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Quantum Optics Maira Amezcua Department of Physics and Astronomy Pomona College May 2, 2012 Page 1 of 55 Page 2 of 55 Contents List of Figures 4 1 Abstract 6 2 Introduction 8 2.1 Background 8 2.1.1 Entangled States 8 2.1.2 Single Photon Interference Experiments 9 2.2 Motivation 10 2.2.1 Experiments 11 2.2.2 Future Work 14 3 Theory 14 3.1 Correlated Photon 14 3.1.1 Calcium Cascade 16 3.1.2 Spontaneous Parametric Down-conversion 17 3.2 Single Photon Interference 19 3.3 Second-order Correlation Function 22 3.4 Quantum Eraser 25 3.4.1 Complementarity 26 3.4.2 Double Slit Experiment 26 3.4.3 Two photon interference 27 3.4.4 Quantum Eraser 28 4 Materials and Methods 32 4.1 Equipment 32 4.1.1 External-cavity Diode Laser 32 4.1.2 The Laser Beam 36 4.1.3 BBO Crystal 37 4.1.4 The Detectors 38 4.1.4 (a) The A-detector 39 Page 3 of 55 4.1.4 (b) The B-detector 41 4.1.4 (c) The B’-detector 41 5 Analysis 43 6 Conclusion 50 Appendix A: Single Counting Photon Module 51 Appendix B: Coincidence Counting Module 52 7 References 53 List of Figures Figure 1: Experimental apparatus used to demonstrate that if a down-converted photon is detected at G, then its pair will be detected at T or R but not at both. DCC: Down-conversion crystal. PBS: polarizing beam splitter. SPCM: single photon counting module. Adapted from Dr. Beck. ..........................................13 Figure 2: Levels of calcium. The atoms are pumped to the upper level by absorption of two photons and emits two photons correlated in polarization [10]. .......................................................................................16 Figure 3: Spontaneous parametric down-conversion....................................................................................17 Figure 4: Type I photon pairs [15]. ..............................................................................................................18 Figure 5: Single photon interference by Grangier, Roger and Aspect [4]. ...................................................19 Figure 6: Anticorrelation parameter as function of number of cascades and trigger rate [4]. ...................20 Figure 7: Mach-Zehnder interferometer [4]. ................................................................................................21 Figure 8: Interference experiment [26]. ........................................................................................................28 Figure 9: Quantum eraser experimental setup [7]. .......................................................................................29 Figure 10: The polarization interferometer. The arrow signifies horizontal polarization and the dot indicates vertical polarization [7]. ................................................................................................................30 Figure 11: The number of coincidence counts between detectors A and B as a function of pathlength difference in the two arms of the interferometer. Detector B measures horizontally polarized photons [7].31 Figure 12: Measure of coincidences (a) AB, and (b) AB’ as a function of the pathlength difference between the arms of the interferometer. The beam splitter is rotated so that detector B measures +45° polarized photons and detector B’ measures -45° polarized photons [7]. ....................................................................31 Figure 13: Schematic of external-cavity laser...............................................................................................33 Figure 14: Spectrum of ECDL.......................................................................................................................34 Figure 15: Spectrum of ECDL.......................................................................................................................35 Page 4 of 55 Figure 16: Experimental apparatus. Components are 405nm cavity-diode laser, a pair of cylindrical lenses, two mirror, two spherical lenses, a half-wave plate (HWP), down-conversion crystal (BBO), and a polarizing beam-splitter (PBS).The half-wave plate before the crystal vertically polarizes the pump beam to match the specifications of the BBO crystal. ................................................................................................36 Figure 17: The collimation lenses are on the right and the RG780 filters are on the right. The orange fiber is the multimode optical fiber. .......................................................................................................................38 Figure 18: Detector A alignment. ..................................................................................................................40 Figure 19: The B' detector alignment setup. .................................................................................................42 Figure 20: Optical table set up with 3 detectors. ..........................................................................................43 Figure 21: Number of coincidences in AB for each 1.0s detection cycle of photon counts in detector A. This data was taken over a window of 26.0 seconds. ............................................................................................44 Figure 22: Counts in A versus coincidences in AB, AB’ and ABB’. The values were integrated over a period of 1.0s. The collection was made over a window of 40s.....................................................................45 Figure 23: Counts for detectors A, B, and B'. Photon counts from B and B’ are on the same order of magnitude of counts from A. ..........................................................................................................................46 Figure 24: Coincidences for a clock rate of 0.00125s and a total integration time of 1.0s. .........................47 Figure 25: Coincidences for a clock cycle of 0.005s and an integration window of 4.0s. ............................48 Figure 26: Coincidences for a period of 0.001s and an integration time of 0.8s. .........................................48 Figure 27: g(0) values for a clock cycle of 800Hz. .......................................................................................49 Figure 28: This is the interface board after removing component C1. .........................................................51 Figure 29: Series 3 CCM which performs up to 4-fold coincidence counting. .............................................52 Page 5 of 55 1 Abstract A pair of photons is said to be entangled if their wave function cannot be factored into the wave functions of the individual photons. These entangled states make it possible to test several quantum mechanical theories such as single and double interference at the quantum level. Due to technological advances in the field of optics, we can create entangled photon pairs through type-I spontaneous parametric down-conversion, in which a photon of a specific frequency is converted into two photons with half the frequency through a non-linear crystal. This thesis focuses on creating and detecting correlated photon pairs with an external-cavity diode laser and a beta-Barium Borate crystal. A second experiment is carried out to find the probability of detecting a photon that passes through a beam splitter at the two output paths. Page 6 of 55 Page 7 of 55 2 Introduction 2.1 Background Particle theory of light has existed since the seventeenth century but it was not until the twentieth century that the particle nature of light was revisited and further developed. The evidence for the particle nature of light is attributed to the photoelectric effect, Compton scattering and Planck’s discovery of quantized energy. With Albert Einstein’s and Max Planck’s contributions, the foundations of quantum mechanics flourished, treating light as a particle. Later, Louis de Broglie made the bold hypothesis that electrons behaved like waves. This wave-particle duality was at the heart of quantum mechanics, which would support a mathematical model to explain both properties. 2.1.1 Entangled States Quantum mechanics uses a probabilistic wave-particle model for determining the behavior of quantum systems. It explains anomalies that cannot be resolved using classical mechanics. With powerful computers and communication security relying on quantum information, advances in technology make it possible test various quantum mechanical theories using correlated photon pairs. Correlated photons are unique because they are multi-particle systems that (a) if measured independently of each other produce random states, and (b) the state of one depends on the state of the other. Entangled photon pairs are used in experiments like the quantum eraser and Bell’s Inequalities. Page 8 of 55 2.1.2 Single Photon Interference Experiments In 1905, Albert Einstein published a paper interpreting the photoelectric effect as light quantum striking the surface of metal and ejecting electrons [1]. His work seemed to support the existence of a photon, but W.E. Lamb and M.O. Scully proved that Einstein’s theory of photons was plausible but not the only one [2]. Their theory treated atoms quantum mechanically and light classically. This semi-classical model described an electromagnetic wave perturbing a quantized atom. The interaction exerted a force on the atom which caused the electrons’ ejection. The field of quantum optics was established in 1956 with the Hanbury-Brown and Twiss experiment [3]. This was the first careful investigation of coincidences, the occurrence of a single photon appearing in two detectors at the same time after passing through a beam splitter. The absence of a coincidence proves photons exist. The task was to prove that for dim light sources single photons arriving on a beam splitter would not split, leading to anticorrelation. The wave theory of light predicts that the light arriving on a beam splitter would be halved. In the experiment, an emission line from mercury was isolated by a system of filters and sent to a half-silvered mirror. Two photomultipliers with interference filters were connected to a coincidence counter. Theoretically there would be no coincidences but the result was just the opposite. The result of Hanbury-Brown and Twiss failed to prove the existence of photons and showed that light traveled in bunches. The results of the experiment are complicated to understand without applying the proper model. In this case the photomultipliers are being treated quantum mechanically and the existence of photons is still dubious. Page 9 of 55 It was the work of Grangier, Roger and Aspect in 1986 that made a major breakthrough in this area of science [4]. Again they would examine the correlations between photon detections at the outputs of a beam splitter. This simple experiment attempted to create an improved photon source by exciting calcium atoms that would decay to the ground state and two photons would be emitted. One photon acts as trigger by sending a signal to the coincidence counter. This alerts the two detectors for the second photon to expect a photon during a brief period. Grangier, Roger and Aspect were fully successful in measuring fewer coincidences than predicted by the classical wave theory. At last the existence of photons was confirmed in their experiment, opening the field to other single photon experiments. 2.2 Motivation The basis of this project is: “Do photons exist?” While we do not need to reprove that photons do exist, this question guides students in understanding the evidence of photons. The photoelectric effect and Compton scattering are attributed to proving photons exist, but both phenomena can be rationalized as quantized matter interacting with a classical electromagnetic field. The experiments presented in this paper allow for an obvious approach to proving photons exist and make the quantum eraser and Bell’s Inequalities easy to understand. Whitman College and Colgate College are the front-runners in establishing quantum mechanics laboratory coursework for undergraduates [5-9]. They have succeeded in producing five quantum mechanics experiments which are the model for the proposed experiments. Other colleges with quantum optic laboratories include Reed Page 10 of 55 College, Trinity College, the University of Rochester, Dickinson College and Harvey Mudd College. Emphasis on quantum mechanical research calls for quantum optics training at the undergraduate level, which is why the Department of Physics and Astronomy at Pomona College has begun to execute these experiments. 2.2.1 Experiments The experiments delineated below are low-cost and simple to build, requiring a standard optical table for the setup. The proposed experiments are simple to setup because they require few components. The nonlinear crystal and photon counter were purchased at the beginning of this project while other equipment was gathered from various sources. Computer software was obtained from Dr. Branning of Trinity College. Our setup differs from other labs because we built an external-cavity diode laser as opposed to buying a laser. We hope that by using an external-cavity diode laser, it can increase the efficiency of pair production. During the summer, our group focused on collimating the laser and observing the spectrum of the laser to make adjustments that would decrease linewidth1. Correlated Photons The first laboratory in the series checks for correlated photons in Type I parametric down-conversion, output beams are polarized parallel to each other. A blue pump laser at wavelength hits a nonlinear Barium Borate crystal, exciting 1 The linewidth is a measure of the spectral coherence of a laser. Small linewidths (~1GHz) correspond to interference of the waves of different frequencies, in the laser, in such a way that they have a fixed relative phase; this creates a pulse. Page 11 of 55 two photons at lower frequencies than the input photon. While there are many photons emitted in different directions and at different frequencies, we focus on correlated photons that are produced in pairs at the same time. If two photons are detected in a short time interval, we assume they are coincident and constitute a pair of correlated photons. This experiment maximizes the coincidence counts by adjusting the vertical and horizontal positions of the collection lenses for two detection channels. Existence of Photons The basic idea behind this experiment is that light displays particle properties at the quantum level. Classically light on a beam splitter can be detected in two directions of polarization but the same does not hold for a photon; thus, a single photon incident on a beam splitter can be transmitted or reflected but not both. Fig 2. illustrates the setup for this experiment which aims to demonstrate that when a down-converted photon is counted at detector G, its pair will be detected at either detector T or R. Since a photon cannot be split, it cannot be detected in two channels at the same time. This experiment also measures the degree of second-order coherence parameter, the probability of measuring a photon at detectors T and R in the same time interval. The purpose of this experiment is to show that quantum mechanically Page 12 of 55 Figure 1: Experimental apparatus used to demonstrate that if a down-converted photon is detected at G, then its pair will be detected at T or R but not at both. DCC: Down-conversion crystal. PBS: polarizing beam splitter. SPCM: single photon counting module. Adapted from Dr. Beck. Quantum Eraser Effectively, if two correlated photons are produced from the same source and the polarization is measured for one of the photons then they will not interfere. However, if the path that was taken by the photon can be erased then the photons should interfere. Here the interference is calculated by looking at the coincidence counts based on the path each photon took. Bell’s Inequalities This experiment tests local realism for two correlated photons. Local realism says that for two photons from the same source have defined polarizations once emitted. Page 13 of 55 However, quantum mechanics is not bound by local realism and one must be abandoned to explain the results in the experiment. Based on coincidence counts for both photons the probability that the photons had a particular set of polarizations can be determined. 2.2.2 Future Work Due to several unforeseen setbacks, only the first two experiments of the sequence are underway. There is still much work that needs to be done to improve the laser, coincidence counts, and begin employing the second set of laboratories. More optical equipment must be purchased, including a second beta-Barium Borate crystal. This work will be continued through the summer of 2012 as part of the Summer Undergraduate Research Program (SURP). 3 Theory 3.1 Correlated Photon Quantum entanglement has made its way into applications such as quantum imaging, quantum information processing, and quantum lithography. Correlated twophoton systems are a statistical mixture of states in which the state of one particle can be determined by the state of the other particle. Another characteristic of entangled photon pairs is that if measured independently of each other the states are random. In this particular project we are concerned with type I correlated photons, down-converted pairs with the same polarization. Page 14 of 55 A two-particle system is described as being in an entangled state if its wave function cannot be factored into a product of individual photon polarization states. One example that displays mutual dependence of polarization orientations with an equal superposition of two states is: (3.1) Where is the horizontally polarized state and is the vertically polarized state, and the subscripts refer to the individual photons. For the wave function given above, the probability of obtaining the result or is . This implies that half the time the photons are horizontally polarized and the other half they are vertically polarized. When the polarization is measured, the photons collapse into one state. Entangled downconverted photons are used to test single-photon interference, the quantum eraser and Bell’s Inequalities. In order to conduct experiments observing single photons it is important to have an adequate single-photon source. The Hanbury-Brown and Twiss experiment [3] attempted to create a stream of photons by focusing on the 435.8-nm light from an emission line of a mercury arc. Mercury atoms collided with electrons; the interactions excited atoms to various states, including one that rapidly decayed to the ground state. This excitation would emit a photon of wavelength 435.8-nm. This cycle produced light that then traveled to the detectors, which had a 435.8-nm interference filter to ensure that only photons of the correct wavelength entered the photomultiplier. The Hanbury-Brown and Twiss correlated photon method had several discrepancies and the mercury arc had several imperfections. The glass bulb lamp introduced differential phase-shifts that Page 15 of 55 reduced correlations. With newer technology available more techniques were presented to create correlated photons. Two methods will be described in the following sub-sections. 3.1.1 Calcium Cascade The work of Grangier, Roger and Aspect [4] used a well-established technique by exciting a stream of collimated calcium atoms by means of two-photon laser excitation. Calcium atoms were excited to a high s-state that decayed to the ground state thru an intermediate p-state: photons of frequency [10]. This cascade in calcium gave two and , correlated in polarization. Figure 2: Levels of calcium. The atoms are pumped to the upper level by absorption of two photons and emits two photons correlated in polarization [10]. An atomic beam of calcium is produced by a tantalum oven, operating at temperature of atoms/cm3. Two lasers with 1000°K [11]. This induces an interaction region of parallel polarizations, a single-mode krypton ion laser ( single-mode Rhodamine 6G dye laser ( Running each laser at 40mW, the cascade rate is Page 16 of 55 ) and a tunable cw ), illuminate the calcium beam at 90°. per second. The calcium cascade proved to be sufficient enough; however, with advancements in technology the amount of equipment to make correlated photons was reduced. 3.1.2 Spontaneous Parametric Down-conversion Parametric down-conversion, or “frequency splitting of light” [12], splits a single pump photon ( ) into two photons of lower frequency ( ) in a nonlinear crystal [13-17]. Figure 3: Spontaneous parametric down-conversion. The light is a monochromatic plane wave with wave vector and frequency .The nonlinear crystal is fabricated to have a similar index of refraction to the linear medium surrounding the crystal. The daughter photons, often named the “signal” and “idler” photons have definitive properties because of the invariance of the medium which conserves energy and momentum. These are the relations of the wave vectors and frequencies of the parent and daughter photons: (3.1.1) Page 17 of 55 (3.1.2) Emission of the products is almost simultaneous. There are two types of parametric down-conversion: type I and type II. Type I down-conversion produces photon pairs with parallel polarization, while type II produces oppositely polarized photon pairs. Any crystal can only support one type of down-conversion and the polarization of the beam must be adjusted accordingly to match the crystal. However, for one crystal there is not a unique solution to the wavelength and direction of the daughter photons. The photons are always produced in pairs but our project focuses on degenerate signal and idler photons, those with the same wavelength. Figure 4: Type I photon pairs [15]. Parametric down-conversion is used because it is easier and cheaper to implement than other methods such as calcium excitation2. 2 Calcium atoms are excited to a high s-state that decayed to the ground state thru an intermediate p-state, by giving off two photons. Page 18 of 55 3.2 Single Photon Interference The issue being addressed is whether a photon can interfere with itself [18-21]. Quantum mechanically a photon can interfere with itself and is the very basis of the theory, but the outcome fails for a single system. To understand the wave-particle duality and the quantum behavior of a photon it is necessary to analyze both single photon experiments executed by Grangier, Roger and Aspect (GRA) [4]. The first experiment is based around the particle nature of light, violating the classical wave model. Fig. 5 shows the experiment setup. Figure 5: Single photon interference by Grangier, Roger and Aspect [4]. The detection of the first photon and for a time duration at photomultiplier 3 triggers photomultipliers in view of the second photon photomultipliers feed into single and coincidence counters for count rate for the gate, and . These two . GRA denote as the as the single rates for the signal photomultipliers, and the coincidence rate. The probabilities of these measurements during the time interval are as follows, 3 Where is the lifetime of the calcium cascade. Page 19 of 55 (3.2.1) A classical approach to this experiment generates the inequality . (3.2.2) This inequality means that the classical coincidence probability product of the single counts is greater than the the probability of accidental coincidences. Any violation of this inequality is a characterization of non-classical behavior. Figure 6: Anticorrelation parameter as function of number of cascades and trigger rate [4]. Page 20 of 55 The experimental values reinforce the particle nature and emphasize that a photon cannot be split at a beam splitter. The second experiment by GRA builds a Mach-Zehnder interferometer around the first beam splitter; this is the actual single photon interference experiment. Figure 7: Mach-Zehnder interferometer [4]. If an intense light source was the input to the Mach-Zehnder interferometer then interference fringes could be constructed based on the outputs of and . The particle aspect would tell us that a single photon in the system could be detected at either output but not both. However, due to the wave-particle duality of quantum mechanics by altering the path difference between the arms of the interferometer and observing over a large time interval one can reconstruct interference fringes. Changing the path difference between the photomultipliers changes the probability of the path the photon traveled and we observe fluctuations in counts of each photomultiplier. Page 21 of 55 3.3 Second-order Correlation Function The results of quantum optics experiments relating to photon interference can be explained classically in terms of coherence. In a classical field, an electromagnetic wave is described perfectly by Maxwell’s equations. The first-order coherence function determines the visibility of interference fringes based on electric field fluctuations. However, Maxwell’s equations cannot be applied in the realm of quantum mechanics so the degree of second-order coherence is applied to quantify the results of Hanbury-Brown and Twiss experiments. The second-order (temporal) correlation function measures coherence of intensity fluctuations with time delay τ [22-23]. (3.3.1) Where and are the electric field and intensity, respectfully at time . The brackets indicate ensemble averages as opposed to time averages. The degree of secondorder coherence can be written as the correlations between the intensities of the transmitted and reflected beams. It is a function of the time delay between measurements of intensity: (3.3.2) Page 22 of 55 The present work is simultaneous intensity measurements. Assuming a 50/50 beam splitter in which the transmitted, reflected and incident intensities are related by, (3.3.3) If we apply this to classical fields, we find that (3.3.4) In our experiments do not measure intensity directly so we use other quantities we can measure experimentally. The function is interpreted as the probability that a coincidence occurs: (3.3.5) where interval and , and are the probabilities of a photodetection at detectors T and R in a time is the joint probability of making detections at both detectors. These probabilities are calculated by finding the average rate of detections multiplied by the time interval . The average rate of detections is just the total number of detections divided by the total counting time . The probabilities for detections and coincidences are as follows: Page 23 of 55 (3.3.6) Substituting equations in (2.6) into (2.5) yields (3.3.7) These functions are used to make two-detector measurements of , but usually three detectors are used. A second source beam incident on a third detector is used as a gate (labeled G). The expression then becomes (3.3.8) The probabilities are calculated a bit differently because they can be normalized. (3.3.9) The expression for (2.8) in terms of number of coincidences is (3.3.10) Page 24 of 55 As Grangier, Roger and Aspect noted, “a photon can only be detected once,” thus the probability of finding any coincidences in detectors G, T and R are zero ( ), and (3.3.11) This means that the photocount at the transmitted and reflected detectors must be anticorrelated because the photon must choose a path. Since the electronics cannot make high speed measurements, often times the time interval coincidences are measured in all three detectors yielding is long enough that . For all classical light, coincidences are measured in all three detectors such that 3.4 Quantum Eraser In quantum mechanics interference is the superposition of states for several indistinguishable paths. Understanding the wave-particle duality requires investigation of mixed states and their impact on interference. The single photon interference experiments demonstrate that photons can interfere with themselves and changing the system destroys interference. The visibility of the fringe pattern is a result of “which-way” information known by the observer. Similar experiments can be performed using correlated photon pairs to show the relationship between interference and entanglement. The development of two photon interference experiments has led to the quantum eraser, which recovers interference by “erasing” the information about the source of the detected photon. Page 25 of 55 3.4.1 Complementarity A general concept in quantum mechanics, complementarity is interpreted as a consequence of the uncertainty principle. Two observables are complementary if knowing the outcome of one observable destroys knowledge of the second. Two examples of complementary will be illustrated [24]. Example 1: For a single particle with position and momentum , if the position is measured at any moment in time it prevents measuring the momentum. The momentum then is in a large range of equally probable states. Example 2: Consider a spin-½ particle with two orthogonal spin components. If the horizontal spin component as a definite value (left or right) then both values of the vertical component (up or down) each have a probability of 50%. For entangled photon pairs produced by type-I spontaneous parametric down-conversion, the signal and idler photons have the same polarization. The two photon interference experiment treats entanglement and interference as complementary observables. Determining the which-way path information of the signal photon by its polarization yields full which-way information for the idler photon and annihilates any interference. However, negating which-way information from both of these photons restores interference. We now turn to the double slit experiment to understand the quantum eraser. 3.4.2 Double Slit Experiment The archetypal double slit experiment [25] demonstrates the wave nature of light and probabilistic nature of quantum mechanics. A photon incident on a thin plate with two parallel slits and is observed on a screen some distance behind the plate. The probability that the photon arrives at any given point on the screen is calculated by the probability of the photon travelling through the top slit or the bottom slit. The interference pattern arises from indistinguishability of which slit the photon crossed. Page 26 of 55 Using the double slit experiment we can derive the relation between the complementary observables: which-way path information and interference. First assume that the incident photon is vertically polarized, or in the vertical polarization state . This does not change the path the photon took nor does it determine which slit it traversed. By adding a half-wave plate (HWP) to the upper slit, we can observe which slit the photon passed through because it rotates the polarization by 90° to Classical mechanics relates the electric fields , the horizontal polarization state. and to the corresponding wave passing through the top and bottom slits, respectively. The interference term is equal to zero because the fields are orthogonal. Interference fringes can be regained by erasing the which-way path information of the photon, thus restoring the photons to the vertical polarization state .A polarizer is placed in front of the screen to mix up the polarization states and the origin of the photon cannot be known; thus, the interference pattern is recovered. The double slit experiment led to two photon interference tests and the next section will discuss the experiment carried out by Zou et al [26]. 3.4.3 Two photon interference Consider the interference experiment by Zou et al. indicated in Fig. 8, in which two mutually coherent pump waves are pumped into two similar nonlinear crystals NL1 and NL2. Page 27 of 55 Figure 8: Interference experiment [26]. Parametric down-conversion occurs at NL1 and NL2, with the emission of a signal photon and an idler photon at each. We observe interference between the signal photons when the paths of the idler photons photodetectors are aligned. The rate of coincidences at display a fourth order interference because the detectors cannot distinguish between the photon pairs. If the connection between is broken by misalignment or a beam-block, it is possible to determine whether the signal photons come from NL1 or NL2. In this case the alignment of the two idlers serves as the quantum eraser. 3.4.4 Quantum Eraser The quantum eraser [27-31] model employed in this project is taken from Gogo et al. experimental arrangement [7]. There are two parts to the setup: the polarization interferometer and the quantum eraser. Fig. 9 indicates the entire experimental layout. Page 28 of 55 Figure 9: Quantum eraser experimental setup [7]. The source used produces entangled photon pairs in the polarization state, (3.4.1) Any photon has a 50% chance that it is horizontally or vertically polarized. The signal photon then goes into a polarization interferometer illustrated in Fig. 10. Page 29 of 55 Figure 10: The polarization interferometer. The arrow signifies horizontal polarization and the dot indicates vertical polarization [7]. The beam displacing prism (BDP) splits the signal beam into vertically and horizontally polarized components and the half-wave plate flips the polarizations. The second BDP realigns the beams, which go through a half-wave plate to interfere on the polarizing beam splitter (PBS). The pathlength difference between the two polarization beams is adjusted by moving the second BDP. Theoretically the photon takes both paths and interference will occur. However, if we can determine the polarization of the photon we can determine which path was taken. The quantum eraser exists in the idler signal; the half-wave plate in front of B,B’ detectors constitutes the eraser. This plate is oriented such that the idler beam horizontally polarized are detected at B and vertically polarized idler photons are detected at B’. If a vertically polarized idler photon is detected at B’ then we know that the signal photon must also be vertically polarized. Page 30 of 55 Figure 11: The number of coincidence counts between detectors A and B as a function of pathlength difference in the two arms of the interferometer. Detector B measures horizontally polarized photons [7]. Rotating the half-wave plate erases any which-way information and interference is renewed. Any knowledge of the original state of polarization is removed, yielding new polarizations. Figure 12: Measure of coincidences (a) AB, and (b) AB’ as a function of the pathlength difference between the arms of the interferometer. The beam splitter is rotated so that detector B measures +45° polarized photons and detector B’ measures -45° polarized photons [7]. Page 31 of 55 4 Materials and Methods 4.1 Equipment 4.1.1 External-cavity Diode Laser Diode lasers are widely used in optical experiments because they are compact, simple and inexpensive [32]. We used an external-cavity diode laser (ECDL) because it is capable of reasonable continuous wave output powers with high efficiency, very stable, and tunable [33]. For our experiments we wanted to tune our laser to a wavelength of because our crystal is cut to produce down-converted photons at that input wavelength. We constructed our own blue diode laser using the essentials for an external-cavity: laser diode, collimating lens and a diffraction grating. All the components are fastened unto a base plate, with adjustable positions for the lens and diffraction grating. The lens is mounted into a threaded tube and its position can be adjusted by rotating the lens cover. This is done by inserting a needle into the pinhole on the rim of the lens cover. The diffraction grating moves by turning either of the two adjustment knobs; one knob translates the grating horizontally and the other changes the vertical dimension. Fig. 11 illustrates the position of the basic units. Page 32 of 55 Figure 13: Schematic of external-cavity laser. The grating is mounted in a Littrow configuration so the first order incident light on the grating is reflected back to the laser diode. This second weaker beam collapses into the main beam, increasing the output power as the feedback is above or reaches the threshold. Because the size of the cavity changes the frequency of the light, mechanical movement and thermal expansion must be avoided. Our laser is mounted on a larger base that is bolted to the optical table. A cooler has been attached to the laser but we have not used it. We use a lens to coarsely collimate the bare diode laser without any grating feedback. We use mirrors to steer the beam across the lab and adjust the position of the lens along the axis of the beam propagation. If the lens is too far from the diode, the beam focuses at 5 feet from the laser diode. To facilitate beam collimation, we use an index card with varying dot sizes so that we can compare the size of the beam up close and far away. Diode lasers are single-mode and the linewidth of a bare diode is [34]. Once the bare laser diode is coarsely collimated, the grating is attached. To maximize the Page 33 of 55 feedback from the grating, we operate the laser at just above threshold and change the length of the cavity by adjusting the horizontal and vertical positions of the grating. At low currents, the laser diode is sensitive to feedback because it cannot lase by itself; the grating is adjusted so a second beam appears near the main output beam. This is the fine adjustment to optimize the feedback and reduce linewidth. Figure 14: Spectrum of ECDL. Page 34 of 55 Figure 15: Spectrum of ECDL. We used a spectrograph to verify that our laser was at the desired wavelength . The first spectrum was taken in November 2011 with our spectrograph and has a linewidth of linewidth of Fig. 13 is a recent spectrum of our ECDL with a Within this four month interval, the linewidth increased and several other wavelengths are present because of the thermal expansion of the cavity and losing feedback of the laser. Both spectra were taken when the laser power output was which was measured with a power meter. Page 35 of 55 4.1.2 The Laser Beam Figure 16: Experimental apparatus. Components are 405nm cavity-diode laser, a pair of cylindrical lenses, two mirror, two spherical lenses, a half-wave plate (HWP), down-conversion crystal (BBO), and a polarizing beam-splitter (PBS).The half-wave plate before the crystal vertically polarizes the pump beam to match the specifications of the BBO crystal. Due to the inherent astigmatism of laser diodes, a laser diode beam is elliptical; several techniques can be used to circularize the beam. We initially placed an anamorphic prism between the ECLD and the first mirror. Anamorphic prisms consist of two wedgeshaped prisms. Anamorphic prisms correct the beam shape by expanding or contracting the beam in one direction without changing the other direction by adjusting the angle of Page 36 of 55 the prisms and the incident light. Even when we used a 3.1x anamorphic prism, the beam was still elliptical when it reached the crystal. Instead we used a pair of cylindrical lenses to circularize the beam. In the case of our laser, the beam was diverging horizontally, while the vertical dimension did not. To minimize aberrations, we placed two cylindrical convex-lenses L1 and L2 with focal lengths respectively, in such a way that the convex surfaces face the collimated beam. We placed L1 a distance from the ECLD module and positioned L2 from the first lens. This resulted in a circularized beam. To increase the intensity of the beam, we used a pair of spherical lenses to form a telescope; thus, increasing the intensity of the beam. A pair of plano-convex spherical lenses L3 and L4 with focal lengths respectively, was placed before the half-wave plate. These lenses were chosen such that the magnification reduced the beam size by a fifth. Based on preliminary tests, reducing the beam increased the number of counts in each channel. 4.1.3 BBO Crystal We used a nonlinear crystal, 5x5 mm aperture and 1-mm-long beta-barium borate (BBO) crystal, cut for type-I spontaneous parametric down-conversion. The direction of the down-converted photons is determined by the angle formed by the optic axis of the crystal and the propagation of the beam, the phase-matching angle .4 Our crystal was cut with a phase-matching angle of so that incoming light at 405nm produces two entangled photons of wavelength 810nm at 4 from the pump beam axis. For an explanation of phase-matching refer to Appendix B of Galvez et al. [6] Page 37 of 55 The crystal was mounted on a stage that would allow us to adjust the orientation of the crystal. Once the collection lenses were set up, we adjusted the horizontal and vertical tilt of the crystal so that it produced maximum counts in both channels. 4.1.4 The Detectors We used a single photon counting module [34] with four channels (SPCM-AQ4C) as our photodetector. The signal and idler beams were collected with fiber optic collimation lenses and directed at different input channels of the counting module. Each channel consists of a fiber optic collimation lens, two optical fibers and RG780 filter between the fibers. The photons passed from a fiber optic collimation lens to a multimode fiber to a RG780 filter and redirected from a second fiber optic collimation lens into a single mode fiber to the photon detector. The RG780 filter acts as a high pass filter to block blue photons from the laser beam, while transmitting down-converted light. The SPCM-AQ4C is highly sensitive to light and all connections must be made before turning on the system. Figure 17: The collimation lenses are on the right and the RG780 filters are on the right. The orange fiber is the multimode optical fiber. Page 38 of 55 The SPCM outputs electrical pulses that went into a coincidence counting module (CCM) designed by Dr. Branning of Trinity College [35]. The CCM takes up to fourinput channels, allowing the user control over the logic and interfaces with a computer over USB. The software attached to the system is a LabVIEW vi that allows the user to control the clock cycle, sample size, bin size, and integration time of the system. The clock cycle is the smallest time interval for a single sample of counts. Most of our experiments have a clock cycle of 800Hz. The sample size is the minimal interval for data collection, written to the screen or file, for an arbitrary number of clock cycles. Each of our sample sizes is only one clock cycle. The bin size is the number of samples the software accumulates before updating the on-screen data. We chose a bin size of 800 samples because it provides a higher count and coincidence rate. Finally, the integration time is based on the time it takes to acquire the 800 samples, which in our case is 1.0s. 4.1.4 (a) The A-detector5 The angle between the signal (and idler) beam at the input pump beam is collection lens for channel A is placed on a translating stage so that it makes a . The angle with the pump beam and faces the crystal. Next, the alignment laser was connected to the single mode fiber for channel A instead of the multi-mode fiber to avoid alignment issues between the multi- and single mode fibers. If the alignment beam is not visible, the tip and tilt of the fiber optic collimation lenses between fibers needs to be adjusted such that the beam from the collection lens is a point source. A coarse alignment is made by shining a laser beam backward through the fiber and onto the crystal. The vertical and 5 The A-detector or detector A refers to the collection lens for channel A. Page 39 of 55 horizontal tilt of the collection lens mount is adjusted so the backward beam hits the center of the crystal. 3° Figure 18: Detector A alignment. This coarse alignment points detector A in the general direction of the down-converted photons. Next, a fine alignment is done by reconnecting the single mode fiber to channel A of the SPCM. Using the LabVIEW program, counts in channel A were measured. The horizontal tilt of the collection lens mount is adjusted until the photon count rate is maximized and then, the vertical tilt of the mount is tweaked until the counts are optimized. The knob of the translation stage for the collection lens is turned to change the angle the detector makes with the crystal. Using the translation stage, detector A was slid left and right on the optical table until the count rate was maximized. Again, the tilt of the collection lens mount was readjusted. This process was repeated several times, always Page 40 of 55 adjusting the tilt before sliding the detector. Once the count rate is maximized the detection optics for channel B can be aligned. 4.1.4 (b) The B-detector6 We used a similar procedure for the coarse alignment of the collection lens for channel B. For the fine alignment, instead of observing the count rates, we maximized the coincidences between channels A and B to optimize the alignment for the collection lens for channel B. We maximize AB coincidences in order to detect as many entangled photon pairs as possible. Next, the horizontal and vertical tilt of detector B was adjusted to maximize AB coincidence counts. Then, detector B was translated to increase AB coincidences. These two steps were repeated until the collection lens for channel B was in the optimal position for AB coincidences. 4.1.4 (c) The B’-detector7 Before setting up the collection lens for channel B’, two irises, a half-wave plate (HWP), and a polarizing beam splitter (PBS) are aligned using a backward beam from the collection lens for channel B. These components are centered such that they are in the path of the down-converted photons traveling to detector B. Since down-converted light is vertically polarized, we can use a half-wave plate and a PBS to fine tune the orientation of polarization so that it splits equally between the two collection lenses. By rotating the HWP, we adjusted the input polarization to be with respect to the axis of the PBS. 6 The B-detector or detector B refers to the collection lens for channel B. 7 The B’-detector or detector B’ refers to the collection lens for channel B’. Page 41 of 55 We can easily transmit or reflect the beam by rotating the HWP, which is useful during the alignment of the third collection lens. Figure 19: The B' detector alignment setup. We placed the B’ collection optics on the reflection side of the beam-splitter and used the coarse alignment method described in the sections above. The alignment laser must shine through the beam-splitter and irises onto the down-conversion crystal. Next, to fine-align detector B’, we observed AB’ coincidences. The half-wave plate was rotated to maximize AB’ counts because the orientation of the wave-plate changes AB and AB’ coincidence counts. We adjusted the horizontal and vertical tilt of the collection lens for channel B’ to increase AB’ coincidences. We slid detector B’ until the coincidence rate was at its highest and repeat the process several times until the collection lens for channel B’ was in the optimal position for AB’ count rates. Finally, we rotated the half-wave plate to equalize AB and AB’ coincidences. Page 42 of 55 5 Analysis Figure 20: Optical table set up with 3 detectors. After aligning detectors A and B using the method described in section 4.1.4, we observed coincidences between channels A and B. These measurements were aimed at verifying that counted photons were indeed down-converted photons, and to compare the number of detected photons against the coincidence rate. First, detection channels were set to a clock period of , but because counts were exceedingly low to detect coincidences the data was integrated over 1.0 second cycles. The graph Page 43 of 55 below displays the number of correlations between channels A and B for each 1.0s cycle of channel A counts when the laser was outputting 1.49W of power. Figure 21: Number of coincidences in AB for each 1.0s detection cycle of photon counts in detector A. This data was taken over a window of 26.0 seconds. The data demonstrates that there are not many coincidences for the amount of photons incident on the detectors. It is likely that we are not completely blocking out the blue photons from the laser. However, comparing the collection data, counts in channel A were higher by a factor of 10. Although correlations were optimized before taking any data, it is possible that the beam block was in the way of path B and caused lower count rates. Several adjustments were made to the position of the beam block and detector B Page 44 of 55 but counts were not improved. In order to measure the second-order correlation function, coincidences AB’ are counted by the addition of the third detector. Once detector B’ is in the optimal position for AB’ coincidences, correlations AB, AB’ and ABB’ were calculated as shown in Fig. 22 below. Figure 22: Counts in A versus coincidences in AB, AB’ and ABB’. The values were integrated over a period of 1.0s. The collection was made over a window of 40s. This demonstrates that the photon must choose path B or path B’ because there are no ABB’ coincidences. The coincidences were significantly higher than our first experiment but not comparable to the total photon count. One salient feature of adding the second half-wave plate and detector B’ is that the B counts, as well as AB coincidences, Page 45 of 55 increased. AB or AB’ coincidences are favored based on the orientation of the half-wave plate after the crystal. First, the wave-plate is oriented to maximize the counts in channel B and detector B is adjusted to amplify AB coincidences. Then, the wave-plate is rotated to favor B’ counts and detector B’ is adjusted to improve AB’ coincidences. Before taking data, the wave-plate was oriented such that AB and AB’ coincidences were even. The photon counts in channel A were comparable to the sum of photon counts in B and B’. We are unsure why the number of counts in channel B increased. Figure 23: Counts for detectors A, B, and B'. Photon counts from B and B’ are on the same order of magnitude of counts from A. The number of coincidences is used to calculate the second-order correlation function for three detectors because there are no correlations between AB and AB’ coincidences. This Page 46 of 55 means that from observing coincidence counts we cannot tell which path the photon took. Instead our observations validate that light is a superposition of two waves can be deconstructed when incident on a beam-splitter. Increasing and decreasing the integration window does not suggest any correlations but increases and decreases the coincidence rates respectively. Figure 24: Coincidences for a clock rate of 0.00125s and a total integration time of 1.0s. Page 47 of 55 Figure 25: Coincidences for a clock cycle of 0.005s and an integration window of 4.0s. Figure 26: Coincidences for a period of 0.001s and an integration time of 0.8s. Page 48 of 55 Once we measured the number of counts and coincidences we calculated the secondorder correlation function for 3-detectors based on equation 3.3.10. Where is the number of photon counts and coincidences. Below is a table of , and are the number of values. NA NA,B NA,B' NA,B,B' g(2)A,B,B'(0) 95876.06 167 163 0 0 96058.06 173 183 0 0 96480.06 155 159 0 0 96340.06 163 155 0 0 96523.06 138 170 0 0 96928.06 166 181 0 0 96308.06 185 170 2 6.124519 96919.06 170 172 0 0 96533.06 170 167 0 0 96362.06 165 193 0 0 96921.06 188 165 1 3.12447 96254.06 146 170 0 0 Figure 27: g(0) values for a clock cycle of 800Hz. Page 49 of 55 Overall, for our data points. The instances in which occur because the sampling rate is not small enough to detect single photons. Thus, within the given sampling rate photons may be detected at B and B’, generating . 6 Conclusion The quantum optics experiments described above are well on their way. While the first two experiments have been executed, there is still much work that must be done before testing Bell’s Inequalities and building the quantum eraser. At the moment efforts are being made to increase coincidence counts and count rates. It is crucial that coincidences are increased to display entanglement and confirm the existence of photons. There are still other issues that must be dealt with. More spectra must be taken and the laser should be tuned to 405nm. A critical issue is automating the data analysis process. LabVIEW saves the data in a text file but getting to large amounts of data takes quite a bit of time. The next step is to create a program that retrieves and analyzes the data. The future endeavor is to produce more calculations of the second order correlation parameter. Under quantum mechanics, means that there are no correlations between all three channels because it would be impossible to detect a photon in channels B and B’ at the same time. Until then the latter experiments cannot progress. Page 50 of 55 Appendix A: Single Counting Photon Module For our apparatus we used Perkin Elmer’s single photon counting module (SPCM-AQ4C) with 760-1-interface board (SPCM-AQ4C-IO). This unit consists of four independently fiber-coupled photon-counting channels, which contain fiber optical receptacles pre-aligned to the detector. The SPCM-AQ4C uses silicon avalanche photodiodes with a peak photon detection efficiency of 60% at 650nm. This module has a dark count rate of 500 cps, but it has little effect on our experiments. The interface board provides power connectors to the power supplies and BNC connectors that output TTL (transistor-transistor logic) signals. The module requires +2Volts, +5 Volt, and +30 Volt power supplies. We used a B&K Precision 1672 triple output DC regulated power supply that provides one 5V fixed output and two variable outputs. During one of the experiments, the wires for the +5V power supply shorted and burned out the capacitor C1 on the interface board. Removing this component from the interface board did not affect the function of our counting module. Figure 28: This is the interface board after removing component C1. Page 51 of 55 Appendix B: Coincidence Counting Module We used a coincidence counting module (CCM) designed by Dr. Branning of Trinity College [36], which uses logical AND gates to count coincidences. The TTL outputs of the SPCM-AQ4C are directed at the CCM via BNC connectors. The detector pulses enter a pulse-shaping circuit that reduces the width between 10-18 ns. This allows the pulses to overlap and reduce the number of accidental coincidences. The pulses are sent to the inputs of an AND gate where the output will read true if both detector pulses arrive at the same time. The output signals are sent over to a LabVIEW program via FTDI USB. 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