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arXiv:0905.0245 · Physical Review A

Fisher information and spin squeezing in the Lipkin-Meshkov-Glick model

Jian Ma, Xiaoguang Wang

Published Jul 17, 2009• 182 citations
arXivHTMLPDF

Abstract

Fisher information, which lies at the heart of parameter estimation theory, was recently found to have a close relation with multipartite entanglement [L. Pezz\'e and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009)]. We use Fisher information to distinguish and characterize behaviors of ground state of the Lipkin-Meskhov-Glick model, which displays a second-order quantum phase transition between the broken and symmetric phases. We find that the parameter sensitivity of the system attains the Heisenberg limit in the broken phase, while it is just around the shot-noise limit in the symmetric phase. Based on parameter estimation, Fisher information provides us a useful approach to the quantum phase transition.

DOI
10.1103/physreva.80.012318

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Fisher information, which lies at the heart of parameter estimation theory, was recently found to have a close relation with multipartite entanglement [L. Pezz'e and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009)]. We use Fisher information to distinguish and characterize behaviors of ground state of the Lipkin-Meskhov-Glick model, which displays a second-order quantum phase transition between the broken and symmetric phases. We find that the parameter sensitivity of the system attains the Heisenberg limit in the broken phase, while it is just around the shot-noise limit in the symmetric phase. Based on parameter estimation, Fisher information provides us a useful approach to the quantum phase transition.
Quantum Fisher information, which plays a central role in parameter estimation theory, is a useful quantity for metrological applications. We use quantum Fisher information to distinguish and characterize behaviors of the evolved state for Bose-Einstein condensates in a symmetric double well, which display a classical bifurcation and a transition from Josephson oscillation to self-trapping. We find that the precision of the parameter estimation on the evolved state in the unstable regime is much higher than that in the stable regime, which lies at the shot-noise level. We also show that the parameter sensitivity of the evolved state approaches the Heisenberg limit in the Josephson oscillation regime, while it is just scaled as the shot-noise limit in the self-trapping regime. Finally, we discuss the relation between the parameter sensitivity of the evolved state and the entanglement entropy.
In terms of quantum Fisher information, a quantity χ2\chi^{2}χ2 was introduced by Pezz\'{e} and Smerzi, which is a multiparticle entanglement measure, and provides a necessary and sufficient condition for sub-shot-noise phase estimation sensitivity. We derive a general expression of χ2\chi ^{2}χ2 for arbitrary symmetric multiqubit states with nonzero mean spins. It is shown that the entangled symmetric states are useful for phase sensitivity beyond the shot-noise limit. Using the expression, we explicitly examine a series of superpositions of spin states. We find that the superpositions of Dicke states perform better than Dicke states themselves in phase esitmation. Although the spin coherent states themselves only have a shot-noise limit phase sensitivity, their superpositions may reach the Heisenberg limit.
We analyze the quantum Fisher information (QFI) and entanglement of the ground state for the XY model. The QFI determines the precision in parameter estimations, moreover, it is a criterion of entanglement. The scaling behaviors of the QFI are studied both at the critical point and in different phases. In the symmetry-broken phase, the QFI scales proportional to the square of the system size, while in the symmetric phase, it is nearly independent of the system size. The study of QFI not only reveals the distinct entanglement properties of the XY model in different phases, but also shows that the ground state of the XY model can be used to perform Heisenberg-limit parameter estimation.
In terms of quantum Fisher information (QFI), a quantity χ 2 was introduced by Pezzé and Smerzi (Phys. Rev. Lett. 102 100401, 2009). They pointed out that the inequality χ 2<1 was a sufficient condition for multiparticle entanglement. For the two-qubit symmetric states, we found that the inequality χ 2<1 is a necessary and sufficient condition for entanglement and spin squeezing, and that χ 2 is equal to the second kind of spin squeezing parameter ξ22\xi _{2}^{2}ξ22​ . For the two-qubit asymmetric states, it is only a sufficient condition. In order to make it a necessary and sufficient condition, we extend the concept of the QFI and χ 2, and generalize the relations among the entanglement measurement, the spin squeezing parameters and χ 2 in symmetric pure states to arbitrary pure states.
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.
Abstract In this paper we pursue the use of information measures (in particular, information diagrams) for the study of entanglement in symmetric multi-quDit systems. We use generalizations to U(D){U}(D)U(D) U ( D ) of spin U(2){U}(2)U(2) U ( 2 ) coherent states and their adaptation to parity (multicomponent Schrödinger cats), and we analyse one- and two-quDit reduced density matrices. We use these correlation measures to characterize quantum phase transitions occurring in Lipkin–Meshkov–Glick models of D=3D=3D=3 D = 3 -level identical atoms, and we propose the rank of the corresponding reduced density matrix as a discrete order parameter.