Generalized quasi mode theory of macroscopic canonical quantization in cavity quantum electrodynamics and quantum optics I. Theory

Abstract

The quasi mode theory of macroscopic quantization in quantum optics and cavity QED developed by Dalton, Barnett and Knight is generalized. This generalization allows for cases in which two or more quasi permittivities, along with their associated mode functions, are needed to describe the classical optics device. It brings problems such as reflection and refraction at a dielectric boundary, the linear coupler, and the coupling of two optical cavities within the scope of the theory. For the most part, the results that are obtained here are simple generalizations of those obtained in previous work. However the coupling constants, which are of great importance in applications of the theory, are shown to contain significant additional terms which cannot be ‘guessed’ from the simpler forms. The expressions for the coupling constants suggest that the critical factor in determining the strength of coupling between a pair of quasi modes is their degree of spatial overlap. In an accompanying paper a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary is given as an illustration of the generalized theory. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes.     

Quasi mode theory of macroscopic canonical quantization in quantum optics and cavity quantum electrodynamics

Abstract

A macroscopic, canonical quantization of the EM field and radiating atom system in quantum optics and cavity QED involving classical, linear optical devices, based on expanding the vector potential in terms of quasi mode functions is presented. The quasi mode functions approximate the true mode functions for the device, and are obtained by solving the Helmholtz equation for an idealized spatially dependent electric permittivity function describing the device. The Hamiltonian for the EM field and radiating atom system is obtained in multipolar form and the quantum EM field is found to be equivalent to a set of quantum harmonic oscillators, one oscillator per quasi mode. However, unlike true mode theory where the quantum harmonic oscillators are uncoupled, in the quasi mode theory they are coupled and photon exchange processes can occur. Explicit expressions for the coupling constants are obtained. The interaction energy between the radiative atoms and the quantum EM field depends on the amplitudes of the quasi mode functions at the positions of the radiating atoms, similar to that for the true mode approach. The simpler forms for the quasi mode functions enable the atom-field interaction energy to be written in a form in which the atoms are only coupled to certain types of modes—for example cavity quasi modes, which are large inside the optical cavity. In such cases the escape of energy from excited atoms in the cavity can be pictured in quasi mode theory as a two step process—the atom de-excites and creates a photon in a cavity quasi mode, the photon in the cavity quasi mode is then lost and appears as a photon in an external quasi mode. In this process the first step occurs via the atom-cavity quasi mode interaction, the second through coupling between cavity and external quasi modes. This may be contrasted with the true mode approach, where the excited atom loses its energy and the photon is created in one of the true modes. As all true modes have non-zero amplitudes outside as well as inside the cavity, the escape of energy from excited atoms in the cavity is seen as a one step process. An application of the quasi mode theory to the quantum theory of the beam splitter is outlined. The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.     

The standard model in cavity quantum electrodynamics. II. Coupling constants and atom field interaction

Abstract

Specific forms of the travelling and trapped vector mode functions for a three-dimensional Fabry-Perot cavity are developed from the general results of the preceding paper, with parameters describing the output cavity mirror chosen for a typical high Q cavity case. Cavity and external quasi-mode functions associated with the quasi-mode theory of macroscopic canonical quantization are then obtained via an idealized choice of output mirror parameters. The coupling constants describing photon exchange processes between the single cavity quasi-mode associated with each Fabry-Perot resonance and various external quasi-modes are calculated, and their slow dependence on the external quasi-mode frequency shows that the conditions for irreversible Markovian damping of the cavity quasi-mode are satisfied. For radiative atoms placed in the cavity the coupling constants for energy exchange processes with sideways travelling external quasi-modes also vary slowely, so that Markovian spontaneous emission damping occurs for the radiative atoms. However, their coupling with the isolated cavity quasi-modes is associated with reversible photon exchanges as represented via one photon Rabi frequencies. The standard model in cavity quantum electrodynamics, in which the basic processes are described by a cavity damping rate, a radiative atom spontaneous decay rate and an atom-cavity mode coupling constant has now been justified in terms of the quasi-mode theory of macroscopic canonical quantization.     

Reflection and refraction of plane waves at interface between two piezoelectric media

This paper analyses reflection and refraction of plane waves at a perfect interface between two anisotropic piezoelectric media. The equations of elastic waves, quasi-static electric field, and constitutive relationships for the piezoelectric media are derived. A solution based on the inhomogeneous wave theory is developed to address the inconsistency between the numbers of independent wave modes in the media and the numbers of interfacial boundary conditions to obtain accurate reflection and refraction coefficients in case of strong piezoelectric media, where all the elastic and electric continuity conditions across the interface are satisfied simultaneously. The study shows that there exist independent and zero energy wave modes satisfying the general Snell’s law and propagating along the interface for any incident wave angle. These waves can be treated as pseudo surface waves. It is further found that all the reflection/refraction waves including the pseudo surface waves obey the energy conservation law at the interface boundary. In addition, the analysis also reveals that the reflection and refraction elastic waves can turn into pseudo surface waves at some critical incident angles.     

Single photon scattering in a pair of coupled-resonator waveguides coupled to a quantum emitter

We theoretically investigate the single photon scattering in a pair of coupled-resonator waveguides coupled to a two-level quantum emitter. It reveals that the incident photon transporting probabilities can be controlled by adjusting the coupling strength between the two-level system and the coupled-resonator waveguide. Two optical nano-devices, quantum optical switch and beam splitter are proposed. In addition, the influence of the dissipation of the two-level system on the photon transporting properties is also analyzed.     

The laws of plane defect wave propagation across the interface between two viscoplastic media

Relations determining the laws of reflection and refraction of a plane wave in the defect field at an interface between two viscoplastic media are obtained based on the dynamic equations of the continuum theory of defects. The reflection and transmission coefficients relating the amplitudes of reflected and transmitted waves to the incident wave amplitude are determined. The obtained relations are applied to a particular case of media with weakly decaying waves.     

Field quantization,photons and non-Hermitean modes

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.     

Ultrasonic guided wave scattering in a plate overlap

Guided wave scattering in a plate overlap is investigated by numerical calculations and experimental measurements of transmission and reflection factors from the overlap region. In the numerical study, a hybrid boundary element-finite element method is used to calculate the guided wave scattered field from the overlap region. Transmission and reflection factors are calculated for incident A0 and S0 Lamb and n0 shear horizontal waves, including higher modes generated through mode conversion phenomena. In addition, parametric studies of transmission and reflection factors in this problem are performed numerically over various incident modes, frequencies, and overlap lengths. For verification and comparison with numerical results, experiments were conducted to measure the transmission and reflection factors for incident Lamb and shear horizontal waves in steel plates with two different overlap areas. The experimental results agree well with the numerical calculations. The numerical and experimental results show that it is highly feasible to carry out efficient Lamb wave nondestructive evaluation (NDE) in overlapped plates and in multilayer structures with various lap joints by selecting various modes and tuning frequency.     

The standard model in cavity quantum electrodynamics. I. General features of mode functions for a Fabry-Perot cavity

Abstract

In the present and the accompanying paper a justification of the standard model of cavity quantum electrodynamics is given in terms of a quasi-mode theory of macroscopic canonical quantization. The coupling of the cavity quasi-mode to external quasi-modes is treated for the representative case of the three-dimensional Fabry-Perot cavity. The general form of the travelling and trapped mode functions for this cavity are derived in this paper and the mode-mode coupling constants are calculated in the accompanying paper. The slow dependence of the coupling constants with the mode frequency difference demonstrates that the conditions for Markovian damping of the cavity quasimode are satisfied. As also discussed in the accompanying paper, the interaction of radiative atoms with cavity quasi-modes is associated with reversible energy exchanges between atom and cavity and represented by Rabi coupling constants. The interaction of radiative atoms located within the cavity with sideways travelling external quasi-modes involves slowly varying coupling constants and is associated with irreversible spontaneous emission dampling. The basic processes represented in the standard cavity quantum electrodynamics model and the associated coupling constant and decay rates thereby follow from the quasi-mode theory.     

Quantum Statistical Properties of Photon and Phonon Fields in Brillouin Scattering with Intense Pumping

The quantum statistical properties of Brillouin scattering of intense laser light are derived including the coupling of Stokes, anti-Stokes and phonon modes, if the anti-Stokes interaction prevails. Making use of the coherent-state technique, the Heisenberg equations of this process are solved neglecting the loss mechanism, and the normal quantum characteristic function and the normal generating function are derived. The time dependences of the photon distribution and its factorial moments are demonstrated if the phonon, Stokes and anti-Stokes modes are initially in a coherent state and periodical anti-bunching of the field is found when the phases of the incident fields fulfil certain phase conditions; the field can also return to the coherent state again.     

Estimation of ultrasonic guided wave mode conversion in a plate with thickness variation

The hybrid boundary element method aimed at analyzing Lamb wave scattering from defects can provide us with an excellent numerical tool for tackling complicated mode conversion phenomena under waveguide thickness variation. In this paper, utilization of hybrid boundary element modeling for specific Lamb wave mode incidence situations with special energy distributions along the structural cross section is proposed for estimating reflection and transmission from various scatterers, such as a step discontinuity and tapered parts of a waveguide, etc. Interaction of individual Lamb wave modes with scatterers that represent arbitrary thickness variation along the direction of guided wave propagation is investigated by calculating the scattered fields for varying incident modes, frequency, and scatterer shape. The mode conversion phenomena through step discontinuity in a plate are also experimentally explored. The theoretical predictions of reflection and transmission by boundary element methods and the utility of dispersion curves are compared with experiments for specific modes. Results in this paper can be used to improve inspection sensitivity and penetration power for a variety of practical NDE applications, notably those in which thickness variation is found. In addition, the feasibility of inspecting sections located behind a waveguide thickness variation region and subsequent mode control will also be discussed.     

Photon density waves scattered from cylindrical inhomogeneities: theory and experiments

We present an analytical solution for the scattering of diffuse photon density waves from an infinite circular, cylindrical inhomogeneity embedded in a homogeneous highly scattering turbid medium. The analytical solution, based on the diffusion approximation of the Boltzmann transport equation, represents the contribution of the cylindrical inhomogeneity as a series of modified Bessel functions integrated from zero to infinity and weighted by different angular dependencies. This series is truncated at the desired precision, similar to the Mie theory. We introduce new boundary conditions that account for specular reflections at the interface between the background medium and the cylindrical inhomogeneity. These new boundary conditions allow the separate recovery of the index of refraction of an object from its absorption and reduced scattering coefficients. The analytical solution is compared with data obtained experimentally to evaluate the predictive capability of the model. Optical properties of known cylindrical objects are recovered accurately. However, as the radius of the cylinder decreases, the required measurement signal-to-noise ratiorapidly increases. Because of the new boundary conditions, an upperlimit can be placed on the recovered size of cylindrical objects with radii below 0.3 cm if they have a substantially different index of refraction from that of the background medium.     

Stimulated Raman Scattering of Squeezed Light with Pump Depletion

Abstract

Using the approximation of small fluctuations around a stationary point, photon statistics and squeezing of vacuum fluctuations are derived in stimulated Raman scattering including the coupling of photon-photon and photon-phonon modes. The depletion of pump light is described and it is shown that a regime of inversion of the process is related to an increase in quantum fluctuations, which are finally reduced again, more in the anti-Stokes mode than in the Stokes mode. If the anti-Stokes coupling is stronger than the Stokes coupling, then an oscillating behaviour of the statistical characteristics occurs and substantial squeezing can be found in photon-phonon modes.     

Law of normal scattering--a comprehensive law for wave propagation at an interface

The fundamental laws of wave propagation at an interface, the laws of reflection, refraction, and diffraction are arrived at from a consideration of wave scattering from an array of scattering centers. It is shown that the number, spacing, and dimension of the scattering centers decide whether the laws of reflection and refraction or the more comprehensive law of normal scattering is obeyed. An exact equation for the summation of the phases and amplitudes of scattered waves arriving at a point is developed. Scattered ray directions are arrived at from the analytical behavior of this equation, and the region of validity of the solutions is discussed. It is shown that the length of the array plays as important a role as the spacing of the scattering centers. The modulation of diffracted ray intensity with an increase in scattering center size is identified and investigated. An expression for far-field distance is arrived at based on numerical simulation and is valid for all scattered angles.     

Highly directional emission and photon beaming from nanocrystal quantum dots embedded in metallic nanoslit arrays

We demonstrate a directional beaming of photons emitted from nanocrystal quantum dots that are embedded in a subwavelength metallic nanoslit array with a divergence angle of less than 4°. We show that the eigenmodes of the structure result in localized electromagnetic field enhancements at the Bragg cavity resonances, which could be controlled and engineered in both real and momentum space. The photon beaming is achieved using the enhanced resonant coupling of the quantum dots to these Bragg cavity modes, which dominates the emission properties of the quantum dots. We show that the emission probability of a quantum dot into the narrow angular mode is 20 times larger than the emission probability to all other modes. Engineering nanocrystal quantum dots with subwavelength metallic nanostructures is a promising way for a range of new types of active optical devices, where spatial control of the optical properties of nanoemitters is essential, on both the single and many photons level.     

Interactions of Phonons and Rotons with Interfaces in Superfluid Helium

We solve the problem of beams of phonons and rotons incident on, and interacting with, solid surfaces. Phonons and rotons are the quasiparticles of superfluid helium and have a unique dispersion curve. The dispersion curve controls the transmission, reflection and mode change of these quasiparticles at the interface with another medium. We develop a non-local hydrodynamic theory in a consistent and unified way. The structure of the solutions in the quantum fluid is discussed. The creation probabilities of all quasiparticles are derived when any one of them is incident on the interface. The dependencies on frequency and angle are analyzed and the backward reflection and refraction for R ^? rotons are discussed.     

A quantum scattering theory approach to quantum-optical measurements

Abstract

The theory of quantum-optical measurements is developed via a Heisenberg picture approach to quantum scattering theory. The multitime quantum correlation functions describing the measurements are expressed in terms of input or output operators, which are related via the scattering operator.     

超声Lamb波在缺陷处的二维散射特性研究

利用散射系数周向分布图研究了超声Lamb波在缺陷处的二维散射特性。建立了超声Lamb波与缺陷交互作用的有限元仿真模型,采用双元激励法产生单一S0模态入射信号,利用吸收边界消除边界回波的影响,采集缺陷所有方向上的散射信号并产生散射系数周向分布图。此模型的计算结果与实验结果一致,证明了模型的正确性。研究了S0模态与通孔和通透裂纹两种缺陷的交互作用,并测量了S0模态以零度角入射缺陷时的周向散射系数分布图,利用其图形特征可以进行缺陷种类识别。分析了散射场中多模态信号的能量分布,指出检测中若激励单一模态,而接收多模态信号,可以防止漏检,更有效地检测缺陷。     

Scattering by a dense layer of infinite cylinders at normal incidence

The solution for scattering by a layer of densely distributed infinite cylinders is presented. The layer is irradiated by an arbitrarily polarized plane wave that propagates in the plane perpendicular to the axes of the cylinders. The theoretical formulation utilized the effective field and quasi-crystalline approximation to treat the multiple scattering interactions in the dense finite medium. Governing equations for the propagation constants and amplitudes of the effective fields are derived for TM and TE mode incident waves, from which the scattered intensity distribution and scattering cross section for arbitrary polarization are obtained. The dense medium gives rise to coherent and incoherent scattered radiation that propagates in the plane normal to the axes of the cylinders. The coherent scattered radiation includes the forward component in the direction of the incident wave and the backward component in the direction of specular reflection. The incoherent scattered intensity distribution shows a pronounced forward peak that coincides with the angle of refraction of the effective waves inside the medium. Numerical results are presented to illustrate the scattering characteristics of a dense layer of cylinders as a function of layer thickness for a given solid volume fraction.     

Macroscopic canonical quantization in quantum optics: Properties of quasi mode annihilation and creation operators

Abstract

Macroscopic canonical quantization of the EM field and radiative atom systems occurring in quantum optics experiments involving linear classical optics devices can be carried out via expansion of the vector potential either in terms of true mode functions for the optical device or in terms of approximate or quasi mode functions. The relationship between the true mode and quasi mode annihilation, creation operators is determined and shown to involve a Bogolubov transformation. Analytic properties are also examined and it is found that the annihilation, creation operators times the square root of the angular frequency are analytic functions of the variables specifying the modes.