# Complete retrieval of attosecond photoelectron dynamics from partially coherent states in entangled photoemission

Morgan Berkane, Richard Taïeb, Gabriel Granveau, Pascal Salières, Charles

Bourassin-Bouchet, Camille Lévêque, Jérémie Caillat

► To cite this version:

Morgan Berkane, Richard Taïeb, Gabriel Granveau, Pascal Salières, Charles Bourassin-Bouchet, et al. Complete retrieval of attosecond photoelectron dynamics from partially coherent states in entangled photoemission. Physical Review A, 2025, 111 (4), pp.L041101. 10.1103/PhysRevA.111.L041101. hal-05026394

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Distributed under a Creative Commons Attribution 4.0 International License
Complete retrieval of attosecond photoelectron dynamics from partially coherent states in entangled photoemission

Morgan Berkane,¹ Richard Taëb,¹ Gabriel Granveau,² Pascal Salières,³ Charles Bourassin-Bouchet,² Camille Lévêque,¹ and Jérémie Caillat¹,*,

¹Sorbonne Université, CNRS, Laboratoire de Chimie Physique-Matière et Rayonnement, LCPMR, 75005 Paris, France

²Université Paris-Saclay, CNRS, Institut d'Optique Graduate School, Laboratoire Charles Fabry, 91127 Palaiseau, France

³Université Paris-Saclay, CEA, LIDYL, 91191 Gif-sur-Yvette, France

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(Received 17 July 2024; revised 22 January 2025; accepted 19 March 2025; published 8 April 2025)

We show that the complete photoemission dynamics in situations of electron-ion entanglement can be retrieved from photoelectron spectral measurements without information on the ion. To this end, we develop an energy-time analysis of the photoelectron's reduced density matrix based on first principles. We test and assess our approach with numerical simulations on a low-dimensional model molecule in interaction with broadband composite pulses concealing the vibrational resolution. Our method is directly applicable to recent experimental schemes measuring the photoelectron reduced density matrices in atomic and molecular photoemission. Therefore, it opens a new window on the dynamics of decoherence and entanglement at the attosecond timescale.

DOI: 10.1103/PhysRevA.111.L041101

Attosecond time-resolved spectroscopies rely to a large extent on interferometric schemes [1–3], where optical and “quantum” phases [4–9] play a central role in the measurements and their interpretations. Standard approaches, such as the RABBIT (reconstruction of attosecond beating by interference of two-photon transitions) [1] and the attosecond streaking [2], as well as their expanding lineage [10–12] assume totally coherent and fully resolved processes and measurements for the time-domain interpretation of experimentally measurable spectral amplitudes. However, the question of decoherence is ubiquitous in this context. It has multiple origins ranging from technical causes to quantum entanglement and statistical mixtures, in the preparation of the studied systems [13–17], the measurements [18], or the investigated processes themselves [19–32]. Decoherence has long been discarded, addressed only through indirect signatures in the observations, or considered as a fundamental lock when applying standard interferometric schemes and their wave-function-based time-domain interpretations. Recently, new pump-probe approaches have been designed to explicitly account for various potential sources of decoherences, in particular the mixed-FROG (frequency-resolved optical gating for mixed states) [18,33] and KRAKEN [34,35]. Yet, they give access to the photoelectron’s final reduced density matrix (RDM) and the problem of recovering the comprehensive photoemission dynamics from an asymptotic RDM remains open. Solving it would give access to the ultrafast buildup of entanglement and decoherence of fundamental processes in real time.

In this Letter, we tackle the intertwined issues of attosecond time-resolved decoherence and entanglement by demonstrating the possibility to reconstruct unambiguously

the entangled photoemission dynamics out of a partially coherent photoelectron's RDM. Our reasoning borrows from quantum information the concept of quantum state purification [36,37] and from attosecond science an energy-time analysis of interferometric measurements in the spectral domain [38]. We illustrate our approach with numerical simulations inspired by an original physical case first proposed in Ref. [23] and further explored in Refs. [25–27]. In this scheme, a diatomic molecule is ionized by a pair of XUV pulses whose temporal separation $\tau_{XUV}$ serves as a knob to tune the final photoelectron-photoion entanglement through its (mis)match with the molecule's vibrational eigenperiods. More specifically, we chose a pulse configuration that creates photoelectron-photoion entanglement by approximately discriminating the odd- and even-parity vibrational channels open in the ionization process. With our original reconstruction scheme applied to this illustrative case, we evidence the possibility to access the ultrafast buildup of entanglement-induced decoherence during light-matter interactions.

Our model is a two-dimensional molecule, denoted $A_2$ in the following, with one degree of freedom accounting for the electron motion, and another one for the molecular vibration [39]. It is therefore, by construction, an idealized “photoelectron+photoion” bipartite system. It has been previously used as a numerical benchmark in various studies (see, e.g., Refs. [40–43]). Here, we use the same parameters as in Ref. [42] which mimic, to some extent, the properties of $H_2$—namely the vibronic energies in its ground electronic state and first ionic state.

Owing to the symmetry of the molecule along the electron's coordinate, and owing to the dipole selection rules in one dimension, the photoelectron state can be uniquely identified by its energy $\epsilon \in \mathbb{R}^+$ (i.e., the photoemission direction is here discarded). Regarding the nuclear motion, we characterize the ion's state by its vibrational level $v \in \mathbb{N}$. Hence,

<footnote>*Contact author: jeremie.caillat@sorbonne-universite.fr</footnote>
the present work is focused on analyzing the outcome of the molecular photoemission process

$$A_2 + \hbar\omega(\tau_{XUV}) \rightarrow A_2^+(v) + e^-(\epsilon). \quad (1)$$

We set the ionizing XUV central frequency to $\omega = 54.29$ eV (35th harmonic of a 800-nm laser). We assigned each of the two pulses a short $\sin^2$ envelope lasting 1.60 fs (full duration), in the perturbative intensity regime. We simulated the process by solving the time-dependent Schrödinger equation (TDSE) with a numerically exact scheme [39] consisting in expanding the vibronic wave function on the orthonormal basis of ionic vibrational states, $\{|\chi_v\rangle\}$, and propagating the associated channel-dependent electron wave packets. Here, we obtained converged results by including all vibrational channels up to $v_{max} = 16$. Using the electronic continuum manifold basis $\{|\phi_\epsilon\rangle\}$, the final state of the “photoelectron+photoion” system can be expressed as<fnref n="1" />

$$|\psi_{\text{end}}\rangle = \int_{0}^{\infty} \sum_{v=0}^{v_{\text{max}}} A_{v}(\epsilon)|\phi_{\epsilon}\rangle \otimes |\chi_{v}\rangle d\epsilon. \quad (2)$$

We fully characterized the photoemission process in our simulations by computing the channel-dependent spectral amplitudes $A_v(\epsilon)$ out of the continuum wave packets [43,44], in the asymptotic region.

To illustrate the interplay between $\tau_{XUV}$ and the final entanglement of the ionized system [23], we simulated two characteristic values of $\tau_{XUV}$: 0 fs (case I) and $\frac{1}{2}T_{01} \simeq 7.62$ fs (case II). The latter is half the vibrational period $T_{01} \equiv 2\pi/\Delta\epsilon_{01}$ associated with the energy gap $\Delta\epsilon_{01} = 271$ meV between the first two channels. For the sake of comparison, we multiplied the field strength by $\sqrt{2}$ in case II to obtain the same final ionization yields as in case I. While previous studies [23,25,27] addressed partial measurements based on photoion spectroscopy, we focus here on the complementary case where only the photoelectron (PE) is detected and characterized through its final RDM.

We first consider the $v$-integrated photoelectron spectra (PES) computed as

$$\sigma(\epsilon) = \sum_{v=0}^{v_{\text{max}}} |A_v(\epsilon)|^2. \quad (3)$$

Figure 1(a) shows the spectrum (solid yellow curve) obtained for case I. It displays a single broad peak whose shape is dominated by the pulse bandwidth (approximatively 5 eV at half maximum) which significantly exceeds the spectral separation of the vibrational channels (few 0.1 eV). The spectrum obtained for case II is shown in Fig. 1(b). It presents the same globally featureless bell shape, with only weak oscillations. Hence, the PE spectra alone reveal no clear signature of the $\tau_{XUV}$-dependent photoelectron-photoion entanglement.

The difference is striking when looking at the photoelectron’s asymptotic RDM,

$$\rho_{\text{end}}(\epsilon, \epsilon') = \sum_{v=0}^{v_{\text{max}}} A_v(\epsilon)[A_v(\epsilon')]^*, \quad (4)$$

<figure><img src="image_3.png" /><figcaption>FIG. 1. Photoelectron characterization in the spectral domain, in case I ($\tau_{XUV} = 0$, left column) and case II ($\tau_{XUV} = \frac{1}{2}T_{01}$, right column), respectively. (a) and (b) Integrated PE spectra $\sigma(\epsilon)$ (solid yellow curve), and its even- and odd-$v$ components (see labels). (c) and (d) Modulus of the photoelectron’s RDM, $|\rho_{\text{end}}(\epsilon, \epsilon')|$. (e) and (f) Spectra of the main Schmidt modes, $s_1(\epsilon)$ and $s_2(\epsilon)$ (see label). The data were normalized to 1 at the maximum of $\sigma(\epsilon)$ in the $\tau_{XUV} = 0$ case.</figcaption></figure>

where $[\cdot]^*$ denotes the complex conjugate. It is implicitly expressed on the photoelectron basis $\{|\phi_\epsilon\rangle\}$, and its diagonal $\rho_{\text{end}}(\epsilon, \epsilon)$ corresponds to the spectrum $\sigma(\epsilon)$, Eq. (3). Relevant additional information is provided by the off-diagonal features, namely the coherences between the populated energies. The modulus of the RDM obtained in case I, shown in Fig. 1(c), is quasicircular. Notably, it displays a similar structure in the $\epsilon = \epsilon'$ and the $\epsilon = -\epsilon'$ directions. This is a direct signature of a weakly entangled state, as could be expected in molecular photoemission with a single broadband pulse: Detecting a photoelectron at a given energy $\epsilon$ provides no information regarding the state $v$ of the ion.

In contrast, the RDM obtained in case II displays a clear banded structure [see Fig. 1(d)]. Here, the structure radically differs in the $\epsilon = \epsilon'$ and the $\epsilon = -\epsilon'$ directions, indicating a strong entanglement. Indeed, a nonvanishing delay between the two identical pulses induces interferences in the process, with a channel-dependent character [23]. With this choice for $\tau_{XUV}$, the spectral alternations of constructive and destructive interferences in the odd- and even-parity channels are nearly out of phase, leading to observed oscillations along the antidiagonal of the RDM. This can be verified by looking at the channel-parity-resolved spectra, which can be isolated in our simulations [see Fig. 1(b) (solid lines, see labels)]. These spectra appear as two interspersed combs, each of them with a $2\pi/\tau_{XUV}$ period. The phase opposition would be exact all through the spectral range if the ionic potential were harmonic. In contrast, the parity-resolved

<footnote n="1">¹Equations are expressed in atomic units (a.u.) throughout the Letter.</footnote>
<table><caption>TABLE I. Weights $W_m$ of the photoelectron Schmidt modes in the (a) $\tau_{XUV} = 0$ and (b) $\tau_{XUV} = \frac{1}{2}T_{01}$ cases. They are expressed in percentages of the final ionization yields $Y_{tot}$.</caption><thead><tr><th>m</th><th>1</th><th>2</th><th>3</th><th>4</th><th>≥5</th></tr></thead><tbody><tr><td>(a)</td><td>97.6%</td><td>2.3%</td><td>&lt;0.1%</td><td>&lt;0.1%</td><td>&lt;10<sup>-3</sup>%</td></tr><tr><td>(b)</td><td>49.4%</td><td>47.9%</td><td>1.4%</td><td>1.1%</td><td>≤0.1%</td></tr></tbody></table>

components of the spectrum for case I are practically undistinguishable and as featureless as the integrated spectrum itself. To summarize, case II creates a significantly entangled photoelectron-photoion state, with nevertheless little signatures of the entanglement in the sole photoelectron spectrum measured without vibrational resolution. From there on, any information regarding the ion is assumed to be irreversibly lost, and our analysis aims at reconstructing the dynamics of the photoelectron.

Among the information encoded in the final RDM, the coherence induced by the entanglement can be characterized by the purity $p_{end} = \text{tr}(\rho_{end}^2)/\text{tr}(\rho_{end})^2$, where tr( ) denotes the trace operation [45]. It is nearly maximum in case I ($p_{end} = 0.95$), while it has approximately half that value in the other case ($p_{end} = 0.48$).

To proceed with the analysis of the RDM, we use the so-called purification, a general procedure applying to mixed quantum states [36,37]. It consists here in considering the Schmidt modes of the photoelectron, i.e., the eigenvectors $B_m(\varepsilon)$ of the RDM with nonvanishing eigenvalues $\Lambda_m \in \mathbb{R}^+$ [28,46], obtained by simple diagonalization of the RDM. They allow for a low-rank reformulation of $|\psi_{end}\rangle$ [Eq. (2)] as

$$|\psi_{\text{end}}\rangle = \int_{0}^{\infty} \sum_{m=1}^{M} B_m(\varepsilon) |\varphi_{\varepsilon}\rangle \otimes |\tilde{\chi}_m\rangle d\varepsilon, \quad (5)$$

with $M \le v_{max} + 1$, and where $\{|\tilde{\chi}_m\rangle\}$ is an effective, a priori unspecified and unknown, orthonormal vibrational basis for the final state. While the eigenmode expansion is by no means a way of recovering an information that was lost in the final-state measurement (as mentioned above), we will see that it allows circumventing this lack of information when accessing the entangled photoelectron-photoion dynamics. Note that the formal expression given in Eq. (5) is very general and holds as well when the nature of the unresolved degree(s) of freedom is unknown. This encompasses a broad range of practical situations. In contrast to the amplitudes $\Lambda_v(\varepsilon)$, the modes $B_m(\varepsilon)$ are orthogonal by construction and they are normalized to $\Lambda_m$, i.e., $\int [B_m(\varepsilon)]^* B_m'(\varepsilon) d\varepsilon = \delta_{mm'} \Lambda_m$. We give in Table I the leading values of $\Lambda_m$ obtained in the two cases, in percentages of the total ionization yields $Y_{tot} = \text{tr}(\rho_{end}) = \sum_{m=1}^{M} \Lambda_m$. The spectra

$$s_m(\varepsilon) = |B_m(\varepsilon)|^2 \quad (6)$$

associated with the first two modes are shown in the lower frames of Fig. 1. The number of effective modes in each case is consistent with the final purities commented above, having in mind that the lower bound for the latter is given by $1/M$. Indeed, in case I, the $m = 1$ mode largely dominates, by more than 97%. As seen in Fig. 1(e), the bell-shaped

dominant mode clearly accounts for the global trend of the integrated spectrum $\sigma(\varepsilon)$ shown in Fig. 1(a). Such a nearly rank-one RDM is an alternative signature of a nearly separable, i.e., nonentangled, state. In case II, the $m = 1$ and 2 modes are dominating with comparable magnitudes. Their spectra $s_m(\varepsilon)$, shown in Fig. 1(f), are shaped as two shifted combs reminiscent of the channel-parity-resolved spectra [47] with significantly more contrasted oscillations [see Fig. 1(b)]. These observable $m$-selective spectra hence contain the entangled physics induced, in the unresolved degree of freedom, by the XUV pulse configuration which approximately discriminates the $v$ parity.

Beyond the individual physical interpretations of the Schmidt modes, their spectra sum up to the diagonal of the RDM, $\sigma(\varepsilon)$. These $B_m(\varepsilon)$ modes are directly accessible from the measurement of $\rho_{end}$ even in partially resolved experiments. Furthermore, they are complex-valued amplitudes that can be analyzed in the time domain, by generalizing tools originally designed for attosecond time-resolved interferometric schemes assuming fully coherent processes [38,48]. With this, we now hence focus on the spectrotemporal analysis and interpretations of the Schmidt modes $B_m(\varepsilon)$. We will use case II as an illustrative case, and validate the “reconstructed” dynamics by confronting them to the actual dynamics extracted from the numerical solution of the TDSE.

To this end, we introduce the temporal amplitudes<fnref n="2" />

$$b_m(t) := \int_{-\infty}^{+\infty} B_m(\varepsilon) e^{-i\varepsilon t} d\varepsilon. \quad (7)$$

Note that, to highlight the Fourier nature of the analysis, we shifted the lower integration bound from 0 [in Eq. (2)] to $-\infty$, the support of $B_m(\varepsilon)$ being restricted to the positive energies. We then build spectrotemporal amplitudes out of $b_m(t)$ as

$$B_m(\varepsilon; t) := \frac{1}{2\pi} \int_{-\infty}^{t} b_m(t') e^{+i\varepsilon t'} dt'. \quad (8)$$

The amplitudes $b_m(t)$ and $B_m(\varepsilon, t)$ are mathematical analogs of the quantities derived out of asymptotic spectral amplitudes in single-channel [11] or fully resolved [49] experimental investigations of atomic photoemission dynamics. Among the various time-energy analyses invoked in similar contexts (see, e.g., Ref. [48]), the considered one presents the advantage of rigorously giving access to fundamental physical quantities, such as ionization rates and transient photoelectron spectra, based on first principles of quantum physics. Hereafter, we generalize these interpretations to the present context of partially resolved photoemission. They were shown in Ref. [38] to be valid for photoemission processes towards continua with negligible spectral variations, i.e., not too close to the threshold. These conditions are met in our simulations. Note also that they encompass the domain of validity of the strong-field approximation on which the mixed-FROG scheme is based [18].

<footnote n="2">²We use here the same phase and normalization conventions as in Ref. [38].</footnote>
<figure><img src="image_4.png" /><figcaption>FIG. 2. Photoelectron characterization in the time domain in the $\tau_{XUV} = \frac{1}{2}T_{01}$ case. (a) Reconstructed ionization rate $I(t)$ (dark violet curve) and instantaneous XUV vector potential squared (solid violet curve). (b) Reconstructed weights $w_m(t)$ for the two dominant modes (see labels). (c) Reconstructed time-dependent purity $p(t)$ (dark green curve) and numerically exact purity (solid light green curve). (d) and (e) Modulus of the reconstructed (d) and exact (e) photoelectron’s RDM $|\rho(\epsilon, \epsilon'; t)|$ at $t = 7.21$ fs. (f) and (g) Same at $t = 7.41$ fs. The RDMs are displayed over the same spectral range as in Figs. 1(c) and 1(d). For each pulse, the central time is indicated by a vertical solid line and the beginning and end times by dashed vertical lines. The time origin $t = 0$ is set to the maximum of the first pulse.</figcaption></figure>

We begin with the ionization rate which, in the present context, can be evaluated as

$$I(t) = \frac{1}{2\pi} \sum_{m=1}^{M} |b_m(t)|^2. \qquad (9)$$

Hereinafter, we included all the modes (i.e., $M = v_{max} + 1$) in the temporal reconstructions for the sake of numerical accuracy. The time-dependent rate reconstructed as such in case II is shown as a dark violet curve in Fig. 2(a). It appears as a succession of two bell-shaped structures coinciding with the pulse profile, also shown as a solid violet curve. Rigorously, the reconstructed profile is time shifted by the Wigner delay associated with the photoelectron scattering in the continuum [38,50]. However, in the considered one-photon process towards a smooth continuum far from threshold, this delay is of a few attoseconds and cannot be resolved in the figure. The excellent agreement between the reconstructed rate and the pulse profile is therefore a first validation of our time-domain reconstruction scheme.

As for the $B_m(\epsilon; t)$ amplitudes, they collectively give access to the temporal buildup of the photoelectron’s RDM itself, reconstructed at each time $t$ as

$$\rho(\epsilon, \epsilon'; t) = \sum_{m=1}^{M} B_m(\epsilon; t)[B_m(\epsilon'; t)]^*. \qquad (10)$$

It is important to note that the physical meaning of the individual $B_m(\epsilon; t)$ is limited. In particular they should not be confused with the time-dependent RDM’s Schmidt modes. We obtained the latter, and the corresponding eigenvalues

$\lambda_m(t)$, by diagonalizing the reconstructed $\rho(\epsilon, \epsilon'; t)$ at each time. We show in Fig. 2(b) the normalized weights $w_m(t) = \lambda_m(t)/Y_{tot}$. Since no significant field-free, vibronically correlated dynamics is likely to take place in this model molecule, they display visible variations only in presence of the XUV pulses. The values of $w_m(t)$ reached between the two pulses are consistent with the final—properly rescaled—weights ($W_m$) obtained in the single pulse configuration (see Table I).

To further illustrate the quality of the temporal reconstruction in terms of entanglement, we also computed the time-dependent purity out of the reconstructed RDM, $p(t) = \text{tr}[\rho(t)^2]/\text{tr}[\rho(t)]^2$. It is shown as a dark green curve in Fig. 2(c). As for the Schmidt mode weights, the purity variations are concomitant with the interactions with the light pulses. Starting slightly below 0.9, $p(t)$ keeps a relatively high value during the first pulse and ends up at 0.95, i.e., the value of $p_{end}$ obtained in case I. Consistently with the scheme design, an important purity variation occurs within the second pulse, where it decays monotonically towards its final value of 0.48. This ultrafast decoherence dynamics is characterized by a maximum entanglement speed of $-dp/dt = 1.07$ fs$^{-1}$ at $t = 7.40$ fs. Interestingly, $p(t)$ evolves mostly in the first half of each pulse. This is the mathematical signature of the quadratic dependency of the purity on the mode weights—the evolution of the mode themselves being properly synchronized with the pulse profile.

To assess the accuracy of the reconstructed $p(t)$, we considered the numerically exact time-dependent RDM of the photoelectron directly obtained in the TDSE simulations. We computed it by analyzing the outgoing photoelectron wave packets using absorbing boundaries [44], shortly after the end
of both pulses to avoid any spurious field dressing artifact while keeping negligible any temporal distortion due to the wave-packet spreading. The corresponding purity is shown in the same figure as a solid light green curve. The agreement with the reconstructed $p(t)$ is remarkable all through the simulated time range, with minor discrepancies showing up only at the earliest times when the ionization probability has reached less than 1% of the final yields.

We verified that the agreement holds also for the whole time-dependent RDM itself, which encodes beyond the mere purity all relevant information on the photoelectron's quantum dynamics. It evolves from the unstructured quasicircular shape shown in Fig. 1(c) (before the arrival of the second pulse) to the striped shape shown in Fig. 1(d) (after the second pulse). This is illustrated at two arbitrary times during the second pulse in Figs. 2(d)–2(g), revealing the progressive buildup of the diagonal stripes. The excellent agreement between the reconstructed and the actual RDM in these two cases is representative of the quality of the procedure at all times (see Supplemental Material [51]). This is a numerical demonstration of the capacities of our time-energy analysis to provide a complete picture of the actual ionization dynamics from the photoelectron perspective, at its natural, intrapulse, attosecond timescale in a broad range of situations. As a straightforward limitation, we anticipate that a transient loss of coherence will not be retrievable out of a significantly coherent final state. Indeed, the dimension of the subspace accessible to our temporal reconstruction scheme is bounded by the rank of the measured final state (the higher the final coherence, the lower the rank). Nevertheless, the reconstruction of nonmonotonic

coherence variations remains possible, as illustrated by the initial increase of purity at the beginning of the present simulations [see Fig. 2(c)].

To conclude, we have introduced an original method to reconstruct the complete photoemission dynamics out of a photoelectron's asymptotic reduced density matrix in the spectral domain. It is based on an *ad hoc* "time-frequency" analysis of the photoelectron RDM's Schmidt modes. We illustrated it with numerical simulations on a model molecule ionized by composite XUV pulses. An excellent agreement is obtained between the RDM-based reconstructed dynamics and the actual ones that are fully accessible in the simulations. From the experimental point of view, the present method requires the measurement of the photoelectron RDM which is now possible with recently developed approaches such as the mixed-FROG [18] or KRAKEN [34]. It can thus be applied to a variety of photoemission processes—that play a key role in attosecond time-resolved spectroscopies—regardless of the nature of the unresolved degree(s) of freedom (e.g., rotational, vibrational, or electronic). It opens the perspective of following experimentally, in “real time,” the ultrafast buildup of entanglement-induced decoherence due to interchannel couplings [15,20], with important consequences for, e.g., charge migration and attochemistry [52].

This research received the financial support of the French National Research Agency through Grant No. ANR-20-CE30-0007-DECAP. The authors wish to acknowledge fruitful and stimulating exchanges with Anne L'Huillier, David Busto, Pascale Senellart, and Michel Brune.

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