Timofeev, Igor; (2025) Superfluidity, Metastability, and Collective Spin Phenomena in Open Quantum Systems. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis covers research in three separate but interlinked areas: superfluidity in driven-dissipative polariton systems, path integral methods for the study of metastability in Markovian systems, and phase space methods for spin master equations. The unifying theme of the three works is their non-equilibrium nature, but other overlaps are also present. The first two topics share a primary analytical tool, the Feynman-Vernon path integral, with the first applying it to quantum field theory and the second to quantum mechanics. The first and third both utilise the TruncatedWigner method, albeit for very different purposes and rather different phase spaces, while the second and third illustrate the twin approaches of geometric and deformation quantisation. The first part is subdivided into three, dealing with different sub-projects undertaken under the broad umbrella of superfluidity. These are, in order, the absence of it in coherently driven polaritons, the application of TruncatedWigner to its study in Markovian polariton systems, and the analysis of whether non-linear dissipation can cause a non-interacting bosonic gas to exhibit it. Non-equilibrium functional integral methods are extensively used for all three but with varying emphasis: the first features extensive diagrammatic calculations, the second focuses on the stochastic semi-classical limit of the integral, while the third foregoes Feynman diagrams for a generating function approach. Conclusive results are obtained in the first project2, which completes a calculation by a predecessor in the group to show that no regime of coherently driven polaritons exhibits superfluidity; in the second I derive a method to extract the linear response tensor used for identifying superfluid components via the Truncated Wigner method derived from the functional integral, which should allow the numerical study of systems too complicated for the analytical tools employed in the first sub-part; in the final, third part I analyse a non-interacting bosonic system with non-linear dissipation, serving as a minimal model of a dye cavity photon Bose-Einstein condensate, to find that at mean-field level it is predicted to exhibit superfluidity. I then perform a significant portion of the associated fluctuation correction calculation, though difficulties with model regularisation prevent a conclusive answer being given at this time. Part II is devoted to calculations of Lindbladian spectra via the coherent state path integral, the initial aim of which was the study of metastability of systems governed by such Markovian superoperators. A working perturbation theory based on the exact propagator for a coherently driven-dissipative harmonic oscillator is derived and tested on some toy models. This perturbation theory in principle can be applied to systems not covered by the existing methods of third quantisation3 and exact expressions for truncated Fock bases. The part also includes a review of the two pre-eminent approaches to the coherent state path integral, emphasizing its links to holomorphic polarisation and highlighting the difficulties associated with it and consequently instanton calculations performed using it. Finally, moving away from bosonic systems, I develop a stochastic Truncated Wigner approach to spin degrees of freedom of arbitrary magnitude. This builds on earlierwork constructing the Stratonovich-Weyl representation of spin as functions on the 2-sphere S2 by calculating the explicit form of the series expansion of the associated star product to higher order than previously available in the literature. This is then used to study nearest-neighbour and long-range anisotropic XY models in two dimensions, obtaining promising results, in particular the direct observation of vortices associated with the BKT transition that destroy the staggered XY phase in the first model. This method differs from existing discrete Truncated Wigner methods and those specialised to spin-1 2, in particular being based around an expansion in a small parameter proportional to ℏ. This allows it to be easily integrated with bosonic phase space methods, since a truncation can then be carried out at a consistent order in ℏ.

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