Effect of hyperfine interactions

The real scattering lengths between specific states _|f₁m_f,1〉 and_|f₂m_f,2〉do not necessarily lie between the associated singlet and triplet scattering lengths, in contradiction to equa- tion (6.14). The major reason for this is the negligence of the hyperfine interaction term,

ˆ

H_hf=Ahf(ˆi1·ˆs1+ ˆi2·ˆs2). Even for elastic collisions, where the internal atomic states are

identical before and after the collision, the internal states are coupled to different states during the collision. This, in combination with the hyperfine level splitting, leads to a change in the phase accumulated during the collision and henceforth a different scattering length. An accurate calculation of scattering lengths including hyperfine interactions requires refined coupled channel models and is not part of this thesis. We will however present a simplified method to estimate the effect of the hyperfine interaction on the scattering length.

Correcting for hyperfine structure effects The use of equation (6.14) is only valid in the limit of negligible hyperfine interactions. For a more refined analysis, we go back to equations (6.6) and (6.7), which express an arbitrary initial two-atom state as a superposition of the eigenstates of the spin exchange interaction, the singlet and triplet states. In the simple model neglecting hyperfine interactions, each of the singlet and triplet states acquires a phase shift during the collision, determined by the singlet and triplet scattering lengths. The situation becomes more involved when the hyperfine coupling termH_hf is taken into account, as the singlet and triplet states are not eigenstates of this part of the Hamiltonian. One can decompose each of the singlet and triplet states in terms of hyperfine eigenstates_|f�

1mf�_,1f₂�m_f�_,2〉, |s_〉|m_i,1m_i,2〉= � f� 1,mf�,1,f2�,mf�,2 〈f� 1mf�_,1f₂�m_f�_,2|s,m_i,1m_i,2〉|f₁�m_f�_,1f₂�m_f�_,2〉 |t_j〉|m_i,1m_i,2〉= � f� 1,mf�,1,f2�,mf�,2 〈f� 1mf�_,1f₂�m_f�_,2|t_j,m_i,1m_i,2〉|f₁�m_f�_,1f₂�m_f�_,2〉. (6.15)

For alkali atoms, the ground state is split into two hyperfine manifolds, and for the exact phase evolution of a singlet or triplet state, one needs to know its decomposition into hyperfine states according to equation (6.15). Whereas in equations (6.6) and (6.7), the initial state

116 Chapter 6. Probing atomic potentials with Bose-condensed sources

Table 6.2:Scattering lengths for iden- tical87_{Rb in different substates of the}

ground state hyperfine manifold, de- termined using the simplified hyper- fine structure correction and the liter- ature values foraF [169]in combina-

tion with equation (6.17). The quoted uncertainties are based on the uncer- tainties in table 6.1.

�

�_f,mf� _{Simplified model Calculated from[169]} |1, 0〉 100.87(6)a0 100.86(9)a0

|1,±1〉 100.40(10)a0 100.40(10)a0

|2, 0_〉 94.41(6)a0 94.57(7)a0

|2,_±1_〉 95.55(5)a0 95.68(9)a0

|2,_±2_〉 98.98₍4₎a₀ 98.98₍4₎a₀

|i_〉is decomposed in terms of singlet and triplet states, equation (6.15) introduces a further decomposition of the singlet and triplet states in terms of the hyperfine eigenstates (which form the basis of possible initial states). Compared to the initial two-atom hyperfine state

|i〉 =|f1mf,1f2mf,2〉, the decomposition in equation (6.15) can contain states whose total

single-atom spin f_i has been raised or lowered by one during the collision, for one or both of the colliding atoms (e.g. f�

i =fi±0, 1). In such a case, the difference in energy between the hyperfine levels leads to a larger or smaller acquired phase shift. We accordingly adjust the phase shift and therefore the scattering length by an addition (subtraction) of�a_t,s to

the triplet and singlet scattering lengths of a state whose f_i has been raised (lowered) within a particular decomposition. This adjustment is specific to each pair of initial states of the two colliding atoms. For each singlet or triplet state we perform a weighted average and obtain a specific scattering length, which depends on the combinations of f_i and f�

i that are possible during the collision. The overall scattering length for the initial two-atom state can be calculated by a weighted average similar to equation (6.14), with the difference being thata_t andδ_tare now specific to the initial state_|i_〉and can depend on the specific triplet state_|t_j〉.

This phenomenological model contains�as a free parameter. We calibrate�for87Rb using

the well known scattering lengtha_1,−1=100.40(10)a0 between two identical atoms in the

|f ₌1,m_{f =−}1_〉state and obtain�₌6.8_·10−2.

Verification of the model We have tested this rather phenomenological model against the scattering lengthsaF given in table 6.1. We consider identical atoms colliding with an initial state_|i_〉=|f m_f f m_f〉. First, we decompose this state in the two-particle angular momentum

{F,M_F}-basis, _� �_{f mf f mf}�₌ � F,MF � f f F MF|f mff mf � �_� f f F MF�. (6.16)

The scattering length can be calculated from equation (6.16) as a weighted average over scattering lengthsa_F to af,mf = � F,MF � � ��f f F MF|f mf f mf���� 2 aF. (6.17)

We calculate a_f,mf with two different methods, using either equation (6.17) or the model described in the previous paragraph. Both methods show reasonable agreement for87_{Rb, as}

shown in table 6.2. The quoted uncertainties are based purely on the uncertainties of the scattering lengths in table 6.1, which form the basis of the derivation leading to the values in table 6.2. The scattering length of atoms in the_|f =1,mf =±1〉state agrees perfectly, as it is used for the calibration of the free parameter�. For the state_|f =2,mf =±2〉, the scattering length is identical to the triplet scattering length in both models. In the experiment described in the following section, we perform a scattering length measurement using a freely

6.2 Probing a Bose-Einstein condensate 117 propagating atom laser colliding with a trapped Bose-Einstein condensate. We measure the scattering length between the states|f ₌2,m_{f =}0〉and|f ₌1,m_{f =−}1〉. From the foregoing derivation, we expect a scattering length ofa=97.6a0.