Aspects of damping in correlated quantum systems

In this work we discuss three problems connected to the damping of oscillations in quantum systems: dynamics and decay in a random matrix model with conserved quantities, energy redistribution and decay of the amplitude mode in a superconductor, and the decay of the gapped mode in a molecular condensate, a bosonic analog to the superconductor. These problems highlight different aspects of damping and energy redistribution in quantum dynamics, while being simple enough that analytic control is possible. The first system captures the idea of a "partially conserved" quantity: in ergodic quantum systems, physical observables have a non-relaxing component if they overlap with a conserved quantity, but how to isolate the non-relaxing component is in general unclear. We compute exact dynamical correlators governed by a Hamiltonian composed of two large interacting random matrices, H=A+B, and we analytically obtain the late-time value of ⟨A(t) A(0)⟩, which quantifies the non-relaxing part of the observable A. We show that the relaxation to this value is governed by a power-law determined by the spectrum of the Hamiltonian H, independent of the observable A, while the long--time value and the amplitude of the oscillations depend on the trace--overlap between the operator and the Hamiltonian. For Gaussian matrices, we further compute out-of-time-ordered-correlators (OTOCs) and find that the existence of a non-relaxing part of A leads to modifications of the late time values and exponents. Our results follow from exact resummation of a diagrammatic expansion and hyperoperator techniques. The above problem deals with energy redistribution in a system with a complex internal structure, but without any spatial dependence nor many--body effects. In the second part of this work we discuss energy relaxation in a system with both: a BCS superconductor. In particular, we study the excitation of the collective Higgs oscillations of the order parameter by incoherent short pulses of light with frequency much larger than the ...