# Wavefunction tomography of topological dimer chains with long-range
couplings

F. Pellerin  Département de Physique, Université de Montréal, C.P. 6128,
Succursale Centre-Ville, Montréal, Québec, Canada H3C 3J7  R. Houvenaghel
Département de Physique, Université de Montréal, C.P. 6128, Succursale Centre-
Ville, Montréal, Québec, Canada H3C 3J7  W. A. Coish  Department of Physics,
McGill University, 3600 rue University, Montreal, Qc H3A 2T8, Canada  I.
Carusotto  Pitaevskii BEC Center, INO-CNR and Dipartimento di Fisica,
Università di Trento, via Sommarive 14, I-38123 Trento, Italy  P. St-Jean
Département de Physique, Université de Montréal, C.P. 6128, Succursale Centre-
Ville, Montréal, Québec, Canada H3C 3J7

###### Abstract

The ability to tailor with a high accuracy the inter-site connectivity in a
lattice is a crucial tool for realizing novel topological phases of matter.
Here, we report the experimental realization of photonic dimer chains with
long-range hopping terms of arbitrary strength and phase, providing a rich
generalization of the celebrated Su-Schrieffer-Heeger model. Our experiment is
based on a synthetic dimension scheme involving the frequency modes of an
optical fiber loop platform. This setup provides direct access to both the
band dispersion and the geometry of the Bloch wavefunctions throughout the
entire Brillouin zone allowing us to extract the winding number for any
possible configuration. Finally, we highlight a topological phase transition
solely driven by a time-reversal-breaking synthetic gauge field associated
with the phase of the long-range hopping, providing a route for engineering
topological bands in photonic lattices belonging to the AIII symmetry class.

Introduction – Engineering materials with specific topological properties
requires an acute control over the hybridization of electronic orbitals
Bradlyn  _et al._ (2017). As was pioneered by the Haldane model Haldane
(1988), introducing next-nearest-neighbor coupling terms with arbitrary phases
strongly enriches the variety of phenomena that can be observed in topological
band models Chiu  _et al._ (2016); Bansil  _et al._ (2016). Furthermore, such
long-range connectivity is expected to facilitate the stabilization of
strongly correlated states of matter Kapit and Mueller (2010).

The experimental implementation and control of sizable hopping terms extending
beyond nearest neighbors is typically not a straightforward task Landig  _et
al._ (2016). In usual realizations of lattice models based on condensed matter
or ultracold atomic systems, hopping typically occurs via tunneling processes
mediated by the spatial overlap of wavefunctions at different sites and is
therefore dominated by short-range processes Jaksch  _et al._ (1998); Bloch
_et al._ (2012).

The situation is very different if we consider lattices extending along
synthetic dimensions Ozawa and Price (2019). Here, one or more of the spatial
coordinates are replaced by some other internal degrees of freedom such as
spin or linear momentum in ultracold atomic gases Celi  _et al._ (2014);
Chalopin  _et al._ (2020); An  _et al._ (2021), or, in photonic systems,
frequency Ozawa  _et al._ (2016); Dutt  _et al._ (2019), angular momentum Luo
_et al._ (2017); Cardano  _et al._ (2017), spatial Lustig  _et al._ (2019), or
temporal modes Regensburger  _et al._ (2011). In the specific case of
synthetic photonic lattices, a wide variety of hopping terms can be engineered
through suitable modulation of the relevant degrees of freedom Lin  _et al._
(2018), which has led to pioneering achievements such as the observation of
the four-dimensional quantum Hall effect Zilberberg  _et al._ (2018), the non-
Hermitian skin effect Weidemann  _et al._ (2020); Xiao  _et al._ (2020), non-
abelian excitations Wang  _et al._ (2021), and treelike photonic networks
Senanian  _et al._ (2022).

In this Letter, we use the frequency of the photon modes in an optical fiber
loop as a synthetic dimension Dutt  _et al._ (2019) to experimentally engineer
lattices with topological bands and tunable long-range hopping terms. In
particular, we realize one-dimensional dimerized lattices where the two sites
within each unit cell are encoded in the symmetric and antisymmetric
combinations of clockwise and counter-clockwise eigenmodes of a single loop.
Selective hopping processes between specific pairs of sites at arbitrary
distance are then introduced through a dynamical modulation of the optical
fiber at a frequency resonant with their frequency difference Dutt  _et al._
(2020a). This offers full control over the magnitude and the phase of the
hopping terms.

When only nearest-neighbor couplings are present, our lattice realizes the
well-known Su-Schrieffer-Heeger (SSH) model Su  _et al._ (1979), which
displays two topologically distinct phases associated with the integer-valued
winding number being $\mathcal{W}=0,1$. A wider variety of phases with
$\mathcal{W}$ ranging from $-1$ to $+2$ is realized by adding
$3^{\mathrm{rd}}$-nearest-neighbor hopping terms with specific amplitudes
Maffei  _et al._ (2018); D’Errico  _et al._ (2020); Vega  _et al._ (2021). The
entire phase diagram of this Hamiltonian is reconstructed in terms of the
winding number that we experimentally extract using a wavefunction tomography
technique for the Bloch modes. Finally, we report the generation of a time-
reversal-breaking synthetic gauge field, which allows to change the band
topology without modifying the strength of the couplings—a stringent
prerequisite in the conventional SSH model.

Figure 1: Description of the experimental setup for realizing topological
dimer chains in the synthetic photonic lattice. (a) Schematic representation
of a topological dimer chain. The unit cell is defined by the dotted
rectangle. (b)-(c) Topological phase diagrams presenting the value of the
winding number as a function of the different couplings for the SSH model (b)
and the dimer chain with long-range couplings (c). (d) Experimental setup: the
clockwise (CW, blue) and counter-clockwise (CCW, red) modes of an optical
fiber loop are coupled with a 75:25 optical fiber coupler (FC) at the center
of the loop. On top, circulators are used to separate CW and CCW modes and to
modulate them independently with a pair of electro-optical phase modulators
(EOMs). To optimize the quality factor of the loop, we use a semiconductor
optical amplifier (SOA) to compensate the losses. The system is probed by
means of an optical fiber acting as a transmission line and coupled to the
main loop with a 99:1 optical fiber coupler (FC). (e) Energy spectrum of the
loops in the uncoupled basis formed by a series of degenerate CW and CCW modes
separated by a free spectral range $\Omega$, and in the coupled basis of
symmetric and anti-symmetric modes $\ket{m,\pm}$ separated by a frequency
splitting $\delta$ and, again, a free spectral range $\Omega$. Figure 2:
Wavefunction tomography of the SSH model. (a)-(b) Band structure measurements
for the trivial (a) and topological phases (b) obtained by probing the
transmitted intensity of the cavity as a function of time for different values
of the laser detuning. Effective hopping terms normalized by $\eta$ are
provided above each panel (in units of $\text{\,}\mathrm{V}$). (c)-(d)
Trajectory of $g(k)$ in the complex plane for $k$ moving across the Brillouin
zone in the trivial and topological cases exhibiting a winding number
$\mathcal{W}=0$ and $1$, respectively. (e) Slow temporal modulation of the
amplitude of the transmitted intensity peaks associated with the lower band in
the trivial $\mathcal{W}=0$ (top) and topological $\mathcal{W}=1$ (bottom)
cases. Vertical dashed lines indicate the periodicity of this modulation.

The topological model – We consider in this work one-dimensional dimerized
chains that present chiral symmetry, i.e. where the two atoms in each unit
cell are identical and hopping processes only connect atoms in different
sublattices (see Fig. 1 (a)). Under the tight-binding approximation, the
generic Hamiltonian describing such lattices is given by:

$H(k)=\left(\begin{array}[]{cc}0&g(k)\\\ g^{*}(k)&0\end{array}\right),$ (1)

where the off-diagonal term $g(k)=|g(k)|e^{i\phi(k)}$ describes the hopping
terms in Fourier space. Diagonalization of this Hamiltonian gives a band
dispersion $E_{\pm}(k)=\pm|g(k)|$, and a phase difference $\phi(k)$ for the
components of the Bloch modes on the two sublattices:

$\ket{k_{\pm}}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\ \pm
e^{-i\phi(k)}\end{bmatrix}.$ (2)

In this framework, the topological phases are characterized by plotting the
trajectory of $g(k)$ in the complex plane throughout the Brillouin zone. The
number of times the trajectory winds around the origin as $k$ spans the
Brillouin zone is called the winding number $\mathcal{W}$ and is linked to the
number of edge states present at the boundaries of the lattice. Although the
winding number depends on the definition of the unit cell and is thus ill-
defined for infinite lattices, it can be identified unambiguously for finite
lattices where the definition of the unit cell is imposed by how the chain is
terminated.

The simplest such configuration, the SSH model, has (potentially complex)
nearest-neighbor hopping amplitudes only: $g(k)=a+b^{*}e^{+ikl}$ with $l$ the
lattice constant. This model hosts a winding number $\mathcal{W}=0$ and $1$
for $|a|>|b|$ and $|a|<|b|$ respectively (Fig. 1 (b)). With the addition of
$3^{\mathrm{rd}}$-nearest-neighbor hopping terms to the SSH Hamiltonian, the
off-diagonal term becomes:

$g(k)=a+b^{*}e^{+ikl}+ce^{-ikl}+d^{*}e^{+2ikl}.$ (3)

As a non-trivial consequence of these longer-range couplings, the winding
number can now take values ranging from -1 to +2 upon varying the ratios
between the different hopping amplitudes, see Fig. 1 (c) for the cases where
$a=b^{*}$, and ratios $c/a$ and $d^{*}/a$ are real, equivalent to a model with
real hopping parameters after a gauge transformation $g(k)\to
e^{-i\phi_{a}}g(k)$ ($\phi_{a}=\mathrm{arg}(a)$). This choice reproduces the
phase diagram from Ref. Maffei  _et al._ , 2018. Our aim is to build a
photonic platform that allows us to simulate arbitrary dimer-chain
configuration, even including complex $c/a$ and $d^{*}/a$ ratios, and to
extract $\mathcal{W}$ by measuring $g(k)$ throughout the Brillouin zone.

The synthetic photonic lattice– In our experiment, we use the frequency of
photons confined in an optical fiber loop as a synthetic dimension. The
underlying principle of this approach, inspired by Refs. Ozawa  _et al._
(2016); Dutt  _et al._ (2019), is to emulate the spatial periodicity of a
lattice by exploiting the periodicity in frequency of the cavity spectrum with
a period given by the free spectral range (FSR) $l=\Omega$. Coupling between
specific eigenmodes is realized by locally modulating the refractive index of
the cavity material with electro-optic phase modulators (EOMs) driven at a
frequency equal to the corresponding mode spacing.

In order to create the alternating hopping terms of a dimer chain, it is
necessary to engineer a cavity with two different frequency splittings and to
drive them independently. To do this, we use a single fiber loop and couple
the degenerate clockwise (CW) and counter-clockwise (CCW) eigenmodes using a
75:25 optical fiber coupler (see Fig. 1 (d)). The resulting hybridized modes
are symmetric and antisymmetric superpositions of the CW and CCW modes:
$\ket{m,\pm}=\frac{1}{\sqrt{2}}\left(\ket{m,CCW}\pm\ket{m,CW}\right)$, where
$m$ is the index of the uncoupled modes. In our setup, the splitting between
$\ket{m,\pm}$ is $\delta/2\pi=3.43~{}\mathrm{MHz}$ and the FSR is
$\Omega/2\pi=10.03~{}\mathrm{MHz}$ (Fig. 1 (e)).

In order to further optimize the coupling efficiency between these eigenmodes,
we use a pair of circulators that spatially separate the CW and CCW modes
allowing for an independent modulation of each of them. In particular, by
driving the EOMs with the same electrical signal amplitude but with a $\pi$
phase shift, $V_{cw}(t)=-V_{ccw}(t)$, we maximize the coupling between states
of opposite parity while suppressing all other couplings Dutt  _et al._
(2020b). This choice enforces chiral symmetry by suppressing hopping processes
between sites belonging to the same sublattice.

Figure 3: Topological phases of dimer chains with long-range hopping
amplitudes. Measurements (top row) and numerical simulations (bottom row) of
the band structure (left column) and of the trajectory of $g(k)$ (central
column) for dimer chains with long-range hopping amplitudes. Each panel shows
results for each possible value of the winding number: $\mathcal{W}=-1$ (a),
$\mathcal{W}=0$ (b), $\mathcal{W}=+1$ (c) and $\mathcal{W}=+2$ (d). On the
right side of each panel, we reproduce the phase diagram of Fig. 1 (c), with a
star indicating in which region of the phase diagram each measurement is
realized, and provide the effective hopping amplitudes, normalized by $\eta$
(in units of $\text{\,}\mathrm{V}$).

SSH lattices – As a first application of our scheme, we realize an SSH
Hamiltonian by driving the EOMs with a bichromatic signal
$V_{ccw/cw}(t)=\pm\frac{i}{2}(V_{a}e^{-i\delta
t}+V_{b}e^{-i(\Omega-\delta)t})+\textrm{c.c.}$ where $\pm$ accounts for the
$\pi$ phase shift between the two modulators on the CCW and CW paths. This
modulation gives rise to effective hopping terms $a=i\eta V_{a}$ and
$b^{*}=i\eta V_{b}^{*}$, where $\eta$ is related to the electro-optical
constant and has units of
$$\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$~{}$\text{\,}\mathrm{s}$^{-1}$\text{\,}\mathrm{V}$^{-1}$
Sup .

To measure the band dispersion, we probe the time-resolved transmission of the
cavity using a high-bandwidth photodiode while scanning the frequency of a
continuous-wave excitation laser. When the laser is in resonance with a
synthetic Bloch mode $\ket{k_{\pm}}$ and the cavity decay rate $\gamma$ is
much smaller than the Bloch bandwidth, the transmitted field intensity
consists of a train of narrow pulses with an overall intensity modulation of
frequency $\delta$:

$\frac{I_{(\pm)}(t)}{|F|^{2}}=\left[1-\frac{\kappa}{\gamma}\left(\frac{2\pi}{\Omega}\right)D_{T}(t-k)\left(1\pm\mathrm{cos}(\delta
t-\phi(k)\right)\right],$ (4)

where $F$ is the input field amplitude, $\kappa$ is the input-output coupling
strength, and $D_{T}(t)=\sum_{n}\,\delta(t-nT)$ is a Dirac comb with period
$T=2\pi/\Omega$. Interestingly, in our synthetic-dimension scheme, the
effective crystal momentum $k$ has units of time and corresponds to the pulse
arrival time Piccioli  _et al._ (2022) and $T$ is the size of the effective
Brillouin zone. A detailed derivation of Eq. (4) and a discussion of the
underlying assumptions can be found in the supplement, Ref. Sup .

The slow modulation arises from the fact that we measure light transmitted
from the CCW mode and the Bloch eigenmodes consist of linear superpositions of
symmetric and anti-symmetric combinations of CW and CCW modes that oscillate
at different frequencies. Hence, the signal exhibits a beating at the
frequency difference $\omega_{-}-\omega_{+}=\delta$. Importantly, the phase of
this modulation is exactly the relative phase of the sublattice amplitudes in
the Bloch modes, allowing experimental access to the phase of the wavefunction
in Eq. (2) at every $k$ point.

Figures 2 (a) and (b) show the transmitted intensity as a function of time
(averaged over multiple Brillouin zones to erase the effect of the modulation)
and as a function of the laser detuning for the cases $|a|>|b|$ and $|a|<|b|$,
corresponding to the trivial and topological phases of the SSH model,
respectively. In both cases, we clearly observe the two bands of the SSH
Hamiltonian with a well-defined gap, on the order of $E_{g}\sim
200~{}\mathrm{kHz}$ for the parameters of the experiment.

The different topology of these two cases can be highlighted by probing the
trajectory of $g(k)$ in the complex plane: $|g(k)|$ is extracted from the
measurement of the band structure, and the phase $\phi(k)=\textrm{arg}(g(k))$
is obtained by Fourier transforming the slow modulation of the output signal
Sup . Figure 2 (e) reports this slow modulation, as a function of time and
$k$-vector, for both the trivial and topological phases. Extracting the phase
of this modulation at every $k$, we can track the trajectory of $g(k)$
throughout the entire Brillouin zone (Fig. 2 (c)-(d)), which winds around the
origin in the topological case (d) but not in the trivial case (c).

Generalized SSH lattices– Having demonstrated the ability to extract both the
band structure and the Bloch wavefunctions for SSH chains, we now examine the
impact of $3^{\mathrm{rd}}$-nearest-neighbor hopping amplitudes. In our
platform, long-range hopping processes are straightforwardly implemented by
adding appropriate higher-frequency components to the signal sent to the EOMs:

$\begin{split}V_{ccw,cw}(t)=&\pm\frac{i}{2}(V_{a}e^{-i\delta
t}+V_{b}e^{-i(\Omega-\delta)t}\\\
&+V_{c}e^{-i(\Omega+\delta)t}+V_{d}e^{-i(2\Omega-\delta)t})+\textrm{c.c.},\end{split}$
(5)

where $c=-i\eta V_{c}$ and $d^{*}=-i\eta V_{d}^{*}$ Sup . Here we consider
cases with $a=b^{*}$ and only the ratios $c/a$ and $d^{*}/a$ are changed.
Keeping these ratios real, this allows exploring the full phase diagram
presented in Fig. 1 (c).

Figure 3 presents the experimental data for different ratios of long-range to
nearest-neighbor hopping amplitudes. Each panel presents a specific case
corresponding to one of the possible values of the winding number in the
topological phase diagram: $\mathcal{W}=-1$ (panel a), $\mathcal{W}=0$ (b),
$\mathcal{W}=+1$ (c) and $\mathcal{W}=+2$ (d). On the right of each panel, we
indicate with a red star the position in the phase diagram of Fig. 1 (c) and
we give the effective hopping amplitudes used in the experiment. The
experimental measurements of the band structure (top left of each panel) and
of the trajectory of $g(k)$ (top center) are obtained in a similar fashion as
for the previous SSH case. Finally, the bottom row presents tight-binding
calculations of the band structure and the $g(k)$ trajectory using the same
hopping ratios as in the experiment. We observe an excellent agreement between
measurements and calculations, which validates our approach to unambiguously
identify the winding number, including the full trajectory of $g(k)$.

Figure 4: Gauge-field induced topological phase transition. (a) Schematic
representation of a topological dimer chain with $a=b^{*}$, $c=0$, and
complex-valued $d^{*}/a$. The two sublattices are identified with $+$ and $-$
as in Fig. 1 (e). (b) Schematic representation of the trajectory in the
topological phase diagram followed by the Hamiltonian as the phase of the
$3^{\mathrm{rd}}$-nearest-neighbor hopping amplitudes ($\theta$) is scanned.
(c)-(e) Measurements of the band structure (top left) and $g(k)$ trajectory
(top right) as the phase of the long-range hopping is tuned from $\theta=0$
(c), to $\theta=\pi/2$ (d), and $\theta=\pi$ (e). The bottom row presents the
corresponding theoretical calculation of the band structure (left) and $g(k)$
(right). For each panel, the effective hopping terms normalized by $\eta$ are
provided (in units of $\text{\,}\mathrm{V}$).

Time-reversal-breaking topological phases – Finally, we demonstrate how the
phases of the $3^{\mathrm{rd}}$-nearest-neighbor couplings can be tuned to
break time-reversal symmetry. Adjusting these phases can induce a topological
phase transition without modifying the hopping strengths. This is in sharp
contrast with the conventional SSH model where it is necessary to adjust the
coupling strengths to change $\mathcal{W}$, because any non-trivial phase of
the hopping parameters can always be gauged away. This is a consequence of the
fact that these dimer chains with broken time-reversal symmetry belong to the
$AIII$ topological class rather than the $BDI$ class to which the SSH model
and its extensions with time-reversal belong.

The relative phase between the different hopping terms of the Hamiltonian is
experimentally realized by adding a phase to $V_{a,b,c,d}$ in Eq. (5). The
relative phase between one of the long-range couplings and the other couplings
effectively induces a synthetic gauge field for photons Dalibard  _et al._
(2011); Ozawa and Carusotto (2017). This is best seen by reformulating the
dimerized chain as a ladder-like lattice (see Fig. 4 (a)). In this equivalent
picture, one clearly sees how the phase of $d=|d|e^{-i\theta}$ gives rise to
flux plaquettes similar to those in ladder models with a staggered magnetic
field. This gauge field provides an additional degree of freedom to explore a
3-dimensional topological phase space that extends the phase diagram presented
in Fig. 1 (c) to the case of complex-valued $d^{*}/a$ ratios. A similar
argument can of course be used for the $c$ couplings.

An example of a trajectory in this extended phase space is schematically
depicted in Fig. 4 (b), where the orange line goes through points where the
magnitude of all hopping amplitudes remains constant, but the phase of
$d^{*}/a$ evolves from $0$ to $\pi$. Figs. 4 (c)-(e) present the measured
(top) and calculated (bottom) band structures (left) and trajectories of
$g(k)$ (right) along this path. Once again, except for the small deviations
observed in Figs. 4(d,e) near gap closures due to the finite decay rate
$\gamma$, an overall excellent agreement is found between the experiment and
the theory. In particular, the breaking of time-reversal induced by the gauge
field leads to a clear asymmetry of the band structure (i.e. $E(k)\neq
E(-k)$). Moreover, as the gauge field is increased, we observe a closing and
re-opening of the energy gap describing a topological phase transition. This
transition is clearly seen through the measurement of the trajectory of $g(k)$
whose winding number changes from 2 to 1.

Conclusions– In this work, we have exploited a synthetic dimension scheme
based on an optical fiber loop to realize a generalized dimer chain model
including long-range and/or time-reversal breaking hopping terms. Experimental
signatures of the non-trivial band topology have been obtained by
reconstructing the geometry of the Bloch wavefunctions throughout the entire
Brillouin zone. The natural next step will be to relate this microscopic
characterization of the topology to macroscopic observables such as a driven-
dissipative version of the mean chiral displacement Villa (2022); Maffei  _et
al._ (2018); Ozawa  _et al._ (2016) and edge states in the presence of
frequency-space potentials, as proposed in Ozawa  _et al._ (2016) and
pioneered in Dutt  _et al._ (2022). Setting up these tools will be
instrumental in enabling investigations of more complex Hamiltonians in
synthetic dimensions, involving non-Hermiticity Gong  _et al._ (2018); Wang
_et al._ (2021), higher dimensions Cheng  _et al._ (2023), quantum states of
light Bello  _et al._ (2019); Kim  _et al._ (2021), non-Markovian dynamics
Ricottone  _et al._ (2020), and/or optical nonlinearities Ozawa and Carusotto
(2017); Pernet  _et al._ (2022).

Note added– During the writing of this letter, this recent work Li  _et al._
(2023) demonstrating the extraction of the Zak phase of SSH lattices in the
synthetic frequency dimension came to our attention.

###### Acknowledgements.

PSJ acknowledges financial support from Québec’s Minstère de l’Économie, de
l’Innovation et de l’Énergie. PSJ and WAC acknowledge financial support from
the Fonds de Recherche–Nature et Technologies (FRQNT) and from the Natural
Sciences and Engineering Research Council (NSERC). FP acknowledges financial
support from FRQNT and RH from MITACS. IC acknowledges continuous
collaboration with Tomoki Ozawa and Greta Villa on this topic. IC acknowledges
financial support from the Provincia Autonoma di Trento and from the Q@TN
initiative.

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##  I Supplementals: Wavefunction spectroscopy in topological dimer chains
with long-range hopping

###  I.1 Description of the experimental method

####  I.1.1 Setup

The experimental setup is shown schematically in figure 5. All components are
polarization maintaining and an in-line polarizer is added inside the loop to
further ensure proper alignment of the propagating electric field polarization
with the optical axis of the electro-optics phase modulators (EOMs). The
cavity free spectral range (FSR) is 10.03 MHz. The staggered spectrum of the
cavity, necessary to emulate dimer chains, is obtained by coupling the
originally degenerate clockwise and counter-clockwise modes of the loop with a
25:75 fiber coupler. The hybridization of these modes lifts their degeneracy,
giving rise to symmetric and anti-symmetric supermodes split by 3.43 MHz. Each
of these pairs of symmetric and anti-symmetric supermodes forms a unit cell
(i.e. a dimer) in the synthetic frequency dimension.

Figure 5: Schematic description of the experimental setup.

Since we are interested in Hamiltonians with chiral symmetry, i.e. where the
hopping between sites on same sublattice vanishes, we must impose a coupling
mechanism that changes the parity of the supermodes. This is achieved by
spatially separating the CW and CCW propagating fields using two circulators
and modulating each path independently with a dedicated EOM. The two EOMs are
driven with the same electrical signal, but with a $\pi$ phase shift that
ensures that symmetric supermodes only couple to anti-symmetric ones and vice-
versa. Explicitly, to emulate dimer chains with long-range hoppings of
amplitudes, $a$, $b$, $c$ and $d$ (as explicited in Fig. 1 (a) of the main
text), the voltages sent to the two EOMs are respectively:

$\displaystyle V_{ccw}(t)=\frac{i}{2}\left[V_{a}e^{-i\delta
t}+V_{b}e^{-i(\Omega-\delta)t}+V_{c}e^{-i(\Omega+\delta)t}+V_{d}e^{-i(2\Omega-\delta)t}\right]+c.c.$
(6) $\displaystyle V_{ccw}(t)=-V_{cw}(t)$

where $V_{a,b,c,d}$ are the voltage amplitudes associated to couplings
$\\{a,b,c,d\\}$ respectively (see subsection below for a derivation of the
exact relationships). Each amplitude can be complex, with the argument
$\phi_{a,b,c,d}$ describing the phase of the corresponding hopping
coefficient. In our setup, the different frequency components are
$f_{a}=\omega_{a}/2\pi=3.43$ MHz, $f_{b}=6.60$ MHz, $f_{c}=13.46$ MHz,
$f_{d}=16.63$ MHz.

A 1:99 input-output fiber coupler is used for realizing transmission
measurements. The transmitted intensity is measured using an InGaAs photodiode
with a bandwidth of $10\text{\,}\mathrm{GHz}$ connected to a
$2\text{\,}\mathrm{GHz}$ oscilloscope. The laser is a Grade 3 Rio Orion laser
with a central wavelength of $1542.27\text{\,}\mathrm{nm}$ and a linewidth of
$3.1\text{\,}\mathrm{kHz}$. The frequency of the laser is modulated at
$40\text{\,}\mathrm{Hz}$, covering a range of more than
$30\text{\,}\mathrm{MHz}$ around the central emission frequency, so to span
several FSR. The two lithium-niobate EOMs from iXblue have a bandwidth of
$150\text{\,}\mathrm{MHz}$ and low insertion losses. Two circulators ensure
that clockwise and counter-clockwise propagating fields are dispatched to EOM
1 and 2, respectively. A semiconductor optical amplifier from Thorlabs
compensates losses from the EOMs and the other components in order to achieve
a high quality factor. An optical filter with a bandwith of 100 GHz is added
to the loop to suppress amplification of modes far from the input laser
frequency.

####  I.1.2 Technique

To obtain the band structure for a given set of parameters $a$, $b$, $c$, $d$,
$\phi_{c}$, and $\phi_{d}$, the laser frequency is modulated by a staircase
triangular waveform of frequency $40\text{\,}\mathrm{Hz}$ and amplitude
$0.25\text{\,}\mathrm{V}$, in which each step has a duration of
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$. On a given step, the laser
frequency is constant. The electrical signal sent to the EOMs (Eq. (6)) is
divided in bursts of $10\text{\,}\mathrm{\SIUnitSymbolMicro s}$ that are
synchronised with the rising edge of every step (see figure 6). For each step,
the signal collected by the photodiode is sent to the oscilloscope: the first
half of this signal (i.e. the first $5\text{\,}\mathrm{\SIUnitSymbolMicro s}$)
is discarded to allow the system to stabilize, and the second half is sliced
in fifty Brillouin zones (one Brillouin zone has a duration
$1/$10.03\text{\,}\mathrm{MHz}$\approx$100\text{\,}\mathrm{ns}$$). Each
horizontal slice of the band structures presented in the main text is obtained
by averaging the signal from these fifty Brillouin zones at every step. This
averaging process ensures that the band structures do not exhibit fringes
associated to the beating modulation of the Dirac comb discussed in the main
text.

Furthermore, for each step, we extract the phase of $g(k)$ at every $k$-point
with the following protocol. Using the band structures measure above, we
determine at what laser detuning a specific time bin (i.e. a specific column
of the band structure corresponding to a given $k$) presents the maximum
transmission. We take the photodiode signal measured at that specific laser
detuning step and keep only the data at that time bin in each Brillouin zone.
This gives a slowly oscillating signal at $3.43\text{\,}\mathrm{MHz}$
(examples of these signals are provided in Fig. 2 (e) of the main text) that
we can Fourier transform to extract the phase. This gives the phase of every
point in the $g(k)$ trajectories presented in the main text.

Figure 6: Example of the electrical signal (in mV) sent to the laser for
adjusting its emission frequency (red line) and to the EOMs for modulating the
loop (blue line).

####  I.1.3 Calibration

##### Laser frequency characterisation

In our system, the frequency of the laser is tuned by applying a time-varying
voltage to the laser. One then needs to perform a careful calibration to link
the applied voltage and the output frequency of the laser. The general
procedure to do this is to consider the simplest possible configuration, i.e.
a single loop with only one EOM and no CW-CCW fiber coupler; the spectrum in
this case is a regular cavity spectrum with equidistant modes. The FSR of this
simple loop is estimated by varying the driving frequency of the EOM until the
measured transmission exhibit a well-defined and symmetric band structure. For
such a simple 1D lattice, the band structure must follow a cosine function and
a small deviation of the driving frequency from the FSR results in an
effective electric field and distorted bands (associated to Bloch
oscillations). Once the driving frequency of the EOM leads to such symmetric
bands, we have a precise estimation of the FSR and we can link the applied
voltage to the laser frequency by doing a power-law fit on the many resonance
peaks in the transmission that occur during half a period of the laser’s
voltage modulation. This calibration is then used to determine the frequency
axis of all our band structures.

##### Quality factor

The quality factor of the cavity is estimated with the ratio between the the
resonance linewidth $\Delta\nu$ and the resonance frequency $\nu_{r}$. A
lorentzian fit yields a FWHM of $\Delta\nu=$0.075\text{\,}\mathrm{MHz}$$. The
quality factor is therefore
$Q=$194.4\text{\,}\mathrm{THz}$/$0.075\text{\,}\mathrm{MHz}$=2.6\times 10^{9}$
and the finesse is $\mathcal{F}=133.7$.

##### EOM signals used

We present below the amplitude and phase of the voltage signal sent to the CCW
EOMs for every experiement present in this work. We use the formalism:
$V_{i}=|V_{i}|e^{i\phi_{i}}$ with $i=\\{a,b,c,d\\}$.

Figure |  Winding |  $|V_{a}|$ ($\phi_{a})$ |  $|V_{b}|$ ($\phi_{b})$ |  $|V_{c}|$ ($\phi_{c})$ |  $|V_{d}|$ ($\phi_{d})$
---|---|---|---|---|---
Figure 2 (SSH) |  $\mathcal{W}=0$ |  $1.33\text{\,}\mathrm{V}$ (0) |  $0.91\text{\,}\mathrm{V}$ (0) |  $0.00\text{\,}\mathrm{V}$ (0) |  $0.00\text{\,}\mathrm{V}$ (0)
$\mathcal{W}=1$ |  $0.86\text{\,}\mathrm{V}$ (0) |  $1.37\text{\,}\mathrm{V}$ (0) |  $0.00\text{\,}\mathrm{V}$ (0) |  $0.00\text{\,}\mathrm{V}$ (0)
Figure 3 (Extended) |  $\mathcal{W}=-1$ |  $0.48\text{\,}\mathrm{V}$ (0) |  $0.48\text{\,}\mathrm{V}$ (0) |  $1.47\text{\,}\mathrm{V}$ (0) |  $0.39\text{\,}\mathrm{V}$ (0)
$\mathcal{W}=0$ |  $0.86\text{\,}\mathrm{V}$ (0) |  $0.86\text{\,}\mathrm{V}$ (0) |  $0.37\text{\,}\mathrm{V}$ (0) |  $0.52\text{\,}\mathrm{V}$ ($\pi$)
$\mathcal{W}=1$ |  $0.95\text{\,}\mathrm{V}$ (0) |  $0.95\text{\,}\mathrm{V}$ (0) |  $0.41\text{\,}\mathrm{V}$ ($\pi$) |  $0.23\text{\,}\mathrm{V}$ (0)
$\mathcal{W}=2$ |  $0.57\text{\,}\mathrm{V}$ (0) |  $0.57\text{\,}\mathrm{V}$ (0) |  $0.00\text{\,}\mathrm{V}$ ($\pi$) |  $1.24\text{\,}\mathrm{V}$ ($\pi$)
Figure 4 (Gauge-field) |  $\mathcal{W}=2$ |  $0.48\text{\,}\mathrm{V}$ (0) |  $0.48\text{\,}\mathrm{V}$ (0) |  $0.41\text{\,}\mathrm{V}$ ($\pi$) |  $1.09\text{\,}\mathrm{V}$ ($\pi$)
Transition |  $0.48\text{\,}\mathrm{V}$ (0) |  $0.48\text{\,}\mathrm{V}$ (0) |  $0.41\text{\,}\mathrm{V}$ ($\pi$) |  $1.09\text{\,}\mathrm{V}$ ($\pi/2$)
$\mathcal{W}=1$ |  $0.47\text{\,}\mathrm{V}$ (0) |  $0.47\text{\,}\mathrm{V}$ (0) |  $0.41\text{\,}\mathrm{V}$ ($\pi$) |  $1.09\text{\,}\mathrm{V}$ (0)

###  I.2 Theoretical model

In this section we analytically derive the effective Hamiltonian of the system
and its equations of motion.

####  I.2.1 Derivation of the dimer chains’ Hamiltonian

Our system consists in an optical fiber loop acting as a cavity. The modes of
this cavity are discrete and equally spaced by a free-spectral range $\Omega$,
defined as $\frac{\Omega}{2\pi}=\frac{c}{nL}$ where $n\sim 1.45$ and
$L\sim$10\text{\,}\mathrm{m}$$ the group index and length of our loop, and $c$
is the speed of light in vacuum. These modes can propagate either in the
clockwise (CW) or counterclockwise (CCW) direction and have thus two degrees
of freedom: their frequency $\omega_{m}=m\Omega$ and their direction of their
propagation (either CW or CCW).

Along the optical fiber the electric field is confined in one dimension, and
can thus be decomposed as a linear combination of CW and CCW modes:

$\mathbf{E}(x,t)=\sum_{m}\alpha_{m}e^{-ik_{m}x}e^{-i\omega_{m}t}+\beta_{m}e^{ik_{m}x}e^{-i\omega_{m}t}+h.c.$
(7)

where $x$ is a periodic coordinate ($x=x+L$) defined along the optical fiber,
$\alpha_{m}$ and $\beta_{m}$ are respectively the amplitudes of the $m$-th CW
and CCW modes, and $k_{m}=m\Omega/c^{\prime}>0$ with $c^{\prime}=c/n$.

Following the usual procedure of quantization of the field, we associated
$\alpha_{m}$ and $\alpha^{*}_{m}$ to bosonic operators $a_{m}$ and
$a^{{\dagger}}_{m}$, and similarly $\beta_{m}$ and $\beta^{*}_{m}$ to $b_{m}$
and $b^{{\dagger}}_{m}$. This quantum notation is useful to derive the
equations of motions but one should keep in mind that the modes considered in
our experiment are classical coherent states.

Figure 7: (a) Schematic representation describing the $x$ coordinate used for
deriving the beams plitter Hamiltonian. (b) Schematic representation of the 4
terms in the beam splitter Hamiltonian in Eq. 8.

The degeneracy between the CW and CCW modes is lifted by coupling them with a
25:75 optical fiber coupler acting as a beam splitter (BS). The definition of
the position coordinates $x$ along the cavity used in the derivation of this
beam splitter Hamilotnian is schematically depicted in Fig. 7 (a). The $x$
axis is periodic, oriented along the CCW direction and follows the outer fiber
(i.e. the path followed by the fiber without the CW/CCW coupler). It first
crosses $d_{1}$ at the bottom input-output ports of the CW/CCW coupler, then
$d_{2}$ at the EOMs’ position, and finally $d_{3}$ at the position of the top
input-output ports of the CW/CCW coupler before reaching the input-output port
of the cavity at $x=L$.

The BS can destroy a photon in the CW $m$-th mode (respectively CCW $m$-th
mode) entering the BS at the position $x=d_{3}$ and reinjects it at the
position $x=d_{1}$ in the CCW $m$-th mode (respectively CW $m$-th mode). It is
the same thing for photons entering the BS at $x=d_{1}$ and exiting at
$x=d_{3}$. All 4 possible processes are schematically depicted in Fig. 7 (b)
where the red and blue arrows respectively describe the CW ($a_{m}$) and CCW
modes ($b_{m}$). The momentum used in the phase factor of each modes are given
by $k_{ccw}=-k_{cw}=+\frac{m\Omega}{c^{\prime}}$.

The Hamiltonian describing the effect of the beam-splitter is given by:

$\displaystyle H_{BS}=-\sum_{m}\Bigl{[}$ $\displaystyle
g\,b_{m}e^{+im\Omega\frac{d_{1}}{c^{\prime}}}\,a^{{\dagger}}_{m}e^{+im\Omega\frac{d_{3}}{c^{\prime}}}+g^{*}\,a_{m}e^{-im\Omega\frac{d_{3}}{c^{\prime}}}\,b^{{\dagger}}_{m}e^{-im\Omega\frac{d_{1}}{c^{\prime}}}$
(8) $\displaystyle+$ $\displaystyle
g\,b_{m}e^{+im\Omega\frac{d_{3}}{c^{\prime}}}a^{{\dagger}}_{m}e^{+im\Omega\frac{d_{1}}{c^{\prime}}}+g^{*}\,a_{m}e^{-im\Omega\frac{d_{1}}{c^{\prime}}}\,b^{{\dagger}}_{m}e^{-im\Omega\frac{d_{3}}{c^{\prime}}}\Bigr{]}$

with g the coupling strength of the coupler. Taking into account that
$e^{i\Omega L/c^{\prime}}=1$ by definition of free-spectral range, we can
rewrite this Hamiltonian as:

$H_{BS}=-2\sum_{m}\left[g^{*}a_{m}b^{{\dagger}}_{m}e^{-im\Omega\frac{\Delta}{c^{\prime}}}+gb_{m}a^{{\dagger}}_{m}e^{+im\Omega\frac{\Delta}{c^{\prime}}}\right]$
(9)

with $\Delta=L-d_{1}-d_{3}$ the difference between the total length of the
loop $L$ and $d_{1}+d_{3}$. If the CW/CCW coupler is located symmetrically to
the input-output coupler (i.e. at the same distance in the $+x$ and $-x$
directions), $\Delta=0$. However, this is not true in general, either because
of thermal fluctuations or due to the difference in fiber lengths. We absorb
the influence of $\Delta$ in the definition of the coupling coefficient:

$ge^{im\Omega\frac{\Delta}{c^{\prime}}}=\tilde{g}.$ (10)

Hence, we can write $H_{BS}$ in a matrix form for the $m^{th}$ pair of modes
as:

$H^{(m)}_{BS}=\begin{bmatrix}m\Omega&-2\tilde{g}\\\
-2\tilde{g}&m\Omega\end{bmatrix}$ (11)

where, with no loss of generality, we have assumed for simplicity that
$\tilde{g}$ is real and positive. The eigenstates are symmetric (+) and
antisymmetric (-) linear combinations of the CCW and CW modes:

$\ket{m,\pm}=\frac{1}{\sqrt{2}}\left(\ket{m,CCW}\pm\ket{m,CW}\right).$ (12)

With this choice for $\tilde{g}$, the eigenfrequencies are
$\omega_{m,\pm}=m\Omega\mp\frac{\delta}{2}$, with $\delta=4g$ proportional to
the coupling strength (in our case $\delta/2\pi=$3.43\text{\,}\mathrm{MHz}$$),
the lower (higher) frequency corresponding to the symmetric $+$ (anti-
symmetric $-$) combination of CW and CCW modes. We define similarly the
associated bosonic operators $c_{m,\pm},c^{{\dagger}}_{m,\pm}$:

$c_{m,\pm}=\frac{1}{\sqrt{2}}\left[b_{m}\pm a_{m}\right].$ (13)

Each of these super-modes acts as a lattice site along the synthetic frequency
dimension. In order to emulate Hamiltonians with chiral symmetry, it is
necessary to couple eigenmodes of opposite parity, i.e. symmetric with anti-
symmetric and vice-versa. This requires modulating the CW and CCW modes
independently; if they are driven with identical, in-phase electrical signals
only eigenmodes of similar parity will couple. As described in the main text,
this sublattice symmetry is achieved by spatially separating CW and CCW modes
with circulators and modulating them independently. The Hamiltonian describing
the effect of the modulators is given by:

$\displaystyle H_{EOM}$ $\displaystyle=H_{cw}+H_{ccw}$ (14)
$\displaystyle=2\eta\sum_{m,n}V_{cw}(t)a^{{\dagger}}_{m}a_{n}e^{2i\pi(n-m)\frac{d_{2}}{L}}+2\eta\sum_{m,n}V_{ccw}(t)b^{{\dagger}}_{m}b_{n}e^{2i\pi(n-m)\frac{d_{2}}{L}},$

where $d_{2}$ is the position of the modulators along the loop (in our case,
$d_{2}\sim L/2$) and $\eta$ is the electro-optical coupling coefficient which
can be approximated (for small eigenmode spacing $m-n$) Dutt  _et al._ (2019):

$\eta=\frac{\Omega}{4V_{\pi}},$ (15)

where $V_{\pi}\sim$2\text{\,}\mathrm{V}$$ is the switching voltage amplitude
of our modulators.

We impose $V_{ccw}(t)=-V_{cw}(t)=V(t)$ to minimize (maximize) coupling between
states of the same (opposite) parity. This reduces the Hamiltonian to:

$H_{EOM}=2\eta
V(t)\sum_{m,n}\left[(-1)^{n-m}b^{{\dagger}}_{m}b_{n}-(-1)^{n-m}a^{{\dagger}}_{m}a_{n}\right].$
(16)

We now express the $H_{EOM}$ with the symmetric and anti-symmetric mode
operators ($c_{m,\pm}$):

$H_{EOM}=2\eta
V(t)\sum_{m,n}(-1)^{n-m}\left[c^{{\dagger}}_{m,+}c_{n,-}+c^{{\dagger}}_{m,-}c_{n,+}\right].$
(17)

Note that the dependence on $\Delta$ has vanished.

The driving voltage $V(t)$ sent to the EOM has a sinusoidal form as given in
the main text:

$V(t)=\frac{i}{2}\left[V_{a}e^{-i\delta
t}+V_{b}e^{-i(\Omega-\delta)t}+V_{c}e^{-i(\Omega+\delta)t}+V_{d}e^{-i(2\Omega-\delta)t}\right]+c.c.,$
(18)

where $V_{a,b,c,d}$ are complex.

Injecting this expression of $V(t)$ in $H_{EOM}$ and neglecting non-resonant
terms in the rotating-wave approximation (RWA), we obtain:

$\tilde{H}_{EOM}=i\eta\sum_{m}\left[V_{a}\tilde{c}^{{\dagger}}_{m,-}\tilde{c}_{m,+}-V_{b}\tilde{c}^{{\dagger}}_{m,+}\tilde{c}_{m-1,-}-V_{c}\tilde{c}^{{\dagger}}_{m+1,-}\tilde{c}_{m,+}+V_{d}\tilde{c}^{{\dagger}}_{m,+}\tilde{c}_{m-2,-}\right]+h.c.,$
(19)

where the operators are expressed in a rotating frame:

$\tilde{c}_{m,\pm}=c_{m,\pm}e^{i\omega_{m,\pm}t}.$ (20)

In particular, note how in Eq.(19) one clearly sees a $\pi$ phase shift
between the nearest-neighbor and long-range couplings that needs to be
considered when defining $V(t)$.

Using the periodicity along the frequency axis, we can rewrite this
Hamiltonian in reciprocal space by defining the Bloch modes:

$d_{k,\pm}=\sum_{m}e^{-ikm\Omega}\tilde{c}_{m,\pm},$ (21)

where $k$ describes the crystal momentum along this synthetic dimension. In
this $k$-space, $H_{EOM}$ becomes:

$\tilde{H}_{EOM}=\sum_{k}H(k)=i\eta\sum_{k}d^{{\dagger}}_{k,-}d_{k,+}\left[V_{a}+V_{b}^{*}e^{ik\Omega}-V_{c}e^{-ik\Omega}-V_{d}^{*}e^{2ik\Omega}\right]+h.c..$
(22)

This Hamiltonian can be expressed in matrix form as:

$\tilde{H}_{EOM}=\sum_{k}\psi_{k}^{\dagger}\cdot h(k)\cdot\psi_{k}$ (23)
$h(k)=\begin{pmatrix}0&g(k)\\\
g^{*}(k)&0\end{pmatrix};\quad\psi_{k}=\begin{pmatrix}d_{k-}\\\
d_{k+}\end{pmatrix},$ (24)

where

$g(k)=|g(k)|e^{i\phi(k)}=i\eta\left[V_{a}+V_{b}^{*}e^{ik\Omega}-V_{c}e^{-ik\Omega}-V_{d}^{*}e^{2ik\Omega}\right].$
(25)

This is precisely the quantity that is plotted in the theory plots in Figs. 3
and 4 of the main text.

Direct comparison with the function $g(k)$ in the extended Hamiltonian
presented in Eq. (3) of the main text, yields the following relationships
between the hopping coefficients ($a,b,c,d$) and EOM voltages ($V_{a,b,c,d}$):

$\displaystyle a$ $\displaystyle=+i\eta V_{a}$ (26) $\displaystyle b^{*}$
$\displaystyle=+i\eta V_{b}^{*}$ $\displaystyle c$ $\displaystyle=-i\eta
V_{c}$ $\displaystyle d^{*}$ $\displaystyle=-i\eta V_{d}^{*}.$

Note the $\pi$ phase shift between the nearest-neighbor and long-range hopping
terms, as in the main text.

These are the relationships used for relating the voltage amplitudes to
effective coupling coefficients, or vice-versa, as presented in all the
figures of the main text. All the voltages used in this work are provided in
Table 1.

####  I.2.2 Derivation of the electromagnetic field time evolution and time-
resolved transmission

The evolution of the electromagnetic field confined in the optical fiber loop,
when driven with a CCW input field is given by the set of Langevin equations:

$\displaystyle\dot{a}_{m}$
$\displaystyle=i\left[H(t),a_{m}\right]-\frac{\gamma}{2}a_{m}$ (27)
$\displaystyle\dot{b}_{m}$
$\displaystyle=i\left[H(t),b_{m}\right]-\frac{\gamma}{2}b_{m}-i\sqrt{\kappa}s_{in}(t)$

where $H(t)=H_{BS}+H_{EOM}(t)$, $\gamma$ is the decay rate in the loop,
$\kappa$ is the input-output coupling strength and $s_{in}=Fe^{-i\omega_{L}t}$
is the input field.

Going to the coupled basis, this set of differential equations becomes, in the
rotating frame:

$\dot{\tilde{c}}_{m,\pm}=i\left[\tilde{H}_{EOM}(t),\tilde{c}_{m,\pm}\right]-\frac{\gamma}{2}\tilde{c}_{m,\pm}-i\sqrt{\frac{\kappa}{2}}e^{i\omega_{m,\pm}t}s_{in}.$
(28)

Then, transferring to $k$-space, we have:

$\dot{d}_{k,\pm}=i\left[H(k),d_{k,\pm}\right]-\frac{\gamma}{2}d_{k,\pm}-i\sqrt{\frac{\kappa}{2}}\sum_{m}e^{-ikm\Omega}e^{i\omega_{m,\pm}t}s_{in}.$
(29)

Finally, we can now go in the basis that diagonalizes $H(k)$:

$\displaystyle u_{k,\pm}=\frac{1}{\sqrt{2}}\left[d_{k,-}\pm
e^{i\phi(k)}d_{k,+}\right]$ (30)

with eigenenergies $\omega_{k,\pm}=\pm|g(k)|$ and $\phi(k)$ defined as the
phase of $g(k)$. In this basis, the equations of motion become:

$\dot{u}_{k,\pm}=\mp
i|g(k)|u_{k,\pm}-\frac{\gamma}{2}u_{k,\pm}-i\frac{\sqrt{\kappa}}{2}s_{in}\left[e^{+i\frac{\delta}{2}t}\pm
e^{-i\frac{\delta}{2}t+i\phi(k)}\right]\sum_{m}e^{im\Omega\left(t-k\right)}$
(31)

Integrating this last equation over time yields:

$u_{k,\pm}(t)=u_{k,\pm}(t=0)-i\frac{\sqrt{\kappa}}{2}\int_{0}^{t}dt^{\prime}e^{(\pm
i|g(k)|+\gamma/2)(t^{\prime}-t)}s_{in}(t^{\prime})\left[e^{+i\frac{\delta}{2}t^{\prime}}\pm
e^{-i\frac{\delta}{2}t^{\prime}}e^{i\phi(k)}\right]\sum_{m}e^{im\Omega\left(t^{\prime}-k\right)}$
(32)

where we can assume $u_{k,\pm}(t=0)=0$.

Under the change of variable $t^{\prime\prime}=t-t^{\prime}$, we can write
this integral as:

$u_{k,\pm}(t)=-i\frac{\sqrt{\kappa}}{2}\left[C_{1}^{(\pm)}(t)\pm
C_{2}^{(\pm)}(t)\right]$ (33)

where

$\displaystyle C_{1}^{(\pm)}(t)$
$\displaystyle=Fe^{-i\omega_{L}t}e^{+i\frac{\delta}{2}t}\sum_{m}\left[e^{im\Omega(t-k)}\int_{0}^{t}dt^{\prime\prime}e^{-\frac{\gamma}{2}t^{\prime\prime}}e^{i(\omega_{L}-(m\Omega+\frac{\delta}{2}\pm|g(k)|))t^{\prime\prime}}\right]$
(34) $\displaystyle C_{2}^{(\pm)}(t)$
$\displaystyle=Fe^{-i\omega_{L}t}e^{-i\frac{\delta}{2}t}e^{+i\phi(k)}\sum_{m}\left[e^{im\Omega(t-k)}\int_{0}^{t}dt^{\prime\prime}e^{-\frac{\gamma}{2}t^{\prime\prime}}e^{i(\omega_{L}-(m\Omega-\frac{\delta}{2}\pm|g(k)|))t^{\prime\prime}}\right]$

These integrals are straightforwardly solved. We can assume that the upper
bound to be much larger than the cavity’s lifetime, i.e. $t\gg\gamma^{-1}$. In
the experiment, this is realized by probing the field at times long enough to
let the field reach a steady-state (in our case several microseconds after the
start of each laser detuning step). This leads to:

$\displaystyle C_{1}^{(\pm)}(t)$
$\displaystyle=-Fe^{-i\omega_{L}t}e^{+i\frac{\delta}{2}t}\sum_{m}\left[e^{im\Omega(t-k)}\frac{1}{i(\omega_{L}-(m\Omega+\frac{\delta}{2}\pm|g(k)|))-\frac{\gamma}{2}}\right]$
(35) $\displaystyle C_{2}^{(\pm)}(t)$
$\displaystyle=-Fe^{-i\omega_{L}t}e^{-i\frac{\delta}{2}t}e^{i\phi(k)}\sum_{m}\left[e^{im\Omega(t-k)}\frac{1}{i(\omega_{L}-(m\Omega-\frac{\delta}{2}\pm|g(k)|))-\frac{\gamma}{2}}\right].$

In this form, $C_{1}^{(\pm)}(t)$ and $C_{2}^{(\pm)}(t)$ exhibits clear
resonances with a linewidth $\gamma/2$ every time the laser frequency reaches
respectively $\omega_{L}=m\Omega+\delta/2\pm|g(k)|$ and
$\omega_{L}=m\Omega-\delta/2\pm|g(k)|$. The interpretation of these periodic
resonance is that the band structure of the dimer chains repeats itself along
the synthetic frequency dimension at every eigenmode of the bare cavity.

Since the linewidths of the laser and of the bare modes are much smaller than
$\Omega$ and $\delta$, these expressions can be further simplified. For
instance, if we consider a drive frequency resonant with a band emerging from
the $\bar{m}^{th}$ symmetric mode (i.e.
$\omega_{L}\simeq\bar{m}\Omega-\delta/2$), we can simplify the equation for
$u_{k,\pm}$ by neglecting completely $C_{1}^{(\pm)}$ and all terms in the
summation in $C_{2}^{(\pm)}$ except the one associated to $\bar{m}$.

####  I.2.3 Input-output relations

Finally, in order to determine the temporal profile of the transmitted field,
we derive the field amplitude at the photodiode with an input-output relation
where only the CCW modes are coupled to the output port:

$\displaystyle s_{pd}$ $\displaystyle=s_{in}-i\sqrt{\kappa}\sum_{m}b_{m}$ (36)
$\displaystyle=s_{in}-i\sqrt{\frac{\kappa}{2}}\sum_{m}\left[\tilde{c}_{m,+}e^{-i(m\Omega-\frac{\delta}{2})t}+\tilde{c}_{m,-}e^{-i(m\Omega+\frac{\delta}{2})t}\right]$
$\displaystyle=s_{in}-i\sqrt{\frac{\kappa}{2}}\sum_{m,k}\left[d_{k,+}e^{-i(m\Omega-\frac{\delta}{2})t}e^{ikm\Omega}+d_{k,-}e^{-i(m\Omega+\frac{\delta}{2})t}e^{ikm\Omega}\right]$
$\displaystyle=s_{in}-i\frac{\sqrt{\kappa}}{2}\sum_{m,k}\left[(u_{k,+}-u_{k,-})e^{-i(m\Omega-\frac{\delta}{2})t}e^{ikm\Omega}e^{-i\phi(k)}+(u_{k,+}+u_{k,-})e^{-i(m\Omega+\frac{\delta}{2})t}e^{ikm\Omega}\right].$

As mentioned above, we assume the laser is in the vicinity of the
$\bar{m}^{th}$ symmetric mode so to only retain a single term in the
expression for $C_{1}^{(\pm)}(t)$. On top of this, we assume that the laser is
resonant with the $\pm$ Bloch band, so that all contributions from the other
band can be neglected. Furthermore, we consider that the laser resonantly
excites a single state in this band, whose wavevector $k$ is determined by a
resonance condition between the laser frequency and the band dispersion
$\omega_{k,\pm}$,

$\omega_{L}=\bar{m}\Omega-\frac{\delta}{2}\pm|g(k)|\,.$ (37)

Putting all together, this gives the final expression for the field amplitude:

$u_{k,\pm}(t)=\mp
i\frac{\sqrt{\kappa}}{\gamma}Fe^{-i\omega_{L}t}e^{-ik\bar{m}\Omega}e^{i(\bar{m}\Omega-\frac{\delta}{2})t}e^{i\phi(k)}\,$
(38)

Note that this approximation is only accurate when the gap
$\omega_{g}=|\omega_{k,+}-\omega_{k,-}|$ is larger than the linewidth
$\gamma$. When the gap is comparable to the linewidth, deviations appear in
the extracted phase of the field and a small kink in the trajectory of $g(k)$
(as observed in Fig. 4d of the main text).

The field at the photodiode is given by:

$s_{pd}^{(\pm)}=s_{in}\mp
i\frac{\sqrt{\kappa}}{2}\sum_{m}u_{k,\pm}e^{ikm\Omega}\left[e^{-i(m\Omega-\frac{\delta}{2})t}e^{-i\phi(k)}\pm
e^{-i(m\Omega+\frac{\delta}{2})t}\right].$ (39)

where we consider a single $k$ in resonance with the laser. We now inject
$u_{k,\pm}$ obtained above and obtain:

$s_{pd}^{(\pm)}=s_{in}-\frac{\kappa}{2\gamma}Fe^{-i\omega_{L}t}\sum_{m^{\prime}}e^{im^{\prime}\Omega(t-k)}\left[1\pm
e^{-i\delta t}e^{i\phi(k)}\right]$ (40)

with $m^{\prime}=\bar{m}-m$.

Then, using the identity

$D_{T}(t-a)=\frac{1}{T}\sum_{m}e^{i\frac{2\pi}{T}m(t-a)}$ (41)

where $D_{T}(t)$ is a Dirac comb of period $T$

$D_{T}(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT)\,,$ (42)

we obtain:

$s_{pd}^{(\pm)}=s_{in}-\frac{\kappa}{2\gamma}F\left(\frac{2\pi}{\Omega}\right)e^{-i\omega_{L}t}D_{T}(t-k)\left[1\pm
e^{-i\delta t}e^{i\phi(k)}\right]$ (43)

with $T=\frac{2\pi}{\Omega}$.

Hence the intensity measured on the photodiode for a mode $\ket{k,\pm}$ is
given by:

$I_{pd}^{(\pm)}=|F|^{2}\left[1-\frac{\kappa}{\gamma}\left(\frac{2\pi}{\Omega}\right)D_{T}(t-k)\left(1\pm\mathrm{cos}(\delta
t-\phi(k)\right)\right]+\mathcal{O}(u_{k}^{2})$ (44)

This clearly demonstrates that the transmission presents a pulse train,
associated to the Dirac comb with period $2\pi/\Omega$, which is modulated at
frequency $2\pi/\delta$ and with a phase $\phi(k)$. Measuring the position of
the Dirac comb within each single period at every laser detuning allows to
extract the band structure, whereas the phase of its slow modulation across
the different periods provides direct information on the relative phase of the
two sublattice components of the Bloch mode wavefunctions. These features are
is used in this work for extracting experimentally the Bloch wavefunction for
every $k$ point and, then, the winding number. This is going to be discussed
in the next Section.

For the sake of completeness, it is important to remind that this discussion
assumes that a single $k$ is on resonance with the laser. If more than one $k$
are resonant with the laser (e.g. $\pm k$ in systems with time-reversal
symmetry), $s_{pd}$ will consist of several pulse trains corresponding to the
different $k$’s. All these pulse trains will have the same period $T$, but
their temporal position will shifted in time according to their respective
values of $k$.

Note also that the shape of each pulse is exactly a $\delta$-function in the
limit of a small cavity linewidth $\gamma$. In the general case, the finite
$\gamma$ leads to a broadening of the pulses in time as visible in Fig.2(a,b).

###  I.3 Experimental extraction of $\phi(k)$

In order to obtain $\phi(k)$ from the experimental data, we perform the
following procedure. For the sake of concreteness, we focus on the lower Bloch
band for these calculations. First, we determine from our experimental band
structure (e.g. in Figs. 2 (a)-(b) of the main text), the laser detuning at
which each $k$ mode is excited. Then, for each $k$ point we perform the
following discrete Fourier transform using the intensity measured on the
photodiode for the laser detuning step corresponding to the chosen $k$ value:

$\mathcal{I}(k)=\sum_{n}e^{i\delta t_{n}^{(k)}}I_{pd}(t_{n}^{(k)})$ (45)

with $t_{n}^{(k)}=k+nT$. In our experiment,
$\delta=$3.43\text{\times}{10}^{6}\text{\,}\mathrm{MHz}$$. The discrete
Fourier transform is realized by considering only the time bins associated to
the specific $k$-point considered, i.e. $t=k+nT$. The phase $\phi(k)$ is
extracted by taking the argument of $\mathcal{I}(k)$:

$\phi(k)=\mathrm{arg}[\mathcal{I}(k)].$ (46)

The complete shape of $\phi(k)$ across the Brillouin zone is obtained by
repeating this procedure for every $k$.