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Properties of finite Gaussians and the discrete-continuous transition

Nicolae Cotfas, Daniela Dragoman

Published Aug 17, 2016• 11 citations
OpenAlex

Abstract

Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger, a self-consistent theoretical framework describing physical systems characterised by a finite-dimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension $d$ of the system increases.

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Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger, a self-consistent theoretical framework describing physical systems characterised by a finite-dimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension ddd of the system increases.
Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger, a self-consistent theoretical framework describing physical systems characterized by a finite-dimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension d of the system increases.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.
The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but here we provide a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete-variable Gaussian states.
The mathematical description of the quantum harmonic oscillator is essentially based on the Gaussian function. In the case of a quantum oscillator with finite-dimensional Hilbert space, the position space consists in a finite number of points, and the ground state can be regarded as a finite version of the Gaussian function. Conversely, each finite Gaussian can be used in order to define some finite oscillators. We investigate certain finite Gaussians as concern their Fourier transform, Wigner function and associated oscillators.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called oscillator model in dimension 2 j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.
Abstract We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group . This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber–Sudarshan functions, as well as future investigations on multipartite quantum states.