This chapter presents the basic principles of quantum optics. It starts from recalling the main elements of quantum mechanics in Heisenberg and Schroedinger interpretations, then the creation and annihilation operators are introduced, and bosonic and fermionic commutation rules are discussed. The secondary quantization theory is used to show differences between coherent, thermal, and Fock states of light; introduce squeezing and antisqueezing; and discuss the statistics of photons. Finally, the polarization and second-order coherence of light are discussed from the point of view of the quantum optics.
This chapter presents the basic principles of quantum optics. It starts from recalling the main elements of quantum mechanics in Heisenberg and Schroedinger interpretations, then the creation and annihilation operators are introduced, and bosonic and fermionic commutation rules are discussed. The secondary quantization theory is used to show differences between coherent, thermal, and Fock states of light; introduce squeezing and antisqueezing; and discuss the statistics of photons. Finally, the polarization and second-order coherence of light are discussed from the point of view of the quantum optics.
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.
In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
Statistical and quantum optics provides a clear, up-to-date and comprehensive introduction to the recent field of modern optics dealing with coherence and fluctuations of light. The first two chapters discuss the statistical frame work needed for later part of the book followed by systematic account of coherence theory with a detailed discussed on photon statistics. Description of quantum theory of coherence is preceded by a introductory chapter on field quantization. A detailed description of non-classical light highlighted by phenomena of anti-bunching, Sub-Poissonian statistics, production and detection of squeezed light and coherent state follows. Next is the discussion on E-P-R paradox on reality, locality and completeness of quantum mechanics bringing out clearly the importance of Bell's inequalities. The concluding two chapters put forward forcefully the new fundamental concepts based on quantum superposition and entanglement leading to two photon interference, quantum cryptography and teleportation.
In this chapter, we derived some elements of quantum mechanics, which are essential for the further development of our theory: the momentum of a system of Fermions in the second quantization, the coordinate and momentum of a harmonic oscillator as a unique operator at two different moments of time, Boson and Fermion operator algebra, coherent states, the electron-field interaction, the quantization of the electromagnetic field, Boson and Fermion distributions, and densities of states in a degenerate, or a non-degenerate system of Fermions. Our starting point is the wave nature of a quantum particle, the Hamiltonian equations were obtained as group velocities in the two conjugate spaces of the wave, of the coordinates and of the momentum. In this way, the Schr¨odinger equation and the electron-field potential of interaction are obtained from quantum equations generated by the particle wave function.
Abstract The theory of the electromagnetic field quantization developed in Chapter 4 simplifies considerably for applications Lo many of the key experiments in quantum optics. Thus many experiments use plane parallel light beams whose transverse intensity profiles are not important for the measured quantities. Again, it is often sufficient in interpreting the data to consider the light beams as exciting a single mode of the field. The present chapter begins with the specialization of the quantization to single-mode excitations. The quantum theory is illustrated by its application to several varieties of single-mode state and the main features of number Cates, coherent states, chaotic statistical mixtures and squeezed states are derived.