Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam.

Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.

This book grew out of our work in the field of the quantum theory of nonlinear optics. Some of this work we have done together and some following our own paths. One major emphasis of this work has been the quantization of electrodynamics in the presence of dielectric media. This is a subject that is often given short shrift in many treatments, and we felt that a book in which it receives a more extensive discussion was warranted.

M.H. would like to thank his thesis advisor, Eyvind Wichmann, for an excellent education in quantum mechanics and quantum field theory, and M. Suhail Zubairy for introducing him to the field of nonlinear optics with quantized fields. Others who played a major role are Leonard Mlodinow, with whom the initial work on quantization in nonlinear media was done, and Janos Bergou and Vladimir Buzek, long-time collaborators with whom it has been a pleasure to work. He also thanks Carol Hutchins for many things.

P.D.D. wishes to acknowledge his parents and family for their invaluable support. The many colleagues who helped form his approach include Crispin Gardiner and the late Dan Walls, who pioneered quantum optics in New Zealand. Subhash Chaturvedi, Howard Carmichael, Steve Carter, Paul Kinsler, Joel Corney, Piotr Deuar and Kaled Dechoum have contributed greatly to this field. He also thanks Margaret Reid, who has played a leading role in some of the developments outlined in the quantum information section, and Qiongyi He, who provided illustrations.

The optical parametric oscillator is one of the most well-characterized devices in nonlinear optics, with many useful applications, especially as a frequency converter and as a wide-band amplifier. Novel discoveries made with these devices in the quantum domain include demonstrations of large amounts of both single-mode and two-mode squeezing, and, in the two-mode case, significant quantum intensity correlations between the two modes and quadrature correlations, which provided the first experimental demonstration of the original Einstein–Podolsky–Rosen (EPR) paradox. In this chapter, a systematic theory is developed of quantum fluctuations in the degenerate parametric oscillator.

The quantum statistical effects produced by the degenerate parametric amplifier and oscillator have been intensely studied. As we have seen, the amount of transient squeezing produced in an ideal lossless parametric amplifier was calculated using asymptotic methods by Crouch and Braunstein. It was found to scale inversely with the square root of the initial pump photon number. When the system is put into a cavity, and driving and dissipation are introduced, we find that the steady state of a driven degenerate parametric oscillator bifurcates. There is a single steady state at low driving strengths, but, as the driving field increases, a threshold value is reached and the single state splits and forms two stable states. Below the threshold, a linearized analysis of squeezing in the quantum quadratures is possible; while above this threshold, or critical point, the tunneling time between the two states due to quantum fluctuations can be calculated.

In this chapter, we will develop methods for mapping operator equations to equivalent c-number equations. This results in a continuous phase-space representation of a many-body quantum system, using phase-space distributions instead of density matrices. In order to do this, we will first find c-number representations of operators. We have already seen one such representation, the Glauber–Sudarshan P-representation for the density matrix. We now introduce several more such representations. The main focus will be on the truncated Wigner representation, valid at large photon number, and the positive P-representation, which uses a double-dimensional phase space and exists as a positive probability for all quantum density matrices.

Phase-space techniques have a great advantage over conventional matrix-type solutions to the Schrödinger equation, in that they do not have an exponential growth in complexity with mode and particle number. Instead, the equations that describe the dynamics of these c-number representations of the density matrix are Fokker–Planck equations. These have equivalent stochastic differential equations, which behave as c-number analogs of Heisenberg-picture equations of motion.

This means that problems that would be essentially impossible to solve using conventional number-state representations can be transformed into readily soluble differential equations. In many cases, no additional approximations, such as perturbation theory or factorization assumptions, are needed.

For the next several chapters we will mainly be focusing on the behavior of a medium with nonlinear susceptibility in an optical cavity. This will entail a detour through the quantum theory of open systems. All physical systems are coupled to the ‘outside world’ to some extent. We call such a system an open system, and the part of the ‘outside world’ that is coupled to it is called a reservoir. If the coupling is very weak, we can ignore it and treat the system as closed.

For the systems of interest here, we must take the coupling to the outside world into account, and, in particular, we need a way to see how this coupling affects the dynamics of the system itself. In the case of a nonlinear medium in a cavity, some of the light leaves the cavity, so the light inside the cavity is coupled to modes outside the cavity. In addition, in order to measure a system, we have to couple it to another system. There are two main ways of treating open quantum systems, namely quantum Langevin equations and master equations. Both methods will be treated here.

Reservoir Hamiltonians

Now let us begin our analysis of open systems. We have a system with degrees of freedom in which we are interested, but these degrees of freedom are coupled to excitations or modes about whose detailed dynamics we do not care. These degrees of freedom in which we are not interested are called reservoirs.

The simplest open, nonlinear optical system consists of a single bosonic mode with nonlinear self-interactions coupled to driving fields and reservoirs. In this chapter, we will examine a number of systems of this type using the methods that have been developed in the preceding chapters. Systems of this type can exhibit a variety of interesting physical effects. For example, in the case of a driven nonlinear oscillator, there can be more than one steady state (bistability), and quantum statistical effects can manifest themselves by affecting the stability of these states. Other quantum statistical effects have also been predicted in a variety of related systems – such as photon anti-bunching, squeezing and changes to spectra. In general, the size of these effects scales inversely with the size of the system. This ‘system size’ refers to a threshold photon number, a number of atoms, or some similar quantity.

The nature of the steady states themselves can become less than straightforward. Nonequilibrium dissipative systems, of which the systems considered in this chapter are examples, have parameters that describe the energy input to the system, and their steady states depend on these parameters. When driven far from equilibrium, such systems can exhibit bifurcations in their steady states. In more complex cases, this eventually leads to periodic oscillations and chaos. Devices like this can be realized with multiple types of nonlinearity, ranging from nonlinear dielectrics through to cavity QED (with near-resonant atoms), cavity optomechanics (with nanomechanical oscillators) and circuit QED (with superconducting Josephson junctions).

We will start with cases where all the interactions are with the reservoirs, which may be linear or nonlinear, while the cavity itself is harmonic in its response.

In this chapter we will consider some simple systems arising out of the Hamiltonians in the previous chapters. Here, we will look at these systems without considering damping or reservoirs. These systems involve only a small number of field modes – typically one, two or three. We will revisit them in later chapters to consider what happens when the fields are confined to a cavity with damping, including possible input and output coupling.

Before discussing the models, however, we need to describe a number of features of fields consisting of a small number of modes. We will begin with a discussion of the quantum theory of photo-detection and optical coherence, due to Mandel and Glauber. This makes use of the microscopic model for atom–field interactions in the previous chapter, to allow us to introduce a theory of coherence and photon counting.

We will also treat the representation for a single-mode field known as a P-representation. This will be the first quasi-distribution function we will encounter. It is a c-number representation of the quantum state of a field mode. As we shall see in subsequent chapters, quasi-distribution functions are very useful in describing the dynamics of interacting quantum systems coupled to reservoirs. We will then go on to use the P-representation to define the notion of a nonclassical state of the field; such states are signified by the fact that the P-representation is no longer a well-behaved, positive distribution. Nonclassical states are natural results of nonlinear optical interactions, and we will show how that comes about for some of our models. It is also possible to define P-representations for multi-mode fields. This leads to a discussion of nonclassical correlations between modes.

We have seen how to quantize the electromagnetic field in free space, so let us now look at how to quantize it in the presence of a medium. We will first consider a linear medium without dispersion, and then move on to a nonlinear medium without dispersion. Incorporating dispersion is not straightforward, because it is nonlocal in time; the value of the field at a given time depends on its values at previous times. This is due to the finite response time of the medium. We will, nonetheless, present a theory that does include the effects of linear dispersion.

It is worth noting that the quantization of a form of nonlinear electrodynamics was first explored in a model of elementary particles by Born and Infeld. Their theory was meant to be a fundamental one, not like the ones we are exploring, which are effective theories for electromagnetic fields in media. However, many of the issues explored in the Born–Infeld theory reappear in nonlinear quantum optics.

The approach adopted in this chapter is to quantize the macroscopic theory, that is, the theory that has the macroscopic Maxwell equations as its equations of motion. A different approach, which will be explored in a subsequent chapter, begins with the electromagnetic field coupled to matter, i.e. the matter degrees of freedom are explicitly included in the theory. One is then able to derive an effective theory whose basic objects are mixed matter–field modes called polaritons.

In a realistic treatment of a three-dimensional nonlinear optical experiment, the complete Maxwell equations in (3 + 1) space-time dimensions should be employed. It is then necessary to utilize a multi-mode Hamiltonian that correctly describes the propagating modes. There is an important difference between these experiments and traditional particle scattering. Quantum field dynamics in nonlinear media is dominated by multiple scattering, which is the reason why perturbation theory is less useful.

Nevertheless, it is interesting to make a link to conventional perturbation theory. Accordingly, we start by considering a perturbative theory of propagation in a one-dimensional nonlinear optical system in a χ(3) medium. While this calculation cannot treat long interaction times, it does give a qualitative understanding of the important features.

This problem is the ‘hydrogen atom’ of quantum field theory: it has fully interacting fields with exact solutions for their energy levels. This is because a photon in a waveguide is an elementary boson in one dimension. The interactions between these bosons are mediated by the Kerr effect, which in quantum field theory is a quartic potential, equivalent to a delta-function interaction.

The quantum field theory involved is the simplest model of a quantum field that has an exact solution. This elementary model is still nontrivial in terms of its dynamics, as the calculation of quantum dynamics using standard eigenfunction techniques would require exponentially complex sums over multi-dimensional overlap integrals. This is not practicable, and accordingly we use other methods including quantum phase-space representations to solve this problem.

Before discussing nonlinear optics with quantized fields, it is useful to have a look at what happens with classical electromagnetic fields in nonlinear dielectric media. The theory of nonlinear optics was originally developed using classical fields by Armstrong, Bloembergen, Ducuing and Pershan in 1962, stimulated by an experiment by Franken, Hill, Peters and Weinreich in which a second harmonic of a laser field was produced by shining the laser into a crystal. This classical theory is sufficient for many applications. For the most part, quantized fields were introduced later, although a quantum theory for the parametric amplifier, a nonlinear device in which three modes are coupled, was developed by Louisell, Yariv and Siegman as early as 1961. In any case, a study of the classical theory will give us an idea of some of the effects to look for when we formulate the more complicated quantum theory.

What we will present here is a very short introduction to the subject. Our intent is to use the classical theory to present some of the basic concepts and methods of nonlinear optics. Further information can be found in the list of additional reading at the end of the chapter. The discussion here is based primarily on the presentations in the books by N. Bloembergen and by R. W. Boyd.

Linear polarizability

We wish to survey some of the effects caused by the linear polarizability of a dielectric medium. When an electric field E(r, t) is applied to a dielectric medium, a polarization, that is, a dipole moment per unit volume, is created in the medium.

Optical fibers are the backbone of the world's communication systems. They also represent the simplest example of a photonic device, where photons in waveguides replace electrons in wires. More sophisticated devices than this can be made, in which all the optical components – including multiple waveguided modes – are integrated on a chip. It is important to understand the quantum noise properties of these systems.

While in some respects these systems can be treated as one-dimensional, there are additional features. Dispersion is due not just to material properties, but also to the waveguide geometry. The nonlinearity comes from a combination of material and geometric properties as well. There is, of course, a quantum aspect to these photonic devices, in which quantum noise occurs due to random effects caused by nonlinearity and gain.

Optical devices based on waveguides are, of course, not just simple fibers. One can have multiple waveguides with various types of linear coupling. The intrinsic nonlinearity of the dielectric leads to four-wave mixing, which can be an important source of correlated photons, both for fundamental tests of quantum mechanics, and for applications in quantum information.

It is often essential to include effects due to anisotropies of various types, as well as material dispersion. In dielectrics, another effect that should be included is the coupling of vibrational modes of the (solid) dielectric to the propagating field, which gives rise to Raman and Brillouin scattering effects. This allows us to obtain a more detailed and correct theory of quantum noise than the simplified case treated in Chapter 3.

The laser

A laser is an oscillator that operates at optical frequencies. These frequencies of operation lie within a spectral region that extends from the very far infrared to the vacuumultraviolet (VUV) or soft-X-ray region. At the lowest frequencies at which they operate, lasers overlap with the frequency coverage of masers, to which they are closely related, and millimeter-wave sources using solid-state or vacuum-tube electronics, such as TRAPATT, IMPATT, and Gunn diodes, klystrons, gyroklystrons, and traveling-wave tube oscillators, whose principles of operation are quite different [1]. In common with electronic-circuit oscillators, a laser is constructed using an amplifier with an appropriate amount of positive feedback. The positive feedback is generally provided by mirrors that re-direct light back and forth through the laser amplifier. The acronym LASER, which stands for light amplification by stimulated emission of radiation, is in reality therefore a slight misnomer.

A little bit of history

The basic physics underlying light emission and absorption by atoms and molecules was first expounded by Albert Einstein (1879–1955) in 1917 [2]. Richard Chace Tolman (1881–1948) observed that stimulated emission could lead to “negative absorption.” In 1928 Rudolph Walther Landenburg (1889–1953) confirmed the existence of stimulated emission and negative absorption. It is interesting to note that a famous spectroscopist, Curtis J. Humphreys (1898–1986), who for most of his career worked at the U.S. Naval Ordnance Laboratory in Corona, California, might have operated the first gas laser without knowing it.

Introduction

The semiconductor laser, in various forms, is the most widely used of all lasers, is manufactured in the largest quantities, and is of the greatest practical importance. Every CD (compact disk), DVD (digital versatile disk or digital video disk), and Blu-ray player contains one. Most of the world's long-, and medium-, distance communication takes place over optical fibers along which propagate the beams from semiconductor lasers. Highpower semiconductor lasers are increasingly part of laser systems for engraving, cutting, welding, and medical applications. Semiconductor lasers operate by using the jumps in energy that can occur when electrons travel between semiconductors containing different types and levels of controlled impurities (called dopants). In this chapter we will discuss the basic semiconductor physics that is necessary to understand how these lasers work, and how various aspects of their operation can be controlled and improved. Central to this discussion will be what goes on at the junction between p- and n-type semiconductors. The ability to grow precisely doped single- and multi-layer semiconductor materials and fabricate devices of various forms – at a level that could be called molecular engineering – has allowed the development of many types of structure with which one can make efficient semiconductor lasers. In some respects the radiation from semiconductor lasers is far from ideal, since its coherence properties are far from perfect, being intermediate between those of a low-pressure gas laser and an incoherent line source.

In this chapter we shall explain how the distortion produced in a crystal lattice by the application of an electric field or by the passage of a sound wave affects the propagation of light through the crystal. These effects – the electro-optic and acousto-optic effects, and related effects such as field-induced changes in the absorption of a material – are of considerable practical importance since they can be used to amplitude- and phase-modulate light beams, shift their frequencies, and alter the direction in which they travel.

Introduction to the electro-optic effect

When an electric field is applied to a crystal, the ionic constituents move to new locations determined by the field strength, the charge on the ions, and the restoring force. As we saw in Chapter 17, unequal restoring forces along three mutually perpendicular axes in the crystal lead to anisotropy in the optical properties of the medium. When an electric field is applied to such a crystal, in general, it causes a change in the anisotropy. These changes can be described in terms of the modification of the indicatrix by the field – both in terms of the principal refractive indices of the medium and in terms of the orientation of the indicatrix. If these effects can be described, to first order, as being linearly proportional to the applied field then the crystal exhibits the linear electro-optic effect. We shall see that this results only if the crystal lattice lacks a center of symmetry. So, some cubic crystals can exhibit the linear electro-optic effect.