Quantum Optics Seminar: Solving nonlinear optomechanical dynamics with a Lie algebra method

Miguel Bello (MPQ):
Hamiltonians that contain products of more than two operators give rise to nonlinear equations of motions.

Miguel Bello (MPQ)
Group seminar (hybrid format: online/seminar room B2.46)
Fri 12 November 2021 2:30 pm (MEZ)

Abstract:

Hamiltonians that contain products of more than two operators give rise to nonlinear equations of motions. These equations and the time-evolution of the Hamiltonian are often notoriously difficult to solve, but the nonlinear dynamics allow for the generation of non-Gaussian states and is a key component for many quantum-information processing schemes.

One example of such a nonlinear Hamiltonian is the radiation-pressure Hamiltonian that arises from light interacting with a mechanical element. Here, the photon number operator couples to the centre-of-mass position of the mechanical mode. The solutions of the dynamics was solved for a constant optomechanical coupling back in 1997 [1,2]. However, certain experimental implementations also allow for time-modulated couplings, which can be tricky to treat.

In my talk, I will provide an introduction to a Lie algebra method [3] that allowed us to solve the dynamics for a time-dependent optomechanical coupling. I will also provide an overview of the results that we derived using the method, such as the effect of optical decoherence on the intra-cavity state.

[1] Bose, S., K. Jacobs, and P. L. Knight. "Preparation of nonclassical states in cavities with a moving mirror." Physical Review A 56.5 (1997): 4175.
[2] Mancini, S., V. I. Man'ko, and P. Tombesi. "Ponderomotive control of quantum macroscopic coherence." Physical Review A 55.4 (1997): 3042.
[3] Wei, James, and Edward Norman. "Lie algebraic solution of linear differential equations." Journal of Mathematical Physics 4.4 (1963): 575-581.