Time evolution of coupled multimode and multiresonator optomechanical systems

Abstract

We study the time evolution of bosonic systems where multiple driven bosonic modes of light interact with multiple mechanical resonators through arbitrary, time-dependent, optomechanical-like interactions. We find the analytical expression for the full time evolution of the system and compute the expectation value of relevant quantities of interest. Among the most interesting ones, we are able to compute the first-order quantum bipartite coherence between pairs of subsystems, and the analytical expression for the mixedness induced by the nonlinear interaction in the reduced state of the mechanical oscillators. Finally, we also compare our results with a linearised version of the system, and we find a regime where there are qualitative and quantitative differences in the behavior of some measurable quantities. Our results can therefore be used to describe the full time-evolution of the system, to characterise its nonlinear character and explore the validity of the linearisation approach.

Time evolution of two harmonic oscillators with cross-Kerr interactions

“…We note that, as expected, the result (D5) reduces to the existing one in the literature forg ′ 2 (τ ) = 0, see [5].…”

“…where we have omitted the expression forâ(τ ) which can be computed but is not illuminating. The change ∆N b (τ ) := N b (τ ) − N b (0) in the number of phonons, given our initial state (5), reads…”

“…Finally, we can also compute the mixedness of the reduced state. The calculations can be found in Appendix D. Lengthy algebra, and the use of a similar approach developed in the literature [5], allow us to find…”

“…Now, we can follow the same procedure introduced in [5] and compute (D4) explicitly. We find S N (ρ m (τ )) =1 − e −2 |µ| 2 n,n ′ |µ| 2 n+2 n ′ n!…”

“…Interestingly, these techniques do not apply to this class of states and transformations only, but it can be generalised to arbitrary states and arbitrary Hamiltonians. This novel approach has led to very interesting preliminary results, where complete and analytical solutions of the time evolution of the system were found for optomechanical-like Hamiltonians of arbitrary numbers of modes of light interacting with arbitrary numbers of mechanical resonators [4,5]. This has allowed, for example, for a quantitative analysis of the nonlinear character of an optomechanical system [6].…”