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Risultati rigorosi riguardanti lo stato fondamentale dell'interazione spin-fonone

Elisabetta Paladino, ANTONINO MESSINA

Published Jan 1, 1994• 0 citations
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Abstract

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.

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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.
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.
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.
Ground states of a spin-boson Hamiltonian, describing one two-level system (a spin) coupled to infinitely many harmonic oscillators (bosons), are studied. This spin-boson Hamiltonian is a prototype for the description of a ‘‘small’’ system (e.g., a molecule) coupled to its environment. The respective ground-state vector(s) are approximated by a linear combination of two coherent-state vectors corresponding to the two levels of the spin. Interest concentrates mainly on phase-transition phenomena (generation of superselection rules) in case the parameters of the Hamiltonian (level splitting of the spin, frequencies of the field modes, and coupling constants) exhibit an infrared singularity. The resulting phase diagrams are shown to satisfy reasonably well the rigorous bounds derived by Spohn, and in particular distinguish between the superohmic, ohmic, and subohmic regime in the sense of Leggett. Nevertheless, the approximation method used is simple enough so that everything can be explicitly calculated. Former results by Pfeifer as well as results by Emery and Luther, Zwerger and Harris and Silbey are extended and discussed.
We demonstrate that the ground energy of a two-level system coupled with a bosonic environment can be labelled with a quantum number related to the total excitation number operator and independent of the coupling strength. Our approach is exact and is based on operator methods combining symmetry considerations with general properties of the lowest energy state. The relevance of our result in connection with the nature of the transition from the weak to strong coupling regime is briefly discussed.
We present a variational method for the ground-state solutions of the spin-Bose models by means of the spin coherent-state transformation. For the Jaynes-Cummings (J-C) models with and without the rotating-wave approximation, the ground-state energies obtained by this method are in perfect agreement with the results from numerical diagonalization in the whole region of the coupling between a light field and an atom. The present variational-method can be directly used to solve the ground-state energies of the Dicke models with arbitrary atom-numbers and furoher study the quantum phase transition; while the variational-method based on the Holstein-Primakoff transformation is only valid in principle for the thermodynamic limit with the atom-number tending to infinity.