The Fermionic Oscillator

Andrey G. Grozin

T. Miyakawa, T. Suzuki, Hiroyuki Yabu

The behavior of collective oscillations of a trapped boson-fermion mixed condensate is studied in the sum-rule approach. Mixing angle of bosonic and fermionic multipole operators is introduced so that the mixing characters of the low-lying collective modes are studied as functions of the boson-fermion interaction strength. For an attractive boson-fermion interaction, the low-lying monopole mode becomes a coherent oscillation of bosons and fermions and shows a rapid decrease in the excitation energy towards the instability point of the ground state. In contrast, the low-lying quadrupole mode keeps a bosonic character over a wide range of the interaction strengths. For the dipole mode the boson-fermion in-phase oscillation remains to be the eigenmode under the external oscillator potential. For weak repulsive values of the boson-fermion interaction strengths we found that an average energy of the out-of-phase dipole mode stays lower than the in-phase oscillation. Physical origin of the behavior of the multipole modes against boson-fermion interaction strength is discussed in some detail.

Arnab Ghosh, Sudarson Sekhar Sinha, Deb Shankar Ray

The quantum dissipation of a fermionic oscillator in an environment of fermionic oscillators is considered. Since fermions anticommute, their eigenvalues are anticommuting numbers. Based on an expansion of the reduced density operator in terms of fermionic coherent states which make use of these anticommuting numbers or Grassmann variables, a Fokker-Planck equation for the associated quasiprobability distribution function is derived. The density operator approach to fermionic fields as introduced by Cahill and Glauber [Phys. Rev. A 59, 1538 (1999)] thus can be extended to the nonequilibrium domain.

A. M. Perelomov

Olivier Henri‐Rousseau, Paul Blaise

Shukla, Dheeraj, Upadhyay, Sudhaker

In this article, we explore the inconsistencies in the physics of fermionic oscillators and propose potential solutions to address them. By rigorously deriving the Hamiltonian and Lagrangian from first principles, we aim to provide a comprehensive and fundamental understanding of the system. Furthermore, we calculate the partition function for a system of fermionic oscillators by drawing a direct analogy to Planck's treatment of energy distribution in bosonic oscillators, offering a parallel approach to this well-established method. Our study extends beyond the conventional framework by investigating the generalized angular momentum algebra within the context of Schwinger's oscillator model. This includes a detailed examination of the algebraic structures for combinations of bosonic-bosonic, bosonic-fermionic, and fermionic-fermionic oscillators. Through this, we delve into these systems' underlying symmetries and algebraic richness, shedding light on the intricate relationships between these different types of oscillators. In addition to these foundational aspects, we explore the broader implications of this generalized Schwinger approach. Our analysis touches upon potential applications and consequences of this formalism, offering insights that could be relevant to various areas of theoretical physics. This work paves the way for a deeper understanding of quantum oscillators and their role in modern physics by bridging the gap between bosonic and fermionic oscillators.