In quantum optics, the Tavis–Cummings model is a theoretical model to describe an ensemble of identical two-level atoms coupled symmetrically to a single-mode quantized bosonic field. The model extends the Jaynes–Cummings model to larger spin numbers that represent collections of multiple atoms. It differs from the Dicke model in its use of the rotating-wave approximation to conserve the number of excitations of the system. Originally introduced by Michael Tavis and Fred Cummings in 1968 to unify representations of atomic gases in electromagnetic fields under a single fully quantum Hamiltonian — as Robert Dicke had done previously using perturbation theory — the Tavis–Cummings model's restriction to a single field-mode with negligible counterrotating interactions simplifies the system's mathematics while preserving the breadth of its dynamics. The model demonstrates superradiance, bright and dark states, Rabi oscillations and spontaneous emission, and other features of interest in quantum electrodynamics, quantum control and computation, atomic and molecular physics, and many-body physics. The model has been experimentally tested to determine the conditions of its viability, and realized in semiconducting and superconducting qubits.
Hamiltonian The Tavis–Cummings model assumes that for the purposes of electromagnetic interactions, atomic structures are dominated by their dipole, as they are for distant neutral atoms in the weak-field limit. Thus the only atomic quantity under consideration is its angular momentum, not its position nor fine electronic structure. Furthermore, the model asserts the atoms to be sufficiently distant that they don't interact with each-other, only with the electromagnetic field, modeled as a bosonic field (since photons are the gauge bosons of electromagnetism).
Formal derivation For two atomic-electronic states separated by a Bohr frequency ω e g {\displaystyle \omega _{eg}} , then transitions between the ground- and excited-states | g ⟩ {\displaystyle |g\rangle } and | e ⟩ {\displaystyle |e\rangle } are mediated by Pauli operators: σ ^ z = | e ⟩ ⟨ e | − | g ⟩ ⟨ g | {\displaystyle {\hat {\sigma }}_{z}=|e\rangle \langle e|-|g\rangle \langle g|} , σ ^ + = | e ⟩ ⟨ g | {\displaystyle {\hat {\sigma }}_{+}=|e\rangle \langle g|} , and σ ^ − = | g ⟩ ⟨ e | {\displaystyle {\hat {\sigma }}_{-}=|g\rangle \langle e|} , and the Hamiltonian separating these energy states in the j {\displaystyle j} th atom is H ^ A ( j ) = ℏ ω e g 2 σ ^ z ( j ) {\displaystyle {\hat {H}}_{A}^{(j)}={\frac {\hbar \omega _{eg}}{2}}{\hat {\sigma }}_{z}^{(j)}} . With N {\displaystyle N} independent atoms each subject to this energy gap, the total atomic Hamiltonian is thus H ^ A = ∑ j = 1 N H ^ A ( j ) = ω e g S ^ z {\displaystyle {\hat {H}}_{A}=\sum _{j=1}^{N}{\hat {H}}_{A}^{(j)}=\omega _{eg}{\hat {S}}_{z}} with total spin operators S ^ α = ℏ 2 ∑ j = 1 N σ ^ α ( j ) {\displaystyle {\hat {S}}_{\alpha }={\frac {\hbar }{2}}\sum _{j=1}^{N}{\hat {\sigma }}_{\alpha }^{(j)}} .
… excerpt ends here. Continue reading the full article.
