Complete Control of Hamiltonian Quantum Systems: Engineering of Floquet Evolution

One encounters the problem of quantum control in several fields of contemporary physics and chemistry, such as molecular dynamics in laser fields [1,2] and quantum optics [3–6]. A few examples of complete control of the quantum state by conditional measurements [7], by adiabatic transport [8], or by unitary evolution [9] have been already proposed for the particular quantum system of atoms interacting with quantized electromagnetic field in a single-mode resonator. But to what extent is it possible to control the quantum dynamics in the general case? In this Letter we show that one can obtain complete control not only over the quantum state but also over the unitary evolution of a generic Hamiltonian system. It can be achieved simply by switching on and off two distinct perturbations V̂A and V̂B in an alternating sequence. For an N-level system [10] the sequence is periodic, and each period consists of N2 time intervals, t1, t2, . . . , tN2 , which are found by solving the “inverse Floquet problem” [11] as described below. In other words, in order to control the system the perturbation V̂A should be applied during time t1 followed by the perturbation V̂B during time t2, and then again V̂A during time t3 followed by V̂B during time t4, and so forth, altogether N2 times. After the last interval tN2 the sequence repeats itself. We emphasize the difference between the control of evolution and the control over a quantum state. The first means free choice of all elements of N 3 N evolution matrix Û, restricted only by the unitarity condition. Inversion or displacement of all wave functions or maintaining a quantum state intact in the course of time