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European Journal of Physics

Contrasting classical probability concepts with quantum mechanical behavior in the undergraduate laboratory

David Guzmán, Leonardo J. Uribe, Alejandra Valencia, Ferney J. Rodríguez, Luis Quiroga

Published Aug 6, 2015• 4 citations
Publisher

Abstract

Abstract In this paper, we report an experimental activity suitable for undergraduate students that makes them question their ideas about locality and realism. The experiment here reported complements previous educational approaches to Bell inequalities, since the usually called S function, that quantifies correlations, is mapped by measuring it for different detection angles. The students themselves worked in a pre-aligned setup that allowed them to test and violate the Clauser–Horne–Shimony–Holt–Bell inequality for two distant photons entangled in polarization.

DOI
10.1088/0143-0807/36/5/055039

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Abstract In this paper, we report an experimental activity suitable for undergraduate students that makes them question their ideas about locality and realism. The experiment here reported complements previous educational approaches to Bell inequalities, since the usually called S function, that quantifies correlations, is mapped by measuring it for different detection angles. The students themselves worked in a pre-aligned setup that allowed them to test and violate the Clauser–Horne–Shimony–Holt–Bell inequality for two distant photons entangled in polarization.
In 1964, J. Bell introduced an inequality that stated a mathematical bound for any physical system that holds both locality and realism; if we violate this inequality, it is clear that we have to reconsider the previous statement. In our work, we report an experimental activity with photons suitable for undergraduate students that makes them question these naïve ideas of nature’s behavior. With a pre-aligned setup, our students tested and violated Bell’s inequality in a two-hour laboratory session, using two distant photons entangled in polarization. In addition, complementing an educational approach to this phenomenon, the usually called S function, that quantifies correlations, was mapped using different detection angles in one of the two locations. In particular, a more complete picture of the S function, allow us to identify the initial state of light. We show in this work that it is possible for undergraduate students to question some of our common sense ideas of nature using experiments with photons.
We demonstrate hybrid entanglement of photon pairs via the experimental violation of a Bell inequality with two different degrees of freedom (DOF), namely, the path (linear momentum) of one photon and the polarization of the other photon. Hybrid entangled photon pairs are created by spontaneous parametric down conversion and coherent polarization to path conversion for one photon. For that photon, path superposition is analyzed, and polarization superposition for its twin photon. The correlations between these two measurements give an SSS parameter of $S=2.653\ifmmode\pm\else\textpm\fi{}0.027$ in a Clauser-Horne-Shimony-Holt inequality and thus violate local realism for two different DOF by more than 24 standard deviations. This experimentally supports the idea that entanglement is a fundamental concept which is indifferent to the specific physical realization of Hilbert space.
The assumption of local realism imposes constraints, such as Bell inequalities, on quantities obtained from measurements. In recent years, various tests of local realism have gained popularity in undergraduate laboratories, giving students the exciting opportunity to experimentally contradict this philosophical assumption. The standard test of the CHSH (Clauser–Horne–Shimony–Holt) Bell inequality requires 16 measurements, whereas a test of Freedman's inequality requires only three measurements. The calculations required to test Freedman's inequality are correspondingly simpler and the theory is less abstract. We suggest that students may benefit from testing Freedman's inequality before proceeding to the CHSH inequality and other more complicated experiments. Our measured data violated Freedman's inequality by more than six standard deviations.
Abstract We report experiments in which the Bell parameter S that enters the Clauser–Horne–Shimony–Holt inequality: ∣ S ∣ ≤ 2 , attains values ∣ S ∣ > 2 . In our experiments, we used two spatially separated optical beams, the electric fields of which were correlated to one another. The amount of correlation was quantified by the spectral degree of coherence η ( α , β ) . This quantity measures the correlation between fields that exist at two distant locations and whose respective polarizations are given in terms of angles α and β , which can be set independently from one another. Such a correlation qualifies for the construction of the Bell parameter S . By changing the amount of field correlation, we could cover a range that goes from ∣ S ∣ < 2 to ∣ S ∣ > 2 . Our experimental findings should provide useful material for the ongoing, theoretical discussions about the quantum–classical border.
We present a violation of the Clauser–Horne–Shimony–Holt and the Clauser–Horne inequalities using heralded single photons entangled in momentum and polarization modes. A Mach–Zehnder interferometer and polarization optics are used to rotate the spatial and polarization bases, respectively. With this setup we were able to test quantum mechanics with the original formulation of the Clauser–Horne inequality. The results rule out a wide class of realistic non-contextual hidden-variable theories.
An entangled state of a two-particle system is a quantum state that cannot be separated, meaning it cannot be written as the product of states of the individual particles. One way to tell if a system is entangled is to use it to violate a Bell inequality (such as the Clauser-Horne-Shimony-Holt, CHSH, inequality), because entanglement is necessary for such a violation. However, there are other, easier-to-perform measurements that determine whether or not a system is entangled. An operator that corresponds to such a measurement is referred to as an entanglement witness. Here, we present the theory of witness operators and an undergraduate experiment that measures entanglement witnesses for the joint polarization state of two photons. We are able to produce states for which the expectation value of a witness operator is entangled by more than 300 standard deviations. In order to further examine the performance of these witness operators, we present a simple way to generate states that closely approximate Werner states, which have a controllable degree of entanglement.