A beyond mean-field study of the Tavis-Cummings (TC) model is developed. This is the simplest model for describing the interaction of a radiation field with a system composed by an array of two-level atoms. The first correction to the mean-field ground state energy and the energy gap between the ground and the first excited states are computed. For the ground-state energy our result improves the mean-field calculation, as expected. For the gap, that cannot be calculated at mean-field level, a drop down to zero is obtained at the critical point where the system undergoes a second-order quantum phase transition.
A beyond mean-field study of the Tavis-Cummings (TC) model is developed. This is the simplest model for describing the interaction of a radiation field with a system composed by an array of two-level atoms. The first correction to the mean-field ground state energy and the energy gap between the ground and the first excited states are computed. For the ground-state energy our result improves the mean-field calculation, as expected. For the gap, that cannot be calculated at mean-field level, a drop down to zero is obtained at the critical point where the system undergoes a second-order quantum phase transition.
A system of N two-level atoms, Tavis-Cummings Dicke (TC-Dicke) model, interacting with a one-mode electromagnetic radiation field in the presence of the Stark shifts is studied, which is expected to predict new phenomena that are not explored in the original TC-Dicke model. We obtained the potential energy surface of the system using a trial state the direct product of coherent states in each subspace. In the frame of mean-field approaches, the variational energy is evaluated as the expectation value of the Hamiltonian for this state. The order of the quantum phase transitions is determined explicitly and numerically. We estimate the ground-state energy and the macroscopic excitations in the superradiant phase. Moreover, we investigated the critical properties of the TC-Dicke model in the classical spin limit and coherent state. We observed that in the thermodynamic limit, the energy surface takes a simple form a direct description of the phase transition. Moreover, it is found that when the microwave amplitude changes the new phase transition occurs with the Stark shift. The analytical solutions and numerical results, which appear in this paper are agreement with our paper which published recently in Int. J. Mod. Phys. B when we studied the same model using a different coherent state.
We show that the ground state of the Tavis-Cummings (TC) model of N 2-level systems in interaction with a 1-mode radiation field can be mirrored by a semi-classical state to an excellent approximation. The latter is obtained through a projection of a variational tensorial product of coherent states onto the value of the constant of motion of the TC Hamiltonian which minimises the (classical) energy of the ground state. This projection restores the Hamiltonian symmetry, and is amiable to analytical calculations. The observed super-radiant phase-transitions are reflected by this projected state, and an excellent agreement with the exact quantum solution for the ground state is obtained as regards to the entanglement entropy between the matter and field, the squeezing parameter of the state, the probability distributions of photons and atoms, and the expectation value of all field and matter observables of interest, including the photon number dispersion. All these are given in analytic form. The fitness between our approximation and the exact quantum solution is measured through the evaluation of the fidelity parameter; this is very close to 1, except in the region of classical phase transitions, as is to be expected. The presented approach may enable novel investigations of super-radiant and sub-radiant states of artificial atoms, where the relevant control parameters can be manipulated to cross the separatrix.
Abstract The interaction between anharmonic quantum emitters (e.g. molecular vibrations) and confined electromagnetic fields gives rise to quantum states with optical and chemical properties that are different from those of their precursors. The exploration of these properties has been typically constrained to the first excitation manifold, the harmonic approximation, ensembles of two-level systems [Tavis–Cummings (TC) model], or the anharmonic single-molecule case. The present work studies, for the first time, a collective ensemble of identical multi-level anharmonic emitters and their dipolar interaction with a photonic cavity mode, which is an exactly solvable many-body problem. The permutational properties of the system allow identifying symmetry classified submanifolds in the energy spectrum. Notably, in this approach, the number of particles, typically in the order of several millions, becomes only a parameter from the operational standpoint, and the size of the dimension of the matrices to diagonalize is independent of it. The formalism capabilities are illustrated by showing the energy spectrum structure, up to the third excitation manifold, and the calculation of the photon contents as a permutationally invariant quantity. Emphasis is placed on (a) the collective (superradiant) scalings of light–matter couplings and the various submanifolds of dark (subradiant) states with no counterpart in the single-molecule case, as well as (b) the delocalized modes containing more than one excitation per molecule with no equivalent in the TC model. We expect these findings to be applicable in the study of non-linear spectroscopy and chemistry of polaritons.
Quantum phase transition in Ξ-configuration Tavis-Cummings model driven by an electromagnetic field for a finite number of atoms
The quantum phase transitions of a three-level Tavis-Cummings model in the \Xi Ξ -configuration driven by a single-mode electromagnetic field in a QED cavity are studied for n_a=1,\,2 n a = 1 , 2 , and 4 4 atoms. Although conventional observables like photon number and energy level populations suffice to locate quantum phase transitions, they often lack sensitivity to finite-size effects. Here, we show that the expectation values of the quadratic Casimir invariants associated with the atomic subsystems offer a more robust measure, exhibiting sharp signatures of criticality that are validated by fidelity calculations. Additionally, the von Neumann entropy is employed to map the entanglement landscape between matter and the quantized field. A key finding is that the external driving field acts as a control parameter that shrinks the normal phase region; as the driving intensity increases, the separable ground state vanishes, leading to the dominance of field-matter entanglement across the parameter space.
We show that an excellent approximation to the exact quantum solution of the ground state of the Tavis–Cummings model is obtained by means of a semi-classical projected state. This state has an analytical form in terms of model parameters and, in contrast to the exact quantum state, allows for an analytical calculation of the expectation values of field and matter observables, entanglement entropy between field and matter, squeezing parameter and population probability distributions. The fidelity between this projected state and the exact quantum ground state is very close to 1, except for the region of classical phase transitions. We compare the analytical results with those of the exact solution obtained through direct Hamiltonian diagonalization as a function of atomic separation energy and matter–field coupling.
We apply the energy surface method to study a system of Na three-level atoms interacting with a one-mode radiation field in the $\ensuremath{\Xi}$, $\ensuremath{\Lambda}$, and V configurations. We obtain an estimation of the ground-state energy, the expectation value of the total number of excitations, and the phase diagram of the model in the interaction parameter space and compare the results with the exact solutions. The phase diagram separates the behavior of, in general, a dominant single state without excitations (M=0) from a collective state with excitations ($M\ensuremath{\ne}0$). We have first- and second-order phase transitions, except for the V configuration, which only presents second-order phase transitions.