Mathematical treatment
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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
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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
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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
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where is the detuning, and the angle is defined as
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Given the eigenstates of the system, the time evolution operator can be written down in the form
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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
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The probability that the two level system is in the excited state as a function of time is then
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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 . Then, the probability that the two level system goes from its ground state to its excited state as a function of time is
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For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning vanishes, and becomes a squared sinusoid with unit amplitude and period
Generalization to N atoms
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The situation in which two level systems are present in a single-mode cavity is described by the Tavis–Cummings model
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, which has Hamiltonian
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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 .[6]
References and notes
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