Scope and aims of this book

Quantum optics is one of the liveliest fields in physics at present. While it has been a dominant research field for at least two decades, with much graduate activity, in the past few years it has started to impact the undergraduate curriculum. This book developed from courses we have taught to final year undergraduates and beginning graduate students at Imperial College London and City University of New York. There are plenty of good research monographs in this field, but we felt that there was a genuine need for a straightforward account for senior undergraduates and beginning postgraduates, which stresses basic concepts. This is a field which attracts the brightest students at present, in part because of the extraordinary progress in the field (e.g. the implementation of teleportation, quantum cryptography, Schrödinger cat states, Bell violations of local realism and the like). We hope that this book provides an accessible introduction to this exciting subject.

Our aim was to write an elementary book on the essentials of quantum optics directed to an audience of upper-level undergraduates, assumed to have suffered through a course in quantum mechanics, and for first-or second-year graduate students interested in eventually pursuing research in this area. The material we introduce is not simple, and will be a challenge for undergraduates and beginning graduate students, but we have tried to use the most straightforward approaches. Nevertheless, there are parts of the text that the reader will find more challenging than others.

“All information is physical”, the slogan advocated over many years by Rolf Landauer of IBM, has recently led to some remarkable changes in the way we view communications, computing and cryptography. By employing quantum physics, several objectives that were thought impossible in a classical world have now proven to be possible. Quantum communications links, for example, become impossible to eavesdrop without detection. Quantum computers (were they to be realized) could turn some algorithms that are labelled “difficult” for a classical machine, no matter how powerful, into ones that become “simple”. The details of what constitutes “difficult” and what “easy” are the subject of mathematical complexity theory, but an example here will illustrate the point and the impact that quantum information processors will have on all of us. The security of many forms of encryption is predicated on the difficulty of factoring large numbers. Finding the factors of a 1024-digit number would take longer than the age of the universe on a computer designed according to the laws of classical physics, and yet can be done in the blink of an eye on a quantum computer were it to have a comparable clock speed. But only if we can build one, and that's the challenge! No one has yet realized a quantum register of the necessary size, or quantum gates with the prerequisite accuracy. Yet it is worth the chase, as a quantum computer with a modest-sized register could out-perform any classical machine.

Over the past three decades or so, experiments of the type called Gedanken have become real. Recall the Schrödinger quote from Chapter 8: “… we never experiment with just one electron or atom or (small) molecule.” This is no longer true. We can do experiments involving single atoms or molecules and even on single photons, and thus it becomes possible to demonstrate that the “ridiculous consequences” alluded to by Schrödinger are, in fact, quite real. We have already discussed some examples of single-photon experiments in Chapter 6, and in Chapter 10 we shall discuss experiments performed with single atoms and single trapped ions. In the present chapter, we shall elaborate further on experimental tests of fundamentals of quantum mechanics involving a small number of photons. By fundamental tests we mean tests of quantum mechanics against the predictions of local realistic theories (i.e. hidden variable theories). Specifically, we discuss optical experiments demonstrating violations of Bell's inequalities, violations originally discussed by Bell in the context of two spin-one-half particles. Such violations, if observed experimentally, falsify local realistic hidden-variable theories. Locality refers to the notion, familiar in classical physics, that there cannot be a causal relationship between events with space-like separations. That is, the events cannot be connected by any signal moving at, or less than, the speed of light; i.e. the events are outside the light-cone. But in quantum mechanics, it appears that nonlocal effects, effects seemly violating the classical notion of locality in a certain restricted sense, are possible.

We have previously emphasized the fact that all states of light are quantum mechanical and are thus nonclassical, deriving some quantum features from the discreteness of the photons. Of course, in practice, the nonclassical features of light are difficult to observe. (We shall use “quantum mechanical” and “nonclassical” more or less interchangeably here.) Already we have discussed what must certainly be the most nonclassical of all nonclassical states of light – the single-photon state. Yet, as we shall see, it is possible to have nonclassical states involving a very large number of photons. But we need a criterion for nonclassicality. Recall that in Chapter 5 we discussed such a criterion in terms of the quasi-probability distribution known as the P function, P(α). States for which P(α) is positive everywhere or no more singular than a delta function, are classical whereas those for which P(α) is negative or more singular than a delta function are nonclassical. We have shown, in fact, that P(α) for a coherent state is a delta function, and Hillery has shown that all other pure states of the field will have functions P(α) that are negative in some regions of phase space and are more singular than a delta. It is evident that the variety of possible nonclassical states of the field is quite large.

At the end of the preceding chapter, we showed that the photon number states |n〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈n|Êx|n〉 = 0 no matter how large the value of n. We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see.

Eigenstates of the annihilation operator and minimum uncertainty states

In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.

In this chapter, we discuss two more experimental realizations of quantum optical phenomena, namely the interaction of an effective two-level atom with a quantized electromagnetic field in a high Q microwave cavity, the subject usually referred to as cavity QED, or sometimes CQED, and in the quantized motion of a trapped ion. Strictly speaking, these experiments are not optical, but they do realize interactions of exactly the type that are of interest in quantum optics, namely the Jaynes–Cummings interaction between a two-level system (an atom) and a bosonic degree of freedom, a single-mode cavity field in the case of a microwave cavity, and a vibrational mode of the center-of-mass motion of a trapped ion, the quanta being phonons in this case. We shall begin with a description of the useful properties of the so-called Rydberg atoms that are used in the microwave CQED experiments, proceed to discuss some general considerations of the radiative behavior of atoms in cavities, the CQED realization of the Jaynes–Cummings model, and then discuss the use of the dispersive, highly off-resonant, version of the model to generate superpositions of coherent states, i.e. the Schrödinger cat states of the type discussed in Chapters 7 and 9 for traveling wave optical fields but this time for a microwave cavity field. Finally, we discuss the realization of the Jaynes–Cummings interaction in the vibrational motion of a trapped ion.

Introduction

So far, we have discussed closed systems involving a single quantized mode of the field interacting with atoms, as for example in the Jaynes–Cummings model in Chapter 4. As we saw in this model, the transition dynamics are coherent and reversible: the atom and field mode exchange excitation to and fro without loss of energy. As we add more modes for the atom to interact with, the coherent dynamics become more complicated as the relevant atom–field states come in and out of phase and beat together to determine the total state occupation probabilities. As time goes on, these beats get out of phase, leading to an apparent decay of the initial state occupation probability. But at later times, the beating eigenfrequencies get back in phase in a manner rather reminiscent of the Jaynes–Cummings revival discussed earlier in this book, and this leads to a partial recurrence or revival of the initial state probability. The time scale for this partial revival depends on the number of participating electromagnetic field modes and as these increase to the level appropriate for an open system in free space the recurrence disappears off to the remote future, and the exponential decay law appropriate for decay is recovered as an excellent approximation.

We have already discussed the origin of spontaneous emission and the Einstein A coefficient using perturbation theory in Chapter 4.

Transmission of a polarization state

The word “teleportation” comes from parapsychology and means transportation of persons or things from one place to another using mental power. It was taken over into science fiction literature, where the transport is imagined to take place instantaneously. However this is still to be invented, and is surely nonsense – relativity theory teaches us that the velocity of light is the upper bound for the motion of an object. Nevertheless, teleportation has occupied a firm place in our fantasies, and when renowned quantum physicists (as has happened) use this word, they can be sure to attract attention. So, what is it all about? The basic idea is that it is not necessary to transport material constituents (ultimately the elementary particles). The same particles already exist at other places; we “simply” need to put them together in the right way. To do this, we need a complete set of building instructions, and this is, according to quantum theory, the quantum mechanical wave function representing the maximum information known about an object. We could imagine the wave function measured on the original system, then transmitted via a conventional (classical) information channel to another place and there used for system reconstruction. Unfortunately, the first step, the determination of the wave function on a single system, is impossible (see Section 10.3). However, quantum mechanics offers us another “magic trick.”

Particle properties of radiation

One of the most important properties of macroscopic material systems is their ability to emit radiation spontaneously. According to quantum mechanics, the emission process is realized in the following way: an atom (or a molecule) makes a transition from a higher lying energy level (to which it was brought, for example, by an electron collision) to a lower lying energy level without any noticeable external influence (in the form of an existing electromagnetic field), and the released energy is emitted in the form of electromagnetic radiation. The discrete energy structure of the atom dictated by the laws of quantum mechanics is imprinted also on the emission process (quantization of the emission energy), since the energy conservation law is also valid for single (individual) transitions. Hence, a single photon, in the sense of a well defined energy quantum, is always emitted.

The emitted quanta can be directly detected by a photodetector. (Strictly speaking, identifying a registered photon with an emitted one is possible only when it is guaranteed that the observed volume contains only a single atom. (For details see Sections 6.8 and 8.1.) Under realistic conditions, the experiment can be performed in the following way. First, a beam of ionized atoms is sent through a thin foil; the emerging beam then consists of excited atoms. (This procedure is known as the beam–foil technique.) A detector is placed at a distance d from the foil to detect light emitted sideways by the atomic beam (Fig. 6.1).