k
∂
∂r[rΨ(r)]
r=0
= ˜C , (1.16)
which yields finally
∇2+k2Ψ(r) = 4πtanδ k δ(~r) ∂
∂r[rΨ(r)]. (1.17)
Assuming that the energy dependence of the scattering length is negligible one may replace a=−tan(δ)/kby a0 defined in Eq. (1.13). Comparing Eqs. (1.14) and (1.17) one finds
Vps(~r) = 4π~2a0 2µ δ(~r) ∂
∂rr , (1.18)
which is known as the Fermi-Huang pseudo potential fors-wave scattering [36].
1.2. Two-channel model of a Feshbach resonance
The s-wave radial wave function of two colliding ground-state alkali-metal atoms in a harmonic trap of frequencyω is determined by the REL-motion Hamiltonian [37]
H =ˆ −~2 2µ
d2 dr2 +1
2µω2r2+
2
X
j=1
( ˆVhfj + ˆVjZ) + ˆVint(r). (1.19)
1The Dirac-delta functionδ(~r) and the phase shiftδ(k) can be well distinguished by their dependence on~randk, respectively. If this is not the case, it is explicitly stated, which of the two is meant.
6
1.2. Two-channel model of a Feshbach resonance
The hyperfine operator ˆVhfj = ajhf/~2 ~sj ·~ij and the Zeeman operator ˆVjZ = (γe~sj − γn~ij)·B~ in the presence of a magnetic field B~ depend on the electronic spin ~sj, the nuclear spin~ij, the hyperfine constantajhf of atomj= 1,2, and on the nuclear and elec-tronic gyromagnetic factors γn and γe. Within Born-Oppenheimer approximation the interaction ˆVint(r) depends on the electronic spin configuration of the colliding atoms, which is determined by the quantum numberS of the total electronic spinS~ =~s1+~s2. LetVS(r) be the spin-singlet (i.e.S= 0) Born-Oppenheimer interaction potential and VT(r) the spin-triplet (i.e. S = 1) interaction potential, then the interaction can be expressed as ˆVint(r) = VS(r) ˆPS +VT(r) ˆPT where ˆPS and ˆPT project on the singlet and triplet components of the scattering wave function, respectively. The exchange interactionVS(r)−VT(r) between the electrons vanishes rapidly for large internuclear distances such that VS(r) and VT(r) become identical. Defining ˆV± = ( ˆVS ±VˆT)/2, the interaction evaluates to
Vˆint(r) =V+(r) +V−(r)( ˆPS−PˆT)r→∞= V+(r). (1.20) Since for large interatomic distances the interaction does not depend of the spin-con-figuration of the atoms, MFRs are conveniently described in the atomic basis {|αi}
consisting of eigenstates of the hyperfine and Zeeman operator ˆVZHf = ˆVhf1 + ˆVZ1 + Vˆhf2 + ˆV2Z. The eigenenergies EZHf(α) of the eigenequation ˆVZHf|αi = EZHf(α)|αi can be directly obtained from the Breit-Rabi formula [38]. For weak magnetic fields, where the hyperfine coupling dominates over the coupling to the magnetic field, the eigenstates of the atomic basis |αi are equal to spin states |f1, mf1i|f2, mf2i, where f~j =~sj +~ij is the total spin of atom j and mf its projection onto the B-field axis. Since each eigenstate for stronger magnetic fields is adiabatically connected to a weak-field basis state, the weak-field basis is used to label the atomic basis states |αi. Independently of the magnetic field the total spin projectionMf =mf1+mf2 is preserved during the collision.
The eigenfunctions of Hamiltonian (1.19) can be written as a superposition
|Φi=X
α
Φα(r)|αi (1.21)
of channel functions Φα(r) belonging to a specific atomic basis state|αi. In the basis of these channel functions the Schrödinger equation evaluates to the multi-channel (MC) differential equation
−~2 2µ
d2 dr2 +1
2µω2r2+Vα,α(r)−E
!
Φα(r) + X
α06=α
Vα,α0(r)Φα(r) = 0 (1.22) with the matrix elements
Vα,α0(r) =V−(r)DαPˆS−PˆTα0E+V+(r) +EZHf(α)δα,α0. (1.23) In the atomic basis different channels are coupled by the exchange interactionVex(r) = 2V−(r) between the two valence electrons of the alkali atoms.
1. Theoretical introduction
10 15 20 30 50 70 100
-0.3 -0.2 -0.1 0.0 0.1 0.2
interatomic distancerin a.u.
arbitraryunits
Singlet potential Triplet potential Exchange coupling Open channel Closed channels
Figure 1.1: Exemplary solution of the MC equation (1.22) for the scattering of 6Li (atom 1) and 87Rb (atom 2) and a total spin projection MF = 3/2 close to an MFR atB = 1067 G. For the chosen scattering energy only one channel
|α0i = |1/2,1/2i|1,1i is open while the seven other channels are closed.
For about r <8 a.u. the triplet potential of the electronic statea3Σ+ has a positive energy so that the triplet admixture to the channel wavefunctions vanishes and the system is well described by a pure singlet interaction of the electronic stateX1Σ+. Forr >8 a.u. the channel wavefunctions show a more complex behavior since they are superpositions of singlet and triplet wave functions. Already for r >16 a.u. the coupling between the channels by the exchange interaction is negligible.
In this thesis solely elastic collisions are considered, where only the channel with the lowest threshold energyEZHf(α0)with spin configuration|α0iis open and all other coupled channels are closed. That is, all channel functions Φα(r) apart from a single channel function decay exponentially for large r. If more than one channel is open, inelastic spin-changing collision are possible. During these inelastic collisions the atoms gain usually a sufficiently large kinetic energy, such that they are immediately lost from the experimental setup.
In Fig. 1.1 a solution of the MC equation (1.22) for the exemplary case of the scat-tering of 6Li-87Rb with one open channel is shown. The numerical MC solutions were provided by Yulian V. Vanne. More details on the calculations are given in Publication VII. 6Li (atom 1) and 87Rb (atom 2) have nuclear spins i1 = 1 and i2 = 3/2, respectively. The channel with the lowest hyperfine and Zeeman energy is
|f1, mf1i|f2, mf2i=|1/2,1/2i|1,1i, which has a total spin projection MF = 3/2. Con-sidering low-energy scattering events, where this is the only open channel, seven other closed channels withMF = 3/2 are coupled to the open channel.
8
1.2. Two-channel model of a Feshbach resonance
Within the TC approximation of the scattering process one projects the full MC Hilbert space onto two subspaces, the one of the open entrance channel (with projection operator ˆP = |α0ihα0|) and the one of the closed channels (with projection operator Q = 1ˆ −P) [39]. The resulting TC Schrödinger equation readsˆ
( ˆHP −E)|ΦPi+ ˆW|ΦQi= 0 (1.24) ( ˆHQ−E)|ΦQi+ ˆW†|ΦPi= 0, (1.25) where ˆHP = ˆP ˆH ˆP, ˆHQ = ˆQ ˆH ˆQ, ˆW = ˆP ˆH ˆQ, |ΦPi= ˆP|Φi and |ΦQi= ˆQ|Φi.
An MFR occurs if the energy E of the system is close to the eigenenergy Eb of a bound state|Φbi of the closed-channel subspace. By a variation of the magnetic field B, Eb =Eb(B) can be brought to resonance with the energy E. In the remainder of the thesis the bound state that is responsible for the occurrence of a specific MFR is denoted as the resonant bound state (RBS).
Eq. (1.25) may be formally solved using the Greens operator ˆGQ = (E+−HˆQ)−1, whereE+=E+i0 is infinitesimally shifted to the positive complex plane. ˆGQ can be expanded in discrete eigenstates|φmi and continuum states|φ(E)i of ˆHQ,
GˆQ=X
m
|φmihφm| E−Em +
Z
d|φ()ihφ()|
E+− . (1.26)
Close to the resonance with then-th bound state the sum is dominated by the contri-bution |φE−Enihφn|
n . Within the one-pole approximation one neglects the contribution of all other eigenstates, such that the closed-channel wave function
|ΦQi= ˆGQWˆ†|ΦPi ≈
DφnWˆ ΦQE E−En
|φni=A|φni (1.27) is equal to a multipleAof a the RBS|Φbi ≡ |φni. Applying the one-pole approximation of Eqs. (1.24) and (1.25) yields the coupled equations
( ˆHP −E)|ΦPi+AW|Φˆ bi= 0 (1.28) (Eb−E)A|Φbi+ ˆW†|ΦPi= 0, (1.29) which are the starting point of the following considerations of free and trapped atoms at an MFR.