Generation of nonclassical states in nonlinear oscillators via Lyapunov control

In this paper, we propose a scheme to generate nonclassical states in a nonlinear oscillator system, such as Fock states, squeezed coherent states, Schr\"odinger-cat states, etc. These nonclassical states are achieved by directly applying the classical field to the nonlinear oscillator, where the amplitude of the field is customized through a state-based Lyapunov control. For generating Fock states, there is no need to precisely control the operation time. As for the generation of other nonclassical states, we resort to a unitary transformation to design the control field, resulting in the requirement of a specific time. The Wigner function of the generated states shows the nonclassical property and shape that matches well with the target. Numerical simulations suggest that the scheme is robust against the field fluctuations and immune to variations in the initial state. In addition, the time-varying control field can be replaced by a square pulse based on the characteristics of the Lyapunov control, which may reduce the difficulty of the experiment.