Bell s inequality Experimental exercise Introduction Purposes of the lab Besides familiarizing yourselves with the important concept of Bell s inequality in relation to the Einstein- Podolsky-Rosen arguments against quantum mechanics 1, this experimental exercise will also allow you to perform an experiment yourselves using modern (at the time of writing) experimental equipment. Lasers, optics, detectors and coincidence counters are the heart of many experiments concerning quantum optics, quantum cryptography, quantum information theory, quantum computation etc.. Preparation The setup used in the lab produces entangled photons. To really understand the background of the lab it helps to read up on optics so that one does not struggle with that. This allows concentrating on the quantum mechanics part while in the lab. A basic understanding of birefringence and half wave plates (λ/2 plates) is of particular interest and refreshing the knowledge about polarizers will also be useful. Apart from that it is very important to read about the Bell s inequality before you come to the lab (compare suggested reading). It is hardly possible to grasp everything when coming to the lab unprepared which will result in a harder time in the lab and a much harder time when writing the lab report as well as a sad lab assistant! Plan what values you need to measure and how to use them to calculate S. Think about which polarization angles (in degrees) give the maximum violation of Bell s inequality. Suggested reading Wikipedia caveat lector: The (English) Wikipedia entry on Bell s inequality and related matters taken with a pinch of salt is a good complement to the papers suggested in the reading list. As this is, by some, considered to be an unsettled matter the content may be colored by other than purely scientific considerations. Information about the Bell inequality and how to disprove it can be found in D. Dehlinger, M. W. Mitchell, Am. J. Phys., 70, 903 (2002). The setup described in this paper serves the same purpose but is different from the one used in the lab. Information about how the entangled photons are produced can be found in P.G. Kwia, et al., PRL, 75, 4337 (1995). Note, that the setup used in this paper serves the same purpose but is also slightly different from the one in the lab. This is primarily due to the used detectors. 1 A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. Lett 47, 777 (1935) 1
In the lab Tasks The laboration will consist of the following tasks: Find the zero position of the half-wave plates. Measure the correlation as a function of angle in steps and compare with the theoretical prediction. Measure the coincidence rates at photon polarization angles which give maximum violation of Bell s inequality. o Calculate S from your measured values. SAFETY CONSIDERATIONS A few simple rules will carry us a long way here: DO NOT LOOK INTO THE LASER BEAM! DO NOT MOVE ANY OBJECTS INTO THE UV-LASER BEAM! THINK before you do anything. Do not touch anything but the two half-wave plates behind the yellow laser fiber. Switch off the detectors before turning on the light. Switch the detectors only on when it is dark. The assistants will show you how to properly turn the half-wave plates when it is dark and how to switch the detectors on and off. Experimental Set-up The photon source will be set-up by the assistants beforehand. You do not need to and you will not touch or adjust anything on it. As one can see in the Figure 1: Sketch of the part of the setup where the entangled photons are produced and guided into the fibers., the experiment is driven by a UV (ultraviolet) laser which generates UV-light. Some of this light is down-converted in a BBO crystal which is basically a birefringent crystal. By down converting a UV photon it is split up into two IR (infra-red) photons. This means that a definite Bell state is produced: 2
Figure 1: Sketch of the part of the setup where the entangled photons are produced and guided into the fibers. These IR photons are spatially separated what allows them to take different paths in the setup. They are passing through λ/2 waveplates (half-wave plates) which turn their polarization by 90 and a BBO crystal with the same orientation as the primary BBO but with half the thickness. This allows to remove any resultant phase/delay owing to different diffractive indices in the primary BBO (which would make a path-selection measurement possible since photons of the one polarization travel slower through the BBO than the ones of the other polarization and are hence delayed)(compare with suggested reading, Kwiat et al.). The IR photons are then caught by a fiber which guides them to the detector part of the experiment. A filter at the entrance to the fiber filters the UV light out such that only IR photons can reach the detector part. The twist of the fibers can be adjusted such that the polarization of the photons passing the fiber gets rotated before they reach the detector part of the setup. The detector part is in a box such that no stray light can reach the detectors. A scheme of it can be seen in Figure 2: Sketch of the part of the setup where the photons leave the fiber and are detected.. The fibers enter the box and one of the two photons created in the initial BBO is emitted from each fiber. They pass half-wave plates which allow turning the photon polarization by any desired value. The beamsplitters are polarizing and let horizontally polarized light through while vertically polarized light is reflected. Depending on which path the photon takes it will end up in either fiber. That will guide it to a single photon detector where it will be detected. Since a disprove of the Bell s inequality can only succeed when measuring the properties of two entangled photons, one has to take precautions to be able to separate entangled photons from normal photons and stray light. This is done by assuming that two entangled photons have to arrive at the detector at the same time but in different arms of the detector system since they originate from the same UV photon. Stray light on the other hand would arrive at the detector at random times primarily as a single photon so that it would not be counted as a coincidence. 3
To be sure that detected photons originate from the same UV photon and are therefore actually entangled, one only considers coincident photons for further analysis. Coincident photons are photons which arrive at the detector at the same time. And since they started in different directions in the primary BBO they have to be detected in different arms of the detector system as well. Thus, four coincidences are measured. H1H2, H1V2, V1H1 and V1V2 where H/V means that the photon is detected in a detector for horizontal/vertical polarization and 1 and 2 denote the arm in which a photon was detected. These coincidences are then shown in a computer program. Figure 2: Sketch of the part of the setup where the photons leave the fiber and are detected. What and how we can measure Counting the coincidences within a certain time allows calculating correlation values which are needed to disprove Bell s inequality. The time interval for counting coincidences can be set in the computer program that also allows reading the number of coincidences. To perform this lab you only have to rotate the half-wave plates of the DETECTOR PART of the setup remember to turn the light or the detectors off before opening the lid! To get a better understanding of the experimental setup it might help, though, to turn the laser on and off or to block the beam paths IN THE DETECTOR PART of the setup. 4
Do NOT block the UV-LASER (compare with safety considerations)! - The first task is to prepare the Bell state that you measure. You can select either of the two states. Selecting a state can be done by rotating the half-wave plate and observing the behavior of the coincidence count rate. After this, note the zero setting of the half-wave plate before continuing with the experiment. - The second task is to measure the value of the correlation E for 5 different photon polarizations to compare with the quantum mechanical expression for the correlation E θ!, θ! = cos 2 θ! θ!. The range of the measurements should span 90. - The third task is to do a Bell test. This requires measurements of the coincidence rates at four different polarization settings. Afterwards, the result can be analyzed in an Octave script on the measurement computer. From this you will get both a value for S and the standard deviation. This value for S is only meant to be used in the lab to get a quick result. When writing the lab report you need to calculate your S value on your own and you need to show the single correlations that were needed for it. The standard deviation from the Octave script can be used. The report Outline Each group needs to hand in one report which should contain: - The intention of the experiment(s) together with some background information. - How the experiment(s) were performed, i.e. describe the main components of the set-up and what their purposes are. - The data obtained from your measurements and the results. o Plot the correlation values for different polarization angles together with the classical and o theoretical predictions. Try to disprove Bell s inequality. Show the data you measured, calculate correlation values that lead to S as well as S itself. The standard deviation can be taken from the Octave script you used in the lab. - Discussion and conclusion. - Key references. Your lab report should be in the order of about 4-5 A4 pages. Tip: Consider writing for a student from your course who did not do this lab. Deadlines You should have your results from the lab with you for the Bell-labfollow-up lecture on the 25 th of November. Please upload this lab report on Studentportalen latest the 3 rd of December. Final deadline 13 th of January Further information will be handed out in the lab or in the lecture on the 9 th of November as well as on Studentportalen. 5