# Flatband slows down polariton dynamics in strongly coupled cavities

Yesenia A García Jomaso  Brenda Vargas  David Ley Domínguez  Román J. Armenta-
Rico  Huziel E. Sauceda  César L Ordoñez-Romero  Hugo A Lara-García  Arturo
Camacho-Guardian  Giuseppe Pirruccio  Instituto de Física, Universidad
Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México C.P.
01000, Mexico  Instituto de Física, Universidad Nacional Autónoma de México,
Apartado Postal 20-364, Ciudad de México C.P. 01000, Mexico

(August 28, 2024)

###### Abstract

Flatbands in condensed-matter, atomic physics, and quantum optics stand as the
basis for several strongly correlated quantum many-body phenomena such as
Wigner crystallization, the fractional quantum Hall effect and Moiré-related
physics. Besides inspiring analogies among diverse physical fields, flatbands
are highly sought-after in photonics because they allow unconventional light
flows such as slow-light. Here, we realize room-temperature slow-light with
Frenkel polaritons excited across two strongly coupled cavities. We
demonstrate the formation of a tuneable flatband appearing in absence of a
periodic in-plane potential. Our simple photonic architecture enables the
unique spatial segregation of photons and excitons in different cavities and
maintains a balanced degree of mixing between them. This unveils a dynamical
competition between many-body scattering processes and the underlying
polariton nature which leads to an increased fluorescence lifetime. The
polariton features are further revealed under appropriate resonant pumping,
where we observe suppression of the flatband polariton fluorescence intensity.

In condensed matter physics, flatbands have led to the achievement of several
breakthroughs, including strongly correlated electronic states Derzhko  _et
al._ (2015), non-conventional superconductivity Balents  _et al._ (2020);
Törmä  _et al._ (2022); Tian  _et al._ (2023), and topological phases of
matter in two-dimensional materials Sun  _et al._ (2011). Similarly, within
photonic systems, the presence of flat optical bands is highly desirable, as
they could prompt the generation of strongly correlated photonic states
Jacqmin  _et al._ (2014); Mukherjee  _et al._ (2015); Diebel  _et al._ (2016);
Mukherjee  _et al._ (2017); Ozawa  _et al._ (2019). This is not only
fundamentally relevant, but could also have an impact on energy harvesting,
sensing, and information processing. Slow-light, the phenomenon of controlling
and manipulating dispersion of light Hau  _et al._ (1999), has received
attention in quantum optics Massou (2005), atomic physics fle (2000);
Fleischhauer and Lukin (2002) and condensed matter Bigelow  _et al._ (2003);
Xiao  _et al._ (2018). The ability to tune the dispersion of light yields new
opportunities to design light-matter interactions and non-linear optical
devices Harris  _et al._ (1990); Lukin and Imamoğlu (2000). In the quantum
domain, slow-light is typically understood in terms of polaritons Basov  _et
al._ (2020), hybrid light-matter quasiparticles that arise from mixing a
photon with an elementary matter excitation. The propagation of light in the
form of a dark-state polariton within a medium can be slowed down and even
stopped by heavily unbalancing the mixture of light and matter. In atomic
gases, slow-light is typically accompanied by the phenomenon of
electromagnetically induced transparency whereby, assisted by a control laser
field, light propagates undamped in a typically opaque medium Boller  _et al._
(1991); Fleischhauer  _et al._ (2005).

Slow-light and intercavity organic polaritons

Polaritons found in both organic and inorganic semiconductors have
demonstrated a high degree of flexibility to control the internal energy
structure and dynamics, unfolding alternative routes, Stranius  _et al._
(2018); Eizner  _et al._ (2019); Xiang  _et al._ (2019); Sánchez-Barquilla
_et al._ (2022); Keeling and Kéna-Cohen (2020); Climent  _et al._ (2021);
Sáez-Blázquez  _et al._ (2019) including the use of photonic lattices to craft
flatbands Su  _et al._ (2020); Berloff  _et al._ (2017); Kavokin  _et al._
(2022); Amano  _et al._ (2022).

In this article, our proposal involves the successful implementation of a
polariton flatband at room temperature through the strong coupling of a
photonic cavity with a polaritonic cavity. The dispersion of the so-formed
intercavity polaritons is tailored by tweaking the three-energy-level diagram.
This adjustment of the diagram allows us to fine-tune the dispersion of the
resulting intercavity polaritons. The observed flatband signals the transition
from a bright to dark polariton state characterized by a stretched polariton
lifetime. The dark nature of such intercavity flat polariton is indirectly
confirmed by its suppressed light-emission obtained under careful resonant
pumping.

Our findings suggest that reducing polariton group velocity is akin to
generating slow-light in atomic gases when the condition for
electromagnetically induced transparency is fulfilled. Furthermore, inter-
cavity polaritons open up the quantum tomography protocols for local and
independent measurements of the photonic and molecular degrees of freedom.

Figure 1: Experimental setup. (a) Representation of the hybrid photonic-
polaritonic coupled system. (b) Relevant energy levels:
$|\omega_{c}^{(L)}\rangle$, $|\omega_{c}^{(R)}\rangle$ and
$|\omega_{X}\rangle$ represent the left photon, right photon and exciton
state, respectively. Photons tunnel from left to right cavity with a hopping
amplitude $t$. $\Omega$ is the light-matter coupling.

Up until now, the exploration of strongly coupled photonic cavities has
primarily centered around photonic crystal cavities Chalcraft  _et al._
(2011), micro-rings Jahangiri  _et al._ (2023), and whispering-gallery modes
Wang  _et al._ (2016). In the context of semiconductor microcavities, double-
well potentials have led to the realization of phenomena like Josephson
oscillations and self-trapping with intercavity polaritons, Abbarchi  _et al._
(2013); Lagoudakis  _et al._ (2010) and have been recently proposed for
quantum chemistry control Du  _et al._ (2019) .

Figure 2: Intercavity polaritons. (a)-(c) s-polarized reflectance as a
function of the angle of incidence and photon energy for (a)
$\delta/\text{eV}=-0.37$, (b) $\delta/\text{eV}=-0.18$ and (c)
$\delta/\text{eV}=0$. The white, red, and black dashed curves correspond to
the theoretical fitting of the energies of the lower, middle, and upper
polariton respectively. (d)-(f) Spectral function, $A(\mathbf{k},\omega)$,
calculated for the same detunings as in (a)-(c). Middle polariton Hopfield
coefficients for (g) $\delta/\text{eV}=-0.37$, (h) $\delta/\text{eV}=-0.18$
and (i) $\delta/\text{eV}=0$. The black, blue and red curves correspond to the
left cavity photon, right cavity photon and exciton component, respectively.

To realize slow-light we designed a three-level energy scheme representing two
strongly coupled cavities. This set-up reminds of the $\Lambda$-scheme
commonly employed in quantum control experiments. Our system, sketched in Fig.
1(a), is composed by two cavities, labelled (Left) and (Right), coupled via a
thin mirror. The left cavity, represented by $|\omega_{c}^{(L)}\rangle$, is
purely photonic, whereas the right cavity host both photons and excitons,
identified by $|\omega_{c}^{(R)}\rangle$ and $|\omega_{c}^{(X)}\rangle$,
respectively. The Hamiltonian of the system is given by

$\displaystyle\hat{H}=\sum_{i=L,R}\omega_{c}^{i}(\theta)\hat{a}_{i}^{\dagger}\hat{a}_{i}-t(\hat{a}^{\dagger}_{L}\hat{a}_{R}+\hat{a}^{\dagger}_{R}\hat{a}_{L})+$
(1)
$\displaystyle+\omega_{X}\hat{x}^{\dagger}\hat{x}+\Omega\left(\hat{x}^{\dagger}\hat{a}_{R}+\hat{a}_{R}^{\dagger}\hat{x}\right)$

where $\omega_{c}^{(L/R)}(\theta)$ indicates the energy of the left/right
cavity photons, which are created with the operator $\hat{a}_{L/R}^{\dagger}$.
Here, the angle $\theta$ corresponds to the incident angle of light injected
to the right cavity. In Fig. 1(b), the photon hopping is characterised by a
tunneling amplitude $t$, while the light-matter coupling is given by the Rabi
frequency $\Omega$.

The sample consists of two vertically stacked nanocavities fabricated on a
glass substrate by a sequence of multiple sputtering and spin-coating steps.
The front, middle and back mirrors are made by Ag and their thickness equal to
20 nm, 20 nm and 300 nm, respectiveley. The middle mirror width determines the
tunneling amplitude $t$. The Left nanocavity is filled with polymethyl
methacrylate (PMMA), whereas the Right one embeds a dye-doped polyvinyl
alcohol (PVA) layer. The excitonic content is provided by a high concentration
of homogeneously dispersed Erithrosine B (ErB) molecules Jomaso  _et al._
(2023). The absorption spectrum of the active medium exhibits a main peak
around $\omega_{X}\approx 2.24\text{eV}$ associated to a principal exciton
resonance and a second peak at $\omega_{v}\approx 2.4\text{eV}$ related to the
first vibron mode.

Figure 3: Short-time dynamics and dark-state polaritons. (a) Normalized
fluorescence lifetime showing the short-time dynamics of the middle and lower
polaritons. The dynamics of the MP is displayed with orange and green
asterisks for $\delta/\text{eV}=-0.37$ and $\delta/\text{eV}=0,$ respectively.
The LP decay is illustrated with blue and purple asterisks for
$\delta/\text{eV}=-0.27$ and $\delta/\text{eV}=0,$ respectively. Shaded pink
depicts the region where the fastest decay component dominates the early
polariton dynamics. (b) Sketch of the relevant energy levels. Fine-tuning the
left cavity photon energy, shifts the energy of the three polariton states.
The grey area symbols the presence of the dark exciton reservoir. The triplet
state of the ErB molecules, $|T_{1}\rangle$, influences the LP dynamics via
intersystem crossing. (c)-(d) s-polarized fluorescence, expressed in counts
per integration time, for $\delta/\text{eV}=-0.37$ and $\delta/\text{eV}=0$,
respectively. The energy of the bare exciton is represented by the dashed
white line. Photons are injected at $\omega_{p}/\text{eV}=2.6$ which lies at
the energy of the upper polariton. (e) Hopfield coefficients of the UP for
$\delta/\text{eV}=0$ showing that it is predominantly formed by right cavity
photons The black, blue and red curves correspond to the left cavity photon,
right cavity photon and exciton component, respectively

.

The polymer thickness of the Left cavity features a slow wedge which provides
us the possibility of fine tuning $|\omega_{c}^{(L)}\rangle$ in a wide photon
energy range. When the resonance frequency of the left cavity at normal
incidence matches the exciton one, an intercavity polariton,
$|D(\theta=0)\rangle$, emerges

$\displaystyle|D(\theta=0)\rangle=\frac{\Omega}{\sqrt{\Omega^{2}+t^{2}}}|\omega_{c}^{(L)}(\theta=0)\rangle-\frac{t}{\sqrt{\Omega^{2}+t^{2}}}|\omega_{X}\rangle,$
(2)

solely formed by the superposition of the left cavity photon and the exciton.
Importantly, the right cavity photon does not participate in the formation of
this polariton state. Further details on the geometrical order of the cavities
are provided in the Supplementary Information.

In Fig. 2(a)-(c), we show the local band structure of the coupled cavity
system measured via Fourier microscopy, which demonstrates that the
$\Lambda$-scheme yields a middle polariton (MP) state with energies close to
the bare exciton one. Fine-tuning of the MP energy is accomplished by
leveraging the wedged PMMA thickness. The MP state shows a reduced dispersive
character compared to the upper (UP) and lower polariton (LP) and flattens for
a specific value of the thickness of the photonic cavity.

To theoretically understand our results, we introduce the detuning,
$\delta=\omega_{c}^{(L)}-\omega_{X}$, and the imaginary-time Green’s function
of the system, $\mathcal{G}_{\alpha,\beta}(\tau)=-\langle
T_{\tau}[\hat{\psi}_{\alpha}(\tau)\hat{\psi}_{\beta}^{\dagger}(0)]\rangle$,
where the subindices $\alpha,\beta$ correspond to the left/right cavity photon
and exciton, respectively. The fields $\psi$ and $\psi^{\dagger}$ evolve
according to Eq. 1. We define the spectral function of the left cavity photon
as $A(\omega)=-2\text{Im}\mathcal{G}_{11}(\omega)$ and we plot it in Fig.
2(d)-(f) for three values of $\delta$, showing a very good agreement with the
experimentally observed reflectance. By continuously varying the left cavity
thickness, its photon energy is driven in resonance with the exciton energy.
Furthermore, the energy of the polariton states obtained from our theory and
plotted with dashed curves on the reflectance maps, Fig. 2(a)-(c), provide an
excellent quantitative understanding of the system.

We obtain a reduction of the photonic dispersion of the left cavity photons

$\displaystyle\omega_{\text{MP}}(\theta)\approx\omega_{X}+\left[\omega_{c}^{L}(\theta)-\omega_{X}\right]\frac{1}{1+\left(\frac{t}{\Omega}\right)^{2}},$
(3)

this means that the dispersion of the MP is controlled by the tunnelling
ratio, $t$, which can be tailored by means of the middle mirror thickness. On
resonance, the MP emerges at the energy of the bare exciton, its dispersion
reduced by a factor $4$-$8$ compared to the energies of the upper and lower
polaritons, this leads to a pronounced flatband over a wide range of incident
angles. While the dispersion can be further reduced, this would be at the
expense of strongly suppressing the photonic component.

The character of the MP is clearly unveiled once it is written in terms of the
bare photon and exciton states
$|\text{MP}\rangle=\sum_{\alpha}\mathcal{C}_{\text{MP}}^{\alpha}|\alpha\rangle$.
The amplitude of its Hopfield coefficients
$Z_{\text{MP}}^{\alpha}=|\mathcal{C}_{\text{MP}}^{\alpha}|^{2}$ in Fig.
2(g)-(i) demonstrates that the MP decouples from the right cavity photons for
$\delta=0$. Furthermore, Fig. 2(i) shows that the Hopfield coefficients remain
invariant along the angular region where the MP is flat. There, the MP has
only non-vanishing Hopfield coefficients of the left cavity photon and the
exciton. For conventional polaritons arising in two-level schemes, suppression
of the polariton dispersion can only be achieved by compromising the degree of
mixing between photons and excitons. This means that only far-detuned
polaritons asymptotically exhibit quasi-flat dispersion and, thus, are formed
by a large exciton component and negligible photon contribution. In contrast,
our three-level design allows for flatband polaritons and slow-light retaining
a significant photonic component of circa $40\%$.

Short-time dynamics.

Now we focus on the effect of the vanishing curvature of the MP, hinting at a
mechanism analogous to the charge population trapping observed in atomic
electromagnetically induced transparency. For this, we measure the prompt
fluorescence decay employing the time-correlated single-photon counting
technique. We explore the dynamics of the LP and MP and relate them with the
corresponding photon component. The coupled cavities are pumped off-resonantly
with laser pulses centered around 2.42 eV.

Figure 3(a) displays decays for two detunings used in Fig. 2, i.e.,
$\delta/\text{eV}=0$ and $\delta/\text{eV}=-0.37$. In the considered time
interval, all decays are well described by two time constants. For our
analysis we focus initial dynamics illustrated in the pink shaded region We
model the evolution assuming a dynamical equation of the form
$C(t)=A\exp(-\Gamma t)+(1-A)\exp(-\eta t),$ normalized to $C(t=0)=1.$ Figure
3(a) shows that this model (solid curves) provides a very good fit to the
experimental data (points). We obtain that the LP dynamics is dominated by a
single exponential with $\Gamma_{\text{LP}}(\delta=0)=3.25\leavevmode\nobreak\
\text{ns}^{-1}$ and $A_{\text{LP}}=0.94$, while
$\Gamma_{\text{LP}}(\delta=-0.37)=3.0\leavevmode\nobreak\ \text{ns}^{-1}$ with
$A=0.86.$ On the other hand, we find a richer dynamics for the MP, whose
dynamics is characterised by
$\Gamma_{\text{MP}}(\delta=0)=0.18\leavevmode\nobreak\ \text{ns}^{-1}$ and
$\eta_{\text{MP}}(\delta=0)=3.25\leavevmode\nobreak\ \text{ns}^{-1}$ for the
flatband condition, and
$\Gamma_{\text{MP}}(\delta=-0.37)=0.19\leavevmode\nobreak\ \text{ns}^{-1}$ and
$\eta_{\text{MP}}(\delta=-0.37)=4.57\leavevmode\nobreak\ \text{ns}^{-1}$ for
the detuned case. In contrast to the LP, the weight of the exponentials for
the MP is almost equally distributed as $A(\delta=0)=0.55$ and
$A(\delta=-0.37)=0.5$. The significantly slower dynamics of the MP shown in
Fig. 3(a) is consequence of the interplay between the two dynamical factors.
Indeed, we observe that at very small times, in the linear regime
$C_{\text{LP}}(t)\approx 1-\Gamma_{\text{LP}}t,$ whereas
$C_{\text{MP}}(t)\approx
1-[A\Gamma_{\text{MP}}+(1-A)\eta_{\text{MP}}]t=1-\gamma_{\text{eff}}t$. On
resonance, $\delta=0$, we obtain $\gamma_{\text{eff}}\approx
1.7\leavevmode\nobreak\ \text{ns}^{-1}$ which is significantly smaller than
the dominant decay rate of the LP. We attribute this effect to a competition
between a reduced photon component of the MP, which suppresses the damping
rate, and the presence of the dark-states reservoir of exciton laying close to
the energy of the MP, which may favour non-radiative scattering and
accelerated decay. We observe that the dynamics of the MP is slower than LP
one, even though the MP energetically lies on top of the reservoir of dark-
state excitons (see Fig. 3(b)). Therefore, we conclude that the vanishing
curvature of the MP and its concomitant dark nature produces a significant
effect in the polariton dynamics in the nanosecond range.

Finally, we see that the decay of the LP becomes slower as it approaches the
energy of the ErB triplet state, $|T_{1}\rangle$. This is evident from the
asymptotic intensity value of the decay curve which does not converge to the
noise floor of the other curves. Here, intersystem crossing not only increases
significantly the slow component of the decay, but also stretches the fast
one, outcompeting the role played by the photonic Hopfield coefficients and
producing long-lived polaritons.

Photoluminescence

To further investigate the nature of the polariton states we now turn our
attention to the steady-state fluorescence. The system is pumped with a CW
laser emitting at $\omega_{p}=2.62\text{eV}$, which corresponds to an off-
resonant excitation for all negative detunings. This means that we inject
photons approximately equally in both cavities and produce excitons in the
right cavity. For $\delta/\text{eV}=-0.37,$ the MP is formed almost equally by
both left and right cavity photons, while LP is formed predominantly by left
cavity photons. Thus, we observe a higher fluorescence intensity from the MP
than from the LP, as seen in Fig. 3(c). In the Supplemental Information we
show that as we decrease the detuning, the LP acquires a larger fraction of
the photon and exciton component of the right cavity, which consistently
conduces to an increased LP fluorescence.

However, as we approach the condition for $\delta/\text{eV}=0,$ the UP energy
shifts until it matches the pump energy at normal incidence. As shown in Fig.
3(e), the UP for $\delta/\text{eV}=0,$ is primarily formed by right cavity
photons. Therefore, this configuration preferentially injects photons into the
right cavity. As we move towards the condition for flatband polariton, the
right cavity photon component of the MP decreases until vanishing exactly.
Decoupling the MP from the right cavity photons results in an inefficient
excitation of the MP and, thus, a strongly suppressed MP fluorescence
intensity, as demonstrated in Fig. 3(d), confirming the transition of the MP
to a dark state.

Conclusions and Outlook

We have experimentally demonstrated the formation of inter-cavity polaritons
composed by the admixture of photons and excitons sitting in physically
separated optical cavities. The three energy level $\Lambda$-scheme
underpinning the physics of this system implies, on resonance, the existence
of a polariton flatband. We observed the flattening of the middle polariton
branch which transits smoothly to a dark state upon tweaking the photon-
exciton detuning. This mechanism effectively decouples the flatband polariton
from free space leading to slower short-time dynamics. On resonance, the
absence of one of the photon states in the composition of the dark polariton
is confirmed by the steady-state fluorescence, whose intensity is suppressed
under resonant pumping of the upper polariton branch. The interplay of the
polaritons with the exciton reservoir needs to be taken into account as this
introduces entropically favored scattering paths that may hamper the
observation of more exotic physics. However, we stress that the short-dynamics
modification hints at that slow-light possibly overcomes the effect of the
reservoir. This motivates further studies on the interplay between the dark-
state dynamics and the possible breakdown of quasiparticle picture Levinsen
_et al._ (2019); Rajabali  _et al._ (2021); Jomaso  _et al._ (2023).

Hybrid photonic-polariton systems are an ideal platform to explore many-body
physics in multi-level coupled cavity systems.

This shares analogy to interlayer excitons observed in stacked 2D materials,
where the indirect character of the interlayer excitons gives rise to strongly
interacting many-body states Wang  _et al._ (2019), long-lived exciton-
polaritons Miller  _et al._ (2017); Jiang  _et al._ (2021), and new classes of
polaritons Zhang  _et al._ (2021); Camacho-Guardian and Cooper (2022).
Inspired by how twistronics in stacked 2D materials gave rise and non-
conventional superconductivity Ciarrocchi  _et al._ (2022), we envisage that
polariton flatbands can be useful to increase polariton correlations and
facilitate the observation of non-trivial quantum phases in lattices-free
systems. Moreover, our strategy to generate slow-light does not compromise the
photonic component of the flatband polariton, making it appealing for strongly
correlated truly polariton states. The reduced in-plane propagation helps
confining spatially polaritons without the additional complication of
fabricating physical boundaries. This may lower the threshold needed for
quantum phases transitions while maintaining the architecture of the system
simple. Furthermore, the quantum entanglement between the photonic and
molecular degrees of freedom may be unraveled by exploiting the spatially
indirect character of the intercavity polariton.

Methods

Sample fabrication

The sample is composed by two vertically stacked Fabry–Pérot cavities
fabricated on a glass substrate $10\times 10$ $\text{mm}^{2}$ by multiple
successive sputtering and spin-coating steps. The bottom 300 nm-thick Ag
mirror was fabricated by magnetron sputtering operated at room temperature and
a base pressure of approximately 10-6 Torr which, during deposition, is
pressurized with argon flow to 3 x 10-3 Torr. We deposited 99.99$\%$ purity Ag
at a rate of 0.08 nm/s. The active layer of the first cavity is obtained
starting from a solution of 25 mg of polyvinyl alcohol (PVA, Mowiol 44-88,
86.7-88.7$\%$ hydrolyzed, Mw $\approx$ 205 000 g/mol) dispersed in 1 mL of
distilled water. Then, 9.8 mg of Erythrosin B (ErB, Sigma Aldrich with dye
content $>$90$\%$) was added to the PVA/water solution, yielding a 0.5 M
concentration. The ErB/PVA thin films were deposited by spin-coating at 2100
rpm using a 0.45 $\mu$m pore PTFE syringe filter, obtaining approximately 120
nm thickness. The first cavity is completed by fabricating a 20 nm-thick
middle mirror on top of the active layer. The second cavity is formed by a
Polymethyl methacrylate (PMMA, Mw $\approx$ 120 000 g/mol) layer embedded
between the middle and top mirror. This layer is obtained starting from a 25
mg/mL solution of PMMA. The solution is spin-coated at 2600 rpm for 60 s using
a 0.45 $\mu$m pore PTFE syringe filter and provides a slow thickness gradient
centered around 140 nm. Using PMMA instead of PVA avoids the formation of
micro-bubbles at the surface of the second cavity.

Experimental set-up

Energy-momentum spectroscopy is performed in a homemade confocal Fourier
optical microscope. Imaging the back-focal plane of a high numerical aperture
microscope objective onto the entrance slit of a spectrograph (Kymera 328i,
Andor) coupled to a sCMOS camera (Zyla 4.2P, Andor) is done by a Bertrand lens
and provides direct access to the angular- and spectral-resolved reflectance.
In our set-up the sample is illuminated through a Plan Fluor 50x/0.8 NA
objective (Nikon) with white light emitted by a halogen lamp. The focal spot
full-width at half-maximum equals 14 $\mu$m. The collected light is dispersed
by a diffraction grating (150 lines mm, blazed at 500 nm). Two linear
polarizers in the excitation and collection path are used to select the s- or
p- polarization. Angular-resolved reflectance is obtained by replacing the
cavity with a commercial mirror, which allows to normalize the spectra
reflected off the cavity at each angle with those obtained with the mirror at
the corresponding angles. Angular-resolved steady-state fluorescence is
measured by pumping the coupled cavity with a 473 nm continuous wave laser
(Excelsior 473, Spectra Physics) coupled to the Fourier microscope in epi-
illumination configuration and focused down to 1 $\mu$m. The laser power is
attenuated to 30 mW to avoid local damaging of the sample. The pump laser is
filtered by a 500 nm long pass filter and its polarization is selected by the
same broadband linear polarizer used for reflectance. The measurements for
different detunings are collected using the same integration time to ensure
comparability.

Lifetime measurements are performed by a homemade time-correlated single-
photon counting module coupled to the Fourier microscope. 100 ps laser pulses
centered around 513 nm (LDH-P-C-520M, PicoQuant) are focused on the sample
surface by the same epi-illumination path used for the steady-state
fluorescence. We use a repetition rate of 10 MHz and an intensity such that
the average photon count rate at the detector is always 0.04% of the
excitation rate. The focus diameter is roughly 1 $\mu$m. The pump beam is
filtered by a 525 nm long pass filter, while the appropriate 10 nm full-width
at half-maximum band pass filter selects the LP or MP normal incidence
wavelength, for each detuning. The emitted light follows the same optical path
as reflectance and steady-state fluorescence but is directed to a single-
photon avalanche photodiode (MPD). The trigger from the laser driver (PDL
800-D, PicoQuant) and the signal from the detector are sent to a time-to-
digital converter (Time Tagger 20, Swabian Instruments). All histograms are
built with 100 ps binwidth. The instrument response function (IRF) has been
measured in several ways to check consistency of the result and ensure a
reliable exponential fit for the short-time dynamics.

The relation between reflectance, fluorescence and decay measurements is
guaranteed by the overlapping focus spots and the slow gradient slope of the
PMMA layer thickness.

Detuning-resolved measurements

In order to access experimentally a large set of detunings in a single sample,
we designed the PMMA layer to exhibit a slow and almost linear gradient
towards the peripheral zones. This radial gradient is controlled by the spin-
coating rotation speed. A radial sample movement of one millimiter corresponds
to increasing the PMMA thickness by approximately 50 nm. On the other hand,
the ErB/PVA layer featured a constant thickness throughout the sample. The
position of the focal spot is controlled by micrometer screws that permit
shifting the sample in the focal plane of the microscope objective.

Theoretical approach.- The Green’s functions follows the Dyson equation
$\mathcal{G}^{-1}(z)=[\mathcal{G}^{(0)}(z)]^{-1}-\Sigma(z),$ with the non-
vanishing terms

$\displaystyle\mathcal{G}^{(0)}_{11}(z)=\frac{1}{z-\omega_{c}^{L}(\theta)},\hskip
28.45274pt\mathcal{G}^{(0)}_{22}(z)=\frac{1}{z-\omega_{c}^{R}(\theta)}$ (4)
$\displaystyle\mathcal{G}^{(0)}_{33}(z)=\frac{1}{z-\omega_{X}},$

and the self-energy given by

$\displaystyle\Sigma_{12}(z)=\Sigma_{12}(z)=-t,$ (5)
$\displaystyle\Sigma_{23}(z)=\Sigma_{32}(z)=\Omega.$

The Green’s function can be obtained analytically in this case, in particular,
we obtain

$\displaystyle\mathcal{G}_{11}(z)=\frac{1}{\mathcal{G}^{(0)}_{11}(z)-t^{2}\mathcal{G}^{(0)}_{11}(z-\Omega^{2}\mathcal{G}^{(0)}_{33}(z)).}$
(6)

After analytic continuation $z\rightarrow\omega+i0^{+},$ the energies of the
polaritons are obtained by the position of the poles of the Green’s function

$\displaystyle\text{Re}[\mathcal{G}^{-1}_{11}(E)]=0,$ (7)

which has the three solutions coined lower, middle, and upper polariton. The
residue

$\displaystyle
Z=\left.\left(\frac{\partial\text{Re}[\mathcal{G}^{-1}_{11}(\omega)]}{\partial\omega}\right)^{-1}\right|_{\omega={E}}.$
(8)

The dispersion of the cavity photons is given
$\omega^{(R/L)}_{c}(\mathbf{k})=\frac{c}{n_{c}^{(R/L)}}\sqrt{k_{z}^{2}+k_{||}^{2}},$
where the incident light is along the $z$ axis, perpendicular to the cavity
mirrors, where the angle $\theta$, is given by
$k_{||}=n^{(R/L)}_{c}\frac{\omega}{c}\sin\theta.$

Acknowledgments.- We thank Joel Yuen-Zhou for the critical reading to our
manuscript and valuable discussions. G. P. acknowledges financial support from
Grants UNAM DGAPA PAPIIT No. IN104522 and CONACyT projects 1564464 and
1098652. H. L. G. acknowledges financial support from Grant UNAM DGAPA PAPIIT
No. IA107023. A. C. G. acknowledges financial support from Grant UNAM DGAPA
PAPIIT No. IN108620. C. L. O-R acknowledges financial support from Grant UNAM
DGAPA PAPIIT IG100521. H.E.S. acknowledges support from DGTIC-UNAM under
Project LANCAD-UNAM-DGTIC-419 and from Grant UNAM DGAPA PAPIIT No. IA106023.
A.C.-G, G. P. and H. E. S acknowledge financial support of PIIF 2023 H.E.S,
acknowledges Carlos Ernesto López Natarén for helping with the high-
performance computing infrastructure. H. A. L.-G, A.C.-G, G. P acknowledge
support from Grant UNAM DGAPA PAPIME No. PE101223.

Contributions

Y.A.G.-C, B. V., D. L. D, C. L. O.-R., H. L.-G, and G. P performed the
experiments. R. A, H. E. S., and A. C.-G provided the theoretical analysis. H.
A. L.-G., A. C. -G and G. P wrote the paper, with input from all authors. A.
C.-G and G. P. designed the project.

Corresponding authors

Correspondence to Arturo Camacho Guardian and Giuseppe Pirruccio.

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Supplementary Information

##  I $\Lambda$-scheme: Theoretical details

We consider the imaginary-time Green’s function defined as
$\mathcal{G}_{\alpha,\beta}(\tau)=-\langle
T_{\tau}[\hat{\psi}_{\alpha}(\tau)\hat{\psi}_{\beta}^{\dagger}(0)]\rangle$,
here the subindices $\alpha,\beta$ correspond to the left/right cavity photon
and exciton, respectively. The Dyson’s equation
$\mathcal{G}^{-1}(z)=[\mathcal{G}^{(0)}(z)]^{-1}-\Sigma(z),$ is given in terms
of the ideal $\mathcal{G}^{(0)}(z)$ Green’s function which define a diagonal
matrix

$\displaystyle\mathcal{G}^{(0)}_{11}(z)=\frac{1}{z-\omega_{c}^{L}(\theta)},\hskip
28.45274pt\mathcal{G}^{(0)}_{22}(z)=\frac{1}{z-\omega_{c}^{R}(\theta)}\hskip
28.45274pt\mathcal{G}^{(0)}_{33}(z)=\frac{1}{z-\omega_{X}}.$ (9)

The self-energy gives the coupling between the left and right cavity photons
$\Sigma_{12}(z)=\Sigma_{12}(z)=-t,$ with $t$ the tunneling coefficient. The
light-matter coupling between the excitons and right cavity photons is given
by $\Sigma_{23}(z)=\Sigma_{32}(z)=\Omega.$

In this case, as mentioned in the main text, the Green’s function acquires an
analytical form, in particular, the Green’s function of the left cavity
photons is given by

$\displaystyle\mathcal{G}_{11}(z)=\frac{1}{z-\omega_{c}^{L}(\theta)-\frac{t^{2}}{z-\omega_{c}^{R}(\theta)-\frac{\Omega^{2}}{z-\omega_{X}}}}.$
(10)

If $\omega_{c}^{R}(\theta)$ is detuned from the exciton energy and following
the branch of the middle polariton we obtain

$\displaystyle\mathcal{G}_{11}(\omega_{X}+\omega)\approx\frac{1}{\omega_{X}+\omega-\omega_{c}^{L}(\theta)-\frac{t^{2}}{\Omega^{2}}(\omega)}.$
(11)

The pole is found at

$\displaystyle\omega(\theta)=\frac{[\omega_{c}^{L}(\theta)-\omega_{X}]}{1+\left(\frac{t}{\Omega}\right)^{2}},$
(12)

that is, the energy of the middle polariton is given by
$\omega_{\text{MP}}(\theta)=\omega_{X}+\frac{[\omega_{c}^{L}(\theta)-\omega_{X}]}{1+\left(\frac{t}{\Omega}\right)^{2}}$
as given in the main text. This holds as long as the energy shift remains
within the energy window
$|\omega(\theta)|\ll\frac{t^{2}}{|\omega_{X}-\omega_{c}^{R}|}.$

The broadening of the photon and exciton lines can be included via additional
terms which account for the losses. Then, the experimental model departs from
the idealized scenario discussed before. The broadening of the right cavity
photons reduces the energy window
$\frac{t^{2}}{\sqrt{(\gamma_{c}^{R})^{2}+|\omega_{X}-\omega_{c}^{R}|^{2}}},$
while the broadening of the exciton lines leads to imperfect slow-light where
the right cavity photons remains coupled to the middle polariton even at
resonance $\omega_{c}^{R}(\theta=0)=\omega_{X}.$ This coupling is small as
long as $\sqrt{(\gamma_{c}^{R})^{2}+|\omega_{X}-\omega_{c}^{R}|^{2}}\ll
t^{2}/\gamma_{X}.$ We estimate the ratio

$\frac{t^{2}}{(\gamma_{X}\sqrt{(\gamma_{c}^{R})^{2}+|\omega_{X}-\omega_{c}^{R}|^{2})}}\approx
10-50$

that is, we expect to see clear signatures of slow-light with right cavity
photons effectively decoupled from the middle polaritons under resonant
conditions.

In Fig. 4 we show the spectral function at normal incidence of the left (a)
and right (b) cavity photons as a function of the detuning
$\delta=\delta_{LX}$. The vanishing value of the spectral function
$A_{22}(\mathbf{0},\omega)$ around zero detuning signals the full decoupling
of the right cavity photon from the middle polariton.

Figure 4: Spectral function of the (a) Left and (b) Right cavity photons as a
function of the left cavity detuning, $\delta=\delta_{LX}$. Around zero
detuning we observe that the middle polariton decouples from the right cavity
photon.

The vanishing of the right cavity photon component on the middle polariton is
equivalent to the transparency window observed in electromagnetically induced
transparency. This implies that the possibility to observe the middle
polariton depends on the arrangement of the stacked cavities. In particular,
we observe the middle polariton in reflectance as light is injected from the
left cavity. If light were injected from the right cavity, then the
reflectance spectrum would not signal the presence of this branch. For the
same reason, transmittance would not reveal the MP branch. In this sense, the
middle polariton is dark under certain circumstances.

##  II Emergence of a flatband

We now show additional experimental data that further illustrate the emergence
of an intercavity polariton with a flatband dispersion. As explained in the
main text, the polariton bands depend on the tunneling ratio, the Rabi
splitting, and the different energy levels. We introduce the cavity detuning
from the exciton energy at normal incidence
$\delta_{LX}=\omega_{c}^{(L)}(\theta=0)-\omega_{X}$ and
$\delta_{RX}=\omega_{c}^{(R)}(\theta=0)-\omega_{X}.$

Figures 5 (a)-(b) show the reflectance spectrum for varying detuning
$\delta_{LX}$ with the remaining parameters as detailed in Table [1]. From
left to right we observe the emergence of a purely middle intercavity
polariton formed by a left cavity photon and an exciton localized in the right
cavity. This is demonstrated by the quasiparticle residues shown in the bottom
of Figs. 5 (a)-(b).

Fig. 4 |  $\delta_{LX}$[eV] |  $\delta_{RX}$[eV] |  $t$[eV] |  $2\Omega$[eV] |  $\omega_{X}$[eV]
---|---|---|---|---|---
a | -0.37 | 0.12 | 0.21 | 0.26 | 2.23
b | -0.18 | 0.09 | 0.21 | 0.26 | 2.23
c | -0.08 | 0.13 | 0.21 | 0.26 | 2.23
d | -0.03 | 0.13 | 0.21 | 0.26 | 2.23
e | 0.00 | 0.17 | 0.23 | 0.245 | 2.25
f | 0.07 | 0.15 | 0.21 | 0.26 | 2.23

Table 1. Theoretical parameters of the polariton branches in Fig. 5.

Figure 5: s-polarized reflectance as a function of the incident angle and the
energy for several cavity detunings. The dashed curves represent the energies
of the three polariton branches, obtained with the parameters given in Table
[1]. In the bootom row, we show the quasiparticle residues, i.e., the Hopfield
coefficients, of the three quasiparticle branches following the color code of
the main text.

The evolution of the MP Hopfield coefficients illustrate that, as the left
cavity detuning is driven to $\delta_{LX}=0$, the right cavity photon
decouples from the MP causing the suppression of its dispersion. For
$\delta_{LX}=0$, although the polariton energy matches the bare exciton, the
MP does not loos its photonic character because it retains significant residue
of the right cavity photon.

In Fig. 6 we show the evolution of the fluorescence as a function of the
detuning. Pumping the system at $\omega_{p}=2.62\text{eV}$ ensures that for
$\delta=\delta_{LX}=0$ we drive coherently the UP which consists primarily of
right cavity photons. The absence of the left photonic component in the driven
UP strongly suppresses the emission of the MP for $\delta=\delta_{LX}=0$

Figure 6: s-polarized fluorescence for the same parameters as in Fig. 5 and
Table 1. From left to right we observe a strong suppression of the MP
fluorescence as the left cavity detuning approaches zero.

We should note that the right cavity photons are the dominant component of the
UP in all of our cavity detunings. However, as we drive at
$\omega_{p}=2.62\text{eV},$ for large negative detuning, this driving is off-
resonant, that is, it lies far above the energy of the upper polariton, and
thus, photons are injected in left and right cavities. Therefore, for negative
detunings, we observe emission from both the middle and lower polariton
branches.

##  III Polariton dynamics

In the main text we focus on the short-time dynamics of the polaritons. Here,
we provide the experimental data in an extended time window. Figure 7 shows
the LP (a) and MP (b) fluorescence decay for $\delta_{LX}/\text{eV}=-0.37$
(blue), on resonance $\delta_{LX}/\text{eV}=0$ (purple) and
$\delta_{LX}/\text{eV}=0.20$ (green). The shaded pink area corresponds to the
time window discussed in the main text where we focused on the initial decay.
As a consequence of the coupling to the triplet state by intersystem crossing
for large negative detunings, or to the bare exciton states for positive
detunings, the decay of the LP for $t>10\text{ns}$ after the laser pulse
arrival becomes more complex. In the former case, where the energy of the LP
approaches the triplet state, we see the presence of a long-living state which
does not decay entirely in the considered time window. A similar effect,
although less pronounced, is seen as the LP shifts toward the bare exciton
energy. An intermediate situation is observed for detunings around
$\delta_{LX}/\text{eV}=-0.2$. The ability to control the dynamics of the LP
was extensively discussed for a single cavity Stranius  _et al._ (2018).

On the other hand, the MP energy lies very far away from the triplet state,
thus the triplet state does not participate in the MP dynamics and its decay
is given by the slow-light physics discussed in the main text. For
$t>10\text{ns}$ after the laser pulse arrival, the MP time evolution resembles
the dynamics of the ErB/PVA thin film in the absence of the cavity. This can
be appreciated by comparing the black curve in Fig. 7(b) with the MP decays.

In Fig. 7(b) we show with red curve the IRF obtained by measuring a 0.5 M
solution of ErB in water which is known to decay mono-exponentially with a
lifetime well below the time resolution of our instrument. This choice ensures
that the detector wavelength-dependent response does not play a role in our
measurements. The same experimental conditions as for the cavity measurements
are used. The full-width at half-maximum of this decay is approximately 350
ps. Having characterized the IRF curve, the short-time fit of the LP and MP
decays can be safely conduct by excluding the first 600 ps after the intensity
maximum. This time value is labelled as $t=0\leavevmode\nobreak\ $ns in
Fig.3(a) of our manuscript.

It is important to highlight that within the shaded pink area, i.e., within
the first 10 ns from the arrival of the laser pulse, the dynamics of the ErB
molecule in absence of the cavity can be excellently modeled by a single
exponential. We find that $A=0.9$, $\Gamma=4.9\leavevmode\nobreak\
\text{ns}^{-1}$ and $\eta=0.24\leavevmode\nobreak\ \text{ns}^{-1}$. Thus the
rich interplay observed for the MP and its overall slow dynamics cannot be
account for with the ErB photophysics.

Figure 7: Fluorescence decays of the LP (a) and MP (b) for different cavity
detunings: $\delta_{LX}/\text{eV}=-0.37$ (blue), on resonance
$\delta_{LX}/\text{eV}=0.00$ (purple) and $\delta_{LX}/\text{eV}=0.20$
(green). In (b), the black curve represents the decay of the ErB/PVA thin film
in the absence of the cavity, while the red curve corresponds to the IRF. The
shaded pink area corresponds to the time window discussed in the main text.


# Simple algorithms to test and learn local Hamiltonians

Francisco Escudero Gutiérrez Qusoft and CWI<EMAIL_ADDRESS>
###### Abstract

We consider the problems of testing and learning an $n$-qubit $k$-local
Hamiltonian from queries to its evolution operator with respect the 2-norm of
the Pauli spectrum, or equivalently, the normalized Frobenius norm. For
testing whether a Hamiltonian is $\varepsilon_{1}$-close to $k$-local or
$\varepsilon_{2}$-far from $k$-local, we show that
$O(1/(\varepsilon_{2}-\varepsilon_{1})^{8})$ queries suffice. This solves two
questions posed in a recent work by Bluhm, Caro and Oufkir. For learning up to
error $\varepsilon$, we show that $\exp(O(k^{2}+k\log(1/\varepsilon)))$
queries suffice. Our proofs are simple, concise and based on Pauli-analytic
techniques.

##  1 Introduction

In this work we consider the problems of testing and learning a local
Hamiltonian from its time evolution operator. These kinds of Hamiltonians
govern the dynamics of many physical systems, which motivates the problem of
testing if an unknown Hamiltonian is local and, if it is local, to learn it in
an efficient way. In fact, there is vast and recent literature about learning
a local Hamiltonian from its time evolution operator [dSLCP11, BAL19, ZYLB21,
HKT22, WKR+22, YSHY23, Car23, DOS23, HTFS23, CW23, LTN+23, MBC+23, SFMD+24,
GCC24], but the first testing algorithm was only proposed recently by Bluhm,
Caro and Oufkir [BCO24a]. There are also many of results about learning a
Hamiltonian from its Gibbs state [AAKS21, HKT22, RSF23, ORSFW23, BLMT23,
GCC24], but in this work we will only consider the model where one accesses
the Hamiltonian through queries to its evolution operator.

An $n$-qubit Hamiltonian $H$ is a self-adjoint operator acting on
$(\mathbb{C}^{2})^{\otimes n}$. As such, it can be expanded in terms of the
Pauli strings as

$H=\sum_{x\in\\{0,1,2,3\\}^{n}}h_{x}\sigma_{x},$

where $h_{x}$ are real numbers and
$\sigma_{x}=\otimes_{i\in[n]}\sigma_{x_{i}},$ where $\sigma_{0}=\mathop{\rm
Id}\nolimits_{2},\,\sigma_{1}=X,\,\ \sigma_{2}=Y,\,\sigma_{3}=Z$. The
coefficients $h_{x}$ are known as the Pauli spectrum of the Hamiltonian. A
Hamiltonian $H$ is $k$-local if it is supported on Pauli strings that act
trivially on all but at most $k$ qubits. In other words, $H$ is $k$-local if
$h_{x}=0$ for all $x\in\\{0,1,2,3\\}^{n}$ that take values different from $0$
on more than $k$ sites. If $H$ is the Hamiltonian describing the dynamics of a
certain physical system, then the states of that system evolve according to
the time evolution operator $U(t)=e^{-iHt}$. This means that if $\rho(0)$ is
the state at time $0$, at time $t$ the state will have evolved to
$\rho(t)=U(t)\rho(0)U^{\dagger}(t).$ Hence, to test and learn a Hamiltonian
one can do the following: prepare a desired state, apply $U(t)$ or tensor
products of $U(t)$ with identity to the state (i.e., query $U(t)$), and
finally measure in a chosen basis. It is usual to impose the normalization
condition $\left\|{H}\right\|_{\infty}\leq 1$ (i.e., that the eigenvalues of
$H$ are bounded in absolute value by $1$), as otherwise the complexities scale
with the norm of the Hamiltonian. Given a distance $d$ in the space of
Hamiltonians, $H$ is $\varepsilon$-far from being $k$-local if for every
$k$-local $H^{\prime}$ it satisfies that $d(H,H^{\prime})>\varepsilon,$ and
otherwise is $\varepsilon$-close. Now we are ready to state the testing and
learning problems.

######  Problem 1 (Tolerant Hamiltonian locality testing).

Let $0\leq\varepsilon_{1}<\varepsilon_{2}$, $\delta\in(0,1)$, $k\in\mathbb{N}$
and $d$ be a distance between Hamiltonians. Let $H$ be an $n$-qubit
Hamiltonian with $\left\|{H}\right\|_{\infty}\leq 1$ that is promised to be
either $\varepsilon_{1}$-close to $k$-local or $\varepsilon_{2}$-far from
$k$-local. The problem is to decide between those cases with success
probability $\geq 1-\delta$ by making queries to $U(t)$.

######  Problem 2 (Local Hamiltonian learning).

Let $\varepsilon>0$, $\delta\in(0,1)$, $k\in\mathbb{N}$ and $d$ be a distance
between Hamiltonians. Let $H$ be a $k$-local $n$-qubit Hamiltonian with
$\left\|{H}\right\|_{\infty}\leq 1$. The problem is to output a classical
description of a $k$-local Hamiltonian $H^{\prime}$ that is
$\varepsilon$-close to $H$ with probability $\geq 1-\delta$ by making queries
to $U(t)$.

In both problems, the main goals are minimizing the number of queries and the
total evolution time. These quantities depend on the distance $d$. The recent
learning literature usually takes $d$ to be the supremum distance of the Pauli
spectrum
($d(H,H^{\prime})=\left\|{H-H^{\prime}}\right\|_{\mathrm{Pauli},\infty}=\max_{x}|h_{x}-h^{\prime}_{x}|$),
and imposes extra constraints on the Pauli spectrum (generally some kind of
geometrical locality, such as assuming that the qubits are displayed in a grid
and the Pauli terms only act on neighboring qubits). However, if one wants to
convert these learning algorithms to learners under a finer distance such as
the 2-norm of the Pauli spectrum
($d(H,H\textasciiacute)=\left\|{H-H^{\prime}}\right\|_{2}=\sqrt{\sum_{x}|h_{x}-h^{\prime}_{x}|^{2}}$,
which equals the normalized Frobenius norm due to Parseval’s identity) it is
not clear how to avoid a $\mbox{\rm poly}(n^{k})$ term appearing, due to the
fact that there are $n^{k}$ Pauli strings that are $k$-local. This 2-norm of
the Pauli spectrum was recently considered by Bluhm, Caro and Oufkir [BCO24a],
who proved that to learn an arbitrary $n$-qubit Hamiltonian under this
distance it is necessary to make $\Omega(2^{2n})$ queries to the time
evolution operator. In the first version of their work, they also proposed a
non-tolerant testing algorithm, meaning that it only works for the case
$\varepsilon_{1}=0,$ whose query complexity is
$O(n^{2k+2}/(\varepsilon_{2}-\varepsilon_{1})^{4})$ and with total evolution
time $O(n^{k+1}/(\varepsilon_{2}-\varepsilon_{1})^{3})$. They posed as open
questions whether the dependence on $n$ could be removed and whether an
efficient tolerant-tester was possible [BCO24a, Section 1.5]. Our first result
gives positive answer to both questions. From now on, unless otherwise
mentioned, we will consider the distance $d$ to be the one induced by the
2-norm of the Pauli spectrum.

######  Theorem 3 (Locality testing).

There is an algorithm that solves the locality testing problem (1) by making
$O(1/(\varepsilon_{2}-\varepsilon_{1})^{8}\cdot\log(1/\delta))$ queries to the
evolution operator and with
$O(1/(\varepsilon_{2}-\varepsilon_{1})^{7}\cdot\log(1/\delta))$ total
evolution time.

Our algorithm to test for locality is simple. It consists of repeating the
following process $1/(\varepsilon_{2}-\varepsilon_{1})^{8}$ times: prepare $n$
EPR pairs, apply $U(\varepsilon_{2}-\varepsilon_{1})\otimes\mathop{\rm
Id}\nolimits_{2^{n}}$ to them and measure in the Bell basis. Each time that we
repeat this process, we sample from the Pauli sprectrum of
$U(\varepsilon_{2}-\varepsilon_{1})$111The Pauli spectrum of a unitary
$U=\sum_{x}u_{x}\sigma_{x}$ determines a probability distribution because
$\sum_{x}|u_{x}|^{2}=1.$. As $\varepsilon_{2}-\varepsilon_{1}$ is small,
Taylor expansion ensures that
$U(\varepsilon_{2}-\varepsilon_{1})\approx\mathop{\rm
Id}\nolimits_{2^{n}}-i(\varepsilon_{2}-\varepsilon_{1})H$, so sampling from
the Pauli spectrum of $U(\varepsilon_{2}-\varepsilon_{1})$ allows us to
estimate the weight of the non-local terms of $H.$ If that weight is big, we
output that $H$ is far from $k$-local, and otherwise we conclude that $H$ is
close to $k$-local.

After sharing Theorem 3 with Bluhm, Caro and Oufkir, they independently
improved the analysis of their testing algorithm. They showed that in a
certain broad regime their tester is tolerant, only makes
$O(1/((\varepsilon_{2}-\varepsilon_{1})^{3}\varepsilon_{2}))$ queries and just
requires $O(1/((\varepsilon_{2}-\varepsilon_{1})^{2.5}\varepsilon_{2}^{0.5}))$
total evolution time [BCO24b, Theorem B.5]. To compare Theorem 3 with the
improved testing result of Bluhm, Caro and Oufkir, we should introduce a
family of problems that has locality testing as an instance. Given a subset
$S$ of $\\{0,1,2,3\\}^{n}$, we define the problem of testing property $S$ as
the problem of testing whether $H$ is $\varepsilon_{1}$-close to be supported
on $S$ or $\varepsilon_{2}$-far from being supported on $S$. By taking $S$
equal to the set of strings that take the value $0$ on at least $n-k$ sites
one recovers the $k$-locality testing problem. Both our Theorem 3 and their
testing result work for testing properties defined by sets $S$. The advantages
of their algorithm are that it uses no auxiliary qubits, while ours requires
$n$, and it is quadratically better than ours with respect to
$\varepsilon_{2}-\varepsilon_{1}$. The advantages of our algorithm are that it
works for any $S$, while theirs only works for
$|S|=O(2^{n}(\varepsilon_{2}^{2}-\varepsilon_{1}^{2})^{2})$, although they can
remove this constraint by allowing access to
$O(\log(|S|^{3}/2^{n})+\log(1/(\varepsilon_{2}-\varepsilon_{1})))$ auxiliary
qubits; that our algorithm just needs
$O(1/(\varepsilon_{2}-\varepsilon_{1})^{8})$ classical post-processing time,
while theirs requires
$O(n^{2}|S|/((\varepsilon_{2}-\varepsilon_{1})^{3}\varepsilon_{2}))$ time; and
that theirs needs $H$ to be traceless, while ours does not. A technical
feature of our result is that the proof is simpler and more concise than
theirs.

Our second result is a learning algorithm for $k$-local Hamiltonians.

######  Theorem 4 (Local Hamiltonian learning).

There is an algorithm that solves the local Hamiltonian learning problem (2)
by making $\exp(O(k^{2}+k\log(1/\varepsilon))\log(1/\delta)$ queries to the
evolution operator with $\exp(O(k^{2}+k\log(1/\varepsilon))\log(1/\delta)$
total evolution time.

The learning algorithm of Theorem 4 has two stages. In the first stage one
samples from the Pauli distribution of $U(\varepsilon)$, as in the testing
algorithm, and from that one can detect which are the big Pauli coefficients
of $H$. In the second stage we learn those big Pauli coefficients using the
SWAP test on $U(\varepsilon)$, as proposed by Montanaro and Osborne [MO08,
Lemma 24]. One can ensure that the coefficients not detected as big in the
first stage of the algorithm can be neglected. To do that we borrow the ideas
of Eskenazis and Ivanisvili for the classical low-degree learning problem
[EI22], combined with the non-commutative Bohnenblust-Hille inequality proved
by Huang, Chen and Preskill [HCP23], and improved by Volberg and Zhang [VZ23].

Most previous work on Hamiltonian learning is done under the distance induced
by the supremum norm of the Pauli spectrum and with extra constraints apart
from locality [dSLCP11, BAL19, ZYLB21, HKT22, WKR+22, YSHY23, Car23, DOS23,
HTFS23, LTN+23, MBC+23, SFMD+24, GCC24]. When transformed into learning
algorithms under the finer distance induced by the 2-norm of the Pauli
spectrum, these proposals yield complexities that depend polynomially on
$n^{k}$ and only work for a restricted family of $k$-local Hamiltonians. A
work that explicitly considers the problem of learning under the 2-norm of the
Pauli spectrum is the one of Castaneda and Wiebe [CW23]. However, their
results require query access to the inverse time evolution operator $e^{-itH}$
and yield complexities of order $O(n^{k})$. Hence, to the best of our
knowledge, Theorem 4 is the first learning result that has no dependence on
$n$ when considering the 2-norm of Pauli spectrum and works for any kind of
$k$-local Hamiltonian. Regarding the tightness of its query complexity, the
$\exp(\Omega(k))$ query lower bound of Bluhm, Caro and Oufkir [BCO24a] to
learn $k$-qubit Hamiltonians shows that our algorithm cannot be improved much.
Closing the gap between this lower bound and our $\exp(O(k^{2}))$ query upper
bound remains an intriguing open problem.

##  2 Preliminaries

In this section we collect a few well-known facts that we will repeatedly use
in our proofs. Given an $n$-qubit operator $A=\sum_{x}a_{x}\sigma_{x}$,
Parseval’s identity states that its normalized Frobenius norm equals the
$2$-norm of its Pauli spectrum. We will denote both by
$\left\|{A}\right\|_{2}$,

$\left\|{A}\right\|_{2}=\sqrt{\frac{\operatorname{Tr}[A^{\dagger}A]}{2^{n}}}=\sqrt{\sum_{x\in\\{0,1,2,3\\}^{2}}|a_{x}|^{2}}.$

Given $x\in\\{0,1,2,3\\}^{n}$, we define $|x|$ as the number of sites where
$x$ does not take the value 0, $A_{>k}$ as $\sum_{|x|>k}a_{x}\sigma_{x}$ and
$A_{\leq k}$ as $\sum_{|x|\leq k}a_{x}\sigma_{x}$. From the formulation of the
2-norm in terms of the Pauli coefficients it follows that
$\left\|{A_{>k}}\right\|_{2}\leq\left\|{A}\right\|_{2}$, while from its
formulation as the normalized Frobenius norm one has that
$\left\|{A}\right\|_{2}\leq\left\|{A}\right\|_{\infty}.$ We recall that
$\left\|{A}\right\|_{\infty}$ is the biggest singular value of $A$. We note
that the distance of a Hamiltonian $H$ from the space of $k$-local
Hamiltonians is given by $\left\|{H_{>k}}\right\|_{2}$, as $H_{\leq k}$ is the
$k$-local Hamiltonian closest to $H$.

It follows from Parseval’s identity that if $U$ is a unitary, then
$\sum|u_{x}|^{2}=1.$ In other words, $(|u_{x}|^{2})_{x}$ is a probability
distribution. Applying $U\otimes\mathop{\rm Id}\nolimits_{2^{n}}$ to $n$ EPR
pairs (i.e., preparing the Choi-Jamiolkowski state of $U$) and measuring in
the Bell basis allows one to sample from this distribution, because

$U\otimes\mathop{\rm
Id}\nolimits_{2^{n}}\ket{\mathrm{EPR}_{n}}=\sum_{x\in\\{0,1,2,3\\}^{n}}u_{x}\otimes_{i\in[n]}(\sigma_{x_{i}}\otimes\mathop{\rm
Id}\nolimits_{2}\ket{\mathrm{EPR}}),$

and the Bell states can be written as $\sigma_{x}\otimes\mathop{\rm
Id}\nolimits_{2}\ket{\mathrm{EPR}}$ for $x\in\\{0,1,2,3\\}$.

We will also use that given a Hamiltonian $H$ with
$\left\|{H}\right\|_{\infty}\leq 1$, the Taylor expansion of the exponential
allows us to approximate the time evolution operator as

$U(t)=e^{-itH}=\mathop{\rm Id}\nolimits_{2^{n}}-itH+ct^{2}R_{2}(t)$

for $t\leq 1/2,$ where the second order remainder $R_{2}(t)$ is bounded
$\left\|{R_{2}(t)}\right\|_{\infty}\leq 1$ and $c>0$ is a universal constant.

##  3 Testing locality

In this section we prove Theorem 3. First, we prove a claim regarding the
discrepancy on the weights of non-local terms of the short-time evolution
operator for close-to-local and far-from-local Hamiltonians.

######  Claim 5.

Let $0\leq\varepsilon_{1}<\varepsilon_{2}$. Let
$\alpha=(\varepsilon_{2}-\varepsilon_{1})/(3c)$ and $H$ be an $n$-qubit
Hamiltonian with $\left\|{H}\right\|_{\infty}\leq 1$. We have that if $H$ is
$\varepsilon_{1}$-close $k$-local, then

$\left\|{U(\alpha)_{>k}}\right\|_{2}\leq(\varepsilon_{2}-\varepsilon_{1})\frac{2\varepsilon_{1}+\varepsilon_{2}}{9c},$

and if $H$ is $\varepsilon_{2}$-far from being $k$-local, then

$\left\|{U(\alpha)_{>k}}\right\|_{2}\geq(\varepsilon_{2}-\varepsilon_{1})\frac{\varepsilon_{1}+2\varepsilon_{2}}{9c}.$

  * Proof:

To save on notation, we set $U=U(\alpha)$ and $R=R_{2}(\alpha)$. First, assume
that $H$ is $\varepsilon_{1}$-close $k$-local. Then

$\left\|{U_{>k}}\right\|_{2}\leq\alpha\left\|{H_{>k}}\right\|_{2}+c\alpha^{2}\left\|{R_{>k}}\right\|_{2}\leq\frac{\varepsilon_{2}-\varepsilon_{1}}{3c}\varepsilon_{1}+c\left(\frac{\varepsilon_{2}-\varepsilon_{1}}{3c}\right)^{2}=(\varepsilon_{2}-\varepsilon_{1})\frac{2\varepsilon_{1}+\varepsilon_{2}}{9c},$

where in the first inequality we have used the triangle inequality and the
Taylor expansion, and in the second that $H$ is $\varepsilon_{1}$-close to
$k$-local and that $\left\|{R_{>k}}\right\|_{2}\leq\left\|{R}\right\|_{2}\leq
1$ because $\left\|{R}\right\|_{2}\leq\left\|{R}\right\|_{\infty}\leq 1.$ Now,
assume that $H$ is $\varepsilon_{2}$-far from being $k$-local. Then

$\left\|{U_{>k}}\right\|_{2}\geq\alpha\left\|{H_{>k}}\right\|_{2}-c\alpha^{2}\left\|{R_{>k}}\right\|_{2}\geq\frac{\varepsilon_{2}-\varepsilon_{1}}{3c}\varepsilon_{2}-c\left(\frac{\varepsilon_{2}-\varepsilon_{1}}{3c}\right)^{2}\geq(\varepsilon_{2}-\varepsilon_{1})\frac{\varepsilon_{1}+2\varepsilon_{2}}{9c},$

where in first inequality we have used again Taylor expansion and the triangle
inequality, and in the second the fact that being $\varepsilon_{2}$-far from
$k$-local implies that $\left\|{H_{>k}}\right\|_{2}\geq\varepsilon_{2}.$
$\Box$

  * Proof of Theorem 3:

Applying $U(\alpha)\otimes\mathop{\rm Id}\nolimits_{2^{n}}$ to
$\ket{\mathrm{EPR}_{n}}$ and measuring in the Bell basis allows one to sample
from $(|U(\alpha)_{x}|^{2})_{x}$. Thus, by the Hoeffding bound, with
$O(1/(\varepsilon_{2}-\varepsilon_{1})^{8}\cdot\log(1/\delta))$ queries to
$U(\alpha)\otimes\mathop{\rm Id}\nolimits_{2^{n}}$ one can estimate
$\sum_{|x|>k}|U(\alpha)_{x}|^{2}$ up to an error
$((\varepsilon_{2}-\varepsilon_{1})^{2}/(18c))^{2}$ with success probability
$1-\delta$. Taking $\alpha=(\varepsilon_{2}-\varepsilon_{1})/(3c)$, thanks to
5, this is enough for testing $k$-locality. $\Box$

######  Remark 6.

The algorithm of Theorem 3 also works to test any property defined by a set of
Pauli strings $S\subseteq\\{0,1,2,3\\}^{n}$, i.e., to test whether
$\sqrt{\sum_{x\notin S}h_{x}^{2}}\leq\varepsilon_{1}$ or $\sqrt{\sum_{x\notin
S}h_{x}^{2}}\geq\varepsilon_{2}$. Also, a union bound allows us to
simultaneously test $M$ properties defined by sets $S_{1},...,S_{M}$ by paying
a factor of $\log M$ in the query complexity and total evolution time.

##  4 Learning local Hamiltonians

In this section we prove Theorem 4. To do that, we need the non-commutative
Bohnenblust-Hille inequality by Volberg and Zhang [VZ23], which they used to
give an algorithm to learn local observables.

######  Theorem 7 (Non-Commutative Bohnenblust-Hille inequality).

Let $H=\sum_{x}h_{x}\sigma_{x}$ be a $k$-local Hamiltonian with
$\left\|{H}\right\|_{\infty}\leq 1$. Then, there is a universal constant $C$
such that

$\sum_{x\in\\{0,1,2,3\\}^{n}}|h_{x}|^{\frac{2k}{k+1}}\leq C^{k}.$

  * Proof of Theorem 4:

Let $\alpha,\beta,\gamma$ be fixed later. Let $U$ be the evolution operator
$U(\alpha)$ at time $\alpha$ and $R$ be the Taylor remainder $R_{2}(\alpha)$.
Our learning algorithm has two stages, the first to detect the big Pauli
coefficients and the second to learn those big Pauli coefficients.

First stage of the algorithm: detect the big Pauli coefficients. In this stage
we prepare $U\otimes\mathop{\rm Id}\nolimits_{2^{n}}\ket{\mathrm{EPR}_{n}}$
and measure in the Bell basis $O(\gamma^{4}\log(1/\delta))$ times. This way we
sample $O(\gamma^{4}\log(1/\delta))$ times from $(|u_{x}|^{2})_{x}$. By
[Can20, Theorem 9] the empirical distribution $|u_{x}^{\prime}|^{2}$ obtained
from these samples approximates $|u_{x}|^{2}$ up to error $\gamma^{2}$ for
every $x\in\\{0,1,2,3\\}^{n}$ with success probability $\geq 1-\delta$. Let
$S_{\gamma}=\\{x:|u_{x}^{\prime}|>\gamma\\}\setminus\\{0^{n}\\}$ be the set of
big Pauli coefficients. Note that if $x$ does not belong to $S_{\gamma}$, then
$u_{x}$ is indeed small since

$x\notin S_{\gamma}\land x\neq
0\implies|u_{x}|\leq|u_{x}^{\prime}|+|u_{x}-u_{x}^{\prime}|\leq 2\gamma.$ (1)

Also note that $u_{0}=1-i\alpha h_{0}+c\alpha^{2}R_{0}$ and that
$u_{x}=-i\alpha h_{x}+c\alpha^{2}R_{x}$ for every $x\neq 0^{n}$. As
$\left\|{R}\right\|_{\infty}\leq 1$, we have that
$\left\|{c\alpha^{2}R}\right\|_{\infty}^{2}\leq c^{2}\alpha^{4}$. As
$\left\|{R}\right\|_{2}\leq\left\|{R}\right\|_{\infty}$, this implies that

$|(u_{0}-1)+i\alpha h_{0}|^{2}+\sum_{x\neq 0}|u_{x}+i\alpha h_{x}|^{2}\leq
c^{2}\alpha^{4}$ (2)

and in particular for every $x\neq 0$

$|u_{x}+i\alpha h_{x}|\leq c\alpha^{2}.$ (3)

Putting Eqs. 1 and 3 together it follows that

$x\notin S_{\gamma}\land x\neq
0\implies|h_{x}|\leq\alpha^{-1}[|u_{x}|+|u_{x}+i\alpha
h_{x}|]\leq\alpha^{-1}(2\gamma+c\alpha^{2}).$ (4)

Moreover, as $\sum_{x}|u_{x}^{\prime}|^{2}=1,$ we have that
$|S_{\gamma}|\leq\gamma^{-2}.$

Second stage of the algorithm: learn the big Pauli coefficients. Montanaro and
Osborne proposed a simple primitive to estimate a given Pauli coefficient of a
unitary up to error $\beta$ with success probability $\geq 1-\delta$ by making
$O((1/\beta)^{2}\log(1/\delta))$ queries [MO08, Lemma 24]. We use this
primitive to learn the Pauli coefficients of $u_{x}$ up to error $\beta$ with
success probability $\geq 1-\delta$ for every $x\in
S_{\gamma}\cup\\{0^{n}\\}$. As, $|S_{\gamma}|\leq\gamma^{-2}$, by a union
bound, this stage requires
$O(\beta^{-2}\gamma^{-2}\log(1/(\gamma^{2}\delta)))$ queries to $U$. Let
$u^{\prime\prime}_{x}$ be these estimates. We output
$H^{\prime\prime}=\mathrm{Re}(i\alpha^{-1}(u^{\prime\prime}_{0}-1))\sigma_{0}+\sum_{x\in
S_{\gamma}}\mathrm{Re}(i\alpha^{-1}u_{x}^{\prime\prime})\sigma_{x}$ as our
approximation of $H$ (we take the real part to ensure that $H^{\prime\prime}$
is self-adjoint).

Correctness of the algorithm. We claim that with an appropriate choice of the
parameters $H^{\prime\prime}$ is a good approximation of $H$. First, as the
Pauli coefficients of $H$ are real we have that

$\displaystyle\left\|{H-H^{\prime\prime}}\right\|_{2}^{2}$
$\displaystyle=\alpha^{-2}|\mathrm{Re}(i(u^{\prime\prime}_{0}-1))-\alpha
h_{0}|^{2}+\alpha^{-2}\sum_{x\in
S_{\gamma}}|\mathrm{Re}(iu_{x}^{\prime\prime})-\alpha h_{x}|^{2}+\sum_{x\notin
S_{\gamma}}|h_{x}|^{2}$
$\displaystyle\leq\underbrace{\alpha^{-2}|(u_{0}^{\prime\prime}-1)+i\alpha
h_{0}|^{2}+\alpha^{-2}\sum_{x\in S_{\gamma}}|u_{x}^{\prime\prime}+i\alpha
h_{x}|^{2}}_{\mathrm{(I})}+\underbrace{\sum_{x\notin
S_{\gamma}}|h_{x}|^{2}}_{\mathrm{(II})}.$

Second, we give an upper bound to term $($I$)$:

$\displaystyle(\mathrm{I})$ $\displaystyle\leq 2\alpha^{-2}[|(u_{0}-1)+i\alpha
h_{0}|^{2}+|u_{0}^{\prime\prime}-u_{0}|^{2}]+2\alpha^{-2}\sum_{x\in
S_{\gamma}}[|u_{x}+i\alpha h_{x}|^{2}+|u_{x}^{\prime\prime}-u_{x}|^{2}]$
$\displaystyle\leq 2c^{2}\alpha^{2}+2\alpha^{-2}\beta^{2}(\gamma^{-2}+1)$

where in the first step we have used the triangle inequality and that
$(a+b)^{2}\leq 2(a^{2}+b^{2})$; and in the second step Eq. 2 to upper bound
the error due to approximating $h_{x}$ from $u_{x}$, and the learning
guarantees of the second stage of the algorithm and that
$|S_{\gamma}|\leq\gamma^{-2}$ to upper bound the error due to approximating
$u_{x}$ by $u_{x}^{\prime\prime}$. Third, we upper bound term $($II$)$

$\displaystyle(\mathrm{II})$ $\displaystyle\leq\max_{x\notin
S_{\gamma}}\\{|h_{x}|^{2/k+1}\\}\sum_{x\notin
S_{\gamma}}|h_{x}|^{2k/k+1}\leq(2\gamma\alpha^{-1}+c\alpha)^{2/k+1}C^{k},$

where in the first step we have used that $2=2/(k+1)+2k/(k+1)$, and in the
second step Theorem 7 and Eq. 2. Finally, if we put everything together we get

$\displaystyle\left\|{H-H^{\prime\prime}}\right\|_{2}^{2}$ $\displaystyle\leq
2c^{2}\alpha^{2}+2\alpha^{-2}\beta^{2}(\gamma^{-2}+1)+(2\gamma\alpha^{-1}+c\alpha)^{2/k+1}C^{k}.$

Thus, if we take $\alpha=\varepsilon^{k+1}C^{-k(k+1)/2},\ \gamma=\alpha^{2},$
and $\beta=\alpha^{3}\varepsilon$ it follows that
$\left\|{H-H^{\prime\prime}}\right\|_{2}^{2}\leq O(\varepsilon^{2}),$ as
desired.

Query complexity and total evolution time. Taking both stages of the algorithm
into account we make
$O(\gamma^{-4}\log(1/\delta)+\gamma^{-2}\beta^{-2}\log(1/(\gamma^{2}\delta)))$
queries to $U(\alpha)$. Hence, by taking $\alpha,\beta$ and $\gamma$ as above,
one obtains the claimed query complexity and total evolution time. $\Box$

### Acknowledgements.

I thank Srinivasan Arunachalam and Matthias Caro for detailed comments and
advice about presentation. I thank Amira Abbas and Jop Briët for useful
comments, discussions and encouragement. I thank Andreas Bluhm, Arkopal Dutt
and Aadil Oufkir for useful comments and discussions. This research was
supported by the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement no. 945045, and by
the NWO Gravitation project NETWORKS under grant no. 024.002.003.

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