Efimovian three-body potential from broad to narrow Feshbach resonances J. van de Kraats1D.J.M. Ahmed-Braun1J -L. Li1 2and S.J.J.M.F. Kokkelmans1 1Eindhoven University of Technology P. O. Box 513 5600 MB Eindhoven The Netherlands

2025-05-03 0 0 1.5MB 12 页 10玖币

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Eﬁmovian three-body potential from broad to narrow Feshbach resonances

J. van de Kraats,1, ∗D.J.M. Ahmed-Braun,1J, -L. Li,1, 2 and S.J.J.M.F. Kokkelmans1

1Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands

2Institut f¨ur Quantenmaterie and Center for Integrated Quantum Science

and Technology IQ ST, Universit¨at Ulm, D-89069 Ulm, Germany

(Dated: September 15, 2023)

We analyse the change in the hyperradial Eﬁmovian three-body potential as the two-body inter-

action is tuned from the broad to narrow Feshbach resonance regime. Here, it is known from both

theory and experiment that the three-body dissociation scattering length a−shifts away from the

universal value of −9.7rvdW, with rvdW =1

2mC6/ℏ21/4the two-body van der Waals range. We

model the three-body system using a separable two-body interaction that takes into account the

full zero-energy behavior of the multichannel wave function. We ﬁnd that the short-range repulsive

barrier in the three-body potential characteristic for single-channel models remains universal for

narrow resonances, whilst the change in the three-body parameter originates from a strong decrease

in the potential depth. From an analysis of the underlying spin structure we further attribute this

behavior to the dominance of the two-body interaction in the resonant channel compared to other

non-resonant interactions.

I. INTRODUCTION

In his seminal papers [1,2], Vitaly Eﬁmov predicted

the appearance of an inﬁnite and geometrically spaced set

of three-particle bound states as the pairwise interaction

becomes resonant. These Eﬁmov states are bound by

a universal attractive potential, decaying asymptotically

as −1/R2for three particles at root mean square sepa-

ration R. In trapped ultracold atomic gases the Eﬁmov

eﬀect induces log-periodic peaks in the atom loss rate,

driven by enhanced three-body recombination when an

Eﬁmov trimer crosses into the three-particle continuum

[3–6]. The position of the loss peak associated with the

ground Eﬁmov state sets a characteristic length scale a−,

commonly referred to as the three-body parameter. In

three-body systems with zero-range interactions, intro-

ducing a three-body parameter is neccessary to regularise

the scale invariant unbounded Eﬁmov spectrum [5].

Despite its short-range nature, experiment has re-

vealed that the three-body parameter in diﬀerent atomic

species attains a value close to a−=−9.7rvdW [7–

11], where rvdW =1

2mC6/ℏ21/4is the van der Waals

length associated with the long-range two-body interac-

tion. Subsequent theoretical studies have found that this

“van der Waals universality” originates from a charac-

teristic suppression of the two-body wave function when

r < rvdW, where ris the two-particle separation [12,13].

This suppression leads to the appearance of a strong

repulsive barrier in the three-body potential at mean

square separations R≈2rvdW, which shields the par-

ticles from probing the non-universal short-range detail

of the atomic species.

The above-mentioned theoretical analyses are based

on single-channel interaction potentials, which are ex-

pected to be accurate provided that the intrinsic length

∗j.v.d.kraats@tue.nl

scale r∗due to the resonance width is much smaller

than the potential range. This broad resonance regime

may be deﬁned by a large resonance strength parameter

sres = ¯a/r∗≫1 [14], where ¯a≈0.955978 rvdW is the

mean scattering length of the van der Waals interaction

[15]. The opposite case of a narrow resonance, where

sres ≪1, is characterized by universal behavior in terms

of the dominant length scale r∗≫rvdW. In this limit,

treatments of the three-body problem which neglect the

details of the van der Waals interaction have found the

three-body parameter to be determined universally as

a−=−10.9r∗[6,16–18]. Connecting the broad and

narrow resonance limits through the intermediate regime

where sres ≈1 with a van der Waals interaction model

remains to be desired, in particular given that recent ex-

periments in this regime in 39K have revealed clear de-

viations from both universal limits [11]. A key aspect of

this problem is the change in structure of the trimer and

its associated potential energy surface as a function of

the resonance strength, which will be the central topic of

this paper.

In this work we study the Eﬁmovian three-body po-

tential using a realistic multichannel two-body van der

Waals interaction, which can be easily tuned to probe a

wide regime of resonance strengths. To solve the three-

body problem we approximate this interaction by a sep-

arable potential which reproduces the zero-energy wave

function of the original interaction. We then derive an ef-

fective three-body potential from the open-channel three-

body wave function, which models the actual three-body

potential that binds the Eﬁmov state. Subsequently we

study the dependence of this potential on the resonance

strength sres, and provide an analysis of our ﬁndings in

terms of the multichannel structure underlying the three-

body dynamics.

This paper will be structured as follows. In Sec. II we

outline our approach at the two-body level, ﬁrst deﬁning

a two-channel model interaction with a Feshbach reso-

nance that can be tuned from the broad to narrow reso-

arXiv:2210.14200v3 [cond-mat.quant-gas] 14 Sep 2023

nance strength limit. Subsequently we formulate a sep-

arable approximation to this interaction. In Sec. III we

move to the three-body level, which we analyse ﬁrst in

momentum space to facilitate our actual computations,

and then subsequently in position space for our analysis

of the three-body potential. In Sec. IV we present and

analyse our results, after which we conclude this paper

in Sec. V.

II. TWO-BODY INTERACTION MODELS

A. Model two-channel interaction

In this section we develop a ﬂexible two-channel model

that can be tuned to produce a Feshbach resonance with

a given Breit-Wigner shape [19–21]. We label the two

scattering channels σ={1,2}, with internal energies εσ,

and deﬁne the two-body Hamiltonian operator,

H=H0+V=H0

1+V1,1V1,2

V2,1H0

2+V2,2.(1)

where H0contains the internal and kinetic energies of

the particles, and Vthe pairwise interactions. Expressed

in the interparticle distance r, the diagonal interactions

are of van der Waals type [22,23],

V1,1(r) = V2,2(r) = C6r4

r10 −1

r6,(2)

with C6the species speciﬁc dispersion coeﬃcient. The

parameter r0controlling the short-range barrier is tuned

such that both channels have 8 uncoupled dimer states,

which we have conﬁrmed is suﬃciently deep such that

the scattering is universally determined by the van der

Waals tail. The oﬀ-diagonal terms of the interaction rep-

resent spin-exchange processes, which we model using a

Gaussian form inspired by Ref. [21],

V1,2(r) = V2,1(r) = βe−α(r−rW)2,(3)

where {β, α, rW}are tuneable parameters. To enforce

the short-range nature of the spin-exchange interaction,

we ﬁx rW= 0.15 rvdW [24]. We will take the channel

σ= 1 to be energetically open, and set its internal energy

as ε1= 0 such that H0

1(r) = −ℏ2∇2

r/m. The channel

σ= 2, referred to as the closed channel, has a magnetic

ﬁeld dependent internal energy, H0

2(r, B) = −ℏ2∇2

r/m +

ε2(B). To model the Feshbach resonance, we deﬁne [19],

ε2(B) = εb+δµ (B−Bres),(4)

where εbis the bare binding energy of the resonant bound

state in the closed channel, δµ the diﬀerential magnetic

moment of the particles which is inferred from experi-

ment, and Bres the bare resonant magnetic ﬁeld. Given

a background scattering length abg, resonance width ∆B

and resonant magnetic ﬁeld B0, we can extract an asso-

ciated set of model parameters {r0, α, β, rW, Bres}. The

details of this mapping are outlined in Appendix A. The

resulting resonance strength is obtained as [14],

sres =m

ℏ2¯aabgδµ∆B, (5)

which quantiﬁes the ratio r∗/¯aas mentioned in Sec. I.

B. EST separable potential

As pointed out in previous studies, the universal van

der Waals three-body parameter and three-body poten-

tial can be reproduced by accounting for the full ﬁnite-

range detail of the van der Waals interaction [12]. Sim-

ilarly it was recently shown that reproducing the three-

body recombination rate for resonances of intermediate

strength in 39K requires an inclusion of the exact three-

body spin structure in the Hamiltonian [25]. Such ap-

proaches however, are complicated numerically, and not

conducive to our goal of developing a simple and ﬂexible

model. Fortunately it was pointed out in Refs. [13,26]

that van der Waals universality can be reproduced using

a much simpler model, based on the Ernst, Shakin and

Thaler (EST) separable potential [27]. In this section we

develop such an approach for our multichannel interac-

tion. The crucial point is that we approximate the inter-

action in such a way that the full two-body wave function

at zero energy is taken into account, whilst retaining the

simplicity of a single-term separable potential.

For an arbitrary multichannel interaction V, one may

deﬁne a separable approximation as Vsep =|g⟩ξ⟨g|. In

the EST formalism, the form factor |g⟩and potential

strength ξare derived from a given eigenfunction |ψ⟩of

the full multichannel Hamiltonian,

|g⟩=V|ψ⟩, ξ−1=⟨ψ|V|ψ⟩.(6)

With these deﬁnitions one may show that |ψ⟩is also an

eigenfunction of the Hamiltonian where Vis replaced

with Vsep, with the exact same eigenvalue [27]. Adopt-

ing the approach of Ref. [13], we take |ψ⟩to be the zero-

energy scattering state, such that our model takes as in-

put the low-energy scattering detail of the actual interac-

tion. The separable interaction has an associated separa-

ble t-matrix, given by the Lipmann-Schwinger equation

[28],

tsep(z) = Vsep +VsepG0(z)tsep(z).(7)

Here G0(z)=(z−H0)−1is the Green’s function in the

absence of interactions. We deﬁne its s-wave eigenstates

as |k, σ⟩, where k=|k|is the relative momentum and σ

the scattering channel introduced in the previous section.

In this basis, the transition matrix may be written as,

tsep

σ′,σ(z, k′, k) = gσ′(k′)τ(z)g∗

σ(k),(8)

where tsep

σ′,σ(z, k′, k) = ⟨k′, σ′|tsep(z)|k, σ⟩and gσ(k) =

⟨k, σ|g⟩. Explicit expressions for τ(z) and gσ(k) are given

0 5 10 15 20 25 30 35

k(units of r−1

vdW)

−40

−20

gσ(k)

sres

1.0 100.0

024

−5

FIG. 1. The form factors gσ(k) as a function of momentum

k, tuned to two diﬀerent resonance strengths. Inset shows a

zoom of the low momentum regime, where one observes the

normalisation g1(0) = 1.

in Appendix B. To obtain the eigenfunction |ψ⟩we ex-

plicitly diagonalise the two-body Hamiltonian, using a

mapped grid discrete variable representation [29,30]. For

the broad resonance limit sres ≫1, the Feshbach reso-

nance is well approximated by a potential resonance in

a single-channel model with the interaction in Eq. (2).

In this case, we obtain |ψ⟩using an eﬃcient Numerov

method [31]. The behavior of the form factor as a func-

tion of resonance width is illustrated in Fig. 1. Note

that the arbitrary normalisation of the form factors is

ﬁxed by taking g1(0) = 1. Since the open-channel com-

ponent of the wave function is independent of sres (see

Sec. IV B), the open-channel form factors are much less

sensitive to changes in the resonance strength than the

closed-channel form factors.

We emphasize that the EST model is based on the zero-

energy wave function, and hence loses accuracy when

used to describe deep bound states. To illustrate this

behavior we have computed the shallow dimer energy

around the 8th potential resonance in the two-channel

model, both by a direct numerical solution of the multi-

channel Schr¨odinger equation, and via the EST potential

of this section. In the latter case a dimer solution is found

through the condition τ−1(ε) = 0, with ε < 0 the binding

energy. The two results are compared in Fig. 2, where

one observes that the EST potential is most accurate near

threshold, and is thus naturally suited to treat states near

resonance. For smaller scattering lengths the EST poten-

tial becomes inaccurate for broad resonances where the

dimer becomes too strongly bound, but remains reason-

ably accurate in describing narrower resonances. In this

paper we only concern ourselves with the near resonant

regime a≫rvdW, where the EST potential is accurate

regardless of the resonance strength.

0.0 0.2 0.4 0.6 0.8

a(units of r−1

vdW)

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

−κ2b (units of r−1

vdW)

sres = 0.1

sres = 100.0

FIG. 2. Binding wavenumber κ2b =pm|ε|/ℏ2of the Fes-

hbach dimer that manifests in the two-channel model of Sec.

II A, as a function of inverse scattering length. Results are

shown both with the EST model potential (solid lines), and

a direct solution of the multichannel Schr¨odinger equation

(dashed lines). We show results for two diﬀerent resonance

strengths, and additionally plot the universal resonant result

κ2b ∼1/a as the black dotted line.

III. THREE-BODY APPROACH

A. Three-body integral equation

To ﬁnd the three-body bound state energy Eand

wave function |Ψ⟩, we adopt the methodology of Ref.

[18]. Three-body states are expressed in the s-wave basis

|k, p, σ, σ3⟩. Here {k, p}are the dimer and atom-dimer

Jacobi momenta respectively, σ=σ1σ2gives the channel

of the two-body system as introduced in Sec. II, where

the underline denotes symmetrization, and σ3the spin of

the third particle (in general in this section we shall use

subscripts to label particles 1,2 and 3). The internal en-

ergy of the three particles as dictated by the spin states

will be denoted as E(σ, σ3). The wave function |Ψ⟩is

written into the Faddeev decomposition [32,33],

|Ψ⟩= (1 + P++P−)|¯

Ψ⟩,(9)

with |¯

Ψ⟩the Faddeev component and P±cyclic permuta-

tion operators of the particle indices. For the EST sepa-

rable two-body transition matrix (see Eq. (8)), the state

|¯

Ψ⟩projected on our three-body basis can be formulated

as,

⟨k, p, σ, σ3|¯

Ψ⟩=gσ(k)Fσ3(p)

E−ℏ2k2

m−3

ℏ2p2

m−E(σ, σ3).(10)

Here the function Fσ3(p) captures the dynamics of the

third particle, and is obtained from the one-dimensional

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Efimovianthree-bodypotentialfrombroadtonarrowFeshbachresonancesJ.vandeKraats,1,∗D.J.M.Ahmed-Braun,1J,-L.Li,1,2andS.J.J.M.F.Kokkelmans11EindhovenUniversityofTechnology,P.O.Box513,5600MBEindhoven,TheNetherlands2Institutf¨urQuantenmaterieandCenterforIntegratedQuantumScienceandTechnologyIQST,Universit¨a...

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Efimovian three-body potential from broad to narrow Feshbach resonances J. van de Kraats1D.J.M. Ahmed-Braun1J -L. Li1 2and S.J.J.M.F. Kokkelmans1 1Eindhoven University of Technology P. O. Box 513 5600 MB Eindhoven The Netherlands.pdf

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Efimovian three-body potential from broad to narrow Feshbach resonances J. van de Kraats1D.J.M. Ahmed-Braun1J -L. Li1 2and S.J.J.M.F. Kokkelmans1 1Eindhoven University of Technology P. O. Box 513 5600 MB Eindhoven The Netherlands

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