A numerical approach was developed that allows the treatment of BEC be-yond mean field in a harmonic anisotropic trap. The mean-field solutions are used in order to construct configurations which in turn can be used as a basis for beyond-mean-field calculations. If the interparticle interaction is represented by a pseudopotential, the configuration-interaction approach breaks down. The exact reason is, however, not yet known.
Unfortunately, the theoretical microscopic investigation of ultracold many-body systems is feasible only within the framework of the pseudopotential approximation. The mean-field study shows that two-body collisions are dominant in ultracold dilute atomic gases. However, in order to understand the physics of the condensate, even a complete knowledge concerning two-body collisions is not sufficient. Nevertheless, a good knowledge of two-two-body collisions should help in understanding the consequences of approximations which must be done when many-body systems are considered.
For the delta potential and where the two atoms are placed in a harmonic trap, the Schrödinger equation possesses an analytical solution [95]. This analytical solution may be used for comparison with the exact CI solution in order to understand and fix the divergency problem. There are many-body studies planned for the future and so this problem will not be considered in the thesis anymore. For these studies, the divergency problem must be solved and the numerical approach developed here must also be further improved for a better convergence.
Chapter 3
Collision of two atoms in the presence of a magnetic field
In ultracold alkali atom gases, the interaction between atoms can be var-ied across a wide range by changing the strength of the magnetic field in the vicinity of a Feshbach resonance. The complete theoretical treatment of magnetic Feshbach resonances requires a multi-channel scattering treatment.
Knowledge of the atomic and molecular structures is important for this treat-ment. In this chapter, the full multi-channel problem is solved numerically for the Feshbach resonances in collisions between generic ultracold 6Li and
87Rb atoms in the absolute ground-state mixture in the presence of a static magnetic field. The radial wave functions for the atomic and molecular basis of the ground-state collisional wave function are analyzed in detail in off-resonant and on-off-resonant points. The solutions obtained in this chapter will be adopted in the Chapter 4 in order to develop approximate single-channel schemes that will be used afterwards in the thesis.
3.1 Atomic properties
The atomic structure of alkali atoms plays an important role for the descrip-tion of ultracold atomic gases. In order to understand the collision properties of two atoms, in both the field-free case and within a field, the atomic struc-ture must be considered.
The ground-state of alkali atoms consists of one valence electron in an outer shell, and a core with closed electronic shells. This structure makes alkali atoms hydrogen-like two-particle systems. For a hydrogen atom, the non-relativistic Schrödinger equation can be solved analytically. The so-lutions are one-electron functions also called atomic orbitals. Due to the
Chapter 3. Collision of two atoms in the presence of a magnetic field
spherical symmetry of the core potential, they can be given as
Ψn,l,ml(re, θ, φ) = Rn,l(re)Yl,ml(θ, φ) . (3.1) Here the principal quantum number n, the angular momentum quantum numberl, and the magnetic quantum numberml, uniquely define the atomic orbital. Rn,l is the radial part, where re is the relative distance between the valence electron and the point-like core. Spherical harmonics Yl,ml are the solutions for the spherical part. The quantum numbers n, l and ml are integers and can have the following values: n= 1,2,3...,l < n,ml= 0, ...,±l.
For alkalis, the shell electrons are tightly bound in a spherically symmetric core. Because the field of the core is spherically symmetric, the orbits of the valence electron are still characterized by the same quantum numbers.
= 0
Figure 3.1: An energy level diagram (not to scale) showing the various levels of the valence electron in atomic87Rb in the absence of the external field.
In the presence of a magnetic field, B~, the electronic energy levels split into 2l+ 1 sublevels, which can be associated with the differentml numbers.
The energy due to the interaction is −~µ·B. The magnetic dipole moment,~
~
µ, is produced by the motion of the electron around an orbital path (in a classical picture). The component of ~µ in the direction of B~ can only take on the integer ml values. So, the splitting of different {l, n} levels in the field is mlµBB where µB is the Bohr magneton (magnetic moment of the electron). If an electron is in the state with l= 0, then its orbital motion is not influenced by the magnetic field.
3.1 Atomic properties
Electrons also possess an internal degree of freedom called spin ~s. The orbital motion of the valence electron is coupled to the spin,s= 1
2, resulting in a fine structure in the splitting of energy levels. The overall angular momentum of the electron is~j =~s+~l. Eachjstate is (2j+1)-fold degenerate in the absence of an external field. For alkali atoms, the splitting between the two possible levels with the samel, i. e.,l−1
2 andl+1
2 is ∆Efs =afs(l+12) where afs is the fine structure constant. However, this spin-orbit interaction is absent for l= 0.
The core of the atom also has a spin determined by the number of protons and neutrons in the nucleus. The nuclear spin~i interacts with the valence electron spin, leading to hyperfine splitting. If the valence electron is in the state with l = 0, its orbital motion does not produce any magnetic field at the nucleus. The coupling arises solely due to the magnetic field produced by the electronic spin. Two spins are combined to a total angular momentum f~=~s+~i where each f state is (2f + 1)-fold degenerate. In a Hamiltonian the coupling is represented by a term Vhf of the form
Vˆhf =ahf~s·~i (3.2)
where ahf is the hyperfine constant. The quantum number f has two pos-sibilities f = i+ 1
2 and f = i− 1
2. The splitting between the f levels is
∆Ehf = ahf(i+ 1
2). The hyperfine quantum numbers can be used to label energy levels of the atom in a magnetic field. The energy level diagram for various levels of the87Rb demonstrating the atomic structure in the absence of the external field is shown as an example in Figure 3.1.
An atom in the presence of a magnetic field B~ experiences Zeeman in-teraction due to coupling between the magnetic field and magnetic moments produced by electronic and nuclear spin. If a magnetic field is taken in the z direction, the Zeeman interaction is described by operator
VˆZ=gµBBˆsz−µ
iBˆiz (3.3)
where µ is the magnetic moment of the nucleus, µB is the Bohr magneton, g ≈ 2 is the electron factor, and B = Bz. Figures 3.2 (a) and (b) show the energy spectrum of the hyperfine states for the individual atoms6Li and
87Rb as a function of magnetic field.
The interaction with nucleus µ
iBˆiz is three orders of magnitude smaller than the interaction with the electron, gµBBˆsz, and for most applications may be neglected. The resulting Hamiltonian contains the hyperfine and
Chapter 3. Collision of two atoms in the presence of a magnetic field
Figure 3.2: Ground-state energies of the hyperfine states for6Li (a) and 87Rb (b) in a magnetic field. The energy curves are labeled by theB= 0 hyperfine quantum numbers.
The diagrams remain the same for all atoms with corresponding values of the nuclear spin, taking into account difference in scaling due to a change in the hyperfine splitting constant ahf.
Zeeman interactions in Equations (3.2) and (3.3). It conserves the total angular momentum, and therefore only couples states with the same total angular momentum. This reflects the invariance of the interactions relative to the magnetic axis.