A dynamical symmetry triad in high-harmonic generation revealed by attosecond recollision control

A key element of optical spectroscopy is the link between observable selection rules and the underlying symmetries of an investigated physical system. Typically, selection rules directly relate to the sample properties probed by light, yielding information on crystalline structure or chirality, for example. Considering light-matter coupling more broadly may extend the scope of detectable symmetries, to also include those directly arising from the interaction. In this letter, we experimentally demonstrate an emerging class of symmetries in the electromagnetic field emitted by a strongly driven atomic system. Specifically, generating high-harmonic radiation with attosecond-controlled two-color fields, we find different sets of allowed and forbidden harmonic orders. Generalizing symmetry considerations of circularly polarized high-harmonic generation, we interpret these selection rules as a complete triad of dynamical symmetries. We expect such emergent symmetries also for multi-atomic and condensed-matter systems, encoded in the spectral and spatial features of the radiation field. Notably, the observed phenomenon gives robust access to chiral processes with few-attosecond time precision.

The concept of emergence describes the appearance of entirely new properties of a coupled system, which only arise from the interaction of its constituents. In light-matter interaction, emergent phenomena are particularly prominent in strongly driven states of matter, as evident in light-induced variants of superconductivity [1], the Hall effect [2], and topological insulators [3] in the condensed phase, as well as exceptional points in molecules [4,5], Kramers-Henneberger states in gases [6,7], and time-crystals in isolated many-body systems [8,9]. Such emergent states are often governed by novel symmetry properties and topologies, which may be probed by external or emitted radiation fields. However, while much attention is drawn to the light-induced creation of novel states of matter, a wider scope should also include possible emergent states of the radiation field.
Here, we report on the experimental observation of an emergent class of symmetries in the electromagnetic field emitted by a strongly driven atomic system. Specifically, we analyze the sets of allowed and forbidden harmonic orders in high-harmonic generation from tailored bi-circular and bi-elliptical fields. Corroborated by theoretical modeling, the identified selection rules correspond to a complete triad of dynamical symmetries. We believe that the general principles underlying our observations will be equally relevant for other systems, including crystalline solids.
For these parametric processes, the selection rules can be straightforwardly interpreted in the time-domain: the destructive interference between multiple light-emission events may lead to a group of forbidden emission frequencies (e.g. two events of equal amplitude and opposite phase). Breaking of the underlying symmetry or changing the timing of the emissions allows these spectral lines to appear. In the case of high-order harmonic generation (HHG), the separate emission events represent recollisions of an electronic wavepacket with an atomic or molecular parent ion [29,30]. The recollision picture facilitates the retrieval of attosecond timing and orientation from spectral features in HHG, as exploited in measurements of the time structure of the tunneling process [31,32], molecular tomography [33], and chiral discrimination [34,35].
A particularly relevant example is the generation of circularly polarized high-harmonics of bi-circular laser fields in isotropic media, which ties the spectrum to the polarization state of the harmonics [36][37][38][39]. The bi-circular field, composed of a counter rotating fundamental field and its second harmonic, reaches its maximum amplitude three times in an optical cycle, where each of those is rotated by 120 • in the polarization plane. A more general representation for the underlying link between temporal delays and rotations of the bi-circular driving field can be written as a three-fold dynamical symmetry, described in more detail in reference [38], Here, E (t) is the overall electric field in the system (both the incident bi-circular driving field and the emitted radiation), T is the period of the fundamental field andR (120 • ) is the operator for 120 • rotation in the polarization plane. Equation (1) originates from the dynamical symmetry of the bi-circular driving laser: a left-rotating field with a cycle T overlaps with its counter-rotating second harmonic three times in an optical period, each of which at a rotated orientation. A temporal shift of T/3, is therefore, equivalent to a rotation by 120 • . In an isotropic medium the resulting Hamiltonian inherits the driving field's symmetry, and thus, the emitted radiation complies with equation (1) as well. The symmetry in equation (1) imposes the distinct suppression of every third harmonic order, q suppr = 3m = . . . 21, 24, . . . , 45, 48, . . . .(with integer m). Notably, the emitted harmonic orders are circularly polarized [36][37][38][39], if effects from the pulse envelope [20,21,40] can be neglected. This three-fold selection rule is presently considered unique, where only a specific set of harmonics is forbidden, and its suppression reveals the Hamiltonian symmetry [11]. Our experimental setup is schematically shown in figure 1. An amplified pulsed beam (Ti:sapphire laser, 800 nm central wavelength, 40 fs pulse duration, 2 mJ pulse energy) is modified by a MAZEL-TOV apparatus [41] into a bi-chromatic field, comprising the fundamental field and its second harmonic with equal intensities at the focus. A tilted glass plate (0.15 mm, fused silica) tunes the bi-chromatic phase, which is the relative phase between the two driving fields. The initial polarization of the two colors is linear, where the second harmonic field is perpendicular to the fundamental. The final polarization state (see inset of figure 1) is controlled by a quarter-wave plate (QWP) for both 800 nm and 400 nm, comprising quartz and MgF 2 slabs. Rotation of the QWP tunes the two-color polarization states from counter-rotating circular at an angle of 45 • , through equal ellipticities with perpendicular major axis, and back to the original perpendicular linear polarizations at QWP angles of 0 • or 90 • . Thus, the polarization is orthogonal in terms of Jones-vector calculus [42] throughout the experiment [see figure 1(b)]. The two colors co-focus in a He-filled gas cell, and the generated high-order harmonic spectrum is analyzed by a toroidal grating and a charge-coupled device (CCD) camera. The spectrum image in figure 1(c) shows the established suppression of every third harmonic order, imposed by the three-fold dynamical symmetry of equation (1).
Beyond this well-known suppression of the q = 3m harmonics by bi-circular fields [36][37][38][39]53], upon controllably varying the bi-elliptical incident polarization state, we have discovered two additional three-fold selection rules, corresponding to the suppression of either the 3m + 1 or the 3m + 2 harmonics. Figure 2(a) plots experimental spectra for selected bi-chromatic phases, and a quarter-wave plate angle of 38 • , corresponding to an ellipticity of 0.78 (ratio of minor to major ellipse-axes) for each of the driving fields. These spectra clearly show that the suppression of every third harmonic order can be shifted in a controllable manner among the sets given by q suppr = 3m, 3m + 1, 3m + 2. The three sets of selection rules can be generalized as the emergence of a family of three dynamical symmetries, that is, a triad,  (1), where the dynamical symmetry is imposed by the bi-circular driving field, the symmetry expressed in equation (2) applies neither to the Hamiltonian nor the driving bi-elliptical field. Thus, the bandwidth of such a symmetry equation is inherently limited. Low frequencies such as the fundamental field and its second harmonic cannot comply with this symmetry. It is surprising, though, that this emerging symmetry applies to a large bandwidth. The experimental spectrum marked red in figure 2(a) shows a selection rule that correspond to equation (2) over the entire observable spectrum, in the range of 20 to 45 harmonic orders (photon energies of 30-70 eV, respectively). The bi-chromatic phase governs whether the 3m + 1 (red) or the 3m + 2 [green line in figure 2(a)] harmonics are suppressed. Figure 2(b) presents the HHG spectrograms recorded by continuously tuning (sweeping) the quarter-wave plate for two bi-chromatic phases [determined to be about 0.1π and 0.7π by comparison with simulations presented in figure 3(d)]. The case of a bi-circular field with a QWP angle of 45 • , is identified by the distinct suppression of harmonic orders q suppr = 3m (e.g., see suppressed harmonic q suppr = 39, dashed black line). The selection rule for the triad index n = 0, as in equation (1), is robust in the sense that it directly follows from the three-fold dynamical symmetry of the incident field (with the isotropic medium) and covers all frequencies. It is therefore independent of the relative phase between the two fields or the relative power. As soon as the quarter-wave plate is detuned from 45 • , the field becomes bi-elliptical, the three-fold symmetry of the driving field is broken, and the suppressed harmonics reappear. Additionally, the cutoff extends due to the generally larger field amplitudes for elliptically polarized fields, compared to circularly polarized fields. At a few-degree offset from circular polarization, namely at QWP = 38 • , a different three-fold symmetric spectrum emerges, with characteristic suppressions of every third harmonic order.
To understand the microscopic origin of the new three-fold symmetries we turn to consider the attosecond emission events contributing to the HHG [43,44]. In the case of a bi-circular driving field, for which equation (1) applies, the HHG radiation can be represented in time domain as a triplet of rotated attosecond pulses, equally spaced within an optical cycle, as illustrated in figure 3(a). Introducing bi-elliptical polarization leads to changes in both the recollision timing, Δt, and the phase, Δφ intr of each burst separately [see figure 3(b)]. Slight changes in the timing and phase of the three bursts have drastic  Tuning the QWP induces a temporal delay, Δt, and an intrinsic phase shift, Δφ intr , to each pulse, originating from the trajectory of the laser-driven electron (details in the supplementary). (c) The overall phase of every attosecond pulse, Δφ total = qω 0 Δt + Δφ intr , vs the QWP angle, for a bi-chromatic phase of 0.7π. At particular QWP angles (vertical arrows), the phase is near a multiplicity of 2π/3, which yields an emergent symmetry (see equation (2) and the supplementary). The effect of the temporal phase alone, qω 0 Δt, is shown in dashed lines. (d) Simulated spectral density vs the QWP angle, for different bi-chromatic phases. The white dashed line marks the parameters for the simulated spectrum shown in (e). effect on the spectrum and polarization of the emitted harmonics, even for changes that correspond to a fraction of an optical cycle.
Using semi-classical simulations for HHG driven by equal bi-chromatic intensities, we follow the phase of the 39th harmonic order for a varying QWP angle (see figures 3(b) and (c)). For these simulations, we find the three electron trajectories during one fundamental optical cycle, in which the electron returns to the origin with a kinetic energy that corresponds to the emission of the selected harmonic order. For each trajectory, we find the initial transverse velocity of the electron, the ejection and recollision times, and the position and kinetic energy along the path. We follow each of these trajectories for varying ellipticity (i.e. QWP angle) and bi-chromatic phases. The orientation, timing, and the intrinsic-phase accumulated during the electron quiver is then used to analyze the HHG radiation (see figure 3). In the case of a bi-circular driving field, the field's properties, e.g., the maximal amplitude, repeats every third of the optical cycle. When the QWP is detuned from 45 • , the peaks of the field are no longer at an equal temporal spacing along the optical cycle. Thus, a particular recollision event can occur later, compared with the bi-circular case, due to delays in the field that brings the electron trajectory back to the origin. Earlier ionization times also correlate with delayed recollisions. In a simplified picture, the intrinsic phase relates to the time the electron spends in the continuum, between its tunnel-ionization and the final recombination. Thus, Δφ intr is inversely correlated to the ionization timing. More accurately, however, the phase is given by the path integral over the entire electron trajectory [30,[45][46][47]. Thus, an imbalance between the laser's electric-field amplitudes during the three trajectories also contributes to a difference in their intrinsic phases. While the dominant contribution to the overall recollision phase is the relative delays, Δt [dashed lines in figure 3(b)], the intrinsic phases provide for important corrections. Specifically, the phase corrections for the first and second recollisions are opposite, reducing and increasing the overall phase of the adjacent recollisions, with respect to a reference recollision. Importantly, for specific bi-chromatic phases, adjacent recollisions acquire an approximately equal phase difference, which simply scales linearly with the QWP rotation ( figure 3(c)). At a particular QWP angle, the radiation phases differ by exact multiples of 2π/3 [see arrows in figure 3(c)]. Thus, the additional phase of 2π/3 between recollisions separated by a delay of T/3 is the origin of the term e 2πi 3 n in equation (2). The simulation shows that the recollision orientations remain separated by 120 • approximately, so its contribution is negligible [see figure S.3(b) in the supplementary]. In the experiment, the emerging selection rules are found for a QWP angle of 38 • , which slightly deviates from the values found by the semi-classical simulations, possibly due to the effect of the screened Coulomb potential [48,49]. The nearly linear relationship between the QWP angle and the suppressed harmonic orders in the experimental spectrogram (figure 2(b)) suggests that for multiple harmonic orders, the tuning of the QWP induces approximately equal relative phases between the recollisions. Indeed, our simulations reproduce this observation, predicting that the phase differences of the three recollisions scale similarly with the QWP rotation for multiple harmonics (see figure S.3 in the supplementary). As with equation (1), the emerging symmetries of equation (2) constrain the harmonics to be circularly polarized. Both numerical simulations and the repeating of the experiment in Ne gas, utilizing chiral p-orbitals [24,[50][51][52]54], indicate that the unsuppressed harmonics are circularly polarized (see supplementary).
The link between the timing of the recollision events and the resulting selection rules can be quantified. With respect to the case of bi-circular fields, the suppression of the 3m ± 1 harmonics corresponds to temporal shifts of the recollisions of only ±23 attoseconds for the 39th harmonic order. Since a transition of one selection rule (QWP = 45 • ) to another (QWP = 38 • ) requires a QWP rotation of 7 degrees, we estimate a timing accuracy of 3.2 attoseconds per degree rotation of the QWP. We believe that this extreme sensitivity of the selection rules will also yield insights into the dynamics of symmetries and chirality in other media, such as crystalline solids. For example, the amount of QWP rotation that is required to reinstate a selection rule provides for a quantitative measure of the symmetry breaking. Although here we used mainly the QWP angle, the same conceptual approach can be applied to experiments with other control parameters.
In conclusion, this work shows experimental evidence for the existence of a triad of three-fold symmetries in high-harmonic spectra, comprised of one known dynamical symmetry and two emergent bandwidth-limited symmetries. These symmetries are unified into a generalized symmetry equation, placing the entire triad on an equal footing. Our findings entail both applied and fundamental aspects. First, sources for the additional phase slips and means of control exist in molecular and condensed-matter systems. Second, the sub-cycle sensitivity of this experiment may allow to resonantly access chiral processes in atoms and molecules with unprecedented attosecond precision and a very high signal to noise ratio. Finally, as dynamical symmetries and their breakdown are broad physical phenomena, the concept of bandwidth-limited symmetries can be extended to other areas of physics and chemistry.