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Journal of Physics Conference Series

Spin squeezing properties of an initial coherent state via a two-axis countertwisting Hamiltonian in the presence and absence of an external fleld

Ahmad Akhound, Mojtaba Jafarpour

Published Aug 1, 2008• 4 citations
Publisher

Abstract

We consider a four-qubit system, initially in a coherent state along the z-axis, and study its time dependence under the influence of the well-known two-axis countertwisting Hamiltonian. Analyzing the time dependence of the spin operators, we observe its squeezing properties, according to the criterion given by Kitagawa et al. We also perform similar calculations for the same Hamiltonian in the presence of an external fleld and compare its squeezing properties with that of the former. It is observed that the details of the squeezing properties are difierent in the two cases. Plots of the squeezing parameters as a function of time and the rotation angle for both cases are presented.

DOI
10.1088/1742-6596/128/1/012021

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We consider a four-qubit system, initially in a coherent state along the z-axis, and study its time dependence under the influence of the well-known two-axis countertwisting Hamiltonian. Analyzing the time dependence of the spin operators, we observe its squeezing properties, according to the criterion given by Kitagawa et al. We also perform similar calculations for the same Hamiltonian in the presence of an external fleld and compare its squeezing properties with that of the former. It is observed that the details of the squeezing properties are difierent in the two cases. Plots of the squeezing parameters as a function of time and the rotation angle for both cases are presented.
We study the quantum dynamics generated by the two-axis countertwisting Hamiltonian from an initial spin coherent state in a spin-1/21/21/2 ensemble. A characteristic feature of the two-axis countertwisting Hamiltonian is the existence of four neutrally stable and two saddle unstable fixed points. The presence of the latter is responsible for a high level of squeezing. The squeezing is accompanied by the appearance of several quantum states of interest in quantum metrology with Heisenberg-limited sensitivity, and we show fidelity functions for some of them. We present exact results for the quantum Fisher information and the squeezing parameter. Although the overall time evolution of both changes strongly with the number of particles, we find that they have regular dynamics for short times. We explain scaling with the system size by using a Gaussian approach.
We study entanglement and squeezing of a cluster of spin systems under the influence of the two-axis countertwisting Hamiltonian. The squeezing parameters given by Wineland et al and also by Kitagawa et al. are chosen as the criteria of spin squeezing. The criterion of pairwise entanglement is chosen to be the concurrence and that of the bipartite entanglement the linear entropy. We also define a new squeezing parameter η, which plays a direct role in the investigation of the relationship between squeezing and entanglement. We observe that if the system is squeezed according to the Wineland’s criterion, it is squeezed according to the Kitagawa’s also, but the reverse is not always true. Moreover, if the system is squeezed according to Kitagawa’s criterion, it is pairwise entangled simultaneously and vice versa. It is also observed that the entropy is a linear function of the parameter η2.
We show that spin squeezing implies pairwise entanglement for arbitrary symmetric multiqubit states. If the squeezing parameter is less than or equal to 1, we demonstrate a quantitative relation between the squeezing parameter and the concurrence for the even and odd states. We prove that the even states generated from the initial state with all qubits being spin down, via the one-axis twisting Hamiltonian, are spin squeezed if and only if they are pairwise entangled. For the states generated via the one-axis twisting Hamiltonian with an external transverse field for any number of qubits greater than 1 or via the two-axis countertwisting Hamiltonian for any even number of qubits, the numerical results suggest that such states are spin squeezed if and only if they are pairwise entangled.
We present a detailed analysis of spin squeezing of the one-axis twisting model with a many-body phase dephasing, which is induced by external field fluctuation in a two-mode Bose-Einstein condensates. Even in the presence of the dephasing, our analytical results show that the optimal initial state corresponds to a coherent spin state ∣θ0,ϕ0⟩|\theta_{0}, \phi_0\rangle∣θ0​,ϕ0​⟩ with the polar angle θ0=π/2\theta_0=\pi/2θ0​=π/2. If the dephasing rate γ≪S−1/3\gamma\ll S^{-1/3}γ≪S−1/3, where SSS is total atomic spin, we find that the smallest value of squeezing parameter (i.e., the strongest squeezing) obeys the same scaling with the ideal one-axis twisting case, namely ξ2∝S−2/3\xi^2\propto S^{-2/3}ξ2∝S−2/3. While for a moderate dephasing, the achievable squeezing obeys the power rule S−2/5S^{-2/5}S−2/5, which is slightly worse than the ideal case. When the dephasing rate γ>S1/2\gamma>S^{1/2}γ>S1/2, we show that the squeezing is weak and neglectable.
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.