Soft-X-ray microscopy using spiral zone plates

Phase sensitive soft-x-ray microscopy methods enable the study of specimens for which phase effects are a prevalent contrast mechanism. One way to detect these phase effects is to optically implement the radial Hilbert transform by using spiral zone plates (SZPs), which results in the isotropic measurement of the amplitude and phase gradient in a sample. Soft-x-ray microscopy using an SZP as a single element objective lens was demonstrated through the imaging of a 1 microm circular aperture at a wavelength of 2.73 nm(454 eV). A regular zone plate, a charge 1 SZP, and a charge 2 SZP were fabricated using electron beam lithography and were used as the imaging optic in the microscopy setup. The charge 1 and charge 2 SZP images exhibited isotropic edge enhancement as a result of radial Hilbert filtering.     

利用光束的轨道角动量实现高密度数据存储的机理研究

光束的轨道角动量本征态可以构建高维Hilbert空间，将之应用于数据存储中，可以实现比传统方法更高的数据存储密度.提出了一种用光束轨道角动量实现高密度数据存储的方法，对存储区域进行相位编码，通过探测透射光束轨道角动量谱(螺旋谱)的特征来区分由相位编码形成的不同数据态.四台阶结构相位编码可以实现57个不同的数据态，有望将数据密度提高为传统方法的近6倍. 关键词： 光束轨道角动量 光学存储 高密度存储     

Continuous symmetry reduction and return maps for high-dimensional flows

We present two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps. In the Hilbert polynomial basis approach, the equivariant dynamics is rewritten in terms of invariant coordinates. In the method of moving frames (or method of slices) the state space is sliced locally in such a way that each group orbit of symmetry-equivalent points is represented by a single point. In either approach, numerical computations can be performed in the original state space representation, and the solutions are then projected onto the symmetry-reduced state space. The two methods are illustrated by reduction of the complex Lorenz system, a five-dimensional dissipative flow with rotational symmetry. While the Hilbert polynomial basis approach appears unfeasible for high-dimensional flows, symmetry reduction by the method of moving frames offers hope.     

基于轨道角动量的多自由度W态纠缠系统

提出了一种制备三光子纠缠W态的方案, 该方案利用携带轨道角动量为lħ的光子(其中l可取(-∞, +∞)的任意整数)可构成无穷维向量空间的特性, 采用两种类型的参量下转换, 产生轨道角动量-自旋角动量纠缠的两对光子和一对偏振纠缠光子, 通过纠缠交换制备三光子多自由度的W态, 实现三光子体系纠缠的高维度、大容量量子信息处理. 方案采用q-plate相位光学器件和单模光纤等器件制备两个不同自由度(轨道角动量与偏振)混合的W态, 并利用计算机全息相位图改进方案制备三个不同自由度(轨道角动量、线动量和偏振)混合的W态. 本方案可稳定产生两种等概率互为对称的W态, 具有高维度、强纠缠特性与抗比特丢失能力, 信息量达log₂^m+2比特(m为l的可取值个数), 有望实现可扩容量子比特的安全通信.     

Connes distance of 2D harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.     

Experimental demonstration of fractional orbital angular momentum entanglement of two photons

The singular nature of a noninteger spiral phase plate allows easy manipulation of spatial degrees of freedom of photon states. Using two such devices, we have observed very high-dimensional spatial entanglement of twin photons generated by spontaneous parametric down-conversion.     

Hamiltonian quantum dynamics with separability constraints

Schroedinger equation on a Hilbert space H, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space PH. Separable states of a bipartite quantum system form a special submanifold of PH. We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.     

Far-infrared cw difference-frequency generation using vertically integrated and planar low temperature grown GaAs photomixers: application to H<Subscript>2</Subscript>S rotational spectrum up to 3 THz

The generation of continuous coherent THz radiation by mixing two cw Ti:Sa laser beams with a well-controlled frequency separation for a new scheme of vertically integrated low temperature grown GaAs (LTG-GaAs) spiral photomixer is reported. For this new photomixer device used in THz emission, the LTG-GaAs active layer is sandwiched between the two parallel metal plates of a high-speed photodetector loaded by a broadband spiral antenna. We have exploited the advantage of a higher delivered power in the low part of the spectrum (<2000 GHz), while a low RC time constant planar interdigitated detector was used at the upper frequency. The performances of the spectroscopic setup in terms of spectral resolution (5 MHz), tunability and frequency capability are assessed by measurements of the pure rotational spectra of hydrogen sulfide (H₂S) up to 3000 GHz. PACS 73.40.Sx; 33.20.Bx; 39.30.+w     

Majorana's stellar representation for the quantum geometric tensor of symmetric states

Majorana's stellar representation provides an intuitive picture in which quantum states in high-dimensional Hilbert space can be observed using the trajectory of Majorana stars. We consider the Majorana's stellar representation of the quantum geometric tensor for a spin state up to spin-3/2. The real and imaginary parts of the quantum geometric tensor, corresponding to the quantum metric tensor and Berry curvature, are therefore obtained in terms of the Majorana stars. Moreover, we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases. Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana's stars.     

Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness

A witness of non-Markovianity based on the Hilbert–Schmidt speed (HSS), a special type of quantum statistical speed, has been recently introduced for low-dimensional quantum systems. Such a non-Markovianity witness is particularly useful, being easily computable since no diagonalization of the system density matrix is required. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in various high-dimensional and multipartite open quantum systems with finite Hilbert spaces. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or quantum correlation measure. These results show that the HSS-based witness is a faithful identifier of the memory effects appearing in the quantum evolution of a high-dimensional system with a finite Hilbert space.     

Entangled phase states via quantum beam splitter

We study the entanglement effect of beam splitter on the temporally stable phase states. Specifically, we consider the eigenstates (phase states) of a unitary phase operator resulting from the polar decomposition of ladder operators of generalized Weyl-Heisenberg algebras possessing finite dimensional representation space. The linear entropy that measures the degree of entanglement at the output of the beam splitter is analytically obtained. We find that the entanglement is not only strongly dependent on the Hilbert space dimension but also quite related to strength the parameter ensuring the temporal stability of the phase states. Finally, we discuss the evolution of the entangled phase states.     

Geometric phases and coherent states

A cyclic evolution of a pure quantum state is characterized by a closed curve γ in the projective Hilbert space , equipped with the Fubini-Study geometry. It is known that the geometric phase for this evolution is given by the integral of the symplectic form of the Fubini-Study geometry over an arbitrary surface spanning γ. This result extends to an infinite-dimensional Hilbert space for a bosonic quantum field. We prove that is bounded above by the infimum area over all surfaces spanning γ, and that the bound is attained if γ can be spanned by a holomorphic curve. Using an earlier result concerning the intrinsic Euclidean geometry of the coherent state submanifold , we derive an expression for the geometric phase for a cyclic evolution amongst coherent states. We indicate how the intensity of a classical configuration can be inferred from the winding number of the exponential geometric phase about the origin in the complex plane. In the case of photon states we present group theoretic and 2-component spinor representations of . We derive an expression for in the case of a sequence of measurements such that the resulting states are coherent at each step, in terms of a sequence of projection operators. The situation in relation to some earlier experiments of Pancharatnam and Tomita–Chiao is explained.     

光子高阶轨道角动量制备、调控及传感应用研究进展

光子既是经典信息也是量子信息的理想载体. 单个光子不仅可以携带自旋角动量(与光波的圆偏振相关), 还可以携带轨道角动量(与光波的螺旋相位相关). 而轨道角动量的重要意义在于可利用单个光子的量子态构建一个高维的Hilbert空间, 从而实现高维量子信息的编码. 自Allen等于1992年确认光子轨道角动量的物理存在以来, 轨道角动量在经典光学和量子光学领域展现了诸多诱人的应用前景, 目前已成为国际光学领域的研究热点之一. 本综述将着重介绍高阶轨道角动量光束的制备与调控技术, 特别是高阶轨道角动量的量子纠缠态操控、旋转Doppler 效应测量及其在远程传感和精密测量技术中的应用.     

Photonic EPR State from Quadratic Waveguide Array with Alternating Positive and Negative Couplings

We propose the generation of photonic EPR state from quadratic waveguide array. Both the propagation constant and the nonlinearity in the array are designed to possess a periodical modulation along the propagation direction.This ensures that the photon pairs can be generated efficiently through the quasi-phase-matching spontaneous parametric down conversion by holding the spatial EPR entanglement in the fashion of correlated position and anticorrelated momentum. The Schmidt number which denotes the degree of EPR entanglement is calculated and it can approach a high value when the number of illuminated waveguide channels and the length of the waveguide array are properly chosen. These results suggest the quadratic waveguide array as a compact platform for engineering photonic quantum states in a high-dimensional Hilbert space.     

States in the Hilbert space formulation and in the phase space formulation of quantum mechanics

We consider the problem of testing whether a given matrix in the Hilbert space formulation of quantum mechanics or a function considered in the phase space formulation of quantum theory represents a quantum state. We propose several practical criteria for recognising states in these two versions of quantum physics. After minor modifications, they can be applied to check positivity of any operators acting in a Hilbert space or positivity of any functions from an algebra with a ∗$\ast$-product of Weyl type.     

Duality, phase structures, and dilemmas in symmetric quantum games

Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners’ Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD, and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.     

Relativistic quantum mechanics of spin-zero and spin-half particles

We consider the quantum mechanics of directly interacting relativistic particles of spin-zero and spin-half. We introduce a scalar product in the vector space of physical states which is finite, positive definite and relativistically invariant and keeps orthogonal eigenstates of total four momentum belonging to different eigenvalues. This allows us to show that the vector space of physical states is, in fact, a Hilbert space. The case of two particles is explicitly considered and the Cauchy problem of physical wave function illustrated. The problem of a spin-1/2 particle interacting with a spin-zero particle is considered and a new equation is proposed for two spin-1/2 particles interacting via the most general form of interaction possible. The restrictions due to Hermiticity, space inversion and time reversal invariance are also considered.     

Quantum approach to a derivation of the second law of thermodynamics

We reinterpret the microcanonical conditions in the quantum domain as constraints for the interaction of the "gas subsystem" under consideration and its environment ("container"). The time average of a purity measure is found to equal the average over the respective path in Hilbert space. We then show that for typical (degenerate or nondegenerate) thermodynamical systems almost all states within the allowed region of Hilbert space have a local von Neumann entropy S close to the maximum and a purity P close to its minimum, respectively. Typically, thermodynamical systems should obey the second law.     

On the role of coherent states in quantum foundations

Coherent states, and the Hilbert space representations they generate, provide ideal tools to discuss classical/quantum relationships. In this paper we analyze three separate classical/quantum problems using coherent states, and show that useful connections arise among them. The topics discussed are: (1) a truly natural formulation of phase space path integrals; (2) how this analysis implies that the usual classical formalism is “simply a subset” of the quantum formalism, and thus demonstrates a universal coexistence of both the classical and quantum formalisms; and (3) how these two insights lead to a complete analytic solution of a formerly insoluble family of nonlinear quantum field theory models.     

Aharonov—Anandan phase gate in sub Hilbert space of a coupling flux qubits system

We study two flux qubits with a parameter coupling scenario. Under the rotating wave approximation, we truncate the 4-dimension Hilbert space of a coupling flux qubits system to a 2-dimension subspace spanned by two dressed states |01> and |10>. In this subspace, we illustrate how to generate an Aharnov--Anandan phase, based on which, we can construct a NOT gate (as effective as a C-NOT gate) in this coupling flux qubits system. Finally, the fidelity of the NOT gate is also calculated in the presence of the simulated classical noise.