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Physical Review A

Temporal Bell-type inequalities for two-level Rydberg atoms coupled to a high-Qresonator

Susana F. Huelga, Trevor W. Marshall, Emilio Santos

Published Sep 1, 1996• 34 citations
Publisher

Abstract

Following the strategy of showing specific quantum effects by means of the violation of a classical inequality, a pair of Bell-type inequalities is derived on the basis of certain additional assumptions, whose plausibility is discussed in detail. Such inequalities are violated by the quantum mechanical predictions for the interaction of a two-level Rydberg atom with a single mode sustained by a high-Q resonator. The experimental conditions required in order to show the existence of forbidden values, according to a hidden variables formalism, in a real experiment are analyzed for various initial field statistics. In particular, the revival dynamics expected for the interaction with a coherent field leads to classically forbidden values, which would indicate a purely quantum effect. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1798

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Following the strategy of showing specific quantum effects by means of the violation of a classical inequality, a pair of Bell-type inequalities is derived on the basis of certain additional assumptions, whose plausibility is discussed in detail. Such inequalities are violated by the quantum mechanical predictions for the interaction of a two-level Rydberg atom with a single mode sustained by a high-Q resonator. The experimental conditions required in order to show the existence of forbidden values, according to a hidden variables formalism, in a real experiment are analyzed for various initial field statistics. In particular, the revival dynamics expected for the interaction with a coherent field leads to classically forbidden values, which would indicate a purely quantum effect. \textcopyright{} 1996 The American Physical Society.
The tests of Bell's inequality to date have failed to be definitive because of the ``detection loophole,'' due to the inefficiency of photodetectors. An idealized experiment using pairs of two-level Rydberg atoms, correlated with the help of a quantized cavity field, was proposed by Oliver and Stroud [J. Opt. Soc. Am. B 4, 1426 (1987)] for the purpose of blocking this loophole. We develop their proposal by removing their idealizations. We carefully define the ensemble of atom pairs by imposing conditions on arrival times at the analyzers of the atoms. We take into account the spread of velocities of the atoms, which implies that the ensemble must be described quantum mechanically by a statistical operator rather than a pure state, and we show that with detector efficiencies of 0.9 or better, quantum mechanics implies a strong violation of Bell's inequality in the ensemble. Upper and lower bounds on ensemble averages can be inferred from experimental averages, thereby permitting an experimental test of Bell's inequality in the ensemble. \textcopyright{} 1996 The American Physical Society.
Rydberg atoms traversing a micromaser cavity one after the other can emerge in correlated states, and according to Copenhagen quantum mechanics this may lead to a violation of Bell's inequality, that is, to a Bell sum S>2. S is here calculated in various ideal physical situations including those with (1) different initial states of the atoms, (2) different free adjustable parameters, being either the phase or the turning angle of a classical radiation field put behind the cavity, and (3) either equal or different Rabi angles for successive atoms. The experiments are crucial for a realist interpretation which predicts S<2 or S=2 because particles of different kinds (photons and atoms) are involved.
The evolution of a two-level system subjected to stimulated transitions which is undergoing a sequence of measurements of the level occupation probability is evaluated. Its time correlation function is compared to the one obtained through the pure Schroedinger evolution. Systems of this kind have been recently proposed for testing the quantum mechanical predictions against those of macrorealistic theories, by means of temporal Bell inequalities. The classical requirement of noninvasivity, needed to define correlation functions in the realistic case, finds a quantum counterpart in the quantum nondemolition condition. The consequences on the observability of quantum mechanically predicted violations to temporal Bell inequalities are drawn and compared to the already dealt case of the rf-SQUID dynamics.
The analogy between two-level atoms and spins is used to show that Rydberg atoms provide a new medium through which Bell’s inequalities may be studied. The low-efficiency particle detectors that have been used in experiments to test Bell’s inequalities are replaced by nearly 100% efficient state-selective ionizers in Rydberg-atom experiments. Two methods for producing correlated states of pairs of atoms are discussed. Each method can be used to create correlated states of more than two atoms. A Bell inequality for a three-atom system is considered and compared with quantum-mechanical predictions. A violation of the inequality for the three-particle system is predicted.
Quantum statistic properties of a two-mode field initially in a coherent state resonantly interacting with a three-level V-type atom are investigated by means of the quantum theory and numerical calculations.The dependence of the sub-Poisson statistics,the coherent properties of two-mode cavity field,photon bunching and anti-bunching,and the violations of Cauchy-Schwarz inequality on the average photon number of the cavity mode,the coupling constant and the interaction time is discussed in three cases:(1) no state-selective measurement on the atom;(2) direct state-selective measurement on the atom;(3) state-selective measurement on the atom after the application of a classical field.The results indicate,when the state-selective measurement is performed on the atom,the sub-Poisson statistic properties is distinctly enhanced,the anti-correlative degree between two-mode is distinctly deadened,and the Cauchy-Schwarz inequality is always violated;the mode a anti-bunching exhibits when the state-selective measurement is performed on the atom and the average photon number difference of the two cavity modes is large.