision.

The analysis through shifts in center-of-mass positions opens up a unique way of analyzing the above-threshold behavior for narrow resonance features. In this study, we applied it to a narrow magnetic Feshbach resonance, but the approach is quite generally applicable and could, for example, be used to locate high-$$\ell$$ shape resonances, which tend to have widths that are very small compared to the resonant collision energy. Our method notably overcomes limitations set by thermal broadening of the collision energy. Such broadening becomes prominent in acceleration schemes46 where the absolute energy spread of the projectile ensemble increases as it is being accelerated; in particular, our optical collider, where each particle of the projectile cloud accrues the same momentum gain, has these characteristics.

The optical collider scheme presented in this article constitutes an advancement toward directly observing the quintessential $$\sqrt E$$ energy-dependent scaling of the elastic width of an s-wave Feshbach resonance at threshold—an effect that in the current experiments still remains too small to be observed. For 87Rb, a more favorable candidate might be sought in the collisions between pairs of atoms in |1, 1〉, which is the absolute ground state and therefore not prone to two-body inelastic decay and furthermore has a magnetic Feshbach resonance at 1007 G of a reasonable width (γ/δμ ≈ 200 mG)47. More generally, and by extending scattering measurements beyond the threshold regime, the functions parametrizing multichannel quantum defect theory as introduced by Mies48 could be directly mapped out49 from careful measurements of resonance widths and positions. Collider studies of Feshbach resonances opens up the possibility of addressing resonances overlapping at threshold27 and has been suggested as a route to provide limits on the time variation of fundamental constants27, 50, 51. Furthermore, it provides an encouraging prospect for an alternative method of observing Efimov physics, where the relative energy dependence of the Efimov state is exploited instead of, as conventional, the dependence on the scattering length through B 26. In conclusion, we believe our optical collider scheme hold considerable promise to be broadly applied.

## Methods

### State preparation and collision procedure

The two 87Rb ensembles, separated by 1 mm in the horizontal plane, are exposed to a magnetic field gradient creating a position-dependent Zeeman shift. This enables site selective preparation of the clouds in the |2, 0〉 and |1, 1〉 hyperfine states, respectively, via an adiabatic rapid passage procedure using microwave fields. Subsequently, the projectile cloud is moved to a position 0.1 mm from the target cloud, from where it ultimately will be accelerated. For the collision process the gradient field is replaced by a homogeneous bias field. This homogeneous field is first ramped to a value close to the threshold resonant value B 0 causing remaining |1, 1〉 impurities in the |2, 0〉 cloud and |2, 0〉 impurities in the |1, 1〉 cloud to leave the traps through resonant Feshbach loss. After 30 ms the bias field is ramped to its final value B and allowed to stabilize before the beginning of the collision. The magnetic field has a stability of 0.13 mG over 3 days.

### Energy spread in the optical collider

We consider the collision between a projectile and target cloud at a nominal relative collision energy E nom. A particle of momentum p p in the projectile cloud collides with a particle with momentum p t in the target cloud at relative energy E = (p p − p t)2/(4m), where m is the mass of the particles. We assume that the momenta of particles in the target cloud follow a Maxwell–Boltzmann distribution

$${f_{\rm{t}}}({\bf{p}}) = \frac{1}{{{{\left( {2\pi m{k_{\rm{B}}}T} \right)}^{3/2}}}}{\rm{exp}}\left[ { - \frac{{{{\bf{p}}^2}}}{{2m{k_{\rm{B}}}T}}} \right],$$
(3)

with temperature T, whereas the action of the optical collider is to prepare the projectile cloud according to

$${f_{\rm{p}}}({\bf{p}}) = \frac{1}{{{{\left( {2\pi m{k_{\rm{B}}}T} \right)}^{3/2}}}}{\rm{exp}}\left[ {\frac{{{{({\bf{p}} - {{\overline p }_{\rm{p}}}\hat {\bf{z}})}^2}}}{{2m{k_{\rm{B}}}T}}} \right],$$
(4)

i.e., an ensemble traveling along the z-axis at mean velocity $${\bf{v}} = {\overline p _{\rm{p}}}\hat {\bf{z}}{\rm{/}}m$$, where $${\overline p _{\rm{p}}} = \sqrt {4m{E_{{\rm{nom}}}}}$$. Assuming $${\overline p _{\rm{p}}} \gg \sqrt {m{k_{\rm{B}}}T}$$, we find that the mean collision energy between the clouds is

$$\left\langle E \right\rangle = \frac{1}{{4m}}\left( {\left\langle {{\bf{p}}_{\rm{p}}^2} \right\rangle + \left\langle {{\bf{p}}_{\rm{t}}^2} \right\rangle } \right) = {E_{{\rm{nom}}}} + \frac{{3{k_{\rm{B}}}T}}{2},$$
(5)

with variance

$${\rm{var}}(E) = \left\langle {{E^2}} \right\rangle - {\left\langle E \right\rangle ^2} = \frac{3}{2}{\left( {{k_{\rm{B}}}T} \right)^2} + 2{E_{{\rm{nom}}}}{k_{\rm{B}}}T.$$
(6)

Hence, for $${E_{{\rm{nom}}}} \gg {k_{\rm{B}}}T$$ the energy spread of the optical collider is

$$\delta E = \sqrt {{\rm{var}}(E)} \approx \sqrt {2{E_{{\rm{nom}}}}{k_{\rm{B}}}T} .$$
(7)

### Modeling center-of-mass shifts of post-collision clouds

The number of unscattered atoms is extracted from absorption images by fitting the two imaged clouds with Gaussian functions. For both the target and projectile cloud the position of the Gaussian, Δz CM(B), is analyzed separately with respect to the applied magnetic field. Figure 5 shows three example data sets of Δz CM(B) for a range of bias fields. Each data set (corresponding to a fixed E nom) is fitted with the derivative of a pseudo-Voigt profile

$$\begin{array}{*{20}{l}}\\ {\Delta {z_{{\rm{CM}}}}(B) = } \hfill & { - \eta A\,{\rm{ln}}(2)\frac{{B - {B_{{\rm{res}}}}}}{{{w^2}}}{{\rm{e}}^{ - {\rm{ln}}(2){{\left( {B - {B_{{\rm{res}}}}} \right)}^2}{\rm{/}}{w^2}}}} \hfill \\ \\ {} \hfill & { + (1 - \eta )\frac{2}{\pi }\frac{{Aw\left( {B - {B_{{\rm{res}}}}} \right)}}{{{{\left[ {{{\left( {B - {B_{{\rm{res}}}}} \right)}^2} + {{\left( {w{\rm{/}}2} \right)}^2}} \right]}^2}}} + \Delta {z_{{\rm{bg}}}},} \hfill \\ \end{array}$$
(8)

where 0 < η < 1, Δz bg is the position of the cloud away from resonance, w is the width, and A is a scaling factor. The data is fitted via a weighted least squares fitting routine, where the weights are determined by the corresponding standard deviations, and η, w, A, B res, and Δz bg are free variables. The field B res, corresponding to the inflection point of Eq. (8), varies within the error of the applied bias field when the first (derivative of a Gaussian) or the second (derivative of a Lorentzian) term of Eq. (8) is omitted for the fit.

### Data availability

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