Controllability on relaxation-free subspaces: On the relationship between adiabatic population transfer and optimal control

We consider the optimal control problem of transferring population between states of a quantum system where the coupling proceeds only via intermediate states that are subject to decay. We pose the question whether it is generally possible to carry out this transfer. For a single intermediate decaying state, we recover the stimulated Raman adiabatic passage process, which we identify as the global optimum in the limit of infinite control time. We also present analytical solutions for the case of transfer that has to proceed via two consecutive intermediate decaying states. We show that in this case, for finite power the optimal control does not approach perfect state transfer even in the infinite time limit. We generalize our findings to characterize the topologies of paths that can be achieved by coherent control under the assumption of finite power. If two or more consecutive states in an N-level chain are subject to decay, complete population transfer with finite-power controls is not possible.