Quantum wave function reconstruction by free-electron spectral shearing interferometry

Abstract

The quantum wave function measurement of a free electron remains challenging in quantum mechanics and is subject to disputes about ψ-ontic/epistemic interpretations of the wave function. Here, we theoretically propose a realistic spectral method for reconstructing quantum wave function of an electron pulse, free-electron spectral shearing interferometry (FESSI). We use a Wien filter to generate two time-delayed replicas of the electron wave packet and then shift one replica in energy using a light-electron modulator driven by a mid-infrared laser. As a direct demonstration, we numerically reconstruct a pulsed electron wave function with a kinetic energy of 10 keV. FESSI is experimentally feasible and enables us to fully determine distinct orders of spectral phases and their physical implications in quantum foundations and quantum technologies, providing a universal approach to characterize ultrashort electron pulses.

An experimentally feasible shearing interferometer approach recovers the quantum wavefunction of a free electron.

INTRODUCTION

A free-space single-electron wave packet perfectly unites in itself the wave-particle duality of quantum mechanics (1). It leads to a description of wave functions for elementary particles such as electrons, bringing tremendous success to quantum mechanics. However, it also brings a long-standing debate that a measurement of wave function would result in the collapse of the wave function, which challenges quantum physicists and philosophers of physics who have spent decades attempting to explain or dispute the wave function collapse (2–4). Given that dualism, at a given point in space, a free electron wave function is readily characterized by a spectral amplitude and phase, much similar to the electric field of a short light pulse. Thus, independent from interpretations of quantum mechanics, a Fourier transformation of both quantities is then able to reveal the full quantum wave function of the electron wave packet. While it is straightforward to measure the spectrum coherently with a precise electron spectrometer, accessing the spectral phase of electron wave packets in free space remains challenging.

Recent research demonstrates that ultrafast laser pulses can generate and manipulate the wave function of free electrons with great coherence and accuracy (5–7). For instance, electrons generated from laser-excited photocathodes (8) and nanoscale metallic tips (7, 9–11) can produce femtosecond electron pulses for a wide range of applications in ultrafast microscopy and diffraction. Control of the photoemission process is possible down to the attosecond time scale (12, 13). Nanotip sources in particular enable high spatial and temporal coherence (14–17). Manipulation of free electrons with light is also advancing rapidly, for example, using photon-induced near-field electron microscopy (PINEM) (15, 18–21), the Kapitza-Dirac effect (22), attosecond electron streaking (23–25), and dielectric laser acceleration (DLA) (26–28). Such manipulation can further reduce the electron pulse duration (24) or lead to the formation of pulse trains (15, 29, 30). Detection methods for determining the duration of femtosecond electron pulses include ponderomotive scattering (31), streaking with infrared and terahertz fields (23, 24, 32), or spectral quantum interference for the regularized reconstruction of free-electron states [SQUIRRELS (29)]. However, in these systems, the electron is seen as an ensemble of either point particles (scattering and streaking methods) or plane waves (SQUIRRELS). Up to now, tracing the quantum phase of a free-electron wave packet has remained elusive.

To construct the full temporal wave function of an electron, the amplitude and phase profiles must be measured, i.e., $\mathrm{\psi }(t)=\sqrt{\mathrm{\rho }(t)}{e}^{i\mathrm{\phi }(t)}$. Using attosecond streaking (23), the longitudinal density distribution ρ(t) of an electron pulse may be determined. However, the quantum phase φ(t) is lost while measuring the amplitude.

Here, notwithstanding interpretations of quantum wave function, we theoretically propose an experimentally feasible spectral shearing interferometry approach for reconstructing the free-electron wave function. The most basic element of spectral shearing interferometry is a spectrometer, which records the electron energy spectrum. With a resolution of 10 meV or better, commercial high-resolution electron spectrometers may directly detect the electron spectrum. Therefore, measuring the spectral phase profile is our primary concern. Specifically, we focus on the ultrafast electron wave function produced from a laser-excited photoelectron gun or from laser-driven modulation of electron pulses. Our approach, which we name free-electron spectral shearing interferometry (FESSI), involves two key ingredients: a time delay by a Wien filter element and an energy shift using a light-electron modulator (LEM). We will discuss the principle and algorithm for FESSI in the following sections. The fundamental origin of the spectral phase is the photoelectron gun configuration and subsequent changes of the spectral phase profile because of free-space propagation, particle acceleration, and laser-driven electron pulse modulation. By using FESSI, we can calibrate and reconstruct femtosecond and attosecond electron pulses in a coherent manner. We believe that our interferometry approach might be widely used for coherent phase control of quantum wave function, characterization of ultrafast electron dynamics, and investigation of quantum foundations and applications.

RESULTS

Time delay and energy shift in the proposed experimental setup

Instead of directly measuring the temporal wave function, we would like to measure the spectral profile in the energy domain

$$\mathrm{\psi }(E)=\sqrt{\mathrm{\rho }(E)}{e}^{i\mathrm{\phi }(E)}$$ (1)

where ρ(E) = ∣ψ(E)∣² is the spectral density distribution measured using an electron spectrometer and φ(E) is the spectral phase. The temporal wave function can be obtained from the Fourier transform, ψ(t) = ∫ dE ψ(E) e^−iEt/ℏ. Now, the primary challenge is measuring the phase φ(E). Inspired by the optical spectral shearing interferometry for measuring the phase profile of ultrafast laser pulses, particularly spectral phase interferometry for direct electric field reconstruction (33, 34), we propose an experimental setup for the direct reconstruction of the quantum phase φ(E) of ultrafast electrons, as shown in Fig. 1A.

Fig. 1. Principle of FESSI.

(A) Setup for measurements of ultrashort electron wave packets using FESSI. An electron pulse generated by a laser-driven electron source is accelerated and split into two replicas. A Wien filter induces a time delay τ and a mid-infrared (MIR) laser pulse irradiating a thin foil shifts the energy of one replica by a small amount, leading to spectral shearing. A spectrometer measures the spectral interference. UV, ultraviolet. (B) In the regime of the acceleration process (σ_E ≫ ℏω_L), the laser-irradiated foil imparts a uniform energy shear of the electron wave packet. Here, the solid and dashed curves represent the spectral amplitude and phase, respectively. (C) In the regime of the PINEM process (σ_E ≪ ℏω_L), sidebands will appear in the electron spectrum after interaction with the near field at the foil.

First, we generate a single electron pulse using a laser-excited photoelectron gun. A high-voltage electron optics system accelerates the electron wave function to 10 keV, sending it to an (optional) laser-driven modulator where the electron pulse can be compressed further. Then, the beam encounters an electron biprism, which splits the electron wave packet into two replicas (35). In the quantum picture, the electron can follow two different quantum paths, resulting in which-way interference when recombined later. Then, a Wien filter creates a time delay between the two replicas to generate spectral interference fringes (36). Furthermore, to imprint the spectral phase information onto the fringes, a LEM accelerates one replica to produce a small energy shift with respect to the other replica. We will introduce the LEM device in the next section. After realizing the time delay between the replicas and the energy shift for one of them, we require one more electron biprism to merge two paths back into one so that the phase information is converted into the spectral intensity. Last, we detect the spectral fringes using a spectrometer and determine φ(E) using our reconstruction algorithm (see Fig. 2).

Fig. 2. Free-electron wave function reconstruction by FESSI.

(A) Temporal amplitude (blue) and phase (red) of an electron wave packet. (B) Two replicas of the wave function after time shift and energy shear. (C) Spectral interference of the wave function replicas. (D) Spectral amplitude (blue) and the reconstructed spectral phase (purple).

The two key ingredients for achieving the spectral phase reconstruction are the time delay τ and a small energy shift ΔE. From the FESSI setup, we obtain the spectral interference signal (see section S1) as

$${\mathrm{\phi }}_{\mathrm{FESSI}}(E)=\mathrm{\phi }(E)-\mathrm{\phi }(E-\text{Δ}E)-\frac{E\mathrm{\tau }}{\hslash }\approx \text{Δ}E\frac{\mathrm{\partial }\mathrm{\phi }}{\mathrm{\partial }E}-\frac{E\mathrm{\tau }}{\hslash }$$ (2)

Consequently, the spectral phase can be deduced from the integral $\mathrm{\phi }(E)={\mathrm{\phi }}_0+\frac{1}{\text{Δ}E}\int \frac{d{E}^{\mathrm{\prime }}}{\hslash }({\mathrm{\phi }}_{\mathrm{FESSI}}+\frac{E^{\mathrm{\prime }}\mathrm{\tau }}{\hslash })$, with an arbitrary constant phase φ₀. Note that the amplitude is directly measured using a spectrometer, allowing both the spectral profile (Eq. 1) and the temporal wave function of the electron pulse to be reconstructed.

To create an accurate time delay, we use a Wien filter. A Wien filter is a device consisting of a static electric field and a magnetic field that are both perpendicular to each other and to the beam path (see Fig. 1A) (36), which has been extensively used to investigate the fundamentals of quantum mechanics (35). However, rather than longitudinal coherence, our spectral interferometry requires spectral coherence, relating to the spectral width of the electron pulse. We can observe the spectral phase signal using a Wien filter to create interference fringes within the spectral coherence region. This indicates that the shorter the pulse duration, the higher the spectral coherence. It also suggests that the spectral interferometry method is ideal for measuring ultrashort pulses (37, 38).

On the other hand, to achieve an energy shift with high accuracy and coherence, we developed a method to precisely regulate the energy shift through the interaction with a mid-infrared (MIR) laser beam, which we refer to as LEM. This LEM device is capable of generating a spectrum with net acceleration for a particle-like electron (see Fig. 1B) and a spectrum with energy-domain sidebands for a wave-like electron (see Fig. 1C) (39). We need the LEM device to operate in the acceleration regime to obtain a net energy shift that can accelerate the electron wave function without affecting its spectral distribution. In our proposal, the LEM device is realized as a 50-nm-thick amorphous Si₃N₄ foil that is illuminated using a MIR laser pulse with a center wavelength of 10.33 μm (25). The role of the foil is to create an optical near field when irradiated by an MIR laser field. We note that no internal vibrational modes of the foil participate in this process. The near field in the foil can realize a phase matching condition for electron group velocity and light phase velocity, serving as a mediator to enhance the interaction between the electron and the MIR laser field. The presence of the foil leads to an abrupt separation between two areas of space with different strengths of the laser field, allowing for an energy modulation of the electron at this field discontinuity. After passing through the optical field on the foil, the electron pulse is accelerated. In other words, both the spectral amplitude and phase of the electron will acquire an energy shift ΔE and a corresponding phase shift, respectively (see Fig. 1B). Through our LEM device, we are able to generate a coherent spectral shearing for the reconstruction.

Light-electron modulation

Next, we demonstrate the mechanism of electron wave packet acceleration using the LEM device. We start by considering the electron to be a monochromatic plane wave ${\mathrm{\psi }}_0=A{e}^{-\frac{i{E}_{0}t}{\hslash }}$, with the center energy E₀ and the amplitude A. The light-electron interaction in the LEM device results in an effective phase modulation of the initial wave function (29); thus, the final electron wave function is given by

$${\mathrm{\psi }}_f(t)=A{e}^{-\frac{i{E}_{0}t}{\hslash }}[\sum _n{J}_n(2\mid g\mid )e^{-in({\mathrm{\omega }}_{\mathrm{L}}t+{\mathrm{\theta }}_0)}]$$ (3)

in which $2\mid g\mid =\frac{e}{\hslash {\mathrm{\omega }}_{\mathrm{L}}}{\int }_{-\frac{L}{2}}^{\frac{L}{2}}F(z)\exp (-\frac{i{\mathrm{\omega }}_{\mathrm{L}}z}{v_0})dz$ is the effective photon number exchange with F(z) as the electric acceleration gradient, ω_L as the laser frequency, L as the interaction length, and v₀ as the central velocity of the electron. The constant phase θ₀ = arg (g) depends on the delay between electron pulse and MIR electric field. For our purpose, we set θ₀ = 0, corresponding to the electron pulse center arriving at the crest of the laser field. Recalling the generating function of Bessel functions, we can rewrite Eq. 3 as ${\mathrm{\psi }}_f(t)=\int \frac{dE}{\hslash }[A\sum _n{J}_n(2\mid g\mid )\mathrm{\delta }(E-E_0-n\hslash {\mathrm{\omega }}_{\mathrm{L}})]e^{-\frac{iEt}{\hslash }}$. Then, the resulting modulated electron energy spectrum reads

$${\mathrm{\psi }}_f(E)=A\sum _n{J}_n(2\mid g\mid )\mathrm{\delta }(E-E_0-n\hslash {\mathrm{\omega }}_{\mathrm{L}})$$ (4)

This corresponds to a typical PINEM spectrum as observed in ultrafast transmission electron microscopy (15, 18). By extending the plane wave electron to a pulsed electron for our purpose, we end up with a final spectrum of the form

$${\mathrm{\psi }}_f(E)=\frac{A}{{(2\mathrm{\pi }{\mathrm{\sigma }}_E^2)}^{\frac{1}{4}}}\sum _n{J}_n(2\mid g\mid )\exp \{-\frac{{(E-E_0-n\hslash {\mathrm{\omega }}_{\mathrm{L}})}^2}{4{\mathrm{\sigma }}_E^2}\}$$ (5)

Each delta-like sideband is replaced by the Gaussian sideband with a finite spectral width 2σ_E. Thus, the initial electron pulse duration is given by 2σ_t = ℏ/σ_E. Under the condition of 2σ_E > ℏΩ_L, these sidebands overlap, and the interference between sidebands would substantially affect the final energy spectrum. To see this explicitly, we assume that ∣g∣ is small and apply the Taylor expansion to these sidebands

$$\begin{array}{c}{\mathrm{\psi }}_f(E)=\frac{A}{{(2\mathrm{\pi }{\mathrm{\sigma }}_E^2)}^{\frac{1}{4}}}\sum _n{J}_n(2\mid g\mid )\left(e^{-n\hslash {\mathrm{\omega }}_{\mathrm{L}}\frac{\mathrm{\partial }}{\mathrm{\partial }E}}\exp \{-\frac{{(E-E_0)}^2}{4{\mathrm{\sigma }}_E^2}\}\right)\\ =\sum _n{J}_n(2\mid g\mid )e^{-n\hslash {\mathrm{\omega }}_{\mathrm{L}}\frac{\mathrm{\partial }}{\mathrm{\partial }E}}{\mathrm{\psi }}_0(E)\approx \sum _n{J}_n(2\mid g\mid )(1-n\hslash {\mathrm{\omega }}_{\mathrm{L}}\frac{\mathrm{\partial }}{\mathrm{\partial }E}){\mathrm{\psi }}_0(E)\\ =(1-2\mid g\mid \hslash {\mathrm{\omega }}_{\mathrm{L}}\frac{\mathrm{\partial }}{\mathrm{\partial }E}){\mathrm{\psi }}_0(E)\approx e^{-2\mid g\mid \hslash {\mathrm{\omega }}_{\mathrm{L}}\frac{\mathrm{\partial }}{\mathrm{\partial }E}}{\mathrm{\psi }}_0(E)={\mathrm{\psi }}_0(E-2\mid g\mid \hslash {\mathrm{\omega }}_{\mathrm{L}})\end{array}$$ (6)

where the initial spectral profile is ${\mathrm{\psi }}_0(E)=\frac{A}{{(2\mathrm{\pi }{\mathrm{\sigma }}_E^2)}^{\frac{1}{4}}}\exp \{-\frac{{(E-E_0)}^2}{4{\mathrm{\sigma }}_E^2}\}$ and the net energy shift is ΔE = 2∣g∣ℏω_L. In the derivation, we use the Taylor expansion formula: $f(x+\mathrm{\epsilon })=e^{\mathrm{\epsilon }\frac{\mathrm{\partial }}{\mathrm{\partial }x}}f(x)\approx f(x)+\mathrm{\epsilon }{f}^{\mathrm{\prime }}(x)$, with ε being a small quantity and the following relations for Bessel functions: $J_{n-1}(z)+J_{n+1}(z)=\frac{2n}{z}{J}_n(z)$ and $\sum _n{J}_n(z)=1$. Therefore, under the conditions of ΔE ≪ σ_E and ℏω_L ≪ σ_E, the LEM device can realize spectral shearing for an electron. After passing through the LEM device, the initial electron energy spectrum given by Eq. 1 obtains a spectral shear: ψ_f(E) = ψ₀(E − ΔE)e^iφ(E−ΔE), as required for FESSI. Notably, the initial spectral amplitude ψ₀(E) is not required to be Gaussian. It can be an arbitrary function with a considerably wide bandwidth. Because it is cumbersome to analytically express a general electron wave function modulation, we use the time-dependent Schrödinger equation to simulate this LEM process (see Materials and Methods). We note that LEM can also realize deceleration of the electron wave packet without any distortion when the electron reaches the negative peaks of the laser electric field. By following the same derivation (Eqs. 3 to 7) with θ₀ = ±π, one can obtain a net deceleration ΔE = −2∣g∣ℏω_L. Experimentally, one can realize an optimal acceleration (θ₀ = 0) or deceleration (θ₀ = ±π) by checking the maximum or minimum energy shift in spectral intensity in the spectrometer (see section S3 and fig. S6 for a complete discussion of the role of θ₀). In the case of σ_E ≪ ℏω_L, we obtain the PINEM regime of electron energy modulation with spectral sidebands instead of small energy shifts. Despite the fact that such a PINEM-type modulation at the LEM foil spectrum also leads to spectral interference within the FESSI scheme, a reconstruction of the phase is not possible because of the absence of a small energy shift (see section S4 for more details).

Reconstruction of the free-electron wave function

Figure 2 illustrates the method of electron wave function reconstruction in FESSI. In Fig. 2A, we assume a pulsed electron wave function with a particular amplitude and phase that is awaiting measurement. The reconstruction uses a spectral interferogram generated by the interference of two replicas of the pulse. The time delay between them is created using the Wien filter. The two replicas are identical except that their energy is displaced relative to one another (Fig. 2B). The resulting interferogram (Fig. 2C), which is the spectral intensity, is given by

$$\begin{array}{c}I(E)={\mid \mathrm{\psi }(E)\mid }^2+{\mid \mathrm{\psi }(E-\text{Δ}E)\mid }^2+2\mid \mathrm{\psi }(E)\mid \mid \mathrm{\psi }(E-\text{Δ}E)\mid \\ \times \cos [\mathrm{\phi }(E)-\mathrm{\phi }(E-\text{Δ}E)-E\mathrm{\tau }/\hslash ]\end{array}$$ (7)

where ψ(E) is the spectral profile, ΔE is the spectral shear, and τ is the time delay. The first two terms are the spectra of the initial pulse and its spectral shifted replica. The third term of Eq. 8 presents the spectral interference. By using a reconstruction algorithm (for details, see section S1 and fig. S1), we can retrieve the spectral phase profile (Fig. 2D). To this end, both the spectral amplitude and phase are measured, so that we can perform the inverse Fourier transform (IFT) to the spectral profile (Eq. 1) and reconstruct the full temporal wave function. Note that the reconstructed wave function is located at the position of the LEM device because the spectral shearing occurs here.

The detection of the spectral phase profile is crucial because it determines the reconstruction of the electron wave function. In this section, we classify the spectral phase to comprehend the underlying physics that might alter or influence the spectral phase. We start with a Taylor expansion of the spectral phase, ${\mathrm{\phi }}_E(E)={\mathrm{\phi }}_0+{\mathrm{\phi }}_1(E-E_0)+\frac{1}{2}{\mathrm{\phi }}_2{(E-E_0)}^2+\frac{1}{6}{\mathrm{\phi }}_3{(E-E_0)}^3+\dots$, where φ_n = ∂ⁿφ_E/∂Eⁿ at E = E₀. The zero-order constant term is only a global phase. The linear phase term φ₁ relates to the arbitrary choice of the initial time t₀ for the electron, which is easily removed by resetting the initial time. Consequently, only higher-order phases (n ≥ 2) dominate the reconstruction of an electron. The second-order phase φ₂ results in pulse chirping in the time domain. It is physically relevant to consider a second-order phase because a propagating electron can accumulate φ₂ because of its nonrelativistic dispersion. The third-order phase φ₃ is also attractive because it can be induced by a high-voltage bias when the emitted electron is pre-accelerated to the final kinetic energy, such as 10 keV. This acceleration process would introduce a third-order phase to the measured electron pulse (40, 41). Our scheme can also provide the fourth and higher-order phases, although they are of minor importance in our case. For reconstructing a wave function, the phases of the second and third orders are most relevant. In our simulation in Fig. 3A, we generate an exemplary electron wave function with a Gaussian envelope function and a spectral phase that combines the second and third orders, φ(E) = 0.34(E − E₀)² + 1.05(E − E₀)³. Here, the spectral width of the pulse is 2σ_E = 0.85 eV.

Fig. 3. Original and reconstructed free-electron wave packet with both amplitude and phase.

(A) The spectral amplitude (blue), the original (orange), and the reconstructed phases (purple dashed curve) of the electron wave function with a spectral width of 0.85 eV and with second- and third-order dispersion. (B) Comparison of the original wave function and the reconstructed one in time. (C) Similar to (A), an attosecond electron pulse wave function with a spectral width of 8.5 eV and accompanied by a second and third dispersion [$\mathrm{\phi }$_E(E) = $\mathrm{\phi }$₂(E − E₀)² + $\mathrm{\phi }$₃(E − E₀)³ = 1.35 × 10⁻²(E − E₀)² + 2.6 × 10⁻³(E − E₀)³]. (D) Comparison of the original attosecond pulse wave function and the reconstructed one in time. Both cases are simulated with time jitter around 0.001% of the time delay; the fidelity value for the reconstruction is F = 99.9%.

DISCUSSION

Two constraints limit the range of time delay introduced by the Wien filter: (i) To observe the spectral interference within the spectral width, the delay has a lower limit, τ > πℏ/σ_E (or $\frac{2\mathrm{\pi }\hslash }{\mathrm{\tau }}<2{\mathrm{\sigma }}_E$). (ii) Limited by the resolution δ_E of the spectrometer, the delay cannot exceed an upper limit, τ < 2πℏ/δ_E. Here, we assume a resolution of δ_E = 10 meV such that the possible time delay ranges from 5 to 400 fs. We choose the delay as τ = 30 fs. Figure 3A shows the numerical result of FESSI, which demonstrates an excellent match between the reconstructed and the original spectral phases.

By a similar token, two constraints restrict the range of the energy shear: (i) We extract the phase from the ac term (eq. S3 in section S1). The ratio of intensity between the ac and the dc term is approximately $\Vert D^{\mathrm{ac}}/D^{\mathrm{dc}}\Vert \propto \exp (-\text{Δ}{E}^2/4{\mathrm{\sigma }}_E^2)$. To acquire an adequate ac signal, we must guarantee ΔE < 2σ_E. (ii) Conversely, the energy shear cannot be too small. Otherwise, the phase contribution of $\text{Δ}E\frac{\mathrm{\partial }\mathrm{\phi }}{\mathrm{\partial }E}$ in Eq. 2 is insignificant. In our simulation, we use ΔE = 0.1 eV. Such a small energy shift for the electron pulse necessitates stable and coherent laser control of the electron acceleration, which is within the capabilities of the LEM device.

For FESSI to work, the measured electron pulse must be local in time for the LEM device to obtain a net energy shift. In analyzing the limits of the time delay and energy shear, we notice that σ_E should be sufficiently large to observe spectral interference. This requirement is consistent with the locality condition of the electron with an intrinsic duration of σ_t0 = ℏ/2σ_E that is temporally small. However, the second and higher-order spectral phases would broaden the pulse although the intrinsic duration is small. We provide a criterion σ_t(φ) < T/4, namely, the pulse duration should be smaller than a quarter of the optical cycle T (see section S2 and fig. S5), so that there is still a possibility of achieving spectral shearing using the LEMs (42). Because of the temporal locality of the electron wave function, the spectral phase variation cannot be too large. As a further test of FESSI, we examine pulse broadening in terms of different-order spectral phases, including oscillatory phases (see section S2 and figs. S3 and S4).

FESSI is strongly sensitive to jittering effects of different kinds. The contribution of energy jitter is negligible because the laser-electron modulator can control it coherently. However, the time jitter δτ is critical because it will lead to a large phase jitter, δφ~E₀δτ. This phase jitter would lead to a strong blur of the interference fringes, rendering the spectral interferometry ineffective. However, state-of-the-art Wien filters can minimize time jitter to a negligible amplitude when using highly stable and precise power supplies (36). To quantify the quality of spectral phase reconstruction, we define a fidelity $F=\frac{\sum {\mid {\mathrm{\phi }}_{\mathrm{o}}(E)\mid }^2}{\sum {\mid {\mathrm{\phi }}_{\mathrm{r}}(E)-{\mathrm{\phi }}_{\mathrm{o}}(E)\mid }^2+\sum {\mid {\mathrm{\phi }}_{\mathrm{o}}(E)\mid }^2}$, with the original phase φ_o and reconstructed phase φ_r. To investigate the potential of experimentally reconstructing the signal, we assume a tiny time jittering of about 0.001% delay time in our simulation (Fig. 3), where the fidelity F is near 1. Increasing the jitter to 0.007%, the spectral interference is notably weaker but we still find that F = 98.1%, indicating the robustness of our method (see fig. S2).

Last, as a spectral interferometry approach, our FESSI setup has the advantage of real-time monitoring ultrashort electron pulses. Figure 3 (C and D) depicts a reconstruction example of an attosecond electron pulse with a spectral width of 8.5 eV [i.e., 10 eV at full width at half maximum (FWHM)], accompanied by a second- and third-order phase dispersion [φ_E(E) = φ₂(E − E₀)² + φ₃(E − E₀)³ = 1.35 × 10⁻²(E − E₀)² + 2.6 × 10⁻³(E − E₀)³], corresponding to φ₂ = 5.9 × 10³ as² and φ₃ = 7.4 × 10⁵ as³ [typical values of second- and third-order phase dispersion for attosecond electron pulse are around 10³ as² and 10⁵ as³, respectively (41)]. The FWHM duration is approximately 250 as. In this case, the time delay and energy shift are calculated as τ = 5 fs and ΔE = 1 eV, respectively. The comparison of original and reconstructed phases, as well as the high fidelity, indicates that our FESSI is capable of attosecond electron pulse reconstruction.

In conclusion, we have proposed a spectral interferometry approach to characterize and reconstruct an ultrashort electron wave packet. The principle, setup, and reconstruction approach of the FESSI enable the measurement of typical electron wave packets in ultrafast science. Because FESSI foots on existing techniques in electron microscopy and spectroscopy, we believe that its experimental realization is feasible. We expect applications of FESSI as a characterization approach for femtosecond and attosecond electron pulses in the photoelectron guns, ultrafast transmission and scanning electron microscopy, DLA, x-ray free-electron lasers, and strong-field physics. These domains will strongly benefit from full detection and manipulation of ultrashort electrons. As our proposed technique is optimized for single-electron pulses, it may encounter limitations in accurately characterizing high-electron flux pulses because of the adverse effects of space charge, which can diminish wave function coherence. Second, free-electron wave function reconstruction can test the foundations of quantum mechanics, for example, through weak measurements (43, 44) and quantum tomography (45), which may shed light on many fundamental issues, such as the wave function collapse and the measurement problem (3, 46, 47), and the dispute of quantum interpretations (48–50).

MATERIALS AND METHODS

Schrödinger equation solution of the LEM

The wave packet acceleration process can be represented by a relativistically modified Schrödinger equation of free electrons in the presence of the electromagnetic field

$$i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\psi }(z,t)=(H_0+H_I)\mathrm{\psi }(z,t)$$ (8)

The free-electron Hamiltonian for one-dimensional relativistic dynamics is $H_0={\mathrm{\epsilon }}_0+v_0(p-p_0)+\frac{{(p-p_0)}^2}{2{\mathrm{\gamma }}^{3}m}$, derived from the Dirac equation in the nonrelativistic approximation when the spin index is ignored. To avoid the time jitter bringing a large phase difference, we choose the initial electron kinetic energy ε₀ = (γ − 1)mc² = 10 keV, the initial momentum p₀ = γmv₀, and the electron velocity v₀ = βc, with the speed of light c, the relative speed β = 0.1949, and the Lorentz factor $\mathrm{\gamma }=1/\sqrt{1-{\mathrm{\beta }}^2}=1.0195$. The pulse has a spectral width 2σ_E = 0.85 eV (an FWHM of 1 eV). The near-field interaction part is $H_I=-\frac{e}{2\mathrm{\gamma }m}(A\cdot p+p\cdot A)$ without gauging. Only the longitudinal component of the near-field vector potential affects ultrafast electron dynamics in the propagation direction (z). In our case, the transverse field components are ignored.

We take a realistic dielectric membrane suggested in the setup in (25, 27), whose longitudinal electric field is given by $E(t)=-\frac{\mathrm{\partial }}{\mathrm{\partial }t}A(t)$, where the vector potential is A(t) = A₀ sin(ω_Lt + φ_L) without the distance-dependent term because we use a thin membrane, with $A_0=-\frac{E_0}{{\mathrm{\omega }}_{\mathrm{L}}}$, the electric field strength E₀, the laser frequency ω_L, and the phase delay φ_L. Here, we choose an MIR laser with central wavelength λ_L = 10.33 μm. The interaction length is assumed to be the same as the thickness of the membrane L = 50 nm.

Supplementary Materials

This PDF file includes:

Sections S1 to S4

Figs. S1 to S6

References