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arXiv:0711.0995 · Journal of Physics A Mathematical and Theoretical

The quadratic spin squeezing operators

P K Pathak, R N Deb, N. Nayak, B Dutta-Roy

Published Mar 26, 2008• 8 citations
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Abstract

We discuss generic spin squeezing operators (quadratic in angular momentum operators) capable of squeezing out quantum-mechanical noise from a system of two-level atoms (spins) in a coherent state. Such systems have been considered by Kitagawa and Ueda (1991 Phys. Rev. Lett. 67 1852, 1993 Phys. Rev. A 47 5138) in this context and a Hamiltonian of this nature governs the Lipkin model (Lipkin et al 1965 Nucl. Phys. 62 188) which is relevant to nuclear physics.

DOI
10.1088/1751-8113/41/14/145302

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We discuss generic spin squeezing operators (quadratic in angular momentum operators) capable of squeezing out quantum-mechanical noise from a system of two-level atoms (spins) in a coherent state. Such systems have been considered by Kitagawa and Ueda (1991 Phys. Rev. Lett. 67 1852, 1993 Phys. Rev. A 47 5138) in this context and a Hamiltonian of this nature governs the Lipkin model (Lipkin et al 1965 Nucl. Phys. 62 188) which is relevant to nuclear physics.
We study the generation of spin squeezing by the repeated action of the angular momentum (Dicke) lowering operator on an atomic coherent state prepared for a collection of NNN two-level atoms or ions. The atoms or ions of the atomic coherent state are not entangled, but the action of the lowering operator (or similarly of the raising operator) generates entanglement among the atoms, and spin squeezing occurs for some ranges of the relevant parameter. Spin squeezing in a collection of two-level atoms or ions is of importance for precision spectroscopy. We discuss methods by which our spin-squeezed states could be generated in the contexts of cavity QED and trapped ions.
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.
There is currently much interest in the two-axis countertwisting spin squeezing Hamiltonian suggested originally by Kitagawa and Ueda, since it is useful for interferometry and metrology. No analytical solution valid for arbitrary spin values seems to be available. In this article we systematically consider the issue of the analytical solvability of this Hamiltonian for various specific spin values. We show that the spin squeezing dynamics can be considered to be analytically solved for angular momentum values upto 21/221/221/2, i.e. for 212121 spin half particles. We also identify the properties of the system responsible for yielding analytic solutions for much higher spin values than based on naive expectations. Our work is relevant for analytic characterization of squeezing experiments with low spin values, and semi-analytic modeling of higher values of spins.
Spin squeezing via atom-field interactions is considered within the context of the Tavis-Cummings model. An ensemble of N two-level atoms interacts with a quantized cavity field. For all the atoms initially in their ground states, it is shown that spin squeezing of both the atoms and the field can be achieved provided the initial state of the cavity field has coherence between number states differing by 2. Most of the discussion is restricted to the case of a cavity field initially in a coherent state, but initial squeezed states for the field are also discussed. Optimal conditions for obtaining squeezing are obtained. An analytic solution is found that is valid in the limit that the number of atoms is much greater than unity and is also much larger than the average number of photons, ${\ensuremath{\alpha}}^{2},$ initially in the coherent state of the cavity field. In this limit, the degree of spin squeezing increases with increasing $\ensuremath{\alpha},$ even though the field more closely resembles a classical field for which no spin squeezing could be achieved.