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AIP conference proceedings

Quantum mechanical techniques in light optics

Sameen Ahmed Khan

Published Jan 1, 2018• 6 citations
Publisher

Abstract

In the traditional prescriptions, the Fermat’s principle of least time or/and the Helmholtz equation are used to derive the beam-optical Hamiltonian. Now, we know that the Maxwell’s electromagnetism gives rise to the correct theory of light. So, we developed a formalism of beam optics starting with an exact matrix-representation of Maxwell’s equations. The resulting beam-optical Hamiltonian is not only exact but also has the mathematical structure of the Dirac equation, enabling us to employ quantum methodologies. The resulting formalism of Maxwell optics successfully describes beam optics along with polarization of light.

DOI
10.1063/1.5042383

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In the traditional prescriptions, the Fermat’s principle of least time or/and the Helmholtz equation are used to derive the beam-optical Hamiltonian. Now, we know that the Maxwell’s electromagnetism gives rise to the correct theory of light. So, we developed a formalism of beam optics starting with an exact matrix-representation of Maxwell’s equations. The resulting beam-optical Hamiltonian is not only exact but also has the mathematical structure of the Dirac equation, enabling us to employ quantum methodologies. The resulting formalism of Maxwell optics successfully describes beam optics along with polarization of light.
Classically, light is an electromagnetic phenomenon, described by Maxwell's equations. However, under certain conditions, such as low intensity or in the presence of certain nonlinear optical materials, light starts to behave differently, and we have to construct a ‘quantum theory of light’. We can exploit this quantum behaviour of light for quantum information processing, which is the subject of this book. In this chapter, we develop the quantum theory of the free electromagnetic quantum field. This means that we do not yet consider the interaction between light and matter; we postpone that to Chapter 7. We start from first principles, using the canonical quantization procedure in the Coulomb gauge: we derive the field equations of motion from the classical Lagrangian density for the vector potential, and promote the field and its canonical momentum to operators and impose the canonical commutation relations. This will lead to the well-known creation and annihilation operators, and ultimately to the concept of the photon. After quantization of the free electromagnetic field we consider transformations of the mode functions of the field. We will demonstrate the intimate relation between these linear mode transformations and the effect of beam splitters, phase shifters, and polarization rotations, and show how they naturally give rise to the concept of squeezing. Finally, we introduce coherent and squeezed states.
The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.
Light in media The backbone of theoretical quantum optics is the quantum field theory of light, the quantum theory of the electromagnetic field. Quantum electrodynamics becomes surprisingly complicated if one insists on formulating it in a relativistically and gauge-invariant form, as one surely should do in elementary particle physics. Moreover, in quantum optics, we are concerned with light quanta in materials such as glass, with quantum electromagnetism in media, which is even more complicated. To cut a long story short, here we follow a minimalistic approach where we develop the essentials of quantum electrodynamics with as little technical effort as possible, sailing around the cliffs of relativistic quantum field theories. We do not assume any prior knowledge of classical field theory; we only borrow a few ideas from classical electromagnetism (Jackson, 1998). Complications and problems still remain, but hopefully these are mostly no longer formal difficulties, but rather conceptual problems of the quantum nature of light. As Dr Samuel Johnson said on poetry “We all know what light is; but it is not easy to tell what it is. ” Maxwell's equations What is light? Light is a quantum object - we describe its properties by quantum observables, Hermitian operators, and its state by a state vector | ψ _ or density matrix ρ . We use the Heisenberg picture of quantum mechanics where the operators evolve in time, but the quantum state does not change.
The behavior of light and its interaction with matter is certainly one of the most interesting phenomena of physics. It has puzzled thinkers and scientists in ancient Greece (Archimedes) and through the Middle Ages, at least since the times of Robert Grosseteste (1168–1235), right up to the theories of Galileo Galilei and Isaac Newton. The classical electromagnetic theory of light (see Tab. 13.1) was beautifully established in the second half of the nineteenth century with the Maxwell equations. Quantum optics is nothing other than the quantum theory of light. It finds its roots in Planck's discovery (1900) that the assumption of energy quantization for a harmonic oscillator allowed us to solve the black-body problem (see Subsec. 1.5.1). The successive early steps that led to the quantization of the electromagnetic field include Einstein's explanation of the photoelectric effect with the introduction of the quantum of light (see Subsec. 1.2.1) and the interpretation of atomic spectra (see Subsec. 1.5.4). Quantum optics is usually not covered by conventional textbooks on quantum mechanics. We include its discussion here for two main reasons: first, because it is a fundamental and advanced application of the basic framework of quantum mechanics, and, second, because in the last 30 years quantum optics has played (and is still playing) a major role in the advancement and understanding of quantum mechanics.
The quantum theory of a magnetic electron lens with rotational symmetry about its straight optic axis has recently been studied entirely on the basis of the Dirac equation [R. Jagannathan, R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Lett. A 134, 457 (1989)]. Following the same type of algebraic approach, the present paper elaborates on the quantum mechanics of beam transport through a general electron-lens system with a straight axis. The basic theory is developed in a general framework, and examples of applications are treated under the paraxial approximation and thin-lens assumption. Extension of the theory to systems with curved optic axes is dealt with very briefly. The present formalism based on a ``beam-optical'' representation of the Dirac equation goes over directly to the Hamiltonian formalism of geometrical electron optics upon classicalization.