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Physica Scripta

A unitary Susskind–Glogower operator based on polarization

Chandravati Prajapati, D. Ranganathan

Published Oct 1, 2012• 0 citations
Publisher

Abstract

We define a Susskind–Glogower operator that is unitary and acts as a true ladder operator for photons of a given frequency and any polarization. The operator acts on a Hilbert space consisting of all number states in both polarizations. The phase operator derived from this ladder operator satisfies the canonical commutation relations with the number operator. The eigenstates of this ladder operator are also found. Our method based on all the observable states enables us to write the operators for objects such as beam splitters, phase shifters or polarizers, as constant matrices; those used to study classical polarized beams. The Pegg–Barnett operator and the two-mode operator of Ban are special cases of our operator. We compared the phase state defined here with that obtained from the original Susskind–Glogower case, as well as that obtained by the method of Pegg and Barnett in the limit of a very small number of photons being present, and found that there are differences which may enable a direct experimental comparison of these operators.

DOI
10.1088/0031-8949/86/04/045007

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We define a Susskind–Glogower operator that is unitary and acts as a true ladder operator for photons of a given frequency and any polarization. The operator acts on a Hilbert space consisting of all number states in both polarizations. The phase operator derived from this ladder operator satisfies the canonical commutation relations with the number operator. The eigenstates of this ladder operator are also found. Our method based on all the observable states enables us to write the operators for objects such as beam splitters, phase shifters or polarizers, as constant matrices; those used to study classical polarized beams. The Pegg–Barnett operator and the two-mode operator of Ban are special cases of our operator. We compared the phase state defined here with that obtained from the original Susskind–Glogower case, as well as that obtained by the method of Pegg and Barnett in the limit of a very small number of photons being present, and found that there are differences which may enable a direct experimental comparison of these operators.
We define a unitary phase operator for photons in a single momentum mode. The operator acts on a Hilbert space with basis consisting of all number states in both polarizations. The Susskind Glogower operator, E^=eiϕ^{\hat{E}} = e^{i \hatϕ}E^=eiϕ^​, thus defined is unitary and acts as a true ladder operator over all the states. It satisfies the commutation relation [E^,n^]=E^[{\hat{E}}, {\hat{n}}] = {\hat{E}}[E^,n^]=E^; consequently, the phase ϕ^\hatϕϕ^​ satisfies the canonical commutation relation [n^,ϕ^]=i[{\hat{n}}, {\hatϕ}] = i[n^,ϕ^​]=i. The eigenstates of the Susskind Glogower operator are found. Our model is able to directly account for phase measurements when polarization changes cannot be neglected and at the same time gives a well defined phase operator.
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.
We define a Hermitian phase operator for zero mass spin one particles (photons) by taking account polarization. The Hilbert space includes the positive helicity states and negative helicity states with opposite circular polarization. We define an operator which corresponds to the physical process of reversing the sense of polarization and acts as a bridge between positive helicity states and negative helicity states. The exponential phase operator obtained using the entire set is unitary and acts as ladder operator over all the states. The phase operator derived from this exponential operator satisfies the canonical commutation relations with the number operator. We have calculated the density matrix and the phase probability distribution of various states like coherent states, squeezed states and thermal states, to illustrate the utility of our operator.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.
By noticing that although single-mode Susskind–Glogower phase operator does not commute with photon number operator, two-mode nonlinear phase operator commutes with number-difference operator. We reveal that the common eigenstate <η| of two-particles' relative coordinate and total momentum involves the entanglement between phase and amplitude. We also reveal entanglement between number difference and amplitude inherent in the state |D,|η|>, where [Formula: see text] is a deductive state of |η>.