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We study the non-equilibrium dissipative dynamics of the center of mass of aparticle coupled to a field via its internal degrees of freedom. We model theinternal and external degrees of freedom of the particle as quantum harmonicoscillators in 1+1 D, with the internal oscillator coupled to a scalar quantumfield at the center of mass position. Such a coupling results in a nonlinearinteraction between the three pertinent degrees of freedom -- the center ofmass, internal degree of freedom, and the field. It is typically assumed thatthe internal dynamics is decoupled from that of the center of mass owing totheir disparate characteristic time scales. Here we use an influence functionalapproach that allows one to account for the self-consistent backaction of thedifferent degrees of freedom on each other, including the couplednon-equilibrium dynamics of the internal degree of freedom and the field, andtheir influence on the dissipation and noise of the center of mass. Consideringa weak nonlinear interaction term, we employ a perturbative generatingfunctional approach to derive a second order effective action and acorresponding quantum Langevin equation describing the non-equilibrium dynamicsof the center of mass. We analyze the resulting dissipation and noise arisingfrom the field and the internal degree of freedom as a composite environment.Furthermore, we establish a generalized fluctuation-dissipation relation forthe late-time dissipation and noise kernels. Our results are pertinent to openquantum systems that possess intermediary degrees of freedom between system andenvironment, such as in the case of optomechanical interactions.

Dissipative dynamics of a particle coupled to field via internal degrees of freedom