# Superoscillations Made Super Simple

Gerard McCaul<EMAIL_ADDRESS>Tulane University, New Orleans, LA 70118,
USA  Peisong Peng<EMAIL_ADDRESS>Tulane University, New Orleans, LA 70118,
USA  Monica Ortiz Martinez<EMAIL_ADDRESS>Tulane University, New
Orleans, LA 70118, USA  Dustin Lindberg<EMAIL_ADDRESS>Tulane
University, New Orleans, LA 70118, USA  Diyar Talbayev<EMAIL_ADDRESS>Tulane University, New Orleans, LA 70118, USA  Denys I. Bondar
<EMAIL_ADDRESS>Tulane University, New Orleans, LA 70118, USA

###### Abstract

In ordinary circumstances the highest frequency present in a wave is the
highest frequency in its Fourier decomposition. It is however possible for
there to be a spatial or temporal region of the wave which locally oscillates
at a still greater frequency, in a phenomenon known as superoscillation [1].
Superoscillations find application in wide range of disciplines [2, 3, 4], but
at present their generation is based upon constructive approaches which are
difficult to implement. Here we address this, exploiting the fact that
superoscillations are a product of destructive interference to produce a
prescription for generating superoscillations from the superposition of
arbitrary waveforms. As a first test of the technique, we use it to combine
four THz laser fields generated by periodically poled Lithium Niobate. From
this, we are able to predict and observe for the first time THz optical
superoscillations in the temporal domain. The ability to generate
superoscillations in this manner has potential application in a wide range of
fields. It may for example contribute to the experimental realization of the
complex pulses required by quantum control, and the generation of attosecond
pulses without resorting to nonlinear processes.

Of all physical phenomena, light is perhaps the most crucial to our perception
and understanding of the world. Its ability to transmit information regarding
matter has long been harnessed as a central tool for the study of the natural
world. Attempts to understand the nature of light stretch back to antiquity,
with the majority of Hellenic philosophers espousing an ‘extramission’ theory
of vision, in which light propagated out from the eyes in order to ‘touch’
objects [5]. This theory was only disproven in the 11th century AD by Alhazen,
who invented the first _camera obscura_ in the process [6]. In later years,
Newton’s determination to confirm his own theory of optics [7, 8, 9] extended
to self-experimentation, using a bodkin to deform the back of his eyeball [10,
11]. A complete description of classical light was however not achieved until
the work of Maxwell in the 1870s [12].

In the modern era, as our understanding of light has progressed, it has
assumed a role of central importance in both industry and research. In this
context, light is not only a subject of fundamental study, but a tool for the
manipulation of matter [13, 14, 15, 16, 17, 18, 19, 20], a medium of
computation [21, 22, 23, 24, 25, 26], and a probe of physical systems. It is
often assumed that the ultimate limitations of light for imaging and other
optical experimentation correspond to the highest frequency in its Fourier
decomposition (e.g. the diffraction limit). In some cases however, it is
possible to construct a _superoscillation_ (SO) - a band-limited signal which
has a temporal or spatial region in which it oscillates at a frequency faster
than its largest Fourier component [1].

The term superoscillation was first coined by Michael Berry [27, 28], in
reference to work by Aharonov, Bergman, and Lebowitz [29]. Today
superoscillations constitute a rapidly developing field of study in both
mathematics [30, 31] and physics [32, 33]. The main hinderance to the
application of superoscillations has been their minute intensity as compared
to full pulse, but advances in the general sensitivity of measurements means
that the relative amplitudes of superoscillations are no longer an impediment
to their study and application [1]. These applications cover a broad range of
topics, including optical metrology [34], super-resolution [35, 36, 37],
super-transmission [2, 3, 4] and free-space plasmonics [38].

In the spatial domain, there have been a number of experimental observations
of superoscillations [39, 28, 40, 41], which extend even into the THz optical
frequency range and allow for imaging below the diffraction limit [42]. This
raises the tantalising possibility that nanoscale imaging with visible light
may be made possible by superoscillation [43]. In the temporal domain THz
acoustic superoscillations have been observed in superlattices [44], while the
envelopes of laser pulses have been manipulated to display superoscillatory
behaviour (relative to the envelope frequency) [45, 46]. Up to this point
however, there has been no direct observation of a time domain THz optical
superoscillation, despite the availability of THz radiation which can be both
characterised and manipulated in the time domain [47]. Much like in the
spatial domain, temporal superoscillations have tremendous potential to
broaden the domain of useful operation for current technology. One could for
example use superoscillations to perform _superspectroscopy_ at higher
frequencies, enhancing the range of spectroscopically accessible frequencies
by currently available light sources.

At present, the prescriptions for directly constructing superoscillations (as
opposed to achieving them via the design of lenses [48] or filters [36]) have
been based on the analysis of band limited functions, but transitioning to
experimentally viable approximations of such functions is fraught with
difficulties [49, 50, 51]. Despite the sensitivity of purposely constructed
superoscillations, there has been a preponderance of ‘accidental’
superoscillations observed both in random functions [27] and more structured
fields [52, 53]. One might therefore ask if there exists a simple heuristic
for generating superoscillatory fields through the superposition of
experimentally available pulses, which avoids the difficulties of explicitly
constructive techniques. Here we present just such a method for the generation
of time domain superoscillations, and apply it to achieve a first realisation
of an optical time-domain superoscillation in the THz regime.

### Superoscillations as destructive interference

The phenomenon of superoscillations has typically been studied in the context
of the function [27, 37]

$F_{n}(a,t)=\left[\cos\frac{t}{n}+ia\sin\frac{t}{n}\right]^{n}.$ (1)

The behaviour of this function for small $t$ (or equivalently large $n$) can
be obtained by Taylor expanding to first order:

$F_{n}(t,a)\approx\left(1+ia\frac{t}{n}\right)^{n}\approx{\rm e}^{iat}.$ (2)

When $a=1$ this expression is exact for any $n$, but in the case of $a>1$, it
produces the counterintuitive behaviour that $F_{n}(a,t)$ oscillates faster
than its highest frequency Fourier component – $e^{it}$. To show this, let us
re-express Eq. (1) as a Fourier series. The most direct route to achieving
this is to express the trigonometric functions as exponentials before
performing a binomial expansion. This leads directly to the Fourier expansion
of $F_{n}(a,t)$:

$\displaystyle F_{n}(a,t)$ $\displaystyle=\sum_{j=0}^{n}A_{j}{\rm
e}^{ik_{j}t},$ (3) $\displaystyle A_{j}$
$\displaystyle=\frac{1}{2^{n}}{n\choose j}(1+a)^{n-j}(1-a)^{j},$ (4)
$\displaystyle k_{j}$ $\displaystyle=1-2\frac{j}{n}.$ (5)

Regardless of the value of $a$ chosen, the highest frequency in the Fourier
decomposition will be $k_{0}=1$. Hence, for $a>1$, the function close to the
origin will “superoscillate” at a frequency greater than its largest Fourier
component. This is illustrated in Fig. 1 (top panel), where the frequency of
the superoscillation relative to its largest Fourier component increases with
$a$.

Figure 1:  Comparison of superoscillating functions (top panel) to their
intensities (bottom panel). Comparison between the two panels readily
demonstrates that the region where the magnitude $\left|F_{n}(a,t)\right|^{2}$
is minimised corresponds to a superoscillatory window where the function
oscillates faster than its greatest Fourier component $k_{0}=1$ (shown as
dashed line). The precise frequency achieved by the superoscillation is a
function of the parameter $a$.

A function of the type described by Eq.(1) is far from the only way to obtain
a superoscillation however. Careful examination of the secondary properties
$F_{n}(a,t)$ can be used to formulate an alternative prescription for
generating super oscillations. Here we focus on the fact that
superoscillations occur in the region where $\left|F_{n}(a,t)\right|^{2}$ is
minimised, as shown by Fig. 1 (bottom panel). This phenomenon can be
understood heuristically by noting that the magnitude of the superoscillation
will be of order 1 by Eq.(2). Examination of the Fourier components via Eq.(4)
however shows individual frequencies may have much larger amplitudes. The size
of these components can only be reconciled with the superoscillation if there
is near total cancellation of the individual field components around $t=0$.
This means that _superoscillations are a product of destructive interference_.
This fact allows us to use the proxy of destructive interference as a figure
of merit for the construction of superoscillations.

## Results

Let us now apply this logic to the problem of generating a superoscillating
optical field $E(t)$ using a set of experimentally accessible, time-limited
$E_{j}(t)$ waveforms with different central frequencies $\omega_{j}$. Each
waveform has an amplitude $a_{j}$ and time delay $\tau_{j}$ such that the
total field is given via

$E(t)=\sum_{j=1}^{N}a_{j}E_{j}(t-\tau_{j}).$ (6)

We wish to select our parameters such that the intensity of this waveform is
minimised over some time window $t\in[-T_{\rm SO},T_{\rm SO}]$ which we shall
refer to as the superoscillatory region. This immediately presents two
difficulties if one varies amplitudes. First, in order to avoid the trivial
solution $a_{j}=0$, it is necessary to condition any minimisation to avoid
this outcome. Moreover, amplitude modulation is guaranteed to reduce the
integrated intensity of the full waveform, making experimental detection more
challenging.

For this reason, we consider only variations of the time delays $\tau_{j}$
when minimising intensity within the superoscillatory region (i.e. setting
$a_{j}=1$). Explicitly, we wish to minimise the objective function

$\displaystyle I(\\{\tau_{j}\\})=\int_{-T_{\rm SO}}^{T_{\rm
SO}}\Bigg{[}\sum_{j=1}^{N}E_{j}(t-\tau_{j})\Bigg{]}^{2}{\rm d}t$ (7)

by varying the time delays $\tau_{j}$. Given the different central frequencies
of the component waveforms, the integrated intensity of the combined beam will
remain roughly constant as time delays are varied, but by minimising that
intensity in some temporal region, we expect to generate superoscillatory
behaviour. Fig.2 shows an example of this, where each component waveform is
shifted in time such that a superoscillation is generated by their
superposition.

Figure 2:  In order to generate superoscillations, each component waveforms is
shifted in time by an amount which minimises Eq.(7). The combination of these
shifted waveforms then creates a superoscillation within the desired region.

This objective function can be minimised in any number of ways, e.g. via
gradient descent [54]. The cost function itself is non-convex, meaning a
number of local minima may be obtained depending on the initial guess for time
delays. Finding the global minimum is not imperative however, as given the
trade-off between the superoscillation amplitude and frequency, such a
solution is likely to correspond to a superoscillation whose amplitude is
beyond the limit of experimental detection. It is therefore useful to have a
number of minima when seeking the best balance between the amplitude and
frequency of the superoscillation.

As a first demonstration of the viability of this method for generating
superoscillations, we combine four near-sinusoidal THz optical beams. Fig. 3
shows the spectrum of each of these component waveforms. Note that while each
measured waveform deviates from monochromaticity, the prescription for
generating superoscillations is insensitive to the precise form of its
constituent beams. Additionally, we set $T_{\rm SO}=0.6$ ps, such that the
superoscillatory region has a length corresponding to approximately one period
of the highest frequency beam.

Figure 3:  Spectra of each of the four component waveforms used to generate
the superoscillation. Each wave is labeled by its nominal frequency in THz.

After measuring each component waveform $E_{j}(t)$ individually, a
minimisation of Eq.(7) is performed. The time delays prescribed by this
procedure are then implemented experimentally. Three example minima obtained
from this process are shown in Fig. 4. As predicted, in each case
superoscillations are clearly observed within the superoscillatory region
[Figs. 4(d,e,f)]. These superoscillations are not only of a sufficient
amplitude for detection, but exhibit excellent agreement with the theoretical
prediction used to generate them.

The precise enhancement in frequency provided by the superoscillation may be
quantified via a calculation of ‘local frequency’. This is defined in analogy
with the local wavenumber used to characterise spatial superoscillations [55],
and is the gradient of the phase $f_{\rm loc}=\frac{{\rm d}\phi}{{\rm d}t}$.
This is obtained by Hilbert transforming the real valued signal $E(t)$ into an
analytic form [56], from which the field phase $\phi$ is extracted. As Figs.
4(g, h, i) show, for most of the duration of the combined waveform, its local
frequency is approximately equal to its highest frequency component. Within
the superoscillatory region however, the combined waveform’s $f_{\rm loc}$
deviates sharply, experiencing a $\sim$ 2 fold increase relative to the
highest frequency component.

Figure 4: Three example superoscillations obtained via the minimisation of Eq.
(7). Left column (a, b, c) shows both the predicted and observed field over
the full measurement window. The middle panels (d, e, f) highlight the
superoscillatory region in which the superoscillation occurs, with comparison
to the highest frequency component waveform. The experimental data is in
excellent agreement with the theoretical 95% confidence limit (shaded blue
area). The right column (g, h, i) plots the local frequencies of both the
total waveform and its highest frequency component. Outside the
superoscillatory region these are roughly equivalent, but within the
superoscillatory region the combined waveform’s local frequency increases
sharply compared to that of the highest frequency component.

## Discussion and Outlook

The ability to probe systems with optical signals underpins the ongoing
development of quantum technologies, and superoscillatory signals have the
potential to greatly expand the toolset available to researchers. The present
work addresses the question of how to practically generate such
superoscillations in the time domain. While the results presented here employ
a specific methodology for producing superoscillations, the ultimate goal of
the method - to minimise pulse intensity over a given time window - can be
achieved with a great variety of techniques. This freedom means that the
prescription for generating superoscillations may be tailored to enforce some
desired secondary properties, depending on the application and specific
technique employed. Indeed, perhaps the principal future challenge will be to
develop methodologies that maximise the control available in the generation of
superoscillations. Nevertheless, as has been shown here, even a relatively
simplistic prescription is capable of producing clearly detectable optical
superoscillations in the time domain.

The potential applications of superoscillations cover a broad range of topics.
Most obviously they extend the range of frequencies accessible by a given
light source. This is of particular interest as researchers begin to explore
dynamics in the attosecond regime [57]. At present the high frequency light
necessary to probe this timescale is generated via highly non-linear effects
such as high harmonic generation [58, 59, 60], but superoscillations offer an
alternative platform for realising these light sources, without recourse to
non-linear effects.

Furthermore, recent developments in quantum control have demonstrated the
highly malleable nature of driven systems, to the extent that one can force
one material to ‘mimic’ the optical response of another [61, 62, 63, 64]. Such
manipulations require complex, broadband fields however, and superoscillations
may facilitate the realisation of these control fields.

Finally, the super-transmissive property of superoscillations [2] means that
they can be used to transmit light through media at frequencies that would
ordinarily be absorbed. This has the potential to allow for the probing of
materials in a range at which they would ordinarily be optically opaque. Taken
together, these applications suggest that superoscillations - and the ability
to generate them - have the opportunity to become vital tools in the armory of
optical physics.

## Data Availability

All experimental data, together with the minimisation procedure required to
construct the superoscillations shown in Fig. 4 may be found at
https://github.com/dibondar/superoscilations/blob/master/superoscillationsfromexperimentaldata/getsuperoscilationsfrom03-11-2022data.ipynb

###### Acknowledgements.

This was was supported by by the W. M. Keck Foundation. D.I.B. and G.M. are
also supported by Army Research Office (ARO) (grant W911NF-19-1-0377; program
manager Dr. James Joseph). The views and conclusions contained in this
document are those of the authors and should not be interpreted as
representing the official policies, either expressed or implied, of ARO or the
U.S. Government. The U.S. Government is authorized to reproduce and distribute
reprints for Government purposes notwithstanding any copyright notation
herein.

### Author Contributions

G.M., D.L. and D.I.B. performed the theoretical analysis, while P.P., M.O.M.
and D.T. designed the experimental setup and performed measurements. D.T. and
D.I.B. supervised and coordinated this project, while G.M. wrote the initial
manuscript draft, with the assistance of all other authors. G.M. and P.P.
contributed equally to this work.

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# Observation of light driven band structure via multi-band high harmonic
spectroscopy

Ayelet J. Uzan-Narovlansky  Department of Complex Systems, Weizmann Institute
of Science, 76100, Rehovot, Israel Department of Physics, Princeton
University, Princeton, NJ, USA  Álvaro Jiménez-Galán  Max-Born-Institut, Max-
Born Strasse 2A, D-12489 Berlin, Germany  Joint Attosecond Science Laboratory,
National Research Council of Canada and University of Ottawa, Ottawa, Canada
Gal Orenstein  SLAC National Accelerator Laboratory, Stanford University,
Stanford, California 94305, USA  Rui E.F. Silva  Instituto de Ciencia de
Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Científicas
(CSIC), Madrid, Spain  Talya Arusi-Parpar  Department of Complex Systems,
Weizmann Institute of Science, 76100, Rehovot, Israel Sergei Shames
Department of Complex Systems, Weizmann Institute of Science, 76100, Rehovot,
Israel Barry D. Bruner  Department of Complex Systems, Weizmann Institute of
Science, 76100, Rehovot, Israel Binghai Yan  Department of Condensed Matter,
Weizmann Institute of Science, 76100, Rehovot, Israel  Olga Smirnova  Max-
Born-Institut, Max-Born Strasse 2A, D-12489 Berlin, Germany  Technische
Universität Berlin, Ernst-Ruska-Gebäude, Hardenbergstr. 36A, D-10623 Berlin,
Germany  Misha Ivanov  Max-Born-Institut, Max-Born Strasse 2A, D-12489 Berlin,
Germany  Blackett Laboratory, Imperial College London, South Kensington
Campus, SW7 2AZ London, United Kingdom  Department of Physics, Humboldt
University, Newtonstrasse 15, 12489 Berlin, Germany  Nirit Dudovich
Department of Complex Systems, Weizmann Institute of Science, 76100, Rehovot,
Israel

Intense light-matter interactions have revolutionized our ability to probe and
manipulate quantum systems at sub-femtosecond time scales1, opening routes to
all-optical control of electronic currents in solids at petahertz rates2, 3,
4, 5, 6, 7. Such control typically requires electric field amplitudes $\sim
V/$\mathrm{\SIUnitSymbolAngstrom}$$, when the voltage drop across a lattice
site becomes comparable to the characteristic band gap energies. In this
regime, intense light-matter interaction induces significant modifications of
electronic and optical properties 8, 9, 10, dramatically modifying the crystal
band structure. Yet, identifying and characterizing such modifications remains
an outstanding problem. As the oscillating electric field changes within the
driving field’s cycle, does the band-structure follow, and how can it be
defined? Here we address this fundamental question, proposing all-optical
spectroscopy to probe laser-induced closing of the band-gap between adjacent
conduction bands. Our work reveals the link between nonlinear light matter
interactions in strongly driven crystals and the sub-cycle modifications in
their effective band structure.

The ability of near-resonant light fields to modify the energetics and the
dynamics of an atom (or a molecule), is central to many phenomena such as
laser cooling, trapping, quantum optics of atoms in a cavity 11, 12, or
extremely efficient generation of hyper-Raman lines13. The combined states of
matter and near-resonant light are generally studied over many laser cycles
using the Floquet formalism, which takes advantage of the periodicity of light
oscillations. In solids, cycle-averaged modification of the hopping rates
between the neighboring sites leads to, e.g., coherent destruction of
tunneling 14 and modified local interaction potentials 10.

The situation changes in light fields with frequencies well below resonance
but with intensities sufficiently high to induce electron-volts-scale voltages
across a lattice site. As the oscillating electric field of the lightwave
changes from zero to its maximum value within a quarter-cycle, rapidly
changing voltages can lead to sub-cycle modifications of the macroscopic
properties, such as the transmittance6, 4 or conductance7. In this regime, a
cycle-averaged frequency-domain Floquet perspective is hardly satisfying.

Here we describe how the effective band structure can be introduced in such
interaction regime within a time-domain perspective. We experimentally
demonstrate all-optical spectroscopy of a strongly driven crystal, revealing
an anomalous spectral intensity response, using high harmonic generation
(HHG). Our theoretical study links these observations to laser-induced closing
of the gap between adjacent conduction bands (fig. 1a).

HHG in solids involves optical tunneling15 across the gap between the valence
and the conduction band. This transition initiates harmonic generation
associated with both intraband currents 16, 17, 10 and electron-hole
recombination, the latter leading to higher-order harmonic emission 18. At
high light intensities and/or low frequencies, one enters a new dynamical
regime. In many systems, as the electron-hole wavepacket approaches the edges
of the Brillouin zone, Landau-Dykhne-type transitions19 can promote electrons
to higher conduction bands, as reflected in the harmonic spectra20, 21 (fig.
1a). However, can the imprint of these transitions on high harmonic emission
be used to observe light-induced modifications of the bands?

One can show that light-induced modifications of the bands are directly linked
to the sub-cycle Landau-Dykhne-type transitions between them (see Methods). In
the low-frequency limit, one begins with the adiabatic approximation, which
treats the phase $\omega t$ of the electric field oscillations as a parameter,
and finds the adiabatic band structure $\epsilon(k,\omega t)$ in the presence
of a quasistatic electric field. When a non-adiabatic Landau-Dykhne type
transition between a pair of bands occurs, typically as the electron is laser-
driven across the minimal band-gap, it modifies the gap as follows:

$\displaystyle\Delta\epsilon_{\rm eff}(t)\simeq\Delta\epsilon_{\rm
ad}(t)\sqrt{\frac{1-w_{\rm LD}(t)}{1+w_{\rm LD}(t)}}$ (1)

where $\Delta\epsilon_{\rm eff}$ is the effective bandgap,
$\Delta\epsilon_{\rm ad}$ is the bandgap in the adiabatic approximation, and
$w_{\rm LD}$ is the probability of the sub-cycle Landau-Dykhne type
transition. The result is intuitive: the effective band gap is closed by the
laser field when the sub-cycle transition approaches unity, $w_{\rm
LD}(t)\simeq 1$. Given the exponential sensitivity of $w_{\rm LD}(t)$ to laser
field strength, the effective bandgap closes soon after the sub-cycle
excitation probability becomes appreciable.

To investigate this phenomenon experimentally, we use the two-color HHG
spectroscopy22, 23 (fig. 1b), augmenting the strong fundamental driver with a
weak second harmonic (SH) field, while controlling their subcycle delay
$\tau$. Electron-hole trajectories responsible for different harmonics 24 are
perturbed by the weak field: each trajectory acquires an additional complex
phase $\sigma(\tau)$, which is accumulated along the entire trajectory,
serving as a sensitive label of its temporal properties. If the fundamental
field generates only odd harmonics, the SH field breaks the symmetry of the
interaction, as $\sigma(\tau)$ changes sign between the two consecutive half
cycles (see SI). This phase is mapped into the harmonic intensity as:

$\begin{split}I_{odd}(\sigma(\tau))\propto(e^{i\sigma}+e^{-i\sigma})\\\
I_{even}(\sigma(\tau))\propto(e^{i\sigma}-e^{-i\sigma})\end{split}$ (2)

Scanning $\tau$ modifies $\sigma(\tau)$ in a periodic manner, modulating the
harmonic spectrum. The modulation phase and contrast encode the dynamical
properties of electron trajectories associated with each harmonic order and
allow their reconstruction with attosecond precision. In previous studies this
method resolved the interband contribution to high-harmonic emission24 and
provided all-optical reconstruction of the field-free band structure upon
exciting a single conduction band 25, 26. In this paper we use this scheme to
study the underlying dynamics of driven multiband currents, probing the
dressed band structure.

Experiments were performed on MgO27, using $\lambda=1.3\mu m$ laser field at
intensities $\sim 10^{13}\text{ }\frac{W}{cm^{2}}$ and a weak SH field,
polarized parallel to the fundamental field. We have measured the HHG
modulations with the two-color delay and extracted the oscillation phase
$\Phi_{N}$ associated with each harmonic order $N$. Figure 2 presents
$\Phi_{N}$ as a function of harmonic order for orientation angles of
$0^{\circ}$ ($\Gamma X$) and $45^{\circ}$ ($\Gamma K$), with respect the
fundamental field’s polarization. For harmonics $N=11-15$ (10.5-14.5 eV),
which are associated with the electron-hole recombination from the first
conduction band, we measure a gradual slope of $\Phi_{N}$ with $N$. The slope
reflects the evolution of the trajectory length with the harmonic order 24,
supported by our semi-classical calculations (see SI). As we approach the edge
of the Brillouin zone, represented by harmonic 17 (16 eV) for $\Gamma K$ and
harmonic 19 for $\Gamma X$ directions, this simple description fails. A clear
phase jump appears in the measurements, associated with the edge of the zone
and the changing bands curvature. At this point the mapping between momentum
and energy becomes singular, leading to the appearance of spectral caustic21.
Thus, the phase measurement serves as sensitive probe of the band gap and
allows its accurate identification.

Beyond the cut-off energy of the first conduction band, the electron dynamics
involve multiple bands 28, 20. The harmonic emission is dictated by electron
currents originating from higher conduction bands, determined by their
structure and coupling. Importantly, these parameters depend on the crystal
orientation, with each orientation offering a new one-dimensional slice of the
band structure27, 29. We can thus track the dependence of electron dynamics on
the band structure and laser-induced couplings by resolving the oscillation
phase of each harmonic as a function of the crystal orientation.

Figure 3a presents the HHG oscillation phase, for various crystal
orientations. The oscillation phases of the lower harmonics (H11,H13,H15)
emitted far from the edge of the Brillouin zone remain unchanged with the
crystal orientation. Indeed, the bottom of the first conduction band, as well
as its distance from the valence band,
$\epsilon_{c1,v}=\epsilon_{c1}-\epsilon_{v}$, is approximately isotropic (fig.
3c).

For higher harmonics ($N=19-29$), a strong variation of the oscillation phase
with crystal orientation is observed (fig. 3a). Figure 3b presents the
oscillation phase as a function of the crystal’s orientation for H21. The
phase shows a sharp increase by more than $\pi$ as the orientation angle
changes towards 45∘. The origin of this sharp variation can be understood by
looking at the angular dependence of the bands. Figure 3c shows the energy
difference between the valence, the first, and the second conduction bands
($\epsilon_{c1,v},\epsilon_{c2,v}$) for different crystal orientations,
together with the energy contours for harmonic 21. The crystal momentum at the
emission point associated with H21 changes quickly with crystal orientation,
as illustrated in fig. 3d. The strong orientation-dependent modifications of
the band gap, $\epsilon_{c2,v}$, lead to significant angular changes of the
corresponding electron trajectories, as captured by the phase measurement. In
contrast, the emission points for H11-H15 are almost constant with the crystal
orientation (fig. 3d). Such nearly isotropic response is captured by the phase
measurement as well (fig. 3b).

We now turn to resolving the light-induced dressing of the band structure.
While in large band gap materials, such as MgO, observing band gap
modifications requires high intensities, the gap between the conduction bands
is relatively small. Therefore, it can be significantly modified at moderate
field intensities. We focus on harmonic emission associated with an energy gap
between the first and second conduction band. In fig. 4a we plot the field
free band structure, which provides a good description of the system at low
laser intensities. At $0^{\circ}$ orientation the minimum energy gap between
two conduction bands, $\epsilon_{c2}-\epsilon_{c1}$, is around $3eV$; harmonic
20 is located inside this gap. As we rotate the crystal, the energy gap
rapidly reduces to zero, and harmonic 20 is emitted from the second conduction
band. In fig. 4b, we plot the oscillation phase of harmonic 20, $\Phi_{20}$,
as a function of crystal orientation, for different laser intensities. At low
intensity (light green), $\Phi_{20}$ shows a dramatic angular dependence,
mainly between $0^{\circ}-25^{\circ}$, as a consequence of the strong angular
dependence of $\epsilon_{c2}-\epsilon_{c1}$. Importantly, as we increase the
field’s intensity, the angular dependence of $\Phi_{20}$ decreases and then
flattens significantly (dark green). These experimental results are confirmed
by our numerical simulations (fig. 4c). Numerically, the flattening of the
angular dependence of the oscillation phase coincides with the onset of strong
sub-cycle Landau-Dykhne transitions, with the probability approaching 50% (see
SI).

Our theoretical and numerical results link these intensity-dependent
observations with the closing of the effective band-gap by the laser field.
According to equation (2), the oscillation phase is dictated by the additional
complex phase, $\sigma$, induced by the SH field23. The imaginary component,
$Im(\sigma)$, is associated with perturbations of the tunneling and
recombination probabilities, while the real component $Re(\sigma)$ reflects
subtle modifications of the electron trajectory as it propagates within the
band. Since the odd and the even harmonics represent constructive and
destructive interference of two subcycle emissions, $Re(\sigma)$ leads to an
oscillation phase difference. However the imaginary part, $Im(\sigma)$,
affects both even and odd harmonics simultaneously, therefore their
oscillation phases coincide (see SI).

In fig. 4d, we plot the relative oscillation phase of harmonics 19 and 20,
$\Phi_{20}-\Phi_{19}$, at $0^{\circ}$ orientation, for different fundamental
field intensities. At sufficient low laser intensity, this phase difference
vanishes reflecting the dominant role of the imaginary component. The origin
of this imaginary component could be associated with an anomalous emission
event of these harmonics, emitted from an energy gap. At higher field
intensities, the band structure is strongly dressed so that the energy gap
between the bands becomes negligible. Therefore, the associated imaginary
component is reduced and the perturbation is dominated by the real component
$Re(\sigma)$, leading to a phase difference between the even and the odd
harmonics (for further discussion see SI)

In summary, our study establishes all-optical spectroscopy of a strongly
driven crystal, revealing laser-induced modification of the band structure. We
identify the dynamical transitions between several conduction bands as well as
probe their structural dependence. Importantly, we resolve the clear signature
of harmonic emission from the energy gap between two conduction bands, probing
its modification by the laser field. This study provides a general framework
for resolving and interpreting attosecond electronic response phenomena in
strongly driven solids. Looking forward, two-color HHG spectroscopy opens a
window into the observation a broad range of electronic phenomena – from sub-
cycle phase transitions to ultrafast dynamics in correlated systems – some of
which have been theoretically predicted decades ago, while others are still
hotly debated.

## Acknowledgments

N.D. is the incumbent of the Robin Chemers Neustein Professorial Chair. N.D.
acknowledges the Minerva Foundation, the Israeli Science Foundation and the
European Research Council for financial support. A.J.N.U. acknowledges
financial support by the Rothschild Foundation and the Zuckerman Foundation.
M.I. acknowledges funding of the DFG QUTIF grant IV152/6-2. Á.J.G.
acknowledges funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement no.
101028938. A.J.G. and M.I. acknowledge funding from the European Union’s
Horizon 2020 research and innovation program under grant agreement No 899794.
R. E. F. S. acknowledges support from the fellowship LCF/BQ/PR21/11840008 from
“La Caixa” Foundation (ID 100010434) and from the European Union’s Horizon
2020 research and innovation program under the Marie Sklodowska-Curie grant
agreement No. 847648.

## Contributions

N.D. and M.I. supervised the study. M.I. supervised the theoretical work and
developed the theoretical model. A.J.N.U. and G.O. conceived and planned the
experiments. A.J.G. and R.S. performed the theoretical study and the numerical
analysis. A.J.N.U., T.A.P., G.O. and B.D.B performed the measurements.
A.J.N.U., G.O. and S.S. analyzed the data. B.Y. performed the DFT
calculations. All authors discussed the results and contributed to writing the
manuscript.

## Competing financial interests

The authors declare no competing financial interests.

## Corresponding authors

Correspondence to Nirit Dudovich (nirit.dudovich@weizmann.ac.il).

Figure 1: All optical spectroscopy of dynamical band structure. a, Non-
adiabatic Landau-Dykhne transition between a pair of bands. b, Two color HHG
spectroscopy probes the internal dynamics, mapping the temporal properties of
electron trajectories, transitions between the bands as well as their laser
driven modifications. Figure 2: HHG spectroscopy beyond the semi-classical
description a,c HHG spectrum (black line) and the oscillation phase (red dots)
as function of photon energy for crystals orientations of $0^{\circ}$(a) and
$45^{\circ}$(c). Calculated oscillation phase using the saddle point
approximation in interband model18 (purple star markers). b, The valence and
the first conduction band for $0^{\circ}$ ($\Gamma$ to $X$) and for
$45^{\circ}$ ($\Gamma$ to $K$). The yellow shaded area emphasizes the energy
range where the semi-classical description fails21, also marked at the
corresponding photon energies in a and c (dashed yellow line). Figure 3:
Probing the structural dependence of multiple conduction bands a, The
oscillation phase as a function of harmonic number, for different crystal
orientation, ranging from $0^{\circ}$ (dark blue) to $45^{\circ}$ (cyan) . The
dash line marks the cutoff of the first conduction band and the dash box
emphasizes H21 oscillation phases for different orientation, plotted at b. b,
Harmonic 21 (purple) and harmonic 11 (gray) oscillation phase as a function of
crystal orientation. c, top: 2D second band gap,
$\epsilon_{c2,v}=\epsilon_{c2}-\epsilon_{v}$, as function of crystal momentum.
The purple contour represents harmonic 21 energy along different crystal’s
orientation. bottom: 2D first band gap,
$\epsilon_{c1,v}=\epsilon_{c1,v}-\epsilon_{v}$, as function of crystal
momentum. The gray contour represents harmonic 11 energy along different
crystal’s orientation. d, 1D cut of the second (yellow) and first (blue) band
gap along $10^{\circ}$ and $45^{\circ}$ crystal orientation. The energy of
Harmonic 21 and 11 is presented by the purple and gray dash lines as well as
their crossing point with the band gaps (cross markers). Figure 4: Dynamical
band structure. a, First (blue) and second (yellow) band gaps,
$\epsilon_{c1,v}$ and $\epsilon_{c2,v}$, for $0^{\circ}$ (right) and
$25^{\circ}$ (left) orientations, as a function of $k/k_{c}$ ($k_{c}=2\pi/a$,
where $a$ the lattice constant). b, Harmonic 20 oscillation phase as function
of crystal orientation, for different fundamental field intensities (light
green to dark green). c, Calculated oscillation phase of H20 as function of
the crystal’s orientation, for different fundamental field intensities (light
green to dark green) d, The oscillation phase difference
,$\Phi_{20}-\Phi_{19}$, at $0^{\circ}$ orientation as function of the
fundamental field intensities. Inset: $\Phi_{18}-\Phi_{17}$ (pink) plot and
$\Phi_{22}-\Phi_{21}$ (blue).

## Data availability

The data and datasets that support the plots within this paper and other
findings of this study are available from the corresponding author upon
reasonable request. The custom code used for the current study has been
described in previous publications, and parts of it can be made available from
the corresponding author on reasonable request.

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## Methods

### Dressed Bands in Strongly Driven Solids

We consider the response of a two-band solid driven by a strong low-frequency
field, and show how an effective band structure can be introduced in this case
on the sub-cycle time-scale, relevant for the low-frequency drivers. In
addition to a simple result which averages the instantaneous adiabatic
energies over the current wavefunction, we introduce an extension of the
Floquet-type analysis to the sub-cycle time-scale.

#### Floquet-type analysis on the sub-cycle time-scale: Effective band gap in
a strong low frequency field

Let us start with some well defined eigenstate, with the wavefunction
$\psi_{\kappa}$, which satisfies the stationary Schroedinger equation,
$\hat{H}\psi_{\kappa}=E_{g}\psi_{\kappa}$. Here $\kappa$ collects the quantum
numbers that label the state. In a solid, $\kappa$ labels the crystal momentum
k together with the band index $n$.

Let the Hamiltonian depend on some external parameter $\beta$,
$\hat{H}(\beta)$. If we change $\beta$ slowly, our eigenstate will slowly
evolve following the stationary Schroedinger equation,

$\displaystyle\hat{H}(\beta)\phi_{\kappa}(\beta)=\epsilon_{\kappa}(\beta)\phi_{\kappa}(\beta)$
(3)

In the low-frequency field, where the driver has frequency $\omega$
substantially lower than the energy gap, such parameter $\beta$ is
$\beta=\omega t$. From now on we will simply refer to the time $t$. In the
low-frequency field, the adiabatic states are the solutions of the stationary
Schroedinger equation with time treated as a parameter:

$\displaystyle\hat{H}(t)\phi_{\kappa}(t)=\epsilon_{\kappa}(t)\phi_{\kappa}(t)$
$\displaystyle\hat{H}(t)=\hat{H}_{0}+\hat{V}(t)$ (4)

where $V(t)$ is the interaction with the low-frequency field. The adiabatic
eigenstates $\phi_{\kappa}(t)$ form complete basis set. Each state has an
associated time-dependent state $\Psi_{\kappa}$ which incorporates the
standard energy phase-factor:

$\Psi_{\kappa}(t)=e^{-i\int_{t_{i}}^{t}\epsilon_{\kappa}(t^{\prime})dt^{\prime}}\phi_{\kappa}(t)$
(5)

Putting this back into the time-dependent Schroedinger equation, we see that
the equation for $\Psi_{\kappa}$ contains extra term proportional to the
derivative of the eigenstate:

$i\dot{\Psi}_{\kappa}=\hat{H}(t)\Psi_{\kappa}+ie^{-i\int_{t_{i}}^{t}\epsilon_{\kappa}(t^{\prime})dt^{\prime}}\dot{\phi}_{\kappa}$
(6)

As long as the non-adiabatic Landau-Dykhne transitions between the adiabatic
states, caused by this term, are small, their instantaneous energies
$\epsilon_{\kappa}(t)$ offer a very good approximation for the effective
instantaneous (and hence sub-cycle) energies of the driven system. However, in
the presence of strong non-adiabatic transitions between the adiabatic states,
these concepts require corrections.

Consider now the specific case of a solid, with bands
$\epsilon_{n}(\textbf{k})$ and the Bloch wavefunctions $\phi_{n,\textbf{k}}$,
interacting with a low-frequency laser field. We note that the analysis below
is not, in fact, limited to such low-frequency case, but the low-frequency
case presents the most natural physical situation where our analysis and its
conclusions are physically transparent.

With the light-solid interaction treated in the dipole approximation and in
the length gauge, the initial crystal momentum k becomes a function of time.
We shall label this time-dependent momentum
$\kappa(t)=\textbf{k}+\textbf{A}(t)$.

It is very useful then to use the Houston states to analyze the interaction.
In this basis the Bloch wavefunctions $\phi_{n,\textbf{k}}$ and the band
energies $\epsilon_{n}(\textbf{k})$ also follow the vector potential,
replacing k with $\kappa(t)=\textbf{k}+\textbf{A}(t)$. The field-free energy
$\epsilon_{n}(\textbf{k})$ goes into
$\epsilon_{n}(\kappa(t))\equiv\epsilon_{n}(\textbf{k}+\textbf{A}(t))$, so that
the band energy “slides” with the instantaneous momentum $\kappa(t)$. The
Bloch wavefunction also “slides” with this instantaneous momentum, becoming
$\phi_{n,\kappa(t)}$.

In this basis, each time-dependent crystal momentum
$\kappa(t)=\textbf{k}+\textbf{A}(t)$ “traces” its own multi-level system, with
its states labeled with the band index $n$ and the time-dependent energies and
couplings. The multi-level system with crystal momentum k is decoupled from
other multi-level systems with momenta k’.

#### Beyond the adiabatic evolution

Consider a two-band solid, with the band indexes $n=1,2$, driven by a low-
frequency field. Thanks to the small frequency, the adiabatic evolution would
be a good approximation for the most part of the Brillouin zone. The adiabatic
evolution will break down to the greatest extent in the regions of the
smallest band gap, with exponential dependence on the bandgap.

Different k will reach these regions at different times and hence with
different instantaneous values of the field $\textbf{F}(t)$. Suppose we start
in the state k, turn on the field, and at a moment $t_{i}$ arrive at some
momentum $\kappa$ in one of the adiabatic states (say with the label
$\ket{1,\kappa}$), approaching a region where the two bands are coming close
to each other.

We shall now look at the propagator across the region of interest. The
analysis is general, but for the physical interpretation to be clear, the
time-interval should be enough to go through the region. For compactness we
will drop the crystal momentum index for the moment.

The propagator, written in the basis of the adiabatic states
$\ket{\Psi^{(1)}},\ket{\Psi^{(2)}}$, with energies $\epsilon^{(1)}(t)$ and
$\epsilon^{(2)}(t)$, has the following form

$\displaystyle\hat{U}(t,t_{i})=e^{-i\frac{1}{2}\left[\lambda^{(1)}+\lambda^{(2)}\right]}\left(\begin{array}[]{cc}\cos\alpha
e^{i\lambda}&-\sin\alpha e^{i\phi}\\\ \sin\alpha e^{-i\phi}&\cos\alpha
e^{-i\lambda}\\\ \end{array}\right)$ (9)
$\displaystyle\lambda=\frac{1}{2}\left[\lambda^{(2)}-\lambda^{(1)}\right]$
(10)

This form of the propagator is general and meets the key requirements:

$\displaystyle|U_{11}|^{2}+|U_{21}|^{2}=1$
$\displaystyle|U_{12}|^{2}+|U_{22}|^{2}=1$
$\displaystyle\hat{U}\hat{U}^{\dagger}=\hat{U}^{\dagger}\hat{U}=\hat{1}$ (11)

The last point also ensures that the wavefunctions remain orthogonal during
the passage, $\bra{\Psi^{(2)}}\hat{U}^{\dagger}\hat{U}\ket{\Psi^{(1)}}=0$.

The meaning of the matrix elements in this propagator is as follows.

  * •

The phases $\lambda^{(1,2)}(t)$ are associated with the adiabatic energies
plus, in general, the geometrical Berry phase:

$\displaystyle\lambda^{(i)}(t)=\int_{t_{i}}^{t}dt^{\prime}\epsilon^{(i)}(t^{\prime})+\gamma_{i}$
(12)

The reason we want to complete the passage across the region of interest is
that we want the geometrical phase, associated with this passage, to
accumulate fully. But, if we treat the problem fully numerically, then, of
course, such requirement is not necessary: the phases $\lambda^{(i)}(t)$ are
simply found numerically.

  * •

The overall factor in front of the propagator sets the zero-energy level as

$\displaystyle\expectationvalue{\epsilon}=\frac{1}{2}\left[\epsilon^{(2)}+\epsilon^{(1)}\right]$
(13)

through the phases associated with the adiabatic energies.

  * •

Irrespective of how the phases $\lambda^{(i)}(t)$ are obtained, their physical
interpretation remains the same: their time-derivatives have to be associated
with the adiabatic energies (which are viable and meaningful in the absence of
non-adiabatic transitions, i.e. when $|\sin\alpha|<<1$):

$\displaystyle\tilde{\epsilon}^{(i)}(t)=\frac{\partial\lambda^{(i)}(t)}{\partial
t}$ (14)

The reason to add “tilde” above the adiabatic energy is to stress that the
adiabatic energies $\tilde{\epsilon}^{(i)}(t)$ obtained in such way may not
always coincide with the adiabatic energies obtained by diagonalizing the
adiabatic Hamiltonian, because of the presence of the geometric phase. Again,
we stress that the phases associated with $\tilde{\epsilon}^{(i)}(t)$ can be
extracted numerically. For the physical interpretation we will need are their
derivatives.

  * •

The off-diagonal elements describe the Landau-Zener-Dykhne non-adiabatic
transitions between the two adiabatic states.

  * •

The phases $\phi$ of these off-diagonal elements are determined by the
landscape of the bands and the Berry connections (couplings). As we shall see
below, $\phi$ will not matter for the effective band structure.

  * •

The probability of staying in the adiabatic state is $\cos^{2}\alpha$ and the
probability of making the transition is $\sin^{2}\alpha$.

We can now develop the sub-cycle version of the Floquet analysis. To this end,
we return to the propagator, dropping the common zero-energy level phase
factor for compactness,

$\displaystyle\hat{U}(t,t_{i})=\left(\begin{array}[]{cc}\cos\alpha
e^{i\lambda}&-\sin\alpha e^{i\phi}\\\ \sin\alpha e^{-i\phi}&\cos\alpha
e^{-i\lambda}\\\ \end{array}\right)$ (17)
$\displaystyle\lambda=\frac{1}{2}\left[\lambda^{(2)}-\lambda^{1)}\right]$ (18)

and look for orthogonal states $\ket{\Psi^{\mu}(t_{i})}$ with time-dependent
quasi-energies $\epsilon^{\mu}(t,t_{i})$, which depend on the crystal momentum
k. We want these states to behave as if they were the Floquet states for this
propagator:

$\displaystyle\hat{U}(t,t_{i})\ket{\Psi^{\mu}}=e^{i\mu}\ket{\Psi^{\mu}}$
$\displaystyle\epsilon^{\mu}=-\frac{\partial\mu}{\partial t}$ (19)

The positive sign in the phase of the exponent is for convenience because we
shall start with the “lower” state, which has negative energy. The minus sign
in the equation for the quasi-energy is related to the positive sign in the
exponent in the first equation.

These quasi-energies and the associated states are as close as one can get to
the effective bands and effective eigenstates in a strongly driven system with
non-adiabatic transitions. As we shall see below, in the absence of non-
adiabatic transitions they, of course, coincide with the adiabatic energies
and states.

In principle, one can try to find such states for any time-interval after
$t_{i}$, but a meaningful time-interval is an interval sufficient to cross the
transition region. Once the region is crossed, each component of the
wavefunction, projected on the adiabatic states, will mostly evolve on the
associated adiabatic bands, and in the absence of non-adiabatic transitions
these bands are fine and, as mentioned above, coincide with the quasi-energies
we shall find below.

The time-dependent “eigenstates” of the propagator have two components
corresponding to the amplitudes in the two adiabatic states
$\ket{\phi^{(i)}(t)}$,

$\displaystyle\ket{\Psi^{\mu}}=\left(\begin{array}[]{cc}a_{\mu}\\\ b_{\mu}\\\
\end{array}\right)$ (22)

The analysis is straightforward. We ask that the determinant of the matrix
$\hat{U}-e^{i\mu}\hat{1}$ is equal to zero, and the solution is found to be

$\displaystyle\cos\mu=\cos\alpha\cos\lambda$ (23)

There are two solutions of this equation, $\mu_{1}=\mu$ and $\mu_{2}=-\mu$,
and the quasienergies $\epsilon^{\mu_{1}},\epsilon^{\mu_{2}}$ are obtained by
differentiating $\mu_{1}$ and $\mu_{2}$ with respect to time.

The first observation is that, in the absence of non-adiabatic transitions,
when $\cos\alpha=1$, $\cos\mu=\cos\lambda$, $\mu_{1}=-\lambda$ and
$\mu_{2}=\lambda$, and the quasi-energies coincide with the adiabatic states.

Eq.(23) is already sufficient to find the effective bandgap, which is equal to

$\displaystyle\Delta\epsilon^{\mu}=\left|\epsilon^{\mu_{1}}-\epsilon^{\mu_{2}}\right|=2\left|\frac{\partial\mu}{\partial
t}\right|$ (24)

We differentiate the two sides of Eq.(23) with respect to time and find

$\displaystyle\Delta\epsilon^{\mu}=\left[\tilde{\epsilon}^{(2)}_{\rm
ad}-\tilde{\epsilon}^{(1)}_{\rm
ad}\right]\left|\cos\alpha\frac{\sin\lambda}{\sin\mu}\right|$ (25)

where we have used that

$\displaystyle 2\dot{\lambda}=\tilde{\epsilon}^{(2)}_{\rm
ad}-\tilde{\epsilon}^{(1)}_{\rm ad}$ (26)

Using the relationship $\cos\mu=\cos\alpha\cos\lambda$, we can re-write

$\displaystyle\left[\cos\alpha\frac{\sin\lambda}{\sin\mu}\right]^{2}=\frac{\cos^{2}\alpha\sin^{2}\lambda}{\cos^{2}\alpha\sin^{2}\lambda+\sin^{2}\alpha}$
(27)

and hence

$\displaystyle\Delta\epsilon^{\mu}=\left[\tilde{\epsilon}^{(2)}_{\rm
ad}-\tilde{\epsilon}^{(1)}_{\rm
ad}\right]\sqrt{\frac{\cos^{2}\alpha\sin^{2}\lambda}{\cos^{2}\alpha\sin^{2}\lambda+\sin^{2}\alpha}}$
(28)

Finally, we can simplify this expression by taking into account that if the
action $\lambda(t)$ is large, which is usually the case in the strong, low-
frequency field, then $\sin^{2}\lambda(t,t_{i})$ is a fast-oscillating
function of $t-t_{i}$. Replacing it with its average,
$\sin^{2}\lambda(t,t_{i})=>1/2$, and introducing the notation
$\sin^{2}\alpha=w_{\rm LD}$, we get the final result for the bandgap between
the two quasi-energies

$\displaystyle\Delta\epsilon^{\mu}=\left[\tilde{\epsilon}^{(2)}_{\rm
ad}-\tilde{\epsilon}^{(1)}_{\rm ad}\right]\sqrt{\frac{1-w_{\rm LD}}{1+w_{\rm
LD}}}$ (29)

This result shows that the band gap collapses when the non-adiabatic Landau-
Dykhne transition approaches unity, $w_{\rm LD}\simeq 1$.

We note that the formalism based on the sub-cycle analogue of the quasi-energy
states of a strongly driven system is especially attractive because it
naturally merges into the Floquet analysis when $t-t_{i}$ is equal to one
period.

One can also find the quasi-eigenergies in a different way, by using only the
relationship $\cos\mu=\cos\alpha\cos\lambda$. Namely, one can use this
relationship to solve the equations for the amplitudes $a^{\mu}$ and
$b^{\mu}$, and then average the full Hamiltonian over the eigenvectors
obtained in such way from Eq.(22). The result is exactly the same, as
expected.