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”.     

Regarding the publication by G. B. Malykin and V. I. Pozdnyakova “On the Velocity of a Light Spot Moving over a Cylindrical Screen”

It has been shown that, in the article by G. B. Malykin and V. I. Pozdnyakova “On the Velocity of a Light Spot Moving over a Cylindrical Screen” [Tech. Phys. 52, 133 (2007)], errors were made in the calculation of the velocity of a light spot moving over a cylindrical screen. As a result, incorrect formulas were obtained to calculate the velocity of the light spot as applied to classical mechanics and the Tangherlini transformation. Correct formulas for calculating the velocity of a light spot moving over a cylindrical in classical mechanics and in the Tangherlini transformation are proposed.     

Multiplication rule of similarity transformations in the coherent state representation and its mapping onto quantum optical generalized ABCD law

We find that classical similarity transformations in the coherent state representation projects onto the similarity transformation operators (STO), these operators constitute a loyal representation of symplectic group. Remarkably, the multiplication rule of the STOs naturally leads to the quantum optical generalized ABCD law, which is the quantum mechanical correspondence of the classical optical ABCD law. Throughout the whole derivation, the technique of integration within an ordered product (IWOP) of operators is employed.     

A New Way of Solving the Wheeler–DeWitt Equation as Applied to Point Sources

A new way of solving the Wheeler–DeWitt equation is proposed which is based on quantization over free parameters of metrics satisfying the Einstein equations. This technique is applied to two point sources described in the classical case by the Tangherlini metric (in an n-dimensional space) and the Reissner–Nordström metric (in the case of the presence of a charge). The results obtained clarify the sense of the Wheeler hypothesis about statistical weights of small dimensionalities and make possible a new approach to the problem of variation of fundamental constants.     

On the relation between the wave aberration function and the phase transfer function for an incoherent imaging system with circular pupil

The consideration is carried out in its general formulation: the wave aberration function is represented in terms of classical aberrations (the Zernike polynomials), the phase transfer function (argument of the complex optical transfer function) is defined by a chain of transformations originating from the generalized pupil function. Quasi-analytical quadrature formulas are derived that link the optical transfer function and the phase transfer function with the aberration terms. It is shown that the phase transfer function contains information on the odd-order aberrations, which can be retrieved from coefficients to the Taylor expansion of the derived quadrature relation. Received 16 June 2001     

Electrostatic self-force in the field of an (<Emphasis Type="Italic">n</Emphasis> + 1)-dimensional black hole: Dimensional regularization

The self-energy of a classical charged particle localized at a relatively large distance outside the event horizon of an (n + 1)-dimensional Schwarzschild–Tangherlini black hole for an arbitrary n ≥ 3 is calculated. An expression for the electrostatic Green function is derived in the first two orders of the perturbation theory. Dimensional regularization is proposed to be used to regularize the corresponding formally divergent expression for the self-energy. The derived expression for the renormalized self-energy is compared with the results of other authors.     

Optical Operator Method Studied via Fresnel Operator Decomposition and Coherent State Representation

We find that the mapping from classical optical transformations to the optical operator method can be realized by using the coherent state representation and the technique of integration within an ordered product of operators. The optical Fresnel operator derived in (Commun. Theor. Phys. (Beijing, China) 38 (2002) 147) can unify those frequently used optical operators. Various decompositions of Fresnel operator into the exponential canonical operators are obtained.     

Optical contrast by laser-induced phase changes: Real time optical measurement of fast transformation times

Real time optical measurements are used to analyse two different kinds of phase changes which generate optical contrast in (In₄₃Sb₅₇)₈₇Ge₁₃ thin films. Amorphous to crystalline and amorphous to amorphous structural transformations are induced by pulsed laser irradiation in micron-sized regions. A two beam configuration is used to follow the evolution of the optical properties of the films in real time. It is shown that real time optical measurements provide a unique tool to analyse laser-induced fast structural transformations leading to optical contrast. Processes occurring via relaxation, solid state crystallization or melting-solidification are clearly distinguished. From the analysis of the optical transients the minimum transformation times are directly determined.On Sabbatical leave from IBM Almaden Research Center, San Jose, CA, USA     

Schlesinger Transformations and Quantum R-Matrices

 Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted -valued connections on the torus. A universal construction is presented which gives the elementary two-point transformations in terms of Belavin's elliptic quantum R-matrix. In particular, the role of the quantum deformation parameter is taken by the difference of the two poles whose residue eigenvalues are shifted. Elementary one-point transformations (acting on the residue eigenvalues at a single pole) are constructed in terms of the classical elliptic r-matrix. The action of these transformations on the τ-function of the system may completely be integrated and we obtain explicit expressions in terms of the parameters of the connection. In the limit of a rational R-matrix, our construction and the τ-quotients reduce to the classical results of Jimbo and Miwa in the complex plane. Received: 19 December 2001 / Accepted: 20 May 2002 Published online: 14 October 2002     

Quantum entanglement and quantum computational algorithms

The existence of entangled quantum states gives extra power to quantum computers over their classical counterparts. Quantum entanglement shows up qualitatively at the level of two qubits. We demonstrate that the one- and the two-bit Deutsch-Jozsa algorithm does not require entanglement and can be mapped onto a classical optical scheme. It is only for three and more input bits that the DJ algorithm requires the implementation of entangling transformations and in these cases it is impossible to implement this algorithm classically.     

BRS gauge and ghost field supersymmetry in gravity and supergravity

Becchi-Rouet-Stora transformations are obtained for the following systems: (i) Pure Einstein gravity in first order form with vierbein and spin connection as independent fields. (ii) First order Einstein gravity coupled to Yang-Mills fields. (iii) Pure supergravity. For the first two systems the results are as in Yang-Mills theory. But for conventional supergravity the BRS transformations leave the effective action invariant only if the classical equations of motion are satisfied. New transformations of the gauge fields of supergravity have been proposed under which the supersymmetry algebra closes. The corresponding BRS transformations do leave the effective action invariant without the need to use the classical equation of motion; moreover, as in Yang-Mills theories, they are nilpotent and have unit Jacobian.     

Efficient classical simulation of optical quantum information circuits

We identify a broad class of physical processes in an optical quantum circuit that can be efficiently simulated on a classical computer: this class includes unitary transformations, amplification, noise, and measurements. This simulatability result places powerful constraints on the capability to realize exponential quantum speedups as well as on inducing an optical nonlinear transformation via linear optics, photodetection-based measurement, and classical feedforward of measurement results, optimal cloning, and a wide range of other processes.     

Gauge-variance of the Dirac bracket

This work is devoted to the study of the change in the functional form of the minus Dirac bracket under pure gauge transformations (in the sense of Levy-Leblond). We found a closed formula which expresses this change and we use it to discuss the relevance of a gauge transformation for the skew-symmetric (Bose-like) quantization procedure of constrained classical models. We found necessary conditions which are to be fulfilled if the gauge transformation is to induce a mere change of representation at the quantum level. It is shown, by considering a simple example, that these conditions can be violated. We conclude then that adding a total time derivative to the Lagrangian of a classical model can drastically change the physical properties of the quantized Bose-like counterpart. A similar result has been detected previously for two particular systems quantized through the symmetric (Fermi-like) rule of quantization.     

Robust Preparation of Atomic Concatenated Greenberger–Horne–Zeilinger States via Shortcuts to Adiabaticity

Here, a protocol for robust preparation of an atomic concatenated Greenberger–Horne–Zeilinger (C‐GHZ) state via shortcuts to adiabaticity (STA) is proposed. The devices for implementing the protocol consist of atoms, cavities, and the optical fibers, which are feasible with current technology. The atoms are trapped in the separated cavities allowing individual control over each atom with classical fields. STA helps to design Rabi frequencies of classical fields so that the atoms can be driven from the initial states to the target states. The numerical simulations show that the protocol holds robustness against atomic spontaneous emissions and photonic leakages. Thus, the protocol may be realized by experiments in the near future.     

One‐electron Optical Properties of a Nanorice with a Nonconcentric Core

We investigate optical properties of nanoshells (small composite clusters composed of a dielectric core and a metal shell; contribution of the shell dominates in the optical properties of the whole cluster) far from plasmon resonance. In particular, nanoshells with the shape of a stretched rotation ellipsoid (nanorice) are considered. For such shell‐type particle the electron wavefunction, the electron energy, the wavenumber spectrum, and the matrix elements of corresponding optical transitions were found. Using these quantities, the classical optical conductivity of such shells (the quantum effects are not considered) and the quantum optical conductivity (relevant addendums for the classical conductivity that are caused by the quantum effects like the electron spectrum discreteness) were found. Oscillating nature of the dependence of these addendums on the incident light frequency is established. The optical conductivity expressions are generalized for the case of a nanorice particle with a weak nonconcentricity.     

Optical absorption and photoluminescence spectroscopy of the growth of silver nanoparticles

Results obtained from the optical absorption and photoluminescence (PL) spectroscopy experiments have shown the formation of excitons in the silver-exchanged glass samples. These findings are reported here for the first time. Further, we investigate the dramatic changes in the photoemission properties of the silver-exchanged glass samples as a function of postannealing temperature. Observed changes are thought to be due to the structural rearrangements of silver and oxygen bonding during the heat treatments of the glass matrix. In fact, photoelectron spectroscopy does reveal these chemical transformations of silver-exchanged soda glass samples caused by the thermal effects of annealing in a high vacuum atmosphere. An important correlation between temperature-induced changes of the PL intensity and thermal growth of the silver nanoparticles has been established in this Letter through precise spectroscopic studies.     

三模共振光学参量振荡腔抽运场的压缩

利用半经典理论分析了由于级联非线性过程,三模共振光学振荡腔反射场的相位压缩,讨论了压缩与各参量的关系,并与结果比较,理论与实验基本吻合。     

The model for self-dual chiral bosons as a Hodge theory

We consider (1+1) dimensional theory for a single self-dual chiral boson as a classical model for gauge theory. Using the Batalin–Fradkin–Vilkovisky (BFV) technique, the nilpotent BRST and anti-BRST symmetry transformations for this theory have been studied. In this model other forms of nilpotent symmetry transformations like co-BRST and anti-co-BRST, which leave the gauge-fixing part of the action invariant, are also explored. We show that the nilpotent charges for these symmetry transformations satisfy the algebra of the de Rham cohomological operators in differential geometry. The Hodge decomposition theorem on compact manifold is also studied in the context of conserved charges.     

Nonlinear distortion generated by semiconductor optical amplifier boosters in analog optical systems

Closed-form theoretical expressions for the second- and third-order harmonic distortions generated by a semi-conductor optical amplifier (SOA) operating under slight saturation conditions are derived for what is believed to be the first time. The distortion results provided by these expressions agree very well with those obtained by means of simulation and experiments, showing that one should carefully adjust the optical power launched into the SOA to obtain both the good linearity and high dynamic range required by analog optical systems.     

Integration of EEG source imaging and fMRI during continuous viewing of natural movies

Electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) are noninvasive neuroimaging tools which can be used to measure brain activity with excellent temporal and spatial resolution, respectively. By combining the neural and hemodynamic recordings from these modalities, we can gain better insight into how and where the brain processes complex stimuli, which may be especially useful in patients with different neural diseases. However, due to their vastly different spatial and temporal resolutions, the integration of EEG and fMRI recordings is not always straightforward. One fundamental obstacle has been that paradigms used for EEG experiments usually rely on event-related paradigms, while fMRI is not limited in this regard. Therefore, here we ask whether one can reliably localize stimulus-driven EEG activity using the continuously varying feature intensities occurring in natural movie stimuli presented over relatively long periods of time. Specifically, we asked whether stimulus-driven aspects in the EEG signal would be co-localized with the corresponding stimulus-driven BOLD signal during free viewing of a movie. Secondly, we wanted to integrate the EEG signal directly with the BOLD signal, by estimating the underlying impulse response function (IRF) that relates the BOLD signal to the underlying current density in the primary visual area (V1). We made sequential fMRI and 64-channel EEG recordings in seven subjects who passively watched 2-min-long segments of a James Bond movie. To analyze EEG data in this natural setting, we developed a method based on independent component analysis (ICA) to reject EEG artifacts due to blinks, subject movement, etc., in a way unbiased by human judgment. We then calculated the EEG source strength of this artifact-free data at each time point of the movie within the entire brain volume using low-resolution electromagnetic tomography (LORETA). This provided for every voxel in the brain (i.e., in 3D space) an estimate of the current density at every time point. We then carried out a correlation between the time series of visual contrast changes in the movie with that of EEG voxels. We found the most significant correlations in visual area V1, just as seen in previous fMRI studies (Bartels A, Zeki, S, Logothetis NK. Natural vision reveals regional specialization to local motion and to contrast-invariant, global flow in the human brain. Cereb Cortex 2008;18(3):705–717), but on the time scale of milliseconds rather than of seconds. To obtain an estimate of how the EEG signal relates to the BOLD signal, we calculated the IRF between the BOLD signal and the estimated current density in area V1. We found that this IRF was very similar to that observed using combined intracortical recordings and fMRI experiments in nonhuman primates. Taken together, these findings open a new approach to noninvasive mapping of the brain. It allows, firstly, the localization of feature-selective brain areas during natural viewing conditions with the temporal resolution of EEG. Secondly, it provides a tool to assess EEG/BOLD transfer functions during processing of more natural stimuli. This is especially useful in combined EEG/fMRI experiments, where one can now potentially study neural-hemodynamic relationships across the whole brain volume in a noninvasive manner.