General solution of the time evolution of two interacting harmonic oscillators

Abstract

We study the time evolution of an ideal system composed of two harmonic oscillators coupled through a quadratic Hamiltonian with arbitrary interaction strength. We solve its dynamics analytically by employing tools from symplectic geometry. In particular, we use this result to completely characterize the dynamics of the two oscillators interacting in the ultrastrong-coupling regime with additional single-mode squeezing on both oscillators, as well as higher-order terms. Furthermore, we compute quantities of interest, such as the average number of excitations and the correlations that are established between the two subsystems due to the evolution. We find that this model predicts a second-order phase transition and we compute the critical exponents and the critical value. We also provide an exact decoupling of the time evolution in terms of simple quantum optical operations, which can be used for practical implementations and studies. Finally, we show how our techniques can be extended to include more oscillators and higher-order interactions.

Ion-laser-like interaction in optomechanical systems with Kerr nonlinearities

“…In order to obtain the more familiar position-position form of the interaction, we have used the Fourier transform operator. This Hamiltonian corresponds to the usual interaction of two oscillators and has been solved in the context of classical-quantum analogies [45], time-dependent coupling [46], and by using tools from simplectic geometry [47].…”

Factorizing time evolution into elementary steps

“…Regardless of the apparently daunting perspective of factorizing the time-evolution operator, in the past decades an increasing number of studies has tackled the problem using mathematical tools stemming from Lie algebra theory. They have addressed both the general theoretical aspects [21,22,11], as well as the concrete model-dependent ones [12,9,7,10]. So far, it has been unclear what are the general assumptions that guarantee that a factorization can be obtained, and when such factorization is finite.…”

“…Finally, a recent revival of interest has brought novel insights [20]. Implementations using symplectic geometry [11] 1 , as well as solutions obtained for specific classes of systems [12,9,7,10] are now available. Nevertheless, only recently a general result on which classes of Hamiltonians can be factorized into a product of finite terms has been obtained [8].…”

Virtual Excitations and Entanglement Dynamics and Polygamy in Three Ultra‐Strongly Coupled Systems

“…As a result, the rotating wave approximation (RWA), where CRT is neglected, breaks down. [2,7,17] Multipartite systems made of coupled oscillators are central in physics [18][19][20] and for reviews we cite the references. [21][22][23] In fact, they lead to the modeling of coupled ions in ion traps, [24] arrayed coupled nano-sized electromechanical devices, [25] light propagation in inhomogeneous media, [26] and a nitrogen vacancy ensemble embedded in a diamond nanobeam.…”