Probing open- and closed-channel p-wave resonances
Denise J. M. Ahmed-Braun, Kenneth G. Jackson, Scott Smale, Colin J. Dale, Ben A. Olsen, Servaas J. J. M. F. Kokkelmans, Paul S. Julienne, Joseph H. Thywissen
PProbing open- and closed-channel p-wave resonances
Denise J. M. Ahmed-Braun, Kenneth G. Jackson, Scott Smale, Colin J. Dale, BenA. Olsen, Servaas J. J. M. F. Kokkelmans, Paul S. Julienne, and Joseph H. Thywissen Department of Physics, Eindhoven University of Technology, The Netherlands Department of Physics, University of Toronto, Canada Yale-NUS College, Singapore Joint Quantum Institute, NIST and the University of Maryland, U.S.A. (Dated: February 1, 2021)We study the near-threshold molecular and collisional physics of a strong K p-wave Feshbachresonance through a combination of measurements, numerical calculations, and modeling. Dimerspectroscopy employs both radio-frequency spin-flip association in the MHz band and resonantassociation in the kHz band. Systematic uncertainty in the measured binding energy is reduced bya model that includes both inhomogeneous broadening and the Franck-Condon overlap amplitude.Coupled-channels calculations based on mass-scaled K potentials compare well to the observedbinding energies and also reveal a low-energy p-wave shape resonance in the open channel. Contraryto conventional expectation, we observe a nonlinear variation of the binding energy with magneticfield, and explain how this arises from strong coupling to the open channel. We develop an analytictwo-channel model that includes both resonances as well as the dipole-dipole interactions which, weshow, become important at low energy. Using this parameterization of the energy dependence of thescattering phase, we can classify the studied K resonance as “broad”. Throughout the paper, wecompare to the well understood s-wave case, and discuss the significant role played by van der Waalsphysics. The resulting understanding of the dimer physics of strong p-wave resonances provides asolid foundation for future exploration of few- and many-body orbital physics.
I. INTRODUCTION
Strong p-wave interactions are rare in nature, so theirextreme tunability in ultracold systems [1, 2] is an op-portunity for discovery [3–5]. Despite recent advancesin understanding, such as universal relations for p-wavesystems [6–10], open questions remain, including the ef-fect of confinement on Feshbach dimers [11–23] and cor-relation strength [24–26]. One-dimensional systems holdthe prospect for duality between strongly interacting oddwaves and weakly interacting even waves [27–30], for atopological phase transition in two-dimensional systems[31, 32], and for engineered states [33–35]. Even in three-dimensional systems, p-wave trimer states have yet to beobserved [36–39].Experimental work on ultracold p-wave alkali systemshas focused on the fermionic isotopes K [1, 40, 41] and Li [2, 20, 42–45], in part because s-wave collisions areeasily suppressed with spin polarization. Experimentalinvestigations have included studies of elastic and inelas-tic [20, 42, 45, 46] collision rates, spectroscopy [8, 41, 44],and low-dimensional confinement [18, 22, 23, 40].In this work, we perform association spectroscopyto determine the binding energies of p-wave Feshbachdimers near a strong resonance of K. To explain thesemeasurements, we offer a new analytic treatment thatbuilds on the commonly used effective-range approxima-tion (ERA) of p-wave scattering [47],cot δ = − ( V k ) − − ( Rk ) − + O{ k } , (1)where δ is the scattering phase, (cid:126) k is the relative momen-tum, V is the ( k -independent) scattering volume, and R > S -matrix element, S = exp(2 iδ ), such that S = V − + R − k + O{ k } − ik − O{ ik } V − + R − k + O{ k } + ik + O{ ik } (2)is an equivalent approximation [48].However, the ERA is invalidated at low energy due to adivergent contribution to the scattering volume from theweak 1 /r dipole-dipole potential. We offer a more com-plete parameterization of terms in the scattering phaseshift by factoring the S -matrix into three terms: S dip fordipole-dipole interactions, S P for the entrance channel,and S FB for the Feshbach mechanism. In the K case S P has a shape resonance and causes the ERA to becomeinaccurate for the largest binding energies we measure.Nonetheless, V and R provide a useful reference, sincethe ERA is appropriate for intermediate energies, and thecorrect low-energy limit for S P S FB . The Feshbach reso-nance [49] tunes the scattering phase primarily throughthe scattering volume, conventionally written as V ( B ) = V bg (cid:18) − ∆ δB (cid:19) (3)where V bg is the background scattering volume, δB = B − B , B is the magnetic field, B is the location ofthe resonance, and ∆ is its magnetic width. We explainhow this form emerges from the low-energy limit of atwo-channel model in the broad- and narrow-resonancecases. We also discuss how the B -field variation of R iscoupled to V ( B ), and in the case of K responsible forthe dominant contribution to V bg .Just below resonance ( V > S (where cot δ = i ) a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n located at k = iκ , with κ > κ = (cid:114) R V sec (cid:20)
13 cos − (cid:18) − / x (cid:19)(cid:21) , (4)where x ≡ ( R / | V | ) / is a dimensionless parameter.The bound-state energy is E = − E b = − (cid:126) κ /m , where m is the atomic mass of K. Its series expansion for x (cid:28) E b ≈ (cid:126) RmV (cid:18) x + 32 x + 218 x + . . . (cid:19) . (5)In contrast to an s-wave Feshbach resonance, where thedimer binding energy curves quadratically with magneticfield towards the collision threshold, the p-wave E b tendsto scale linearly across threshold, as the Feshbach dimerstate is confined by the centrifugal barrier. One can seethis from the near-linearity of V − in Eq. (3), for | δB | (cid:28) ∆. Also in contrast to the s-wave case, the binding energydepends on the effective range to lowest order.At the other side of the Feshbach resonance ( V < k plane, adding a width to the resonance[50]. Although this pole controls the scattering phase, itdoes not correspond to a true molecular state. Instead, itcreates a positive-energy scattering resonance (cot δ = 0)at E = − (cid:126) m RV (6)with which the resonant contribution to the phase shiftcan be written cot δ ( E ) = − E / R ( E − E ) E − / , where E R = (cid:126) m − R − . An approximate form of the angle-averaged scattering cross section, 4 πk − sin δ in each M L channel, is a Lorentzian with half-widthΓ E / E / R . (7)We see that unlike s-wave scattering, ultracold p-wavescattering is energetically narrow: Γ /E → E → K around 198.5 G. In Sec. II wedetermine the dimer resonance locations as a function ofmagnetic field using analytic models for the lineshapes.These measurements extend the pioneering work of Gae-bler et al. [41] to higher precision and to a wider rangeof magnetic fields. Energies are compared to a fullcoupled-channels calculation (Sec. III) that updates priorwork [51], enables us to identify the molecular physicsthat creates the Feshbach resonance, and allows us toidentify the range of validity of simplified models. Wefind a clear nonlinearity of the p-wave binding energy versus magnetic field and explain its origin in the strongcoupling to a broad shape resonance above threshold. InSec. IV, we develop an analytic two-channel model thattreats both resonances in the open and closed channels.Here, strong coupling manifests as a field-dependence ofthe effective coupling between the channels. In Sec. Vwe provide a new parameterization of p-wave scatteringbased on this model. We summarize in Sec. VI.
II. DIMER SPECTROSCOPY
Fermionic K has a nuclear spin of four and a S ground state, giving rise to two hyperfine states in theground state manifold with total spins f = 9 / / m f [52]. We use the convention of labelingthese states | a (cid:105) , | b (cid:105) , | c (cid:105) , . . . in order of increasing energyat nonzero magnetic field. Due to the inverted hyperfinestructure of the K atom, the lowest energy state | a (cid:105) is | f, m f (cid:105) = | / , − / (cid:105) , the next state | b (cid:105) is | / , − / (cid:105) ,and so forth. The p-wave ( L = 1) Feshbach resonances ofinterest here live in the | bb (cid:105) entrance channel near 198.3 Gand 198.8 G. A. Association spectra
Dimer spectroscopy typically begins with a gas of3 × atoms held in a crossed optical dipole trap witha mean trap frequency of 2 π ×
320 Hz, at temperaturesranging between 0 . µ K and 0 . µ K. Microfabricated sil-ver wires on an atom chip several hundred microns fromthe atomic cloud create oscillating magnetic fields thatdrive molecular association. Since the average collisionalenergy is comparable to 10 kHz, the order-100-kHz dipo-lar splitting between the M L = 0 and | M L | = 1 scatteringchannels is well resolved, which is an advantage for thestudy of p-waves over Li, for which | M L | channels arean order of magnitude closer [2, 18, 42, 44, 53]. In nei-ther species can the splitting between M L = +1 and − | b (cid:105) state, and B is tuned to thedesired value, between 195 G and 200 G. The oscillatingfield direction is aligned with the Feshbach field, driving∆ m f = 0 transitions from free pairs of atoms to dimers,as illustrated in Figs. 1(b) and 1(d). Since the initialand final states share the same continuum threshold, freeatoms with energy E k are able to associate into eitherbound dimers with energy E k − (cid:126) ω osc , or quasi-bounddimers with energy E k + (cid:126) ω osc , where ω osc is the oscilla- ( ω – ω )/2 π (kHz) osc ab RA b bb b bb E bb abbb bbabab ab bb ba bb b bb b ba
120 200140 160 220180 240– E (a) (h)(g)(f)(e) (d)(b) SFASFARA ( ω – ω )/2 π (kHz) osc ab ω /2 π (kHz) osc A t o m nu m be r , N ( a r b . ) b A t o m nu m be r , N + N ( a r b . ) a b k ћω osc ћω osc ћω osc ћω osc EE k – E b E ω /2 π (kHz) osc A t o m nu m be r , N ( a r b . ) b b E – ħω k ab E – ħω k ab A t o m nu m be r , N ( a r b . ) a (c) -340 -280-300-320 -260-360 -24080 140120100 18016060 FIG. 1.
Dimer association spectroscopy. (a, b) Resonant association to a true bound state: atom number remaining versusRA oscillation frequency at 197 . | M L | = 1 true bound state with E b /h = 92 . .
5) kHz,indicated by a green vertical line. The shaded bands around both the lineshape and vertical line represent 1-sigma confidenceintervals. (c, d) Resonant association to a quasi-bound state at 199 . E /h = 210(4) kHz in the | M L | = 1 channel. (e, f) Spin-flip association to a true bound state: atom number remaining versus the difference betweenSFA oscillation frequency and the spin-flip frequency at 197 . | M L | = 1 true bound statewith E b /h = 262 . . | M L | = 1 channel with energy E /h = 182(5) kHz. Insets (b, d, f, h) illustrate each spectroscopy protocol. Dimerdecay (represented by the blue arrow) produces untrapped final states and the loss signature. tion frequency of the field. The cloud is released from thetrap and imaged after Stern-Gerlach separation to countatoms remaining in each spin state. Typical frequency-dependent loss curves are shown in Figs. 1(a) and 1(c).Atom number, imaged here at 209 G, is a signature ofmolecular association since dimers decay on a millisec-ond time scale, through several mechanisms [1, 20, 42].At low density, loss is due to dipolar relaxation [42, 51]to the open | ab (cid:105) channel (see Sec. III and Fig. 4), whoserelease energy ejects the pair from the optical trap (ofdepth (cid:46) . | a (cid:105) and | b (cid:105) atoms. Spin-flip transitions between these statesare induced by the σ + -polarization component of theradio-frequency (rf) field near 44 MHz, as illustrated inFigs. 1(f) and 1(h). Typical SFA spectroscopic curvesare shown in Figs. 1(e) and 1(g).Comparing Fig. 1(e) to 1(a), we see that the asym-metry of bound spectra inverts, since for SFA the dimerenergy is always E k + (cid:126) ω osc − (cid:126) ω ab , where ω ab is the barespin-flip transition frequency, whereas for RA the dimerenergy is E k − (cid:126) ω osc for true bound states.The loss signatures shown in Figs. 1(c) and 1(g) differ, for the following reason. In the SFA protocol, the cre-ation of a quasi-bound dimer is tagged by the conversionof a | a (cid:105) to a | b (cid:105) atom, which is not reversed by dissocia-tion into a pair of | b (cid:105) atoms. In the RA protocol, quasi-bound dimers that decay through the centrifugal barrierare not necessarily lost, but can simply re-convert to twofree atoms. We correct for this in the lineshape model,as presented in Sec. II B 2. Despite these differences, thedimer energies determined by these two protocols agreewithin experimental uncertainty. B. Lineshapes and atom loss
In order to fit the spectral lines, we start with an anal-ysis of the transition rate γ from an initial free state | i (cid:105) to a final (quasi-)bound state | f (cid:105) . We assume the role ofatom-dimer coherence is negligible, since the pulses arelong compared to the dimer lifetime. (This is supportedby calculations of the dimer lifetime, shown in Fig. 4 anddiscussed in Sec. III.)To first order, the transition rate is γ = 2 π (cid:126) |(cid:104) f | H (cid:48) | i (cid:105)| , (8)with the perturbing Hamiltonian H (cid:48) that drives the tran-sition. Considering H (cid:48) to correspond to the rf interactionwith Rabi frequency Ω and rotating wave component (cid:126) Ω / γ = h Ω F fi , (9)where we have introduced the Franck-Condon (FC) fac-tor F fi , defined as F fi = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ψ i ( r ) ψ f ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) , (10)with energy-normalized incident wave function ψ i ( r ) liv-ing in the entrance channel and (quasi-)bound-state wavefunction ψ f ( r ) living in the outgoing channel. As out-lined in Sec. II A, the RA protocol does not involve chan-nel transitions and both ψ i ( r ) and ψ f ( r ) correspond towave functions in the | bb (cid:105) channel. For the SFA protocolon the other hand, the entrance state corresponds to the | ab (cid:105) channel, which is coupled to the outgoing state inthe | bb (cid:105) channel.Knowing how to compute the lineshape for both proto-cols, we are now left with the task to relate the formationof (quasi-)bound states to the experimentally observedatom-loss. To ease this procedure, we note the analogybetween the protocols used here and the photoassociation(PA) process as discussed in Ref. [56]. Using the approx-imations outlined in App. A, we find that in the caseof free-to-bound-state transitions, the number of atomsthat remain trapped is N ( ω osc ) = N in − A N P ( T, E k ) F fi , (11)where N in is initial number of trapped atoms, A N is anundetermined proportionality coefficient, P ( T, E k ) rep-resents the thermal distribution of the incoming (free)particles, and E k = − E b + (cid:126) ω osc in the RA case, or E k = − E b − (cid:126) ω osc + (cid:126) ω ab in the SFA case, as discussedabove (Sec. II A). We analyze our data with a Fermi-Dirac distributed P ( T, E k ), and compare to the use of aMaxwell-Boltzmann distribution in Sec. II C.In the case of free-to-quasi-bound transitions, we needto consider off-resonant transitions. For the SFA proto-col, Eq. (11) is replaced with N SFA = N in − A (cid:48) N (cid:90) ∞ k min P ( T, E ) F fi ( k q ) dk q , (12)where E = E q − ( (cid:126) ω osc − (cid:126) ω ab ), E q = (cid:126) k /m , and k min = (cid:112) m ( ω osc − ω ab ) / (cid:126) . For the RA protocol, N RA = N in − A (cid:48) N (cid:90) ∞ k min [ P ( T, E ) − P ( T, E q )] F fi ( k q ) dk q , (13)where E = E q − (cid:126) ω osc and k min = (cid:112) mω osc / (cid:126) , such thatwe have corrected for the zero-transitions in the | bb (cid:105) chan-nel. The implicitly assumed hierarchy of spectral widths is discussed in more detail in App. A. Equations (11)through (13) indicate how the atom loss is directly re-lated to the lineshape. The calculation of the lineshapein the case of free-to-bound and free-to-quasi-bound tran-sitions will be discussed in the following two subsections.
1. Free-to-bound transitions
Starting our analysis of the Franck-Condon (FC) fac-tor for free-to-bound transitions, we consider the over-lap between the radial component of the bound-statewave function ψ f ( r ) = u b ( r ) /r and the scattering state ψ i ( r ) = u sc ( r ) /r . In the asymptotic region, r ≥ r c , withshort-range cutoff r c , the bound state is [7] u b ( r ) = √ r c (cid:18) r + κ (cid:19) e − κr . (14)The radial component of the energy-normalized scatter-ing state in the asymptotic region in turn can be ex-pressed as u sc ( r ) = (cid:114) mπ (cid:126) k [cos( δ ) ˆ ( kr ) + sin( δ ) ˆ n ( kr )] , (15)with k -dependent scattering phase δ and Ricatti-Besselfunctions ˆ and ˆ n . Whereas the above asymptotic wavefunctions do not capture the correct behavior at shortrange, we approximate the rapid and out-of-phase oscil-lations of u sc ( r ) and u b ( r ) that occurs for deep potentialsto lead to a vanishingly small overlap. Consequently, weneglect the short-range contribution to the FC overlap.Substitution of Eqs. (14) and (15) into Eq. (10) resultsin the overlap F fi = mπ (cid:126) e − κ k r c ( k + κ ) (cid:12)(cid:12)(cid:12) kr c κ cos( kr c + δ ) − (cid:2) κ + k (1 + r c κ ) (cid:3) sin( kr c + δ ) (cid:12)(cid:12)(cid:12) . (16)In the experimentally relevant regime, we can simplifyEq. (16) significantly. In our measurements, a typicalcollision energy is 5 kHz, and binding energies range be-tween 14 kHz and 700 kHz. The maximum value of R at the Feshbach resonance is ≈ . r vdW [37, 57, 58],where r vdW ≈ . a for K [59]; for this resonance, R ∼ a , as we discuss later. Since R and r vdW arecomparable, so are the energy scales that correspond tothem, E R and E vdW , both on the 20 MHz scale. (See fur-ther discussion in Sec. III A and Sec. V.) We furthermoreassume that the binding energy of the lowest-lying dimerin the open channel, E (cid:48) b , is set only by van der Waalsphysics. In sum, the typical hierarchy of energy scales inour experiment is E k , | E b | (cid:28) { E R ∼ E vdW ∼ | E (cid:48) b |} (17)Equivalently, there is a separation of length scales andmomenta: k and { κ ∼ (cid:112) R/ | V |} (cid:28) { R − ∼ r − } . (18)These inequalities give us two small parameters: kR (cid:28) (cid:115) R | V | (cid:28) (cid:112) R / | V | was defined as x in the Introduction.A natural scale for the cutoff r c is also set by theshort-range length scale, R ∼ r vdW . The asymptoticwave functions used have neglected the van der Waals C /r potential, but not the centrifugal 1 /r barrier. Theinterplay between these is indicated by the first zero-crossing of the combined effective potential, − C r − +(2 (cid:126) /m ) r − , which is 2 / r vdw . This is, for instance, theinner classical turning point of the centrifugal barrier inthe low-energy limit. Since the resulting lineshape is in-dependent of the precise cutoff chosen, and r vdW is com-parable to R , we choose to fix r c = R from here forward.Applying the small parameters to the computation ofthe overlap resulting from Eqs. (14) and (15), we findthat the FC overlap for a free-to-bound transition can beapproximated as F fi ≈ mπ (cid:126) Rk ( k + κ ) = 1 π E − / R E / k ( E k + E b ) (20)In the high kinetic energy limit E k (cid:29) E b , Eq. (20) scalesas E − / k , in agreement with the scaling law of the p-wave contact and with a numerical prefactor of 2 R , cor-responding to the value of the p-wave contact of a Fesh-bach dimer [6–10]. Using the obtained expression forthe FC overlap and fixing the thermal factor P ( T, E k ) asanalyzed in detail in Sec. II C, we can straightforwardlyapply Eq. (11) in order to extract the binding energy forfree-to-bound transitions for various values of the appliedexternal magnetic field.
2. Free-to-quasi-bound transitions
Once the closed-channel dimer crosses threshold, it ac-quires a finite resonance width. As outlined in App. A,this width has to be taken into account in the computa-tion of the atom-loss, meaning that we need to considerthe possibility of resonant as well as non-resonant transi-tions to the quasi-bound state. Additionally, we requirethe quasi-bound wave function u q ( r ) to be of the formof the scattering wave function as presented in Eq. (15),as the wave function beyond the range of the potentialbarrier behaves as a scattered wave with wave number k q and phase shift δ q . The phase shift of this scattered waveprovides us with the scattering volume and the effectiverange of the | bb (cid:105) channel, such that we can compute theenergy of the positive-energy scattering resonance E asdetailed in Sec. I.Analogously to the method outlined in the previoussection, the implementation of Eq. (10) allows for the for-mulation of an analytic expression of the FC overlap for afree-to-quasi-bound transition as presented in Eq. (B1). In the case of the SFA protocol, we can additionally usethe two small parameters as presented in Eq. (19) in or-der to obtain the simplified expression F fi ( k q ) ≈ m π (cid:126) k sin ( δ q ) k (cid:0) k − k (cid:1) for k (cid:54) = k q . (21)Analogous to the computation of the bound-state atomloss, we can straightforwardly substitute Eqs. (B1) and(21) in Eqs. (12) and (13) respectively and fix P ( T, k )in order to fit the experimental atom-loss and find theresonant energy E . C. Determination of the Feshbach dimer energy
Dimer energies are determined by a fit of spectroscopicdata to Eqs. (11), (12), and (13). We use the MonteCarlo Bootstrap method [60, 61] for fitting and uncer-tainty estimation. In brief, this method works as follows:we randomly sample M times from the M data pointsin each data set with replacement. The resulting collec-tion of M points (with individual data points randomlyomitted or repeated) is fit, yielding best-fit parameters E b or E , N , T , and A N . This procedure is repeated5000 times for bound-state data and 500 for quasi-bounddata (due to constraints on computing time and the in-creased complexity of the quasi-bound fitting function).The strength of the method is that it does not rely upon aprior assumption of a probability distribution. Data setsare excluded when the distribution of best-fit parametersis significantly skewed or non-Gaussian. Otherwise, wetake the median of the resulting parameter distributionsas the overall best-fit parameters, and use 1 σ confidenceintervals for uncertainties.The vertical green bars in Fig. 1 show examples of bind-ing energies determined by this procedure. It is striking,especially for Fig. 1(a) and Fig. 1(e), how far E b /h isfrom the loss peak. The accuracy of the determinationdepends upon the FC factor, which adds a significantasymmetry to the thermally broadened lineshape. Bycomparison, we find that extrapolation of the frequencyof maximum loss at finite E F to zero E F , as used inRef. [41], over-estimates the binding energy by roughly10 kHz in our typical experimental conditions. This em-phasizes the critical role of the lineshape functions foundin Sec. II B.For these fits, we use a collisional factor based on theFermi-Dirac distribution ¯ n ( µ, (cid:15) ), where µ is the chemicalpotential and (cid:15) is the single-particle energy. The proba-bility distribution of relative momentum k averaged overthe inhomogeneous density distribution in the trap is P FD ( k ) = N k (cid:90) d k av (cid:90) d r ¯ n ( µ L , (cid:15) A ) ¯ n ( µ L , (cid:15) B ) (22)where k av is the average momentum, r = ( x ω x + y ω y + z ω z ) / ¯ ω is the rescaled position in the trap, µ L = µ − m ¯ ω r / (cid:15) A,B = M L = 1 M L = 0| |195 196 197 198 199 200 E ne r g y ( k H z ) (a) B (G) B – B (G) -600-400-2002004000 R e s i dua l s ( k H z ) (d) -3 -2 -1 0-10010 (c) R e s i dua l s ( k H z ) -3 -2 -1 0 1-20020 D e v i a t i on ( k H z ) (b) B – B (G) -40 FIG. 2. p-wave dimer energy versus magnetic field. (a) Experimental best-fit results for M L = 0 ( | M L | = 1) are shown inred (blue), with 1 σ error bars. The dashed lines show a linear fit, and the solid lines show coupled-channels calculations of thedimer energy. The shaded region above resonance shows ± Γ /h , where Γ is the calculated energetic width of the scatteringresonance. (b) The deviation from a linear fit clearly shows the bending of the bound state near resonance as explained inSec. III B. (c,d) The residual deviation between the data and the coupled channels calculation. The shaded gray band givesthe energetic width ± Γ /h of the quasi-bound states. Since Γ / → E / E − / R near threshold and the two components havesimilar R , the widths are indistinguishable on this plot. ( (cid:126) / m )( k ± k av ) . Since the energy of the cloud is ro-tationally symmetric in free space, all three M L chan-nels see the same distribution, and one can choose anarbitrary axis for k so long as | k | = k . The lead-ing factor of k accounts for relative-velocity weightingof the the event rate [56]. Here N is a normaliza-tion factor, chosen so that (cid:82) P ( T, E k ) dE k = 1, where P ( T, E k ) dE k = 4 πk P FD ( k ) dk . This treatment takes asemiclassical isotropic limit, which should be valid dueto the large number of fermions in the trap, and rela-tively weak trap. We use P FD distributions generatedat the measured T = 0 . T F ; re-fitting with T = 0 . T F shifts binding energies by less than 1 kHz. By compari-son, a Maxwell-Boltzmann distribution was found to givea ∼ E b / (cid:126) and E / (cid:126) are much larger than the trap frequencies, confine-ment has a negligible effect on the dimer wave functions[62]. Similarly, the discretization of the continuum willaffect the thermal model at the sub-kHz scale, smallerthan statistical uncertainties. Our two-body FC coef-ficient does not take into account possible three-bodyprocesses [1, 42, 63] or many-body correlations. We re- strict our data collection to fields below or (cid:38) δB above and be-low threshold did not lead to the same zero-energy Fesh-bach resonance location. Since our measurements probea wider range of fields, the curvature in binding energyappears clearly. We find good agreement, especially forbound-state energies, with new coupled-channels calcu-lations (shown as black lines and discussed in Sec. III).Allowing for an overall magnetic-field shift in calculatedbinding energies, we find a best-fit − . .
0) mG. Figure2(c) and (d) show residuals of this comparison, with anrms scatter in E b of 4 kHz, comparable to the statisticalerrors of the individual spectra.Figure 2(d) shows increased scatter and a possibletrend in the difference between measured and calculated E . This could be explained in part by heating and po-larization of the cloud during spectroscopy. Quasi-bounddimers that decay through the centrifugal barrier createatoms with a relatively large kinetic energy, which rapidlyheat the cloud. For SFA of quasi-bound atoms, a simi-lar process also spin-polarizes the cloud since | a (cid:105) atomsare irreversibly converted into high energy | b (cid:105) atoms. Wemitigate this effect by fitting just the remaining | a (cid:105) atomsafter our rf pulse [see Fig. 1(g)]. Another systematicmay come from increasing overlap between the M L = 0and | M L | = 1 channels, which restricts the spectroscopicrange. The linewidth of the scattering resonances in-creases roughly as ( B − B ) / , causing the features tooverlap beyond ∼
201 G, as illustrated in Fig. 3(d) anddiscussed in the next section.
III. COUPLED-CHANNELS CALCULATIONS
We carry out standard coupled-channels (CC) calcula-tions [64–66] based on the known atomic matrix elementsof the full spin Hamiltonian [67] and the Σ + g (singlet)and Σ + u (triplet) molecular potentials for the K dimermolecule. We mass-scale the singlet and triplet potentialsof Falke et al. [59] without Born-Oppenheimer correctionsand use the effective spin-dipolar coupling determined byRef. [68]. We use the molscat [69, 70] and bound [71, 72]packages to calculate the needed scattering S -matrix andnear-threshold bound-state energies for two K atoms,as illustrated in Fig. 3. These are calculated without ad-justable parameters. The excellent agreement betweenthe CC predictions and the measured p-wave resonanceis much better than the same comparison for the s-waveresonance (see App. D). This agreement in the p-wavecase gives us confidence in the accuracy of CC predic-tions over wider range of field and energy near threshold.A scattering channel f m f f m f LM L is specified bythe f, m f values of the two atoms that are interactingand the “partial wave” quantum numbers L, M L of theirrelative angular momentum. The total angular momen-tum projection quantum number M tot = m f + m f + M L (23)is conserved, allowing us to block the Hamiltonian ac-cording to the M tot value.Since we are interested in collisions of two | b (cid:105) -stateatoms, and identical fermions only collide with odd L val-ues, the threshold channels of interest are the | bb (cid:105) p-waveones with L = 1 and M L = −
1, 0, or 1, for which M tot = − −
7, and − M tot values inturn give rise to Hamiltonian blocks with 8, 13, and 20 p-wave spin channels for the separated atom states | ij, M L (cid:105) ,where i and j represent the states ( a , b , c , . . . ) consistentwith Eq. (23) for a given M L . Figures 3(a) and 3(b) showthe channel energies and magnetic moments of the fivelowest channels of the M tot = − | bb, (cid:105) channel. The only open p-wave channels for lowcollision energy E/h less than 1 MHz in the 200 G regionare | aa, +1 (cid:105) , | ab, (cid:105) and | bb, − (cid:105) for M tot = − | ab, +1 (cid:105) and | bb, (cid:105) for M tot = −
7, and | bb, +1 (cid:105) for M tot = −
6; allother channels are closed at such low energy (meaningthe channel energy is larger than E ).Bound states below the | bb (cid:105) threshold with M tot = − − B ; however, bound states with M tot = − L = 3 f-wave open channel. Weinclude f-wave basis functions in the CC calculations forthe decay rates, but they are not necessary for the energypositions, which change negligibly (less than 1 kHz) whenf-waves are introduced. For simplicity in the following,when M tot is specified, we suppress the implied M L inthe ket notation. A. van der Waals character of threshold scattering
The CC calculations reveal the spin character of thebound and quasi-bound states of the K dimer. Due tothe relatively low mass of the K atom, these states aresparse near threshold. Furthermore, they are relativelyeasy to understand, although complicated by the spinmixing among the various spin channels. It is easiest firstto sketch out the vibrational and rotational character ofthese states in terms of the long-range van der Waalscharacter of the long-range singlet and triplet potentialwith a leading term that varies as − C /r . This potentialis characterized by the length [73–75]¯ a = 2 π Γ(1 / (cid:18) m r C (cid:126) (cid:19) = 4 π Γ(1 / r vdW (24)and corresponding energy¯ E = (cid:126) m r ¯ a , (25)where m r is the reduced mass, such that ¯ a ≈ . r vdW . Since both atoms are in electronic S / ground states, the coefficient C ≈ . E h a [59] isthe same for all channels, where E h is the Hartree en-ergy, and a is the Bohr radius. For two K atoms,¯ a = 62 . a and ¯ E/h = 23 .
375 MHz.The spectrum of a van der Waals potential gives muchinsight into the states near threshold [49, 76]. Quan-tum defect theory shows that given the s-wave scatteringlength a of a van der Waals potential, the states andscattering properties near threshold of the other partialwaves are also determined [57, 75, 77]. When a (cid:29) ¯ a thebinding energy of the last s-wave bound state is universal, E − = ¯ E/ ( a/ ¯ a ) , and the p-wave phase shift is [78]tan δ L =1 ( k ) = 2 k ¯ V a − ¯ aa − a (26)where ¯ V is the van der Waals volume,¯ V = Γ(1 / π Γ(3 / ¯ a = 4 π / r . (27)For two K atoms, ¯ V ≈ (0 . r vdW ) ≈ (63 . a ) .One sees from Eq. (26) that tan δ L =1 diverges atthreshold when a = 2¯ a . This implies there is a p-wavebound state at E = 0. For a range of a (cid:38) a that bound (c) (d)(b)
120 140 160 180 200 220 B (G) -100-50050100 E / h ( M H z )
194 196 198 200 202 204-1.00.01.0 E / h ( M H z ) B (G) B (G) -2.6-2.4-2.2-2.0-1.8 µ ij / h ( M H z / G ) abbbacadbc-2.8160 180 200 220 240-600-500-400-300 E / h ( M H z ) abbbacbcad (a) M L = 0 narrow res. c.c.“bare” c.c. M L = 0 shape res.region M L | | = 1 FIG. 3. p-wave properties versus magnetic field. (a) Separated-atom energies of the five lowest spin channels of f =9 / , f = 9 / M tot = −
7. (b) The magnetic moments of the five spinchannels of lowest energy. (c) Calculated bound-state energies and scattering over a wide scale of E and B of ±
100 MHz and100 G, where E = 0 represents the separated atoms energy of the | bb (cid:105) channel. The solid red line shows the M L = 0 level thatmakes the 198.8 G Feshbach pole studied in the experiment; the dashed red line approximates the corresponding “bare” closedchannel state. The green line shows the closed channel bound state that makes a narrow resonance near 170 G (see App. C).The color contours above threshold show the loss 1 − | S bb,bb | from the | bb (cid:105) channel; red indicates near unitary maximum lossand blue indicates minimal loss. The dashed black line indicates the region of the ≈ | bb (cid:105) channel.Loss from | bb (cid:105) is almost entirely due to strong decay of the closed channel resonance to the | ac (cid:105) channel. (d) Near-thresholdproperties over a scale of ± M L = 0 (lower) and | M L | = 1 (upper) levels that cross threshold at 198.8 G and 198.3 G respectively. The color contours show the near-thresholdelastic scattering of the three M L components, Eq. (28). state becomes an open-channel “shape resonance”, whichis a quasi-bound state above threshold trapped inside thecentrifugal barrier of the p-wave potential. This leads toenhanced amplitude of the scattering wave function in-side the barrier near the broadened energy of the quasi-bound state, which manifests in the broad loss featureshown in Fig. 3(c). If we approximate the “background”scattering length for the fictitious | bb (cid:105) s-wave channelto be that of the K triplet potential, 169.2a , then a/ ¯ a ≈ . | bb (cid:105) channel tohave such a shape resonance. In fact, the quantum defecttheory predicts a broad maximum in p-wave scatteringamplitude inside the barrier around a collision energy of E/h ≈ µ K. The CC calculations demon-strate that such a p-wave shape resonance actually existsin this region, as indicated by the black dashed line inFig. 3(c). The location and width of the shape reso-nance becomes a key parameter in the model developedin Sec. IV.
B. Near-threshold molecular physics
Since the “last” p-wave bound state in the | bb (cid:105) channelis an above-threshold shape resonance, there are no other | bb (cid:105) levels near threshold. The actual last level with dom-inant | bb (cid:105) character lies around 1.2 GHz below thresh-old. There is a cluster of p-wave components of mixedsinglet-triplet character starting around −
120 MHz near B = 0 and crossing threshold in the 200 G region. Thesolid red line in Fig. 3(c) shows the M tot = − | bb (cid:105) and | ac (cid:105) channelsthrough short range spin-exchange to make the M L = 0p-wave bound and quasi-bound levels studied in this ex-periment. Figure 3(d) shows the calculated energies ofthe M L = 0 and degenerate M L = ± E due totunneling through the p-wave centrifugal barrier. Theselevels are approximately 60% singlet in character, and80% of their norm comes from a mixture of | aq (cid:105) and | br (cid:105) spin channels associated with one ground 9 / / M tot = − | bb (cid:105) channel through spin-dipolarinteractions. It is a M L = −
194 195 196 197 198 199 200 B (G) Г / ħ ( s - ) M L = –1 M L = 0 M = +1 L FIG. 4.
Quasi-bound decay rates versus magneticfield B . Calculated decay widths of the M L = −
1, 0, and 1quasi-bound levels in Fig. 3(d). The M L = ± | bb (cid:105) threshold to | ab (cid:105) or | aa (cid:105) exit channelsthrough dipolar spin relaxation. The decay rate of the M L =+1 level below the | bb (cid:105) threshold is smaller than the M L = − | bb (cid:105) threshold. The level lifetimes are found by τ = 1 / Γ .Dashed lines show decay rates calculated from Eq. (7) usingparameters from the bottom row of Tab. II. the | ap (cid:105) and | cr (cid:105) channels. Both bound levels in Fig. 3(c)have similar slopes at B = 120 G, δµ/h = 1 .
59 MHz / Gfor the lower and δµ/h = 1 .
61 MHz / G for the upper.The upper level is barely curved, having a slope of1 .
51 MHz / G where it crosses threshold near 170.6 G tomake a very weak, narrow p-wave resonance (see App. C).By contrast, the bent lower level has a rapidly decreasingslope as it takes on more | bb (cid:105) character in approachingthreshold, having a value of 0 .
20 MHz / G at 198 G just be-low its threshold crossing. We estimate the approximateposition of the “bare” (or “undressed”) closed channelbound state in the dashed red line of Fig. 3(c) by dis-placing the weakly interacting dashed red line by 2.5 Gto higher B , which is the separation of the two states at120 G. This gives B n ≈
173 G, roughly 26 G below B .The parameter ∆ res = δµ ( B − B n ) is discussed furtherin Sec. IV B.While our experiment concentrates on the bound andquasi-bound levels within less than ± | bb (cid:105) and | ac (cid:105) channels indicated by strong | bb (cid:105) to | ac (cid:105) loss mediated through the resonance, the qual-itative picture in Fig. 3 suggests an “avoided crossing”between this ramping closed channel state and the abovethreshold | bb (cid:105) shape resonance in the | bb (cid:105) channel. The “lower branch” of the “crossing” connects the lower curv-ing bound states Figs. 3(c) and 3(d) with the shape res-onance at high B , whereas the “upper branch” of the“crossing” distorts the shape resonance at low B to joininto a broad above-threshold p-wave Feshbach resonanceof dominant singlet character extending to high E and B .This broad resonance ( ≈
30 MHz) of the upper branchshows up prominently in the sloping broad red contourof unitary loss where 1 − | S bb,bb | ≈ | S bb,ac | ≈ − . [59] lying below thecluster of channels with separated atom spins of 9 / / | bb (cid:105) to the | ac (cid:105) channel below the | ac (cid:105) threshold, which lies approxi-mately 2 MHz above | bb (cid:105) in this region of B , Fig. 3(d)shows the elastic scattering in the near- bb -threshold re-gion shown by the sum0 ≤ (cid:88) M L = − | − S bb,bb ( E, B ; M L ) | ≤ . (28)In the limit of no loss (see Sec. IV), | − S bb,bb | =sin δ bb . Just above threshold, the M L = 0 bound stateand the two degenerate M L = ± τ = Γ − (cid:46) . M L = −
1, 12 ms for M L = 0, and 8 . M L = +1. These lifetimes areroughly 30% higher than the original predictions madein Ref. [41], primarily due to the inclusion of an effectivespin coupling from Ref. [68], which results in a smaller de-cay rate and consequently longer lifetime. The M L = 0lifetime is also consistent with the experimental lowerbound given in Ref. [80].Figure 5 shows elastic and inelastic scattering proper-ties versus E for cuts at constant B . Both cuts showthat the Feshbach resonances below the | ac (cid:105) thresholdfeature prominently in the unitary peak in elastic scat-tering. The shape resonance shows up prominently ininelastic loss, with a rapid onset versus energy when the | ac (cid:105) channel opens near 2.5 MHz in Fig. 5(b). The logplot in panel (a) shows that the weak dipolar loss from | bb (cid:105) to the | ab (cid:105) channel also mirrors the Feshbach peak;a similar loss to | ab (cid:105) would show up in a log plot of panel(b), but approximately one hundred times smaller in peakmagnitude and much broader in resonance width due tothe hundred-fold larger width at 210 G as compared to199.3 G. The larger width is due to much faster tunnelingthrough the centrifugal barrier.0 E / h (MHz) 1 – | S bb,bb | E / h (MHz) S bb,bb | ac threshold p-waveshape resonanceFeshbachresonance (a)(b) -5 -4 -3 -2 -1 |1 – S bb,bb | |1 – S bb,bb |
41_ 22
FIG. 5. p-wave scattering versus collision energy E . (a)At 199.3 G, the elastic scattering ( | − S bb,bb | , dashed line)and inelastic loss (1 − | S bb,bb | , solid line) on a log scale versuscollision energy for the the relatively narrow resonance with M tot = − M L = 0 at E/h = 87 . /h =8 . | ab (cid:105) channel shows a peak atthe same location as the unitary elastic scattering. (b) Similarfigure for the very broad resonances at 210 G. While the broadFeshbach resonance near 1.5 MHz with a width of 0.75 MHzshows up prominently in elastic scattering, the p-wave shaperesonance is prominent in the strong unitary inelastic lossfrom the | bb (cid:105) channel to the | ac (cid:105) channel. IV. MODEL FOR P-WAVE SCATTERING
The coupled-channels analysis has allowed us to delin-eate an experimentally relevant regime of energies andfields in which | bb (cid:105) is quite weakly coupled to loss chan-nels, i.e. the limit in which the diagonal | S bb,bb | → S -matrix, S = S P S FB S dip . (29) In other words, this breaks the scattering phase into threecontributions, δ P + δ FB + δ dip . Our model is elastic, withreal δ and unitary S for each component.In the following section, we derive simple analytic ex-pressions that allow us to parameterize the field andenergy dependence of the p-wave scattering phase. Al-though we fit the model parameters to the CC results for K, it should also provide a useful framework to under-stand elastic p-wave scattering in Li or any other coldgas.
A. Open-channel shape resonance
With the | bb (cid:105) channel taking the role of open channel,we need to carefully incorporate the effect of the near-threshold p-wave shape resonance. As seen in Fig. 5(b),this 7 MHz resonance is broad and asymmetric, in partbecause it lies near the top of the p-wave centrifugal bar-rier, and in part because of its strong coupling to the | ac (cid:105) channel. Our model aims to capture only its effecton low-energy scattering, which is still essential to un-derstand the energy dependence of the scattering phase,as illustrated in Fig. 6(a).Inspired by the general form of Ref. [81], we describethe open-channel resonance as S P = e − ikr ( k − k ∗ s )( k + k s )( k − k s )( k + k ∗ s ) , (30)where k s = k R + ik I and − k ∗ s are the locations of theshape-resonance poles in the fourth and third quadrantsof the complex k -plane ( k I < k s , this form satisfies S ∗ P S P = 1, S P ( − k ) = S ∗ P ( k ).The background term exp( − ikr ) allows for an essen-tial singularity at infinity and takes into account theeffect of short range physics. We naturally expect r to be on the order of r vdW . However, if we constrain S P to follow the low-energy p-wave threshold law, then r = 2 Im { /k s } = − k I / | k s | . One can then match the k → S P to find a scattering volume V P = − r | k s | − (cid:0) − | r k s | / (cid:1) (31)and an effective range R P = r (1 − | r k s | / − | r k s | + | r k s | / V P ≈ − r | k s | − and R P ≈ r for | k I | (cid:28) k R ,as expected for a narrow low-energy resonance [82]. Asimilar form of the S -matrix is found in p-wave scatteringfrom a square-well potential whose range is r [83]. Thus S P = e k/k ∗ s e k/k s ( k − k ∗ s )( k + k s )( k − k s )( k + k ∗ s ) . (33)1 B. Feshbach Resonance
Generally, the energy of ultracold collisions in the openchannel is much smaller than the asymptotic energy ofthe closed channel to which the open channel is coupled.In addition, the bound states in the closed channel aretypically spaced such that only a single bound state af-fects the open-channel state. Therefore, we can approx-imate the closed channel as a single bound state withbare energy (cid:15) Q [84] that crosses the energy threshold at B = B n , and with a magnetic moment δµ relative to theopen channel, such that (cid:15) Q = δµ ( B − B n ).Under this assumption, we can exploit the Feshbachformalism [85–87] to write a two-channel model: S FB = 1 − i Γ( E ) E − δµ ( B − B n ) − A ( E ) . (34)Here A ( E ) represents the complex-energy shift of thebare bound state | φ b (cid:105) , A ( E ) = (cid:104) φ b | H QP E − H PP H PQ | φ b (cid:105) , (35)where we have introduced the coupling matrix strengths H QP = H ∗ PQ and the open-channel Hamiltonian H PP .By inserting a complete set of eigenstates for the open-channel Hamiltonian, we can decompose Eq. (35) as A ( E ) = ∆ res ( E ) − i E ) , (36)where the real part ∆ res ( E ) corresponds to the real en-ergy shift of the bound state (estimated to be 26 G inSec. III), and Γ( E ) adds a width to the Feshbach reso-nance. Since the propagator ( E − H P P ) − in Eq. (35)shares its poles with the open-channel S -matrix S P as in-troduced in Eq. (33), we expect the complex-energy shift A ( E ) to be well described in terms of a Mittag-Leffer se-ries which runs over the poles of S P [88–90], such that(suppressing factors of (cid:126) /m for now) A ( E ) = (cid:104) φ Q | H QP | Ω s (cid:105) (cid:104) Ω Ds | H PQ | φ Q (cid:105) k s ( k − k s ) − (cid:104) φ Q | H QP | Ω Ds (cid:105) (cid:104) Ω s | H PQ | φ Q (cid:105) k ∗ s ( k + k ∗ s ) , (37)where we have introduced the Gamow state | Ω s (cid:105) , as wellas its dual state | Ω Ds (cid:105) ≡ | Ω s (cid:105) ∗ . Gamow states corre-spond to eigenstates of the open-channel Schr¨odingerequation with purely outgoing boundary conditions [91].Together with their dual states, Gamow states form abiorthogonal set, such that (cid:104) Ω s | Ω Ds (cid:48) (cid:105) = δ s , s (cid:48) [89, 92]. Since | Ω s (cid:105) is an eigenstate of H PP with eigenvalue E s and | Ω Ds (cid:105) is an eigenstate of H † PP with eigenvalue E ∗ s , thesestates follow the typical low-energy p-wave threshold be-havior as implied by Wigner’s threshold law [93]. Forenergy-normalized states, this implies that Ω s ∝ k / s and Ω Ds ∝ ( k / s ) ∗ , such that we can approximate the matrix-element (cid:104) φ Q | H QP | Ω s (cid:105) (cid:104) Ω Ds | H PQ | φ Q (cid:105) as g (cid:48) k s , withmomentum-independent coupling parameter g (cid:48) . Substi-tuting this approximation into Eq. (37) and separat-ing the real and imaginary contributions according toEq. (36), we find that∆ res ( E ) ≈ g (cid:48) Re (cid:40) E / s E − E s (cid:41) (38)and Γ( E ) ≈ − g (cid:48) E / | E − E s | Im { E s } . (39)Since the pole of the shape resonance is independent ofthe external magnetic field, the full magnetic-field depen-dence of ∆ res ( E ) and Γ( E ) is contained in the single pa-rameter g (cid:48) . Retaining only the lowest-order momentumdependence of the previous two expressions in the near-resonant regime, we find that ∆ res ( E ) ≈ ∆ res (0) + O ( k )and Γ( E ) ≈ gk + O ( k ), where ∆ res (0) = − g (cid:48) k R and g = 4 g (cid:48) k I k R ( k + k ) − only depend on the complexmomentum k s and the parameter g (cid:48) . Keeping only thelowest-energy terms, Eq. (34) can be conveniently recastinto the following simplified form around resonance S FB ≈ − igk E − δµ ( B − B n ) − ∆ res (0) + i gk . (40)We see that the momentum dependence of S FB with thisapproximation follows the threshold scaling of Eq. (2),with effective range R FB = m (cid:126) g V FB = − g/ δµ ( B − B n ) + ∆ res (0) . (42)At B = B , there is a resonance in V FB . In addition, since g is B-field-dependent, Eq. (42) also has a backgroundterm. Keeping only the linear variation g ≈ g (1 + δB/ ∆ g ), one recovers Eq. (3), V FB = V bg (1 − ∆ /δB ),with V − ≈ − m ∆ g δµ (cid:126) R − | k s | r and ∆ ≈ − ∆ g (43)where here R is the value of R FB at B = B .When fitting the combined S P S FB to coupled-channelsdata in Sec. V, we allow ∆ res and Γ to have independentvariations with magnetic field. Relaxing this constraintcan capture weak coupling to other channels with correc-tions to the positions of the poles of the S-matrix. Wewrite this as S P S FB = e k/k ∗ s ( k − k ∗ s )( k + k s ) e k/k s ( k − k s )( k + k ∗ s ) E − c − i gk E − c + i gk , (44)2 Collision energy (MHz) R e ( k c o t δ ) ( a ) - - CC calculationERA (b)(a)
FIG. 6.
Scattering at high and low energy for M L = +1at 200 G. The phase shift δ from CC calculations is plottedas the real part of k cot δ (blue solid line) versus collisionalenergy (cid:126) k /m . (a) k cot δ becomes nonlinear at higher col-lision energies, showing the necessity of including the shaperesonance in the experimentally relevant regime. The best-fit model S (blue squares) captures the deviation from theeffective-range approximation (black dashed line) up to sev-eral MHz. (b) At low energy, k cot δ has an apparent diver-gence due to the dipole-dipole interactions, as explained inthe text. S dip captures this effect with no fitting parameters. where δµ ( B − B n ) + ∆ res (0) has been replaced by thefitting parameter c ( B ), and E = (cid:126) k /m . This is the p-wave analogue of the dual-resonant S -matrix presentedin Ref. [84] for s-waves.However, even this parameterized two-channel modelwill break down if we move sufficiently far away fromresonance. As discussed in Sec. III, especially the cou-pling to the | ac (cid:105) channel becomes increasingly importantand we expect the need to update Eq. (44) to (at least)a three-channel model in order to accurately model theCC data if we are further away from resonance.Figure 6(a) compares S P S FB to CC calculations. TheERA captures only the linear variation of k cot δ withscattering energy, and requires a significant correctionon the MHz scale [94]. Using k s as a free parameter, ourmodel captures the effect of the shape resonance out toseveral MHz. The results from fitting for a variety ofmagnetic fields near resonance are discussed further inSec. V. C. Dipole-dipole interaction
A third contribution to scattering is the long-rangedipole-dipole interaction (DDI). As shown in Fig. 6(b),the DDI dominates at low energy, causing a divergence inthe scattering volume and invalidating the ERA [95, 96].However, the DDI phase shift δ dip itself is small and iswell treated by the Born approximation. In the limit k → δ L,M L dip ≈ − π (cid:104) ijLM L | V ( r ) | ijLM L (cid:105) , (45)where we have introduced the dipole-dipole potential V ( r ) acting on a channel-state | ijL (cid:105) , with atoms in theinternal states | i (cid:105) and | j (cid:105) interacting with relative angu-lar momentum L and partial-wave projection M L . Con-sidering an external magnetic field oriented along the z -direction, we can express the dipole-dipole potential V ( r ) as V ( r ) = − d − θr = − d Y r , (46)where d is the magnetic dipole moment and Y isa Racah-normalized spherical harmonic. SubstitutingEq. (46) into Eq. (45), one finds δ L,M L dip ≈ πmd (cid:126) (cid:104) LM L | Y | LM L (cid:105) (cid:90) ∞ J L +1 / ( kr ) r dr, (47)where J L +1 / ( kr ) represents the Bessel function of thefirst kind. For L = 1, we then obtain δ ,M L dip ≈ − a ,M L dip k (48)and S dip ≈ exp (cid:16) iδ ,M L dip (cid:17) , where a , = − md (cid:126) and a , ± = md (cid:126) . (49)The same linear-in- k scaling is found for all L ≥ δ L,M L dip = − ( m r C L,M L k/ (cid:126) ) / ( L + L ) for scattering eventswith reduced mass m r from a potential V = C L,M L /r [97]. The characteristic length scale has been variouslydefined as D ∗ = md / (cid:126) [98, 99], or a d = m r C L,M L / (cid:126) [100], which differ only by numerical factors calculatedas in Eq. (47).In the | bb (cid:105) channel, at B = 198 . d/µ B = − . a , = − . a and a , ± = 0 . a .The phase shift is small, and furthermore cancels whensummed over all three M L channels. For these reasons, δ dip has been neglected in previous discussions of p-wavescattering of ultracold alkali atoms [57, 94], althoughit is this term that permits the evaporative cooling ofspin-polarized Fermi gases in strongly dipolar species[101, 102].However, since the M L = 0 is well separated from the M L = ± K , δ dip does not vanish, and infact becomes the leading phase shift at sufficiently lowenergy, as shown in Fig. 6. In the low- k limit, thresh-old scaling would give δ → − V k , which vanishes fasterthan δ dip → a ,M L dip k . One sees that the threshold law isinvalidated, and instead for k (cid:28) | a ,M L dip /V | ,cot δ → − a ,M L dip k + V ( a ,M L dip ) k + O ( k ) . (50)3 Re k s . /r vdW Im k s − . /r vdW Re E s .
295 ¯ E Im E s / − .
038 ¯ Er . r vdW V P − .
39 ¯ V TABLE I.
Shape resonance parameterization.
The best-fit location of the poles of S P are k s and − k ∗ s , with values givenabove in terms of the van der Waals length r vdW = 65 . a .Also given for reference are the pole locations in the complexenergy plane, E s and E ∗ s , in terms of the van der Waals energy¯ E/h = 23 .
375 MHz. The associated r and V P are also givenin van der Waals units. With the usual V = − lim k → tan δ/k , one would find adivergent V ( k ) = a ,M L dip /k .We see that a ,M L dip /V determines a collision energy E ,M L dip = − (cid:126) a ,M L dip mV , (51)at which there is a zero-crossing in the scattering phasewhen E ,M L dip >
0. Since | E ,M L dip | (cid:28) E F , this appearsas a low-energy divergence in cot δ [seen in Fig. 6(b)]for either M L state. For the background scatteringlengths (see Sec. V), E , /h ≈ −
11 kHz and E , ± /h ≈ E ,M L dip /E = a ,M L dip /R , which is roughly − . . M L = 0 and | M L | = 1, respectively.In analyses of the strong elastic scattering near thep-wave Feshbach resonance, this weak dipolar effect ismerely a nuisance that causes a low-energy divergence incot δ . We remedy this by subtracting its contribution, fit-ting S P S FB to S − S for the energy- and field-dependent S from CC calculations. In general, we find that theBorn approximation captures the DDI phase shift quan-titatively below 10 microkelvin [see Fig. 6(b)], with nofitting parameters. We can then discuss the low-energylimit of the reduced phase shift, δ − δ dip , in terms of a welldefined scattering volume and effective range, recoveringEq. (1).However outside of the resonant regime, the dipolarterm can dominate. For instance | E , | /k B (cid:38) µ K acrossa 10-gauss range around the zero-crossing of V . Also,within molecular orbitals of the closed channel, the samedipolar interaction physics is responsible for the split-ting of the Feshbach resonance into distinct M L = 0 and | M L | = 1 features [51]. V. FIELD DEPENDENCE OF THESCATTERING PARAMETERS
Having understood both low- and high-energy depen-dence of the scattering phase, we can now fit our model B (G) V bg ( a ) ∆ (G) R ( a ) Source M L = 0198.85 -(101.6) -21.95 47.2 [51]198.81(5) [41]198.79(1) [8]198.796(1)(5) exp198.803 -(108.0) -19.89 49.4 th | M L | = 1198.37 -(96.7) -24.99 46.2 [51]198.30(2) [41]198.30(1) [8]198.293(1)(5) exp198.300 -(107.35) -19.54 48.9 thTABLE II. The K p-wave Feshbach resonance param-eters.
Comparison of experimental determinations of reso-nance location B , background scattering length V bg , width∆, and on-resonant effective range R . The results fromSec. II C are listed as “exp”; the fits discussed in Sec. V arelisted as “th”. When two uncertainties are listed, the first isstatistical, and the second is systematic. to the scattering phase found by coupled-channels calcu-lations. Using Eq. (44), the variables k s , c , and g are fitto numerically generated S − S across a range of collisionenergy and fields. We find that both g and c are approx-imately linear in magnetic field, while k s can remain ata fixed, field-independent value across the range of inter-est (see Tab. I). This gives the collisional energy of theshape resonance as Re E s /h = 6 .
88 MHz, and its effectivewidth − Im E s /h = 0 .
89 MHz. The remaining degreesof freedom in the model are constrained by lower-energyscattering, and are represented in Fig. 7 and Tab. II interms of the ERA parameters.When combining two p-wave S -matrices with well de-fined scattering volumes V and V , and effective ranges R and R , the resulting scatting matrix has V = V + V and 1 R = V R V + V R V . (52)The scattering volume of S P S FB is therefore V ( B ) = V FB ( B ) + V P , (53)where V FB ( B ) = − g ( B ) /c ( B ) and V P is given byEq. (31). As discussed in Sec. IV B, the first term createsthe resonance of Eq. (3) and contributes a backgrounddue to the field dependence of g . The second term is in-dependent of magnetic field, and here contributes ∼ − level, shown in the inset of Fig. 7(a),4 R vdW B − B (G) CC calculationCC calc. ζ = −1.85linear E / h ( k H z ) Eq. (3) B (G) -0.50.00.51.52.0-2.0-1.5-1.01.0 30405070800102060 -2000-1500-10000500-500 -505 4-202 -8 -6 -4 -2 0 2 4 6 8 10 12 V ( a ) -5 0 5 10 -5 0 5 10 B − B (G) -8 -6 -4 -2 0 2 4 6 8 192 194 196 198 200 202 R ( a V ( a ) - - - R e s i dua l ( a ) (c) (d)(a) (b) ) M = 1 M L = 0| | CC calculationERA2 ch. model L FIG. 7.
Effective range parameters. (a) Scattering volume V and (b) inverse scattering volume V − versus field-detuningfrom from Feshbach resonance. Blue squares are fits to the CC phase shift versus energy at each magnetic field; the black lineis Eq. (3) with values given in Tab. II. The inset shows the fit residuals. (c) The effective range R versus field-detuning fromresonance. Values determined through fits to CC phase shift versus energy (blue squares) are compared to the van der Waalslimit of a broad resonance (black dashed and Eq. (55)), a single-parameter fit with the width parameter ζ (black solid lineand Eq. (56)), and a linear fit (green dashed). Panels (a) and (b) show the | M L | = 1 channel, but apart from a shift in B ,the three channels have a similar behavior. (d) Dashed lines show bound-state poles from Eq. (4) below resonance, and thescattering resonance (6) above resonance, using the parameterization of Eq. (3) and Eq. (56), with values given in Tab. II. Thepoints (open squares) are the poles of S FB . They are both compared to the CC binding energies (solid lines). where the uncertainties correspond to the difference be-tween fitting to the real and imaginary components ofthe S -matrix. We bound the residual field-dependentcorrections to Eq. (3) by looking at symmetric and an-tisymmetric combinations of V ( B ) around B . For in-stance, V sym ( x ) = V ( B + x ) / V ( B − x ) /
2, whichshould give an x -independent V sym ( x ) = V bg if Eq. (3)is exact, or reveal corrections even in δB . Similarly, V as ( x ) = V ( B + x ) / − V ( B − x ) / x -independent V as ( x ) = V bg ∆ /x . Here, we find a resid-ual background slope of (cid:46) a / G across the plotted ± − precision of the numericaldetermination. In sum, Eq. (3) is an excellent parame-terization of the CC-determined V ( B ) near the p-waveFeshbach resonances of K.The best-fit values of B , V bg , and ∆ are given in Ta-ble II. The accuracy of the resonance positions, ± V bg estimatedby using the triplet scattering length with Eq. (26) is ≈ − .
77 ¯ V , and here we find V bg / ¯ V ≈ − .
84. The reso-nance width ∆ predicts a zero-crossing of V at B − ∆ ≈
179 G. However, as discussed in Sec. IV C, dipolar physicswill dominate for ultracold collisions with small V , suchthat the ERA will no longer be valid. As pointed out byRefs. [57, 58, 94], the long-range nature of the van derWaals potential also becomes relevant within one gaussof the zero-crossing.The effective range of S P S FB is given by Eq. (52), andplotted in Figure 7(c) for | M L | = 1 versus field. A linearfit to R ( B ) = R (1 + δB/ ∆ R ) gives ∆ R = 21 . R = 21 . | M L | = 0 and 1 respectively, and isshown as a green dashed line in Fig. 7(c). As anticipatedby Eq. (43), ∆ R ≈ − ∆, since both are due primarily tothe field-dependence of the coupling g .Insight into the relationship between R ( B ) and V ( B )can be gained by considering a p-wave Feshbach reso-nance with B -field independent R FB , = mg / (2 (cid:126) ), suchthat V FB , = − g / (2 δµ δB ). The background scatteringvolume will then be solely due to the open channel, with V bg and R bg . Using Eq. (52), one finds that the scatteringvolume, V bg + V FB , , now matches the form of Eq. (3) with∆ = g / (2 δµ V bg ). In the limit V /R bg (cid:28) V , /R FB , ,the effective range is controlled by S FB , and is R ≈ R FB , (cid:18) − δB ∆ (cid:19) = R FB , (cid:18) − V bg V ( B ) (cid:19) − . (54)A similar form is found [103] in the s-wave case for nar-row resonances, where the effective range r e scales withscattering length a as (1 − a bg /a ( B )) . For a resonanceof mixed open- and closed-channel character, more gener-ally it has been found that a quadratic variation of r e a across the resonance is widely applicable to resonancesof various widths and partial waves [104]. Equation (54)shows the limit in which similar functional coupling of R ( V ) arises from a two-channel p-wave model. The ef-fective range on resonance is not expected to be universal,but can be taken as a fit parameter to experimental dataor a CC calculation.In the opposite limit, when the effective range is con-trolled by the open channel, one would expect R to ap-5proach the van der Waals limit, R vdW , where [105] R vdW ( B ) = R max (cid:18) VV ( B ) + ¯ V V ( B ) (cid:19) − . (55)Here R max ≡ V / (4 r ) and ¯ V is given by Eq. (27). Inthe case of K, R max ≈ . r vdW ≈ a is the maxi-mum on-resonance value of the effective range allowed bycausality [37, 58, 106] and the long-range van der Waalstail.One can define a dimensionless parameter ζ that char-acterizes the strength of the resonance by interpolatingbetween Eqs. (55) and (54) in the broad | ζ | (cid:29) | ζ | (cid:28) R is given by R − ( B ) = − R − ζ (cid:18) − V bg V ( B ) (cid:19) + R − ( B ) , (56)in which − ζ = R R max − R = mV bg ∆ δµ (cid:126) R max = 0 . V bg ∆ δµ ¯ V ¯ E . (57)Here the factor 0 . ) / (20 π ), ∆ is the reso-nance width defined in Eq. (3), and δµ is the effective dif-ferential magnetic moment between the open and closedchannels [109]. This dimensionless parameter serves asa p-wave analog of the similar dimensionless resonancestrength parameter s res = a bg ∆ δµ/ (¯ a ¯ E ) introduced byRef. [49] for s-wave resonances [110]. Figure 7(d) showsthat this single-parameter fit can match both the reso-nant value of R and its linear slope R / ∆ R , where fromEq. (56), ∆ R = − ∆ / [2 − R /R max )( ¯ V /V bg )].The best-fit values are ζ ≈ {− . , − . } for the M L = 0 and | M L | = 1 channels, respectively. Since | ζ | > ζ ≈ − . × − for the narrow res-onance at 170 G discussed in App. C, ζ ≈ − . Li p-wave Feshbach resonance[2, 20, 22, 23, 42–45], and ζ as large as −
560 for bosonic Rb/ Rb mixtures [108].To be clear, even in the limit | ζ | → ∞ , however, ultra-cold p-wave scattering is energetically narrow: Γ /E → E →
0, unlike s-wave scattering but similar to any L ≥ S -matrix can be found directly from S FB , instead of theERA parameters. The pole below resonance can be foundby solving E b − c + igk / E ≈ c is a goodapproximation of the scattering resonance. As shown inFig. 7(d), isolating this term in the two-channel modelmatches the ERA analytic result at very low energy ( E (cid:28) E s , for which S P can be taken to be unity) but the two-channel model is required to capture fully the bending ofthe bound state further from threshold. This is evidentat energies (cid:38) | ac (cid:105) channel (seeFig. 5). However, low-energy scatting remains well de-scribed by the ERA parameters of the S -matrix across awider range, as shown in Figs. 7(a,b). VI. CONCLUSION
We have used experimental and theoretical tools toprobe the physics of ultracold p-wave collisions and near-threshold Feshbach dimer states.A salient new phenomenon is the “bending” of thedimer energy, seen as a deviation from field-linearity in E b (below resonance) and E (above resonance). Theclear observation of this nonlinearity is enabled by newanalytic lineshape functions, resulting in excellent agree-ment with coupled-channels calculations, at a surprisingmilligauss scale. We explain, through both numerical andanalytical models, that the origin of the curvature is thestrong coupling between the ramping closed channel andthe open channel at this Feshbach resonance. This pro-vides a satisfying resolution to the discrepancy found inpioneering spectroscopy of the same resonance [41].Coupled channels calculations based on the full Hamil-tonian proves to be an effective and accurate tool inunderstanding the collision physics of this dimer sys-tem. Calculations beyond the range of experimental mea-surements illustrate the complexity of collisions near theshape resonance, for which coupling between channelsreach their unitary limit in this system. However, forultracold (here, sub-MHz) collisions, cross-channel cou-plings become small and the physics is dominated byelastic scattering. In this regime we develop an ana-lytic treatment based on a three-component S -matrix,which includes the effect of the p-wave Feshbach res-onance, the open-channel p-wave shape resonance, andlong-range dipole-dipole interactions.The dipolar phase shift causes the scattering volumeto diverge at low energy due to its long-range character.This is strikingly different from the s-wave case. Howeversince the absolute phase shift is small, it can be treatedanalytically, allowing us to isolate its contribution to the S -matrix and recover the conventional threshold scalingof scattering from the short-range component of the po-tential.Applied to the K p-wave resonances near 198.5 G,we fit our model to the low-energy scattering phase gen-6erated by coupled-channels calculations. This allows animproved parametrization of the resonances, includinglocation and effective range parameters ( V and R ), aswell as an effective shape-resonance location ( k s ), and aparameterization of the Feshbach pole ( c and g ).Several aspects of the collision physics can be under-stood approximately in terms of the open-channel van derWaals potential and quantum defect theory: the energyof the p-wave shape resonance, the background scatter-ing volume (26), the relations (38) and (39) between thechannel coupling and the complex-energy shift of the barebound state, and the broad-resonance limit of the effec-tive range (55). The comparison between the inferred R at resonance and its maximum causality-limited valueleads to a quantification of the “width” of the p-waveresonance.Our work sets the stage for exploration in several ways.We have benchmarked coupled-channels calculations fortwo-body p-wave states in the continuum, building confi-dence in their extension to three-body states or strongly confined geometries. These calculations also make an up-dated prediction for the lifetime of the p-wave Feshbachdimers, relevant to feasibility of p-wave Fermi liquids[111]. We have a better understanding of the limitationsof the ERA, which is relevant to theoretical predictionsbased on this parameterization of the phase. ACKNOWLEDGMENTS
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As mentioned in Sec. II B, we exploit the analogy be-tween the PA protocol as discussed in Ref. [56] and theSFA and RA protocols in order to find a relation forthe atom-loss. Reference [56] used a field-dressed scat-tering formalism to obtain the two-body rate constant K loss for atom loss driven on-resonance by an optical fre-quency laser. This formalism is readily adapted to ourcase where the field-dressed states are coupled by a low-frequency oscillatory magnetic field instead of an opticalfrequency laser [112, 113]. We find that the number ofatoms lost due to a near-resonant pulse of such radiationis N loss = a N (cid:90) P ( T, E ) (cid:126) γ d γ ( E )( E − E k ) + ( (cid:126) γ tot ( E ) / dE, (A1)with a scaling factor a N that depends on experimentaldetails (e.g., the time of the association pulse and thespatial distribution of the atoms in the trap) and a ther-mal factor P ( T, E ). The necessity to average over a ther-mal distribution arises from the non-zero temperature ofatoms in the the experimental set-up and will be dis-cussed in detail in Sec. II C. We furthermore recognizethe presence of three different rate constants γ ( E ), γ d and γ tot ( E ) in Eq. (A1). Here γ d represents the decay rate ofthe (quasi-)bound state due to all 1-, 2-, and 3-body de-cay processes that result in trap-loss, whereas γ ( E ) repre-sents the stimulated emission rate of (quasi-)bound-stateatoms as presented by Eq. (8) back to the initial state | i (cid:105) .The total decay rate γ tot ( E ) = γ ( E ) + γ d + γ oth , where γ oth represents any remaining part of the total decay ratethat does not correspond to stimulated emission and thatdoes not result in trap-loss.As previously discussed in Sec. II A, we are workingwith experimental temperatures ranging from 0 . µ K to0 . µ K. This means that the thermal factor P ( T, E ) inEq. (A1) will have an energetic width in the order of k B T /h ∼
10 kHz. For free-to-bound transitions, thisthermal width k B T is much larger than the decay width,such that k B T / (cid:126) (cid:29) γ tot ( E ). Consequently, the lineshapein Eq. (A1) looks like a delta-function centered at E k withrespect to the thermal spread in the energy, such that wecan integrate over the Lorentzian and find [114] N loss = A N P ( T, E k ) γ d γ ( E k ) γ tot ( E k ) , (A2)where we have introduced the new scaling constant A N and where the energy is now fixed to E k defined by E b and (cid:126) ω osc . In both the SFA and RA protocol, we as-sume that the decay rate γ d of the (quasi-)bound statethat leads to atom-loss is much larger than the rate ofstimulated emission γ ( E ) and the non-loss-related decayrate γ oth . This implies that the total decay rate can beapproximated as γ tot ≈ γ d , such that Eq. (A2) reduces to N loss = A N P ( T, E k ) γ ( E k ) , (A3)and the total number of atoms N = N in − N loss can becalculated in accordance with Eq. (11) as presented inthe main text.Whereas we can still approximate γ tot ≈ γ d in the caseof free-to-quasi-bound transitions, we are no longer inthe regime where the limit k B T / (cid:126) (cid:29) γ d applies. As ana-lyzed in Sec. III B, the width of the bound state rapidlyincreases once the energy threshold is crossed. Conse-quently, the lineshape can no longer be regarded to benarrow with respect to the thermal distribution and wehave to consider off-resonant transitions to contribute tothe atom-loss. When γ d ≈ Γ > k B T / (cid:126) , the width termin the denominator exceeds the energy term in the lowenergy regime where the thermal factor is large, the vari-ation with E in the denominator is weak, and Eq. (A1)can be approximated by a form similar to Eq. (A3), N loss = A (cid:48) N (cid:90) P ( T, E ) γ ( E ) dE, (A4)where A (cid:48) N once more represents an energy-independentconstant and where the total number of atoms N can nowbe calculated in accordance with Eq. (12) as presentedin the main text. Appendix B: Franck-Condon factor for quasi-boundtransitions
As outlined in Sec. II B 1, we neglect the short-rangecontribution to the overlap by cutting off all integrals atthe length scale r c . Beyond this, we assume that theasymptotic wave functions can be used to compute theFranck-Condon overlap.For quasi-bound transitions, we find that F fi ( k q ) = m π (cid:126) kk q (cid:12)(cid:12)(cid:12) cos( δ ) cos( δ q )I + sin( δ ) sin( δ q )II+ cos( δ ) sin( δ q )III + sin( δ ) cos( δ q )IV (cid:12)(cid:12)(cid:12) , (B1)where the integrals I-IV areI ≡ (cid:90) ∞ r c ˆ ( k q r )ˆ ( kr ) dr = π δ ( k − k q ) (B2)+ k sin( kr c )ˆ ( k q r c ) − k q sin( k q r c )ˆ ( kr c ) k − k II ≡ (cid:90) ∞ r c ˆ n ( k q r )ˆ n ( kr ) dr = π δ ( k − k q ) (B3)+ k cos( kr c )ˆ n ( k q r c ) − k q cos( k q r c )ˆ n ( kr c ) k − k ≡ (cid:90) ∞ r c ˆ ( kr )ˆ n ( k q r ) dr = (B4) k sin( kr c )ˆ n ( k q r c ) − k q cos( k q r c )ˆ ( kr c ) k − k IV ≡ (cid:90) ∞ r c ˆ n ( kr )ˆ ( k q r ) dr = (B5) k cos( kr c )ˆ ( k q r c ) − k q sin( k q r c )ˆ n ( kr c ) k − k Here we have used Ref. [115] and the recursion rela-tions ˆ (cid:96) +1 ( z ) = (1 + 2 (cid:96) )ˆ (cid:96) ( z ) − ˆ (cid:96) − ( z ), and ˆ n (cid:96) +1 ( z ) =(1 + 2 (cid:96) )ˆ n (cid:96) ( z ) − ˆ n (cid:96) − ( z ). As outlined in Sec. II B 2,Eq. (B1) reduces to the simplified expression Eq. (21)if the parameters in the SFA experiment satisfy Eq. (19).This reduces the computational time of the fitting to theexperimental data significantly. Appendix C: Narrow p-wave resonance at 170 G
Figure 8 expands the region of the narrow resonancenear 171 G in Figure 3(c) to illustrate how the very weakresonance differs dramatically from the much strongerone near 198 G studied in this paper. The spin proper-ties of the bound state making this narrow resonance areexplained in Sec. III B. The contour plot above thresh-old shows how this resonance shows up in the weak lossgiven by | S bb,ab | (cid:28)
1. It is clear from the Figure thatthe quasi-bound state for
E > B as it crosses the broad horizontal shape- -5-5 narrow res.closed channel M L = 0 B (G) E / h ( M H z ) FIG. 8.
Bound and scattering properties for the nar-row p-wave resonance with M tot = − . The two boundstates below the | bb (cid:105) threshold are the same as in Fig. 3(c)with the same color coding; the energy zero is the separatedatom energy of the | bb (cid:105) channel. The contours above thresholdshow | S bb,ab | for weak decay from | bb (cid:105) to | ab (cid:105) , with red andblue respectively indicating maximum and minimum magni-tudes of 2 × − and zero respectively. The horizontal bandcentered around 7 MHz represents enhanced loss due to the | bb (cid:105) channel shape resonance. resonance region of enhanced background loss to | ab (cid:105) cen-tered around 7 MHz.Fitting the CC elastic scattering S-matrix element near171 G yields a small value of V bg ∆ ≈
149 a G, which withthe value δµ/h = 1 .
51 MHz / G in Sec. III B gives a verysmall ζ = − . × − . This is five orders of magni-tude smaller than the value ζ ≈ − Appendix D: Discussion of the s-wave Feshbachresonance near 202 G
As an ancillary result, our updated K coupled-channels calculations provide some insight into thephysics and parameterization of the commonly used s-wave Feshbach resonance, near 202 G. First, we elucidatethe relation of the s-wave resonance in the | ab (cid:105) channelto the p-wave resonance in the | bb (cid:105) channel that is themain topic of this paper. Then we comment on the dis-crepancy in parameterization of this resonance, and itspossible resolution.
1. Coupled channels theory
The same coupled channels theory as described in Sec-tion III is used to calculate the bound and scatteringproperties of the s-wave resonance in the | ab (cid:105) channel, forwhich M tot = −
8. There are only two s-wave channelswith this M tot value, | ab (cid:105) and | ar (cid:105) , which are stronglycoupled by spin-exchange coupling. Adding d-waves tothe calculation makes only mG-scale shifts in the calcu-lated resonance positions.Figure 9, analogous to Fig. 3(c), shows the avoidedcrossing of the last bound state of the | ab (cid:105) channel andthe next-to-last bound state of the | ar (cid:105) channel. Thisavoided crossing “pushes out” the ab bound state at itsthreshold crossing of 202.110 G. The lower branch of thecrossing “returns” to the ab level below threshold at high B above the crossing. The contour plot in the contin-uum shows that the ramping state reappears in the upperbranch above threshold as a broad resonance feature slop-ing to the right. The last | ab (cid:105) bound state has entrance-channel character ( > | ar (cid:105) character( ≈ | ab (cid:105) or | bb (cid:105) ) is different: in the s-wave case, it isa real bound state below threshold; in the p-wave case,it is a shape resonance above threshold, as discussed inSec. III. Otherwise, the spin nature of the ramping statesand the “avoided crossing” with the emergence of a broad2
150 200 250 300 B (G) E / h ( M H z ) -2, ar -1, ab-1, ab 0.00.51.0 FIG. 9.
Bound and scattering properties for s-waveresonance with M tot = − . The two bound states belowthreshold are the last bound state (labeled − , ab ) of the | ab (cid:105) open s-wave channel and the next-to-last bound state (labeled − , ar ) of the closed | ar (cid:105) s-wave channel; the energy zero isthe separated atom energy of the ab channel. The dashed lineshows the last bound state of a van der Waals potential withthe triplet scattering length. The contours above thresholdshow sin δ bb ( E, B ), with red and blue respectively indicatingunity and zero. resonance above threshold are similar in the s-wave andp-wave cases, with the threshold specifics being quite dif-ferent because of the different nature of the thresholdproperties for the s-wave and p-wave cases. The universalquantum defect theory for a van der Waals potential pre-dicts both the -9 MHz s-wave “last” bound state and the+7 MHz p-wave shape resonance connected with a vander Waals potential with the triplet scattering length. Inthe 200 G region, both the | ab (cid:105) and | bb (cid:105) channels are pre-dominately triplet in character, which is why the tripletscattering length makes a good background parameter forboth channels (discussed in more detail in Sec. V and thenext section), whereas the ramping state in both cases ispredominantly singlet in character.
2. Discrepancy question
The scattering length is typically parameterized as a ( B ) = a bg (cid:18) − ∆ B − B (cid:19) (D1)where a bg is the background scattering length, B is thelocation of the Feshbach resonance, and ∆ is the widthof the resonance. This implies a zero-crossing of the scat-tering length of B zc = B + ∆. This resonance has beenstudied in numerous previous works, its parameteriza-tion improved [59, 80, 116–121]. However, as precisionhas improved, a discrepancy has developed. As shown inTab. III, B determined by Refs. [80] and [120] disagreeby 60 mG, or at least three sigma, and B zc determined by B (G) ∆ (G) B zc (G) a bg ( a ) Source [116]
174 [1]
174 [80] [118]7.0 [119] [121]202.15(2) 6.910(3) 209.06(2) 166.978(2) this workTABLE III.
The K s-wave Feshbach resonance pa-rameters.
Comparison of experimental determinations ofresonance location B , width ∆, zero-crossing field B zc , andthe background scattering length a bg . Directly measuredquantities are indicated in bold face, while inferred or as-sumed values are in regular font. Refs. [120] and our previous work [121] differ by 200 mG,or at least six sigma.By comparison, based on the potentials we use, ourCC calculations find B = 202 . B zc =209 . B is -30(10) mG fromRef. [120], and the value for B zc is -50(10) mG fromRef. [121]. This suggests that a +40(20) mG correctionto the CC theory would give good agreement with bothrecent measurements. We show this updated parameterset as the last line of Tab. III: B = 202 . B zc = 209 . a ( B ), the background scatteringlength and quadratic correction terms can be extractedby plotting a sym ( x ) = a ( B + x ) / a ( B − x ) / x , which should give an x -independent value of a bg if Eq. (D1) is correct. We find a small curvature( ≈ × − a / G ) leads to a 10 − a variation in a sym ( x )across ±
10 G. The best-fit a bg is 166.978(2) a . Thisvalue is roughly 2.5 a (2%) lower than the model-A andmodel-B triplet scattering lengths, a T , A = 169 . a and a T , B = 169 . a , from Ref. [59]. As discussed in the previ-ous section, the closed-channel state is primarily, but notpurely, triplet. We estimate that roughly half of the dis-crepancy between the ∆ = 6 . a bg = a T , B , since near-resonant spectroscopy constrainsthe product a bg ∆.A similar analysis using a asym ( x ) = a ( B + x ) / − a ( B − x ) / x -independent value of xa asym ( x ) = a bg ∆ if Eq. (D1) is accurate. We find aresidual background slope of ∼ . × − a / G acrossa ±
10 G range. Including this term would decrease thebest-fit ∆ in Eq. (D1) by 3 mG. We use this systematicas an estimate of its uncertainty in the values in Tab. III.In sum, Eq. (D1) is an excellent parameterization of theCC-determined a ( B ) resonance across ±±