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Classification of spin and multipolar squeezing

Emi Yukawa, Kae Nemoto

Published May 17, 2016• 6 citations
OpenAlex

Abstract

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.

Related papers

Classification of spin and multipolar squeezing

Emi Yukawa, Kae Nemoto

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.

arXiv:1602.06047•Journal of Physics A Mathematical and Theoretical•May 16, 2016•11 citations

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

Emi Yukawa, Masahito Ueda, Kae Nemoto

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.

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Squeezing of spin-1 quantum states via a one-axis twisting Hamiltonian

Tuguldur Begzjav, G. S. Agarwal

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 $\text{SU}(3)$ 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 $\text{su}(3)$ Lie algebra. All these subalgebras are identical with $\text{su}(2)$ Lie algebra but not all of them have the same anticommutation relations as $\text{su}(2)$. 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.

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Two-mode quantum systems: Invariant classification of squeezing transformations and squeezed states

Arvind, Biswajit Dutta, Narasimhaiengar Mukunda +1 more

A general anaylsis of squeezing transformations for two-mode systems is given based on the four-dimensional real symplectic group Sp(4,openR). Within the framework of the unitary (metaplectic) representation of this group, a distinction between compact photon-number-conserving and noncompact photon-number-nonconserving squeezing transformations is made. We exploit the U(2) invariant squeezing criterion to divide the set of all squeezing transformations into a two-parameter family of distinct equivalence classes with representative elements chosen for each class. Familiar two-mode squeezing transformations in the literature are recognized in our framework and seen to form a set of measure zero. Examples of squeezed coherent and thermal states are worked out. The need to extend the heterodyne detection scheme to encompass all of U(2) is emphasized, and known experimental situations where all U(2) elements can be reproduced are briefly described.

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Semiclassical approach to squeezing-like transformations in quantum systems with higher symmetries

A. B. Klimov, Hossein T. Dinani, Hubert de Guise

We provide a coarse but intuitive classification of squeezing in quantum systems with SU ( n ) symmetries. This classification is based on the non-equivalent paths (classical trajectories) in the corresponding phase-space. The example of SU (3) is studied in details.

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s u ( d ) -squeezing and many-body entanglement geometry in finite-dimensional systems

Giuseppe Vitagliano, Otfried Gühne, Géza Tóth

Generalizing the well-known spin-squeezing inequalities, we study the relation between squeezing of collective N -particle s u ( d ) operators and many-body entanglement geometry in multi-particle systems. For that aim, we define the set of pseudo-separable states, which are mixtures of products of single-particle states that lie in the ( d 2 − 1 ) -dimensional Bloch sphere but are not necessarily positive semidefinite. We obtain a set of necessary conditions for states of N qudits to be of the above form. Any state that violates these conditions is entangled. We also define a corresponding s u ( d ) -squeezing parameter that can be used to detect entanglement in large particle ensembles. Geometrically, this set of conditions defines a convex set of points in the space of first and second moments of the collective N -particle s u ( d ) operators. We prove that, in the limit N ≫ 1 , such set is filled by pseudo-separable states, while any state corresponding to a point outside of this set is necessarily entangled. We also study states that are detected by these inequalities: We show that states with a bosonic symmetry are detected if and only if the two-body reduced state violates the positive partial transpose (PPT) criterion. On the other hand, highly mixed states states close to the s u ( d ) singlet are detected which have a separable two-body reduced state and are also PPT with respect to all possible bipartitions. We also provide numerical examples of thermal equilibrium states that are detected by our set of inequalities, comparing the spin-squeezing inequalities with the s u ( 3 ) -squeezing inequalities.

arXiv:2406.13338•Quantum•Sep 3, 2025•3 citations
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.
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.
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 general anaylsis of squeezing transformations for two-mode systems is given based on the four-dimensional real symplectic group Sp(4,openR). Within the framework of the unitary (metaplectic) representation of this group, a distinction between compact photon-number-conserving and noncompact photon-number-nonconserving squeezing transformations is made. We exploit the U(2) invariant squeezing criterion to divide the set of all squeezing transformations into a two-parameter family of distinct equivalence classes with representative elements chosen for each class. Familiar two-mode squeezing transformations in the literature are recognized in our framework and seen to form a set of measure zero. Examples of squeezed coherent and thermal states are worked out. The need to extend the heterodyne detection scheme to encompass all of U(2) is emphasized, and known experimental situations where all U(2) elements can be reproduced are briefly described.
Generalizing the well-known spin-squeezing inequalities, we study the relation between squeezing of collective N -particle s u ( d ) operators and many-body entanglement geometry in multi-particle systems. For that aim, we define the set of pseudo-separable states, which are mixtures of products of single-particle states that lie in the ( d 2 − 1 ) -dimensional Bloch sphere but are not necessarily positive semidefinite. We obtain a set of necessary conditions for states of N qudits to be of the above form. Any state that violates these conditions is entangled. We also define a corresponding s u ( d ) -squeezing parameter that can be used to detect entanglement in large particle ensembles. Geometrically, this set of conditions defines a convex set of points in the space of first and second moments of the collective N -particle s u ( d ) operators. We prove that, in the limit N ≫ 1 , such set is filled by pseudo-separable states, while any state corresponding to a point outside of this set is necessarily entangled. We also study states that are detected by these inequalities: We show that states with a bosonic symmetry are detected if and only if the two-body reduced state violates the positive partial transpose (PPT) criterion. On the other hand, highly mixed states states close to the s u ( d ) singlet are detected which have a separable two-body reduced state and are also PPT with respect to all possible bipartitions. We also provide numerical examples of thermal equilibrium states that are detected by our set of inequalities, comparing the spin-squeezing inequalities with the s u ( 3 ) -squeezing inequalities.