# Dirac mass induced by optical gain and loss

Letian Yu  Division of Physics and Applied Physics, School of Physical and
Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  Haoran Xue  Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University,
Singapore 637371, Singapore  Ruixiang Guo  Division of Physics and Applied
Physics, School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore 637371, Singapore  Centre for Disruptive Photonic
Technologies, Nanyang Technological University, Singapore 637371, Singapore
Eng Aik Chan  Division of Physics and Applied Physics, School of Physical and
Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  Centre for Disruptive Photonic Technologies, Nanyang Technological
University, Singapore 637371, Singapore  Yun Yong Terh  Division of Physics
and Applied Physics, School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore  Cesare Soci
<EMAIL_ADDRESS>Division of Physics and Applied Physics, School of Physical
and Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  Centre for Disruptive Photonic Technologies, Nanyang Technological
University, Singapore 637371, Singapore  Baile Zhang<EMAIL_ADDRESS>Division of Physics and Applied Physics, School of Physical and Mathematical
Sciences, Nanyang Technological University, Singapore 637371, Singapore
Centre for Disruptive Photonic Technologies, Nanyang Technological University,
Singapore 637371, Singapore  Y. D. Chong<EMAIL_ADDRESS>Division of
Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371, Singapore  Centre for
Disruptive Photonic Technologies, Nanyang Technological University, Singapore
637371, Singapore

Mass is commonly regarded as an intrinsic property of matter, but modern
physics shows that particle masses have complex origins [1]. Elementary
particles acquire mass from couplings to other fields: most fermions and
bosons receive mass from the Higgs field [2], as well as other interactions
(e.g., quarks gain mass from interactions with gluons) [1]. In low-energy
physics, quasiparticles behaving like fundamental particles can arise in
crystalline lattices, such as relativistic Dirac quasiparticles in graphene
[3]. Mass can be imparted to these quasiparticles by various lattice
perturbations [4, 5, 6, 7]. By tailoring lattice properties, we can explore
otherwise-inacessible phenomena, such as how particles behave when
Hermiticity, the symmetry responsible for energy conservation, is violated [8,
9]. Non-Hermiticity has long seemed incompatible with mass generation; when
Dirac points are subjected to energy-nonconserving perturbations, they
typically become exceptional points [10, 11, 12] instead of opening a mass
gap. Here, we show experimentally that Dirac masses can be generated via non-
Hermiticity. We implement a photonic synthetic lattice with gain and loss
engineered to produce Dirac quasiparticles with real mass [13, 14]. By tuning
this mass, we demonstrate a crossover from conical to non-conical diffraction
[15], topological boundary states between domains of opposite Dirac mass [16,
17, 3], and anomalous tunneling into potential barriers [18].

It is a remarkable fact that quasiparticles in periodic media can exhibit many
key features of fundamental particles. Famously, graphene-like lattices host
quasiparticles that behave like massless relativistic 2D particles, tied to
Dirac point degeneracies in the bandstructure [3]. These quasiparticles have
been realized in solid-state materials [19, 20], and in synthetic structures
such as photonic lattices [21, 22]; they have enabled the experimental
exploration of various universal properties of massless Dirac particles, such
as the half-integer quantum Hall effect [23], Zitterbewegung [24], Klein
tunneling [18], and more [25]. Adding mass to the quasiparticles is equivalent
to lifting the Dirac point degeneracy, and may be accomplished by breaking
inversion symmetry [4], breaking time-reversal symmetry [5, 6], or spin-orbit
coupling [7]. A profound consequence of a nonzero Dirac mass is that its sign
influences the topology of the bandstructure [16, 17, 3]; hence, each of the
three aforementioned gap-opening mechanisms is the basis for a class of
topological materials: valley Hall insulators [26], Chern insulators [5, 6],
and quantum spin Hall insulators [7].

Are there any other mechanisms for adding mass to a relativistic particle or
quasiparticle? Quasiparticles in synthetic structures are a promising
candidate for exploring this question, as they can be engineered with physical
properties not easily achieved in natural settings [27]. For example, one can
design structures that deliberately break Hermiticity, the symmetry normally
responsible for the conservation of energy in detail, by incorporating gain
and/or loss (i.e., energy transfer with an unmonitored external system). It is
notable that some non-Hermitian lattices, such as those possessing “unbroken
parity/time reversal symmetry”, can host stable quasiparticles that do not
blow up or decay over time [8, 9].

Unfortunately, Dirac quasiparticles are usually spoiled by gain and loss.
Previous studies have found that the introduction of gain/loss turns Dirac
points into pairs of exceptional points (EPs) or rings of EPs [10, 11, 12].
EPs are the most generic type of non-Hermitian degeneracy, and behave very
differently from Hermitian degeneracies like Dirac points [28, 29, 30]. Band
energies do not remain real everywhere in the vicinity of an EP, destabilizing
quasiparticle wavefunctions [10, 28]. EPs also fuse bands together,
invalidating the assumptions behind standard theories of band topology, and
necessitating alternative formulations of non-Hermitian band topology in which
Dirac quasiparticles play no special role [31, 32, 30, 33].

Figure 1: Scheme for realizing a non-Hermitian synthetic lattice. a, Schematic
of the experiment. Two optical fiber loops are coupled by a 50/50 beamsplitter
(BS). Each fiber loop has an intensity modulator (IM), which can alter the
amplitude of individual pulses. The short loop has an additional phase
modulator (PM) to vary the phase of incoming signals. The pulses are measured
by photodetectors (PDs) placed just before the BS. b, The synthetic lattice
that the pulse train evolution maps onto. The index $n$ in each pulse train
maps to a spatial position, and the number of round trips $m$ maps to time.
Inset: definitions of wavefunction amplitudes in the synthetic lattice. c–e,
Schematic of the distribution of gain/loss in the synthetic lattice (top), and
the resulting quasienergy spectrum (bottom). The Hermitian case (c) exhibits a
Dirac point at $k=0$, $\phi=\pi$. Gain/loss can turn this into a pair of
exceptional points (d) or a real mass gap (e). In d and e, the gain/loss level
is $g=0.41$. Only the two bands closest to quasienergy $\theta=0$ are plotted.

In this paper, we experimentally demonstrate that gain and loss can be used to
create stable massive relativistic quasiparticles. We implement a looped
optical fiber apparatus hosting a synthetic lattice [34] whose bandstructure
has a Dirac point in the Hermitian limit. Applying gain and loss, according to
a specific “semi-Hermitian” symmetry [13, 14], opens a relativistic mass gap
(see Supplementary Information). While the overall Hamiltonian is non-
Hermitian, the two participating bands obey an effective Dirac Hamiltonian
with a real relativistic mass, whose sign and magnitude can be controlled via
the gain/loss level. We verify this by experimentally observing a transition
from conical to non-conical diffraction, without long-run amplification or
decay. Although it was previously known that certain non-Hermitian bands can
exhibit real band energies (e.g., through unbroken parity/time reversal
symmetry [8, 9]), the use of gain/loss to generate a real mass for a Dirac
quasiparticle has never been demonstrated, to our knowledge.

Furthermore, we demonstrate that the gain/loss induced Dirac mass affects the
topology of the non-Hermitian bandstructure. Applying different gain/loss
distributions to adjacent lattice segments, corresponding to Dirac masses of
opposite signs, we observe the emergence of a topological boundary state [16,
35, 13, 14] not present in the Hermitian limit. Recently, there have been
various theoretical proposals for using gain/loss to induce topological
transitions [36, 37, 38, 39] in terms of novel non-Hermitian formulations of
band topology [31, 32, 30, 33], which have been experimentally implemented
with acoustic crystals [40], plasmonic waveguides [41], elastic and mechanical
materials [42] and electric circuits [43]. The present topological transition,
however, takes place through the standard route of a real Dirac mass
determining the Chern-type topology of the participating bands [16, 17].

Finally, we investigate the interplay between non-Hermiticity and the physics
of Dirac quasiparticles. In the Hermitian regime, one of the distinguishing
features of Dirac quasiparticles is the ability to efficiently “Klein tunnel”
across a scalar potential barrier [18]. Our experimental and theoretical
results show that the non-Hermitian massive Dirac quasiparticles can exhibit
Klein tunneling. However, this is contingent on semi-Hermiticity remaining
unbroken at the boundary. In certain lattice configurations, the interface
violates this symmetry (although the bulk domains individually preserve semi-
Hermitian symmetry and thus host real Dirac masses), giving rise to an
anomalous variant of Klein tunneling in which flux is not conserved at the
interface.

Our experiment is based on a pair of coupled looped optical fibers (Fig. 1a
and Extended Data Fig. 1). A single optical pulse is initially injected into
Loop 1, which is connected to the shorter Loop 2 by a 50/50 beamsplitter. The
initial pulse evolves into two pulse trains (one in each loop), which undergo
repeated splitting and recombination at the beamsplitter. The evolution of the
pulse train maps onto a discrete-time “quantum walk” of a particle in a 1D
synthetic lattice (Fig. 1b), as established in previous works [34, 44]. The
discrete position along each pulse train maps to the discrete lattice position
$n$; the number of round trips the pulses have undergone map to the discrete
time step $m$; and for each $(n,m)$, the pulses in Loop 1 and 2 represent
right- and left-moving wave amplitudes in the synthetic lattice, denoted by
$v_{n}^{m}$ and $u_{n}^{m}$ respectively (Fig. 1b, inset). For further details
of the experimental setup, see Methods.

Figure 2: Evolution of a gaussian wavepacket in the synthetic lattice. a,
Evolution in the Hermitian lattice ($g=0$), for an initial wavepacket centered
at $k=0$, $\phi=\pi$. Conical diffraction (red arrows) occurs in both
simulations (left panel) and experimental data (right panel). b, Evolution in
the non-Hermitian lattice ($g=0.41$), corresponding to the massive
relativistic quasiparticles of Fig. 1e. For the same initial wavepacket as in
a, the wavefunction spreads throughout the light cone (red arrows). Red dashes
in a–b have slopes equal to the effective light speed $c=1/\sqrt{2}$ (see
Supplementary Information). c, Similar to b, but with the initial wavepacket
centered at $k=\phi=\pi$, where the band energies are non-real. d–f, Total
intensity of the pulse train versus time step $m$. Conservation of total
intensity in e shows that the non-Hermitian quasiparticles behave like
Hermitian particles, unlike the exponential intensity increase seen in f. In c
and f, we perform the experiment with an additional loss on both fiber loops
to avoid blow-up, and compensate for the loss when plotting.

Several electro-optical modulators, attached to the fiber loops, act as phase
modulators (PMs) and intensity modulators (IMs) (Fig. 1a). The PM in Loop 2
applies a time-dependent phase modulation

$\Phi(m)=(-1)^{m}\phi,$ (1)

where $\phi$ is a tunable phase parameter. The IMs, acting alongside fiber
amplifiers [34], apply a spatially varying gain/loss factor $F(n)$ to each
loop. The resulting evolution equations are

$\displaystyle\begin{aligned}
u_{n}^{m+1}&=\frac{1}{\sqrt{2}}(u_{n+1}^{m}+iv_{n+1}^{m})\,F(n)\,e^{i\Phi(m)},\\\
v_{n}^{m+1}&=\frac{1}{\sqrt{2}}(iu_{n-1}^{m}+v_{n-1}^{m})\,F(n-1).\end{aligned}$
(2)

For each $n$, we let $F(n)$ take two possible values,

$F(n)=\begin{cases}e^{g/2},&\textrm{(gain)}\\\
e^{-g/2},&\textrm{(loss)},\end{cases}$ (3)

where $g$ parameterizes the gain/loss level.

Consider the Hermitian case, $g=0$. This lattice is translationally symmetric
in both space and time, with period $\Delta n=\Delta m=2$ (Fig. 1b, inset). We
take a Floquet-Bloch ansatz [45]

$\begin{pmatrix}u_{n}^{m}\\\ v_{n}^{m}\end{pmatrix}=\begin{pmatrix}U\\\
V\end{pmatrix}\exp\left[-\frac{im\theta}{\Delta m}\,+\frac{ikn}{\Delta
n}\right],$ (4)

where $k$ is the quasimomentum and $\theta$ the quasienergy (both real).
Substituting into Eq. (2), we obtain the Floquet bandstructure shown in Fig.
1c, where $\theta$ is plotted against $k$ and the phase parameter $\phi$
(which serves as an additional parametric dimension [44]). It exhibits a Dirac
point at $k=\theta=0$, $\phi=\pi$. (There is another Dirac point at
$\theta=k=\pi$, $\phi=0$, which we do not use.) Here, we plot only the
quasienergy bands near the Dirac point; a full band diagram is shown in
Extended Data Fig. 2.

Next, we apply gain/loss to the lattice. We first consider a trivial case as
shown in the upper panel of Fig. 1d, the spatial period remains $\Delta n=2$.
The Dirac point turns into a pair of exceptional points (Fig. 1d, lower
panel), joined by a “bulk Fermi arc” along $\mathrm{Re}[\theta]=0$, $\phi=\pi$
[46]. Similar non-Hermitian bandstructures have been found in earlier works
[47]. Near the exceptional line, the quasiparticles have
$\mathrm{Im}[\theta]\neq 0$, and blow up or decay over time.

Figure 3: Boundary states induced by gain and loss. a, Schematic of a
synthetic lattice with two domains separated by a boundary (dashes). The
effective Dirac Hamiltonians have mass $M<0$ and $M>0$ in the left and right
domains, respectively. b, Floquet band diagram for a finite sample of the
lattice shown in a, with total size $N=59$ and gain/loss level $g=0.59$. A
chiral dispersion relation, corresponding to gain/loss induced boundary
states, spans the gap. c, Spatial distribution of $|u_{n}|^{2}+|v_{n}|^{2}$
for the mid-gap boundary state, calculated with different values of $g$. d,
Spatial profiles of $|u_{n}^{m}|^{2}+|v_{n}^{m}|^{2}$ for the theoretical
boundary eigenstate for $g=0.41$ (red line), and the corresponding
experimental data time-averaged over $40\leq m\leq 70$ (blue dots). In c–d, we
normalize the maximum intensity for individual subplots to 1. e–g, Time
evolution of a pulse injected at $n=0$, for $g=0.22$, $0.41$, and $0.59$. In
simulations (left panels) and experimental data (right panels), the pulse
spreads into a wavefunction localized at the boundary, with localization
length following the trend shown in c. In c–g, we set $\phi=\pi$.

Another gain/loss distribution with $\Delta n=4$ is shown in the upper panel
of Fig. 1e (the temporal period remains $\Delta m=2$). The Floquet-Bloch
ansatz still has the form (4), but with a four-component wavefunction (see
Supplementary Information). The spectrum now has a real quasienergy gap (Fig.
1e, lower panel). In the vicinity of $k=0$, $\phi=\pi$, the two plotted bands
are numerically verified to have exactly real $\theta$. Moreover, they are
governed by an effective Dirac Hamiltonian

$H_{\mathrm{eff}}=M(g)\sigma_{1}+\frac{1}{2}v_{D}k\sigma_{2}-v_{D}(\phi-\pi)\sigma_{3},$
(5)

where $\sigma_{1,2,3}$ are Pauli matrices, $v_{D}$ is the Dirac velocity, and
$M(g)=\cosh g-1$ is a real Dirac mass that varies with the gain/loss level
(see Extended Data Fig. 3 and Supplementary Information). We also verified
that the two participating bands are orthogonal (using the standard definition
of the wavefunction inner product) for each $k$ and $\phi$. These properties
result from a “semi-Hermitian” symmetry which protects both the real-
valuedness of eigenvalues and the pairwise orthogonality of eigenvectors, as
shown in previous theoretical works [13, 14] (see Methods).

For Fig. 1e, note that the bands are not real everywhere: around $k=\pm\pi$,
semi-Hermiticity is spontaneously broken [29] and the $\theta$’s are complex.

To study how quasiparticles evolve in the synthetic lattice, we use a
diffusion protocol to prepare pulse trains corresponding to gaussian
wavepackets centered around some $k$, with $\phi$ and other parameters fixed
(see Methods). We do not excite a specific band (which would involve specially
preparing initial pulse sequences [48]); instead, the initial wavepacket
excites all Bloch states in all bands within a wavenumber window $\Delta
k\approx 0.12\pi$.

For the Hermitian case ($g=0$), with the initial wavepacket centered at $k=0$,
the wavefunction spreads in two well-defined beams (Fig. 2a). Simulation
results obtained via Eq. (2) (left panel) agree well with the experimental
data (right panel). Such “conical diffraction” is a signature of a massless
relativistic quasiparticle [15]. The propagation speed is consistent with the
effective light speed $c=1/\sqrt{2}$ calculated from the Floquet bandstructure
(see Supplementary Information). Note that the $g=0$ case has two distinct
bands, which form a Dirac cone. Fig. 2d shows a logarithmic plot of the total
intensity $I(m)=\sum_{n}\left(|u_{n}^{m}|^{2}+|v_{n}^{m}|^{2}\right)$ versus
$m$. The graphs for both simulation and experiment are approximately constant,
consistent with energy conservation.

Next, we apply the semi-Hermitian gain/loss distribution of Fig. 1e, which is
predicted to generate a real Dirac mass. For the same initial conditions, we
observe that the conical diffraction disppears, and the wavefunction instead
spreads over the entire light cone (Fig. 2b). The overall results are
consistent with theoretical expectations: in the two participating bands, the
quasiparticles become massive and thus possess a range of group velocities
below the effective light speed. There are also two other bands, whose
quasienergies are real near $k=0$ (Extended Data Fig. 2d). In Fig. 2e, we see
that the total intensity remains stable over long times, similar to the
Hermitian case of Fig. 2d and consistent with the Bloch states having real
quasienergies.

For comparison, Fig. 2c shows the evolution for a wavepacket centered at
$k=\pi$, where the bands have non-real quasienergies. All other parameters,
including the initial conditions, are the same as in Fig. 2d. In the
experiment, we apply an additional uniform loss to both fiber loops to prevent
an intensity blow-up, and compensate for it when plotting (see Methods). The
results reveal exponential amplification over time (Fig. 2c, f) [49].

One of the most remarkable features of massive Dirac quasiparticles is that
they act as “building blocks” for topological band insulators. A massive 2D
Dirac Hamiltonian, Eq. (5), contributes $\pm\textrm{sgn}(m)/2$ to the Chern
numbers of the two participating bands [17]. When there are two adjacent
domains with opposite signs of $m$, topological states appear at the boundary
[16, 35].

To show that the gain/loss induced Dirac masses can similarly create
topological boundary states, we implement a lattice consisting of two domains,
whose gain/loss distributions are displaced relative to each other by a
quarter of a unit cell (Fig. 3a). From the derivation of the Floquet
bandstructure, we can show that this displacement flips the sign of $M$ (see
Supplementary Information). (In an earlier theoretical analysis of a non-
Hermitian 1D topological lattice, a similar displacement was also shown to
induce a topological transition [38].) In Fig. 3b, the quasienergy bands for a
finite sample are plotted against $\phi$ (which serves as a momentum-like
variable). This reveals a branch of boundary states spanning the gain/loss
induced band gap. They have a chiral dispersion relation, as expected of the
topological boundary states generated by a Dirac mass inversion, even though
$\mathrm{Im}(\theta)$ does not strictly vanish because the domain wall breaks
the semi-Hermitian symmetry [38]. From the spatial distributions (Fig. 3c), we
see that the localization length is affected by the gain/loss level $g$, which
governs the size of the bulk gap. We emphasize that these topological boundary
states are solely induced by gain/loss, since the bulk gap is closed for
$g=0$.

To probe this phenomenon experimentally, we inject a single initial pulse at
the boundary ($n=0$), which has strong spatial overlap with the boundary
states. In Fig. 3d, we plot the intensity distribution for $g=0.41$ (blue
circles), obtained by a normalized time average of
$|u_{n}^{m}|^{2}+|v_{n}^{m}|^{2}$ (see Methods). The profile fits the
theoretically calculated spatial distribution for the boundary state (red
line). Finally, from the time evolution data for different values of $g$ (Fig.
3e–g), we observe that the localization length decreases with $g$, in
agreement with Fig. 3c.

Figure 4: Klein tunneling of non-Hermitian quasiparticles. a, Schematic of
Klein tunneling with massive Dirac cones. An incident quasiparticle of energy
$E$ (left) impinges on a domain with scalar potential $V$ (right). b,
Schematic of a synthetic lattice with a gain/loss distribution supporting a
massive Dirac cone. In the right domain, shaded in pink, a uniform scalar
potential $V$ is applied. c, Experimentally measured (cross) and simulated
(dot) transmittance for the lattice configurations shown in d–f. d–f,
Scattering results. In all cases, the right domain has an additional scalar
potential $V$, but the sites around the interface are laid out differently
(top). In the simulated (bottom left) and experimentally-measured (bottom
right) intensity plots, reflection is found to be suppressed, while the
transmittance is unity (d), damped (e), and amplified (f) respectively. The
lattice parameters are $\phi=\pi$, $g=0.24$, $k=0.1\pi$, and $V=0.2\pi$.

Finally, we investigate the scattering properties of the non-Hermitian Dirac
quasiparticles. In the Hermitian setting, Dirac particles can efficiently
traverse strong potential barriers (Fig. 4a), an intriguing phenomenon known
as Klein tunneling [18]. However, it is also possible for semi-Hermitian Dirac
quasiparticles to exhibit an anomalous form of Klein tunneling, whereby energy
is conserved in the bulk domains but not at the domain wall [14]. Here, we
demonstrate that the non-Hermitian synthetic lattice can exhibit both standard
Klein tunneling and its anomalous counterpart.

Our experiment, shown schematically in Fig. 4b, involves an effective scalar
potential barrier (pink background) implemented by a uniform phase shift
applied at those $n$ [50] (see Methods). There are several possibilities for
the lattice configuration at the interface. In the baseline case (Fig. 4d),
the wavepacket incident from the left experiences nearly unity transmission,
with reflection strongly suppressed. Alternatively, we consider configurations
where the gain/loss is flipped across the interface (keeping the sign of $M$
unchanged). In such cases, we find that the reflection is still strongly
suppressed, but the transmitted beam is either amplified (Fig. 4e) or damped
(Fig. 4e). This amplification or damping depends on the gain/loss of the
interface’s central site, and occurs only while the wavepacket is impinging on
the interface; within the bulk domains, the intensity is conserved since the
effective mass $M$ is real. Our experimental results for the net transmittance
are in excellent agreement with simulations, as shown in Fig. 4c.

In conclusion, we have demonstrated a novel mechanism for generating real
Dirac masses, using a synthetic photonic lattice containing a specific
gain/loss distribution and Floquet modulations. Whereas previous studies of
non-Hermitian lattices showed Dirac points turning into exceptional points,
the quasiparticles in this lattice possess the dynamical and topological
features of massive Dirac particles. Future studies could further explore
their similarities and differences relative to real relativistic particles.
Aside from the anomalous Klein tunneling studied in this work, it would be
interesting to investigate how they form Landau levels [13]. Different
varieties of massive non-Hermitian quasiparticles may also be realizable using
more complicated Floquet lattices [51].

Methods

Experimental setup

A schematic of the experiment is shown in Extended Data Fig. 1. A distributed
feedback laser diode operating at $\lambda=1550\,\textrm{nm}$ sends a
continuous-wave (CW) signal into our system. To prevent backscattering into
the laser source, an isolator is placed immediately after it. The input signal
is shaped into rectangular pulses of width $100\,\textrm{ns}$ by an electro-
optical modulator (EOM). Subsequently, the signal is enhanced by an erbium-
doped fiber amplifier (EDFA), and then cleaned up by a band pass filter (BPF)
and a polarizer.

Before developing the signal into a synthetic lattice, we take additional
steps to stabilize the amplification provided by the EDFA, based on the
protocol reported in Ref. 34. A quasi-continuous train of pulses, each
separated by roughly one round trip time, is injected into both loops prior to
the actual experiment. During this initial phase, the EOMs in both loops are
set to 0% transmission, allowing the EDFAs to adapt to the input power level.
After about $8\,\textrm{ms}$, we set the EOM at the signal generation side to
0%, so that only the last pulse from the warm-up phase is retained for the
experiment.

After the warm-up phase, the pulses begin to propagate within the coupled
fiber loops. The average round trip time is about $20\,\mu\textrm{s}$, with a
difference of about $300\,\textrm{ns}$ between the loops. To further stablize
the pulses, a strong pilot signal at $1535\,\textrm{nm}$ (the wavelength
corresponding to the peak of EDFA emission spectrum) is applied in both loops
right after the fiber spools. This signal can be cleaned by a following BPF
due to its large wavelength difference with our operating wavelength of
$\lambda=1550\,\textrm{nm}$. Polarizers and polarizing beamsplitters are
employed to ensure that only a single polarization is present in both loops.

We remotely control the electro-optical modulator (EOM) and phase modulators
(PM) to implement the gain/loss and phase modulation profiles described in the
main text. We also use these to implement absorbing boundary conditions for
the synthetic lattice, by setting the transmission in both loops to 0% at the
start or near the end of each round trip. These settings also help prevent
noise build-up inside the loops.

To observe the wavefunction dynamics, we monitor the pulses circulating in
each loop with photodiodes (PDs) placed before the central coupler.

Semi-Hermitian Floquet bandstructures

The non-Hermitian Floquet bandstructures we consider (configuration in Fig.
1e) can host a massive Dirac Hamiltonian because of a pair of symmetries we
call “semi-Hermiticity” [13, 14]. In terms of their action on a $4\times 4$
evolution operator $U$, the symmetries are:

$\displaystyle\Sigma_{0}U\Sigma_{0}$ $\displaystyle=(U^{-1})^{\dagger}$ (6)
$\displaystyle\\{iU,\Sigma_{3}\Sigma_{1}T\\}$ $\displaystyle=0,$ (7)

where $T$ is the complex conjugation operator and

$\Sigma_{\mu}=\begin{bmatrix}0&\sigma_{\mu}\\\
\sigma_{\mu}&0\end{bmatrix},\quad\mu=0,1,2,3.$ (8)

Here, $\sigma_{0}$ is the $2\times 2$ identity matrix, and $\sigma_{1}$,
$\sigma_{2}$, and $\sigma_{3}$ are the Pauli matrices.

We first consider Eq. (S11), which is an instance of pseudo-Hermiticity [9].
$U$ has a set of right eigenvectors, obtained from the eigenproblem

$U|\psi_{n}\rangle=e^{-i\theta_{n}}|\psi_{n}\rangle,$ (9)

where $\theta_{n}\in\mathbb{C}$ is the quasienergy. The corresponding left
eigenvectors satisfy

$\displaystyle\langle\phi_{n}|U$
$\displaystyle=\langle\phi_{n}|e^{-i\theta_{n}}$ (10)
$\displaystyle\Rightarrow\quad\left(U^{-1}\right)^{\dagger}|\phi_{n}\rangle$
$\displaystyle=e^{-i\theta_{n}^{*}}|\phi_{n}\rangle.$ (11)

Therefore Eq. (S11) implies that

$U\Big{(}\Sigma_{0}|\phi_{n}\rangle\Big{)}=\Sigma_{0}(U^{-1})^{\dagger}|\phi_{n}\rangle=e^{-i\theta_{n}^{*}}\Big{(}\Sigma_{0}|\phi_{n}\rangle\Big{)}.$
(12)

Thus, for each solution to Eq. (9) with quasienergy $\theta_{n}$, there is a
solution with quasienergy $\theta_{n}^{*}$. The quasienergies are either real,
or come in complex conjugate pairs.

Eq. (S12) is an instance of anti-PT symmetry. It is equivalent to

$U^{*}\Sigma_{3}\Sigma_{1}=\Sigma_{3}\Sigma_{1}U.$ (13)

Using this with Eq. (9) gives

$\displaystyle\Sigma_{3}\Sigma_{1}U|\psi_{n}\rangle$
$\displaystyle=U^{*}\Sigma_{3}\Sigma_{1}|\psi_{n}\rangle=e^{-i\theta_{n}}\Sigma_{3}\Sigma_{1}|\psi_{n}\rangle$
(14) $\displaystyle UT\Sigma_{3}\Sigma_{1}|\psi_{n}\rangle$
$\displaystyle=e^{i\theta_{n}^{*}}T\Sigma_{3}\Sigma_{1}|\psi_{n}\rangle.$ (15)

Hence, there is an eigenstate $T\Sigma_{3}\Sigma_{1}|\psi_{n}\rangle$ with
quasienergy $-\theta_{n}^{*}$. Moreover, we can show that this is orthogonal
to the original eigenstate, as follows: let

$|\psi_{n}\rangle=\begin{pmatrix}p\\\ q\end{pmatrix},$ (16)

where $p$ and $q$ are two-component vectors. Then

$\displaystyle\langle\psi_{n}|T\Sigma_{3}\Sigma_{1}|\psi_{n}\rangle$
$\displaystyle=\begin{pmatrix}p^{\dagger}&q^{\dagger}\end{pmatrix}\begin{pmatrix}-i\sigma_{2}&0\\\
0&-i\sigma_{2}\end{pmatrix}\begin{pmatrix}p^{*}\\\ q^{*}\end{pmatrix}$ (17)
$\displaystyle=-i\left(p^{\dagger}\sigma_{2}p^{*}+q^{\dagger}\sigma_{2}q^{*}\right)$
(18) $\displaystyle=0\;\;\;\textrm{for any $p$, $q$}.$ (19)

Thus, the combination of Eqs. (S11) and (S12) can produce behavior analogous
to Hermiticity. If the pseudo-Hermiticity (S11) is unbroken, then (S12)
implies the existence of pairs of orthogonal eigenstates with real eigenvalues
$(\theta_{n},-\theta_{n})$ where $\theta_{n}\in\mathbb{R}$ [13].

We thus see that the combination of two non-Hermitian symmetry
conditions—unbroken pseudo-Hermiticity and anti-PT symmetry—yield the two
features (real eigenvalues and pairwise-orthogonal eigenvectors) consistent
with the existence of an effective Dirac Hamiltonian. Satisfying both
conditions simultaneously requires at least a $4\times 4$ matrix; hence, aside
from the two orthogonal bands participating in the Dirac cone, there are two
other auxiliary bands (which are likewise orthogonal to each other).

Note that a $2\times 2$ matrix cannot give this behavior. In the $2\times 2$
case, having two real eigenvalues with orthogonal eigenvectors implies that
the Hamiltonian is Hermitian (for a 2D vector space, pairwise orthogonality
simply implies that the two eigenvectors form an orthogonal basis).

State preparation

For the experiments of Fig. 2, we prepare initial wavepackets designed to
excite quasiparticles around a target momentum $k$. This is achieved using the
procedure described in Ref. 48. We start with a single pulse produced by the
previously-described signal generation system. This is allowed to evolve
according to the evolution equations

$\displaystyle u_{n}^{m+1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(u_{n+1}^{m}+iv_{n+1}^{m}\right)\,e^{i\phi_{0}}\,F(m)$
(20) $\displaystyle v_{n}^{m+1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(iu_{n-1}^{m}+v_{n-1}^{m}\right),$ (21)

where

$F(m)=m\;(\textrm{mod}~{}2).$ (22)

This is equivalent to fully absorbing the signal inside the short loop in
every two time steps. To select the Bloch modes at wavenumber $k$, we choose
the phase modulation $\phi_{0}$ such that

$k=2\pi+2\phi_{0}.$ (23)

Extended Data Fig. 5a shows the evolution of the pulses, and Extended Data
Fig. 5b shows the intensity distribution at the end of the preparation stage
($m=55$). The experimental results are clearly in excellent agreement with
simulations, as well as fitting a Gaussian envelope

$f(n)=A\exp\left[-\frac{(n-n_{0})^{2}}{2\sigma^{2}}\right],$ (24)

where $n_{0}$ is the center of the wavepacket and $\sigma$ is its spatial
width. In our case, $\sigma\approx 3.7$, corresponding to a spectral width of
$\delta\theta\approx 0.12\pi$.

Probing states with complex quasienergies

In Fig. 2c,f, we plot the time evolution of a wavepacket centered at $k=\pi$,
where the band quasienergies are non-real. In contrast to the $k=0$ case,
where the quasiparticles behave like energy-conserving Dirac particles, in
this case the wavefunction is predicted to undergo exponential amplification.

Experimentally, it is not desirable to access the amplifying regime, as the
amplification of the EDFAs will saturate [34]. To bypass this problem, we
employ a simple compensation method previously used, e.g., in Ref. 52. We
apply an additional loss, which is uniform in both space and time, to both
loops, at a loss level that maintains nearly constant overall power throughout
the experiment (based on the PD observations). The evolution equations are
modified to

$\displaystyle u_{n}^{m+1}$
$\displaystyle=\frac{1}{\sqrt{2}}(u_{n+1}^{m}+iv_{n+1}^{m})e^{i\Phi(m)}F(n)e^{-\gamma/2}$
(25) $\displaystyle v_{n}^{m+1}$
$\displaystyle=\frac{1}{\sqrt{2}}(iu_{n-1}^{m}+v_{n-1}^{m})F(n-1)e^{-\gamma/2},$
(26)

where $\exp(-\gamma/2)$ is the damping factor over one time step. The uniform
loss does not alter the bandstructure, apart from adding a constant imaginary
part to the quasienergy of every eigenstate, equivalent to a spatially uniform
decay factor $\exp(-\gamma m/2)$.

In Extended Data Fig. 4, we plot the Fourier spectra obtained from the
numerically simulated evolving wavefunctions, for the three different unit
cell configurations of Fig. 1c–e with $\phi=\pi$. The results show good
agreement with the theoretical Floquet bandstructures (the imaginary parts of
the bands manifest as bright regions in the Fourier spectra). Experimentally
accessing the corresponding phase information is impractical with our current
system parameters; with a 20 $\mu\textrm{s}$ average round trip time and 8 ms
warm-up interval, a reference field with coherence length of up to 200 km
would be required.

In Extended Data Fig. 6a, we plot the raw (uncompensated) time evolution of a
wavepacket subject to this uniform loss. The total intensity does not undergo
exponential amplification, as shown in Extended Data Fig. 6b. We then multiply
the wavefunctions by $\exp(\gamma m/2)$ to produce the experimental data shown
in Fig. 2.

The boundary states in Fig. 3 also have complex quasienergies, arising because
the lattice’s semi-Hermiticity is locally spoiled by the domain wall. The
boundary state experiences a weak decay rate, which we compensate by applying
a different gain $g^{\prime}$ to a single link at the boundary (see Extended
Data Fig. 7a),

$F(0)=e^{g^{\prime}/2}.$ (27)

This does not significantly alter $\mathrm{Re}[\theta]$ or the boundary state
wavefunction, as shown in Extended Data Fig. 7b,c.

Time-averaging of wavefunction intensities

In Fig. 3, we seek to compare the wavefunction intensity from a time evolution
experiment to the intensity of a numerically-calculated Floquet eigenstate.
Given that Floquet lattices are temporally modulated, such comparisons are
customarily done by time averaging the intensity of the dynamical
wavefunction. In our case, we take

$\frac{1}{T}\sum_{m}\frac{|u_{n}^{m}|^{2}}{I_{m}},$ (28)

where $T$ denotes the number of time steps being averaged over and
$I_{m}=\sum_{n}|u_{n}^{m}|^{2}+|v_{n}^{m}|^{2}$ denotes the total intensity at
time step $m$. For Fig. 3, the average is taken over $m\in[40,70]$.

Klein tunneling setup

Here, we provide additional details about the Klein tunneling results shown in
Fig. 4. In these experiments, we use an initial wavepacket at $k=0.1\pi$ with
a preparation stage of time step $m=105$. To form the barrier, we follow the
method of Ref. [50]: instead of having a single phase modulator in the short
loop, another phase modulator is added to the long loop. The evolution
equation is then modified to

$\displaystyle\begin{aligned}
u_{n}^{m+1}&=\frac{1}{\sqrt{2}}(u_{n+1}^{m}+iv_{n+1}^{m})\,F(n)\,e^{i\Phi(m)+iV/2},\\\
v_{n}^{m+1}&=\frac{1}{\sqrt{2}}(iu_{n-1}^{m}+v_{n-1}^{m})\,F(n-1)\,e^{iV/2},\end{aligned}$
(29)

where $V$ denotes the scalar potential (in Fig. 4, we use $V=0.2\pi$). The
factor of $1/2$ is due to the Floquet unit cell having a time step of $2$. The
transmittance and intensity plots for other values of $V$ are shown in
Extended Data Fig. 8.

In Fig. 4c and Extended Data Fig. 8b, we plot the transmittance and
reflectance, which respectively correspond to the intensities summed over
sites to the right and left of the interface. These intensities are evaluated
at time step $m=60$, after the scattering process has completed. One
complication is that during the wavepacket preparation stage, in addition to
the right-propagating wavepacket (residing in the upper band), we also excite
pulses of the opposite propagation direction (residing in the lower band), as
shown in the short loop intensity evolution in Extended Data Fig. 8c–e. These
additional excitations do not participate in the scattering process, and are
excluded when finding the reflectance and tramittance.

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Acknowledgements This work was supported by the Singapore Ministry of
Education (MOE) Tier 1 Grant No. RG148/20, and by the National Research
Foundation (NRF), Singapore under Competitive Research Programme NRF-
CRP23-2019-0005 and NRF-CRP23-2019-0007, and NRF Investigatorship NRF-
NRFI08-2022-0001. C.S. and R.G. acknowledge the support of the Quantum
Engineering Programme of the Singapore National Research Foundation, grant
number NRF2021-QEP2-01-P01.

Author Contributions H.X., B.Z., and Y.D.C. conceived the idea. H.X., L.Y.,
R.G., and E.A.C. designed and performed the experiment. L.Y. and R.G. analysed
the data. C.S., B.Z., Y.D.C. supervised the project. H.X., L.Y., R.G., Y.Y.T,
C.S., B.Z., and Y.D.C. contributed to the discussion of the results and
writing of the manuscript.

Data availability Correspondence and requests for materials should be
addressed to Cesare Soci, Baile Zhang or Y.D. Chong.

Code availability Correspondence and requests for materials should be
addressed to Cesare Soci, Baile Zhang or Y.D. Chong.

Competing Interests The authors declare no competing interests.

Additional Information Correspondence and requests for materials should be
addressed to Cesare Soci, Baile Zhang or Y.D. Chong.

Extended Data Figure 1: Schematic of the experiment. The abbreviations are as
follows: continuous-wave laser (CW), electro-optical modulator (EOM), erbium-
doped fiber amplifier (EDFA), photodiode detector (PD), band-pass filter
(BPF), polarization beam splitter (PBS), and phase modulator (PM). The boxes
containing arrows denote isolators, and the ovals labeled ‘50/50’ and ‘90/10’
denote optical couplers with the indicated splitting ratios. Extended Data
Figure 2: Floquet band diagrams of the synthetic lattice for different
gain/loss and phase distributions. a, No gain/loss ($g=0$). b,
Gain/loss/gain/loss. The Dirac point turns into a pair of EPs. The band
energies are non-real at $k=0$, $\phi=\pi$, while the band energies close to
$k=\pm\pi$, $\phi=\pi$ are real. c, Gain/loss/loss/gain. At $k=0$, $\phi=\pi$,
there are two orthogonal bands with real band energies, governed by a massive
Dirac Hamiltonian. In b–c, the gain/loss level is $g=0.41$. Extended Data
Figure 3: Dispersion relation for the Dirac quasiparticles. These Floquet band
diagrams are calculated using the evolution equations (orange) and the
effective Dirac Hamiltonian (blue; see Supplementary Information), for $g=0$
(left) and $g=0.41$ (right). a, Quasienergies versus $k$. b, Quasienergies
versus $\phi$.  Extended Data Figure 4: Optical energy evolution and Fourier
spectra for different gain/loss and phase distributions. a, No gain/loss
($g=0$), $\phi=\pi$. b, Gain/loss/gain/loss, $\phi=\pi$, $g=0.4$. c,
Gain/loss/loss/gain. At $k=0$, $\phi=\pi$, $g=0.4$. In a–c, the simulated
intensity evolutions are obtained with a single pulse injection in the long
loop at time $m=0$. In each subplot, the Fourier spectra is derived from
Fourier transformation of the simulation results on the left. All the other
parameters are the same as these in experiments. Extended Data Figure 5:
Preparation of pulse train for exciting quasiparticles. a, Time evolution of a
single initial pulse under Eqs. (20)–(23), for $\phi_{0}=-\pi$. b, Intensity
profile at $m=55$. Extended Data Figure 6: Pulse propagation in the symmetry-
broken regime. a, Raw data showing the evolution of a wavepacket centered at
$k=\pi$, with $g=0.4055$, $\phi=\pi$, and uniform loss rate $\gamma=0.223$. b,
Plot of the total intensity in the short loop versus $m$. Extended Data Figure
7: Behavior of boundary states. a, Schematic of the lattice with a domain
wall. Additional gain is added to $F(0)$ to compensate for the imaginary
component of the boundary state’s quasienergy. b, Floquet band diagrams for a
finite sample of the lattice (with $N=59$ sites and bulk gain/loss level
$g=0.59$), for different values of the compensating gain level
$g^{\prime}=0.22,0.41,0.59$. c, Distribution of $|u_{n}^{m}|^{2}$ for the
boundary eigenstate at these values of $g^{\prime}$. Extended Data Figure 8:
Pulse propagation at different potential barrier heights. a, Schematic of the
lattice configuration near the interface. b, Transmission and reflection
coefficient as a function of potential $V$. c-e, simulated intensity evolution
as the potential barrier varies from $0$ to $0.2\pi$. In c–e, we set
$\phi=\pi$, $g=0.4$, $k=0.1\pi$.

Supplementary Information for
“Dirac mass induced by optical gain and loss”

##  S1 Evolution equations for synthetic lattice

In this section, we discuss the derivation of the bandstructure for the non-
Hermitian synthetic lattice. This begins with the following evolution equation
from the main text:

$\displaystyle u_{n}^{m+1}$
$\displaystyle=\frac{1}{\sqrt{2}}(u_{n+1}^{m}+iv_{n+1}^{m})\,e^{i\Phi(m)}\,F(n)$
(S1) $\displaystyle v_{n}^{m+1}$
$\displaystyle=\frac{1}{\sqrt{2}}(iu_{n-1}^{m}+v_{n-1}^{m})\,F(n-1),$ (S2)

where

$\displaystyle\Phi(m)$ $\displaystyle=(-1)^{m}\phi,$ (S3) $\displaystyle F(n)$
$\displaystyle=\begin{cases}e^{g/2},&\textrm{(gain)}\\\
e^{-g/2},&\textrm{(loss)}.\end{cases}$ (S4)

The unit cell is shown in Fig. S1. This corresponds to the gain/loss
distribution corresponding to Fig. 1e of the main text.

Evidently, the lattice has a temporal period of $\Delta m=2$ and a spatial
period of $\Delta n=4$. Hence, let us expand Eqs. (S1)–(S2) into the following
two-step equations valid for $n\equiv 1~{}(\textrm{mod}~{}4)$ and $m\equiv
1~{}(\textrm{mod}~{}2)$:

$\displaystyle u_{n}^{m+2}$
$\displaystyle=\frac{1}{2}\left[u_{n+2}^{m}+iv_{n+2}^{m}+\left(-u_{n}^{m}+iv_{n}^{m}\right)e^{g+i\phi}\right]$
(S5) $\displaystyle v_{n}^{m+2}$
$\displaystyle=\frac{1}{2}\left[(iu_{n-2}^{m}+v_{n-2}^{m}+\left(iu_{n}^{m}-v_{n}^{m}\right)e^{g-i\phi}\right]$
(S6) $\displaystyle u_{n+2}^{m+2}$
$\displaystyle=\frac{1}{2}\left[u_{n+4}^{m}+iv_{n+4}^{m}+\left(-u_{n+2}^{m}+iv_{n+2}^{m}\right)e^{-g+i\phi}\right]$
(S7) $\displaystyle v_{n+2}^{m+2}$
$\displaystyle=\frac{1}{2}\left[iu_{n}^{m}+v_{n}^{m}+\left(u_{n+2}^{m}-v_{n+2}^{m}\right)e^{-g-i\phi}\right].$
(S8)

Extended Data Figure S1: Schematic of the synthetic lattice.

We then take the following ansatz:

$\begin{pmatrix}u_{n}^{m}\\\ v_{n}^{m}\\\ u_{n+2}^{m}\\\
v_{n+2}^{m}\end{pmatrix}=\begin{pmatrix}U_{0}\\\ V_{0}\\\ U_{1}\\\
V_{1}\end{pmatrix}e^{-\frac{im\theta}{2}}e^{\frac{ikn}{4}},$ (S9)

where $\theta$ is the Floquet quasienergy and $k$ is the Bloch wavenumber. The
Floquet quasienergy plays a role similar to the energy in the more familiar
Hamiltonian-based eigenvalue problems, except that $\mathrm{Re}[\theta]$ is a
periodic variable [45]. Substituting this into Eqs. (S5)–(S8) yields the
following eigenvalue problem:

$\frac{1}{2}\begin{pmatrix}-e^{g+i\phi}&ie^{g+i\phi}&e^{\frac{ik}{2}}&ie^{\frac{ik}{2}}\\\
ie^{g-i\phi}&-e^{g-i\phi}&ie^{-\frac{ik}{2}}&e^{-\frac{ik}{2}}\\\
e^{\frac{ik}{2}}&ie^{\frac{ik}{2}}&-e^{-g+i\phi}&ie^{-g+i\phi}\\\
ie^{-\frac{ik}{2}}&e^{-\frac{ik}{2}}&ie^{-g-i\phi}&-e^{-g-i\phi}\\\
\end{pmatrix}\begin{pmatrix}U_{0}\\\ V_{0}\\\ U_{1}\\\
V_{1}\end{pmatrix}=e^{-i\theta}\begin{pmatrix}U_{0}\\\ V_{0}\\\ U_{1}\\\
V_{1}\end{pmatrix}.$ (S10)

By solving this numerically, we can plot the Floquet band diagrams shown in
Fig. 1c–e of the main text. For the gain/loss distribution that we focus on
(Fig. S1), the evolution matrix satisfy the semi-Hermiticity symmetries
described in the Methods section of the main text. We can check numerically
that the Dirac-like eigenstates around $k=0$, $\phi=\pi$ have the stated
properties discussed in the main text—i.e., the energies are precisely real
and the two relevant bands closest to $\theta=0$ are orthogonal to each other.

In Extended Data Fig. 2, we plot Floquet band diagrams for different choices
of gain/loss distribution, obtained using such evolution matrix calculations:

  * •

In the Hermitian case, in addition to the Dirac point at $\theta=k=0$,
$\phi=\pi$, there is another Dirac point at $\theta=k=\pi$, $\phi=0$, which
does not play a significant role in the present work.

  * •

For the gain/loss/gain/loss configuration, similar to Fig. 1d of the main
text. The Dirac point turns into a pair of exceptionl points (EPs). The band
energies are non-real at $k=0$, $\phi=\pi$, while the band energies close to
$k=\pm\pi$, $\phi=\pi$ are real. If $g$ is further increased, all the band
quasienergies become complex.

  * •

For the semi-Hermitian gain/loss/loss/gain configuration, which adds a real
mass to the Dirac point at $k=0$, $\phi=\pi$. Note that the band energies are
non-real for $k=\pm\pi$; this case is used as a comparison in Fig. 2c,f of the
main text.

##  S2 Effective Dirac Hamiltonian

In this section, we consider the effective Dirac Hamiltonian arising from the
synthetic lattice’s evolution equation, Eq. (S10). As noted in the Methods
section of the main text, the semi-Hermitian symmetry

$\displaystyle\Sigma_{0}U\Sigma_{0}$ $\displaystyle=(U^{-1})^{\dagger}$ (S11)
$\displaystyle\\{iU,\Sigma_{3}\Sigma_{1}T\\}$ $\displaystyle=0,$ (S12)

ensures that the evolution matrix has exactly real eigenvalues with pairwise-
orthogonal eigenstates (in the unbroken-symmetry regime). When $g=k=0$ and
$\phi=\pi$, there is a Dirac point at $\theta=0$; for small but nonzero $g$,
the Dirac point degeneracy is lifted, but the semi-Hermitian symmetry remains
unbroken. Hence, the bandstructure near $k=0$, $\phi=\pi$ must be governed by
an effective Dirac equation with nonzero mass.

We can use Eq. (S10) to derive the Dirac mass. For $k=0$ and $\phi=\pi$, the
evolution matrix reduces to

$U_{0}(g)=\frac{1}{2}\begin{pmatrix}e^{g}&-ie^{g}&1&i\\\ -ie^{g}&e^{g}&i&1\\\
1&i&e^{-g}&-ie^{-g}\\\ i&1&-ie^{-g}&e^{-g}\end{pmatrix}.$ (S13)

Take the ansatz

$|\psi_{\pm}\rangle=\begin{pmatrix}1\\\ \pm 1\\\ a\\\ \pm a\end{pmatrix}.$
(S14)

Plugging this into the evolution equation
$U_{0}(g)|\psi_{\pm}\rangle=\exp(-i\theta)|\psi_{\pm}\rangle$, we obtain

$\displaystyle\frac{1}{2}\left[e^{g}\mp ie^{g}+a+ia\right]$
$\displaystyle=e^{-i\theta}$ (S15) $\displaystyle\frac{1}{2}\left[1\pm
i+e^{-g}a\mp ie^{-g}a\right]$ $\displaystyle=e^{-i\theta}a.$ (S16)

The two other rows of the eigenproblem give exactly the same equations.
Eliminating $a$, we obtain the following equation for $\theta$:

$e^{-2i\theta}-e^{-i\theta}(1\mp i)\cosh g\mp i=0.$ (S17)

This has quasienergy solutions

$\displaystyle\theta$
$\displaystyle=\pm\left[\frac{\pi}{4}-\cos^{-1}\left(\frac{\cosh
g}{\sqrt{2}}\right)\right]$ (S18) $\displaystyle=\pm\frac{g^{2}}{2}+O(g^{4}),$
(S19)

which have the property of being purely real for small $g$ (specifically,
$g<\cosh^{-1}\sqrt{2}$), and vanishing when $g=0$.

To develop an approximate expression for the effective Dirac equation, let us
take small deviations from the Dirac cone,

$k=\delta k,\quad\phi=\pi+\delta\phi,\quad\theta=\delta\theta,$ (S20)

and retain only terms up to first order in $\delta k$, $\delta\phi$, and
$\delta\theta$ in Eq. (S10):

$\frac{1}{2}\begin{pmatrix}e^{g}(1+i\delta\phi)&-ie^{g}(1+i\delta\phi)&1+\frac{i\delta
k}{2}&i-\frac{\delta k}{2}\\\
-ie^{g}(1-i\delta\phi)&e^{g}(1-i\delta\phi)&i+\frac{\delta
k}{2}&1-\frac{i\delta k}{2}\\\ 1+\frac{i\delta k}{2}&i-\frac{\delta
k}{2}&e^{-g}(1+i\delta\phi)&-ie^{-g}(1+i\delta\phi)\\\ i+\frac{\delta
k}{2}&1-\frac{i\delta k}{2}&-ie^{-g}(1-i\delta\phi)&e^{-g}(1-i\delta\phi)\\\
\end{pmatrix}|\Psi\rangle\approx(1-i\delta\theta)|\Psi\rangle.$ (S21)

Note that the evolution matrix on the left side of Eq. (S21), derived from a
Taylor expansion, spoils the semi-Hermiticity symmetry (S11)–(S12). Hence, our
results for the Dirac parameters will only be approximations. Now, let us take

$|\Psi\rangle=\begin{pmatrix}|\psi_{0}\rangle\\\
|\psi_{1}\rangle\end{pmatrix},$ (S22)

where $|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ are two-component vectors. We
can rewrite Eq. (S21) into

$\displaystyle
e^{g}\Big{[}(\sigma_{0}-i\sigma_{1})+i\delta\phi(\sigma_{2}+\sigma_{3})\Big{]}|\psi_{0}\rangle+\Big{[}(\sigma_{0}+i\sigma_{1})-\frac{i\delta
k}{2}(\sigma_{2}-\sigma_{3})\Big{]}\psi_{1}$ $\displaystyle\approx
2(1-i\delta\theta)|\psi_{0}\rangle$ (S23)
$\displaystyle\Big{[}(\sigma_{0}+i\sigma_{1})-\frac{i\delta
k}{2}(\sigma_{2}-\sigma_{3})\Big{]}|\psi_{0}\rangle+e^{-g}\Big{[}(\sigma_{0}-i\sigma_{1})+i\delta\phi(\sigma_{2}+\sigma_{3})\Big{]}|\psi_{1}\rangle$
$\displaystyle\approx 2(1-i\delta\theta)|\psi_{1}\rangle.$ (S24)

Multiplying Eq. (S23) by $\sigma_{0}-i\sigma_{1}+\frac{i\delta
k}{2}(\sigma_{2}-\sigma_{3})$, and again retaining only terms up to first
order, we obtain:

$|\psi_{1}\rangle\approx\left[(1-i\delta\theta)(\sigma_{0}-i\sigma_{1})+ie^{g}\sigma_{1}+\frac{i\delta
k}{2}(\sigma_{2}-\sigma_{3})+ie^{g}\left(-\delta\phi+\frac{\delta
k}{2}\right)\sigma_{3}\right]|\psi_{0}\rangle.$ (S25)

This can be substituted into Eq. (S24) to eliminate $|\psi_{1}\rangle$, thus
reducing the problem into

$\displaystyle H_{D}|\psi_{0}\rangle$
$\displaystyle\approx\delta\theta\left(\sigma_{0}+iM\sigma_{1}\right)|\psi_{0}\rangle,$
(S26) $\displaystyle H_{D}$
$\displaystyle=M(g)\sigma_{1}+\frac{1}{2}\left[\left(1-\frac{e^{-g}}{2}\right)\delta
k-e^{-g}\delta\phi\right]\sigma_{2}+\frac{1}{2}\left[\left(\frac{e^{g}}{2}-1\right)\delta
k-e^{g}\delta\phi\right]\sigma_{3},$ (S27) $\displaystyle M(g)$
$\displaystyle=\cosh g-1.$ (S28)

If we let

$\sigma_{0}+iM\sigma_{1}\approx\exp(iM\sigma_{1}),$ (S29)

which is valid for small $g$ (i.e., small $M$), then

$H_{D}\,\big{(}e^{iM\sigma_{1}/2}|\psi^{\prime}\rangle\big{)}\approx\delta\theta\,\big{(}e^{iM\sigma_{1}/2}|\psi^{\prime}\rangle\big{)},$
(S30)

to lowest order in $M$ and the other perturbative parameters ($\theta$,
$\delta k$, and $\delta\theta$).

$H_{D}$ has the form of an anisotropic Dirac equation. The Dirac mass is $M$,
given by Eq. (S28), is consistent with out previous result (S19):

$M=\frac{g^{2}}{2}+O(g^{4}).$ (S31)

To determine the anisotropy factors, we can take the dispersion relation of
Eq. (S27) when $M$ is negligible:

$\displaystyle\delta\theta^{2}$
$\displaystyle=\frac{1}{4}\left[\left(1-\frac{e^{-g}}{2}\right)\delta
k-e^{-g}\delta\phi\right]^{2}+\frac{1}{4}\left[\left(\frac{e^{g}}{2}-1\right)\delta
k-e^{g}\delta\phi\right]^{2}$ (S32) $\displaystyle=\frac{\delta
k^{2}}{8}+\frac{\delta\phi^{2}}{2}+\,O(g^{3}).$ (S33)

The $\delta k\delta\phi$ term is also of order $g^{4}$, and thus also
neglected. Hence, by appropriately rotating around $\sigma_{1}$, we can bring
$H_{D}$ into the form

$H_{D}^{\prime}\approx M(g)\sigma_{1}+\frac{1}{\sqrt{8}}\delta
k\,\sigma_{2}-\frac{1}{\sqrt{2}}\delta\phi\,\sigma_{3},$ (S34)

where corrections to $M$ and the Dirac velocities are $O(g^{3})$.

It is important to note that the wavenumber $k$ was defined in Eq. (S9) such
that advancing by one unit cell (which consists of 4 sites) advances the phase
by $\exp(ik)$. Likewise, $\theta$ is defined such that two time steps advances
the phase by $\exp(i\theta)$. Hence, the Dirac velocity along $k$, expressed
in terms of sites per time step, is

$v_{D}=\frac{1}{\sqrt{2}}.$ (S35)

In Extended Data Fig. 3, we plot the Floquet band diagrams in the $\phi$ and
$k$ directions, for different values of $g$. For $g=0$ and small nonzero
values of $g$, the quasienergies calculated directly from Eq. (S10) are well
approximated by the Dirac equation (S34).

The above derivation above applies to the unit cell of configuration
gain/loss/loss/gain shown in Fig. S1. In the main text, we discuss an
alternative lattice whose unit cell is gain/gain/loss/loss. Repeating the
above derivation yields a mass $-M$.

##  S3 Evolution of the boundary modes

In Fig. 3a of the main text, we depicted a lattice consisting of two domains,
whose unit cells have the loss/loss/gain/gain (left) and gain/loss/loss/gain
(right) configurations. As discussed in Supplementary Information Section S2,
these unit cells correspond to the Dirac masses $M_{0}$ (right) and $-M_{0}$
(left), where $M_{0}(g)=\frac{1}{2}\left(e^{g}+e^{-g}-2\right)$.

Extended Data Figure S2: Schematic of the synthetic lattice with a domain
wall.

To derive the Floquet bandstructure for this two-domain lattice, we apply the
evolution equations (S1)–(S2) to the layout depicted in Fig. S2. For $n\geq 1$
and $n\equiv 1(\textrm{mod}~{}4)$, Eqs. (S5)–(S8) hold (right domain). For
$n\leq 0$ and $n\equiv 1(\textrm{mod}~{}4)$, we plug in the corresponding
evolution equations for the loss/loss/gain/gain lattice. We then take the
ansatz

$\begin{pmatrix}\vdots\\\ u_{0}^{m}\\\ v_{0}^{m}\\\ u_{1}^{m}\\\ v_{1}^{m}\\\
u_{2}^{m}\\\ v_{2}^{m}\\\ \vdots\end{pmatrix}=\begin{pmatrix}\vdots\\\
U_{0}\\\ V_{0}\\\ U_{1}\\\ V_{1}\\\ U_{2}\\\ V_{2}\\\
\vdots\end{pmatrix}e^{-\frac{im\theta}{2}},\;\;\;m\equiv 1(\textrm{mod}~{}2).$
(S36)

(This utilizes the fact that the lattice retains discrete time translation
symmetry.) The evolution equations then reduce to an eigenvalue problem, which
can be solved numerically to obtain the Floquet band diagram shown in Fig. 3b
of the main text.

The occurrence of boundary states in this system is consistent with the view
of the quasiparticles behaving as though they obey the Hermitian Dirac
equation (there is one complication, which is that the boundary state energies
are not exactly real, as discussed below). It is well known that 1D and 2D
Dirac equations host boundary states at domain walls where the Dirac mass
flips sign [16, 35, 13, 14]. Plugging an exponentially localized ansatz
$\psi(n)\propto\exp(-4v_{D}M_{0}n)$ into the effective Dirac Hamiltonian
derived in Supplementary Information Section S2, we obtain the new dispersion
relation:

$\delta\theta^{2}=M_{0}^{2}+\left(2iv_{D}M_{0}-iv_{D}M_{0}e^{-g}-\frac{\delta\phi
e^{-g}}{2}\right)^{2}+\left(iv_{D}M_{0}e^{g}-2iv_{D}M_{0}-\frac{\delta\phi
e^{g}}{2}\right)^{2}.$ (S37)

Here, we have made the substitution $\delta k\rightarrow-i\partial_{n}$.
Hence,

$\displaystyle\delta\theta^{2}$
$\displaystyle=-4i\delta\phi\,v_{D}\,M_{0}\,\left(\frac{e^{-g}}{2}-\frac{e^{-2g}}{4}+\frac{e^{2g}}{4}-\frac{e^{g}}{2}\right)$
$\displaystyle\quad-8M_{0}^{2}\left[\left(\frac{1}{2}-\frac{e^{-g}}{4}\right)^{2}-\left(\frac{e^{g}}{4}-\frac{1}{2}\right)^{2}\right]+\delta\phi^{2}\left(\frac{e^{2g}}{4}+\frac{e^{-2g}}{4}\right)+M_{0}^{2}$
(S38)
$\displaystyle=\delta\phi\,\cdot\,8iv_{D}M_{0}^{2}\sqrt{M_{0}^{2}+2M_{0}}-M_{0}^{2}\left(2M_{0}^{2}+1\right)+\delta\phi^{2}\left(\frac{e^{2g}}{4}+\frac{e^{-2g}}{4}\right)+M_{0}^{2}.$
(S39)

Dropping the higher-order terms, we obtain

$\delta\theta^{2}=\frac{1}{2}\cosh 2g\;\delta\phi^{2}.$ (S40)

The decay rate is $4v_{D}M_{0}(g)$, which increases with $g$ in agreement with
Fig. 3c of the main text.

Extended Data Figure S3: Optical energy evolution for different types of
heterojunctions. On the top of each subplot shows the unit cell configurations
at two sides of the interface. In all the plots, we set $\phi=\pi$, $g=0.3$,
$k=0.1\pi$, $V=0.2\pi$.

##  S4 Scattering from a potential step

In Fig. 4d–e of the main text, we plot the intensity evolution for scattering
at heterojunctions between two uniform lattice domains. The results show that
the Dirac quasiparticle crosses the interface with either perfect
transmission, amplification, or damping, depending on the configuration of the
interface.

We noted that since the bulk domains have unbroken semi-Hermiticity, the
interface itself acts as a source or sink of flux. This is verified in Fig.
S3, which shows the evolution of the total intensity inside the lattice, as
well as the intensity summed over the left and right sides of the interface.
Before and after the incident wavepacket strikes the interface (around
$m=30$), the intensities are approximately constant.

Extended Data Figure S4: Comparison of wavepacket evolution with $H_{tbm}$ and
$U$. a, beam dynamics for the lattice governed by Equation. (S41), we set
$g=0.32$, $t=-\frac{1}{\sqrt{2}}$. b, beam dynamics for the lattice governed
by the semi-Hermitian Hamiltonian $U$, we set $\phi=\pi$, $g=0.32$. The
corresponding lattices schematic for a and b are shown on the top of each
subplot, respectively. For clarity, here we only show regions with the
scattered wave packets. c, comparison of band structures between these two
models. The lattice governed by $H_{tbm}$ is denoted by blue dots and by
$H_{F}$ is denoted by black dots. Only the real part of the spectrum is
plotted. d, transmission spectra for the barrier structure in a. e,
transmission spectra for the barrier structure in b. Transmission is denoted
by the blue dots and reflection is denoted by the orange dots.

##  S5 Scattering in an alternative non-Hermitian lattice

In this section, we compare the reflection and transmission behavior of the
semi-Hermitian lattice with the non-Hermitian 1D model proposed by Takata and
Notomi [38]. Notably, the bands in the Takata-Notomi model are not orthogonal,
and are therefore not described by a Dirac equation. Its momentum space
Hamiltonian is

$H_{tbm}=\begin{pmatrix}ig&t&0&te^{-ik}\\\ t&ig&t&0\\\ 0&t&-ig&t\\\
te^{ik}&0&t&-ig\end{pmatrix},$ (S41)

where $g$ is the gain/loss level and $t$ is a coupling constant. To apply an
additional scalar potential, we add $V$ to the on-site entries of the lattice
Hamiltonian.

Fig. S4 compares the behavior of the two types of lattices, with the same
gain/loss level $g=0.32$. The scattering results are shown in Fig. S4a–b. In
Fig. S4c, we plot the band structures for the two lattices, which look
superficially similar. However, the scattering behavior is quite different. As
shown in Fig. S4d, for large potential barriers the reflectance of the Takata-
Notomi model is not suppressed to zero. For the semi-Hermitian lattice, shown
in Fig. S4e, when the potential barrier is strong the reflectance approaches
zero and the transmittance approaches unity, consistent with the Klein
tunneling of Hermitian Dirac particles. The behavior for larger $g$ is also
qualitatively similar.