On the exponential form of the displacement operator for different systems

Václav Potoček,Stephen M Barnett

doi:10.1088/0031-8949/90/6/065208

Václav Potoček, Stephen M Barnett

Open Access

https://doi.org/10.1088/0031-8949/90/6/065208

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Abstract

The family of displacement operators , a central concept in the theory of coherent states of a quantum mechanical harmonic oscillator, has been successfully generalized to systems of quantized, cyclic or finite position coordinates. However, out of the plethora of mutually equivalent expressions for the displacement operators valid in the continuous case, only few are directly applicable in the other systems of interest. The aim of this paper is to strengthen the analogy between the different cases by identifying the root cause of the issues accompanying the straightforward generalization of certain important expressions and, more importantly, offering alternative ones of general validity. Ultimately we arrive at an algorithm allowing one to express any displacement operator as an exponential of a pure imaginary multiple of a generalized ‘quadrature’ observable that is not obtained by a linear combination of position and momentum observables but rather by a shear transform of one of them in the system's phase space.

Highlights

The displacement operators represent a powerful tool in quantum optics as well as other parts of quantum theory
Originating in the study of a quantum mechanical harmonic oscillator, the Heisenberg unitary group of continuous variable displacement operators and their complex unit multiples plays a key role in the theory of coherent states, semiclassical methods and quasiprobability distributions
We have presented a number of alternative expressions used for the displacement operator in the continuous variable systems and discussed the possibilities of their generalization to infinite lattice, angle—angular momentum systems, and finite dimensional systems

Summary

Introduction

The displacement operators represent a powerful tool in quantum optics as well as other parts of quantum theory Among other uses, they can be employed in generating coherent states from the vacuum, defining the characteristic function of a quantum state, converting expressions for measurement probabilities involving coherent states to vacuum expectation values, reconstruction of a state from its P-representation, provides an integral resolution of unity, and more [1]. The aim of this paper is to establish such a framework by introducing a number of mathematical expressions for the displacement operator in each of the cases and comparing them with the reference continuous-variable system side by side, avoiding the reliance on the Lie algebraic methods where they are inaccessible.

Displacement in continuous variable systems—an overview

Finite-dimensional systems

Example: a qutrit

Conclusions