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Cambridge University Press eBooks

What is a qubit?

Michel Le Bellac

Published Jun 15, 2006• 0 citations
Publisher

Abstract

The polarization of light Our first example of a qubit will be the polarization of a photon. First we briefly review the subject of light polarization. The polarization of light was demonstrated for the first time by the Chevalier Malus in 1809. He observed the light of the setting sun reflected by the glass of a window in the Luxembourg Palace in Paris through a crystal of Iceland spar. He showed that when the crystal was rotated, one of the two images of the sun disappeared. Iceland spar is a birefringent crystal which, as we shall see below, decomposes a light ray into two rays polarized in perpendicular directions, while the ray reflected from the glass is (partially) polarized. When the crystal is suitably oriented one then observes the disappearance (or strong attenuation) of one of the two rays. The phenomenon of polarization displays the vector nature of light waves, a property which is shared by shear sound waves: in an isotropic crystal, a sound wave can correspond either to a vibration transverse to the direction of propagation, i.e., a shear wave, or to a longitudinal vibration, i.e., a compression wave. In the case of light the vibration is only transverse: the electric field of a light wave is orthogonal to the propagation direction.

DOI
10.1017/cbo9780511755361.004

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In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.

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The aim of this chapter is to present the vector nature of quantum states. We shall also become acquainted with the concepts of projectors and the mathematical tool of matrix calculation. In this chapter we shall elaborate the concepts presented in the previous chapter. To this end, let us consider another important property of the photon: polarization. Light is an electromagnetic wave and ordinarily it oscillates in all directions orthogonal to its propagation direction, as it happens for the light coming from the sun or spreading out of a normal electric bulb. However, when passing through a polarization filter (e.g., polarized sunglasses) light may acquire a privileged oscillation direction, and in this casewe say that it is polarized in that direction (in this way polarized sunglasses are able to eliminate reflected glare).

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Maxwell showed that light consists of electric and magnetic fields that oscillate in directions perpendicular to the direction of propagation. Associated with this picture of light as an electromagnetic wave is an important property—the polarization of light. The polarization of light is related to the direction of oscillation of the electric field in an electromagnetic wave. In this chapter, the basic principles of quantum mechanics are discussed by studying the polarization property of a single photon. First the properties of a polarizer are presented and Malus’ law for polarized light is derived. Next it is shown that the basic features of quantum mechanics can be understood via an analysis of a single photon passing through a polarizer. This simple system allows an introduction of Dirac’s ket and bra notations for a quantum state. Finally the transformation properties of the quantum beam splitter and the polarization beam splitters are discussed.

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Quantitative assessment of polarization properties of waves opens up the way for effective exploitation of them in many amazing applications. Tamm surface waves (TSW) that propagate on the interface of periodic dielectric media are proposed for many applications in numerous reports. The polarization state of TSW is not simply intuitive and would not be extracted from reflection spectra. Here considering orientation sensitive nature of the interaction between polarized electromagnetic wave and atom, we try to quantitatively characterize the polarization state of TSWs, excited on the surface of the 1D photonic crystal. To do this we performed direct contact between TSW and rubidium atomic gas by fabrication of a one-dimensional photonic crystal-atomic vapor cell and applied a moderate external magnetic field to create geometrical meaning and a sense of directionality to dark lines in reflection intensity. Our experimental results indicate that transition lines in the reflection spectrum of our hybrid system modify dependent on the orientation of the applied magnetic field and the transverse spin of TSW. We have used these changes to redefine the geometry of Voigt and Faraday for evanescent waves, especially Tamm surface waves. In the end, we performed simple mathematical operations on absorption spectra and extract the ratio of longitudinal and transverse electric field components of the polarization vector of TSW equal to [Formula: see text].

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Conceptual and operational aspects of the description of the polarization of light in classical and quantum optics are discussed. Primary emphasis is placed on the concept of P -quasispin, which ensures an adequate analysis for arbitrary quantum light fields. In terms of this concept, the polarization density matrix, which is obtained by reduction of the total field density operator and which completely describes the polarization state of quantum radiation, as well as the operational (measured in polarization experiments) characteristics and the transformations of the polarization dynamics, are determined. Applications of the formalism developed to biphoton optics are discussed.

Optics and Spectroscopy•Jul 1, 2007•6 citations
The polarization of light Our first example of a qubit will be the polarization of a photon. First we briefly review the subject of light polarization. The polarization of light was demonstrated for the first time by the Chevalier Malus in 1809. He observed the light of the setting sun reflected by the glass of a window in the Luxembourg Palace in Paris through a crystal of Iceland spar. He showed that when the crystal was rotated, one of the two images of the sun disappeared. Iceland spar is a birefringent crystal which, as we shall see below, decomposes a light ray into two rays polarized in perpendicular directions, while the ray reflected from the glass is (partially) polarized. When the crystal is suitably oriented one then observes the disappearance (or strong attenuation) of one of the two rays. The phenomenon of polarization displays the vector nature of light waves, a property which is shared by shear sound waves: in an isotropic crystal, a sound wave can correspond either to a vibration transverse to the direction of propagation, i.e., a shear wave, or to a longitudinal vibration, i.e., a compression wave. In the case of light the vibration is only transverse: the electric field of a light wave is orthogonal to the propagation direction.
In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.
The aim of this chapter is to present the vector nature of quantum states. We shall also become acquainted with the concepts of projectors and the mathematical tool of matrix calculation. In this chapter we shall elaborate the concepts presented in the previous chapter. To this end, let us consider another important property of the photon: polarization. Light is an electromagnetic wave and ordinarily it oscillates in all directions orthogonal to its propagation direction, as it happens for the light coming from the sun or spreading out of a normal electric bulb. However, when passing through a polarization filter (e.g., polarized sunglasses) light may acquire a privileged oscillation direction, and in this casewe say that it is polarized in that direction (in this way polarized sunglasses are able to eliminate reflected glare).
Maxwell showed that light consists of electric and magnetic fields that oscillate in directions perpendicular to the direction of propagation. Associated with this picture of light as an electromagnetic wave is an important property—the polarization of light. The polarization of light is related to the direction of oscillation of the electric field in an electromagnetic wave. In this chapter, the basic principles of quantum mechanics are discussed by studying the polarization property of a single photon. First the properties of a polarizer are presented and Malus’ law for polarized light is derived. Next it is shown that the basic features of quantum mechanics can be understood via an analysis of a single photon passing through a polarizer. This simple system allows an introduction of Dirac’s ket and bra notations for a quantum state. Finally the transformation properties of the quantum beam splitter and the polarization beam splitters are discussed.
Quantitative assessment of polarization properties of waves opens up the way for effective exploitation of them in many amazing applications. Tamm surface waves (TSW) that propagate on the interface of periodic dielectric media are proposed for many applications in numerous reports. The polarization state of TSW is not simply intuitive and would not be extracted from reflection spectra. Here considering orientation sensitive nature of the interaction between polarized electromagnetic wave and atom, we try to quantitatively characterize the polarization state of TSWs, excited on the surface of the 1D photonic crystal. To do this we performed direct contact between TSW and rubidium atomic gas by fabrication of a one-dimensional photonic crystal-atomic vapor cell and applied a moderate external magnetic field to create geometrical meaning and a sense of directionality to dark lines in reflection intensity. Our experimental results indicate that transition lines in the reflection spectrum of our hybrid system modify dependent on the orientation of the applied magnetic field and the transverse spin of TSW. We have used these changes to redefine the geometry of Voigt and Faraday for evanescent waves, especially Tamm surface waves. In the end, we performed simple mathematical operations on absorption spectra and extract the ratio of longitudinal and transverse electric field components of the polarization vector of TSW equal to [Formula: see text].
Conceptual and operational aspects of the description of the polarization of light in classical and quantum optics are discussed. Primary emphasis is placed on the concept of P -quasispin, which ensures an adequate analysis for arbitrary quantum light fields. In terms of this concept, the polarization density matrix, which is obtained by reduction of the total field density operator and which completely describes the polarization state of quantum radiation, as well as the operational (measured in polarization experiments) characteristics and the transformations of the polarization dynamics, are determined. Applications of the formalism developed to biphoton optics are discussed.