As a prerequisite for further investigation, it is convenient to perform a gauge transformation

$$\mathrm{|}\psi \mathrm{(}t\mathrm{)}\u3009\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\mathrm{|}{\psi}^{\prime}\mathrm{(}t\mathrm{)}\u3009\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\widehat{U}}^{\u2020}\mathrm{(}t\mathrm{)}|\psi \mathrm{(}t\mathrm{)}\u3009$$(15)

$$\widehat{H}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}{\widehat{H}}^{\prime}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\widehat{U}}^{\u2020}\mathrm{(}t\mathrm{)}\widehat{H}\mathrm{(}t\mathrm{)}\widehat{U}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}i\hslash {\widehat{U}}^{\u2020}\mathrm{(}t\mathrm{)}\dot{\widehat{U}}\mathrm{(}t\mathrm{)}$$(16)

with the time-periodic unitary operator

$$\widehat{U}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{q,b}\mathrm{|}bq\mathrm{,}\text{\hspace{0.17em}}t\u3009\u3008bq\mathrm{|}}\mathrm{.}$$(17)

Here we have introduced the normalised instantaneous eigenstates |*bq*, *t*〉 of the time-dependent Hamiltonian,

$$\widehat{H}\mathrm{(}t\mathrm{)}|bq\mathrm{,}\text{\hspace{0.17em}}t\u3009\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{E}_{b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}|bq\mathrm{,}\text{\hspace{0.17em}}t\u3009\mathrm{.}$$(18)

They are Bloch waves of the lattice system at the instantaneous lattice depth *V*_{0}[1+*α* sin(*ωt*)] labeled by the same quantum numbers, quasimomentum *q* and band index *b*, as the eigenstates of the undriven system. The transformed Hamiltonian reads

$${\widehat{H}}^{\prime}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{q}{\widehat{H}}^{\prime}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}}$$(19)

with

$${\widehat{H}}^{\prime}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{b}\mathrm{|}bq\u3009{E}_{b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\u3008bq\mathrm{|}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\displaystyle \sum _{b{b}^{\prime}}\mathrm{|}{b}^{\prime}q\u3009{M}_{{b}^{\prime}b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\u3008bq\mathrm{|}}$$(20)

and matrix elements

$${M}_{{b}^{\prime}b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-i\hslash \u3008{b}^{\prime}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}{\partial}_{t}\mathrm{|}bq\mathrm{,}\text{\hspace{0.17em}}t\u3009\mathrm{.}$$(21)

For the sake of a light notation, in the following we will suppress the quasimomentum label *q*, when denoting states, energies, and matrix elements. Applying the transformation (17) is a standard procedure when treating slow parameter variations in quantum systems. Following this standard procedure further, we can bring the matrix elements *M*_{b′b}(*t*) in a more convenient form. Let us first discuss the diagonal matrix elements. They describe Berry phase effects and can, in the present case, be removed by a simple gauge transformation, since we are varying a single parameter, the lattice depth, during each driving cycle only. Namely, we can write the diagonal matrix elements like

$${M}_{bb}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-i\hslash \u3008b\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}{\partial}_{t}\mathrm{|}b\mathrm{,}\text{\hspace{0.17em}}t\u3009\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\hslash {A}_{b}\mathrm{(}V\mathrm{)}\dot{V}\mathrm{(}t\mathrm{)}$$(22)

in terms of the Berry connection *A*_{b}(*t*)=*i*〈*b*, *V*|∂_{V}|*b*, *V*〉 for a variation of the lattice depth *V*. Here we have introduced the eigenstates |*b*, *V*〉 for a lattice of depth *V*, so that |*b*, *t*〉=|*b*, *V*(*t*)〉 with *V*(*t*)=*V*_{0}[1+*α* sin(*ωt*)]. A gauge transformation $\mathrm{|}b\mathrm{,}\text{\hspace{0.17em}}V{\u3009}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}^{i{\theta}_{b}\mathrm{(}V\mathrm{)}}\mathrm{|}b\mathrm{,}\text{\hspace{0.17em}}V\u3009$ changes the Berry curvature to *A*_{b′}(*V*)=*A*_{b}(*V*)–∂_{V}*θ*_{b}(*V*), which vanishes for the choice ${\theta}_{b}\mathrm{(}V\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{V}\text{d}W{A}_{b}\mathrm{(}W\mathrm{)}}.$ Thus, for a suitable definition of the phase of the instantaneous eigenstates, the diagonal matrix elements vanish

$${M}_{bb}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.$$(23)

Berry phase effects can matter, however, in more complicated driving scenarios involving the variation of several parameters.

In order to evaluate the off-diagonal matrix elements *M*_{b′b}(*t*) with *b*′≠*b*, we consider the quantity $\u3008{b}^{\prime}\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}\frac{\text{d}}{\text{d}t}\mathrm{(}{\widehat{H}}^{\prime}\mathrm{(}t\mathrm{)}|b\mathrm{,}\text{\hspace{0.17em}}t\u3009\mathrm{)},$ which can be evaluated to both $\u3008{b}^{\prime}\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}\dot{\widehat{H}}\mathrm{(}t\mathrm{)}|b\mathrm{,}\text{\hspace{0.17em}}t\u3009\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{E}_{{b}^{\prime}}\mathrm{(}t\mathrm{)}\u3008{b}^{\prime}\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}{\partial}_{t}\mathrm{|}b\mathrm{,}\text{\hspace{0.17em}}t\u3009$ and *E*_{b}(*t*)〈*b*′, *t*|∂_{t}|*b*, *t*〉. Equating both provides an expression for 〈*b*′, *t*|∂_{t}|*b*, *t*〉 that gives

$${M}_{{b}^{\prime}b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-i\frac{\hslash \u3008{b}^{\prime}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}\dot{\widehat{H}}\mathrm{(}t\mathrm{)}|bq\mathrm{,}\text{\hspace{0.17em}}t\u3009}{{E}_{{b}^{\prime}}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{E}_{b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}}$$(24)

as long as *E*_{b′}(*q*, *t*)≠*E*_{b}(*q*, *t*). Here we have reintroduced the quasimomentum *q*.

All in all, the system is described by the time-periodic Hamiltonian

$${\widehat{H}}^{\prime}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{b}\left[\mathrm{|}bq\u3009{E}_{b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\u3008bq\mathrm{|}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\displaystyle \sum _{{b}^{\prime}\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}b}\mathrm{|}{b}^{\prime}q\u3009{M}_{{b}^{\prime}b}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\u3008bq\mathrm{|}}\right]}\mathrm{.}$$(25)

So far, no approximation has been made.

The properties of the matrix elements (24) become more transparent, when expressing the instantaneous Bloch waves in terms of instantaneous Wannier states |*b**ℓ*, *t*〉,

$$\mathrm{|}bq\mathrm{,}\text{\hspace{0.17em}}t\u3009\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{\sqrt{M}}{\displaystyle \sum _{\ell}{e}^{iqa\ell}\mathrm{|}b\ell \mathrm{,}\text{\hspace{0.17em}}t\u3009}\mathrm{.}$$(26)

Their wave functions

$$\u3008x\mathrm{|}b\ell \mathrm{,}\text{\hspace{0.17em}}t\u3009\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{w}_{b}\mathrm{(}x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\ell a\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}$$(27)

are real and exponentially localised at the lattice minima *x*=*ℓ**a* with integer *ℓ*; moreover, *w*_{b}(*x*) is even (odd) for *b* even (odd), *w*_{b}(–*x*)=(–1)^{b}*w*_{b}(*x*) [47]. The time dependence describes a breathing motion of the Wannier functions, since the width of the Wannier orbitals decreases slightly with increasing lattice depth. The numerator on the right-hand side of (24) can then be expressed like

$$\hslash \u3008{b}^{\prime}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}\dot{\widehat{H}}\mathrm{(}t\mathrm{)}|bq\mathrm{,}\text{\hspace{0.17em}}t\u3009=\alpha {V}_{0}\hslash \omega \mathrm{cos}\mathrm{(}\omega t\mathrm{)}{\displaystyle \sum _{\ell}{e}^{iqa\ell}{W}_{{b}^{\prime}b}^{\mathrm{(}\ell \mathrm{)}}\mathrm{(}t\mathrm{)}}\mathrm{,}$$(28)

with matrix elements (see Figure 4)

$${W}_{{b}^{\prime}b}^{\mathrm{(}\ell \mathrm{)}}\mathrm{(}t\mathrm{)}={\displaystyle \int \text{d}x{w}_{{b}^{\prime}}\mathrm{(}x+\ell a\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}{\mathrm{sin}}^{2}\mathrm{(}{k}_{L}x\mathrm{)}{w}_{b}\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}}$$(29)

that obey

$${W}_{{b}^{\prime}b}^{\mathrm{(}-\ell \mathrm{)}}\mathrm{(}t\mathrm{)}={\mathrm{(}-1\mathrm{)}}^{b\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{b}^{\prime}}{W}_{{b}^{\prime}b}^{\mathrm{(}\ell \mathrm{)}}\mathrm{(}t\mathrm{)}\mathrm{.}$$(30)

Figure 4: Coupling matrix elements ${\nu}_{b}^{\ell}={W}_{b0}^{\mathrm{(}\ell \mathrm{)}}$ as defined in (29) for a static lattice of depth *V*_{0}/*E*_{R}.

Thus, for even (*b*′+*b*) the sum on the right-hand side of (28) reads

$${W}_{{b}^{\prime}b}^{\mathrm{(}0\mathrm{)}}\mathrm{(}t\mathrm{)}+2{W}_{{b}^{\prime}b}^{\mathrm{(}1\mathrm{)}}\mathrm{(}t\mathrm{)}\mathrm{cos}\mathrm{(}qa\mathrm{)}+2{W}_{{b}^{\prime}b}^{\mathrm{(}2\mathrm{)}}\mathrm{(}t\mathrm{)}\mathrm{cos}\mathrm{(}2qa\mathrm{)}+\cdots \mathrm{,}$$(31)

whereas for odd (*b*′+*b*) the leading *ℓ*=0 term vanishes, and one finds

$$2i{W}_{{b}^{\prime}b}^{\mathrm{(}1\mathrm{)}}\mathrm{(}t\mathrm{)}\mathrm{sin}\mathrm{(}qa\mathrm{)}+2i{W}_{{b}^{\prime}b}^{\mathrm{(}2\mathrm{)}}\mathrm{(}t\mathrm{)}\mathrm{sin}\mathrm{(}2qa\mathrm{)}+\cdots \mathrm{.}$$(32)

These equations explain why transitions to odd bands are suppressed completely in the spectra of Figure 2 for *q*=0 and *q*=*π*/*a*. The missing *ℓ*=0 term for odd transitions, which is related to parity conservation within a single lattice site, explains also the observed relative suppression of transitions from the lowest to odd bands for other values of *q*. Namely, due to the exponential localisation of the Wannier functions, the matrix elements ${W}_{{b}^{\prime}b}^{\mathrm{(}\ell \mathrm{)}}\mathrm{(}t\mathrm{)}$ drop rapidly with *ℓ*. It is, therefore, reasonable to keep only the leading term and to approximate

$$\hslash \u3008{b}^{\prime}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}\dot{\widehat{H}}\mathrm{(}t\mathrm{)}|bq\mathrm{,}\text{\hspace{0.17em}}t\u3009=\alpha {V}_{0}\hslash \omega \mathrm{cos}\mathrm{(}\omega t\mathrm{)}{W}_{{b}^{\prime}b}^{\mathrm{(}0\mathrm{)}}\mathrm{(}t\mathrm{)}$$(33)

for even (*b*′+*b*) and

$$\hslash \u3008{b}^{\prime}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{|}\dot{\widehat{H}}\mathrm{(}t\mathrm{)}|bq\mathrm{,}\text{\hspace{0.17em}}t\u3009=i2\alpha {V}_{0}\hslash \omega \mathrm{cos}\mathrm{(}\omega t\mathrm{)}{W}_{{b}^{\prime}b}^{\mathrm{(}1\mathrm{)}}\mathrm{(}t\mathrm{)}\mathrm{sin}\mathrm{(}qa\mathrm{)}$$(34)

for odd (*b*′+*b*).

In order to be explicit, in the following we will focus on transitions from the lowest to the second excited band. For small quasimomenta *k*≪*π*/*a* these transitions constitute the dominant heating channel. The relevant matrix element ${W}_{20}^{\mathrm{(}0\mathrm{)}}$ has a rather weak dependence on the lattice depth, as can be seen from Figure 4 where this parameter is plotted for a lattice of static depth *V*_{0}. Thus, when the lattice depth is modulated, *V*_{0}→*V*_{0}[1+*α* sin(*ωt*)], we can approximate

$${W}_{20}^{\mathrm{(}0\mathrm{)}}\mathrm{(}t\mathrm{)}\approx W-\alpha {W}^{\prime}\mathrm{sin}\mathrm{(}\omega t\mathrm{)}\mathrm{,}$$(35)

neglecting higher harmonics. Both coefficients *W* and *W*′ have a very weak dependence on *α* only, and one has *W*′≪*W*~1. At an *n* “photon” resonance, we can likewise approximate the instantaneous energy difference between both bands like

$${E}_{2}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}-{E}_{0}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\approx \Delta \mathrm{(}q\mathrm{)}+\alpha F\mathrm{(}q\mathrm{)}\mathrm{sin}\mathrm{(}\omega t\mathrm{)}$$(36)

and its inverse like

$$\frac{1}{{E}_{2}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}-{E}_{0}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}}\approx \frac{1}{\Delta \mathrm{(}q\mathrm{)}}-\alpha \frac{F\mathrm{(}q\mathrm{)}}{{\Delta}^{2}\mathrm{(}q\mathrm{)}}\mathrm{sin}\mathrm{(}\omega t\mathrm{)}\mathrm{.}$$(37)

Taking terms up to *α*^{2}, the matrix element *M*_{20}(*q*, *t*) then reads

$${M}_{20}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\approx -i\frac{{V}_{0}}{n}\mathrm{[}\alpha W\mathrm{cos}\mathrm{(}\omega t\mathrm{)}-{\alpha}^{2}X\mathrm{(}q\mathrm{)}\mathrm{sin}\mathrm{(}2\omega t\mathrm{)}]\mathrm{,}$$(38)

where we have used sin(*a*)cos(*a*)=sin(*2*a)/2, employed the resonance condition

$$\Delta \mathrm{(}q\mathrm{)}=n\hslash \omega \mathrm{,}$$(39)

and defined

$$X\mathrm{(}q\mathrm{)}=\frac{1}{2}\left[{W}^{\prime}+\frac{F\mathrm{(}q\mathrm{)}}{\Delta \mathrm{(}q\mathrm{)}}\right]\mathrm{,}$$(40)

where *X*(*q*)≪*W*.

At the resonance (2, *n*), we will describe the system within the the subspace spanned by the bands *b*=0 and *b*=2. Up to a time-dependent energy constant, the relevant Hamiltonian is given by

$$\begin{array}{c}{\widehat{H}}^{\prime}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}\mathrm{[}n\hslash \omega \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\alpha F\mathrm{(}q\mathrm{)}\mathrm{sin}\mathrm{(}\omega t\mathrm{)}]|2q\u3009\u30082q\mathrm{|}\\ +\text{\hspace{0.17em}}{M}_{20}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}|2q\u3009\u30080q\mathrm{|}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{M}_{20}^{\ast}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}|0q\u3009\u30082q|\mathrm{.}\end{array}$$(41)

The Fourier decomposition of the Hamiltonian is given by

$${\widehat{H}}^{\prime}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}={\displaystyle \sum _{m}{{\widehat{H}}^{\prime}}_{m}\mathrm{(}q\mathrm{)}{e}^{im\omega t}}\mathrm{,}$$(42)

$${{\widehat{H}}^{\prime}}_{m}\mathrm{(}q\mathrm{)}=\frac{1}{T}{\displaystyle {\int}_{0}^{T}\text{d}t{e}^{-im\omega t}\widehat{H}\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}}\mathrm{,}$$(43)

with driving period *T*=2*π*/*ω*, we find

$${{\widehat{H}}^{\prime}}_{0}\mathrm{(}q\mathrm{)}=n\hslash \omega |2q\u3009\u30082q|\mathrm{,}$$(44)

${{\widehat{H}}^{\prime}}_{1}\mathrm{(}q\mathrm{)}=-i\frac{\alpha F\mathrm{(}q\mathrm{)}}{2}|2q\u3009\u30082q\mathrm{|}$

$$-i\frac{\alpha {V}_{0}W}{2n}\mathrm{(}|2q\u3009\u30080q\mathrm{|}-|0q\u3009\u30082q|\mathrm{)}\mathrm{,}$$(45)

$${{\widehat{H}}^{\prime}}_{2}\mathrm{(}q\mathrm{)}=\frac{{\alpha}^{2}{V}_{0}X\mathrm{(}q\mathrm{)}}{2n}\mathrm{(}|2q\u3009\u30080q\mathrm{|}-|0q\u3009\u30082q|\mathrm{)}\mathrm{,}$$(46)

as well as the conjugated terms ${\widehat{H}}_{-m}={\widehat{H}}_{m}^{\u2020}.$ The terms ${\widehat{H}}_{m}$ become smaller with increasing *m* and depend on the driving strength like *α*^{|m|}. This applies also to the higher harmonics that we neglected.

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