Paired Phonon Analysis of Many-Body Boson Systems.

T. Nishiyama

Enrico Perfetto, Michele Cini

We investigate the interplay of phonons and correlations in superconducting pairing by introducing a model Hamiltonian with on-site repulsion and couplings to several vibration branches having the Cu-O plane of the cuprates as a paradigm. We express the electron-phonon coupling (EP) through two force constants for O-Cu and O-O bond stretchings. Without phonons, this reduces to the Hubbard Model, and allows purely electronic W=0 pairing. A W=0 pair is a two-body singlet eigenstate of the Hubbard Hamiltonian, with no double occupancy, which gets bound from interactions with background particles. Indeed, this mechanism produces a Kohn-Luttinger-like pairing from the Hubbard repulsion, provided that its symmetry is not severely distorted. From the many-body theory, a canonical transformation extracts the effective two-body problem, which lends itself to numerical analysis in case studies. As a test, we use as a prototype system the $\cu$ cluster. We show analytically that at weak EP coupling the additive contributions of the half-breathing modes reinforce the electronic pairing. At intermediate and strong EP coupling and $U \sim t$, the model behaves in a complex and intriguing way.

R. D. Mattuck

Roberto P. L. Caporali

In this work, we developed a study of a set of Cooper pairs in a superconducting system and the control dynamics of the corresponding Quantum Control System. This paper deals specifically with the interaction in a crystal between a bosonic field given by the Cooper pairs and another bosonic field given by the System of vibrational modes of the phonons in the crystal, regarding applications related to Quantum Controllers. We consider an interaction of the bosons-bosons type, which specifically refers to the possible coupling between the phonon field and the bosonic field constituted by Cooper pairs. We develop the Hamiltonian of the whole system, giving the Hamiltonian of the Cooper Pairs, the Hamiltonian of the interacting phonon system, and the interaction Hamiltonian. In our study, we define, for the Quantum Control System relative to the bosons-bosons interaction, a Quantum Kalman Filter. We obtain, as a first result of this work, the expression that makes explicit the Quantum Kushner-Stratonovich equation in our case of interaction between Cooper pairs and phonons of the crystal lattice of a superconductor. It expresses the density of states ρ of the Cooper pairs as a function of time in the non-Markovian statistical system described by us. Furthermore, we obtain a second fundamental result of this work, expressing the Hamiltonian of the overall open system as a function of the Pauli matrix operator. This operator allows us to represent the same Hamiltonian of the open system described by us through the logic gates of a quantum controller. So, in the end, we obtain the control law of the temporal evolution of the bosonic system and the possibility of representing it through the logic gates of a quantum control system.

H. Aiba, K. Matsuyanagi

The selfconsistent-collective-coordinate method is applied to anharmonic quadrupole vibrations in a degenerate many j-shell model with the pairing-plus-quadrupole force. By comparing the result obtained by this method with that obtained by the boson-expansion method, it is shown that 1) the selfconsistent-collective-coordinate method yields, after a suitable canonical quantization, the result that agrees to order (j + 1/2)-1 with that obtained by the boson-expansion method, in the limit that the time-dependent Hartree-Bogoliubov space is truncated to the multiphonon space built out of the 2+ phonons, and 2) the couplings between the RPA 2+ phonons and the other modes (the pairing rotations and the noncollective modes) considerably modify the fourth-order boson Hamiltonian obtained by the conventional boson-expansion method.

R. D. Mattuck