# Evidence for universal relations describing a gas with -wave interactions

###### Abstract

Thermodynamics provides powerful constraints on physical and chemical systems in equilibrium. However, non-equilibrium dynamics depends explicitly on microscopic properties, requiring an understanding beyond thermodynamics. Remarkably, in dilute gases, a set of universal relations is known to connect thermodynamics directly with microscopic properties. So far, these “contact” relations have been established only for interactions with -wave symmetry, i.e., without relative angular momentum. We report measurements of two new physical quantities, the “-wave contacts”, and present evidence that they encode the universal aspects of -wave interactions through recently proposed relations. Our experiments use an ultracold Fermi gas of K, in which -wave interactions are suppressed by polarising the sample, while -wave interactions are enhanced by working near a scattering resonance. Using time-resolved spectroscopy, we study how correlations in the system develop after “quenching” the atoms into an interacting state. Combining quasi-steady-state measurements with new contact relations, we infer an attractive -wave interaction energy as large as half the Fermi energy. Our results reveal new ways to understand and characterise the properties of a resonant -wave quantum gas.

A fundamental question provoked by observation of natural systems is how macroscopic and collective properties depend on microscopic few-body interactions. Ultracold neutral atoms provide a model system in which to explore this question, since in certain conditions, few-body interactions can be tuned and characterised precisely. Over the last decade, a direct link has been made in these systems between thermodynamic properties and the underlying isotropic (-wave) interactions. At the centre stage is a quantity called the “contact” Tan2008:01 (); Tan2008:02 (); Tan2008:03 (); Werner2009 (); Zhang2009 (); Braaten:2012gh (), which describes how the energy of a system changes when the interaction strength is changed. Surprisingly, the contact is also the pivot of a set of universal relations, that constrain numerous microscopic properties, including the two-particle correlation function at short range. These relations apply regardless of temperature, density, or interaction strength Tan2008:01 (); Tan2008:02 (); Tan2008:03 (); Zhang2009 (), to fermions and bosons Werner2012 (); Werner:2012hy (); Wild:2012fi (), and in one-, two-, and three-dimensional systems Olshanii:2003kn (); Combescot:2009gw (); Frohlich:2012ic (); Werner2012 (). Contact relations have also been extended to Coulomb gases Barth:2011ky () and neutron-proton interactions Weiss:2015gf (). Despite the breadth of this discussion, measurements of the contact have so far been restricted to systems with -wave interactions.

In general, the relative wave function of any pair of particles can be decomposed into components with angular momentum equal to an integer multiple of quanta. In a spin-polarised Fermi gas, quantum statistics forbids short-range interactions with even values of . Therefore, the first allowed scattering channel has (-wave), which is typically weak due to the centrifugal barrier (see Fig. 1): the scattering cross section decreases with the square of the collision energy DeMarco1999 (). However, resonant enhancement of -wave collisions has been observed in K and Li Regal2003 (); Zhang2004 (); Schunck:2005cf (), raising the possibility of studying a gas with strong, tuneable -wave interactions.

Interest in -wave systems originated with liquid He, which at low temperature is a superfluid of pairs with in a spin triplet state. It is thought that SrRuO realises a chiral superconductor, although definitive evidence is still elusive Kallin:2012kx (). An ultracold gas with tuneable -wave interactions could be employed to explore the evolution from weak- to strong-coupling superfluids, including the topological quantum phase transition predicted in two-dimensional samples Read:2000iq (); Levinsen:2007gy (). In certain conditions, a -wave superfluid could host Majorana modes that exhibit non-Abelian statistics, which is important for topological quantum computing Kallin:2012kx (); MajoranaReview ().

Here we report on time-resolved spectroscopic characterisation of an ultracold spin-polarised Fermi gas with near-resonant -wave interactions. We find that both the momentum distribution and the radio-frequency (rf) response follow asymptotic scaling consistent with -wave contact relations Inotani2012 (); Yoshida2015 (); Yu2015 (); Zhou:2016 (), Furthermore, contact values measured with these two independent methods show good agreement. Then, relying upon the validity of the complete contact theory, we interpret the dynamical evolution of the contact as the population dynamics in the closed channel.

Near-resonant -wave interactions

For -wave scattering, the van der Waals interaction between atoms can be regarded as short-ranged Zhang2010 (): i.e., acting only when , where is the inter-nuclear separation and is comparable to the van der Waals length, about nm for potassium. In ultracold Fermi gases, the inter-particle separation is much larger: is typically nm, where is the Fermi momentum. As a result, the low-energy interaction can be described completely by the -wave scattering phase-shift , , where is the relative wave vector, is the scattering volume, is the effective range, and labels the projection of angular momentum onto the magnetic-field axis Ticknor2004 (); Zhang2010 ().

Figure 1 depicts the control introduced by a Feshbach resonance Chin2010 (), where a closed-channel molecular bound state is tuned near energetic resonance with an open-channel scattering state of a pair of free atoms. Tuning is accomplished with a magnetic field due to a differential magnetic moment between the closed and open channels. Far from resonance, and take on background values and , but near resonance, the scattering volume is resonantly enhanced: , where and are the position and width of the resonance (see Methods and Fig. 1). In contrast, is only weakly field-dependent Ticknor2004 (); Levinsen:2007gy (); Jona2008 (). Since the resonance for is split from the degenerate and resonances, both the strength and the anisotropy of -wave interactions can be controlled with the magnetic field Regal2003 (); Ticknor2004 (); Gunter2005 (); Peng2014 ().

For , there is a Feshbach dimer state, which is a superposition of and , at energy , where is the atomic mass. For , the dimer state rises above threshold () and decays at a rate (see Supplementary Information). At low energy, all -wave resonances are narrow (, where is the Fermi energy) since decreases with energy. In some ways, they resemble narrow -wave resonances Hazlett:2012dt (); Kohstall:2012kg (), which also have a quasi-bound dimer state above threshold. For the particular -wave resonance we use, dipolar relaxation of to more deeply bound states limits the lifetime of the Feshbach dimer to ms Ticknor2004 (); Gaebler2007 ().

Universal -wave contact relations

That -wave interactions might be described with an analogue of the contact was conjectured in ref Inotani2012, , and recently followed by a full theory in refs Yoshida2015, ; Yu2015, . The most significant structural difference from the -wave contact theory is that each scattering channel has two contacts, and , which are the thermodynamic “forces” conjugate to and respectively. We summarize here what can be learned from the contacts, once they are measured or calculated.

A. Thermodynamic identity. The change in free energy for a uniform system is

(1) |

where is entropy, is temperature, is pressure, is volume, is the total atom number, and is chemical potential. Note that has units of length, has units of inverse length, and that both are extensive variables. In a harmonic trap, is replaced by , where is the potential energy of the cloud and is the trapping frequency.

B. Correlations. At short range, , the many-body wave function has a form that is controlled by two-body physics, but a normalisation that is controlled by the contacts Zhang2009 (). For instance, the pair correlation function is

(2) |

in the regime , where is the relative coordinate, , and are the spherical harmonics for .

C. Momentum distribution. The contacts also constrain the asymptotic form of the momentum distribution, . Indeed the -wave contact is often defined to be the high- limit of Tan2008:01 (); Tan2008:02 (). For -waves, in the asymptotic regime , has two components Inotani2012 (); Yoshida2015 (); Yu2015 (); Zhou:2016 ():

(3) |

where and is the volume of the system.

D. Spectral response. The spectral weight of excitations that probe the high-energy or short-range sector of the many-body wave function are also controlled by the contacts Chin2005 (); Pieri2009 (); Schneider2010 (); Braaten2010 (); Stewart2010 (). For rf transfer to a non-interacting probe state, the high-frequency tail of the spectral density is

(4) |

where is the detuning of the probe frequency from resonance, , and .

E. Fraction of the closed-channel molecules . Close to the Feshbach resonance, where , is proportional to :

(5) |

where Yoshida2015 (). In this aspect, is similar to the -wave contact Werner2009 (); Zhang2009 (). In contrast, is an energy-weighted quantity that in the two-channel model also involves atom-dimer interactions (see Supplementary Information).

Observation of the -wave contacts

The primary impediment to the exploration of -wave many-body physics in trapped quantum gases has been atom loss that is faster or comparable to trap-wide equilibration Chevy2005 (); Gaebler2007 (); Inada2008 (); Nakasuji2013 (). Our experimental approach is to study the gas after a “quench” that quickly initiates enhanced -wave interactions, accomplished with rf pulses. Before each pulse sequence, K atoms are confined in a crossed-beam optical dipole trap, spin-polarised in the lowest hyperfine-Zeeman state and cooled to nK, which is , above the superfluid critical temperature Ohashi2005 (); Gurarie:2007gs (); Inotani2012 (). A uniform magnetic field is stabilised at , in the vicinity of a -wave Feshbach resonance for state . A resonant 40-s -pulse transfers all atoms to , initiating tuneable -wave interactions. After a variable hold time , the gas is characterised either with rf spectroscopy or with time-of-flight (TOF) imaging, allowing contacts to be measured through relations (4) or (3) respectively, as shown in Fig. 2. Losses restrict to be short compared to thermalisation times of low-energy or long-wavelength degrees of freedom. However, we find that spectra reach a quasi-steady-state, which likely reflects a local equilibrium.

Radio-frequency spectroscopy probes the gas by transferring a fraction of atoms in to the third-lowest energy state , which (like ) does not have resonantly enhanced interactions. The fractional transfer to , , is measured by state-selective absorption imaging, after a magnetic field jump that dissociates any Feshbach dimers. Figure 2a shows an rf spectrum taken at G. The transfer to is given as a rescaled rate , where is the probe frequency rescaled by , is the transition matrix element and is the pulse area. The latter is chosen to be small enough to probe the transition in the linear regime for , where . The high-frequency tail fits well to equation (4), and is used to determine and (see Methods and SI for details).

The momentum distribution is measured by resonant absorption imaging of the cloud after a ms time-of-flight expansion. For these measurements, no rf pulse is applied at ; instead, the field is rapidly jumped away from the -wave resonance, preserving the interacting momentum distribution, which determines the ballistic flight after release from the trap. Figure 2b shows the normalized distribution observed at versus , after azimuthal averaging in the image plane. Inherent to imaging is also a line-of-sight integration of , so that the high-momentum scaling of is for the term and for the term (Methods). Figure 2c shows that the leading order appears as an asymptotic plateau in . A full fit with both terms is used to determine and .

Figure 2d compares the -wave contacts determined from and , across a range of magnetic field values. The dimensionless and are scaled by calculated using the peak density of a non-interacting gas, but results should be understood as an average over an inhomogeneous trapped ensemble Sagi2012 (). Since the contacts are only revealed in the asymptotic part of the distribution, analysis involves a low-energy cutoff, the systematic effect of which is studied in SI. Our analysis also assumes both and are nonnegative, but the possibility of is also discussed in the SI.

The correlation between the two observables is , as determined by the slope of a best-fit line with no offset. This agreement, in addition to the observation of the predicted asymptotic scaling of equations (3) and (4), is strong evidence that the -wave contact relations are valid.

Field dependence of the -wave contacts

Figures 3a,b show the -wave contacts versus near both the and the resonances. The data includes contacts determined at from , at from , and asymptotic values from versus . The variable-t data (discussed in more detail below) also identifies a loss-dominated regime for G, outside of which contacts reach a steady-state value despite atom loss of up to 20%.

We observe a pronounced asymmetry about each Feshbach resonance: significant contacts are only observed for . is largest close to resonance, decreases with , and vanishes beyond G, where . instead peaks at G before abruptly falling to zero for larger fields.

Some of these salient features can be explained by a simple model, in which non-interacting closed-channel dimers are in equilibrium with free fermions. Each dimer has and , but free fermions make no contribution to the contacts. Since the and resonances are well separated, and . The assumption of equilibrium gives in a harmonic trap at zero temperature Gurarie:2007gs (). This model would predict that both and are the same near the and the resonances, that and as , and that a fully dimerised gas would have , since in typical conditions. The additional factor of in gives at resonance and at peak at .

Although this model does explain the peak value of and the range of at which significant contacts are seen, it does not explain the value or location of the maximum in . A more realistic model would include finite temperature, and interactions between dimers, between atoms, and/or between atoms and dimers. For instance, resonant enhancement of atom-dimer interactions have been seen in a three-body calculation Levinsen:2007gy (); Jona2008 ().

Independent of any particular microscopic model, but assuming adiabaticity, we can understand the thermodynamic implications of the observed contacts using equation (1). The change in free energy versus is given by the integral of over , assuming all other variables are constant. The contribution of is not significant (Methods). The inferred is shown in Fig. 3c. The values shown have several possible systematic errors. First, some of the other variables that determine are varied by : decreases due to loss, and increases by near resonance. A second and more significant error may lie in the calibration of number and rf power, which combine to give a 30% systematic uncertainty in . Finally, equilibration is likely to be only local, and not trap-wide. Despite these uncertainties, the integrated data is sufficient to demonstrate several qualitative regimes:

(i) Below resonance (), the gas is weakly repulsive, with . Here, resonant scattering is inaccessible to free particles, and the gas remains on the “upper branch” Pricoupenko2006 (); Shenoy:2011dm (). Few or no dimers are formed, because energy-conserving two-body collisions cannot produce a dimer with a finite binding energy. Instead, the gas has weakly repulsive -wave interactions.

(ii) At resonance, we do not extract a value for , because a steady-state in is not achieved, as discussed in the next section. However, the discontinuity in between regime (i) and regime (iii) implies that the systems shifts from upper to lower branch in this region.

(iii) Above resonance, in the range , we infer a reduction in per particle approaching half the Fermi energy near resonance. The significant reduction of free energy is partially explained by the formation of dimers, whose binding energy could contribute up to in a harmonic trap Gurarie:2007gs (). Additional contributions to include dimer-dimer or atom-dimer interactions. Accompanying the large in this regime are the largest observed contacts, and therefore the strongest -wave correlations, as described be equation (2).

(iv) Farther above resonance, the scattering resonance at exceeds the maximum collision energy in a zero-temperature Fermi sea, leaving primarily non-resonant interactions between atoms. In this regime, -wave interactions are weakly attractive: .

Dynamics of after the quench

Evolution of the many-body wave function is required to adjust to the -wave interactions initiated by transferring atoms from into . We are able to observe these dynamics in the range , by varying the time at which we measure the contacts. For an isolated pair of atoms, the lifetime of the quasi-bound state would set the key time scale (between 0.2 ms and 1 ms here). However, we find that the contact develops at a rate twenty to thirty times larger than .

From Fig. 3 we know that only contributes to for . Measurement of at a single rf frequency (typically ) is used to determine . We also record the atom number at each . Figures 4a–f show the time-resolved measurements of and at various . We observe that rises to an apparent steady-state value, while decays relatively slowly. The initial growth rate of increases with , until it can no longer be resolved, at .

Near resonance, the wave function of each Feshbach dimer is dominated by Fuchs2008 (); Gubbels2007 (), allowing us to interpret the contact dynamics through a multi-channel model and equation (5). Three closed channels of -wave dimers are initially empty, but come to equilibrium with the initially populated open channel of atoms, but all contribute to the observed atom number . We calculate the time evolution of and with rate equations that omit coherence between the channels but include collisional loss:

(6) |

where is the dimer association rate, and is the fermion-dimer loss coefficient. We constrain the model to have near the resonance, and near the resonance. We furthermore combine the degenerate modes at the resonance to , , and . We fit the solution of equation (6) to the data, finding good agreement (see Figs. 4a–f). The rapid growth of for was observed directly in refs Jin2008, ; Inada2008, .

With , equation (6) would lead to a dimer population that asymptotically tends towards and . The associated equilibration rates are

(7) |

With loss present, we can distinguish between regimes (ii) and (iii). If , the system reaches a quasi-steady state from which it decays slowly. This scenario is realised in Fig. 4b,e and Fig. 4c,f, and corresponds to regime (iii) in Fig. 3c. If, however, , the loss rate of the system is faster than the association rate, and equilibration between and is inhibited. As a consequence, reaches a significantly reduced steady-state value on a time scale that is dominated by the loss rate. In turn, atom loss occurs on the time-scale of the dimer association. This case, corresponding to regime (ii) in Fig. 3c, occurs closer to resonance (see Fig. 4a,d and data at G and G in Fig. 4g), because for . The SI provides further details of how these two regimes are distinguished, and demonstrates robustness against details of the loss model.

Figure 3a includes the asymptotic values of (smaller points). They are within uncertainty of at for G, but provide an upwards correction for G. With the inclusion of this correction, decreases monotonically with throughout regime (iii).

Using only data from regime (iii), we find that are proportional to , with best-fit ratios and (see lines in Fig. 4g). These ratios are consistent with a perturbative treatment of resonant closed-channel molecular formation in a Fermi cloud that predicts as (SI). The consistency between the dynamical response of and a model of dimer population supports the validity of equation (5).

Conclusions

While studied here for a Fermi gas in a metallic state, the -wave contacts are expected to be “universal” in that they hold for any type of particle, whether boson or fermion, in any dimensionality, and in any state (superfluid or normal), so long as interactions are short-range and -wave. Further tests of universality should include comparison to direct measurements of energy, structure factor, and dimer number, as has been done for the -wave contact Partridge2005 (); Stewart2010 (); Kuhnle2010 (); Navon:2010ix (); Kuhnle2011 ().

We have also identified a regime where the atom-dimer equilibration is much faster than loss. After equilibration, strong -wave correlations persist for at least half a millisecond, eventually limited by dipolar and three-body loss rates. Searching for the onset of pair condensation in this dynamical window at a low temperature would be of great interest, especially in two-dimensional systems, where some loss mechanisms are less significant Levinsen2008 (), and the possibility of chiral order exists Read:2000iq (); Levinsen:2007gy ().

Methods

Sample preparation. Spin-polarised K atoms are cooled sympathetically with bosonic Rb atoms. The , and states of K refer to the high-field states adiabatically connected to the low-field , and states of the hyperfine manifold of the electronic ground state, where and denote the total angular momentum and the corresponding magnetic quantum number respectively. At the end of cooling, residual Rb atoms are removed with a resonant light pulse leaving only K atoms in state , held in a cross-beam optical dipole trap. For spectroscopic data, , while dynamics data was taken with . The pulse initialising dynamics is efficient, so that we assume . The mean oscillation frequency in the trap is Hz.

The ideal-gas Fermi energy in a harmonic trap is , here kHz, and is also the local Fermi energy at the centre of the trap. Thermometry is based on fitting an ideal-gas Fermi distribution to an absorption image taken at high magnetic field after release from the trap. To measure the temperature after a s hold time in , we jump the field rapidly to 209 G, turn off the trap, and transfer all atoms to during time-of-flight. At G, the apparent temperature rise is nK. Accompanied with a 25% number loss, this is an increase in reduced temperature of 0.05. We note that with such short hold times, the gas is likely out of thermal equilibrium.

State-resolved absorption images are taken at high magnetic field, after Stern-Gerlach separation of states , , and . Subsequent state transfer in a residual gradient field is used to map any population of interest to , since neither of the other states have an accessible cycling transition. For spectroscopic measurements, we optimise and calibrate this sequence to determine and .

The magnetic field values are calibrated with the frequency of the – transition, measured spectroscopically with a 120- pulse to an accuracy of kHz. The short-term accuracy of field values is mG. Over one spectroscopic or dynamic data series, the magnetic field drifts by less than mG.

Analysis of rf spectra. The high-frequency tail in the spectral density of the rf transition to a non-interacting probe state () is given by equation (4). We measure the transfer rate , with the length of the spectroscopy pulse. The transfer rate is proportional to the spectral density and obeys the sum rule . We therefore identify . The scaled frequency and normalised transfer rate as defined in the main text are normalised to .

With these definitions, the high-frequency tail of the normalised transfer rate is

(8) |

Close to the -wave resonances where , we measure using at a single frequency value :

(9) |

This single-frequency measurement technique is used in the data presented in Fig. 4.

We calibrate the strength of rf fields by driving Rabi oscillations between and at 209 G. At the peak power used in this experiment, we find a Rabi frequency of approximately 70 kHz with a typical uncertainly of 15%.

The spectroscopic pulse has a Blackman envelope to minimise the frequency sidebands in the spectra. For a given pulse length , the Blackman pulse area is a factor of 0.4266 different from that of a square pulse with the same length and peak power. The effective Rabi frequency that is used to normalise our rf spectra is kHz. Within the measurement window, we find for 0.05 G (-0.10 G), which agrees with the sum rule to within systematic error. Note that when we take normalisable spectra the Rabi frequency must be reduced for , to keep the coupled fraction small.

Analysis of momentum distributions. The momentum distribution in Fig. 2b,c is obtained from a time-of-flight absorption image. The imaging beam propagates along the direction, parallel to the Feshbach field. The optical density of the cloud is proportional to the column-integrated density . For long time of flight, the initial cloud size is rendered unimportant and . We normalize by its pixel sum to obtain with . Signal-to-noise is further improved by azimuthal averaging, yielding a distribution , where . Thirty to forty images are taken, and averaged. The contacts are determined with a fit to

(10) |

While the coefficient of the leading order is proportional to , just as in rf spectra, the sub-leading order is not simply proportional to . We interpret the data assuming that near the resonance, and that near the resonance.

-wave scattering parameters. For the -wave Feshbach resonance in state near 198.5 G we parameterise and as a function of magnetic field magnitude using

(11) | ||||

(12) |

where . From ref Ticknor2004, , we use the values , , , , , , , and . From our own measurements, we use and . Equations (11) and (12) are matched to ref Ticknor2004, , neglecting the next order in . However, since G in this work, the next-order term has a relative magnitude for and for . The binding energies measured in ref Gaebler2007, matches the calculated to 3%, but further measurements are needed to constrain, for instance, . Our measurements of and are consistent, within error, to those found in ref Gaebler2007, . Both results may share a systematic offset due to the differential polarisability of the open and closed channel.

Numerical integration to find . As a function of magnetic field, the reversible change in free energy is

Assuming the resonance is well isolated from the resonance, all terms in this equation can be evaluated with equation (1), measurements of and , and the parameterisations (11) and (12). The dominant contribution comes from the scattering volume:

(13) |

The change in energy between two magnetic field values, and , is given by . To evaluate the shift in energy due to near-resonant interactions, we calculate the numerical integrals

(14) | ||||

(15) |

where we have suppressed the subscripts for clarity, but are assuming that the and resonances are isolated. To estimate the statistical uncertainty, we repeat the integration with data normally distributed around the mean values of at each . The reported error is the standard deviation of the resultant . The next order in given in equation (13) can easily be included, but is when is above the noise floor of spectroscopy, so the resultant shift is not significant.

The -dependence of due to the change in is

(16) |

Across the range in which we observe -wave contacts, this contribution to is small. One can understand this by comparing equations (13) to (16): since , even if is comparable to , the ratio of contributions to near resonance is roughly , which is for typical experimental parameters.

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Acknowledgements We thank F. Chevy for discussion and shared notes concerning versus . We also thank N. Zuber for experimental assistance, and J. Bohn, B. Ruzic, Shina Tan, Edward Taylor, Pengfei Zhang, and Qi Zhou for discussion. This work was supported by AFOSR under FA9550-13-1-0063, ARO under W911NF-15-0603, the Croucher Foundation, RGC under 17306414, NKBRSFC, NSERC, and NSFC under 11474179, and the Tsinghua University Initiative Scientific Research Program.

Author Contributions C. L., S. S., and S. T. performed the experiments. C. L., S. T., and J. T. analysed the data. All authors contributed to the understanding the spectra. Z. Y., S. T., and S. Z. developed the two-channel model of the dynamics. All authors contributed to the preparation of the manuscript.

Additional Information Supplementary Information is available online, discussing the two-channel model and fitting systematics. Correspondence and requests for materials should be addressed to J. T. and S. Z.

Competing Financial Interests The authors declare no competing financial interests.

## Appendix A Supplementary Material

Connection between closed-channel molecules and p-wave contacts

Let the creation operators for molecules and fermions at position be and , respectively, and the internal wave function of the molecules be given by , with the relative coordinate. Then the interaction Hamiltonian that converts scattering fermions to molecules can be written as

(S1) |

where is the (un-renormalised) coupling constant. Transforming to Fourier space and noticing that is an odd function, which requires its Fourier transform to be linearly proportional to , the effective coupling Hamiltonian can be written as Ohashi2005 (); Gurarie:2005it (); Cheng:2005kv ()

(S2) |

where () creates a molecule (fermion) with momentum , and is the coupling constant. We shall relate the latter to the scattering volume and also the effective range , for the scattering channel characterised by the magnetic quantum number . The free Hamiltonian, on the other hand, can be written as

(S3) |

where , where is the detuning of the closed-channel molecule. is the magnetic moment difference between the open-channel scattering state and the closed-channel -wave dimer.

The connection of and to and can be established by calculating the fermionic scattering -matrix. The two-body Schrödinger equation can be written as

(S4) |

and can be solved conveniently as , where is the asymptotic state in the absence of interaction and we have defined the scattering -matrix, . Developing, as usual, the perturbative series for , one obtains, in the centre-of-mass frame

(S5) |

where we note that the factor of arises from the exchange symmetry of the identical fermions. To evaluate the integral, we need to impose an ultra-violet cutoff in momentum. By further matching the relation with , we find

(S6) | ||||

(S7) |

These two equations establish the relation between the two parameters in the Hamiltonian and the physical parameters .

To obtain the number of closed channel molecules, it is simplest to use the Hellmann-Feynman theorem, by taking the derivative with respect to . That is

(S8) |

Using the thermodynamic relation for , derived from equation (1) in the main text, and the fact that close to resonance, , we find that the number of dimers can be written as Yoshida2015 ()

(S9) |

In fact, close to resonance, since with being the size of the closed channel molecules, the coefficient is just of order . Namely,

(S10) |

such that is directly proportional to the number of -wave dimers in the closed channel, and the proportionality constant is of the order of the effective range . As a result, monitoring the value of through rf spectroscopy is equivalent to monitoring the number of dimers in the closed channel.