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arXiv:1305.2560 · Physical Review A

Classification of spin-nematic squeezing in spin-1 collective atomic systems

Emi Yukawa, Masahito Ueda, Kae Nemoto

Published Sep 26, 2013• 26 citations
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Abstract

In spin-1 collective atomic systems, the spin and nematic-tensor operators constitute the su(3) Lie algebra, whose su(2) subalgebras are shown to give two distinct classes of squeezing which are unitarily equivalent to spin squeezing and spin-nematic squeezing. We explicitly construct a unitary operator that generates an arbitrary squeezed spin-nematic state from an arbitrary spin-1 condensate state. In particular, we demonstrate that squeezed spin states can be generated from a polar state and that squeezed spin-nematic states can be generated from a fully spin-polarized state.

DOI
10.1103/physreva.88.033629

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In spin-1 collective atomic systems, the spin and nematic-tensor operators constitute the su(3) Lie algebra, whose su(2) subalgebras are shown to give two distinct classes of squeezing which are unitarily equivalent to spin squeezing and spin-nematic squeezing. We explicitly construct a unitary operator that generates an arbitrary squeezed spin-nematic state from an arbitrary spin-1 condensate state. In particular, we demonstrate that squeezed spin states can be generated from a polar state and that squeezed spin-nematic states can be generated from a fully spin-polarized state.
We investigate various types of squeezing in a collective su(2J + 1) system consisting of spin-J particles (J > 1 2). We show that squeezing in the collective su(2J + 1) system can be classified into unitary equivalence classes, each of which is characterized by a set of squeezed and anti-squeezed observables forming an su(2) subalgebra in the su(2J + 1) algebra. The dimensionality of the unitary equivalence class is found to be fundamentally related to its squeezing limit. We also demonstrate the classification of squeezing among the spin and multipolar observables in a collective su(4) system.
Abstract We investigate various types of squeezing in a collective su( 2 J + 1 ) system consisting of spin- J particles ( J > 1 / 2 ). We show that squeezing in the collective su( 2 J + 1 ) system can be classified into unitary equivalence classes, each of which is characterized by a set of squeezed and anti-squeezed observables forming an su(2) subalgebra in the su( 2 J + 1 ) algebra. The dimensionality of the unitary equivalence class is found to be fundamentally related to its squeezing limit. We also demonstrate the classification of squeezing among the spin and multipolar observables in a collective su(4) system.
Squeezing of quantum states is of great interest due to its application in high-precision measurement that can exceed standard quantum noise limit described by Heisenberg uncertainty relations. Here, we study the squeezing of quantum systems with SU(3)\text{SU}(3)SU(3) symmetry via a one-axis twisting Hamiltonian and examine the intriguing connections between the squeezing of spin-1/2 and spin-1 systems. There are seven subalgebras of su(3)\text{su}(3)su(3) Lie algebra. All these subalgebras are identical with su(2)\text{su}(2)su(2) Lie algebra but not all of them have the same anticommutation relations as su(2)\text{su}(2)su(2). Interestingly, squeezing parameters corresponding to spin-1 subalgebras depend not only on structure constants but also on anticommutation relations of the subalgebras. Our results are reported for the subalgebras with vanishing anticommutators and nematic squeezing, while in other cases our first-principle calculation recovers known results.
A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-12\frac{1}{2}21​ particles to entanglement conditions for spin-jjj particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in S\o{}rensen et al. [Nature (London) 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.
Exotic types of magnetic order and phases resulting from collective behaviour of quantum spins are an important focus of many-body physics. Nematic or quadrupolar ordering of spins is one such example [1], which breaks O(3) rotational symmetry, has no magnetic moment and is analogous to the well-known ordering of molecules in nematic phases of liquid crystals [2]. Spin nematic phases have been posited for a variety of condensed matter systems including frustrated quantum magnets [3–5], and heavyfermion [6] and iron-based superconductors [7], although they are challenging to detect directly [8]. Spin-1 atomic Bose-Einstein condensates [9– 11] provide a natural system to investigate spinnematic quantum phases with a key advantage that the nematic tensor is directly measurable. Here, we measure spin-nematic fluctuations in a spin-1 condensate following a quench through a nematic to ferromagnetic quantum phase transition and observe quadrature squeezing in the variance of the fluctuations up to −8.3 −0.7 dB (−10.3 −0.9 dB corrected for detection noise) below the standard quantum limit. Previous work in atomic squeezing has employed two-level systems exhibiting SU(2) symmetry on the Bloch sphere [12, 13], while the squeezing observed here is an example of squeezing in an SU(3) system. These results demonstrate spin-nematic dynamics in the quantum regime and could form the basis of a quantum-enhanced magnetometer.
Abstract SU(1, 1) coherent and squeezed states have been widely studied in optical, hybrid atom–light, and spinor Bose–Einstein condensate experiments. Traditionally, the definition of SU(1, 1) squeezing refers to the uncertainty relationship, which implies that the SU(1, 1) coherent state is already squeezed if it is placed in an appropriate system of coordinates. In this report we investigate the basic concept of SU(1, 1) squeezing and discuss the principle for the generation of squeezed states. We establish a new definition of the squeezing criterion for the SU(1, 1) states. We study the dynamical properties in an atom–molecule system and show that the SU(1, 1) squeezed state can be generated with a nonlinear atomic interaction, which is similar to the one-axis twisting mechanism in spin squeezing. We also discuss the effect of the atomic number on squeezing, and show that the optimal squeezing and optimal squeezing angle exhibit a sudden transition at a fixed atomic number value.