Abstract
I begin this chapter with a very brief review of atomic structure, covering the fine and hyperfine structures as well as the magnetic sublevels within the hyperfine manifold; these ideas will be used later to discuss trapping and cooling of atoms. The following sections describe the density matrix approach and build up to a derivation of the Optical Bloch Equations. These equations are the tools necessary to examine the interaction between an atom and the electromagnetic field and, ultimately, to derive an expression for the forces acting on an atom, as parametrised by the polarisability of that atom. The chapter continues with a note on the fluctuation–dissipation theorem and shows how the calculation of the full force acting on the atom allows the prediction of the equilibrium temperature a population of such atoms will tend to, and concludes with a short discussion on multi-level atoms. The reader is referred to Refs. [1–7], and references therein, for a more in-depth and complete treatment of certain parts of this chapter.
[...] [T]he semiclassical theory,when extended to take into account both the effect of the field on the molecules and the effect of the molecules on the field,reproduces almost quantitatively the same laws of energy exchange and coherence properties as the quantised field theory,even in the limit of one or a few quanta in the field mode.
E. T. Jaynes and F. W. Cummings, Proceedings of the IEEE 51, 89 (1963)
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Notes
- 1.
- 2.
It is perhaps interesting to note that Eq. (2.15) differs in sign from the Heisenberg equation of motion of an operator [1, Complement G\(_{\text{ III}}\)]. Mathematically, this is due to the operator ordering, of the form \(\hat{U}^{\dagger }\hat{H}\hat{U}\), used to transform from the Schrödinger picture, where the state vector is time-dependent and observables correspond to time-independent operators, to the Heisenberg picture, where only the observables are time-dependent.
- 3.
Formally, the Lindblad terms arise upon tracing the master equation Eq. (2.16) over the bath variables.
- 4.
- 5.
The dipole operator has odd parity and its diagonal elements are therefore zero.
- 6.
Since the two systems of equations generally describe time evolution on two vastly different timescales, it is reasonable to assume that the physical processes they describe do not interfere, and that the time derivatives can simply be added.
- 7.
We will refer to \(\zeta \) as ‘polarisability’ throughout—the linear relation between \(\chi \) and \(\zeta \), as well as the context, allows us to do this without giving rise to any ambiguity. \(\chi \) is perhaps more correctly referred to as a ‘susceptibility’.
- 8.
A note about notation is due: in this section we use the angle brackets, \(\langle \ \cdot \ \rangle \), to represent the average for a classical force or the expectation value for a quantised force, depending on the nature of the force.
- 9.
Considerations related to the general quantum case imply the failure of the Onsager hypothesis, which states that “the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process” [32], in cases such as where strong coupling between the system and the noise bath is allowed. Put differently, the quantum regression theorem cannot be called a ‘theorem’ in the mathematical sense. See the discussions in Refs. [33] and [34] for further details.
- 10.
In other words, an atom in one of the \(|g_i\rangle \) states will remain in that state, justifying their designation as ‘ground’ states.
- 11.
JB set up the problem, solved the three-level system case, provided the interpretation and wrote the paper; AX produced the analytical formulation of the eigensystem of the general \((N+2)\)-level Hamiltonian.
- 12.
The interaction picture, essentially, removes the fast time evolution from the state vectors and is accomplished by a transformation of the type in Eq. (2.69) where the transformation matrix is the diagonal matrix of eigenvalues of the time-independent part of the Hamiltonian. It lies in between the Schrödinger and the Heisenberg pictures. See Ref. [1, Complement G\(_{\text{ III}}\)].
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Xuereb, A. (2012). Atom–Field Interactions. In: Optical Cooling Using the Dipole Force. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29715-1_2
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