Phonon renormalization in the single-mode spin-boson Hamiltonian

Phonon Renormalization in the single-mode spin-boson Hamiltonian
The single-mode spin-boson Hamiltonian reads as follows :
: H = -h.sigma_x + omega.a^+.a + C.sigma_z(a + a^+)
With sigma_x en sigma_z the Pauli spin 1/2 matrices, a^+ and a are the boson creation and annihilation operators ;
2h is the tunnel splitting, omega is the boson frequency and C is the binding energy. The expression single-mode
means that only one phonon mode is considered, i.e., there is just one boson creation (and annihilation) operator
in the Hamiltonian.
The Nobel laureate Anthony J. Leggett has extensively studied the spin dynamics for a large sum or a continuum of
bosons (Ref.1). This is important because the two-level system can represent quantum bits and their coherence properties matter to design a quantum computer.
The present contribution points out that for particular physical systems it is also important to study the boson
or phonon dynamics. This is the case for tunneling impurities in solids of which the tunnel splitting 2h and the
frequency omega of an impurity-induced resonant mode are known. One such system is KCl:Li (Ref.2), but there exist many more of them. To the best of my knowledge no efforts have been done to study the spin and phonon dynamics together for the given model Hamiltonian.
In order to find the renormalized phonon frequency and tunnel splitting one has to diagonalize a finite matrix
representation of the Hamiltonian. The first numerical results are the following : As C increases the phonon is
hardening while the effective tunnel frequency goes to zero. Depending on the adiabaticity h/omega also phonon
softening accompanied by faster tunnelling can occur.

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