Quantum mechanics cements the intimate relationship between the nature of light and its detection. Historically, quantum theories of photodetection have generally fallen into two categories: the first tries to determine what quantum field observable is measured when photoelectrons are detected, laying the theoretical groundwork for photodetection being possible. The second type are phenomenological theories, which take great care to model the details of specific photodetectors. In this dissertation, we fill in the gap between these two models in the modern literature on photodetection by constructing a fully quantum-mechanical and sufficiently realistic model that includes all stages of the photodetection process. We accomplish this within the framework of quantum information theory using the language of positive operator valued measures (POVMs). \n\nA POVM provides the most general description of a quantum measurement. In the context of single-photon detection, the photodetector POVM provides a complete characterization of the single-photon detector (SPD) from which all figures of merit can be calculated. Each element of the POVM is comprised of a weighted sum over single-photon state projectors. To construct SPD POVMs, we identify the states projected onto by a measurement and the associated weights. We first accomplish this for a simple time-dependent system, laying out how to efficiently project onto arbitrary single-photon states. Then, we describe the three stages of a realistic SPD: transmission, amplification, and measurement.\n\nIn the transmission stage of photodetection a photon enters a network of discrete energy levels, its energy propagating through that network and finally exiting into another output continuum of modes, a process entirely described by a single complex function. In the amplification stage of photodetection, a single excitation is amplified into a macroscopic signal via a nonlinear amplification mechanism. In the final stage, an inefficient ``classical'' measurement is made on the macroscopic signal. By combining these three stages we form a chain of inference from the final ``click'' to the input photon and construct a POVM projecting onto the input Hilbert space. Lastly, we discuss limits and tradeoffs that arise at each stage and implications for photodetection applications.\n\nThis dissertation contains previously published and unpublished material.
Quantum mechanics cements the intimate relationship between the nature of light and its detection. Historically, quantum theories of photodetection have generally fallen into two categories: the first tries to determine what quantum field observable is measured when photoelectrons are detected, laying the theoretical groundwork for photodetection being possible. The second type are phenomenological theories, which take great care to model the details of specific photodetectors. In this dissertation, we fill in the gap between these two models in the modern literature on photodetection by constructing a fully quantum-mechanical and sufficiently realistic model that includes all stages of the photodetection process. We accomplish this within the framework of quantum information theory using the language of positive operator valued measures (POVMs). \n\nA POVM provides the most general description of a quantum measurement. In the context of single-photon detection, the photodetector POVM provides a complete characterization of the single-photon detector (SPD) from which all figures of merit can be calculated. Each element of the POVM is comprised of a weighted sum over single-photon state projectors. To construct SPD POVMs, we identify the states projected onto by a measurement and the associated weights. We first accomplish this for a simple time-dependent system, laying out how to efficiently project onto arbitrary single-photon states. Then, we describe the three stages of a realistic SPD: transmission, amplification, and measurement.\n\nIn the transmission stage of photodetection a photon enters a network of discrete energy levels, its energy propagating through that network and finally exiting into another output continuum of modes, a process entirely described by a single complex function. In the amplification stage of photodetection, a single excitation is amplified into a macroscopic signal via a nonlinear amplification mechanism. In the final stage, an inefficient classical'' measurement is made on the macroscopic signal. By combining these three stages we form a chain of inference from the final click'' to the input photon and construct a POVM projecting onto the input Hilbert space. Lastly, we discuss limits and tradeoffs that arise at each stage and implications for photodetection applications.\n\nThis dissertation contains previously published and unpublished material.
Quantum mechanics cements the intimate relationship between the nature of light and its detection. Historically, quantum theories of photodetection have generally fallen into two categories: the first tries to determine what quantum field observable is measured when photoelectrons are detected, laying the theoretical groundwork for photodetection being possible. The second type are phenomenological theories, which take great care to model the details of specific photodetectors. In this dissertation, we fill in the gap between these two models in the modern literature on photodetection by constructing a fully quantum mechanical and sufficiently realistic model that includes all stages of the photodetection process: transmission, amplification, and a final classical measurement. We accomplish this within the framework of quantum information theory using the language of positive operator valued measures (POVMs). This dissertation contains material previously published in three papers: Propp, Tz. B & van Enk, S. J. (2019). On nonlinear amplification: improved quantum limits for photon counting. Optics Express 27, 16, 23454-23463. Propp, Tz. B & van Enk, S. J. (2019). Quantum networks for single photon detection. Physical Review A, 100, 033836. Propp, Tz. B & van Enk, S. J. (2020). How to project onto an arbitrary single-photon wavepacket. Physical Review A, 102, 053707.
Single photon detection generally consists of several stages: the photon has to interact with one or more charged particles, its excitation energy will be converted into other forms of energy, and amplification to a macroscopic signal must occur, thus leading to a ``click.'' We focus here on the part of the detection process before amplification (which we have studied in a separate publication). We discuss how networks consisting of coupled discrete quantum states and structured continua (e.g., band gaps) provide generic models for that first part of the detection process. The input to the network is a single continuum (the continuum of single photon states); the output is again a single continuum describing the next irreversible step. The process of a single photon entering the network, its energy propagating through that network, and finally exiting into another output continuum of modes can be described by a single dimensionless complex transmission amplitude, $T(\ensuremath{\omega})$. We discuss how to obtain from $T(\ensuremath{\omega})$ the photodetection efficiency, how to find sets of parameters that maximize this efficiency, as well as expressions for other input-independent quantities such as the frequency-dependent group delay and spectral bandwidth. We then study a variety of networks, discuss how to engineer different transmission functions $T(\ensuremath{\omega})$ amenable to photodetection, and discuss implications for single photon detection technology.
The time-frequency degree of freedom of the electromagnetic field is the final frontier for single-photon measurements. The temporal and spectral distribution a measurement retrodicts (that is, the state it projects onto) is determined by the detector's intrinsic resonance structure. In this paper, we construct ideal and more realistic positive operator-valued measures (POVMs) that project onto arbitrary single-photon wave packets with high efficiency and low noise. We discuss applications to super-resolved measurements and quantum communication. In doing so we will give a fully quantum description of the entire photodetection process, give prescriptions for (in principle) performing single-shot Heisenberg-limited time-frequency measurements of single photons, and discuss fundamental limits and trade-offs inherent to single-photon detection.
Photodetection plays a key role in basic science and technology, with exquisite performance having been achieved down to the single-photon level. Further improvements in photodetectors would open new possibilities across a broad range of scientific disciplines and enable new types of applications. However, it is still unclear what is possible in terms of ultimate performance and what properties are needed for a photodetector to achieve such performance. Here, we present a general modeling framework for photodetectors whereby the photon field, the absorption process, and the amplification process are all treated as one coupled quantum system. The formalism naturally handles field states with single or multiple photons as well as a variety of detector configurations and includes a mathematical definition of ideal photodetector performance. The framework reveals how specific photodetector architectures introduce limitations and tradeoffs for various performance metrics, providing guidance for optimization and design.
A photodetector may be characterized by various figures of merit such as response time, bandwidth, dark count rate, efficiency, wavelength resolution, and photon-number resolution. On the other hand, quantum theory says that any measurement device is fully described by its positive-operator-valued measure (POVM) which generalizes the textbook notion of the eigenstates of the appropriate hermitian operator (the 'observable') as measurement outcomes. Here we show how to define a multitude of photodetector figures of merit in terms of a given POVM. We distinguish classical and quantum figures of merit and issue a conjecture regarding trade-off relations between them. We discuss the relationship between POVM elements and photodetector clicks, and how models of photodetectors may be tested by measuring either POVM elements or figures of merit. Finally, the POVM is advertised as a platform-independent way of comparing different types of photodetectors, since any such POVM refers to the Hilbert space of the incoming light, and not to any Hilbert space internal to the detector.
Photodetection plays a key role in basic science and technology, with exquisite performance having been achieved down to the single photon level. Further improvements would open new possibilities across a broad range of scientific disciplines, and enable new types of applications. However, it is still unclear what is possible in terms of ultimate performance, and what properties are needed to achieve such performance. Here, we present a general modeling framework for photodetectors whereby the photon field, the absorption process, and the amplification process are all treated as one coupled quantum system. The formalism naturally handles field states with single or multiple photons as well as a variety of detector configurations, and includes a mathematical definition of ideal photodetector performance. The framework reveals how specific photodetector architectures introduce limitations and tradeoffs for various performance metrics, providing guidance for optimization and design.