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Quantum Imaging and Information by P. Benjamin Dixon Submittted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor John C. Howell Department of Physics and Astronomy Arts, Sciences and Engineering School of Arts and Sciences University of Rochester Rochester, New York 2011 Dedicated to my parents and grandparents. iii Curriculum Vitae P. Ben Dixon was born on October 16, 1981 in Hanover, NH. He attended the University of Florida as a Florida Academic Scholar from 2000 to 2005 and graduated with a Bachelor of Science in Mechanical Engineering in 2005. He came to the University of Rochester in the Summer of 2006 and received a Master of Arts in Physics in 2008. He pursued his doctoral research in experimental quantum optics under the supervision of John C. Howell. CURRICULUM VITAE iv Publications 1. Quantum Mutual Information Capacity for High Dimensional Entangled States, P. Ben Dixon, Gregory A. Howland, James Schneeloch, and John C. Howell, arXiv:1107.5245v1 [quant-ph], in submission. 2. A theoretical analysis of quantum ghost imaging through turbulence, Kam Wai Clifford Chan, D. S. Simon, A. V. Sergienko, Nicholas D. Hardy, Jeffrey H. Shapiro, P. Ben Dixon, Gregory A. Howland, John C. Howell, Joseph H. Eberly, Malcolm N. O’Sullivan, Brandon Rodenburg, and Robert W. Boyd, Physical Review A 84, 04807 (2011). 3. Photon-Counting Compressive Sensing Lidar for 3D Imaging, Gregory A. Howland, P. Ben Dixon, and John C. Howell, Applied Optics, 50, 5917 − 5920 (2011). 4. Quantum ghost imaging through turbulence, P. Ben Dixon, Gregory A. Howland, Kam Wai Clifford Chan, Colin O’Sullivan-Hale, Brandon Rodenburg, Nicholas D. Hardy, Jeffrey H. Shapiro, D. S. Simon, A. V. Sergienko, R. W. Boyd and John C. Howell, Physical Review A 83, 051803(R) (2011). 5. Precision frequency measurements with interferometric weak values, David J. Starling , P. Ben Dixon, Andrew N. Jordan and John C. Howell, Physical Review A 82, 063822 (2010). 6. Heralded single-photon partial coherence, P. Ben Dixon, Gregory Howland, Mehul Malik, David J. Starling, R. W. Boyd, and John C. Howell, Physical Review A 82, 023801 (2010). 7. Continuous phase amplification with a Sagnac interferometer, David J. Starling, P. Ben Dixon, Nathan S. Williams, Andrew N. Jordan, and John C. Howell, Physical Review A 82, 011802(R) (2010). 8. Interferometric weak value deflections: Quantum and classical treatments, John C. Howell, David J. Starling, P. Ben Dixon, Praveen K. Vudyasetu, and Andrew N. Jordan, Physical Review A 81, 033813 (2010). CURRICULUM VITAE v 9. Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values, David J. Starling, P. Ben Dixon, Andrew N. Jordan, and John C. Howell, Physical Review A 80, 041803(R) (2009). 10. Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification, P. Ben Dixon, David J. Starling, Andrew N. Jordan, and John C. Howell, Physical Review Letters 102, 173601 (2009). 11. Realization of an All-Optical Zero to π Cross-Phase Modulation Jump, Ryan M. Camacho, P. Ben Dixon, Ryan T. Glasser, Andrew N. Jordan, and John C. Howell, Physical Review Letters 102, 013902 (2009). 12. On the feasibility of detection and identification of individual bioaerosols using laser-induced breakdown spectroscopy, P. Ben Dixon and D. W. Hahn, Analytical Chemistry, 77:631-638 (2005). CURRICULUM VITAE vi Conference Proceedings 1. Quantum Ghost Imaging through Turbulence, P. B. Dixon, G. A. Howland, K. W. C. Chan, C. O’Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shapiro, D. S. Simon, A. V. Sergienko, R. W. Boyd and J. C. Howell, in Advanced Photonics Congress: Optical Sensors (Optical Society of America, 2011), p. SWD3. 2. Weak Values and Beam Deflection Measurements, P. B. Dixon, D. J. Starling, N. S. Williams, P. K. Vudyasetu, A. N. Jordan, and J. C. Howell, in Frontiers in Optics (Optical Society of America, 2010), p. FTuE4. 3. Heralded Single Photon Partial Coherence, P. B. Dixon, G. A. Howland, M. Malik, D. J. Starling, R. W. Boyd, and J. C. Howell, in Conference on Lasers and Electro-Optics (Optical Society of America, 2010). p. CMCC2. 4. All Optical Zero to π Cross Phase Modulation, P. B. Dixon, R. M. Camacho, R. T. Glasser, A. N. Jordan, and J. C. Howell, in Frontiers in Optics (Optical Society of America, 2008), p. FTuI6. vii Acknowledgments It is a pleasure to thank the people who helped me in the process of my graduate studies. I thank my advisor, John C. Howell, for giving me help and support in many aspects of my life, including guiding my research. In addition to John’s guidance there has been a sense of humor and a fantastic sense of camaraderie in the laboratory work environment—for this I would like to thank the graduate students that I have worked closely with including: Irfan Ali Khan, Curtis J. Broadbent, Ryan M. Camacho, Michael V. Pack, David J. Starling, Gregory A. Howland, and the visiting Ryan T. Glasser. The physics department faculty and staff has helped me navigate the University rules and regulations, for this I thank: Barbara Warren, Sondra Anderson, Janet Fogg-Twichell, Michie Brown, Connie M. Hendricks, Connie Jones, Ali DeLeon, Patricia T. Sulouff, Eric Blackman, Dan Watson, Arie Bodek, and Nicholas Bigelow. I thank the collaborators outside of my lab who have I have worked and who helped my investigations. These people include, at the University of Rochester: Kam Wai Cliff Chan, Justin Dressel, Nathan S. Williams, Colin O’Sullivan-Hale, Brandon Rodenburg, Andrew N. Jordan, Robert W. Boyd, Joseph Eberly, and Emil Wolf, and at other institutions, Alexander V. Sergienko, David Simon, Jeffrey Shapiro, and Nicholas Hardy. I would like to thank the entire University community and larger Rochester community for providing me with a wonderful place in which to live and study. Finally, I would like to thank Ellie Rose Adair for her support and care. viii Abstract Quantum optics provides a unique avenue to investigate quantum mechanical effects. Typically, it is easier to observe the particle-like behavior of a physical object than it is to observe wave-like behavior. Optics presents us with the reverse case, observing the particle-like behavior of light is difficult. I investigate the utility and limitations of two quantum mechanical effects—weak values and spatial entanglement—in the context of experimental quantum optical communication channels. I show that weak values can be used to increase the signal power and effectively decrease the noise power in a physical communication channel, up to the standard quantum limit for signal to noise ratio. I also show show a method for decreasing the negative environmental effects on a communication channel using spatial entanglement and show that such a channel can be used to transmit over 7 bits of information per joint photon detection event. ix Table of Contents Foreword 1 Chapter 1. Introduction 1.1 Information Theory . . . . . . . . . . . . . . . . . 1.1.1 Discrete Probabilities . . . . . . . . . . . . 1.1.2 Continuous Probability Densities . . . . . . 1.1.3 Channel Limitations . . . . . . . . . . . . . 1.2 Low Dimensional Images and Weak values . . . . 1.2.1 Weak Values . . . . . . . . . . . . . . . . . 1.2.2 Deflection Amplification . . . . . . . . . . . 1.2.3 Controversy . . . . . . . . . . . . . . . . . . 1.2.4 What Is Classical and What Isn’t . . . . . . 1.2.5 Weak Value Investigations . . . . . . . . . . 1.3 High Dimensional Images and Entanglement . . . 1.3.1 Entanglement . . . . . . . . . . . . . . . . . 1.3.2 Paradoxes . . . . . . . . . . . . . . . . . . . 1.3.3 Bell Inequalities . . . . . . . . . . . . . . . 1.3.4 Nonlocality . . . . . . . . . . . . . . . . . . 1.3.4.1 Entropic Uncertainty . . . . . . . . 1.3.5 Related Concepts . . . . . . . . . . . . . . 1.3.6 Spontaneous Parametric Down-Conversion . 1.3.7 Entanglement in High Dimensional Imaging Chapter 2. Weak Values and Deflection 2.1 Introduction . . . . . . . . . . . . . . . 2.2 Theoretical Description . . . . . . . . . 2.3 Experiment and Results . . . . . . . . 2.4 Channel Analysis . . . . . . . . . . . . 2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 5 8 10 13 15 17 18 19 20 20 21 22 23 26 27 30 30 35 . . . . . 36 36 37 40 45 47 x Chapter 3. Weak Values SNR for Deflections 3.1 Theoretical Description . . . . . . . . . . . . 3.2 Technical Noise . . . . . . . . . . . . . . . . 3.3 Experimental Setup . . . . . . . . . . . . . . 3.4 Channel Analysis . . . . . . . . . . . . . . . 3.5 Concluding Remarks . . . . . . . . . . . . . Chapter 4. Ghost Imaging 4.1 Introduction . . . . . . 4.2 Theoretical Description 4.3 Experiment . . . . . . 4.4 Channel Analysis . . . 4.5 Concluding remarks . . Through . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5. Mutual Information 5.1 Introduction . . . . . . . . . . 5.2 Theoretical description . . . . 5.3 Experiment . . . . . . . . . . 5.4 Concluding Remarks . . . . . Chapter 6. Conclusion . . . . . . . . . . . . . . . . . . Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 49 53 54 58 60 . . . . . 61 61 64 66 72 74 . . . . 76 76 78 82 89 90 xi List of Figures 1.1 1.2 1.3 1.4 Relation between mutual information and marginal entropies . . . Relation between mutual information and conditional entropies . Electric field and intensity plots of low order TEM modes . . . . . Electric field and intensity plots of a sum of low order TEM modes as a beam deflection . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Example weak value experiment using beam deflection and polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Effect on electric fields in weak value experiment using polarization and beam deflection . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 An EPR experiment using photon polarization measurement . . . 1.8 A Bell experiment using photon polarization measurements . . . . 1.9 A “map” of related concepts in quantum mechanics . . . . . . . . 1.10 A conceptual SPDC interaction in a nonlinear crystal . . . . . . . 1.11 Probability density for a position-momentum entangled state . . . 1.12 Probability density for a separable state . . . . . . . . . . . . . . 2.1 2.2 2.3 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 Experimental setup for interferometric weak values beam deflection measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of beam radius on interferometric weak values beam deflection measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular mirror displacement in interferometric weak values beam deflection measurement . . . . . . . . . . . . . . . . . . . . . . . . 6 7 13 14 17 18 23 24 31 32 33 34 41 43 44 Experimental setup for interferometric weak values signal to noise measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal to noise ratio for interferometric weak values metrology and standard metrology techniques for different deflections . . . . . . . Signal to noise ratio for interferometric weak values metrology and standard metrology techniques for different beam sizes . . . . . . 55 Experimental setup for ghost imaging through turbulence measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual setup for ghost imaging through turbulence measurement Conceptual setup for ghost imaging through turbulence measurement Representative ghost image profiles . . . . . . . . . . . . . . . . . Ghost image visibilities turbulence near the object . . . . . . . . . 62 63 63 69 70 50 56 xii 4.6 4.7 Ghost image visibilities for turbulence near the illumination source 71 Increased mutual information in the novel configuration . . . . . . 73 5.1 Experimental setup for high dimensional quantum mutual information characterization . . . . . . . . . . . . . . . . . . . . . . . . High dimensional mutual information capacity data for position correlation measurements . . . . . . . . . . . . . . . . . . . . . . . High dimensional mutual information capacity data for momentum correlation measurements . . . . . . . . . . . . . . . . . . . . . . . 5.2 5.3 78 83 84 1 Foreword This dissertation investigates two quantum mechanical effects: weak values and spatial entanglement. These effects are investigated in the context of quantum imaging experiments. I introduce the main concepts of images, weak values, spatial entanglement, and information theory in chapter 1. This broad topic introduction summarizes the standard understanding of these topics and introduces no new research or results. New research is described in chapters 2 through 5. Experiments investigating weak value metrology are described in chapters 2 and 3. Experiments involving ghost imaging relying on spatial entanglement are described in chapters 4 and 5. In all of these chapters, relevant theoretical descriptions are given along with discussions of the results and their meaning. The research is experimental in nature and the light sources, optics, and detection equipment used in the experiments were commercially available. The scope of the research was not to create new detectors or sources, but rather to create or observe new physical effects with the equipment available. The experiments were small in scale, taking up no more than several square feet of optical table space, with additional space for supporting detection equipment. In all of the experiments presented, I built the experiment and took the data as part of a small team of two to three graduate students. The original ideas for each experiment came mainly from John C. Howell in discussions with the graduate student teams. For the results in chapter 2 the experimental team of graduate students consisted of David J. Starling and myself; Andrew N. Jordan 2 was the main contributor to the theoretical analysis, and I was the main contributor to the data analysis and manuscript preparation. For the results in chapter 3, the experimental team of graduate students again consisted of David J. Starling and myself; John C. Howell, David J. Starling, and Andrew N. Jordan were the main contributors to the theoretical description, data analysis and manuscript preparation. For the results in chapter 4 the experimental team of graduate students consisted of Gregory A. Howland and myself; I was the main contributor to the data analysis and manuscript preparation and the theoretical analysis was performed mainly by Kam Wai Cliff Chan and myself. For the results in chapter 5, the experimental team of graduate students consisted of Gregory A. Howland, James Schneeloch, and myself; I was the main contributor to the data analysis and manuscript preparation, the theoretical analysis was performed mainly by John C. Howell and myself. Additional helpful contributions were made by Nathan S. Williams, Justin Dressel, Curtis Broadbent, Joseph Eberly, and Emil Wolf, all from the Department of Physics and Astronomy here at the University of Rochester; Robert W. Boyd, Colin O’Sullivan-Hale, and Brandon Rodenburg from the Institute of Optics here at the University of Rochester; Alexander V. Sergienko and David Simon from Boston University; and Jeffrey Shapiro and Nicholas Hardy from the Massachusetts Institute of Technology. 3 Chapter 1 Introduction The field of quantum optics—concerned with the quantum mechanical properties of light—begins with the start of quantum mechanics itself. Max Planck was investigating electromagnetic radiation when he made his initial quantum conjecture; that light could only be emitted with a quanta of energy E = hν [1, 2]. Einstein added to the beginnings of quantum mechanics with his work on the photo-electric effect in which he again discussed the quanta of light energy. Einstein’s work proposed the idea that the quantized energy of light was a fundamental aspect of light itself, not just the emission process [3]. These quanta of light are what modern physicists call “photons”—a term coined in the 1920’s. The stellar interferometry experiment of Robert Hanbury Brown and Richard Q. Twiss, using intensity correlations led to much debate and research that clarified aspects of quantum optics [4]. As Brown and Twiss pointed out in response to the controversy, the predictions from Maxwell’s equations for classical electrodynamics are identical to the predictions of quantum optics with quantized photo-detections (i.e. detecting photons) [5]. Their experiment inspired the formulation of a quantum mechanical theory of optical coherence and detection [6–10], extending the earlier work in classical coherence to the quantum optical and single photon domain [11]. More recently, the quantum mechanical properties of images and imaging techniques have been investigated [12–14]. Imaging, a subfield of optics, deals 4 primarily with the transverse spatial degrees of freedom of an optical field. Research has included quantum ghost imaging [15–17], quantum communication [18, 19], and quantum enhanced sensing and lithography [20–23]. Two interesting features of quantum mechanics are entanglement and weak values. These concepts are connected to quantum measurement theory and for this reason have the potential for practical use in metrology or measurement based communication technology. The research I present aims to answer the question: What are practical applications of these effects and what are the limits of those applications? It is often the case when dealing with complex problems that consideration of a related but simplified problem is useful. In imaging applications this corresponds to considering low dimensional images first, and then progressing to higher dimensional images. I use the term “dimension” here to refer not to the spatial dimensions of the image (width and height for example), but rather to the number of free parameters needed to fully characterize the image. The concept of a low dimensional image may seem counter-intuitive, but is actually very useful in target tracking or in scanning [24, 25]; although a full image is usually collected, the location of a target in that image can be accurately described by a small number of free parameters. High dimensional images are more in line with what we think of as an image. These images are in principle infinite dimensional, however in many cases continuous locations and intensities are digitized, reducing them to a high but finite dimension. Both low and high dimensional signals can be thought of as communication channels. Techniques to improve aspects of these images can then be cast in terms of improved channel capacity. This approach allows seemingly different experiments to be treated on the same footing, making their similarities and connections more transparent. 5 In this chapter I review both low and high dimensional imaging systems and their characterization in terms of communication channels. I also review two non-classical concepts—weak values and spatial entanglement—and discuss their role in these imaging systems. 1.1 Information Theory Information theory is a relatively young field, beginning in the 1940’s with the seminal work of Claude Shannon [26, 27]. It concerns itself with the communication over a noisy channel and uses entropy to quantify information. Communication or information transfer can be quantified in terms of mutual information, here I describe the concept of mutual information for continuous and discrete probabilities. 1.1.1 Discrete Probabilities Consider random variables X (the sent message) and Y (the received message) which take the values x and y, characterized by the discrete probabilities P (x) and P (y). The mutual information between these variables is the sum of their individual marginal entropies minus their joint entropy I(X; Y ) = H(X) + H(Y ) − H(X, Y ), (1.1) where H(X) is the marginal entropy. This is shown conceptually in Fig. 1.1. The marginal entropy of X is given by H(X) = − X P (x) log P (x) , (1.2) x∈X the joint entropy of X and Y is given by H(X, Y ) = − X x∈X y∈Y P (x, y) log P (x, y) , (1.3) 6 Figure 1.1: The concept of mutual information is shown visually. The system defined by the variable X is represented by the blue circle on the left, the system defined by the variable Y is represented by the pink circle on the right. Marginal entropies for each system H(X) and H(Y ), respectively, and the entropy of the joint system H(X, Y ) can be calculated. The mutual information is the overlap of the marginal entropies of systems X and Y . The sum of the marginal entropies for system X and system Y counts this overlap region twice, and by subtracting off the joint system entropy we are left only with the mutual information as described by Eq. 1.1. 7 Figure 1.2: An alternative concept of mutual information is shown visually. The system defined by the variable X is again represented by the blue circle on the left and the system defined by the variable Y is again represented by the pink circle on the right. Non-overlapping conditional entropies for each system H(X|Y ) and H(Y |X), respectively, can be calculated. The mutual information is the overlap of the entropies of systems X and Y . The joint system entropy H(X, Y ) minus these conditional entropies gives the mutual information as described by Eq. 1.4. where the function P (x, y) is the joint probability which characterizes the correlation between X and Y . An alternative formulation of the mutual information is I(X; Y ) = H(Y ) − H(Y |X) = H(X) − H(X|Y ), (1.4) where H(X|Y ) is the conditional entropy of X given Y : H(X|Y ) = − X P (x, y) log P (x|y) , (1.5) x∈X y∈Y where P (x|y) is the probability of X = x given that Y = y. This is shown conceptually in Fig. 1.2. This formulation of the mutual information can be useful when the correlation between the random variables is known. 8 1.1.2 Continuous Probability Densities In considering continuous probability densities, it is common to replace the discrete probabilities in the entropic calculations with the continuous probability density functions p(x), p(y), and p(x, y), resulting in Z Hc (X) = − Z Hc (X, Y ) = − p(x) log p(x) dx, (1.6) p(x, y) log p(x, y) dxdy, (1.7) p(x, y) log p(x|y) dx, (1.8) and Z Hc (X|Y ) = − where the subscript c indicates the quantity uses continuous probability distributions. These continuous probability densities however, present somewhat of a problem for several reasons, including the fact that the probability density functions p(x), p(y), and p(x, y) can exceed 1 (over a sufficiently narrow domain). Additionally, the densities are no longer unitless and one cannot sensibly take the logarithm of anything with units. The solution to this problem is to introduce a unit magnitude dimension-conversion constant whose units are the same as that of the probability amplitude functions. These units will depend on the nature of what is being measured, but examples include probability per unit time and probability per unit area, and it is common to suppress the writing of this conversion term. The fact that the continuous probability distribution can exceed 1 indicates the continuous and discrete entropic formulas may not converge in the limit of small discretization widths. The probability density function is related to the 9 discrete probabilities in the following manner: P (x) , ∆x→0 ∆x p(x) = lim (1.9) where the region of x with significant probability is discretized into b bins of width ∆x. We now compare the discrete formulas to the continuous ones: X p(x)∆x log p(x)∆x lim H(X) = − lim ∆x→0 ∆x→0 x∈X ! = − lim ∆x→0 Z X p(x) log p(x) ∆x ! − lim ∆x→0 x∈X p(x) log p(x) dx + lim log ∆x→0 1 =Hc (X) + lim log . ∆x→0 ∆x =− 1 ∆x X p(x) log ∆x ∆x x∈X (1.10) The entropy formulas for discrete and continuous probabilities therefore do not converge to the same value for small discretization width. They are offset by the logarithm of the number of discretization bins b; H(X) ≈ Hc (X) + log b . (1.11) This is not the case for the mutual information formula however; because the mutual information involves adding and subtracting entropies I(X; Y ) = H(X) + H(Y ) − H(X, Y ), the divergent offset terms cancel out, resulting in I(X; Y ) = Ic (X; Y ). (1.12) The experiments that I describe involve either communication or measurement schemes, both of these types of experiments can be described using mutual information. For a communication scheme, the variables are the sent message and the received message. For a measurement scheme, we can think of the measurement apparatus as the communication channel between the system and the 10 observer. The variable X is then the true value of what is being measured, and the variable Y is the measured value. All of the measurements I make use discrete probabilities. 1.1.3 Channel Limitations Physical channels have several limitations, one of them is that the signal power itself is limited. This manifests itself as a limitation of the possible values that X can take. If we assume that the form of this channel limitation is that it has finite variance hx2 i = S, then, using calculus of variations, the p(x) that maximizes entropy (and thus the channel capacity) satisfies the equation Z Z Z d 2 p(x) log (p(x) dx + λ1 x p(x)dx − S + λ2 p(x)dx − 1 = 0. dx (1.13) This simplifies to the requirement that p(x) log p(x) = −λ1 x2 p(x) − λ2 p(x). (1.14) Assuming a nonzero p(x), this requires p(x) = exp(−λ1 x2 − λ2 ) = A exp(−λ1 x2 ), (1.15) R where A = exp(λ2 ). The conditions that hx2 i = S and p(x)dx = 1 require √ λ1 = 1/(2S) and A = 1/ 2πS. The result is that the channel probability distribution that maximizes the entropy, subject to an average power limitation, is a Gaussian distribution: 1 p(x) = √ exp 2πS −x2 2S , and the corresponding entropy is: 2 2 Z 1 −x 1 −x H(X) = − √ exp log √ exp dx 2S 2S 2πS 2πS 1 = log 2πeS . 2 (1.16) (1.17) 11 The multiple spatial dimensions in images act as independent channels. Entropies of independent channels are additive resulting in H(X) = n log 2πeS 2 (1.18) for images in n spatial dimensions. Another common model for channel limitation, rather than an average power limitation, is a peak power limitation. This type of limitation means there is a finite range of values the signal variable can take. The same type of process shows that a channel with this limitation maximizes its mutual information when its probability distribution p(x) is flat across the possible range of values. For this type of channel limitation, a Gaussian distribution is not possible, however such a channel will still have a variance in signal power. The maximum entropy from Eq. 1.18 is then an upper bound that cannot be reached. In addition to signal power limitations, another fundamental limitation is that noise is present in the channel. Noise in the channel manifests itself in the joint probability function p(x, y), or alternatively in p(x|y), the conditional probability function—the sent message is not perfectly correlated to the received message. We can model this noise as additive and uncorrelated to the sent message, such that the received message distribution is the sent message distribution plus noise Y = X + Z. The noise distribution Z is a random variable taking values z with a probability distribution p(z). The mutual information is given by: I(X, Y ) = H(Y ) − H(Y |X) = H(Y ) − H(X + Z|X) = H(Y ) − H(Z|X). (1.19) But the noise Z is assumed to be uncorrelated to the signal X so this reduces to I(X, Y ) = H(Y ) − H(Z). (1.20) 12 The mutual information capacity of a noisy channel is the capacity of the measured variable minus the capacity of the noise. If we assume, in addition to the sent message variable being Gaussian distributed with variance hx2 i = S, that the noise is Gaussian distributed (meaning its deleterious effect is maximized) with variance hz 2 i = N , then hy 2 i = h(x + z)2 i = hx2 i + hz 2 i + hxihzi = S + N . The entropy of Y is then H(Y ) = n log 2πe(S + N ) , 2 (1.21) n log 2πeN , 2 (1.22) and the entropy of the noise is H(Z) = resulting in a mutual information capacity of the channel of n S I(X; Y ) = log 1 + . 2 N (1.23) The units of a channel’s mutual information is the somewhat nebulous “information per transmission.” The logarithm base determines what units are used for information; for base 2 logarithm information is measured in units of bits. Meanwhile, the concept of “transmission” depends on the nature of the channel. Commonly, a transmission is considered to be an interval of time—resulting in units of bits/second, or an amount of received power—resulting in units of bits/watt. It should be noted that changing what is considered a “transmission” changes the noise power. It is in this way that this description accounts for the standard practice of averaging out noise. In this thesis, we concern ourselves with “transmissions” consisting of detected single photons or joint detections of pairs of photons—yielding mutual information in units of bits/photon. 13 Figure 1.3: A profile of the TEM00 mode as defined in Eq. 1.24 is shown in (a), representing the electric field (in arbitrary units). Plots in (b) and (c) show a profile and a density plot of |TEM00 |2 , representing the intensity of this field (in arbitrary units). A profile of the TEM10 mode as defined in Eq. 1.25 is shown in (d), representing the electric field (in arbitrary units). Plots in (e) and (f) show a profile and a density plot of |TEM10 |2 , representing the intensity of this field (in arbitrary units). 1.2 Low Dimensional Images and Weak values Transverse beam deflections are a simple type of a low dimensional image. The beam profile makes up the image and the moving beam centroid then makes up a one or possibly two dimensional characterization. A common mathematical description of this type of image is to expand the field into transverse electromagnetic (TEM) modes. An undeflected beam is simply the Gaussian TEM00 mode given by TEM00 = 2 πw2 1/4 exp −x2 w2 , (1.24) where w is the 1/e2 beam radius (in power). Deflections are represented by combining the TEM00 with nonsymmetric higher order modes such as the TEM10 mode, given by TEM10 = 2x w 2 πw2 1/4 exp −x2 w2 . (1.25) 14 Figure 1.4: A sum of TEM modes as a beam deflection is shown. In the function displayed, 85% of the power is from the TEM00 mode and the remaining 15% is from the TEM00 mode. A cross section of the sum of modes, representing the electric field (in arbitrary units), is displayed in (a). A cross section and a density plot of the magnitude square of the mode sum, representing the intensity (in arbitrary units), is shown in (b) and (c), respectively. This sum of modes approximates a deflection of the beam by almost half the beam radius. These modes are shown in Fig. 1.3. An example of how these modes can be added to cause a beam shift is shown in Fig. 1.4 displaying a beam with 85% of the power in the TEM00 mode and 15% of the power in the TEM10 mode. In addition to target sensing and tracking, beam deflection measurements have applications in metrology fields as diverse as positioning, microcantilever cooling, and atomic force microscopy [24, 28, 29]. The physics of beam deflection metrology and the ultimate measurement sensitivities of such low dimensional images have been studied extensively [25, 29–31]. I use standard methods of creating these low dimensional images, namely using a mirror to tilt a beam, but use a novel technique—weak values—to increase the channel capacity of this imaging communication channel. Weak values are a recent and striking development in quantum physics. The concept was introduced in 1989 in the context of time symmetric quantum mechanics involving both forward and reverse causality [32, 33]. The basic premise is as follows: A pre-selected quantum state with multiple degrees of freedom is weakly perturbed such that two degrees of freedom are linked (or 15 entangled)—the resulting state is then post-selected on only one degree of freedom and the remaining degree of freedom is measured. This pre- and post-selected (thus time-symmetric) state can exhibit strange behavior when measured. The strange behavior can aid in aspects of beam deflection metrology [34], where the beam is the quantum system and the beam centroid is one of the degrees of freedom that we use. Before getting into the details of weak values it is perhaps advantageous to point out that in many situations involving optics, weak values can be described classically [35, 36], a point that is discussed more fully in section 1.2.4. 1.2.1 Weak Values A typical example of weak values using light is as follows: A beam of light is pre-selected on a polarization state by using a polarizer. The polarized beam then passes through a thin calcite crystal that introduces a slight relative displacement between certain orthogonal polarization (dependent on how the crystal is oriented). This is the weak perturbation—entangling the polarization degree of freedom to the position degree of freedom. The beam is then post-selected on a polarization state using a polarizer and finally the centroid position is measured. Strange behavior happens when the post-selection state is nearly orthogonal to the pre-selection state. Because the position degree of freedom has a higher dimensionality (infinite dimensional) than the polarization degree of freedom (two dimensional), there is ambiguity in assigning specific position measurements to specific polarization states—it can appear that measured polarization value lies far outside the normal eigenvalue range. More formally, we preselect an initial quantum state involving two degrees of freedom with different dimensionalities, here I use a two dimensional discrete 16 variable P and an infinite dimensional continuous variable x for definiteness: Z |ψi i = |Pi i ⊗ fi (x)|xidx, (1.26) where fi (x) is is the initial wavefunction of the state in the variable x. We then weakly perturb the system, linking the degrees of freedom with the interaction Hamiltonian H = k P̂ x where the modulus of k is the strength parameter for the perturbation, and P̂ describes how the different polarizations are affected differently. This results in an intermediate state of Z |ψi = eikP x/~ |Pi i ⊗ fi (x)|xidx. (1.27) p hx2 i, meaning the x shifts induced by this perturbation As long as kP/~ are small compared with the initial wavefunction width, we can approximate the intermediate state as Z |ψi = ikP x 1− ~ |Pi i ⊗ fi (x)|xidx. We then post-select on a final polarization state |Pf i: Z ikP x |Pi i ⊗ fi (x)|xidx |ψi = hPf | 1 − ~ ! Z k hPf |P̂ |Pi i = hPf |Pi i 1−i x ⊗ fi (x)|xidx. ~ hPf |Pi i (1.28) (1.29) The overlap term hPf |Pi i describes attenuation and is eliminated by renormalizing the state. This results in the final state: Z |ψf i = kPw x exp −i h ⊗ fi (x)|xidx, (1.30) where Pw = hPf |P̂ |Pi i/hPf |Pi i is the weak value. This final form is valid assuming p kPw x/~ hx2 i. In this final form we can see that the weak, shifting perturbation is modulated by the weak value, which can in principle be larger than 1. 17 Figure 1.5: An example weak value experiment, described in section 1.2.2, is shown. An unpolarized beam of light is incident on a pre-selection polarizer oriented to pass only vertically polarized light. The beam then passes through a calcite crystal oriented to slightly displace diagonal and anti-diagonal polarizations. The pre-selected and perturbed beam then passes through a post-selection polarizer oriented to pass a polarization slightly off of horizontal. A measurement of the beam deflection is then made. Viewing the weak value in this way—of modulating a shift of the wavefunction— allows us to see that the assumption used in the final simplification means simply that we cannot post-select on a portion of the wavefunction that was not there to begin with. 1.2.2 Deflection Amplification For beam deflections we use an initial state involving optical beam polar- ization and spatial beam profile, pre-selecting on vertical polarization |Pi i = |−i and post-selecting on slightly off of horizontal polarization |Pf i = |+i + δ|−i, where δ 1. This experimental setup is shown in Fig. 1.5. We use a calcite crystal oriented to weakly displace diagonal |Di = |+i + |−i and antidiagonal |Ai = |+i−|−i polarizations, described mathematically by P̂ = |AihA|−|DihD|. The post-selection probability is then hPf |Pi i = δ and the weak value is Pw = 2/δ 1. The small displacement caused by the calcite crystal has been ampli- 18 Figure 1.6: The electric fields at several points in the weak value beam deflection experiment are shown. An initial pre-selected Gaussian beam in arbitrary units with radius w is shown in (a). The beam that has been perturbed by the calcite crystal along with the individual diagonal and antidiagonal polarization components of the beam are shown in (b). The vertical lines show the locations of the centroids of the diagonal and antidiagonal polarized beam components. The calcite crystal perturbs the beam by 5% of the radius. The perturbed beam along with the final post-selected beam is shown in (c). The vertical lines representing the polarization component centroids is shown again. The post-selected beam is seen to have decreased intensity but its deflection is seen to lie outside the range allowed by the pure polarizations. The post-selection is set to give a post selection probability of Pps = 20% and a weak value amplification of A = 10 . fied by the weak value, however the remaining beam power is also attenuated. The amplification and attenuation in this case are inversely related. The electric field at various points in the experiment is shown in Fig. 1.6. 1.2.3 Controversy Because of this strange behavior, weak values have been a controversial subject in physics—beginning with their introduction in the provocatively titled paper “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100” [32]. Various objections to weak values have been made including that weak values violate the uncertainty principle [37], and include an incorrect description of the measurement process [37, 38]. Despite the objections, there has been considerable research into the foun- 19 dations of quantum mechanics involving weak values, including an attempt to resolve quantum mechanical paradoxes [39], the simultaneous observation of the wave and the particle nature of a photon [40, 41], and research clarifying the mathematical description of the measurement process [42]. There has also been more applied weak value research involving amplifying optical nonlinearities [43, 44], measuring phase and frequency shifts [45–48], and even incorporating the amplification effect into gravimeters [49]. 1.2.4 What Is Classical and What Isn’t An interesting aspect of weak values is that although they were introduced in the context of an esoteric quantum phenomenon, in many quantum optics implementations their description reduces to a purely classical one [35, 36]. This is due to the fact that superposition and interference—features thought to be purely quantum mechanical in many cases—are considered to be classical effects in optics. Indeed, by using a combination of classical tilt and lead interferometry, one can implement a weak value apparatus. There are several cases when a weak value experiment is non-classical and must use a quantum mechanical description. These cases include: when the particles used in the experiment are massive particles for which wave-like behavior is inherently quantum, when the entangling interaction leads to non-local effects, and when the quantity being measured requires a quantum description. This final case is considered in chapter 3 where light fluctuations, which require a quantum mechanical description, are measured. 20 1.2.5 Weak Value Investigations I study the amplification effect of weak values. For low dimensional imag- ing applications that involve beam deflections, simple analysis suggests that amplifying a deflection by a factor of A could boost the noisy channel’s mutual information from S 1 I(X; Y ) = log 1 + 2 N (1.31) 1 2S I(X; Y ) = log 1 + A . 2 N (1.32) to This fails to include the possibility of a changed noise spectrum, however. Additionally, it neglects the fact that the measurement is made on a small percentage of the photons, affecting a measure of information per detected photon. A more detailed investigation of the effects of weak values on beam deflection measurements is given in chapters 3 and 4. Chapter 3 investigates the smallest disturbance that can be measured using weak values. Chapter 4 considers the amplification as well as well as the effect on noise, determining the optimum signal to noise ratio that weak values can be used to achieve. 1.3 High Dimensional Images and Entanglement In the common usage of the term, an image is a continuous spatial distri- bution of intensities. In this sense, even an image that is small in size is infinite dimensional—practically however, the continuous distribution can be discretized into pixels and the intensities can be digitized. This results in a very high but finite dimensional representation of the image. The applications of high dimensional imaging are incredibly diverse, ranging from television and movie applications, to free space communication and 21 cryptography. In the experiments described in this thesis, I create quantum mechanical images for image based communication channels using entanglement and then use classical techniques to quantify the capabilities of these channels, or to improve their capabilities. This is the inverse of the low dimensional image systems where I use standard techniques for creating images and novel techniques to improve their performance. 1.3.1 Entanglement The concept of entanglement in quantum mechanics is older than weak val- ues but is similarly striking. Two quantum sub-systems are said to be entangled if one sub-system cannot be accurately described independent of the other one (see for example ref. [50]). A typical example of entanglement involves the decay of a spin-zero particle into two spin-1/2 particles, particles A and B. Because this interaction must conserve angular momentum, if particle A has spin “up” in some axis, the other particle must have spin “down”in that axis and vice versa. We can write the quantum state of the composite system as |ψi = |+iA |−iB + |−iA |+iB . We see that the spin degree of freedom of particle A cannot be accurately described without considering the spin of particle B as well—the composite state is entangled. A key aspect of entanglement is that this correlation is independent of measurement axis, referred to as “rotational invariance” for spin based entanglement. The generalization of this aspect to other variables is the presence of simultaneous quantum correlations in conjugate variables. Quantum systems entangled in position are also entangled in momentum; systems entangled in energy are also entangled in time. 22 1.3.2 Paradoxes Although entangled systems do not necessarily involve two spatially sep- arated particles, this case has led to famous paradoxical “thought experiments” including the Einstein, Podolsky, and Rosen (EPR) paradox [51], Karl Popper’s paradox [52], and Hardy’s Paradox [53]. All paradoxes invoke entangled states in an attempt to show quantum theory to be either incomplete or in disagreement with other aspects of the physical world. The EPR paradox is particularly famous and involves a system of two particles that have interacted such that they are position-momentum entangled— meaning the two particles have correlated locations as well as momenta. The paradox is as follows: the two particles travel in opposite directions, moving far from each other and the momentum of particle A is measured. Because of the correlations, once we know the momentum of particle A we also know the momentum of particle B. Relativity prevents signals from particle A being transmitted instantaneously to particle B, and since the distances between the particles can in principle be as large as we like, they argued that the now known momentum of B must have been that value even before the measurement on particle A took place. At the same time that the momentum of particle A is measured, the position of particle B is measured. Therefore it seems that we have simultaneously measured both position and momentum of particle B—a violation of Heisenberg’s quantum mechanical uncertainty principle. An equivalent formulation of the EPR thought experiment can be made with photon polarization [54], the setup is shown in Fig. 1.7. In this formulation there is a source that creates two photons with correlated polarizations in the state |ψi = |+iA |−iB + |−iA |+iB where |+i and |−i indicate horizontal and vertical polarizations respectively. The measurement on photon A is done in 23 Figure 1.7: An EPR experiment using photon polarization is shown. Photon A goes to the left where it is measured in the horizontal/vertical polarization basis. Photon B goes to the right where is is measured in the diagonal/anti-diagonal polarization basis. the horizontal/vertical polarization basis, while the measurement on photon B is done in the diagonal/anti-diagonal polarization basis, and by the same argument, simultaneous values of these non-commuting variables have been made—a seeming violation of the uncertainty principle. EPR were arguing that all variables have definite values at all times, a concept known as “realism,” and that these values are described locally, a concept known as “locality.” EPR did not believe that quantum theory was necessarily wrong, just that it was incomplete—that a locally realistic variable could be added to the theory to make it more complete. The idea was that this added “hidden variable” would not modify predictions of standard quantum theory, but simply allow the modified theory to have a greater range of applicability than the standard quantum theory [55]. 1.3.3 Bell Inequalities Paradoxical thought experiments and conflicting interpretations of quan- tum mechanics were thought to have little potential for experimental investigation. This was shown to be incorrect in 1964 when John Bell derived an inequality obeyed by all locally realistic “hidden variable” theories and violated by standard quantum theory [56]. Bell proposed a straightforward experiment to test 24 Figure 1.8: A Bell experiment using photon polarization is shown. The measurement bases for photons A and B are changed by rotating the corresponding detection apparatus. the inequality using a pair of spatially separated, entangled particles. Like the EPR experiment, Bell’s experiment can be cast in terms of photon polarization. This formulation is identical to the polarization EPR experiment with one key difference—each measurement apparatus is rotated through several measurement bases. The experimental setup is shown in Fig. 1.8 Following the presentation of Sakurai [50], I label the measurement bases by their rotation angle θ1 , θ2 , and θ3 . According to local realistic hidden variable theories, photons have definite polarizations in each of these three measurement bases. For any ensemble then, the mutually exclusive possibilities are: Number N1 N2 N3 N4 N5 N6 N7 N8 Left Photon θ1 +, θ2 +, θ3 + θ1 +, θ2 +, θ3 − θ1 +, θ2 −, θ3 + θ1 +, θ2 −, θ3 − θ1 −, θ2 +, θ3 + θ1 −, θ2 +, θ3 − θ1 −, θ2 −, θ3 + θ1 −, θ2 −, θ3 − Right Photon θ1 −, θ2 −, θ3 − θ1 −, θ2 −, θ3 + θ1 −, θ2 +, θ3 − θ1 −, θ2 +, θ3 + θ1 +, θ2 −, θ3 − θ1 +, θ2 −, θ3 + θ1 +, θ2 +, θ3 − θ1 +, θ2 +, θ3 + The probability of jointly measuring the left traveling photon as horizontally polarized in the θ1 rotated measurement basis and the right traveling photon as 25 horizontally polarized in the θ2 rotated measurement basis is N3 + N4 , P (θ1 +, θ2 +) = P i Ni (1.33) N2 + N4 P (θ1 +, θ3 +) = P , i Ni (1.34) N3 + N7 P (θ3 +, θ2 +) = P . i Ni (1.35) similarly and Since N3 + N4 ≤ (N2 + N4 ) + (N3 + N7 ) we can write P (θ1 +, θ2 +) ≤ P (θ1 +, θ3 +) + P (θ3 +, θ2 +). (1.36) This equation holds for any locally realistic hidden variable description of the experiment. We now compare this to the quantum mechanical prediction. Using the state (in the unrotated basis) |ψi = |+iA |−iB + |−iA |+iB , the probability that photon A is measured as horizontal in the θ1 basis is 1/2. Because the polarizations are perfectly correlated, photon B is known to be vertically polarized in this same basis. Given that this measurement is made on photon A, the probability that photon B is measured as horizontal in the θ2 basis is given by Malus’s law as sin2 (θ12 ), where θ12 = θ1 − θ2 . This results in P (θ1 +, θ2 +) = 1 2 sin (θ12 ) . 2 (1.37) P (θ1 +, θ3 +) = 1 2 sin (θ13 ) , 2 (1.38) P (θ3 +, θ2 +) = 1 2 sin (θ32 ) . 2 (1.39) Similarly and 26 The local hidden variable inequality then becomes sin2 (θ12 ) ≤ sin2 (θ13 ) + sin2 (θ32 ) . (1.40) This inequality is violated for a range of values. For example if we let θ1 = 0, θ2 = π/4, and θ3 = π/8 the inequality becomes 0.50 ≤ 0.29. One can use similar reasoning to come up with many types of Bell like inequalities, involving different types of particles besides photons and different degrees of freedom besides polarization [57, 58]. Experiments measuring Bell type inequalities have been performed. The results support the quantum mechanical prediction [59, 60]. This indicates that it very unlikely that the the observed correlations can be accounted for by a locally realistic hidden variable theory. 1.3.4 Nonlocality Bell inequalities show that quantum theory violates local realism. How- ever, the concepts of locality and realism are distinct and in fact, when considering light polarization, realism is violated by classical optics. It is therefore of interest to be able to determine when a system violates locality as an independent concept. If we assume a system is described by quantum mechanics we allow for realism to be violated, we also accept the Heisenberg uncertainty principle for conjugate variables position and momentum, denoted by x and p, respectively: ∆x∆p ≥ where, for example ∆x = ~ 2 (1.41) p h(x − hxi)2 i. We consider a joint system in spatially separated regions—described by position variables x1 and x2 , and corresponding 27 momentum variables p1 and p2 . If the system obeys locality, then measurements relating to x1 are independent of conditioning measurements made on x2 —indicating that the variance of x1 conditioned upon a measurement of x2 , written as ∆x1 |x2 , is equal to the unconditioned variance of x1 . Using the same reasoning for momentum we can see that, by assuming locality, a system obeys the uncertainty principle ∆x1 |x2 ∆p1 |p2 ≥ ~ . 2 (1.42) Violating this inequality shows that the system is nonlocal [61, 62]. Moreover, since a violation indicates the system cannot be factored into spatially separated x1 and x2 subsystems, the system is entangled. 1.3.4.1 Entropic Uncertainty Heisenberg’s uncertainty principle is the most well known uncertainty principle, however it is possible to derive other uncertainty principles using quantum theory. Because of the connections to information theory and channel capacity it is useful, for our purposes, to make use of an entropic uncertainty principle. I follow the presentation of Bialynicki-Birula and Mycielski [63] and derive an entropic uncertainty principle by considering the position and momentum wavefunctions of a particle, h~x|ψi = ψ(~x) and h~k|ψi = φ(~k) respectively. These wavefunctions are related through the Fourier transform φ(~k) = 1 (2π)n/2 Z ~ ψ(~x)eik·~x d~x (1.43) where n is the spatial dimension (images are two dimensional). The (p, q)-norm of a Fourier transform is defined as the smallest number k(p, q) that satisfies the relation kψkq ≤ k(p, q)kφkp , (1.44) 28 where, for example, Z kψkq = 1/q |ψ(~x)| d~x . q (1.45) The form of the (p, q)-norm is as follows: k(p, q) = 2π q n/2q 2π p −n/2p . (1.46) Of course we are interested in the case when p = q = 2, such that the norms relate to probabilities, but for the purposes of the derivation we allow p and q to vary. Similarly, we impose the restriction that 1 1 + = 1, p q (1.47) such that p is a function of q, leaving only one free parameter and enforcing p = 2 when we set q = 2. We rewrite the definition of the (p, q)-norm as follows W (q; p) ≡ k(q; p)kφkp − kψkq ≥ 0. (1.48) For q = 2 it is clear that W (q; p) is simply subtracting unit probabilities, indicating that W (2, 2) = 0. This, along with the fact that W (q; p) ≥ 0, requires the derivative at q = 2 to be non-negative as well: d W (q; p)|q=2 ≥ 0. dq (1.49) The derivative is: nkψkp 2πe nkψkp 2πe d log log W (q; p) = − − dq 2kq 2 q 2kp2 (q − 1)2 p Z kkψkq kkψkp p p − |ψ(~x)| log |ψ(~x)| d~x + log kψkp 2 2 p(q − 1) p(q − 1) Z kφkq kφkq − |φ(~k)|q log |φ(~k)|q d~k + log kφkq . q q (1.50) 29 At q = 2 we have p = 2, k = 1, and kψkq = kφkp = 1, as well as W (2, 2) = 1, so the derivative becomes: Z d −1 W (q; p)|q=2 = n log πe + |φ(~k)|2 log |φ(~k)|2 d~k dq 2 Z 2 2 + |ψ(~x)| log |ψ(~x)| d~x , (1.51) which is non-negative so, Z Z 2 2 2 2 − |φ(~k)| log |φ(~k)| d~k + |ψ(~x)| log |ψ(~x)| d~x ≥ n log πe . (1.52) The terms |ψ(~x)|2 and |φ(~k)|2 are recognized to be probability densities, indicating the integrals are continuous variable entropies, resulting in Hc (X) + Hc (K) ≥ n log πe , (1.53) where X and K represent the position and momentum variables in this case, but more generally can represent any two conjugate variables related through Eq. 1.43. Changing the base of the logarithms introduces only a multiplicative factor to both sides, so the equation is valid for any logarithm base as long as it is the same for all terms in the inequality. This formula uses continuous probability distributions, whereas measured data will be discrete probabilities. As shown previously, in the limit of small discretization widths, entropies for continuous and discrete probabilites are equal up to an additive offset. The offset serves to increase the entropic bound H(X) + H(K) ≥ n log(πe) + log b1 b2 , where b1 and b2 are the number of position and momentum discretization bins, respectively. As a result, neglecting this offset does not affect the validity of the inequality, and the inequality for discrete probabilities holds, independent of discretization scheme, H(X) + H(K) ≥ n log πe . (1.54) 30 As noted earlier, n = 2 for images, so the entropic uncertainty relation for images is H(X) + H(K) ≥ 2 log πe ≈ 6.18. By the same logic used for Heisenberg’s uncertainty relation, violating this entropic inequality shows that the system is nonlocal (and therefore entangled). 1.3.5 Related Concepts It should be stressed that entanglement, nonlocality, and non-realism are all distinct concepts. Non-realism is more general than entanglement and nonlocality. It is possible to violate a Bell like inequality relying solely on non-realistic properties of quantum systems. The violation results from the modified probability laws required for non-realistic descriptions of reality. Entanglement and nonlocality are very similar and both require non-realistic systems. A system is entangled if it is non-factorable in any two degrees of freedom; a system is only nonlocal if it is non-factorable in spatially separate degrees of freedom. All nonlocal systems are therefore entangled however not all entangled systems are nonlocal. The relationship between these concepts is shown conceptually in Fig. 1.9 1.3.6 Spontaneous Parametric Down-Conversion A common method of generating entangled particles is the process of spon- taneous parametric down-conversion (SPDC) [64–67]. This process, which involves a nonlinear optical interaction to create pairs of entangled photons, is well understood and reliably creates entangled photons that are easily manipulated in the lab. Nonlinear optical materials are those where the polarization P does not vary linearly with the electric field E—it can often be represented as a power 31 Figure 1.9: A “map” of related concepts in quantum mechanics is shown. Classical particle systems and quantum mechanical systems exist on separate islands. All quantum mechanical systems can exhibit non-realism, a subset of these systems exhibit entanglement. A subset of the entangled systems exhibit nonlocality. series in field strength. In Gaussian-cgs units this can be writen as [68]: P ∝ χ(1) E + χ(2) E 2 + χ(3) E 3 + ... (1.55) The process of SPDC is a three wave mixing process and so requires a medium with a χ(2) nonlinearity. It involves a field at frequency ωi producing two fields at ω2 and ω3 , such that ωi = ω2 + ω3 . Quantum mechanically this is a photon of a specific energy being converted into two photons with their energy sum equal to the original photon’s energy. This conservation of energy results in entanglement. Other conservation principles apply as well and indeed photons from SPDC can be entangled in polarization, time and energy, and position and momentum, which is the entanglement I will focus on in this thesis. Momentum conservation restrains the wave-vector output of the SPDC interaction. This so-called “phase matching” requirement is common in nonlinear optical interactions [68]. In SPDC the result is that there are two relevant k- 32 Figure 1.10: A conceptual SPDC interaction is shown where a beam of frequency ωi is incident upon a crystal of length L with a χ(2) nonlinearity. The SPDC interaction in the crystal produces a beam at frequency ωo . Quantum mechanically this is a single ωi photon being transformed into two ωo photons. vector widths: the spread in SPDC output k-vectors (governed by phase-matching considerations) and the initial spread of k-vectors from the input beam (governed by laser specifications). In many cases, including the experiments I perform, we can approximate the resulting two photon state as a double Gaussian with two widths [69, 70]. In the momentum basis this is: Z Z |φo i ∝ exp −(k1 − k2 )2 2b2 exp −(k1 + k2 )2 2a2 (†) (†) a1 a2 |0idk1 dk2 , (1.56) where subscripts label the two SPDC output photons, a is the k-vector spread in the k1 + k2 direction, and b is the k-vector width in the k1 − k2 direction. By Fourier transforming this state we can represent it in the position basis: Z Z |ψo i ∝ exp −(x1 − x2 )2 2/b2 exp −(x1 + x2 )2 2/a2 (†) (†) a1 a2 |0idx1 dx2 . (1.57) This double Gaussian state is shown in both the position and momentum basis in Fig. 1.11. A separable state with the same single photon widths as the entangled state is shown in Fig. 1.12. It is seen that the conditional widths of the separable state are different than those of the entangled state. 33 Figure 1.11: A double Gaussian entangled state from Eq. 1.57 and Eq. 1.56 is shown in (a) and (b) respectively. The state has width in the k1 − k2 direction of a = 2, and width in the k1 + k2 direction of b = 1/2. The Fourier transformed state has the inverse of this: the width in the x1 + x2 direction is 1/a = 2, and the width in the x1 − x2 direction is 1/b = 2. The experimentally accessible single photon width and conditional width are shown in (a). 34 Figure 1.12: A separable state with similarities to the state shown in Fig. 1.11 is shown. The state in the position basis is shown in (a) and the state in the momentum basis is shown in (b). The single photon widths for this state are the same as those of the state shown in Fig. 1.11, however the conditional widths are different. 35 1.3.7 Entanglement in High Dimensional Imaging Applications In an entangled state the conditioned width is only accessible in correlation measurements, and it is this feature that we take advantage of for creating secure communication channels. Considering the mutual information of this channel with the probability density in position given by p(x1 , x2 ) = |hx1 , x2 |ψo i|2 suggests experimentally realizable parameters give mutual information ranging up to 10 or more bits of information per joint photon detection event. I investigate this in chapter 6, where the experimental realization of over 7 bits/photon using position-momentum entangled photons is described. The use of positionmomentum entangled states however requires free space propagation. In chapter 5 I use position-momentum entangled states and investigate possibly negative effects from free space propagation and how to minimize them. 36 Chapter 2 Weak Values and Deflection Aside from the fundamental physics interest in weak values, they also are useful. If we consider the spin of the system as a small signal, the fact that the use of weak values maps this small signal onto a large shift of a measuring device’s pointer may be seen as an amplification effect. Like any amplifier, something must be sacrificed in order to achieve the enhancement of the signal. For weak values the sacrifice comes in the form of throwing away most of the data in the post-selection process. If the detector is shot noise limited, then the advantage gained in the amplification via the weak value is negated by the loss of data in the post-selection. However, most experiments are limited by technical noise, and the weak value technique can improve existing experimental set-ups by orders of magnitude. While this feature was briefly noted in the original weak value paper [32], its utility has been dramatically demonstrated by Hosten and Kwiat [34] who were able to detect a polarization-dependent beam deflection of 1 Å. 2.1 Introduction This chapter describes the development of a weak value amplification tech- nique for any optical deflection. In particular, our weak value measurement uses the which-path information of a Sagnac interferometer, and can obtain dramatically enhanced resolution of the deflection of an optical beam. This technique has several advantages for amplification: the post-selection consists of a photon 37 emerging from the interferometer, and the post-selection attenuation originates from the destructive interference between the two paths, it is therefore completely independent of the source of the optical deflection. In the experiment reported here, the weak measurement consists of monitoring the transverse position of the photon, which gives partial information about the system. The deflection is caused by a slight mirror motion, which, for this geometry, causes opposite deflections for the two interferometer paths. For other geometries the beams can be deflected in the same way. However, this is not a problem for this proposal, because the source of the deflection may be placed asymmetrically in the interferometer, causing one path to be longer (corresponding to a larger spatial shift), and the other path to be shorter (corresponding to a shorter spatial shift). 2.2 Theoretical Description Consider the schematic of the weak value amplification scheme shown in Fig. 2.1. A light beam enters an optical Sagnac interferometer composed of a 50/50 beam splitter and mirrors to cause the beam to take one of two paths and eventually exit the 50/50 beam splitter. For an ideal, perfectly aligned Sagnac interferometer, all of the light exits the input port of the interferometer. The port that all of the light exits is referred to as the bright port, the other port as the dark port. The constructive interference at the entrance port occurs due to the sum of two π/2 phase shifts which occur on reflection in the beam splitter. This symmetry is broken with the presence of a Soleil-Babinet compensator (SBC), which introduces a relative phase φ between the paths, allowing one to continuously change the dark port to a bright port. In presenting the theory, we assume a single photon undergoes a weak measurement. The beam travels through the interferometer, and the spatial shift of the 38 beam exiting the dark-port is monitored. We refer to the which-path information of the interferometer as the system, described with the states {|i, |i}. The transverse position degree of freedom, labeled by the states |xi, is referred to as the meter. A slight periodic tilt is given to the mirror at the symmetric point in the interferometer. This tilt corresponds to a shift of the transverse momentum of the beam. Importantly, the tilt also breaks the symmetry of the Sagnac interferometer, with one propagation direction being deflected to the left, and the other being deflected to the right. This effect entangles the system with the meter via the impulsive interaction Hamiltonian Hi = xAk, where x is the transverse position of the meter, k is the transverse momentum shift given to the beam by the mirror, and the system operator A = |ih| − |ih| describes the fact that the momentum-shift is opposite, depending on the propagation direction. The splitting of the beam at the 50/50 beam splitter, plus the SoleilBabinet compensator (causing the phase-shift φ) results in an initial system state √ of |ψi i = (ieiφ/2 |i + e−iφ/2 |i)/ 2. The entangling of the position degree of freedom with the which-path degree of freedom results in the state Z |Ψi = dxψ(x)|xi exp(−ikAx)|ψi i. (2.1) Where ψ(x) is the wavefunction of the meter in the position basis. This evolution is part of a standard analysis on quantum measurement, where the above transformation would result in a momentum-space shift of the meter, Φ(p) → Φ(p±k), if the initial state is an eigenstate of A. The weak value analysis then consists of expanding exp(−iAkx) to first p order (assuming ka < 1, where a = hx2 i is the beam initial size) and post√ selecting with a final state |ψf i = (|i + i|i)/ 2 (describing the dark-port of 39 the interferometer). This leaves the state as Z hψf |Ψi = dxψ(x)|xi[hψf |ψi i − ikxhψf |A|ψi i]. (2.2) We now assume that ka|hψf |A|ψi i| < |hψf |ψi i| < 1, and can therefore factor out the dominant state overlap term to find Z hψf |Ψi = hψf |ψi i dxψ(x)|xi exp(−ixkAw ), (2.3) where we have re-exponentiated to find an amplification of the momentum shift by the weak value Aw = hψf |A|ψi i hψf |ψi i (2.4) with a post-selection probability of Pps = |hψf |ψi i|2 = sin2 φ/2. The new momentum shift kAw will be smaller than the width of the momentum-space wavefunction, 1/a, but the weak value can greatly exceed the [−1, 1] eigenvalue range of A. In the case at hand, the weak value is purely imaginary, Aw = −i cot φ/2 ≈ −2i/φ for small φ. This has the effect of causing a shift in the position expectation, hxi = 2ka2 |Aw | = 4ka2 /φ, (2.5) assuming a symmetric spatial wavefunction. In these investigations, further enhancement is possible by extending beyond the collimated beam analysis described above, and putting a lens before the interferometer, with a negative image distance si , corresponding to a diverging beam (we neglect diffractive effects). Taking paraxial beam propagation into account, the result analogous to Eq. (2.5) is found to have an additional factor of F = `im (`im + `md )/s2i , where `im is the distance from the lens image to the moving mirror, and `md is the distance from the moving mirror to the detector [71]. 40 From an experimental point of view, it is convenient to express the deflection in terms of easily measurable quantities. This can be done through using the beam size at the detector, σ = a(`im + `md )/si , and the initial beam size at the lens, a, to eliminate si from the equation, and express it in terms of `lm , the distance from lens to the moving mirror. This gives hxi = 2k|Aw | σ 2 `lm + σa`md . `lm + `md (2.6) Finally, we compare this result to the unamplified deflection (without the interferometer) of δ = k`md /k0 , where k0 is the wavenumber of the light so that θ = k/k0 is the small angle the mirror imparts to the light beam. This gives an amplification factor of A = hxi/δ which is independent of k. Under the conditions described below, this gives a magnification about about M = 70|Aw |. However, we expect that it is possible to further amplify the signal with modifications to the optics. 2.3 Experiment and Results A fiber coupled 780 nm laser beam is collimated using a 10× microscope objective. Just after the objective, the beam has a Gaussian radius (denoted by a in the theory) of 640 µm. The beam can be made to be converging or diverging by moving the fiber end relative to the microscope objective. After collimation, the beam passes through a polarizing beam splitter giving a pure horizontal polarization. Half and quarter wave plates are used to adjust the intensity of the beam passing through the polarizing beam splitter. The beam then enters a Sagnac interferometer input port (the pre-selection process). Passing through the interferometer in the clockwise direction, the beam first passes through a half wave plate which rotates the polarization to vertical, the beam then passes 41 Figure 2.1: Experimental setup for interferometric weak values beam deflection measurement. The objective lens collimates a 780 nm beam. After passing through polarization optics, the beam enters a Sagnac interferometer consisting of three mirrors and a 50/50 beamsplitter arranged in a square. The output port is monitored by both a quadrant detector and a CCD camera. The SBC and halfwave plate in the interferometer allow the output intensity of the interferometer to be tuned. The piezo mirror gives a small beam deflection. through a Soleil-Babinet compensator which adds a tunable phase to the beam (the compensator is set to add this phase to vertically polarized beams relative to those polarized horizontally). Passing counterclockwise, the beam first passes through the Soleil-Babinet compensator which now has no relative effect, then through the half wave plate, changing the polarization to vertical. A piezo electric actuator scans the tilt of one of the interferometer mirrors back and forth. A gimbal mount is used so the center of the mirror is the fulcrum. The tilt of the mirror gives the two propagation directions opposite deflections. The small beam deflection is the weak interaction between transverse beam deflection (meter) and which path degree of freedom (system). Post-selection is achieved simply by monitoring the light that exits the dark port of the interferometer. Tuning the Soleil-Babinet compensator to add a 42 small but nonzero relative phase allows a small amount of light out of the dark port. This light is split by a 50/50 beamsplitter and sent to a CCD camera (Newport model LBP-2-USB) which monitors the beam structure, and to a quadrant detector (New Focus model 2921) which monitors beam deflection as well as total power. The interferometer is roughly square with sides of approximately 15 cm. The distance from the microscope objective to the piezo driven mirror is `lm = 48cm. The distance from the piezo driven mirror to the detectors is `md = 114cm (the same distance to both the CCD camera and the quadrant detector). The piezo driven mirror has a lever arm of 3.5 cm. Piezo deflection was calibrated by removing the 50/50 beam splitter from the interferometer and observing beam centroid position on the CCD camera. In this configuration the beam experiences no interference and ray optics describes the beam deflection. Driving the mirror, the piezo response was found to be 91 pm/mV. The piezo response was verified from 500 Hz down to D.C. To characterize the system the interferometer was first aligned well, minimizing the light exiting the dark port. The Soleil-Babinet compensator relative phase was then tuned away from zero, allowing light to exit the interferometer. The piezo driven mirror was given a 500 mV peak to peak amplitude, 100 Hz, sinusoidal driving voltage and the beam deflection was observed using the quadrant detector connected to an oscilloscope. This was done over a range of beam sizes, σ, for three values of Soleil-Babinet compensator phase difference. These measurements, as well as the corresponding theoretical prediction curves given by Eq. (2.6) are shown in Fig. 2.2. The measured data is, in general, well described by the theory. At the smallest Soleil-Babinet compensator angle (7.2◦ ) the small overlap 43 Figure 2.2: Effect of beam radius on interferometric weak values beam deflection measurement. Measured beam deflection is plotted as a function of beam radius, σ. The value of SBC angle (φ) for each data set is labeled. The scale on the left is the measured beam deflection, hxi. The scale on the right is the amplification factor, A. The unamplified deflection is δ = 2.95 µm. The solid lines are theoretical predictions based on Eq. 2.6. between pre and post-selected states allows only a small amount of light to exit the dark port. With this light at low intensities it begins to be of roughly equal intensity to stray light incident on the quadrant detector. This leads to less than ideal amplification, as shown in Fig. 2.2. For fixed interferometer output intensity, the range of detectable deflections was also explored. The interferometer was again aligned such that the beams only had a small phase offset from each other. The alignment was adjusted to give the maximum quadrant detector output while still having a large weak value amplification factor. For these measurements the beam size at the detector was σ = 1235 µm and the weak value amplification factor was 86. The amplification factor was found by driving the piezo with a 500 mV peak to peak 44 Figure 2.3: Angular mirror displacement in interferometric weak values beam deflection measurement. The angular displacement of the mirror is plotted versus the piezo driving voltage. Weak value signal amplification allows small deflections to be measured. The solid line shows the expected deflection based on an interpolation of calibrated measurements of the piezo actuator’s linear travel at higher voltages. These data were taken using a weak value amplification of approximately 86. 45 signal and comparing the measured beam deflection with the aligned interferometer to the measured beam deflection with the interferometer beam splitter removed. The piezo driving voltage was varied over five orders of magnitude while the output of the quadrant detector was sent to a lock-in amplifier and the signal was observed. The smallest driving voltage that yielded measurable beam deflection was 220 nV corresponding to an angular deflection of the mirror of 560 femtoradians (the mirror angle is half the beam deflection angle). These measurements are shown in Fig. 2.3. There is another, perhaps, more interesting point; the deflection indirectly measured the linear travel resolution of the piezo electric actuator. The piezo actuator moved approximately 20 fm in making this measurement. This distance is on the order of large atomic nucleus diameters (Uranium is 15 fm) and is almost six orders of magnitude more resolution than the manufacturer’s specifications of 10 nm. Steps to achieve this resolution increase include: using a quadrant detector with a larger active area which allows a larger beam size to be used, decreasing stray light on the detector by carefully minimizing any back reflections from optics, and aligning the interferometer to have an improved dark port, possibly by using a deformable mirror. 2.4 Channel Analysis By considering the deflecting mirror as the source of the message and the measured signal as the received message, through a noisy communication channel, we can analyze these experimental investigations in terms of communication theory. It is reasonable to model the mirror deflection as a Gaussian random distribution with variance S, and the total noise as an additive Gaussian random distribution with variance N . The maximum channel capacity I is then described 46 by the equation: S I(X; Y ) = log 1 + . N (2.7) In this case, since we are not detecting single photons, but rather classical photocurrents over an extended period of time, it makes sense to use mutual information units of bits per optical power. There is an additionally ambiguity as one can either use the detected optical power, or the sent optical power. The advantage of the bits per detected power measure is that it is invariant to changes in detector efficiencies or other sources of loss that could in principle be overcome by sufficient engineering. The downside is that it neglects any sources of loss that are fundamental—possibly skewing the utility of the measure as a meaningful system characterization. In this experiment the power that is not post-selected on is not lost and can still be utilized, indicating that a measure of bits per detected power may be more natural. It should be noted that these different measures will have different noise powers associated with them. If we naively neglect the effect of the weak value amplification on the noise power, the improvement of the mutual information can be characterized in the following way: By amplifying the signal variable by a factor A we amplify its variance by a factor A2 , similarly through post-selection the detected optical power is decreased by the factor Pps . The amplified mutual information capacity Ia in bits per detected optical power is then: 1 2S Ia (X; Y ) = log 1 + A . Pps N (2.8) In the limit of small signal to noise ratio S N —which was the case for the small deflections in this experiment—the mutual information is amplified by the factor AI = Ia /I = A2 . Pps (2.9) 47 In the measurement sensitivity experiment with A = 86 and Pps = sin2 (2◦ ) ≈ 0.02, resulting in an amplification of the mutual information of AI ≈ 400, 000. This extends to the single photon level as well. The weak value experiment amplified the mutual information per detected photon by a factor of 400, 000, however the initial mutual information per photon was so low that the amplified value is still very small and requires many photon measurements to get appreciable information. It should be noted that this analysis is incomplete in that it ignores the changing noise spectrum between I and Ia . The changed noise will affect the value of A, and it is this topic that we explore in depth in chapter 3. This channel analysis, despite its lack of noise analysis, does point to the utility of the interferometric weak value amplification technique. 2.5 Concluding Remarks In this chapter we have described and demonstrated a method of amplify- ing small beam deflections using weak values. The amplification is independent of the source of the deflection. In this experiment a small mirror deflection in a Sagnac interferometer provides the beam deflection. By tuning the interferometer misalignment and monitoring the resulting small amount of light exiting the interferometer dark port, weak value amplification factors of over 100 are achieved. The weak-value experimental setup, in conjunction with a lock-in amplifier, allows the measurement of 560 femtoradians of beam deflection which is caused by 20 femtometers of piezo actuator travel. Analysis of the experiment as a communication channel suggests the weak value technique can amplify the channel capacity by several orders of magnitude. 48 Chapter 3 Weak Values SNR for Deflections In this chapter the amplification obtained using weak values is quantified through a detailed investigation of the signal to noise ratio for an optical beam deflection measurement. We show that for a given deflection, input power and beam radius, the use of interferometric weak values allows one to obtain the optimum signal to noise ratio using a coherent beam. This method has the advantage of reduced technical noise and allows for the use of detectors with a low saturation intensity. We report on an experiment which improves the signal to noise ratio for a beam deflection measurement by a factor of 54 when compared to a measurement using the same beam size and a quantum limited detector. The ultimate limit of the sensitivity of a beam deflection measurement is of great interest in physics. The signal to noise ratio (SNR) of such measurements is limited by the power fluctuations of coherent light sources such as a laser, providing a theoretical bound known as the standard quantum limit [30]. It was found that interferometric measurements of longitudinal displacements and splitdetection of transverse deflections have essentially the same ultimate sensitivity [25]. Standard techniques to optimize the SNR of a beam deflection measurement include focusing the beam onto a split detector or focusing the beam onto the source of the deflection. The improvement of the SNR is of great interest in not only deflection and interferometric phase measurements but also in 49 spectroscopy and metrology [72, 73], anemometry [74], positioning [24], microcantilever cooling [28], and atomic force microscopy [29, 75]. In particular, atomic force microscopes are capable of reaching atomic scale resolution using either a direct beam deflection measurement [29] or a fiber interferometric method [75]. We show that for any given beam radius, interferometric weak value amplification (WVA) can improve (or, at least match) the SNR of such beam deflection measurements. It has also been pointed out by Hosten and Kwiat that WVA reduces technical noise [34], which combined with our result provides a powerful technique. 3.1 Theoretical Description In chapter 2 I described an interferometric weak value setup measuring beam deflection (caused by a piezo-actuated (PA) mirror) that used the whichpath degree of freedom (the system observable) of a Sagnac interferometer coupled with the transverse degree of freedom (the meter variable) of a laser beam (see Fig. 3.1). This chapter is concerned with the same type of experimental setup. The analogy between interferometry and beam deflection described in a paper by Barnett et al. [31] allows one to predict the SNR for a deflection of an arbitrary optical beam (a coherent beam or a squeezed beam for example). For a coherent beam with a horizontal Gaussian intensity profile at the detector of I(x) = √ 1 2 2 e−x /2σ , 2πσ Barnett et al. show that the SNR is given by r √ 2 Nd R= , π σ (3.1) (3.2) where N is the total number of photons incident on the detector, d is the transverse deflection, and σ is the beam radius defined in Eq. (3.1). Equation 3.2 50 Figure 3.1: Experimental setup for interferometric weak values signal to noise measurement. A fiber coupled laser beam is launched into free space before passing through a polarizer, producing a horizontally polarized single mode Gaussian beam. The laser enters the input port of a Sagnac interferometer via a 50/50 beamsplitter (BS). The light is divided equally and travels through the interferometer clockwise and counterclockwise, encountering three mirrors before returning to the BS. The piezo-actuated mirror (PA), positioned symmetrically in the interferometer, causes a slight opposite deflection for the two different paths, altering the interference at the BS. The dark port is monitored with both a CCD camera and a quadrant cell detector (QCD) positioned at equal lengths from the second BS. The CCD is used only to verify the mode quality of the dark port. 51 represents the ultimate limit of the SNR for position detection with a coherent Gaussian beam. We now incorporate weak values by describing the amplification of a deflection at a split detector as a multiplicative factor A. Thus, da = Ad is the amplified deflection caused by the weak value. Also, the post-selection probability Pps modifies the number of photons incident on the detector such that Na = Pps N. The beam radius is not altered. In chapter 2 I showed that for a collimated Gaussian beam passing through a Sagnac interferometer (see Fig. 3.1) the WVA factor and the post-selection probability are given by A= 2k0 σ 2 cot(φ/2), Pps = sin2 (φ/2), lmd (3.3) where lmd is the distance from the piezo-actuated mirror to the detector, k0 is the wave number of the light and φ is the relative phase of the two paths in the interferometer. Using Eqs. (3.3) and making the substitutions d → Ad and N → Pps N into Eq. (3.2), we find the weak value amplified SNR, RA = α R, (3.4) where α = 2k0 σ 2 cos(φ/2)/lmd . For a typical value of φ we note that cos(φ/2) ≈ 1. The experiment can be modified by inserting a negative focal length lens before the interferometer, creating a diverging beam. This changes the WVA such that the new SNR is given by R0A where C = llm + almd /σ lmd = αR =C σ+a , llm + lmd llm (3.5) p (8N)/π(k0 llm d cos(φ/2))/(lmd (llm + lmd )) and a is the radius of the beam at the lens which is a distance llm from the piezo-actuated mirror. It is 52 interesting to note that the dependence of the SNR is proportional to the beam radius at the detector in the amplified case [Eq. (3.5)] but inversely proportional when there is no amplification (Eq. 3.2). Equations 3.4 and 3.5 are the main theoretical results of this chapter. We see that it is possible to greatly improve the SNR in a deflection measurement with experimentally realizable parameters. Typical values for the experiment to follow are φ/2 = 25◦ , σ = 1.7 mm, lmd = 14 cm and k0 = 8 × 106 m−1 such that the expected SNR amplification is α ≈ 300. We notice that for small φ, the value of α is the ratio of the SNR for a beam deflection measurement in the far-field and the near-field. The far-field measurement can be obtained at the focal plane of a lens. This is recognized as a typical method to reach the ultimate precision for a beam deflection measurement [25]. Consider a collimated Gaussian beam with a large beam radius σ which acquires a transverse momentum shift k given by a movable mirror. The beam then passes through a lens with focal length f followed by a split detector. The total distance from the source of the deflection to the detector is lmd , and the detector is at the focal plane of the lens. This results in a new deflection d0 = f k/k0 and a new beam radius σ 0 = f /2k0 σ at the detector. Making the substitutions d → d0 and σ → σ 0 into Eq. 3.2, we see that when the beam is focused onto a split detector the SNR is amplified: Rf = αf R, (3.6) where αf = 2k0 σ 2 /lmd is the improvement in the SNR relative to the case with no lens [i.e. Eq. (3.2)]. Yet this is identical to the improvement obtained using interferometric weak values, up to a factor of cos(φ/2) ≈ 1 for small φ. Thus we see that the improvement factors are equal using either WVA or a lens focusing 53 the beam onto a split detector, resulting in the same ultimate limit of precision. However, WVA has three important advantages: the reduction in technical noise, the ability to use a large beam radius and lower intensity at the detector due to the post selection probability Pps = sin2 (φ/2). 3.2 Technical Noise We now consider the contribution of technical noise to the SNR of a beam deflection measurement. Suppose that there are N photons contributing to the measurement of a deflection of distance d. In addition to the Poisson shot noise ηi , there is technical noise ξ(t) that we model as a white noise process with zero mean and correlation function hξ(t)ξ(0)i = Sξ2 δ(t). The measured signal x = d + ηi + ξ(t) then has contributions from the signal, the shot noise, and the technical noise. The variance of the time-averaged signal x̄ is given by R t 0 00 P 2 0 00 ∆x̄2 = (1/N2 ) N i,j=1 hηi ηj i + (1/t ) 0 dt dt hξ(t )ξ(t )i, where the shot noise and technical noise are assumed to be uncorrelated with each other. For a coherent beam described in Eq. (3.1), the shot noise variance is hηi ηj i = σ 2 δij . Therefore, given a photon rate Γ (so N = Γt), the measured distance (after integrating for a time t) is given by σ Sξ hxi = d ± √ ± √ . t Γt (3.7) We now compare this with the weak value case. Given the same number of original photons N, we will only have Pps N post-selected photons, while the technical noise stays the same. Taking d → Ad this gives p ! S Pps 1 σ ξ hxi = p αd ± √ ± √ . Pps t Γt (3.8) In other words, once we re-scale, we have the same enhancement of the SNR by α as discussed in Eq. 3.4, but additionally the technical noise contribution is 54 reduced by p Pps from using the weak value post-selection. Therein lies the power of weak value amplification for reducing the technical noise of a measurement. 3.3 Experimental Setup The experimental setup is shown in Fig. 3.1. A 780 nm fiber-coupled laser is launched and collimated using a 20× objective lens followed by a spherical lens with f = 500 mm (not shown) to produce a collimated beam radius of σ = 1.7 mm. For smaller beam radiuses, the lens is removed and the 20× objective is replaced with a 10× objective. A polarizer is used to produce a pure horizontal linear polarization. The beam enters the interferometer (this is the pre-selection) and is divided, traveling clockwise and counterclockwise, before returning to the beamsplitter (BS). A piezo-actuated mirror on a gimbal mount at a symmetric point in the interferometer is driven (horizontally) with a 10 kHz sine wave with a flat peak of duration 10 µs. The piezo actuator moves 127 pm/mV at this frequency with a lever arm of 3.5 cm. Due to a slight vertical misalignment of one of the interferometer mirrors, the output port does not experience total destructive interference (this is the post-selection on a nearly orthogonal state) and contains approximately 20% of the total input power, corresponding to φ/2 = 25◦ . A second beamsplitter sends this light to a quadrant cell detector (QCD) (New Focus model 2921) and a charge coupled device (CCD) camera (Newport model LBP-2-USB). The output from the CCD camera is monitored and the output from the quadrant cell detector is fed into two low-noise preamplifiers with frequency filters (Stanford Research Systems model SR560) in series. The first preamplifier is ac coupled with the filter set to 6 dB/oct band-pass between 3 and 30 kHz with no amplification. The second preamplifier is dc coupled with the filter set to 12 dB/oct low-pass at 30 kHz and 55 Figure 3.2: A comparison of signal to noise ratio of interferometric weak values metrology to standard metrology techniques is shown. The blue curve shows the quantum limited SNR for SD setup calculated using Eq. (3.2). The red circles show the measured SNR was measured for SD setup. The black diamonds show the measured SNR for the WVA setup. As predicted by Eq. (3.4) the data shows linear dependence on driving voltage (and hence deflection d). The data for the SD setup (blue and red) use the right axis whereas the data for WVA setup (black) use the left axis. The lines are linear fits to the measured data with y-intercepts forced to zero. The statistical variations are smaller than the data points. an amplification ranging from 100 to 2000. The low-pass filter limits the laser noise to the 10–90% risetime of a 30 kHz sine wave τ = 10.5 µs) and so we take this limit as our integration time such that the number of photons incident on the detector is N = P τ /Eλ where P is the power of the laser and Eλ is the energy of a single photon at λ = 780 nm. In what follows, we compare measurements using two separate configurations: WVA setup is shown in Fig. 3.1 and produces the weak value amplification SNR found in Eq. 3.4; SD setup (for standard detection) is the same as WVA setup except with the first 50/50 beamsplitter removed, resulting in the SNR given by Eq. 3.2. The theoretical SNR curves in figures 3.2 and 3.3 assume a 56 Figure 3.3: A comparison of signal to noise ratio of interferometric weak values metrology to standard metrology techniques is shown. The blue curve shows the quantum limited SNR for SD setup calculated using Eq. (3.2). The red circles show the measured SNR was measured for SD setup. The black diamonds show the measured SNR for the WVA setup. The data show the linear vs. inverse dependence on beam radius as predicted by Eqs. (3.2) and (3.5). The lines are linear or inverse fits to the measured data. The statistical variations are smaller than the data points. 57 noiseless detector with perfect detecton efficiency; this is what we refer to as an “ideal measurement.” We see reasonable agreement of the data with theory by noting the trends in figures 3.2 and 3.3 follow the predictions of Eqs. 3.4 and 3.5. The quoted error comes from the measured data’s standard deviation from the linear fits. Data were taken for various piezo voltage amplitudes with a beam radius σ = 1.7 mm, a detector distance lmd = 14 cm, and an input power of 1.32 mW (Fig. 3.2). Using SD setup, we measured a SNR that was 1.77 ± 0.07 times worse than an ideal measurement; with WVA, i.e. WVA setup, we measured a SNR that was 21.8 ± 0.5 times better than an ideal measurement, corresponding to an amplification of the SNR by α = 39 ± 3. Next, the beam radius at the detector σ was varied from 0.38 mm to 1.1 mm while the beam radius at the lens was roughly constant at a = 850 µm. For these measurements, the input power was 1.32 mW, the distances were llm = 0.51 m and lmd = 0.63 m, and the driving voltage amplitude was 12.8 mV. The results are shown in Fig. 3.3. Using SD setup, we find that the SNR varies inversely with beam radius as predicted by Eq. 3.2. However, using WVA setup, we see a linear increase in the SNR as the beam radius is increased as predicted by Eq. 3.5. It is interesting to note that equation 3.4 is independent of the detector location lmd . The deflection in the numerator d = lmd ∆θ depends on lmd which cancels with the lmd in the denominator, leaving only the angular deflection ∆θ. this was verified experimentally and suggests a much smaller interferometer can be advantageous by combining amplification effects with increased stability. To demonstrate this we constructed an interferometer with a lmd = 42 mm and a smaller beam radius σ = 850 µm. For this geometry with 2.9 mW of input light and 390 µW of output light, the predicted amplification is α = 260, however the 58 measured value was only α = 150. An important note is that although the small interferometer did not achieve the expected value of α = 260, the smaller size allowed it to approach this expected value much better than the larger interferometer. Indeed, the small interferometer’s measured α was only 260/150 = 1.7 times below the expected value, while the large interferometer’s measured α was 300/39 = 7.7 times below the expected value. 3.4 Channel Analysis Just as I did in chapter 2, by considering the deflecting mirror as the source of the message and the measured signal as the received message, through a noisy communication channel, I can analyze this set of experiments in terms of communication theory. Again, I model the mirror deflection as a Gaussian random variable X with variance S. The noise can in general come from several sources including the beam itself (in the form of shot noise) and the detection system (technical noise), which I model as independent additive Gaussian noises with variances Ns and Nt , respectively. The mutual information for such a system without weak value amplification is I(X; Y ) = log 1 + S (Ns + Nt . (3.9) By incorporating weak value amplification we can increase this mutual information. The effect is slightly different based on which noise terms dominates. For the case of technical noise being the dominant noise term Nt Ns , weak values amplifies the deflection by the factor A, leaves the noise unaffected, and post-selects on a small portion of the beam power given by Pps . Because the noise is unaffected, this results in the mutual information in bits per detected 59 power being amplified by the factor A2 /Pps , as described in by Eq. 2.9 (assuming a small initial SNR). For the case of shot noise being the dominant noise term Ns Nt , weak values amplifies the SNR by the factor α ≈ A/φ meaning S/Ns is amplified by α2 , again the post selection probability is Pps , resulting in 1 2 S Ia (X; Y ) = log 1 + α . Pps Ns (3.10) In the limit of small initial SNR the mutual information is amplified by the factor AI = Ia /I = α2 . Pps (3.11) The small interferometer had an expected SNR amplification of α = 260, and the post-selection probability of Pps = 0.14. Assuming this system was shot noise limited, these values give an expected mutual information amplification of AI = 480, 000. The measured SNR amplification was smaller than expected however, with α = 150, resulting in a mutual information amplification of AI = 160, 000. It should be noted that, for these interferometers, having a shot noise limited detection system will only gain a factor of 10 in the mutual information amplification factor AI . This is a result of the fact that φ ≈ 1/10 in these experiments and α ≈ A/φ. This limitation would not apply to more sophisticated interferometers or other weak value experiments where Pps could be much smaller. A more important effect related to technical noise is that the detector itself has different requirements. As a result of the post-selection, the optical power incident on the detector is reduced. This allows for a different, possibly lower noise detector to be used. In this way, the technique allows for the detection system to more easily (and cheaply) approach being shot noise limited. 60 3.5 Concluding Remarks While the interferometric weak value technique does not beat the ultimate limit for a beam deflection measurement, it does have a number of improvements over other schemes: it reduces technical noise, it allows for the use of high power lasers with low power detectors while maintaining the optimal SNR, and it allows one to obtain the ultimate limit in deflection measurement with a large beam radius. Additionally, we point out that, while weak values can be understood semi-classically, the SNR in a deflection measurement requires a quantum mechanical understanding of the laser and its fluctuations. It is interesting to note that interferometry and split detection have been competing technologies in measuring a beam deflection [25]. Here we show that the combination of the two technologies leads to an improvement that can not be observed using only one, i.e. that measurements of the position of a large radius laser beam with WVA allows for better precision than with a quantum limited system using split detection for the same beam radius. In terms of communication channels, the signal amplification as well as the altered noise is analyzed. The technique is found to amplify the channel capacity in bits/photon by 5 orders of magnitude for standard laboratory optical setups. 61 Chapter 4 Ghost Imaging Through Turbulence This chapter begins my investigation of position-momentum entangled states. I use these states not in metrology settings (as with weak values) but in direct two party communication settings. Before characterizing the capabilities of such states, it is important to know whether they have utility in realistic communication settings. In this chapter I investigate the effect of turbulence on quantum ghost imaging using position-momentum entangled photons. 4.1 Introduction The phenomenon of ghost imaging, first observed by Pittman et al. in 1995 [15], is a method of generating the image of an object from correlation measurements. Pittman’s experiment made use of pairs of entangled photons. One of the photons passed through a transmission object and then to a photon counter with no spatial resolution. The other photon passed directly to a spatially resolving photon counter. When looking at coincident photon detections, the detectors were able to see the object despite the fact that the object and the spatially resolving detector were in different arms of the experiment. While it was initially thought to be a quantum mechanical effect reliant upon the entanglement between the two photons, similar results were later obtained using classical sources [76]. In addition to clarifying the boundary between quantum and classical 62 Figure 4.1: Experimental setup for ghost imaging through turbulence measurement. A pump beam undergoes SPDC at a nonlinear crystal (NLC), the output passes a beamsplitter (BS). One beam is sent through a lens and onto a transmission object. The other beam is sent through a lens and onto a scanning slit. The ghost image of the object is profiled by the slit. Photons are detected with single-photon avalanche diodes (SPAD). Detection events are then correlated. effects [77–79], ghost imaging has been used for lensless imaging [80], superresolution [22, 81], and entanglement detection [82]. More recently, research has recognized connections between ghost imaging and compressive sensing [20, 21]. For many optical applications, imaging through turbulence is unavoidable [83, 84], and research on the effect of turbulence on ghost imaging has recently witnessed a surge of interest [85–88]. In this chapter, I experimentally investigate the effect of turbulence on ghost imaging using position-momentum entangled photons. I present the first experimental demonstration that entangled-photon ghost imaging is affected by turbulence and how the effect can be reduced. 63 Figure 4.2: Conceptual setup for ghost imaging through turbulence measurement. The experiment is shown using the Klyshko picture [64], the object (on the right) is ghost imaged onto the scanning slit (on the left). The nonlinear crystal is offset from the central image plane by a distance ∆. The top picture shows the turbulence—represented by wavy lines—between the crystal and the lens. The bottom picture shows the turbulence located between the lens and the object. Experimentally relevant distances are labelled. Figure 4.3: Conceptual setup for ghost imaging through turbulence measurement. The experiment is shown using the Klyshko picture [64], the object (on the right) is ghost imaged onto the scanning slit (on the left). The nonlinear crystal is offset from the central image plane by a distance ∆. The top picture shows the turbulence—represented by wavy lines—between the crystal and the lens. The bottom picture shows the turbulence located between the lens and the object. Experimentally relevant distances are labelled. 64 4.2 Theoretical Description The experimental apparatus is depicted in Fig. 4.1. A biphoton state |ψi is created at a nonlinear crystal [65] and then split by a 50/50 beamsplitter, sending the biphoton into two arms of the apparatus. In the object arm, the biphoton travels a distance 2f + ∆, to a lens which has focal length f . The biphoton then travels a distance 2f to a photon detector with no spatial resolution (a “bucket” detector). A transmission object— consisting of alternating opaque and clear vertical bars—is placed just in front of the detector. In the image arm, the biphoton travels a distance 2f − ∆ to a lens which again has focal length f . The biphoton then travels a distance 2f to a spatially-resolving detector. For ∆ = 0 the detectors and crystal are all located at image planes of each other. As one arm’s lens/detector is moved towards the crystal by a distance ∆, the other arm’s lens/detector is moved away by the same distance, keeping the sum of the arm’s length constant, see figures 4.2 and 4.3. Turbulent air flow is introduced into the beam path of the object arm. For turbulence between the crystal and the lens, it is a distance l1 from the crystal— or a distance l1 − ∆ from the central image plane. For turbulence between the lens and the object, it is a distance ∆ − l1 from the object. The relevant function for GI is the second order degree of coherence G(2) (x1 , x2 ), where x1 is a transverse position variable in the plane of the spatiallyresolving detector and x2 is a transverse position variable in the plane of the bucket detector. We begin with the standard quantum mechanical form and include an additional ensemble averaging—represented by large outer brackets—to 65 account for the statistical effect of turbulence: † (2) † G (x1 , x2 ) = hψ|Êi (x1 )Ês (x2 )Ês (x2 )Êi (x1 )|ψi . (4.1) Neglecting overall normalization, this can be represented in the following way: Z (4) (2) G (x1 , x2 ) = ψ ? (x̃s , x̃i )H? (x̃i , x1 )H? (x̃s , x2 ; x̃t ) (4.2) ×H(xs , x2 ; xt )H(xi , x1 )ψ(xs , xi )dx̃i dx̃s dxs dxi . Subscript s and i indicate variables in the crystal plane and subscript t indicates variables in the plane of the turbulence. The function ψ(xs , xi ) is the transverse biphoton wavefunction which we approximate as a plane-wave with delta function correlations ψ(xs , x1 ) = δ(xs − xi ). The function H(xs , x2 ; xt ) is a propagation operator going from the crystal plane to the object arm detection plane, passing through the plane of turbulence; H(xi , x1 ) is a propagation operator going from the crystal plane to the image arm detection plane. These operators can be represented in the following way: Z −ik (x2 − xt )2 T̂(xt ) H(xs , x2 ; xt ) = exp 2(l1 − ∆) ik(xt − xs )2 × exp dxt , 2l1 −ik 2 H(xi , x1 ) = exp (xi − x1 ) . 2∆ (4.3) (4.4) In our theoretical treatment, we assume a narrow sheet of turbulent air, whose effect on propagation can be characterized by a multiplicative operator T̂(xt ). We also assume that the lenses are sufficiently large that they capture all of the light from the SPDC source. As a result, both turbulence locations (shown in Fig. 4.2 and Fig. 4.3) are governed by the same operators. We model the turbulence as a 6/3 scaling law effect: T̂? (x̃t )T̂(xt ) = exp [−α (xt − x̃t )2 /2], where α parameterizes the strength of the turbulence and 66 has units 1/m2 [83, 84]. The resulting expression for G(2) (x1 , x2 ) is: " 2 # 2 x − x −k 1 2 G(2) (x1 , x2 ) = exp . 2α (l1 − ∆)2 (4.5) The ghost image I(x1 ) is then the product of the object and G(2) (x1 , x2 ), integrated over x2 . We represent the object as: O(x2 ) = exp [−x22 /2w2 ] 1 + cos(ko x2 ) . Here w is the spatial width of the illuminating beam and ko is √ wavenumber for the object’s pattern spacing. Assuming (l1 − ∆) α k w, the ghost image is found to be 1 x 1 2 I(x1 ) = exp − 1 + V cos(ko x1 ) . 2 w (4.6) I(x1 ) has the same form as O(x1 ) with the object’s unity visibility replaced by the GI visibility V: " V = g × exp 2 # −α l1 − ∆ 2 (k/k0 )2 . (4.7) Where g is the the optimum GI visibility with no turbulence. As either the turbulence increases in strength (increasing α) or the turbulence is moved away from the central image plane or the detector (increasing l1 − ∆), the detected visibility V decreases—thus obscuring the detected pattern. 4.3 Experiment Collimated light from a 3 mW, 325 nm HeCd laser with a 1/e2 full width of approximately 1600 µm pumped a 10 mm thick BBO nonlinear crystal. The crystal was oriented for degenerate type-I collinear spontaneous parametric downconversion (SPDC). After the crystal, the pump beam was blocked by colored glass filters and the SPDC bandwidth was limited by a 3nm wide spectral filter centered at 650 nm. The remaining SPDC beam was split into two arms by a 50:50 beamsplitter. 67 In the image arm, a lens was located 1000 mm − ∆ from the crystal; in the object arm, a lens was located 1000 mm + ∆ from the crystal. Both lenses had focal length f = 500 mm. Detectors were located 1000 mm from the lenses. The transmission object was a test pattern located 1000 mm from the lens. The bucket detector consisted of a 10× microscope objective which collected the transmitted light into a multimode optical fiber. The pattern had unit visibility and 3.6 cycles per mm, which resulted in an object pattern wavenumber of ko = 7.2×π mm−2 . The spatially-resolving detector consisted of a computer controlled scanning slit located 1000 mm from the lens, which was again followed by a 10× microscope objective which collected light into a multimode optical fiber. The slit was approximately 40 µm wide and was scanned in 5 µm increments, giving spatial resolution. The optical fibers were connected to Perkin Elmer single-photon detectors. The outputs of these detectors were time correlated using a PicoHarp 300 from PicoQuant. Photon counts were integrated at each slit location for between 1 and 4 seconds. The spatially resolved coincident detections made up the ghost image profiles. A heat gun was mounted above the setup, providing turbulent air flow across the beam path. The effect of the turbulence was fitted to the model’s wave structure function α x2 [84]. From the fit we determined α = 2.5 ± 1.5 mm−2 . It should be noted that although our theoretical model makes use of a thin sheet of turbulence, experimentally the turbulent region was approximately 10 cm wide. The turbulence was therefore present for a significant portion of the apparatus arm. Data was taken for an unshifted configuration with ∆ = 0, and for a shifted configuration with ∆ = 330 mm. In each configuration, ghost images were 68 recorded with turbulence present in the object arm: both between the crystal and lens, and between the lens and the object. Ghost images were also recorded with no turbulence. The recorded ghost image profiles were fitted to I(x1 ) from Eq. 4.6. The fit included a visibility term which constituted our measurement of the visibility V. While allowing access to the central image plane of the apparatus, the shifted configuration introduced two experimental limitations: the detected flux decreased significantly as a result of the detectors being away from the beam focus, and fewer spatial frequencies contributed to the ghost image as a result of the nonlinear crystal having a stronger aperturing effect. Representative ghost images are shown in Fig. 4.4. With no turbulence, the unshifted configuration produced GI visibilities of 1.00 ± 0.05. The shifted configuration produced GI visibilities of only 0.65 ± 0.05, which can be explained by the true correlation area having finite extent (this is explained in the detailed theoretical analysis of ref. [89]). The scans also show the decreased flux and the broader beam profile associated with the shifted configuration. Visibilities for turbulence between the lens and the object are shown in Fig. 4.5. When turbulence was close to the object, the observed visibility was near its no turbulence levels. As the turbulence was moved away from the object, the GI visibility decreased. The visibility for the unshifted configuration remained above the visibility for the shifted configuration for all turbulence locations. Visibilities for turbulence between the crystal and the lens are shown in Fig. 4.6. This is the main result of the experiment. Visibilities decreased as the turbulence was moved away from the crystal, however, the unshifted configuration had lower fringe visibility than the shifted configuration. Indeed, for turbulence located 432 mm from the crystal, the visibility was V = 0.15 ± 0.04 for the 69 Figure 4.4: Representative ghost images for the unshifted configuration (left), and shifted configuration (right). The top row shows images with no turbulence. The middle row shows images for turbulence between the lens and the object, 203 mm (right) and 229 mm (left) from the object. The bottom row shows images for turbulence between the crystal and the lens, 432 mm from the crystal. Points are experimental data while curves are fits to the data. Counts are measured in coincident photon detections per second. 70 Figure 4.5: Ghost image visibilities turbulence near the object. GI visibilities are shown for turbulence between the lens and the object. Visibilities are plotted as a function of distance from the object to the turbulence (l1 − ∆ in Eq. 4.7). Data for the unshifted configuration are shown as blue circles. Data for the shifted configuration are shown as purple squares. Curves are plots from Eq. 4.7. The solid curve is for the unshifted configuration, with g = 1.00. The dashed curve is for the shifted configuration, with g = 0.65. For both curves α = 2.0 mm−2 . 71 Figure 4.6: Ghost image visibilities for turbulence near the illumination source. GI visibilities are shown for turbulence between the crystal and the lens. Visibilities are plotted as a function of distance from the crystal to the turbulence (l1 in Eq. 4.7). Data for the unshifted configuration are shown as blue circles while data for the shifted configuration are shown as purple squares. Curves are plots from Eq. 4.7. The solid curve is for the unshifted configuration, with g = 1.00. The dashed curve is for the shifted configuration, with g = 0.65. For both curves α = 2.0 mm−2 . The vertical line marks the location of the central image plane. 72 unshifted configuration, while for the shifted configuration it was V = 0.42±0.04. Moving to the shifted configuration tripled the visibility. This effect can be understood physically by recognizing that the image of an object is unaffected by perturbing the phase of the illumination source—images consist of intensities only. By placing the turbulence near one of the image planes, it is as if we are perturbing the phase of the illumination source only, not the propagation. 4.4 Channel Analysis In this experiment I can make direct use of the probability distributions that give rise to the coincident photon detections to calculate the mutual information. When normalized, the G(2) (x1 , x2 ) function, along with the overall beam profile exp [−x22 /2w2 ] make up the joint measurement probability density: 2 1 −(x1 − x2 )2 −x2 p(x1 , x2 ) = exp exp , (4.8) 2 CW π C W2 √ where C is the width from the G(2) (x1 , x2 ) function: C = k/( 2α(l1 − ∆)). Using this probability distribution to directly calculate the mutual information (as explained in chapter one), where the x1 position corresponds to the sent message and the x2 position corresponds to the received message, results in: 1 W2 I = log 1 + 2 . (4.9) 2 C This measures bits of information per joint photon detection event. We can see that by removing the turbulence entirely (α → 0), or offsetting the crystal such that l1 = ∆, the mutual information tends to infinity. This is a consequence of assuming delta function correlations in the quantum state as well as assuming infinitely large lenses. In reality, the finite correlation area and the optical system’s point spread function will limit the mutual information to a finite but still large number. 73 Figure 4.7: The additional mutual information is shown as a function of l1 − ∆ for the case of turbulence near the crystal (shown in figure 4.3), with the crystal shifted by ∆ = 330 mm. This behavior is described by Eq. 4.10 For the situation with turbulence near the crystal, one can analyze the increased mutual information inherent in the shifted configuration. Unlike in the weak value experiments, the experimental parameters here are such that W C, resulting in additional mutual information per biphoton given by W2 W2 1 1 Is − I = log 1 + 2 − log 1 + 2 2 Cs 2 Cs 1 l1 = log , 2 l1 − ∆ (4.10) where Cs is the width of the G(2) (x1 , x2 ) function for the shifted configuration and C is the width for the unshifted configuration. The effect is shown in Fig. 4.7 using the experimentally implemented shift of ∆ = 330 mm. At the l1 = 432 mm point that resulted in a tripling of the fringe visibility, there is 1 additional bit per biphoton resulting from the shifted configuration benefit. This counteracts much of the loss of bits due to the turbulence and a sufficiently engineered system should be able to completely cancel turbulence induced loss of information. 74 4.5 Concluding remarks By moving the crystal from the central image plane we were able to place turbulence in this plane. This decreased the observed effect of turbulence, in fact it more than made up for the inherent loss of visibility associated with the shifted configuration. This technique has use in free space GI applications where turbulence is involved. By arranging detectors to place an image plane at the location of the turbulence, image degradation from the turbulence can be diminished. Although we used optical fields and turbulent airflow, our result applies to any type of propagating wave and a broad class of random or complex media including, for example, biological tissue or metamaterials. Although we have used entangled photons, similar results are expected for thermal light ghost imaging. It should also be noted that the theoretical description assumes delta function correlations for the biphoton state and a thin region, non-Kolmogorov turbulence model [83, 84, 90]. The limitations of the theoretical do not extend to the experimental results, indeed the biphoton state had a correlation size of approximately 50 µm. The turbulence was in reality volume turbulence approximately 10 cm in length, and it did not truly have the Gaussian structure function of our approximation. Indeed, a more sophisticated theoretical description of the experiment has been developed [89]. This model takes into account volume turbulence and more realistic quantum states and agrees better with the experimental data than the theoretical description provided here. In this chapter I have demonstrated a method of ameliorating the effects of turbulence on GI systems, and have provided a theoretical model which accurately describes the experimental data. I shift the source of entangled photons away from a quantum ghost imaging system’s central image plane, and place 75 turbulence near this plane. This dramatically increases the ghost image contrast. For turbulence located 432 mm from the crystal, this technique took the observed pattern visibility from V = 0.15 ± 0.04 to V = 0.42 ± 0.04, tripling the system’s imaging visibility. Analyzed in terms of communication channel and mutual information—the novel configuration allows for additional bits of information per biphoton to be transmitted. For turbulence located 432 mm from the crystal, each biphoton transmitted 1 additional bit of information, thus counteracting much of the loss of information due to turbulence. 76 Chapter 5 Mutual Information In this chapter I investigate the capabilities of position-momentum entangled states as a communication channels. High dimensional Hilbert spaces used for quantum communication channels offer the possibility of large data transmission capabilities. I propose a method of characterizing the channel capacity of an entangled photonic state in high dimensional position and momentum bases. I use this method to measure the channel capacity of a parametric downconversion state, achieving a channel capacity over 7 bits/photon in either the position or momentum basis, by measuring in up to 576 dimensions per detector. The channel violated an entropic separability bound, suggesting the performance cannot be replicated classically. 5.1 Introduction Quantum systems can be entangled in various degrees of freedom. Typi- cal examples include photonic polarization states [91] and atomic or ionic energy levels [92]. These systems enable various technologies such as quantum communication, quantum cryptography, and quantum computation [93–95], however the systems are not in principle limited to two states [18, 19, 96–99]. Indeed, for communication purposes—such as quantum key distribution [100, 101] or dense coding communication [102]—higher dimensional states increase quantum communication channel capacity and offer additional benefits such as increased 77 security [103, 104]. The photonic position degree of freedom is a good candidate for practical high-dimensional entangled systems due to the wide availability of off the shelf technology for manipulating this degree of freedom [105]. In this chapter I describe a quantum channel capacity characterization that considers both the quantum state and the measurement apparatus. I use this method to measure the channel capacity of a high dimensional position and momentum entangled photonic state. The measurements include up to 576 dimensions per detector and demonstrate channel capacities of over 7 bits/photon. The capacity of a quantum channel using entangled photons characterizes information transfer in joint detection events. The locations where joint photon detections can take place can be thought of as characters in an alphabet; the size of this alphabet is the number of distinguishable joint detection locations available within the beam envelopes (see for example [26]). Theoretically, the channel capacity calculation the best possible measurement for a given state used as the channel. Practically however, the measurement technique is not necessarily optimal and the channel capacity depends on the details of this measurement. Experimental characterization of channel capacities has consisted of performing quantum state tomography and then using the result in channel capacity calculations. Recently Pors et al. [106] proposed a more direct way of characterizing the channel capacity without recreateding the full quantum state. By considering the quantum state in conjunction with the measurement apparatus, they define an effective channel dimensionality called the Shannon dimensionality. We propose a similar measure: an effective entropic channel capacity—rather than dimensionality—that considers both the state and the measurement apparatus, and measures bits of information per detection event. This quantity characterizes the information capacity that can be effectively probed for a given system. 78 Figure 5.1: Experimental setup for high dimensional quantum mutual information characterization. A collimated laser beam undergoes spontaneous parametric down-conversion at a nonlinear crystal. The output passes a focusing lens followed by a beam-splitter. The outputs from the beam-splitter are sent to digital micro-mirror devices at either image planes or Fourier planes of the crystal. The micro-mirror devices are set to retro-reflect the beams, a quarter wave plate and a polarizing beam-splitter send the retro-reflected beam to a single-photon detector. A coincidence circuit correlates these measurements. It depends solely on each party’s measurements and is independent of character coding scheme. 5.2 Theoretical description I use mutual information to quantify the channel capacity. Mutual in- formation describes how much information can be determined about a random variable A, by knowing the value of a correlated random variable B [26, 107]. Discrete variables A and B are characterized by the values they take a and b, respectively, and the probability of these discrete values P (a) and P (b), respectively. The mutual information can be written as I(A; B) = H(A) + H(B) − H(A, B), (5.1) 79 where, for example H(A) = − X P (a) log P (a) (5.2) P (a, b) log P (a, b) (5.3) a∈A is the marginal entropy of A, and H(A, B) = − X a∈A b∈B is the joint entropy of A and B. The function P (a, b) is the joint probability distribution which characterizes the correlation between A and B. We created a position-momentum entangled state using spontaneous parametricdownconversion (SPDC). The state, represented both in position and in momentum, is approximated as [12] Z d~xa d~xb f (~xa , ~xb ) â†a â†b |0i Z d~ka d~kb f˜(~ka , ~kb ) â†a â†b |0i |ψi = = (5.4) where â† is the photon creation operator. Subscript a or b indicates the photon is created in the signal or idler mode, which are sent to Alice or Bob, respectively. The function f (~xa , ~xb ) = N exp −(~xa − ~xb )2 4w12 exp −(~xa + ~xb )2 16w22 (5.5) is the entangled biphoton wavefunction in the position basis, and f˜(~ka , ~kb ) = (4w1 w1 )2 N exp(−w12 (~ka − ~kb )2 ) (5.6) × exp(−4w22 (~ka + ~kb )2 ) is the biphoton wavefunction in the momentum basis. In these equations N = (2πw1 w2 )−1 is a normalization constant, w2 is Gaussian width in the x1 + x2 direction, and 2w1 is the Gaussian width in the x1 − x2 direction. These representations are related through a Fourier transform, which inverts the relative 80 Gaussian function widths, such that position correlations become momentum anti-correlations. Experimentally measured widths however are the single photon width sigmap and the conditional width σc . These widths are related to w1 and w2 in the following way: σp2 = σc2 = w22 + w 2 1 2 4w12 w22 4w22 + w12 (5.7) (5.8) For w1 w2 we can approximate the single photon width as σp = w2 and the conditional photon width as σc = w1 . To measure position correlations we put spatially-resolving single-photon detectors at image planes of the SPDC source: to measure momentum correlations we put the detectors at Fourier transform planes of the source. For our purposes then, random variable A corresponds either to the position or momentum of Alice’s photon and B corresponds either to the position or momentum of Bob’s photon. The theoretical maximum mutual information for the wavefunction in Eq. 5.5 (measuring in the position basis), is: Z p(~xa , ~xb ) I(A; B) = − p(~xa , ~xb ) log d~xa d~xb (5.9) p(~xa )p(~xb ) R where p(~xa , ~xi ) = |f (~xa , ~xb )|2 and p(~xa ) = |f (~xa , ~xb )|2 d ~xb . are discrete probability densities For the momentum basis, the same relations hold, but the position variables are replaced by the momentum variables and the position wavefunction is replaced by the momentum wavefunction of Eq. 5.6. For either basis, this theoretical maximum simplifies to 2 2 2 σp 4w2 + w12 I(A; B) = log = log , σc 4w1 w2 (5.10) 81 which is independent of detector characteristics. The ratio σp /σc is the familiar Fedorov ratio for quantifying entanglement [69]. Our physical SPDC state had a beam envelope width of σp = 1500 µm and a correlation width of approximately σc = 40 µm, resulting in an optimum mutual information from Eq. 5.10 of Ic ∼ = 10 bits/photon. This mutual information is slightly different than what we might expect from Shannon’s noisy channel calculations: if we think of σp as the signal channel width and σc as the noise width, then we would expect I(A; B) = log(1 + (σp /σc )2 ). The reason for the discrepancy is that the σc noise is not additive as Shannon’s formula requires. Additive noise can be reduced by signal averaging over time, and indeed for σc = σp this is not the case. We can cast the formula in terms of an effective additive noise N as follows: σ 2 p I(A; B) = log 1 + N σp2 . N = 1 − σc2 /σp2 (5.11) (5.12) We can see that N ≈ σc for σc σp but diverges as σc → σp . The measurement apparatus consisted of a digital micromirror device (DMD) chip reflecting a portion of the signal or idler beam onto a single photon counting module. The DMD chip allowed us to raster scan over the face of the beam in a controllable number of detection pixels, giving varying detector resolution. To incorporate the effects of the measurement apparatus, we integrate the probability density over the pixel area giving a true porbability (rather than a probability density) corresponding to the value of a discrete variable. For position correlations between the mth pixel on Alice’s detector, and the nth pixel on Bob’s detector, the joint detection probability is: Z Z P (m, n) = d~xa d~xb |f (~xa , ~xb )|2 . m n (5.13) 82 Similarly, for momentum correlations between the mth pixel on Alice’s detector, and the nth pixel on Bob’s detector, the joint detection probability is: Z Z ~ P (m, n) = dka d~kb |f˜(~ka , ~kb )|2 . (5.14) m n The detected mutual information in either position or momentum is then X X I(A; B) = P (m) log P (m) + P (n) log P (n) m − n X (5.15) P (m, n) log P (m, n), m,n where, for example, P (m) = X P (m, n) (5.16) n is the marginal probability for a pixel on Alice’s detector. It should be noted that the mutual information calculation only takes into account coincident photon detections. Thus a non-ideal detection efficiency does not change the form of the mutual information equations, rather it reduces the effective SPDC beam intensity. This effect reduces the system’s information per unit time, but not the information per detected photon pair. 5.3 Experiment The experimental setup is shown in Fig. 5.1. A 325 nm wavelength laser beam with diameter a diameter of σp = 1500 µm pumped a 10 mm long BBO nonlinear crystal aligned for Type-I degenerate collinear SPDC. The SPDC output had a correlation width of approximately σc = 40 µm. For these parameters the optimum mutual information from Eq. 5.9 is I ∼ = 10 bits/photon. For measuring position correlations the SPDC output passed through a 125 mm focal length lens followed by a beamsplitter and a spatially resolving detectors were located at the resulting image planes of the crystal. For measuring momentum 83 Figure 5.2: Mutual information for position correlation measurements are shown as a function of detector resolution. Data for detector resolutions of 8 × 8 pixels, 16 × 16 pixels, and 24 × 24 pixels are shown. The dark blue points with error bars are experimental data. The light blue bars are numerical simulations based on Eq. 5.15 both for the case of perfectly relative transversely aligned detectors and the case of a relative transverse misalignment of half a pixel. The red curve is the maximum mutual information that can be detected for the corresponding number pixels per detector. 84 Figure 5.3: Mutual information for momentum correlation measurements are shown as a function of detector resolution. Data for detector resolutions of 8 × 8 pixels, 16×16 pixels, and 24×24 pixels are shown. The dark blue points with error bars are experimental data. The light blue bars are numerical simulations based on Eq. 5.15 both for the case of perfectly relative transversely aligned detectors and the case of a relative transverse misalignment of half a pixel. The red curve is the maximum mutual information that can be detected for the corresponding number pixels per detector. 85 correlations the SPDC output passed through a 150 mm focal length lens followed by a beamsplitter and a spatially resolving detector was placed at the resulting Fourier transform planes of the crystal. The spatially resolving single-photon detectors consisted of a computer controlled digital micro-mirror device from Texas Instruments in conjunction with a Perkin Elmer single photon avalanche diode (SPAD) running in geiger mode. The micro-mirror displays had 1064 × 768 resolution which selectively reflected portions of the SPDC signal and idler beams to SPADs. Photon detection events were correlated with a PicoHarp 300 from PicoQuant with a 3 ns coincident window. The micro-mirror displays were each raster scanned and counts for each pixel pair were recorded for between 1 and 5 seconds. These double raster scans were set such that they were centered on the signal and idler beams, divided into 8 × 8 pixels, 16 × 16 pixels, and 24 × 24 pixels, encompassing 80% of each beam. For ideal alignment, a given pixel on Alice’s detector will correlate very well to only one pixel on Bob’s detector. In practice however, a relative lateral shift of pixels between Alice’s and Bob’s detectors—both vertically and horizontally—spreads correlations to four pixels at best. However, pixels far from the correlated pixel will still have no correlation. This was verified experimentally and it allowed us, for a given pixel on Alice’s detector, to scan only in a region of interest around the correlated pixel on Bob’s detector, thus reducing the time required to complete a double raster scan. Both the predicted and experimentally measured values for mutual information are shown in figures 5.2 and 5.3. Values for position correlation measurements are shown in 5.2 and values for momentum correlation measurements are shown in 5.2. Uncertainties in detected photon number N for each point in the 86 double raster scan were assumed to be √ N. This uncertainty was then propa- gated through the entropy calculations, giving the uncertainties for the measured mutual information values. These values were found to be in agreement with the statistics found by taking multiple data scans for a given detector resolution. It should be noted that this uncertainty calculation method does not take into account detector dark counts. Since the dark counts from each detector are uncorrelated, the dark coincident rate is much less than the coincident rate from the highly correlated SPDC state. Light blue bars represent predicted mutual information values from numerical calculations of Eq. 5.15. The tops of the bars correspond to perfect lateral pixel alignment between Alice and Bob, and the bottoms correspond to relative lateral shifts, both horizontally and vertically, of half a pixel. These cases represent the maximum and minimum mutual information possible for a given number of detector pixels. The dark blue circles represent experimentally measured channel capacities. The red curve gives the maximum mutual information that can be detected I = log(n × n) for n × n pixel resolution per detector. Data for the detectors scanning in 8 × 8 pixels, 16 × 16 pixels, and 24 × 24 pixels are shown. The experimental data agrees well with the theoretically predicted values. For momentum correlations, a maximum mutual information of 7.2 ± 0.3 bits/photon was achieved; for position correlations only 7.1 ± 0.7 bits/photon were achieved. In principle, the two measurement bases should give the same mutual information. However, the alignment for position correlations was more sensitive—the reduction of mutual information for this basis most likely resulted from slight system misalignments. The 576 dimensional measurement space is 16 times larger than the previous maximum for position-momentum entangled photons, and had an increase 87 of bit capacity of more than 50% [99]. It should be noted that channel capacity characterization is different from using the channel for communication. When used for key distribution or communication, the characterized channel will indeed transmit 7 bits of information for a single joint detection event, despite the fact that our characterization method requires many photon detection events. The use of this channel for key distribution or communication does however require some additional structure (such as in ref. [100] or ref. [102]). We are further investigating the ultimate experimental realization these structures. Although our entropic channel characterization measure is similar to the Shannon dimensionality of Pors et al. it has several important differences: the units of the measures are not the same and the the weighting of the different pixel probabilities differs between the measures—however for a flat probability distribution the measures have identical magnitudes. By measuring bits of information rather than dimensionality, our measure is more directly linked to channel information capacity. By taking data in two mutually unbiased bases (position and momentum) I am able to test the separability of the state used as the communication channel. A separable state satisfies the inequality H(A|B)P + H(A|B)M ≥ 2 log2 (πe) ≈ 6.18 where subscripts P or M indicate measurements in the position or momentum bases respectively, and H(A|B) = H(A, B) − H(B) is the conditional entropy of A given B [63, 108, 109]. From the 8 × 8 pixel scan data, we calculate H(A|B)P + H(A|B)M = 3.7 ± 0.3 (5.17) H(B|A)P + H(B|A)M = 3.9 ± 0.2, (5.18) 88 from the 16 × 16 pixel scan data, we calculate H(A|B)P + H(A|B)M = 3.2 ± 0.8 (5.19) H(B|A)P + H(B|A)M = 3.1 ± 0.8, (5.20) and from the 24 × 24 pixel scan data, we calculate H(A|B)P + H(A|B)M = 2.2 ± 0.7 (5.21) H(B|A)P + H(B|A)M = 2.2 ± 0.6. (5.22) As the scan resolution increases, we probe stronger correlations and the separability sums decrease. For 24 × 24 resolution the separability bound is violated by more than 5 standard deviations, indicating it is unlikely that the channel performance can be replicated classically. Although our characterization method is independent of character coding scheme, it suggests a simple one: we assign an alphabet character to each of the pixels. This scheme achieves the measured channel capacities, and errors can be minimized by reducing the size (but not location) of pixels on Alice’s detector, such that the system is unaffected by small relative lateral pixel misalignments between Alice and Bob. The experiment was limited only by pump laser flux. Higher resolution scans are in principle possible, however they are not practical for our setup due to the scan time exceeding the relaxation time of the optical setup. A more powerful pump laser would enable higher resolution scans while maintaining feasible times. Such scans would come closer to experimentally demonstrating the optimum capacity for a position momentum entangled state. 89 5.4 Concluding Remarks In this chapter I have proposed and demonstrated a simple method of char- acterizing the quantum mutual information based channel capacity of a quantum communication channel using position and momentum entangled photons and a controllable pixel mirror. I measured up to 576 dimensions per detector, in both the position and the momentum basis, which resulted in a measured channel capacity of more than 7 bits/photon for either basis. The channel violated an entropic separability bound, indicating the performance cannot be replicated classically. 90 Chapter 6 Conclusion In this dissertation I have demonstrated the utility of several nonclassical quantum imaging techniques—weak values and position-momentum entanglement. I carefully investigated their uses and limitations and found that both have very promising practical applications. The broad applicability of classical information theory allows for the analysis of seemingly disparate ideas under a common theoretical framework. Weak values in the field of metrology, and entangled states in the field of data transmission, are two such disparate ideas linked by their relationship to information. When analyzed in these information terms, natural similarities emerge. The two experiments described in chapters 2 and 3 show that interferometric weak values is a method for dramatically amplifying information transmission rates. Chapter 2 shows this method can be used to amplify incredibly small signals and both chapter 2 and 3 show that the subsequent information transmission rate can be amplified by over 5 orders of magnitude. The applicability of interferometric weak values is limited however to cases where the initial information transmission rate is very low—useful information is gained only over a large ensemble of detected photons. Although weak values are thought of as a quantum phenomenon, the signal amplification obtained in chapter 2 can be described classically. This is because the quantum behavior is related to interference, which is thought of as being 91 a classical phenomenon for light. Noise calculations in optics however are an inherently quantum process and subsequently the amplification of signal to noise ratio described in chapter 3 cannot be described classically. The main benefit of interferometric weak values in practical realizations however may not be this amplification itself, but rather the way in which the amplification is realized. Indeed, as described in chapter 3, the amplification limit does not exceed standard amplification techniques, however it does allow for this amplification to be realized while only making a destructive measurement on a small subset of the photons and with fewer requirements on the measuring detector. On the other hand, the two experiments described in chapters 4 and 5 show that position-momentum entanglement can be utilized to create communication channels with incredibly large information transmission rates with no amplification needed, where many bits of information are transmitted for a single photon pair. Chapter 4 shows the negative effects of the communication environment can be overcome in some cases by a novel experimental configuration. Chapter 5 shows that over 7 bits of information transmitted per photon pair is experimentally achievable. The use of a more sophisticated optical system or the implementation of a multimode quantum dense coding apparatus [102] should bring this number to over 10 bits per photon pair. An additional benefit of a quantum dense coding apparatus is that the communicated message could be controlled rather than being random as in the apparatus in chapter 5. Although the novel configuration in chapter 4 was demonstrated using non-classical, entangled states, it is expected that the benefits would apply to classical states as in thermal ghost imaging experiments. In this way, chapter 4 demonstrates the utility of a classical technique on a quantum system. The 92 channel analysis in chapter 5 however is characterizing this quantum system. The channel performance explicitly violated a separability bound, indicating that it cannot be described classically. 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