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Journal of Optics B Quantum and Semiclassical Optics

Simultaneous squeezing of two orthogonal spin components

Hari Prakash, Rakesh Kumar Rai

Published Nov 23, 2005• 17 citations
Publisher

Abstract

Squeezing of spin is usually discussed for x and y components ( S x and S y ), considering the commutation relation [ S x , S y ] = i S Z for the triad ( S x , S y , S Z ) of three mutually perpendicular spin components. We generalize and discuss squeezing of the spin component S θ = S x cosθ+ S y sinθ considering all possible such triads, which include S θ . For an explicit example involving the interaction of an assembly of two excited two-level atoms with single-mode coherent radiation, we find that S θ and S θ+π/2 can be squeezed simultaneously.

DOI
10.1088/1464-4266/7/12/046

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Squeezing of spin is usually discussed for x and y components ( S x and S y ), considering the commutation relation [ S x , S y ] = i S Z for the triad ( S x , S y , S Z ) of three mutually perpendicular spin components. We generalize and discuss squeezing of the spin component S θ = S x cosθ+ S y sinθ considering all possible such triads, which include S θ . For an explicit example involving the interaction of an assembly of two excited two-level atoms with single-mode coherent radiation, we find that S θ and S θ+π/2 can be squeezed simultaneously.
In contrast to the canonically conjugate variates q, p representing the position and momentum of a particle in the phase space distributions, the three Cartesian components, Jx, Jy, Jz of a spin-j system constitute the mutually non-commuting variates in the quasi-probabilistic spin distributions. It can be shown that a univariate spin distribution is never squeezed and one needs to look into either bivariate or trivariate distributions for signatures of squeezing. Several such distributions result if one considers different characteristic functions or moments based on various correspondence rules. As an example, discrete probability distribution for an arbitrary spin-1 assembly is constructed using Wigner-Weyl and Margenau-Hill correspondence rules. It is also shown that a trivariate spin-1 assembly resulting from the exposure of nucleus with non-zero quadrupole moment to combined electric quadrupole field and dipole magnetic field exhibits squeezing in certain cases.
Using a generalization of the q-commutation relations, we develop a formalism in which it is possible to define generalized q-bosonic operators. This formalism includes both types of the usual q-deformed bosons as special cases. The coherent states of these operators show interesting and novel noise reduction properties including simultaneous squeezing in both field components, unlike the conventional case in which squeezing is permitted in only one component. This also contrasts with the usual quantum group deformation which also only permits one component squeezing.
In the present paper, we consider the interaction of a single mode radiation initially in a coherent state with an assembly of two two-level atoms in some different states using the Hamiltonian, H = ω(a+a + Sz)+g(a S+ + a+ S-) in the natural units, where a+ and a are the creation and annihilation operators for radiation, Sz, S± are the collective Dicke operators, g is the coupling constant, ω is the energy of the photons and energy difference between the two atomic levels. We solve it exactly. We study ordinary squeezing and amplitude-squared squeezing of radiation for the general operators, [Formula: see text] and Yθ=1/2(a2e-iθ+a+2eiθ), when atoms are fully excited, super-radiant or in the ground state. We get a large ordinary squeezing (nearly 88% squeezing) for the super-radiant atomic state, at a time, which is 0.451 of the first revival time, and this should be easily observable. We also get a large amplitude-squared squeezing (nearly 20%) at the time, which is about 0.033 of the first revival time. Variation of variances near their minima with coupling time gt, square root of mean photon number |α| and phase are also discussed.
Planar squeezed states, i.e. quantum states which are squeezed in two orthogonal spin components, have recently attracted attention due to their applications in atomic interferometry and quantum information (He et al 2012 New J. Phys. 14 093012). While canonical variables such as quadratures of the radiation field can be squeezed in at most one component, simultaneous squeezing in two orthogonal spin components can be achieved due to the angular momentum commutation relations. We present a novel scheme for planar squeezing via quantum non-demolition (QND) measurements in spin-1 systems. The QND measurement is achieved via near-resonant paramagnetic Faraday rotation probing, and the planar squeezing is obtained by sequential QND measurement of two orthogonal spin components. We compute the achievable squeezing for a variety of optical depths, initial conditions and probing strategies. The planar squeezed states generated in this way contain entanglement detectable by spin-squeezing inequalities and give an advantage relative to non-squeezed states for any precession phase angle, a benefit for single-shot and high-bandwidth magnetometry.
We investigate the presence of spin- and planar-squeezing in generalized superpositions of atomic (or spin) coherent states. Spin-squeezing has been shown to be a useful tool in determining the presence of entanglement in multipartite systems, such as collections of two-level atoms, as well as being an indication of reduced projection noise and sub-shot-noise-limited phase uncertainty in Ramsey spectroscopy, suitable for measuring phases ϕ ∼ 0 . On the other hand, planar-squeezed states display reduced projection noise in two directions simultaneously and have been shown to lead to enhanced metrological precision in measuring phases without the need for explicit prior knowledge of the phase value. In this paper, we show that the generalized superposition state can be parametrized to display both spin-squeezing along all orthogonal axes and planar-squeezing along all orthogonal planes for all values of J > 1 / 2 . We close with an application of the maximally spin- and planar-squeezed states to quantum metrology.
The nature of spin squeezing has been studied earlier for a coupled state of two spinors by Usha Devi et. al. (J. Phys. A: Math. Gen. 36 5333 (2003)). In this paper, we extend this study to a coupled state of two spin-1 systems. Here, we show that the direct product of two spin-1 states exhibits squeezing unlike the direct product of two spinors. We investigate the nature of squeezing for non-separable coupled state of two spin-1 systems and identify the parameters which are responsible for the existence of squeezing. We also provide ways to generate them by external interactions.