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Physical Review A

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

Arnab Ghosh

Published Jan 23, 2015• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.91.013835

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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.
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.
The analysis of the quantum-mechanical parametric amplifier begun in the preceding paper is extended to describe the joint quantum state of the two interacting modes of oscillation of the system. Equations are derived for the time evolution of the Schr\"odinger density operator. Comparison with classical theory is made by describing the state of the quantum-mechanical system by means of the PPP representation. The time evolution of the weight function $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ for this representation, and of the corresponding Wigner distribution function $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$, is found. A decoupled form for the equations of motion for the field variables is exhibited which simplifies the solutions for these functions. It is shown that the amplification process gives rise to a correlation between the two mode amplitudes which becomes exceedingly strong with increasing time. The relation of this correlation to the time development of the functions $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ and $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ is discussed.
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.
Fermion-like coherent states are obtained as projections from the Glauber coherent states for a boson field, and the two sets of states have an interesting correspondence. It is shown that because of the correspondence the equation of motion of the density operator for a fermion interacting with a radiation field maps into a classical-like transport equation in a hypothetical boson phase space. Besides being useful in the description of systems of fermions involved in cooperative phenomena, it is shown that the use of the continuous basis offers some unique advantages even for systems which are essentially Markovian. As an illustration, the method is applied to the description of the time evolution of a two-level atom interacting with a blackbody radiation field.
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.
The description of the statistical dynamics of quantum oscillators is formulated in terms of the Wigner distribution, analogous to the more commonly used PPP distribution, with explicit formulas being obtained for its time evolution and for average values. This formulation is desirable because, e.g., the Wigner distribution always exists whereas the PPP distribution does not. The formalism is applied to the process of parametric amplification in a single mode, which may be considered as the degenerate form of the well-known two-mode case. This degeneracy gives rise to significantly different properties; for example, the PPP distribution for the single mode of interest evolves from a circularly symmetric two-dimensional Gaussian into an elliptically symmetric form and ceases to exist after a finite time, even for amplification in the presence of losses. This is contrary to the two-mode case. The corresponding Wigner distribution is found to exist as a well-behaved function for all time as expected, regardless of the amount of losses, and is used to calculate average values of various quantities of interest. It is found, e.g., that in the lossless case the average number of photons in the signal mode always becomes infinite as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. This is in contrast to the corresponding classical result for the lossless case which allows the signal to decay rather than to grow with time, depending on the relative phase between the signal and the pump. Field fluctuations are discussed and found to have some unusual properties. The combination of frequency up-conversion with single-mode amplification is also described briefly. The effect of the quantization of the pump oscillator is considered in an Appendix.