This thesis presents a summary of the foundation and background of the field of quantum optics, and an analysis of some recent discoveries in various fields of which I have aided in furthering investigative research and advancement through publications. Such topics include numerical optimization of generalized quantum states used in phase sensitive quantum metrology, an analysis of object detection through the use of quantum interferometry in the presence of lossy conditions, and the use of the latter technique to propose an invisible quantum tripwire. First is a collaborative effort to numerically optimize quantum optical states for quantum metrological applications. We optimize two-mode, entangled, number states of light in the presence of loss in order to maximize the Fisher information, which is equivalent to minimizing the phase uncertainty. We find that in the limit of zero loss the optimal state is the so-called N00N state, for small loss, the optimal state gradually deviates from the N00N state, and in the limit of large loss the optimal state converges to a generalized two-mode coherent state, with a finite total number of photons. The results provide a general protocol for optimizing the performance of a quantum optical interferometer in the presence of photon loss. The next topic is statistical hypothesis testing of interaction free measurement and a quantitative limit on the obtainable error. Previous analyses have been based solely on detection probabilities known only in the infinite photon limit. Our analysis assumes a finite number of photons, and an investigation of reliability in the presence of photon loss and phase fluctuations. We use symmetric hypothesis testing and the Chernoff bound to provide error estimation after N independent, single photon trials. Finally, we present a quantum optical interrogation technique capable of detecting an intrusion with very low probability of the quantum “tripwire” being revealed to the intruder. The tripwire exploits a curious nonlinear behavior of the quantum Zeno effect we discovered, which occurs in a lossy system. We also employ statistical hypothesis testing, allowing us to calculate a confidence level of interaction-free measurement after a given number of trials.
This thesis presents a summary of the foundation and background of the field of quantum optics, and an analysis of some recent discoveries in various fields of which I have aided in furthering investigative research and advancement through publications. Such topics include numerical optimization of generalized quantum states used in phase sensitive quantum metrology, an analysis of object detection through the use of quantum interferometry in the presence of lossy conditions, and the use of the latter technique to propose an invisible quantum tripwire. First is a collaborative effort to numerically optimize quantum optical states for quantum metrological applications. We optimize two-mode, entangled, number states of light in the presence of loss in order to maximize the Fisher information, which is equivalent to minimizing the phase uncertainty. We find that in the limit of zero loss the optimal state is the so-called N00N state, for small loss, the optimal state gradually deviates from the N00N state, and in the limit of large loss the optimal state converges to a generalized two-mode coherent state, with a finite total number of photons. The results provide a general protocol for optimizing the performance of a quantum optical interferometer in the presence of photon loss. The next topic is statistical hypothesis testing of interaction free measurement and a quantitative limit on the obtainable error. Previous analyses have been based solely on detection probabilities known only in the infinite photon limit. Our analysis assumes a finite number of photons, and an investigation of reliability in the presence of photon loss and phase fluctuations. We use symmetric hypothesis testing and the Chernoff bound to provide error estimation after N independent, single photon trials. Finally, we present a quantum optical interrogation technique capable of detecting an intrusion with very low probability of the quantum “tripwire” being revealed to the intruder. The tripwire exploits a curious nonlinear behavior of the quantum Zeno effect we discovered, which occurs in a lossy system. We also employ statistical hypothesis testing, allowing us to calculate a confidence level of interaction-free measurement after a given number of trials.
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.
Quantum optical metrology deals with estimation of an unknown parameter by exploiting the non-classical properties of the light. The unknown parameter that we are trying to estimate is the optical phase. Precise optical phase measurement has been a well-known problem and has many applications, most notably the gravitational wave detection. In this thesis we investigate the interferometric measurement schemes. We consider the parity detection for a class of input states that have been shown to exhibit sub-shot noise limited phase estimate with their respective detection schemes. Our results indicate that the parity detection applies to all these strategies with various input states and thus acts as a unified detection scheme towards the goal of interferometric phase estimates beyond the shot-noise limit. We also consider the performance of the so-called optimal state with the canonical phase measurement scheme that was proposed by Sanders and Milburn [Phys. Rev. Lett. 75, 2944 (1995)] in presence of photon loss. The model for photon loss is a generic fictitious beam splitter and the analytical treatment requires density matrix approach rather than the state-vector formalism. We present full density-matrix calculations. Our results indicate that, for a given amount of loss, the phase estimate saturates but does not diverge as one would expect with increasing the loss. Finally, we study the continuous measurement and feedback scheme with optical homodyne detection for a single optical qubit. We found a protocol that speeds up the rate of increase of the average purity of the system and generates a deterministic evolution for the purity in the limit of strong feedback.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
One important problem in quantum optics is to resolve an extremely small change of phase shift. The complementarity between photon number and phase sets an ultimate limit, the so-called Heisenberg limit, on the phase measurement sensitivity. The precise phase estimation has many technological applications, such as optical gyroscopes, gravitational wave detection, quantum imaging and sensing. In this thesis I show that the utilization of the parity measurement in the optical interferometry is actually applicable to a wide range of quantum entangled input states. Comparison of the performance of the various quantum states then can be made within such a unified output measurement scheme. Based on such a universal detection scheme, we present a comparison of the phase sensitivity reduction for various quantum states of light in the presence of photon loss. I also provide a simple condition that could be used to check whether an arbitrary state can achieve the Heisenberg limit independently of the detection scheme. It implies that the fidelity between the two output states with zero phase and minimal detectable phase applied respectively should significantly different from unity as the minimal phase shift scaling as 1/N, whereas the average number of input photons N goes to infinity. Next I give several measures to characterize the which-way information in the interference experiment. We define a new distinguishability associated with the fidelity between two density matrices to measure the which-way information. We demonstrate that the changes of mutual entropy as well as the entanglement of formation of the whole system, i.e. the physical system plus the which-way detector can also be used to describe the which-way information. With such quantities, we show that as the fringe visibility of the interference pattern gets larger, the less which-way information is obtained. Finally, I show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. This scheme would have applications to laser radar, given the difficulty in making entangled states of light, as well as their susceptibility to atmospheric absorption.
Optical interferometry is a cornerstone technique for precise phase measurements across various fields. In many applications, for example, biological imaging, it often necessitates stringent limits on light intensity to prevent adverse effects on light-sensitive samples, a condition known as dose-limited regimes. Maximizing the precision per dose is therefore crucial. In quantum metrology, quantum correlations enable high precision in phase estimation while adhering to dose constraints. Nevertheless, photon loss, including absorption by a sample, substantially diminishes the benefits of quantum enhancement in interferometry. In this work, we experimentally investigate a dose-efficient approach to quantum phase estimation using sequential strategies in the presence of loss. Performance of sequential strategies with and without control is evaluated through quantum Fisher information (QFI) per dose. Experimental results show that both sequential strategies exceed the classical limit and outperform the parallel strategy using unbalanced N00N states. Notably, the control-enhanced sequential strategy attains superior QFI per dose, approaching the quantum limit. These results highlight the promise of sequential strategy for imaging and sensing in resource-constrained scenarios, marking a significant step toward practical and efficient quantum metrology in lossy environments.
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.