Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Parametric control in coupled fermionic oscillators

Arnab Ghosh

Published Oct 14, 2014• 5 citations
Publisher

Abstract

A simple model of parametric coupling between two fermionic oscillators is considered. Statistical properties, in particular the mean and variance of quanta for a single mode, are described by means of a time-dependent reduced density operator for the system and the associated $P$ function. The density operator for fermionic fields as introduced by Cahill and Glauber [K. E. Cahill and R. J. Glauber, Phys. Rev. A 59, 1538 (1999)] thus can be shown to provide a quantum mechanical description of the fields closely resembling their bosonic counterpart. In doing so, special emphasis is given to population trapping, and quantum control over the states of the system.

DOI
10.1103/physreva.90.043820

Related papers

Parametrically coupled fermionic oscillators: Correlation functions and phase-space description

Arnab Ghosh

A fermionic analog of a parametric amplifier is used to describe the joint quantum state of the two interacting fermionic modes. Based on a two-mode generalization of the time-dependent density operator, time evolution of the fermionic density operator is determined in terms of its two-mode Wigner and $P$ function. It is shown that the equation of motion of the Wigner function corresponds to a fermionic analog of Liouville's equation. The equilibrium density operator for fermionic fields developed by Cahill and Glauber is thus extended to a dynamical context to show that the mathematical structures of both the correlation functions and the weight factors closely resemble their bosonic counterpart. It has been shown that the fermionic correlation functions are marked by a characteristic upper bound due to Fermi statistics, which can be verified in the matter wave counterpart of photon down-conversion experiments.

Physical Review A•Jan 23, 2015•3 citations

Fermionic oscillator in a fermionic bath

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.

Physical Review E•Jul 30, 2012•21 citations

Quantum Statistics of Coupled Oscillator Systems

B. R. Mollow

The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the $P$ representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the $P$ representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.

Physical Review•Oct 25, 1967•95 citations

Quantum Theory of Parametric Amplification. I

B. R. Mollow, R. J. Glauber

The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the $P$ representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and $n$-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative $P$ representation for the density operator for a single mode of oscillation.

Physical Review•Aug 25, 1967•537 citations

Glauber P-representations for fermions

Stephen Mark Barnett, Bryan J. Dalton

Abstract The Glauber-Sudarshan P-representation for bosons is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart, introduced by Cahill and Glauber. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. Expressions for the moments involving products of mode annihilation and creation operators are obtained. For simplicity only one type of boson or fermion will be considered, but generalising to more types is straightforward.

Physica Scripta•Mar 14, 2023•1 citation

Density operators for fermions

Kevin E. Cahill, Roy J. Glauber

The mathematical methods that have been used to analyze the statistical properties of boson fields, and in particular the coherence of photons in quantum optics, have their counterparts for Fermi fields. The coherent states, the displacement operators, the P representation, and the other operator expansions all possess surprisingly close fermionic analogues. These methods for describing the statistical properties of fermions are based upon a practical calculus of anticommuting variables. They are used to calculate correlation functions and counting distributions for general systems of fermions.

arXiv:physics/9808029•Physical Review A•Feb 1, 1999•163 citations
A simple model of parametric coupling between two fermionic oscillators is considered. Statistical properties, in particular the mean and variance of quanta for a single mode, are described by means of a time-dependent reduced density operator for the system and the associated PPP function. The density operator for fermionic fields as introduced by Cahill and Glauber [K. E. Cahill and R. J. Glauber, Phys. Rev. A 59, 1538 (1999)] thus can be shown to provide a quantum mechanical description of the fields closely resembling their bosonic counterpart. In doing so, special emphasis is given to population trapping, and quantum control over the states of the system.
A fermionic analog of a parametric amplifier is used to describe the joint quantum state of the two interacting fermionic modes. Based on a two-mode generalization of the time-dependent density operator, time evolution of the fermionic density operator is determined in terms of its two-mode Wigner and PPP function. It is shown that the equation of motion of the Wigner function corresponds to a fermionic analog of Liouville's equation. The equilibrium density operator for fermionic fields developed by Cahill and Glauber is thus extended to a dynamical context to show that the mathematical structures of both the correlation functions and the weight factors closely resemble their bosonic counterpart. It has been shown that the fermionic correlation functions are marked by a characteristic upper bound due to Fermi statistics, which can be verified in the matter wave counterpart of photon down-conversion experiments.
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.
The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the PPP representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the PPP representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.
The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the PPP representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and nnn-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative PPP representation for the density operator for a single mode of oscillation.
Abstract The Glauber-Sudarshan P-representation for bosons is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart, introduced by Cahill and Glauber. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. Expressions for the moments involving products of mode annihilation and creation operators are obtained. For simplicity only one type of boson or fermion will be considered, but generalising to more types is straightforward.
The mathematical methods that have been used to analyze the statistical properties of boson fields, and in particular the coherence of photons in quantum optics, have their counterparts for Fermi fields. The coherent states, the displacement operators, the P representation, and the other operator expansions all possess surprisingly close fermionic analogues. These methods for describing the statistical properties of fermions are based upon a practical calculus of anticommuting variables. They are used to calculate correlation functions and counting distributions for general systems of fermions.