figurec

# Supplemental material
Topological charges of periodically kicked molecules

Volker Karle, Areg Ghazaryan, Mikhail Lemeshko  Institute of Science and
Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria

##  I Technical details

###  I.1 The sudden approximation for short half-cycle pulses

In this section we justify the use of the sudden approximation for short half-
cycle pulses. For intense laser pulses that are much shorter than the
rotational timescale, it is possible to model the laser potential as a delta-
pulse Dion  _et al._ (2001); Seideman (1999); Dion  _et al._ (2002).
Separating the time scale of the rotation of the molecule $\tau_{B}=\pi/B$
from the fast oscillations of a single laser pulse simplifies the time-
dependent Schrödinger equation. In this work we make use of two kinds of
intense off-resonant laser pulses: (i) short multi-cycle infrared pulses that
couple to the molecular polarizability and (ii) half-cycle terahertz pulses
that couple to the permanent dipole moment of the molecule. The half-cycle
pulse is typically treated as a strong unipolar pulse $\propto
P_{\text{eff}}\cos(\theta)$, since one intense peak is followed by a long,
negative and weak tail Dion  _et al._ (1999); Mirahmadi  _et al._ (2018), see
Fig. S1. However, while the validity of the sudden approximation for infrared
pulse trains (i) is well studied in theory and experiment Bitter and Milner
(2016, 2017); Lin  _et al._ (2015), it remains to be shown for half-cycle
pulses (ii). Theoretically, the question is whether the time-evolution
operator that includes the full time-evolution of the laser intensity can be
approximated by a simplified propagator. For a rigid rotor driven by a
linearly polarized laser pulse with field amplitude $\mathcal{E}(t)$, the
Schrödinger equation reads Dion  _et al._ (1999, 2001):

$\mathrm{i}\mkern
1.0mu\partial_{t}|\psi(t)\rangle=\hat{H}(t)|\psi(t)\rangle\quad\text{with}\quad\hat{H}(t)=B\hat{\mathbf{L}}^{2}-\mu_{0}\mathcal{E}(t)\cos(\hat{\theta}),$
(S1)

with rotational constant $B$, angular momentum operator $\hat{\mathbf{L}}^{2}$
and permanent dipole moment $\mu_{0}$. In many works, the field amplitude is
assumed to take the form of a Gaussian. However, for realistic experiments
with molecular beams, the integral of field has to vanish in the far-field
regime Rauch and Mourou (2006); Arkhipov  _et al._ (2022); Barth  _et al._
(2008), i. e. $\int^{\infty}_{-\infty}\mathcal{E}(t)\mathrm{d}t=0$. In that
case, it is not trivial to show that such a pulse can be approximated by an
instantaneous pulse given the long negative tail of the field. It turns out
that the validity of the approximation strongly depends on the initial state,
the energy transferred to the molecule and the typical timescales (i.e. the
pulse duration and the decay time) of the laser. To elaborate further on this
matter, we choose a parametrization of the field amplitude with zero
integration measure Barth  _et al._ (2008),

$\mathcal{E}(t)=\begin{cases}0&(t\leq 0)\\\
\mathcal{E}_{1}\cos^{2}(\omega_{L}(t-t_{p})/2)\sin(\omega_{L}(t-t_{p}))&(0\leq
t<t_{p})\\\
\mathcal{E}_{2}\left(1-e^{-(t-t_{p})/\tau_{1}}\right)e^{-(t-t_{p})/\tau_{2}}&(t\geq
t_{p}),\end{cases}$ (S2)

with electric field amplitudes $\mathcal{E}_{1},\mathcal{E}_{2}>0$
(dimensionless in units of $B\equiv\mu_{0}\equiv\hbar\equiv 1$), the laser
frequency $\omega_{L}$, the pulse duration of the first part of the laser
pulse $t_{p}$, the switch-on and switch-off times $\tau_{1},\tau_{2}$. Here
all quantities are dimensionless, but we easily can convert the dimensionless
$\mathcal{E}$ to physical units by noting that $B=\hbar=1$ rescales time as
$\tilde{t}=B\cdot t/\hbar$. Then, the time-evolution of the potential reads
$\exp[-i\hat{V}\tilde{t}/B]$, hence the conversion factor of a field $E$ with
units of kV/cm is

$E\,[\mathrm{kV/cm}]=59.55\,[\mathrm{kV}\cdot\mathrm{Debye}]\cdot
B[\mathrm{cm}^{-1}]/\mu_{0}[\mathrm{Debye}]\cdot\mathcal{E}[\mathrm{dimensionless}]$
(S3)

For the OCS molecule frequently used in short-pulse alignment experiments
Chatterley  _et al._ (2020) ($B=0.203$ cm-1, $\mu_{0}=0.715$ D), this is for
example around $\approx 17\leavevmode\nobreak\ \mathrm{kV}/\mathrm{cm}$. That
implies that in order to get the physical units, one needs to multiply the
dimensionless $\mathcal{E}$ (e.g. from the $x$-axis of Fig. S3(a)) by this
factor.

As far as laser frequencies are concerned, the field has to be far-off-
resonant with respect to all rotational and vibrational transitions in the
molecule, and have a much lower frequency as compared to the electronic
excitations. In this case, at every moment of time during the pulse, the
electrons in the molecule instantaneously adjust to the laser field and the
treatment can be reduced to the effective interaction between the laser’s
electric field and the dipole moment (and, in the next order, dipole
polarizability) of the molecule. In order to reach the short-pulse regime,
$t_{p}\approx 1\%\tau_{B}$, a terahertz laser with $\omega_{L}\approx
4\leavevmode\nobreak\ \mathrm{Thz}$ can be used. The condition that the
electric field is smooth at $t=t_{p}$ further leads to
$\tau_{1}=\frac{\mathcal{E}_{2}}{\omega_{L}\mathcal{E}_{1}}$ and the zero
integration measure leads to
$\tau_{2}=(2\omega_{L}^{2}\tau_{1})^{-1}+\sqrt{(2\omega_{L}\tau_{1})^{-2}+(\omega_{L})^{-2}}$
(for more details see Ref. Barth  _et al._ (2008)). Clearly, the integration
measure of the positive peak scales as
$P_{\mathrm{eff}}\propto\mathcal{E}_{1}\cdot t_{p}$ (the negative tail
compensates for exactly this value). The decay time is determined by
$\tau_{2}$.

In order to study the validity of the sudden approximation, we numerically
integrated the full differential equation of the time-evolution operator
$\hat{U}(t)$ with $\mathrm{i}\mkern
1.0mu\partial_{t}\hat{U}(t)=\hat{H}(t)\hat{U}(t)$ for a reasonable cut-off
$l<l_{\mathrm{max}}$ and various parameters. Generally speaking, each angular
momentum eigenstate oscillates with frequency
$\omega_{\mathrm{rot}}(l)=\pi[l(l+1)]$ (in units of rotational times
$\tau_{B}$), which leads to a natural cut-off scale; the approximation can
only succeed for states with $\langle l|\psi\rangle\approx 0$ for $l$ with
$\omega_{L}<\omega_{\mathrm{rot}}(l)$. The eigenstates $l$ with
$\omega_{L}<\omega_{\mathrm{rot}}(l)$ oscillate with a frequency equal or
higher than the laser frequency and a separation of timescales is not
possible. For the sudden approximation with $\hat{U}^{\mathrm{sudd}}(t)\approx
e^{-\mathrm{i}\mkern
1.0mu(t-t_{p})\hat{H}_{0}}e^{i\hat{V}_{\mathrm{sudd}}}e^{-\mathrm{i}\mkern
1.0mut_{p}\hat{H}_{0}}$ where $\hat{V}_{\mathrm{sudd}}=P\cos(\hat{\theta})$,
only the off-diagonal matrix element $\langle l\pm
1|\hat{V}_{\mathrm{sudd}}|l\rangle$ are non-zero (c.f. Eq. (S4)). Whether or
not the full time evolution can be approximated by that expression depends
therefore on the question to what extend the matrix elements of
$V_{\mathrm{eff}}=-\mathrm{i}\mkern 1.0mu\log[e^{+\mathrm{i}\mkern
1.0mu(t-t_{p})\hat{H}_{0}}\hat{U}(t)e^{+\mathrm{i}\mkern
1.0mut_{p}\hat{H}_{0}}]$ resemble the ones of $V_{\mathrm{sudd}}$ for $t\gg
t_{p}$ and $l<l_{\mathrm{max}}$. In Fig. S2 we calculate the time-evolution of
a rotational wavepacket with (a) the full propagator and (b) the sudden
approximation propagator. In that case, the approximation works very well and
the two trajectories are practically the same.

To analyze in detail in what regime the sudden approximation works well is out
of the scope of this work, however, we would like to shortly discuss the
convergence of the full propagator to the propagator of the sudden
approximation in certain limits. To this end, we compare the matrix element of
the full propagator $v_{l}=\langle l+1|V_{\mathrm{eff}}|l\rangle$ with the
exact term $w_{l}\equiv\langle l+1|\cos(\hat{\theta})|l\rangle$ and calculate
the ratio $p_{l}=v_{l}/w_{l}$. For realistic scenarios, we care about the low
angular momentum regime $l<l_{\mathrm{mean}}\ll l_{\mathrm{max}}$ and estimate
$P_{\mathrm{eff}}$ with the average $P_{\mathrm{eff}}=\langle{p_{l}}\rangle$
and the error with $\delta P_{\mathrm{eff}}=\sqrt{\langle
p_{l}^{2}\rangle-{\langle p_{l}\rangle}^{2}}$, where the averages are defined
as sums up to some $l_{\mathrm{mean}}$, depending on the experimental setup
and the number of kicks. In the case when the matrix-elements of the full
propagator converges to the approximation, we expect $\delta
P_{\mathrm{eff}}/P_{\mathrm{eff}}\rightarrow 0$. Numerically, this is what we
numerically observe for the limits $\mathcal{E}_{1}\rightarrow 0$,
$\mathcal{E}_{2}/\mathcal{E}_{1}\rightarrow 0$, and $t_{p}\rightarrow 0$, see
Fig. S3. These results indicate that the sudden approximation is a valid
assumption in a number of cases. Specifically, the ratio
$\mathcal{E}_{2}/\mathcal{E}_{1}$ strongly depends on the specific mechanism
of the pulse generation and the switch-off time. On top of that, the
characteristics of the laser with frequency $\omega_{L}$ already govern for
which electric field amplitudes the sudden approximation can be used. However,
the existence of Dirac cones in the Floquet spectrum does not rely on the
strength of laser pulses alone, but on the ratio $P_{1}/P_{2}$, see Fig. S6
and the corresponding text. Therefore, even for small field amplitudes it is
possible to observe Dirac cones, given that the IR pulse strength is of the
same order of magnitude as the Terahertz pulse in terms of transferred $\hbar$
per pulse (which does not depend on $\mathcal{E}_{1,2}$ alone, but also on
$t_{p}$).

Figure S1: The pulse shape of (S2) taken from Ref. Barth  _et al._ (2008) for
different parameters. The field amplitude $\mathcal{E}_{1}$ determines the
height of the positive pulse, and $\mathcal{E}_{2}$ the depth of the negative
tail. The pulse duration is determined by $t_{p}=\pi/\omega_{L}$ with the
laser frequency $\omega_{L}$ and the decay time is given by the ratio
$\mathcal{E}_{2}/\mathcal{E}_{1}$ (see main text); a larger ratio leads to
faster decay time, but larger negative peak, since the integral over
$\mathcal{E}(t)$ is zero. For the sudden approximation to hold, this ratio
needs to be small enough (i. e. $\mathcal{E}_{1}\gg\mathcal{E}_{2}$) to ensure
an approximately positive Gaussian peak, otherwise the potential will vary
considerably. Further, it is important that the pulse duration
$t_{p}\ll\tau_{B}$ to ensure a separation of the timescales of the laser and
the rotation. Figure S2: Time-evolution of a rotational wavepacket with
$\langle l=0|\psi(t=0)\rangle=1$, driven by a half-cycle pulse train,
calculated with (a) the exact time-evolution operator $U(t)$ and (b) the-
evolution operator $U^{\mathrm{sudd}}(t)$ of the sudden approximation. The
parameters of the half-cycle pulse are
$\epsilon_{1}=30,\epsilon_{2}=2,\tau_{p}\cdot\pi=0.3\tau_{B}$ and
$T=\tau_{B}/3$. The effective kicking strength (i. e. the number of
transferred $\hbar$ per kick) is estimated (optimizing the overlap) as
$P_{\mathrm{eff}}\approx 2.879$. Since the pulse duration is comparatively
long, one has to correct the effective pulse position for the sudden
approximation to $t_{p,\mathrm{eff}}\cdot\pi\approx 0.222\tau_{B}$ (i. e. the
operator is $U=e^{-\mathrm{i}\mkern
1.0mu(t-t_{p})\hat{H}_{0}}e^{+i\hat{V}}e^{-\mathrm{i}\mkern
1.0mut_{p}\hat{H}_{0}}$). We observe that at this level of accuracy it is
impossible to see a difference between the two different time evolutions. For
initial states at large $l$, it is possible distinguish the two operators.
However, for realistic experimental setups it is sufficient to understand the
qualitative behavior of the sudden approximation. Figure S3: In (a) and (b) we
show the dependence of the effective kicking strength $P_{\mathrm{eff}}$ on
the half-cycle field amplitudes $\mathcal{E}_{1},\mathcal{E}_{1}$ and the
pulse duration $t_{p}$ (in units of the rotational period $\tau_{B}=\pi/B$),
see (S2) for the parametrization of the half-cycle pulse. The effective
kicking strength is calculated using the method in the text with
$P_{\mathrm{eff}}=\langle p_{l}\rangle$ with the error $\delta
P^{2}_{\mathrm{eff}}=\langle p_{l}^{2}\rangle-{\langle p_{l}\rangle}^{2}$ with
a summation up to $l=50$ and a maximum $l_{\mathrm{max}}=150$. We observe that
for the parameters shown here the errors are relatively small and the
agreement with the sudden approximation is therefore very good. In (a), for
increasing $\mathcal{E}_{1}$, the errors become larger in agreement with the
expectation for large field amplitudes the exact time evolution becomes
important. The same reasoning applies to (b), which shows that for vanishing
pulse duration, the sudden approximation becomes exact. In (c) we show how the
relative error $\Delta P_{\mathrm{eff}}/P_{\mathrm{eff}}$ scales with the
ratio $\mathcal{E}_{2}/\mathcal{E}_{1}$. We see that it falls off
algebraically in the limit $\mathcal{E}_{2}/\mathcal{E}_{1}\rightarrow 0$.
This shows that in this limit the full time-evolution converges to the sudden
approximation. The reason is that the decay time is controlled by exactly this
ratio, and for decreasing ratios the decay time goes to infinity. The
reasoning is that for decay times larger than the pulse duration $t_{p}$, the
sudden limit is valid. A similar convergence can be observed for the limits
$t_{p}/\tau_{B}\rightarrow 0$ and the trivial limit
$\mathcal{E}_{1,2}\rightarrow 0$. We note that in order to observe the
convergence of the $\mathcal{E}_{2}/\mathcal{E}_{1}\rightarrow 0$ limit, the
pulse duration needs to be small enough, i.e. $t_{p}\ll\tau_{B}$. Further, the
limitations of the approximation which were mentioned in the main text remain,
i. e. the approximation only holds up to $l$ with
$\omega_{\mathrm{rot}}(l)\ll\omega_{L}$.

###  I.2 Fourier transforms of angular momentum lattices

Here we show that the Fourier transform of the angular momentum lattice
provides a viable tool to analyze the Floquet spectra at quantum resonances of
a periodically driven molecule. As elaborated in the main text, at a quantum
resonance, the time-translational operator takes the form
$\hat{U}=e^{-\pi\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}/N}e^{\mathrm{i}\mkern 1.0mu\hat{V}(P_{1},P_{2})}$.
For odd $N$, the rotational part is periodic with $N$, while for even $N$ the
periodicity is $2N$, which becomes evident when looking in the $l$-basis with
$\langle l|e^{-\pi\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}/N}|l^{\prime}\rangle=e^{-i\pi
l(l+1)/N}\delta_{ll^{\prime}}$. Our approach works for all laser potentials
which converge to a constant value, i. e. $V_{l,l^{\prime}}\approx
V_{l+1,l^{\prime}+1}$, which is in particular true for the combination of a
“regular” multi-cycle laser pulse (the $\cos^{2}(\hat{\theta})$ term) and a
half-cycle laser pulse (the $\cos(\hat{\theta})$ term), which can be proven
using the Edmonds asymptotic formula for the $3j$-symbols Flude (1998). The
asymptotic expressions take the simple form i. e.

$\displaystyle\langle l^{\prime}m^{\prime}|\cos(\theta)|lm\rangle$
$\displaystyle=-\delta_{mm^{\prime}}C^{l^{\prime}m}_{lm10}C^{l0}_{l^{\prime}010}\xrightarrow[]{\text{for
$l,l^{\prime}\gg
0$}}\delta_{mm^{\prime}}\left(\delta_{l,l^{\prime}+1}+\delta_{l,l^{\prime}-1}\right)/2$
(S4) $\displaystyle\langle l^{\prime}m^{\prime}|\cos^{2}(\theta)|lm\rangle$
$\displaystyle=\delta_{mm^{\prime}}\left(\tfrac{2}{3}C^{l^{\prime}m}_{lm20}C^{l0}_{l^{\prime}020}+\tfrac{1}{3}\delta_{ll^{\prime}}\right)\xrightarrow{\text{for
$l,l^{\prime}\gg
0$}}\delta_{mm^{\prime}}\left(\delta_{l,l^{\prime}}+(\delta_{l,l^{\prime}+2}+\delta_{l,l^{\prime}-2})/2\right)/2$
(S5)

with the usual Clebsch–Gordan coefficients
$C^{LM}_{l^{\prime}m^{\prime}lm}=\langle
l^{\prime},l,;m^{\prime},m|l^{\prime},l;L,M\rangle$. To prove this asymptotic,
one can use the Edmonds asymptotic formula for 3j-symbols Flude (1998)

${\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}}\xrightarrow[]{\text{for $l_{2},l_{3}\gg
l_{1}$}}(-1)^{l_{3}+m_{3}}{\frac{d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta)}{\sqrt{l_{2}+l_{3}+1}}}\quad\text{with}\quad\cos(\theta)=\frac{m_{2}-m_{3}}{l_{2}+l_{3}+1},$
(S6)

where $d^{l}_{m,m^{\prime}}(\theta)$ is the Wigner function and 3j-symbols are
related to the Clebsch-Gordan coefficients varshalovich_quantum_1988

$\langle
j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle=(-1)^{-j_{1}+j_{2}-M}{\sqrt{2J+1}}{\begin{pmatrix}j_{1}&j_{2}&J\\\
m_{1}&m_{2}&-M\end{pmatrix}}.$ (S7)

For Eq. (S4) this leads to

$C^{l^{\prime}m}_{lm10}C^{l0}_{l^{\prime}010}=(-1)^{-m}\sqrt{(2l^{\prime}+1)(2l+1)}\begin{pmatrix}1&l^{\prime}&l\\\
0&m&m\end{pmatrix}\begin{pmatrix}1&l&l^{\prime}\\\
0&0&0\end{pmatrix}\xrightarrow[]{\text{for $l,l^{\prime}\gg
0$}}(-1)^{l+l^{\prime}}d^{1}_{0,l-l^{\prime}}(\nicefrac{{\pi}}{{2}})d^{1}_{0,l^{\prime}-l}(\nicefrac{{\pi}}{{2}})$
(S8)

where we used that $\theta=\pi/2$ for both terms. Since the Clebsch-Gordan
coefficients allow only $l-l^{\prime}=\pm 1$ and $d^{1}_{0,\pm
1}(\nicefrac{{\pi}}{{2}})d^{1}_{0,\mp
1}(\nicefrac{{\pi}}{{2}})=\nicefrac{{1}}{{2}}$ this leads to the desired
relation. For Eq. (S5) this derivation proceeds analogously, with the
difference that $d^{2}_{0,\pm 2}(\nicefrac{{\pi}}{{2}})d^{2}_{0,\mp
2}(\nicefrac{{\pi}}{{2}})=\nicefrac{{3}}{{8}}$ and $d^{2}_{0,\pm
2}(\nicefrac{{\pi}}{{2}})d^{2}_{0,\mp
2}(\nicefrac{{\pi}}{{2}})=\nicefrac{{3}}{{8}}$ and
$d^{2}_{0,0}(\nicefrac{{\pi}}{{2}})d^{2}_{0,0}(\nicefrac{{\pi}}{{2}})=\nicefrac{{1}}{{4}}$,
which in total leads to the desired result. For the projection of angular
momentum $m=0$ these coefficients converge fast, see Fig. S4, and the overall
behavior of the system is very well described by an approximation where we
assume a constant value. In this work we only consider $m=0$, but for other
$m$ the approximations work out as well when considering larger $l$.

Figure S4: Convergence of the hopping elements for $l,l^{\prime}\gg 0$ to a
constant value. In (a) we show the element which is relevant for the
$\cos(\theta)$ potential and in (b) for the $\cos^{2}(\theta)$. In (c) we show
the real part of the effective hopping elements $\langle
l^{\prime}|U|l\rangle$ where the time-translation operator is
$\hat{U}=e^{-\pi\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}/N}e^{\mathrm{i}\mkern 1.0mu\hat{V}(\hat{\theta})}$,
with $N=3,P=2.5$ and $\alpha=\pi/3$ for the potential
$V(\hat{\theta})=P\cdot\left(\cos^{2}(\alpha)\cos^{2}(\hat{\theta})+\sin^{2}(\alpha)\cos(\hat{\theta})\right)$
as in the rest of the letter. We marked the unit-cell size $N=3$ by gray
lines. Note that the hopping elements are inherently complex, as the driven
molecule is a non-hermitian system.

The $N$-periodicity allows us to define a Fourier transform which approximates
the periodic behavior for $l,l^{\prime}\gg 0$. Any operator $\hat{A}$ with
that periodicity, and the effective Hamiltonian in particular, follows

$\langle l^{\prime}|\hat{A}|l\rangle\approx\langle
l^{\prime}+N|\hat{A}|l+N\rangle$ (S9)

for $l,l^{\prime}\gg 0$. Let us parametrize each $l^{\prime},l$ by $l=n\cdot
N+i,l^{\prime}=n^{\prime}\cdot N+j$ with
$n,n^{\prime}\in\mathbb{N}_{0},i,j\in\\{0,1,\dots,N\\}$. Then,

$A_{l^{\prime}l}=A_{i,j}(n^{\prime},n)=A_{i,j}(n^{\prime}-n)=A_{i,j}(\Delta
n)$ (S10)

i. e. $A$ only depends on the off-diagonal index $\Delta n$ because of the
periodicity. The Fourier transform from $l$\- to $k$-space then becomes (for a
dimensionless momentum variable $k\in[0,2\pi]$)

$\displaystyle\underbrace{A_{ij}(k^{\prime},k)}_{N\text{x}N\text{Matrix}}$
$\displaystyle=\sum_{n^{\prime},n}e^{-\mathrm{i}\mkern 1.0mu(n^{\prime}\cdot
k^{\prime}-n\cdot k)}A_{ij}(n^{\prime},n)=\sum_{\Delta
n,\bar{n}}e^{-\mathrm{i}\mkern 1.0mu(\bar{n}(k^{\prime}-k)/2+\Delta
n(k^{\prime}+k)/2)}A_{ij}(\Delta n)=\delta_{kk^{\prime}}\sum_{\Delta
n}e^{-\mathrm{i}\mkern 1.0mu\Delta n\cdot k}A_{ij}(\Delta n),$ (S11)
$\displaystyle\text{i.e.}\quad A_{ij}(k)$ $\displaystyle=\sum_{\Delta
n}e^{-\mathrm{i}\mkern 1.0mu\Delta n\cdot k}A_{ij}(\Delta n)\quad\text{and
correspondingly}\quad A_{ij}(\Delta
n)=\frac{1}{2\pi}\int_{0}^{2\pi}A_{ij}(k)\mathrm{d}k.$

where we used $\bar{n}=n^{\prime}+n,\,\Delta n=n^{\prime}-n$ and
$A_{ij}(n^{\prime},n)=A_{ij}(\Delta n)$, which shows that operator $\hat{A}$
conserves momentum $k$. Since the Fourier transform is only well defined for
an infinite system, for a finite system this implies practically an infrared
cutoff in momentum which is defined by the system size. When calculating the
Fourier transform of an operator, one needs to choose a unit-cell in the
middle of the lattice to avoid edge effects. Practically, we choose
$A_{ij}(\Delta n)=A_{N\cdot n_{0}+i,N\cdot(n_{0}+\Delta n)+j}$ with a large
enough $n_{0}$ to ensure convergence. For an infinite and perfectly
translationally invariant system, we would sum over all $\Delta
n\in\mathbb{Z}$. However, here we need to choose $-n_{0}\ll\Delta n\ll n_{0}$.
For small to moderate $P$, only few off-diagonals suffice to achieve high
agreement with the “real-space” diagonalization. A state transforms with

$f_{i}(k)=\sum_{\Delta n}e^{-\mathrm{i}\mkern 1.0mu\Delta n\cdot
k}f(i+N(n_{0}+\Delta n)),\quad f(i+N(n_{0}+\Delta
n))=\frac{1}{2\pi}\int_{0}^{\infty}f_{i}(k)e^{+\mathrm{i}\mkern 1.0mu\Delta
n\cdot k}\mathrm{d}k.$ (S12)

The fourier transform of the operators at $l^{\prime},l\gg 0$ in (S4) and (S5)
is given by

$(\cos(\hat{\theta}))_{ij}(k)=\frac{1}{2}\underbrace{\begin{pmatrix}0&1&&&&&\dots&&e^{-ik}\\\
1&0&1&&&&&&\\\ &1&0&&&&&&\\\ \vdots&&&\ddots&&&&&\vdots\\\ &&&&&&&1&\\\
&&&&&&1&0&1\\\ e^{+ik}&&\dots&&&&&1&0\\\ \end{pmatrix}}_{N\times N\text{
Matrix}}$ (S13)
$\text{and}\quad(\cos^{2}(\hat{\theta}))_{ij}(k)=\frac{1}{2}\underbrace{\begin{pmatrix}1&0&\nicefrac{{1}}{{2}}&&&&\dots&e^{-ik}/2&0\\\
0&1&0&\nicefrac{{1}}{{2}}&&&&\dots&e^{-ik}/2\\\
\nicefrac{{1}}{{2}}&0&1&0&\nicefrac{{1}}{{2}}&&&&\\\
&\nicefrac{{1}}{{2}}&0&1&0&&&&\\\ \vdots&&&\ddots&&&&&\vdots\\\
&&&&&&1&0&\nicefrac{{1}}{{2}}\\\ e^{+ik}/2&&&&&&0&1&0\\\
0&e^{+ik}/2&\dots&&&&\nicefrac{{1}}{{2}}&0&1\\\ \end{pmatrix}.}_{N\times
N\text{ Matrix}}$ (S14)

To calculate $e^{\mathrm{i}\mkern 1.0mu\hat{V}}$ it is sufficient to
diagonalize $V_{i,j}(k)$ and express the exponential in terms of its
eigenstates $v_{n}(k)$ and eigenvalues $\lambda_{n}(k)$, i. e.
$e^{\mathrm{i}\mkern 1.0muV(k)}=\sum_{n}e^{\mathrm{i}\mkern
1.0mu\lambda_{n}(k)}\left(v_{n}(k)\otimes v^{\dagger}_{n}(k)\right).$

###  I.3 Symmetries

Figure S5: Comparison of two diagonalization methods for
$P_{1}=1.5,P_{2}=1.8$. The left figure shows the quasienergies sorted by their
relative index, and in the right figure depending on their momentum $k$. The
agreement of the two diagonalization methods are high, implying that the bulk
behavior dominates the spectrum. In the left figure the diagonalization in
$l-$space gives rise to two in-gap states, which are the edge modes mentioned
in the main text. They cannot be obtained using the momentum space
diagonalization for obvious reasons. Also visible in the spectrum obtained in
$k-$space is the reflection symmetry, which guarantees
$\epsilon_{k}=\epsilon_{-k}$.

The rotational phases, $\phi_{l}=e^{-\pi l(l+1)/N}$, give rise to reflection
symmetry for all $N$ with translationally-invariant potentials (i. e. in our
case for $l^{\prime},l\gg 0$). For an odd $N$, the reflection center of the
first unit cell is $n_{c}=(N+1)/2\in\mathbb{N}$. Then, for a $0\leq j<n_{c}$
we have $\phi_{n_{c}+j}=\phi_{n_{c}-j}$ (for even $N$, the unit-cell is $2N$
and $n_{c}=N$ with $\phi_{n_{c}+j+1}=\phi_{n_{c}-j}$). If we limit ourselves
to a finite Hilbert space with maximum $l_{\mathrm{max}}$, we can write the
symmetry in the $l$-basis as

$\mathcal{R}=\underbrace{\begin{pmatrix}&&\dots&0&1\\\ &&&1&0\\\
\vdots&&\iddots&&\vdots\\\ 0&1&&&\\\
1&0&\dots&\end{pmatrix}}_{(l_{\mathrm{max}}+1)\times(l_{\mathrm{max}}+1)\text{
Matrix}}$ (S15)

with $\mathcal{R}^{\dagger}\mathcal{R}=\mathcal{R}^{2}=\mathbb{I}$. It
commutes with the time-translation operator, $[U,\mathcal{R}]=0$. In terms of
the effective Hamiltonian, $\hat{H}=\mathrm{i}\mkern 1.0mu\log\hat{U}$, this
symmetry reads in $k$-space $\mathcal{R}H_{k}=H_{-k}\mathcal{R}$ with

$\mathcal{R}=\underbrace{\begin{pmatrix}&&\dots&0&1\\\ &&&1&0\\\
\vdots&&\iddots&&\vdots\\\ 0&1&&&\\\ 1&0&\dots&\end{pmatrix}}_{N\times N\text{
Matrix}}\mathcal{P}\quad{\text{and the parity operator
with}}\quad\mathcal{P}\phi_{k}=\phi_{-k}.$ (S16)

For the one-kick system, shifting the position of the kick does not change the
model, i. e.

$U(\beta)=e^{-\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}\pi(1-\beta)/N}e^{\mathrm{i}\mkern
1.0muV_{1}}e^{-\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}\pi\beta/N}\quad\sim\quad U=e^{-\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}\pi/N}e^{\mathrm{i}\mkern 1.0muV_{1}}$ (S17)

where the $\sim$ implies that the two operators have the same spectrum up to a
constant for any $0\leq\beta\leq 1$. However, as we have seen above, a finite
$\beta$ leads to different $N-$periodicity. Let us write $\beta=\frac{a}{b}$
with $a,b\in\mathbb{N},a<b$. Then, for incommensurable $a,b\cdot N$, the
periodicity of the operator will be $b\cdot N$ (correspondingly, for
commensurable $a$ and $N\cdot b$ it will be the lowest common denominator).
The equivalence of the spectra then leads to $bN$ bands in place of $N$ bands
which compensates for the larger unit-cell. This is a well-known phenomenon
commonly known as “band folding” in solid-state physics.

Figure S6: Smallest quasienergy differences (i. e. smallest band gap) for
$k=\pi$ (left figure) and $k=0$ (right figure). For small to intermediate
couplings the only way to achieve gap closings is with $P_{1}\neq 0,P_{2}\neq
0$.

In addition to that, we find a time-reflection symmetry present for all models
with one kick in the Floquet operator (where we use that our laser potentials
follow $V^{*}=V=V^{T})$. More specifically, the second kick (for instance,
$U_{\text{2-kick}}=e^{-\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}\pi/N}e^{\mathrm{i}\mkern
1.0muV_{1}}e^{-\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}\pi/N}e^{\mathrm{i}\mkern 1.0muV_{2}}$ with
$V_{1}\neq V_{2}$ ) would break the time-reflection invariance. The time-
reflection invariance is proportional to unity times conjugation when the
pulse is at the center, i. e. $\beta=\nicefrac{{1}}{{2}}$. In that gauge, we
find

$\langle
l^{\prime}|U^{*}(\nicefrac{{1}}{{2}})U(\nicefrac{{1}}{{2}})|l\rangle=\sum_{l^{\prime\prime}}\langle
l^{\prime}|e^{+\mathrm{i}\mkern 1.0mu\pi
l^{\prime}(l^{\prime}+1)/2N}e^{-\mathrm{i}\mkern
1.0muV}|l^{\prime\prime}\rangle e^{+\mathrm{i}\mkern 1.0mu\pi
l^{\prime\prime}(l^{\prime\prime}+1)/2N}e^{-\mathrm{i}\mkern 1.0mu\pi
l^{\prime\prime}(l^{\prime\prime}+1)/2N}\langle
l^{\prime\prime}|e^{\mathrm{i}\mkern 1.0muV}e^{-\mathrm{i}\mkern 1.0mu\pi
l(l+1)/2N}|l\rangle=\delta_{l^{\prime}l}$ (S18)

which is equivalent to the time-reflection symmetry of the effective
Hamiltonian $H(\beta)=H^{*}(\beta)$. To construct $\mathcal{T}$ for arbitrary
gauges $\beta$, we only need to bring $U$ (or $H$, respectively) back to the
$\beta=\nicefrac{{1}}{{2}}$ gauge. This is accomplished by the gauge
transformation $\mathcal{U}(\beta)$ with

$\mathcal{U}(\beta)=e^{-\mathrm{i}\mkern
1.0mu\hat{\mathbf{L}}^{2}\pi(2\beta-1)/(2N)}\quad\text{with}\quad\mathcal{U}^{\dagger}(\beta)U(\beta)\mathcal{U}(\beta)=U(\nicefrac{{1}}{{2}}).$
(S19)

The (anti-unitary) time-reflection operator then takes the form
$\mathcal{T}(\beta)=\mathcal{U}(\beta)\,\mathcal{C}\,\mathcal{U}^{\dagger}(\beta)$
with conjugation $\mathcal{C}$. The time-translation operator then fulfills
$\mathcal{T}(\beta)U(\beta)=U^{\dagger}(\beta)\mathcal{T}(\beta)$, while the
Hamiltonian $[H(\beta),\mathcal{T}(\beta)]=0$. In $k$-space, this leads to

$\mathcal{T}(\beta)H_{k}(\beta)=H_{-k}^{*}(\beta)\mathcal{T}(\beta).$ (S20)

###  I.4 $N=3$ case: Dirac cones

For $N=3$, the Fourier transform of the laser potential becomes

$\displaystyle V(k)$
$\displaystyle=\frac{P_{1}}{2}\begin{pmatrix}1&e^{-ik}/2&\nicefrac{{1}}{{2}}\\\
e^{ik}/2&1&e^{-ik}/2\\\
\nicefrac{{1}}{{2}}&e^{ik}/2&1\end{pmatrix}+\frac{P_{2}}{2}\begin{pmatrix}0&1&e^{-ik}\\\
1&0&1\\\ e^{ik}&1&0\end{pmatrix}$ (S21)
$\displaystyle=\frac{1}{2}\begin{pmatrix}P_{1}&P_{1}e^{-ik}/2+P_{2}&P_{1}/2+P_{2}e^{-ik}\\\
P_{1}e^{ik}/2+P_{2}&P_{1}&P_{1}e^{-ik}/2+P_{2}\\\
P_{1}/2+P_{2}e^{ik}&P_{1}e^{ik}/2+P_{2}&P_{1}\end{pmatrix}.$

The reflection symmetry $1\leftrightarrow 3,k\leftrightarrow-k$ is clearly
visible here. In Fig. S5 we show a comparison of the diagonalization in
$l-$space and $k-$space. For small to intermediate coupling ($P\sim 5$) the
agreement is very good. If we look at the smallest energy difference
$d(P_{1},P_{2})=\min_{m,n\in\\{1,2,3\\}}\left(|\epsilon_{n}-\epsilon_{m}|_{2\pi}\right)$
with the $2\pi-$periodic distance, i. e.

$|x-x^{\prime}|_{2\pi}=\min(|x-x^{\prime}|,|x-x^{\prime}+2\pi|,|x-x^{\prime}-2\pi|),$
(S22)

then we observe that the gap closings occur for $k=0$ and $k=\pi$ on straight
lines, see Fig. S6. Note that the linearity of the gap closings can be turned
into non linear curves using multiple pulse schemes and more complicated
potentials.

Figure S7: The three energy bands, $\epsilon_{n}(k,\alpha)$ at $P=2.5$, from
(a) lowest to (c) highest. The dashed lines mark exemplary integration
contours around the Dirac cones, which provide a way to evaluate the
topological charge of the cones, see text. The color of the lines denote the
result of the integration with $+\pi$ (red lines) and $-\pi$ (blue lines).

Furthermore, we mention in the main text that the topological charge of the
Dirac cones can be evaluated numerically. To this end, we combine the two
variables as in the main text into a vector $\mathbf{k}=(k,\alpha),$ and
compute the Berry connection, which is well-defined if we choose a gauge such
that the eigenstates form a smooth manifold. Choosing a path $\gamma$ around
the cones (in the second or third band), the integration of the Berry
connection leads to an integer-$\pi$ Berry phase (see Fig. S7 for exemplary
contours and the result of the numerical integration Fukui  _et al._ (2005))

$\mathcal{A}_{n}(\mathbf{k})=i\langle\psi_{n}(\mathbf{k})|\nabla_{\mathbf{k}}\psi_{n}(\mathbf{k})\rangle,\quad\int_{\gamma}\mathcal{A}(\mathbf{k})_{n}\mathrm{d}\mathbf{k}=\pm\pi.$
(S23)

##  II Possible experimental realizations of Dirac cones

The origin of this behavior can be explained by analyzing the eigenstates of
the Hamiltonian. At each moment of time, the eigenstates
$\psi(k,\alpha)\in\mathbb{C}^{3}$ can be obtained by

$H(k,\alpha(t))\psi(k,\alpha(t))=\epsilon(k,\alpha(t))\psi(k,\alpha(t))$ (S24)

with $\alpha(t)=t\cdot(\alpha_{c}/t_{c})$ and the critical value $\alpha_{c}$
with the Dirac cone, see Fig. S8. In Fig. S9, we recognize the Dirac cone as a
vortex in the expectation value of the eigenstate
$\langle\psi_{k}|\cos^{2}(\hat{\theta})|\psi_{k}\rangle$. The Dirac cone
causes a drastic change in the orientation and alignment of the eigenstates. A
state $\phi(t)$ which consists of a superposition of different momentum
eigenstates mimics their evolution accordingly. The drastic change is only
observed for states with large occupation at $k=\pi$, since that is the
position of the Dirac cone at $t=t_{c}$. The states for the time-evolution in
the main text are obtained as follows. We begin with a Gaussian state in
angular momentum space, $\phi_{0}\propto e^{-(l-l_{0})^{2}/\Delta l^{2}}$ and
project it into the third band, i.e.
$|\phi_{0}\rangle\rightarrow\sum_{i\in\mathcal{N}_{3}}|\psi_{i}\rangle\langle\psi_{i}|\phi_{0}\rangle$
with all Floquet states $i\in\mathcal{N}_{i}$ of the third band. Then, in
order to guarantee a peak at $k=\pi$, we convolve this state with a Gaussian
distribution centered at $k_{0}=\pi$, that is

$\tilde{\phi}_{k}\propto\int_{0}^{2\pi}e^{-(k-k_{0})/2\Delta
k^{2}}(\phi_{k}\cdot\psi^{*}_{k})\psi_{k}\,\mathrm{d}k$ (S25)

where we choose $\Delta k=0.5$. Since this convolution removes overlap with
the original Gaussian distribution in $l$-space, it can lead to a
delocalization of the former localized state. Clearly, this is a very simple
procedure and choosing a more complicated initial state with a well-defined
momentum distribution is also possible. Further, in the main text we also show
the results for a generic state that was created using only one pulse pulse
from $l=0$ (see parameters of this state in the caption of Fig. S10). As
already mentioned in the main text, in order to see a difference in the
alignment/orientation signal, one needs to start from a state which is
somewhat aligned or oriented, that is one need to choose an appropriate
strength in the pulse.

Figure S8: The quench protocol of the main text and the corresponding changes
of the spectrum in time. For the chosen parameters, $P_{1}=1.5,P_{2}=1.6$, the
critical value of the parameter is $\alpha_{c}\approx 0.6221$. The
parametrization reads $\alpha(t)=t\cdot(\alpha_{c}/t_{c})$, where we choose
$t_{c}=15$. As for the spectrum in the main text, for $\alpha>\alpha_{c}$ two
edge states emerge in the middle of the two upper bands. They can only be
removed by a second cone in between the two bands. Figure S9: The orientation
and alignment signals of the instantaneous eigenstates of the Hamiltonian at a
given moment of time. The panels (a),(b),(c) correspond to the first, second
and third band, respectively. In the first row we show the expectation value
of $\cos(\theta)$ and in the second of $\cos^{2}(\theta)$. The red dashed line
marks the occurrence of the Dirac cone. The Dirac cone at $t_{c}=15$ leads to
a vortex-like structure in the phases of the eigenstates, which translates
into a similar vortex in orientation/ alignment signals. For $k=\pi$, the
change is most drastic, e.g.
$\langle\psi_{k=\pi}|\cos(\hat{\theta})|\phi_{k=\pi}\rangle$ changes sign
after passing $t_{c}$. This figure explains the pronounced changes in the
signals of Gaussian wavepackets peaked at $k=\pi$, as shown in the main text,
when assuming a quasi-adiabatic time evolution. The change of the behavior of
the eigenstates can be explained using the inversion quantum number $n_{i}=\pm
1$, given by the eigenvalues of the inversion operator defined earlier for
inversion-invariant momenta $k=\pi$ and $k=0$. At these momenta, the
Hamiltonian becomes block-diagonal. The Dirac cones furnish an exchange of a
$n=+1$ state with the $n=-1$ between the bands while maintaining the band
structure. This exchange of the states leads to a drastic change of the
alignment/orientation signal of the whole band after the cone. Figure S10:
Time evolution of the states discussed in the main text, also see (S25). (a)
and (b) show Gaussian states with $\Delta l=\Delta k=0.5$ with (a) $l_{0}=30$
and (b) $l_{0}=0$. (c) shows a “generic” state created by a laser pulse
starting from $l=0$, $|\phi\rangle=e^{-i\Delta
t\hat{L}^{2}}e^{iV(P_{1},P_{2})}|l=0\rangle$, with $\Delta t\approx
0.911,P_{1}\approx 6.67,P_{2}\approx 2.67$. These parameters ensure that the
state has a large overlap with the third band and is sufficiently occupied
around $k=\pi$. We observe that in all three exemplary time evolutions the
behavior changes drastically in the vicinity of the Dirac cone. This can be
explained by the change of the state due to the band crossing and its spread
over different bands, see Fig. S9 and Fig. S11. Ultimately, the red line marks
a topological transition, where also edge states appear in the middle of the
gap from the cone, and they can also be populated.

In Fig. S10 we show the time evolution of the states discussed in the main
text. In the beginning of the time evolution, the states do not spread but
remain more or less localized, since only a single band is occupied. After
crossing the Dirac cone, more bands are occupied (this is partly a non-
adiabatic effect) and the wavepacket spreads. However, the change of the
alignment/orientation measures is due to the change of the eigenstate, not due
to the change of bands, as can be inferred from Fig. S9 (see also caption
within, where we explain in detail the role of the quantum number of the
inversion symmetry at the transition). The Dirac cone marks the position where
these measures suddenly change, and henceforth also the states which depend on
them. This implies that by tuning the position of the Dirac cone in terms of
the laser intensities, we can also tune when this changes happen for an
arbitrary wave function. In Fig. S11 we show the momentum-dependent time-
evolution of a wavepacket, determined by a numerical Fourier transform, and in
Fig. S12 we show the time evolution for different critical times $t_{c}$ (i.
e. when the Dirac cone occurs). See the captions within for detailed
explanations.

Figure S11: Time-evolution a wavepacket (see Fig. S10) in $k-$space for the
two important bands $n=2$ and $n=3$. The parameters are as before. The
Fourier-transform is calculated numerically and cannot capture the complete
physics, as it misses finite-size contributions of the lattice. In (a) and (b)
we show the time-evolution of the gaussian wavepacket with $l_{0}=30$. We note
that after the Dirac cone (red line) the second band gets occupied. In (c) and
(d) we show the total populations in the respective bands. Due to the fast
quench over the cone at momenta different from $k=\pi$, there are persistent
oscillations in both bands. They originate from the non-adiabatic transitions
that occur for small gap differences (in terms of a Landau–Zener transition
between two energy states). Only for states that are perfectly occupying
$k=\pi$ (and are therefore completely delocalized in angular momentum, such as
a angle eigenstate), a clean transition from one band to the other is
possible. Figure S12: The alignment (a) and orientation signal (b), similar to
Fig. 2 in the main text, for the gaussian wavepacket at $l_{0}=30$ and for
different critical times $t_{c}$ (at which the Dirac cone occurs). Here, we
set $t_{c}=t_{\mathrm{final}}/2$, i. e. the cone is exactly in the middle of
the time-evolution. In different colors we show different trajectories for
different final times, see colorbar, with the number of kicks ranging from
$10\dots 100$. Clearly, the total number of kicks is not significantly
changing the overall behavior. We note that for longer times the transition of
band $n=3$ to $n=2$ is smoother and can occur henceforth in a more controlled
way.

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