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).

The development of new sources of radiation that produce nonclassical and entangled light has changed the landscape of quantum optics. The production, characterization, and detection of single photons is important not only in understanding fundamental issues but also in the transfer of quantum information. Entangled light and matter sources as well as ones possessing squeezing are used for precision interferometry and for implementing quantum communication protocols. Furthermore, quantum optics is making inroads in a number of interdisciplinary areas, such as quantum information science and nano systems.

These new developments require a book which covers both the basic principles and the many emerging applications. We therefore emphasize fundamental concepts and illustrate many of the ideas with typical applications. We make every possible attempt to indicate the experimental work if an idea has already been tested. Other applications are left as exercises which contain enough guidance so that the reader can easily work them out. Important references are given, although the bibliography is hardly complete. Thus students and postdocs can use the material in the book to do independent research. We have presented the material in a self-contained manner. The book can be used for a two-semester course in quantum optics after the students have covered quantum mechanics and classical electrodynamics at a level taught in the first year of graduate courses. Some advanced topics in the book, such as exact non-Markovian dynamics of open systems, quantum walks, and nano-mechanical mirrors, can be used for seminars in quantum optics.

It is well known that electromagnetic fields are important probes of the properties of matter. We can learn about atomic molecular energy levels by studying the absorption, emission, and scattering of electromagnetic waves. For example, the rate at which a system absorbs energy and its dependence on the frequency of the electromagnetic field gives information on the allowed transitions. From such studies one can determine the energy levels and their lifetimes. Similarly, scattering processes provide a wealth of information. The traditional probing of matter is restricted to weak fields; however, in Chapter 11 we saw how strong fields dress the energy levels of a system. Strong fields also modify the transition rates. In order to study the characteristics of such modifications we need to probe a coherently driven system using a probe field. In this chapter we study the absorption, emission, and scattering processes in strongly driven systems. A novel characteristic of radiation from strongly driven systems is its nonclassical nature.

Effects of relaxation: optical Bloch equations

So far we have considered only interactions with external electromagnetic fields. In reality, one has to account for various sources of decay of the atomic population and coherences. For example, an atom can decay radiatively by emitting a photon. The resulting collisions change the populations and coherences. In Chapter 9 we discussed in detail how various relaxation processes can be included from first principles in the master equation framework.

Dipole–dipole interactions between atoms or molecules profoundly affect the light absorption that occurs in matter. The spectral characteristics of light absorption can be strongly modified. For example, the weak field absorption spectrum splits into a doublet [1]. The separation of the doublet depends on the strength of the dipole–dipole interaction. Furthermore, the photon antibunching exhibited by a single-atom fluorescence starts becoming bunching due to a nearby atom [2]. The dipole–dipole interactions also give rise to fascinating applications in quantum information science, such as quantum logic operations in neutral atoms [3]. The dipole–dipole interaction can transfer excitation from one atom to the other and this transfer process produces entanglement between two atoms. For two atoms with the first one in the excited state ∣eA, gB⟩, the excitation would be on the atom B, i.e. ∣eA, gB⟩ → ∣gA, eB⟩ after a certain time. Clearly halfway through one would expect the state of the two atom system would be of the form (∣eA, gB⟩ + ∣gA, eB⟩)/√2, which is a state of maximum entanglement The dipole–dipole interaction is known to aid the process of simultaneous excitation of two atoms leading to the possibility of nanometric resolution of atoms [2, 4–6]. There are other types of dipole–dipole interactions, such as van der Waal interaction [7] which involves two atoms each in a state, which could be an excited state or the ground state.

This chapter is devoted to the dynamical evolution of open quantum system [1–3]. An open quantum system is one where it interacts with the environment. A system undergoing relaxation is an example of an open quantum system. We have already come across an example of open quantum system in Chapter 7 where we have discussed spontaneous emission from a two-level system. The two-level system interacts with the vacuum of the electromagnetic field. The vacuum consists of infinite number of modes and is a large system. The vacuum in this case is the environment. The population in the excited state decays. A photon is emitted and the emitted photon leaves the vicinity of the atom, i.e. the emitted photon is not reabsorbed by the atom. Another example of an open system is the case of atoms colliding with the atoms of a buffer gas. Here the buffer gas is the environment. Other examples of open systems are the fields confined in the cavities. The case of ideal cavities, i.e. cavities bounded by mirrors with 100% reflectivity, is uninteresting. We need the photons from the cavity to leak out in order to learn about the photons in the cavity. Thus we need to have mirrors with nonzero transmission. In this case, the electromagnetic field inside the cavity couples to the vacuum modes outside the cavity; thus the vacuum outside is the environment.