This chapter presents a short introduction to relativistic quantum mechanics that conforms to the style and methods of this book. In this context, relativity is a special kind of symmetry. It turns out that it is a fundamental symmetry of our world, and any understanding of elementary particles and fields should be necessarily relativistic – even if the particles do not move with velocities close to the speed of light. The symmetry constraints imposed by relativity are so overwhelming that one is able to construct, or to put it better, guess the correct theories just by using the criteria of beauty and simplicity. This method is certainly of value although not that frequently used nowadays. Examples of the method are worth seeing, and we provide some.

We start this chapter with a brief overview of the basics of relativity, refresh your knowledge of Lorentz transformations, and present the framework of Minkowski spacetime. We apply these concepts first to relativistic classical mechanics and see how Lorentz covariance allows us to see correspondences between quantities that are absent in non-relativistic mechanics, making the basic equations way more elegant. The Schrödinger equation is not Lorentz covariant, and the search for its relativistic form leads us to the Dirac equation for the electron. We find its solutions for the case of free electrons, plane waves, and understand how it predicted the existence of positrons. This naturally brings us to the second quantization of the Dirac equation. We combine the Dirac and Maxwell equations and derive a Hamiltonian model of quantum electrodynamics that encompasses the interaction between electrons and photons. The scale of the interaction is given by a small dimensionless parameter which we have encountered before: the fine structure constant α ≈ 1/137 ≪ 1. One thus expects perturbation theory to work well. However, the perturbation series suffer from ultraviolet divergences of fundamental significance. To show the ways to handle those, we provide a short introduction to renormalization. The chapter is summarized in Table 13.1.

In this chapter we explain the procedure for quantizing classical fields, based on the preparatory work we did in the previous chapter. We have seen that classical fields can generally be treated as an (infinite) set of oscillator modes. In fact, for all fields we considered in the previous chapter, we were able to rewrite the Hamiltonian in a way such that it looked identical to the Hamiltonian for a generic (mechanical) oscillator, or a set of such oscillators. We therefore understand that, if we can find an unambiguous way to quantize the mechanical oscillator, we get the quantization of all classical fields for free!

We thus start considering the mechanical harmonic oscillator and refresh the quantum mechanics of this system. The new idea is to associate the oscillator with a level for bosonic particles representing the quantized excitations of the oscillator. We then turn to a concrete example and show how the idea can be used to quantize the deformation field of the elastic string discussed in Section 7.2. In this case, the bosonic particles describing the excitations of the field are called phonons: propagating wave-like deformations of the string. As a next example, we revisit the low-lying spinfull excitations in a ferromagnet, already described in Section 4.5.3. Here, we start from a classical field theory for the magnetization of the magnet, and show how the quantization procedure leads to the same bosonic magnons we found in Chapter 4. Finally, we implement the quantization technique for the electromagnetic field, leading us to the concept of photons – the quanta of the electromagnetic field.

This chapter is devoted to coherent states. Generally speaking, a wave function of a system of identical particles does not have to have a certain number of particles. It can be a superposition of states with different numbers of particles. We have already encountered this in Chapters 5 and 6 when we studied superconductivity and superfluidity. Another important example of states with no well-defined number of particles is given by coherent states. In a way, these states are those which most resemble classical ones: the uncertainty in conjugated variables (such as position and momentum) is minimal for both variables, and their time-evolution is as close to classical trajectories as one can get.

Coherent states of radiation are relatively easy to achieve. They arise if we excite the electromagnetic field with classical currents. We look in detail at the coherent state of a single-mode oscillator and all its properties. The coherent state turns out to be an eigenfunction of the annihilation operator, the distribution of photon numbers is Poissonian, and it provides an optimal wave function describing a classical electromagnetic wave. We come back to our simple model of the laser and critically revise it with our knowledge of coherent states, and estimate the time at which the laser retains its optical coherence. We then derive Maxwell–Bloch equations that combine the master equation approach to the lasing (as outlined in Chapter 9) with the concept of coherence.

The next example of second quantization techniques we consider concerns superconductivity. Superconductivity was discovered in 1911 in Leiden by Kamerlingh Onnes, in the course of his quest to achieve ultra-low temperatures. He was applying freshly invented refrigeration techniques to study the low-temperature behavior of materials using every method available at that time. The most striking discovery he made, was that the electrical resistance of mercury samples abruptly disappeared below a critical temperature of 4.2 K – the metal underwent a transition into a novel phase.

After this first discovery, superconductivity has since been observed in a large number of metals and metallic compounds. Despite long-lasting efforts to find a material that is superconducting at room temperature, today superconductivity still remains a low-temperature phenomenon, the critical temperature varying from compound to compound. A huge and expensive quest for so-called high-temperature superconductivity took place rather recently, in the 1980s and 1990s. Although the record temperature has been increased by a factor of 5 during this time, it still remains as low as 138 K.

The most important manifestation of superconductivity is of course the absence of electrical resistance. In a superconductor, electric current can flow without any dissipation. Such a supercurrent, once excited in a superconducting ring, would in principle persist forever, at least as long as the superconductivity of the ring is preserved.

The foundations of the quantum theory of radiation were laid by the work of Planck, Einstein, Dirac, Bose, Wigner, and many others. Historically Planck's [1] work on black body radiation is the foundation of any work on the quantum theory of radiation. Einstein's [2] work on the photoelectric effect established the particle nature of the radiation field. These particles were named as photons by Lewis [3]. Einstein [4] also introduced the A and B coefficients to describe the interaction of radiation and matter. He characterized stimulated emission using the B coefficient. Using thermodynamic arguments, he could also extract the A coefficient describing spontaneous emission which is at the heart of the origin of all spectral lines. This was quite a remarkable achievement. Dirac [5] implemented the quantization of the electromagnetic field and showed how Einstein's A coefficient emerges naturally from the quantization of the radiation field. It should be remembered that stimulated emission is the key to the working of any laser system. Following Dirac's quantization of the radiation field, Weisskopf and Wigner [6] were able explain in a very fundamental way the decay of the excited states of a system and hence derive the remarkable law of exponential decay. Bose [7] discovered a quantitative explanation for Planck's law. He introduced a new way of counting statistics relevant to quantum particles with zero mass. This was the beginning of quantum statistics. Bose's work was followed by Einstein [8] who produced a counting statistics for particles with finite mass (now known as Bosons).

In Chapter 13 we saw how the optical properties of a two-level system can be modified by the application of an additional strong coherent field. For example, the absorption of light by a two-level system depends on the strength and frequency of the driving field. Figure 13.5 showed that in certain frequency regions we can amplify a probe beam. We assumed in Chapter 13 that the coherent light beam was acting on the same optical transition as the weak probe beam. However, the atomic/molecular systems have many energy levels and we can take advantage of this to produce a variety of ways of controlling the optical properties. This would offer much more flexibility as different optical transitions would have different frequencies and hence one could use a variety of sources. In this chapter we present results for the optical properties of a multilevel system. We show that coherent control can make an opaque medium transparent. We also show that the dispersive properties, which are important for the linear and nonlinear propagation of light, can be manipulated by light fields [1–3].

Having discussed in the previous chapters many different aspects of single photons and nonclassical light, we are now ready to discuss interferometry with single photons. We first discuss traditional interferometers and their performance if classical light beams are replaced by quantum fields.

The earliest interferometer is the Young's double slit interferometer. Young's work on interference confirmed the wave nature of light and was a turning point in optics. A complete description of Young's double slit is more complicated as it involves the propagation of wavefronts through the slits and hence we will take it up in Chapter 8. Michelson designed an interferometer which he very successfully used in measurements of spectral lines and the diameter of stars. However, in this chapter we focus on the Mach–Zehnder and Sagnac interferometers which are currently used extensively. We note that all interferometers use the interference between light beams arising from at least two paths. In what follows we assume that the beams arising from the two paths are coherent with respect to each other. This would be the case if the path difference was short compared to the coherence length of the light sources at the input ports of the interferometer. In Chapter 8, we will consider more general cases, which will allow us to relax this assumption somewhat.

All optical interferometers use optical devices such as beam splitters, mirrors, and phase shifters. The action of all such optical devices is very well understood for classical beams of light.

In this chapter we discuss a variety of physical effects which primarily depend on the dispersive properties of the medium, i.e. how the real part of the refractive index depends on the frequency of light. For example, it is well known that the efficiency of nonlinear optical processes such as harmonic generation depends on the phase matching, which in turn depends on the refractive index at the fundamental and harmonic frequencies [1]. Thus a control of dispersion will enable us to obtain more efficient harmonic generation [2–5]. This in fact was the starting point of the work on control of dispersion [2]. Another subject where the dispersion is very important is in the propagation of the pulses which generally are distorted [6] by the dispersion of the medium and hence one needs to tailor the dispersion to obtain nearly distortionless propagation [7]. In Section 17.1, we have already shown how an appropriately chosen control field leads to a significant modification of the dispersion (Figure 17.4). We will now discuss some applications of this. We will also discuss how hole burning physics (Section 13.2) can be used to obtain very significant control of the dispersion.

Group velocity and propagation in a dispersive medium

Let us consider the one-dimensional propagation of an electromagnetic pulse in a dispersive medium characterized by susceptibility χ(ω) and refractive index n(ω).