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Eﬁmov physics beyond universality
Article in The European Physical Journal B · January 2012
DOI: 10.1140/epjb/e2012-30841-3 · Source: arXiv

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Eﬁmov physics beyond universality
Richard Schmidt, Steﬀen Patrick Rath, and Wilhelm Zwerger Physik Department, Technische Universita¨t Mu¨nchen, 85747 Garching, Germany
We provide an exact solution of the Eﬁmov spectrum in ultracold gases within the standard twochannel model for Feshbach resonances. It is shown that the ﬁnite range in the Feshbach coupling makes the introduction of an adjustable three-body parameter obsolete. The solution explains the empirical relation between the scattering length a− where the ﬁrst Eﬁmov state appears at the atom threshold and the van der Waals length lvdw for open channel dominated resonances. There is a continuous crossover to the closed channel dominated limit, where the scale in the energy level diagram as a function of the inverse scattering length 1/a is set by the intrinsic length r⋆ associated with the Feshbach coupling. Our results provide a number of predictions for non-universal ratios between energies and scattering lengths that can be tested in future experiments.
PACS numbers: 03.65Nk, 11.10.Hi, 34.50.-s, 67.85.-d

arXiv:1201.4310v2 [cond-mat.quant-gas] 27 Feb 2012

Most of the basic features that distinguish quantum from classical physics show up already at the single particle level. Genuine two-particle eﬀects like the HongOu-Mandel two-photon interference are typically a consequence of particle statistics, not of interactions [1]. Surprisingly, novel quantum eﬀects in which statistics and interactions are combined appear at the level of three particles. As shown by Eﬁmov in 1970 [2], three particles which interact via a resonant short-range attractive interaction exhibit an inﬁnite sequence of three-body bound states or trimers. Remarkably, the trimers exist even in a regime where the two-body interaction does not have a bound state. Eﬁmov trimers thus behave like Borromean rings: three of them are bound together but cutting one of the bonds makes the whole system ﬂy apart. While theoretically predicted in a nuclear matter context, Eﬁmov states have ﬁnally been observed with ultracold atoms [3]. The assumption of short range interactions is perfectly valid in this case and, moreover, the associated scattering lengths can be tuned by an external magnetic ﬁeld, exploiting a Feshbach resonance [4]. One key signature of Eﬁmov physics is the resonant enhancement of the three-body recombination rate when the nth Eﬁmov state meets the atom threshold at a scattering length a(−n). In most experiments, only the lowest Eﬁmov state at a− = a(−0) can be observed because of large atom losses as the scattering length increases. An important feature of the Eﬁmov trimers is that the binding energies exhibit universal scaling behavior. In the limit where the twobody interaction is just at the threshold to form a bound state, the ratio E(n)/E(n+1) of consecutive binding energies approaches the universal value e2π/s0 ≃ 515.028 for n ≫ 1 with the Eﬁmov number s0 ≈ 1.00624. Similarly, the ratio of consecutive values of the scattering length a(−n) approaches a(−n+1)/a(−n) → eπ/s0 ≃ 22.6942. The origin of this universality can be understood from an effective ﬁeld theory approach to the three-body problem [5, 6]. There remains, however, a non-universal aspect in the theory: Although the relative position of the trimer

states is universal, their exact position in the (a, E) plane is not ﬁxed and is determined by the so-called three-body parameter (3BP). It is presumed that the 3BP is highly sensitive to microscopic details of the underlying twobody potential as well as genuine three-body forces [7].
As more experimental data have been accumulated in recent years [8–15], a puzzling observation came to light: In most experiments, no matter which alkali atoms were used, the measured values for a− clustered around a− ≈ −9.45 lvdW. Where does this apparent ‘universality of the three-body parameter’ come from? A possible answer to this question is based on the observation that, typically, Eﬁmov trimers that are accessible with ultracold atoms appear in a situation where the scattering length is tuned via an open channel dominated Feshbach resonance. The observed universality may thus be a simple consequence of the fact that in all cases the interactions are, up to a scale factor, almost identical. Indeed, within a singlechannel description, potentials with a van der Waals tail have lvdw as the only relevant length scale at energies much smaller than the depth of the potential well [16]. It is then plausible that lvdw provides the characteristic scale for the 3BP. This has been conﬁrmed in recent, independent work on this problem by Chin [17] and by Wang et al. [18], using single-channel potentials with a van der Waals tail.
It is an open question, however, to which extent a single-channel description captures the physics near a Feshbach resonance. In the following, we will show that a simple extension of the standard two-channel model for Feshbach resonances [4], which takes into account the ﬁnite range of the Feshbach coupling, provides a complete description of the Eﬁmov spectrum in terms of only two experimentally accessible parameters: the van der Waals length lvdW and the intrinsic length r⋆. Depending on the dimensionless resonance strength sres = 0.956 lvdw/r⋆ [4], there is a continuous change in the relation between the trimer energy spectrum and the scattering length, with the lowest Eﬁmov state appearing at a− ≈ −8.3 lvdW as sres ≫ 1 while a− ≈ −10.3 r⋆ in the opposite limit

2

sres ≪ 1. In addition, we show that for the experimentally accessible lowest Eﬁmov states there are strong deviations from universality which have apparently been observed in recent experiments. Speciﬁcally, we consider non-relativistic bosons described by the microscopic action (in units where 2m = = 1)

S=

ψ∗(r, t)[i∂t − ∇2]ψ(r, t)

r,t

+

φ∗(r, t)Pφclφ(r, t)

+

g 2

χ(r2 − r1) ×
r1 ,r2 ,t

φ( r1

+ 2

r2

,

t)ψ∗(r1

,

t)ψ∗

(r2,

t)

+

c.c.

,

(1)

where ψ denotes the atoms and φ the molecule in the closed channel. Here Pφcl = i∂t − ∇2/2 + ν with ν(B) = µ(B − Bres) the bare detuning from the resonance and µ is the diﬀerence in the magnetic moment between the
molecule and the open-channel atoms. For a description
of universal features of Eﬁmov trimers like the asymptotic ratio a(−n+1)/a(−n) → eπ/s0 , the atom-molecule conversion amplitude ∼ g may be taken as pointlike in co-
ordinate space [5, 6]. In reality, however, the coupling
has a ﬁnite range σ which is determined by the scale of the wave function overlap between the open and closed-
channel states. As has been pointed out by a num-
ber of authors [19–24], this can be accounted for by a form factor χ(r) in Eq. (1). A convenient choice is χ(r) ∼ e−r/σ/r which leads to χ(p) = 1/(1 + σ2p2) in
momentum space. It is important to note that the action (1) can also be used to describe the situation where
the interaction is dominated by a large background scattering length abg. Indeed, integrating out the closedchannel ﬁeld φ, one obtains a contribution ∼ (ψ∗ψ)2 that
properly describes background scattering of range σ and scattering length ∼ g2/ν˜, provided the Feshbach coupling g2 = 32π/r⋆ ≫ 1/lvdw is strong enough that the momentum dependence of Pφcl can be neglected. As has been discussed in [20, 22–24], this does not work, however, in
the limit of closed channel dominated resonances. In our model the scattering of two atoms is mediated
by the exchange of the closed-channel or dimer ﬁeld φ.
The two-body problem is thus solved by computing the renormalization of the inverse propagator of the dimer Gφ−1. Evaluation of the standard ladder diagram yields

Gφ−1(E, q) = Pφcl(E, q) −

g2/(32π)

2 (2)

σ 1+σ

−

E 2

+

q2 4

− iǫ

with Pφcl(E, q) = −E + q2/2 − ν(B) − iǫ. The twoatom scattering amplitude now follows from f (k) = g2χ(k)2Gφ(2k2, 0)/(16π). Its standard low-energy expansion then determines the scattering length a and the ef-
fective range re via

1 a

=

1 2σ

−

16π g2

ν

(B),

re = −2r∗ + 3σ

1

−

4σ 3a

. (3)

FIG. 1: Feynman diagrams contributing to the renormalization of the atom-dimer vertex λ(3k) (large circle). The small black circle represents the atom-dimer coupling ∼ g and the
solid (dashed) line denotes the atom (dimer) propagator.

This allows to express the bare parameters g, σ, and µBres which appear in (1) in terms of ﬁxed, experimental parameters. Close to a Feshbach resonance at magnetic ﬁeld B0, the scattering length can be written as a(B) = −1/r∗ν˜(B) where ν˜(B) = µ(B − B0) is the renormalized detuning in units of a wavenumber squared, while r⋆ > 0 is the intrinsic length scale which characterizes the strength of the Feshbach coupling [4]. This ﬁxes g2 = 32π/r∗. Moreover, the resonance shift is given by µ(B0 − Bres) = 1/(r⋆σ), which is always positive in our model. Comparing our result for the resonance shift with the corresponding expression obtained for Feshbach resonances with interaction potentials that have a van der Waals tail [25] yields the identiﬁcation σ = a¯, with the so-called mean scattering length a¯ = 4π/Γ(1/4)2lvdw ≈ 0.956 lvdw [26].
Based on the knowledge of the full two-body scattering amplitude, the three-body problem can be solved exactly, keeping only s-wave interactions. In particular, the three-boson scattering can be expressed in terms of an atom-dimer interaction ∼ φ∗ψ∗φψ. The corresponding one particle irreducible atom-dimer vertex λ3(Q1, Q2, Q3) [Qi = (Ei, qi)] develops a complicated energy and momentum dependence which determines the full Eﬁmov spectrum for arbitrary values of the scattering length. The derivation becomes particularly simple using the functional renormalization group (fRG) [27]. The central quantity of the fRG is an RG scale k dependent eﬀective action Γk which interpolates between the microscopic action S = Γk=Λ and the full quantum eﬀective action Γ = Γk=0 by successively including quantum ﬂuctuations on momentum scales q k. Here, we adopt an RG strategy adjusted to the few-body problem as discussed in [28, 29], where the ﬂowing action Γk is of the form of S in (1) but with Pφcl replaced by 1/Gφ from (2) and an additional term

Γ3kB = −

λ3(k)(Q1, Q2, Q3) ×

Q1 ,Q2 ,Q3

φ∗(Q1)ψ∗(Q2)φ(Q3)ψ(Q1 + Q2 − Q3). (4)

Since we do not consider a microscopic three-body force
here, we have λ(3Λ) = 0 at the UV scale Λ. The atomdimer vertex λ3(k) is then the only running coupling in Γk. It is important to note that the truncation of Γk

3

is complete for the solution of the three-body problem
as no additional couplings can be generated in the RG
ﬂow [28, 29]. In our scheme the propagator of the bosons
ψ is not regularized and the dimer φ is supplemented
with a sharp momentum regulator. In Fig. 1 we show the Feynman diagrams contributing to the ﬂow of λ(3k). The number of independent energies and momenta is re-
duced by working in the center-of-mass frame and by not-
ing that the loop frequency integration puts one internal
atom on mass-shell [6]. After performing the s-wave projection λ(3k)(q1, q2; E) = 1/(2g) d cos θλ(3k)(q1, q2; E), θ = ∠(q1, q2), one ﬁnds the RG equation

∂k λ(3k) (q1 ,

q2;

E)

=

−

g2k2Gφ(E − 2π2

k2,

k)

×

λ3(k)(q1, k; E)λ(3k)(k, q2; E) + λ(3k)(q1, k; E)GE (k, q2)

+GE(q1, k)λ3(k)(k, q2; E) + GE (q1, k)GE (k, q2) , (5)

where

GE (p, q)

≡

1 2

1 −1

d

cos

θ

−E

χ( p + p2

+ +

q 2

)χ(

q

q2 + (p

+ +

p 2

)

q)2

−

iǫ

.

(6)

Making use of the binomial form of Eq. (5) the ﬂow can

be integrated analytically and yields

Λ
fE(q1, q2) = gE(q1, q2) − dl gE(q1, l) ζE(l) fE(l, q2),
0
(7)
which is a modiﬁed form of the well-known STM equation [30] with fE(q1, q2) = gE(q1, q2)+λ˜E(q1, q2), gE(q1, q2) = 16q1q2GE(q1, q2), λ˜E(q1, q2) = 16q1q2 λ3(q1, q2; E), and ζE(l) = −g2ξGφ(E − l2, l)/(32π). Due to the presence of the form factor χ in gE and the ﬁnite range corrections in Eq. (2) the UV limit Λ → ∞ can safely be taken. This
explicitly demonstrates the independence of the cutoﬀ and leads to the disappearance of the 3BP.

0.0

- 0.5

- 0.5

0.0

0.5

1.0

- 1.0

0.0

-0.5

-1.0
- 1.5 -1.5

- 0.5

0.0

0.5

1.0

FIG. 2: (color online). The Eﬁmov spectrum in dimensionless units for a broad Feshbach resonance of strength sres = 100. The inset shows the spectrum for a resonance of intermediate strength sres = 1. The dimer binding energy is shown in blue.

The knowledge of the full vertex λ3 gives all information about the scattering of three bosons, such as bound states, recombination rates, and lifetimes, by evaluating the corresponding tree-level diagrams [6]. In the following we compute the trimer bound state spectrum by identifying the poles of λ3 as a function of the energy E. In the vicinity of a bound state pole the atomdimer vertex can be parametrized as λ3(q1, q2; E) ≈ B(q1, q2)/[E + ET(n) + iΓT(n)]. When inserted into Eq. (7) an integral equation for B is obtained which is solved by discretization and amounts to evaluating the determinant det[C − I] = 0 with C(q1, q2) = g2gE(q1, q2)Gφ(E − q22, q2)/(32π). C has a log-periodic structure where lowmomentum modes are suppressed by any ﬁnite 1/a = 0 and energy E < 0 below the atom-dimer threshold. Highmomentum modes are suppressed due to the ﬁnite range potential of our model.
In Fig. 2 we show the Eﬁmov spectrum including the atom-dimer threshold for a broad and a Feshbach resonance of intermediate strength in dimensionless units. The position of the trimer states in the (1/a, E) plane is completely ﬁxed by our calculation. The overall appearance of the spectrum remains similar as the strength of the resonance is varied. For narrow resonances it gets pushed towards the unitarity point E = 1/a = 0 while for open channel dominated resonances it reaches a maximal extent in the (1/a, E) plane. The detailed position of the lowest energy levels depends on both the value of the van der Waals length and the resonance strength sres. Universality is only reached in the experimentally hardly accessible limit n ≫ 1 where the ratios of a−(n), a∗(n) (the scattering length for which the trimer meets the atom-dimer threshold), and E(n) for consecutive lev-

sres

n

0

1

2 UT

E(n)/E(n+1) 530.871 515.206 515.035 515.028

100

a(−n+1) /a(−n) a(∗n+1) /a(∗n)

17.083 21.827 22.654 22.694 3.980 40.033 23.345 22.694

κ(∗n) a(−n)

2.121 1.573 1.512 1.5076

E(n)/E(n+1) 515.830 515.039 515.035 515.028

1

a(−n+1)/a(−n) 22.869 22.650 22.690 22.694 a(∗n+1)/a(∗n) 17.183 22.303 22.716 22.694

κ(∗n)

a(n) −

1.500 1.511 1.508 1.5076

E(n)/E(n+1) 521.273 515.059 515.010 515.028

0.1

a(−n+1) /a(−n) a(∗n+1) /a(∗n)

κ(∗n) a(−n)

26.230 22.964 26.965 21.286
1.296 1.489

22.71 22.694 22.48 22.694 1.506 1.5076

TABLE I: The ratio between consecutive energies E(n) = 2(κ(∗n))2/m and threshold scattering lengths (a(−n,)∗) as well
as the product a(−n)κ(∗n) for a broad (sres = 100), intermediate (sres = 1) and narrow (sres = 0.1) Feshbach resonance and
the three lowest-lying Eﬁmov states n = 0, 1, 2. The right-
most column shows the predictions of universal theory.

4

0.2

0.1

0.0

- 0.1

-3

-2

-1

0

1

2

3

FIG. 3: (color online). Inverse threshold scattering length a− (solid line) and wavenumber κ∗ (dashed line) in units of a¯ as functions of the resonance strength sres. The dots with error bars show the experimental results for 7Li [10, 13], 39K [11], 85Rb [15] and 133Cs [14].

els approach their universal values. It is instructive to
quantify to which extent the lowest states deviate from the universal prediction. In Table I our results for vari-
ous quantities for a broad (sres = 100), an intermediate (sres = 1), and a narrow (sres = 0.1) Feshbach resonance are shown (for details cf. [31]). Although the third state
already follows almost universal behavior regardless of the value of sres, the experimentally most relevant lowest states exhibit large deviations. In fact, our prediction for
open channel dominated resonances is supported by recent measurements of the position of the second Eﬁmov trimer in 6Li who ﬁnd a ratio near 19.7 [32, 33]. Remarkably, for intermediate Feshbach resonances (sres ≈ 1), the interplay between the scales r∗ and σ leads to ratios close
to their universal ones even for the lowest states. Note that the values of a(∗n) for small n are highly sensitive to the precise form of the two-body bound state spectrum
which is strongly non-universal [37]. The ratios between the lowest a(∗n) are therefore in general not suitable for a measurement of universal ratios. By contrast, this is not
the case for the Eﬁmov spectrum in the regime a ≤ 0, which is determined by energies on the order of or below
the scale set by the van der Waals length.
We ﬁnally study the dependence of a(−n) and κ(∗n) on the strength of the Feshbach resonance. In Fig. 3 the behavior for the lowest, experimentally accessible, state
is shown. For open channel dominated resonances a−/a¯ and a¯κ∗ become independent of sres and thus of r∗ and we ﬁnd a− ≈ −8.27 lvdw and κ∗lvdw = 0.26. In the limit of closed channel dominated resonances, the van der Waals length becomes irrelevant and the scale for the full Eﬁ-
mov spectrum is set by r∗ only. Speciﬁcally, we ﬁnd a(−n) = ξ(n)r∗ and κ(∗n)r∗ = η(n) with numbers ξ(n) and η(n) which approach universal values as n → ∞. In fact, as n ≫ 1, we accurately reproduce the predictions for
narrow Feshbach resonances which were derived previ-

ously within a zero range model where σ = 0 [34, 35]. Note, however, that the low-lying Eﬁmov states deviate from these universal predictions. While universal theory predicts for example a− = −12.90 r∗ and κ∗r∗ = 0.117 [34, 35], we ﬁnd a− = −10.3 r∗ and κ∗r∗ = 0.125. Comparing to the experimental data, we ﬁnd that the open channel dominated resonances in 85Rb [15] and in 133Cs [14], which also exhibit a large background scattering length, ﬁt well into our prediction, apart from a 12 % deviation. The regime of Feshbach resonances of intermediate strength sres ≈ 1 is particularly interesting, since both scales r∗ and σ are relevant. Our approach equally applies to this regime, which is realized, e.g., in the case of 39K, where sres ≃ 2.1 [11]. As shown in Fig. 3, the observation [11] of a considerable deviation from the apparent ‘universal’ result a− ≈ −9.45 lvdW in this case is qualitatively explained by our theory. Unfortunately, the case of 7Li, which seems to follow nicely the result a− ≈ −9.45 lvdW [10, 13] despite the even smaller value sres ≃ 0.58 [13] of the resonance strength is not consistent with our prediction. A possible origin of this discrepancy might be that the identiﬁcation of the scale σ of the Feshbach coupling with the van der Waals length does not apply in this case [38]. Note, however, that the proportionality between a− and lvdW is certainly invalid in the limit sres ≪ 1. In conclusion, we have presented a simple model containing only r∗ and lvdw as experimentally accessible parameters, which allows to predict the full Eﬁmov spectrum in quantitative terms for Feshbach resonances of arbitrary strength without an adjustable 3BP. Our results provide an explanation for why the 3BP appears to be a ‘universal’ number in terms of the van der Waals length, which applies for open channel dominated resonances. It is important to stress, however, that for the lowest Eﬁmov states there is no universality in its standard meaning. Beyond the dependence on the resonance strength, the value of the ratio a−/lvdW exhibits a weak dependence on the precise form of the cutoﬀ function χ(r) in our model (1). Moreover, these numbers will be changed by taking into account three-body forces, e.g., of the Axilrod-Teller type, that are present for neutral atoms with van der Waals interactions [36]. Still, our exact solution of the standard two-channel model for Feshbach resonances of ultracold atoms captures the qualitative features necessary for an understanding of Eﬁmov physics beyond the universal regime.
We thank Francesca Ferlaino, Rudi Grimm, Selim Jochim, Felix Werner and Matteo Zaccanti for useful discussions and acknowledge support by the DFG through FOR 801.

5

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[31] For the detailed dependence of the quantities shown in Table I on the resonance strength we refer to the supplementary material.
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(1943). [37] For example, a positive background scattering length abg
leads to a open-channel bound state which is not included in our model. Similarily, due to the absence of a ﬁnite background scattering length, the lowest trimer eventually always meets the atom-dimer threshold. [38] Note that for sres 1 and a ﬁnite background scattering length, abg also becomes a relevant scale as shown by Pricoupenko [22].

6

Supplementary material

530

: UT

In this supplementary material we give results for the

: n=1

dependence of various ‘universal’ ratios on the continuous

: n=2

varied strength sres of the Feshbach resonance.

525

2.5

: UT

: n=1

2.0

: n=2

: n=3

1.5

520

515

510

0.01

0.1

1

10

100

1.0

FIG. 6: (color online). The ratio E(n)/E(n+1) of the consec-

utive trimer energies at unitarity 1/a = 0 as function of the

resonance strength sres for the ﬁrst two states (black). The

0.5

universal prediction is shown in red.

0.01

0.1

1

10

100

FIG. 4: (color online). The ratio κ(∗n)a(−n), which can be viewed as a measure of the distortion of the trimer levels from their universal shape in the (a, E) plane, as function of the resonance strength sres. Shown are the results for the lowest three levels (black). The universal result is shown in red.
While the third state almost behaves universally the lowest state shows strong deviations from the universal prediction.

26

24

22

20
: UT
: n=1
18
: n=2

16

0.01

0.1

1

10

100

FIG. 5: (color online). The ratio of scattering lengths a(−n+1)/a(−n) where the consecutive trimer states meet the atom threshold at E = 0 as function of the resonance strength sres for the ﬁrst two states (black). The universal prediction is
shown in red.

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