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arXiv:quant-ph/0106065 · Physical Review A

Spin squeezing in the Ising model

Xiaoguang Wang, Anders Søndberg Sørensen, Klaus Mølmer

Published Oct 12, 2001• 36 citations
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Abstract

We analyze the collective spin noise in interacting spin systems. General expressions are derived for the short-time behavior of spin systems with general spin-spin interactions, and we suggest optimum experimental conditions for the detection of spin squeezing. For Ising models with site-dependent nearest-neighbor interactions, general expressions are presented for the spin squeezing parameter for all times. The reduction of collective spin noise can be used to verify the entangling powers of quantum computer architectures based on interacting spins.

DOI
10.1103/physreva.64.053815

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What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized spin squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular momentum components. They can be used for the experimental detection of entanglement in a system of spin‐12 particles in which the spins cannot be individually addressed. We also show how to obtain the complete set of entanglement conditions if the mean spin and the spin correlation matrix are known. Finally, we apply our criteria for entanglement detection in spin models, showing that they can be used to detect bound entanglement in the thermal state of these systems.
We investigate spin squeezing effects of trapped ions in an off-resonance optical potential system using the arbitrary range spin—spin interaction and transverse field model. The collective spin noises at any time are analyzed exactly. The general expression of spin squeezing factor is presented for arbitrary-range spin interaction. For the nearest-neighbor and next-nearest neighbor spin interaction model, the analytic solutions are reduced from the general expressions. It is shown that the maximum spin squeezing is enhanced for the general arbitrary-range spin interaction compared with the nearest-neighbor interaction model as the long-range interaction with arbitrary sites enforces stronger correlation.
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.
A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-12\frac{1}{2}21​ particles to entanglement conditions for spin-jjj particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in S\o{}rensen et al. [Nature (London) 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.
We present criteria to detect the depth of entanglement in macroscopic ensembles of spin- j particles using the variance and second moments of the collective spin components. The class of states detected goes beyond traditional spin-squeezed states by including Dicke states and other unpolarized states. The criteria derived are easy to evaluate numerically even for systems of very many particles and outperform past approaches, especially in practical situations where noise is present. We also derive analytic lower bounds based on the linearization of our criteria, which make it possible to define spin-squeezing parameters for Dicke states. In addition, we obtain spin squeezing parameters also from the condition derived in (Sørensen and Mølmer 2001 Phys. Rev. Lett. 86 4431 ). We also extend our results to systems with fluctuating number of particles.