Tavis–Cummings model
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,[2] bright and dark states,[3] 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.
[4] The model has been experimentally tested to determine the conditions of its viability,[5][6] and realized in semiconducting[7] and superconducting qubits.
[1] Thus the only atomic quantity under consideration is its angular momentum, not its position nor fine electronic structure.
Similarly, in a free field with no modal restrictions, creation and annihilation operators dictate the presence of photons in each mode:
specifies the coupling strength of the total dipole to each electric field mode, and functioning as a Rabi frequency that scales with ensemble size
In total, the Tavis–Cummings Hamiltonian includes the atomic and photonic self-energies and the atom-field interaction:
The Tavis–Cummings model as described above exhibits two symmetries arising from the Hamiltonian's commutation[1] with excitation number
The size of each of these smallest blocks (irreps of SU(2)) determine the bounds of the final quantum number that specifies the eigenenergy:
From these elements, one can express Schrödinger equations of motion to demonstrate the photon field's ability to mediate entanglement formation between atoms without atom-atom interactions:[8]
, for which the fine-tuned, multivariate dependence on quantum numbers demonstrates the difficulty of solving the Tavis-Cumming model's eigensystem.
Here, a few approximate methods, and an exact solution involving Stark shifts and Kerr nonlinearities follow.
In 1996, Nikolay Bogoliubov (son of the 1992 Dirac Medalist of the same name), Robin Bullough, and Jussi Timonen found that adding quadratic excitation-dependent terms to the Tavis–Cummings Hamiltonian allowed for an exact analytic eigensystem.
[10] In the limit where these Kerr and Stark shifts vanish, this solution can recover the eigensystem of the unmodified Tavis–Cummings system.
complex-parametrized operator matrices (that is, matrices whose elements are operators), one acting on the bosonic degrees of freedom and the other on the spin degrees of freedom produces a monodromy matrix whose determinant is directly proportional to
Manipulating the monodromy matrix allows its spectral parameter to determine the Hamiltonian eigenstates and eigenenergies[11] as the complex roots
) through global interactions,[8] as was explored in the 2003 paper by Tessier et al. One realization by Tuchman et al., in 2006, used a stream of ultracold Rubidium-87 atoms (
, or 12% its maximum possible value,[12] indicating very high interatomic coherence relative to experimental capabilities of the time.
A seminal result from Fink et al. in 2009 involved 3 transmons as virtual "atoms"[3] with qubit-dependent Bohr frequencies
[3] To ensure symmetric coupling of the qubits to the field, each transmon was placed at an antinode of the standing wave, and to best conserve excitations by minimizing photon leakage, the resonator was kept ultracold (20mK) which ensured a high quality factor.
[7] The micrometer regime is a far greater distance than that over which semiconducting qubits had previously achieved entanglement, and the difficulty of long-range interactions in semiconducting qubits was at the time a major weakness compared to other quantum computing platforms, for which the Tavis–Cummings model's ability to form entanglement through global atom-field interactions is one solution.
[8] By observing the reflection amplitude of field waves between the SQUID array and the DQDs, the team isolated the photon number states as they smoothly coupled to the first qubit to form superpositional Jaynes-Cummings eigenstates when the first qubit tuned to the resonator.
signalling qubits interacting with "virtual" photons, measured by the phase shift of the field rather than the reflection amplitude.
[7] Recent investigations by Johnson, Blaha, et al., have verified and explained two major regimes where the Tavis–Cummings model fails to predict physical reality,[5] both following from systemic parameters approaching or exceeding the free spectral range
then the coupling enters the so-called "superstrong" regime and atom-light interactions must consider multiple field-modes.
As photons cross the waveguide and interact sequentially with the atoms in the ensemble, they accumulate phase at phenomenon-dependent rates.
The total phase accumulated by electromagnetic waves in one round-trip of the waveguide may manifest resonances causing high transmission rates under specific dephasings
Using a fluid of ultracool Cesium atoms surrounding a nanofiber-section of a 30m fiber-ring resonator, the team coupled the atoms to the light passing through the nanofiber via an evanescent field, measuring the light's transmission for variable
[6] The data from the nanofiber-Cesium experiment agreed better with the cascade model's predictions than with the Tavis–Cummings', specifically in the parametrically violating regimes above.
