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arXiv:1211.3697

Replay to "Comment on 'Quantum phase for an arbitrary system with finite-dimensional Hilbert space'

D. Aesenovic, Nikola Burić, Dragomir M Davidović, Slobodan Prvanović

Published Nov 15, 2012• 0 citations
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Abstract

We point out the crucial difference between the relative and absolute phase observables treated in our contribution \cite{1} and in the Comment by Hall and Pegg \cite{HP} respectively. The main contribution of our work is to show that the quantum expectation of the relative phase is highly discontinuous function of the frequency and to point out interesting dependence of the phase on the number-theoretic nature of the frequencies.

DOI
10.48550/arxiv.1211.3697

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We point out the crucial difference between the relative and absolute phase observables treated in our contribution \cite{1} and in the Comment by Hall and Pegg \cite{HP} respectively. The main contribution of our work is to show that the quantum expectation of the relative phase is highly discontinuous function of the frequency and to point out interesting dependence of the phase on the number-theoretic nature of the frequencies.
A construction of covariant quantum phase observables, for Hamiltonians with a finite number of energy eigenvalues, has been recently given by D. Arsenovi\ifmmode \acute{c}\else \'{c}\fi{} et al. [Phys. Rev. A 85, 044103 (2012)]. For Hamiltonians generating periodic evolution, we show that this construction is just a simple rescaling of the known canonical ``time'' or ``age'' observable, with the period TTT rescaled to $2\ensuremath{\pi}$. Further, for Hamiltonians generating quasiperiodic evolution, we note that the construction leads to a phase observable having several undesirable features, including (i) having a trivially uniform probability density for any state of the system, (ii) not reducing to the periodic case in an appropriate limit, and (iii) not having any clear generalization to an infinite energy spectrum. In contrast, we note that a covariant time observable has been previously defined for such Hamiltonians, which avoids these features. We also show how this ``quasiperiodic'' time observable can be represented as the well-defined limit of a sequence of periodic time observables.
Abstract In quantum mechanics the position and momentum operators are related to each other via the Fourier transform. In the same way, here we show that the so-called Pegg–Barnett phase operator can be obtained by the application of the discrete Fourier transform to the number operators defined in a finite-dimensional Hilbert space. Furthermore, we show that the structure of the London–Susskind–Glogower phase operator, whose natural logarithm gives rise to the Pegg–Barnett phase operator, is contained in the Hamiltonian of circular waveguide arrays. Our results may find applications in the development of new finite-dimensional photonic systems with interesting phase-dependent properties.
In this paper, we study the non-Gaussianity of the eigenstates of the Pegg-Barnett phase observable. By computing the Wigner functions of the eigenstates, we confirm that they take negative values in specific regions of the phase space. The Pegg-Barnett phase-operator eigenstates are defined in a finite-dimensional Hilbert space. Thus, we examine how their negativity volumes depend on the dimension of the Hilbert space. Moreover, we present a quantum-optical circuit that generates these eigenstates and identify single-photon detection as the origin of their non-Gaussianity. To investigate a more realistic experimental implementation, we introduce imperfect single-photon detectors with non-unit efficiency into the circuit. We then examine how the detection probability, the output-ideal fidelity, and the negativity volume of the approximate eigenstate output from the circuit change with the detector efficiency. Finally, as a practical application, we consider a phase-estimation experiment of an arbitrary unknown state by injecting both the unknown state and a known Pegg-Barnett eigenstate into a 50-50 beam splitter and individually counting the numbers of photons emitted from its two output ports.