Perhaps no other subject in the history of physics has captured the human imagination more than quantum mechanics has. This captivation has extended beyond physics, and science, and well into the realm of popular culture. This is because quantum mechanics, also known as quantum physics, correctly describes the microworld and the nanoworld in a mathematical way that appears to be mysterious to us, the inhabitants of the classical world.
Perhaps one of the best, and most succinct, descriptions of quantum mechanics has been given by the well-known quantum and particle physicist John Clive Ward: “The inner mysteries of quantum mechanics require a willingness to extend one’s mental processes into a strange world of phantom possibilities, endlessly branching into more and more abstruse chains of coupled logical networks, endlessly extending themselves forward and even backwards in time” (Ward, 2004).
1.2 Brief Historical Perspective
Quantum mechanics came to light via the work of Max Planck, published in 1901. In that contribution, Planck used concepts of thermodynamics to explain the energy distribution of light sources as a function of wavelength (Planck, 1901). In doing so he introduced, without derivation, an equation where the energy of the emission was a function of frequency ν, that is,
(1.1) |
where
the units of the energy E is the joule (J)
the units of the frequency ν is the Hz
h is known as Planck’s constant (h = 6.62606957 × 10−34 J s).
That was the birth of quantum mechanics. It was born from the experiment; it was an empirical birth.
Another important experimental observation relevant to the development of quantum mechanics was the photoelectric effect (Hertz, 1887). This effect, of fundamental significance to modern photomultipliers, and photo detectors in general, means that when a surface composed of charged particles is irradiated with light of frequency ν, there is a probability that electrons will be emitted from that surface. An explanation to the photoelectric effect was provided by Einstein (1905) via the relationship
(1.2) |
where W is defined as the work function or energy required to emit an electron from the irradiated surface. In this contribution, Einstein also proposed that light behaves as a stream of localized units of energy that he called lightquanta.
A few years later, Bohr (1913) postulated that electrons in an atom can only populate well-defined orbits at discrete energies Wn. When the electron jumps from one orbit of energy Wn to another one at Wn+1, it does so emitting radiation at a frequency ν, so that (Bohr, 1913)
(1.3) |
The developments introduced earlier were the preamble to the 1925–1927 revolution that yielded the quantum mechanics we know today. Heisenberg (1925), Born and Jordan (1925), and Born et al. (1926) introduced the quantum mechanics in matrix form. Schrödinger (1926) introduced his quantum wave equation. Dirac (1925) first established that there was a correspondence between Heisenberg’s non-commuting dynamical variables and the Poisson bracket (Dirac, 1925). Then he discovered that there was an equivalence between the Born–Jordan formulation and Schrödinger’s equation (Dirac, 1926). Further, he demonstrated that there was a direct correspondence between the Heisenberg–Dirac quantum mechanics and Schrödinger’s wave mechanics (Dirac, 1927).
In addition to the three formulations just mentioned, Dirac (1939) further introduced his bra–ket notation, also known as the Dirac notation, which is the preferred formulation of quantum mechanics used in this book.
Further approaches to quantum mechanics include the Feynman formulation via integral paths (Feynman and Hibbs, 1965) and the phase-space formulation (Moyal, 1949). There are also other lesser-known formulations.
Post quantum mechanical developments include quantum electrodynamics (Tomonaga, 1946; Schwinger, 1948; Dyson, 1949; Feynman, 1949), renormalization theory (Ward, 1950), Feynman diagrams (Feynman, 1949), and the standard model of particle physics (see, e.g., Salam and Ward, 1959, 1964; Glashow, 1961; Weinberg, 1967). The Higgs boson was theorized in 1964 (Higgs, 1964).
FIGURE 1.1
Time line depicting important developments in the quantum era.
Figure 1.1 provides a time line of important developments in the quantum era.
1.3 Principles of Quantum Mechanics
The Principles of Quantum Mechanics is the landmark book written by one of the creators of quantum mechanics Paul Adrien Maurice Dirac. The first edition of this masterpiece was published in 1930, the second edition in 1935, and the third edition in 1947. The fourth edition was released in 1958, and it is this edition that gives origin to the 1978 version, its ninth revised printing, used as the standard reference in this book.
An interesting aspect of this book is that the Dirac bra–ket notation was introduced in its third edition (1947). This is explained by the Australian particle physicist R. H. Dalitz (known of the Dalitz plot and the Dalitz pair) whom in 1947 was taking lectures from Dirac in Cambridge (Dalitz, 1987).
The Principles of Quantum Mechanics, third and fourth editions, are the vehicles by which the Dirac notation was introduced to physicists although Dirac first disclosed the notation in a paper entitled A new notation for quantum mechanics (Dirac, 1939). This paper, in a fairly mechanistic style, limits itself to introduce the new notation and to provide a correspondence between it and the “old notation.” The paper does not explain how Dirac discovered or created the new notation. Nor does he explain it in the book. At one time I did ask Dick Dalitz if Dirac had explained in his lectures how he created, or discovered, his bra–ket notation, and his reply was “no.”
Here we should make a necessary point: albeit we use as reference in this book, a revised version of the fourth edition of The Principles of Quantum Mechanics, we should be very much aware that the first edition was published in 1930 and that the Dirac notation was incorporated in 1947. Thus, given Dirac’s famous precision as a communicator, we should assume that our version of this masterpiece goes back to 1947.
Dirac’s book, The Principles of Quantum Mechanics, includes 12 chapters. The most relevant of those chapters to our immediate interest are
The principle of superposition
Dynamical variables and observables
Representations
The quantum conditions
The equations of motion
Perturbation theory
Systems containing several similar particles
Theory of radiation
Throughout the book he does use his bra–ket notation extensively albeit it is not the only type of notation he utilizes.
Besides issues of notation, the Dirac book is remarkable in that it provides probably the very first discussion of optics in a quantum context. It does so via a brilliant and prophetic discussion of interferometry. He begins by considering a beam of “roughly monochromatic light” and continues by referring to this beam of light as “consisting of a large number of photons,” and the beam is then “split up into two components of equal intensity” (Dirac, 1978). In today’s terminology Dirac’s discussion applies perfectly well to a high-power narrow-linewidth laser beam undergoing interference in a Mach–Zehnder interferometer (Duarte, 1998). This discussion qualifies Dirac as the father of quantum optics and laser optics (Duarte, 2003).
It is also apparent that The Principles of Quantum Mechanics served as inspiration to Feynman for his lectures on physics not only on the central topic of the Dirac notation but also on the fundamental ideas on interference and other various mathematical formalisms.
1.4 The Feynman Lectures on Physics
Volume III of The Feynman Lectures on Physics (Feynman et al., 1965) offers a brilliant discussion of quantum mechanics via the Dirac notation. From a fascinating discussion of the two-slit interference thought experiments, using electrons, to practical applications of two-state systems, and beyond, this book is a physics treasure.
At a basic level, The Feynman Lectures on Physics, Volume III, is a valued introduction to the use and practice of the Dirac notation in quantum mechanics.
At this stage it is also instructive to mention that in his 1965 book on the path integral approach to quantum mechanics, Feynman applies quantum mechanics directly to the classical problem of diffraction (Feynman and Hibbs, 1965). It is necessary to make this observation for the benefit of some practitioners that insist in imposing the use of classical tools only to describe macroscopic diffraction and interference.
In this section first we explore the opinion on this subject given by a few luminaries of quantum physics: Dirac, Feynman, Haken, and Lamb. Then, our own opinion on the subject is examined.
Dirac (1978): “Quantum mechanics is able to effect a reconciliation of the wave and corpuscular properties of light. The essential point is the association of each of the translational states of a photon with one of the wave functions of ordinary wave optics… the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place.”
Feynman (1965): “Newton thought that light was made up of particles, but then it was discovered that it behaves like a wave.…” We say: “It is like neither.”
Haken (1981): “In quantum mechanics we attribute an infinite extended wave to a freely moving particle with momentum p so that λ = h/p. The wave must be of infinite extent, otherwise it would not have a definite wavelength.”
Lamb (1995): “Photons cannot be localized in any meaningful manner, and they do not behave at all like particles, whether described by a wave function or not.” Indeed, the nonlocality of the photon is intuitive to experimentalists experienced in optics: “All the indistinguishable photons illuminate the array of N slits, or grating, simultaneously. If only one photon propagates, at any given time, then that individual photon illuminates the whole array of N slits simultaneously” (Duarte, 2003).
So, those are some the written definitions of a photon. As can be observed, there is no complete conceptual convergence on the meaning of a photon. Here, rather than offering yet another language-based concept of the photon, we shall examine in a pragmatic approach what we know about the basic features of the photon:
1. A single photon moves, in vacuum, at the speed of light c.
2. A single photon has a wavelength λ, which is related to its frequency ν by
(1.4) |
3. A single photon exhibits a quantum energy of
or
(1.5) |
4. A single photon exhibits a quantum momentum of
(1.6) |
5. A single photon is associated with the wave functions of ordinary wave optics, such as
(1.7) |
6. Single photons are nonlocal and can exhibit enormous coherence lengths as described by
(1.8) |
Under these premises we can attempt a conceptual description of a photon as a unique entity that can be mathematically described using the wave functions of ordinary wave optics (Dirac, 1978) while exhibiting a quantum energy E = hν and a quantum momentum of p = ℏk. As of now, limitations in the existing language prevent us from a more definite description other than this tenuous outline. Therefore, when we refer to a photon, or quanta, we refer to a unique energetic entity, which is the basic component of light.
Notice that in the case of emission resulting in the generation of an ensemble of coherent photons, as in the case of a narrow-linewidth lasers, a refinement on the wave description of Haken (1981) should refer to a near-infinite wave since the wavelength, in practice, would always exhibit a measurable linewidth; in other words the wavelength would be λ ± ∆λ and not just λ.
One further point of extreme importance is the following: quantum mechanically indistinguishable photons are the same photon. In other words, two photons coming from different narrow-linewidth lasers with energies E = hν1 and E = hν1 are the same photon and will interfere precisely as described by Dirac (1978) even though they originate from different sources. Thus, a seventh item to be added to the list earlier becomes
7. Ensembles of indistinguishable photons exhibiting very narrow-linewidth ∆ν originating from nearly monochromatic sources, such as narrow-linewidth lasers, approximates the behavior of a single photon.
Finally, in this terminology a monochromatic source is an ideal laser source with an emission linewidth, which is extremely narrow, so that ∆ν → 1 Hz or less.
As described previously, the first known discussion of quantum optics was provided by Dirac in his book. He did so via interference. Furthermore, and very importantly, he did so considering a macroscopic interferometric experiment involving a beam of “roughly monochromatic light” and continues by referring to this beam of light as “consisting of a large number of photons,” and the beam is then “split up into two components of equal intensity” (Dirac, 1978). In other words, Dirac applies his quantum concepts directly to a macroscopic interferometric experiment.
The use of quantum physics in macroscopic optics is not unique to Dirac. In 1965, Feynman used his path integrals to describe divergence and diffraction resulting from the passage through a Gaussian slit (Feynman and Hibbs, 1965). Even further, Feynman in his Feynman Lectures on Physics (problem book to Feynman et al., 1965) gives credit to Hanbury Brown and Twiss (1956) as performing an early experiment in quantum optics.
Hanbury Brown and Twiss collected light from the star Sirius in two separate detectors, and the signals from these detectors are then made to interfere. Building on Feynman’s description of double-slit electron interference, the Dirac quantum notation was applied to N-slit interference (Duarte, 1991).
In addition to applications to macroscopic interference, a clear and intrinsic quantum physics development was the derivation of probability amplitude equations associated with counterpropagating photons with entangled polarizations (Pryce and Ward, 1947; Snyder et al., 1948; Ward, 1949)
(1.9) |
and the subsequent experimental confirmation provided via the measurements of polarization correlations by Wu and Shaknov (1950). A development directly related to photon entanglement was the introduction of Bell’s inequality (1964). All-optical experiments on polarization entanglement were reported by Aspect et al. (1982).
A further development in quantum optics was the introduction of quantum cryptography (Bennett, 1992). An advance directly related to the physics of entanglement is quantum teleportation (Bennett et al., 1993). Figure 1.2 highlights the time line of important developments in quantum optics while emphasizing the application of the Dirac notation.
FIGURE 1.2
Time line depicting important developments in quantum optics while emphasizing the application of the Dirac notation.
1.7 Quantum Optics for Engineers
Quantum Optics for Engineers is designed as a textbook, primarily utilizing the Dirac quantum notation, to describe optics in a unified and coherent approach. The emphasis is practical. This approach uses a minimum of mathematical sophistication. In other words, the reader should be able to use the tools provided primarily with the knowledge of first-year courses in calculus and algebra.
The subject matter is contained in Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, while a set of companion Appendices A–K provide additional necessary information relevant to the chapter material. The concept here is to offer the student a self-contained book, thus minimizing the need to refer to additional texts except for those who would like to expand their knowledge of a particular subject.
The reader will also notice that some of the equations, and figures, in this book are reproduced in several of the chapters. In other words, they are repeated. This has been done quite deliberately to avoid having to go back in the text to find a particular equation and then forward again to continue the work. Besides highlighting the importance of some concepts, this approach should facilitate remembering those equations and easing the lecture process. Hopefully, this will enhance the learning process according to the old Roman saying repetitio est mater studiorum (approximately translated as “repetition is the mother of learning”).
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