# Quantized topological energy pumping and Weyl points in Floquet synthetic
dimensions with a driven-dissipative photonic molecule

Sashank Kaushik Sridhar  Department of Mechanical Engineering, University of
Maryland, College Park, Maryland 20742, USA  Sayan Ghosh  Department of
Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur, West Bengal 741246, India  Avik Dutt  Department of Mechanical
Engineering, University of Maryland, College Park, Maryland 20742, USA
Institute for Physical Science and Technology, University of Maryland, College
Park, Maryland 20742, USA

###### Abstract

Topological effects manifest in a wide range of physical systems, such as
solid crystals, acoustic waves, photonic materials and cold atoms. These
effects are characterized by ‘topological invariants’ which are typically
integer-valued, and lead to robust quantized channels of transport in space,
time, and other degrees of freedom. The temporal channel, in particular,
allows one to achieve higher-dimensional topological effects, by driving the
system with multiple incommensurate frequencies. However, dissipation is
generally detrimental to such topological effects, particularly when the
systems consist of quantum spins or qubits. Here we introduce a photonic
molecule subjected to multiple RF/optical drives and dissipation as a
promising candidate system to observe quantized transport along Floquet
synthetic dimensions. Topological energy pumping in the incommensurately
modulated photonic molecule is enhanced by the driven-dissipative nature of
our platform. Furthermore, we provide a path to realizing Weyl points and
measuring the Berry curvature emanating from these reciprocal-space
($k$-space) magnetic monopoles, illustrating the capabilities for higher-
dimensional topological Hamiltonian simulation in this platform. Our approach
enables direct $k$-space engineering of a wide variety of Hamiltonians using
modulation bandwidths that are well below the free-spectral range (FSR) of
integrated photonic cavities.

††preprint: APS/123-QED

Quantized transport is a hallmark of topological insulators (TIs). Initially
explored for electronic transport in condensed matter systems (i.e. quantized
Hall conductance [1, 2], Hamiltonians supporting nontrivial topology have now
been experimentally simulated in a wide variety of systems such as ultracold
atoms [3, 4], photonics [5], acoustics [6, 7] and topolectrical circuits [8].
In such simulators, the Hamiltonians are typically created by controlling the
coupling between a lattice of real-space sites encoded, for example, in large
arrays of atoms, photonic resonators or photonic waveguides [9, 10, 11].
Experiments have reported robust unidirectional edge states [12, 9, 13] in
these real-space emulators, but the theoretical and experimental evidence for
quantized topological transport in photonic TIs remains scant – especially
when compared to the near-perfect quantized conductivity [2] in electronic
quantum Hall systems to one part in $10^{9}$, which was later used to define
the resistance standard [14]. A prime reason for this is the difficulty in
defining transport properties and the analog of conductivity for neutral
particles such as atoms or photons as they do not naturally respond to
electromagnetic fields. Other impediments to ideal transport quantization in
real-space simulators include the inescapable effects of dissipation and
external driving [15].

In recent years, the concept of synthetic dimensions has emerged by
repurposing internal degrees of freedom of atoms and photons as extra
dimensions, thus realizing high-dimensional topological phenomena on compact,
low-dimensional physical structures [16, 17]. Synthetic dimensions have
enabled lattice Hamiltonians with straightforward reconfigurability and
tunability, long-range coupling, and artificial magnetic gauge fields for
neutral particles, through precise control of coupling between modes labeled
by degrees of freedom such as frequency, temporal modes, orbital angular
momentum, spin and transverse spatial modes [18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30]. Photonic synthetic frequency dimensions, in particular,
have successfully demonstrated both Hermitian and non-Hermitian topology,
electromagnetic gauge fields, unidirectional edge states, Bloch oscillations,
and bulk as well as boundary phenomena [18, 22, 31, 32, 16, 33, 34, 35, 36,
37]. However, the aforementioned limitations of quantized topological
transport in real-space photonic emulators – that of neutral particle
transport in electromagnetic fields, the presence of dissipation and drive –
also apply to synthetic-space systems.

Figure 1: Schematic of the proposed system. The non-degenerate eigenmodes
separated by the coupling rate $\mu$ of the photonic molecule are the
symmetric (S) and anti-symmetric (AS) supermodes, i.e., single-ring azimuthal
modes that are in-phase (S) or $\pi$-phase separated (AS). Coherent evolution
in the subspace of a single pair of these resonance modes can be equated to
the motion on a Bloch sphere, which is controlled by driving the electro-optic
modulator (EOM) with an RF signal near resonance. The transmitted power at the
photodetector (PD) is shown on the top right for the photonic molecule without
any RF drive on the EOMs. Due to the finite resonance linewidths, the
symmetric decay in mode amplitudes is given by $\gamma$. We can generate an RF
signal with amplitude I-Q and frequency modulation, as defined in Eq. 5, using
an arbitrary waveform generator (AWG), allowing us to implement the half-BHZ
Hamiltonian on a Floquet lattice [Eq. (1)].

Here we show how the effects of multiple drives and dissipation can support
quantized topological transport of photons, by using the concept of Floquet
synthetic dimensions in a pair of modulated cavities. Our system consists of
two identical coupled photonic cavities, often called a “photonic molecule”
[38, 39], that are modulated to induce transitions between their symmetric and
antisymmetric supermodes. This modulation is itself driven by two or more
drives at incommensurate frequencies, each of which realizes an orthogonal
synthetic dimension. We refer to the lattice sites created by each
incommensurate drive as being along a “Floquet” synthetic dimension. We
explicitly construct a 2D Chern insulator that exhibits quantized topological
energy pumping, by realizing a driven-dissipative analog of the conservative
protocol by Martin, Refael and Halperin [40]. The rate of energy pumping not
only survives the effects of external laser drive but is abetted by the
presence of finite dissipation. Moreover, the driven-dissipative nature of our
protocol obviates the limitations imposed by finite qubit coherence times and
the need for complicated state initialization in other platforms [41, 42],
attesting to the remarkable topological robustness of the 2D Chern TI.

Note that our usage of the term Floquet dimensions distinguishes it from
synthetic frequency dimensions although both require modulated photonic
cavities, as the modulation frequencies in the former case are significantly
below the modulation at the free-spectral-range (FSR) required in the latter
case. This eases on-chip realization of our approach in integrated photonics
by reducing the demanding bandwidths of low-loss electro-optic modulation. The
reduction is particularly beneficial for the photonic construction of high-
dimensional models as that would require modulation with frequencies
10-100$\times$ the FSR in the synthetic frequency dimension case.

As an illustration of high-dimensional topology, we use our effectively 0D
system to construct a three-dimensional (3D) topological Hamiltonian
supporting Weyl points, which act as magnetic monopoles in the reciprocal
($k$) space of the Floquet lattice [43]. To observe this monopole behaviour,
we require a reconstruction of the Berry curvature’s ‘field’ lines, measured
around these Weyl points. We conclude by showing how the Berry curvature can
be experimentally measured throughout the bands for any general two-band
Hamiltonian.

More generally, harnessing Floquet synthetic dimensions offers a powerful tool
for direct $k$-space Hamiltonian engineering in high dimensions using simple,
compact geometries with experimentally realizable excitation and measurement
protocols.

### Half-BHZ model in a photonic molecule and topological energy pumping

To illustrate the analogous topological behavior of temporally modulated
systems in Floquet synthetic dimensions, we consider the Bloch form of the Qi-
Wu-Zhang model [44, 45, 46] (equivalently, the half–Bernevig-Hughes-Zhang
(half-BHZ) model). When implemented on a Floquet lattice by driving a spin (or
a two-level system) with two incommensurate (irrationally related) frequencies
[40], the Hamiltonian is

$\displaystyle\mathcal{H}=\Omega_{R}[\mathrm{sin}(\Omega_{1}t+\phi_{1})\sigma_{x}+\mathrm{sin}(\Omega_{2}t+\phi_{2})\sigma_{y}$
$\displaystyle+\\{m-\mathrm{cos}(\Omega_{1}t+\phi_{1})-\mathrm{cos}(\Omega_{2}t+\phi_{2})\\}\sigma_{z}]$
(1)

where $\Omega_{1}t+\phi_{1}\rightarrow k_{x}$ and
$\Omega_{2}t+\phi_{2}\rightarrow k_{y}$ give us the $\vec{k}$-space
representation of the half-BHZ model on a real lattice [47]. This duality
between the drive phases and Bloch quasimomentum $\mathbf{k}$ also implies
that the linear evolution of the phase with time emulates the effect of a
charge moving under an electric field in a 2D lattice. The half-BHZ
Hamiltonian breaks both time-reversal and chiral symmetries but possesses
particle-hole symmetry with $\hat{C}=\sigma_{x}K$ and inversion symmetry with
$\hat{P}=\sigma_{z}$, $\hat{C}\mathcal{H}(k)\hat{C}^{-1}=-\mathcal{H}(-k)$ and
$\hat{P}\mathcal{H}(k)\hat{P}^{-1}=\mathcal{H}(-k)$. Thus
$\mathcal{H}(\vec{k})$ belongs to class D in Altland-Zirnbauer’s tenfold way
of classification of TIs, and supports chiral transport [48, 49, 50]. It
models the behavior of a 2D Chern insulator with a Chern number ($C$)
determined by the value of $m$; for $|m|>2$ (trivial phase), $C=0$, and for
$|m|<2$ (topological phase), $C=1$. These Chern insulators exhibit an
anomalous current that is proportional to $C$, and the current flows
perpendicular to the applied field $\vec{\Omega}$.

Quantized chiral transport is advantageous to probe in Floquet synthetic
dimensions as the effective electric field naturally arises here [40], while
other real-space and synthetic-space systems with neutral particles require
effective electric fields to be explicitly introduced. On the Floquet lattice,
this leads to topological energy pumping that is quantized by $C$, and we
therefore expect to see quantized energy transfer from one incommensurate
drive to the other. While implementing this Hamiltonian with qubits [41, 42]
can produce exotic phenomena such as engineering cat states [51] and quantum
state boosting [52], they encounter constraints in demonstrating topological
pumping on long timescales due to decoherence. However, barring quantum
measurements, this Hamiltonian can be simulated by classical systems, such as
magnetic nanoparticles in a time-dependent magnetic field, which allows for a
demonstration of topological energy pumping that is unimpeded by the coherence
times of qubits. We now look at how a photonic molecule driven by two
incommensurate frequencies can implement the same physics encapsulated by Eq.
(1).

A photonic molecule comprises two identical optical ring resonators
evanescently coupled at a rate $\mu$. [38, 39]. The eigenmodes of the molecule
are the symmetric (S) and anti-symmetric (AS) supermodes of various azimuthal
orders as shown in Fig. 1. Each ring has an electro-optic phase modulator
(EOM) that couples the eigenmodes when driven near the splitting, i.e.,
$\omega_{m}\simeq\mu$, with opposite polarities of the RF drive signal $V(t)$.
We isolate a single pair of eigenmodes as our two-level system, and work in
the single-photon subspace. Taking the bosonic annihilation operators for the
uncoupled modes of the two rings to be $a_{1}$ and $a_{2}$ respectively, we
define the S and AS eigenmode annihilation operators as
$c_{1}=\frac{1}{\sqrt{2}}(a_{1}+a_{2})$ and
$c_{2}=\frac{1}{\sqrt{2}}(a_{1}-a_{2})$, giving us the Hamiltonian,

$\mathcal{H}=\omega_{+}c_{1}^{\dagger}c_{1}+\omega_{-}c_{2}^{\dagger}c_{2}+gV(t)(c_{1}^{\dagger}c_{2}+c_{2}^{\dagger}c_{1})$
(2)

where $g$ is the electro-optic coupling strength. Measuring all frequencies
relative to the uncoupled ring resonance frequency $\omega_{0}$, we set
$\omega_{\pm}=\pm\frac{\mu}{2}$. In this single-photon supermode subspace, the
Pauli operators are defined as usual:
$\sigma_{x}=c_{1}^{\dagger}c_{2}+c_{2}^{\dagger}c_{1},\
i\sigma_{y}=c_{1}^{\dagger}c_{2}-c_{2}^{\dagger}c_{1},\
\sigma_{z}=c_{1}^{\dagger}c_{1}-c_{2}^{\dagger}c_{2}\ $, giving us the
Hamiltonian

$\mathcal{H}=-\sigma_{z}\,\mu/2+\sigma_{x}\,gV(t)$ (3)

We now consider a specific form of $V(t)$ as an RF carrier at $\omega_{m}$
with I-Q amplitude modulations (AM) $V_{x}(t)$ and $V_{y}(t)$ respectively, as
well as frequency modulation (FM) $\lambda(t)={\rm d/d}t[\Delta(t)]$, leading
to,

$\mathcal{H}=\sigma_{z}\mu/2+\sigma_{x}g{\rm
Re}\left[\\{V_{x}-iV_{y}\\}\times\exp\\{i\omega_{m}t+i\Delta(t)\\}\right]$

where $\Delta(t)=\int^{t}\lambda(t^{\prime})\mathrm{d}t^{\prime}$. Taking the
interaction picture with $|gV(t)|\ll\mu\,\forall\,t$ (weak driving), the
effective Hamiltonian under the rotating-wave approximation is (Supp. I),

$\mathcal{H}=-\frac{\delta+\lambda(t)}{2}\sigma_{z}+\frac{gV_{x}(t)}{2}\sigma_{x}+\frac{gV_{y}(t)}{2}\sigma_{y}$
(4)

where $\delta=\omega_{m}-\mu$. Comparing with Eq. (1) gives us the necessary
amplitude and frequency modulation signals to be applied

$\displaystyle\lambda(t)$ $\displaystyle=$ $\displaystyle-
gV_{0}\\{\mathrm{cos}(\Omega_{1}t+\phi_{1})+\mathrm{cos}(\Omega_{2}t+\phi_{2})\\}$
$\displaystyle V_{x}(t)$ $\displaystyle=$ $\displaystyle
V_{0}\,\mathrm{sin}(\Omega_{1}t+\phi_{1})$ $\displaystyle V_{y}(t)$
$\displaystyle=$ $\displaystyle V_{0}\,\mathrm{sin}(\Omega_{2}t+\phi_{2})$ (5)

The tunable topological parameter $m=\delta/(2gV_{0})$ is now the normalized
detuning of the RF drive $\omega_{m}$ from the resonance of the two-level
system and maps to a static $\sigma_{z}$ coefficient in Eq. (1). Thus,
$\omega_{m}$ can be readily controlled from $0<m<2$ to $m>2$ to engender a
topological phase transition. The I-Q amplitude modulation maps to
$\sigma_{x}$ and $\sigma_{y}$ components, whereas the frequency modulation
maps to the $\sigma_{z}$ component.

We note that this system can be made to evolve adiabatically, i.e.,
$\Omega_{1},\Omega_{2}\ll gV_{0}$, by choosing the frequency scales of our
modulation accordingly. We also emphasize that all frequency scales in our
proposed implementation are much below the ring’s free spectral range
($\Omega_{1},\Omega_{2}\ll\mu\ll\omega_{\rm FSR}$), thus easing bandwidth
requirements for integrated photonic modulators. Thus, we have showed how the
2D half-BHZ Hamiltonian can be realized in a photonic molecule, an effectively
0D system, with appropriately engineered drives and detunings, but in an as-
of-yet conservative system. Before introducing dissipation and external drive,
we briefly summarize the phenomenon of quantized topological energy pumping as
introduced in [40]. Splitting the Hamiltonian into the respective $\Omega_{1}$
and $\Omega_{2}$ drive contributions:

$\mathcal{H}=h_{1}(t)+h_{2}(t)+\sigma_{z}\,\delta/2,$ (6)

the work done by each drive $i=1,2$ over time $T$ is given by

$W_{i}(T)=\int_{0}^{T}dt\left\langle\frac{dh_{i}}{dt}\right\rangle$ (7)

For $\Omega_{1}/\Omega_{2}$ irrational, the system samples the full Brillouin
zone, leading to quantized energy pumping:

$W_{1}=-W_{2}=C\frac{\Omega_{1}\Omega_{2}T}{2\pi}$ (8)

where $C$ is the Chern number.

### Driven-dissipative quantized pumping in a photonic molecule

Figure 2: Spectral and temporal signatures of topological dynamics, in the
presence of optical drive and dissipation. (a) Normalized spectral amplitude
of the symmetric super-mode $c_{1}$. In the topological regime ($m=1$) we see
the dense spectrum showing a continuous floor, indicative of aperiodic and
highly delocalized dynamics that are characterized by harmonics of the
incommensurate drives, $\Omega_{1}$ and $\Omega_{2}$. In the trivial regime
($m=3$), the spectrum is discrete with sharp peaks, indicating periodic orbits
and localization in the Bloch sphere. (b) Normalized work done by the drives
$\Omega_{1}$ and $\Omega_{2}$ in the topological ($m=1$) and trivial ($m=3$)
regimes. $W_{1}$ and $W_{2}$ respectively show slopes of 1.022 and -1.010 in
the topological regime over $\sim 5$ photon lifetimes, and almost no pumping
(slopes of -0.053 and 0.070 respectively for $W_{1}$ and $W_{2}$) in the
trivial regime, clearly exhibiting the linear dependence on the Chern number
$C$. Insets show Bloch sphere trajectories. (c) Dynamics for commensurate
drives ($\Omega_{2}/\Omega_{1}=1.5$) shows no quantization and the possibility
of higher pumping rates in the trivial regime (see text).

We next explore how the addition of an external laser drive and cavity
dissipation enable quantized topological energy pumping in a quasi-steady
state regime without stringent requirements on initialization of the
molecule’s state. Dissipation is natural as all optical ring resonators have
finite Q-factors, which define the photon decay rate $\gamma$ for the system.
Nevertheless, this decay affects both modes symmetrically, and they can be
renormalized to look at the dynamics within the two-level subspace. This is a
unique advantage that photonic systems provide, and motivates us to verify
persistent topological pumping when adding an external optical drive. The
driven-dissipative equations of motion for $c_{1}$ and $c_{2}$ are

$\dot{c}_{1,2}=i[\mathcal{H},c_{1,2}]-\gamma
c_{1,2}+\sqrt{\gamma_{e}}s_{in}(t)e^{\pm i((\mu+\delta)t+\Delta(t))/2}$ (9)

where $\gamma_{e}$ is the coupling rate into the bus waveguide, and
$s_{in}(t)$ is the external laser drive (note that from this point onward in
the text, we distinguish between the Hamiltonian drives $\Omega_{1}$ and
$\Omega_{2}$ and the external optical/laser drive $s_{in}(t)$). We physically
motivate the chosen laser frequency with the adiabatically varying
eigenspectrum of the Hamiltonian in Eq. (4), as a function of $m$.

Figure 3: Work done by the drives, simulated for $\gamma=$ (a)
$0.1\Omega_{1}/\pi$, (b) $\Omega_{1}/\pi$, (c) $10\Omega_{1}/\pi$, and
(d)$100\Omega_{1}/\pi$. We see close to quantized pumping in (a) with slopes
of 1.261 and -1.342 for $W_{1}$ and $W_{2}$ respectively, which increases and
starts to disappear in (b) and (c). Almost no observable phase transition
behavior occurs in (d), due to overdamped oscillator dynamics. The increase in
slope is not an anomaly, however, as the actual work done, without
normalization depends on $|c_{1}|^{2}+|c_{2}|^{2}$, which reduces by orders of
magnitude and kills the effect of the increased slope (Supp. Fig. S3) .

### Topological signatures in temporal and spectral dynamics

This system shows a variety of topological signatures even with drive and
dissipation added into the mix [Fig. 2], reinforcing the remarkable robust
topology of the conservative half-BHZ model. In Fig. 2(a), a clear qualitative
difference is seen in the spectral amplitudes of one of the super-modes
(denoted by $c_{1}$): The topological regime is characterized by a dense
spectrum with a continuous floor, indicative of the aperiodic evolution of the
system, while the trivial regime shows several discrete peaks that represents
the periodic localization in the dynamics. Our numerical simulations reproduce
results calculated in a conservative system averaged over initial states [53].
Note that no such averaging over initial states is required in our model, as
the signatures presented are long-time quasi-steady-state simulations; they
are fairly independent of the initial state transients which have decayed at
long times.

As noted previously, the hallmark of this Hamiltonian is the presence of
quantized topological pumping, which is absent in the trivial regime. In Fig.
2(b), we see that the work done by the $\Omega_{1}$ drive ($W_{1}$) increases
at the quantized rate stated in Eq. (8), while the $\Omega_{2}$ drive
($W_{2}$) decreases correspondingly, reflecting that there is energy being
pumped from one drive to the other. The linear slopes reinforce the
quantization by the Chern number $C$, with the pumping rate being 1.022 and
-1.010 for $W_{1}$ and $W_{2}$ respectively in the topological phase (top
figure) and correspondingly -0.053 and 0.070 in the trivial phase (bottom).
The Bloch sphere plots (Fig. 2(b) insets, SI video 1 and SI video 2) reflect
the ergodic behavior of the system in the topological regime, while the
trivial regime dynamics remains localised near the $\ket{0}$ state.
Remarkably, we are not limited by the photon lifetime ($\tau_{p}=1/\gamma$,
with $\gamma\approx 0.01\Omega_{1}/\pi$), and see sustained pumping for $\sim
5$ photon lifetimes for $m=1$. Almost no pumping is observed for $m=3$ in the
trivial regime. An important detail here is that the pumping is being observed
under a quasi-steady state condition, due to the drive and dissipation, thus
obviating the need for complex Floquet eigenstate initialization in previous
protocols [42, 41] and allowing us to start with vacuum states in both
resonators.

One way to verify the need for incommensurateness in the Hamiltonian’s drive
frequencies is to observe the dynamics for rational values of
$\Omega_{1}/\Omega_{2}=p/q,\ p,q\in\mathbb{Z}$. We see the absence of a
transition in this case, as the half-BHZ model effectively becomes a 1D tight-
binding Hamiltonian with long-range couplings [54, 18, 55]. The pumping rate
no longer depends on the Chern number $C$, but on the integrated Berry
curvature over specific periodic trajectories in the Brillouin Zone. We
therefore also see periodic orbits on the Bloch sphere (SI video 3 and SI
video 4).

### Impact of dissipation on topological pumping

A unique feature of our system is the interplay between the dynamics and the
dissipation. We show in Fig. 3 that the topological pumping loses quantization
for higher values of $\gamma$, but the striking qualitative contrast between
the topological and trivial regimes still persists up to
$\gamma>\Omega_{1},\Omega_{2}$. Although the normalized pumping rate seems to
increase, the total power $|c_{1}|^{2}+|c_{2}|^{2}$ reduces on average in the
new quasi-steady state, leading to a net reduction in pump power. Physical
intuition for the loss of pumping beyond a certain $\gamma\gg
10\Omega_{1}/\pi$ comes from the coupled oscillator system becoming
overdamped, washing out the topological dynamics and the pumping effects.

### Scaling to higher dimensions: Weyl points

Figure 4: The band structure for a Weyl point Hamiltonian along the (a)
$k_{x}k_{y}$-plane, and the (b) $k_{y}k_{z}$-plane. The band-touching Weyl
points show isotropic linear dispersion in their vicinity, with their
topological robustness quantified by the Berry curvature flux normal to a
surface near the Weyl point being quantized. Neighbouring Weyl points show
opposite signs in the quantized flux, indicated by the insets, which show the
Berry curvature field lines in 3-D. This behaviour is indicative of synthetic
magnetic field lines originating from a Weyl point and converging at the
neighbouring Weyl point, leading to the physical picture of Weyl points as
monopoles of the synthetic magnetic field.

Now we look at the question of scalability. The key advantage of Floquet
synthetic dimensions is the ability to engineer higher-dimensional
Hamiltonians, with relatively simple additions to the current setup with a
photonic molecule. By simply adding a third incommensurate frequency to the
modulation, we can create bulk 3-D non-trivial topology, leading to phenomena
such as Weyl points [43]. These points have garnered great interest in the
community for the fundamentally unique phenomena they lead to, such as Fermi
arc surface states [56], which also hold potential for applications in next-
generation electronics [57, 58].

Furthermore, Weyl points behave as monopoles of the Berry curvature (akin to a
magnetic field in momentum space), and the Berry curvature surrounding a Weyl
point has not been measured experimentally before. With the photonic molecule,
there are two routes to increasing the dimensionality of the system and
obtaining Weyl points - introducing a third incommensurate frequency, or
harnessing the frequency synthetic dimension of the rings by modulating at the
free spectral range [31, 22]. While modulation at $\omega_{FSR}$ offers the
capability to create boundaries on the frequency lattice [59, 37, 60, 61],
adding a third incommensurate frequency gives us the $k$-space Hamiltonian
directly. This allows us to simulate Weyl Hamiltonians such as [62]:

$\mathcal{H}(\mathbf{k})=\Omega_{R}\\{\mathrm{sin}(k_{x})\sigma_{x}+\mathrm{cos}(k_{y})\sigma_{y}+\mathrm{cos}(k_{z})\sigma_{z}\\}$
(10)

The band structure of this Hamiltonian is shown in Fig. 4, where we see linear
dispersion in all three directions near the band-touching points at
($0,\pm\pi/2,\pm\pi/2$) and ($\pi,\pm\pi/2,\pm\pi/2$).

What makes our method especially powerful is that we can now experimentally
map out the Berry curvature throughout the bands, for any given two-band
Hamiltonian $\mathcal{H}(\mathbf{k})$, by tracking the dynamics of the
artificial spin encoded in the photonic molecule. This allows us to generate
Berry curvature data similar to what the insets in Fig. 4 display in the
vicinity of each Weyl point, thus verifying their synthetic monopole
character. To buttress this claim, we calculate $\mathbf{B}(\mathbf{k})$ in a
gauge-invariant fashion as [63]:

$\mathbf{B}(\mathbf{k})=\sum_{m\neq
n}\mathrm{Im}\left[\frac{\bra{n(\mathbf{k})}\\{\mathbf{\nabla}_{\mathbf{k}}\mathcal{H}(\mathbf{k})\\}\ket{m(\mathbf{k})}\times\bra{m(\mathbf{k})}\\{\mathbf{\nabla}_{\mathbf{k}}\mathcal{H}(\mathbf{k})\\}\ket{n(\mathbf{k})}}{(E_{m}-E_{n})^{2}}\right]$
(11)

where $\ket{m(\mathbf{k})}$ and $\ket{n(\mathbf{k})}$ are the eigenvectors of
$\mathcal{H}(\mathbf{k})$, $E_{m}$ and $E_{n}$ being the corresponding
eigenvalues. Solving for the general eigenvalues and eigenvectors of the
$2\times 2$ Hamiltonian (Supp II), this simplifies to

$\displaystyle\mathbf{B}(\mathbf{k})$
$\displaystyle=\frac{\Omega_{R}^{2}}{4E_{m}^{2}}\Bigg{[}\hat{k}_{x}\sin(k_{y})\sin(k_{z})\left(\bra{n}\sigma_{x}\ket{n}-\frac{\Omega_{R}\sin(k_{x})}{E_{m}}\right)$
$\displaystyle\;\;-\hat{k}_{y}\cos(k_{x})\sin(k_{z})\left(\bra{n}\sigma_{y}\ket{n}-\frac{\Omega_{R}\cos(k_{y})}{E_{m}}\right)$
$\displaystyle\;\;-\hat{k}_{z}\cos(k_{x})\sin(k_{y})\left(\bra{n}\sigma_{z}\ket{n}-\frac{\Omega_{R}\cos(k_{z})}{E_{m}}\right)\Bigg{]}$

With Floquet synthetic dimensions in a photonic molecule, we can envision the
following protocol: Initialize to the eigenstate $\ket{n}$ for a given initial
phase and drive adiabatically with three incommensurate frequencies, while
measuring the spin expectation values for all $k$ except at the Weyl points.
Here, the eigenstate flips due to the band-touching, and the Berry curvature
diverges, so we can threshold this measurement and accumulate experimental
runs over multiple initial phases to reconstruct the insets in Fig. 4. Thus,
the photonic molecule can provide a novel experimental probe for the Berry
curvature of a topological Floquet system.

### Discussion

In summary, we have proposed a new candidate system to engineer topological
Hamiltonians in a 2-D Floquet synthetic lattice, and have shown that it
exhibits topological energy pumping that persists in the presence of
dissipation and an external drive, over multiple parameter regimes. The
pumping persists over multiple photon lifetimes and stays quantized in the
normalized spin subspace. Furthermore, this system can be extended to simulate
higher-dimensional Hamiltonians with the straightforward addition of an extra
incommensurate frequency. As an example, we look at Weyl points and provide a
path forward to observing them in a photonic molecule. While synthetic
dimensions offer many advantages by circumventing the complexity of
constructing analogous material systems, such as in cold-atom experiments
[62], the complexity inherently shifts to the control signal RF engineering
requirements. Furthermore, measuring the topological signatures requires
information about the complex amplitudes of $c_{1}$ and $c_{2}$, which
necessitate dynamically frequency-tuned phase-sensitive detection schemes.
However, on-chip electro-optically modulated ring resonators with quality
factors upwards of $10^{6}$, driven by state-of-the-art electronics, have been
demonstrated [64, 39]. A key advantage lies in the frequencies we modulate at,
since $\omega_{m}\ll\omega_{FSR}$ for Floquet synthetic dimensions, which
eases up the specifications on electronics for driving on-chip devices. One
can thus envision a fully integrated device that can achieve high-dimensional
Hamiltonian analog simulation in this framework for considerable periods of
time.

## Acknowledgements

This work was supported by a Northrop Grumman seed grant and a National Q-Lab
joint seed grant from IonQ and the University of Maryland.

## References

  * Klitzing  _et al._ [1980a] K. v. Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494 (1980a).
  * Thouless  _et al._ [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982).
  * Miyake  _et al._ [2013] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices, Phys. Rev. Lett. 111, 185302 (2013).
  * Aidelsburger  _et al._ [2013] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realization of the hofstadter hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett. 111, 185301 (2013).
  * Price  _et al._ [2022] H. Price, Y. Chong, A. Khanikaev, H. Schomerus, L. J. Maczewsky, M. Kremer, M. Heinrich, A. Szameit, O. Zilberberg, Y. Yang, B. Zhang, A. Alù, R. Thomale, I. Carusotto, P. St-Jean, A. Amo, A. Dutt, L. Yuan, S. Fan, X. Yin, C. Peng, T. Ozawa, and A. Blanco-Redondo, Roadmap on topological photonics, J. Phys. Photonics 4, 032501 (2022).
  * Khanikaev  _et al._ [2015] A. B. Khanikaev, R. Fleury, S. H. Mousavi, and A. Alu, Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice, Nature communications 6, 8260 (2015).
  * Ma  _et al._ [2019] G. Ma, M. Xiao, and C. T. Chan, Topological phases in acoustic and mechanical systems, Nature Reviews Physics 1, 281 (2019).
  * Imhof  _et al._ [2018] S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, F. Schindler, C. H. Lee, M. Greiter, T. Neupert, and R. Thomale, Topolectrical-circuit realization of topological corner modes, Nature Physics 14, 925 (2018).
  * Hafezi  _et al._ [2013] M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Imaging topological edge states in silicon photonics, Nature Photonics 7, 1001 (2013).
  * Rechtsman  _et al._ [2013] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature 496, 196 (2013).
  * Aidelsburger  _et al._ [2015] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbène, N. R. Cooper, I. Bloch, and N. Goldman, Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms, Nature Physics 11, 162 (2015).
  * Wang  _et al._ [2009] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature 461, 772 (2009).
  * Atala  _et al._ [2014] M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Observation of chiral currents with ultracold atoms in bosonic ladders, Nature Physics 10, 588 (2014).
  * Klitzing  _et al._ [1980b] K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance, Phys. Rev. Lett. 45, 494 (1980b).
  * Ozawa and Carusotto [2014] T. Ozawa and I. Carusotto, Anomalous and Quantum Hall Effects in Lossy Photonic Lattices, Phys. Rev. Lett. 112, 133902 (2014).
  * Yuan  _et al._ [2021] L. Yuan, A. Dutt, and S. Fan, Synthetic frequency dimensions in dynamically modulated ring resonators, APL Photonics 6, 071102 (2021).
  * Ozawa and Price [2019] T. Ozawa and H. M. Price, Topological quantum matter in synthetic dimensions, Nature Reviews Physics 1, 349 (2019).
  * Dutt  _et al._ [2019] A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, Experimental Demonstration of Dynamical Input Isolation in Nonadiabatically Modulated Photonic Cavities, ACS Photonics 6, 162 (2019).
  * Leefmans  _et al._ [2022] C. Leefmans, A. Dutt, J. Williams, L. Yuan, M. Parto, F. Nori, S. Fan, and A. Marandi, Topological dissipation in a time-multiplexed photonic resonator network, Nat. Phys. 18, 442 (2022).
  * Bartlett  _et al._ [2021] B. Bartlett, A. Dutt, and S. Fan, Deterministic photonic quantum computation in a synthetic time dimension, Optica 8, 1515 (2021).
  * Lustig  _et al._ [2019] E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, Photonic topological insulator in synthetic dimensions, Nature 567, 356 (2019).
  * Dutt  _et al._ [2020] A. Dutt, Q. Lin, L. Yuan, M. Minkov, M. Xiao, and S. Fan, A single photonic cavity with two independent physical synthetic dimensions, Science 367, 59 (2020).
  * Boada  _et al._ [2012] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Quantum Simulation of an Extra Dimension, Phys. Rev. Lett. 108, 133001 (2012).
  * Stuhl  _et al._ [2015] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Visualizing edge states with an atomic Bose gas in the quantum Hall regime, Science 349, 1514 (2015).
  * Mancini  _et al._ [2015] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015).
  * Sundar  _et al._ [2018] B. Sundar, B. Gadway, and K. R. A. Hazzard, Synthetic dimensions in ultracold polar molecules, Scientific Reports 8, 3422 (2018).
  * Luo  _et al._ [2015] X.-W. Luo, X. Zhou, C.-F. Li, J.-S. Xu, G.-C. Guo, and Z.-W. Zhou, Quantum simulation of 2D topological physics in a 1D array of optical cavities, Nature Communications 6, 7704 (2015).
  * Kanungo  _et al._ [2022] S. K. Kanungo, J. D. Whalen, Y. Lu, M. Yuan, S. Dasgupta, F. B. Dunning, K. R. A. Hazzard, and T. C. Killian, Realizing topological edge states with Rydberg-atom synthetic dimensions, Nat Commun 13, 972 (2022).
  * Yang  _et al._ [2023] M. Yang, H.-Q. Zhang, Y.-W. Liao, Z.-H. Liu, Z.-W. Zhou, X.-X. Zhou, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, Realization of exceptional points along a synthetic orbital angular momentum dimension, Science Advances 9, eabp8943 (2023).
  * Hu  _et al._ [2020] Y. Hu, C. Reimer, A. Shams-Ansari, M. Zhang, and M. Loncar, Realization of high-dimensional frequency crystals in electro-optic microcombs, Optica 7, 1189 (2020).
  * Yuan  _et al._ [2016] L. Yuan, Y. Shi, and S. Fan, Photonic gauge potential in a system with a synthetic frequency dimension, Opt. Lett. 41, 741 (2016).
  * Yuan and Fan [2016] L. Yuan and S. Fan, Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator, Optica 3, 1014 (2016).
  * Wang  _et al._ [2021a] K. Wang, A. Dutt, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, Generating arbitrary topological windings of a non-Hermitian band, Science 371, 1240 (2021a).
  * Wang  _et al._ [2021b] K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Topological complex-energy braiding of non-Hermitian bands, Nature 598, 59 (2021b).
  * Wang  _et al._ [2018] J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, Multidimensional quantum entanglement with large-scale integrated optics, Science 360, 285 (2018).
  * Dutt  _et al._ [2022a] A. Dutt, L. Yuan, K. Y. Yang, K. Wang, S. Buddhiraju, J. Vučković, and S. Fan, Creating boundaries along a synthetic frequency dimension, Nat Commun 13, 3377 (2022a).
  * Lu  _et al._ [2020] X. Lu, A. Rao, G. Moille, D. A. Westly, and K. Srinivasan, Universal frequency engineering tool for microcavity nonlinear optics: multiple selective mode splitting of whispering-gallery resonances, Photon. Res., PRJ 8, 1676 (2020).
  * Spreeuw  _et al._ [1990] R. J. C. Spreeuw, N. J. van Druten, M. W. Beijersbergen, E. R. Eliel, and J. P. Woerdman, Classical realization of a strongly driven two-level system, Phys. Rev. Lett. 65, 2642 (1990).
  * Zhang  _et al._ [2018] M. Zhang, C. Wang, Y. Hu, A. Shams-Ansari, T. Ren, S. Fan, and M. Lončar, Electronically programmable photonic molecule, Nature Photonics 13, 36 (2018).
  * Martin  _et al._ [2017] I. Martin, G. Refael, and B. Halperin, Topological frequency conversion in strongly driven quantum systems, Phys. Rev. X 7, 041008 (2017).
  * Boyers  _et al._ [2020] E. Boyers, P. J. D. Crowley, A. Chandran, and A. O. Sushkov, Exploring 2d synthetic quantum hall physics with a quasiperiodically driven qubit, Phys. Rev. Lett. 125, 160505 (2020).
  * Malz and Smith [2021] D. Malz and A. Smith, Topological two-dimensional floquet lattice on a single superconducting qubit, Phys. Rev. Lett. 126, 163602 (2021).
  * Weyl [1929] H. Weyl, Z. Physik 56, 330 (1929).
  * Qi  _et al._ [2006] X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors, Phys. Rev. B 74, 085308 (2006).
  * Che  _et al._ [2020] Y. Che, C. Gneiting, T. Liu, and F. Nori, Topological quantum phase transitions retrieved through unsupervised machine learning, Phys. Rev. B 102, 134213 (2020).
  * Liang  _et al._ [2023] M.-C. Liang, Y.-D. Wei, L. Zhang, X.-J. Wang, H. Zhang, W.-W. Wang, W. Qi, X.-J. Liu, and X. Zhang, Realization of Qi-Wu-Zhang model in spin-orbit-coupled ultracold fermions, Phys. Rev. Res. 5, L012006 (2023).
  * Bernevig  _et al._ [2006] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science 314, 1757 (2006).
  * Schnyder  _et al._ [2008] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  * Altland and Zirnbauer [1997] A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  * Kitaev [2009] A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conference Proceedings 1134, 22 (2009).
  * Nathan  _et al._ [2019] F. Nathan, I. Martin, and G. Refael, Topological frequency conversion in a driven dissipative quantum cavity, Phys. Rev. B 99, 094311 (2019).
  * Long  _et al._ [2022] D. M. Long, P. J. Crowley, A. J. Kollár, and A. Chandran, Boosting the quantum state of a cavity with floquet driving, Physical Review Letters 128, 183602 (2022).
  * Crowley  _et al._ [2019] P. J. D. Crowley, I. Martin, and A. Chandran, Topological classification of quasiperiodically driven quantum systems, Physical Review B 99, 10.1103/physrevb.99.064306 (2019).
  * Bell  _et al._ [2017] B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, Spectral photonic lattices with complex long-range coupling, Optica 4, 1433 (2017).
  * Wang  _et al._ [2020] K. Wang, B. A. Bell, A. S. Solntsev, D. N. Neshev, B. J. Eggleton, and A. A. Sukhorukov, Multidimensional synthetic chiral-tube lattices via nonlinear frequency conversion, Light: Science & Applications 9, 132 (2020).
  * Turner  _et al._ [2013] A. M. Turner, A. Vishwanath, and C. O. Head, Beyond band insulators: topology of semimetals and interacting phases, Topological Insulators 6, 293 (2013).
  * Xu  _et al._ [2015] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, _et al._ , Discovery of a weyl fermion semimetal and topological fermi arcs, Science 349, 613 (2015).
  * Osterhoudt  _et al._ [2019] G. B. Osterhoudt, L. K. Diebel, M. J. Gray, X. Yang, J. Stanco, X. Huang, B. Shen, N. Ni, P. J. Moll, Y. Ran, _et al._ , Colossal mid-infrared bulk photovoltaic effect in a type-i weyl semimetal, Nature materials 18, 471 (2019).
  * Dutt  _et al._ [2022b] A. Dutt, L. Yuan, K. Y. Yang, K. Wang, S. Buddhiraju, J. Vučković, and S. Fan, Creating boundaries along a synthetic frequency dimension, Nature Communications 13, 3377 (2022b).
  * Yuan  _et al._ [2019] L. Yuan, Q. Lin, A. Zhang, M. Xiao, X. Chen, and S. Fan, Photonic Gauge Potential in One Cavity with Synthetic Frequency and Orbital Angular Momentum Dimensions, Physical Review Letters 122, 083903 (2019).
  * Buddhiraju  _et al._ [2021] S. Buddhiraju, A. Dutt, M. Minkov, I. A. D. Williamson, and S. Fan, Arbitrary linear transformations for photons in the frequency synthetic dimension, Nature Communications 12, 2401 (2021).
  * Dubček  _et al._ [2015] T. Dubček, C. J. Kennedy, L. Lu, W. Ketterle, M. Soljačić, and H. Buljan, Weyl Points in Three-Dimensional Optical Lattices: Synthetic Magnetic Monopoles in Momentum Space, Phys. Rev. Lett. 114, 225301 (2015).
  * Bernevig [2013] B. A. Bernevig, Topological insulators and topological superconductors, in  _Topological Insulators and Topological Superconductors_ (Princeton university press, 2013).
  * Zhang  _et al._ [2017] M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and M. Lonč ar, Monolithic ultra-high-q lithium niobate microring resonator, Optica 4, 1536 (2017).

Supplemental Materials: Quantized topological energy pumping and Weyl points
in
Floquet synthetic dimensions with a driven-dissipative photonic molecule

##  I Half-BHZ Hamiltonian for a Photonic Molecule

Consider the two-resonator system with a Hamiltonian

$\displaystyle
H=\omega_{0}\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)+\frac{\mu}{2}\left(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}\right)+gV(t)(a_{1}^{\dagger}a_{1}-a_{2}a_{2}^{\dagger})$
(S1)

where $a_{1}$ and $a_{2}$ are the respective bosonic annihilation operators
for the individual ring resonator modes at the resonance $\omega_{0}$, with a
coupling rate $\mu$, and an electro-optic coupling coefficient $g$. Upon
diagonalizing the time-independent terms, the Hamiltonian becomes

$\displaystyle
H=\omega_{+}c_{1}^{\dagger}c_{1}+\omega_{-}c_{2}^{\dagger}c_{2}+gV(t)(c_{1}^{\dagger}c_{2}+c_{2}^{\dagger}c_{1})$
(S2)

with $\displaystyle{\omega_{\pm}=\omega_{0}\pm\frac{\mu}{2}}$ and
$\displaystyle{c_{1}=\frac{1}{\sqrt{2}}\left(a_{1}+a_{2}\right),c_{2}=\frac{1}{\sqrt{2}}\left(a_{1}-a_{2}\right)}$.
With a variable change, we can rotate out the
$\omega_{0}\left(c_{1}^{\dagger}c_{1}+c_{2}^{\dagger}c_{2}\right)$ term
(within the single photon subspace) and consider
$\omega_{\pm}=\pm\dfrac{\mu}{2}$. We specify $V(t)$ to have a carrier tone
$\omega_{m}$ with independent amplitude I-Q modulations and frequency
modulation, giving us

$\displaystyle H$
$\displaystyle=\omega_{+}c_{1}^{\dagger}c_{1}+\omega_{-}c_{2}^{\dagger}c_{2}+g\left[V_{x}(t)\cos\left(\omega_{m}t+\Delta(t)\right)+V_{y}(t)\sin\left(\omega_{m}t+\Delta(t)\right)\right]\,(c_{1}^{\dagger}c_{2}+c_{2}^{\dagger}c_{1})$
(S3)

We can define our spin-matrices to be

$\displaystyle\sigma_{x}$
$\displaystyle=\left(c_{1}^{\dagger}c_{2}+c_{2}^{\dagger}c_{1}\right)$
$\displaystyle\sigma_{y}$
$\displaystyle=-i\left(c_{1}^{\dagger}c_{2}-c_{2}^{\dagger}c_{1}\right)$
$\displaystyle\sigma_{z}$
$\displaystyle=\left(c_{1}^{\dagger}c_{1}-c_{2}^{\dagger}c_{2}\right)$

in the single-photon subspace. These operators now form a Pauli group and obey
the same commutation relations. In terms of these new spin operators, the
Hamiltonian is

$\displaystyle
H=\frac{\mu}{2}\sigma_{z}+gV(t)\sigma_{x}=\frac{\mu}{2}\sigma_{z}+g\left[V_{x}(t)\cos\left\\{\omega_{m}t+\Delta(t)\right\\}+V_{y}(t)\sin\left\\{\omega_{m}t+\Delta(t)\right\\}\right]\sigma_{x}$
(S4)

In the interaction picture, we can consider
$U=\exp\left[i\sigma_{z}\\{\omega_{m}t+\Delta(t)\\}/2\right]$, and the dynamic
Hamiltonian becomes

$\displaystyle
V_{I}(t)=\exp\left[i\sigma_{z}\\{\omega_{m}t+\Delta(t)\\}/2\right]\,\left(gV(t)\sigma_{x}\right)\exp\left[-i\sigma_{z}\\{\omega_{m}t+\Delta(t)\\}/2\right]=gV(t)\sum_{k=0}^{\infty}\frac{(i\omega_{m}t+\Delta(t))^{k}}{2^{k}k!}\left[\sigma_{z},\sigma_{x}\right]_{(k)}$
(S5)

where

$\displaystyle\left[A,B\right]_{0}$ $\displaystyle=B$
$\displaystyle\left[A,B\right]_{1}$ $\displaystyle=[A,B]$
$\displaystyle\left[A,B\right]_{2}$ $\displaystyle=[A,[A,B]]$
$\displaystyle\left[A,B\right]_{3}$ $\displaystyle=[A,[A,[A,B]]]$

and so on. Since
$[\sigma_{z},\sigma_{x}]=2i\sigma_{y},[\sigma_{z},\sigma_{y}]=-2i\sigma_{x}$,

$\displaystyle
V_{I}(t)=gV(t)\left[-\sum_{k}\frac{(-1)^{k}(\omega_{m}t+\Delta(t))^{(2k+1)}}{(2k+1)!}\sigma_{y}+\sum_{k}\frac{(-1)^{k}(\omega_{m}t+\Delta(t))^{2k}}{(2k)!}\sigma_{x}\right]$
$\displaystyle=gV(t)\left[-\sin\\{\omega_{m}t+\Delta(t)\\}\sigma_{y}+\cos\\{\omega_{m}t+\Delta(t)\\}\sigma_{x}\right]$
$\displaystyle=g\left[V_{x}(t)\cos\left\\{\omega_{m}t+\Delta(t)\right\\}+V_{y}(t)\sin\left\\{\omega_{m}t+\Delta(t)\right\\}\right]\left[-\sin\\{\omega_{m}t+\Delta(t)\\}\sigma_{y}+\cos\\{\omega_{m}t+\Delta(t)\\}\sigma_{x}\right]$
(S6)

For $|gV_{x}(t)|,|gV_{y}(t)|\ll\omega_{m}$, we can expand and apply the
rotating wave approximation to $V_{I}$, keeping only the slowly rotating
terms.

$\displaystyle V_{I}(t)$
$\displaystyle=g\left(e^{i\left(\omega_{m}t+\Delta(t)\right)}\left(V_{x}(t)-iV_{y}(t)\right)+c.c.\right)\left(e^{i(\omega_{m}t+\Delta(t))}(\sigma_{x}+i\sigma_{y})/2+h.c.\right)$
$\displaystyle\approx\frac{gV_{x}(t)}{2}\sigma_{x}+\frac{gV_{y}(t)}{2}\sigma_{y}$
(S7)

In this rotated frame, the Hamiltonian becomes,

$\displaystyle\mathcal{H}$ $\displaystyle=UHU^{\dagger}+i\frac{\partial
U}{\partial t}U^{\dagger}$
$\displaystyle=-\frac{\delta+\lambda(t)}{2}\sigma_{z}+\frac{gV_{x}(t)}{2}\sigma_{x}+\frac{gV_{y}(t)}{2}\sigma_{y}$
(S8)

##  II Weyl points in Floquet Synthetic Dimensions

Figure S1: Topological energy pumping for
$\Omega_{2}/\Omega_{1}=(1+\sqrt{5})/2$. We see a qualitative change in
behavior around $m=2$, indicative of the topological transition. We simulate
this with $gV_{0}=40\Omega_{1}$ and $\gamma=0.01\Omega_{1}/\pi$. Figure S2:
Topological energy pumping for $\Omega_{2}/\Omega_{1}=\sqrt{\pi}$. The
transition persists even for a different choice of irrational ratio. We
simulate this with $gV_{0}=40\Omega_{1}$ and $\gamma=0.01\Omega_{1}/\pi$.

Floquet synthetic dimensions allow for the direct simulation of $k$-space
Hamiltonians by emulating their evolution with incommensurate drive phases.
Thus, we can achieve Hamiltonians such as

$\mathcal{H}(\mathbf{k})=\Omega_{R}\\{\mathrm{sin}(k_{x})\sigma_{x}+\mathrm{cos}(k_{y})\sigma_{y}+\mathrm{cos}(k_{z})\sigma_{z}\\}$
(S9)

by simply performing the substitutions, $\Omega_{1}t+\phi_{1}\rightarrow
k_{x}$, $\Omega_{2}t+\phi_{2}\rightarrow k_{y}$ and
$\Omega_{3}t+\phi_{3}\rightarrow k_{z}$. The linear evolution of these phases
also emulates the effect of a synthetic electric field, which naturally allows
us to study charge transport effects in these systems. We now study this
Hamiltonian and its topological effects in detail. The gauge-invariant Berry
curvature of this system can be written as

$\mathbf{B}(\mathbf{k})=\mathrm{Im}\left[\frac{\bra{n(\mathbf{k})}\\{\mathbf{\nabla}_{\mathbf{k}}\mathcal{H}(\mathbf{k})\\}\ket{m(\mathbf{k})}\times\bra{m(\mathbf{k})}\\{\mathbf{\nabla}_{\mathbf{k}}\mathcal{H}(\mathbf{k})\\}\ket{n(\mathbf{k})}}{(E_{m}-E_{n})^{2}}\right]$
(S10)

We evaluate $\mathbf{\nabla}_{\mathbf{k}}\mathcal{H}(\mathbf{k})$ to be

$\mathbf{\nabla}_{\mathbf{k}}\mathcal{H}(\mathbf{k})=\Omega_{R}[\hat{k}_{x}\cos(k_{x})\sigma_{x}-\hat{k}_{y}\sin(k_{y})\sigma_{y}-\hat{k}_{z}\sin(k_{z})\sigma_{z}]$

where $\hat{k}_{i}\,(i=x,y,z)$ is the unit vector in the Floquet momentum
space. Another important quantity to calculate are the eigenvalues and
eigenvectors of the Hamiltonian as functions of $\mathbf{k}$. Using the
properties of spin-1/2 Hamiltonians, we get

$\displaystyle E_{m}$ $\displaystyle=$
$\displaystyle\Omega_{R}\sqrt{\sin^{2}k_{x}+\cos^{2}k_{y}+\cos^{2}k_{z}}=-E_{n}=E$
$\displaystyle\rho_{m}$ $\displaystyle=$ $\displaystyle|m\rangle\langle
m|=\frac{1}{2}\left(\mathbb{I}+\frac{\mathcal{H}(\mathbf{k})}{E}\right)$
$\displaystyle\rho_{n}$ $\displaystyle=$ $\displaystyle|n\rangle\langle
n|=\frac{1}{2}\left(\mathbb{I}-\frac{\mathcal{H}(\mathbf{k})}{E}\right)$ (S11)

Substituting these into Eq. S10 and using properties of Pauli matrices,

$\displaystyle\mathbf{B}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\frac{\Omega_{R}^{2}}{4E^{2}}\mathrm{Im}\big{[}\hat{k}_{x}\sin(k_{y})\sin(k_{z})\bra{n}\sigma_{y}\rho_{m}\sigma_{z}-\sigma_{z}\rho_{m}\sigma_{y}\ket{n}-\hat{k}_{y}\cos(k_{x})\sin(k_{z})\bra{n}\sigma_{z}\rho_{m}\sigma_{x}-\sigma_{x}\rho_{m}\sigma_{z}\ket{n}$
(S12)
$\displaystyle\;\;-\hat{k}_{z}\cos(k_{x})\sin(k_{y})\bra{n}\sigma_{x}\rho_{m}\sigma_{y}-\sigma_{y}\rho_{m}\sigma_{x}\ket{n}\big{]}$
$\displaystyle=$
$\displaystyle\frac{\Omega_{R}^{2}}{8E^{2}}\mathrm{Im}\Bigg{[}\hat{k}_{x}\sin(k_{y})\sin(k_{z})\bra{n}\left([\sigma_{y},\sigma_{z}]-2i\sin(k_{x})\frac{\Omega_{R}\mathbb{I}}{E}\right)\ket{n}$
$\displaystyle\;\;-\hat{k}_{y}\cos(k_{x})\sin(k_{z})\bra{n}\left([\sigma_{z},\sigma_{x}]-2i\cos(k_{y})\frac{\Omega_{R}\mathbb{I}}{E}\right)\ket{n}$
$\displaystyle\;\;-\hat{k}_{z}\cos(k_{x})\sin(k_{y})\bra{n}\left([\sigma_{x},\sigma_{y}]-2i\cos(k_{z})\frac{\Omega_{R}\mathbb{I}}{E}\right)\ket{n}\Bigg{]}$
$\displaystyle=$
$\displaystyle\frac{\Omega_{R}^{2}}{4E^{2}}\Bigg{[}\hat{k}_{x}\sin(k_{y})\sin(k_{z})\left(\bra{n}\sigma_{x}\ket{n}-\frac{\Omega_{R}\sin(k_{x})}{E}\right)-\hat{k}_{y}\cos(k_{x})\sin(k_{z})\left(\bra{n}\sigma_{y}\ket{n}-\frac{\Omega_{R}\cos(k_{y})}{E}\right)$
$\displaystyle\;\;-\hat{k}_{z}\cos(k_{x})\sin(k_{y})\left(\bra{n}\sigma_{z}\ket{n}-\frac{\Omega_{R}\cos(k_{z})}{E}\right)\Bigg{]}$

This is Eq. 11 in the main text, and the Berry curvature in this form can be
measured experimentally, when adiabaticity is maintained. Simplifying further,
we get

$\displaystyle\mathbf{B}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\frac{\Omega_{R}^{2}}{4E^{2}}\Bigg{[}\hat{k}_{x}\sin(k_{y})\sin(k_{z})\left(\mathrm{Tr}\\{\rho_{n}\sigma_{x}\\}-\frac{\Omega_{R}\sin(k_{x})}{E}\right)-\hat{k}_{y}\cos(k_{x})\sin(k_{z})\left(\mathrm{Tr}\\{\rho_{n}\sigma_{y}\\}-\frac{\Omega_{R}\cos(k_{y})}{E}\right)$
(S13)
$\displaystyle\;\;-\hat{k}_{z}\cos(k_{x})\sin(k_{y})\left(\mathrm{Tr}\\{\rho_{n}\sigma_{z}\\}-\frac{\Omega_{R}\cos(k_{z})}{E}\right)\Bigg{]}$
$\displaystyle=$
$\displaystyle\frac{-\hat{k}_{x}\sin(k_{x})\sin(k_{y})\sin(k_{z})\sigma_{x}+\hat{k}_{y}\cos(k_{x})\cos(k_{y})\sin(k_{z})\sigma_{y}+\hat{k}_{z}\cos(k_{x})\sin(k_{y})\cos(k_{z})\sigma_{z}}{2\left(\sin^{2}k_{x}+\cos^{2}k_{y}+\cos^{2}k_{z}\right)^{3/2}}$

where the final closed form expression for the Berry curvature comes from
substituting Eq. S11, containing the expression for $\rho_{n}$. We then plot
this in the main text to see the monopole behavior of Weyl points. The
curvature at the Weyl point diverges, and integrating around this singularity
leads to the quantized Chern number that characterizes the topological effects
of this system.

Figure S3: Total power in the photonic molecule for different photon loss
rates. The driven-dissipative system reaches a quasi-steady state, with the
intra-cavity power displaying oscillations, but not fully decaying or
diverging.