Induced Emission in a Strong Monochromatic Field

Abstract

Of most interest for laser theory is the case were the frequency ω of the external field differs from the frequency ω _mn of the transition between the levels m, n by the order of the width of the corresponding spectral line. For gas systems at moderate pressures and electron densities a typical value of line width in the optical and infrared regions of the spectrum is 10⁹-10¹⁰ sec^-1. The transition frequencies are approximately 10¹⁵ sec^-1. Such a relationship between the line widths and transition frequencies allows the use of the so-called two-level scheme. Let the atom* in the absence of an external electromagnetic field have stationary states 1, 2,…, n,.., m,… with energies E₁,…, E_n,…, E_m,… ¡and wave functions ψ ₁,…, ψ _n,…, ψ _m,…. If the field spectrum is concentrated close to some natural frequency ω _mn, the field in these conditions will lead to transitions only between the states n, m. Hence, the wave function of the atom in the presence of a field can be sought in the form

$$ \Psi (t) = {a_m}(t){\psi_m}{e^{{ - i\frac{{{E_m}}}{\hbar }t}}} + {a_n}(t){\psi_n}{e^{{ - i\frac{{{E_n}}}{\hbar }t}}}. $$

((3.1))

All the other states of the atom are manifested either in excitation of the levels m, n, which is reflected in the initial conditions for α _m(t), α _n(t), or in the decay of states m, n due to spontaneous transitions. The Hamiltonian of the system

$$ \mathop{H}\limits^{ \wedge } = {\mathop{H}\limits^{ \wedge }_0} + \hbar \mathop{V}\limits^{ \wedge }, \quad \mathop{V}\limits^{ \wedge } = \mathop{p}\limits^{ \wedge } E/\hbar $$

((3.2))

contains the Hamiltonian Ĥ₀ of the atom in the absence of an external field and the term ħV̂, which describes the interaction of the atom with the field E; p̂ denotes the dipole moment operator.