3. Two-channel solution in a harmonic trap 23
3.1.4. Bound-state admixture at narrow and broad Feshbach resonances 32
open-channel dominated and closed-channel dominated MFRs [9]. For an open-channel dominated MFR the closed channel admixtureAis negligible over a large range of the MFR. In this case the MFR is called universal since its properties are solely determined by the scattering length. On the other hand, at a closed-channel dominated MFR the bound-state admixture cannot be neglected and thus influences the properties of the MFR. Closed and open-channel dominated MFRs are conventionally called narrow MFR and broad MFR, respectively.
The concept of narrow and broad MFRs can be also extended to MFRs in harmonic traps. In the case of a broad MFR the coupling strength to the bound state is rela-tively large such that it is admixed to unbound trap states in a large energy domain.
Consequently, its admixture to a specific eigenstate is small. On the other hand, in the case of a narrow resonance the RBS is only admixed to the background state that is in resonance. If no background state is in resonance, the RBS is almost an eigenstate of the system and can be strongly occupied (see Fig. 3.5).
In the formal limitEbg →E of Eq. (3.13) withf(Ebg) =f(E) + (Ebg−E)f0(E) one
On the other hand, in the same limit Eq. (3.2) and its short-range approximation give (E−Eb)(E−Ebg) =hΦE|W|Φˆ bi2 =γγbgΦ˜2E(0)
By combining the two results one can obtain an expression for ˜ΦE(0) as a function of f(E) and f0(E),
Φ˜2E(0) = 2 aho~ω
f2(E)
f0(E) (3.29)
Using Eqs. (3.13) and (3.29) the short-range approximation of Eq. (3.1) can be written as
3.1. Derivation of the analytic model
The derivation of this result is equally valid in an anisotropic trap. Therefore, as in the case of the eigenenergy relation, Eq. (3.30) can be applied to traps withωx=ωy =ηωz
by replacingahof(E) by −√
πd/F(u) defined in Eqs. (3.24) and (3.25).
The eigenenergies and RBS admixtures of two atoms in an isotropic harmonic trap at two exemplary narrow and broad MFRs are shown Fig. 3.5.
-2
Figure 3.5:Top: Energies of the eigenstates at a narrow MFR (left) and a broad MFR (right). The chosen parameters of the MFRs are given in the graphs.
Bottom: Corresponding RBS admixtures of several eigenstates. At a nar-row MFR the weak coupling of the bound state to the trap states leads to small avoided crossings in the spectrum. Away from any avoided crossing the RBS is almost an eigenstate of the system and can be strongly occu-pied. At a broad MFR the RBS couples strongly to all trap states and is admixed to many eigenstates. Hence, the admixture to a specific eigenstate is smaller.
In order to get a quantitative estimate of the RBS admixture, Eq. (3.30) is approx-imated for E >~ω/2 using f(E) = f>(E) [see Eqs. (3.5) and (3.6)]. With the short
3. Two-channel solution in a harmonic trap
two limits of large or small background scattering length one finds tan2δRBS≈ γ
Of course for negative energies the bound state is stronger and stronger occupied.
ForE <−~ω/2 one can approximate Eq.(3.30) by discussion below Eq.(3.8)] one can summarize that an MFR is broad (narrow) if for small background scattering length ∆Eabg ~ωaho (∆Eabg ~ωaho) and for large background scattering length ∆E/abg ~ω/aho (∆E/abg ~ω/aho).
Especially for largeabg the properties of the MFR are quite different in free space and in the trap. According to Eq. (2.8) the bound-state admixture to the scattering wave function in free space is maximal at the resonance positionE=Eb+δEof the scattering length. According to Eq. (3.32) for|abg| aho the bound-state admixture is maximal where sin(π) =−1 or equivalently for E = (3/2 + 2n)~ω. However, at these energies the scattering length vanishes (see Fig. 3.2). In contrast, for |abg| aho the bound-state admixture is maximal where sin(π) = 1 or accordingly for E = (1/2 + 2n)~ω which coincides with the resonance position of the scattering length.
Of course, the case |abg| aho is much more common since background scattering lengths are usually on the order of 100 a.u. while trap lengths are at least about 1000 a.u.
For some MFRs of 6Li and 133Cs the background scattering length can reach up to
−1727 a.u. and 2500 a.u., respectively [46]. For these MFRs features of the formulas for the limit|abg| ahocan become apparent. That is, although the MFRs are broad in free space, they become in fact significantly narrower in sufficiently strong confinement.
The width of the avoided crossings and thus the width of the MFR are crucial pa-rameters, which determines the behavior of the atoms if the magnetic field and thus the energy of the RBS is varied. Provided that at an avoided crossing the energy difference between the two corresponding eigenstates is small compared to the energy difference to other eigenstates, the dynamical behavior of the system is mainly determined by these two states while all other eigenstates can be neglected.
Expandingf(E)≈f(Ebg) +f0(Ebg)(E−Ebg) in Eq. (3.13) around some background energyEbg yields the eigenenergy equation
(E−Eb−δE)(E−Ebg) =δ2, (3.34) which describes the coupling of a molecular eigenstate with energy E1=Eb+δE to a background state with energyE2=Ebgwith coupling strengthδ2 = ∆Ef(Ebg)/f0(Ebg).
Approximating f(Ebg) by f>(Ebg) [see Eq. (3.6)] one obtains for the nth avoided
34
3.1. Derivation of the analytic model According to the Landau-Zener formula the probability of a diabatic transition through the avoided crossing, i.e. the probability of finding the system after the tran-sition still in its initial state, is given asP =e−2πGwith
G=
The experimentally selectable value of ˙B thus determines whether a transition is di-abatic (G 1) or adiabatic (G 1). Ideally, sequences of diabatic and adiabatic transitions allow for bringing the system to an arbitrary eigenstate. Of course, only for
|δ| ~ω the Landau-Zener theory can give exact results while otherwise two coupled states offer only a quantitative approximation.
3.1.5. Summary of the model
In order to provide a better overview of the model of MFRs in harmonic traps, the most important equations derived in the previous sections shall be shortly summarized for the special case of an isotropic harmonic trap.
The eigenenergies as a function of the external magnetic fieldB are determined by ahof(E) =abg For many MFRs the mean scattering length a, the relative magnetic moment σ, the zero-energy resonance width ∆B|E=0, the zero-energy resonance position BR =B0− δB|E=0, the zero-energy background scattering length a0, and the energy-dependent background scattering lengthabg in effective-range approximation are known [46]. The values ofγγbg and B0 can then be easily determined from ∆B|E=0 and BR, such that there is often no free parameter in the eigenenergy relation.
For a given eigenenergy the ratio of the RBS admixture A and the open-channel admixtureC is given as
A2
bg is not directly related to the energy-dependence of the scattering length
3. Two-channel solution in a harmonic trap
and is usually unknown. However, for many MFRs the one-pole approximation is well applicable such that the coupling strength of the RBS to the background eigenstate and to the eigenstate at the resonance are almost equal and γγ
bg ≈ 1. Below in Sec. 3.2.2 the model is compared to full numerical MC calculations for the 6Li-87Rb MFR at 1066.9 G. In this case it is found that γγ
bg = 1.05.