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arXiv:0909.3322 · Physica Scripta

Analytic approximation of the Tavis–Cummings ground state via projected states

Octavio Castaños, Eduardo Nahmad-Achar, Ramón López-Peña, Jorge G Hirsch

Published Oct 27, 2009• 26 citations
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Abstract

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.

DOI
10.1088/0031-8949/80/05/055401

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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 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.
We show that semiclassical states adapted to the symmetry of the Hamiltonian are an excellent approximation to the exact quantum solution of the ground and first excited states of the Dicke model. Their overlap with the exact quantum states is very close to 1 except in a close vicinity of the quantum phase transition. Furthermore, they have analytic forms in terms of the model parameters and allow us to calculate analytically the expectation values of field and matter observables. Some of these differ considerably from results obtained via the standard coherent states and by means of Holstein-Primakoff series expansion of the Dicke Hamiltonian. Comparison with exact solutions obtained numerically supports our results. In particular, it is shown that the expectation values of the number of photons and of the number of excited atoms have no singularities at the phase transition. We comment on why other authors have previously found otherwise.
A system of three-level atoms interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations is studied. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. A semi-classical approximation to the exact quantum solution is built by projecting coherent field and matter states to a specific value of the total number of excitations. The states in parameter-space and the expectation values of relevant observables are obtained explicitly, and it is shown that those for the projected states offer an excellent approximation to the exact quantum solution. The exact form of the ground state near the triple point in the Ξ configuration is given.
Quantum phase transition in Ξ\XiΞ-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.
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 general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for N a = 1 and N a = 4 particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with 2 modes of electromagnetic field. For N a = 1, the reduced density matrix of the matter allows us to determine the phase regions in a 2-simplex (associated to a general three dimensional density matrix), on the different 3-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established.