This dissertation serves as a general introduction to Wigner functions, phase space, and quantum metrology but also strives to be useful as a how-to guide for those who wish to delve into the realm of using continuous variables, to describe quantum states of light and optical interferometry. We include many of the introductory elements one needs to appreciate the advantages of this treatment as well as show many examples in an effort to make this dissertation more friendly. In the initial segment of this dissertation, we focus on the advantages of Wigner functions and their use to describe many quantum states of light. We focus on coherent states and squeezed vacuum with a Mach Zehnder Interferometer for many of our examples, also used by experiments such as advanced LIGO. Later, we will also analyze this setup in more detail with a full example including the effects of many noise sources such as phase drift, photon loss, inefficient detectors, and thermal noise. In this setup, we also show the optimal measurement scheme, which is currently not employed in experiment. Throughout our metrology discussions, we will also discuss various quantum limits and use quantum Fisher information to show optimal bounds. When applicable, we also discuss the use of quantum Gaussian information and how it relates to our Wigner function treatment. The remainder of our discussion focuses on investigating the effects of photon addition and subtraction to various states of light and analyze the nondeterministic nature of this process. We use examples of m photon additions to a coherent state as well as discuss the properties of an m photon subtracted thermal state. We also provide an argument that this process must always be a nondeterministic one, or the ability to violate quantum limits becomes apparent. We show that using phase measurement as one's metric is much more restrictive, which limits the usefulness of photon addition and subtraction. When we consider SNR however, we show improved SNR statistics, at the cost of increased measurement time. In this case of SNR, we also quantify the efficiency of the photon addition and subtraction process.
This dissertation serves as a general introduction to Wigner functions, phase space, and quantum metrology but also strives to be useful as a how-to guide for those who wish to delve into the realm of using continuous variables, to describe quantum states of light and optical interferometry. We include many of the introductory elements one needs to appreciate the advantages of this treatment as well as show many examples in an effort to make this dissertation more friendly. In the initial segment of this dissertation, we focus on the advantages of Wigner functions and their use to describe many quantum states of light. We focus on coherent states and squeezed vacuum with a Mach Zehnder Interferometer for many of our examples, also used by experiments such as advanced LIGO. Later, we will also analyze this setup in more detail with a full example including the effects of many noise sources such as phase drift, photon loss, inefficient detectors, and thermal noise. In this setup, we also show the optimal measurement scheme, which is currently not employed in experiment. Throughout our metrology discussions, we will also discuss various quantum limits and use quantum Fisher information to show optimal bounds. When applicable, we also discuss the use of quantum Gaussian information and how it relates to our Wigner function treatment. The remainder of our discussion focuses on investigating the effects of photon addition and subtraction to various states of light and analyze the nondeterministic nature of this process. We use examples of m photon additions to a coherent state as well as discuss the properties of an m photon subtracted thermal state. We also provide an argument that this process must always be a nondeterministic one, or the ability to violate quantum limits becomes apparent. We show that using phase measurement as one's metric is much more restrictive, which limits the usefulness of photon addition and subtraction. When we consider SNR however, we show improved SNR statistics, at the cost of increased measurement time. In this case of SNR, we also quantify the efficiency of the photon addition and subtraction process.
Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.
In 2009, physicists at the National Institute of Standards and Technology in Colorado, Boulder developed what could arguable be called the first rudimentary quantum computer [1]. The past decade has seen unprecedented improvements in quantum information science culminating in the demonstration of quantum supremacy --- that quantum computers can solve problems that are impractical to be solved on the best supercomputers [2]. This remarkable progress necessitates the development of techniques to characterize the quantum devices that are being developed. In my thesis, I will focus on such devices that manipulate and detect light. In Chapter 1, I will introduce the reader to the historical underpinnings of the study of light. After surveying the history of light since time immemorial, I will delve into the developments in the study of light in the 19th and the early 20th century. I will then present a brief introduction to quantum mechanics. I end this chapter by demonstrating how light is quantized. In Chapter 2, I will introduce the various tools necessary to delve into this dissertation. After defining quadrature operators and Wigner functions, I will introduce various states of light used in this thesis. I will then discuss how the evolution, as well as detection of quantum optical states, are modelled. I will then present the formalism of Gaussian quantum information that simplifies the manipulation of certain states of light. Finally, I will briefly talk about quantum metrology and show its advantage in phase estimation. In Chapter 3, I will introduce a method to characterize linear and quadratically nonlinear optical systems. After motivating the need for this work, I will introduce the modifications necessary to the standard Mach-Zehnder interferometer in order to characterize optical systems with it. I will then show how to characterize linear optical systems with coherent probes. I will also show that this scheme is shot-noise limited. I will then show that by using single photons in addition, we can characterize quadratically nonlinear optical systems. Finally, I will show that no advantage is gained in sensitivity by using squeezed light along with coherent light as probes. In Chapter 4, I will introduce a method to characterize photodetectors. This is done by developing an experimental method to find the Wigner functions of the POVM set corresponding to a photodetector. After motivating the necessity of this work as well as describing the proposed experimental setup, I will show how this scheme can
In this dissertation we begin with a brief introduction to quantum optics concentrating on the topics of the noise of quantum optical states, quantum estimation theory, quantum interferometry and the atom-field interaction. This background is necessary for understanding the discussions in later chapters. In particular, quantum interferometry, which is optical interferometry when the light source is a quantum mechanical state, plays a central role in this dissertation. In Chapter 2 we discuss the phase estimation sensitivity of quantum metrology when photon loss is present. In Chapter 3 we extend the discussion to include the phase fluctuation of the system caused by the environment. We model our metrological system with the Mach-Zehnder interferometer (MZI) and use a light field in the symmetric number-path entangled state as the source. In both chapters we use the parity operator as the detection scheme and show that it is optimal under pure dephasing. In Chapter 4 we discuss the application of quantum optical states in remote sensing and propose a new scheme for a quantum radar. Again, our scheme consists of a MZI and a coherent light source. It is shown that using only coherent states of light and quantum homodyne detection, super-resolving ranging and angle determination are achievable. Chapter 5 is devoted to the generation of a super-resolving single-photon number-path entangled state which may serve as a proof-of-principle prototype for quantum lithography. The repeated implementation of MZIs is shown to be able to remove photons coherently from both modes of a symmetric number-path entangled state with arbitrarily high photon number. Lastly, in Chapter 6 we introduce the phenomenon known as polarization self-rotation and discuss its potential in generating a squeezed vacuum state, which has a huge impact in quantum interferometry.
The fundamental precision limit of an interferometer is crucial since it bounds the best possible sensitivity one could achieve using such a device. This thesis will focus on several different interferometers and try to give the ultimate precision bounds by carefully counting all the resources used in the interferometers. The thesis begins with the basics of the quantum state of light. The fundamentals of quantum metrology are also reviewed and discussed. More specifically, the terminology of classical and quantum Cram\'er-Rao bound and classical and quantum Fisher information are introduced. Chapter 3 discusses the conclusive precision bounds in two-mode interferometer such as Mach-Zehnder interferometer (MZI) and SU(1,1) interferometer. I revisit the quantum Fisher information approach of these two interferometers and show the discrepancy of phase sensitivity on the physically same setup. Then I establish fundamental precision estimation bounds for such device, due to the reason that many works of literature fail to accurately count the resources and knowledge of phase-to-be-estimated used in the interferometers. The analysis suggests that for a MZI, one can never do better than SNL in phase sensitivity, when an input to one of the two ports is the vacuum. If one does not allow the detector to use any external phase reference or power resource, then the precision is limited by the SNL. For a SU(1,1) interferometer, firstly, when one of the input states is restricted to be a vacuum state, I showed that by using either the phase-averaging method or the quantum Fisher information matrix method, different phase configurations of the SU(1,1) interferometer result in the same QFI. Secondly, I compared the results of the phase-averaging method and the quantum Fisher information matrix method, and then I argued that for an SU(1,1) interferometer, phase averaging or quantum Fisher information matrix method is generally required, and they are equivalent. Finally, I used the quantum Fisher information matrix method to calculate the precision limit for other common input states, such as two coherent state inputs or coherent state with squeezed vacuum inputs. In chapter 4, I will consider a passive multi-mode interferometer for multiparameter phase estimation. It was suggested that optical networks with relatively inexpensive overhead---single photon Fock states, passive optical elements, and single photon detection---can show significant improvements over classical strategies for single-parameter estimation, when the number of modes in the network is small. In this chapter, I analytically compute the quantum Cramer-Rao bound to show these
In this thesis I study optimal strategies to estimate or characterize quantum states and processes (quantum channels). I consider the problem of estimating a phase encoded on a gaussian state. Phase estimation is at the heart of many quantum metrology applications, such as gravitational wave detectors or clock synchronization. I address the problem to the phase estimation fidelity for pure and mixed gaussian states and study how temperature affects the precision of estimation. Although the case of pure states has been studied before, very little is known about estimation in mixed states. The interest in mixed gaussian states relies on the fact that these are the states one has in real experiments with light due to the unavoidable presence of noise. Displaced and squeezed thermal states are studied. For the case of displaced states, I find that fidelity estimation gets worse when the temperature increases. On the contrary, for squeezed thermal states I find the surprising result that a larger temperature yields better estimation. This apparently paradoxical feature, also appears when I derive the estimation accuracy that can be obtained when an asymptotically large number of copies of the state is available. I also study quantum benchmarks that can be applied to a wide variety of protocols, such us quantum storage or quantum teleportation. A very important question in QIT is to know whether a particular quantum protocol can be realized by classical means with the same efficiency. After all, QIT lives from the promise of this quantum advantage. This question becomes specially relevant in experimental (real) implementations of quantum schemes. In real experiments there are always losses and the ideal quantum protocols become imperfect. It is then necessary to assess whether the same experimental result could be obtained by classical (less costly) means. So, for example, in a particular teleportation experiment -which involves generation of entanglement, complicated Bell measurements, fighting decoherence, etc.- one might ask whether one could achieve the same (i.e. mapping the state of a system to a second system located in a different place or time) by measuring the system, transmitting the collected information to the other location, and finally preparing the system accordingly. It is hence of paramount importance to give bounds on the efficiency of these measure and prepare (or entanglement breaking) channels. For this purpose we propose a threshold based on a phase-covariant family of states. Besides of being a family with a straightforward group-theoretical structure, which gives obvious computational advantages, this family is interesting because it can be tested experimentally also in a straightforward way. A crucial feature for the practical use of our benchmark is that it does not require pure (ideal) input states. I finally consider the problem of estimating the parameter of the amplification channel. The amplification channel is interesting by itself but also by the fact that the Hawking-Unruh effect can be formally modeled by it. The Unruh-Hawking effect is a relativistic effect and as such it is hard to measure and detect in current experiments. I investigate whether quantum strategies can provide an improvement over classical protocols in the estimation of the amplification parameter. If the field is inertially prepared in the vacuum one could measure its temperature as detected by a noninertial observer and extract information about the parameter. But I would like to know if the vacuum state is the optimal probe state in an inertial frame for such a task or a better sensitivity could be achieved if the field were in a state of nonzero energy (mean photon number) from an inertial perspective. The aim is to determine the optimal input states, so that, by optimally measuring the output, one can construct an estimator for amplification channel parameter which is unbiased and has the minimum possible variance.
Quantum metrology aims to achieve measurement precision that surpasses classical limits in the estimation of physical parameters. In realistic conditions, however, its advantage can be diminished by interactions between a quantum system and its environment. This necessitates careful design of quantum resources that can sustain enhanced performance even under noise. Here, we investigate how a quantum non-Gaussian operation can provide an advantage in noisy quantum metrology. Specifically, we examine the effect of single-photon subtraction applied to a two-mode squeezed vacuum state in the task of phase estimation using a Mach–Zehnder interferometer under photon loss. The single-photon subtraction can be implemented either before or after the first beam splitter in the interferometer. By evaluating the quantum Fisher information for both the original and the photon-subtracted states, we assess the potential enhancement in sensing precision. We show that the enhancement occurs when photon subtraction is performed before the first beam splitter, provided that loss acts only on the signal mode. Our findings shed light on the role of non-Gaussian operations in noisy quantum metrology, clarifying when photon subtraction can yield a genuine advantage over Gaussian resources.