Comparison with numerical calculations

3.2.1. Comparison to a coupled square-well resonance model

The energy-dependent scattering lengtha(E, Eb) of Eq. (3.17), a(E, E_b) =a_bg

1− ∆E

Eb+δE−E

differs significantly from that of the theory of the scattering of free atoms, where a(E, Eb) =abg



1−

1 +a²_bgk²∆E_f E_b+δE_f −E+a²_bgk²∆E_f





[see Eq. (2.11)]. The index “f” shall indicate that the resonance width and resonance detuning in free space is defined differently from those in the harmonic trap. The energy dependence of the total resonance width ∆E_tot = 1 +a²_bgk²∆E_f and the total resonance detuning δEtot = δEf +a²_bgk²∆Ef in the case of free scattering is considered in Eqs. (2.14) and (2.15), respectively. They depend on three parameters γ₁,γ₂, andr₀. Remarkably, within the introduced model in the harmonic trapa(E, E_b) depends effectively only on two parameters, the product of the coupling strengthγγbg

and the length a^∗ that has initially replaced the interaction range r₀ [see Eq. (3.8)].

Indeed replacingr₀ bya^∗also in Eq. (2.14) both theories agree, i.e. the resonance width in free space and in the harmonic trap are parametrized in the same way. Nevertheless, both models disagree regarding the parametrization of the total resonance detuning given in Eqs. (2.15). In the free-scattering theory the detuning is proportional to (abgk)² for large background scattering length, which is in stark contrast to the theory of MFRs in harmonic traps where no dependence on (a_bgk)² appears [see Eq. (3.19)].

Also for other two-channel models of MFRs in the presence of a trapping potential the dependence on (abgk)² is absent [55, 64, 65].

In order to determine whether the description in free space or in the trap offers a more accurate determination of the scattering length, a simple model of an MFR is considered. It consists of an open and a closed channel whose potentials are square wells of ranged. In the remainder of this section all lengths are given in units ofdand all energies in units of E0 = ~²/(2µd²). The square-well potentials have the general form

V_a,b(ρ) =

(−a for ρ <1

b for ρ≥1. (3.37)

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3.2. Comparison with numerical calculations

interatomic distance energy,

wavefunction

Figure 3.6: Sketch of the used coupled square-well potentials and wave functions of the closed channel (blue) and the open channel (red). Both channels are coupled by a constant coupling potential (orange) within the interaction range.

Also the coupling is considered to be constant within the interaction range 0≤ρ < 1 (see Fig. 3.6). The MFR is described by the dimensionless coupled equations

−Φ⁰⁰_P(ρ) + (V_aP,0(ρ)−E) Φ_P(ρ) +V_δ,0(ρ)Φ_Q(ρ) = 0

−Φ⁰⁰_Q(ρ) +V_aQ,bQ(ρ)−EΦ_Q(ρ) +V_δ,0(ρ)Φ_P(ρ) = 0, (3.38) which are visualized in Fig. 3.6. For scattering energies E aP the background scattering length of the open channel is given asa_bg =d[1−tan(√

aP)/√

aP] and can be chosen arbitrarily large. The width of the MFR is determined by the coupling strength δ. In the following the case δ =−0.1E₀ and a_bg =−400d(i.e. a_P = 22.2016E₀) are considered. The depth of the closed-channel potential is chosen to beaQ+bQ= 100E0. One bound state of the closed-channel potential has an energyE_b=b_Q−29.05E₀.

As discussed above, the theory of free scattering suggests that for a given scattering energy E the energy of the bound state E_b^(res) = E −δEf −(abgk)²∆Ef for which the scattering length diverges [see Eq. (2.11)] can strongly depend on the energy if the background scattering length is large. On the other hand, the theory of an MFR in the harmonic trap suggest no such strong energy dependence, i.e. according to Eq. (3.17) it holdsE_b^(res) =E−δE.

In Fig. 3.7 the results of a numerical solution of Eq. (3.38) are shown. As one can see, the value of E_b^(res)−E does hardly depend on the energy, although the value of (a_bgk)² is on the order of one. The invisible change on the order of 10⁻⁵E₀ can be well explained by the energy dependence of the background scattering length, which changes over the shown energy range on the order of 10⁻⁴ relative to the value ofa_bg forE= 0.

To ensure that the stability of the resonance position is no coincidence of the chosen parameters, the same calculations have been performed for other potential depths of the closed channel, always showing similar results. The numerical results imply that

3. Two-channel solution in a harmonic trap

one has to choseγ1 =γ2 in Eq. (2.14) in order to realistically describe MFRs. In this case both models parametrize consistently not only the resonance width but also the resonance detuning in the same way. Replacingr₀ bya^∗in the theory of free scattering the ratio of resonance width and resonance detuning is given as (1− _a^a∗

bg) which is in accordance with Eqs. (3.18) and (3.19).

As one can also see in Fig. 3.7, the numerically determined scattering lengtha(E_b, E = 0) can be very well described by the parametrization of the scattering length according to Eq. (3.17) with ∆E=−0.0565E₀,δE=−0.0563E₀ and a_bg =−383d. The value of the observed background scattering length does not exactly agree with the background scattering length of the uncoupled open channel. Since the scattering length can be influenced by relatively small perturbations, the coupling to the closed channel has a significant influence even if any bound state is far from resonance. For real MFRs this significant off-resonant coupling is one of the reasons for a slight magnetic-field dependence of the background scattering length (see also Fig. 2.2).

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.00 0.02 0.04 0.06 0.08 0.10

HabgkL² HEbHresL -ELE0

-0.2 -0.1 0.0 0.1 0.2 0.3 -2000

-1500 -1000 -500 0 500 1000

EbE0

aHEb,E=0Ld

Figure 3.7:Left: Value ofE_b^(res)−E (for the bound-state energy E_b^(res) the scattering length diverges) as a function of (ka_bg)². Right: Scattering length for E = 0 as a function of the bound-state energy E_b. The numerical results are shown as dots while the red solid line shows the behavior according to Eq. (3.17) with ∆E =−0.0565E₀,δE=−0.0563E₀ and a_bg=−383d.

3.2.2. Comparison to multi-channel calculations for Li-Rb

For the case of ⁶Li-⁸⁷Rb that was already regarded in Sec. 2.2 Yulian V. Vanne has performed full numerical MC calculations in order to obtain eigenenergies and channel admixtures in isotropic harmonic traps with different trap frequenciesω as a function of the magnetic field².

In order to compare the results with the introduced model, the model parameters have to be determined. The zero-energy background scattering length is determined

2Both in the numerical calculations and in the model a possible coupling of REL and COM motion for different atomic species appearing in Eq. (1.4) is ignored.

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