The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It provides a simple and generally solvable example of light–atom interactions and is named after Isidor Isaac Rabi.
Classical Rabi problem In the classical approach, the Rabi problem can be represented by the solution to the driven damped harmonic oscillator with the electric part of the Lorentz force as the driving term:
x ¨ a + 2 τ 0 x ˙ a + ω a 2 x a = e m E ( t , r a ) , {\displaystyle {\ddot {x}}_{a}+{\frac {2}{\tau _{0}}}{\dot {x}}_{a}+\omega _{a}^{2}x_{a}={\frac {e}{m}}E(t,\mathbf {r} _{a}),}
where it has been assumed that the atom can be treated as a charged particle (of charge e) oscillating about its equilibrium position around a neutral atom. Here xa is its instantaneous magnitude of oscillation, ω a {\displaystyle \omega _{a}} its natural oscillation frequency, and τ 0 {\displaystyle \tau _{0}} its natural lifetime:
2 τ 0 = 2 e 2 ω a 2 3 m c 3 , {\displaystyle {\frac {2}{\tau _{0}}}={\frac {2e^{2}\omega _{a}^{2}}{3mc^{3}}},}
which has been calculated based on the dipole oscillator's energy loss from electromagnetic radiation. To apply this to the Rabi problem, one assumes that the electric field E is oscillatory in time and constant in space:
E = E 0 [ e i ω t + e − i ω t ] = 2 E 0 cos ω t , {\displaystyle E=E_{0}[e^{i\omega t}+e^{-i\omega t}]=2E_{0}\cos \omega t,}
and xa is decomposed into a part ua that is in-phase with the driving E field (corresponding to dispersion) and a part va that is out of phase (corresponding to absorption):
x a = x 0 ( u a cos ω t + v a sin ω t ) . {\displaystyle x_{a}=x_{0}(u_{a}\cos \omega t+v_{a}\sin \omega t).}
Here x0 is assumed to be constant, but ua and va are allowed to vary in time. However, if the system is very close to resonance ( ω ≈ ω a {\displaystyle \omega \approx \omega _{a}} ), then these values will be slowly varying in time, and we can make the assumption that u ˙ a ≪ ω u a {\displaystyle {\dot {u}}_{a}\ll \omega u_{a}} , v ˙ a ≪ ω v a {\displaystyle {\dot {v}}_{a}\ll \omega v_{a}} and u ¨ a ≪ ω 2 u a {\displaystyle {\ddot {u}}_{a}\ll \omega ^{2}u_{a}} , v ¨ a ≪ ω 2 v a {\displaystyle {\ddot {v}}_{a}\ll \omega ^{2}v_{a}} . With these assumptions, the Lorentz force equations for the in-phase and out-of-phase parts can be rewritten as
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