Quantum entanglement for systems of identical bosons: II. Spin squeezing and other entanglement tests

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dc.contributor.author Dalton, B. J.
dc.contributor.author Goold, J.
dc.contributor.author Garraway, B. M.
dc.contributor.author Reid, M. D.
dc.date.accessioned 2018-05-02T10:16:26Z
dc.date.available 2018-05-02T10:16:26Z
dc.date.issued 2017
dc.identifier.citation Dalton, B. J., Goold, J., Garraway, B. M. and Reid, M. D. (2017) 'Quantum entanglement for systems of identical bosons: II. Spin squeezing and other entanglement tests', Physica Scripta, 92(2), 023005 (34pp). doi: 10.1088/1402-4896/92/2/023005 en
dc.identifier.volume 92
dc.identifier.issued 2
dc.identifier.issn 0031-8949
dc.identifier.uri http://hdl.handle.net/10468/5943
dc.identifier.doi 10.1088/1402-4896/92/2/023005
dc.description.abstract These two accompanying papers are concerned with entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. The main focus is on two mode entanglement, but multi-mode entanglement is also considered. The bosons may be atoms or molecules as in cold quantum gases. The previous paper I dealt with the general features of quantum entanglement and its specific definition in the case of systems of identical bosons. Entanglement is a property shared between two (or more) quantum subsystems. In defining entanglement for systems of identical massive particles, it was concluded that the single particle states or modes are the most appropriate choice for sub-systems that are distinguishable, that the general quantum states must comply both with the symmetrization principle and the super-selection rules (SSR) that forbid quantum superpositions of states with differing total particle number (global SSR compliance). Further, it was concluded that (in the separable states) quantum superpositions of sub-system states with differing sub-system particle number (local SSR compliance) also do not occur. The present paper II determines possible tests for entanglement based on the treatment of entanglement set out in paper I. Several inequalities involving variances and mean values of operators have been previously proposed as tests for entanglement between two sub-systems. These inequalities generally involve mode annihilation and creation operators and include the inequalities that define spin squeezing. In this paper, spin squeezing criteria for two mode systems are examined, and spin squeezing is also considered for principle spin operator components where the covariance matrix is diagonal. The proof, which is based on our SSR compliant approach shows that the presence of spin squeezing in any one of the spin components requires entanglement of the relevant pair of modes. A simple Bloch vector test for entanglement is also derived. Thus we show that spin squeezing becomes a rigorous test for entanglement in a system of massive bosons, when viewed as a test for entanglement between two modes. In addition, other previously proposed tests for entanglement involving spin operators are considered, including those based on the sum of the variances for two spin components. All of the tests are still valid when the present concept of entanglement based on the symmetrization and SSR criteria is applied. These tests also apply in cases of multi-mode entanglement, though with restrictions in the case of sub-systems each consisting of pairs of modes. Tests involving quantum correlation functions are also considered and for global SSR compliant states these are shown to be equivalent to tests involving spin operators. A new weak correlation test is derived for entanglement based on local SSR compliance for separable states, complementing the stronger correlation test obtained previously when this is ignored. The Bloch vector test is equivalent to one case of this weak correlation test. Quadrature squeezing for single modes is also examined but not found to yield a useful entanglement test, whereas two mode quadrature squeezing proves to be a valid entanglement test, though not as useful as the Bloch vector test. The various entanglement tests are considered for well-known entangled states, such as binomial states, relative phase eigenstates and NOON states-sometimes the new tests are satisfied while than those obtained in other papers are not. The present paper II then outlines the theory for a simple two mode interferometer showing that such an interferometer can be used to measure the mean values and covariance matrix for the spin operators involved in entanglement tests for the two mode bosonic system. The treatment is also generalized to cover multi-mode interferometry. The interferometer involves a pulsed classical field characterized by a phase variable and an area variable defined by the time integral of the field amplitude, and leads to a coupling between the two modes. For simplicity the center frequency was chosen to be resonant with the inter-mode transition frequency. Measuring the mean and variance of the population difference between the two modes for the output state of the interferometer for various choices of interferometer variables is shown to enable the mean values and covariance matrix for the spin operators for the input quantum state of the two mode system to be determined. The paper concludes with a discussion of several key experimental papers on spin squeezing. en
dc.description.sponsorship Science Foundation Ireland (E T S Walton Visiting Fellowship, 11/W.1/11959); Engineering and Physical Sciences Research Council (EP/1010394/1); Australian Research Council (Discovery Project Grant) en
dc.format.mimetype application/pdf en
dc.language.iso en en
dc.publisher IOP Publishing Ltd. en
dc.relation.uri http://iopscience.iop.org/article/10.1088/1402-4896/92/2/023005
dc.rights © 2017, Royal Swedish Academy of Sciences. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. en
dc.rights.uri https://creativecommons.org/licenses/by/3.0/
dc.subject Entanglement en
dc.subject Identical massive bosons en
dc.subject Super-selection rules en
dc.subject Spin squeezing en
dc.subject Correlation en
dc.subject Quadrature squeezing en
dc.subject Phase reference en
dc.title Quantum entanglement for systems of identical bosons: II. Spin squeezing and other entanglement tests en
dc.type Article (peer-reviewed) en
dc.internal.authorcontactother John Goold, Physics, University College Cork, Cork, Ireland. +353-21-490-3000 Email: jgoold@phys.ucc.ie en
dc.internal.availability Full text available en
dc.description.version Published Version en
dc.contributor.funder Australian Research Council
dc.contributor.funder Engineering and Physical Sciences Research Council
dc.contributor.funder Science Foundation Ireland
dc.description.status Peer reviewed en
dc.identifier.journaltitle Physica Scripta en
dc.internal.IRISemailaddress jgoold@phys.ucc.ie en
dc.identifier.articleid 23005


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© 2017, Royal Swedish Academy of Sciences. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Except where otherwise noted, this item's license is described as © 2017, Royal Swedish Academy of Sciences. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
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