Quantizing Ulam’s Control Conjecture

Like classically chaotic systems, can quantum systems be nudged from a given initial state to any other by arbitrarily weak control fields? We explore a scaling theory that sets fundamental limits on the time required for population transfer with weak control fields. Our results depend on the size of the quantum state space and on the rate and extent of energy flow within the system. When the unperturbed dynamics is quantum localized, the total distance in state space over which control can be achieved grows linearly with the number of photons that are expended.

Figure 1

State space showing origin and target states (red balls) on their respective energy shells. Unassisted dynamics under $H^{(0)}$ is indicated by the tubes ($d_f$ can be smaller than the total dimensionality $D$ of the energy shell because of approximately good quantum numbers). When the dynamics are assisted by the $\mu \mathit{\epsilon }$ term, the goal is to transfer the population from the original to the target state via states $|i⟩$ and $|j⟩$. This process may occur anywhere (bolts), although it is most likely where the evolved states approach one another by quantum diffusion, and the connecting dipole operator is large.

Figure 2

(a) The surface $\overline{M^2}$ is shown as a function of distance $|{\mathbf{n}}^{(O)}-{\mathbf{n}}^{(T)}|$ in state space, and as a function of time elapsed, in terms of scaled units. (b) Relative distance an optimized multiphoton control field can transport an initial state population, scaled by the required time and fluence. The horizontal axis is the ratio of quantum diffusion and dipole localization lengths. Three different dimensionalities of the state space are shown. ${\varphi }_{\mathrm{sc}}$ contains all constant factors in Eq. (21).