Deterministic Generation of Arbitrary Fock States via Resonant Subspace Engineering

Abstract

Deterministic preparation of high-excitation Fock states is a central challenge in bosonic quantum information, with control complexity that generically explodes as the Hilbert space dimension grows. Here we introduce resonant subspace engineering (RSE), a protocol that analytically confines the infinite-dimensional bosonic dynamics to a two-dimensional invariant subspace spanned by an initial coherent state and the target state. State transfer then reduces to a geodesic rotation on a synthetic Bloch sphere, governed by resonance and phase-matching conditions we derive in closed form. For single Fock states, RSE achieves $O(n^{1/4})$ scaling in both evolution time and gate depth, showing a fundamental improvement over existing deterministic schemes. The construction generalizes to $K$-component superpositions via a $(K{+}1)$-dimensional invariant subspace with full $\mathrm{SU}(K{+}1)$ controllability, requiring only 3-5 iterations of operations for superpositions spanning photon numbers 70--100. RSE provides a scalable and analytically transparent framework for large-scale bosonic state engineering and gate synthesis across single- and multimode platforms.

Figures (3)

• Figure 1: Principle of resonant subspace engineering (RSE). (a) Traditional quantum control approaches engineer complex transition paths that connect the initial and target states (left). In contrast, RSE reduces the dynamics to a two-dimensional system spanned by the initial coherent state $|\alpha\rangle$ and the target state $|\psi\rangle$. (b) Illustration of rotations that carry the initial coherent $|\alpha\rangle$ to the target state on the synthetic Bloch sphere. The resonance condition of the generator Hamiltonian ensures that the quantum state evolves along a geodesic line on the Bloch sphere with the axis in the XY plane.
• Figure 2: (a) Fidelity dynamics for preparing $| n=100 \rangle$ from $| \alpha=10 \rangle$. The blue solid (orange dashed) curve shows the evolution for matched $H_{11}=H_{22}$ (mismatched $H_{11}=0.8H_{22}$). (b) Required evolution time $T$ (blue solid, left axis) and optimized iteration number $N$ (orange markers, right axis) versus target photon number $n$ for a coherent input $| \alpha=\sqrt{n} \rangle$, illustrating the sublinear $\mathcal{O}(n^{1/4})$ scaling.
• Figure 3: Preparation of Fock-state superpositions from $| \alpha=\sqrt{88} \rangle$ with optimized generalized oracle operation angles. Target states: $| \varphi_1 \rangle=\sqrt{0.3}| 70 \rangle+\sqrt{0.7}| 100 \rangle$, $| \varphi_2 \rangle=\sqrt{0.2}| 70 \rangle+\sqrt{0.5}| 85 \rangle+\sqrt{0.3}| 100 \rangle$, and $| \varphi_3 \rangle=\sqrt{0.1}| 70 \rangle+\sqrt{0.3}| 80 \rangle+\sqrt{0.4}| 90 \rangle+\sqrt{0.2}| 100 \rangle$. $F=1$ is obtained with $N=5, 4, 3$ iterations, respectively.