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.

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].

Introduction

Helical phase fronts are commonly associated with Laguerre-Gaussian (LG) beams and optical vortices. These beams are characterized by an input field normally expressed as exp(−ilϕ), where l is an integer called the topological charge, which describes the magnitude and handedness of the helical phase profile. In quantum theory, the topological charge relates to a quantized orbital angular momentum (OAM) of lℏ per photon [1–2]. When projected in the far field, the azimuthal components of the phase create destructive interference giving rise to a dark centre surrounded by a high-intensity ring of light via constructive interference. As the topological charge is increased, the steeper phase gradients deflect light farther off-axis, thereby enlarging the light ring surrounding the dark centre.

When focused using a lens, the propagation of an LG beam along the optical axis follows a conical ray until it reaches a minimum ring radius at the focus and then conically increases after the focus. A high concentration of photons is maintained at the outskirts of the conical beam forming a ring at the transverse plane. Increasing the topological charge (and consequently the OAM) disperses the distribution of photons around a larger ring. Aside from circularly symmetric beams, higher-order LG beams bring about unique laser transverse modes, which display unique three-dimensional (3D) light patterns. Hence, the interplay of helical phase fronts results in far field beam profiles that can be uniquely utilized in optical tweezers [3–7] or as means of engineering the point spread function in laser microscopy [8].

Introduction

Higher-order modes of optical beams have been the subject of many studies over the past 20 years [1, 2]. Because laser resonators deliver in principle quite complex spatial patterns, spatial modes of beams were studied intensively when lasers were first developed [3]. Beams in higher-order spatial modes are solutions of the paraxial wave equation, with Hermite-Gauss and Laguerre-Gauss beams being the most important families of beams. These are solutions of the wave equation in Cartesian and cylindrical coordinates, respectively. The latter beams have been at the heart of a revival of research on higher-order modes due to the orbital angular momentum that they carry [4]. They have also received much attention due to the phase singularities present in their transverse amplitude [5]. Higher-order modes have stimulated much research in the application of forces and torques to objects in optical tweezers [1, 2]. Their usefulness has carried higher-order spatial modes further into new paths, in studies with non-classical sources of light and applications in quantum information [6].

The beams mentioned above are scalar solutions of the wave equation and thus independent of the polarization of the light. Vector beams are formed by the non-separable combinations of spatial and polarization modes. This enhanced modal space produces an interesting set of beams that offer new effects and applications. The origin of these beams is not recent either, as the possibility of combining higher-order spatial modes and polarization started with those early studies of modes as well.

For optical fields the notion of a total angular momentum has long been known. The concept of a light beam carrying orbital angular momentum, however, was unfamiliar until it was discovered that Laguerre-Gaussian beams, within the paraxial approximation, carry a well-defined orbital angular momentum [1, 2]. This discovery started the modern interest in orbital angular momentum of light. In this chapter we discuss the theoretical framework of orbital angular momentum of light in terms of fields and light beams and how to generate these. The material in this chapter is based in parts on the PhD thesis of Götte [3].

Introduction

A quantitative treatment of the mechanical effects of light became possible only after light had been integrated into Maxwell's dynamical theory of electromagnetic waves. With this theory Poynting [4] derived a continuity equation for the energy in the electromagnetic field. After Heaviside [5, 6] introduced the vectorial notation for the Maxwell equations this continuity equation could be written in its modern form using the Poynting vector. Interestingly, the linear momentum density in the electromagnetic field is also given by the Poynting vector apart from constant factors depending on the chosen system of units. Poynting [7] also derived an expression for the angular momentum of circularly polarised light by means of a mechanical analogue in the form of a rotating shaft. Later, Poynting's expression was verified by measuring the torque on a quarter wave-plate due to circularly polarised light [8].

Introduction

Vortices and vorticity are common objects of study in the field of fluid dynamics [1]. In a fluid, vortices appear as spinning, often turbulent, flows, giving rise to common phenomena such as tornadoes and whirlpools. In their more discreet manifestations, fluid vortices are thought to be responsible for the destruction of bridges due to the formation of vortex streets and structural resonance [2] and for the flight of insects [3], ending a long debate in the community about the reversibility of the movement of insect wings. Because of the importance of all those natural phenomena, the study of vortex dynamics and the creation and annihilation of hydrodynamical vortices is a matter of intense study.

When the first properties of superfluids were found, the question of whether vortices could appear in such irrotational fluids was addressed by prominent scientists [4, 5]. It was found that vortices could also appear in such systems, but they could only appear as point vortices, i.e. singular locations where the vorticity of the fluid was infinite, whereas everywhere else the vorticity was strictly zero. This property was also found later on in Bose-Einstein condensates (BECs), where the quest for exciting and controlling the point vortices of quantum systems is still a hot topic. The dynamics of and processes related to the birth and destruction of vortex-antivortex pairs play an important role in the physics of these exotic systems.

Introduction

There exist two well-established methods to trap charged particles: the Penning trap [1] and the Paul trap [2]. In the Penning trap the particle is confined in space by a combination of static magnetic and electric fields. In the Paul trap the trapping is caused by a high frequency electric quadrupole field. The subject of this article is to present a third mechanism for trapping charged particles – trapping by beams of electromagnetic radiation. It was discovered some time ago [3–6] that a properly prepared beam acts as a “waveguide” for particles, confining their motion in the transverse directions. Similar phenomena occur in RF fields [7]. In all these cases, the essential role is played by the electric field configuration in the plane perpendicular to the beam axis (for nonrelativistic electrons, the magnetic field is less important). Particles are confined to the vicinity of the minimum-energy points. In particular, for beams of electromagnetic radiation carrying orbital angular momentum such points lie on the beam axis. One beam may confine particles only in the transverse direction. Two or three crossing beams may fully confine particles, acting as a substitute for the Paul trap.

Trapping of charged particles by beams of electromagnetic radiation is based on the same general principles as the Paul trap. In the Paul trap the quadrupole radio-frequency field is produced by a system of electrodes. In our case, the field is that of freely propagating electromagnetic beams. In both cases the essential role is played by two factors: the proper shape of the force field and fast oscillation or rotation. These features are clearly seen when we invoke the notion of the ponderomotive potential.

The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure.

The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found.

Introduction

The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2π radian range or whether it evolves continuously without bound.

Introduction

When light engages with matter, the interactions that take place at the photon level are subject to the operation of powerful underlying symmetry laws. Such principles underpin the physics of even the simplest photon interactions, as for instance in the familiar Planck- Einstein relation E = hν for the absorption or emission of radiation. As emerged from Noether's work [1], this manifestation of overall conservation of energy is a direct consequence of a system invariance under temporal translation [2]. In connection with specific atomic photophysics, the term ‘selection rule’ is often used in connection with other space or time symmetries, these frequently being manifest as constraints over the conservation of quantized angular momentum. Obvious examples are the rules that govern the ‘allowed’ and ‘forbidden’ lines in atomic spectra, where the associated conditions over the geometric disposition and flow of charge emerge in the form of transition multipoles [3, 4].

The angular momentum attributes of light are most familiar in connection with the integer spin of the photon. Circularly polarized states have well-defined spin angular momentum along the direction of propagation [5], and numerous chiral or gyrotropic interactions exploit the differences in behaviour – observed in a material that is itself chirally constituted – between light beams of left- and right-handedness [6–8]. The principles are well known, and their applications have a surprisingly wide compass.

Introduction

Although the orbital angular momentum (OAM) of light is often quantified in terms of ℓℏ per photon, one should recognise that this result can be derived directly from Maxwell's equations and is most certainly not an exclusively quantum property. Following the recognition of optical OAM in 1992 [1], the immediate thrust of experimental work was to observe OAM transfer to microscopic objects. These exciting experiments by Rubinsztein- Dunlop [2, 3] and others established that the predicted angular momentum was truly mechanical, but did not provide insight on OAM at the level of single photons. Laguerre- Gaussian modes were identified as being a natural basis set for describing beams with OAM, where their phase terms describe the helical phasefronts corresponding to an OAM of ℓℏ per photon.

The polarisation of light has played an important historical role in the experimental investigation of quantum entanglement, quantum gate operations and quantum cryptography. A potential advantage of OAM is that it resides in a higher-dimensional state space which, as follows from the theory, is infinite-dimensional. One of the experimental challenges is to access a large number of these dimensions. In quantum communication, the most obvious advantage of high n-dimensional systems is the increased bandwidth [4], but qunits also lead to stronger security [5, 6]. An increase in the number of entangled quantum modes has also been shown to improve measures of nonlocality [7], to allow quantum computation with better fault tolerance [8] and to simplify quantum logic structures [9].

Introduction

The orbital angular momentum (OAM) carried by light is widely seen as an extremely useful optical characteristic, with applications in many areas of optics. It was Allen et al. [1] who recognised that a helically phased light beam with a phase cross-section of exp(iℓϕ) carries an OAM, with a value of ℓℏ per photon. Such a light beam contains an optical vortex line of ℓ on its axis. One issue that is yet to be completely resolved is the development of a simple and 100% efficient method for the measurement of OAM.

A better known case of optical angular momentum is spin angular momentum (SAM). SAM is associated with the polarisation state of the light; the spin angular momentum in a left and right circularly polarised beam is σℏ=±1, per photon, respectively [2]. The SAM can be easily determined through the use of a polarising beam splitter, where a π/4 waveplate converts circular polarised light into a p- or s-polarised state which is then transmitted or reflected to give one of two outputs, as shown in Fig. 13.1(a).

OAM arises from the amplitude cross-section of the beam and is therefore independent of the spin angular momentum. One key characteristic of beams carrying OAM is that whereas SAM has only two orthogonal states, the OAM is described by an unbounded state space, i.e. ℓ (as in exp(iℓϕ) can take any integer value [3].