Nonlinear Stimulated Raman Exact Passage by Resonance-Locked Inverse Engineering V. Dorier, 1 M. Gevorgyan, 1,2 A. Ishkhanyan, 2,3 C. Leroy, 1 H. R. Jauslin, 1 and S. Gu´ erin 1,* 1 Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS UMR 6303, Universit´ e Bourgogne Franche-Comt´ e, BP 47870, 21078 Dijon, France 2 Institute for Physical Research NAS of Armenia, 0203 Ashtarak-2, Armenia 3 Institute of Physics and Technology, National Research Tomsk Polytechnic University, Tomsk 634050, Russia (Received 11 August 2017; published 13 December 2017) We derive an exact and robust stimulated Raman process for nonlinear quantum systems driven by pulsed external fields. The external fields are designed with closed-form expressions from the inverse engineering of a given efficient and stable dynamics. This technique allows one to induce a controlled population inversion which surpasses the usual nonlinear stimulated Raman adiabatic passage efficiency. DOI: 10.1103/PhysRevLett.119.243902 Introduction.—Quantum control methods for driving a quantum system from an initial to a target state by external fields is at the heart of modern applications of quantum physics and chemistry [1,2]. The effective description of many-particle systems in a mean field [3] or of nonlinear phenomena in optics [4,5] often leads to a dynamics described by a nonlinear Schrödinger equation, e.g., in atoms to molecule conversion in Bose-Einstein condensates (BEC) [6–13] or in nonlinear optics [14–17]. Adapting the tools of quantum control, such as composite pulses [18,19], adiabatic passage [20], optimal control [21,22], shortcuts to adiabaticity ] 23 ], and single-shot shaped pulses [24,25], to such nonlinear problems is a natural but challenging question. In addition to controllability issues, the general instability of the dynamics prevents the direct application of simple strategies [9,11]. Besides a few optimal control studies [26,27], nonlinear quantum systems has been mostly treated by adiabatic techniques transposed from their linear counterparts. The formulation of adiabatic passage techniques is not straightfor- ward for nonlinear systems since it needs a classical Hamiltonian formulation and a generalized Bloch sphere. One can make it explicit for two-state problems [28–30], but its formulation for more complex systems is highly ques- tionable since the underlying dynamics is in general not integrable [11]. In nonlinear Λ systems with coupling only between each ground state, denoted j1i and j3i, and the excited state j2i (via pump and Stokes fields, respectively), nonlinear stimulated Raman adiabatic passage (nonlinear STIRAP), aiming at population inversion (defined from j1i and j3i) with a low transient population in state j2i, which can be lossy in real experiments, has been proposed [6,8] from its linear counterpart [20]. An alternative derivation of adiabatic condition through a linearization procedure was introduced in Refs. [9,11,31,32], based on the existence of a stable coherent population trapping (or dark) state [6]. Under similar con- ditions of the usual linear STIRAP, the resulting nonlinear inversion is less efficient, typically 80%. Only very large Rabi pulse areas (typically 1 order of magnitude larger than for its linear counterpart) can compensate the nonlinearities [6,8,11]. In this Letter, we establish an exact inverse-engineering technique for such Λ systems featuring second- and third- order nonlinearities, which allows an efficient (almost 100%) and robust population inversion with Rabi pulse areas comparable to the ones of their linear counterpart. We show first an important preliminary result: the second-order non- linearities prevent, as in the case of the two-level problem [29], the complete population inversion. We thus define a target state as close as desired to the final state j3i and derive a family of dynamics that connect exactly and in a robust way the initial and target states. Our formulation does not need any adiabatic theorem to support the dynamics since the derived solutions are exact, but it has to be stable (or robust) to be realistically implementable. Our solution shows the following salient features: (i) only the Rabi frequency associated to the second-order nonlinearity features a larger strength which is only a few times larger than its linear counterpart; (ii) the third-order nonlinearities are fully dynamically compensated by appropriate detunings, which is referred to as resonance- locked inverse engineering; (iii) the transient transfer in the upper state is controlled: less transient population in the upper state needs larger pulse areas; (iv) it is more robust with respect to deviations of the various parameters for lower transient population in the excited state (or equivalently for larger pulse areas), similarly to the linear STIRAP. Model and the strategy.—The general state in the con- sidered Λ system is written as a vector ψ ¼½c 1 c 2 c 3  t , where c j is the amplitude in state jji. We focus our study on the controlled dynamics described by the following general nonlinear Schrödinger equation: i _ c 1 ¼ K 1 c 1 þ P ¯ c 1 c 2 ; ð1aÞ i _ c 2 ¼ K 2 c 2 þ Δ P c 2 þ Pc 2 1 þ Sc 3 ; ð1bÞ i _ c 3 ¼ K 3 c 3 þðΔ P − Δ S Þc 3 þ Sc 2 ; ð1cÞ where P, S are the Rabi frequencies of the pump and Stokes fields, Δ P;S are the detunings of the fields. The second-order nonlinearity appears via the pump coupling, PRL 119, 243902 (2017) PHYSICAL REVIEW LETTERS week ending 15 DECEMBER 2017 0031-9007=17=119(24)=243902(5) 243902-1 © 2017 American Physical Society