chapter6 - Chapter 6 Interaction of Light and Matter Atomic...

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Chapter 6 Interaction of Light and Matter Atomic or molecular gases in low concentration show sharp energy eigen - spectra. This was shown for the hydrogen atom. Usually, there are in fi nitely many energy eigenstates in an atomic, molecular or solid-state medium and the spectral lines are associated with allowed transitions between two of these energy eigenstates. For many physical considerations it is already su cient to take only two of these possible energy eigenstates into account, for exam - ple those which are related to the laser transition. The pumping of the laser can be later described by phenomenological relaxation processes into the up - per laser level and out of the lower laser level. The resulting simple model is often called a two-level atom, which is mathematically also equivalent to a spin 1/2 particle in an external magnetic fi eld, because the spin can only be parallel or anti-parallel to the fi eld, i.e. it has two energy levels and energy eigenstates [4]. The interaction of the two-level atom with the electric fi eld of an electromagnetic wave is described by the Bloch equations. 6.1 The Two-Level Model An atom with only two energy eigenvalues is described by a two-dimensional state space spanned by the two energy eigenstates | e i and | g i . The two states constitute a complete orthonormal system. The corresponding energy eigenvalues are E e and E g , see Fig. 6.1. In the position-, i.e. x -representation, these states correspond to the wave functions ψ g ( x ) = h x | g i , and ψ e ( x ) = h x | e i . (6.1) 271
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272 CHAPTER 6. INTERACTION OF LIGHT AND MATTER Figure 6.1: Two-level atom The Hamiltonian operator of the two-level atom is in the energy representa - tion H A = E e | e i h e | + E g | e i h g | . (6.2) In this two-dimensional state space only 2 × 2 = 4 linearly independent linear operators are possible. A possible choice for an operator base in this space is 1 = | e i h e | + | g i h g | , (6.3) σ z = | e i h e | | g i h g | , (6.4) σ + = | e i h g | , (6.5) σ = | e i h e | . (6.6) The non-Hermitian operators σ ± could be replaced by the Hermitian oper - ators σ x,y σ x = σ + + σ , (6.7) σ y = j σ + + j σ . (6.8) The physical meaning of these operators becomes obvious, if we look at the action when applied to an arbitrary state | ψ i = c g | g i + c e | e i . (6.9) We obtain σ + | ψ i = c g | e i , (6.10) σ | ψ i = c e | g i , (6.11) σ z | ψ i = c e | e i c g | g i . (6.12)
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273 6.1. THE TWO-LEVEL MODEL The operator σ + generates a transition from the ground to the excited state, and σ does the opposite. In contrast to σ + and σ , σ z is a Hermitian operator, and its expectation value is an observable physical quantity with expectation value h ψ | σ z | ψ i = | c e | 2 | c g | 2 = w, (6.13) the inversion w of the atom, since | c e | 2 and | c g | 2 are the probabilities for fi nding the atom in state | e i or | g i upon a corresponding measurement. If we consider an ensemble of N atoms the total inversion would be W = N h ψ | σ z | ψ i . If we separate from the Hamiltonian (6.1) the term ( E e + E g ) / 2 1 , where 1 denotes the unity matrix, we rescale the energy values · correspondingly and obtain for the Hamiltonian of the two-level system 1
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