Atomic properties of the alkali atoms play a key role in experiments on cold atomic gases, and we shall discuss them briefly in the present chapter. Basic atomic structure is the subject of Sec. 3.1. Two effects exploited to trap and cool atoms are the influence of a magnetic field on atomic energy levels, and the response of an atom to radiation. In Sec. 3.2 we describe the combined influence of the hyperfine interaction and the Zeeman effect on the energy levels of an atom, and in Sec. 3.3 we review the calculation of the atomic polarizability. In Sec. 3.4 we summarize and compare some energy scales. Atomic structure The total spin of a Bose particle must be an integer, and therefore a boson made up of fermions must contain an even number of them. Neutral atoms contain equal numbers of electrons and protons, and therefore the statistics that an atom obeys is determined solely by the number of neutrons N: if N is even, the atom is a boson, and if it is odd, a fermion. Since the alkalis have odd atomic number Z, boson alkali atoms have odd mass numbers A. In Table 3.1 we list N, Z, and the nuclear spin quantum number I for some alkali atoms and hydrogen.
Atomic properties of the alkali atoms play a key role in experiments on cold atomic gases, and we shall discuss them briefly in the present chapter. Basic atomic structure is the subject of Sec. 3.1. Two effects exploited to trap and cool atoms are the influence of a magnetic field on atomic energy levels, and the response of an atom to radiation. In Sec. 3.2 we describe the combined influence of the hyperfine interaction and the Zeeman effect on the energy levels of an atom, and in Sec. 3.3 we review the calculation of the atomic polarizability. In Sec. 3.4 we summarize and compare some energy scales. Atomic structure The total spin of a Bose particle must be an integer, and therefore a boson made up of fermions must contain an even number of them. Neutral atoms contain equal numbers of electrons and protons, and therefore the statistics that an atom obeys is determined solely by the number of neutrons N: if N is even, the atom is a boson, and if it is odd, a fermion. Since the alkalis have odd atomic number Z, boson alkali atoms have odd mass numbers A. In Table 3.1 we list N, Z, and the nuclear spin quantum number I for some alkali atoms and hydrogen.
A number of atomic properties play a key role in experiments on cold atomic gases, and we shall discuss them briefly in the present chapter with particular emphasis on alkali atoms. Basic atomic structure is the subject of Sec. 3.1. Two effects exploited to trap and cool atoms are the influence of a magnetic field on atomic energy levels, and the response of an atom to radiation. In Sec. 3.2 we describe the combined influence of the hyperfine interaction and the Zeeman effect on the energy levels of an atom, and in Sec. 3.3 we review the calculation of the atomic polarizability. In Sec. 3.4 we summarize and compare some energy scales. Atomic structure The total spin of a Bose particle must be an integer, and therefore a boson made up of fermions must contain an even number of them. Neutral atoms contain equal numbers of electrons and protons, and therefore the statistics that an atom obeys is determined solely by the number of neutrons N : if N is even, the atom is a boson, and if it is odd, a fermion. Since the alkalis have odd atomic number Z , boson alkali atoms have odd mass numbers A . Likewise for atoms with even Z , bosonic isotopes have even A . In Table 3.1 we list N, Z , and the nuclear spin quantum number I for some alkali atoms and hydrogen.
The laser cooling mechanisms described in Chapter 4 operate irrespective of the statistics of the atom, and they can therefore be used to cool Fermi species. The statistics of a neutral atom is determined by the number of neutrons in the nucleus, which must be odd for a fermionic atom. Since alkali atoms have odd atomic number Z , their fermionic isotopes have even mass number A . Such isotopes are relatively less abundant than those with odd A since they have both an unpaired neutron and an unpaired proton, which increases their energy by virtue of the odd–even effect. In early experiments, 40 K and 6 Li atoms were cooled to about one-quarter of the Fermi temperature. More recently, fermionic alkali atoms have been cooled to temperatures well below one-tenth of their Fermi temperature, and in addition a degenerate gas of the fermionic species 173 Yb (with I = 5/2) has been prepared. In the classical limit, at low densities and/or high temperatures, clouds of fermions and bosons behave alike. The factor governing the importance of quantum degeneracy is the phase-space density ϖ introduced in Eq. (2.24), and in the classical limit ϖ ≪ 1. When ϖ becomes comparable with unity, gases become degenerate: bosons condense in the lowest single-particle state, while fermions tend towards a state with a filled Fermi sea. As one would expect on dimensional grounds, the degeneracy temperature for fermions – the Fermi temperature T F – is given by the same expression as the Bose–Einstein transition temperature for bosons, apart from a numerical factor of order unity.
From a theoretical point of view, one of the appealing features of clouds of alkali atom vapours is that particle separations, which are typically of order 10 2 nm, are large compared with the scattering length a which characterizes the strength of interactions. Scattering lengths for alkali atoms are of the order of 100 a 0 , where a 0 , is the Bohr radius, and therefore alkali atom vapours are dilute, in the sense that the dominant effects of interaction are due to two-body encounters. It is therefore possible to calculate properties of the gas reliably from a knowledge of two-body scattering at low energies, which implies that information about atomic scattering is a key ingredient in work on Bose–Einstein condensates. An alkali atom in its electronic ground state has several different hyperfine states, as we have seen in Secs. 3.1 and 3.2. Interatomic interactions give rise to transitions between these states and, as we described in Sec. 4.6, such processes are a major mechanism for loss of trapped atoms. In a scattering process, the internal states of the particles in the initial or final states are described by a set of quantum numbers, such as those for the spin, the atomic species, and their state of excitation. We shall refer to a possible choice of these quantum numbers as a channel.1 At the temperatures of interest for Bose-Einstein condensation, atoms are in their electronic ground states, and the only relevant internal states are therefore the hyperfine states. Because of the existence of several hyperfine states for a single atom, the scattering of cold alkali atoms is a multi-channel problem.
We begin our study of atomic physics by relying on the phenomenological notions summarized in chapter 1 and by fully exploiting the quantum mechanical formalism outlined in chapter 2. A hierarchy of approximations will be introduced in the first instance to describe the hydrogen atom, which allows one to focus on the main features of its energy spectrum and quantum properties, including the key ‘atomic orbital’ concept. Next, these results will be extended to the more general case of hydrogenic atoms. The physical picture will be made more rigorous by introducing the magnetic interactions, included those linked to the intrinsic spin degree of freedom, leading to the full description of the fine and hyperfine structure of one-electron atoms. The effects of a static external magnetic or electric field will be eventually discussed by calculating the corresponding energy shifts and splittings of the energy levels.
1. Discuss the Coulomb interaction, the spin-orbit coupling, and the hyperfine coupling as the three effects that determine atomic structure. 2. Highlight the importance of the separation of energy scales of these three effects. 3. Introduce different atomic structures of alkali-metal, alkaline-earth-metal, and magnetic atoms. 4. Introduce the long-lived excited states in alkaline-earth atom and their applications, such as in atomic optical clocks. 5. Discuss the Zeeman structure of atoms in a magnetic field. 6. Discuss the idea of magnetic trapping that can naturally lead to the emergence of a synthetic gauge field. 7. Introduce the scalar light shift and its applications, such as laser trapping, optical lattices, and laser cooling. 8. Introduce the vector light shift and its applications, such as the light-induced Zeeman field and synthetic spin-orbit coupling. 9. Discuss the synthetic spin-orbit coupling and various kinds of gauge fields generated by the vector light shift. 10. Introduce the basic idea of the stimulated Raman adiabatic passage.
Introduction Helium is the natural connection between the hydrogen-like atoms (one electron outside a spherical core, such as the alkali-metal atoms of Chap. 10) and all the others in the periodic table. For these one-electron atoms the field-free description of the electron motion in terms of three quantum numbers n , ℓ , and m suffices. The other electrons are considered to form the “core” and its effect on the energy levels is reduced to the quantum defects δ ℓ , as discussed in Chap. 10. The case of helium is very different since the two ground-state electrons are in the same electronic state and differ only in the orientation of their spins. Thus spin is of paramount importance for the description of all but the hydrogenic atoms. Since helium has only two spins to deal with, their roles can be written explicitly. Because the two electrons are identical, quantum mechanics requires that the wavefunction undergoes nothing more than an unobservable phase shift when they are exchanged. This consequence is called the Pauli symmetrization principle, and is sometimes referred to as “the source of our individuality”. Moreover, it has a profound influence on the wavefunction and thus on the allowed energy states. It will be shown that this symmetry requirement dictates that two electrons cannot be in states with identical quantum numbers, and this topic will be discussed in detail in this chapter. Symmetry The profound difference between the hydrogen and helium wavefunctions arises because the two electrons are indistinguishable, and this results in a fundamental change of the energy-level structure. The concept of “spin”, introduced as a shortcut for the intrinsic magnetic moment of the electron in Chap. 7, now becomes a dominating feature of the description. Before discussing the helium energy levels, the effects of this symmetry with respect to exchange of identical particles is presented. The exchange operator In quantum mechanics, identical particles are inherently indistinguishable, so the exchange of the two electrons cannot change any observable and there should be no measurable effect on the overall wavefunction. However, there can be an overall phase change whose nature can easily be found by defining an operator P 12 that exchanges the electrons and applying it to the wavefunction. The exchange applies to all coordinates of the electron including its spin coordinate, but for simplicity only the spatial coordinate will be indicated here.