Aspects of damping in correlated quantum systems
Matteo Bellitti
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 superconducting gap. We find that the excitation amplitude of the Higgs mode is controlled by a single parameter, determined by the total number of quasiparticles excited by the pulse, which we trace back to the universality of the shape of the light-induced quasiparticle cascade at energy below the Debye frequency and above the gap. Our analysis is primarily based on the Keldysh technique for non--equilibrium field theory and the Boltzmann kinetic equation. Finally, we study the damping of the gapped mode in a molecular Bose--Einstein condensate, the Bosonic analogue of a BCS superconductor. This system has the advantage of giving the experimentalist fine control over the interatomic interactions using Feshbach resonances, and is the object of renewed interest as the molecular superfluid phase has only very recently been realized in the lab. We discuss damping in the nontrivial thermodynamic phases: in the molecular superfluid phase the gapped excitation is protected by parity, and is damped only above a threshold momentum --as in the Cherenkov effect--, while in the atomic superfluid phase the gapped mode is damped at all momentum scales. We propose a class of experiments where our results are measurable: transmission (and reflection) of an atom beam through a molecular condensate cloud.
•Jan 1, 2023•0 citations