This dissertation studies the statistics and modeling of a quantum system probed by a coherent laser field. We focus on an ensemble of qubits dispersively coupled to a traveling wave light field. The first research topic explores the quantum measurement statistics of a quasi-monochromatic laser probe. We identify the shortest timescale that successive measurements approximately commute. Our model predicts that for a probe in the near infrared, noncommuting measurement effects are apparent for subpicosecond times. The second dissertation topic attempts to find an approximation to a conditional master equation, which maps identical product states to identical product states. Through a technique known as projection filtering, we find such a equation for an ensemble of qubits experiencing a diffusive measurement of a collective angular momentum projection, and global rotations. We then test the quality of the approximation through numerical simulations. In the presence of strong randomized rotations, the approximation reproduces the exact expectation values to within 95%. The final topic applies the projection filter to the problem of state reconstruction. We find an initial state estimate based on a single continuous measurement of an identically prepared atomic ensemble. Given the ability to make a continuous collective measurement and simultaneously applying time varying controls, it is possible to find an accurate estimate given based upon a single measurement realization. Here we explore the fundamental limits of this protocol by studying an idealized model for pure qubits, which is limited only by measurement backaction. Using the exact dynamics to produce simulated measurements, we then numerically search for a maximum likelihood estimate based on the approximate expression. Our estimation technique nearly achieves an average fidelity bound set by an optimum POVM.
This dissertation studies the statistics and modeling of a quantum system probed by a coherent laser field. We focus on an ensemble of qubits dispersively coupled to a traveling wave light field. The first research topic explores the quantum measurement statistics of a quasi-monochromatic laser probe. We identify the shortest timescale that successive measurements approximately commute. Our model predicts that for a probe in the near infrared, noncommuting measurement effects are apparent for subpicosecond times. The second dissertation topic attempts to find an approximation to a conditional master equation, which maps identical product states to identical product states. Through a technique known as projection filtering, we find such a equation for an ensemble of qubits experiencing a diffusive measurement of a collective angular momentum projection, and global rotations. We then test the quality of the approximation through numerical simulations. In the presence of strong randomized rotations, the approximation reproduces the exact expectation values to within 95%. The final topic applies the projection filter to the problem of state reconstruction. We find an initial state estimate based on a single continuous measurement of an identically prepared atomic ensemble. Given the ability to make a continuous collective measurement and simultaneously applying time varying controls, it is possible to find an accurate estimate given based upon a single measurement realization. Here we explore the fundamental limits of this protocol by studying an idealized model for pure qubits, which is limited only by measurement backaction. Using the exact dynamics to produce simulated measurements, we then numerically search for a maximum likelihood estimate based on the approximate expression. Our estimation technique nearly achieves an average fidelity bound set by an optimum POVM.
This thesis explores the topics of parameter estimation and model reduction in the context of quantum filtering. The last is a mathematically rigorous formulation of continuous quantum measurement, in which a stream of auxiliary quantum systems is used to infer the state of a target quantum system. Fundamental quantum uncertainties appear as noise which corrupts the probe observations and therefore must be filtered in order to extract information about the target system. This is analogous to the classical filtering problem in which techniques of inference are used to process noisy observations of a system in order to estimate its state. Given the clear similarities between the two filtering problems, I devote the beginning of this thesis to a review of classical and quantum probability theory, stochastic calculus and filtering. This allows for a mathematically rigorous and technically adroit presentation of the quantum filtering problem and solution. Given this foundation, I next consider the related problem of quantum parameter estimation, in which one seeks to infer the strength of a parameter that drives the evolution of a probe quantum system. By embedding this problem in the state estimation problem solved by the quantum filter, I present the optimal Bayesian estimator for a parameter when given continuous measurements of the probe system to which it couples. For cases when the probe takes on a finite number of values, I review a set of sufficient conditions for asymptotic convergence of the estimator. For a continuous-valued parameter, I present a computational method called quantum particle filtering for practical estimation of the parameter. Using these methods, I then study the particular problem of atomic magnetometry and review an experimental method for potentially reducing the uncertainty in the estimate of the magnetic field beyond the standard quantum limit. The technique involves double-passing a probe laser field through the atomic system, giving rise to effective non-linearities which enhance the effect of Larmor precession allowing for improved magnetic field estimation. I then turn to the topic of model reduction, which is the search for a reduced computational model of a dynamical system. This is a particularly important task for quantum mechanical systems, whose state grows exponentially in the number of subsystems. In the quantum filtering setting, I study the use of model reduction in developing a feedback controller for continuous-time quantum error correction. By studying the propagation of errors in a noisy quantum memory, I present a computation model which scales polynomially, rather than exponentially, in the number of physical qubits of the system. Although inexact, a feedback controller using this model performs almost indistinguishably from one using the full model. I finally review an exact but polynomial model of collective qubit systems undergoing arbitrary symmetric dynamics which allows for the efficient simulation of spontaneous-emission and related open quantum system phenomenon.
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3⊕F=4), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3⊕F=4), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.
The field of quantum metrology concerns the physical measurement of sensors with a precision comparable to fundamental limits set by quantum mechanics. It is possible to outperform naive interpretations of these limits by using entangled states of the sensor system. One example is that of a spin-squeezed state, in which the uncertainty of one variable is decreased at the expense of another while still obeying Heisenberg's uncertainty principle, improving rotation sensitivity along a chosen axis. These states are potentially useful in devices including atomic clocks, inertial sensors, and magnetometers. Any model of a quantum metrology device must respect the fact that physical measurements are not passive, as imagined classically, but necessarily invasive. Far from being a negative feature, well-understood quantum measurement can conditionally drive a system into desirable entangled states, including spin-squeezed states. Furthermore, the fundamental randomness of this process can, in principle, be removed with real-time feedback control, motivating an adaptation of classical feedback concepts to the quantum realm. In this thesis, I describe these ideas in the context of one experimental example. A laser-cooled cloud of cesium spins is polarized along one axis via optical pumping and, subsequently, a linearly polarized far-off resonant probe beam traverses the sample. Due to the interaction Hamiltonian, the optical polarization rotates by an amount nominally proportional to one spin component of the collective spin state, enacting a weak, continuous, nondemolition measurement of that collective variable. This optical Faraday rotation is then measured with a polarimeter and the inherently noisy result used to condition the collective atomic state via a quantum filter, or stochastic master equation. Ideally, this process is capable of producing spin-squeezed states via the measurement itself. The details of this measurement are investigated in depth, including a derivation of the nonideal polarizability Hamiltonian, an analysis of the projection process with control, and a derivation of the magnetometry sensitivity. Experimentally, we demonstrate continuous measurement of the collective spin state with a large single-shot signal-to-noise ratio and verify many predictions of the model. Finally, we describe attempts to observe the atomic projection noise, which would infer the preparation of spin-squeezed states.
Quantum systems are the foundation of quantum technology development. To harness these systems, one must extract information from them and be able to control their dynamics. This thesis presents a body of work, containing both published and unpublished papers, that focuses on two key research areas: quantum continuous measurement and quantum bath sensing. While these topics may seem distinct, they are fundamentally connected through the framework of open quantum systems. As such, this thesis is structured into two parts. The first part of this thesis is dedicated to diffusive continuous quantum measurement, where the system of interest is continuously monitored through a coupled Markovian bath or meter. A key challenge in this context is the numerical errors introduced in quantum trajectory simulations caused by the finite time resolution ( 𝑡) in real experiments. To address this, we propose a systematic framework to assess theoretical tools used in processing measurement records. Focusing on a single Lindblad channel, we analyze the errors of existing measurement (or Kraus) operators in literature, in orders of 𝑡 under conditions that ensure valid average evolution (complete positivity, convex linearity, and trace preservation). We also analyze a stronger condition: finding the order at which these existing measurement operators match the Lindblad equation. Based on these analyzes, we propose a Kraus operator that satisfies the conditions for valid quantum evolution and matches the Lindblad master equation to second order in 𝑡, surpassing all existing approaches. Moreover, to evaluate how well our operator reproduces exact quantum trajectories, we analyze two qubit measurement examples where exact maps can be derived: 𝑧-basis measurement and fluorescence (dissipative) measurement. We show analytically that our proposed operator achieves the smallest average trace distance to the exact quantum trajectories compared to existing methods. Finally, we extend this Kraus operator to multiple Lindblad channels with a driving Hamiltonian. We test our analytical findings numerically using the two qubit examples, considering both single and multiple Lindblad channels with a driving Hamiltonian. The numerical results confirm our theoretical predictions. The second part of this thesis focuses on quantum bath sensing, where we use a quantum system (qubit) to extract information about its surrounding baths. Specifically, we develop protocols to detect nonclassical pure-dephasing noise in a single-qubit system coupled to temporally correlated (non-Markovian) baths. Inspired by the Ramsey protocol, we introduce a coherent intervention (𝜋 2 -pulse) within the protocol, which can reveal the quantum noise - noise correlations affecting the probe qubit that originate from quantum degrees of freedom. Our protocols are simple, experimentally viable, and rigorously derived for general bath models. Using an exactly solvable bosonic bath model, we analyse how key noise parameters respond to fixed control schemes, identifying the optimal strategy for detecting the quantum noise. From the perspective of bath dynamics, we further decompose the qubit-bath state, demonstrating that only Gaussian components of the bath contribute to quantum noise, highlighting the bath's role in system dynamics. Finally, we extend the modified Ramsey protocol by incorporating multi-axis 𝜋 2 pulse and demonstrate that it can emulate the modified Ramsey protocol with any completely positive trace-preserving intervention. This applies to both Gaussian and non-Gaussian baths under the pure-dephasing assumption.
This thesis presents a collection of projects spanning different sub-fields of quantum science, undertaken with the goal of understanding and advancing quantum technologies. Broadly, it can be divided into two thematic parts — parameter estimation and open system dynamics. Each project addresses inverse and ill-posed problems in Physics, which lie at the heart of developing robust and scalable quantum devices.A central theme in quantum technology is the challenge of inference of microscopic system parameters from limited, indirect observations. The first part of the thesis investigates this through the problem of Hamiltonian reconstruction via local measurements. Indirect estimation of couplings in quantum systems relies on the measurement of the spectrum and the overlap of eigenvectors with some reference state. For a one-dimensional chain, this data can be obtained by local measurement on one end of the chain, thereby reducing the need for full Hamiltonian tomography. Information about couplings is extracted from the temporal data obtained by measuring the expectation values of the time evolution 𝑒−𝑖𝐻𝑡 for a reference state. In Chapter 3, we analyze the robustness of this estimation scheme under perturbations arising from beyond nearest neighbour interactions. We show that the error scales linearly with the strength of such interactions and on average, the error increases with the distance of interaction between sites.Chapter 4 examines whether coupling information can be extracted from local thermal measurements, using expectation values of the Gibbs state at varying temperatures rather than temporal data. We develop a procedure to obtain the coefficients of the Taylor expansion of these expectation values and relate them to the Hamiltonian parameters. For very small chains, this mapping can be carried out analytically, while for larger systems it reveals the algebraic structure that links local measurements to the underlying couplings and highlights the trade-offs between measurement resources, numerical stability, and identifiability. The analysis also clarifies how additional local fields or self-interaction terms enhance the information content of thermal data, suggesting a role for local thermal probes as practical tools for Hamiltonian verification and parameter estimation.Building on the ideas of controlling and inferring information about a quantum system, we now turn to active intervention through adaptive measurement. In Chapter 5, we develop a process tensor driven procedure for cooling the spin bath that is coupled to a nitrogen vacancy (NV) center in diamond. By modeling the sequence of intermediate quantum instruments acting on a correlated system-environment process, we optimize a pair of completely positive (CP) maps to maximize the population in a lower-energy state. Unlike existing approaches that rely solely on positive operator valued measures (POVMs), our method encompasses a broader class of CP maps, including adaptive feed-forward and mixed operations. We explicitly account for initial correlation between system and environment and distinguish between the controllable and uncontrollable parts of the dynamics. Our framework supports non-Markovian operations and benchmarks them against optimal Markovian strategies. The optimization is performed via Semi-definite Programming (SDP), ensuring global optimality. We also assess robustness to Hamiltonian noise and provide a proposal for implementation using NV center in diamonds.Finally, Chapter 6 explore the relaxation dynamics of topological qubits, focusing on a tetron qubit formed from two uncoupled Kitaev chains coupled to a finite fermionic bath. Using the correlation matrix formalism for purely quadratic couplings, we show that the relaxation is fundamentally independent of temperature when either the bath is large, or the zero-mode energy of the system is close to zero. This challenges a key assumption in the field that cooling enhances coherence. Interestingly, we find that introducing glassy disorder into the system slows down relaxation, offering a potential route to prolong qubit lifetimes.Collectively, these studies form a coherent narrative around extraction, control and utilization of quantum information under fundamental constraints. By tackling such inverse and often ill-posed problems, this thesis contributes new tools and insights essential for the realization of future quantum technologies.