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arXiv:quant-ph/9811042

Comment on testing Bell's inequality with Rydberg atoms entangled through cavity photons

Arthur Jabs

Published Nov 17, 1998• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/9811042

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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 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.
An experimental test of Bell’s inequalities using two-level atoms is proposed. The test is analogous to that proposed by Bohm but, instead of using spins and Stern-Gerlach magnets, two-level atoms and resonant pulses are used. Two atoms excited to a common Rydberg state, but traveling in different directions, are introduced into a microwave cavity. The microwave field resonantly couples two Rydberg states and excites the atoms into a correlated state. The atoms then pass out of the cavity and into separate interaction regions where resonant microwave fields rotate their Bloch vectors through angles directly corresponding to the angles of orientation of the Stern-Gerlach magnets in the spin experiment. State selective ionization is then used to determine whether each atom is in the upper or lower Rydberg state. State selective ionization of Rydberg states has been demonstrated with an efficiency approaching 100% so that tests of Bell’s inequalities which are even more stringent than those using cascade photon polarization may be possible.
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.
In recent years the fulfillment of the inequalities established by Bell has been tested in different laboratories in the world. These inequalities make it possible to establish which of the interpretations of quantum mechanics is correct—the interpretation due to Einstein, according to which the properties of a quantum system exist as elements of physical reality irrespective of their observation, or the Copenhagen interpretation of Bohr and Fock, according to which the properties of a microscopic system described by noncommuting operators do not exist independently of the means of measurement. The experiments can be divided into three classes: experiments with optical photons, experiments with γ rays, and experiments with nucleons. The experiments that have been performed convincingly indicate violation of Bell's inequalities. Thus, the Copenhagen interpretation of quantum mechanics and its associated principle of relativity with respect to the means of measuring the properties of a microscopic system give at the present the only description of quantum phenomena that does not contradict experiment.
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We show that a micromaser can work as an effective source of highly correlated atoms. We consider a one-photon micromaser pumped by a Poissonian beam of atomic pairs. We show that the atoms forming the pairs leave the micromaser's cavity in entangled states and that they can violate the Bell inequality. We consider two aspects of the violation of the Bell inequality: we study the maximal value Bmax{\mathbf{B}}_{\mathrm{max}}Bmax​ of an expression appearing in the Bell inequality and we evaluate a vector-dependent $\mathbf{B}(\stackrel{\ensuremath{\rightarrow}}{a},{a}^{\ensuremath{'}},\stackrel{\ensuremath{\rightarrow}}{b},{b}^{\ensuremath{'}})$ depending upon experimental setup. We calculate the entanglement of the formation of states of the atomic pairs. We show that the pairs of atoms, which fly out from the micromaser cavity, are entangled for almost all values of control parameters characterizing the considered two-atom micromaser. This happens even if they do not violate the Bell inequality.