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Concept
Vacuum Rabi oscillation
Summary
A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom interacts with a single-mode field confined to a limited volume V in an optical cavity. Spontaneous emission is a consequence of coupling between the atom and the vacuum fluctuations of the cavity field.
A mathematical description of vacuum Rabi oscillation begins with the Jaynes–Cummings model, which describes the interaction between a single mode of a quantized field and a two level system inside an optical cavity. The Hamiltonian for this model in the rotating wave approximation is
where is the Pauli z spin operator for the two eigenstates and of the isolated two level system separated in energy by ; and are the raising and lowering operators of the two level system; and are the creation and annihilation operators for photons of energy in the cavity mode; and
is the strength of the coupling between the dipole moment of the two level system and the cavity mode with volume and electric field polarized along .
The energy eigenvalues and eigenstates for this model are
where is the detuning, and the angle is defined as
Given the eigenstates of the system, the time evolution operator can be written down in the form
If the system starts in the state , where the atom is in the ground state of the two level system and there are photons in the cavity mode, the application of the time evolution operator yields
The probability that the two level system is in the excited state as a function of time is then
where is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number is zero, the Rabi frequency becomes .
Official source
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