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Physical Review A

Rotation angle, phases of oscillators with definite circular polarizations, and the composite ideal phase operator

A. Lukš, Vlasta Peřinová, J. Křepelka

Published Jul 1, 1994• 11 citations
Publisher

Abstract

The exposition of the formalisms of the quantum optical phase has been provided commencing from the quantum mechanics of a massive particle. The clockwise and counterclockwise components of the planar motion are shown to correspond to the signal and idler modes in the heterodyne detection scheme. The appropriate ideal phase proposal has been embodied in this scheme.

DOI
10.1103/physreva.50.818

Related papers

Quantum phase and phase dependent measurements

W. P. Schleich, Stephen Mark Barnett

The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.

Physica Scripta•Jan 1, 1993•74 citations

Sending quantum bits with rotating photons

Songhao Liu, Changjun Liao

Linearly polarized photon can be encoded by phase modulation of half wavelets composing the photon to form rotating photon that is circularly polarized clockwise or counterclockwise. Quantum key distribution can be performed by the rotating photons. Theoretical analysis indicates that this new scheme of quantum key distribution system uses less optical components and therefore less insert loses. The rotating photon is insensitive to birefringence. The experimental procedure is also described.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Sep 11, 2002•0 citations

Conceptions of quantum optical phase

Stephen Mark Barnett, Bryan J. Dalton

Much confusion has resulted from the failure to recognise the different concepts embodied in the single word "phase" as applied to a quantized electromagnetic field mode. In this paper we describe three conceptions of phase and the connections between them.

Physica Scripta•Jan 1, 1993•34 citations

Photon Phase Operator

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We have already shown that the coherent states are very close to the classical state of the well-defined amplitude and phase. A question not answered yet is: Does there exist an analogous quantum mechanical observable for the phase that is given by some Hermitian operator? We know from classical optics that phase of an EM field is an observable quantity. Hence, in quantum physics it should be associated with a Hermitian operator. In order to set up a quantitative description of the phase operator, we will introduce some of the ideas presented in the literature which led to the formulation of the Hermitian phase operator.

•Apr 19, 2016•0 citations

4.3 - Heterodyne ”Weak Measurements“ of Nanorad Beam Deflections

M. Pfeifer, P. Fischer

Generally a measurement yields (eigen)values that can be distinguished and recorded by a suitable measuring device. Aharanov, Albert, and Vaidman (AAV) considered the case of weak measurements in quantum mechanics, where the eigenvaluespectrum is not properly resolved. AAV showed that in this case it is helpful to choose a suitable postselection state, which should be almost orthogonal to the initial state, and which is able to give rise to arbitrarily large expectation values. Here, we discuss how weak value amplification can be adapted to measure the optical activity of chiral liquids. The hallmark of optical activity is that the liquid has different refractive indices for right and left circularly polarized light, which causes the rotation of the plane of polarization of a linearly polarized light beam traversing the solution. This small difference (<< 10-7) in refractive indices may, not only be detected in transmission (polarimetry), but can also be detected in refraction. A linearly polarized light beam incident at an interface between a chiral and an achiral medium will split into its two circular polarization components, as the two components refract with different angles of refraction. A position sensitive detector may be used to register the difference in beam positions. For chemical and pharmaceutical applications it is of interest to measure small optical activities (with correspondingly small refractive index differences) in minute liquid samples. In order to increase the sensitivity of the measurement it becomes necessary to increase the separation between the two split circular beam components. We show that conventional weak value amplification, as reported in the literature before, does not provide sufficient information to determine the handedness of a chiral liquid. Instead we describe how weak value amplification combined with a heterodyne detection scheme can measure both the concentration and the handedness of the solution. The Faraday Effect in a glass prism is used as a model system for optical activity. The linear polarized beam is split into its circular polarization components at the interface.. To amplify these small angular displacements (typically a few nanoradians) of the two circular beam components we have implemented a weak measurement with a suitable post-selection scheme. It is shown, that the signal is proportional to the circular birefringence and furthermore sensitive to the sign of the optical activity.

AMA Conferences OPTO and IRS² (International Conference on Optical Technologies for Sensing and Measurement and International Conference on Infrared Sensors and Systems)•Jan 1, 2011•0 citations

Hamiltonians for Quantum Optics

Z. Ficek, Mohamed Ridza Wahiddin

We often hear at seminars and presentations ‘Show me the Hamiltonian of your problem and I will tell you what problem you are talking about’. Therefore, to understand what quantum optics is about, we start from the description of a standard Hamiltonian of the problems considered in quantum optics. We illustrate the method of derivation of the explicit form of the Hamiltonian for a simple system that is composed of two subsystems that interact (communicate) with each other. As we shall see, the explicit form of the Hamiltonian is essential to explicitly calculate energy levels of a given combined system and the temporal evolution of an arbitrary operator representing the system.

•Apr 19, 2016•0 citations
The exposition of the formalisms of the quantum optical phase has been provided commencing from the quantum mechanics of a massive particle. The clockwise and counterclockwise components of the planar motion are shown to correspond to the signal and idler modes in the heterodyne detection scheme. The appropriate ideal phase proposal has been embodied in this scheme.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.
We have already shown that the coherent states are very close to the classical state of the well-defined amplitude and phase. A question not answered yet is: Does there exist an analogous quantum mechanical observable for the phase that is given by some Hermitian operator? We know from classical optics that phase of an EM field is an observable quantity. Hence, in quantum physics it should be associated with a Hermitian operator. In order to set up a quantitative description of the phase operator, we will introduce some of the ideas presented in the literature which led to the formulation of the Hermitian phase operator.
Generally a measurement yields (eigen)values that can be distinguished and recorded by a suitable measuring device. Aharanov, Albert, and Vaidman (AAV) considered the case of weak measurements in quantum mechanics, where the eigenvaluespectrum is not properly resolved. AAV showed that in this case it is helpful to choose a suitable postselection state, which should be almost orthogonal to the initial state, and which is able to give rise to arbitrarily large expectation values. Here, we discuss how weak value amplification can be adapted to measure the optical activity of chiral liquids. The hallmark of optical activity is that the liquid has different refractive indices for right and left circularly polarized light, which causes the rotation of the plane of polarization of a linearly polarized light beam traversing the solution. This small difference (<< 10-7) in refractive indices may, not only be detected in transmission (polarimetry), but can also be detected in refraction. A linearly polarized light beam incident at an interface between a chiral and an achiral medium will split into its two circular polarization components, as the two components refract with different angles of refraction. A position sensitive detector may be used to register the difference in beam positions. For chemical and pharmaceutical applications it is of interest to measure small optical activities (with correspondingly small refractive index differences) in minute liquid samples. In order to increase the sensitivity of the measurement it becomes necessary to increase the separation between the two split circular beam components. We show that conventional weak value amplification, as reported in the literature before, does not provide sufficient information to determine the handedness of a chiral liquid. Instead we describe how weak value amplification combined with a heterodyne detection scheme can measure both the concentration and the handedness of the solution. The Faraday Effect in a glass prism is used as a model system for optical activity. The linear polarized beam is split into its circular polarization components at the interface.. To amplify these small angular displacements (typically a few nanoradians) of the two circular beam components we have implemented a weak measurement with a suitable post-selection scheme. It is shown, that the signal is proportional to the circular birefringence and furthermore sensitive to the sign of the optical activity.
We often hear at seminars and presentations ‘Show me the Hamiltonian of your problem and I will tell you what problem you are talking about’. Therefore, to understand what quantum optics is about, we start from the description of a standard Hamiltonian of the problems considered in quantum optics. We illustrate the method of derivation of the explicit form of the Hamiltonian for a simple system that is composed of two subsystems that interact (communicate) with each other. As we shall see, the explicit form of the Hamiltonian is essential to explicitly calculate energy levels of a given combined system and the temporal evolution of an arbitrary operator representing the system.