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arXiv:1211.6004 · International Journal of Quantum Information

SPIN SQUEEZING AND ENTANGLEMENT VIA FINITE-DIMENSIONAL DISCRETE PHASE-SPACE DESCRIPTION

Marcelo A. Marchiolli, D. Galetti, Tiago Debarba

Published Feb 1, 2013• 19 citations
arXivHTMLPDF

Abstract

We show how mapping techniques inherent to N2-dimensional discrete phase spaces can be used to treat a wide family of spin systems which exhibits squeezing and entanglement effects. This algebraic framework is then applied to the modified Lipkin–Meshkov–Glick (LMG) model in order to obtain the time evolution of certain special parameters related to the Robertson–Schrödinger (RS) uncertainty principle and some particular proposals of entanglement measure based on collective angular-momentum generators. Our results reinforce the connection between both the squeezing and entanglement effects, as well as allow to investigate the basic role of spin correlations through the discrete representatives of quasiprobability distribution functions. Entropy functionals are also discussed in this context. The main sequence correlations ↦ entanglement ↦ squeezing of quantum effects embraces a new set of insights and interpretations in this framework, which represents an effective gain for future researches in different spin systems.

DOI
10.1142/s0219749913300015

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We show how mapping techniques inherent to N2-dimensional discrete phase spaces can be used to treat a wide family of spin systems which exhibits squeezing and entanglement effects. This algebraic framework is then applied to the modified Lipkin–Meshkov–Glick (LMG) model in order to obtain the time evolution of certain special parameters related to the Robertson–Schrödinger (RS) uncertainty principle and some particular proposals of entanglement measure based on collective angular-momentum generators. Our results reinforce the connection between both the squeezing and entanglement effects, as well as allow to investigate the basic role of spin correlations through the discrete representatives of quasiprobability distribution functions. Entropy functionals are also discussed in this context. The main sequence correlations ↦ entanglement ↦ squeezing of quantum effects embraces a new set of insights and interpretations in this framework, which represents an effective gain for future researches in different spin systems.
Abstract Collective spin operators for symmetric multi-quDit (namely identical D -level atom) systems generate a U( D ) symmetry. We explore generalizations to arbitrary D of SU(2)-spin coherent states and their adaptation to parity (multi-component Schrödinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric N -quDit states, expressed in the last two cases in terms of collective U( D )-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin–Meshkov–Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU( D ) and relate it to pairwise D -level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting D -level atom models.
We show how quasiprobability distribution functions defined over N2{N}^{2}N2-dimensional discrete phase spaces can be used to treat physical systems described by a finite space of states which exhibit spin-tunneling effects. This particular approach is then applied to the Lipkin-Meshkov-Glick model in order to obtain the time evolution of the discrete Husimi function, and as a by-product the energy gap for a symmetric combination of ground and first excited states. Moreover, we also show how an angle-based potential approach can be efficiently employed to explain qualitatively certain features of the energy gap in terms of a spin tunneling. Entropy functionals are also discussed in this context. Such results reinforce not only the formalism per se but also the possibility of some future potential applications in other branches of physics.
Abstract For general symmetric multi-qubit systems, the behavior of one- and two-qubit entanglement for Dicke, spin coherent and parity-adapted (even and odd) spin coherent states is determined. These quantum correlations are quantified by linear and von Neumann entropies of the corresponding one- and two-qubit reduced density matrices of the multi-qubit system. These states play a fundamental role in the study of Hamiltonian systems written in terms of collective generators of the angular momentum algebra like, for example, the Lipkin–Meshkov–Glick (LMG) model. Here we shall use these entanglement measures as a signature to characterize the different quantum phases that appear in these models.
Spin squeezing is vitally important in quantum metrology and quantum information science. The noise reduction resulting from spin squeezing can surpass the standard quantum limit and even reach the Heisenberg limit (HL) in some special circumstances. However, systems that can reach the HL are very limited. Here, we study the spin squeezing in atomic systems with a generic form of the quadratic collective-spin interaction, which can be described by the Lipkin-Meshkov-Glick model. We find that the squeezing properties are determined by the initial states and the anisotropic parameters. Moreover, we propose a pulse rotation scheme to transform the model into a two-axis twisting model with Heisenberg-limited spin squeezing. Our study paves the way for reaching HL in a broad variety of systems.
Spin squeezing, as a crucial quantum resource, plays a pivotal role in quantum metrology, enabling us to achieve high-precision parameter estimation schemes. Here, we investigate the spin squeezing and the quantum phase transition in an anisotropic central spin system. We find that this kind of central spin system can be mapped to the anisotropic Lipkin-Meshkov-Glick model in the limit where the ratio of transition frequencies between the central spin and the spin bath tends towards infinity. This property can induce a one-axis twisting interaction and provides another possibility for generating spin squeezing. We consider the generation of spin-squeezed states through the ground state and the dynamical evolution of the central spin model, respectively. The results show that the dynamical approach is more effective, and the spin-squeezing parameter improves as the anisotropy parameter decreases, while its value scales with system size as ${N}^{\ensuremath{-}2/3}$. Furthermore, we obtain the critical exponent of the quantum Fisher information around the critical point by numerical simulation, and find its value tends to 4/34/34/3 as the frequency ratio and the system size approach infinity. This paper offers a promising scheme for generating spin-squeezed state and paves the way for potential advancements in quantum sensing.
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.