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arXiv:2011.05677 · Journal of Physics A Mathematical and Theoretical

Spin-entangled squeezed state on a Bloch four-hyperboloid

Kazuki Hasebe

Published May 21, 2021• 2 citations
arXivHTMLPDF

Abstract

Abstract The Bloch hyperboloid H 2 underlies the quantum geometry of the original SO (2, 1) squeezed states. In Hasebe (2020 J. Phys. A: Math. Theor. 53 055303), the author utilized a non-compact 2nd Hopf map and a Bloch four-hyperboloid H 2,2 to explore an SO (2, 3) extension of the squeezed states. In the present paper, we further pursue the idea to derive an SO (4, 1) version of squeezed vacuum based on the other Bloch four-hyperboloid H 4 . We show that the obtained SO (4, 1) squeezed vacuum is a particular four-mode squeezed state not quite similar to the previous SO (2, 3) squeezed vacuum. In view of the Schwinger’s formulation of angular momentum, the SO (4, 1) squeezed vacuum is interpreted as a superposition of an infinite number of maximally entangled spin-pairs of all integer spins. We clarify basic properties of the SO (4, 1) squeezed vacuum, such as von Neumann entropy of spin entanglement, spin correlations and uncertainty relations with emphasis on their distinctions to the original SO (2, 1) case.

DOI
10.1088/1751-8121/abee9c

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Abstract The Bloch hyperboloid H 2 underlies the quantum geometry of the original SO (2, 1) squeezed states. In Hasebe (2020 J. Phys. A: Math. Theor. 53 055303), the author utilized a non-compact 2nd Hopf map and a Bloch four-hyperboloid H 2,2 to explore an SO (2, 3) extension of the squeezed states. In the present paper, we further pursue the idea to derive an SO (4, 1) version of squeezed vacuum based on the other Bloch four-hyperboloid H 4 . We show that the obtained SO (4, 1) squeezed vacuum is a particular four-mode squeezed state not quite similar to the previous SO (2, 3) squeezed vacuum. In view of the Schwinger’s formulation of angular momentum, the SO (4, 1) squeezed vacuum is interpreted as a superposition of an infinite number of maximally entangled spin-pairs of all integer spins. We clarify basic properties of the SO (4, 1) squeezed vacuum, such as von Neumann entropy of spin entanglement, spin correlations and uncertainty relations with emphasis on their distinctions to the original SO (2, 1) case.
Abstract We explore the hyperbolic geometry of squeezed states in the perspective of the non-compact Hopf map. Based on analogies between the squeeze operation and hyperbolic rotation, two types of the squeeze operators, the (usual) Dirac and the Schwinger types, are introduced. We clarify the underlying hyperbolic geometry and representations of the squeezed states along the line of the first non-compact Hopf map. Following the geometric hierarchy of the non-compact Hopf maps, we extend the analysis to —the isometry of a split-signature four-hyperboloid. We explicitly construct the squeeze operators in the Dirac and Schwinger types and investigate the physical meaning of the four-hyperboloid coordinates in the context of the Schwinger-type squeezed states. It is shown that the Schwinger-type squeezed one-photon state is equal to an entangled superposition state of two squeezed states and the corresponding concurrence has a clear geometric meaning. Taking advantage of the group theoretical formulation, basic properties of the squeezed coherent states are also investigated. In particular, we show that the squeezed vacuum naturally realizes a generalized squeezing in a 4D manner.
Spin- s assemblies are classified into two mutually exclusive classes: oriented and non-oriented systems. The density matrix ρ, describing oriented systems, can assume diagonal form in the angular momentum basis ( ) defined with respect to the axis of quantization, whereas the eigenstates of ρ for the non-oriented assembly cannot all be identified with states. A new scheme for constructing a mixed, non-oriented spin- s state using s (2 s +1) spinors all pointing in different directions in space and 2 s weights is discussed. Such a construction takes its inspiration from Schwinger's idea of realizing an state as being made up of ( s + m ) 'up' spinors and ( s − m ) 'down' spinors, all defined with respect to a single axis in space. Since the oriented systems are never squeezed, non-oriented spin-1 assemblies which can be prepared in the laboratory with the available NQR technology are examined for signatures of squeezing using our scheme in a frame of reference where the Heisenberg–Robertson uncertainty relation has the same form as the Schrödinger uncertainty relation. It is shown that unlike in the case of the pure spin-1 state where squeezing is synonymous with non-orientedness, a non-oriented spin-1 system need not be squeezed and the existence of entanglement is a necessary but not sufficient condition for the system to be squeezed.
Atkins and Dobson [Proc. R. Soc. London Ser. A 321, 321 (1971)] have constructed coherent angular-momentum states using the Schwinger bosonic representation and operating the displacement (coherent) operator on both of the bosonic vacuum states. By performing a squeezing transformation on the bosonic coherent states we generalize the idea of Schwinger angular-momentum coherent (SAMC) states to Schwinger angular-momentum squeezed (SAMS) states. We show that these SAMS states, which are constructed from the bosonic squeezed coherent states, exhibit squeezing of Schwinger angular-momentum operators under a certain range of parameters. The properties of these states are studied and compared with their single-mode bosonic counterpart. Application of these states in increasing the accuracy of an interferometer is discussed.
Abstract These two accompanying papers are concerned with entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. The main focus is on two mode entanglement, but multi-mode entanglement is also considered. The bosons may be atoms or molecules as in cold quantum gases. The previous paper I dealt with the general features of quantum entanglement and its specific definition in the case of systems of identical bosons. Entanglement is a property shared between two (or more) quantum sub-systems. In defining entanglement for systems of identical massive particles, it was concluded that the single particle states or modes are the most appropriate choice for sub-systems that are distinguishable, that the general quantum states must comply both with the symmetrization principle and the super-selection rules (SSR) that forbid quantum superpositions of states with differing total particle number (global SSR compliance). Further, it was concluded that (in the separable states) quantum superpositions of sub-system states with differing sub-system particle number (local SSR compliance) also do not occur. The present paper II determines possible tests for entanglement based on the treatment of entanglement set out in paper I. Several inequalities involving variances and mean values of operators have been previously proposed as tests for entanglement between two sub-systems. These inequalities generally involve mode annihilation and creation operators and include the inequalities that define spin squeezing. In this paper, spin squeezing criteria for two mode systems are examined, and spin squeezing is also considered for principle spin operator components where the covariance matrix is diagonal. The proof, which is based on our SSR compliant approach shows that the presence of spin squeezing in any one of the spin components requires entanglement of the relevant pair of modes. A simple Bloch vector test for entanglement is also derived. Thus we show that spin squeezing becomes a rigorous test for entanglement in a system of massive bosons, when viewed as a test for entanglement between two modes. In addition, other previously proposed tests for entanglement involving spin operators are considered, including those based on the sum of the variances for two spin components. All of the tests are still valid when the present concept of entanglement based on the symmetrization and SSR criteria is applied. These tests also apply in cases of multi-mode entanglement, though with restrictions in the case of sub-systems each consisting of pairs of modes. Tests involving quantum correlation functions are also considered and for global SSR compliant states these are shown to be equivalent to tests involving spin operators. A new weak correlation test is derived for entanglement based on local SSR compliance for separable states, complementing the stronger correlation test obtained previously when this is ignored. The Bloch vector test is equivalent to one case of this weak correlation test. Quadrature squeezing for single modes is also examined but not found to yield a useful entanglement test, whereas two mode quadrature squeezing proves to be a valid entanglement test, though not as useful as the Bloch vector test. The various entanglement tests are considered for well-known entangled states, such as binomial states, relative phase eigenstates and NOON states—sometimes the new tests are satisfied while than those obtained in other papers are not. The present paper II then outlines the theory for a simple two mode interferometer showing that such an interferometer can be used to measure the mean values and covariance matrix for the spin operators involved in entanglement tests for the two mode bosonic system. The treatment is also generalized to cover multi-mode interferometry. The interferometer involves a pulsed classical field characterized by a phase variable and an area variable defined by the time integral of the field amplitude, and leads to a coupling between the two modes. For simplicity the center frequency was chosen to be resonant with the inter-mode transition frequency. Measuring the mean and variance of the population difference between the two modes for the output state of the interferometer for various choices of interferometer variables is shown to enable the mean values and covariance matrix for the spin operators for the input quantum state of the two mode system to be determined. The paper concludes with a discussion of several key experimental papers on spin squeezing.
Recent years have witnessed revolutionary improvement in the production, manipulation, characterization and quantification of multiatom (multiqubit) states - because of their promising applications in high precision atomic clocks, atomic interferrometry, quantum metrology and quantum information protocols. In the characterization and quantification of non-classical atomic correlations, two concepts namely, spin squeezing and quantum entanglement emerge. While spin squeezing originates from the uncertainty relation between atomic collective angular momentum operators, entanglement is a peculiar phenomena arising due to superpositions in the multiparticle Hilbert space. Here, we outline different quantitative measures of spin squeezing and discuss the intrinsic relation of spin squeezing with quantum entanglement in multiqubit systems.
Recently, investigations of the quantum properties of an antiferromagnet in the spin wave approximation have identified the eigenstates as two-mode squeezed sublattice-magnon states. The uniform magnon states were shown to display a massive sublattice entanglement. Here we extend this idea and study the squeezing properties of all sublattice Fock states throughout the magnetic Brillouin zone. We derive the full statistics of the sublattice magnon number with wave number $\stackrel{P\vec}{k}$ in the ground state and show that sublattice magnons occur in pairs with opposite wave vectors, hence, resulting in entanglement of both modes. To quantify the degree of entanglement we apply the Duan-Giedke-Cirac-Zoller inequality and show that it can be violated for all modes. The degree of entanglement decreases towards the corners of the Brillouin zone. We relate the entanglement to measurable correlations of components of the N\'eel and the magnetization vectors, thus allowing one to experimentally test the quantum nature of the squeezed vacuum. The distinct kkk-space structure of the entanglement shows that the squeezed vacuum has a nonuniform shape that is revealed through the $\stackrel{P\vec}{k}$-dependent correlators for the magnetization and the N\'eel vectors.