Bridges between quantum and classical mechanics: Directed polymers, flocking and transitionless quanum driving

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Loewe Yanez, Benjamin Andres
Goldbart, Paul M.
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Often considered separated worlds, classical and quantum mechanics share numerous connections with one another. Indeed, as classical mechanics corresponds to a limiting case of quantum mechanics, certain concepts and elements of physical intuition developed in one theory can be and have been used to tackle issues in the other. Nevertheless, these connections do not only cover conceptual issues but also numerous techniques. Indeed, the inherently probabilistic nature of quantum mechanics and its close resemblance to Markovian stochastic processes opens the door to the application of a broad range of powerful methods, initially developed for quantum mechanics, in classical equilibrium and non-equilibrium statistical mechanics. In this thesis we develop progress in three subjects by taking advantage of either a conceptual or a methodological connection between quantum and classical mechanics. First, we develop the well-known mapping between systems of strongly repelling, two-dimensional directed lines and systems of one-dimensional fermions in order to extend the directed polymer problem to richer geometries. By expressing path integrals in generalized curvilinear coordinates, we successfully generalize the model to settings such as polymers anchored to curved edges, polymers constrained to uneven walls, and polymers constrained to curved surfaces. In each case, we identify the Hamiltonian of an analogous quantum system, which, because of the new geometry of each setting, develops features such as a time- and position-dependent mass and an external electromagnetic vector potential. Along the way, we perform an in-depth analysis of the approximations made and establish regimes of their validity. Finally, in order to obtain analytical results, we employ an extension of the time-dependent perturbation theory scheme of quantum mechanics to imaginary time. Complementing this with the assumption of ground-state dominance, we obtain compact expressions for universal shifts in the free energies of the various systems that allows for isolation of the effects of the distinct geometrical properties of each system. The second piece of work presented into this thesis relates to the non-equilibrium setting of active particles. Strongly interacting, self-propelled particles can form a spontaneously flowing, polar (i.e. motionally aligned), active fluid. The study of the connection between the microscopic dynamics of a single, self-propelled particle and the macroscopic dynamics of a liquid comprising such particles can yield insights into experimentally realizable active flows, but this connection is well understood in only a few select cases. Here, we introduce a model of self-propelled particles that is based on an analogy with the motion of an electron subject to strong spin-orbit coupling. We find that, within our model, self-propelled particles experience an analog of the Heisenberg uncertainly principle that instead relates positional and rotational noise. An extension to many-component (and hence more classical) spinors, under which this uncertainty relation vanishes, contributes to the justification of this interpretation. Furthermore, by coarse-graining the microscopic model, we find expressions for the coefficients of the hydrodynamic Toner-Tu equations, established some time ago to describe an active liquid composed of “active spins.” The connection between self-propelled particles and quantum spins may possibly provide a route for realizing exotic phases of matter using active liquids, via inspiration hailing from systems composed of strongly correlated electrons. The third and final piece of work to be presented on this thesis consists of a semi-classical approach to Transitionless Quantum Driving (TQD). TQD is a method, developed by means of a reverse engineering strategy, under which non-adiabatic transitions in time-dependent quantum systems are, in M. V. Berry's words, “stifled” through the introduction of a specific auxiliary Hamiltonian. This Hamiltonian comes, however, expressed as a formal sum of outer products of the original instantaneous eigenstates and their time-derivatives. Generically, how to actually create such an operator in the laboratory is thus rarely evident. The operator may even be non-local. By following a semi-classical approach, we obtain a recipe that yields the auxiliary Hamiltonian explicitly, in terms of the fundamental operators of the system (e.g., position and momentum). By using this formalism, we are able to ascertain criteria for the locality of the auxiliary Hamiltonian, and also to determine its exact form in certain special cases. Lastly, the explicit connection between the auxiliary Hamiltonian and the observables of the system allows for a perturbation scheme in cases in which an exact solution is not easily achievable. This scheme shows that, even in situations in which an exact local auxiliary term cannot be achieved, under special circumstances it is possible to achieve an operator that is local and still approximately stifles non-adiabatic transitions.
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