Adiabatic Condition for Nonlinear Systems

The adiabatic approximation is an important concept in quantum mechanics. In linear systems, the adiabatic condition is derived with the help of the instantaneous eigenvalues and eigenstates of the Hamiltonian, a procedure that breaks down in the presence of nonlinearity. Using an explicit example relevant to photoassociation of atoms into diatomic molecules, we demonstrate that the proper way to derive the adiabatic condition for nonlinear mean-field (or classical) systems is through a linearization procedure, using which an analytic adiabatic condition is obtained for the nonlinear model under study.

Figure 1

The three-level system. ${\Omega }_p$ and ${\Omega }_d$ are the two coupling strengths, $\Delta$ and $\delta$ are one and two-photon detunings, respectively.

Figure 2

Population dynamics. Solid lines: population in state $|a⟩$ ($|{\psi }_a{|}^2$); dashed lines: population in state $|g⟩$ ($|{\psi }_g{|}^2$ for the linear case and $2|{\psi }_g{|}^2$ for the nonlinear case). The parameters are $\Delta =0$, ${\Omega }_0=5$, $t_d=3$, and $t_p=3.8$.

Figure 3

Adiabaticity parameter as defined in (3, 11) for the linear and nonlinear systems, respectively. Same parameters as in Fig. 2.