Principles of Optics in the Fock Space: Scalable Manipulation of Giant Quantum States

Abstract

The manipulation of distinct degrees of freedom of photons plays a critical role in both classical and quantum information processing. While the principles of wave optics provide elegant and scalable control over classical light in spatial and temporal domains, engineering quantum states in Fock space has been largely restricted to few-photon regimes, hindered by the computational and experimental challenges of large Hilbert spaces. Here, we introduce ``Fock-space optics", establishing a conceptual framework of wave propagation in the quantum domain by treating photon number as a synthetic dimension. Using a superconducting microwave resonator, we experimentally demonstrate Fock-space analogues of optical propagation, refraction, lensing, dispersion, and interference with up to 180 photons. These results establish a fundamental correspondence between Schrödinger evolution in a single bosonic mode and classical paraxial wave propagation. By mapping intuitive optical concepts onto high-dimensional quantum state engineering, our work opens a path toward scalable control of large-scale quantum systems with thousands of photons and advanced bosonic information processing.

Figures (5)

• Figure 1: Concept and experimental implementation of Fock-space optics.a-c, Illustration of optical propagation in different degrees of freedom: spatial (e.g., a, Gaussian beams), temporal (e.g., b, optical pulses), and the discrete photon-number dimensions (c, Fock-space wavefunctions). d, Schematic of the experimental platform: a superconducting resonator coupled to an ancilla transmon qubit. The resonator encodes quantum states in Fock space, while the ancilla qubit enables both phase manipulation and state-selective readout. e, Calibration of the qubit frequency versus photon number in the resonator, which establishes a Fock-space camera for detecting population along the photon-number axis. Insets: photon-number-splitting peaks of coherent states with different mean photon numbers, $\bar{n}$.
• Figure 2: Prism and lens elements in Fock space.a, Sequence and b, concept of the prism. A coherent state (e.g., $\bar{n}=150$) is prepared via a strong displacement, followed by a weak single-photon pump. A linear phase accumulation is imprinted on each Fock state, controlled by either the phase of the weak pump or a period of detuned free evolution. Finally, the population distribution across the Fock states is measured. c, Simulation and d, experimental results for the Fock-space prism. e, Sequence and f (i), concept of the convex (concave) lens. After preparing a coherent state (e.g., $\bar{n}=150$), the resonator evolves freely under the self-Kerr effect, followed by a weak single-photon pump. During the intermediate free evolution period, the combined effects of self-Kerr nonlinearity and detuning lead to a quadratic phase accumulation centered around $n=150$, analogous to a classical lens. By inverting the phase of the pump, the process is transformed to a concave lens. g (j), Simulations and h (k), experiments demonstrating the focusing (diverging) effect on quantum states. Using the highly focused state from the convex lens, Fock state $\left|150\right\rangle$ can be prepared with a success rate up to 17%.
• Figure 3: Newton's prism experiment in Fock space.a, Concept of Newton's prism experiment. By combining a prism and a lens, light of different colors is deflected and focuses onto different points on the focal plane. In Fock space, the microwave frequency $\omega_\mathrm{p}$ of the single-photon pump, corresponding to different $\Delta$ in Eq. (\ref{['eq:Hamitonian']}), acts as the "color", determining the deflection angle of the prism. b, c, Experimental results demonstrating the focusing of a coherent state ($\bar{n}=150$) at different Fock numbers for different $\tilde{\Delta}$. For $\tilde{\Delta}/2\pi=-23\,\mathrm{kHz}$ (b) and $23\,\mathrm{kHz}$ (c), the quantum state is focused to $n=140$ and $n=160$, respectively, at time $t_l=144\,\mathrm{ns}$. d, Fock state populations at the focal plane for different detunings. The red and blue dashed lines correspond to the cases in b and c, respectively.
• Figure 4: Young's double-slit interference in Fock space.a, Concept of Young's double-slit interference experiment. Coherent light diffracts from two slits and creates interference fringes in the overlapping region on a distant screen. The fringe spacing is inversely proportional to the slit separation $d$ and the pattern shifts with the relative phase $\theta$ between the two slits. b, Ideal and experimental results of the double-Gaussian (DG) state $\left|\psi_\mathrm{DG}\right\rangle$, analogous to the double slits. The state consists of two Gaussian peaks of equal height centered at $n_1=130$ and $n_2=170$, shown for $\theta=0$ and $\pi$. c, Simulated and experimental interference fringes at time $t_l=1.1\,\mathrm{\mu s}$ for $\theta=0$ and $\theta=\pi$. d, Experimental interference fringes for different phases $\theta$ between the slits. The change of phase results in an overall shift in the fringe pattern with a $2\pi$ period, causing the central Fock state to transition from coherent enhancement to subtraction and back again. e, Fringe spacing as a function of the distance between the two peaks of the DG state (analogous to the slit separation). The spacings, extracted by fitting the data to a Gaussian-envelope cosine function, confirm the inverse proportionality. Experimental data and fits are provided in Fig. S8 of the Supplementary Materials.
• Figure 5: Imaging in Fock space.a, Concept of imaging in classical optics for the case $f<u<2f$, where $u$ is the object distance and $f$ is the focal length. b, Left: Fock-space distribution of the "object" state $\left|\psi_\mathrm{DG}\right\rangle$, featuring two Gaussian peaks (standard deviation $\sigma=5$) centered at $n_1=135$ and $n_2=165$ with a peak height ratio of $1:4$. Right: Fock-space distribution of the inverted, magnified, real "image" state observed at the image distance predicted by Eq. (\ref{['eq:image']}). The separation between the peak centers confirms the magnification factor $t_v/t_u$. c, "Object" and "image" states from a similar experiment using a DG state with a peak height ratio of $4:1$. The cosine similarity between the ideal and measured photon number distributions of the image is 84% (b) and 90% (c), respectively.