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Physical Review A

Structure and properties of the ground state of a two-level system arbitrarily coupled to a boson mode including the counter-rotating terms

G. Benivegna, ANTONINO MESSINA

Published Apr 1, 1987• 19 citations
Publisher

Abstract

The structure of the ground state of the Hamiltonian of a two-level system linearly coupled to a single mode of a radiation Bose-like field is studied for arbitrary values of the coupling constant including the counter-rotating terms. First of all the Hamiltonian is canonically transformed, introducing a new set of unitary operators in such a way that the eigenstates of the transformed Hamiltonian are exactly factorizable into eigenstates of the new pseudospin (1/2) and of the new field. The ground state is then found by a variational procedure. The validity of the results obtained by this approach is shown introducing a suitable class of canonical transformations by which it is possible to see that this variational ground state differs from the exact one only for perturbative contributions for which we give explicit expressions. Furthermore, we present investigations on the properties of this system in its ground state based on the calculation of the covariance of suitable pairs of operators. In this way we succeed in obtaining, among other things, a physically transparent meaning for the mathematical variational condition which determines the ground state.

DOI
10.1103/physreva.35.3313

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The structure of the ground state of the Hamiltonian of a two-level system linearly coupled to a single mode of a radiation Bose-like field is studied for arbitrary values of the coupling constant including the counter-rotating terms. First of all the Hamiltonian is canonically transformed, introducing a new set of unitary operators in such a way that the eigenstates of the transformed Hamiltonian are exactly factorizable into eigenstates of the new pseudospin (1/2) and of the new field. The ground state is then found by a variational procedure. The validity of the results obtained by this approach is shown introducing a suitable class of canonical transformations by which it is possible to see that this variational ground state differs from the exact one only for perturbative contributions for which we give explicit expressions. Furthermore, we present investigations on the properties of this system in its ground state based on the calculation of the covariance of suitable pairs of operators. In this way we succeed in obtaining, among other things, a physically transparent meaning for the mathematical variational condition which determines the ground state.
We present some characteristic properties of the ground state of a two-level unit linearly coupled to a bosonic field, exactly deduced without previously solving the eigenvalue problem for the physical system. Such properties are valid in the whole range of the characteristic parameters appearing in the Hamiltonian modelling this system. Our treatment is based on operatorial methods which combine symmetry considerations with some general peculiarities of the lowest energy state. In particular we present rigorous conclusions concerning the expectation value of the two-level unit energy and the mean value of the interaction energy for arbitrary strenght of the coupling constant. The relevance of these rigorous results in connection with an approximate variational determination of the ground state of the system is evidenced.
The Hamiltonian describing, without the rotating-wave approximation (RWA), the linear interaction between M bosonic modes with an Einstein spectrum and a single two-level atom is exactly and canonically transformed introducing M suitable collective independent field modes, in such a way that only one among them is coupled to the atom. Some physical consequences of this fact are analyzed and, in particular, the existence of radiation-trapping phenomena together with the possibility of atomic absorption suppression is established. The applicability of the RWA to this system is discussed and the importance of the effective-field statistics for the time evolution of the system is pointed out.
We present some characteristic properties of the ground state of a two-level system linearly coupled to a phonon field. Such properties are valid over the whole range of the parameters appearing in the Hamiltonian of the system. The knowledge of these exact results provides a physical basis for an accurate variational determination of the ground state. A physical picture of the self-trapping transition is discussed in terms of particle-field correlations. Our treatment is based on operator methods combining symmetry considerations with some general properties of the lowest energy state.
It is well known that, under the rotating-wave approximation, the Rabi Hamiltonian, which describes the interaction between a two-level atom and a single field mode, reduces to the Jaynes-Cummings model through exclusion of its counter-rotating term. In this work, it is found that, since photons in the mode are indistinguishable, there exist two conditions where the counter-rotating term can never be ignored. These conditions, all subject to the requirements that the atom-mode coupling is weak and the atom is initially in its ground state, are obtained in the stationary perturbation theory and confirmed through an analysis of the atom-mode system's evolution.
We introduce a combination of coherent states as variational test functions for the atomic and radiation sectors to describe a system of N a three-level atoms interacting with a one-mode quantised electromagnetic field, with and without the rotating wave approximation, which preserves the symmetry presented by the Hamiltonian. These provide us with the possibility of finding analytical solutions for the ground and first excited states. We study the properties of these solutions for the V -configuration in the double resonance condition, and calculate the expectation values of the number of photons, the atomic populations, the total number of excitations, and their corresponding fluctuations. We also calculate the photon number distribution and the linear entropy of the reduced density matrix to estimate the entanglement between matter and radiation. For the first time, we exhibit analytical expressions for all of these quantities, as well as an analytical description for the phase diagram in parameter space, which distinguishes the normal and collective regions, and which gives us all the quantum phase transitions of the ground state from one region to the other as we vary the interaction parameters (the matter-field coupling constants) of the model, in functional form.
In this article, by using the perturbation theory, we analytically solve the eigenvalue problem for the Hamiltonian describing the interaction of a Λ -type three-level atom with a single-mode radiation field without the rotating wave approximation (RWA). For this purpose, the atom–field interaction Hamiltonian, which contains the counter-rotating terms (CRTs), is transformed to an analytically solvable Hamiltonian by applying three successive unitary transformations. According to our calculations, the contribution of CRTs within the transformed Hamiltonian is in fact replaced by transforming the 'constant detuning' with the 'intensity-dependent detuning' in the first order, and the 'constant atom–field coupling' with the intensity-dependent coupling in the second order of the perturbation parameters. Then, by solving the eigenvalue problem for the transformed Hamiltonian, the eigenvector of the considered atom–field Hamiltonian is obtained analytically. Finally, after achieving the state vector of the atom–field system at an arbitrary time, a few nonclassical properties of the system state are investigated numerically. Meanwhile, we compare our results with the presence of RWA, from which the role of CRTs will be established.