Time Minimal Trajectories for two-level Quantum Systems with Drift

Abstract

Abstract-On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds.Let (−E, E) be the two energy levels, and |Ω(t)| ≤ M the bound on the field amplitude. For each values of E and M , we provide the explicit expression of the time optimal trajectory steering the state one to the state two in terms of a parameter that should be computed numerically.For M << E, every time optimal trajectory is periodic (and in particular bang-bang) with frequency of the order of the resonance frequency ωR = 2E.On the other side, for M > E the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed E we also prove that for M → ∞ the time needed to reach the state two tends to zero.Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained in the Rotating Wave Approximation.

Comparison of time optimal control for two level quantum systems

“…As the external control laws in [16] and [17] are different, H c is different and the specific forms are given in Section 3 and subsection 4.1, respectively.…”

“…where T is the time from North Pole to South Pole under bang-bang control, namely the minimum time T BB1 is [16] T…”

“…For the initial state and the target state, we need to determine a set of parameters {s i , n, s f } first, and then the control in (16) can be determined. However, there is no fixed method to determine the parameters {s i , n, s f }.…”

“…For a two level quantum system, the trajectories of state transfer from North Pole to South Pole on the Bloch sphere are shown explicitly. Boscain et al studied the time optimal transfer of state with bounded bang-bang control [16]. Lou et al also studied the state transfer problem for two level systems by using the geometric control in the x − y plane [17].…”

Control Methods of Closed Quantum Systems

“…Boscain and Mason studied the time optimal transfer of state with bounded bang-bang control (Boscain and Mason, 2005). Although the two control strategies are for the same state transfer problem, their design principles are quite different, which leads to different characteristics, so the purpose of this section is to compare the time characteristics of steering the state from the north pole to the south pole on the Bloch sphere by means of the control strategies proposed by Boscain and Mason (2005) with bounded control. Although the two control strategies are for the same state transfer problem, their design principles are quite different, which leads to different characteristics, so the purpose of this section is to compare the time characteristics of steering the state from the north pole to the south pole on the Bloch sphere by means of the control strategies proposed by Boscain and Mason (2005) with bounded control.…”