Quantifying Decoherence in Attosecond Metrology

Laser-dressed photoemission spectroscopy has established itself as the gold standard of attosecond temporal metrology. In this technique, the attosecond structure of an extreme-ultraviolet pulse is retrieved from the wave function of the electron wave packet released during photoionization. Here, we show that this electron wave packet should rather be described using the density matrix formalism, thus allowing one to account for all processes that can affect its coherence, from the attosecond pulse generation to the photoemission and the measurement processes. Using this approach, we reconstruct experimentally a partially coherent electron wave packet with a purity of 0.11 (1 for full coherence). Comparison with theoretical models then allows us to identify the origins of this decoherence and to overcome several limitations such as beam-line instabilities or spectrometer resolution. Furthermore, we show numerically how this method gives access to the coherence and eigencomponents of complex photoelectron wave packets. It thus goes beyond the current measurement of photoionization time delays and provides a general framework for the analysis and understanding of complex photoemission processes.

Popular Summary

When an attosecond light pulse ionizes an atom, the latter releases an electron that behaves quantum mechanically, that is, as a wave packet. Historically, attoscience started when scientists figured out how to access experimentally the amplitude and phase of this wave packet to obtain information on the attosecond light pulse and on the probed atom. However, this “amplitude-and-phase”-based description leaves aside the loss of coherence that may alter the wave packet’s quantum state. Such a decoherence arises from all the degrees of freedom that remain unobserved in the experiment, owing to technical limitations or quantum-mechanical reasons. Here, we rethink the attosecond photoionization process using von Neumann’s concept of a density matrix—a mathematical quantity that includes the influence of decoherence—and we show how to measure this density matrix in practice.

We first identify the experimental conditions required to access the density matrix and then measure it using laser-dressed photoelectron spectroscopy. Surprisingly, we reconstruct an electron wave packet with a very low purity. To attach a physical origin to this loss of coherence, we compare the experimental quantum state to several models, each representing the influence of a different decoherence process. Finally, we numerically replicate a historical experiment in attosecond physics, namely, the measurement of photoemission time delays in neon. This allows us to illustrate on a famous test case the benefits of measuring the density matrix.

Accessing experimentally the density matrix of a photoelectron wave packet is an important paradigm shift in attosecond physics. In addition to providing a sensitive technique to diagnose beam imperfections, this novel approach will allow one to probe ultrafast decoherence phenomena in ionization processes.

Figure 1

Principle of a Mixed FROG measurement. (a) An XUV attosecond pulse train (purple curve) frees an electron wave packet (gray filled line) through photoionization. During this step, a laser pulse (red curve) modifies the wave packet, the kinetic energy spectrum of which is then measured with a time-of-flight spectrometer. (b) In the absence of a laser pulse, this spectrum gives the populations (the diagonal elements in black) of the laser-free density matrix $\rho$. Adding the laser pulse gives access to the off-diagonal elements of $\rho$ (in red) through the emission or absorption of laser photons. (c) The state of the photoelectron wave packet can equivalently be described by a Wigner distribution $W$ in the energy or time phase space. Measuring the energy spectrum for various XUV and laser delays provides a set of projections of $W$, which makes it possible to access the Wigner distribution by tomography.

Figure 2

Implementation of Mixed FROG for an attosecond pulse train generated in neon and photoionizing neon in the $2p$ valence shell. (a) The left and right show the photoelectron spectra for laser intensities of $4.5\times {10}^{11}$ and $6\times {10}^{12}\mathrm{W}·{\mathrm{cm}}^{-2}$, respectively. The central panel gives the evolution of the spectrum between these two intensities: The linear drift in energy of the peaks observed when the intensity is increased corresponds to the laser-induced ponderomotive shift [43]. Increasing the dressing intensity enables the electron wave packet to emit or absorb more laser photons, which is confirmed by the appearance of numerous “outer” sidebands in the photoelectron spectrum [40, 44]; see the right. When multi-laser-photon transitions allow the first and last populated levels to interfere, the highest-order coherence becomes encoded into the spectrum, and the full density matrix can be obtained. (b) Retrieved density matrix in the energy domain ${\rho }_{\exp }(\epsilon ,{\epsilon }^{\prime })$ (only the modulus shown) for a dressing laser intensity of $6\times {10}^{12}\mathrm{W}·{\mathrm{cm}}^{-2}$. The dark gray shaded curve on the left gives the diagonal elements, that is, the laser-free photoelectron spectrum. (c) Reconstructed Wigner distribution $W_{\exp }(\epsilon ,t)$. Its projections onto the temporal (upper) and energy axes (left), respectively, correspond to the average time profile of the electron wave packet and the laser-free photoelectron spectrum.

Figure 3

Comparison to decoherence models. (a) Fourier transforming ${\rho }_{\exp }(\epsilon ,{\epsilon }^{\prime })$ provides the experimental density matrix in the time domain ${\rho }_{\exp }(t,t^{\prime })$; only the modulus is shown. Inset: Enlargement of the area delimited by the black square in the main panel. Its diagonal elements (upper) give the time structure of the electron wave packet. Off-diagonal-element disappearance, also called decoherence, can be observed at the femtosecond and attosecond timescales as emphasized by the red dashed lines. (b) To identify the physical origin of decoherence, several density matrix models ${\rho }_{\text{model}}$ are fitted onto ${\rho }_{\exp }$. The best agreement (shown here) is obtained with model 5: The femtosecond decoherence comes from the limited spectrometer resolution, while the attosecond decoherence arises from fluctuations of the synchronization between the laser and XUV pulses in the spectrometer. (c) Evolution of the least-squares error $\iint {|{\rho }_{\exp }-{\rho }_{\text{model}}|}^{2}dtd{t}^{\prime }$ with the number of iterations of the fitting algorithm for various decoherence models.

Figure 4

Parameters of the most probable decoherence scenario. (a) Spectrum (blue shaded curve) and spectral phase (segmented blue line) of the coherent electron wave packet ${\chi }_0$ free from arrival-time variations and from the spectrometer’s influence. The gray shaded curve shows the experimental spectrum also shown in Figs. 2 and 2. The red dots (connected by a dashed line) stand for the spectral phase obtained with a RABBIT measurement in the same experimental conditions except for the dressing laser intensity fixed at $4.5\times {10}^{11}\mathrm{W}·{\mathrm{cm}}^{-2}$. (b) Time profile ${|{\chi }_0(t)|}^2$ of the coherent electron wave packet. (c) Envelope $P$ of the fluctuations of the synchronization between the laser and XUV pulses (gray shaded curve) compared with an attosecond pulse composing the coherent electron wave packet ${\chi }_0$ (blue shaded curve) and the average pulse in the train retrieved with RABBIT (red shaded curve). (d) Response function $R$ of the time-of-flight spectrometer. All widths indicated in the figure correspond to full widths at half maximum.

Figure 5

Simulation of a multichannel photoelectron wave packet reconstructed with Mixed FROG. (a) Electron energy spectrum obtained after ionization of neon by high-order harmonics. The spectrum shows three ionization pathways: direct ionization from the $2p$ and $2s$ subshells and SU ionization; see the main text for details. (b) Simulated spectrograms in the 30–60 eV spectral region with and without interchannel coherences between the $2s$ and SU wave packets (respectively labeled as the coherent and incoherent cases). The dressing laser pulse has an intensity of $1.5\times {10}^{12}\mathrm{W}·{\mathrm{cm}}^{-2}$ and a duration of 27 fs. Inset: Enlargement of the area delimited by the black square in the main panel. (c) Density matrices in the coherent and incoherent scenarios (only the modulus shown). In both cases, the left side corresponds to the original density matrix used to simulate the spectrogram in (b), and the right side shows the density matrix retrieved with Mixed FROG.

Figure 6

Eigenstate decomposition of the multichannel density matrix in the coherent and incoherent cases. (a),(b) Eigenvalues ${\lambda }_i$ of $\rho (\epsilon ,{\epsilon }^{\prime })$ sorted in decreasing order (left) and modulus of the associated eigenstates $|{\chi }_i(\epsilon )|$ (right). When ${\lambda }_i$ is negligible, the eigenstate is shaded in gray. (c),(d) After Fourier transforming the dominant eigenstates, one obtains the time-domain wave packets ${|{\chi }_i(t)|}^2$ (shaded curves); the black dashed lines show the time profile of the exact wave packets used to perform the simulation. (e) The photoionization time delays can easily be recovered from the retrieved density matrices (dots connected by dashed lines); see the main text for details. The original delays are taken from Ref. [19]. The high-harmonic photon spectrum is also shown (gray shaded curve).

Figure 7

Simulated density matrices in various cases. (a) The electron wave packet is pure, (b) the XUV pulses suffer from arrival-time variations, or (c) attochirp variations. (d) The XUV pulse ensemble is fully coherent, but the limited resolution of the spectrometer is accounted for.

Figure 8

For each decoherence model (a)–(e), the density matrix ${\rho }_{\text{model}}$ is fitted onto the experimental density matrix ${\rho }_{\exp }$ reported in (f). In each case, the density matrix is depicted in the energy domain (left) and in the time domain (right); only the modulus is shown.