Xi-Feng Ren, Guo-Ping Guo, Jian Li, Guang-Can Guo
We found that the Hermit-Gaussian(HG) modes of the down converted beams from the spontaneous parametric down conversion are quasi-conserved and the generated photon pairs are HG modes entangled for some special cases. This is valuable for either the investigation of fundament properties of multi-dimensional entanglement or quantum information applications.
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We show that the transfer of the angular spectrum of the pump beam to the two-photon state in spontaneous parametric down-conversion enables the generation of entangled Hermite-Gaussian modes. We derive an analytical expression for the two-photon state in terms of these modes and show that there are restrictions on both the parity and order of the down-converted Hermite-Gaussian fields. Using these results, we show that the two-photon state is indeed entangled in Hermite-Gaussian modes. We propose experimental methods of creating maximally-entangled Bell states and non-maximally entangled pure…
On the role of Hermite-like polynomials in the Fock representations of Gaussian states
本文利用厄米及类厄米多项式的运算性质，为一般单模高斯算符和高斯噪声态提供了闭式Fock表示，展示了这些多项式在连续变量量子力学中的基础作用。
Exact Bateman-Hillion solutions of the wave equation are applied to Hermite-Gaussian beams using a space-time constraint condition that requires the field density to fall as the inverse square of distance from the focal point of the beam at large distances from it. Following a familiar practice, the constraint is implemented in integrals through the use of a Dirac delta function. It is shown the Hermite-Gaussian functions evolve to become pure functions of angular position on the fully developed spherical phase fronts. Under the paraxial approximation it is further shown the wave equation and …
We provide a general decomposition of the two-photon state produced by spontaneous parametric down-conversion in Hermite-Gaussian modes, in the case that the pump beam is described by a Hermite-Gaussian beam of any order. We show that the spatial correlations depend explicitly on the order of the pump beam, as well as other experimental parameters. We use the decomposition to demonstrate a few interesting cases. Our results are applicable to the engineering of two-photon spatial entanglement, in particular for non-Gaussian states.
We demonstrate quantum correlations in the transverse plane of continuous wave light beams by producing -4.0 dB, -2.6 dB and -1.5 dB of squeezing in the TEM00, TEM10 and TEM20 Hermite- Gauss modes with an optical parametric amplifier, respectively. This has potential applications in quantum information networking, enabling parallel quantum information processing. We describe the setup for the generation of squeezing and analyze the effects of various experimental issues such as mode overlap between pump and seed and nonlinear losses.
The spatial modes of light have grasped great research interests because of its great potentials in optical communications, optical manipulation and trapping, optical metrology and quantum information processing. Here we report on generating of photon pairs in Hermite-Gaussian (HG) mode in a type-I optical parametric oscillator (OPO) operated far below threshold. The bandwidths of the photon pairs are 11.4 MHz and 20.8MHz for two different HG modes respectively, which is capable to be stored in cold Rubidium atomic ensembles. From correlation measurements, non-classical properties of HG modes …
Pump-tailored Alternative Bell State Generation in the First-Order Hermite-Gaussian basis
该研究通过定制泵浦光子空间结构，在非线性晶体中直接实现一阶厄米-高斯基的纠缠态交换，产生平衡HG模式纠缠态，适用于几何相位对称累积及方位角灵敏的量子传感成像，且可扩展至其他模式基。
The Hermite-Gaussian (HG) modes, sometimes also referred to as transverse electromagnetic modes in free space, form a complete and orthonormal basis that have been extensively used to describe optical fields. In addition, these modes have been shown to be helpful to enhance information capacity of optical communications as well as to achieve super-resolution imaging in microscopy. Here we propose and present the realization of an efficient, robust mode sorter that can sort a large number of HG modes based on the relation between HG modes and Laguerre-Gaussian (LG) modes. We experimentally demo…
We study a class of optical circuits with vacuum input states consisting of Gaussian sources without coherent displacements such as down-converters and squeezers, together with detectors and passive interferometry (beam-splitters, polarisation rotations, phase-shifters etc.). We show that the outgoing state leaving the optical circuit can be expressed in terms of so-called multi-dimensional Hermite polynomials and give their recursion and orthogonality relations. We show how quantum teleportation of photon polarisation can be modelled using this description.
High Precision Multi-parameter Weak Measurement with Hermite-Gaussian Pointer
该研究借助高阶厄米-高斯指针和量子Fisher信息矩阵构建多参数弱测量方案，理论上精度提升√(2n+1)倍，可通过最大似然估计和零差法逼近极限，还给出了原理验证装置以探索精密计量应用。
We calculate the differential cross sections for Compton scattering of photons described by Hermite Gaussian (HG) wave function in the framework of relativistic quantum mechanics. The HG wave gamma-rays propagating along the z-direction have quantum numbers of nodes of nx and ny in the x- and y-directions, respectively. The calculated differential cross section is symmetric with respect to both the zx- and zy-planes. The nodes whose number is identical with nx and ny appear in the energy spectrum measured in zx- and zy-planes, respectively. These results indicate that it is possible to identif…
Exploring high sensitivity on the measurement of angular rotations is an outstanding challenge in optics and metrology. In this work, we employ the mn-order Hermite-Gaussian beam in the weak measurement scheme with an angular rotation interaction, where the rotation information is taken by another HG mode state completely after the post-selection. By taking a projective measurement on the final light beam, the precision of angular rotation is improved by a factor of 2mn+m+n. For verification, we perform an optical experiment where the minimum detectable angular rotation improves $\sqrt{15}$-fo…
Efficient Quantum Hermite Transform
本研究提出可高效实现离散厄米变换的量子算法新基元，基于量子谐振子演化加速方法，在性质测试与学习中展现量子查询优势，还可用于连续量子系统动力学模拟。
Multiple-degrees-of-freedom free-space communication combining polarization and high-order spatial modes promises high-capacity communication channel. While high-order spatial modes have been widely exploited for dense coding and high-dimensional quantum information processing, the properties of polarization preservation of high-order spatial beams propagating in turbulent atmosphere have not been comprehensively investigated yet. Here we focus on the properties of polarization preservation of partially coherent Hermite-Gaussian beams propagating along different atmospheric turbulence paths. T…
We propose an all-optical setup, which couples different degrees of freedom of a single photon, to investigate entanglement generation by a common environment. The two qubits are represented by the photon polarization and Hermite-Gauss transverse modes, while the environment corresponds to the photon path. For an initially two-qubit separable state, the increase of entanglement is analyzed, as the probability of an environment-induced transition ranges from zero to one. An entanglement witness that is invariant throughout the evolution of the system yields a direct measurement of the concurren…
Exact Gaussian--Hermite closure of the mapping quantum--classical Liouville equation
该研究针对映射量子-经典刘维尔方程，提出精确高斯-厄米闭合方法，其在Tully双避免交叉模型上的表现优于PBME，精度更高。
Through the von Neumann interaction followed by post-selection, we can extract not only the eigenvalue of an observable of the measured system but also the weak value. In this post-selected von Neumann measurement, the initial pointer state of the measuring device is assumed to be a fundamental Gaussian wave function. By considering the optical implementation of the post-selected von Neumann measurement, higher-order Gaussian modes can be used. In this paper, we consider the Hermite--Gaussian (HG) and Laguerre--Gaussian (LG) modes as pointer states and calculate the average shift of the pointe…
The connection between Poincaré spheres for polariz-ation and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic 2-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and Generalized Hermite-Laguerre Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the 2-dimensional oscillator, which acquire an extra significance as fa…
The optimization of entanglement evolution for two-mode Gaussian pure states under one-side Gaussian map is studied. Even there isn't complete information about the one-side Gaussian noisy channel, one can still maximize the entanglement distribution by testing the channel with only two specific states.
In a $m$ particle quantum system, one can have $k=1,\,2,\,\ldots,\,m$ body interactions. The rank of interactions and the nature of particles (fermions or bosons) can strongly affect the dynamics of the system. To explore this in detail, we study quenched quantum dynamics in many particle systems varying rank of interactions, both for fermionic and bosonic particles. We represent the system Hamiltonian by Fermionic Embedded Gaussian Orthogonal Ensembles (FEGOE) and Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE) respectively. We show that generating function for $q$-Hermite polynomials …
We investigate localization of entanglement of multimode Gaussian states into a pair of modes by local Gaussian measurements on the remaining modes and classical communication. We find that for pure states and for mixed symmetric states maximum entanglement between two modes can be localized by local homodyne detections, i.e. projections onto infinitely squeezed states. We also show that non-Gaussian measurements allow to localize more entanglement than Gaussian ones.
Engineering of self-bending surface plasmon polaritons through Hermite-Gaussian mode expansion
本文提出一种基于厄米-高斯模式展开的理论框架，用于设计自弯曲表面等离激元光束并控制其能量位置，数值验证了其在空气-银界面的可行性。
The recently developed Kraus representation for bosonic Gaussian channels is employed to study analytically the robustness of non-Gaussian entanglement against evolution under noisy attenuator and amplifier environments, and compare it with the robustness of Gaussian entanglement. Our results show that some non-Gaussian states with one ebit of entanglement are more robust than all Gaussian states, even the ones with arbitrarily large entanglement, a conclusion of direct consequence to the recent conjecture by Allegra et al. [PRL, 105, 100503 (2010)].
Segal-Bargmann type spaces related to non-rotational measure, and entanglement of bipartite squeezed coherent states
本文利用基于Wigner分布的相空间方法，研究了双组份压缩态纠缠，揭示了压缩参数与非旋转测度参数的关系，并借助Peres-Horodecki判据和对数负性确定了纠缠的定量特征。
Simplified scheme for continuous-variable entanglement distillation: multicopy distillation of Gaussian entanglement without heralding Gaussian measurements
本文提出一种简化连续变量纠缠纯化协议，通过消除 heralding 预备的 Gaussian 测量并整合到输入状态中，减少探测器数量并提高成功率，同时保持等效输出。
We write the optimal pure-state decomposition of any two-mode Gaussian state and show that its entanglement of formation coincides with the Gaussian one. This enables us to develop an insightful approach of evaluating the exact entanglement of formation. Its additivity is finally proven.
We demonstrate that the entanglement in a class of two-mode non-Gaussian states obtained by subtracting photons from Gaussian twin beams can be bounded from above and from below by functionals of the second moments only. Knowledge of the covariance matrix thus suffices for an entanglement quantification with appreciable precision. The absolute error in the entanglement estimation scales with the non-Gaussianity of the considered states.
We present a formalism to derive entanglement criteria beyond the Gaussian regime that can be readily tested by only homodyne detection. The measured observable is the Einstein-Podolsky-Rosen (EPR) correlation. Its arbitrary functional form enables us to detect non-Gaussian entanglement even when an entanglement test based on second-order moments fails. We illustrate the power of our experimentally friendly criteria for a broad class of non-Gaussian states under realistic conditions. We also show rigorously that quantum teleportation for continuous variables employs a specific functional form …
We establish tight upper and lower bounds for the Entanglement of Formation of an arbitrary two-mode Gaussian state employing necessary properties of Gaussian channels. Both bounds are strictly given by the Entanglement of Formation of symmetric Gaussian states, which are simply constructed from the reduced states obtained by partial trace of the original one.
For two gaussian states with given correlation matrices, in order that relative entropy between them is practically calculable, I in this paper describe the ways of transforming the correlation matrix to matrix in the exponential density operator. Gaussian relative entropy of entanglement is proposed as the minimal relative entropy of the gaussian state with respect to separable gaussian state set. I prove that gaussian relative entropy of entanglement achieves when the separable gaussian state is at the border of separable gaussian state set and inseparable gaussian state set. For two mode ga…
Entanglement distillation is an essential ingredient for long distance quantum communications. In the continuous variable setting, Gaussian states play major roles in quantum teleportation, quantum cloning and quantum cryptography. However, entanglement distillation from Gaussian states has not yet been demonstrated. It is made difficult by the no-go theorem stating that no Gaussian operation can distill Gaussian states. Here we demonstrate the entanglement distillation from Gaussian states by using measurement-induced non-Gaussian operations, circumventing the fundamental restriction of the n…
We suggest an experimentally realizable scheme to test entanglement of a mixed Gaussian continuous variable state. We find that the entanglement condition is simplified for the family of Gaussian states which are relevant to experimental realization. The entanglement condition is then shown to be directly related to joint homodyne measurements. We show how robust the proposed test of entanglement is against imperfect detection efficiency.
A Gaussian degree of entanglement for a symmetric two-mode Gaussian state can be defined as its distance to the set of all separable two-mode Gaussian states. The principal property that enables us to evaluate both Bures distance and relative entropy between symmetric two-mode Gaussian states is the diagonalization of their covariance matrices under the same beam-splitter transformation. The multiplicativity property of the Uhlmann fidelity and the additivity of the relative entropy allow one to finally deal with a single-mode optimization problem in both cases. We find that only the Bures-dis…
We introduce a Gaussian version of the entanglement of formation adapted to bipartite Gaussian states by considering decompositions into pure Gaussian states only. We show that this quantity is an entanglement monotone under Gaussian operations and provide a simplified computation for states of arbitrary many modes. For the case of one mode per site the remaining variational problem can be solved analytically. If the considered state is in addition symmetric with respect to interchanging the two modes, we prove additivity of the considered entanglement measure. Moreover, in this case and consi…
A review of probability representation of quantum states in given for optical and photon number tomography approaches. Explicit connection of photon number tomogram with measurable by homodyne detector optical tomogram is obtained. New integral relations connecting Hermite polynomials of two variables with Laguerre polynomials are found. Examples of generic Gaussian photon states (squeezed and correlated states) are studied in detail.
Intrinsic entanglement (IE) is a quantity which aims at quantifying bipartite entanglement carried by a quantum state as an optimal amount of the intrinsic information that can be extracted from the state by measurement. We investigate in detail the properties of a Gaussian version of IE, the so-called Gaussian intrinsic entanglement (GIE). We show explicitly how GIE simplifies to the mutual information of a distribution of outcomes of measurements on a conditional state obtained by a measurement on a purifying subsystem of the analyzed state, which is first minimized over all measurements on …
Entanglement of formation is a fundamental measure that quantifies the entanglement of bipartite quantum states. This measure has recently been extended into multipartite states taking the name $α$-entanglement of formation. In this work, we follow an analogous multipartite extension for the Gaussian version of entanglement of formation, and focusing on the the finest partition of a multipartite Gaussian state we show this measure is fully additive and computable for 3-mode Gaussian states.
We develop an entanglement criterion with third- and fourth-order cumulants to detect the entanglement of non-Gaussian states. The efficiency of the entanglement criterion is investigated for gravitating mirrors in optomechanical systems. We show that the entangled regime of the mirrors is enlarged by the third- and fourth-order cumulants. We also discuss the limitations of the entanglement criterion for mirrors in a highly non-Gaussian state.
Optimal beam displacement measurements using high-order structured light modes
本文提出一种利用高阶厄米-高斯和拉盖尔-高斯模式测量光束微小位移的新技术，通过干涉奇偶性分类器实现最优测量，显著提升信噪比，为结构光计量提供了无需量子光或零差检测的可行方案。
We determine the maximum squashed entanglement achievable between sender and receiver of the noiseless quantum Gaussian attenuators and amplifiers and we prove that it is achieved sending half of an infinitely squeezed two-mode vacuum state. The key ingredient of the proof is a lower bound to the squashed entanglement of the quantum Gaussian states obtained applying a two-mode squeezing operation to a quantum thermal Gaussian state tensored with the vacuum state. This is the first lower bound to the squashed entanglement of a quantum Gaussian state and opens the way to determine the squashed e…
In this work we focus on entanglement of two--mode Gaussian states of continuous variable systems. We first review the formalism of Gaussian measures of entanglement, adopting the framework developed in [M. M. Wolf {\em et al.}, Phys. Rev. A {\bf 69}, 052320 (2004)], where the Gaussian entanglement of formation was defined. We compute Gaussian measures explicitely for two important families of nonsymmetric two--mode Gaussian states, namely the states of extremal (maximal and minimal) negativities at fixed global and local purities, introduced in [G. Adesso {\em et al.}, Phys. Rev. Lett. {\bf 9…
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