In Chapter 1, we treated the electromagnetic field as classical. Henceforth, we will want to treat it as a quantum field. In order to do so, we will first present some of the formalism of quantum field theory. This formalism is very useful in describing many-particle systems and processes in which the number of particles changes. Why this is important to a quantum description of nonlinear optics can be seen by considering a parametric amplifier of the type discussed in the last chapter. A pump field, consisting of many photons, amplifies idler and signal fields by means of a process in which a pump photon splits into two lower-energy photons, one at the idler frequency and one at the signal frequency. Therefore, what we would like to do in this chapter is to provide a discussion of some of the basics of quantum field theory that will be useful in the treatment of the quantization of the electromagnetic field.

In particular, we will begin with a summary of quantum theory notation, and a discussion of many-particle Hilbert spaces. These provide the arena in which all of the action takes place. We will then move on to a treatment of the canonical quantization procedure for fields. This will allow us to develop a scattering theory for fields, which is ideally what we need. This relates the properties of a field entering a medium to those of the field leaving it, and this corresponds to what is done in an experiment.

Quantum information, which studies the representation of information by quantum mechanical systems and the type of information processing this makes possible, is a relatively new field. Its roots, however, go back to early discussions of the interpretation of the quantum mechanical formalism. In 1935 Einstein, Podolsky and Rosen suggested that there were interpretational issues with quantum mechanics, having to do with local realism. Einstein was puzzled by an apparent lack of locality in quantum mechanical descriptions of reality, and suggested that quantum mechanics was not a complete theory. This stimulated a subsequent series of theoretical and experimental investigations. In particular, John Bell realized that Einstein's proposal for a ‘completion’ of quantum mechanics by the addition of more variables – called ‘hidden’ variables – was not consistent with quantum predictions, and could not be carried out.

While these interpretational issues set the stage for what followed, the modern field of quantum information arose in the 1980s stimulated by the ever decreasing size of the elements in information processing circuits. It was realized that at some point a threshold would be crossed and the devices would start to exhibit quantum mechanical behavior. It was first determined that, if they did, it would not present a problem, and, in addition, it could even be advantageous. This led to the idea of a quantum computer that stores information in quantum states, or ‘qubits’, rather than as binary digits, or ‘bits’, in a standard digital computer.

The field took off when, in 1994, Peter Shor found an efficient quantum algorithm for finding the prime factors of an integer.

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.