… there are certain situations in which the peculiarities of quantum mechanics can come out in a special way on a large scale. Richard Feynman Laser light Nowadays, everyone has heard of lasers, and laser light displays are a frequent ingredient of modern rock concerts. Laser light has many applications, ranging from astronomy to hydrogen fusion. What is the special feature of laser light that makes it so useful? The answer to this question involves a property of wave motion known as ‘coherence’, with light photons acting together in a special form of quantum mechanical co-operation. This type of quantum co-operation will turn out to be vital for an understanding of the peculiar behaviour of quantum ‘superfluids’. To understand the special nature of laser light, however, we must first explain what is meant by coherence. Consider the simple wave motion shown in Fig. 7.1. We see that the pattern repeats itself after one wavelength and the frequency of the wave corresponds to the number of wavelengths sent out per second. If this wave is a wave on a string, each point on the string just moves up and down with a certain amplitude: the maximum distance the point can move out to before it starts to come back. Up to now, this is really all we have needed to know about waves. Now consider two waves of the same wavelength but started at slightly different times, as shown in Fig. 7.2.
… there are certain situations in which the peculiarities of quantum mechanics can come out in a special way on a large scale. Richard Feynman Laser light Nowadays, everyone has heard of lasers, and laser light displays are a frequent ingredient of modern rock concerts. Laser light has many applications, ranging from astronomy to hydrogen fusion. What is the special feature of laser light that makes it so useful? The answer to this question involves a property of wave motion known as ‘coherence’, with light photons acting together in a special form of quantum mechanical co-operation. This type of quantum co-operation will turn out to be vital for an understanding of the peculiar behaviour of quantum ‘superfluids’. To understand the special nature of laser light, however, we must first explain what is meant by coherence. Consider the simple wave motion shown in Fig. 7.1. We see that the pattern repeats itself after one wavelength and the frequency of the wave corresponds to the number of wavelengths sent out per second. If this wave is a wave on a string, each point on the string just moves up and down with a certain amplitude: the maximum distance the point can move out to before it starts to come back. Up to now, this is really all we have needed to know about waves. Now consider two waves of the same wavelength but started at slightly different times, as shown in Fig. 7.2.
Lasers are a relatively recent invention. Microwave lasers were in existence in the 1950s and the first visible-light laser appeared in 1960. In the early 1960s some people were of the opinion that lasers were just an interesting curiosity but with no significant use. Nowadays we find lasers in all sorts of everyday situations: CD players, rock concert light shows, bar code scanners, etc. They are also extensively used in technological applications such as eye surgery and welding: modem car plants use robot-controlled lasers to weld sheets of metal together (Figure 3.1) and they are used in experiments at the forefront of research in spectroscopy (Figure 3.2). Lasers have found these many applications because of their particular properties, which are useful in a variety of situations. A laser beam is intense, monochromatic (has a very small range of wavelengths), narrow and well directed. How do these features arise? Laser light originates in the transitions made by electrons in atoms from one energy level to a lower one. Its properties depend on many atoms being stimulated to make the same transition at the same time. So, laser operation depends crucially on the electronic structure of atoms as well as their interaction with electromagnetic radiation. In this chapter, we hope to show you how quantum mechanics can help us to understand atomic structure.
T hink about coherent systems, and odds are that the first example to cross your mind will be a laser: a device that emits monochromatic light homogenized so that the photons propagate with their phase in lockstep. The well-defined wavefront of the laser beam makes it possible to focus the light to spot sizes on the order of the wavelength of the light without the aberrations usually observed with the random wavefront of incoherent light. The well-defined phase also provides a medium capable of storing and carrying information more efficiently than the intensity-modulated signals typical of incoherent sources. These useful properties of coherence have made lasers almost ubiquitous in science and technology, and that “almost” is shrinking by the day.
In general, coherence comes about when individual components of a system act in unison, either in phase or with a fixed phase difference between the components. More accurately, the system can be described as a local oscillator whose wavefunction contains phase and amplitude terms that define how the system evolves in space and time. The collection of articles in this special issue describes the recent progress and developments of coherence in optical, atomic, and semiconductor systems, and how manipulating that coherence may lead to useful applications and to experimental systems for studying the elements of quantum mechanics.
Since the first demonstration of the laser in 1960, progress in improving the extent of the spatial and temporal coherence of laser light has been steady. Psaltis (p. [1359][1]) reviews developments in optical coherence and highlights examples of the key role coherence has played in applications in imaging, communication, and information processing and storage. Just as lasers revolutionized the field of optics, many researchers have high expectations that other coherent systems will provide similar advances in their respective fields. Exploiting the fundamental tenet of quantum mechanics whereby particles have the properties of waves, Kasevich (p. [1363][2]) reviews coherence effects in collections of cold atoms and describes how the shorter wavelengths associated with atoms and the recent accessibility of bright coherent atom sources may lead to precision measurements beyond anything currently available to the optical regime. For integration and potential applications, the ideal system would be based in semiconductors. Snoke (p. [1368][3]) describes recent work toward the formation of coherent semiconductor systems.
With coherence in the atomic and optical systems generally comprising macroscopic systems, Mabuchi and Doherty (p. [1372][4]) look at coherence effects at the quantum limit, probing the interactions between single atoms and single photons confined to high-quality cavities. In addition to providing invaluable information on the limits of quantum mechanics, such effects might also shed new light on information science in the quantum regime.
The section opens with a pair of News features by staff writers Charles Seife and Robert Service. Seife presents a smattering of common applications that scientists and engineers have found for coherent waves, whether in light, sound, or subatomic particles. Service describes efforts to create new coherent sources of the type of radiation known as “hard” x-rays—a long-sought range of high frequencies with enormous potential for scientific research, technology, and medicine.
[1]: /lookup/doi/10.1126/science.1078823
[2]: /lookup/doi/10.1126/science.1079430
[3]: /lookup/doi/10.1126/science.1078082
[4]: /lookup/doi/10.1126/science.1078446
Waves of light Light behaves like a wave. In the laboratory, we can measure the vibrations that are set up in matter when a light wave comes by. We can also observe the peculiar light-and-dark patterns that occur when two light rays are made to act simultaneously, an unmistakable sign of wave behaviour. Let us make a small excursion into wave motion. Throughout this exposition, you are encouraged to experiment as much as possible with waves (preferably real ones, or at least those in the diagrams), to become familiar with their fascinating behaviour. Water waves in the bathtub, or those seen on open water from a high tower or an airplane, are especially instructive. Waves are periodic; if you pick a point in space (e.g. the surface of a pond at the point where a reed pokes through) and you watch the motion of the wave carrier (the water that bobs up and down), then you will see that the state of motion of the carrier repeats itself at regular intervals of time. This interval is the period of the wave. A free wave has a velocity, i.e. a direction and a speed. At a given point, a wave alternates periodically between a certain maximum and a minimum.
This chapter is devoted to coherent states. Generally speaking, a wave function of a system of identical particles does not have to have a certain number of particles. It can be a superposition of states with different numbers of particles. We have already encountered this in Chapters 5 and 6 when we studied superconductivity and superfluidity. Another important example of states with no well-defined number of particles is given by coherent states . In a way, these states are those which most resemble classical ones: the uncertainty in conjugated variables (such as position and momentum) is minimal for both variables, and their time-evolution is as close to classical trajectories as one can get. Coherent states of radiation are relatively easy to achieve. They arise if we excite the electromagnetic field with classical currents. We look in detail at the coherent state of a single-mode oscillator and all its properties. The coherent state turns out to be an eigenfunction of the annihilation operator, the distribution of photon numbers is Poissonian, and it provides an optimal wave function describing a classical electromagnetic wave. We come back to our simple model of the laser and critically revise it with our knowledge of coherent states, and estimate the time at which the laser retains its optical coherence. We then derive Maxwell–Bloch equations that combine the master equation approach to the lasing (as outlined in Chapter 9) with the concept of coherence.
Abstract Quantum optics did not, and could not, flourish without the laser. The present paper is not about the principles of laser construction, still less a history of how the laser was invented. Rather, it addresses the question: what are the fundamental features that distinguish laser light from thermal light? The obvious answer, ‘laser light is coherent’, is, I argue, so vague that it must be put aside at the start, albeit to revisit later. A more specific, quantum theoretic, version, ‘laser light is in a coherent state’, is simply wrong in this context: both laser light and thermal light can equally well be described by coherent states, with amplitudes that vary stochastically in space. Instead, my answer to the titular question is that four principles are needed: high directionality, monochromaticity, high brightness, and stable intensity. Combining the first three of these principles suffices to show, in a quantitative way—involving, indeed, very large dimensionless quantities (up to )—that a laser must be constructed very differently from a light bulb. This quantitative analysis is quite simple, and is easily relatable to ‘coherence’, yet is not to be found in any textbooks on quantum optics to my knowledge. The fourth principle is the most subtle and, perhaps surprisingly, is the only one related to coherent states in the quantum optics sense: it implies that the description in terms of coherent states is the only simple description of a laser beam. Interestingly, this leads to the (not, as it turns out, entirely new) prediction that narrowly filtered laser beams are indistinguishable from similarly filtered thermal beams. I hope that other educators find this material useful; it may contain surprises even for researchers who have been in the field longer than I have.
The previous chapter surveyed part of the rich variety of physical phenomena that can be understood using the ideas of quantum physics. Now that we are beginning the task of looking more deeply into the subject we shall find it useful to concentrate on examples that are comparatively simple to understand but which still illustrate the fundamental principles and highlight the basic conceptual problems. Some years ago most writers discussing such topics would probably have turned to the example of a ‘particle’ passing through a two-slit apparatus (as in Figure 1.2), whose wave properties are revealed in the interference pattern. Much of the discussion would have been in terms of wave–particle duality and the problems involved in position and momentum measurements, as in the discussion of the uncertainty principle in the last chapter. However, there are essentially an infinite number of places where the particle can be and an infinite number of possible values of its momentum, and this complicates the discussion considerably. We can illustrate all the points of principle we want to discuss by considering situations where a measurement has only a small number of possible outcomes. One such quantity relating to the physics of light beams and photons is known as polarisation . In the next section we discuss it in the context of the classical wave theory of light, and the rest of the chapter extends the concept to situations where the photon nature of light is important.