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\title{Quantum Entanglement}
\author{Emmanuel Billias}
\affiliation{State University of New York at Stony Brook}
%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy
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\maketitle
This Bell test experiment utilizes a process known as spontaneous parametric down conversion in order to form entangled photons. The entangled photons are passed through two adjustable polarizers in order to demonstrate a violation of the CHSH inequality which concludes that \(|S| \leq 2\). A brief history of the development of local hidden variable theory is provided which is followed by derivations of Bell's original inequality as well as the CHSH inequality. Examples of Bell test experiments are given leading up to this very experiment as a demonstration of the importance of thorough experimental physics and the avoidance of Bell test experiment loopholes. Visuals of the quantum Entanglement Demonstrator (quED) are provided along with an explination of its operations and use of non linear optical components. Experimentally, the CHSH inequality is violated wih a value of \(S = 2.670 \pm 0.006\) as measured by the quED. By our own data analysis we calculate \(S = 2.648 \pm 0.033\).
\section*{Introduction}
A fascinating result of quantum mechanics leads us to the idea of entangled particles. When measuring one of two entangled photons, the measurment gives us data about both particles. Measuring one particle tells us about the quantum state of the other particle instantaneously. This instantaneous transfer of information is a serious locality violation as proposed by Albert Einstein. As a result, hidden variable theory was developed. The idea of hidden variables formulated by Einstein, Podolsky, and Rosen (EPR) was meant to explain a quantum mechanical locality violation. \par
As we will explore in greater detail, quantum entanglement seems to violate the idea of locality. In his 1905 papers\cite{Einstein_1905,Einstein_1905a}, Einstein had proposed that information (or some form of physical influence) is bound to travel at or below the speed of light. In quantum entanglement if we know the original spin of the parent particle and one of the new spins, then we immediately know the spin of the second particle upon measurement of the first. It seems that we would receive information about the spin state of the second particle instantaneously! \par
The main concept of the EPR argument was that a "hidden variable" would account for this violation and show how a quantum system does in fact obey locality. John Bell had taken this idea of hidden variables and applied it to a system of entangled photons and polarizers. He formulated an inequality under the assumption that hidden variable theory is true. Through matheatics and experimentation we will show how this inequality is violated. It appears that the idea of hidden variables must also be false and that a quantum system does indeed violate locality.\par
Following Bell's inequaity comes a series of experiments in which we may test the inequality. From these tests follow the ideas of "loopholes". Loopholes in Bell test experiments provide a path in which we may question the validity of the results of these experiments. As an example, the aforementioned locality issue must be broken in order to show that the experiment is valid. For if information has been transfered in accordance with locality, then why preform the experiment at all? We will explore a recent Bell test experiment which suggests that the locality conditions are indeed violated.
~
\section*{Review of Preveious Work}
Two entangled particles must obey conservation of spin, therefore the total spin of the two entangled particles, \(A\) and \(B\), must add up to the spin of the original "parent particle". With a parent particle of spin 0, \(A\) and \(B\) will have spins of either \((+1,-1)\) or \((-1,+1)\) in obeying conservation. This phenomenon produced a particular dilemma to some physicists in that once a particle is observed, the spin state of other entangled particle could be determined instantenously. This was in violation of locality which states that no information can travel faster than the speed of light.
A group of physicists comprised of Einstein, Podolsky, and Rosen (EPR) attempted to solve this paradox \cite{Einstein_1935} by introducing a hidden variable, denoted as \(\lambda\), that may determine the spin state of each particle. However, the propostion eventually led to the finding that locality is not compatible with quantum mechanics.\par
John Bell had used this idea of hidden variables to formulate an explination of how entangled particles may behave. In a system of entangled photons and polarizers, Bell creates an inequality compatible with hidden variable theory. He then shows how is inequality may be violated. John Bell effectively takes the idea of hidden variables, assumes them to be true, applies them to an entangled system, and then breaks his own system. \par
For our particular experiment we will be showing a violation of the CHSH inequality.\cite{Clauser_1969} Although Bell produced his theory in 1964, it was not able to be tested until 1972 \cite{Freedman_1972} by two scientists named Freedman and Clauser. Even without the use of lasers or non linear optics, they were able to show that a violation of Bell's inequality is certainly possible. \par
It may be important to note that in our version of the experiment we will be dealing with entangled photons. However, Bell's inequality or local hidden variables are not strictly limited to photons. A paper published in 2001 shows that even with the use of heavy Beryllium atoms a violation of the inequality is still possible\cite{Rowe_2001}. This experiment also eludes several major arguments, or "loopholes", against the validity of Bell test experiments. For the first time, an experiment challenged the "detection loophole". The detection loophole works around statistics. It states that for a large number of coincidences, Bell's inequality will hold true. However, for a very small number of councidences the inequality may not always hold to be true. The use of heavy entangled Beryllium atoms allowed for the negation of error that will usually occur in the generalization of statistics for many small particles such as photons. This single experiment was able to cover several areas of concern of Bell test experiments. \par
A relatively famous experiment was carried out in 2015 which accounted for a second Bell loophole. The loophole tested is known as the "lightcone loophole". The idea here is that some data is able to be sent between entangled particles such that quantum mechanics does not violate locality. In this particular experiment, the two entangled electrons were separated by 1.25km\cite{Hensen_2015} in order to assure the locality conditions of Bell's hypothesis remain true. The experiment also avoids the aforementioned detection loophole. The experiment resulted in a violation of Bell's inequality as well as a evidence against the validity of several loopholes.
Less than two months after Hensen's experiment, a paper was published which put to test the "strong loophole"\cite{Shalm_2015}. This large collaberation of physicists explored the Clauser-Horne (CHSH) inequality. The researchers used the entanglement process of spontaneous parametric down conversion, just as we have done with the quED. Two detectors named Alice and Bob were separated at a spacelike distance from each other. Using single photon detection for 6 seperate tests, the researchers stated in their article that "We therefore reject the hypothesis that local realism governs our experiment". An important difference between this experiment and ours is that we are limited with the distance of our detectors using the quED.
\section*{Theoretical Model}
Consider two detectors, \(A\) and \(B\), oriented at angles \(a\) and \(b\), respectively. In the case of EPR, the detectors are set parallel which leads to \((s_{A},s_{B}) = (\pm1,\mp1)\) with \(s_{A}\) and \(s_{B}\) denoting the spin states of the particles. Bell considers a more generalized cases and compares the averaged product of the two particles spins, \(P(a,b)\). In the case when the detectors are not arranged in parrallel, it is possible to create a case with \((s_{A},s_{B}) = (\pm1,\pm1)\) which would produce \(P(a,b) = \pm1\). The local hidden variable theory proposes that there exists a hidden variable that determies the state of the particles in the system characterized by the variable \(\lambda\) acting on the results of the detector. This seperates the measurement of the spin state of one particle from the state of the other particle, thus preserving the locality. Let the measurement from each detector be characterized as \(A(\lambda,a)\) and \(B(\lambda,b)\) which can attain values of \(\pm1\). With a function accounting for the probability distribution of the hidden variable, \(\rho(\lambda)\), we may write the average value of the product for parallel detectors as:
\begin{equation} P(a,b) = \int \rho(\lambda)A(\lambda,a)B(\lambda,b)d\lambda \end{equation}
Recall that when the detectors are parallel, \(A(\lambda,a)=-B(\lambda,b)\) therefore we may write \(A(\lambda,b)=-B(\lambda,b)\).
\begin{equation} P(a,b) = -\int \rho(\lambda)A(\lambda,a)A(\lambda,b)d\lambda \end{equation}
If we were to consider a differen orientation of the detectors, say, angles \(a\) and \(c\) we will have:
\begin{equation} P(a,c) = -\int\rho(\lambda)A(\lambda,a)A(\lambda,c)d\lambda \end{equation}
Subtracting:
\begin{equation} P(a,b)-P(a,c)=-\int [\rho(\lambda)A(\lambda,a)A(\lambda,b)-\rho(\lambda)A(\lambda,a)A(\lambda,c)]d\lambda \end{equation}
If we consider that \(|A(\lambda,b)|^{2} = +1\) then we may rewrite equation 4 as:
\begin{equation} P(a,b) - P(a,c) = -\int\rho(\lambda)A(\lambda,a)A(\lambda,b)[1-A(\lambda,b)A(\lambda,c)]d\lambda \end{equation}
With \(A(\lambda,a)A(\lambda,b)\) taking a maximum value of +1, we may rewrite equation 5 as an inequality:
\begin{equation} P(a,b)-P(a,c) \leq -\int\rho(\lambda)[1-A(\lambda,b)A(\lambda,c)]d\lambda \end{equation}
Where \(\int \rho(\lambda)d\lambda =1\) and \(-\int\rho(\lambda)A(\lambda,b)A(\lambda,c)=P(b,c)\), therefore:
\begin{equation} P(a,b)-P(a,c) \leq 1 + P(b,c) \end{equation}
This is Bell's inequality. It is not all that difficult to violate if we orient the detectors correctly. Consider \(a \perp b \) and c is at a \(45^{\circ}\) angle between the two. From quantum mechanics we know that \(P(a,b) = 0\) and \(P(a,c)=P(b,c) = -1/ \sqrt{2}\). We find that \(2/\sqrt{2} \nleq 1\). \par
An implication of this is that the hidden variable theory is incompatible with quantum mechanics. Bell took local hidden variable theory, derived an inequality, then showed why it cannot be correct.\par
For this experiment, we will be proving a violation of the CHSH inequality which states that,
\begin{equation} |S| \leq 2 \end{equation}
We must now introduce the terms \(E(\alpha,\beta)\) which are known as the quantum correlation terms for the entangled particles. Defined as:
\begin{equation}
\begin{split}
E(\alpha, \beta) = P_{VV}(\alpha, \beta) + P_{VV}(\alpha_{\perp},\beta_{\perp})
\\
- P_{VV}(\alpha_{\perp},\beta) - P_{VV}(\alpha, \beta_{\perp})
\end{split}
\end{equation}
Note that \(\alpha_{\perp} = \alpha + \pi/2\) and \(\beta_{\perp} = \beta+\pi/2\).
To derive the CHSH inequality, we first assume that:
\begin{equation} E(\alpha, \beta) = \int \overline{A}(\alpha,\lambda) \overline{B}(\beta,\lambda) \rho(\lambda) d\lambda
\end{equation}
where \(\overline{A},\overline{B}\) denote the average probability of detection from polarizers A and B, respectively.
If the polarizers are rotated to some arbitary angle \(\alpha'\) and \(\beta'\):
\begin{equation}
E(\alpha,\beta)-E(\alpha,\beta') = \int [\overline{A}(\alpha,\lambda) \overline{B}(\beta,\lambda) - \overline{A}(\alpha,\lambda) \overline{B}(\beta',\lambda)]\rho(\lambda) d\lambda
\end{equation}
\begin{equation}
\begin{split}
E(\alpha,\beta)-E(\alpha,\beta') = \int \overline{A}(\alpha,\lambda)\overline{B}(\beta,\lambda)[1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta',\lambda)]
\\
- \overline{B}(\beta',\lambda)[1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta,\lambda)] \rho(\lambda) d\lambda
\end{split}
\end{equation}
Note that because of normalization, \(|A| < 1\) and \(|B| < 1\). Therefore, due to the triangle inequality:
\begin{equation}
\begin{split}
|E(\alpha,\beta)-E(\alpha,\beta')| \leq
\\
\int |\overline{A}(\alpha,\lambda) \overline{B}(\beta,\lambda)[1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta',\lambda)] \rho(\lambda)d\lambda|
\\
- \int |\overline{B}(\beta',\lambda)[1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta,\lambda)] \rho(\lambda)|d\lambda
\end{split}
\end{equation}
Since \([1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta',\lambda)], [1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta,\lambda)]>0\):
\begin{equation}
\begin{split}
|E(\alpha,\beta)-E(\alpha,\beta')| \leq
\\
\int [1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta',\lambda)] \rho(\lambda)d\lambda
\\
+ \int \overline{B}(\beta',\lambda)[1 \pm \overline{A}(\alpha',\lambda)\overline{B}(\beta,\lambda)] \rho(\lambda)
\\
= 2 \pm [E(\alpha',\beta')+E(\alpha',\beta)]
\end{split}
\end{equation}
Since S is defined to be:
\begin{equation} S = E(\alpha,\beta) + E(\alpha ',\beta) - E(\alpha,\beta ') + E(\alpha ',\beta ') \end{equation}
CHSH inequality can be established as:
\begin{equation} S \leq |E(\alpha,\beta)-E(\alpha,\beta')| + E(\alpha',\beta')+E(\alpha',\beta) \leq 2
\end{equation}
In the context of the experiment, the expected outcome of S can be re-expressed as
\begin{equation}
\begin{split}
S = P_{VV}(\alpha,\beta) + P_{VV}(\alpha_{\perp},\beta_{\perp})
\\
+ P_{VV}(\alpha_{\perp},\beta) - P_{VV}(\alpha,\beta_{\perp}) \leq 2
\end{split}
\end{equation}
With \(P_{VV}\) being the probability of simultaneous photon detection from two polarizers at angles \(\alpha\) and \(\beta\). Through the process of spontaneous parametric down conversion, we may create photons entangled by their polarizations, rather than spins. Consider diagonal bases of polarization. Two entangled photons may be described by the four possible Bell states:
\begin{equation} |\phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|HH\rangle \pm |VV\rangle)\end{equation}
\begin{equation} |\psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|HV\rangle \pm |VH\rangle) \end{equation}
With \(H\) and \(V\) being the horizontal \(0^{\circ}\) and vertical \(90^{\circ}\) bases. We wish to find the probability of two photons being detected simultaneously by the detectors set at angles\(\alpha\) and \(\beta\). As polarizers allow only orthogonally polarized photons to pass, the probability of simultaneous detection is written as:
\begin{equation} P_{VV}(\alpha,\beta) = |\langle V_{\alpha}| \langle V_{\beta}| |\psi_{ent}\rangle|^{2} \end{equation}
In the \(|H\rangle\) and \(|V\rangle\) bases, it is not difficult to see that at angle \(\alpha\):
\begin{equation} |H_{\alpha}\rangle = cos(\alpha)|H\rangle + sin(\alpha)|V\rangle \end{equation}
\begin{equation} |V_{\alpha}\rangle = -sin(\alpha)|H\rangle + cos(\alpha)|V\rangle \end{equation}
The same equations are easily modified for angle \(\beta\) by replacing all of the \(\alpha\)'s. For \(|\psi_{ent}\rangle = |\phi^{-}\rangle\), substitution into the probability yields:
\begin{equation} P_{VV}(\alpha,\beta) = \frac{1}{2}cos^{2}(\alpha + \beta) \end{equation}
So for a constant number of incoming photons, \(N_{0}\), we can write our coincidence count, \(N(\alpha, \beta)\), as:
\begin{equation} N(\alpha,\beta)=\frac{N_{0}}{2}cos^{2}(\alpha + \beta) \end{equation}
using the same method, the \(P_{VV}\) for each of the bell states are calculated to be:
\begin{equation} |\langle V_{\alpha}|\langle V_{\beta}| |\phi^{+}\rangle|^{2} = \frac{1}{2} cos^{2}(\alpha - \beta) \end{equation}
\begin{equation} |\langle V_{\alpha}|\langle V_{\beta}| |\psi^{+}\rangle|^{2} = \frac{1}{2} sin^{2}(\alpha + \beta) \end{equation}
\begin{equation} |\langle V_{\alpha}|\langle V_{\beta}| |\psi^{-}\rangle|^{2} = \frac{1}{2} sin^{2}(\alpha - \beta) \end{equation}
Assuming that \(P_{VV}(\alpha,\beta) = \frac{1}{2}cos^{2}(\alpha + \beta)\), the violation of CHSH inequality can be calculated by determining the local maxima of S by taking the derivative, dS, and finding the root. By trignometric identity:
\begin{equation} E(\alpha,\beta) = cos^{2}(\alpha+\beta)-sin^{2}(\alpha+\beta)=cos(2(\alpha+\beta))
\end{equation}
Then, the derivative of S is:
\begin{equation}
\begin{split}
dS = -4sin(2[\alpha+\beta]) - 4sin(2[\alpha'+\beta])
\\
+ 4sin(2[\alpha+\beta']) - 4sin(2[\alpha'+\beta'])
\end{split}
\end{equation}
Among the roots of dS, a notable case is observed when:
\begin{equation} \alpha = 0 , \beta = 22.5\degree, \alpha' = 45\degree, \beta' = 67.5\degree
\end{equation}
Plugging in the values to the equation for S results in the maximum S value since:
\begin{equation}
\begin{split}
\(E(\alpha,\beta) = \dfrac{1}{\sqrt{2}}
\\
E(\alpha',\beta') = \dfrac{1}{\sqrt{2}}
\\
E(\alpha',\beta) = \dfrac{1}{\sqrt{2}}
\\
E(\alpha,\beta') = \dfrac{-1}{\sqrt{2}} \)
\end{split}
\end{equation}
Therefore:
\begin{equation}
S_{max} = \dfrac{4}{\sqrt{2}} = 2\sqrt{2}
\end{equation}
\section*{Experimental Setup}
In order to create entangled photons, the quantum Entanglement Demonstrator (quED) utilizes a process known as spontaneous parametric down conversion\cite{Burnham_1970} or SPDC. A photon from a laser pump directed through a non linear crystal has a very low, but non negligable, probability of being spontaneously converted into two low frequency entangled photons. Energy and momentum conservation laws then become responsible for the wavelengths and directions of the newly emitted photons. In type-I phase matching the emitted photons are projected in the shape of concentric cones.\cite{2016}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Screenshot-20170419-1/Screenshot-20170419-1}
\caption{{Emitted photons can be mapped as concentric rings on a plane P which is
perpendicular to the laser beam pump.
{\label{981200}}{\label{981200}}%
}}
\end{center}
\end{figure}
The cone size is dependant on the emitted wavelength and on the angle between the laser beam and the optical axis, \(\Theta_{p}\). \par
The quED is also used to obtain polarized entanglment through the process of SPDC. Two non linear crystals of type-I phase matching are overlapped. Their optical axises are also mutually perpendicular to one another, one vertical and one horizontal. So by pumping light with a linear polarization of \(45^{\circ}\), there is an equal probability of a photon being spontaneously converted in either crystal. Without knowing which crystal a pair of photons were originally down coverted in, the following entangled state is:
\begin{equation} |\Phi \rangle = \frac{1}{\sqrt{2}}[|H\rangle_{1}|H\rangle_{2}+e^{i\phi}|V\rangle_{1}|V\rangle_{2}] \end{equation}
where the relative phase \(\phi\) can be adjusted by changing the horizontal and vertical components of the light pump. \par
The use of thin crystals is paramount to the function of the quED. With thin crystals, almost all of the spacial information that the photon carried before detection is effectively lost. We could never know in which crystal the photon experienced down conversion. This results in "pure polarization-entangled photon pairs"\cite{2016}. This removes their spacial distinguishability. \par
A second issue with the crystals is a small time delay. The arrival times of non-degenerate photons is predictable due to physical properties of the crystals themselves. The photons down converted in the first crystal are advanced relative to those of the second. Additionally, the photons from the first crystal experience a higher dispersive delay because they must also travel through the second crystal. To counter these effects, the quED has in place two additional birefringent crystals in order to remove their temporal distinguishability. This crystal may also be used to adjust the relative phase of the entangled photons.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Screenshot-20170421-103029/Screenshot-20170421-103029}
\caption{{Non-degenerate photons \(\lambda_{1} \neq \lambda_{2}\) have time delays
where~\(\tau^{1}_{+}\) \textless{}~\(\tau^{2}_{+}\). They also
experience a dispursive delay where~\(\tau^{1}_{-}\) \textgreater{}
\(\tau^{2}_{-}\).
{\label{856833}}%
}}
\end{center}
\end{figure}
The quED setup is labled in figure \ref{974464}. The laser diode head (1) is comrpised of the necessary blue laser diode with an aspheric lense. A mirror (2) sends the laser through a variety of beam shaping optics (3a is the collimating aspheric lense and 3b is the negative spherical lense) then through a half-wave retarder (4). The half-wave retarder is used to adjust the linear polarization of light to be a relative \(45^_{\circ}\) to the horizontal and vertical basises of the SPDC crystals (6).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Screenshot-20170421-105339/Screenshot-20170421-105339}
\caption{{quED set up labled for the convenience of a descriptive analysis.
{\label{974464}}%
}}
\end{center}
\end{figure}
The beam is then directed to the birefringent crystals (5) then through an additional cylindrical lense (3c) as compensation for the elliptical shape of the beam. The polarized beam then passes through the SDC crystals (6).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/Screenshot-20170421-112505/Screenshot-20170421-112505}
\caption{{Engineering schematic of the quED.
{\label{535443}}%
}}
\end{center}
\end{figure}
The entangled photons are each sent to a mirror (7) and through an iris diaphragm (8). The adjustable polarizers (9) are our primary experimental tools of this experiment, discussed in the Theoretical Model section. Long pass filters (10) are used in order to block any residual light from entering into the optical collimator (11). Finally, the down-converted photons enter a single mode fibre (12) which is used to send information to the quED electronic unit.
\section*{Measurements}
To experimentally measure the probability of detecting entangled particles, a BBO crystal is set up so that it produces two photons in the vertical polarization. Each photon is passed through one of the two polarizers (A or B). The detector records any two photons detected within a time frame as one coincidence count.
In addition, the number of detected photons that passes the polarizer A is called single 0 and the number of photons that passes through the polarizer B is called single 1. A sample plot of single 1 (S1) and single 0 (S0) shows a relatively constant S0 and a periodic function on S1.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/coincidence-alpha-1/coincidence-alpha-1}
\caption{{The plot of single counts vs. beta degree. During the measurement the
alpha angle is set to 0. The plot of ~single 1 is proportional
to~\(-\cos^{ }\theta\)
{\label{112916}}%
}}
\end{center}
\end{figure}
The stability of S0 is due to the constant configuration of \(\alpha\) while the periodic behavior of S1 is due to the polarizer allowing photons that are parallel and anti parallel to its polarization to pass.
The coincidence count with fixed \(\alpha\) and varying \(\beta\) is plotted and fit to sine squared and cosine squared curve. As the data point at \(\beta = 0\) is a peak, the coincidence count is determined to be related to cosine sqaured.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/coincidence-alpha-0/coincidence-alpha-0}
\caption{{Polarizer~\(\alpha\) set to 0 degrees and
polarizer~\(\beta\) varying with 10 degree increments
exhibiting a cosine squared function.
{\label{951093}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Coincidence-alpha-45/Coincidence-alpha-45}
\caption{{Coincidence plot with polarizer A at 45 degree. Compared to the previous
graph, the cosine graph is shifted to left.~
{\label{893425}}%
}}
\end{center}
\end{figure}
Varying \(\alpha\) shifts the cosine squared curve to the left. Therefore, the sign of \(\alpha\) must be positive. As a result, the entangled state is determined to be \(\phi^{-}\).
Then, \(\alpha\) and \(\beta\) are set to be equal as the angles varied from 0 to \(360\degree\) in the increment of \(10 \degree\). The result exhibits two periodic function.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/a-=-B/a-=-B}
\end{center}
\end{figure}
The behavior of both the S0 and S1 are remarkably similar to the cosine wave in that the maximum is acheived at 0 degree and the minimum at 90 degree. The counts for S0 and S1 are also simliar in magnitude. The behavior of the single counts are similar due to the identical contribution to probability of detection as indicated by the entangled state of \(\phi^{-}\).
The coincidence count when \(\alpha = \beta\) is generally higher than the coincidence counts for constant \(\alpha\). The highest coincidecne count when \(\alpha\) is kept at constant is exhibited when \(\alpha\) is equal to \(315^{\circ}\). The peak at the \(\alpha\) is around 23328 counts while the peak when \(\alpha = \beta\) is around 29729 counts. The minimum in the case of \(\alpha = \beta\) is also generally higher at 17026 counts while the minimum for constant \(\alpha\) is aronud 674 count, a negligible amount compared to 17000. At all angles, the coincidence count is higher when
\(\alpha = \beta\).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/coincidence-comparison/coincidence-comparison}
\caption{{Comparison of data for when the polarizers were adjusted and kept equal
as well as polarizer~\(\alpha\) being fixed at 315 degrees with
a variable polarizer~\(\beta\)
{\label{563364}}%
}}
\end{center}
\end{figure}
Polarizer \(\alpha\) was removed and \(\beta\) was varied from \(0 \degree\) to \(360 \degree\). Consider figure \ref{722469}. The S1 count shows a constant value with much higher count than with polarizer \(\alpha\). Since the removal of the polarizer does not filter photons with orthogonal polarization, there are more photons detected than with polarizer.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/no-A-single/no-A-single}
\caption{{Polarizer~\(\alpha\) was removed entirely with
polarizer~\(\beta\) adjusted as before.
{\label{722469}}%
}}
\end{center}
\end{figure}
The coincidence count, CO1, is identical to single count S1, since single 0 is always detected without the polarizer. Then, there is a coincidence whenever S1 is detected.
The experimental measurment of S was made by measuring the coincidence count for each basis: H, V, +, and -. Since the basis indicate the values of \(\alpha\) for S to be maximum, it is possible to demonstrate the violation of CHSH by setting \(\beta\) accordingly. The table * shows the measured value and the measured S value is \(2.670 \pm 0.006\).
\begin{tabular}{ccccc}
{$ $}&{$\beta$}&{$\beta_{\perp}$}&{$\beta^{\prime}$}&{$\beta^{\prime}_{\perp}$} \\ \midrule
$H$ & 35916 & 5699 & 7659 & 34748\\
$V$ & 3611 & 26871 & 24264 & 5839 \\
$+$ &30210 & 4796 & 27675 & 7735 \\
$-$ & 10102 & 27720 & 3439 & 32614 \\
\end{tabular}
\section*{Comparison of Data and Theoretical Model}
\subsection*{Visibility}
The visibility for a fixed \(\alpha\) is defined to be
\begin{equation}
V = \dfrac{V_{max}-V_{min}}{V_{max}+V_{min}}*100 \%
\end{equation}
where \(V_{max}\) denotes the maximum peak in coincidence and \(V_{min}\) represents the minimum peak of the coincidence. Several measurments of coincidence counts were made at \(\alpha = 0, 45\degree, 90\degree,\) and \(135\degree \).
As the coincidence counts are statistically independent, the uncertainty of each coincidence counts are determined to be \(\sqrt{C01}\) by the poisson distribution. Then, the uncertainty for visibility is determined by the error propogation for multiplcation.
\begin{equation}
\Delta V = V * \sqrt{(\dfrac{\sqrt{V_{max}}}{V_{max}})^{2}+(\dfrac{\sqrt{V_{min}}}{V_{min}})^{2}}
\end{equation}
\begin{tabular}{ccccc}
{$\alpha$}&{$V_{max}$}&{$V_{min}$}&{$Visibility$}&{$\Delta V$} \\ \midrule
0 & 6734 & 620 & 83\% & 3.5 \% \\
45 & 11257 & 689 & 88\% & 3.5 \% \\
90 & 16576 & 334 & 96\% & 5.3 \% \\
135 & 45249 & 800 & 97\% & 3.5 \% \\
\end{tabular}
Therefore, the average of the calculated visibility is 91\% \pm 3.95\%
In the experiment, the visibility was measured by setting the polarizers first to be in parallel arrangement and then be in orthogonal arrangement for the measurement of maximum and minimum coincidence count. The measured value of raw visibility is 94.9\% and the corrected visibility is 99.3\%. The raw visibility is consistent with the calculated visibility within the uncertainty.
\subsection*{Violation of CHSH}
The maximum value of S can be measured from the experimentally measured values of E(\alpha,\beta), E(\alpha ',\beta), E(\alpha,\beta '), E(\alpha ',\beta ').
From the eq.9, the expectation value, E(\alpha,\beta), can be expreseed as
\begin{equation}
E(\alpha,\beta) = \dfrac{C(\alpha,\beta) + C(\alpha_{\perp}, \beta_{\perp}) - C(\alpha_{\perp}, \beta) - C(\alpha, \beta_{\perp})}{C(\alpha,\beta) + C(\alpha_{\perp}, \beta_{\perp}) + C(\alpha_{\perp}, \beta) + C(\alpha, \beta_{\perp})}
\end{equation}
The terms in denominator is included in order to take account for normalization. The calculated values of Es for \(\alpha = 0, \beta = 22.5 \degree, \alpha'= 45 \degree,\) and \(\beta' = 67.5 \degree\) are: \(E(\alpha, \beta) = 0.74174, E(\alpha, \beta^{\prime}) = -0.62769, E(\alpha^{\prime}, \beta) = 0.59087,\) and \(E(\alpha^{\prime}, \beta^{\prime}) = 0.68728 \).
The uncertainty for expectation value can be determined by applying the gaussian error propogation for multiplication of the numerator and denominator that depends on coincidence counts. The uncertainty of the numerator and the denominator is the sum of individual uncertainties for each coincidence count, \(\Delta CO = \sqrt{CO}\).
\begin{tabular}{ccc}
{$ $}&{$E$}&{$\Delta E$} \\ \midrule \\
$\alpha,\beta$ & 0.74174 & \pm 0.00844 \\
$\alpha,\beta^{\prime}$ & -0.62769 & \pm 0.00816 \\
$\alpha^{\prime}, \beta$ & 0.59087 & \pm 0.00813 \\
$\alpha^{\prime}, \beta^{\prime}$ & 0.68728 & \pm 0.00838
\end{tabular}
With all values of E known, S can be calculated from the eq. 15. As S only contains additions, the uncertainty of S is sum of the uncertainties for each E. Thus, the calculated value of S is \(2.648 \pm 0.033\) which is consistent with the measured S value, 2.67.
\section*{Discussion and Conclusion}
The result obtained from the experiment indicates the S value of \(2.648 \pm 0.033\) which is smaller but close to the theoretical value of the maximum S, \(2\sqrt{2} \approx 2.83\). The smaller value of S may be due to the inaccurate arrangement of \(\alpha\) and \(\beta\) caused by the wide distance between the markings of polarizer angle which was marked every \(20 \degree\). As a result, it was particularly challenging to accuratly place the polariers to angle such as \(67.5 \degree\).
It may be well to also note that our calulated \(S\) value of \(2.648 \pm 0.033\) does in fact fall within range of the quED's calulated \(S\) value of \(2.670 \pm 0.006\).
Nevertheless, this finding experimently confirms that the CHSH inquality does not hold true and disproves the viability of the local hidden variable as a explanation of quantum mechanical systems. In turn, it is implied that locality must also be violated in quantum mechanical systems. In other words, it could be possible for a particle to exchange information with another particle within a small distance instantanously, which could possibly explain the discontinuity of the quantum state. Although some inperpretation has risen to evade the result of Bell's Theorem, the non-locality of the quantum system has come to be accepted in most modern interpretation of quantum mechanics.
As a point of discussion, the comparison of the quED with other experimental set ups may be beneficial. Most similarly, the quED is comprable to the December 2015, Shalm experiment \cite{Shalm_2015} mentioned earlier. Both of these experiments created entangled photon pairs through the process of SPDC. The photons were then sent through a variety of optical equipment and eventually to the detectors. As shown in the schematic in figure \ref{535443}, our detectors were only a few inches apart. In the Shalm experiment, the detectors were over 100 meters apart. The additional 100 meters of separation was to ensure spacelike separation. We cannot boast that separation with the quED. In the experimental setup section we had explored a possible solution to remove spacial and temporal distinguishability from the photons. We may conclude that these separations did indeed work well. With the results from the Shalm experiment, we may \(\textit{infer}\) that our experiment does not abide by local realism as well.
As for the other experiments mentioned it would be difficult to draw conclusions about the quED's ability in itself to avoid certain loopholes. To have spacelike distances separating the detectors is simply not possilble with the quED. The detection loophole cannot be thoroughly tested unless some serious engineering adjustments were made to the device. We are limited to the use of photons in a compact space. Even so, it seems that the disproof of certain proposed loopholes over five decades of experimentation may be enough for us to trust in the abilities of the quED. We may presume it to have the relatively high accuracy that we have both measured and calculated.
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