Here we propose a tracking quantum control protocol for arbitrary N-level systems. The goal is to make the expected value of an observable $${\mathcal O}$$O to follow a predetermined trajectory S(t). For so, we drive the quantum state $$|\varPsi (t) \rangle $$|?(t)? evolution through an external potential V which depends on $$M_V$$MV tunable parameters (e.g., the amplitude and phase (thus $$M_V = 2$$MV=2) of a laser field in the dipolar condition). At instants $$t_n$$tn, these parameters can be rapidly switched to specific values and then kept constant during time intervals $$\Delta t$$Δt. The method determines which sets of parameters values can result in $$\langle \varPsi (t) | {\mathcal O} |\varPsi (t) \rangle = S(t)$$??(t)|O|?(t)?=S(t). It is numerically robust (no intrinsic divergences) and relatively fast since we need to solve only nonlinear algebraic (instead of a system of coupled nonlinear differential) equations to obtain the parameters at the successive $$\Delta t$$Δt's. For a given S(t), the required minimum $$M_V = M_{\min }$$MV=Mmin `degrees of freedom' of V attaining the control is a good figure of merit of the problem difficulty. For instance, the control cannot be unconditionally realizable if $$M_{\min } > 2$$Mmin>2 and V is due to a laser field (the usual context in real applications). As it is discussed and exemplified, in these cases a possible procedure is to relax the control in certain problematic (but short) time intervals. Finally, when existing the approach can systematically access distinct possible solutions, thereby allowing a relatively simple way to search for the best implementation conditions. Illustrations for 3-, 4-, and 5-level systems and some comparisons with calculations in the literature are presented.
['G.J. Delben', 'Marcos G. E. da Luz']
General tracking control of arbitrary N-level quantum systems using piecewise time-independent potentials