Interpolated Collision Model Formalism

The dynamics of open quantum systems (i.e., of quantum systems interacting with an uncontrolled environment) forms the basis of numerous active areas of research from quantum thermodynamics to quantum computing. One approach to modeling open quantum systems is via a Collision Model. For instance, one could model the environment as being composed of many small quantum systems (ancillas) which interact with the target system sequentially, in a series of "collisions". In this thesis I will discuss a novel method for constructing a continuous-time master equation from the discrete-time dynamics given by any such collision model. This new approach works for any interaction duration, $\delta t$, by interpolating the dynamics between the time-points $t = n\,\delta t$. I will contrast this with previous methods which only work in the continuum limit (as $\delta t\to 0$). Moreover, I will show that any continuum-limit-based approach will always yield unitary dynamics unless it is fine-tuned in some way. For instance, it is common to find non-unitary dynamics in the continuum limit by taking an (I will argue unphysical) divergence in the interaction strengths, $g$, such that $g^2 \delta t$ is constant as $\delta t \to 0$.Comment: 121 pages, 3 figures, Daniel Grimmer's PhD Thesis University of Waterloo 202

A Discrete Analog of General Covariance: Could the world be fundamentally set on a lattice?

A crucial step in the history of General Relativity was Einstein's adoption of the principle of general covariance which demands a coordinate independent formulation for our spacetime theories. General covariance helps us to disentangle a theory's substantive content from its merely representational artifacts. It is an indispensable tool for a modern understanding of spacetime theories. Motivated by quantum gravity, one may wish to extend these notions to quantum spacetime theories (whatever those are). Relatedly, one might want to extend these notions to discrete spacetime theories (i.e., lattice theories). This paper delivers such an extension with surprising consequences. One's first intuition regarding discrete spacetime theories may be that they introduce a great deal of fixed background structure (i.e., a lattice) and thereby limit our theory's possible symmetries down to those which preserve this fixed structure (i.e., discrete symmetries). However, as I will discuss, these intuitions are doubly wrong and overhasty. Discrete spacetime theories can and do have continuous translation and rotation symmetries. Moreover, the exact same theory can be given a wide variety of lattice structures and can even be described with no lattice at all. As my discrete analog of general covariance will reveal: lattice structure is rather less like a fixed background structure or part of an underlying manifold and rather more like a coordinate system, i.e., merely a representational artifact. Thus, the world cannot be "fundamentally set on a square lattice" (or any other lattice) any more than it could be "fundamentally set in a certain coordinate system". Like coordinate systems, lattice structures are just not the sort of thing that can be fundamental; they are both thoroughly representational. Spacetime cannot be discrete (even when it might be representable as such).Comment: 36 pages, 10 figures. Video abstract: https://youtu.be/dc58WyWX-z

Collisional quantum thermometry

We introduce a general framework for thermometry based on collisional models, where ancillas probe the temperature of the environment through an intermediary system. This allows for the generation of correlated ancillas even if they are initially independent. Using tools from parameter estimation theory, we show through a minimal qubit model that individual ancillas can already outperform the thermal Cramer-Rao bound. In addition, due to the steady-state nature of our model, when measured collectively the ancillas always exhibit superlinear scalings of the Fisher information. This means that even collective measurements on pairs of ancillas will already lead to an advantage. As we find in our qubit model, such a feature may be particularly valuable for weak system-ancilla interactions. Our approach sets forth the notion of metrology in a sequential interactions setting, and may inspire further advances in quantum thermometry

Interpolated Collision Model Formalism

The dynamics of open quantum systems (i.e., of quantum systems interacting with an uncontrolled environment) forms the basis of numerous active areas of research from quantum thermodynamics to quantum computing. One approach to modeling open quantum systems is via a \textit{Collision Model. For instance, one could model the environment as being composed of many small quantum systems (ancillas) which interact with the target system sequentially, in a series of ``collisions''. In this thesis I will discuss a novel method for constructing a continuous-time master equation from the discrete-time dynamics given by any such collision model. This new approach works for any interaction duration, δt, by interpolating the dynamics between the time-points t = n δt. I will contrast this with previous methods which only work in the continuum limit (as δt → 0). Moreover, I will show that any continuum-limit-based approach will always yield unitary dynamics unless it is fine-tuned in some way. Given the central role of information flow between the system and environment plays in open quantum systems, unitary models are wholly insufficient. Thus continuum limit master equations must be fine-tuned to even function as valid models of open quantum systems. For instance, it is common to find non-unitary dynamics in the continuum limit by taking an (I will argue unphysical) divergence in the interaction strengths, g, such that g^2 δt is constant as δt →0. In addition to overcoming the above limitations, the new interpolation-based approach allows for the straightforward treatment of essentially any representation of a quantum system (e.g., Hilbert space vector, density matrix, Bloch vector, probability vector, in addition to a Gaussian state's mean vector and covariance matrix). Examples of each of these representations will be given throughout this thesis. Moreover, the new interpolation-based approach allows for an order-by-order analysis of the dynamics as a series in $\delta t$. This allows us to identify which types of dynamics are ``fast'' and which are ``slow'' as well as how this ``speed'' depends on the interaction Hamiltonian between the system and ancilla. For instance, we can (and will) investigate under what conditions we can see purification effects at first order in δt. As I will show the ``speed'' of the purification effects are tied to the complexity of the interaction; Purification at first order in δt requires the interaction Hamiltonian to be at least Schmidt rank-2. A necessary condition for thermalization is also discussed. In addition to this purification study, I will present a complete analysis of Gaussian dynamics regarding which types of dynamics appear at which orders in δt under which Hamiltonians. Given a Hamiltonian (either designed or fixed by fundamental considerations e.g., the light-matter interaction) we can determine what dynamics are supported at what orders in δt. Conversely, given some dynamics (e.g., from experiments) we can determine what class of interaction Hamiltonians could support it

The Unruh effect in slow motion

We show under what conditions an accelerated detector (e.g., an atom/ion/molecule) thermalizes while interacting with the vacuum state of a quantum field in a setup where the detector's acceleration alternates sign across multiple optical cavities. We show (non-perturbatively) in what regimes the probe `forgets' that it is traversing cavities and thermalizes to a temperature proportional to its acceleration. Then we analyze in detail how this thermalization relates to the renowned Unruh effect. Finally, we use these results to propose an experimental testbed for the direct detection of the Unruh effect at relatively low probe speeds and accelerations, potentially orders of magnitude below previous proposals.Comment: 5 pages (+ 8 pages of accessory appendices) 4 figures. RevTeX 4.

Machine learning quantum field theory with local probes

We propose the use of machine learning techniques to address the problem of local measurements in quantum field theory. In particular we discuss how neural networks can efficiently process measurement outcomes from local probes to determine both local and non-local features of the quantum field. As toy examples we show: a) how a particle detector distinguishes boundary conditions imposed on the field without the need of signals propagating from them, and b) how detectors can determine the temperature of the quantum field without thermalizing with it. We discuss how the formalism proposed can be used for any kind of local measurement on a quantum field and, by extension, to local measurements of non-local features in many-body quantum systems.Comment: 9 pages (4 pages of appendices), 2 Figures, RevTeX 4.