It is well known that electromagnetic fields are important probes of the properties of matter. We can learn about atomic molecular energy levels by studying the absorption, emission, and scattering of electromagnetic waves. For example, the rate at which a system absorbs energy and its dependence on the frequency of the electromagnetic field gives information on the allowed transitions. From such studies one can determine the energy levels and their lifetimes. Similarly, scattering processes provide a wealth of information. The traditional probing of matter is restricted to weak fields; however, in Chapter 11 we saw how strong fields dress the energy levels of a system. Strong fields also modify the transition rates. In order to study the characteristics of such modifications we need to probe a coherently driven system using a probe field. In this chapter we study the absorption, emission, and scattering processes in strongly driven systems. A novel characteristic of radiation from strongly driven systems is its nonclassical nature.

Effects of relaxation: optical Bloch equations

So far we have considered only interactions with external electromagnetic fields. In reality, one has to account for various sources of decay of the atomic population and coherences. For example, an atom can decay radiatively by emitting a photon. The resulting collisions change the populations and coherences. In Chapter 9 we discussed in detail how various relaxation processes can be included from first principles in the master equation framework.

Dipole–dipole interactions between atoms or molecules profoundly affect the light absorption that occurs in matter. The spectral characteristics of light absorption can be strongly modified. For example, the weak field absorption spectrum splits into a doublet [1]. The separation of the doublet depends on the strength of the dipole–dipole interaction. Furthermore, the photon antibunching exhibited by a single-atom fluorescence starts becoming bunching due to a nearby atom [2]. The dipole–dipole interactions also give rise to fascinating applications in quantum information science, such as quantum logic operations in neutral atoms [3]. The dipole–dipole interaction can transfer excitation from one atom to the other and this transfer process produces entanglement between two atoms. For two atoms with the first one in the excited state ∣eA, gB⟩, the excitation would be on the atom B, i.e. ∣eA, gB⟩ → ∣gA, eB⟩ after a certain time. Clearly halfway through one would expect the state of the two atom system would be of the form (∣eA, gB⟩ + ∣gA, eB⟩)/√2, which is a state of maximum entanglement The dipole–dipole interaction is known to aid the process of simultaneous excitation of two atoms leading to the possibility of nanometric resolution of atoms [2, 4–6]. There are other types of dipole–dipole interactions, such as van der Waal interaction [7] which involves two atoms each in a state, which could be an excited state or the ground state.

This chapter is devoted to the dynamical evolution of open quantum system [1–3]. An open quantum system is one where it interacts with the environment. A system undergoing relaxation is an example of an open quantum system. We have already come across an example of open quantum system in Chapter 7 where we have discussed spontaneous emission from a two-level system. The two-level system interacts with the vacuum of the electromagnetic field. The vacuum consists of infinite number of modes and is a large system. The vacuum in this case is the environment. The population in the excited state decays. A photon is emitted and the emitted photon leaves the vicinity of the atom, i.e. the emitted photon is not reabsorbed by the atom. Another example of an open system is the case of atoms colliding with the atoms of a buffer gas. Here the buffer gas is the environment. Other examples of open systems are the fields confined in the cavities. The case of ideal cavities, i.e. cavities bounded by mirrors with 100% reflectivity, is uninteresting. We need the photons from the cavity to leak out in order to learn about the photons in the cavity. Thus we need to have mirrors with nonzero transmission. In this case, the electromagnetic field inside the cavity couples to the vacuum modes outside the cavity; thus the vacuum outside is the environment.

In this chapter we will show how many concepts from quantum optics, such as squeezing, nonclassicality, and quantum entanglement, can be applied to nano-mechanical systems leading to the possibility of realizing the quantized behavior of macroscopic systems [1]. Furthermore, nano-mechanical systems can exhibit a variety of rich nonlinear phenomena as the basic interaction between the nano-mechanical system and the radiation fields is via radiation pressure [2]. This interaction is nonlinear. Thus many nonlinear processes such as electromagnetically induced transparency, optical bistability, and up-conversion of radiation are expected to occur for nano-mechanical systems. Similarly, cavity QED effects such as vacuum Rabi splittings are also expected to occur provided one can design systems such that the interaction of a single photon with the nano-mechanical mirror is large. We note that the work on nano-mechanical systems originated with the discussion of Braginsky and collaborators [3] on how to measure small forces accurately. In this chapter, we will discuss only the fundamental quantum and nonlinear optical effects in nano-mechanical systems interacting with quantized and semiclassical fields.

Introduction

The term rotational frequency shift (RFS) has been used in different contexts and given different meanings [1]-[8]. Other terms have also been used (e.g. azimuthal Doppler shift, angular Doppler shift) to describe various related phenomena. In this article we stick to the meaning of the rotational frequency shift given by us in [9]. In order to make a clear distinction between our RFS and other related shifts we use the term dynamical RFS (DRFS). We will study the spectral properties of radiation emitted by rotating quantum sources.

Radiation emitted by sources in motion looks different when observed in the laboratory frame. Frequency shift can only be determined for monochromatic waves. In general, a monochromatic field will lose this property when observed from a moving frame. Therefore, to observe a frequency shift we have to restrict ourselves to some special forms of radiation, which fully preserve their monochromaticity when the frame of reference is changed.

When the source moves with constant velocity, one observes the well-known Doppler shift. In this case a special role is played by plane waves characterized by their wave vectors. We may invoke the relativistic transformation properties of a wave vector to derive the change of frequency. As a result, all inertial observers see such waves as monochromatic plane waves with a shifted frequency and a transformed direction of propagation. This effect can also be deduced from the transformation laws of the photon energy-momentum four-vector pμ = ℏkμ. For uniform motion this kinematical Doppler shift is the only effect.

The twentieth century saw four astonishing revolutions in physics: relativity, quantum mechanics, elementary particles, and cosmology. Each one radically changed our understanding of the Universe. There were also, of course, extraordinary breakthroughs in technology (electronics, lasers, computers) that had a much larger influence on our daily lives, but did not carry the same conceptual impact.

This is a book about those four revolutions. It is intended for anyone with a serious interest in the great ideas that have shaped modern physics: advanced high school students or freshman physics majors who would like a taste of what lies ahead; undergraduates who do not intend to major in the sciences but are curious to know about some of the most profound intellectual achievements of our time; general readers who have heard about quarks and quanta, Albert Einstein and the Big Bang, and would like to know what all the fuss is about.

I should tell you up-front what this book is not. It is not another breathless account of the fantastic speculations that seem to dominate much of contemporary theoretical physics – things you may have read about, or seen on NOVA. Apart from a few footnotes and an occasional parenthetical remark, there is nothing here about superstrings or extra dimensions or multiple universes. We're dealing with well-established, robustly confirmed “facts.” In a way, modern physics has been a victim of its own success. The revolutions described in this book account so perfectly for everything that is known about our world, that anyone hoping to come up with the next “great idea” is forced to rely more on imagination than on observation.

For any sceptic of the continued capacity of science to uncover new truth, to pave the way for previously unimagined applications, there is hardly a better corrective than to invite reflection on recent discoveries in the science of light. It may be unscientific to say that light is unfathomable, but it certainly is a characteristic of the subject that there is always more to be learned, just when the utmost depths seem within grasp. There is no better illustration than the specific subject of the volume before you.

It has long been known that light conveys energy, and the associated linear momentum has also been understood since the days of Maxwell and Bartoli. With angular momentum the history is more recent, and the property a little less straightforward. What we quickly learned is that light has a propensity to convey angular momentum, depending on its state. The pioneering work in which Beth established a link with circular polarisation is nonetheless already three-quarters of a century old. Once the quantum theory of light was developed, many would have surmised that the science was complete, the concept of angular momentum so beautifully related to the unit spin of the photon-the hallmark of a boson. But what has been discovered in the past quarter century has shown that the spin angular momentum is only half the story – and the other half has no ending yet in sight.

Recent developments in the angular momentum of light are leading to new and wideranging applications, even as the subject presents fresh challenges to long established and cherished concepts.

Introduction

Spin-orbit interaction in quantum physics

The term spin-orbit interaction (SOI) is known from quantum physics, where it describes the coupling between the spin and orbital angular momentum (AM) of electrons or other quantum particles [1]. The SOI is usually interpreted as an electromagnetic interaction of the moving magnetic moment of the electron with an external electric field. However, in central fields the SOI Hamiltonian becomes proportional to the product of the spin AM (SAM) and orbital AM (OAM). Furthermore, the geometric Berry-phase description of the SOI uncovered that it is basically related to the intrinsic AM properties of the particle and is largely independent of the particular character of interaction with the external field [2–4]. The SOI can take various forms in different systems, but the unifying feature in all cases is the coupling between the spin and momentum of the particle [5].

There are two basic manifestations of the SOI of electrons. First, the SOI brings about the fine splitting of the energy levels in the finite orbital motions, e.g., in potential wells or magnetic fields [1, 6, 7]. This can be regarded as the Berry-phase contribution to the quantization of orbits [8–11]. Second, due to the SOI upon free motion in an external field the electron undergoes a transverse spin-dependent deflection known as the spin Hall effect [4, 5, 12–14]. This effect is a dynamical (transport) manifestation of the Berry phase closely related to the Coriolis effect [15, 16], and conservation of the total AM of the particle [12].