# Probing phonon-driven symmetry alterations in graphene via high-harmonic
spectroscopy

Navdeep Rana  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076 India  Gopal Dixit<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076 India

###### Abstract

High-harmonic spectroscopy has become an essential ingredient in probing
various ultrafast electronic processes in solids with sub-cycle temporal
resolution. Despite its immense importance, sensitivity of high-harmonic
spectroscopy to phonon dynamics in solids is not well known. This work
addresses this critical question and demonstrates the potential of high-
harmonic spectroscopy in probing intertwined phonon-electron dynamics in
solids. A pump pulse excites in-plane optical phonon modes in monolayer
graphene and a circularly polarised pulse is employed to probe the excited
phonon dynamics that generates higher-order harmonics. We show that the
coherent phonon dynamics alters the dynamical symmetry of graphene with the
probe pulse and leads the generations of the symmetry-forbidden harmonics.
Moreover, sidebands associated with the prominent harmonic peaks are generated
as a result of the coherent dynamics. It is found that the symmetries and the
characteristic timescale of the excited phonon mode determine the polarisation
and positions of these sidebands. Present work opens an avenue in time-
resolved probing of phonon-driven processes and dynamical symmetries in solids
with sub-cycle temporal resolution.

##  I Introduction

Vibrations of atoms within molecules and solids are fundamental processes that
regulate several physical, optical and chemical properties of matter. When
light triggers atomic vibrations, atoms exhibit periodic oscillations in a
particular fashion and these light-induced vibrations could potentially lead
to the modifications in various symmetries of solids. These modifications are
dynamic in nature and result in several transient phenomena, such as light-
induced superconductivity Hu  _et al._ (2014); Mitrano  _et al._ (2016),
vibrationally-induced magnetism Rini  _et al._ (2007); Nova  _et al._ (2017),
and switching of electrical polarisation Mankowsky  _et al._ (2017), to name
but a few. Thus, time-resolved mapping of the interplay of lattice vibration
with electronic motion on electronic timescale is essential to comprehend
several ubiquitous phenomena in solids, such as structural phase transition
Bansal  _et al._ (2020); Hase  _et al._ (2015), thermal Niedziela  _et al._
(2019); Bansal  _et al._ (2018) and optical properties Katsuki  _et al._
(2013); Fultz (2010); Gambetta  _et al._ (2006); and predicting new concepts
in solids. Several spectroscopy and imaging-based methods are employed to
probe lattice vibrations in solids Dhar  _et al._ (1994); Debnath  _et al._
(2021); Graf  _et al._ (2007); Virga  _et al._ (2019); Koivistoinen  _et al._
(2017); Rana  _et al._ (2021); Brown  _et al._ (2019); Flannigan (2018).
However, probing transiently-evolving intertwined lattice-electronic dynamics
and dynamical symmetries of solids in the presence of light in a single
experimental setup are challenging. Present work addresses this crucial
problem.

High-harmonic generation (HHG) is a non-perturbative nonlinear frequency up-
conversion process and is sensitive to the sub-cycle electron dynamics driven
by intense laser. Over the last decade, high-harmonic spectroscopy became an
emerging method to interrogate various equilibrium and non-equilibrium
properties of solids by investigating the emitted spectrum during HHG Luu  _et
al._ (2015); Schubert  _et al._ (2014); Mrudul  _et al._ (2021); Mrudul and
Dixit (2021a); Hohenleutner  _et al._ (2015); Zaks  _et al._ (2012);
Pattanayak  _et al._ (2020); Langer  _et al._ (2018); Mrudul  _et al._ (2020);
Luu and Wörner (2018); Banks  _et al._ (2017); Mrudul  _et al._ (2019); Bai
_et al._ (2020); Imai  _et al._ (2020); Borsch  _et al._ (2020). In spite of
the tremendous applications of high-harmonic spectroscopy, impact of lattice
vibration on HHG from solids remains uncharted territory, except recent work
Ginsberg  _et al._ (2021); Neufeld  _et al._ (2022). Present work focuses to
highlight the abilities of high-harmonic spectroscopy in time-resolved mapping
of the interplay of coherent lattice vibrations and electronic motions; and
transiently-evolving symmetries of solids during the dynamics.

In the following, we will demonstrate that the high-harmonic spectroscopy of
the coherent lattice dynamics leads to the generations of the higher-order
sidebands along with the main harmonic peaks in the high-harmonic spectrum.
The frequency and symmetry of the coherently excited phonon mode are imprinted
in the position and polarisation of the sidebands, respectively. Moreover,
symmetry-forbidden harmonics are allowed due to the symmetry alterations
caused by the coherent lattice dynamics.

To illustrates the sensitivity of the coherent phonon dynamics to high-
harmonic spectroscopy, two-dimensional graphene with
$\textbf{D}_{6\textrm{h}}$ point group symmetry is chosen. The phonon spectrum
of graphene is consist of three acoustic and three optical phonon branches.
Out of the three optical phonon modes, one phonon mode is out-of-plane where
the vibrations are out of the two-dimensional plane of graphene, and other two
are in-plane modes in which lattice vibrations are confined within the plane
of graphene Kim  _et al._ (2013). In-plane optical phonon modes are considered
in this work. At the $\mathsf{\Gamma}$ point, there are two degenerate in-
plane optical modes with phonon frequency of 194 meV, which corresponds to
phonon oscillation period of $\sim$ 21 femtoseconds Kim  _et al._ (2013).
These phonon modes are represented as E${}_{2\textrm{g}}$ or $\mathsf{G}$
modes and are Raman active. It is known that light can couple to a phonon mode
at the Brillouin zone centre and therefore E${}_{2\textrm{g}}$ Raman-active
mode can be excited by stimulated Raman excitation either via a broad pulse
that covers a bandwidth of 194 meV or via using two laser pulses with a
difference of 194 meV in photon energy. By tuning the polarisation of the
phonon exciting pulse either along $\mathsf{\Gamma-K}$ or $\mathsf{\Gamma-M}$
direction, one of the two in-plane phonon modes can be excited selectively.

##  II Computational Methods

To incorporate coherent phonon dynamics, nearest-neighbour tight-binding
Hamiltonian is extended from static to time domain as

$\mathcal{\hat{H}}_{\textbf{k}}(t)=-\gamma(t)\sum_{i\in
nn}e^{i\textbf{k}\cdot\textbf{d}_{i}(t)}\hat{a}_{\textbf{k}}^{\dagger}\hat{b}_{\textbf{k}}+\textrm{H.
c.}$ (1)

Here, $\gamma(t)$ = $\gamma_{0}~{}e^{-(|\textbf{d}_{i}(t)|-a)/\delta}$ is the
time-dependent nearest-neighbour hopping energy, which is modelled to capture
the temporal variations in the relative distance between nearest-neighbour
atoms ($\textbf{d}_{i}=a=1.42$ Å) Mohanty and Heller (2019). $\gamma_{0}=$ 2.7
eV is the nearest-neighbour hopping energy and $\delta$ = 0.184a0 is the width
of the decay function with a0 = 2.46 Å as the lattice parameter of the
equilibrium structure Moon and Koshino (2013).

Semiconductor-Bloch equations corresponding to time-dependent Hamiltonian
given in Eq. (1) are solved as

$\displaystyle\frac{d}{dt}\rho_{cv}^{\textbf{k}}$
$\displaystyle=\left[-i\varepsilon_{cv}(\textbf{k}_{t},t)+\frac{1}{\textrm{T}_{2}}\right]\rho_{cv}^{\textbf{k}}+i\textbf{E}(t)\cdot\textbf{d}_{cv}(\textbf{k}_{t},t)\left[\rho_{vv}^{\textbf{k}}-\rho_{cc}^{\textbf{k}}\right]~{}~{}\textrm{and}$
(2a) $\displaystyle\frac{d}{dt}\rho_{vv}^{\textbf{k}}$
$\displaystyle=i\textbf{E}(t)\cdot\textbf{d}_{vc}(\textbf{k}_{t},t)\rho_{cv}^{\textbf{k}}+\textrm{c.c.}$
(2b)

Here, the vector potential and the electric field corresponding to the driving
laser field are represented as $\textbf{A}(t)$ and $\textbf{E}(t)$,
respectively, and are related as $\textbf{E}(t)$ = $-d\textbf{A}(t)/dt$.
$\textbf{d}_{cv}(\textbf{k})$ and $\varepsilon_{cv}(\textbf{k})$ are,
respectively, the dipole matrix elements and the band-gap energy between
conduction and valence bands at given k with
$\textbf{d}_{cv}(\textbf{k})=i\langle
c,\textbf{k}|\nabla_{\textbf{k}}|v,\textbf{k}\rangle$. The matrix elements
become time-dependent due to coherent phonon dynamics and are updated smoothly
at each consecutive time-steps during the phonon dynamics. To account the
interband decoherence, a phenomenological term with a constant dephasing time
$\textrm{T}_{2}$ is introduced. Here,
$\textbf{k}_{t}=\textbf{k}+\textbf{A}(t)$.

High-harmonic spectrum is calculated as

$\mathcal{I}(\omega)=\left|\mathcal{FT}\left(\dfrac{\rm{d}}{{\rm
d}t}\textbf{J}(t)\right)\right|^{2},$ (3)

where $\mathcal{FT}$ stands for the Fourier transform. The total current J(t)
is calculated by integrating $\textbf{J}(\textbf{k},t)$ over the entire
Brillouin zone as

$\textbf{J}(\textbf{k},t)=\sum_{m,n\in\\{c,v\\}}\rho_{mn}^{\textbf{k}}(t)\textbf{p}_{nm}(\textbf{k}_{t},t),$
(4)

where $\textbf{p}_{nm}(\textbf{k})=\langle
n,\textbf{k}|\nabla_{\textbf{{k}}}\mathcal{\hat{H}}_{\textbf{k}}|m,\textbf{k}\rangle$
is momentum matrix-element. We have adopted the same methodology for HHG from
a solid with phonon dynamics as given in our earlier work Rana  _et al._
(2022). The ellipticity and the phases of harmonics are estimated by following
the recipe given in Ref. Tancogne-Dejean  _et al._ (2017).

It is assumed that a pump-pulse initiate the coherent excitation of an in-
plane Raman active phonon mode in graphene. The excited phonon mode is probed
by the high-harmonic generating pulse. The excitation of the phonon mode is
approximated by direct initiation of the coherent vibrations of carbon atoms
within adiabatic approximation. Circularly-polarised pulse with a wavelength
of 2.0 $\mu$m and peak intensity of 1$\times 10^{11}$ W/cm2 is used to
generate high-harmonics in graphene with and without coherent phonon dynamics.
The pulse duration is 100 fs, which is much longer than an oscillation period
of in-plane phonon dynamics $\simeq$ 21 fs. Moreover, the coherence times of
these phonon modes of around 1 ps are reported Jeong  _et al._ (2015); Hubener
_et al._ (2018). Parameters of the harmonic generating laser are similar to
the ones used earlier for probing electron dynamics in graphene Heide  _et
al._ (2018); Yoshikawa  _et al._ (2017).

##  III Results and Discussion

High-harmonic spectrum corresponding to monolayer graphene without phonon
dynamics is presented in Fig. 1(a). We have employed left-handed circularly
polarised laser pulse for HHG in graphene with and without phonon dynamics. As
dictated by the symmetry constraints and selections rules, it is expected that
circularly polarised pulse yields $(6m\pm 1)$-orders of harmonics from an
inversion-symmetric graphene with the six-fold symmetry Alon  _et al._ (1998);
Chen and Qin (2019). Here, $m=0,1,2,\ldots$ is a positive integer. In this
case, third harmonic is symmetry forbidden. On the other hand, linearly
polarised laser pulse leads to $(2m+1)$-orders of harmonics as shown earlier
Mrudul and Dixit (2021b). Our results shown in Fig. 1(a) are consistent with
the selection rules and earlier report Alon  _et al._ (1998); Chen and Qin
(2019). Graphene is not chiral in nature, so left- and right-handed circular
laser pulses yield same harmonic spectra.

Figure 1: High-harmonic spectra, generated by left-handed circularly polarised
laser pulse, of two-dimensional graphene with and without coherent phonon
dynamics. (a) The spectra of the graphene without phonon dynamics. The spectra
of graphene with the coherent in-plane (b) longitudinal optical (iLO) phonon
mode and (c) transverse optical (iTO) phonon mode. In all the spectra,
sidebands corresponding to the prominent harmonic peaks are identified at
frequencies ($\omega_{0}\pm n\omega_{\textrm{ph}}$) with
$\omega_{\textrm{ph}}$ as the phonon frequency, $\omega_{0}$ as the frequency
of the probe laser pulse and $n$ is an integer. The unit cell of the graphene
with the eigenvector of a particular phonon mode and polarisation of the
harmonic generating probe pulse are shown in the respective insets. Results
are presented for $\textrm{T}_{2}=$ 10 fs and a maximum 0.03a0 displacement of
atoms from their equilibrium positions during coherent phonon dynamics where
a0 is the lattice parameter of the equilibrium structure. Our findings remain
qualitatively same for $\textrm{T}_{2}$ ranging from 5 to 30 fs and for
displacements ranging from 0.01a0 to 0.05a0 with respect to the equilibrium
positions of the carbon atoms.

After discussing HHG from graphene without phonon dynamics, let us investigate
how the in-plane phonon modes affect the harmonic spectrum shown in Fig. 1(a).
For this purpose, we coherently excite one of the two degenerate in-plane
phonon modes and assume that the excitation is done prior to the probe
harmonic pulse. Figure 1(b) presents the harmonic spectrum corresponding to
coherently excited longitudinal optical (iLO) phonon mode. The spectrum in
Fig. 1(b) is drastically different from the one without phonon dynamics [see
Fig. 1(a)]. There are only odd harmonics in the spectrum as the iLO phonon
mode preserves the inversion symmetry in graphene Rana  _et al._ (2022).
Moreover, the spectrum exhibits multiple sidebands along with the main odd
harmonics as evident from Fig. 1(b). The coherent excitation of the in-plane
transverse optical (iTO) phonon mode also leads to multiple sidebands along
with the odd harmonics as visible from Fig. 1(c). In both cases, the energy
separation between the successive sidebands is equal to the energy of the
excited phonon (iLO or iTO) mode, i.e., 194 meV. Therefore, the energy of the
excited phonon mode is encoded in the spectra. Apparently, it seems that the
spectra is insensitive to the symmetry of the excited phonon mode as both iLO
and iTO phonon modes yields similar harmonic spectra [see Figs. 1(b) and
1(c)]. In the following, we will show that this is not the case and the
symmetry of the excited phonon mode is encoded in the polarisation properties
of the spectra.

Not only coherent phonon dynamics leads to the generation of the multiple
sidebands but also the forbidden harmonics become allowed. As stated earlier,
third harmonic is absent for the circular laser driven HHG from graphene
without phonon [see Fig. 1(a)]. However, dynamics of the coherent
E${}_{2\textrm{g}}$ phonon mode reduces graphene’s six-fold symmetry into two-
fold dynamically, which allows the generation of $(2m\pm 1)$ harmonic orders.
The presence of the third harmonic in both cases, graphene with (iLO or iTO)
phonon mode, is a signature of the symmetry reduction as evident from Fig.
1(b) and 1(c). At a glance, it seems that the criteria for HHG is same for
linearly polarised laser pulse [$(2m+1)$ orders and third harmonic] and the
combination of the phonon-driven symmetry reduction with circularly polarised
laser pulse [$(2m\pm 1)$ orders and third harmonic]. To distinguish the two
situations, let us analyse the polarisation properties of the emitted
harmonics.

Figure 2 displays the time-domain representations of the $x$ and $y$
components of the first and fifth harmonics of the spectrum in Fig. 1(a). It
is known that the polarisation of a given harmonic for a material with
$l$-fold symmetry is determined by $lm+\sigma$, where $\sigma=+(-)1$
represents the $m^{\textrm{th}}$ harmonic’s polarisation, which is same
(opposite) as the helicity of the driving laser pulse Alon  _et al._ (1998).
It is straightforward to see that $(lm-1)^{\textrm{th}}$ and
$(lm+1)^{\textrm{th}}$ harmonics are circularly polarised with $\sigma=-1$ and
$\sigma=1$, respectively. In present case, the first and fifth harmonics are
circularly polarised with opposite helicity and is consistent with earlier
findings Saito  _et al._ (2017); Chen and Qin (2019).

Figure 2: Lissajous figure (in cyan), total (in red), $x$ (in blue) and $y$
(in green) components of the first and the fifth harmonics in time-domain
corresponding to the spectrum shown in Fig. 1(a) for graphene without phonon
dynamics.

As stated above, $(2m\pm 1)$ harmonic orders are allowed due to phonon-driven
dynamical symmetry reduction from six-fold to two-fold, which leads to the
generation of the third harmonic. Moreover, this dynamical symmetry reduction
also alters the polarisation properties of the emitted harmonics. The $x$ and
$y$ components of the first, third, and fifth harmonics in time domain for
graphene with iLO and iTO phonon modes are presented in Figs. 3(a) and 3(b),
respectively. As evident from the figure, the ellipticity of the first
harmonic reduces drastically from 1 for graphene without phonon to 0.63 for
graphene with phonon. The change in the ellipticity can be understood as
follow: When iLO phonon mode is excited, carbon atoms vibrate along the X
direction, which increases the velocity of the electrons in the X direction.
It is known that the intraband current is proportional to the velocity, and
low-order harmonics in graphene are dominated by the intraband current Mrudul
and Dixit (2021b); Vampa  _et al._ (2015). Thus, the major axis of the ellipse
is along the X direction in the case of the first harmonic, which reduces the
ellipticity from 1 to 0.63.

Figure 3: Same as Fig. 2 for the first, third and fifth harmonics in time-
domain for graphene with (a) iLO and (b) iTO phonon modes. In the case of iLO
(iTO) phonon mode, the ellipticities of the first, third and fifth harmonics
are 0.63 (0.62), 0.77 (0.79), and 0.93 (0.96), respectively. Also, the phase
differences between $x$ and $y$ components of the first, third and fifth
harmonics are 90∘ (90∘), 40∘ (35∘), and 125∘ (45∘), respectively.

Similarly, the excitation of the iTO mode leads the vibrations of the atoms
along Y direction. This provides an additional velocity component to electrons
in the Y direction, which translate to the major axis of the ellipse along the
Y direction. The ellipticity of the fifth harmonic, corresponding to graphene
without phonon to with iLO (iTO) phonon mode, changes significantly, i.e.,
from 1 to 0.93 (0.96). Not only the phonon dynamics modifies the ellipticity
of the harmonics significantly but also changes the phase between the $x$ and
$y$ components of the harmonics. In the case of iLO (iTO) mode, the phase
differences between the components for first and fifth harmonics are 90∘ (90∘)
and 125∘ (45∘), respectively (see Fig. 3). The ellipticity and the phase
difference of the third harmonic for graphene with iLO (iTO) mode is 0.77
(0.79), and 40∘ (35∘), respectively. Thus, the changes in the ellipticity and
phase indicate that the harmonics are sensitive to the symmetry of the excited
phonon mode as reflected from Fig. 3.

After demonstrating how the information of the excited phonon mode and its
symmetry are imprinted in the main harmonics and their polarisation
properties, let us analyse what informations are encoded in the sidebands
associated with prominent harmonics. Time-domain picture of the $x$ and $y$
components of the first, second and third sidebands corresponding to the first
harmonic corresponding to graphene with iLO phonon mode is shown in Fig. 4(a).
All three sidebands have nonzero $x$ and $y$ components as evident from the
figure. The same is true for the sidebands associated with iTO phonon
excitation as reflected from Fig. 4(b). Moreover, the ellipticities of the
first, second and third sidebands of the first harmonic corresponding to
graphene with iLO (iTO) phonon mode read as 0.98 (0.56), 0.72 (0.95), and 0.59
(0.67), respectively. Also, the phases between the $x$ and $y$ components of
the first, second and third sidebands of the iLO (iTO) mode are 85∘ (90∘), 70∘
(135∘), and 60∘ (75∘), respectively. Thus, the analysis of Fig. 4 establishes
that the polarisation and the phase properties of the sidebands are different
for different phonon mode. However, it is not obvious why the sidebands have
nonzero $x$ and $y$ components, whereas a particular phonon mode (iLO or iTO)
induces atomic vibrations along a particular direction (X or Y).

Figure 4: Same as Fig. 2 for the first, second and third sidebands in time-
domain associated with the first harmonic for graphene with (a) iLO phonon
mode corresponding to Fig. 1(b), and (b) iTO phonon mode corresponding to Fig.
1(c). The ellipticities of the first, second and third sidebands corresponding
to iLO (iTO) phonon mode are 0.98 (0.56), 0.72 (0.95), and 0.59 (0.67),
respectively. The phase differences between $x$ and $y$ components of the
first, second and third sidebands corresponding to iLO (iTO) phonon mode are
85∘ (90∘), 70∘ (135∘), and 60∘ (75∘), respectively.

To know the origin of the nonzero $x$ and $y$ components of the sidebands, we
employe Floquet formalism to graphene with coherently excited phonon mode and
circularly polarised probe pulse. Floquet formalism determines dynamical
(spatio-temporal) symmetries (DSs) of the system, which dictate selection
rules for the sidebands. The $n^{\rm th}$-order sideband obeys the symmetry
constraint as
$\hat{X}^{t}\textbf{E}_{s,n}(t)[\hat{X}^{t}\textbf{E}(t)]^{\dagger}$ =
$\textbf{E}_{s,n}\textbf{E}^{\dagger}(t)$ with the condition that the spatial
symmetries of $\hat{X}^{t}$ and the probe pulse are same Nagai  _et al._
(2020). Here, $\hat{X}^{t}$ is a DS, and the electric fields associated with
$n^{\rm th}$-order sideband and the probe laser are represented by
$\textbf{E}_{s,n}(t)$ and $\textbf{E}(t)$, respectively. The Raman tensor,
denoted as $\mathcal{R}_{n}(t)=\textbf{E}_{s,n}(t)\textbf{E}(t)^{\dagger}$,
has to be invariance under the operation of the DSs Nagai  _et al._ (2020). In
the following, we will follow the same treatment as given in Refs. Nagai  _et
al._ (2020); Rana  _et al._ (2022) to investigate the properties of the
sidebands using DSs within Floquet formalism.

In the present case, $\mathcal{D}_{1}=\hat{\sigma}_{y}\cdot\mathcal{T}$ is the
DS, which leaves the system with iLO phonon mode invariant. Here,
$\hat{\sigma}_{y}$ is the reflection with respect to $y$-axis, and
$\hat{\mathcal{T}}$ is the time-reversal operator. Thus, the condition
associated with the sidebands is determined as
$\hat{\mathcal{D}_{1}}\mathcal{R}_{n}(t)=\mathcal{R}_{n}(t)$ with

$\mathcal{R}_{n}(t)=\mathbf{E}_{s,n}(t)\mathbf{E}(t)^{\dagger}=\begin{bmatrix}E_{s,n_{x}}E^{*}_{x}&E_{s,n_{x}}E^{*}_{y}\\\
E_{s,n_{y}}E^{*}_{x}&E_{s,n_{y}}E^{*}_{y}\end{bmatrix}.$ (5)

Let us substitute the expression of
$\mathbf{E}^{\dagger}(t)=[\cos(\omega_{0}t)E^{\dagger}_{x}~{}~{}\sin(\omega_{0}t)E^{\dagger}_{y}]$
with $\omega_{0}$ is the frequency of the probe pulse in the above equation,
the expression of the Raman tensor reads as

$\mathcal{R}_{n}(t)=\sin[(n\omega_{\textrm{ph}}+\omega_{0})t]\begin{bmatrix}\cos(\omega_{0}t)E_{s,n_{x}}\\\
\sin(\omega_{0}t)E_{s,n_{y}}\end{bmatrix}.$ (6)

Here, $\omega_{\textrm{ph}}$ is the frequency of the phonon mode. On operating
$\mathcal{D}_{1}$ on $\mathcal{R}_{n}(t)$ leads the following expression of
the invariant $\mathcal{R}_{n}(t)$ as

$\sin[(m\omega_{\textrm{ph}}+\omega_{0})t]\begin{bmatrix}\cos(\omega_{0}t)E_{s,n_{x}}\\\
\sin(\omega_{0}t)E_{s,n_{y}}\end{bmatrix}=\sin[-(n\omega_{\textrm{ph}}+\omega_{0})t]\begin{bmatrix}-\cos(\omega_{0}t)E_{s,n_{x}}\\\
-\sin(\omega_{0}t)E_{s,n_{y}}\end{bmatrix}.$ (7)

Now it is straightforward to notice that all the sidebands exhibit nonzero $x$
and $y$ components. The ellipticity and the phase difference are estimated
from the nonzero components and our numerical results shown in Fig. 4 are
consistent with the present analysis. If linearly polarised probe pulse is
used instead of the circularly polarised pulse, the odd- and even-order
sidebands are, respectively, polarised perpendicular and parallel to the
polarisation of probe pulse in the case of graphene with iLO phonon mode. On
the other hand, iTO phonon mode leads all sidebands polarised along the
direction of the probe pulse Rana  _et al._ (2022).

Our findings are drastically different from the recent reported work where HHG
is employed to probe phonon dynamics in hexagonal boron nitride (h-BN). In
Ref. Ginsberg  _et al._ (2021), it has been reported that the phonon dynamics
leads the generation of the forbidden second harmonic and enhancement of the
third harmonic in few-layer h-BN. Moreover, coherent phonon in h-BN leads to
the attenuation of the harmonic spectrum with no discrete harmonics Neufeld
_et al._ (2022). HHG becomes sensitive to the carrier-envelope phase of the
probe pulse when the pulse duration and the period of the excited phonon are
similar Neufeld  _et al._ (2022).

##  IV Conclusion

In conclusion, we have explored the potential of high-harmonic spectroscopy in
probing intertwined coherent-phonon electron dynamics in solids. To this end,
coherent excitation of both in-plane iLO and iTO Raman-active phonon modes in
graphene are considered. The six-fold symmetry of the graphene reduces to two-
fold dynamically due to the coherent phonon excitation. As a result of this
symmetry alteration, symmetry-forbidden third harmonic of circularly polarised
probe pulse is generated. Moreover, coherent phonon leads to the generation of
the sidebands corresponding to the prominent harmonic peaks. Floquet formalism
of the dynamical symmetries of the system is applied to understand the
properties of the sidebands. It is found that the positions of the sidebands
are determined by the energy of the excited phonon modes. Moreover, the
dynamical symmetries of the system, consists of graphene with an excited
phonon mode and probe pulse, determine the polarisation of the sidebands.
Thus, polarisation properties of the sidebands are a sensitive probe of the
dynamical symmetries. Present study could be extended to bilayer graphene
where infrared-active phonon modes can be expressed in terms of double
degenerate in-plane Raman-active phonon modes of monolayer graphene Gierz  _et
al._ (2015); Pomarico  _et al._ (2017). Moreover, this work provides a
platform to study non-linear phononics with sub-cycle temporal resolution.

## Acknowledgements

We acknowledge fruitful discussion with M S Mrudul (Uppsala University), and
Klaus Reimann (MBI Berlin). G. D. acknowledges support from Science and
Engineering Research Board (SERB) India (Project No. MTR/2021/000138).

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# Power based adaptive compensator of output oscillations

Michael Ruderman  University of Agder, Department of Engineering Sciences,
Norway
email<EMAIL_ADDRESS>
###### Abstract

Power-based output feedback compensator for oscillatory systems is proposed.
The average input-output power of an oscillatory signal serves as an
equivalent control effort, while the unknown oscillation’s amplitude and
frequency are detected at each half-period. This makes the compensator
adaptive and discrete, while the measured oscillatory output is the single
available signal in use. The proposed compensator is derived for second-order
systems, while an extension to higher-order dynamics, like e.g. in case of
two-inertia systems, is also provided. An illustrative experimental case study
of the fifth-order oscillatory system is provided.

##  1 Introduction

Controlling the oscillating outputs in various systems is relevant to
different types of the applications. One can find those in e.g. active and
flexible structures, Preumont (2018), robotics with elastic elements, De Luca
and Book (2016), power electronics (like e.g. inverters), Wu et al. (2017),
suspension systems, Tseng and Hrovat (2015), for mentioning here the few. For
a control-based rejection (or at lest attenuation) of the oscillatory
disturbances, an adaptation or online estimation of the dynamic state(s),
usually more than one, can be required, see e.g. De Wit and Praly (2000),
Aranovskiy and Freidovich (2013), and Landau (2020) with references therein.
Often, a robust online estimation, see e.g. Hsu et al. (1999), Bobtsov et al.
(2012), Ruderman (2022), of the corresponding unknown (or uncertain)
oscillation frequency is also required. When an estimation of the required
parameters (correspondingly states) become a part of the compensation scheme
itself, an often nontrivial proof of stability can arise as strictly
necessary. Consequently, the overall oscillations control can be sensitive to
the measured output noise, uncertain system parameters, and phase lag
(correspondingly delay) associated with the possible use of additional
filtering.

Another way of looking at the system, which is motivating for the present
work, is based on energy-, correspondingly power-balancing, see e.g. García-
Canseco et al. (2010). For an input-output system representation, a power-flow
and -shaping represent an easily interpretable and mathematically
comprehensible way to analysis and control synthesis. The basics of the energy
principles in control can be looked in e.g. lecture notes of Ortega et al.
(2001). It is worth emphasizing, at that point, that the oscillatory signals
are particularly straightforward for a power-based representation,
correspondingly calculations.

While most power- and energy-based control methods imply the continuous
control laws and, therefore, assume a one-to-one correspondence (respectively
mapping) between a vector of the system states used and the control variables,
another argument motivates also for the approach proposed in this work. The
goal of compensating for a specific oscillating quantity makes it possible to
significantly reduce the commutation rate of the controller and to accommodate
feedback control logic that will commutate for only fractions of the
oscillation period. This will yield the proposed compensation scheme to some
type of an event-triggered feedback control, see e.g. Heemels et al. (2012)
and references therein. For the related basics on hybrid (continuous and
switching) systems control we also refer to Liberzon (2003), and Lunze and
Lamnabhi-Lagarrigue (2009). Examples of the hybrid event-switching controls
are well known, for instance a bang-bang funnel controller, Liberzon and Trenn
(2013), and event-triggered controller for saturated linear systems, Seuret et
al. (2016).

Against the above background, a novel discrete-valued power-based adaptive
control of the output oscillations is proposed. The main advantage of the
proposed control scheme, over other output feedback based oscillation
compensation approaches, is a largely reduced communication effort in case of
a digitally connected system framework, i.e. remote location of the sensing
and control (or actuating) units. In fact, any continuous feedback control
requires a communication effort of $\>2\pi\omega^{-1}f_{s}\times const\>$ per
oscillations period, where $\omega$ is the oscillation’s frequency and (most
importantly) $f_{s}$ is the digital sampling frequency. On the contrary, the
proposed feedback compensation scheme has a communication effort of $\,2\times
const$, since assigning the updated control values only twice per oscillations
period. Obviously, for some large $f_{s}$ values (like in kHz range or even
larger) such reduction of the communication effort can be beneficial in
multiple regards.

The rest of the paper is organized as follows. The main results are given in
section 2. First, the power-based control is developed for second-order
systems using the input-output power balance of oscillatory signals. Then, the
proposed control is extended for higher-order systems assuming the
compensating signal’s propagation through the known forward dynamics. An
extrema detection algorithm, which constitutes an inherent part of the
proposed control, is also given in detail. An experimental case study of
oscillations compensation in the fifth-order system, with two-masses connected
by a low-damped spring, is provided in section 3. The case of an additional
excitation of the oscillating behavior by the external (mechanical)
disturbances is also shown, in favor of robustness of the control scheme.
Short summary is given in section 4.

##  2 Main results

###  2.1 Power-based compensator

Consider next the class of second-order systems with the measurable
oscillatory output

$y(t)=A\sin(\omega t+\varphi)+\Psi.$ (1)

Assume the oscillations amplitude and frequency $A,\omega>0$ are uncertain,
and only the frequency upper bound $\Omega_{\max}>\omega$ is known. The phase
shift $\varphi$ is insignificant, as shown later, especially since the
formulated power-based control operates on every period and $y(t)$ is
available. A non-oscillatory term $\Psi$ is assumed to be known and
$|d\Psi/dt|\ll A\omega$, i.e. the oscillation dynamics in $\dot{y}(t)$
dominates over the dynamics of $\Psi$. Moreover, the oscillations amplitude
itself can be a slow-varying process; we notice that ’slow’ here is also in
comparison with the oscillations dynamics i.e. $|dA/dt|<<A\omega$. Therefore,
in the following developments, $A$ will be considered as a ’frozen’ (to say
quasi-constant) process parameter, at least over one period, while the
resulted control behavior yields effective for a slowly changing $A(t)$ as
well. The system with output (1) has a structure depicted in Fig. 1, while the
control input $u(t)$ is available for compensating the oscillations in $y(t)$.
Note that the driving input $f(t)$ is unknown here.

Figure 1: Signals flow in 2nd order system with output (1).

Differentiating twice the signal (1) we obtain

$\ddot{y}(t)=-A^{2}\omega^{2}\sin(\omega t+\varphi)+\ddot{\Psi}.$ (2)

Denoting the oscillatory part of (1) by $\tilde{y}$, we obtain the
corresponding part of the (instantaneous) power flow as

$\tilde{P}(t)=\ddot{\tilde{y}}(t)\,\tilde{y}(t)=-A^{2}\omega^{2}\sin(\omega
t+\varphi)^{2}.$ (3)

It is obvious that for one period, equally as for an average with
$t\rightarrow\infty$, the input-output power flow yields

$P=-\frac{1}{2}A^{2}\omega^{2}.$ (4)

In order to compensate for (4), the corresponding control power $P^{*}=-P$ is
required, and an equivalent control effort (per one period) can be obtained as

$U\propto\frac{|P|}{A}=\frac{1}{2}\,\omega^{2}A.$

When using an oscillations-suppressing control $u(t)$, the corresponding
amplitude $A$ and so the input-output power (4) reduces. Thus, the control
power $P^{*}(\tilde{A},\tilde{\omega})$ needs an instantaneous amplitude
$\tilde{A}$ to be estimated, but also an instantaneous frequency estimate
$\tilde{\omega}$; the latter due to $\omega$ is assumed uncertain. Since both
are detectable between two successive extrema in $y(t)$, an estimate
(correspondingly update) of the $(\tilde{A},\tilde{\omega})$ pair appears
twice per period.

Denoting by $t^{*}$ the time instant of the last extrema, i.e. either minimum
of maximum of the oscillating $y(t)$, the proposed power-based control has the
form

$u(t)=K\,\tilde{\omega}^{2}\tilde{A}(t^{*})$ (5)

with

$\tilde{A}(t^{*})\equiv\tilde{A}\,\mathrm{sign}\bigl{(}y(t^{*})-\Psi(t^{*})\bigr{)}.$

We next determine an optimal (over one period) gain $K>0$, while emphasizing
that the control (5) keeps a constant value between two successive extrema.

While the input-output power of the oscillatory output, which has to be
compensated, is given by (3), the input-output power of a constant input $u=U$
(if zeroing the oscillations-driving input $f=0$) is given by

$P^{*}=U\iint Udt=\frac{1}{2}\,U^{2}t^{2}.$ (6)

Integrating (3) and (6) over one period, we obtain the corresponding energies
that can be balanced as

$\int\limits_{0}^{2\pi/\omega}\bigl{(}-A^{2}\omega^{2}\sin(\omega
t+\varphi)^{2}\bigr{)}dt+\frac{1}{2}\int\limits_{0}^{2\pi/\omega}U^{2}t^{2}dt=0.$
(7)

Solving (7) with respect to $U$ we obtain

$U=\frac{\sqrt{3}}{2\pi}\,\omega^{2}A,$

cf. with the control (5). Following to that, an optimal gain for compensating
the oscillations in $y(t)$ with (5) is

$K=\frac{\sqrt{3}}{2\pi}.$ (8)

Note that the determined gain (8) is rather conservative. Indeed, the energy
balance (7) is considered over one full period, while the extrema detection
and, correspondingly, update of the oscillation parameters in (5) take place
every half-period. However, the balancing of input-output power (respectively
of energy) turns out more reasonable over a full period $[t,\,2\pi/\omega+t]$,
due to a possible time delay of the extrema detection, cf. later in section
2.3. This allows avoiding an overcompensation which, otherwise, can
additionally excite the oscillations in $y(t)$.

####  2.1.1 Numerical example:

A second-order oscillatory system

$\ddot{y}(t)+a\dot{y}(t)+by(t)=u(t),\quad y(0)=c\neq 0$ (9)

is used in the numerical simulation, once without feedback compensator (i.e.
$u=0$) and once using the power-based control law (5). The assigned
coefficient $b=100$ results in a natural frequency $\omega=10$ rad/s. The
initial value constant is assigned to $c=2$. The damping coefficient is
assigned as $a=\\{2,\,-1\\}$, thus resulting in a homogenous solution of (9)
which is oscillatory and exponentially converging in the first case, and
oscillatory and diverging in the second case, respectively, see the red dashed
lines in Fig. 2 (a) and (b).

Figure 2: Simulation results of the second-order oscillatory system with and
without use of compensator (5). The output response and control value, in (a)
and (c) for $a=2$, in (b) and (d) for $a=-1$.

The compensated output response, i.e. when the control (5) is on, is shown for
comparison by the blue solid line, while the corresponding control values are
depicted in Fig. 2 (c) and (d), respectively.

###  2.2 Extension to higher-order systems

For dynamic systems with order higher than two, the structure is assumed as
shown in Fig. 3. The assumptions made in section 2.1 remain valid, while a
rational stable transfer function $G(s)$ is assumed to be known.

Figure 3: Signals flow in higher-order system with output (1).

Furthermore, it is assumed that the unknown driving input signal $h(t)$
provides as asymptotically stable response

$\int\limits_{0}^{t}g(t-\tau)h(t)d\tau,$

where $g(t)$ is the corresponding inverse Laplace transform of $G(s)$, i.e.
the impulse response function. Also assume that the time constant(s) of $g(t)$
are significantly lower than $\pi\omega^{-1}$, i.e. half-period of the output
oscillations.

Since the compensator output $u(t)$, cf. (5), is no longer matched with
$f(t)$, cf. with Fig. 1, its propagation through $G(s)$ requires the following
processing. The available for control input $\hat{u}(t)$ will be modified by
the amplitude response $|G(j\omega)|$ at the characteristic frequency
$\omega$. Moreover, an additional phase lag $\arg\bigl{[}G(j2\omega)\bigr{]}$
appears at twice the oscillation frequency. Here we recall that the control
$u(t)$ is of the discrete type and switches at half of the oscillation period.
Following to that, the compensator signal (5) undergoes the proposed
transformation

$\hat{u}(t)=L\,\Bigl{|}G(s)^{-1}\Bigr{|}_{s=j\omega}\,u(t-T).$ (10)

The corresponding time delay factor

$T=\Bigl{(}2\pi+\arg\bigl{[}G(j2\omega)\bigr{]}\Bigr{)}\,\omega^{-1}$ (11)

shifts the control value by the negative phase lag with respect to a full
period $2\pi\omega^{-1}$. This synchronizes $u(t)$, propagated through (10),
again with the input of double integrator, cf. Fig. 1. Note, that the
rectangular pulse signal $u(t)$ is reshaped by $G(j\omega)$, thus losing its
energetic content, correspondingly for the impulse

$U\int\limits_{0}^{\pi\omega^{-1}}dt\>>\int\limits_{0}^{\pi\omega^{-1}}U\int\limits_{0}^{t}g(t-\tau)d\tau
dt.$

In order to regard for this inequality, a tunable impulse weighting factor

$1<L<3$ (12)

is additionally used in (10).

####  2.2.1 Numerical example:

A fifth-order oscillatory system, the same as shown later in the experimental
case study in section 3, is numerically simulated with the compensator (10).
Note that for keeping a known $\Psi$ value, cf. (1), a simple proportional
feedback control

$v(t)=70\bigl{(}R_{1}-y(t)\bigr{)}+R_{2}$

is additionally applied, which also leads to destabilization of the closed-
loop when not using (10). The first constant $R_{1}$ constitutes a reference
value. The second constant $R_{2}$ is used to compensate for the total gravity
term, cf. with section 3.1. Further we note that for approaching the real
systems in a numerical simulation, the output $y(t)$ is additionally subject
to a band-limited white noise. The diverging oscillatory behavior, when (10)
is switched off, is shown in Fig. 4.

Figure 4: Simulation results of the fifth-order oscillatory system with and
without use of compensator (10).

When switching on the compensator (10) at time $t=4$ sec, the otherwise
unstable oscillations become stabilized, as shown in Fig. 4 for two weighting
factors $L=\\{1,\,2\\}$. Obviously $L$ affects the convergence rate of the
power-based oscillations compensation.

###  2.3 Extrema detection

Figure 5: Example of extrema detection from noisy output.

An extrema detecting algorithm, required for $(\tilde{A},\tilde{\omega})$
estimation at each $t^{*}_{i}$, cf. (5), is summarized below. The discrete
time sampling index is $n$, the size of the trapped delay buffer for smoothing
filter is $N$, and the index of recent extrema is $i$. An example of extrema
detection from the simulated $y(t)$, which is affected by the band-limited
white noise, is shown in Fig. 5 for $f_{s}=1$ kHz and $N=30$.

$\displaystyle 1)$ $\displaystyle\hbox{{Initialization}:
}\tilde{A}(t^{*}_{i-1})=y_{0};\;\tilde{\omega}(t^{*}_{i-1})=0.5\Omega_{\max}$
$\displaystyle 2)$ $\displaystyle\hbox{{For } }n=1\hbox{{ to }
}\infty\>\hbox{{ do } }$ $\displaystyle 3)$
$\displaystyle\quad\bar{y}_{n}=\max\bigl{(}y_{n},y_{n-1},\ldots,y_{n-N}\bigr{)};$
$\displaystyle 4)$ $\displaystyle\quad
S_{n}=\mathrm{sign}\bigl{(}\bar{y}_{n}-\bar{y}_{n-1}\bigr{)};$ $\displaystyle
5)$ $\displaystyle\quad\hbox{{If } }S_{n}\neq 0\,\wedge\,S_{n}\neq
S_{i-1}\,\wedge\,\pi/\bigl{(}nf_{s}-t^{*}_{i-1}\bigr{)}<\Omega_{\max}$
$\displaystyle 6)$ $\displaystyle\qquad\hbox{{Update}:
}\>S_{i}=S_{n};\;t^{*}_{i}=nf_{s};$ $\displaystyle 7)$
$\displaystyle\qquad\qquad\qquad\tilde{A}(t^{*}_{i})=\bar{y}_{n}-\Psi;\;\tilde{\omega}(t^{*}_{i})=\pi/\bigl{(}nf_{s}-t^{*}_{i-1}\bigr{)}$
$\displaystyle 8)$ $\displaystyle\qquad\qquad\qquad i=1+1;$ $\displaystyle 9)$
End  $\displaystyle 10)$ End

##  3 Experimental case study

###  3.1 Oscillatory system

The following experimental case study is accomplished on the two-mass
oscillatory system under gravity, see Ruderman (2021, 2022), Voß et al. (2022)
for details. The setup shown in Fig. 6 consists one one free hanging load and
one

Figure 6: Experimental setup of oscillatory system.

linear actuator, based on the voice-coil-motor. Both moving bodies with one
vertical degree of freedom are connected via an elastic spring with the
relatively high stiffness and hardening effects subject to unknown
uncertainties. The actuator input voltage $v\in[0,\,10]$ V is the available
control channel. The single (contact-less) measured output value is the
relative displacement of the load $y(t)$. Note that the load is passive, while
the actuator displacement $z\in[0,\,0.021]$ m remains an unmeasurable state.
Both the input and output values are real-time available with the sampling
rate set to $f_{s}=5$ kHz.

The system structure is known, being given by

$f(s)=\frac{3.2811}{0.0012s+1}\,v(s),$ (13) $\displaystyle\dot{x}(t)$
$\displaystyle=$ $\displaystyle A\,x(t)+B\,f(t)+D,$ (14) $\displaystyle y(t)$
$\displaystyle=$ $\displaystyle C\,x(t),$

with the state vector $x\in\mathbb{R}^{4}$ and

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}-333.35&-333.33&0.015&333.33\\\
1&0&0&0\\\ 0.012&266.66&-0.012&-266.66\\\ 0&0&1&0\end{array}\right),$ (19)
$\displaystyle B$ $\displaystyle=$ $\displaystyle(1.667,0,0,0)^{\top},\quad
C\;=\;(0,0,0,1),\;\hbox{ and}$ $\displaystyle D$ $\displaystyle=$
$\displaystyle(-9.806,0,-9.806,0)^{\top}.$

At the same time, the identified parameters in (13), (14) are not explicitly
used in $u(t)$. Also worth noting is that the disturbance $D$ is composed by
the constant gravity acting on both, the moving actuator and load.

For exposing the low damped output oscillations, a free fall scenario is
performed, see Fig. 7. Starting from non-zeros initial conditions with
$v(t)=\mathrm{const}$ which compensates for the total gravity, the control
signal is switched off at $t=20$ sec. Due to $v(t>20)=0$ both moving masses
fall down, while $|\dot{y}|$ is larger than the actuator velocity due to the
bearing.

Figure 7: Measured and computed oscillatory response.

Once the actuator displacement experiences a hard impact at its down
mechanical limiter, the $y(t)$ becomes largely excited. It begins to oscillate
and is structurally damped by the spring only, cf. Fig. 7. Note that the
oscillation amplitude is sensitive to the initial conditions and exact
knowledge of the moving masses and stiffness. This leads to a visible
amplitude difference between the measured and computed output, while the
frequency and damping ratio match well for both.

###  3.2 Experimental control results

The experimental control results are obtained with the PI (proportional-
integral) output feedback control

$v(t)=150\bigl{(}R_{1}-y(t)\bigr{)}+170\int\bigl{(}R_{1}-y(t)\bigr{)}dt+R_{2}+\hat{u}(t),$

once without (i.e. $\hat{u}=0$) and once with the use of the oscillations
compensator (10). Note that a PI control structure is necessary for keeping
$\Psi\approx R_{1}$ despite all additional uncertainties and disturbances
(like for example friction), which are not captured in (13), (14), cf. section
2.2. Here again, a constant term $R_{2}$ is compensating for the total known
gravity of the actuator and load masses. An unstable diverging output response
for $\hat{u}=0$ is shown in Fig. 8.

Figure 8: Experimental results of PI-controlled load position without use of
the compensator (10).

On the contrary, the compensator (10), switched on at time $t=4$ sec, is
suppressing the unstable oscillations, see Fig. 9. Moreover, an impulse-like
external mechanical disturbance was manually injected, once to the actuator
(at time around 17 sec), and once to the load (at time around 30 sec). Note
that the latter leads also to some vertical misalignments of the hanging load
and, in consequence, to un increase of the measurement and process noise and,
thus, less accurate extrema detection. Still in both external disturbance
cases, the output is stabilized again after some transient oscillations.

Figure 9: Experimental results of PI-controlled load position with use of the
compensator (11). Mechanical disturbance is injected once to actuator and once
to load.

##  4 Summary

A power-based discrete-valued adaptive compensator is proposed for oscillatory
outputs. Comparing to other oscillation-compensating approaches, like e.g. a
recent time-delay-based one in Ruderman (2023), the presented control scheme
commutates only twice per oscillations’s period. This yields a control
efficiency in terms of the communication efforts, in case of a digitally
connected system framework, i.e. when the output sensing and control
(correspondingly actuator) units are remote. The proposed control is derived
and discussed for the second-order systems, while an extension to the higher-
order systems is also given. Proposed control requires a robust extrema
detection, which then allows operating also with noisy output signals. An
experimental case study accomplished on the fifth-order oscillatory mechanical
system is shown.

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# A Proof of the Kahn–Kalai Conjecture

Jinyoung Park and Huy Tuan Pham<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics, Stanford University
450 Jane Stanford Way, Building 380, Stanford, CA 94305

###### Abstract.

We prove the “expectation-threshold” conjecture of Kahn and Kalai.

##  1\. Introduction

Given a finite set $X$, write $2^{X}$ for the power set of $X$. For
$p\in[0,1]$, let $\mu_{p}$ be the product measure on $2^{X}$ given by
$\mu_{p}(A)=p^{|A|}(1-p)^{|X\setminus A|}$. In this paper we always denote
$\mathcal{F}\subseteq 2^{X}$ an increasing family, meaning that if $B\supseteq
A\in\mathcal{F}$, then $B\in\mathcal{F}$. It is a well-known fact that
$\mu_{p}(\mathcal{F})(:=\sum_{A\in\mathcal{F}}\mu_{p}(A))$ is strictly
increasing in $p$ for any $\mathcal{F}\neq\emptyset,2^{X}$. The threshold,
$p_{c}(\mathcal{F})$, is then the unique $p$ for which
$\mu_{p}(\mathcal{F})=1/2$.

Following [6], we say $\mathcal{F}$ is $p$-small if there is
$\mathcal{G}\subseteq 2^{X}$ such that

$\mathcal{F}\subseteq\langle\mathcal{G}\rangle:=\bigcup_{S\in\mathcal{G}}\\{T:T\supseteq
S\\}$ (1)

and

$\sum_{S\in\mathcal{G}}p^{|S|}\leq 1/2.$ (2)

We say that $\mathcal{G}$ is a cover of $\mathcal{F}$ if (1) holds. The
expectation-threshold of $\mathcal{F}$, $q(\mathcal{F})$, is defined to be the
maximum $p$ such that $\mathcal{F}$ is $p$-small. Observe that
$q(\mathcal{F})$ is a trivial lower bound on $p_{c}(\mathcal{F})$, since

$\mu_{p}(\mathcal{F})\leq\mu_{p}(\langle\mathcal{G}\rangle)\leq\sum_{S\in\mathcal{G}}p^{|S|}.$
(3)

Note that, with $X_{p}$ the random variable whose distribution is $\mu_{p}$,
the right-hand side of (3) is $\mathbb{E}[|\\{S\in\mathcal{G}:S\subseteq
X_{p}\\}|]$.

Given an increasing family $\mathcal{F}$, write $\ell(\mathcal{F})$ for the
size of a largest minimal element of $\mathcal{F}$. Our main theorem resolves
the expectation-threshold conjecture of Kahn and Kalai [3].

######  Theorem 1.1 (The Kahn-Kalai Conjecture).

There is a universal constant $K$ such that for every finite set $X$ and
increasing family $\mathcal{F}\subseteq 2^{X}$,

$p_{c}(\mathcal{F})\leq Kq(\mathcal{F})\log\ell(\mathcal{F}).$

For the discussion about the significance and applications of this theorem, we
refer the readers to [2].

Part of our proof is inspired by the algorithm in [1] and the analysis of the
algorithm in [2, 4], though our implementation is significantly different from
the ideas in those papers. In particular, our analysis completely avoids the
use of “spread,” which was a key ingredient in the aforementioned papers. We
remark that our proof technique can also be adapted to simplify the proof of
the main lemma of [2]; this derivation is straightforward and we omit the
details.

In the upcoming paper [5], using a more elaborate argument that shares some
ideas with those in this paper, we resolve a conjecture of Talagrand ([6,
Conjecture 5.7]).

Reformulation. In Section 2 we prove Theorem 1.2 below, which implies Theorem
1.1.

A hypergraph on $X$ is a collection $\mathcal{H}$ of subsets of $X$, and a
member of $\mathcal{H}$ is called an edge of $\mathcal{H}$. We say
$\mathcal{H}$ is $\ell$-bounded if each of its edges has size at most $\ell$.
Recall that
$\langle\mathcal{H}\rangle=\bigcup_{S\in\mathcal{H}}\\{T:T\supseteq S\\}$.
Note that we can extend the definition of $p$-small to $\mathcal{H}$ without
any modification. For an integer $m$, we use an $m$-subset of $X$ for a subset
of $X$ of size $m$, and $X_{m}$ for a uniformly random $m$-subset of $X$.

######  Theorem 1.2.

There is a universal constant $L$ such that for any $\ell$-bounded hypergraph
$\mathcal{H}$ on $X$ that is not $p$-small,

a uniformly random $((Lp\log\ell)|X|)$-element subset of $X$ belongs to
$\langle\mathcal{H}\rangle$ with probability $1-o_{\ell\to\infty}(1)$. (4)

###### Derivation of Theorem 1.1 from Theorem 1.2..

Let $\mathcal{F}$ be as in Theorem 1.1. We assume Theorem 1.2 and derive that
if $q>q(\mathcal{F})$ then, with $p=Kq\log\ell(\mathcal{F})$ ($K$ is a
universal constant to be determined), we have
$\mathbb{P}(X_{p}\in\mathcal{F})>1/2$. Here, we recall that for $0\leq p\leq
1$, we denote by $X_{p}$ the random variable with distribution $\mu_{p}$.

Let $\mathcal{H}$ be the set of minimal elements of $\mathcal{F}$ (so
$\langle\mathcal{H}\rangle=\mathcal{F}$). Then $\mathcal{H}$ is
$\ell(\mathcal{F})$-bounded and not $q$-small (since $q>q(\mathcal{F})$). Let
$C$ be a (universal) constant for which, with $\ell=C\ell(\mathcal{F})$, the
exceptional probability in Theorem 1.2 is less than $1/4$.

Now, with $m=(Lq\log\ell)|X|$ and $p^{\prime}=2m/|X|$,

$\mathbb{P}(X_{p^{\prime}}\in\langle\mathcal{H}\rangle)\geq\mathbb{P}(|X_{p^{\prime}}|\geq
m)\;\mathbb{P}(X_{m}\in\langle\mathcal{H}\rangle)\geq(3/4)\;\mathbb{P}(|X_{p^{\prime}}|\geq
m)>1/2,$

where the last inequality follows from standard concentration bound, upon
noting that $\mathcal{H}$ is not $q$-small implies $|X|q>1/2$ and hence
$m>(L\log\ell)/2$. Finally, letting $K$ be such that $p\geq p^{\prime}$
concludes the derivation. ∎

Notations and Conventions. All logarithms are base $2$ unless specified
otherwise. We did not attempt to optimize the absolute constants.

##  2\. Proof of Theorem 1.2

Before going through the proof in detail, we first give an informal overview
of our strategy. Roughly speaking, a hypergraph $\mathcal{H}$ is $p$-small if
$\mathcal{H}$ admits a “cheap” cover, where being cheap refers to the
condition in (2). Our proof uses a randomized process, in which we iteratively
construct a cover of $\mathcal{H}=\mathcal{H}_{0}$. At the $i$th step, we
start with some hypergraph $\mathcal{H}_{i-1}$ produced from the $(i-1)$th
step. We will show that there is a sub-hypergraph $\mathcal{G}_{i}$ (the $i$th
good set) of $\mathcal{H}_{i-1}$ that admits a cheap cover $\mathcal{U}_{i}$.
In the $(i+1)$th step we repeat our process with an updated hypergraph
$\mathcal{H}_{i}$. Here, the crucial property of $\mathcal{H}_{i}$ is that

$\mathcal{H}_{i-1}\setminus\mathcal{G}_{i}\subseteq\langle\mathcal{H}_{i}\rangle,$
(5)

which enables us to reduce the task of finding a cover of
$\mathcal{H}_{i-1}\setminus\mathcal{G}_{i}$ (the “leftover”) to finding a
cover of $\mathcal{H}_{i}$.

We will show that if we assume the failure of (4) then this randomized process
“successfully terminates” with positive probability, producing a cover
$\bigcup_{i}\mathcal{U}_{i}$ of $\mathcal{H}$ that satisfies (2). This
contradicts the assumption that $\mathcal{H}$ is not $p$-small, from which
Theorem 1.2 follows.

In Section 2.1 we describe our construction of the cheap cover
$\mathcal{U}=\mathcal{U}_{i}$ (in each step), and in Section 2.2 we analyze
our iteration, concluding our proof.

###  2.1. Constructing a cover

We use $n$ for $|X|$. Let $L\geq 1024$ and $\mathcal{H}$ be $\ell$-bounded. In
the following argument, we always assume that
$S,S^{\prime},\hat{S}\in\mathcal{H}$ and $W\in{X\choose w}$, where $w:=Lpn$
(as usual, ${X\choose w}$ is the collection of $w$-subsets of $X$).

Following [4], given $S$ and $W$, we call a set of the form
$S^{\prime}\setminus W$ with $S^{\prime}$ contained in $S\cup W$ an
$(S,W)$-fragment. Given $S$ and $W$, define $T=T(S,W)$ to be a minimum
$(S,W)$-fragment; that is, $T=S^{\prime}\setminus W$ for some
$S^{\prime}\subseteq W\cup S$ with the property that $|S^{\prime}\setminus W|$
is the minimum among all the $(S,W)$-fragments (breaking ties arbitrarily). We
use $t=t(S,W)$ for $|T(S,W)|$.

Given $W$, the good set, $\mathcal{G}=\mathcal{G}(W)$, is the collection of
$S$ whose minimum fragment with respect to $W$ is “large;” formally,

$\mathcal{G}(W):=\\{S\in\mathcal{H}:t(S,W)\geq.9\ell\\}.$

Then we define $\mathcal{U}(W)$, a cover of $\mathcal{G}(W)$, as

$\mathcal{U}(W):=\\{T(S,W):S\in\mathcal{G}(W)\\}$

(the fact that $\mathcal{U}(W)$ covers $\mathcal{G}(W)$ follows from the
definition of fragment).

Note that the edges in $\mathcal{H}\setminus\mathcal{G}(W)$ are not
necessarily covered by $\mathcal{U}(W)$. We define

$\mathcal{H}^{\prime}=\mathcal{H}^{\prime}(W)=\\{T(S,W):S\in\mathcal{H}\setminus\mathcal{G}(W)\\};$
(6)

this $\mathcal{H}^{\prime}$, which is $.9\ell$-bounded, will be the host
hypergraph in the next iteration step (see (13)). Note that
$\mathcal{H}\setminus\mathcal{G}(W)\subseteq\langle\mathcal{H}^{\prime}\rangle$
(as promised in (5)), so in particular,

a cover of $\mathcal{H}^{\prime}$ also covers
$\mathcal{H}\setminus\mathcal{G}(W)$. (7)

Of course, different $W$ produce different good sets and covers, and sometimes
using $\mathcal{U}(W)$ can be expensive. However, we claim that for a typical
choice of $W$, the corresponding cover $\mathcal{U}(W)$ is cheap. Formally:

######  Lemma 2.1.

For $W$ uniformly chosen from ${X\choose w}$, with probability at least
$1-L^{-.1\ell}$,

$\sum_{U\in\mathcal{U}(W)}p^{|U|}<L^{-.5\ell}.$

Observe that, to prove Lemma 2.1, it suffices to show that

$\sum_{W\in{X\choose w}}\sum_{U\in\mathcal{U}(W)}p^{|U|}<{n\choose
w}L^{-.6\ell},$ (8)

since then we have

$\mathbb{E}\left[\sum_{U\in\mathcal{U}(W)}p^{|U|}\right]<L^{-.6\ell}$

(where the expectation is over the choice of $W$), from which the lemma
follows using Markov’s Inequality.

###### Proof of (8).

Given $W$ and $m\geq.9\ell$, let

$\mathcal{G}_{m}(W):=\\{S\in\mathcal{H}:t(S,W)=m\\}$

and

$\mathcal{U}_{m}(W):=\\{T(S,W):S\in\mathcal{G}_{m}(W)\\}.$

Note that for any $U\in\mathcal{U}_{m}(W)$ we have $|U|=m$, so
$\sum_{W\in{X\choose w}}\sum_{U\in\mathcal{U}_{m}(W)}p^{|U|}$ is equal to
$p^{m}$ multiplied by

$\left|\left\\{(W,T(S,W)):W\in{X\choose w},S\in\mathcal{H},\mbox{ and
}t(S,W)=m\right\\}\right|.$ (9)

We bound the number of choices of $W$ and $T=T(S,W)$’s in the collection in
(9) using the following specification steps.

  1. Step 1.

Pick $Z:=W\cup T$. Since $|Z|=w+m$ (note $W$ and $T$ are always disjoint), the
number of possibilities for $Z$ is at most (recalling $w=Lpn$)

${n\choose w+m}={n\choose w}\cdot\prod_{j=1}^{m}\frac{n-w-j}{w+j}\leq{n\choose
w}(Lp)^{-m}.$

  2. Step 2.

Pick any $\hat{S}\subseteq Z$. Note that $Z\;(=W\cup T)$ must contain an edge
of $\mathcal{H}$ by the definition of fragment. The choice of $\hat{S}$ is
free. Here a crucial observation is that, since $T(S,W)$ is a minimum
fragment,

$T\subseteq\hat{S};$ (10)

indeed, since $\hat{S}$ is contained in $T\cup W\subseteq S\cup W$, the
failure of (10) implies that $\hat{S}\setminus W$ is an $(S,W)$-fragment that
is smaller than $T$, contradicting the minimality of $T$.

The property (10) enables us to specify $T$ as a subset of $\hat{S}$, whose
number of possibilities is at most $2^{\ell}$.

Note that $(W,T)$ is determined upon fixing a choice of $Z$ and $T$. In sum,
we have

$\sum_{W\in{X\choose w}}\sum_{U\in\mathcal{U}_{m}(W)}p^{|U|}\leq
p^{m}{n\choose w}(Lp)^{-m}2^{\ell}={n\choose w}L^{-m}2^{\ell},$

and the left hand side of (8) is at most

$\sum_{m\geq.9\ell}{n\choose w}L^{-m}2^{\ell}\leq{n\choose w}L^{-.6\ell}$

for $L\geq 1024$. ∎

###  2.2. Iteration

Recall that $n=|X|,\ell\rightarrow\infty,$ and $L\geq 1024$. In the following
definitions, $i=1,2,\ldots,\lfloor\log_{.9}(1/\ell)\rfloor+1=:\gamma$. Let
$\ell_{i}=.9^{i}\ell$ and note that

$0<\ell_{\gamma}<1.$ (11)

Let $X_{0}=X$ and $W_{i}$ be uniform from ${X_{i-1}\choose w_{i}}$, where
$X_{i}=X_{i-1}\setminus W_{i}$ and $w_{i}=L_{i}pn$ with

$L_{i}=\begin{cases}L&\mbox{if }\quad i<\gamma-\sqrt{\log_{.9}(1/\ell)}\\\
L\sqrt{\log\ell}\;&\mbox{if }\quad\gamma-\sqrt{\log_{.9}(1/\ell)}\leq
i\leq\gamma.\end{cases}$

At the end, $W:=\bigcup_{i=1}^{\gamma}W_{i}$ is a uniformly random
$(CLp\log\ell)n$-subset of $X$ where $C\leq C^{\prime}$ for some absolute
constant $C^{\prime}>0$. Note that there is an absolute constant $c>0$ for
which

$\ell_{i}>\exp(c\sqrt{\log\ell})\quad\forall
i<\gamma-\sqrt{\log_{.9}(1/\ell)}$. (12)

By iteratively applying our argument in Section 2.1, we produce a sequence
$\\{\mathcal{H}_{i}\\}$ with $\mathcal{H}_{0}=\mathcal{H}$ and

$\mathcal{H}_{i}=\mathcal{H}_{i-1}^{\prime}$ (13)

(see (6) to recall the definition of $\mathcal{H}^{\prime}$). Note that each
$\mathcal{H}_{i}$ is $\ell_{i}$-bounded, and associated to each set $W_{i}$ in
step $i$, we have a good set $\mathcal{G}_{i}=\mathcal{G}_{i}(W_{i})$ and a
cover $\mathcal{U}_{i}=\mathcal{U}_{i}(W_{i})$ of $\mathcal{G}_{i}$.

For $i=1,2,\ldots,\gamma$, call $W_{i}$ successful if

$\sum_{U\in\mathcal{U}(W_{i})}p^{|U|}<L_{i}^{-.5\ell_{i}};\mbox{ and}$ (14)
$\mathcal{H}_{i}$ does not contain $\emptyset$. (15)

######  Remark 2.2.

Having (15) is essential for our purpose – constructing a cheap cover; the
cost of covering the emptyset is 1 (see (2)), which is not affordable.

We terminate our process as soon as we have

$\mathcal{G}_{i}=\mathcal{H}_{i-1}\setminus\\{\emptyset\\}$

(or equivalently, $\mathcal{H}_{i}\subseteq\\{\emptyset\\}$) for some
$i(=:i_{\max})$. Note that $i_{\max}\leq\gamma$ because of the upper bound on
$\ell_{\gamma}$ in (11). We say our process terminates successfully if $W_{i}$
are successful for all $i\leq i_{\max}$.

######  Proposition 2.3.

If our process terminates successfully, then $\mathcal{U}:=\bigcup_{i\leq
i_{\max}}\mathcal{U}(W_{i})$ covers $\mathcal{H}$.

###### Proof.

Suppose there is some $S\in\mathcal{H}$ that is not covered by $\mathcal{U}$.
Let $S=S_{0},S_{1},S_{2},\ldots$ ($S_{i}\in\mathcal{H}_{i}$) be the evolution
of $S$ in the iteration process, i.e., $S_{i}:=T(S_{i-1},W_{i})$. Observe that
$S$ not being covered by $\mathcal{U}$ means there is some $i_{0}$ for which
$S_{i_{0}}=\emptyset$. But this means (15) is violated for $i_{0}$, which is a
contradiction. ∎

If we have a successful termination, the cost for the cover $\mathcal{U}$ is

$\sum_{U\in\mathcal{U}}p^{|U|}\stackrel{{\scriptstyle\eqref{success.1}}}{{<}}\sum_{i\leq
i_{\max}}L_{i}^{-.5\ell_{i}}\stackrel{{\scriptstyle\eqref{i.bound}}}{{\leq}}2L^{-.5\exp(c\sqrt{\log\ell})}+O((L\sqrt{\log\ell})^{-c^{\prime}})\stackrel{{\scriptstyle(\star)}}{{\leq}}1/2,$

where $c^{\prime}>0$ is some constant, and ($\star$) holds for $\ell$
sufficiently large. This, combined with Proposition 2.3, implies that
$\mathcal{H}$ is $p$-small. Therefore, Theorem 1.2 follows if we show that,
assuming the failure of (4),

$\mathbb{P}\left(\bigwedge_{i\leq i_{\max}}\\{W_{i}\mbox{
successful}\\}\right)>0.$ (16)

######  Proposition 2.4.

If (15) fails for some $i$, then $W\in\langle\mathcal{H}\rangle$.

###### Proof.

Note $W\in\langle\mathcal{H}\rangle$ is another way of saying that $W$
contains some $S\in\mathcal{H}$. Suppose (15) fails for $j$ so
$\mathcal{H}_{j}$ contains $\emptyset$. By the construction of
$\\{\mathcal{H}_{i}\\}$, there are
$\emptyset=S_{j},S_{j-1},\ldots,S_{1},S_{0}:=S$ with $S_{i}\in\mathcal{H}_{i}$
and $S_{i}=S_{i-1}\setminus W_{i}$, whence $S_{i}=S\setminus(\bigcup_{k\leq
i}W_{k})$ for $i\in[j]$. This means $S\subseteq\bigcup_{k\leq j}W_{k}\subseteq
W$. ∎

With $\mathcal{E}_{i}$ the event that (14) fails, using Proposition 2.4 we
have

$\begin{split}\mathbb{P}\left(\bigvee_{i\leq i_{\max}}\\{W_{i}\mbox{ not
successful}\\}\right)&\leq\sum_{i\leq
i_{\max}}\mathbb{P}(\mathcal{E}_{i})+\mathbb{P}(W\in\langle\mathcal{H}\rangle)\\\
&\leq\sum_{i\leq\gamma}L_{i}^{-.1\ell_{i}}+\mathbb{P}(W\in\langle\mathcal{H}\rangle).\end{split}$

Note that

$\begin{split}\sum_{i\leq\gamma}L_{i}^{-.1\ell_{i}}&=\sum_{i<\gamma-\sqrt{\log_{.9}(1/\ell)}}L_{i}^{-.1\ell_{i}}+\sum_{i\geq\gamma-\sqrt{\log_{.9}(1/\ell)}}^{\gamma}L_{i}^{-.1\ell_{i}}\\\
&\stackrel{{\scriptstyle\eqref{gamma1},\eqref{i.bound}}}{{\leq}}2L^{-.1\exp(c\sqrt{\log\ell})}+O((L\sqrt{\log\ell})^{-c^{\prime\prime}})=o_{\ell}(1)\end{split}$
(17)

for some constant $c^{\prime\prime}>0$. Finally, the failure of (4) implies
that there is a fixed $\varepsilon>0$ such that

$\mathbb{P}(W\in\langle\mathcal{H}\rangle)\leq 1-\varepsilon,$

which, combined with (17), gives (16).

##  3\. Acknowledgements

The authors would like to thank Jacob Fox, Jeff Kahn and David Conlon for
their support and helpful comments on the paper. The second author is
supported by a Two Sigma Fellowship.

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