UNIVERSITY OF THE WITWATERSRAND DOCTORAL THESIS Time-Efficient Quantum Imaging Author: Chané Simone Moodley Supervisor: Professor Andrew Forbes A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Structured Light Laboratory School of Physics June 7, 2023 http://www.wits.ac.za https://structured-light.org/ https://www.wits.ac.za/physics/ iii Declaration of Authorship I, Chané Simone Moodley, declare that this thesis titled, “Time-Efficient Quantum Imaging” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research de- gree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed my- self. Signed: Date: Chane Moodley Chane Moodley 08 June 2023 v “If you can’t explain it simply, you don’t understand it well enough.” Albert Einstein vii UNIVERSITY OF THE WITWATERSRAND Abstract School of Physics Doctor of Philosophy Time-Efficient Quantum Imaging by Chané Simone Moodley Quantum ghost imaging offers many advantages over classical imaging, includ- ing the ability to probe an object with one wavelength and record the image with another. The low photon fluxes, which are characteristic to quantum optics, offer the ability to probe objects with fewer photons thereby avoiding photo-damage to light sensitive structures, such as biological matter. Unfortunately, quantum ghost imaging suffers from slow image reconstruction due to sparsity and the probabilistic arrival positions of photons. Progressively, quantum ghost imaging has advanced from single-pixel scanning systems to 2-dimensional (2D) digital projective masks which offer a reduction in image reconstruction times through shorter integration times. The focus of this thesis was time-efficient quantum ghost imaging, the neces- sary literature is presented and discussed followed by a focus on the technical details and components required for quantum ghost imaging. Several image reconstruction algorithms using two different 2D projective mask types are showcased and the util- ity of each is discussed. Furthermore, a notable artefact of a specific reconstruction algorithm and projective mask combination is presented and how this artefact can be used to retrieve an image signals heavily buried under artefacts is discussed. Tests to confirm the presence of quantum entanglement were conducted and dis- cussed along with the necessary results to confirm the of quantum entanglement. The Bell inequality was successfully violated with a Bell parameter S > 2, while a full quantum state tomography was performed with an almost perfect fidelity. This thesis was aimed at time-efficient quantum ghost imaging which was achieved through implementing a series of neural network and machine learning based ap- proaches. These approaches consisted of speeding up the image reconstruction pro- cess, enhancing images early on in the reconstruction, establishing an optimal early stopping point and achieving image resolutions that are impractical-to-measure in real time. HTTP://WWW.WITS.AC.ZA https://www.wits.ac.za/physics/ viii First a two-step deep learning approach was proposed to establish an optimal early stopping point based on object recognition, even for sparsely filled images. In step one the reconstructed image was enhanced after every measurement by a deep con- volutional auto-encoder, followed by step two in which a neural classifier was used to recognise the image. This approach was tested on a non-degenerate ghost imaging setup while physical parameters such as the mask type and resolution were varied. A 5-fold decrease in image acquisition time at a recognition confidence of 75% was achieved. The significant reduction in experimental running time is an important step towards real-time ghost imaging, as well as object recognition with few pho- tons, especially in the detection of light sensitive structures. Many computationally intense deep-learning methods have been implemented in an effort to speed up image acquisition times by retrieving image information. Often over-looked, machine learning methods can offer the same, if not better, reduction up in image acquisition time by an object recognition process. Four machine learn- ing algorithms were implemented and trained on a uniquely generated, noised and blurred dataset of numerical digits 1 through 9. Of the tested recognition algorithms, logistic regression showed a 10× speed up in image acquisition time with a 99% prediction accuracy. Additionally, this reduction in acquisition time was achieved without any image denoising or enhancement prior to recognition thereby reduc- ing training and implementation time, as well as the computational intensity of the approach. This method can be implemented in real-time, requiring only 1/10th of the measurements needed for a general solution, making it ideal for quantum ghost imaging and the recognition of light sensitive structures. In quantum ghost imaging the image reconstruction time depends on the resolution of the required image which scale quadratically with the image resolution. A super- resolved imaging approach was proposed based on neural networks where a low resolution image was reconstructed. The low resolution image was subsequently denoised and then super-resolved to a higher image resolution. To test the approach, both a generative adversarial network as well as a super-resolving autoencoder net- work were implemented in conjunction with an experimental quantum ghost imag- ing setup, demonstrating its efficacy across a range of object and imaging projective mask types. A super-resolving enhancement of 4× the measured resolution was achieved with a fidelity close to 90% at an acquisition time of N2 measurements, required for a complete N × N pixel image solution. This significant resolution en- hancement is a step closer to a common ghost imaging goal, to reconstruct images with the highest resolution and the shortest possible acquisition time. The approaches detailed here prove valuable to the community working towards time-efficient quantum ghost imaging. Not only has the image reconstruction pro- cess been reduced by up to 10×, but general image solutions have been enhanced through denoising and super-resolving capabilities. Through the introduction of these techniques and algorithms via machine intelligence, a significant step in time- efficient quantum imaging has been achieved. ix Acknowledgements To my supervisor, Distinguished Professor Andrew Forbes, saying thank you is not enough. Your supervision, guidance, enthusiasm and ability to address un- favourable conditions and circumstances has played the most pivotal role in shaping me as a budding researcher. You are, and always will, be a role model to me. I have learnt so much from you, both about physics and life. I could not have asked for a better supervisor. Thank you for all the opportunities, understanding and assistance. My seniors: Valeria, Angela, Wagner and Najmeh. Thank you for the technical chats, help with code, every morning technical catch ups while on lockdown, always be- ing available on speed dial for chats and a morale boost. Valeria, you are one of the strongest women I know, knowing you is a privilege. Thank you for believing in me even when I didn’t believe in myself. Thank you for assisting me, for always being available and for always keeping in touch. Angela thank you for always listening to me, thank you for allowing me a place to be myself and for always being there for me to lean on. My colleagues: Bereneice, Isaac, Nick and Keshaan. The laughs, the support, the terrible senses of humour, the fun times, the random jokes, the good taste in music and the never ending supply of round-the-clock technical support was so appreci- ated and very necessary. Thank you for the role you have played in my journey. To the newer students: Pedro, Leerin, Lehloa and Cade. Your constant technical questions have made me better at what we do. Thank you for creating that drive and level of enthusiasm, I hope you will carry this energy throughout your careers. Keep asking those difficult questions. To my family, my siblings, and in particular my mother and husband. Thank you for providing round-the-clock care, support, advice, chats, sustenance, snacks, kind- ness and love. You have gotten me to this point and for that I could not be more appreciative. To my dad. Physically you aren’t here but I feel your presence, this PhD was your dream for me and I hope I will make you proud. I miss you and I would not have gotten here without you. Finally, thank you Lord for giving me the physical and mental strength I needed when I didn’t have it. xi Contents Declaration of Authorship iii Abstract vii Acknowledgements ix 1 Introduction to quantum ghost imaging 1 1.1 Quantum ghost imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review and progress in quantum ghost imaging . . . . . . . . . . . . . 2 1.3 Quantum correlations and the phase-matching condition . . . . . . . . 4 1.4 Confirming quantum entanglement . . . . . . . . . . . . . . . . . . . . 8 1.4.1 Quantum state tomography . . . . . . . . . . . . . . . . . . . . . 9 Spatial modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Projective measurements . . . . . . . . . . . . . . . . . . . . . . 10 Experimental measurements . . . . . . . . . . . . . . . . . . . . 11 Results of a two-qubit QST . . . . . . . . . . . . . . . . . . . . . 13 1.4.2 Violating the Bell inequality . . . . . . . . . . . . . . . . . . . . . 15 Quantifying entanglement . . . . . . . . . . . . . . . . . . . . . . 15 Experimental measurements and the Bell parameter . . . . . . . 16 1.5 Position and momentum configurations of quantum ghost imaging . . 17 1.6 From single-pixel scanning to projective masks . . . . . . . . . . . . . . 20 1.7 Spatial resolution and field of view . . . . . . . . . . . . . . . . . . . . . 22 1.8 Machine intelligence in ghost imaging . . . . . . . . . . . . . . . . . . . 23 1.9 Present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Components and considerations for quantum ghost imaging 29 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Practical considerations for the quantum optical setup . . . . . . . . . . 29 2.2.1 SPDC wavelengths and detection . . . . . . . . . . . . . . . . . 30 2.2.2 Pump laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.3 Non-linear crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.4 Spatial light modulators . . . . . . . . . . . . . . . . . . . . . . . 34 Phase-only modulation . . . . . . . . . . . . . . . . . . . . . . . 35 Complex amplitude modulation . . . . . . . . . . . . . . . . . . 36 2.2.5 Projective masks . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.6 Optical fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.7 Simulating the image . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3 Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.1 Experimental specifics . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.3 Photon collection . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.4 Coincidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.5 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 xii 2.4 Image reconstruction and quality factor indices . . . . . . . . . . . . . . 50 2.4.1 Algorithms commonly used in quantum ghost imaging . . . . . 50 2.4.2 Algorithms commonly used in classical ghost imaging . . . . . 51 2.4.3 Image quality factor indices . . . . . . . . . . . . . . . . . . . . . 52 2.5 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5.1 Mask artefact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5.2 Image reconstruction examples . . . . . . . . . . . . . . . . . . . 56 2.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Efficient quantum imaging by enhancement and recognition 61 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Optimal early stopping strategy . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.2 Deep learning network training details . . . . . . . . . . . . . . 62 3.3 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 Neural network architecture . . . . . . . . . . . . . . . . . . . . 65 3.3.3 Physical experimental parameters . . . . . . . . . . . . . . . . . 66 3.3.4 Application of neural networks to the experimental process . . 67 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4 Time-efficient quantum imaging by machine recognition 75 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Concept and strategy using machine learning . . . . . . . . . . . . . . . 76 4.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.2 Machine learning algorithms . . . . . . . . . . . . . . . . . . . . 76 4.3 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.2 Projective mask summary . . . . . . . . . . . . . . . . . . . . . . 79 4.3.3 Dataset generation . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3.4 Machine learning algorithm implementation . . . . . . . . . . . 80 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5 Achieving impractical-to-measure image resolutions 89 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Image reconstruction methods . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.2 Summary of quantum imaging reconstruction algorithms . . . 90 5.3 Super-resolving network details . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.1 Generative adversarial network . . . . . . . . . . . . . . . . . . 92 5.3.2 Super-resolving autoencoder . . . . . . . . . . . . . . . . . . . . 93 5.4 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6 Concluding remarks 101 A MATLAB codes 105 xiii B Python codes 109 B.1 Generative adversarial netowork . . . . . . . . . . . . . . . . . . . . . . 110 B.2 Autoencoder network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Bibliography 117 xv List of Figures 1.1 In a conventional imaging system the photons used to image an object will physically interact with the object, travel to the image plane and transfer information by virtue of position correlations that are estab- lished by the optical system itself. . . . . . . . . . . . . . . . . . . . . . . 2 1.2 In the quantum ghost imaging regime entangled photons are used to image the object. One photon is sent to the object while the twin photon, which has not interacted with the object, is sent to an imaging detector. By measuring both photons in coincidence it is possible to reconstruct an image of the object (an adaptation of the cover artwork for Advanced Quantum Technologies, February 2023 6(2)). . . . . . . . . . 3 1.3 Schematic representation of degenerate spontaneous parametric down- conversion (SPDC) with non-collinear geometry. (a) A high-energy pump photon is incident on a non linear crystal producing two lower energy entangled daughter photons. (b) The energy level description and (c) the wavevector description for the degenerate SPDC process. The insets show the corresponding diagrams for the non-degenerate case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Diagrams of SPDC interaction geometries for (a) collinear and (b) non- collinear phase-matching. Insets show the required momentum con- servation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Intensities of OAM carrying modes, from left to right, ℓ = -3, -2, ..., +2 and +3 units of OAM. Insets show the corresponding phase profile. . . 10 1.6 Analogy between polarisation and OAM can be seen using (a) the Poincaré sphere and (b) the Bloch sphere. . . . . . . . . . . . . . . . . . 11 1.7 The orthogonal and mutually unbiased basis (MUB) states employed for QST projections in the OAM basis for a state space of ℓ = ±1. While both the intensity and phase (as insets) profiles are shown, it is usually the phase profiles that are employed for projective measure- ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Conceptual experimental setup and holograms. (a) A schematic of the experimental setup, where the NLC, SLMs and APDs are placed in conjugate planes. SLMs are imaged onto APDs to perform projective measurements of (b) the six states as shown by the holograms which were displayed on the SLMs. . . . . . . . . . . . . . . . . . . . . . . . . 13 1.9 A two-qubit quantum state tomography. (a) A two-qubit QST where the projected state of each photon in the entangled pair is indicated by its phase map. The color of each box represents the normalised coinci- dence counts for a given set of projections on the two-photon state. (b) Using the tomographic data, the density matrix was computed, and its real component is shown. . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.10 Phase profiles of the holograms used to define the sector state for the ℓ = ±2 OAM subspace. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 xvi 1.11 Violation of the Bell inequality. The normalised coincidence count as a function of relative orientation of the superpositions of orbital angular momenutum (OAM) states for the entangled state |ψ⟩2. . . . . . . . . . 17 1.12 Conceptual sketch of quantum ghost imaging optical setups in the position and momentum configurations. Entangled photons, gener- ated by a high energy pump photon at a non linear crystal (NLC) by spontaneous parametric downconversion (SPDC), are spatially sep- arated along two arms. One photon interacts with the object and is collected by a bucket detector - the idler photon. The other photon (signal photon) is collected by a spatially resolving detector consist- ing of a projective mask and a bucket detector. Each detector is con- nected to a coincidence counting (CC) device to perform coincidence measurements. (a) An illustration of a ghost imaging optical setup in which the object and projective mask are placed in the near-field of the crystal. (b) An illustration of a ghost imaging optical setup in which the object and projective mask are placed in the far-field of the crystal. The insets show the respective ghost images obtained for the different experimental configurations. fi is indicative of the different focal lengths for lenses required for either the position of momentum configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.13 Entangled photons produced, by a high energy pump photon, at the NLC are spatially separated. The signal photon has not interacted with the object and is spatially resolved by a single-pixel detector which mechanically scans the transverse plane to record the posi- tional information of the signal photon. The idler photon interacts with the object and is collected by a bucket detector without record- ing any positional information. Both photons are detected in coin- cidence by a coincidence counter (CC), the coincidences are used to reconstruct the image. Examples of the single-pixel scan are shown on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.14 Entangled photons produced, by a high energy pump photon, at the non-linear crystal (NLC) are spatially separated. The signal photon has not interacted with the object and is spatially resolved by a se- quence of patterned projective masks displayed on a spatial light mod- ulator (SLM) and collected by a bucket detector. The idler photon in- teracts with the object and is collected by a bucket detector without recording any positional information. Both photons are detected in coincidence by a coincidence counter (CC), the coincidences are used to reconstruct the image. Examples of the projective masks are shown on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.15 While there are many factors that affect time-efficiency in quantum ghost imaging, the main factors are the number of measurements, the projective mask type, the mask resolution, and the integration time per measurement. The aforementioned factors are addressed in this thesis, through machine intelligence, to provide a time-efficient ap- proach to quantum ghost imaging. . . . . . . . . . . . . . . . . . . . . . 25 xvii 1.16 A schematic representation of the layers employed in an autoencoder type neural network. The autoencoder represented here is constructed from an encoder section and a decoder section. The encoder com- prises a series of user defined 2D convolutional layers separated by max-pooling layers which lead to the latent space. The decoder com- prises a series of 2D transpose convolutional layers separated by up- sampling layers which lead to the final output image. While this is representative of a common autoencoder network there are many other types as each neural network type can be user defined. . . . . . . 26 2.1 Transmission graph of a bandpass filter centered at λ = 810 nm with a 10 nm FWHM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Beam profiles of a diode laser centered at a wavelength of λ = 405 nm, (a) before and (b) after spatial filtering. . . . . . . . . . . . . . . . . . . . 32 2.3 Methods to achieve optimal phase-matching by (a) angle tuning and (b) periodic poling (temperature tuning). . . . . . . . . . . . . . . . . . 33 2.4 Far-field images of the SPDC transitions (from left to right) of a collinear geometry to a non-collinear geometry from a Type-I BBO sandwich NLC tilted at different angles. In this case the tilt angle determines the geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Image of a typical Holoeye PLUTO 2 spatial light modulator. The inset shows the schematic structure of the liquid crystal display for a reflective SLM, taken from Ref. [133]. . . . . . . . . . . . . . . . . . . . . 35 2.6 To generate the hologram to be displayed on the SLM, the ampli- tude object (or projective mask) is combined with a phase grating also known as a blazed grating. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.7 Example holograms displayed on a spatial light modulator (SLM) for amplitude modulation. The grating is either turned off or on depend- ing on the amplitude distribution shown in the insets. The grating depth, when turned on, is not varied to ensure optimal modulation of photons at all transverse positions on the SLM. . . . . . . . . . . . . . . 37 2.8 The digital object to be imaged is superimposed on a Gaussian dis- tribution which is representative of the SPDC distribution containing the idler photons (left). A projective mask is superimposed on a Gaus- sian distribution representative of the SPDC distribution containing the signal photons (right). The mask is smaller than the Gaussian dis- tribution and the object is smaller than the mask, in this example the SPDC geometry is collinear. Both SPDC distributions are the same size. 38 2.9 Example of a discretised 4 × 4 pixel basis superimposed on the SPDC distribution containing the signal photons. . . . . . . . . . . . . . . . . 39 2.10 Example of a 4 × 4 pixel basis, with a single pixel activated at a time (per frame) to indicate how a single-pixel scan works when the signal photons in a ghost imaging experiment are scanned in the transverse plane. The discretised pixels are superimposed on the SPDC distribu- tion containing the signal photons. . . . . . . . . . . . . . . . . . . . . . 40 2.11 Examples of 32 × 32 pixel resolution random patterned projective mask types used to spatially resolve the signal photon. . . . . . . . . . 41 2.12 Examples of 32 × 32 pixel resolution Walsh-Hadamard patterned pro- jective mask types used to spatially resolve the signal photon. . . . . . 42 2.13 Schematic illustration of different optical fibre core sizes. SMF, MMF and larger mode field diameter MMF (from left to right). . . . . . . . . 43 xviii 2.14 A simulated image (I(x, y)), with a resolution of 32 × 32 pixels, is re- constructed as a linear combination of each projective mask (Pi(x, y)) weighted by a coefficient determined by the detection probability. The calculated detection probability is proportional to the experimental coincidence counts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.15 Schematic diagram of the implemented quantum optical setup for de- generate quantum ghost imaging. Degenerate entangled photons are produced at the NLC . A bandpass filter (BPF) centered at λ = 810 nm filters out any unconverted photons while a half-wave plate (HWP) rotates the polarisation for optimal modulation by the SLMs. A 50:50 beamsplitter is used to spatially separate the entangled signal and idler photons. Each photon impinges on a SLM displaying either the object or projective mask. The photons are collected by coupling each beam to a fibre connected to an APD, the photons that are detected in coincidence are counted by a coincidence counting device (CC). Li are lenses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.16 A histogram of the photons detected in coincidence due to the time correlation, the entangled photons are found to have the same time difference and the counts in that specific time bin of the histogram increase while uncorrelated or ambient photons that are not entangled or correlated are spread randomly across all the time bins and do not contribute to any specific signal build up. . . . . . . . . . . . . . . . . . 49 2.17 The artefact arising from the TGI reconstruction algorithm when cou- pled with the Walsh-Hadamard mask type. (a) Image reconstruction by a 32 × 32 pixel Walsh-Hadamard mask type, before contrast ad- justment. The zoomed-in image shows the mask artefact visible at the top-right corner, in the form of a single activated pixel. (b) Image reconstruction by a 32 × 32 pixel Walsh-Hadamard mask type, after contrast adjustment. Insets in each reconstructed image show the dig- ital object used in the experiment. . . . . . . . . . . . . . . . . . . . . . . 55 2.18 When TGI is coupled with the random mask type no artefacts are present, as such there is no visual difference between (a) the raw re- constructed image and (b) the image which has undergone contrast adjustment. Insets in each reconstructed image show the digital ob- ject used in the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.19 Contrast adjusted reconstructed images for (a) input digital objects, using 32 × 32 pixel (b) random and (c) Walsh-Hadamard masks for re- construction algorithms: TGI, TGIDC, ASGI, ASGIDC, DGI, LGI and NGI, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.20 Statistical test conducted on the contrast adjusted images showing the fidelity per image, per algorithm, for both random and Walsh- Hadamard mask types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.21 Statistical test conducted on the contrast adjusted images showing the PSNR per image, per algorithm, for both random and Walsh- Hadamard mask types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 xix 3.1 Conceptual sketch of an all-digital ghost imaging setup powered by a two-step deep learning approach. Non-degenerate entangled photons are spatially separated along two arms. One photon interacts with the object and is collected by a bucket detector. The other photon is spa- tially resolved where the detector comprises a projective mask and a bucket detector. Each detector is connected to a coincidence counting (CC) device to perform coincidence measurements. The image is re- constructed by a linear combination of the projective masks weighted by the coincidences. The reconstructed image, after each measure- ment, is passed through both the deep learning algorithms for image enhancement and recognition. . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 The architecture of the deep convolutional auto-encoder showing the convolutional and deconvolution layers used. . . . . . . . . . . . . . . 63 3.3 Images below showing steps of dataset distortion achieved by vary- ing a hyper-parameter until the dataset was no longer accurately recog- nised. The corresponding auto-encoder enhanced counterparts are shown directly below. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 The architecture of the classifier showing each of the fully connected layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5 Schematic diagram of the implemented quantum optical setup for non-degenerate ghost imaging. Non-degenerate entangled photons were produced at the NLC. A long-pass filter (LPF) centered at λ = 800 nm filtered out any unconverted photons while a half-wave plate (HWP) rotated the polarisation for optimal modulation by the SLMs. A short-pass dichroic mirror (DM) was used to spatially separate the non-degenerate entangled signal and idler photons, the infrared pho- tons were reflected while the near infrared photons were transmitted. BPFs centered at λ = 810 nm and 1550 nm respectively were used to improve visibility by filtering out any photons of different energies that were produced during SPDC. Each photon impinged on a SLM displaying either the object or projective mask. The photons were col- lected by coupling each beam to a fibre connected to an APD, the sub- set of photons that were detected in coincidence were counted by a coincidence counting device. Li are lenses. . . . . . . . . . . . . . . . . 66 3.6 Results of the deep learning recognition algorithm. The reconstructed raw images for objects (a) four and (b) nine at 20% intervals of the im- age reconstruction time. Followed by the corresponding confidence predictions for all digits and iterations. Vertical dashed lines indicate when the stopping criteria is achieved. . . . . . . . . . . . . . . . . . . . 71 3.7 Results of the two-step deep learning approach. The reconstructed raw images for objects (a) four and (b) nine at 20% intervals of the image reconstruction time, are passed through the auto-encoder for image enhancement, displayed are the corresponding enhanced im- ages. Followed by the corresponding confidence predictions for all digits and iterations. Vertical dashed lines indicate when the stopping criteria is achieved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 xx 3.8 Comparison of the two-step approach for Walsh-Hadamard and Ran- dom masks. (a) The reconstructed images at 20% intervals of the im- age reconstruction time, for 48 x 48 pixel Walsh-Hadamard masks. Followed by the corresponding enhanced images from the auto-encoder output. The confidence level, predicted by the recognition algorithm after enhancement, only for digit 9 is shown beneath. (b) Here the same aspects as (a) are shown however 10000 random masks were used in comparison to the Walsh-Hadamard masks. The solid black lines show the imposed stopping point for each case. It can be seen that the Walsh-Hadamard masks converge 10× faster than the ran- dom masks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.9 Comparison of the two step approach for different resolutions of the Walsh-Hadamard masks. (a) The reconstructed images at 20% inter- vals of the image reconstruction time, for 48 x 48 pixel Walsh-Hadamard masks. Followed by the corresponding enhanced images from the auto-encoder output. The confidence level, predicted by the recogni- tion algorithm after enhancement, only for digit 9 is shown beneath. (b) Similarly for 96 × 96 pixel Walsh-Hadamard masks. The solid black lines show the imposed stopping point for each case at around 20% of image acquisition time. . . . . . . . . . . . . . . . . . . . . . . . 73 4.1 Conceptual sketch of a quantum ghost imaging setup using machine learning algorithms for early object recognition. Entangled photons are generated at the non-linear crystal (NLC) and are spatially sepa- rated along two arms. One photon interacts with the object and is col- lected by a bucket detector using an avalanche-photo diode (APD). The other photon is collected by a spatially resolving detector con- sisting of a binary patterned mask and an APD. Each detector is con- nected to a coincidence counting (CC) device to perform coincidence measurements. The image is reconstructed by a linear combination of the patterned masks weighted by the coincidences. The reconstructed image, after each measurement, is passed through each of the four algorithms for object recognition. . . . . . . . . . . . . . . . . . . . . . 76 4.2 Examples of 32 × 32 pixel resolution Walsh-Hadamard patterned mask types that are used in the ghost imaging experiments to spatially re- solve the idler photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 Uniquely generated and normalised dataset, on which the four ma- chine learning algorithms were trained for 50 - 450 masks in intervals of 100 masks, for digits 2 and 4. The digital objects are shown as insets in each frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Experimental image reconstructions for 50 - 450 masks in intervals of 100 masks, for digits 2 and 4. The digital objects are shown as insets in each frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 Confidence predictions of the machine learning algorithms for input digits 2 and 4, normalised to the same scale for comparative pur- poses. All algorithms were tested on images reconstructed by Walsh- Hadamard masks and are shown as follows: (a) SVM, and (b) LR, for objects digit 2 and digit 4 (indicated by the red outline on the y-axis), respectively. The dashed lines represent the point at which a confi- dence prediction greater than 75% is achieved for the input digit of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 xxi 4.6 Confidence predictions of the machine learning algorithms for input digits 2 and 4, normalised to the same scale for comparative pur- poses. All algorithms were tested on images reconstructed by Walsh- Hadamard masks and are shown as follows: (a) NN, and (b) NB, for objects digit 2 and digit 4 (indicated by the red outline on the y-axis), respectively. The dashed lines represent the point at which a confi- dence prediction greater than 75% is achieved for the input digit of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 Bar graph indicating the average number of masks required to achieve the recognition criteria, for early stopping, across all digits using the Walsh-Hadamard masks for the experimental image reconstructions across all the classifiers tested. . . . . . . . . . . . . . . . . . . . . . . . 86 4.8 The work covered in this chapter was featured on the cover of Ad- vanced Quantum Technologies, February, 2023. . . . . . . . . . . . . . . . 87 5.1 Conceptual sketch of the quantum ghost imaging optical setup. En- tangled photons are spatially separated along two arms. One photon interacts with the object and is collected by a bucket detector. The other photon is collected by a spatially resolving detector comprising a projective mask and a bucket detector. Each detector is connected to a coincidence counting (CC) device to perform coincidence measure- ments. The reconstructed low res image is then sent through a series of neural networks for denoising and super-resolving to a higher res- olution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2 Schematic diagrams of the implemented neural network architectures showing the series of implemented 2-dimensional convolutional lay- ers. (a) Schematic diagram of the implemented generative adversar- ial network (GAN), trained on simulated data to denoise the recon- structed image. (b) Schematic diagram of the implemented super- resolving autoencoder (SR AutoE), trained to enhance the resolution of an image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Fidelity results of the different reconstruction algorithms, insets show the image after a complete reconstruction. . . . . . . . . . . . . . . . . . 96 5.4 Results of the image reconstructions for both digits that were recon- structed, denoised and super-resolved. (a) The digital objects that were used in the experiment. (b) Results of the image reconstruction by random masks - from left to right: the image obtained from the ex- periment, the denoised image from the GAN, the super-resolved im- age output from the SR AutoE. (c) Results of the image reconstruction by Walsh-Hadamard masks - from left to right: the image obtained from the experiment, the denoised image from the GAN, the super- resolved image output from the SR AutoE. . . . . . . . . . . . . . . . . 97 5.5 The results of the object-image fidelity calculated from the SR AutoE output for objects of several resolutions for a 32 × 32 pixel object. . . . 98 5.6 The results of the average object-image for the different reconstruction algorithms, the GAN output, and the SR AutoE output for a 32 × 32 pixel object. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 xxiii List of Tables 1.1 Summary of the different phase-matching regimes for SPDC, assum- ing that the pump is horizontally (H) polarised and whether collinear or non-collinear geomertries are possible. . . . . . . . . . . . . . . . . . 7 2.1 Specifications of different avalanche photo-diodes (APDs) employed. . 31 3.1 Summary of experimental parameters that were varied during the ex- periment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 The number of masks required to satisfy the early stopping criteria, for each tested digit per classifier. . . . . . . . . . . . . . . . . . . . . . . 85 xxv List of Publications 1. Moodley, C., Sephton, B., Rodríguez-Fajardo, V. and Forbes, A., 2021. Deep learning early stopping for non-degenerate ghost imaging. Scientific Reports, 11(1), p.8561. 2. Moodley, C. and Forbes, A., 2022. Super-resolved quantum ghost imaging. Scientific Reports, 12(1), pp.1-9. 3. Moodley, C., Ruget, A., Leach, J. and Forbes, A., 2023. Time-Efficient Object Recognition in Quantum Ghost Imaging. Advanced Quantum Technologies, p.2200109. 4. Moodley, C. and Forbes, A., 2023. All-digital quantum ghost imaging: tutorial. In review at JOSA B. 5. Ruget, A., Moodley, C., Forbes, A., and Leach, J., 2023. Translated object iden- tification for efficient quantum ghost imaging. In preparation. 6. Sroor, H., Moodley, C., Rodríguez-Fajardo, V., Zhan, Q. and Forbes, A., 2021. Modal description of paraxial structured light propagation: tutorial. JOSA A, 38(10), pp.1443-1449. 7. Sephton, B., Nape, I., Moodley, C., Francis, J. and Forbes, A., 2023. Revealing the embedded phase in single-pixel quantum ghost imaging, Optica 10, 286- 291. 8. Nape, I., Sephton, B., Ornelas, P., Moodley, C., and Forbes, A., 2023. Quantum structured light in high dimensions. In review at APL Photonics. xxvii List of Proceedings 1. Moodley, C. and Forbes, A., 2022, October. Does quantum ghost imaging need a camera to image an object?. In Laser Beam Shaping XXII (Vol. 12218, pp. 7- 17). SPIE. 2. Moodley, C., Sephton, B., Rodríguez-Fajardo, V. and Forbes, A., 2021, August. Time-efficient non-degenerate ghost imaging powered by deep learning. In Quantum Communications and Quantum Imaging XIX (Vol. 11835, pp. 55- 65). SPIE. 3. Moodley, C., Sephton, B., Rodríguez-Fajardo, V. and Forbes, A., 2022, May. Fast neural-network-enhanced quantum imaging. In CLEO: QEL - Fundamen- tal Science (pp. FM1C-5). Optica Publishing Group. 4. Moodley, C. and Forbes, A., 2022, October. Quantum imaging gets super smart. In Society of Photo-Optical Instrumentation Engineers (SPIE) Confer- ence Series (Vol. 12238, p. 1223803). 5. Moodley, C., Sroor, H., Rodríguez-Fajardo, V., Zhan, Q. and Forbes, A., 2021, September. Demonstrating a modal approach to paraxial light propagation for photonics education. In Education and Training in Optics and Photonics (pp. W2A-6). Optica Publishing Group. xxix Dedicated to my mother and husband I could not have done it without you. To my father - I miss you. 1 Chapter 1 Introduction to quantum ghost imaging In this chapter quantum ghost imaging is introduced, where a review and the cur- rent progress in quantum ghost imaging is presented. Quantum correlations and the phase-matching condition for spontaneous parametric downconversion are in- troduced, followed by a brief discussion on the components required for testing and confirming the presence of entanglement. Two possible optical configurations for quantum ghost imaging are detailed and important considerations such as the spa- tial resolution and field of view are presented. Finally, the use of machine intel- ligence through machine learning and deep learning methods in ghost imaging is discussed, which forms the basis for the work presented in this thesis. 1.1 Quantum ghost imaging Before delving into the practicality of quantum ghost imaging, it is important to in- troduce the physics of why quantum ghost imaging occurs. To introduce the physics, a conventional imaging system is first outlined. In a conventional imaging system, the photons used to illuminate the scene will interact with the object, travel to the image plane and transfer information by virtue of the position correlations which are established by the optical system itself. A schematic example of a conventional imaging system is shown in Fig. 1.1. In a conventional imaging system, as such, if the object is assumed to be either externally illuminated or self illuminating, each point on the surface of the object can then be thought of as a point radiation sub-source. An imaging lens can map the object plane to the image plane, point to point, estab- lishing the necessary position correlations. If the positions are not carefully mapped out the image that is retrieved is blurred and unclear, and if the position correlations are carefully mapped out a sharp, clear image is retrieved. An imaging lens is used to focus the light scattered or reflected by the object onto an image plane, defined by the Gaussian thin lens equation: 1 so + 1 si = 1 f , (1.1) where so is the distance between the object and the imaging lens, si is the distance be- tween the imaging lens and the image plane and f is the focal length of the imaging lens. The Gaussian thin lens equation defines a point-to-point relationship between the object plane and the image plane. Each point on the object plane is mapped to a unique point on the image plane. By exploiting correlations unique to the quantum world, it is possible to establish the 2 Chapter 1. Introduction to quantum ghost imaging Object Lens Image s! s" 𝑓𝑓 FIGURE 1.1: In a conventional imaging system the photons used to image an object will physically interact with the object, travel to the image plane and transfer information by virtue of position correlations that are established by the optical system itself. required position correlations by the principles of quantum mechanics and in par- ticular quantum entanglement. Quantum correlations, of an entangled photon pair, are established at the source and are carried forward through the optical system. The position correlations are present in both the near- and far-field of the source and are indicative of a realisation of Einstein-Podolsky-Rosen (EPR) type correlations in the transverse spatial degree of freedom of the photon pairs [1, 2]. In quantum ghost imaging, the entangled photon pair is spatially separated. One photon is sent to the object and the other (which has not interacted with the object) is sent to an imaging detector which detects the transverse position of the photon. By measur- ing both photons in coincidence, it is possible to reconstruct an image of the object. By measuring the spatial information of one photon it is possible to infer the spa- tial information of the entangled counterpart. The technique known as quantum ghost imaging is conceptually illustrated in Fig. 1.2. Quantum ghost imaging is a sub-discipline in the larger disciplines of both quantum imaging and ghost imaging. 1.2 Review and progress in quantum ghost imaging A consequence of quantum mechanics is the non-local correlation of a multi-particle system which is observable in the joint-detection of spatially separated particle- detectors [3]. Quantum ghost imaging is one such phenomenon that saw its in- ception in 1995 when Pittman and co-workers first demonstrated this imaging ap- proach, which was informed by the theoretical work of Klyshko conducted in 1988 [4]. Quantum ghost imaging employs entangled signal and idler photon pairs. Ini- tially demonstrated in the experiment undertaken by Pittman et al. [5], these pho- ton pairs were produced by spontaneous parametric downconversion (SPDC). The beauty of quantum ghost imaging is that no individual photon, of the entangled pair, can reveal the image information, rather the image is revealed through the mutual correlations of the entangled photon pair by detecting the pair in coincidence [6–8]. 1.2. Review and progress in quantum ghost imaging 3 FIGURE 1.2: In the quantum ghost imaging regime entangled photons are used to image the object. One photon is sent to the object while the twin photon, which has not interacted with the object, is sent to an imaging detector. By measuring both photons in coincidence it is possible to reconstruct an image of the object (an adaptation of the cover artwork for Advanced Quantum Technologies, February 2023 6(2)). Pittman et al. concluded that although they had utilised entangled photons, it was indeed possible to employ a classical source that would reproduce the ghost imag- ing phenomenon. Subsequently, Abouraddy and co-workers developed a theoretical model that showed an entangled photon source produced images where the charac- teristics of the image differed fundamentally from images obtained by a correlated, but not entangled, photon source [9]. In 2002 Bennink et al. experimentally demon- strated that a classical source yielded a ghost image [10], albeit with higher photon numbers than in the quantum regime [11, 12]. A theoretical paper in 2003 by Gatti et al. showed that classically it is not possible to reproduce a key feature of quantum ghost imaging, where a SPDC source was used [13]. This key feature was that entan- glement allows for ghost images to be reconstructed in both the near- and far-field of the entanglement source, while the classical source can form an image in either the near- or far-field but not both [13]. This translates back to the rules of quantum mechanics where it is possible to first emit a photon and only after it has been emit- ted to decide whether to measure its position or transverse momentum. In 2004 this was confirmed, where it was shown that a source of entangled photons could form ghost images in either its near- or far-field [14]. In preparation for physical objects, such as biological samples, digital objects are used in a sub-field known as all-digital quantum ghost imaging. Digital objects are displayed on a light modulating device, this also serves the purpose of know- ing the ground truth so as to establish image quality. Recent years have seen the 4 Chapter 1. Introduction to quantum ghost imaging emergence of all-digital quantum ghost imaging in an effort to develop and test ad- vancements to technique and efficiency. The goal being to realise the most efficient state, of quantum ghost imaging, before utilising these advancements to image light sensitive biological structures. Quantum ghost imaging was also recently demon- strated with entanglement swapped photons [15] and symmetry engineered quan- tum states [16] and has seen significant growth in the past two decades [11, 17–19], with the promise of enhanced resolution [20, 21], while expanding the application to the imaging of photo-sensitive structures [11, 22], and moving into the x-ray [23, 24] and electron wavelength spectrum [25, 26]. Imaging quality has steadily im- proved while acquisition times have steadily decreased, these advancements are fu- eled by progress in computational imaging [27, 28], compressive sensing [29, 30] and deep learning based methods [31–34]. While there have been considerable advance- ments to date, quantum ghost imaging still requires substantial development before a commercially viable product becomes available. All-digital quantum ghost imag- ing provides a platform for efficient tests which will pave the way towards quantum microscopy. Quantum ghost imaging is therefore a powerful alternative imaging technique to image light sensitive structures. In the quantum regime, low photon numbers allow for little to no damage to photo-sensitive structures which importantly paves the way towards quantum microscopy of biological samples [17], while also permitting measurements in either the near- or far-field of the entangled photon source which is not reproducible by a classical source. Additionally, characteristic to quantum ghost imaging is the special performance it offers, where it is possible to produce non- degenerate signal and idler photons. The object is probed or illuminated with one wavelength while the spatial information is recorded at another wavelength where the imaging detector is less noisy, more sensitive or cost effective [22]. 1.3 Quantum correlations and the phase-matching condition At the core of quantum optics and specifically quantum ghost imaging, lies the pro- duction of entangled photon pairs. A commonly employed method to generate en- tangled photon pairs is that of SPDC by a non linear crystal (NLC). It has been well established that SPDC leads to the creation of entangled photons and is therefore a hallmark of quantum [35–42]. As the SPDC process is produced entirely by the elec- tromagnetic vacuum, it is a purely a quantum process [37–40, 43]. Non linear optical processes, such as SPDC, comprise light interacting with materials whose properties are modified in the presence of photons [43]. The strength of the optical electric field determines the polarisation response of the material. This is often described as: P(t) = ϵ0[χ (1)E(t) + χ(2)E2(t) + χ(3)E3(t) + ...], (1.2) where P is the polarisation of the material (dipole per unit volume), E is the strength of the optical electric field, ϵ0 is the permittivity of free space and χ is a tensor known as the optical susceptibility of the order of non linearity, expressed in the superscript inside the brackets. Accordingly, different responses are promoted based on the term in the expansion. As examples, the second term (ϵ0χ(2)E2(t)) would refer to a second-order non linear process and would favour occurrences such as second har- monic generation. Correspondingly, the third term (ϵ0χ(3)E3(t)) is then a third-order non linear process and would favour occurrences such as third harmonic generation. By fabricating the non linear susceptibility of a material for a specific order, while 1.3. Quantum correlations and the phase-matching condition 5 diminishing that of the other orders, a specific non linear process order is favoured while the process of particular interest (such as SPDC) is promoted. To promote a second order non linear optical process, such as that of SPDC, non- centrosymmetric crystals are required where the inversion symmetry is broken. Im- portantly, while these crystals have a high second order non linear sensitivity, the process still has a limited efficiency due to the conversion of the input light field to the output of the desired second order non linear process. Non-centrosymmetric crystals can be used to generate SPDC. While there exists several sources to generate entangled photons, in this thesis SPDC is employed due to the ease of its accessibil- ity. The SPDC process is governed by Poissonian statistics, as each photon pair that is produced is time-independent of the next pair. Additionally, due to the quan- tum nature, the process is stimulated by vacuum fluctuations whereby any coupled field may result in radiation affecting the spontaneous production of an output pair without an input field [44, 45]. In the SPDC process, a high-energy pump photon is downconverted into pairs of lower energy daughter photons. These pairs of lower energy photons conserve the energy and momentum of the input photon. The pho- ton pairs that are produced are restricted by conditions determined by the non linear crystal and the pump field, as such the photon pairs are generated by any energy and momenta combinations that are within the input conditions [46–48]. Accord- ingly, SPDC is one of the most commonly utilised resources in quantum optics for producing pairs of entangled photons with several degrees of freedom in the entan- glement to select between such as spatial modes [49–54], path [55], and energy-time [56, 57]. A complete description of the SPDC process would require a full quantum descrip- tion of the electromagnetic field. Such a description is well known and therefore heuristic arguments are presented here to describe the correlations of the fields pro- duced by SPDC. Figure 1.3 (a) is a schematic diagram of the SPDC process for the degenerate case with the non-degenerate case shown in the inset. A pump beam of angular frequency ωp, is incident on a non linear crystal χ(2), this interaction pro- duces two daughter photons of ωs and ωi, respectively. The daughter photons are commonly distinguished as the signal and idler photons. As the rules of conserva- tion are followed, the angular frequencies of the signal and idler photons should add to the angular frequency of the pump photon that produced them: ωp = ωs + ωi. (1.3) In the degenerate case: ωs = ωi, while in the non-degenerate case ωs ̸= ωi as il- lustrated in the energy level diagram in Fig. 1.3 (b) and the corresponding inset. Additionally, the linear momentum of the daughter photons must add to that of the pump photon, given by: kp = ks + ki, (1.4) where kp,s,i are the wavevectors of the pump, signal, and idler photons, respectively as illustrated in Fig. 1.3 (c). The efficiency of emission for the photons is greatest when energy and linear momentum relations are obeyed and the properties are con- served. This is known as the phase-matching condition and the limitations imposed by meeting this holds for the entire length of the crystal. 6 Chapter 1. Introduction to quantum ghost imaging a. b. c. ω! ω" ω# k! k" k# ω! ω" ω# χ(%) ω! ω"χ(%) ω# ω! ω" ω# k! k" k# FIGURE 1.3: Schematic representation of degenerate spontaneous parametric down- conversion (SPDC) with non-collinear geometry. (a) A high-energy pump photon is incident on a non linear crystal producing two lower energy entangled daughter photons. (b) The en- ergy level description and (c) the wavevector description for the degenerate SPDC process. The insets show the corresponding diagrams for the non-degenerate case. Mathematically the phase-matching condition states that the daughter photon emis- sion efficiency is at it’s greatest when Eq. 1.3 and Eq. 1.4 are obeyed. Furthermore, it is only the fields that are in phase with the pump field for the length of the crystal that will be generated and produce a reasonably observable output field. A phase mismatch will result due to chromatic dispersion, as different frequencies propa- gate at different velocities in a medium, as the length of the crystal is traversed. It is, however, possible to attenuate or mitigate the dispersion through the use of the birefringence properties of the crystal. The phase-matching condition is thus sensi- tive to both the crystal geometry and polarisation. Polarisation sensitivity and phase-matching conditions result in different SPDC phase- matching regimes, namely Type 0, I and II. In Type-0 the downconverted photons are produced with the same polarisation basis as the pump photon. Types I and II are the more commonly used types in quantum optics. In Type-I, the down-converted photons are produced with the same polarisation basis which is orthogonal to the polarisation of the pump photon. Daughter photons are emitted on concentric cones centred around the pump beam’s axis of propagation. Type-II SPDC emits one pho- ton with the same polarisation as the pump and the other with orthogonal polar- isation, these photons are polarisation entangled. SPDC, additionally, consists of two geometric types: these are collinear and non-collinear. In the collinear geome- try the output field wavevectors propagate in the same direction as the pump field, as shown in Fig. 1.4 (a). Conversely, in the non-collinear SPDC geometric case, the output fields propagate off-axis with respect to the pump field, however their tra- jectories are equal although on opposite sides as illustrated in Fig. 1.4 (b). A good review in Ref. [58] covers the theoretical derivation of SPDC which is beyond the scope of this thesis. The SPDC details presented above are, however, sufficient for an understanding of how quantum ghost imaging works. The different types of SPDC phase-matching and geometric types are summarised in Table 1.1. 1.3. Quantum correlations and the phase-matching condition 7 b. k! k" k# NLC ω! ω" ω# χ(%) k! k" k# ω! ω" ω# a. NLC χ(%) FIGURE 1.4: Diagrams of SPDC interaction geometries for (a) collinear and (b) non-collinear phase-matching. Insets show the required momentum conservation. TABLE 1.1: Summary of the different phase-matching regimes for SPDC, assuming that the pump is horizontally (H) polarised and whether collinear or non-collinear geomertries are possible. Type Pump polarisation Signal Idler Collinear Non-collinear 0 H H H Y Y I H V V Y Y II H V H N Y If the profile of the pump beam is Gaussian then the state of the two-photon field produced by SPDC is represented as: |ψ⟩ = n=+∞ ∑ n=−∞ cn|n⟩s| − n⟩i, (1.5) where s and i represent properties pertaining to the signal and idler photons re- spectively. |n⟩ is the orbital angular momentum (OAM) eigenmode of order n and |cn|2 represents the probability of generating a photon pair of OAM ±n. In this case energy and momentum are conserved as follows: if the signal photon is in mode |n⟩ then the idler photon can only be in mode | − n⟩, if the pump beam profile is a Gaussian distribution [36, 37]. Hence, by exploiting the symmetry (or rather anti- symmetry) of OAM in SPDC it is possible to test for, confirm and quantify the degree of entanglement within a quantum optical setup. Spatial modes carrying OAM are briefly discussed in Section 1.4.1. While a detailed discussion is beyond the scope of this thesis, a comprehensive discussion can be found in Ref. [59]. 8 Chapter 1. Introduction to quantum ghost imaging 1.4 Confirming quantum entanglement A defining feature of quantum mechanics is entanglement [60, 61], which is a valu- able resource in quantum information, quantum communication [42, 62], and par- ticularly in quantum ghost imaging [17]. It is therefore important to characterise a quantum optical setup to test for, confirm and determine the presence and degree of entanglement [63]. EPR correlations allow for the distinction between quantum entanglement from classical correlations [1] and is a phenomenon that can be tested (in it’s most precise form) by violating what is known as the Bell-type inequality [60, 61, 64]. A quantum system is fully characterised by calculating the density matrix which predicts the outcome of any measurement. The density matrix of a bi-partite system allows for the description of the actual outcome of the measurements for both pure and mixed states [36, 65, 66]. A measurement based reconstruction of the density matrix with a high enough fidelity is, therefore, a viable method to charac- terise any possible system. This is usually a tedious experimental task as the number of measurements scale unfavourably with dimensions. Recently, a fast and direct method was reported that simultaneously returns the dimensionality and purity of high-dimensional entangled states while the number of measurements scale linearly with the dimensions [67]. While such dimensionality and purity measurements are beyond the scope of this thesis, characterising the optical system and confirming the presence of entanglement is a necessity prior to conducting the quantum ghost imaging experiments reported in this thesis. A challenge within quantum optics is to fully unravel an unknown state, where the difficulty arises from the measurement problem within quantum mechanics [68–71]. A measurement will destroy the information of the state or in the simplest case per- turb it. It is therefore not possible to perform multiple measurements on the same state, and it is also not possible to clone a quantum state [72, 73]. As these are the limitations that exist, it is only possible to infer information about the state by prob- ing the state for a specific bit of information at a time, i.e., only bits of information are gained at a time by probing an individual aspect of a quantum state. Simply put - asking one question achieves one bit of information. The standard approach is, therefore, to perform multiple tomographic measurements (ask many questions - one question at a time) in what is known as a Quantum State Tomography (QST) [21]. While it is possible to determine the Bell parameter through a full QST [66], it is also common practice to test for a violation of the Bell inequality separately. In terms of EPR correlations a source emits entangled photons, by spatially separating these particles and performing correlation measurements a correlation coefficient on the two particles can be determined [74]. In 1964, Bell postulated that such correlations are due to common properties of both particles from the same pair [75]. By adding a locality assumption he showed that these correlations are constrained by certain inequalities that are not always obeyed by quantum mechanics which lead to gener- alised Bell inequalities [76]. In the proceeding sub-sections the components required for testing and confirm- ing the presence of entanglement are briefly discussed and the results are presented. The components required consist of a full Quantum State Tomography (QST) and violation of a suitable Clauser-Horne-Shimony-Holt (CHSH) Bell inequality. The proceeding sections serve as a brief summary of the tests conducted to confirm the 1.4. Confirming quantum entanglement 9 presence of quantum entanglement prior to conducting the quantum ghost imaging experiments. It must be noted that these tests were carried out prior to each quan- tum ghost imaging experiment carried out within this thesis, but are summarised here to reduce redundancy. 1.4.1 Quantum state tomography For a QST, many projective measurements are made where each measurement probes a certain aspect of the state. The complete quantum state is then built up through a tomographic process. Each unknown quantum state is “sliced” and each slice is probed per measurement. By doing so, the state is completely characterised through a series of structured projective measurements [66]. The outcome is a complete set of observables where the weighted probabilistic outcome fully describes the state [66]. At this point it becomes an inverse problem, all the answers to each question are available and through these answers or outcomes it is then possible to fully de- termine the quantum state. This translates to a reconstruction of what is known as a density matrix, through which all required information can be inferred. The density matrix of a pure quantum state is formed by the outer product of the state vector with itself described by: ρ = |ψ⟩⟨ψ|. (1.6) Spatial modes A quantum bit (often referred to as a qubit) is the fundamental unit of quantum information. A classical bit assumes one of two states (0 or 1), while a quantum bit is a weighted superposition of two orthogonal states for a chosen degree of freedom. A weighted superposition of two orthogonal states would look like: |ψ⟩ = α|0⟩+ β|1⟩, (1.7) where α and β are probability amplitudes so that |α|2 + |β|2 = 1. Traditionally, QST is carried out in terms of the polarisation degree of freedom, where the states rep- resented in Eq. 1.7 can correspond to left and right circular polarisation states. The state of a polarisation qubit can be expressed as: |ψ⟩ = cos(θ/2)|R⟩+ exp(iφ) sin(θ/2)|L⟩, (1.8) where |R⟩ and |L⟩ represent the right and left circular polarisation states, respec- tively. φ represents the phase difference between the polarisation states and θ de- fines the weighting factor, by altering these parameters arbitrary polarisation states may be produced. Intuitively, Eq. 1.8 is representative of a point on a sphere com- monly known as the Poincaré sphere. This sphere describes a two-dimensional state space and is illustrated in Fig. 1.6 (a). The choice of degree of freedom in which to perform a QST is arbitrary, however an alternative and topical choice is to use the spatial mode of the photon [66, 77]. Spatial modes of photons refer to transverse solutions of the paraxial wave equa- tion, Laguerre-Gaussian modes are one of these solutions and carry discrete units of OAM which is a fundamental quantum number (ℓ). In contrast to polarisation, the OAM state space is infinitely large allowing for increased encoding capacity for photons [78–82]. 10 Chapter 1. Introduction to quantum ghost imaging In analogy to polarisation one may write the OAM description of a weighted su- perposition of two orthogonal states as: |ψ⟩ = cos(θ/2) |ℓ1⟩+ exp(iφ) sin(θ/2) |ℓ2⟩ , (1.9) where |ℓi⟩ refers to a paraxial field that carries ℓh̄ units of OAM. The field, when written in cylindrical co-ordinates (r, ϕ, z), is expressed as: |ℓ⟩ ≡ A(r, z) exp(iℓϕ), (1.10) where A(r, z) is an amplitude term that varies transversely and longitudinally. The intensity and phase profiles of OAM modes (-3, -2, -1, 0, +1, +2, +3 OAM units) of the Laguerre-Gaussian basis are shown in Fig. 1.5. 0 1 −𝜋 𝜋 FIGURE 1.5: Intensities of OAM carrying modes, from left to right, ℓ = -3, -2, ..., +2 and +3 units of OAM. Insets show the corresponding phase profile. Analogous to polarisation, OAM qubits may also be represented on the surface of a sphere - the OAM Bloch sphere [83] illustrated in Fig. 1.6 (b). The analogy between the Poincaré sphere and Bloch sphere is illustrated in Fig. 1.6. A QST of an OAM qubit is similar to that of a polarisation qubit where density matrices are expanded in terms of the Pauli matrices and where the eigenvectors instead correspond to OAM modes and their superpositions thereof. Using the OAM degree of freedom for such projective measurements has become highly topical in quantum optics [63, 66, 67]. As such, for the purposes of this thesis, the projective measurements required for testing and confirming the presence of quantum entanglement were carried out in the OAM basis for an all-digital realisation of QST with spatial modes. Projective measurements To perform a QST it is important to decide on the projective measurements to make, i.e., what questions to ask about the state in order to gain the adequate information. The objective of a QST is to reconstruct the density matrix using an appropriate set of measurements. For a full QST of a two dimensional quantum state, one needs to locate the qubit state on the OAM Bloch sphere. The density matrix is then expanded in terms of the Pauli matrices and the identity, as is done with polarisation. While full details on QST using the OAM basis are contained in Ref. [66] these details are beyond the scope of this thesis, as such only the basics are covered here. The density matrix is important as it contains all necessary information about the quantum state and is reconstructed post measurement from the various projective measurements. The eigenstates of a two qubit OAM density matrix, when expressed in terms of the OAM basis (|ℓ, ℓ⟩, |ℓ,−ℓ⟩, | − ℓ, ℓ⟩, | − ℓ,−ℓ⟩), were employed in this 1.4. Confirming quantum entanglement 11 𝜙 𝜃|𝑉⟩|𝐻⟩ |𝐿⟩ |𝑅⟩ 𝐻 + |𝑉⟩ 𝐻 − |𝑉⟩ |ℓ⟩ | − ℓ⟩ ℓ + | − ℓ ⟩ ℓ − | − ℓ ⟩ ℓ + 𝑖| − ℓ ⟩ ℓ − 𝑖| − ℓ ⟩ a. b. FIGURE 1.6: Analogy between polarisation and OAM can be seen using (a) the Poincaré sphere and (b) the Bloch sphere. thesis to test for the presence of quantum entanglement. The density matrix, ρ, is written as: ρ =  A11 A12eiϕ12 A13eiϕ13 A14eiϕ14 A12e−iϕ12 A22 A23eiϕ23 A24eiϕ24 A13e−iϕ13 A23e−iϕ23 A33 A34eiϕ34 A14eiϕ14 A24eiϕ24 A34eiϕ34 A44  , (1.11) where Aij and ϕij are the respective amplitudes and phases of the matrix elements [63]. In the density matrix, the diagonal elements describe the probability of simul- taneously detecting each of the entangled photons in one of the states |ℓ⟩ and | − ℓ⟩. The off-diagonal elements are, however, determined by superpositions of the previ- ously mentioned states. Projections are therefore made onto six states, where four of these states are superpositions of |ℓ⟩ and | − ℓ⟩. It is useful to visualise these projec- tive measurements as projections into two orthogonal bases |ℓ⟩ and | − ℓ⟩ and four mutually unbiased bases (MUBs) which are constructed from superpositions of the two orthogonal bases and calculated as: MUB = 1√ d (|ℓ⟩+ exp(iℓθ)| − ℓ⟩), (1.12) for θ = [0, π/2, π, 3π/2] for a QST in two dimensions, where d is the dimension of the Hilbert space. Such MUBs require amplitude modulation and are therefore approximated in the projective measurement as arg[ℓ⟩ + exp(iθ)| − ℓ⟩] to produce binary phase patterns instead of amplitude functions [66]. The intensities and phases are shown in Fig. 1.7 for orthogonal bases |1⟩ and | − 1⟩ in the OAM basis. The overlap of MUBs with one of the orthogonal states always yields an outcome with a probability of 1/d, in this thesis d = 2. The probability of simultaneously finding each of the entangled photons in one of the six states is measured, with a total of 36 projective measurements conducted. Experimental measurements While experimental details for quantum ghost imaging are presented in each chap- ter, here a brief description of the optical setup is outlined for completeness. The 12 Chapter 1. Introduction to quantum ghost imaging Orthogonal states MUB states FIGURE 1.7: The orthogonal and mutually unbiased basis (MUB) states employed for QST projections in the OAM basis for a state space of ℓ = ±1. While both the intensity and phase (as insets) profiles are shown, it is usually the phase profiles that are employed for projective measurements. quantum optical setups do not differ however several elements within the opti- cal setup differs in each chapter, i.e., the laser, NLC, fibres for photon collection. For the purposes of this thesis, periodically poled potassium titanyl phosphate (PP- KTP) NLCs are used, which utilise an additional method to avoid a mismatch in the wavevectors, however in Section 2.2.3 different types of NLCs are discussed. Peri- odic poling is a technique employed to achieve quasi-phase matching in the non lin- ear interactions of a transparent crystal which consists of several periodically poled domains. A wavevector mismatch, ∆k, is rectified by alternating the poling within each domain (Λ) of the crystal. This alternation is done such that the mismatch in the adjoining domain has the opposite sign, −∆k. By using this approach, the alter- nation in each domain allows for the mismatch to approximately cancel out across each domain, after having done so quasi-phase matching is achieved. Entangled photons were generated by SPDC in a NLC with collinear phase match- ing in both the degenerate and non-degenerate regimes. Entangled photons were spatially separated, each impinging on a SLM. The NLC was imaged to a SLM in each arm, which were subsequently imaged to APDs and connected to a coinci- dence counter (CC) to perform coincidence measurements. The crystal (PPKTP - two types: one for degenerate and one for non-degenerate SPDC) was pumped by a diode laser with wavelengths of either λ = 405 nm or λ = 532 nm. The downcon- verted light impinged on a SLM in each arm before being collected by single mode fibres (SMFs) with a core size of 5 µm in each arm. Photons from the first diffractive order were coupled into the SMFs. The SMFs were of the same length and each one was connected to an APD which, in turn, was connected to the CC. Coincidences were recorded within a 25 ns window. A schematic of the experimental setup is shown in Fig. 1.8 (a). By using appropriate holograms displayed on the SLMs in each arm, it was pos- sible to generate two-photon states entangled within a two-dimensional OAM sub- space. The spatial profile of the downconverted photons was modulated by the SLMs placed in the image plane of the NLC. Computer generated holograms as shown schematically (with false colour) in Fig. 1.8 (b) were used to perform the pro- jective measurements. In each arm, each of the six holograms were cycled through resulting in a total of 36 projective measurements. These 36 measurements are the basis for a 2D QST. A 2D QST was performed using the 36 projective measurements 1.4. Confirming quantum entanglement 13 for the l = ± subspace in the OAM basis. SLMA SLMB APD1 APD2 CC NLC −𝜋 𝜋OAM Holograms Superpositions of OAM holograms a. b. FIGURE 1.8: Conceptual experimental setup and holograms. (a) A schematic of the exper- imental setup, where the NLC, SLMs and APDs are placed in conjugate planes. SLMs are imaged onto APDs to perform projective measurements of (b) the six states as shown by the holograms which were displayed on the SLMs. Results of a two-qubit QST For a physically allowed density matrix certain criteria need to be met, the trace must be one and the eigenvalues need to be non-negative as they comprise true probabilities [63]. Through proper normalisation it was possible to meet the criteria to reconstruct a density matrix. Figure 1.9 (a) shows the experimental results for a two-qubit full QST where the projected state of each photon in the entangled pair is indicated by the relevant phase map. The results show the normalised coincidence counts for the 36 projective measurements. To reconstruct the density matrix a minimisation procedure (as detailed in Ref. [63]) is followed which minimises χ2, described by [63, 84]: χ2 = 36 ∑ i=1 C(M) i − C(P) i√ C(M) i + 1 2 . (1.13) This optimisation procedure minimises the difference between the predicted coinci- dence count rates C(P) and the experimentally determined coincidence count rates C(M) as detailed in Ref. [63]. Importantly the fidelity is a measure determined from the density matrix which quantifies the similarity between the measured density matrix and the predicted density matrix and can be expressed as [66, 84, 85]: F = Tr (√√ ρ1ρ2 √ ρ1 )2 , (1.14) where ρ1 is the density matrix of the target state and ρ2 is the reconstructed density matrix. If the matrices are identical then the fidelity = 1. The target state is often a maximally entangled state where the fidelity is an indication of whether or not the measured state is maximally entangled, or rather is the comparison of the measured state to the maximally entangled state. Other important measures are calculated 14 Chapter 1. Introduction to quantum ghost imaging from the density matrix (ρ) such as, the linear entropy: SL = 4 3 (1 − Tr(ρ2), ) (1.15) which quantifies the mixture of the measured state [86]. The concurrence, which is also a measure of entanglement, where a value of 1 indicates a maximally entangled state given by: C(ρ) = max { 0, √ λ1 − ∑ i=2 √ λi } , (1.16) where λi are the eigenvalues of the operator R = √ ρ √ ρ̃, in descending order [66]. Additionally, although not calculated from the density matrix, the quantum contrast is an important measure which is the ratio of the measured coincidence counts and the classical (or accidental) coincidence counts. The reconstructed density matrix is shown in Fig. 1.9 (b) where the presence of quantum entanglement for the ℓ = ±1 OAM subspace is confirmed by a calculated fidelity of F = 0.962. Additionally, the concurrence was C = 0.927, and a linear entropy of SL = 0.061, while the quantum contrast was Q = 20.18 ± 0.44. Thereby confirming the presence of entanglement, while quantifying that the state is close to a maximally entangled state with a high purity. a. Re(ρ) 0 0.1 0.2 0.3 0.4 Ph ot on A Photon B 0.0 1.0 b. FIGURE 1.9: A two-qubit quantum state tomography. (a) A two-qubit QST where the pro- jected state of each photon in the entangled pair is indicated by its phase map. The color of each box represents the normalised coincidence counts for a given set of projections on the two-photon state. (b) Using the tomographic data, the density matrix was computed, and its real component is shown. The tomographic reconstruction for a 2-dimensional quantum state of entangled photon pairs was shown for the OAM basis. The density matrix was reconstructed which showed a high fidelity with the maximally entangled state and a low linear entropy. The tomographic measurements and reconstruction of the density matrix as well as subsequent calculations are important to characterise the state prior to quantum ghost imaging. 1.4. Confirming quantum entanglement 15 1.4.2 Violating the Bell inequality Now that the presence of entanglement is confirmed with a high fidelity between the measured density matrix and the predicted one, it is important to quantify the degree of entanglement. This was done by testing for the entangled nature of OAM states generated by SPDC through the violation of a suitable Clauser-Horne-Shimony- Holt (CHSH) Bell inequality [60, 74, 76, 87]. Quantifying entanglement As aforementioned, there is an analogy that exists between polarisation and OAM. Demonstrated in earlier sections, this analogy was utilised to confirm the presence of quantum entanglement. In this section the degree of entanglement is quantified. In 2001, it was holographically shown that the conservation of transverse momentum within SPDC leads to a correlation between the OAM of the signal and idler photons [36]. By using holograms similar to those in Fig. 1.8 (b) it was possible to generate two-photon entangled states within a 2D OAM subspace. Using the OAM carrying modes, the surface of the OAM Bloch sphere maps out all superpositions of these modes in a 2D OAM subspace. An equally weighted superposition of |ℓ⟩ and | − ℓ⟩ with a relative phase is represented by a point along the equator of the OAM Bloch sphere and is given by: |ψ⟩ = 1√ 2 |ℓ⟩+ exp(iℓθ)| − ℓ⟩) . (1.17) The angle θ relates to the orientation of these superpositions and are often referred to as sector states [41, 64]. The sector states are equivalent to linear polarisation states that lie on the equator of the Poincaré sphere. To quantify the entanglement it is necessary to demonstrate the correlations of the signal and idler photons that exist for superposition states. Therefore one must de- tect the photons in sector states orientated at different angles. The coincidence count rate (C(θA, θB)) for detecting one photon in sector state |θA⟩ and the other in sector state |θB⟩ is given by: C (θA, θB) = |⟨θA| ⟨θB|Ψ⟩ |2 ∝ cos2[ℓ(θA − θB)]. (1.18) High-visibility sinusoidal fringes are the signature of 2D entanglement for such a joint probability detection. Modal impurities degrade the entangled nature of the state and reduce the visibility of these fringes [64]. Sinusoidal behaviour of the coincidence rate by overlapping different sector states follows directly from quantum mechanics, such behaviour cannot be simulated by any classical theory on local hidden-variables [75]. It is, however, possible to sim- ulate correlations between orthogonal states by classical correlations while correla- tions between certain superposition states may be stronger than that which is al- lowed classically (i.e., > 2). Bell’s inequality measures the deviation of a quantum experiment from classical local theory, where the CHSH Bell inequality is regularly used. Experimentally the Bell parameter (S) is defined as[64]: S = E (θA, θB)− E ( θA, θ′B ) + E ( θ′A, θB ) + E ( θ′A, θ′B ) (1.19) 16 Chapter 1. Introduction to quantum ghost imaging where θA and θB define the rotation angles of the OAM holograms. The inequality is violated for values of |S| which are greater than 2. It follows that E(θA, θB) is cal- culated from the coincidence rates at particular hologram orientations, as described by [64]: E (θA, θB) = C (θA, θB) + C ( θA + π 2ℓ , θB + π 2ℓ ) − C ( θA + π 2ℓ , θB ) − C ( θA, θB + π 2ℓ ) C (θA, θB) + C ( θA + π 2ℓ , θB + π 2ℓ ) + C ( θA + π 2ℓ , θB ) + C ( θA, θB + π 2ℓ ) . (1.20) For a given entangled state |ψ⟩ℓ, the inequality is maximally violated when θA = 0, θB = π 8ℓ , θ′A = π 4ℓ and θ′B = 3π 8ℓ [64]. Experimental measurements and the Bell parameter The experimental setup is the same as that outlined in Section 1.4.1 and as schemat- ically shown in Fig. 1.8 (a). Although the experimental setup is the same, the holo- grams displayed on each SLM vary where sector states are used to conduct a Bell- type measurement. The sector states are defined by phase apertures which are dis- played on each SLM and are shown in Fig. 1.10. The phase apertures select the 2D subspace and additionally act as analysers for these states as their relative orienta- tion changes. 0 1 −𝜋 𝜋 𝑙 = −2 𝑙 = 2 FIGURE 1.10: Phase profiles of the holograms used to define the sector state for the ℓ = ±2 OAM subspace. Figure 1.11 shows the recorded sinusoidal fringes for the ℓ = ±2 OAM subspace representative of the entangled state |ψ⟩2. The phase aperture on SLMB was ro- tated relative to the phase aperture on SLMA, by fixing θA the hologram on SLMB was scanned for angles θB = [0, 2π] and the coincidence count for each scan was recorded. The coincidence rate shows a sinusoidal variation which is a signature of Bell-type measurements [64]. Interestingly, the periodicity of the fringe patterns is indicative of the ℓ dependency of the 2-dimensional entangled state. A Bell param- eter was found, in accordance with Eq. 1.19, for the ℓ = ±2 OAM subspace to be S = 2.59 ± 0.08, thereby successfully violating the Bell inequality. Entanglement between sector states which are superpositions of OAM states was observed for the ℓ = ±2 OAM subspace. Sinusoidal fringes are characteristic of Bell-type measurements and vary with the ℓ dependency. Finally, the calculated Bell parameter successfully violated the Bell inequality to confirm and quantify the degree of entanglement. This is a crucial step to quantify entanglement prior to util- ising the quantum optical setup for quantum ghost imaging. 1.5. Position and momentum configurations of quantum ghost imaging 17 0.25 0.5 !π 4 !π 2 !3π 4 π Angle of hologram B, 𝜃! N or m al ise d co in ci de nc e co un ts FIGURE 1.11: Violation of the Bell inequality. The normalised coincidence count as a function of relative orientation of the superpositions of orbital angular momenutum (OAM) states for the entangled state |ψ⟩2. While dimensionality and purity measurements are not necessarily the focus of this thesis, conducting a QST and confirming that the Bell inequality was violated forms the basis for quantum ghost imaging. It was important to test for and confirm the presence of entanglement within all optical setups employed in this thesis. Although only one optical setup was presented in this section, it must be noted that these tests were successfully conducted for the different optical setups reported in this thesis where entanglement was confirmed and the Bell inequality was successfully vio- lated. 1.5 Position and momentum configurations of quantum ghost imaging The ghost imaging protocol utilises the spatial correlation of photon pairs which can, in principle, be accomplished by a classical source [88]. However, in the context of EPR studies one needs to demonstrate two-photon correlations in each of the two conjugate variables, such as both in position and transverse momentum [1]. In the position configuration of the quantum ghost imaging protocol (shown in Fig. 1.12 (a)), the plane of the NLC is imaged to both the object (O(x, y)) and projective mask (Pi(x, y)), thereby relying on the near-perfect spatial correlations between the entan- gled photons generated by SPDC to reconstruct the image. However in the plane of the crystal, the signal and idler photons are anti-correlated in their transverse mo- menta. As a consequence, in the far-field of both photons the positions of the signal and idler photons are anti-correlated. By placing both the object and the imaging de- tector in the far-field of the crystal one will achieve a momentum configured quan- tum ghost imaging system with the image inverted, as shown in Fig. 1.12 (b). The image type (upright or inverted) therefore depends on the system configuration and 18 Chapter 1. Introduction to quantum ghost imaging is a manifestation of EPR entanglement. Although both results could be obtained independently, no single classical system could produce both [2, 89]. An additional embodiment of ghost imaging is when the SPDC is phase-matched to produce non- degenerate signal and idler wavelengths [22], this depends on the combination of the pump photon’s energy as well as the NLC in use. The wavelength non-degeneracy offers special performance in applications to the biological sciences as the object is il- luminated with one wavelength while the spatial information is recorded at another wavelength, especially when mitigating the risk of photo-damage to light sensitive organisms [90, 91]. Figure 1.12 conceptually illustrates how a quantum ghost imaging optical setup is implemented in both the (a) position and (b) momentum configurations. It is known that the spatial correlations between signal and idler photon pairs, particularly those produced by SPDC, can be used within a quantum imaging system [5, 92–94]. Im- age information is revealed by the correlations between the signal and idler photons and is not present in the detection of each individual photon. A large-area, single- pixel detector (or a bucket detector) collects the photons that have interacted with the object (idler photons) yet no image information of these photons is recorded. In the signal arm the positional information of the detected signal photons is recorded. The signal photons are therefore spatially resolved, yet similarly no image informa- tion is recorded. The imaging protocol requires that the coincidence counts between signal and idler photons are recorded. Specifically, the position or momentum (de- pending on the quantum ghost imaging configuration) of the photon impinging on the spatially resolving detector is recorded only if its detection is coincident with the recording of a photon by the bucket detector. In essence, the idler photons have il- luminated the object, and the signal photons have been recorded by a detector with spatial resolution. The subset of those position-measured signal photons that is coin- cident with detection of the position of the signal photon reveals the image (I(x, y)). An image is reconstructed by a linear combination of each weighted position. In terms of Klyshko’s advanced wave model [4] it is possible to use classical optics to predict the spatial distribution of the quantum photon correlations in a setup which employs SPDC. The Klyshko advanced wave model can be implemented in either the position or momentum configuration as discussed in Ref. [95]. The bucket de- tector is replaced with a laser diode at a central wavelength that is the same as the SPDC wavelength. The diode laser is coupled to a multimode fibre and the light is projected backwards through the object arm to the NLC. The NLC is replaced by a mirror, the light then bounces off the mirror and travels through the other arm of the experimental setup towards the imaging detector [17]. This allows for one to experimentally simulate the outcome of a quantum optical setup with the use of classical light. Klyshko showed that backprojection in this manner generates a classical analogue to predicting what the measured quantum correlations from the quantum optical system would look like [4]. The intensity measurements will reflect what the quantum correlations (coincidence counts) would look like when detecting the entangled photons in coincidence. In order to understand how the image reconstruction process occurs, it is first im- portant to understand the physics of the coincidence counts employed in image re- construction. Importantly, the coincidence counts used to reconstruct the image are 1.5. Position and momentum configurations of quantum ghost imaging 19 𝑓!𝑓! a. NLC CC Object Pump photon Mask Position configuration b. Detector NLC CC Object Pump photon Signal photon Idler photon Mask Momentum configuration Detector Ghost image Ghost image 𝑓" 𝑓" 𝑓" 𝑓" 𝑓! 𝑓! 𝑓! Detector Signal photon Idler photon Detector FIGURE 1.12: Conceptual sketch of quantum ghost imaging optical setups in the position and momentum configurations. Entangled photons, generated by a high energy pump pho- ton at a non linear crystal (NLC) by spontaneous parametric downconversion (SPDC), are spatially separated along two arms. One photon interacts with the object and is collected by a bucket detector - the idler photon. The other photon (signal photon) is collected by a spa- tially resolving detector consisting of a projective mask and a bucket detector. Each detector is connected to a coincidence counting (CC) device to perform coincidence measurements. (a) An illustration of a ghost imaging optical setup in which the object and projective mask are placed in the near-field of the crystal. (b) An illustration of a ghost imaging optical setup in which the object and projective mask are placed in the far-field of the crystal. The insets show the respective ghost images obtained for the different experimental configurations. fi is indicative of the different focal lengths for lenses required for either the position of mo- mentum configuration. proportional to the normally ordered coherence function [96, 97]: C (x1, x2) = 〈 ψ ∣∣∣Ê(−) (x1) Ê(−) (x2) Ê(+) (x2) Ê(+) (x1) ∣∣∣ ψ 〉 = ∣∣∣〈0 ∣∣∣Ê(+) (x1) Ê(+) (x2) ∣∣∣ ψ 〉∣∣∣2 (1.21) where Ê(+) (x) and Ê(−) (x) are the parts of the electric-field containing the respec- tive creation and annihilation operators at position x, while x1 and x2 represent the detectors in the object (idler photon) and scanning (signal photon) arms, respectively [30]. |ψ⟩ is representative of the biphoton state as entangled photons are employed. As there are now bucket detectors in each arm (as illustrated in Fig. 1.14) the object information is contained in the integrated coincidence signal defined as: ci = ∫ dx1dx2C (x1, x2) (1.22) where i refers to the ith measurement in a sequence of projective masks (Pi(x, y)) discussed further in Section 2.2.5. The two-photon amplitude for a ghost imaging 20 Chapter 1. Introduction to quantum ghost imaging optical setup as shown in Fig. 1.14 is then given by [30]:〈 0 ∣∣∣Ê(+) (x2) Ê(+) (x1) ∣∣∣ ψ 〉 = ∫ dxsdξdηdζdxih (x2, ζ) Pi(ζ)h(ζ, η)L(η) × h (η, xi)ψ (xs, xi) h (ξ, xs) T(ξ)h (x1, ξ) , (1.23) where h (x, x′) is the paraxial Fresnel free-space propagation kernel, L(x) is the trans- fer function of a lens, T(x) is the transmission function of the object. The object transmission function (usually denoted O(x, y) in later chapters), is the quantity in question in quantum ghost imaging, as it is the quantity one wishes to determine. Finally, Pi is the 2D projective mask displayed per measurement for the ith mea- surement, in later chapters this is referred to as Pi(x, y). The projective masks and bucket detector accomplish a spatially resolving detector whereby the spatial infor- mation contained within the object is mapped to a sequence of coincidence signals determined by the different iterations of the projective masks. If the biphoton state generated by the SPDC process is approximated by ψ (xs, xi) ∝ δ (xs − xi) for a thin non linear crystal then the integrated coincidence signal becomes ci = ∫ dx1dx2 ∣∣∣〈0 ∣∣∣Ê(+) (x1) Ê(+) (x2) ∣∣∣ ψ 〉∣∣∣2 ∝ ∑ n |Pi (−ξn)|2 |T (ξn)|2 . (1.24) The image is reconstructed as a linear combination of each projective mask in the se- quence weighted by a factor (χ) determined by the measured integrated coincidence signal I(x, y) = N ∑ i=1 χciPi(x, y), (1.25) where I(x, y) is the image. The weightings, defined by χ, are discussed further in Section 2.4 which covers several image reconstruction algorithms. Simulating the overlap between the object and projective mask to calculate what the theoretical image would look like is explained in Section 2.2.7. The projective masks must form a complete spatial basis in order to reconstruct a general image solution. In principle, to reconstruct an image of N pixels, a sequence of N different projective masks is required [12]. If, however, the set of projective masks contains patterns that are non- orthogonal and do not form a complete spatial basis, reconstructing an image would require a larger number of basis elements, i.e., an over-complete set. 1.6 From single-pixel scanning to projective masks Common camera systems typically consist of an array of millions of detector pix- els that are used to capture the image. In contrast, a single-pixel detector uses a sequence of projective (or patterned) masks to filter the image or scene and the cor- responding measured intensity is then recorded by the single-pixel detector [18]. Quantum ghost imaging is no different to single-pixel imaging from the optical per- spective. The signal photon that has not interacted with the object is spatially re- solved, i.e., one measures the position of photons that have not interacted with the object. The imaging detector used to spatially resolve the reference photon and the 1.6. From single-pixel scanning to projective masks 21 methodology behind this has seen significant growth in the past decade [18]. The most simple scanning method uses a single-pixel detector to sequentially scan a single pixel at a time (for a specific time interval) and is also known as raster scan- ning, originally used in the mechanical televisor [98] and first demonstrated in ghost imaging by Pittman and co-workers [5, 99]. Raster scanning requires that a sin- gle pixel, in the transverse plane, is scanned for an extended amount of time (also known as the integration time per mask/scan) to accumulate enough signal photons to effectively distinguish between signal photons and ambient photons contributing to noise, the scanning process is illustrated in Fig. 1.13. Importantly the smaller the pixel size the longer the amount of time spent on the individual scan, resulting in lengthy and inefficient imaging times. This is an inefficient use of photons, espe- cially in the quantum regime, as there already exists lower levels of photon numbers available to image with, more efficient scanning methods were developed. Scanning detector Bucket detector NLC CC ObjectPump photon Signal photon Idler photon FIGURE 1.13: Entangled photons produced, by a high energy pump photon, at the NLC are spatially separated. The signal photon has not interacted with the object and is spatially resolved by a single-pixel detector which mechanically scans the transverse plane to record the positional information of the signal photon. The idler photon interacts with the object and is collected by a bucket detector without recording any positional information. Both photons are detected in coincidence by a coincidence counter (CC), the coincidences are used to reconstruct the image. Examples of the single-pixel scan are shown on the right. A more efficient use of photons is to implement a scanning strategy that requires a sequence of spatially resolving projective masks and record the photon correlations between each projective mask (spatial scan) and the object. These spatial projective masks were first implemented by Duart et al. in the Hadamard basis [100] and then by Shapiro et al. in the random basis [27]. As more than one pixel is activated at a time with these spatial masks, they are also known as 2-dimensional (2D) spatial projective masks [34]. The added advantage of using 2D spatial projective masks is the significant reduction in image reconstruction times, as integration time per scan is significantly reduced. Figures 1.13 and 1.14 illustrate the move from single- scanning to projective masks, respectively. As can be deduced, quantum ghost imaging faces rather unsatisfactory imaging 22 Chapter 1. Introduction to quantum ghost imaging Bucket detector Projective mask Bucket detector NLC CC ObjectPump photon Signal photon Idler photon FIGURE 1.14: Entangled photons produced, by a high energy pump photon, at the non- linear crystal (NLC) are spatially separated. The signal photon has not interacted with the object and is spatially resolved by a sequence of patterned projective masks displayed on a spatial light modulator (SLM) and collected by a bucket detector. The idler photon inter- acts with the object and is collected by a bucket detector without recording any positional information. Both photons are detected in coincidence by a coincidence counter (CC), the coincidences are