A Theoretical Solution for a Nonlinear Master Equation

Sofia Qvarfort, Michael Ross Vanner, P. F. Barker +1 more

Open-system dynamics play a key role in the experimental and theoretical study of cavity optomechanical systems. In many cases, the quantum Langevin equations have enabled excellent models for optical decoherence, yet a master-equation approach to the fully nonlinear optomechanical Hamiltonian has thus far proven more elusive. To address this outstanding question and broaden the mathematical tool set available, we derive a solution to the Lindblad master equation that models optical decoherence for a system evolving with the nonlinear optomechanical Hamiltonian. The method combines a Lie-algebra solution to the unitary dynamics with a vectorization of the Lindblad equation, and we demonstrate its applicability by considering the preparation of optical cat states via the optomechanical nonlinearity in the presence of optical loss. Our results provide a direct way of analytically assessing the impact of optical decoherence on the optomechanical intracavity state.

A. Fuliński, W. J. Kramarczyk

S. Chaturvedi, Venkatraman Srinivasan

Using the thermo-field dynamics notation, an elegant solution of the master equation for an attenuated or amplified nonlinear oscillator is presented. It is found that, in this notation, the solution of the master equation under consideration amounts to a simple modification of that for the linear oscillator.

G. L. Braglia

Hans Christian Öttinger

The quantum master equation obtained from two different thermodynamic arguments is seriously nonlinear. We argue that, for quantum systems, nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature and consequences of the nonlinear contribution. The thermodynamic nonlinearity naturally leads to canonical equilibrium solutions and extends the range of validity to lower temperatures. We discuss the Markovian character of the thermodynamic quantum master equation and introduce a solution strategy based on coupled evolution equations for the eigenstates and eigenvalues of the density matrix. The general ideas are illustrated for the two-level system and for the damped harmonic oscillator. Several conceptual implications of the nonlinearity of the thermodynamic quantum master equation are pointed out, including the absence of a Heisenberg picture and the resulting difficulties with defining multitime correlations.

P. Schwendimann