We revisit the adiabatic criterion in stimulated Raman adiabatic passage for the three-level Λ-system, and compare the situation with and without nonlinearity. 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 the nonlinear systems is through a linearization procedure.PACS numbers: 03.75. Mn, 05.30.Jp, 32.80.Qk According to the adiabatic theorem of quantum mechanics [1], if the Hamiltonian is changed sufficiently slowly, then a system in a given non-degenerate eigenstate of the initial Hamiltonian (say, |i ) evolves into the corresponding eigenstate of the instantaneous Hamiltonian without making any transitions. Here "sufficiently slowly" means that the rate of change of the Hamiltonian is much smaller compared to the level spacings:where ω f i denotes the transition frequency between the instantaneous eigenstates |i(t) and |f (t) . Condition (1) is referred to as the adiabatic condition. The standard version of the adiabatic theorem, which has many applications in quantum state preparation and manipulation, applies to linear quantum systems. We, however, have noticed that the adiabatic condition (1) have been used in recent studies of nonlinear quantum gases. This may be problematic since the proof of the adiabatic condition makes explicit use of the concept of an orthonormal set of energy eigenstates and the linear superposition principle involving these states, both of which become invalid when nonlinearity is introduced into the system. The purpose of this paper is to provide a general method for deriving the adiabatic condition in nonlinear systems. This is achieved by considering a specific example of coherent population transfer in three-level quantum systems using the stimulated Raman adiabatic passage (STIRAP) method [2, 3].Consider the three-level Λ-system schematically shown in Fig. 1. The excited state |e is coupled to two ground states |a and |g , with the coupling strengths (Rabi frequencies) denoted as Ω p and Ω d , respectively. Here the subscripts "p" ("d") stand for "pump" ("dump"). In the interaction picture, under the two-photon resonance condition (δ = 0), the Hamiltonian of this system readsWithout loss of generality, we will assume that the Rabi frequencies Ω p,d are real and positive. Hamiltonian (2) can be easily diagonalized. The particular energy eigenstatewhere Ω eff = Ω 2 p + Ω 2 d , is known as the coherent population trapping (CPT) state or the dark state. The state |CPT has zero eigenenergy and is decoupled from the excited state. When ∆ = 0, the other two eigenstates possess energies ± Ω eff /2. When Ω p,d are varied adiabatically, an initially prepared CPT state will remain in the instantaneous CPT state. A straightforward application of (1) leads to the adiabatic condi...

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We study the possibility of creating a coherent population trapping (CPT) state, involving free atomic and ground molecular condensates, during the process of associating atomic condensate into molecular condensate. We generalize the Bogoliubov approach to this multi-component system and study the collective excitations of the CPT state in the homogeneous limit. We develop a set of analytical criteria based on the relationship among collisions involving atoms and ground molecules, which are found to strongly affect the stability properties of the CPT state, and use it to find the stability diagram and to systematically classify various instabilities in the long-wavelength limit.

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