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).

Measuring the diameter of stars

As mentioned several times already, the particle character of light is best illustrated by the photoelectric effect. This effect can be exploited in the detection of single photons by photocounting. The analysis of such counting data allows us, as will be discussed in detail in this chapter, to gain a deeper insight into the properties of electromagnetic fields. We can recognize the “fine structure” of the radiation field – in the form of fluctuation processes – which was hidden from us when using previous techniques relying only on the eye or a photographic plate, i.e. techniques limited to time averaged intensity measurements.

The credit for developing the basic technique for intensity fluctuation measurements goes to the British scientists R. Hanbury Brown and R. Q. Twiss, who became the fathers of a new optical discipline which investigates statistical laws valid for photocounting under various physical situations. When we talk of studies of “photon statistics” it is these investigations that we are referring to.

Interestingly enough, it was a practical need, namely the improvement in experimental possibilities of measuring the (apparent) diameters of fixed stars, that gave rise to the pioneering work by Hanbury Brown and Twiss. Because the topic is physically exciting, we will go into more detail.

It is well known that the angular diameters of fixed stars – observed from Earth – appear to be so small that the available telescopes are not able to resolve the stars spatially.

The rapid technological development initiated by the invention of the laser, on the one hand, and the perfection attained in the fabrication of photodetectors, on the other hand, gave birth to a new physical discipline known as quantum optics. A variety of exciting experiments suggested by ingenious quantum theorists were performed that showed specific quantum features of light. What we can learn from those experiments about the miraculous constituents of light, the photons, is a central question in this book. Remarkably, the famous paradox of Einstein, Podolsky and Rosen became a subject of actual experiments too. Here photon pairs produced in entangled states are the actors.

The book gives an account of important achievements in quantum optics. My primary goal was to contribute to a physical understanding of the observed phenomena that often defy the intuition we acquired from our experience with classical physics. So, unlike conventional textbooks, the book contains much more explaining text than formulas. (Elements of the mathematical description can be found in the Appendix.) The translation gave me a welcome opportunity to update the book. In particular, chapters on the Franson experiment and on quantum teleportation have been included.

Quantum mechanical uncertainty

After reviewing the main characteristics of the classical description of light, let us discuss those aspects of the quantization of the electromagnetic field which are of relevance for the analysis of the phenomena we are interested in. It seems a reasonable place to start to make clear the fundamental difference between the classical and the quantum mechanical description of nature; we will come across this difference many times when discussing experiments, and it will often give us a headache. We have to deal with the physical meaning of what is called uncertainty.

The starting point of the classical description is the conviction that natural processes have a “factual” character. This means that physical variables such as the position or momentum of a particle have, in each single case, a well defined (in general, time dependent) value. However, it will not always be possible to measure all the appropriate variables (for instance, the instantaneous electric field strength of a radiation field); furthermore under normal circumstances we are able to measure only with a finite precision. Hence the basic credo of classical physics should be given in the following form: we are justified in imagining a world with variables possessing well defined values which are not known precisely (or not known at all). In doing this we are not forming any conclusions that contradict our everyday experiences.

This is the fundamental concept of classical statistics: we are satisfied with the use of probability distributions for the variables we are interested in, not from fundamental but purely practical reasons.

Most probably all people, even though they belong to different cultures, would agree on the extraordinary role that light – the gift of the Sun-god – plays in nature and in their own existence. Optical impressions mediated by light enable us to form our views of the surrounding world and to adapt to it. The warming power of the sun's rays is a phenomenon experienced in ancient times and still appreciated today. We now know that the sun's radiation is the energy source for the life cycles on Earth. Indeed, it is photosynthesis in plants, a complicated chemical reaction mediated by chlorophyll, that forms the basis for organic life. In photosynthesis carbon dioxide and water are transformed into carbohydrates and oxygen with the help of light. Our main energy resources, coal, oil and gas, are basically nothing other than stored solar energy.

Finally, we should not forget how strongly seeing things influences our concepts of and the ways in which we pursue science. We can only speculate whether the current state of science could have been achieved without sight, without our ability to comprehend complicated equations, or to recognize structures at one glance and illustrate them graphically, and record them in written form.

The most amazing properties, some of which are completely alien to our common experiences with solid bodies, can be ascribed to light: it is weightless; it is able to traverse enormous distances of space with incredible speed (Descartes thought that light spreads out instantaneously); without being visible itself, it creates, in our minds, via our eyes, a world of colors and forms, thus “reflecting” the outside world.

Light waves à la Huygens

Christian Huygens (1629–1695) is rightfully considered to be the founder of the wave theory of light. The fundamental principle enabling us to understand the propagation of light bears his name. It has found its way into textbooks together with the descriptions of reflection and refraction which are based on it.

However, when we make the effort and read Huygens' Treatise of Light (Huygens, 1690) we find to our surprise that his wave concept differs considerably from ours. When we speak of a wave we mean a motion periodic in space and time: at each position the displacement (think about a water wave, for instance) realizes a harmonic oscillation with a certain frequency ν, and an instantaneous picture of the whole wave shows a continuous sequence of hills and valleys. However, this periodicity property which seems to us to be a characteristic of a wave is completely absent in Huygens' wave concept. His waves do not have either a frequency or a wavelength! Huygens' concept of wave generation is that of a (point-like) source which is, at the same time, the wave center inducing, through “collisions” that “do not succeed one another at regular intervals,” a “tremor” of the ether particles.

Beamsplitting

Interference phenomena are certainly among the most exciting phenomena in the whole of physics. In the following we will concentrate mainly on interference of weak fields; i.e. the beams contain, on average, only a few photons.

The principle of classical interference is as follows: a light beam is split by an optical element, for example by a semitransparent mirror or a screen with several very small apertures, into two or more partial beams. These beams will take different paths and are then reunited and form interference patterns. The first step, the splitting of the beam into partial beams, plays a decisive role; light beams coming from different sources (or from different spatial areas of the same source) do not interfere with each other!

We start our discussion of interference with an analysis of the action of a beamsplitter. To form a realistic idea of this device, let us imagine a semitransparent mirror. (Our considerations apply equally well to a screen with two apertures. We could also generalize to cases of unbalanced mirrors, with reflectivity different from 1/2, or screens with apertures of different size.)

The classical wave picture can describe interference phenomena without any great effort: the incoming beam is split into the reflected and the transmitted partial wave, and each of these waves contains half of the energy. The process of splitting becomes conceptually difficult only when we think of the beam as consisting of spatially localized energy packets, or photons.