Coherent Dynamics of Low Dimensional Quantum Fluids of Light and Matter

In this Thesis we apply several theoretical techniques developed in the fields of Quantum and Nonlinear Optics, Statistical Mechanics and Condensed Matter to the study of a few relevant systems where the coupling of radiation and matter degrees of freedom plays a central role. While the original results presented here are of analytical or computational origin, the experimental aspects of the available platforms are thoroughly discussed all over the manuscript. One main approach underlying many parts of the Thesis is to describe a quantum fluid via a classical field; this is possible when the fluid possesses a high degree of coherence, as a result of the transition to a lasing state or because the fluid is coherently created by an external drive. Focusing on the coherent component of the dynamics of the quantum fluid allows to obtain an effective description of many interesting phenomena without the formidable effort of dealing with the full quantum problem. As a consequence, the starting point of such a semiclassical analysis will typically be a nonlinear Schroedinger equation: the Gross-Pitaevskii equation for weakly interacting quantum gases and its driven-dissipative extensions and the Complex Ginzburg Landau equation for the dynamics of laser systems will be two important declination of this concept. Another element which plays a major role is dimensionality. Basically all the devices that we will review are implementions of one or two dimensional models. Correspondingly, polariton hydrodynamics will be investigated in one and two dimensions; even more importantly, the low dimensionality of the lattice determines the lack of long-range order of the field emitted from a 1D laser array or from the edge of a 2D topological device, resulting in a broadening of the linewidth. In contrast to these unifying methodological elements, the range of experimental platforms discussed is quite wide. For this reason, the division in Chapters has been chosen based on the physical system, while the separation between known and original results has been performed via the use of the Sections, the material contained in a Section marked by an asterisk being mostly due to ourselves. Having stated the general scope and features of the Thesis, let's briefly introduce the contents of the individual parts.More specifically: Chapter 1 deals with hydrodynamics and superfluidity of resonantly injected polariton fluids (Carusotto and Ciuti [2013]). In semiconductor microcavities, a mode of the electromagnetic field can be strongly coupled to the excitonic transitions of the embedded quantum well, resulting in quasi-particle excitations called exciton-polaritons (Yu and Cardona [2010]). These have bosonic nature, light mass and are weakly interacting. Depending on the pumping scheme, polariton condensation can be achieved (Kasprzak et al. [2006]), which brings many analogies with the physics of lasers; alternatively, polaritons can be injected quasi-resonantly, giving rise to a rich phenomenology which can be described in terms of the generalized Gross-Pitaevskii equation introduced by (Carusotto and Ciuti [2004]) and which includes bistable behaviours and flow without scattering. After reviewing these well known results, we report some interesting features of the generalized Gross-Pitaevskii equation. First, we provide, via a formal argument based on Galilean boosts, a rederivation of the Doppler shift and of the link between critical velocity and speed of sound. In particular, under an infinite excitation spot, the flow of a polariton fluid against a static defect and the displacement of a moving defect in a fluid at rest are related by a mathematical boost and in some sense are the same situation pictured in two different reference frames (Amelio et al.[2020b]). The other important finidings spring from a careful reconsideration of the results by (Pigeon et al. [2011]), which suggested that, in analogy to weakly interacting atomic gases, a polariton fluid ejected from a finite spot against a static defect would alternate a superfluid, turbulent and solitonic behaviour for increasing injection velocity. Instead, we show that the dynamics of the ejected flow is an extremly nonlinear one and it is not possible to directly control the flow velocity via the pump wavevector (Amelio and Carusotto [2020a]). Correspondingly, the flow against a defect always features a solitonic pattern, which can be shallower or deeper, but never displays clean superfluidity and never breaks down in vortical or turbulent dynamics. These numerical observations raise important questions on the validity of the generalized Gross-Pitaevskii description in the conditions in which the experiment by (Amo et al. [2011]) yielded a scenario compatible with the atomic gas predictions. Chapter 2 is devoted to the study of the physics of resonantly injected polariton fluids in the presence of a dark excitonic reservoir. Since such reservoir introduces an important bias when trying to estimate the polariton-polariton interaction constant from the observed blueshift (Sun et al. [2017], Estrecho et al. [2019]), we start reviewing the theory of exciton-exciton interactions (Ciuti et al. [1998]). The experimental efforts directed at measuring the dispersion of the collective excitations of the polariton fluids are then summarized, and the experiment by Stepanov et al. [2019], which is the leading thread of the Chapter, is discussed in details. In this experiment, the excitation laser is linearly polarized and the detection occurs in the opposite polarization. The energy and angle resolved photoluminescence from the fluid is collected as close as possible to the acoustic regime. Looking at the measured dispersion of the excitations on top of the fluid, as presumably induced by thermal phonons, a much reduced slope is observed with respect to the standard theory of a fully coherent polariton fluid, where the speed of sound is determined by the blueshift. This fact, together with other recent findings (Sarkar et al. [2010], Walker et al. [2017]), suggests the existence of dark excitonic states which are populated by polariton-polariton scatterings, contribute to the blueshift of the fluid and have a slower dynamics than the polaritons, so that they impact on the dynamics only at frequencies lower than the linewidth. As a consequence, our theoretical analysis highlights that the speed of sound is mostly determined by the fraction of the Hartree energy due to polariton-polariton scattering, while exciton-polariton interactions mainly produce a blueshift of the fluid. While these concepts are first illustrated for a circularly polarized fluid, a careful quantitative analysis has been performed to include the interplay of the two linear polarizations and to model the excitation of the photoluminescence by thermal phonons. Keeping into account all these features allows for an estimate of the reservoir parameters and for the polariton-polariton interaction constant. In the final Section, the influence of the resevoir on the superfluid phenomenology of the polaritons is highlighted (Amelio et al. [2020b]). In particular, the breaking of the formal Galilean invariance argument discussed in Chapter 1, entails that the critical velocity is not in general given by the speed of sound. The two are equivalent for a defect moving in a fluid at rest, while in the steady state situation of a fluid flowing past a static defect the critical velocity is determined by the total blueshift. Chapter 3 addresses the spatial and temporal coherence of 1D nonequilibium quasi-condensates, or equivalently of tight binding models of 1D laser arrays. In the first Section the basic concepts underlying LASER operation are illustrated. We show how the celebrated Schawlow-Townes linewidth (Schawlow and Townes [1958]) can be derived in a simple semi-classical analysis: due to the phase invariance of the laser equations, spontaneous emission events (or other sources of white noise) lead to a diffusion of the phase of the laser field and an exponential decay of the coherence in time (Henry [1982]). A brief overview of different laser devices is also provided. While the Schawlow-Townes treatment is adequate for an effectively zero dimensional laser field, the statistical mechanics literature focuses on the spatio-temporal correlation functions of infinitely extended nonequilibrium fluids described by the Complex Ginzburg Landau equation (Gladilin et al. [2014], He et al. [2015], Ji et al. [2015], Squizzato et al. [2018]), of which polariton condensates are an example. In 1D in particular, any linearized approached is bound to fail due to the absence of long-range order; a nonlinear Kuramoto-Sivashinskii equation for the dynamics of the phase can be derived by eliminating the fast density fluctuations. Then, the renormalization group approach predicts that the infrared physics falls into the Kardar-Parisi-Zhang (KPZ) universality class (Kardar et al. [1986]), which was originally proposed to describe the stochastic growth process of a classical interface. Our contribution (Amelio and Carusotto [2019]) consists in elucidating the connection between the Schawlow-Townes linewidth and KPZ physics. The former can be viewed as a finite-size effect that will dominate at very long times, while KPZ universality is visible at intemediate times and for large enough systems. Moreover, while when the system size is small enough the linewidth is the one described by the Bogoliubov-Schawlow-Townes theory, a marked broadening occurs for large enough systems, where corrections due to the nonlinear phase dynamics are expected. More specifically, the coherence time is expected to scale as the length of the system in the linear regime and as its square root in the nonlinear one. Chapter 4 presents the first theoretical study of the coherence properties of a topological laser. First, we review the impetuous developement of the topological photonics field and how lasing in the edge modes of a topological lattice for photons has been achieved by several groups in the last three years (Bahari et al. [2017], Harari et al. [2018], Zeng et al. [2020]). A theoretical model of a topological laser is provided by the 2D Harper-Hofstadter laser (Longhi et al. [2018]). The core of this Chapter reports the semiclassical solution of this model in the case of a class-A device and for small intensity fluctuations (Amelio and Carusotto [2019]). To this end, the Bogoliubov modes are computed and explained via a dimensional reduction argument (Loirette-Pelous [2020]). Moreover, the structure of the correlation functions reflects the fact that fluctuations travel in a chiral way around the edge of the system. The linewidth is computed via the Bogoliubov method. We generalize the notion of the transverse Petermann factor and show that it is close to 1, clarifying the nature of topological lasing as opposed to gain guiding. For large enough arrays, the field on the edge brings the signatures of KPZ universality and of the linewidth broadening due to the nonlinear phase dynamics. Then, we demostrate the robustness of the coherence properties to the presence of static onsite disorder: more precisely, the disorder threshold for the topological array is roughly set by the band-gap and is orders of magnitude higher than for a topologically trivial array. The robustness of the KPZ universality in the correlation functions is instead mostly determined by the density of the edge states. Remarkably, the coherence time is enhanced by an intermediate amount of disorder, since the Bogliubov modes of the Goldstone dome have a shorter lifetime and the nonlinear phase dynamics is hampered. These results have been derived within a very simple model. We expect that including nonlinearities and the carrier dynamics within the tight binding framework should not introduce major qualitative modifications if no instabilities arise; the time and length scales entering the correlation functions will instead change, per example yielding a Henry factor for the linewidth (Henry [1982]). Finally, the tight binding approximation has no predictive power in determining whether the emission is single or multi-moded in the very relevant case of semiconductor laser arrays; however, if single moded emission is verified experimentally, our guess is that the qualitative structure of the correlation functions described here should apply also to that situation.