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.
[1][2][3] 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
of the isolated two level system separated in energy by
are the raising and lowering operators of the two level system;
are the creation and annihilation operators for photons of energy
ℏ ω
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
[4] 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 identified as the Rabi frequency.
For the case that there is no electric field in the cavity, that is, the photon number
Then, the probability that the two level system goes from its ground state to its excited state as a function of time
is For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning
becomes a squared sinusoid with unit amplitude and period
two level systems are present in a single-mode cavity is described by the Tavis–Cummings model [5] , which has Hamiltonian Under the assumption that all two level systems have equal individual coupling strength
to the field, the ensemble as a whole will have enhanced coupling strength
As a result, the vacuum Rabi splitting is correspondingly enhanced by a factor of