Enhancing the controllability of quantum systems via a static field

TL;DR

This work addresses controllability of bilinear quantum systems under a static field plus a time-dependent control by introducing the weakly conically connected spectrum as a spectral criterion. It proves that, for two-input systems with this spectral structure and rationally unrelated eigenvalue germs, fixing one input yields controllability for almost every fixed static value of the other input, via non-resonant spectra and nonzero couplings between consecutive eigenstates. The main result provides a precise theorem: for almost all fixed values, the single-input subsystem is exactly controllable (finite dimension) or approximately controllable (infinite dimension). The theory is demonstrated on two physically relevant models—the enantio-selective three-level system for chiral molecules and the driven Jaynes–Cummings Hamiltonian—showing that static-field settings combined with a single time-dependent drive suffice to realize a broad class of unitary evolutions, with implications for molecular control and quantum optics.

Abstract

We provide a sufficient condition for the controllability of a bilinear closed quantum system steered by a static field and a time-varying field, based on the notion of weakly conically connected spectrum. More precisely, we show that if a controlled Hamiltonian with two inputs has a weakly conically connected spectrum, then, freezing one of the two inputs at almost every constant value, the obtained single-input system is controllable. The result is illustrated with two examples, enantio-selective excitation in a chiral molecule and the driven Jaynes-Cummings Hamiltonian.

Figures (4)

• Figure 1: Intersections between eigenvalues of $H(u)$ seen as functions of $u=(u_1,u_2)$, with $m=2$. For the conical intersection, the red lines show the separation of the eigenvalues along a conical direction. For the weakly conical intersection, the red lines show the separation of the eigenvalues along a conical direction and the green lines show the separation along a non-conical direction.
• Figure 2: Three-level cyclic excitation scheme for enantiomer selective excitation of ro-vibrational states. The three states $|1\rangle$, $|2\rangle$, $|3\rangle$ with energies $\epsilon_1$, $\epsilon_2$, and $\epsilon_3$, respectively represent three ro-vibrational states of a chiral molecule. The central frequency of the pulses driving the transition $i \leftrightarrow j$, $i,j=1,2,3$, is detuned from the energy difference $\epsilon_j - \epsilon_i$ by $\delta_{ij}$. Here we choose $\delta_{12}=\delta_{23}$.
• Figure 3: Case $(E_1,E_2,E_3)=(-1.5,0.5,1)$ and $\bar{v}=3$: the spectrum of $H^{+}(u,\bar{v},w)$ as a function of $(u,w)$ is conically connected
• Figure 4: Plot of some eigenvalues $E_{(n,\nu)}(\cdot)$ when $(\omega,\Omega)=(2/5,\sqrt{2}).$ This illustrative example shows that, for each eigenvalue $E_{n,\nu}(\cdot)$, it is possible to find another eigenvalue $E_{m,\mu}(\cdot)$ such that $E_{n,\nu}(g)=E_{m,\mu}(g)$ at some positive $g$. For $(n,\nu)=(10,+)$, we can choose, for example, $(m,\mu)=(15,-)$ or $(m,\mu)=(20,-)$. Similarly, for $(n,\nu)=(10,-)$, we can choose $(m,\mu)=(2,-)$ or $(m,\mu)=(2,+)$.

Theorems & Definitions (34)

• Definition 3.1: Controllability on the unitary group
• Remark 3.2
• Definition 3.3: Approximate controllability on the unitary group
• Remark 3.4
• Definition 3.5: Conical intersection
• Definition 3.6: Conical direction
• Definition 3.7: Weakly conical intersection
• Definition 3.8: Weakly conical connected spectrum
• Definition 3.9: Rationally unrelated germs
• Theorem 3.10: Theorem 1.1 in liang:hal-04174206