In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schrödinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described by a stochastic equation for the state vector. We present a method of deriving explicit evolution operators for linear quantum trajectories, and apply this to a number of physical examples of varying mathematical complexity. In the Heisenberg picture evolution resulting from continuous observation may be described by quantum Langevin equations. We use this method to examine the noise spectrum that results from a continuous observation of the position of a moving mirror, and examine the possibility of detecting the noise resulting from the quantum back-action of the measurement. In addition to the work on continuous measurement theory, we also consider the problem of reconstructing the state of a quantum system from a set of measurements. We present a scheme for determining the state of a single cavity mode from the photon statistics measured both before and after an interaction with one or two two-level atoms.
In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schrödinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described by a stochastic equation for the state vector. We present a method of deriving explicit evolution operators for linear quantum trajectories, and apply this to a number of physical examples of varying mathematical complexity. In the Heisenberg picture evolution resulting from continuous observation may be described by quantum Langevin equations. We use this method to examine the noise spectrum that results from a continuous observation of the position of a moving mirror, and examine the possibility of detecting the noise resulting from the quantum back-action of the measurement. In addition to the work on continuous measurement theory, we also consider the problem of reconstructing the state of a quantum system from a set of measurements. We present a scheme for determining the state of a single cavity mode from the photon statistics measured both before and after an interaction with one or two two-level atoms.
This is a theoretical thesis in the area of quantum measurement theory. Due to the extensive breadth of this field we choose to narrow our focus to examine a particular problem - the quantum Zeno effect (defined below). Quantum measurement theory is introduced in Chap. 1, using the terminology of effects and operations. This approach allows an operational definition of such terms as a state vector, an ensemble, and a measurement device (for instance), and a consideration of interactions between quantum systems and inaccurate measurement devices. We further introduce the quantum trajectories approach to consider the evolution of an individual quantum system subject to measurement. The quantum Zeno effect is introduced in Chap. 2. Any quantum treatment of a measurement interaction must consider the measurement backaction onto the measured system and this backaction will disrupt the free evolution of the system. The quantum Zeno effect occurs in the strong measurement limit where the measurement backaction totally freezes the evolution of the system, thus rendering the measurement useless. The effect is introduced via projective measurements of two level systems subject to measurement of level populations. At this stage we are able to discuss the main questions addressed by this thesis, and present its structure in Chap. 2. We then develop a new measurement model for the interaction between a system and a measurement device in Chap. 3. Our motivation in doing this is to better model the usual laboratory meter, and in our approach the meter dynamics are such that it relaxes towards an appropriate readout of the system parameter of interest. The irreducible quantum noise of the meter introduces fluctuations that drive the stochastic dynamical collapse of the system wavefunction. In our model, the measured system dynamics (if treated selectively) are described by a stochastic, nonlinear Schroedinger equation. A double well system subject to position measurement provides a natural first application for this model. This is done in Chap. 4 where we monitor the coherent tunnelling of a particle from one well to the other. The advantage afforded by considering this system is that it displays differing regimes where the measurement observable (position) is approximated as possessing either, respectively, a continuous or a discrete eigenvalue structure. Thus, we use this one model to explore the quantum Zeno effect in both measurement regimes. The above treatment is of a theoretical measurement model. In Chap. 5 we turn to consider a recent experimental test of the quantum Zeno effect which examined the dynamics of a two level atom subject to pulsed measurements of atomic level populations. We treat a slightly modified experiment in a fully continuous measurement regime. By first unravelling the optical Bloch equations, and second, using the quantum trajectories approach we demonstrate the existence of certain measurement regimes where there is a quantum Zeno effect, and other regimes where no measurement of the atomic populations is being effected at all. Through these results we demonstrate the importance of making a full analysis of the system-detector interaction before any conclusions can be made. In the remainder of the thesis we propose further possible tests of the quantum Zeno effect. In Chap. 6 the evolution of a Rydberg atom exchanging one photon with a single cavity mode subject to measurement is examined. The measurement is made by monitoring the photon number occupancy of the cavity mode using a beam of Rydberg atoms configured so as to perform phase sensitive detection. In the limit of frequent monitoring we show that the free oscillation of the atomic inversion is disrupted, and the atom is trapped close to its initial state. This is the quantum Zeno effect. In Chap. 7 we realize the Zeno effect on two possible systems. We consider first, a two level Jaynes-Cumming atom interacting with a cavity mode, and second, two electromagnetic modes configured as a multi-level parametric frequency converter. These systems interact with another cavity mode via a quadratic coupling system based on four wave mixing, and constructed to be a quantum nondemolition measurement of the photon number. This mode is damped to the environment thus effecting a measurement of the system populations. Again we show that this interaction, can manifest the quantum Zeno effect. Our explicit modelling of the system-detector interaction enables us to show how the effect depends on the resolution time of the detector. Finally, we consider a proposed measurement of the square of the quadrature phase of an electromagnetic mode in Chap. 8. Here, a three mode interaction mediated by a second order nonlinear susceptibility is considered. One mode, the pump, is prepared in a feedback generated photon number state to give insight into the role of pump noise. The other two modes are treated as an angular momentum system, and we show that photon counting on the two mode rotation system effects the above mentioned measurement. In addition, this measurement provides a direct measure of the second order squeezing of the signal. With that we finish our investigation of the quantum Zeno effect using the techniques of quantum measurement theory. However, in the epilogue [Chap. 9] we note that no thesis in quantum measurement theory would be complete without some consideration of the ``meaning attributed to the theory. In the epilogue we take a novel historical approach and examine the method by which metaphysical theories are formed to draw conclusions regarding quantum metaphysics.
The possibility of determining the state of a quantum system after a continuous measurement of position is discussed in the framework of quantum trajectory theory. The initial lack of knowledge of the system and external noises are accounted for by considering the evolution of conditioned density matrices under a stochastic master equation. It is shown that after a finite time the state of the system is a pure state, and can be inferred from the measurement record alone. The relation to emerging possibilities for the continuous experimental observation of single quanta, as for example in cavity quantum electrodynamics, is discussed.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
Starting from a generalization of the quantum trajectory theory [based on the stochastic Schr\"odinger equation (SSE)], non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows a consistent theory of quantum measurement in continuous time to be developed for these non-Markovian quantum trajectory models. In this context, the notions of ``instrument,'' ``a priori,'' and ``a posteriori'' states can be introduced. The key point is that by starting from a stochastic equation on the Hilbert space of the system, we are able to respect the complete positivity of the mean dynamics for the statistical operator and the requirements of the axioms of quantum measurement theory. The flexibility of the theory is next illustrated by a concrete physical model of a noisy oscillator where non-Markovian effects come from the random environment, colored noises, randomness in the stimulating light, and delay effects. The statistics of the emitted photons and the heterodyne and homodyne spectra are studied, and we show how these quantities are sensitive to the non-Markovian features of the system dynamics, so that, in principle, the observation and analysis of the fluorescent light could reveal the presence of non-Markovian effects and allow for a measure of the spectra of the noises affecting the system dynamics.
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.