Integrated photonics enables continuous-beam electron phase modulation

Integrated photonics facilitates extensive control over fundamental light–matter interactions in manifold quantum systems including atoms1, trapped ions2,3, quantum dots4 and defect centres5. Ultrafast electron microscopy has recently made free-electron beams the subject of laser-based quantum manipulation and characterization6–11, enabling the observation of free-electron quantum walks12–14, attosecond electron pulses10,15–17 and holographic electromagnetic imaging18. Chip-based photonics19,20 promises unique applications in nanoscale quantum control and sensing but remains to be realized in electron microscopy. Here we merge integrated photonics with electron microscopy, demonstrating coherent phase modulation of a continuous electron beam using a silicon nitride microresonator. The high-finesse (Q0 ≈ 106) cavity enhancement and a waveguide designed for phase matching lead to efficient electron–light scattering at extremely low, continuous-wave optical powers. Specifically, we fully deplete the initial electron state at a cavity-coupled power of only 5.35 microwatts and generate >500 electron energy sidebands for several milliwatts. Moreover, we probe unidirectional intracavity fields with microelectronvolt resolution in electron-energy-gain spectroscopy21. The fibre-coupled photonic structures feature single-optical-mode electron–light interaction with full control over the input and output light. This approach establishes a versatile and highly efficient framework for enhanced electron beam control in the context of laser phase plates22, beam modulators and continuous-wave attosecond pulse trains23, resonantly enhanced spectroscopy24–26 and dielectric laser acceleration19,20,27. Our work introduces a universal platform for exploring free-electron quantum optics28–31, with potential future developments in strong coupling, local quantum probing and electron–photon entanglement.


Analytical equations used for fitting optical resonance
The surface Rayleigh or bulk scattering in the microresonator leads to the coupling of clockwise (a cw ) and counter-clockwise (a ccw ) modes [1]. Assuming both modes have degenerate frequencies ω with modal coupling rate γ, the Hamiltonian of the microresonator system reads H = ω(a † cw a cw + a † ccw a ccw ) + γ(a cw + a † cw )(a ccw + a † ccw ).
Considering the laser's frame with frequency ω l = ω + ∆, under the rotating wave approximation, the Hamiltonian reads H = − ∆(a † cw a cw + a † ccw a ccw ) + γ(a cw a † ccw + a † cw a ccw ).
After considering coupling to the bus waveguide with rate κ ex and losses to the environment with rate κ 0 , this results in the Langevin equation (ignoring vacuum fluctuation) a cw = (−κ/2 + i∆)a cw − iγa ccw + √ ηκa cw,in a ccw = (−κ/2 + i∆)a ccw − iγa cw , where κ = κ ex +κ 0 describes the total loss rate and η = κ ex /κ denotes the coupling efficiency. The stationary solution of the intracavity fields can be easily obtained as The cavity transmission, reflection and dissipation are then obtained from the input-output formalism  Figure 1: a) An optical transmission scan measured at the output of the chip is fitted using Eq.(7). The sidebands generated via an electro-optic modulator are used to calibrate the frequency. b) Fitting to the fitted g frequency sweep using Eq. (10). Note that the frequency sweep line shape difference near the resonance is due to the coupling to the frequency degenerate counter-clockwise optical mode. c,d) The Markov chain Monte Carlo random walk corner plot of the fitting to the optical data and the fitted g data. The fitted system key parameters (cavity decay rate κ, splitting ratio γ/κ, sideband ratio A sb ) of the two frequency sweeps are all within 7% discrepancy, indicating great consistency between the optical and electron spectroscopic measurements. Also strong correlation between the fitted γ and κ is observed for both fittings, indicating the necessity of applying the coupled modes model to correctly extract the cavity decay rate.
And the intracavity photon numbers are simply d) e) f) Figure 2: a) Simulated power distribution in the optical system when pump is along clockwise direction, generated using parameters fitted from the frequency sweep measurement shown in Fig.1. b) |g| 2 scan where each point is obtained after fitting the electron energy distribution. c) Experimentally measured optical signal of the cavity transmission (blue), reflection (red), and dissipation (green). The resonance shaped (black dashed) curve shows the inferred cold cavity transmission without thermal absorption and Kerr nonlinearity induced cavity frequency shift, which is present in the triangular shaped (blue) curve due to high input power (∼ > 50µW ). d) Calibrated detuning ∆(t) based on the experimentally measured optical signal in (c). e) Calibrated correction factor P diss,cw /P diss from the detuning plot (d). The reflection calibration is empirically erroneous. This effect can be attributed to the etalon formed by the chip facets, by which the transmission signal is less affected. f ) The calibrated dissipated power in the clockwise mode shows linear relation with |g| 2 from different measurement channels (transmission and dissipation).
Eq. (7) and (10) are used for fitting the frequency sweep of the optical transmission P t (∆). The frequency sweep fitting was done using the Markov chain Monte Carlo (MCMC) methods [2], with the optical sidebands ±Ω sb and absorption induced cavity frequency shift χ th P diss (∆) included in the models. The fitting function where F i is either P t for optical measurement or n a for measurement of the electron-light coupling g. The fitting results are shown in Fig.1, and the fitted system parameters show great consistency between the two distinct measurements. For the power sweep calibration, Eq. (7), (10) and (11) are used for calibrating the clockwise dissipated powerP diss,cw (t). The detuning∆(t) was extracted from the experimentally measuredP t (t) (Fig.2(b), suffers the least from background noise) using the fitted resonator parameters (κ, η, γ) from the frequency sweep ( Fig.2(a)(c)). Then the clockwise dissipated powerP diss,cw (t) (Fig.2(e)) was calibrated from the experimentally measured transmission powerP t (t) and the calibrated∆(t) byP diss,cw (t) = P diss,cw (∆(t)) Pt(∆(t))P t (t). We later calculate the characteristic coupled optical power P = n cw ωκ by scaling the dissipated power P = κ κ0 ×P diss,cw , and plot it against the fitted coupling constant |g(t)| 2 ( fig. 2f), one could find the linear relation as is expected in theory. The observable oscillations in the linear dependence of |g| 2 on the clockwise dissipated power are related to a 50 Hz noise in the beam position leading to variations in electron-light coupling strength. To eliminate this 50 Hz noise, we binned the retrieved coupling strength in time intervals of 20 ms. The resulting power dependence is shown in main text figure 2.e).

Quantum optical description of electron-photon interaction
As discussed in the main manuscript, for electron-photon interaction, it is more natural to work in the velocity gauge [3]. For simplicity, we reduce the problem to one dimension (ẑ) along the electron propagation direction. The vector potential is quantized as [4] A where is the optical permittivity, ω is the optical frequency, V is the effective optical mode volume, and u(z) is the z projection (along the propagation direction of electron k e ) of the vector mode function u( r) which satisfies We choose the electron plane wave states as the basis, |k e − ∆k e k e |. Here, L eff (∆k e ) = Lc dze −i∆ke·z u(z) is the effective interaction length, and represents the phase matching condition between the optical field profile function u(z) along the electron propagation trajectory and the electron wavevector change e −i∆ke·z upon the absorption/emission of a photon. Under conditions of phase matching, the expression can be reduced to the order of the physical length over which the optical field and the electron interact.
In the interaction picture, the scattering matrix, derived from the interaction Hamiltonian, is with a re-defined expression ofb = = ω, is the electron wave-vector change due to the absorption/emission of a photon. The vacuum coupling rate is also re-defined asg 0 = η e 2 v 2 e 2 ωV , so that the interaction time τ int = Lint ve has a very clear physical meaning, which is the fly-by time of the electron over the interaction region L int . Here the phase matching coefficient η = dze −i∆k·z u(z)/L int is re-introduced in the vacuum coupling rateg 0 .
In the presence of a strong coherent laser drive |α , the scattering matrix reduces to a displacement operator on the electron state S ≈ exp −ig 0 τ int αb † − h.c. , which results in the probability distribution on the N th sideband where the complex fieldẼ(z) is related to the physical electric field E(z, t) via E(z, t) = Real[Ẽ(z)e −iωt ].

Estimation of transverse beam deflection
Since inelastic electron-light scattering is typically accompanied by a three-dimensional momentum transfer, we here estimate the transverse deflection of the electron beam upon passing by the resonator structure.
For an electron energy of 120 keV, a photon energy of about 0.8 eV (corresponding to a wavelength of 1550 nm) and assuming E ϕ = E z (i.e. electron-photon scattering above a straight waveguide), a single photon scattering would lead to a deflection of ∼ 1.5 µrad. Accordingly, in the case of multi-photon exchange with N = 250, as shown in Fig. 2d of the main text, a deflection of < 0.4 mrad is expected. For the measurements presented here, the STEM camera length was chosen such that it enables full beam transmission through the angle-limiting aperture into the spectrometer.

Integrated photonics platforms for electron beam modulation
Various integrated photonics platforms have been used for controlling manifold quantum systems [5][6][7][8][9][10][11][12][13][14][15]. The current Si 3 N 4 -based electron beam phase modulation demonstration could be potentially extended to other well-established photonic integrated platforms showing the broader applicability. Finite element method based simulations are performed to calculate the n eff of the microresonators with different materials by sweeping the frequency and changing the waveguide height. The width of the waveguide is kept at 1.5 µm for all materials (Fig. 3). Different shaded regions are calculated by performing simulations at two different waveguide heights (e.g., solid light-red line GaP: 1500 x 800 nm 2 , dashed light-red line GaP: 1500 x 400 nm 2 ). It is possible to achieve phase matching for electron kinetic energies ranging from 30 to 180 keV by using different established integrated photonic platforms (AlGaAs [16,17], GaP [18], 4H-SiC [19], Diamond [20], LiNbO 3 [21], Si 3 N 4 [22], Hydex [22,23]). Some other prominent platforms, such as Si [24], AlN [25], Ta 2 O 5 [26], and SiO 2 [27] are not shown due to overlapping phase matching, but could also be considered for implementing the electron beam phase modulation. In addition to phase matching, other important parameters that also need to be considered are linear propagation loss, light coupling into the chip and optical power handling.  a) For each electron, the electron velocity determines a phase-matched refractive index value, which is accessible through a wide range of optical frequencies and material platforms by engineering of the waveguide geometry. The possible simulated regions of the microresonators' effective index (quasi-TM mode) with 50 µm radius for different materials are achieved by varying either the waveguide dimensions (height) or the optical frequency. Each shaded region shows a possible operating regime for a single integrated platform. Si 3 N 4 could provide phase matching from 95 to 145 keV, while phase matching below 80 keV could be implemented using 4H − SiC or LiNbO 3 . b) The two shaded regions show phase matching windows for geometry or frequency tuning. The light purple region can be reached by changing the Si3N4 waveguide height from 800 nm to 500 nm while operating at a single frequency ( 193 THz). The gray-shaded region can be reached by changing the optical frequency from ∼193 THz to 230 THz.