Quantum theory of continuous measurements and its applications in quantum optics

We present an overview of the quantum theory of continuous measurements and discuss some of its important applications in quantum optics. Quantum theory of continuous measurements is the appropriate generalization of the conventional formulation of quantum theory, which is adequate to deal with counting experiments where a detector monitors a system continuously over an interval of time and records the times of occurrence of a given type of event, such as the emission or arrival of a particle. We first discuss the classical theory of counting processes and indicate how one arrives at the celebrated photon counting formula of Mandel for classical optical fields. We then discuss the inadequacies of the so called quantum Mandel formula. We explain how the unphysical results that arise from the quantum Mandel formula are due to the fact that the formula is obtained on the basis of an erroneous identification of the coincidence probability densities associated with a continuous measurement situation. We then summarize the basic framework of the quantum theory of continuous measurements as developed by Davies. We explain how a complete characterization of the counting process can be achieved by specifying merely the measurement transformation associated with the change in the state of the system when a single event is observed in an infinitesimal interval of time. In order to illustrate the applications of the quantum theory of continuoius measurements in quantum optics, we first derive the photon counting probabilities of a single-mode free field and also of a single-mode field in interaction with an external source. We then discuss the general quantum counting formula of Chmara for a multi-mode electromagnetic field coupled to an external source. We explain how the Chmara counting formula is indeed the appropriate quantum generalization of the classical Mandel formula. To illustrate the fact that the quantum theory of continuous measurements has other diverse applications in quantum optics, besides the theory of photodetection, we summarize the theory of ‘quantum jumps’ developed by Zoller, Marte and Walls and Barchielli, where the continuous measurements framework is employed to evaluate the statistics of photon emission events in the resonance fluorescence of an atomic system.     

Casimir effects: An optical approach II. Local observables and thermal corrections

We recently proposed a new approach to the Casimir effect based on classical ray optics (the “optical approximation”). In this paper we show how to use it to calculate the local observables of the field theory. In particular, we study the energy–momentum tensor and the Casimir pressure. We work three examples in detail: parallel plates, the Casimir pendulum and a sphere opposite a plate. We also show how to calculate thermal corrections, proving that the high temperature ‘classical limit’ is indeed valid for any smooth geometry.     

Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the “Minkowski vacuum” for a box at rest in an inertial frame and a “Rindler vacuum” for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. The classical analysis gives no grounds for the “heating effects of acceleration through the vacuum” which appear in the literature of quantum field theory. Thus this distinction provides (in principle) an experimental test to distinguish the two theories.     

Unified Field Theory with EINSTEINian Photons and Heavy Bosons as Field Quants

After discussing in two previous papers [1, 2] the classical electrodynamics which corresponds to the quantum electrodynamics with two sorts of photons (photons with zero rest mass and nonvanishing rest mass), in the present paper the general field theory of a vector field A^v with two sorts if field quanta is given. It is shown that the postulate of the “unity of the four-current” determining the physical contents of this theory makes it possible to regard it as a classical ansatz of a unified theory of the electromagnetic and the weak interactions. From the “unity of the currents” results that the electrons are δ-like point-particles with a finite self-potential and finite field masses M = ε²/2 kc^?2. The COMPTON wave-length of the heavy photons k^?1 = h/mc has the meaning of an “elementary length” for the electromagnetic interactions and the rest mass m = khc^?1 of these bosons is of the order of a baryon mass.     

Maxwell's equations from geometrical optics

Maxwell's electromagnetic equations are derived from Fermat's principle in geometrical optics by the process of “wavization” analogous to quantization of classical mechanics.     

Quantum chaos of the two-level atom

The quantum theory of the two-level atom coupled to a single mode of the electromagnetic field is considered as a simple example of “quantum chaos”, defined as the quantum behavior of a dynamical system which is non-integrable in the classical limit.     

麦克斯韦方程组的建立及其作用

麦克斯韦提出了描述经典电磁场运动及其与带电粒子相互作用规律的完备方程组,将电学、磁学和光学统一为电磁场动力学理论。这一理论具有洛仑兹协变性和U(1)局域规范不变性,成为构造粒子物理标准模型的经典模板,在物理理论和实验发展中起着不可估量的巨大作用。     

Apertureless near-field laser nanotechnology

We consider apertureless near-field optics that provides subwavelength resolution. We study the enhancement of the electromagnetic field near nanospheres and under the tip of a scanning probe microscope using the finite difference time-domain (FDTD) method. We discuss the mechanisms of field enhancement connected with the system geometry (“lightning rod effect”) and resonance excitation of local plasmon eigenmodes for different materials of the tip and various geometrical parameters of the system. We describe the possible applications in nano-optics and nanotechnology. We present the experimental achievements in apertureless near-field nanolithography.     

Interaction of Photons with Electrons in Dielectric Media

The question of the field energy-momentum tensor in a medium is not new and many workers, in the past, attempted to find the answer to it. Nevertheless, there was no general agreement about the form of such a tensor, thus resulting in a confusion involving the very fundaments of physics. Although the present work uses well established theories, the underlying philosophy is completely novel. Investigations are mainly centered around the collisionless plasma as, in that state, development of the argument is most transparent. On the basis of a simple theoretical reasoning, it is, first of all, postulated that the momentum of the photon in a plasma is given by the Minkowski expression. This hypothesis becomes more viable as it directly leads to the accepted form for the field energy density. Furthermore, utilizing the concepts of stimulated and spontaneous emission and absorption, it is confirmed that, in the equilibrium (between radiation and plasma), the transferred momentum has the value given by the Minkowski theory. However, as will be seen, this result is valid only in the region of high frequencies. Put otherwise, when conditions are such that the laws of geometrical optics apply, the momentum of the photon is, to a good approximation, given by the Minkowski theory. In order to arrive at a more general result, valid in the domain of wave optics, a similarity between the dispersion relation (describing the wave propagation through a medium) and the relativistic energy-momentum of a particle moving through vacuum, is explored. Accepting the equivalence of these two relations implies that some of the properties of radiation in a medium, can also be described by a massive particle travelling through empty space. In other words, the task of analysing the behaviour of electromagnetic waves in a medium, can be replaced by the analysis of the free neutral vector meson field. Adoption of this equivalence results in the field energy-momentum tensor being symmetrical. A thorough study of the field theory then provides the basis for interpreting the Abraham tensor as the sum of the “orbital” and “spin” tensors. The former is directly connected with the energy transport, whereas the latter one is not. Realising that the magnitude of the spin term increases towards the lower end of the frequency spectrum provides the resolution of the Abraham-Minkowski dilemma: the energy-momentum tensor, corresponding to the electromagnetic wave in a medium, is that given by Abraham, while the one of Minkowski is only an approximation valid at high frequencies. Although this conclusion stems from studies involving collisionless isotropic plasma, it is in excellent agreement with the experimental data.     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

The geometrical aspect of the gauge fields and the possibility of a general description of local and nonlocal interaction in QED

It is shown that the use of analogues of the QCD gauge-invariant structures (type of “strings” and “stars”) in QED enables one to consider the generalized gauge-invariant amplitude, which satisfies the requirements of the quantum theory of gauge fields and allows one to describe electromagnetic (EM) interactions both with local and nonlocal charged matter fields beyond the scope of the Lagrange approach. This causes no negative consequences for the theory as a whole. The invariant character of the amplitude structure in relation to hierarchical evolution of the structural forces and structural elements of the nonlocal field allows its use in an unchanged form to describe EM interaction processes in different scales of matter structure. The generalized amplitude features a continuous limit in transition from nonlocal to local fields.     

The compatibility between quantum mechanics and local realist theories in atomic cascade experiments

We show that the divergence between the predictions of quantum optics and the local realist theory known as stochastic optics, for the extended type of photon-coincidence experiment described recently by de Caro, is of the same order of magnitude as for Aspect-type experiments. This means that, in such new experiments, as in those so far performed, counting statistics will have to be greatly improved before a discrimination between the two theories becomes possible.We also show that the outstanding difference between the two theories is that, while stochastic optics uses a genuine, that is, positive, detection probability, the corresponding quantum formalism leads to a pseudoprobability. Nevertheless, there is a striking parallel, in that both theories recognize the phenomenon known as enhancement, and both describe it as having its origin in a mixing of the signal field with the zero-point field by a polarizing device.     

开孔矩形腔体电磁泄漏特性的解析研究

本文提出了一种用于计算开孔矩形腔体电磁泄漏场的解析理论模型.该理论模型先基于模式展开法求解封闭腔场,进而依据Bethe小孔耦合理论将泄漏场与封闭腔场用等效偶极子关联.该模型可以考虑波频率、场源位置、开孔位置及场强观测点位置等因素的影响,计算结果与全波仿真结果一致.本文计算分析了相关因素对电磁屏蔽效能的影响规律,并给出了物理解释.结果表明近场屏蔽效能小于远场屏蔽效能,且近场区电场屏蔽效能与磁场屏蔽效能并不相同.     

Optical Super-Resonances in Mesoscale Dielectric Cenosphere: Giant Magnetic Field Generations

Resonant light scattering by mesoscale dielectric spheres has received enormous attention and found many interesting applications. The recently emerged field of Mesotronics provides novel opportunities for wavelength-scaled optics and new fundamental aspects are still being uncovered. It has recently been demonstrated that high-order Mie resonances can be excited in homogeneous low-dissipation mesoscale dielectric spheres, leading to the generation of intense magnetic fields. This article describes a simple and effective way to drastically enhance the superresonance effect. Proof-of-principle results for the first time show that yet one more novel phenomenon of increasing the intensity of the magnetic field without changing the resonant Mie size parameter of the sphere by introducing an air cavity. In such a dielectric cenosphere (from two Greek words “kenos”-hollow and “sphaira”-sphere), by correct choice of the air cavity size, it is possible to increase the intensity of the electromagnetic fields at the poles of the sphere by an order of magnitude due to increasing of the amplitude of resonant partial wave coefficient.     

The van Cittert-Zernike theorem in atom optics

The van Cittert-Zernike theorem of classical optics is generalized to the situation of matter-wave optics. Applying a Green's function method the first- and second-order correlation functions of the matter field emitted from an incoherent source are calculated, and the results are compared to the case of electromagnetic waves.     

Scattering from a penetrable sphere at short wavelengths

A scalar plane wave incident on a penetrable sphere is considered in the short wavelength limit. A new representation of the scattering amplitude is introduced which is particularly appropriate in this limit, and which requires only the evaluation of certain integrals. Some of these may be evaluated asymptotically by the method of steepest descent and lead to the geometrical optics field contribution. Included in this is the bow field. The remainder of the integrals are evaluated by the method of residues and lead to the diffracted field contribution. This “diffracted ray” field is known from recent investigations in diffraction theory. An essential part of the analysis is the introduction of the parameter p, the number of internal refractions that a ray which hits the sphere undergoes. The results obtained are all in agreement with that which would be expected on the basis of geometrical diffraction theory.     

String loop divergences and effective lagrangians

We isolate logarithmic divergences from bosonic string amplitudes on a disc. These divergences are compared with “tadpole” divergences in the effective field theory, with a covariant cosmological term implied by the counting of string coupling constants. We find an inconsistency between the two. This might be a problem in eliminating divergences from the bosonic string.