# Yet Another Comparison of SAT Encodings for the At-Most-K Constraint

Neng-Fa Zhou CUNY Brooklyn College & Graduate Center, email:
<EMAIL_ADDRESS>
###### Abstract

The at-most-k constraint is ubiquitous in combinatorial problems, and numerous
SAT encodings are available for the constraint. Prior experiments have shown
the competitiveness of the sequential-counter encoding for k $>$ 1, and have
excluded the parallel-counter encoding, which is more compact that the binary-
adder encoding, from consideration due to its incapability of enforcing arc
consistency through unit propagation. This paper presents an experiment that
shows astounding performance of the binary-adder encoding for the at-most-k
constraint.

##  1 INTRODUCTION

The at-most-k constraint, which is denoted as
$\leq_{k}(x_{1},x_{2},\ldots,x_{n})$ in this paper, commonly occurs in
combinatorial problems. Numerous encodings have been proposed for the
constraint. Prior experiments [2, 7] have shown that Sinz’s sequential-counter
(SC) encoding [6], which uses unary presentation for counters, is competitive
for $k>1$. SC introduces $O(nk)$ new variables, and generates $O(nk)$ clauses.
This encoding is not scalable for large $n$ and $k$. As a sequel, Sinz also
proposed a parallel-counter (PC) encoding, which uses binary representation
for counters, and is more compact than SC. PC uses incomplete adders that only
propagate 1’s. The binary-adder (BA) encoding, which uses complete adders
capable of propagating both 0’s and 1’s, has been shown to be competitive for
encoding integer-domain variables and arithmetic constraints [8]. The at-
most-k constraint can be treated as a special linear constraint. One question
arises: how competitive is the binary-adder encoding for the at-most-k
constraint?

This paper addresses the above question. It presents an experiment comparing
the BA, PC, and SC encodings. As the benchmark used in the experiment, the
pigeonhole problem, includes the at-most-one constraint, this paper also
surveys SAT encodings for the at-most-one constraint, and presents comparison
results of these encodings.

##  2 SAT ENCODINGS for AT-MOST-K

This section gives an overview of each of the SAT encodings for the at-most-k
constraint used in the experiment.

The  pairwise (PW) encoding for $\leq_{1}(x_{1},x_{2},\ldots,x_{n})$
decomposes the constraint into $\neg x_{i}\lor\neg x_{j}$, for $i\in 1..n-1$
and $j\in i+1..n$. PW introduces no new variables. However, it generates
$O(n^{2})$ clauses, and is therefore not viable for large $n$. PW is utilized
to handle the base case when $n\leq 4$ in the recursive encodings.

The  bisect (BS) encoding for $\leq_{1}(x_{1},x_{2},\ldots,x_{n})$ splits the
variables into two groups $G_{1}=\\{x_{1},x_{2},\ldots,x_{m}\\}$ and
$G_{2}=\\{x_{m+1},\ldots,x_{n}\\}$ when $n>4$, where
$m=\lfloor\frac{n}{2}\rfloor$. It introduces a new variable $b$ as the
commander variable for $G_{1}$, and uses $\neg b$ as the commander for
$G_{2}$. BS decomposes the constraint into the following:

(BS-1) | For $i\in 1..m$: $x_{i}\Rightarrow b$
---|---
(BS-2) | For $i\in m+1..n$: $x_{i}\Rightarrow\neg b$
(BS-3) |  $\leq_{1}(x_{1},x_{2},\ldots,x_{m})$
(BS-4) |  $\leq_{1}(x_{m+1},\ldots,x_{n})$

Constraint BS-1 forces $b$ to be 1 if any of the variables in $G_{1}$ is 1.
Constraint BS-2 forces $b$ to be 0 if any of the variables in $G_{2}$ is 1.
Since $b$ cannot be both 0 and 1 at the same time, it is impossible for one
variable in $G_{1}$ and another variable in $G_{2}$ to be 1 simultaneously.
Constraints BS-3 and BS-4 recursively enforce at-most-one on the two groups.
The BS is a special case of the bimander encoding [5], which generalizes the
binary encoding [3] and the commander encoding[4]. The number of clauses
generated by BS is $O(n\log_{2}(n))$, and the number of new variables
introduced is $O(n)$.

The  product (PD) encoding [1] for $\leq_{1}(x_{1},x_{2},\ldots,x_{n})$
arranges the variables on an $m\times m$ matrix $M$ when $n>4$, where
$m=\sqrt{n}$. It introduces two vectors of new variables
$<$$u_{1},u_{2},\ldots,u_{m}$$>$ and $<$$v_{1},v_{2},\ldots,v_{m}$$>$, where
$u_{i}$ represents row $i$ and $v_{j}$ represents column $j$. In case $n$ is
not a square number, the extra entries of $M$ are filled with 0. PD decomposes
the constraint into the following:

(PD-1) | For $i\in 1..m$, $j\in 1..m$: $M_{ij}\Rightarrow u_{i}\land v_{j}$
---|---
(PD-2) |  $\leq_{1}(u_{1},u_{2},\ldots,u_{m})$
(PD-3) |  $\leq_{1}(v_{1},v_{2},\ldots,v_{m})$

The number of clauses generated by PD is characterized by
$f(n)=2n+2f(\sqrt{n})$, and the number of new variables is characterized by
$g(n)=2\sqrt{n}+2g(\sqrt{n})$.

The  sequential-counter (SC) encoding [6] for
$\leq_{k}(x_{1},x_{2},\ldots,x_{n})$ successively counts the number of
$x_{i}$’s that are 1, in the following fashion: $c_{1}=x_{1}$,
$c_{2}=c_{1}+x_{2}$, $\ldots$, $c_{n}=c_{n-1}+x_{n}$. Each count is a unary
(base-1) number with $k$ bits: $c_{i}=$$<$$c_{i}^{1}c_{i}^{2}\ldots
c_{i}^{k}$$>$, where $c_{i}^{j}=0$ for $j\in i+1..k$, and $c_{i}^{j}=1$
entails $c_{i}^{j-1}=1$ for $j\in 2..k$. SC decomposes the constraint into the
following:

(SC-1) | For $i\in 2..n-1$: $x_{i}\Rightarrow c_{i}^{1}$
---|---
(SC-2) | For $i\in 2..n-1$, $j\in 1..k$: $c_{i-1}^{j}\Rightarrow c_{i}^{j}$
(SC-3) | For $i\in 2..n-1$, $j\in 2..k$: $x_{i}\land c_{i-1}^{j-1}\Rightarrow c_{i}^{j}$
(SC-4) | For $i\in k+1..n$: $x_{i}\Rightarrow\neg c_{i-1}^{k}$

Constraint SC-1 ensures that if $x_{i}=1$ then the first bit of $c_{i}$ is 1.
Constraint SC-2 ensures that monotonicity of addition: if the $j$th bit of
$c_{i-1}$ is 1, then the $j$th bit of $c_{i}$ is also 1. Constraint SC-3
ensures that $c_{i}=c_{i-1}+1$ if $x_{i}=1$. Constraint SC-4 ensures that no
count exceeds $k$. The number of clauses and the number of new variables
generated by SC are both $O(nk)$.

The  parallel-counter (PC) encoding [6] decomposes
$\leq_{k}(x_{1},x_{2},\ldots,x_{n})$ into the following:

(PC-1) |  $t\ =\ $sum$(x_{1},x_{2},\ldots,x_{n})$
---|---
(PC-2) |  $t\ \leq\ k$

The function sum$(x_{1},x_{2},\ldots,x_{n})$ returns a binary counter that
represents the number of 1’s in $x_{i}$’s with
$m=\lfloor\log_{2}(k)+0.5\rceil$ bits. Constraint PC-2 is enforced using a
binary comparator, which compares the sum and the binary representation of $k$
from the highest bit to the lowest bit. The function
sum$(x_{1},x_{2},\ldots,x_{n})$ is defined as follows: if $n=2$, then the two
variables $x_{1}$ and $x_{2}$ are added using a half adder; if $n=3$, then the
three variables are added using a full adder; otherwise, the variables are
split into two halves, each is summed recursively, and the results are added
using a ripple-carry adder. Since the ultimate goal is to ensure that the
final count never exceeds k, the binary counter only needs enough bits to
count up to $k$, and incomplete adders that only propagate 1’s are used. A
half adder is implemented with 3 clauses, and a full adder is implemented with
7 clauses. The number of clauses and the number of variables generated by PC
are both $O(mn)$.

The  binary-adder (BA) encoding implemented in Picat [8] treats
$\leq_{k}(x_{1},x_{2},\ldots,x_{n})$ as a linear arithmetic constraint for
$k>1$. It repeatedly combines variables that have the smallest domains into a
new variable until the constraint is reduced to the primitive form $t\ \leq\
k$. It uses log encoding for all the newly introduced variables, uses complete
binary adders for addition constraints, and enforces the constraint $t_{i}\
\leq\ k$ on all new variables. Unlike incomplete adders used in PC that only
propagate 1’s, complete adders used in BA propagate both 1’s and 0’s. A
complete half adder is implemented with 7 clauses, and a complete full adder
is implemented with 10 clauses.

##  3 EXPERIMENTAL RESULTS

Experiments were conducted to evaluate the encodings using the pigeonhole
problem as the benchmark. Given $P$ pigeons and $H$ holes, each of which can
hold $K$ pigeons, the goal of the problem is to put the pigeons into the holes
such that every pigeon is assigned a hole and no more than $K$ pigeons are put
into any hole. The benchmark uses a $P\times H$ matrix of variables, $B$, and
enforces the following constraints:

| (C-1) | For $p\in 1..P$: $\sum_{h=1}^{H}B_{ph}=1$
---|---|---
| (C-2) | For $h\in 1..H$: $\sum_{p=1}^{P}B_{ph}\leq K$

Constraint C-1 ensures that every pigeon is assigned a hole, and constraint
C-2 ensures that every hole receives at most $K$ pigeons. Obviously, if
$P>H\times K$, then the constraints are unsatisfiable.

The Maple SAT solver 111shorturl.at/joCIK was used in the experiments. All the
CPU times reported below were measured on Linux Ubuntu with an Intel i7
3.30GHz CPU and 32G RAM.

Table 1 compares the encodings on CPU time (the sum of the compilation and
solving times) for the at-most-one constraint. The first five instances are
small and unsatisfiable, and the remaining five instances are large and
satisfiable. While there are no significant differences in the speed for the
small instances, PD is outstandingly fast for the large instances.

Table 1: A comparison of encodings for $\leq_{1}$ (CPU time, seconds) P-H-K | BS | PC | PD | SC
---|---|---|---|---
12-11-1 | 4 | 9 | 9 | 5
13-12-1 | 13 | 15 | 8 | 10
14-13-1 | 75 | 28 | 46 | 16
15-14-1 | 123 | 47 | 119 | 42
16-15-1 | 511 | 229 | 549 | 216
100-100-1 | 901 | 1 | 1 | 76
200-200-1 |  $>$1200 | 4 | 1 |  $>$1200
300-300-1 |  $>$1200 | 25 | 2 |  $>$1200
400-400-1 |  $>$1200 |  $>$1200 | 6 |  $>$1200
500-500-1 |  $>$1200 |  $>$1200 | 10 |  $>$1200

Table 2 compares the BA, PC, and SC encodings on CPU time for the at-most-k
constraint (k $>$ 1). For constraint C-1, PD was used in all the runs, so this
experiment evaluates the encodings for constraint C-2. BA is significantly
faster than PC and SC, especially on the large instances.

Table 2: A comparison of encodings for $\leq_{k}$ (CPU time, seconds) P-H-K | BA | PC | SC
---|---|---|---
19-9-2 | 47 | 50 | 53
21-5-4 | 4 | 9 | 4
22-7-3 | 49 | 177 | 51
25-6-4 | 50 | 459 | 325
26-5-5 | 9 | 41 | 501
100-20-5 | 7 | 49 | 45
200-40-5 | 19 |  $>$1200 |  $>$1200
300-60-5 | 41 |  $>$1200 |  $>$1200
400-80-5 | 72 |  $>$1200 |  $>$1200
500-100-5 | 105 |  $>$1200 |  $>$1200

##  4 CONCLUSION

This paper has presented an experiment that shows astounding performance of BA
for the at-most-k (k $>$ 1) constraint in comparison with SC and PC. The major
differences between BA and PC are that, BA uses complete adders and enforces
cardinality on all new variables, while PC uses incomplete adders and only
enforces cardinality on the final variable. The result entails that the
clauses added by BA, while redundant, are helpful in enhancing the speed.

## References

  * [1] Jingchao Chen, ‘A new SAT encoding of the at-most-one constraint’, in Proc. of the Int. Workshop of Constraint Modeling and Reformulation, (2010).
  * [2] Alan M. Frisch and Paul A. Giannaros, ‘Sat encodings of the at-most-k constraint some old , some new , some fast , some slow’, in Proc. of the Int. Workshop of Constraint Modeling and Reformulation, (2010).
  * [3] Alan M. Frisch, Timothy J. Peugniez, Anthony J. Doggett, and Peter Nightingale, ‘Solving non-Boolean satisfiability problems with stochastic local search: A comparison of encodings’, J. Autom. Reasoning, 35(1-3), 143–179, (2005).
  * [4] Will Klieber and Gihwon Kwon, ‘Efficient CNF encoding for selecting 1 from n objects’, in the Fourth Workshop on Constraints in Formal Verification(CFV), (2007).
  * [5] Van-Hau Nguyen and Son T. Mai, ‘A new method to encode the at-most-one constraint into SAT’, in Proceedings of the Sixth International Symposium on Information and Communication Technology, SoICT 2015, pp. 46–53, (2015).
  * [6] Carsten Sinz, ‘Towards an optimal CNF encoding of Boolean cardinality constraints’, in CP, pp. 827–831, (2005).
  * [7] Ed Wynn, ‘A comparison of encodings for cardinality constraints in a SAT solver’, CoRR, abs/1810.12975, (2018).
  * [8] Neng-Fa Zhou and Håkan Kjellerstrand, ‘Optimizing SAT encodings for arithmetic constraints’, in CP, pp. 671–686, (2017).


# High harmonic generation in fullerene molecules

H. K. Avetissian  A. G. Ghazaryan  G. F. Mkrtchian<EMAIL_ADDRESS>Centre of
Strong Fields Physics, Yerevan State University, 0025, Yerevan, Armenia

###### Abstract

Using dynamical Hartree-Fock mean-field theory, we study the high-harmonic
generation (HHG) in the fullerene molecules C60 and C70 under strong pump wave
driving. We consider a strong-field regime and show that the output harmonic
radiation exhibits multiple plateaus, whose borders are defined by the
molecular excitonic lines and cutoff energies within each plateau scale
linearly with the field strength amplitude. In contrast to atomic cases for
the fullerene molecule, with the increase of the pump wave photon energy the
cutoff harmonic energy is increased. We also show that with the increase of
the electron-electron interaction energy overall the HHG rate is suppressed.
We demonstrate that the C70 molecule shows richer HHG spectra and a stronger
high-harmonic intensity than the C60.

Introduction– An intense light interaction with a quantum system can excite
the system’s electrons towards extreme nonequilibrium states during a fraction
of its cycle Avetissian ; Piazza . Excited by the wavefield and subjected to
the internal forces inside the system, the electrons emit coherent
electromagnetic (EM) radiation that can contain from tens to many hundreds of
harmonics of an incident light Agostini2004 ; Kohler2012 . This is one of the
fundamental processes in the intense laser-matter interaction called high
harmonic generation (HHG) Corkum . The HHG process in atoms or molecules with
the three-step model three-step  explanation is a well-demonstrated method for
producing coherent extreme ultraviolet radiation. The coherent spectrum of HHG
implies access to the extreme time resolution of the underlying quantum
dynamics that opens the way for attosecond physics Atto1 ; Atto2  and
ultrafast imaging methods for emitters themselves. In particular, using HHG
spectroscopy one can reconstruct the crystal potential Lakhotia , observe Mott
Mott  and Peierls Peierls  transitions, retrieve the band structure Vampa ;
Dejean .

For HHG it is crucial to increase HHG conversion efficiency and to extend the
harmonics cutoff Wah . Specifically, the conversion efficiency of the HHG
process strongly depends on the density of emitters and the density of states
of emitters. The use of molecular systems, clusters, and crystals can
significantly increase the harmonic intensity by utilizing multiple excitation
channels Donnelly96 ; Vozzi05 ; Smirnova09 . Thus, in the last decade, there
has been a growing interest to extend HHG to crystals sol1 ; Zaks ; Langer ;
sol2 ; sol3 ; sol4 ; sol5 ; sol6 ; sol7  and two-dimensional nanostructures,
such as semimetallic graphene gr1 ; gr2 ; gr3 ; gr4 ; gr5 ; gr6 ; gr7 ; gr8 ;
gr9 ; gr10 ; gr11 ; gr12 , semiconductor transition metal dichalcogenides TMD1
; TMD2 , and dielectric hexagonal boron nitride BN . Currently, this is a new
growing research field – extreme nonlinear optics of nanostructured materials.

Among the variety of nanostructured materials, carbon allotropes play a
central role. The discovery of fullerene C60 c60  through laser evaporation of
graphite and its synthesis in macroscopic amounts c60mc  was triggered the
study of many other carbon nanostructures, such as carbon nanotubes nanotub ,
graphene and its derivatives derivative . Currently, carbon nanomaterials are
promising materials for many applications, and in particular for extreme
nonlinear optics. Being the member of the carbon allotropes, it is expected a
strong HHG from fullerene molecules. Experimentally, in Refs. Ganeev1 ;
Ganeev2  it is reported a strong harmonic signal from C60 plasma. Theoretical
works predicted a strong HHG from a C60 molecule Zhang1 ; Zhang2 ; Zhang3  and
solid C60 Zhang4 . The theoretical analyses in Refs. Zhang1 ; Zhang2 ; Zhang3
are dominated by a single-particle picture, but it is unclear how the
electron-electron Coulomb interaction leaves its mark on HHG and sub-cycle
electronic response in these materials. Besides, in Zhang1 , moderately strong
fields were considered, and excitonic lines were termed as noninteger
harmonics. However, these intrinsic exciton lines are the result of Raman
scattering of light, not harmonic radiation. Another problem is how the
symmetry groups of the most abundant fullerenes C60 and C70, namely the
icosahedron group and the dihedral group, affect the HHG process in these
materials.

In the present work, we develop a microscopic theory of a fullerene molecule
nonlinear interaction with strong EM radiation of linear polarization taking
into account electron-electron interaction (EEI). In particular, we consider
C60 and C70 molecules as the most abundant examples of fullerene molecules
with different point group symmetries. By means of the dynamical Hartree-Fock
approximation, we reveal the general and basal structure of the HHG spectrum
and its relation to the molecular excitations.

Figure 1: The HHG spectra in the strong-field regime in logarithmic scale via
the normalized dipole acceleration Fourier transformation
$a\left(\Omega\right)/a_{0}$ (in arbitrary units) for C60 (a, b, c) and for
C70 (d, e, f). The wave is assumed to be linearly polarized along the
different axes. The frequency is $\omega=0.1\ \mathrm{eV}/\hbar$ and the field
strength is taken to be $E_{0}=0.5\ \mathrm{V/\mathring{A}.}$The relaxation
rate is taken to be $\hbar\gamma=50\ \mathrm{meV}$. It is also shown the
molecular excitonic lines (lower curves, except the line near $0.5\
\mathrm{eV}$). The latter is obtaind at the excitation of C60 and C70 with
probe laser pule of frequency $0.5\ \mathrm{eV}/\hbar$ and $E_{0}=10^{-4}\
\mathrm{V/\mathring{A}}$. The relaxation rate is taken to be $\hbar\gamma=0.5\
\mathrm{meV}$. The spectra are shown for moderate EEI energy: $U=2\
\mathrm{eV}$.

Model– Let a fullerene molecule, C60 or C70, interact with strong coherent EM
radiation that results in HHG. We assume neutral fullerene molecules, which
will be described in the scope of the tight-binding theory where the interball
hopping is much smaller than the on-ball hopping, and EEI is described in the
extended Hubbard approximation HV1 ; HV2 ; HV3 ; HV4 . Hence, the total
Hamiltonian reads:

$\widehat{H}=\widehat{H}_{0}+\widehat{H}_{\mathrm{int}},$ (1)

where

$\widehat{H}_{0}=-\sum_{\left\langle
i,j\right\rangle\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+\frac{U}{2}\sum_{i\sigma}n_{i\sigma}n_{i\overline{\sigma}}+\frac{1}{2}\sum_{\left\langle
i,j\right\rangle}V_{ij}n_{i}n_{j}$ (2)

is the free fullerene Hamiltonian. Here $c_{i\sigma}^{\dagger}$ creates an
electron with the spin polarization
$\sigma=\left\\{\uparrow,\downarrow\right\\}$ at site $i$ ($\overline{\sigma}$
is the opposite to $\sigma$ spin polarization), and $\left\langle
i,j\right\rangle$ runs over all the first nearest-neighbor hopping sites with
the transfer energy $t_{ij}$. The density operator is:
$n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$, and the total electron density
for site $i$ is: $n_{i}=n_{i\uparrow}+n_{i\downarrow}$. The second and third
terms in (2) describe the EEI Hamiltonian with on-site and inter-site Coulomb
repulsion energies $U$ and $V_{ij}$, respectively. Since the distance $d_{ij}$
between the nearest-neighbour pairs varies over the system, we scale inter-
site Coulomb repulsion: $V_{ij}=Vd_{\min}/d_{ij}$, where $d_{\min}$ is the
minimal nearest-neighbor distance. For the all calculations we use the ratio
$V=0.4U$ HV1 ; HV3 . In the Hamiltonian, we neglected the lattice vibrations.
In the calculations, the light-matter interaction is described in the length-
gauge via the pure scalar potential

$\widehat{H}_{\mathrm{int}}=e\sum_{i\sigma}\mathbf{r}_{i}\cdot\mathbf{E}\left(t\right)c_{i\sigma}^{\dagger}c_{i\sigma},$
(3)

with the elementary charge $e$, position vector $\mathbf{r}_{i}$, and the
electric field strength
$\mathbf{E}\left(t\right)=f\left(t\right)E_{0}\hat{\mathbf{e}}\cos\omega t$,
with the frequency $\omega$, polarization $\hat{\mathbf{e}}$ unit vector, and
pulse envelope $f\left(t\right)=\sin^{2}\left(\pi t/\mathcal{T}\right)$. The
pulse duration $\mathcal{T}$ is taken to be $20$ wave cycles:
$\mathcal{T}=40\pi/\omega$. From the Heisenberg equation one can obtain
evolutionary equations for the single-particle density matrix
$\rho_{ij}^{\left(\sigma\right)}=\left\langle
c_{j\sigma}^{\dagger}c_{i\sigma}\right\rangle$. In addition, we will assume
that the system relaxes at a rate $\gamma$ to the equilibrium
$\rho_{0ij}^{\left(\sigma\right)}$ distribution. To obtain a closed set of
equations for the single-particle density matrix
$\rho_{ij}^{\left(\sigma\right)}=\left\langle
c_{j\sigma}^{\dagger}c_{i\sigma}\right\rangle$, EEI will be considered under
the Hartree-Fock approximation Supplement .

Figure 2: The envelopes of HHG spectra (only peaks are connected) in the
strong-field regime for different EEI energies in logarithmic scale via the
normalized dipole acceleration Fourier transformation
$a\left(\Omega\right)/a_{0}$ (in arbitrary units) for C60 (a) and C70 (b). The
frequency is $\omega=0.1\ \mathrm{eV}/\hbar$ and the field strength is taken
to be $E_{0}=0.5\ \mathrm{V/\mathring{A}}$. The relaxation rate is taken to be
$\hbar\gamma=50\ \mathrm{meV}$. In (c) we show HHG for C60 versus C70.

Thus, we obtain the following equation for the density matrix:

$i\hbar\frac{\partial\rho_{ij}^{\left(\sigma\right)}}{\partial
t}=\sum_{k}\left(\tau_{kj\sigma}\rho_{ik}^{\left(\sigma\right)}-\tau_{ik\sigma}\rho_{kj}^{\left(\sigma\right)}\right)+\left(V_{i\sigma}-V_{j\sigma}\right)\rho_{ij}^{\left(\sigma\right)}$

$+e\mathbf{E}\left(t\right)\left(\mathbf{r}_{i}-\mathbf{r}_{j}\right)\rho_{ij}^{\left(\sigma\right)}-i\hbar\gamma\left(\rho_{ij}^{\left(\sigma\right)}-\rho_{0ij}^{\left(\sigma\right)}\right),$
(4)

where
$V_{i\sigma}=\sum_{j\alpha}V_{ij}\left(\rho_{jj}^{\left(\alpha\right)}-\rho_{0jj}^{\left(\alpha\right)}\right)+U\left(\rho_{ii}^{\left(\overline{\sigma}\right)}-\rho_{0ii}^{\left(\overline{\sigma}\right)}\right)$
and
$\tau_{ij\sigma}=t_{ij}+V_{ij}\left(\rho_{ji}^{\left(\sigma\right)}-\rho_{0ji}^{\left(\sigma\right)}\right)$
are defined via the density matrix $\rho_{ij}^{\left(\sigma\right)}$ and its
initial value.

We numerically diagonalize the tight-binding Hamiltonian $\widehat{H}_{0}$
with the parameters that provide molecular orbitals close to experiment H3 ;
gap1 ; Supplement , and construct the initial density matrix
$\rho_{0ij}^{\left(\sigma\right)}$ via the filling of electron states in the
valence band according to the zero temperature Fermi–Dirac-distribution
$\rho_{0ij}^{\left(\sigma\right)}=\sum_{\mu=N/2}^{N-1}\psi_{\mu}^{\ast}\left(j\right)\psi_{\mu}\left(i\right)$,
where $\psi_{\mu}\left(i\right)$ is the eigenstate of $\widehat{H}_{0}$.

Results– The HHG spectrum is evaluated from the Fourier transformation
$\mathbf{a}\left(\Omega\right)$ of the dipole acceleration
$\mathbf{a}\left(t\right)=d^{2}\mathbf{d}/dt^{2}$. The dipole is defined as
$\mathbf{d}\left(t\right)=e\sum_{i\sigma}\mathbf{r}_{i}\rho_{ii}^{\left(\sigma\right)}\left(t\right)$.
For convenience, we normalize the dipole acceleration by the factor
$a_{0}=e\overline{\omega}^{2}\overline{d},$ where $\overline{\omega}=1\
\mathrm{eV}/\hbar$ and $\overline{d}=1\ \mathrm{\mathring{A}}$. The power
radiated at the given frequency is proportional to
$\left|\mathbf{a}\left(\Omega\right)\right|^{2}$.

Figure 3: The site occupations $n_{i}$ in 3D color mapped molecular structures
for C70. The wave is assumed to be linearly polarized along the z-axis. The
first configuration is the equilibrium, the next 3 configurations are at the
peak of the laser field $t=\mathcal{T}/2$ for different EEI energies.

In order to clarify main aspects of HHG in C60 and C70, we assume the
excitation frequency is much smaller than the typical gap $\sim 2\ eV$. For
all calculations the wave is assumed to be linearly polarized. Orienting the
linearly polarized pump wave along different axes results in different
harmonics spectra. The difference is essential for C70. This is because for
C70 the inversion symmetry takes place only for the 5-fold rotation axes
(z-axis). In Fig. 1(a-f), we show the HHG spectra in the strong field regime
($E_{0}=0.5\ V/\mathring{A}$) for different orientation of the wave’s
polarization. For C60 molecule, because of the inversion symmetry only the odd
harmonics appear in the HHG spectrum. For C70 molecule we have both even and
odd harmonics except at the z-polarization of the pump wave. The insets in
Figs. 1(a) and 1(d) show low harmonics, where the difference in symmetry for
C60 and C70 is clearly visible (see, also Supplement  for a relatively weak
field). As is seen from Fig. 1, in both cases HHG spectra have a multi plateau
structure that is connected with the intrinsic molecular excitations between
the occupied molecular orbitals and the unoccupied molecular orbital. To show
this in a more transparent way, in Fig. 1 it is also shown the molecular
excitonic lines. The latter is obtained at the excitation of C60 and C70 with
probe weak laser pulse. In particular, the lines near 2.7 $eV$ and 1.7 $eV$
for C60 and C70, respectively, are the first dipole-allowed transition from
highest occupied molecular orbital to the lowest unoccupied molecular orbital.
These excitonic lines were termed as noninteger harmonics in Zhang1  have
their fingerprints in the multi plateau structure for the HHG spectra in the
strong-field regime. As is seen, plateaus’ borders are defined by these lines.
Besides for C70 there are close lines which enhance the HHG yield compared
with C60.

Since the boundaries of the plateaus are determined by dipole-allowed multiple
electronic transitions between the molecular orbitals, it can be argued that
for the frequencies $\hbar\omega<<$ $t_{ij}$ these boundaries are almost
invariant with respect to the pump wave frequency. They are determined through
the intrinsic features of the free fullerene. Note also that the position of
excitonic lines and relative intensities depend also on EEI. Besides, as was
shown in Zhang5 , the on-site EEI suppresses the charge fluctuation and
reduces the absorbed energy. It is also expected HHG yield suppression due to
EEI. The latter is shown in Fig. 2, where the HHG spectra in the strong-field
regime for different EEI energies are shown. To obtain the mean picture which
does not depend on the orientation of the molecule with respect to laser
polarization we take the wave polarization unit vector as
$\hat{\mathbf{e}}=1/\sqrt{3}\left\\{1,1,1\right\\}$. To make the plateaus more
visible in Figs. 2(a) and 2(b) we show the envelopes of HHG spectra. As is
seen from Figs. 2(a) and 2(b), with the increase of the EEI energy overall the
HHG rate is suppressed. Another interesting aspect of HHG in fullerene is the
qualitative and quantitative difference between both molecules. As is seen,
the C70 molecule shows more pronounced nonlinear properties (Fig. 2(c)). Due
to broken inversion symmetry in the case of C70, we have even harmonics,
besides due to the smaller energy gap and the larger density of states the HHG
rate is larger by one to two orders compared with C60. For comparison, in Fig.
2(c) we show the first four plateaus for both molecules. The suppression of
HHG with the increase of EEI energy is connected with the fact that at strong
on-site and inter-site electron-electron repulsion the polarizability of
molecules or in other words electrons migration from the equilibrium states is
suppressed. To show this visually, in Fig. 3 we display site occupations
$n_{i}=\left\langle
c_{i\uparrow}^{\dagger}c_{i\uparrow}\right\rangle+\left\langle
c_{i\downarrow}^{\dagger}c_{i\downarrow}\right\rangle$ via 3D color mapped
molecular structures at the peak of the laser field for z-polarized wave for
the same interaction parameters. The color bar represents site occupation. As
can be seen from Fig. 3, the deviation from the equilibrium position and,
therefore, the polarization is maximum for vanishing EEI.

Figure 4: The HHG spectra versus pump wave frequency (a,b) and intensity
(c,d). The color bar represents the emission rate via dipole acceleration
Fourier transformation $a\left(\Omega\right)/a_{0}$ in the logarithmic scale
for C60 (a, c) and C70 (b,d). The wave is assumed to be linearly polarized
with polarization unit vector
$\hat{\mathbf{e}}=\frac{1}{\sqrt{3}}\left\\{1,1,1\right\\}$. The field
strength is fixed $E_{0}=0.5\ \mathrm{V/\mathring{A}}$ for (a) and (b), while
for (c) and (d) the frequency is fixed $\omega=0.1\
\mathrm{eV}/\hbar\mathrm{.}$The spectra are shown for moderate EEI energy:
$U=2\ \mathrm{eV}$. The relaxation rate is taken to be $\hbar\gamma=50\
\mathrm{meV}$. The white lines in (a) and (b) are envelopes
$\alpha/\sqrt{\omega}$ which show the cutoff harmonic positions. The white
horizontal lines in (c) and (d) are excitonic resonances that show each
plateau’s borders. Figure 5: The interband and itraband contribution in HHG
spectra (a) and the spectrogram (b,c) of the HHG process via the windowed
Fourier transform of the dipole acceleration for C60. The wave is assumed to
be linearly polarized with polarization unit vector
$\hat{\mathbf{e}}=\frac{1}{\sqrt{3}}\left\\{1,1,1\right\\}$. The frequency is
$\omega=0.1\ \mathrm{eV}/\hbar$ and the field strength is taken to be
$E_{0}=0.5\ \mathrm{V/\mathring{A}.}$ The spectra are shown for $U=2\
\mathrm{eV}$ and $\hbar\gamma=50\ \mathrm{meV}$. (b) low frequency part and
(c) high frequency part. Then, the white curve on the density plots is scaled
by the factor 10 population of the initially unoccupied molecular orbitals
versus time.

We also investigated the HHG spectra dependence versus pump wave frequency and
intensity. In Figs. 4(a) and 4(b) we plot the HHG spectra versus pump wave
frequency at moderate EEI energy $U=2$ $\mathrm{eV}$, for C60 and C70,
respectively. For both molecules, the cutoff harmonic position is well
approximated by the dependence $N_{\mathrm{cut}}\sim\omega^{-1/2}$ which is
plotted along with density plot. Note that for atomic HHG via free continuum
$N_{\mathrm{cut}}\sim\omega^{-3}$ three-step . In case of two-level atom
$N_{\mathrm{cut}}\sim\omega^{-1}$ 2Lhhg . Thus, in contrast to atomic cases
for fullerene molecule with the increase of the pump wave photon energy the
cutoff harmonic energy ($\hbar\omega N_{\mathrm{cut}}$) is increased.

Next, we consider the HHG spectra as a function of pump wave intensity. In
Figs. 4(c) and 4(d), we show the HHG spectra as a function of field amplitude
and the harmonic order for a fixed frequency. The HHG spectra have interesting
structures. First of all, it is clearly seen excitonic lines (marked by the
white horizontal lines). These excitonic lines define the borders of the
plateaus and within each plateau, the cutoff harmonic linearly increases with
increasing the field strength. Then, reaching the harmonic $\sim$160 which
corresponds to the transition of the lowest occupied molecular orbital to the
highest unoccupied molecular orbital, the HHG rate is saturated. Note that
linear dependence of the cutoff harmonics on the field strength is inherent to
HHG via discrete levels 2Lhhg , or in crystals with linear energy dispersion
sol3 ; sol4 ; gr11 .

We now consider the origin of the HHG in fullerene molecules. There are two
contributions to the current: the electron/hole transitions within
unoccupied/occupied molecular orbitals and the electron-hole creation
(transitions from occupied molecular orbitals to unoccupied ones) and
subsequent recombination. The former makes contribution only for low harmonics
and is analogous to the intraband current in a semiconductor, while the latter
makes the main contribution in the high frequency part corresponds to the
interband current, which represents recombination/creation of electron-hole
pairs. This picture is analogous to HHG in solid state systems. To separate
these contributions in the dipole acceleration spectrum we made change of the
basis via formula
$\rho_{ij}=\sum_{\mu^{\prime}}\sum_{\mu}\psi_{\mu^{\prime}}^{\ast}\left(j\right)\rho_{\mu\mu^{\prime}}\psi_{\mu}\left(i\right)$,
where $\rho_{\mu\mu^{\prime}}$ is the density matrix in the energetic
representation. Hence, we define interband part of dipole acceleration, as

$\mathbf{d}_{\mathrm{inter}}\left(t\right)=2\sum_{\mu^{\prime}=N/2}^{N-1}\sum_{\mu=0}^{N/2-1}\mathrm{Re}\left(\rho_{\mu\mu^{\prime}}\left(t\right)\mathbf{d}_{\mu^{\prime}\mu}\right),$
(5)

and intraband part will be

$\mathbf{d}_{\mathrm{intra}}\left(t\right)=\sum_{\mu,\mu^{\prime}=N/2}^{N-1}\rho_{\mu\mu^{\prime}}\left(t\right)\mathbf{d}_{\mu^{\prime}\mu}+\sum_{\mu,\mu=0}^{N/2-1}\rho_{\mu\mu^{\prime}}\left(t\right)\mathbf{d}_{\mu^{\prime}\mu},$
(6)

where dipole transition matrix elements are
$\mathbf{d}_{\mu^{\prime}\mu}=e\sum_{i}\psi_{\mu^{\prime}}^{\ast}\left(i\right)\mathbf{r}_{i}\psi_{\mu}\left(i\right)$.
In Fig. 5(a), we show the interband/itraband contribution in HHG spectra for
C60. The similar picture we have for C70. As is seen, intraband dipole
acceleration is significant for low frequency part of the spectrum, while in
the high frequency part the main contribution is caused by the electron-hole
creation and subsequent recombination. This information can also be extracted
from the evolution of the high harmonic spectrum as a function of time. For
this propose a Blackman window of width $1.2\pi/\omega$ is scanned across $20$
optical cycles. The results along with the population of the conduction band
(unoccupied molecular orbitals)
$W\left(t\right)=\sum_{\mu=0}^{N/2-1}\rho_{\mu\mu}\left(t\right)$ are
displayed in Figs. 5(b) and 5(c). As is seen from these figures, the emission
of high harmonics takes place two times per wave cycle, corresponding to two
maxima or minima of the population. The low frequency harmonic bursts take
place in-between maxima and minima of the population Fig. 5(b), while higher
harmonics are the results of the recombination and the bursts take place at
minima of the population (Fig. 5(c)). There are also a domain of harmonics
where we have interplay between intra- and interband emission.

Conclusion– We revealed the general features of the HHG in fullerene molecule
under strong-field driving. The HHG spectra show multiple plateaus, which is
explained by the recombination of electrons and holes from molecular orbitals.
Those are intrinsic molecular excitations between the unoccupied molecular
orbitals and the occupied molecular orbital. These intrinsic molecular
excitations – so-called excitonic lines define the borders of the plateaus.
Within the each plateau, the cutoff harmonic linearly increases with
increasing the field strength. In contrast to atomic cases, for fullerene
molecule with the increase of the pump wave photon energy the cutoff harmonic
energy is increased. The HHG spectra strongly depends on the molecule symmetry
qualitatively as well as quantitatively. The C70 molecule shows more
pronounced nonlinear properties due to degradation of molecular symmetry
compared with C60. We also revealed the role of EEI. With the increase of the
EEI energy overall the HHG rate is suppressed. The fullerene molecules are
known to have different isomers with other point-group symmetries. Therefore,
they are interesting systems as a new sources of HHG, and spectroscopy based
on HHG might be useful to reveal the symmetries and electron dynamics
involved. Developing a detailed understanding of the HHG in different classes
of fullerene molecule is an interesting topic for the future work.

This work was supported by the RA State Committee of Science in the frame of
the research project 20TTWS-1C010.

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