Nonseparable States Research Articles

Experimental generation of scalar and vector vortex Pearcey-Gauss Beams

Abstract In this manuscript, we put forward two new types of structured light beams, the vortex Pearcey-Gauss (VPeG) beam, with a homogeneous polarisation distribution, and the vector vortex Pearcey-Gauss (VVPeG) beam, with a non-homogeneous polarisation distribution. The latter is generated as a non-separable superposition of the spatial and polarisation degrees of freedom of light. We achieve their experimental realization through the combination of a spatial light modulator, which creates a scalar Pearcey-Gauss beam, and a q-plate which transforms it into a vortex or a vortex vector beam, depending on its input polarisation state. Their intensity and polarisation distributions along the propagation direction were determined through Stokes polarimetry, which was compared with numerical simulations. As demonstrated, the VVPeG beam evolves from an all-linear polarisation distribution to an approximately full Poincar'e beam. The proposed vector beams add to the extensive family of non-separable states of light. We anticipate that both types of beams will find applications in fields as diverse as optical metrology and tweezers, amongst others.

Optical orbital angular momentum analogy to the Stern-Gerlach experiment.

Symmetry breaking has been shown to reveal interesting phenomena in physical systems. A notable example is the fundamental work of Otto Stern and Walther Gerlach [Stern and Zerlach, Z. Physik9, 349 (1922)10.1007/BF01326983] nearly 100 years ago demonstrating a spin angular momentum (SAM) deflection that differed from classical theory. Here we use non-separable states of SAM and orbital angular momentum (OAM), known as vector vortex modes, to demonstrate how a classical optics analogy can be used to reveal this non-separability, reminiscent of the work carried out by Stern and Gerlach. We show that by implementing a polarization insensitive device to measure the OAM, the SAM states can be deflected to spatially resolved positions.

Generation of polarization and coherence non-separable states in twisted partially coherent vector light

Generation of polarization and coherence non-separable states in twisted partially coherent vector light

Evidence of a significant full tripartite entanglement in a mixed spin-(1/2,1) Heisenberg tetramer

We rigorously analyze the full tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer in the presence of an external magnetic field. The tetramer consists of two mixed spin-(1/2,1) dimers arranged to form a square plaquette, with distinct exchange couplings J and J1 between spins from the same and different dimers, respectively. The full tripartite entanglement is classified according to the tripartite negativity NABC defined as a geometric mean of all possible bipartite negativities NA|BC, NB|AC, and NC|AB, when the degrees of freedom of the fourth spin D of the tetrapartite system ABCD are traced out. We demonstrate that for J1/J>1, both trimeric subsystems 1/2-1-1 and 1/2-1/2-1 exhibit tripartite entanglement mediated through the central spin. This entanglement becomes more robust upon the application of an external magnetic field. The opposite limit J1/J<1 leads to the spontaneous emergence of full tripartite entanglement solely within the subsystem 1/2-1-1. However, the increasing magnetic field swiftly disrupts the three-particle correlation across the relevant ground-state phase boundary. Additionally, we identify that in the isotropic limit J1/J=1, the ground-state degeneracy significantly reduces the strength of full tripartite entanglement and favors nontrivial subtypes of full nonseparable states with none or just one spin-dimer entanglement. Finally, we discuss in detail the thermal stability of all fully tripartite nonseparable states.

Enhanced Photo-excitation and Angular-Momentum Imprint of Gray Excitons in WSe2 Monolayers by Spin-Orbit-Coupled Vector Vortex Beams.

A light beam can be spatially structured in the complex amplitude to possess orbital angular momentum (OAM), which introduces an extra degree of freedom alongside the intrinsic spin angular momentum (SAM) associated with circular polarization. Furthermore, superimposing two such twisted light (TL) beams with distinct SAM and OAM produces a vector vortex beam (VVB) in nonseparable states where not only complex amplitude but also polarization is spatially structured and entangled with each other. In addition to the nonseparability, the SAM and OAM in a VVB are intrinsically coupled by the optical spin-orbit interaction and constitute the profound spin-orbit physics in photonics. In this work, we present a comprehensive theoretical investigation, implemented on the first-principles base, of the intriguing light-matter interaction between VVBs and WSe2 monolayers (WSe2-MLs), one of the best-known and promising two-dimensional (2D) materials in optoelectronics dictated by excitons, encompassing bright exciton (BX) as well as various dark excitons (DXs). One of the key findings of our study is that a substantial enhancement of the photoexcitation of gray excitons (GXs), a type of spin-forbidden DX, in a WSe2-ML can be achieved through the utilization of a 3D-structured TL with the optical spin-orbit interaction. Moreover, we show that a spin-orbit-coupled VVB surprisingly allows for the imprinting of the carried optical information onto GXs in 2D materials, which is robust against the decoherence mechanisms in the materials. This suggests a promising method for deciphering the transferred angular momentum from structured light to excitons.

Spatiotemporal sculpturing of light: a tutorial

Spatiotemporal sculpturing of light pulses with sophisticated structures on demand is one major goal of the everlasting pursuit of ultrafast information transmission and processing as well as ultraintense energy concentration and extraction using light. It may hold the key to unlocking new extraordinary fundamental physical effects. Traditionally, spatiotemporal light pulses are treated as spatiotemporally separable wave packets as a solution to Maxwell’s equations. In the past decade, more generalized forms of spatiotemporally nonseparable solution started to emerge with growing importance for their striking physical effects. This tutorial intends to provide the necessary basics on how to sculpture light in the spatiotemporal domain to realize spatiotemporal structures on demand and highlight some of the recent advances in the creation and characterization of increasingly complex spatiotemporal wave packets. These spatiotemporally separable to complex nonseparable states with diverse geometric and topological structures exhibit unique physical properties during propagation, focusing, and interaction with matter. The broad potential applications as well as outlook and future trends and open challenges in this field are presented.

Logical rotation of non-separable states via uniformly self-assembled chiral superstructures

The next generation of high-capacity, multi-task optical informatics requires sophisticated manipulation of multiple degrees of freedom (DoFs) of light, especially when they are coupled in a non-separable way. Vector beam, as a typical non-separable state between the spin and orbital angular momentum DoFs, mathematically akin to entangled qubits, has inspired multifarious theories and applications in both quantum and classical regimes. Although qubit rotation is a vital and ubiquitous operation in quantum informatics, its classical analogue is rarely studied. Here, we demonstrate the logical rotation of vectorial non-separable states via the uniform self-assembled chiral superstructures, with favorable controllability, high compactness and exemption from formidable alignment. Photonic band engineering of such 1D chiral photonic crystal renders the incident-angle-dependent evolution of the spatially-variant polarizations. The logical rotation angle of a non-separable state can be tuned in a wide range over 4π by this single homogeneous device, flexibly providing a set of distinguished logic gates. Potential applications, including angular motion tracking and proof-of-principle logic network, are demonstrated by specific configuration. This work brings important insight into soft matter photonics and present an elegant strategy to harness high-dimensional photonic states.

Generating arbitrary non-separable states with polarization and orbital angular momentum of light

We demonstrate an experimental method to generate arbitrary non-separable states of light in two degrees of freedom i.e., polarization and orbital angular momentum (OAM). We observe the intensity distribution corresponding to OAM modes of the light beam by projecting the non-separable state into different polarization states. We further verify the presence of non-separability by measuring the degree of polarization and linear entropy. This classical non-separability can be easily transferred to the quantum domain using spontaneous parametric down-conversion for applications in quantum communication and quantum sensing.

Distribution of Bipartite and Tripartite Entanglement within a Spin-1/2 Heisenberg Star in a Magnetic Field.

The spatial distribution of entanglement within a spin-1/2 Heisenberg star composed from a single central spin and three peripheral spins is examined in the presence of an external magnetic field using the Kambe projection method, which allows an exact calculation of the bipartite and tripartite negativity serving as a measure of the bipartite and tripartite entanglement. Apart from a fully separable polarized ground state emergent at high-enough magnetic fields, the spin-1/2 Heisenberg star exhibits at lower magnetic fields three outstanding nonseparable ground states. The first quantum ground state exhibits the bipartite and tripartite entanglement over all possible decompositions of the spin star into any pair or triad of spins, whereby the bipartite and tripartite entanglement between the central and peripheral spins dominates over that between the peripheral spins. The second quantum ground state has a remarkably strong tripartite entanglement between any triad of spins in spite of the lack of bipartite entanglement. The central spin of the spin star is separable from the remaining three peripheral spins within the third quantum ground state, where the peripheral spins are subject to the strongest tripartite entanglement arising from a two-fold degenerate W-state.

Experimentally accessible nonseparability criteria for multipartite-entanglement-structure detection

The description of the complex separability structure of quantum states in terms of partially ordered sets has been recently put forward. In this work, we address the question of how to efficiently determine these structures for unknown states. We propose an experimentally accessible and scalable iterative methodology that identifies, on solid statistical grounds, sufficient conditions for nonseparability with respect to certain partitions. In addition, we propose an algorithm to determine the minimal partitions (those that do not admit further splitting) consistent with the experimental observations. We test our methodology experimentally on a 20-qubit IBM quantum computer by inferring the structure of the 4-qubit Smolin and an 8-qubit W state. In the first case, our results reveal that, while the fidelity of the state is low, it nevertheless exhibits the partitioning structure expected from the theory. In the case of the W state, we obtain very disparate results in different runs on the device, which range from nonseparable states to very fragmented minimal partitions with little entanglement in the system. Furthermore, our work demonstrates the applicability of informationally complete positive operator-valued measurements for practical purposes on current noisy intermediate-scale quantum devices.

Quantum analogue information processing using “classical entanglement” of acoustic waves

Various quantum analogues have emerged due to recent advances in the design and fabrication of acoustic structures. Acoustic analogous to “entangled” states of quantum bits or qubits offer a transformative new solution to reach some of the goals of quantum information science. Beams of linear acoustic waves support degrees of freedom (DOF), analogous to quantum mechanics and light beams. The states associated with these DOFs—that we named phase-bits or phi-bits—are algebraically non-separable, like the DOF of a single quantum particle. The strong coupling and nonlinearity of acoustic waves in our waveguides offer new ways to realize non-separable states for classical multipartite systems that span exponentially complex Hilbert spaces with much larger dimensions [1,2]. These acoustic structures enable quantum analogue computing even at room temperature and for long periods of coherence. [Work supported by NSF.] [1] Hasan, M A, et al. “Experimental classical entanglement in a 16 acoustic qubit-analogue.” Scientific Reports 11, 2021. [2] Runge, K, et al. “Demonstration of a two-bit controlled-NOT quantum-like gate using classical acoustic qubit-analogues.” Scientific Reports 12, 2022.

Nonseparable States of Light: From Quantum to Classical

AbstractControlling the various degrees‐of‐freedom (DoFs) of structured light at both quantum and classical levels is of paramount importance in optics. It is a conventional paradigm to treat diverse DoFs separately in light shaping. While, the more general case of nonseparable states of light, in which two or more DoFs are coupled in a nonseparable way, has become topical recently. Importantly, classical nonseparable states of light are mathematically analogue to quantum entangled states. Such similarity has hatched attractive studies in structured light, for example, the spin‐orbit coupling in vector beams. However, nonseparable classical states of light are still treated in a fragmented fashion, while its forms are not limited by vector beams and its potential is certainly not fully exploited. For instance, exotic space‐time coupled pulses open nontrivial light shaping toward ultrafast time scales, and ray‐wave geometric beams provide new dimensions in optical manipulations. Here, a bird's eye view on the rapidly growing but incoherent body of work on general nonseparable states involving various DoFs of light is provided and a unified framework for their classification and tailoring, providing a perspective on new opportunities for both fundamental science and applications, is introduced.

Bounding the separable rank via polynomial optimization

We investigate questions related to the set SEPd consisting of the linear maps ρ acting on Cd⊗Cd that can be written as a convex combination of rank one matrices of the form xx⁎⊗yy⁎. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In particular we introduce bounds for the separable rank ranksep(ρ), defined as the smallest number of rank one states xx⁎⊗yy⁎ entering the decomposition of a separable state ρ. Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter τsep(ρ), a natural convexification of the combinatorial parameter ranksep(ρ). A distinguishing feature is exploiting the positivity constraint ρ−xx⁎⊗yy⁎⪰0 to impose positivity of a polynomial matrix localizing map, the dual notion of the notion of sum-of-squares polynomial matrices. Our approach extends naturally to the multipartite setting and to the real separable rank, and it permits strengthening some known bounds for the completely positive rank. In addition, we indicate how the moment approach also applies to define hierarchies of semidefinite relaxations for the set SEPd and permits to give new proofs, using only tools from moment theory, for convergence results on the DPS hierarchy from Doherty et al. (2002) [16].

Multipartite classically entangled scalar beams.

Classically entangled light is used to refer to a class of structured beams with space-polarization, polarization-time, and space-time non-separable states akin to entangled states, which enable novel quantum-analog methods and applications in structured light. Here, we argue that classical entanglement is also available for pure scalar beams with multiple non-separable spatial degrees of freedom (DoFs). We theoretically and experimentally demonstrate a class of scalar ray-wave structured light with multiple controllable local DoFs to emulate multipartite entangled states, including the Greenberger-Horne-Zeilinger states. Our work unveils a rich parameter space for high-dimensional and multi-DoF control of structured light to extend applications in classical-quantum regimes.

Publisher’s Note: “Navigating the Hilbert space of nonseparable elastic states in arrays of periodically coupled one-dimensional waveguides” [AIP Adv. 10, 095105 (2020)

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Representing Quantum Information with Digital Coding Metasurfaces.

With the development of science and technology, the way to represent information becomes more powerful and diversified. Recent research on digital coding metasurfaces has built an alternative bridge between wave‐behaviors and information science. Different from the logic information in traditional circuits, the digital bit in coding metasurfaces is based on wave‐structure interaction, which is capable of exploiting multiple degrees of freedom (DoFs). However, to what extent the digital coding metasurface can expand the information representation has not been discussed. In this work, it is shown that classical metasurfaces have the ability to mimic qubit and quantum information. An approach for simulating a two‐level spin system with meta‐atoms is proposed, from which the superposition for two optical spin states is constructed. It is further proposed that using geometric‐phase elements with nonseparable coding states can induce the classical entanglement between polarization and spatial modes, and give the condition to achieve the maximal entanglement. This study expands the information representing range of coding metasurfaces and provides an ultrathin platform to mimic quantum information.

Navigating the Hilbert space of nonseparable elastic states in arrays of periodically coupled one-dimensional waveguides

A planar array of three one-dimensional elastic waveguides mutually coupled periodically along their length and driven externally is shown theoretically and numerically to support nonseparable superpositions of states. These states are the product of Bloch waves describing the elastic displacement along the waveguides and spatial modes representing the displacement across the array of waveguides. For a system composed of finite length waveguides, the frequency, relative amplitude, and phase of the external drivers can be employed to selectively excite specific groups of discrete product modes. The periodicity of the coupling is used to fold bands enabling superpositions of states that span the complete Hilbert space of product states. We show that we can realize a transformation from one type of nonseparable superposition to another one that is analogous to a nontrivial quantum gate. This transformation is also interpreted as the complex conjugation operator in the space of the complex amplitudes of individual waveguides.

Exponentially complex nonseparable states in planar arrays of nonlinearly coupled one-dimensional elastic waveguides

Externally driven parallel arrays of elastic waveguides inelastically coupled along their length are shown to support inelastic modes that span exponentially complex Hilbert spaces. The nonlinear coupling takes the form of power functions of the relative displacement between waveguides. First order perturbation theory is employed to obtained closed form solutions for the contribution of nonlinearity to the displacement field in the form of nonseparable superpositions of product states of plane waves. When the system is dissipative, the coefficients of the superposition of product states are complex and can be varied by controlling the characteristics of the way the array is driven externally. The entropy of ‘entanglement’ of these nonseparable superpositions of product states is calculated from the complex amplitudes through the density matrix. Navigation of the exponentially complex Hilbert space of product states enables manipulation of the degree of nonseparability of the superpositions.

Joint effects of entanglement and symmetrization: Physical properties and exclusion

Entanglement and symmetrization lead to non-separable states that can modify physical properties. Using the example of atomic absorption we compare both types of effects when they are relevant at once. The presence of multi-particle superpositions largely alters the absorption rates of identical atoms, even inhibiting the dependence on overlapping for fermions. We also identify a set of non-standard excluded states related to multi-fermion superposition that naturally emerge in this context. We propose an arrangement based on the dissociation of molecules to test these ideas.

Finite elements computational modeling of coupled elastic waveguides

The theoretical study of one-dimensional-infinite systems of elastically coupled parallel waveguides has established the existence of band structures with pseudo-spin characteristics. Those systems, which are named ϕ-bits, have been shown to exhibit a spinor character associated with directional degrees of freedom, which makes them potential quantum mechanical analogs. The realization of such systems is challenged by the three-dimensional and finite nature of physical elastic waveguides. We address this problem, and with it the design of ϕ-bits in general, by developing finite elements models based on COMSOL Multiphysics®. We model systems of one or more coupled finite length Al rods. The analysis of their dispersion relations, transmission spectra, and amplitudes establishes their ϕ-bit character. For three coupled finite length Al rods, the elastic field is associated with wavefunctions, tensor products of a spinor part related to the directional degrees of freedom, and an orbital angular momentum part representing the phase of the coupled waveguides. We demonstrate the possibility of creating non-separable states between these degrees of freedom.