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arXiv:quant-ph/0509080 · International Journal of Modern Physics B

SQUEEZING IN MULTIVARIATE SPIN SYSTEMS

Swarnamala Sirsi

Published May 20, 2006• 0 citations
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Abstract

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.

DOI
10.1142/s0217979206034054

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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.
Spin- s assemblies are classified into two mutually exclusive classes: oriented and non-oriented systems. The density matrix ρ, describing oriented systems, can assume diagonal form in the angular momentum basis ( ) defined with respect to the axis of quantization, whereas the eigenstates of ρ for the non-oriented assembly cannot all be identified with states. A new scheme for constructing a mixed, non-oriented spin- s state using s (2 s +1) spinors all pointing in different directions in space and 2 s weights is discussed. Such a construction takes its inspiration from Schwinger's idea of realizing an state as being made up of ( s + m ) 'up' spinors and ( s − m ) 'down' spinors, all defined with respect to a single axis in space. Since the oriented systems are never squeezed, non-oriented spin-1 assemblies which can be prepared in the laboratory with the available NQR technology are examined for signatures of squeezing using our scheme in a frame of reference where the Heisenberg–Robertson uncertainty relation has the same form as the Schrödinger uncertainty relation. It is shown that unlike in the case of the pure spin-1 state where squeezing is synonymous with non-orientedness, a non-oriented spin-1 system need not be squeezed and the existence of entanglement is a necessary but not sufficient condition for the system to be squeezed.
We have produced and characterized spin-squeezed states at a temperature of 26 °C in a nuclear magnetic resonance quadrupolar system. The experiment was carried out on 133Cs nuclei of spin I=7/2 in a sample of lyotropic liquid crystal. The source of spin squeezing was identified as the interaction between the quadrupole moment of the nuclei and the electric field gradients present within the molecules. We use the spin angular momentum representation to describe formally the nonlinear operators that produce the spin squeezing on a Hilbert space of dimension 2I+1=8. The quantitative and qualitative characterization of this spin-squeezing phenomenon is expressed by a squeezing parameter and squeezing angle developed for the two-mode Bose-Einstein condensate system, as well as by the Wigner quasiprobability distribution function. The generality of the present experimental scheme points to potential applications in solid-state physics.
We show how mapping techniques inherent to N2-dimensional discrete phase spaces can be used to treat a wide family of spin systems which exhibits squeezing and entanglement effects. This algebraic framework is then applied to the modified Lipkin–Meshkov–Glick (LMG) model in order to obtain the time evolution of certain special parameters related to the Robertson–Schrödinger (RS) uncertainty principle and some particular proposals of entanglement measure based on collective angular-momentum generators. Our results reinforce the connection between both the squeezing and entanglement effects, as well as allow to investigate the basic role of spin correlations through the discrete representatives of quasiprobability distribution functions. Entropy functionals are also discussed in this context. The main sequence correlations ↦ entanglement ↦ squeezing of quantum effects embraces a new set of insights and interpretations in this framework, which represents an effective gain for future researches in different spin systems.
The notion of spin squeezing has been discussed in this paper using the density matrix formalism. Extending the definition of squeezing for pure states given by Kitagawa and Ueda in an appropriate manner and employing the spherical tensor representation, we show that mixed spin states which are non-oriented and possess vector polarization indeed exhibit squeezing. We construct a mixed state of a spin 1 system using two spin ½ states and study its squeezing behaviour as a function of the individual polarizations of the two spinors.
The quantum analogue of the classical characteristic function for a spin 1/2 assembly is considered and the probability mass function of the random vector associated with the assembly is derived. It is seen that the positive regions of Wigner and Margenau-Hill quasi distributions for the three components of spin, correspond to a trivariate probability mass function. We identify the domain of these positive regions as an Octahedron inscribed in the Bloch sphere with its vertices on the surface of the sphere. It is in this domain that a quantum characteristic function characterizing the quasi distribution, admits a probability mass function in IR^3 . It is also observed that the classical variates X1, X2, X3 corresponding to the 3 spin operators σ_1,σ_2,σ_3 in the domain, are independent iff the Bloch vector lies on any one of the axes.