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.

Mathematical treatment

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 |e \rangle and |g\rangle of the isolated two level system separated in energy by ; and are the raising and lowering operators of the two level system; and \hat{a} 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 \mathbf{d} of the two level system and the cavity mode with volume V and electric field polarized along. The energy eigenvalues and eigenstates for this model are where is the detuning, and the angle \theta_n 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 n+1 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 |e,n\rangle as a function of time t 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 n is zero, the Rabi frequency becomes. Then, the probability that the two level system goes from its ground state to its excited state as a function of time t is For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning \delta vanishes, and P_e(t) becomes a squared sinusoid with unit amplitude and period

Generalization to N atoms

The situation in which N two level systems are present in a single-mode cavity is described by the Tavis–Cummings model , which has Hamiltonian Under the assumption that all two level systems have equal individual coupling strength g 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 \sqrt{N}.

References and notes