Entangled Photons and Bell’s Inequality

Christopher Marsh Entanglement is a phenomenon where two particles are linked by some sort of characteristic. A particle such as an electron can be entangled by its spin. Photons can be entangled through its polarization. The aim of this lab is to generate and detect photon entanglement. This was accomplished by subjecting an incident beam to spontaneous parametric down conversion, a process where one photon produces two daughter polarization entangled photons. Entangled photons were sent to polarizers which were placed in front of two avalanche photodiodes; by changing the angles of these polarizers we observed how the orientation of the polarizers was linked to the number of photons coincident on the photodiodes. We dabbled into how aligning and misaligning a phase correcting quartz plate affected data. We also set the polarizers to specific angles to have the maximum S value for the Clauser-Horne-Shimony-Holt inequality. This inequality states that S is no greater than 2 for a system obeying classical physics. In our experiment an S value of 2.5 ± 0.1 was calculated and therefore in violation of classical mechanics. Introduction – Graham Jensen We report on an effort to verify quantum nonlocality through a violation of Bell’s inequality using polarization-entangled photons. When light is directed through a type 1 beta barium borate (BBO) crystal, a small fraction of the incident photons (on the order of 10−10) undergo spontaneous parametric down conversion. In spontaneous parametric down conversion, a single pump photon is split into two new photons called the signal and idler photons. In order to maintain a conservation of energy and momentum, these photons will have longer wavelengths than the pump, i.e., less energy, and have paths deflected at an angle relative to the pump photon propagation direction. Another result of spontaneous parametric down conversion is that the signal and idler photons will be entangled in polarization: if the polarization state of one photon is measured, their entangled wavefunction collapses and the state of the other is known with absolute certainty, no matter the distance between them [1]. Entanglement in polarization can be experimentally measured through a violation of Bell’s inequality. Using polarization-entangled photons, an experiment to violate Bell’s inequality is performed by measuring the coincidence of entangled photons as a pair of polarizers in their paths are adjusted though various relative polarization differences. If the down converted photons behave as predicted by a supplemental hidden variable theory, their coincidence counts will not result in a violation of Bell’s inequality. Conversely, if the down-converted photons follow the predictions of quantum mechanics, their coincidence counts will show a contradiction with any hidden variable theory by violating Bell’s inequality. Theory – Graham Jensen The notion of quantum entanglement was first proposed by Einstein, Podolsky, and Rosen in 1935. The group proposed a paradox (known as the EPR paradox) which was designed to show a flaw in the current theory of quantum mechanics. David Bohm proposed a simplified version of this situation where a pi meson decays down to an electron and positron. Quantum theory predicts that if the spin of one of the particles is measured the other must be the opposite, no matter the distance between them. Einstein believed this “spooky action at a distance” is in violation of the theory of relativity, citing the impossibility of faster than light information travel. His conclusion was that quantum mechanics is an incomplete theory and there lies a hidden variable that was yet to be discovered [2]. In the years following, several hidden variable theories were proposed to supplement quantum mechanics. However, these theories were dismissed in 1964 when John Bell mathematically proved that any hidden variable theory is incompatible with quantum mechanics [2]. In Bell’s original paper, particles entangled in spin are considered, but, following the generalized formulation by Clauser, Horne, Shimony, and Holt, the same principle can be applied to photons entangled in polarization [2-4]. We may consider the situation seen in Figure 1. A pump photon undergoes spontaneous parametric down conversion and produces a pair of photons (named signal and idler) with paths leading to polarizers with polarizations of αα and ββ respectively. Fig. 1. A sketch of the proposed situation. An unpolarized pump photon interacts with a type 1 BBO crystal and undergoes spontaneous parametric downconversion (SPDC). As a result of this process, two photons of identical polarization are produced: a signal photon and an idler photon. These photons are directed in separate paths towards the polarizers. If we assume that the realist interpretation of quantum mechanics is true and apply a hidden variable theory, then the outcomes of measurements at each polarizer are independent of each other and are determined only by the angle of their own respective polarizers and a hidden variable, λλ. In order to satisfy any hidden variable theory, λλ will be described by the distribution ρρ(λλ) with the conditions that ρρ(λλ) ≥ 0 and∫ρρ(λλ)ddλλ = 1. The detection of vertically and horizontally polarized photons with αα and ββ bases may be expressed as AA(αα, λλ) = ±1 BB(ββ, λλ) = ±1 respectively. The probability of measuring each detectable outcome is given by the following set out integrals: PPVVVV(αα,ββ) = � 1 + AA(αα, λλ)