key: cord-0940270-j6klzezr
authors: Tzur, Matan Even; Neufeld, Ofer; Fleischer, Avner; Cohen, Oren
title: Selection rules in symmetry-broken systems by symmetries in synthetic dimensions
date: 2021-06-08
journal: Nature communications
DOI: 10.1038/s41467-022-29080-3
sha: f428531b3acc7c06da2e081958741eb70b04bb07
doc_id: 940270
cord_uid: j6klzezr

Selection rules are often considered a hallmark of symmetry. When a symmetry is broken, e.g., by an external perturbation, the system exhibits selection rule deviations which are often analyzed by perturbation theory. Here, we employ symmetry-breaking degrees of freedom as synthetic dimensions, to demonstrate that symmetry-broken systems systematically exhibit a new class of symmetries and selection rules. These selection rules determine the scaling of a system's observables (to all orders in the strength of the symmetry-breaking perturbation) as it transitions from symmetric to symmetry-broken. We specifically analyze periodically driven (Floquet) systems subject to two driving fields, where the first field imposes a spatio-temporal symmetry, and the second field breaks it, imposing a symmetry in synthetic dimensions. We tabulate the resulting synthetic symmetries for (2+1)D Floquet group symmetries and derive the corresponding selection rules for high harmonic generation (HHG) and above-threshold ionization (ATI). Finally, we observe experimentally HHG selection rules imposed by symmetries in synthetic dimensions. The new class of symmetries&selection rules extends the scope of existing symmetry breaking spectroscopy techniques, opening new routes for ultrafast spectroscopy of phonon-polarization, spin-orbit coupling, and more.

powers and polarization states as the system transitions from the symmetric state to the strongly perturbed state.

We start by considering a general Floquet system with period = 2 ⁄ , and a DS denoted by ̂. The operation ̂ is a (2+1)D spatio-temporal symmetry, jointly imposed by the symmetries of the target material and the first driving laser (or by any other periodic excitation of the system 25, 26 ). The operations ̂ were comprehensively tabulated within the framework of Floquet group theory 8 For the exemplary scenario we focus on here, this translates to

In 

Here, ̅ , are complex conjugates of , respectively, and , , ℎ, are non-negative integers.

The symmetry ̂⋅̆ results in selection rules on nx/y (klhj) to all orders in , , ℎ and , and thus provides a non-perturbative formula for the scaling of the HHG spectrum, which are laid out in Table 1 . For and expanded discussion on the derivation and application of ̂⋅̆ and its selection rules, see SI, section III. Similar rules were derived for the ATI spectrum (see SI, section II). We further emphasize that the only necessary condition for this construction is that the system exhibits a broken-symmetry, and that Eq. (1) holds. For example, in section V of the SI we derive ̂⋅̆ for a system whose real-symmetry ̂ is broken by spin-orbit coupling, instead of the second laser (i.e., ̂ is different). In this case, ̆ operates on the synthetic space ( , ) spanned by the Rashba ( ) and Dresselhaus ( ) coupling strengths.

Next, we demonstrate experimentally synthetic symmetries and their corresponding selection rules in a symmetry-broken system by driving HHG with a bi-chromatic field composed of incommensurate frequencies. In our set-up 27 ( Fig.1.(a) ), a bi-chromatic laser beam with frequencies − 1.95 (corresponding to the wavelengths 800nm and 410nm, respectively) is passed through an achromatic zero-order quarter-wave plate (QWP). The rotation angle θ of the QWP controls the ellipticities (and the DSs) of the driving field, which is explicitly given by

The bi-chromatic beam is focused onto a supersonic jet of argon gas, yielding an intensity of 2 × 10 14 W cm 2 ⁄ at the focus, where 10% of the intensity is in the redshifted SH driver. The measured HHG spectrum ( Fig.1. (b)) exhibits two types of selection rules, imposed by standard and synthetic DSs.

Firstly, for = 0° and = 45°, the driving field exhibits standard HHG selection rules in the form of forbidden harmonics, due to the spatio-temporal DSs ̂′ =̂2 ′ ⋅̂ and ̂5 9,20 ′ =̂2 0 ′ ⋅ 59,20 , respectively. Here, ̂′ is a ′ ⁄ time translation where ′ = 20 , ̂ is a reflection relative to the Cartesian basis vector ̂, and ̂5 9,20 is a 20 × 2 59 ⁄ spatial rotation. Hence 8 , even harmonic generation is forbidden for = 0° and 3 harmonic generation is forbidden for = 45° ( Fig.1.(b) ). As is detuned from 0° and 45°, the DSs ̂′ and ̂′ 59,20 are broken, and instead, can be thought of as arising due to the changing ellipticity of the original beams when the QWP is rotatedany deviation from the original Floquet Hamiltonian can be assigned to a perturbative term). By plugging in either ( = 2 , = 0, = 39) or ( = 0, = 2 , = 40) into Table 1 and employing ′ as the fundamental frequency, we identify that the system exhibits ̂′ ⋅̆ symmetry where ̆( ) = − (Fig.1.(c) ). The corresponding selection rules forbids the amplitudes of harmonic H18.8 and H20.7 to have a contribution proportional to = 2 in their scaling, hence, they scale quadratically, as 2 . Similarly, ̂′ ⋅̆ forbids H19.75 to have a contribution proportional to 2 ∝ 2 , hence it scales linearly with . See section VI of the SI for an explicit derivation of these predictions using Table 1 . To compare the scaling of each harmonic amplitude to the analytical predictions, we fitted it to a general polynomial model ( Fig.1(e-g) , yellow shaded regions). The ratio between the coefficients of the polynomial fit indicates whether the emission scales linearly or quadratically, e.g., in Fig.1.(g) , H19.75 scales as 9.5(θ − 0.1) + 0.05(θ − 0.1) 2 in the yellow-shaded region, in accordance with the analytically derived selection rule that forbids quadratic (∝ 2 ) contributions (because the coefficient for 2 is ~200 times weaker than the coefficient). Additionally, it was verified that when fitting the scaling of each harmonic amplitude to its specific prediction, i.e., to a linear or quadratic powerlaw, all fits result in R 2 >0.95.

As θ is detuned away from 45°, we probe the breaking of Ĉ ′ 59,20 DS (to 1 st order in θ) by two linearly polarized fields sin(20ω ′ t)̂ and cos(39ω ′ t) √10 ⁄̂ with amplitude 2(θ°− 45°). As a result, the ̂5 9,20 ′ is broken, and synthetic symmetries of the form ̂5 9,20 ′ ⋅̆ are imposed ( Fig.1 . 4 ). These predictions ( Fig.1. (e) ) are clearly observed in the experimental measurement (Fig.3(f-h) , blue shaded regions).

To summarize, we have demonstrated that systems that are traditionally regarded as symmetrybroken, can systematically exhibit a new class of symmetries and selection rules through synthetic dimensions. These symmetries and selection rules determine how the system's observables scale as the system transitions out of its original symmetric state, showing the role of the broken symmetries in the dynamics of the symmetry-broken systems. We have tabulated these symmetries for periodically driven Floquet systems subject to two driving fields, and derived the corresponding selection rules for HHG, ATI and more (see section VII in the SI). We observed experimentally selection rules rooted in synthetic dimensions by driving HHG with a bi-chromatic laser field that consists of incommensurate frequencies. We highlight that our theory is a non-perturbative theory, that applies to all orders of the perturbation's strength. We further emphasize that real-synthetic symmetries and their associated selection rules are general concepts, relevant to all systems with a broken symmetry in real space. For example, one (or both) of the lasers we have employed in our construction, may be replaced by a different periodically oscillating (or static) element of the system (either extrinsic or intrinsic), e.g., spin orbit coupling strengths 28 (see section V in the SI) or lattice excitations 25, 26, 29 . Specifically, by reformulating them as effective gauge fields, the derived symmetries and selection rules ( Table   1 ) can be directly applied to dynamical symmetry breaking by phonons and magnons, opening new opportunities for all-optical time-resolved spectroscopy (and control) of their dynamics.

Overall, the presented approach provides a unified framework for the analysis of symmetrybroken systems, complementary to perturbation theory, hence we expect it to be used throughout science and engineering. in the x and y amplitude components of the symmetry breaking part of the field, i.e., they are proportional to ( − 45°).(e) summary of analytical predictions for the scaling of each spectral component (f-h) The scaling of harmonics 18.8,19 .75 and 20.7. The scaling of each harmonic amplitude was fit to a general polynomial model within the yellow & blue shaded regions, and the resulting polynomials appear in the corresponding color above each subfigure. The ratio between the coefficients of the polynomial indicates whether the emission scales linearly or quadratically.

The data supporting the findings of this study are available from the corresponding author upon reasonable request. 

Group Theory and Chemistry

Optical absorption properties of laser-driven matter

Dynamical Symmetries and Symmetry-Protected Selection Rules in Periodically Driven Quantum Systems

The effect of Hamiltonian symmetry on generation of odd and even harmonics

Selection rules for the high harmonic generation spectra

Floquet group theory and its application to selection rules in harmonic generation

Interference carpets in above-threshold ionization: From the coulomb-free to the coulomb-dominated regime

Generation and control of high-order harmonics by the interaction of an infrared laser with a thin graphite layer

Spin angular momentum and tunable polarization in high-harmonic generation

Generation of bright phase-matched circularlypolarized extreme ultraviolet high harmonics

Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism

Observation of selection rules for circularly polarized fields in high-harmonic generation from a crystalline solid

Ionization and high-order harmonic generation in aligned benzene by a short intense circularly polarized laser pulse

Bicircular High-Harmonic Spectroscopy Reveals Dynamical Symmetries of Atoms and Molecules

Probing polar molecules with high harmonic spectroscopy

Ultrasensitive Chiral Spectroscopy by Dynamical Symmetry Breaking in High Harmonic Generation

Synthetic chiral light for efficient control of chiral light-matter interaction

Chiral high-harmonic generation and spectroscopy on solid surfaces using polarization-tailored strong fields

Background-Free Measurement of Ring Currents by Symmetry-Breaking High-Harmonic Spectroscopy

Measurement of the Berry curvature of solids using high-harmonic spectroscopy

Topological strong-field physics on sub-laser-cycle timescale

Dimensional crossover in a layered ferromagnet detected by spin correlation driven distortions

Phonon Driven Floquet Matter

Floquet analysis of excitations in materials

In-line production of a bi-circular field for generation of helically polarized high-order harmonics

All authors made substantial contributions to all aspects of the work.

The authors declare no competing interest