# Enhancement of High Harmonic Generation in Bulk Floquet Systems

Abhishek Kumar  Department of Physics, Indiana University, Bloomington,
Indiana 47405, USA  Yantao Li  Department of Physics, Indiana University,
Bloomington, Indiana 47405, USA  Babak Seradjeh  Department of Physics,
Indiana University, Bloomington, Indiana 47405, USA  IU Center for Spacetime
symmetries, Indiana University, Bloomington, Indiana 47405, USA  Quantum
Science and Engineering center, Indiana University, Bloomington, Indiana
47405, USA

###### Abstract

We formulate a theory of bulk optical current for a periodically driven
system, which accounts for the mixing of external drive and laser field
frequencies and, therefore, the broadening of the harmonic spectrum compared
to the undriven system. We express the current in terms of Floquet-Bloch bands
and their non-adiabatic Berry connection and curvature. Using this expression,
we relate spatio-temporal symmetries of the driven model to selection rules
for current harmonics. We illustrate the application of this theory by
studying high harmonic generation in the periodically driven Su-Schrieffer-
Heeger model. At high frequencies and low field amplitudes, we find analytical
expressions for current harmonics. We also calculate the current numerically
beyond the high frequency limit and verify that when the drive breaks a
temporal symmetry, harmonics forbidden in the undriven model become available.
Moreover, we find significant enhancement in higher harmonics when the system
is driven, even for low field amplitudes. Our work offers a unified Floquet
approach to nonlinear optical properties of solids, which is useful for
realistic calculations of high harmonic spectra of electronic systems subject
to multiple periodic drives.

##  I Introduction

High harmonic generation (HHG) plays an important role in extreme ultraviolet
and attosecond physics [1]. The HHG in gaseous system, first observed
experimentally in 1961 [2], is usually understood within a three-step re-
collision model [3]. More recently, with its experimental realization [4, 5,
6, 7, 8, 9, 10, 11] and theoretical modeling of inter-band polarization and
intra-band nonlinear current in solids [12, 13, 14, 15, 16], HHG has emerged
as a useful nonlinear probe of electronic properties of condensed matter
systems, including topological materials [17, 18, 19, 20, 21, 22, 23, 24, 25,
26].

Along with this renewed interests in HHG, periodically driven quantum systems
have gained attention in recent years, in part due to the high degree of
control they can offer in experiments [27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55,
56, 57, 58, 59]. In contrast to their equilibrium counterparts, driven systems
have rich nonequilibrium dynamics and can support nontrivial topology with no
equilibrium analogue. Due to its inherently nonlinear character, it is
important to study of the effect of external drive on HHG and its relationship
with topology in periodically driven crystalline systems. To do so, we need a
non-perturbative theoretical formulation of HHG in such systems.

In this paper, we aim to advance our understanding of the interplay among
external periodic drive, topology, and HHG by employing the power of Floquet
theory [60, 61, 62], applied to periodically driven solids in the bulk under
intense light. We derive an expression for optical current in a periodically
driven quantum system in terms of the occupation of Floquet-Bloch bands and
relate it to quantum geometrical quantities like the nonadiabatic Berry
connection and curvature. Along the way, we reveal some subtleties in the
relation between HHG spectra and nonequilirbium topology. We demonstrate the
utility of this formulation by obtaining HHG selection rules due to
spatiotemporal symmetries. More importantly, our approach naturally extends
recent work on Floquet linear response theory [63] to general frequency mixing
between the drive and the probe fields, thus revealing a mechanism for
enhancement of HHG spectra in periodically driven systems.

As a concrete illustration, we study HHG spectra in the one-dimensional driven
Su-Shrieffer-Higger (SSH) lattice model [43], which admits various trivial and
topological Floquet topological phases. We confirm the general HHG selection
rules in this model and demonstrate that by breaking the temporal part of a
spatio-temporal symmetry, one can obtain high harmonics that are forbidden in
the undriven model. This is only possible in the driven system. We also obtain
analytical expressions for the optical current and HHG spectra in the high
frequency approximation. Assuming ideal and projected occupations of Floquet-
Bloch bands, numerical calculation shows that there is a significant
enhancement in higher harmonics when the system is driven, even at lower field
amplitudes. This means HHG can be generated even at low pulse intensities in a
Floquet system.

The paper is organized as follows. In Sec. II, along with a primer on Floquet
theory, we present a general formulation of optical current in a periodically
driven system and discuss its selection rules. In Sec. III, we consider the
periodically driven SSH model and apply our formalism to obtain the optical
current and its high harmonics. In Sec. IV, we present and discuss numerical
calculations of HHG spectra in the driven SSH model. We conclude in Sec. V
with a summary and outlook. Some details of our calculations are provided in
the Appendix. Throughout the paper, we shall use the natural units
$\hbar=e/c=a=1$, where $a$ is the relevant length scale, such as the lattice
spacing.

##  II Optical current in Floquet Systems

###  II.1 Primer on Floquet theory

For a time-periodic Hamiltonian, $\hat{H}(t)=\hat{H}(t+T)$, with period $T$,
the solution of Schrödinger equation takes the form
$\mathinner{|{\Psi_{\alpha}(t)}\rangle}=e^{-i\Xi_{\alpha}t}\mathinner{|{\Phi_{\alpha}(t)}\rangle}$,
where $\Xi_{\alpha}\in(-\pi/T,\pi/T]$ is the quasienergy, a conserved quantity
and the Floquet state, $\mathinner{|{\Phi_{\alpha}(t)}\rangle}$, is also
periodic in time with the same period $T$, and satisfies the Floquet-
Schrödinger equation

$[\hat{H}(t)-i\partial_{t}]\mathinner{|{\Phi_{\alpha}(t)}\rangle}=\Xi_{\alpha}\mathinner{|{\Phi_{\alpha}(t)}\rangle}.$
(1)

Also, the time-ordered evolution operator

$\displaystyle\hat{U}(t,t_{0})$
$\displaystyle:=\mathrm{T}\exp{\left[-i\int_{t_{0}}^{t}\hat{H}(s)ds\right]}$
(2) $\displaystyle=e^{-i(t-t_{0})\hat{H}_{F}(t)}\hat{P}(t,t_{0}),$ (3)

is decomposed into a periodic micromotion operator

$\displaystyle\hat{P}(t,t_{0})$
$\displaystyle=\hat{P}(t+T,t_{0})=\hat{P}(t,t_{0}+T)$
$\displaystyle\equiv\sum_{\alpha}\mathinner{|{\Phi_{\alpha}(t)}\rangle}\mathinner{\langle{\Phi_{\alpha}(t_{0})}|}$
(4)

and the evolution under the Floquet Hamiltonian

$\displaystyle\hat{H}_{F}(t)=\sum_{\alpha}\Xi_{\alpha}\mathinner{|{\Phi_{\alpha}(t)}\rangle}\mathinner{\langle{\Phi_{\alpha}(t)}|}.$
(5)

For a Hamiltonian that depends on two commensurate frequencies ($\omega$,
$\Omega$), we can define $\gamma=\omega t$ and $\Gamma=\Omega t$ so that
$\hat{H}(\gamma,\Gamma)=\hat{H}(\gamma+2\pi,\Gamma)=\hat{H}(\gamma,\Gamma+2\pi)$.
We can then write the multi-mode Floquet-Schrödinger equation

$\left[\hat{H}(\gamma,\Gamma)-i(\omega\partial_{\gamma}+\Omega\partial_{\Gamma})\right]\mathinner{|{\Phi_{\alpha}(\gamma,\Gamma)}\rangle}=\Xi_{\alpha}\mathinner{|{\Phi_{\alpha}(\gamma,\Gamma)}\rangle},$
(6)

where $\mathinner{|{\Phi_{\alpha}(\gamma,\Gamma)}\rangle}$ is periodic in
$\gamma$ and $\Gamma$ with period $2\pi$.

We may expand any such function, $f(\gamma,\Gamma)$ in Fourier modes,
$f(\gamma,\Gamma)=\sum_{m,M}e^{-im\gamma-iM\Gamma}f^{(m,M)}$ with

$f^{(m,M)}=\oint
e^{im\gamma+iM\Gamma}f(\gamma,\Gamma)\frac{d\gamma}{2\pi}\frac{d\Gamma}{2\pi},$
(7)

where $\oint$ is over a cycle of each integration variable. The single-time
Schrödinger equation and its solutions are obtained simply by replacing
$\gamma=\omega t$ and $\Gamma=\Omega t$ in the multi-mode Schrödinger equation
and its solutions.

Figure 1: Sketch for high harmonic generation in a Floquet system. The system
is driven by a drive frequency $\Omega$ and irradiated by field frequency
$\omega$. Here the output frequency is $m\omega+M\Omega$.

###  II.2 Optical current and its harmonics

We now derive an expression for optical current in a time periodic system
using Floquet theory. The density matrix of the system at some initial time
$t_{0}$ is $\hat{\rho}(t_{0})$, which evolves in time through a unitary
operator $\hat{U}(t,t_{0})$ as
$\hat{\rho}(t)=\hat{U}(t,t_{0})\hat{\rho}(t_{0})\hat{U}^{\dagger}(t,t_{0})$.

For further simplification, we assume that all the operators can be written in
the single-particle fermionic basis and take periodic boundary conditions. So,
the Hamiltonian
$\hat{H}(t)=\sum_{krs}\hat{c}^{\dagger}_{kr}h_{rs}(k,t)\hat{c}_{ks}$, where
$\hat{c}^{\dagger}_{kr}$ and $\hat{c}_{kr}$ are the creation and annihilation
operators, respectively, at lattice momentum $k$ and quantum number $r$, and
$h(k,t)$ is the time-periodic Bloch Hamiltonian. Similarly, the current
operator $\hat{J}(t)=\sum_{krs}\hat{c}^{\dagger}_{kr}j_{rs}(k,t)\hat{c}_{ks}$,
where $j(k,t)=\partial{h}(k,t)/\partial k$.

The current is

$\displaystyle J(t,t_{0})$
$\displaystyle=\mathrm{Tr}\left[\hat{\rho}(t)\hat{J}(t)\right]$ (8)
$\displaystyle=\sum_{k}\mathrm{tr}\left[g(k)j^{H}(k,t,t_{0})\right],$ (9)

where
$g_{rs}(k,t_{0})=\mathrm{Tr}[\hat{\rho}(t_{0})c^{\dagger}_{ks}c^{\vphantom{{\dagger}}}_{kr}]$,
‘$\mathrm{Tr}$’ is taken over the many-body Hilbert space, and ‘$\mathrm{tr}$’
is over the single-particle Hilbert space for each momentum $k$. Here,
$j^{H}(k,t,t_{0})=u^{\dagger}(k,t,t_{0})j(k,t)u(k,t,t_{0})$ is the (single-
particle) current in the Heisenberg picture and
$u(k,t,t_{0})=\mathrm{T}\exp[-i\int_{t_{0}}^{t}h(k,s)ds]$. In the Floquet
basis $\mathinner{|{\phi_{\alpha}(k,t)}\rangle}$ of $h(k,t)$, we find

$J(t,t_{0})=\sum_{k\alpha\beta}g_{\alpha\beta}(k,t_{0})j^{F}_{\beta\alpha}(k,t)e^{-i(\epsilon_{\alpha}-\epsilon_{\beta})(t-t_{0})},$
(10)

where
$j^{F}_{\beta\alpha}(k,t)=\mathinner{\langle{\phi_{\beta}(k,t)}|}j(k,t)\mathinner{|{\phi_{\alpha}(k,t)}\rangle}$
is periodic in time, and $\epsilon_{\alpha}$ are the single-particle
quasienergies.

In the following, we specify the time dependence of the Hamiltonian through a
periodically driven internal parameter
$\delta(t)=\sum_{M}\delta^{(M)}e^{-iM\Omega t}$ and an external electric field
via the gauge potential $A(t)=\sum_{m}A^{(m)}e^{-im\omega t}$. Thus, the
Hamiltonian $h(k,t)$ depends on the two frequencies $\Omega$ of the drive and
$\omega$ of the external field. Therefore, we may employ the Fourier expansion
$j^{F}_{\beta\alpha}(k,t)=\sum_{m,M}j_{\beta\alpha}^{(m,M)}(k)e^{-i(m\omega+M\Omega)t}$
to write

$\displaystyle J(t,t_{0})$
$\displaystyle=\sum_{\begin{subarray}{c}k\alpha\beta\\\
mM\end{subarray}}g_{\alpha\beta}(k,t_{0})j^{(m,M)}_{\beta\alpha}(k)e^{-i(m\omega+M\Omega)t}$
$\displaystyle\hskip 71.13188pt\times
e^{-i(\epsilon_{\alpha}-\epsilon_{\beta})(t-t_{0})}.$ (11)

The dependence on the initial time $t_{0}$ in Eqs. (10) and (II.2), where we
set the occupation of the quasi-energy bands, is the result of our assumption
that the ensuing dynamics is fully coherent. However, in reality the
occupation of the quasienergy bands, $g_{\alpha\beta}(k,t_{0})$, depends on a
specific relaxation process for a given system. We are interested in
situations in which the relaxation process, e.g. through suitable coupling to
a thermal bath, results in a diagonal population
$g_{\alpha\beta}=g_{\alpha}\delta_{\alpha\beta}$ independent of $t_{0}$. This
removes the dependence of the optical current on the initial time. Physically,
this means the system approaches a Floquet steady state. Formally, we can
obtain this result by assuming $g_{\alpha\beta}$ is independent of $t_{0}$ and
average the current over $t_{0}$, to which only the diagonal occupations in
Floquet states contribute.

Then the current becomes periodic in time $J(t)=J(t+T)$, and

$J(t)=\sum_{k\alpha
mM}g_{\alpha}(k)j^{(m,M)}_{\alpha\alpha}(k)e^{-i(m\omega+M\Omega)t}.$ (12)

This form of the optical current clearly shows the frequency mixing in the
generation of its harmonics. For example, if the drive frequency is an integer
multiple of the field frequency, $\Omega=N\omega$, the current acquires
harmonics $m+NM$ of the field frequency. Conversely, when $\omega=n\Omega$ for
integer $n$, the current acquires fractional harmonics $m+M/n$ of the field
frequency. As we shall demonstrate in concrete examples, some of these
harmonics may vanish or be enhanced depending on the order of the process and
selection rules imposed by spatiotemporal symmetries.

We also note that this diagonal optical current may appear to correspond to
only the intra-band contributions. However, it is important to bear in mind
that each quasienergy band is obtained by a reconstruction of the original
energy bands of static system. Therefore, the optical current, Eq. (12),
already includes both intra- and inter-band contributions of the static
system.

###  II.3 Optical current from Berry connection and curvature

We would like to recast the optical current in terms of geometrical quantities
like Berry curvature and Berry connection. First, we note that in the Floquet
basis, the current matrix elements,

$\displaystyle j^{F}_{\beta\alpha}(k,t)$
$\displaystyle\equiv\mathinner{\langle{\phi_{\beta}(k,t)}|}\frac{\partial{h(k,t)}}{\partial
k}\mathinner{|{\phi_{\alpha}(k,t)}\rangle}$
$\displaystyle=[\epsilon_{\beta}(k)-\epsilon_{\alpha}(k)+i\partial_{t}]\mathinner{\langle{\phi_{\beta}(t)}|{\partial_{k}\phi_{\alpha}(k,t)}\rangle}$
$\displaystyle\quad+\delta_{\alpha\beta}\partial_{k}\epsilon_{\beta}(k).$ (13)

So the full expression of the current, Eq. (10), with diagonal Floquet
population becomes

$J(t)=\sum_{k\alpha}g_{\alpha}(k)\left[\partial_{t}\mathscr{A}_{\alpha,k}(k,t)+\partial_{k}\epsilon_{\alpha}(k)\right],$
(14)

where
$\mathscr{A}_{\alpha,k}(k,t)=\mathinner{\langle{\phi_{\alpha}(k,t)}|{i\partial_{k}\phi_{\alpha}(k,t)}\rangle}$
is the Berry connection of the Floquet band $\alpha$. This expression makes it
clear that all time dependence and, therefore, all higher harmonics of the
optical current result from the time dependence of the Floquet Berry
connection. As in the static case, this also shows that a completely filled
Floquet band cannot contribute to DC current.

On the other hand, the gauge invariant of the optical current is not
manifested in Eq. (14), since the Berry connection
$\mathscr{A}_{\alpha,k}(k,t)$ itself is not a gauge invariant quantity.
Therefore, we now further recast this expression in terms of the gauge-
invariant Floquet Berry curvature,

$\displaystyle\mathscr{E}_{\alpha}$
$\displaystyle=\partial_{t}\mathscr{A}_{\alpha,k}-\partial_{k}\mathscr{A}_{\alpha,t},$
(15)
$\displaystyle=i\mathinner{\langle{\partial_{t}\phi_{\alpha}}|{\partial_{k}\phi_{\alpha}}\rangle}-i\mathinner{\langle{\partial_{k}\phi_{\alpha}}|{\partial_{t}\phi_{\alpha}}\rangle}$
(16)

where
$\mathscr{A}_{\alpha,t}(k,t)=\mathinner{\langle{\phi_{\alpha}(k,t)}|{i\partial_{t}\phi_{\alpha}(k,t)}\rangle}$
is the non-adiabatic (Aharonov-Anandan) connection of the states in Floquet
band $\alpha$. [64] Numerically, we can also calculate the Berry curvature
more efficiently and accurately without the need of fixing a special gauge.
[65]

To proceed, note that

$\displaystyle\partial_{t}\mathscr{A}_{\alpha,k}$
$\displaystyle=\mathscr{E}_{\alpha}+\partial_{k}{\mathinner{\langle{\phi_{\alpha}}|{i\partial_{t}\phi_{\alpha}}\rangle}}$
(17)
$\displaystyle=\mathscr{E}_{\alpha}+\partial_{k}\mathinner{\langle{\phi_{\alpha}}|{h}|{\phi_{\alpha}}\rangle}{-\partial_{k}\epsilon_{\alpha}}.$
(18)

Therefore, we find

$J(t)=\sum_{k\alpha}g_{\alpha}(k)\left[\mathscr{E}_{\alpha}(k,t)+\partial_{k}\eta_{\alpha}(k,t)\right],$
(19)

where
$\eta_{\alpha}(k,t)=\mathinner{\langle{\phi_{\alpha}(k,t)}|}h(k,t)\mathinner{|{\phi_{\alpha}(k,t)}\rangle}$
is the expectation value of the instantaneous Hamiltonian in the Floquet band
$\alpha$. As before, due to periodicity in crystal momentum $k$, this last
term does not contribute in a completely filled band. However, in partially
filled bands, it can contribute to DC as well as the higher harmonics of
optical current.

We note that these expressions can be readily extended to higher dimensions,

$\displaystyle\mathbf{J}(t)$
$\displaystyle=\sum_{\mathbf{k}\alpha}g_{\alpha}(\mathbf{k})\left[\partial_{t}{\mathscr{A}}_{\alpha,\mathbf{k}}(\mathbf{k},t)+\partial_{\mathbf{k}}\epsilon(\mathbf{k})\right]$
(20)
$\displaystyle=\sum_{\mathbf{k}\alpha}g_{\alpha}(\mathbf{k})\left[\boldsymbol{\mathscr{E}}_{\alpha}(\mathbf{k},t)+\partial_{\mathbf{k}}\eta(\mathbf{k},t)\right],$
(21)

where
$\boldsymbol{\mathscr{E}}_{\alpha}=\partial_{t}{\mathscr{A}}_{\alpha,\mathbf{k}}-\partial_{\mathbf{k}}{\mathscr{A}}_{\alpha
t}$ and the Floquet Berry connection
${\mathscr{A}}_{\alpha,\mathbf{k}}=\mathinner{\langle{\phi_{\alpha}}|{i\partial_{\mathbf{k}}\phi_{\alpha}}\rangle}$
and
${\mathscr{A}}_{\alpha,t}=\mathinner{\langle{\phi_{\alpha}}|{i\partial_{t}\phi_{\alpha}}\rangle}$.

###  II.4 Selection rules

Here, we analyze the constraints on the current and its harmonics due to
symmetries for the ideal Floquet occupation with each band either fully
occupied or empty. In particular, we consider a unitary mirror ($I_{F}$) and
an antiunitary time-reflection ($\Theta_{F}$) symmetries, which include time
glide operations,

$\displaystyle h(k,t)$ $\displaystyle=I_{F}^{\dagger}h(-k,t+t_{I})I_{F},$ (22)
$\displaystyle h(k,t)$
$\displaystyle=\Theta_{F}^{\dagger}h(-k,-t+t_{R})\Theta_{F}.$ (23)

Under $I_{F}$ symmetry, the occupied Floquet bands $\alpha\in\text{occ}$
satisfy
$I_{F}\mathinner{|{\phi_{\alpha}(k,t)}\rangle}=\sum_{\beta\in\text{occ}}\mathfrak{I}_{\beta\alpha}(k,t)\mathinner{|{\phi_{\beta}(-k,t+t_{I})}\rangle}$,
where $\mathfrak{I}(k,t)$ is a unitary matrix. This leads to a relation
between the current at different times,

$J(t)=-J(t+t_{I}).$ (24)

Therefore, the current vanishes at an odd number of times in any interval of
length $t_{I}$ in the cycle, at the boundary of which the current is nonzero.
For $t_{I}=\pi/\omega$, we get the following condition for the harmonics of
the current,

$J^{(p)}=(-1)^{p+1}J^{(p)},$ (25)

which means in the presence of $I_{F}$ symmetry, only odd harmonics will
survive.

Under $\Theta_{F}$ symmetry, the occupied Floquet bands satisfy
$\Theta_{F}\mathinner{|{\phi_{\alpha}(k,t)}\rangle}=\sum_{\beta\in\text{occ}}\mathfrak{T}_{\beta\alpha}(k,t)\mathinner{|{\phi_{\alpha}(-k,-t+t_{R})}\rangle}$,
where $\mathfrak{T}(k,t)$ is an orthogonal matrix. This also leads to a
relation between current at different times,

$J(t)=-J(-t+t_{R}).$ (26)

As a result, the current must vanish at $J(t_{R}/2)=J(T/2+t_{R}/2)=0$. For
$t_{R}=\pi/\omega$ we have the following properties of the harmonics of the
current,

$J^{(p)}=(-1)^{p+1}J^{(-p)},$ (27)

which means the odd harmonics of the current will be real and even harmonics
of the current will be imaginary, and the DC component $J^{(0)}=0$. If
$t_{R}=0$ instead, the current $J(t)=-J(-t)$ is an odd function of time and we
find

$J^{(p)}=-J^{(-p)},$ (28)

which means $J^{(0)}=0$ and all the other harmonics will be imaginary.

When both $I_{F}$ and $\Theta_{F}$ symmetries are present, and taking
$t_{I}=t_{R}=\pi/\omega$, we have

$J^{(p)}=J^{(-p)}=(-1)^{p+1}J^{(p)},$ (29)

which means that only odd harmonics are nonzero and they are real.

###  II.5 Relation to topology

The expressions of the current in terms of the Floquet Berry connection and
curvature, Eqs. (14) and (19), provide a link between the optical current and
the quantum geometry of Floquet bands. This geometry is also responsible for
nontrivial topological properties of the system [66]. Here, we examine the
connection between the optical current and nontrivial topology in different
frequency regimes.

In the low-frequency limit, the Floquet states
$\mathinner{|{\phi_{\alpha}(k,t)}\rangle}$ approach, up to a smooth gauge
[67], the adiabatic states
$\mathinner{|{\psi_{\alpha}^{\text{ad}}(k,t)}\rangle}$, i.e. the instantaneous
eigenstates of the driven Hamiltonian,
$h(k,t)\mathinner{|{\psi_{\alpha}^{\text{ad}}(k,t)}\rangle}=E_{\alpha}(k,t)\mathinner{|{\psi_{\alpha}^{\text{ad}}(t)}\rangle}$,
assuming the instantaneous energy bands $E_{\alpha}(k,t)$ remain gapped. Then,
the optical current of fully occupied bands approaches
$J(t)\to\sum_{\alpha\in\text{occ}}\oint\mathscr{E}^{\text{ad}}_{\alpha}(k,t)dk$,
where $\mathscr{E}^{\text{ad}}_{\alpha}(k,t)$ is the adiabatic Berry
curvature. This is, of course, the Thouless pump, in which a charge

$Q=\oint
J(t)dt=\sum_{\alpha\in\text{occ}}\oint\mathscr{E}_{\alpha}^{\text{ad}}(k,t)dkdt\equiv\text{Ch}^{\text{ad}}_{\text{occ}},$
(30)

equal to the Chern number $\text{Ch}^{\text{ad}}_{\text{occ}}$ of the occupied
adiabatic bands in $(k,t)$ space, is pumped in each cycle.

While Eqs. (14) and (19) hold formally at all frequencies, the gap structure
of Floquet bands deviates significantly away from the adiabatic limit. As a
result, the Floquet bands in the intermediate frequency range become partially
filled. In the high frequency limit, other approximations such as rotating-
wave frame and Floquet Magnus expansion become available. In certain cases,
one can also justify or specifically engineer the Floquet bands to be be fully
occupied or empty. For example, for high frequencies that are off-resonant
with gapped static bands, one may take the occupation of Floquet bands to be
nearly the same as the original static bands. In such case, the optical
current is obtained from the non-adiabatic Berry connection and curvature of
Floquet bands.

We would like to note a subtlety related to the choice of gauge in using Eqs.
(14) and (19). Consider a driven one-dimensional Hamiltonian $h(k,t)$ with
time-reflection symmetry, e.g. $h(k,-t)=h(k,t)$, and chiral symmetry
$\\{C,h(k,t)\\}=0$ satisfying $C=C^{\dagger}=C^{-1}$. Then, the Floquet
Hamiltonians $h_{F}(k,t_{*})$ at time-reflection symmetric times
$t_{*}=0,T/2$, inherit the chiral symmetry $C$. The topology of the Floquet
bands is then characterized by two gauge-invariant winding numbers $w(t_{*})$,

$w(t_{*})=\frac{1}{\pi}\sum_{\alpha\in\text{occ}}\oint\mathinner{\langle{C\phi_{\alpha}(k,t_{*})}|{i\partial_{k}\phi_{\alpha}(k,t_{*})}\rangle}dk\in\mathbb{Z}.$
(31)

There is a bulk-boundary correspondence between $w(t_{*})$ and the number of
midgap bound states, $\nu_{0}$ and $\nu_{\pi}$, respectively, at quasienergies
$0$ and $\pi/T$,

$\nu_{0}=\frac{w(T/2)+w(0)}{2},\quad\nu_{\pi}=\frac{w(T/2)-w(0)}{2}.$ (32)

It is possible to choose a smooth gauge at each $t_{*}$,
$\mathinner{|{\phi_{\alpha}(k,t_{*})}\rangle}\to
e^{i\Lambda_{\alpha}(k,t_{*})}\mathinner{|{\phi_{\alpha}(k,t_{*})}\rangle}$,
such that

$w(t_{*})=\frac{1}{\pi}\sum_{\alpha\in\text{occ}}\oint\mathinner{\langle{\phi_{\alpha}(k,t_{*})}|{i\partial_{k}\phi_{\alpha}(k,t_{*})}\rangle}dk.$

It is then tempting to related $w(T/2)-w(0)$ to $\int_{0}^{T/2}j(t)dt$.
However, while both of these quantities are gauge-invariant, this would not be
in general correct because it is not in general possible to find a gauge
$\mathinner{|{\phi_{\alpha}(k,t)}\rangle}\to
e^{i\Lambda_{\alpha}(k,t)}\mathinner{|{\phi_{\alpha}(k,t)}\rangle}$ that
smoothly connects $\Lambda_{\alpha}(k,0)$ to $\Lambda_{\alpha}(k,T/2)$ in the
cycle. The degree to which this smooth gauge condition is broken is indeed
what $\nu_{\pi}$ quantifies.

##  III HHG in Periodically Driven Su-Schrieffer-Heeger model

###  III.1 Model and symmetries

In order to illustrate the physics of high harmonic generation in Floquet
systems concretely, we consider the Su-Schrieffer-Heeger (SSH) model with
time-periodic hopping amplitudes. In addition, an external time-periodic and
spatially uniform gauge field, $A(t)$, is introduced via Peierls substitution
$k\rightarrow k-A(t)$. The full loch Hamiltonian is

$\displaystyle h(k,t)$
$\displaystyle=w\left\\{1+m(t)+[1-m(t)]\cos[k-A(t)]\right\\}\sigma_{x}$
$\displaystyle\quad+w\left[1-m(t)\right]\sin[k-A(t)]\sigma_{y}.$ (33)

Here, $w$ is the average hopping amplitude. In the following, we take the
driven field $A(t)=A_{0}\sin(\omega t+\theta)$ and the driven hopping
modulation $m(t)=m_{0}+m_{1}\cos(\Omega t)$.

The static SSH model ($m_{1}=0,A_{0}=0$) has a trivial ($m_{0}>0$) and a
topological phase ($m_{0}<0$) phase protected by the chiral symmetry,
$\\{h,C\\}=0$ with $C=\sigma_{z}$. In addition, the static model has both
unitary inversion (or mirror) symmetry ($I=\sigma_{x},k\to-k$) and antiunitary
time-reversal symmetry ($\Theta=K,k\to-k$ , where $K$ is the complex
conjugation).

In the driven model, these symmetries are in general lifted by the external
field. However, for certain drive protocols, we may recover these symmetries
upon a suitable mapping within the cycle. That is, the driven system may have
related _spatiotemporal_ symmetries, in which the temporal part of the
Hamiltonian is also transformed. Indeed, the inversion symmetry can be
restored for the driven system if there exists a time glide
$\tau(t_{I}):t\mapsto t+t_{I}$ within the cycle, for which $A(t+t_{I})=-A(t)$
is odd and $m(t+t_{I})=m(t)$ is even. Similarly, the time-reversal symmetry is
restored if there exists a reflection time $t_{R}$, for which
$A(t_{R}-t)=-A(t)$ is odd and $m(t_{R}-t)=m(t)$ is even. The spatiotemporal
symmetries of the driven system are, then,

$I_{F}^{-1}h(-k,t+t_{I})I_{F}=h(k,t),\quad I_{F}=\tau(t_{I})I,$ (34a) and
$\Theta_{F}^{-1}h(-k,-t+t_{R})\Theta_{F}=h(k,t),\quad\Theta_{F}=\tau(t_{R})\Theta.$
(34b)

When the drive frequency is an even multiple of the field frequency
($\Omega=2n\omega$, $n\in\mathbb{Z}$), we may choose $t_{I}=\pi/\omega$ for
any $\theta$ to obtain symmetry under $I_{F}$, yielding $J(t+T/2)=-J(t).$ For
$\theta=\pm\frac{\pi}{2}$, we can take $t_{R}=\pi/\omega$ to obtain symmetry
under $\Theta_{F}$, whereby $J(t+T/2)=-J(-t).$ Similarly, if $\theta=0$ or
$\pi$, we can take $t_{R}=0$ to find $J(t)=-J(-t)$. Therefore, selection rules
for the ideal occupation of the Floquet bands apply to the harmonics of the
current in these cases as discussed in Sec. II.4.

###  III.2 Current in the high frequency limit

In this section we derive an analytical expression for the current in diagonal
Floquet occupation using high frequency approximation. We will calculate the
Floquet Hamiltonian and states as well as micromotion operators using high
frequency expansion. We assume $\Omega=N\omega$ for an integer $N$.

In the high frequency limit, the Floquet Hamiltonian and micromotion operator
are

$\displaystyle{h}_{F}$
$\displaystyle={h}^{(0)}+\sum_{n>0}\frac{[{h}^{(-n)},{h}^{(n)}]}{n\omega}+O\left(\frac{1}{\omega^{2}}\right),$
(35) $\displaystyle P(t)$
$\displaystyle=\exp\left[i\sum_{n>0}\frac{{h}^{(-n)}e^{in\omega
t}-{h}^{(n)}e^{-in\omega
t}}{in\omega}+O\left(\frac{1}{\omega^{2}}\right)\right],$ (36)

respectively, where $h^{(n)}(k)$ are the Fourier components of the Bloch
Hamiltonian $h(k,t)$,

$\displaystyle{h}^{(0)}(k)$
$\displaystyle=\big{[}(1+m_{0})+\mathrm{Re}\,[f_{0}(k)]\big{]}\sigma_{x}-\mathrm{Im}\,[f_{0}(k)]\sigma_{y},$
(37) $\displaystyle{h}^{(n)}(k)$
$\displaystyle=\frac{f_{-n}(k)+f^{*}_{n}(k)}{2}\sigma_{x}+i\frac{f_{-n}(k)-f^{*}_{n}(k)}{2}\sigma_{y},$
(38)

where,

$f_{n}(k)=\frac{m_{1}}{2}(\delta_{n,N}+\delta_{n,-N})+e^{-ik}\left\\{(1-m_{0})\mathcal{J}_{n}(A_{0})e^{in\theta}-\frac{m_{1}}{2}\left[\mathcal{J}_{n-N}(A_{0})e^{i(n-N)\theta}+\mathcal{J}_{n+N}(A_{0})e^{i(n+N)\theta}\right]\right\\},$
(39)

and $\mathcal{J}_{n}$ are Bessel functions. Thus, we may write the Floquet
Hamiltonian as ${h}_{F}(k)=\mathbf{d}_{F}(k)\cdot\boldsymbol{\sigma},$ with

$\mathbf{d}_{F}(k)=\left((1+m_{0})+\mathrm{Re}\,[f_{0}(k)],-\mathrm{Im}\,[f_{0}(k)],\sum_{n>0}\frac{|f_{n}(k)|^{2}-|f_{-n}(k)|^{2}}{n\omega}\right),$
(40)

and the micromotion as
$P(k,t)=\exp\left[i\mathbf{V}(k,t)\cdot\boldsymbol{\sigma}\right],$ with

$\mathbf{V}(k,t)=\sum_{n>0}\frac{1}{n\omega}\Big{(}\mathrm{Im}\,[f_{n}^{+}(k,t)],\mathrm{Re}\,[f_{n}^{-}(k,t)],0\Big{)},\vspace{-0.1cm}$
(41)

where $f_{n}^{\pm}(k,t)=[f^{*}_{-n}(k)\pm f_{n}(k)]e^{in\omega t}$.

$N=\Omega/\omega$ | harmonic $p$ | $\tilde{W}_{1}$ | $\tilde{W}_{2}$ | $\tilde{W}_{3}$ | $\tilde{W}_{4}$
---|---|---|---|---|---
even | odd | $iW_{1}$ | $iW_{3}$ | $iW_{2}$ | $iW_{3}$
even | even | $W_{2}$ | $W_{4}$ | $W_{2}$ | $W_{4}$
odd | odd | $iW_{1}$ | $iW_{3}$ | $W_{2}$ | $W_{4}$
odd | even | $W_{2}$ | $W_{4}$ | $iW_{1}$ | $iW_{3}$
Table 1: Choice of $\tilde{W}_{j}$ in Eq. (53), in terms of $W_{j}$, Eqs.
(49).

We can now use Eq. (10) to calculate the optical current by writing

$j^{F}_{\alpha\alpha}=\mathinner{\langle{\phi_{\alpha}(k,0)}|{P^{\dagger}(k,t)\frac{\partial
h(k,t)}{\partial k}P(k,t)}|{\phi_{\alpha}(k,0)}\rangle},$ (42)

where $\mathinner{|{\phi_{\alpha}(k,0)}\rangle}$ are the eigenstates of the
Floquet Hamiltonian $h_{F}(k)$. The current operator $\partial h/\partial
k=\mathbf{j}\cdot\boldsymbol{\sigma}$, where

$\mathbf{j}(k,t)=w[1-m(t)]\Big{(}\sin[A(t)-k],\cos[k-A(t)],0\Big{)}.$ (43)

Then, $P^{\dagger}(k,t)[{\partial h(k,t)}/{\partial
k}]P(k,t)=\mathbf{j}_{F}\cdot\boldsymbol{\sigma}$ with

$\mathbf{j}_{F}=\mathbf{j}_{\parallel}+\cos(2V)\mathbf{j}_{\perp}-\sin(2V)\,\mathbf{j}\times\hat{\mathbf{V}},$
(44)

where $V(k,t)=|\mathbf{V}(k,t)|$,
$\hat{\mathbf{V}}(k,t)=\mathbf{V}(k,t)/V(k,t)$,
$\mathbf{j}_{\parallel}(k,t)=\mathbf{j}(k,t)\cdot\hat{\mathbf{V}}(k,t)\hat{\mathbf{V}}(k,t)$
and $\mathbf{j}_{\perp}(k,t)=\mathbf{j}(k,t)-\mathbf{j}_{\parallel}(k,t)$ are
the components of $\mathbf{j}(k,t)$, respectively, parallel and perpendicular
to $\mathbf{V}(k,t)$. The eigenstates of $h_{F}(k)$ with quasienergies
$\pm|\mathbf{d}_{F}(k)|$ are oriented along
$\pm\hat{\mathbf{d}}_{F}(k,t)=\pm\mathbf{d}_{F}(k,t)/|\mathbf{d}_{F}(k,t)|$ on
the Bloch sphere.

Thus, after some algebra, we find

$J(t)=\oint\left[g_{+}(k)-g_{-}(k)\right]\left[\mathbf{j}_{F}(k,t)\cdot\hat{\mathbf{d}}_{F}(k)\right]dk,\vspace{-0.1cm}$
(45)

where we denote the occupation of $\pm|\mathbf{d}_{F}(k)|$ quasienergies by
$g_{\pm}(k)$.

Figure 2: Comparison of current harmonics obtained using the analytical
expression in the high frequency approximation (“HFA”) in Eq. (53) with the
full numerical calculation (“Full”). The parameters are chosen as
$m_{0}=0.3,m_{1}=0.17,A_{0}=0.1,\Omega=\omega$, and $\theta=0$.

We can present explicit expressions for the harmonics for sufficiently small
$A_{0}$, such that $\mathbf{d}_{F}$ and $\mathbf{V}$ can be approximated well
by neglecting $\mathcal{J}_{n>0}(A_{0})$. Then,
$\mathbf{V}=m_{1}\mathbf{v}_{0}\sin(N\omega t)/(N\omega)$ with

$\mathbf{v}_{0}=\Big{(}1-\mathcal{J}_{0}(A_{0})\cos
k,\mathcal{J}_{0}(A_{0})\sin k,0\Big{)},$ (46)

and

$\mathbf{d}_{F}=\Big{(}1+m_{0}+(1-m_{0})\mathcal{J}_{0}(A_{0})\cos k,\\\
(1-m_{0})\mathcal{J}_{0}(A_{0})\sin k,0\Big{)},$ (47)

and Then, $\mathbf{d}_{F}$, $\mathbf{V}$ and $\mathbf{j}$ are coplanar and
$\mathbf{j}\times\hat{\mathbf{V}}\cdot\hat{\mathbf{d}}_{F}=0$. So,

$\mathbf{j}_{F}\cdot\hat{\mathbf{d}}_{F}\approx\left[1-m(t)\right]\Big{\\{}W_{1}(k)\sin
A(t)+W_{2}(k)\cos A(t)\\\ +\cos(2V)\left[W_{3}(k)\sin A(t)+W_{4}(k)\cos
A(t)\right]\Big{\\}},$ (48)

where

$\displaystyle W_{1}$
$\displaystyle=+\frac{\mathbf{v}_{0}\cdot\mathbf{d}_{F}}{v_{0}^{2}d_{F}}\left[\cos
k-\mathcal{J}_{0}(A_{0})\cos 2k\right],$ (49a) $\displaystyle W_{2}$
$\displaystyle=-\frac{\mathbf{v}_{0}\cdot\mathbf{d}_{F}}{v_{0}^{2}d_{F}}\left[\sin
k-\mathcal{J}_{0}(A_{0})\sin 2k\right],$ (49b) $\displaystyle W_{3}$
$\displaystyle=\frac{(1+m_{0})\cos
k+(1-m_{0})\mathcal{J}_{0}(A_{0})}{d_{F}}-W_{1},$ (49c) $\displaystyle W_{4}$
$\displaystyle=-\frac{(1+m_{0})\sin k}{d_{F}}-W_{2},$ (49d)

and

$\displaystyle v_{0}^{2}$
$\displaystyle=1+\mathcal{J}_{0}^{2}(A_{0})-2\mathcal{J}_{0}(A_{0})\cos k,$
(50) $\displaystyle d_{F}^{2}$
$\displaystyle=(1+m_{0})^{2}+(1-m_{0})^{2}\mathcal{J}_{0}^{2}(A_{0})$
$\displaystyle\qquad\qquad\qquad+2(1-m^{2}_{0})\mathcal{J}_{0}(A_{0})\cos k.$
(51) $\displaystyle\mathbf{v}_{0}\cdot\mathbf{d}_{F}$
$\displaystyle=1+m_{0}[1+\mathcal{J}_{0}^{2}(A_{0})\cos
2k-2\mathcal{J}_{0}(A_{0})\cos k]$
$\displaystyle\qquad\qquad\qquad-\mathcal{J}_{0}^{2}(A_{0})\cos 2k.$ (52)

Altogether, we find $J^{(p)}=\oint[g_{+}(k)-g_{-}(k)]J^{(p)}(k)dk$,

$J^{(p)}(k)=(1-m_{0})\left[\tilde{\mathcal{J}}_{p}(A_{0})\tilde{W}_{1}+\sum_{n\in\mathbb{Z}}\mathcal{J}_{2n}(2\tilde{v}_{0})\tilde{\mathcal{J}}_{p-2nN}(A_{0})\tilde{W}_{2}\right]\\\
-\frac{m_{1}}{2}\sum_{q=\pm
N}\left[\tilde{\mathcal{J}}_{p-q}(A_{0})\tilde{W}_{3}+\sum_{n\in\mathbb{Z}}\mathcal{J}_{2n}(2\tilde{v}_{0})\tilde{\mathcal{J}}_{p-2nN-q}(A_{0})\tilde{W}_{4}\right],$
(53)

where we use the shorthands $\tilde{v}_{0}=m_{1}v_{0}/(N\omega)$,
$\tilde{\mathcal{J}}_{n}(A_{0})=\mathcal{J}_{n}(A_{0})e^{-in\theta}$, and
$\tilde{W}_{j}$ are listed in Table. 1. In Fig. 2, we compare the harmonics
obtained from the numerical integration of the analytical expression in Eq.
(53) to the full numerical calculation reported in the next Section for the
ideal Floquet occupations, $g_{-}=1,g_{+}=0$, showing excellent agreement for
sufficiently large frequencies.

##  IV Numerical Results

In this section, we present our numerical results for the current in the
driven SSH model obtained by exact diagonalization of Floquet Hamiltonian and
time evolution operators. We have used the Floquet Berry curvature method as
given in Eq. 19 to numerically compute the current. We have ensured our
results converge both in lattice momentum $k$ and time $t$ by taking a mesh in
$k$ with 200 points as well as a mesh in $t$ with 256 points for
$\Omega=N\omega$ and 512 points for $\omega=n\Omega$. In order to investigate
the effects of occupation of Floquet bands we compare our results for the
ideal Floquet and thermal occupation projected to Floquet bands.

###  IV.1 Ideal Floquet occupation

First, we consider an ideal Floquet occupation where the Floquet state with
lowest quasi-energy band is fully occupied and the higher quasi-energy band is
fully empty. The quasienergies are plotted in Fig. 3(a)-(c) for the undriven
model with $\delta=0$, and for the driven model with $\delta/w=0.17$ and two
values of drive frequency $\Omega/\omega=1$ and $\Omega/\omega=2$. Below, we
provide illustrative results for $\Omega=N\omega$ as well as $\omega=n\Omega$
with $N$ and $n$ positive integers.

Figure 3: Quasienergy bands $\epsilon_{\alpha}(k)$, (a)-(c), of the driven SSH
model and their projected thermal occupations $g_{\alpha}(k)$, (d)-(f). Here,
$\omega/w=3,m_{0}=0.3$ and $\theta=0$ in all cases, and $m_{1}=0.17$, for the
driven cases in (b), (c), (e), and (f). For the undriven cases in (a) and (d)
$m_{1}=0$.  Figure 4: Optical current and its harmonics in the driven SSH
model with ideal Floquet occupation. For different values of field amplitude
$A_{0}$, (a)-(c) show the optical current as a function of time $t$ and
(d)-(f) show the absolute value of current harmonics with respect to harmonic
order $p$. Here, $\omega/w=3$, $m_{0}=0.3$ and $\theta=0$ for all cases, and
$m_{1}=0.17$ for the driven cases.  Figure 5: Floquet gain of the high
harmonics of the current in the driven SSH model with ideal Floquet
occupation. Here, $\omega/w=3$, $m_{0}=0.3$, and $\theta=0$ for all cases; and
$m_{1}=0.17$ for the driven cases. The white background color denotes values
too small to show in our precision.

We show the Floquet current within a period corresponding to the field
frequency in Fig. 4(a)-(c) and the absolute values of its harmonics in Fig.
4(d)-(f) for different values of field amplitude $A_{0}$ and $\theta=0$. As
expected from our symmetry analysis leading to Eq. (29) with
$t_{I}=T/2,t_{R}=0$, we find that for the undriven model, Fig. 4(a) and 4(c),
and the driven model with $\Omega/\omega=2$, Fig. 3(d) and 4(f), the Floquet
currents at times $t$ and $t+T/2$ have the same magnitude but opposite sign.
Consequently, we also see that only odd harmonics are nonzero in these cases.
However, in the driven model with $\Omega/\omega=1$, Fig. 4(b) and 4(e), these
selection rules are lifted and all current harmonics $n\neq 0$ become nonzero.

Fig. 5 shows the gain in harmonics of the current in the driven system for
$\Omega=N\omega$. We call this the “Floquet gain” because it becomes available
when the system is periodically driven along with the external field. We see
that as we increase the ratio of drive to field frequencies, we can obtain
more and more harmonics. This Floquet gain holds even for low value of the
field amplitude as shown in Fig. 5(a). As expected, increasing the field
amplitude $A_{0}$ leads to higher harmonic generation both for undriven and
driven cases. Interestingly, the Floquet gain at frequencies
$M\Omega\pm|m|\omega$, or equivalently at harmonic order
$p=M(\Omega/\omega)\pm|m|$, is prominent even at low field amplitudes. In all
panels, and especially in (a), Floquet gain can be observed for $m=1,2,\cdots$
up to the nonlinear order with significant value in the undriven model
(leftmost column of each panel) and several values of $M=1,2,3$ and $4$. Due
to the quick suppression of nonlinear contributions with increasing $m$ at low
field amplitudes, the Floquet gain can be distinguished easily as streaks of
high harmonic generation with slopes equal to $M$ at order
$p=M(\Omega/\omega)\pm 1$.

In Fig. 6, we plot the current harmonics as a function of the field amplitude,
$A_{0}$, and the drive amplitude $m_{1}$. For small amplitudes, the dependence
follows a power law with an exponent $\zeta$, which increases as these
amplitudes increases. We note that at small field amplitude all HHG
components, even those with $p>1$, show a linear dependence on $A_{0}$ due to
frequency mixing with the drive.

We also investigate the dependence of HHG on the phase difference
$\theta-\pi/2$ between the field and the drive. Optical current and its
harmonics are calculated numerically and plotted in Fig. 7 for several values
of $\theta$. For all $\theta$ when $\Omega=2\omega$ and, separately, for
$\theta=\pi/2$ and $\pi$, we can explicitly observe the symmetries under
$I_{F}$ and $\Theta_{F}$, respectively, discussed under Eqs. (34). In
particular, for $\Omega=2\omega$ in Fig. 7(b) and (d), the symmetry under
$I_{F}$ forces even harmonics of the current to vanish. Our results show that,
while these symmetry properties influence the shape of the current as a
function of time, they do not significantly affect the magnitude of its
harmonics.

When the drive frequency is a fraction of the field frequency so that
$\omega=n\Omega$ for integer $n$, then the optical current for diagonal
Floquet occupation produces fractional harmonics of the field frequency. To
calculate the fractional harmonics numerically, we consider an ideal Floquet
occupation with the lower Floquet band fully occupied. In Fig. 8 we show the
absolute value of the harmonics of the current as a function of harmonic order
for different field amplitudes. In general, we find harmonics order
$p=m+M(\Omega/\omega)$ in agreement with Eq. (12). Of course, we can also view
this as the generation of high harmonics of the _drive_ frequency. From this
perspective, the harmonic generation in the undriven system, Fig. 8(a)
provides the amplitudes needed for frequency mixing with the drive.

We note that unlike the previous case, when the drive frequency is the
principal Floquet frequency neither the inversion nor the time-reversal
symmetry can be restored, since there is no time glide $t\to t+t_{0}$ within
the cycle for which we can ensure $m(t+t_{0})=m(t)$ and $A(t+t_{0})=-A(t)$,
nor $m(-t+t_{0})=m(t)$ and $A(-t+t_{0})=-A(t)$. Therefore all the harmonics of
the drive that were forbidden in undriven case are now generated.

Figure 6: Current harmonics vs. the field amplitudes (a) and vs. driven
hopping term (b) in the driven SSH model with ideal Floquet occupation. Here,
$\omega/w=3$, $m_{0}=0.5$, $\Omega=\omega$, and $\theta=0$. We have taken
$m_{1}=0.3$ in (a) and $A_{0}=0.1$ in (b). At the bottom of panel (a), we show
lines corresponding to power-laws with exponent $\zeta$ for easy comparison.
Figure 7: Dependence of optical current and its harmonics on $\theta$, with
$\theta-\pi/2$ as the phase difference between the field and the drive, for
ideal Floquet occupation. Here, $\omega/w=3$, $m_{0}=0.3,m_{1}=0.17$, and
$A_{0}=0.71$.

###  IV.2 Projected Floquet thermal occupation

We now compare the optical current harmonics obtained for a thermal density
matrix $\rho_{0}(k)=\frac{1}{Z}e^{-{h}_{0}(k)/T_{0}}$ projected on the Floquet
states, where $T_{0}$ is the temperature (we set $k_{B}=1$),
$Z=\mathrm{tr}[e^{-{h}_{0}(k)/T_{0}}]$, and the static Hamiltonian
${h}_{0}(k):={h}(k,t_{0})|_{A_{0}=0,m_{1}=0}$. That is, we retain only the
diagonal elements

$\displaystyle g_{\alpha}(k,t_{0})$
$\displaystyle=\mathinner{\langle{\phi_{\alpha}(k,t_{0})}|{\rho_{0}(k)}|{\phi_{\alpha}(k,t_{0})}\rangle}$
$\displaystyle{=\frac{1}{Z}\sum_{\mu}e^{-E_{\mu}^{0}(k)/T_{0}}|\langle\phi^{0}_{\mu}(k)|\phi_{\alpha}(k,t_{0})\rangle|^{2},}$
(54)

where $E_{\mu}^{0}(k)$ and $\mathinner{|{\phi^{0}_{\mu}(k)}\rangle}$ are the
energy eigenvalues and eigenstates of ${h}_{0}(k)$, respectively. We see the
dependence of occupation on $t_{0}$ remains in Eq. (54) because the Floquet
Hamiltonian and, therefore, the Floquet state depend on $t_{0}$. We would
average the current over $t_{0}$; however, in our numerics we have seen only
tiny differences in the occupations of Floquet-Bloch bands for different
values of $t_{0}$. So, we set $t_{0}=0$ for this calculation.

For $T_{0}\to 0$, this corresponds to a quenched occupation where the initial
state is the ground state of the static Hamiltonian $h_{0}(k)$ formed by the
lowest static energy band $\mathinner{|{\phi^{0}_{\text{GS}}}\rangle}$, i.e.
$g_{\alpha}(k)=|\mathinner{\langle{\phi^{0}_{\text{GS}}(k)}|{\phi_{\alpha}(k)}\rangle}|^{2}$.
We illustrate the projected Floquet occupation of quasienergy bands in Fig.
3(d)-(f). The Floquet principal frequency in this case is the field frequency
$\omega$, which is resonant with the energy bands $E^{0}_{\mu}$. Therefore,
there is a qausienergy band inversion and quasienergy gap induced by the field
and the drive. Note also that the projected Floquet occupations are not
symmetric in $k$ and that the asymmetry increases with with the field
amplitude $A_{0}$.

In Fig. 9, we plot the Floquet current within a drive cycle and show its
harmonics . Even though the Hamiltonian is symmetric under $I_{F}$ and
$\Theta_{F}$, the asymmetry of the projected Floquet occupation under $k\to-k$
now breaks the corresponding selection rules of the optical current.
Therefore, we obtain many of the previously forbidden harmonics, such as
$J^{(0)}$, even for the undriven system. Interestingly, we find significant
Floquet gain for $\Omega/\omega=2$ even at low field amplitudes.

Figure 8: Fractional harmonics of the current in the driven SSH model with
ideal Floquet occupation. Here, $\omega/w=3,m_{0}=0.3,m_{1}=0.17$, and
$\theta=0$.  Figure 9: Current harmonics for the projected Floquet thermal
occupation for different field amplitudes $A_{0}$ (a)-(c), and for different
temperatures $T_{0}$ (d)-(f). Here, $\omega/w=3,m_{0}=0.3$ and $\theta=0$ for
all cases, and $m_{1}=0.17$ for the driven cases. We have taken $T_{0}/w=0.01$
in (a)-(c), and $A_{0}=0.1$ in (d)-(f).

##  V Conclusions and Outlook

We have developed a theoretical framework using Floquet theory to calculate
the optical current in a periodically driven system. The Floquet optical
current mixes the field and drive frequencies, $\omega$ and $\Omega$,
respectively, generating harmonics $m\omega+M\Omega$. The frequency mixing
broadens the spectrum of harmonic orders and enhances the values of harmonics
even for a low field amplitude.

This formulation naturally contains the Floquet theoretic approach to
understanding high harmonic generation in initially undriven systems and the
Floquet linear response theory [63] as limiting cases when the drive amplitude
and the field amplitude, respectively, approach zero. In the latter case this
work provides a nonlinear Floquet theoretic framework for optical current for
any time periodic field.

The optical current can be recast in terms of the occupation of Floquet-Bloch
bands and their non-adiabatic Berry connection and curvature. This formulation
naturally brings out a relation between topology and optical current for
sufficiently low drive and field frequencies and exposes and clarifies
subtleties of gauge fixing that prevent a direct relation between current
harmonics and Floquet topological invariants for higher frequencies.

In the presence of spatio-temporal symmetries like inversion and time reversal
with time glide, we obtained the properties of optical current and selection
rules for the harmonics. As an application of our formulation, we studied the
optical current and in the driven SSH model and found analytical expressions
for its harmonics at sufficiently high frequency and weak field amplitude.
Symmetry analysis and numerical calculations show that by altering the
temporal part of the spatio-temporal symmetry, we can obtain harmonics that
are forbidden in undriven cases. The broader implication of this phenomenon is
that without changing the spatial structure or symmetry of a material, we can
procure forbidden harmonics by driving the system.

Importantly, frequency mixing between the drive and the field opens a way to
enhance the HHG spectrum. This Floquet enhancement can be accessed naturally
in pump-probe setups, where the pump and drive frequencies are commensurate.
For non-optical drives, a more realistic setup is when the drive frequency is
a small commensurate fraction of the field frequency, which always breaks the
temporal symmetry and leads to a broad-spectrum HHG. Additionally, our
calculations pertaining to the driven SSH model may be tested in synthetic
quantum simulators. [68]

Our results demonstrates that we can broaden and enhance the spectrum of
harmonic order by driving the system. This work encourages further theoretical
and experimental investigations of harmonic generation in bulk driven quantum
systems.

###### Acknowledgements.

This work has been supported in part by the NSF CAREER award DMR-1350663, the
U.S. Department of Energy, Office of Science, Basic Energy Sciences, under
Award No. DE-SC0020343, and the Vice Provost for Research at Indiana
University, Bloomington through the Faculty Research Support Program.

*

##  Appendix A Symmetries of the Floquet Hamiltonian

In this appendix, we derive the symmetries of the current from fully occupied
Floquet bands, Eqs. (24) and (26), under the unitary mirror and antiunitary
time-reversal symmetries, Eqs. (22) and (23), respectively.

The unitary symmetry $I_{F}$ transforms the Floquet unitary
$U_{F}(k,t)=\text{Texp}[-i\int_{t}^{t+T}h(k,s)ds]=e^{-iTh_{F}(k,t)}$ as

$\displaystyle I_{F}U_{F}(k,t)I_{F}^{\dagger}$
$\displaystyle=\text{Texp}\left[-i\int h(-k,s+t_{I})ds\right]$
$\displaystyle=U_{F}(-k,t+t_{I}).$ (55)

Thus, $I_{F}^{\dagger}h_{F}(-k,t+t_{I})I_{F}=h_{F}(k,t)$ and the projector
$P_{\text{occ}}(k,t)=\sum_{\alpha\in\text{occ}}\mathinner{|{\phi_{\alpha}(k,t)}\rangle}\mathinner{\langle{\phi_{\alpha}(k,t)}|}$
onto the occupied Floquet bands is mapped as

$I_{F}P_{\text{occ}}(k,t)I_{F}^{\dagger}=P_{\text{occ}}(-k,t+t_{I}).$ (56)

The occupied Floquet states are mapped explicitly as
$I_{F}\mathinner{|{\phi_{\alpha}(k,t)}\rangle}=\sum_{\beta\in\text{occ}}\mathfrak{I}_{\beta\alpha}(k,t)\mathinner{|{\phi_{\alpha}(-k,t+t_{I})}\rangle}$,
with a unitary matrix $\mathfrak{I}(k,t)$: for $\lambda,\beta\in\text{occ}$,

$\displaystyle[\mathfrak{I}(k,t)\mathfrak{I}^{\dagger}(k,t)]_{\lambda\beta}$
$\displaystyle=\sum_{\alpha\in\text{occ}}\mathfrak{I}_{\lambda\alpha}(k,t)\mathfrak{I}_{\beta\alpha}^{*}(k,t)$
$\displaystyle=\sum_{\alpha\in\text{occ}}\mathinner{\langle{\phi_{\lambda}(-k,t+t_{I})}|{I_{F}}|{\phi_{\alpha}(k,t)}\rangle}\mathinner{\langle{\phi_{\beta}(-k,t+t_{I})}|{I_{F}}|{\phi_{\alpha}(k,t)}\rangle}^{*}$
$\displaystyle=\sum_{\alpha\in\text{occ}}\mathinner{\langle{\phi_{\lambda}(-k,t+t_{I})}|{I_{F}}|{\phi_{\alpha}(k,t)}\rangle}\mathinner{\langle{\phi_{\alpha}(k,t)}|{I_{F}^{\dagger}}|{\phi_{\beta}(-k,t+t_{I})}\rangle}$
$\displaystyle=\mathinner{\langle{\phi_{\lambda}(-k,t+t_{I})}|{I_{F}P_{\text{occ}}(k,t)I_{F}^{\dagger}}|{\phi_{\beta}(-k,t+t_{I})}\rangle}$
$\displaystyle=\mathinner{\langle{\phi_{\lambda}(-k,t+t_{I})}|{P_{\text{occ}}(-k,t+t_{I})}|{\phi_{\beta}(-k,t+t_{I})}\rangle}$
$\displaystyle=\delta_{\lambda\beta}.$ (57)

Then, the current has the symmetry

$\displaystyle J(t)$
$\displaystyle=\sum_{\alpha\in\text{occ}}\oint\mathinner{\langle{\phi_{\alpha}(k,t)}|{\partial_{k}h(k,t)}|{\phi_{\alpha}(k,t)}\rangle}dk=\oint\mathrm{tr}[P_{\text{occ}}(k,t)\partial_{k}h(k,t)]dk$
$\displaystyle=\oint\mathrm{tr}[P_{\text{occ}}(k,t)I_{F}^{\dagger}\partial_{k}h(-k,t+t_{I})I_{F}]dk=\oint\mathrm{tr}[I_{F}P_{\text{occ}}(k,t)I_{F}^{\dagger}\partial_{k}h(-k,t+t_{I})]dk$
$\displaystyle=\oint\mathrm{tr}[P_{\text{occ}}(-k,t+t_{I})\partial_{k}h(-k,t+t_{I})]dk=-\oint\mathrm{tr}[P_{\text{occ}}(k,t+t_{I})\partial_{k}h(k,t+t_{I})]dk$
$\displaystyle=-J(t+t_{I}).$ (58)

The antiunitary symmetry $\Theta_{F}$ transforms the Floquet unitary as

$\displaystyle\Theta_{F}U_{F}(k,t)\Theta_{F}^{\dagger}$
$\displaystyle=\prod_{s:t\to t+T}\Theta_{F}e^{-ih(k,s)ds}\Theta_{F}^{\dagger}$
$\displaystyle=\prod_{s:t\to t+T}e^{+ih(-k,-s+t_{R})ds}$
$\displaystyle=\left(\prod_{s:-t-T+t_{R}\to-t+t_{R}}e^{-ih(-k,s)ds}\right)^{\dagger}$
$\displaystyle=U_{F}^{\dagger}(-k,-t+t_{R}).$ (59)

Here, $\prod_{s:t_{1}\to t_{2}}$ denotes the time-ordered product over
$t_{1}<t_{2}$. Thus,
$\Theta_{F}^{\dagger}h_{F}(-k,-t+t_{R})\Theta_{F}=h_{F}(k,t)$ and

$\Theta_{F}P_{\text{occ}}(k,t)\Theta_{F}^{\dagger}=P_{\text{occ}}(-k,-t+t_{R}).$
(60)

Explicitly, the occupied Floquet states are mapped as
$\Theta_{F}\mathinner{|{\phi_{\alpha}(k,t)}\rangle}=\sum_{\beta\in\text{occ}}\mathfrak{T}_{\beta\alpha}(k,t)\mathinner{|{\phi_{\alpha}(-k,-t+t_{R})}\rangle}$,
with an orthogonal matrix $\mathfrak{T}(k,t)$: for
$\lambda,\beta\in\text{occ}$,

$\displaystyle[\mathfrak{T}(k,t)\mathfrak{T}^{\raisebox{-1.0pt}{{\scriptsize{T}}}}(k,t)]_{\lambda\beta}$
$\displaystyle=\sum_{\alpha\in\text{occ}}\mathfrak{T}_{\lambda\alpha}(k,t)\mathfrak{T}_{\beta\alpha}(k,t)$
$\displaystyle=\sum_{\alpha\in\text{occ}}\mathinner{\langle{\phi_{\lambda}(-k,-t+t_{R})}|{\Theta_{F}}|{\phi_{\alpha}(k,t)}\rangle}\mathinner{\langle{\phi_{\beta}(-k,-t+t_{R})}|{\Theta_{F}}|{\phi_{\alpha}(k,t)}\rangle}$
$\displaystyle=\sum_{\alpha\in\text{occ}}\mathinner{\langle{\phi_{\lambda}(-k,-t+t_{R})}|{\Theta_{F}}|{\phi_{\alpha}(k,t)}\rangle}\mathinner{\langle{\phi_{\alpha}(k,t)}|{\Theta_{F}^{\dagger}}|{\phi_{\beta}(-k,-t+t_{I})}\rangle}$
$\displaystyle=\mathinner{\langle{\phi_{\lambda}(-k,-t+t_{R})}|{\Theta_{F}P_{\text{occ}}(k,t)\Theta_{F}^{\dagger}}|{\phi_{\beta}(-k,-t+t_{R})}\rangle}$
$\displaystyle=\mathinner{\langle{\phi_{\lambda}(-k,-t+t_{R})}|{P_{\text{occ}}(-k,-t+t_{R})}|{\phi_{\beta}(-k,-t+t_{R})}\rangle}$
$\displaystyle=\delta_{\lambda\beta},$ (61)

where we have used the identity
$\mathinner{\langle{\psi}|{\Theta_{F}}|{\chi}\rangle}=\mathinner{\langle{\chi}|{\Theta_{F}^{\dagger}}|{\psi}\rangle}$
for antiunitary $\Theta_{F}$. Similarly, the current has the symmetry

$\displaystyle J(t)$
$\displaystyle=\oint\mathrm{tr}[P_{\text{occ}}(k,t)\partial_{k}h(k,t)]dk$
$\displaystyle=\oint\mathrm{tr}[P_{\text{occ}}(k,t)\Theta_{F}^{\dagger}\partial_{k}h(-k,-t+t_{R})\Theta_{F}]dk=\oint\mathrm{tr}[\Theta_{F}P_{\text{occ}}(k,t)\Theta_{F}^{\dagger}\partial_{k}h(-k,-t+t_{R})]dk$
$\displaystyle=\oint\mathrm{tr}[P_{\text{occ}}(-k,-t+t_{R})\partial_{k}h(-k,-t+t_{R})]dk=-\oint\mathrm{tr}[P_{\text{occ}}(k,-t+t_{R})\partial_{k}h(k,-t+t_{R})]dk$
$\displaystyle=-J(-t+t_{R}),$ (62)

as stated in he main text.

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