In 1964, physicist John Bell proposed that quantum entanglement could be demonstrated by separating the particles at a great enough distance that any correlating effect on both particles could not possibly be caused by local environmental factors. These were called the Bell Test experiments. For these experiments Bell derived an inequality. If this inequality is violated it would imply that the entangled particles would correlate in a way that cannot be explained by hidden variables.
We have performed a Bell test experiment to study whether we can violate Bell’s inequality. To achieve this we have created entangled photon pairs that both passed through a different polarization filter and were captured by two single-photon detectors. The data was collected by a quED control unit from where the polarization of the entangled photon pairs can be analyzed. Using this data we can test Bell’s inequality $-2 leq S leq 2$ and found $S = 2.51 pm 0.03$, where $S$ quantifies the correlation between measurement outcomes. This implies that Bell’s inequality is violated which means that the entangled particles correlate in a way that cannot be explained by hidden variables. There is, however, a loophole in this experiment. This loophole is called the “communication” or “locality” loophole, where the particles could be communicating via some hidden means at light-speed. To overcome this loophole, a same sort experiment should be performed over a much larger distance (in the order of $10^3$ meters) so that the particles couldn’t have been able to communicate with each other.
end{abstract}
newpage
tableofcontents % Include a table of contents
newpage
addcontentsline{toc}{section}{List of symbols}
section*{List of symbols}
hfill\
begin{center}
begin{tabular*}{textwidth}{l@{extracolsep{fill}}lll}
textbf{symbol}& tectextbf{Description} & textbf{Units}\
hline
\
$mathbf{E}$ & Electric field & $frac{V}{m}$\
$mathbf{B}$ & Magnetic Field & $T$\
$alpha, beta, phi$ & angles & $circ$\
$P$ & probability & – \
$S$ & parameter from Bell’s inequality & – \
$E$ & correlator from Bell’s inequailty & – \
$ket{psi}, ket{Psi},ket{Phi}$ & Quantum states & – \
$A, B$ & are arbitrary real numbers & – \
end{tabular*}
end{center}
newpage
section{Introduction}
label{sec:introduction}
Quantum entanglement is a physical phenomenon in quantum mechanics where two or more particles interact in such a way that the quantum state of each particle cannot be described independently from the state of the others, even though the particles are separated from each other. Instead the quantum state must be described for the system as a whole. Einstein did not accept the theory of quantum entanglement, as he called it “spooky action at a distance”. Albert Einstein, Boris Podolsky and Nathan Rosen wrote a paper on this subject in 1935 cite{EPR}, in which they concluded that the quantum-mechanical description of physical reality is not complete. In 1964, physicist John Bell proposed that quantum entanglement could be demonstrated by separating the particles at a great enough distance that any correlating effect on both particles could not possibly be caused by local environmental factors cite{Bell1964}. These were called the Bell Test experiments. For these experiments Bell derived an inequality. If this inequality is violated it would imply that the entangled particles would correlate in a way that cannot be explained by hidden variables. In 2015 Ronald Hanson and his team where the first to perform a loophole free Bell test experiment that violated Bell’s inequality thus indicates rejection of all hidden variable theories cite{hensen2015}.\
We have performed a Bell test experiment to study whether we can violate Bell’s inequality. To achieve this we have created entangled photon pairs that both passed through a different polarization filter and were captured by two single-photon detectors.\
In section ref{sec:Theory} of this report, the theory used for this research is explained. In section ref{sec:setup} is a description of the experimental set up. Section ref{sec:results} shows the result and discussion. In section ref{sec:conclusion} a conclusion of the research is stated. This research is part of the course Research Practicum, which is part of the Bachelor Applied Physics at the Delft University of Technology.
newpage
section{Theory}
label{sec:Theory}
subsection{Bell’s Theorem}
label{sec:Bell’s Theorem}
According to quantum mechanics two experiments that have exactly the same set-up, can still result in different results. For example, when a vertically polarized photon is emitted through a polarization filter that has an angle of 45degree with the vertical axis. According to this experiment there is a 50% chance the light particle passes through the filter and a 50% chance the light will be absorbed by the filter. However, it’s not possible to predict the faith of an independent photon, because all photons are absolutely identical before they pass the filter. They have no distinctive property that determines whether they pass the filter or not. In the early days of quantum mechanics Einstein, Podolsky and Rosen thought that the theory must have been incomplete. They concluded that the result of the experiment must be predetermined by until then unknown “hidden variables”.\\
In the sixties, the physicist John Bell derived an inequality called Bell’s Inequality. This was an inequality that must be satisfied if the result of the experiment where actually predetermined by hidden variables cite{Bell1964}. In 1982 the physicist Alain Aspect performed an experiment that strongly indicated that the theory of hidden variables can be rejected cite{Aspect1982}.
subsection{EPR-Paradox} % Sub-section
In 1935 Einstein, Podolsky and Rosen wrote an article in which they questioned the correctness and completeness of current theory about quantum mechanics. They did this by postulating two conditions, one was the condition of completeness:
begin{quote}
textit{“Every element of the physical reality must have a counterpart in the physical theory” cite{EPR}.}
end{quote}
The second one was the condition of physical reality:
begin{quote}
textit{“If without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity” cite{EPR}.}
end{quote}
The conclusion of this article is that two physical quantities, with non-commuting operators can have simultaneous reality, while the current theory stated that if two operators corresponding to two physical quantities do not commute, than the precise knowledge of one of them precludes such knowledge from the other. The assumption on which the EPR-paradox relies is that two systems can influence each other faster than the speed of light. This is called the principle of locality cite{REL}. To solve the paradox the authors assumed that there had to be some kind of hidden variable and that the quantum theory was not complete.\
Later Bohr stated that the classical view on reality used in the EPR-paradox was not sufficient to tear down the quantum theory and that with new physical theories, also new ways of defining reality have to arise cite{EPRB}.
subsection{Polarization Light Waves} % Sub-sub-section
label{sec:Polarization Light Waves}
The classical theory about electromagnetics tells us that light waves are electromagnetic waves and therefore transverse. This implies that the displacement of the wave will be perpendicular to the direction of propagation cite{GEM}. The direction of propagation can be described as a single vector ($ hat{n}$) in classical physics. The displacement of the electromagnetic waves, the electric field $mathbf{E}$ and the magnetic field $mathbf{B}$ have to be perpendicular to each other as well as to the direction of propagation so:
begin{equation} label{vect}
mathbf{E} perp mathbf{B} perp hat{n} perp mathbf{E}.
end{equation}
%
Equation (ref{vect}) together with the algebraic identity that n vectors from $ mathbb{R}^n $ with $mathbf{u_1} perp mathbf{u_2} ldots perp mathbf{u_n}$ will construct $ mathbb{R}^n $. There can be concluded that when an arbitrary $ hat{n}$ from $ mathbb{R}^3 $ is given, there are infinitely many ways to construct $mathbf{E}$ and $mathbf{B}$. This means that the orientation of the light wave can be in any arbitrary direction as long as equation (ref{vect}) is satisfied. By polarizing a light wave, a certain type of orientation will be extracted from the wave. Therefore, when this light passes a polarization filter it will oscillate in only one direction, depending on the direction of the filter. The remaining light is then polarized. An example of the polarization of a light wave is shown in figure~ref{fig:wave} and figure~ref{fig:pol}.
begin{figure}[h] % Example image
center{includegraphics[width=0.8linewidth]{LiWave}}
caption{Unpolarized light wave traveling in the $hat{z}$-direction cite{GEM}}.
label{fig:wave}
end{figure}
begin{figure}[h] % Example image
center{includegraphics[width=0.8linewidth]{HorVerPol}}
caption{Polarized light waves left: in the vertical direction
right: in the horizontal direction cite{GEM}}.
label{fig:pol}
end{figure}\
The intensity of a polarized light is then half of the original intensity, i.e. half of the photons will pass through the filter, while the other half will be absorbed by the filter. If a second polarization filter is placed behind the first one, with an angle of $90degree$ relative to the first one, all the photons will be absorbed. In general if the second filter makes an angle of $phi$ with respect to the first one, the chance that a photon will pass through both filters will be equal to $cos^2phi$ cite{GEM}.
subsection{Entanglement}
label{sec:Entanglement}
Entangled photons are two photons that are in exactly the same quantum state. It’s possible to create such a pair and shoot them in opposite positions. If two polarization filters are placed on both sides, orientated parallel to each other, so both photons would propagate through one of the filters, then the photons will either both pass through the filter or both be absorbed by the filter. However, if the filters are placed with perpendicular orientation, relative to each other, one of the photons will pass through the filter, while the other one is absorbed. There is no way of knowing if the photon will pass through the filter or will be absorbed on forehand. If the filters are placed with a relative angle of $phi$ with respect to each other, the chance that both photons will either pass or be absorbed by the filter is equal to $cos^2phi$, and the chance that the photons have an opposite fate is equal to $sin^2phi$ cite{bengtsson2017}. To ease this problem, lets say that the photons are either vertically or horizontally polarized, and designate the polarization states as $ket{1}$ and $ket{0}$, respectively, to it. Then the correlated photon pairs have the following properties:
begin{enumerate}
item The polarization of either photon 1 or photon 2 measured independently of the other is random
item The polarization of the pair of photons is perfectly correlated; that is, if the first detector measures $ket{0}$, then the second detector will always measure $ket{0}$, and if the first detector measures $ket{1}$, the second detector always measures $ket{1}$. Alternatively if detector 1 measures $ket{0}$, then the detector 2 always measures $ket{1}$, and vice versa.
end{enumerate}
A multi-particle system is described as being in an entangled state if its wave function cannot be factorized into a product of the wave functions of the individual particles. The mutual dependence of the results of the polarization measurements on the correlated photon pair means that the wave function has to be written in the form:
begin{equation}
label{eq:waveposcor}
ket{Phi^pm} = frac{1}{sqrt{2}} (ket{0_1,0_2} pm ket{1_1,1_2}),
end{equation}
for the case of perfect positive correlation, and:
begin{equation}
label{eq:wavenegcor}
ket{Psi^pm} = frac{1}{sqrt{2}} (ket{0_1,1_2} pm ket{1_1,0_2}),
end{equation}
for perfect negative correlation, with the subscripts referring to the individual photons. The wave function in equations eqref{eq:waveposcor} and eqref{eq:wavenegcor} are thus examples of entangled states. These states are called Bell states. The entangled form of the wave functions in equations eqref{eq:waveposcor} and eqref{eq:wavenegcor} implies that a measurement of the polarization of one photon determines the result of a polarization measurement on the other. Thus for the function given in equation eqref{eq:waveposcor} the result will be (0,0) or (1,1) each with equal probability. Similarly, equation eqref{eq:wavenegcor} implies result of (0,1) or (1,0) each with 50% probability. In both cases a measurement on one photon allows us to predict the result of the measurement on the other with 100% certainty cite{fox2006}.
subsection{Bell’s Inequality}
label{Bell’s Inequality}
Bell’s theorem states that the inequality is always obeyed if the local hidden variables picture of the microscopic world is correct. Quantum mechanics, by contrast, predicts violations of Bell’s inequality cite{fox2006}. To test Bell’s inequality, an experiment has to be performed, where two entangled photons are sent through polarization filters, $DBS1$ and $DBS2$, as was described in section ref{sec:Entanglement}, and then collected by two single-photon detectors $D_1$ and $D_2$. Let $alpha$ and $beta$ be the angles of the filters with respect to the vertical. For each setting of the angles $alpha$ and $beta$ the Bell experiment has four possible results, which are characterized by their respective probabilities:\
$P_{11}(alpha,beta)$ is the probability that both $D_1$(1) and $D_2$(1) detect the photon,\
$P_{10}(alpha,beta)$ is the probability that $D_1$(1) detects the photon but $D_2$(0) doesn’t,\
$P_{01}(alpha,beta)$ is the probability that $D_2$(1) detects the photon but $D_1$(0) doesn’t,\
$P_{00}(alpha,beta)$ is the probability that both $D_1$(0) and $D_2$(0) don’t detect the photon.\
The probabilities must satisfy two simple check rules. First, the total probability of getting a 1 or a 0 result for each photon must be exactly 50%, implying that:
begin{align}
begin{split}
P_{11}(alpha,beta) + P_{10}(alpha,beta) = 0.5,\
P_{01}(alpha,beta) + P_{00}(alpha,beta) = 0.5,\
P_{11}(alpha,beta) + P_{01}(alpha,beta) = 0.5,\
P_{10}(alpha,beta) + P_{00}(alpha,beta) = 0.5.
end{split}
end{align}
Second, the perfect correlations must be reproduced when $alpha = beta$. Implying for the case of positive correlations that:
begin{align}
begin{split}
P_{11}(alpha,beta) &= 0.5,\
P_{10}(alpha,beta) &= 0,\
P_{01}(alpha,beta) &= 0,\
P_{00}(alpha,beta) &= 0.5,
end{split}
end{align}
and vice versa for negative correlations.
First the case with $beta = 0$ will be analyzed, when the source emits positively correlated Bell states of the type given in equation eqref{eq:waveposcor}. A horizontal/vertical measurement basis is chosen that coincides with the axes of $PBS2$. Suppose the result $D_2(0)$ is obtained. This means that a vertically polarized photon is sent to $PBS1$ and thus the result $D_1(0)$ is obtained with probability $cos^2(alpha)$ and $D_1(1)$ with probability $sin^2(alpha)$. Similarly, if the result $D_2(1)$ is obtained,then a horizontally polarized photon is going to $PSB1$ meaning that the results $D_1(0)$ and $D_1(1)$ are obtained with probabilities of $sin^2(alpha)$ and $cos^2(alpha)$, respectively. Now the results $D_2(0)$ and $D_2(1)$ both occur with probability 50% and so:
begin{align}
begin{split}
P_{11}(alpha,0) = frac{1}{2} cos^2(alpha),\
P_{10}(alpha,0) = frac{1}{2} sin^2(alpha),\
P_{01}(alpha,0) = frac{1}{2} sin^2(alpha),\
P_{00}(alpha,0) = frac{1}{2} cos^2(alpha).
end{split}
end{align}
Now suppose that $beta$ is also arbitrary. We are free to choose any pair of orthogonal axes as our measurement basis. Therefore axes are chosen at angles of $beta$ and $beta + 90degree$ which coincide with those of $PBS2$. The argument is then identical, except that the probabilities now depend on $(alpha – beta)$ rather than just $alpha$, giving:
begin{align}
label{eq:prob}
begin{split}
P_{11}(alpha,beta) = frac{1}{2} cos^2(alpha – beta),\
P_{10}(alpha,beta) = frac{1}{2} sin^2(alpha – beta),\
P_{01}(alpha,beta) = frac{1}{2} sin^2(alpha – beta),\
P_{00}(alpha,beta) = frac{1}{2} cos^2(alpha – beta).
end{split}
end{align}
In the case of negative correlation, the sine and cosine functions are reversed.
The beauty of Bell’s theorem is that it is completely general and applies to all local hidden variables models. There are several different forms of Bell’s theorem. One of this versions was derived by Clauser, Horne, Shimony and Holt in 1969 cite{clauser1969}. They introduced an experimentally determinable parameter $S$ defined by:
begin{equation}
label{eq:Spar}
S = E(alpha_1,beta_1) – E(alpha_1,beta_2) + E(alpha_2,beta_1) + E(alpha_2,beta_2),
end{equation}
where
begin{equation}
label{eq:E}
E(alpha,beta) = P_{11}(alpha,beta) + P_{00}(alpha,beta) – P_{10}(alpha,beta) – P_{01}(alpha,beta),
end{equation}
and proved that the following Bell inequality:
begin{equation}
label{eq:Bell}
-2 leq S leq 2,
end{equation}
holds for all possible local hidden variables theories. It is not hard to find examples where the quantum predictions violate equation eqref{eq:Bell}. For example, if $alpha_1 = 0degree, beta_1 = 22.5degree, alpha_2 = 45degree$ and $beta_2 = 67.5degree$ are the values that are used in equation eqref{eq:prob}, there will be found that $S = 2sqrt{2}$, which violates equation eqref{eq:Bell} cite{fox2006}. A full derivation for $S$ is given in appendix ref{appendix:Sder}.
newpage
section{Experimental Set Up}
label{sec:setup}
This experiment is a modern approach to Aspects research done in 1981 cite{Aspect1982}. Where Aspect was the first to succeed in experimentally violating Bell’s inequality. Figure ref{fig:setup} describes the experimental set-up that was used in our approach to violate Bell’s inequality.
begin{figure}[h]
includegraphics[width=textwidth]{ExperimentalSetup}
caption{Schematic image of the experimental set-up used for this experiment cite{experimentalreading}.}
label{fig:setup}
end{figure}\
To generate entangled photon pairs, a second-order nonlinear process, usually referred to as spontaneous parametric down-conversion (SPDC), is used. A laser, controlled by the quED control unit, emits unpolarized photons through a series of non-linear crystals which will create the entangled photon pairs that are either horizontally or vertically polarized. Single photon detectors in combination with polarizing filters are used to detect the entangled photon pairs, analyze their polarization and verify their non-classical polarization correlations. The corresponding counting rates are displayed on an integrated display on the quED cite{experimentalreading}.\
As is described in equation eqref{eq:Spar} there are four different angle combinations needed to provide a value for $S$. The angles chosen for this experiment are:
begin{gather*}
alpha_1 = 0degree, quad beta_1 = 22.5degree\
alpha_2 = 45degree, quad beta_2 = 67.5degree
end{gather*}
These angles are chosen because for these angles the theoretical value of $S$ is bigger than 2, more specific, equal to $2sqrt{2}$, which indicates that there are no hidden variables.
In real life there is no unit detection efficiency. When a ’01’ event is measured, it could also be a ’11’ event, but one photon was lost or failed to detect. When no photons are measured, it could be possible that one or two photons were lost. To solve this issue, only ’11’ events have been measured. With this method it’s certain that it was a ’11’ event originally. To correct for the other events, the polarization filters will be either rotated by $0degree$ (Horizontal basis measurement, or $90degree$ (Vertical basis measurement), and the probabilities will be normalized to the total amount of coincident-photon detection. So
begin{align*}
P_{00} rightarrow frac{N_{VV}(alpha,beta)}{mathcal{N}}, quad P_{01} rightarrow frac{N_{VH}(alpha,beta)}{mathcal{N}},\
P_{10} rightarrow frac{N_{HV}(alpha,beta)}{mathcal{N}}, quad P_{11} rightarrow frac{N_{HH}(alpha,beta)}{mathcal{N}},
end{align*}
with normalization:
begin{equation}
mathcal{N} = N_{HH}(alpha,beta) + N_{VH}(alpha,beta) + N_{HV}(alpha,beta) + N_{VV}(alpha,beta).
end{equation}
Here $N_{HH}