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Journal of Modern Optics

Proposal for a macro/meso-scopic Greenberger–Horne–Zeilinger test of quantum mechanics with entangled optical travelling-wave coherent states

Christopher C. Gerry, Adil Benmoussa, K. Elalouani

Published Jun 2, 2006• 1 citation
Publisher

Abstract

An optical realization of the Greenberger, Horne, Zeilinger (GHZ) state is proposed for testing local realistic theories. The optical GHZ states in this case are three-mode entangled coherent states and the relevant observable used to perform tests of quantum mechanics against local realistic theories is photon number parity. The amplitudes of the coherent states need to be mesoscopic if significant losses are involved but may be macroscopic if losses are minimal. A method is proposed for generating the required GHZ state that amounts to an extension of a proposal previously discussed (C.C. Gerry, Phys. Rev. A 59 4095 (1999)) for generating macroscopic superposition states of travelling wave fields.

DOI
10.1080/09500340600572685

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An optical realization of the Greenberger, Horne, Zeilinger (GHZ) state is proposed for testing local realistic theories. The optical GHZ states in this case are three-mode entangled coherent states and the relevant observable used to perform tests of quantum mechanics against local realistic theories is photon number parity. The amplitudes of the coherent states need to be mesoscopic if significant losses are involved but may be macroscopic if losses are minimal. A method is proposed for generating the required GHZ state that amounts to an extension of a proposal previously discussed (C.C. Gerry, Phys. Rev. A 59 4095 (1999)) for generating macroscopic superposition states of travelling wave fields.
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We study Greenberger-Horne-Zeilinger--type (GHZ-type) and WWW-type three-mode entangled coherent states. Both types of entangled coherent states violate Mermin's version of the Bell inequality with threshold photon detection (i.e., without photon counting). Such an experiment can be performed using linear optics elements and threshold detectors with significant Bell violations for GHZ-type entangled coherent states. However, to demonstrate Bell-type inequality violations for WWW-type entangled coherent states, additional nonlinear interactions are needed. We also propose an optical scheme to generate WWW-type entangled coherent states in free-traveling optical fields. The required resources for the generation are a single-photon source, a coherent state source, beam splitters, phase shifters, photodetectors, and Kerr nonlinearities. Our scheme does not necessarily require strong Kerr nonlinear interactions; i.e., weak nonlinearities can be used for the generation of the WWW-type entangled coherent states. Furthermore, it is also robust against inefficiencies of the single-photon source and the photon detectors.
As a development of our previous work, this paper is concerned with the Greenberger-Horne-Zeilinger (GHZ) nonlocality for continuous-variable cases. The discussion is based on the introduction of a pseudospin operator, which has the same algebra as the Pauli operator, for each of the N modes of a light field. Then the Bell-Clauser-Horne-Shimony-Holt inequality is presented for the N modes, each of which has a continuous degree of freedom. Following Mermin's argument, it is demonstrated that for N-mode parity-entangled GHZ states (in an infinite-dimensional Hilbert space) of the light field, the contradictions between quantum mechanics and local realism grow exponentially with N, similarly to the usual N-spin cases.
A scheme is proposed for generating maximally entangled Greenberger-Horne-Zeilinger (GHZ) atomic states for testing quantum nonlocality. In the scheme, three atoms are simultaneously sent through a nonresonant cavity in a vacuum state. They are initially in the same state, and thus there is no energy exchange in the process. The cavity-assisted collision results in a phase-shift which depends upon the collective atomic excitations. In principle, the scheme can be generalized to generate N-atom GHZ states. The scheme is insensitive to cavity decay and requires only one cavity, providing new prospects for testing fundamental aspects of quantum mechanics and for quantum information processing.
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A scheme is proposed for the generation of superpositions of multi-mode coherent states, i.e. multi-mode Schrödinger cat states. In the scheme a -type three-level atom is sent through many cavities filled with coherent fields. The whole system is entangled via the Raman atom-field interaction. Then the detection of the atom leaves the cavity system in a multi-mode cat state. Under certain conditions, such a cat state is identical to a Greenberger-Horne-Zeilinger state and thus can be used to test local realistic theories.