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Physical Review A

Proposal for a mesoscopic cavity QED realization of the Greenberger-Horne-Zeilinger state

Christopher C. Gerry

Published Oct 1, 1996• 86 citations
Publisher

Abstract

A proposal is made for obtaining a mesoscopic realization of the Greenberger-Horne-Zeilinger (GHZ) "all or nothing" contradiction between quantum mechanics and local realism in the context of cavity QED. The proposed experiment involves three separated micromaser cavities, each initially prepared in a coherent state, which are then entangled by dispersive atom-field interactions followed by selective ionization of a single atom that has passed through all three cavities. The cavities are then probed individually by single atoms manipulated to obtain a measurement of the parity of the cavity fields. Using an isomorphism between coherent-state cavity fields and a spin-\textonehalf{} system, we show how this experiment gives rise to violations of local realism in the manner of GHZ.

DOI
10.1103/physreva.54.r2529

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A proposal is made for obtaining a mesoscopic realization of the Greenberger-Horne-Zeilinger (GHZ) "all or nothing" contradiction between quantum mechanics and local realism in the context of cavity QED. The proposed experiment involves three separated micromaser cavities, each initially prepared in a coherent state, which are then entangled by dispersive atom-field interactions followed by selective ionization of a single atom that has passed through all three cavities. The cavities are then probed individually by single atoms manipulated to obtain a measurement of the parity of the cavity fields. Using an isomorphism between coherent-state cavity fields and a spin-\textonehalf{} system, we show how this experiment gives rise to violations of local realism in the manner of GHZ.
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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.
Using the even and odd coherent states, we show that a single mode cavity field, prepared in a coherent state by a classical source and manipulated by both dispersive and resonant interactions with atoms, is analogous to a spin one-half particle interacting with Stern–Gerlach magnets where the parity of the field is the analog of spin. Because the number of photons in the cavity may be large, the system we describe can exhibit quantum effects on at least a mesoscopic scale. We show that entangled two and three cavity systems can be generated. The three cavity case can be used to demonstrate the contradiction between local realistic theories and quantum mechanics in the manner proposed by Greenberger, Horne and Zeilinger in 1989 [13].
The Greenberger-Horne-Zeilinger (GHZ) theorem exhibits an ``all versus nothing'' contradiction between Ernstein, Podelsky, and Rosen's local realism and quantum mechanics by using a maximally entangled state of three qubits. However, it is shown in this paper that no extension of GHZ's proof is possible for any entangled state of two qubits, including Bell states. The proof is obtained by showing the fact that there are no more than three distinct elements (except for a different sign) in a set of mutually commuting nonlocal spin observables of two qubits. This also leads to a generic setup of the Kochen-Specker theorem for the two-qubit system.