# Self acceleration from spectral geometry in dissipative quantum-walk
dynamics

Peng Xue<EMAIL_ADDRESS>Department of Physics, Southeast University,
Nanjing 211189, China  Beijing Computational Science Research Center, Beijing
100084, China  Quan Lin  Beijing Computational Science Research Center,
Beijing 100084, China  Kunkun Wang  School of Physics and Optoelectronic
Engineering, Anhui University, Hefei 230601, China  Lei Xiao  Department of
Physics, Southeast University, Nanjing 211189, China  Beijing Computational
Science Research Center, Beijing 100084, China  Stefano Longhi
<EMAIL_ADDRESS>Dipartimento di Fisica, Politecnico di Milano, Piazza
L. da Vinci 32, I-20133 Milano, Italy  IFISC (UIB-CSIC), Instituto de Fisica
Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain  Wei
Yi<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei 230026, China  CAS Center For
Excellence in Quantum Information and Quantum Physics, Hefei 230026, China

###### Abstract

Dynamic behaviors of a physical system often originate from its spectral
properties. In open systems, where the effective non-Hermitian description
enables a wealth of spectral structures on the complex plane, the concomitant
dynamics is significantly enriched, whereas the identification and
comprehension of the underlying connections are challenging. Here we
experimentally demonstrate the correspondence between the transient self
acceleration of local excitations and the non-Hermitian spectral topology
using lossy photonic quantum walks. Focusing first on one-dimensional quantum
walks, we show that the measured short-time acceleration of the wave function
is proportional to the area enclosed by the eigenspectrum. We then reveal
similar correspondence in two-dimension quantum walks, where the self
acceleration is proportional to the volume enclosed by the eigenspectrum in
the complex parameter space. In both dimensions, the transient self
acceleration crosses over to a long-time behavior dominated by a constant flow
at the drift velocity. Our results unveil the universal correspondence between
spectral topology and transient dynamics, and offer a sensitive probe for
phenomena in non-Hermitian systems that originate from spectral geometry.

In both classical and quantum mechanics, the dynamics of a system is
intimately connected with its spectral features through the equations of
motion sc ; kre ; upre ; Eberly . Just as the energy of a celestial body
impacts its trajectory genov , so the energy quantization accounts for the
spontaneous collapse and revival in quantum models Eberly . In solid
materials, transport of electrons depends on the lattice dispersion neto ;
TGU12 ; jacq , with strict connections between spectral and dynamical features
of transport. For example, in a clean lattice with absolutely continuous
spectrum, transport is ballistic, while in disordered lattices, the different
nature of the energy spectrum greatly impacts the spreading dynamics of the
wave function, leading to distinct behaviors ranging from the Anderson
localization to diffusive and intermittent quantum dynamics Mantica . These
examples, however, all concern isolated systems with completely real energy
spectra. For open systems that exchange energy or particles with its
environment, an effective non-Hermitian description is often adopted, where
the underpinning non-Hermitian Hamiltonians feature complex eigenspectra bend
; openbook ; AGU20 . This enables a rich variety of spectral geometries on the
complex plane, with non-trivial consequences in the system behavior GongPRX ;
fangchenskin ; fangchenskin2 ; kawabataskin ; BBK21 . The dynamics generated
by non-Hermitian Hamiltonians are often less intuitive than those of
conventional Hermitian systems. For example, the semiclassical equations of
non-Hermitian Hamiltonians generalize the Ehrenfest theorem in a nontrivial
way AJP , leading to phase-space dynamics with a changing metric structure AJP
; Graefe1 ; Graefe2 . Beyond the semiclassical models, recent studies of non-
Hermitian Hamiltonians with the non-Hermitian skin effect BBK21 ; WZ1
unravelled that, spectral features such as the closing of the imaginary gap on
the complex plane nonHtopo1 ; nonHtopo2 ; WZ1 ; JD23 ; XDW+21 ; ed ; edx ;
YR23 , or the overall spectral topology Budich ; murakami ; yzsgbz ; GLZ21 ;
stefano2 ; WL23 ; triplep ; wangk ; lli , can have detectable dynamic
consequences, including anomalous relaxation dynamics WZ3 ; haga ; MSS23 ,
boundary accumulations of loss in the dynamics (known as the edge burst) ed ;
edx ; YR23 , and the persistent directional flow which has served as an
experimental signature for the non-Hermitian skin effect photonskin ; teskin ;
metaskin ; scienceskin ; WWM22 ; coldatom ; teskin2d ; dzou ; zhangx ; LTL23 .
However, as most of such dynamic behaviors only dominate at long times and
require post selection to avoid quantum jumps, their experimental
identification in genuinely quantum systems may be challenging.

In this work, we experimentally reveal the impact of non-Hermitian spectral
topology in transient dynamics, by studying the propagation of a local
excitation along a dissipative lattice using photonic quantum walks. We show
that the short-time, center-of-mass acceleration of the wave function, dubbed
self acceleration stefano3  because of the absence of any external force, is
proportional to the area enclosed by the eigenenergy spectrum of the system on
the complex plane. While the direction of the propagation is given by the
spectral winding number, the self acceleration vanishes at long times, giving
way to a directional flow with a constant drift velocity. The correspondence
between the spectral geometry and bulk dynamics also persists in a wide class
of two-dimensional systems, for which we demonstrate that the self
acceleration becomes proportional to the volume enclosed by the eigenspectrum
in the complex parameter space.

Our experiment establishes a fundamental correspondence between the spectral
geometry and short-time dynamics in non-Hermitian systems, complementing
existing experiments on the long-time dynamics stefano ; ql ; ql2 . As the
spectral topology is intimately connected with the non-Hermtian skin effect
fangchenskin ; fangchenskin2 ; kawabataskin , self acceleration offers a
practical and sensitive dynamic signal for its detection, particularly in
quantum systems where decoherence dominates at long times.

Results

Figure 1: Illustration of the connection between self acceleration and
spectral geometry. a A schematic of the PBC energy spectra in a two-
dimensional system with $k_{y}$ (upper panel) and $k_{x}$ (lower panel) as a
parameter. b The corresponding two different volumes lead to distinct dynamic
behaviors, i.e., different accelerated speeds of the motion of the wave
packets in two directions, and eventually approach to a constant. c A
schematic with a finite energy spectrum area in a one-dimensional system. d
The motion of the wave packet shifts from accelerated to constant velocity as
time increases.

Time-multiplexed quantum walk. We simulate the dynamics of a local excitation
along a dissipative lattice using photonic quantum walks. Following the well-
established protocol in Refs. ql ; ql2 ; ql3 , we implement both one- and two-
dimensional quantum walks by employing a time-multiplexed configuration,
sending attenuated single-photon pulses (with a wavelength of $808$ nm and a
pulse width of $88$ ps) through a fibre network. While each full cycle around
the fibre loop represents a discrete time step, the built-in optical elements
within the loop, such as the half-wave plates (HWPs), polarization beam
splitters (PBSs), quarter-wave plates (QWPs), realize the time-evolution
operator $U$ within each step.

Taking the more general two-dimensional quantum walk as an example, we
implement the non-unitary Floquet operator

$U=M_{y}S_{y}R(\theta_{2})M_{x}S_{x}R(\theta_{1}),$ (1)

where shift operators are defined as
$S_{j}=\sum_{\bm{r}}\ket{0}\bra{0}\otimes\ket{\bm{r}-\bm{e}_{j}}\bra{\bm{r}}+\ket{1}\bra{1}\otimes\ket{\bm{r}+\bm{e}_{j}}\bra{\bm{r}}$
with $\bm{r}=(x,y)\in\mathbb{Z}^{2}$ labeling the coordinates of the lattice
sites, $j\in\\{x,y\\}$, and $\bm{e}_{x}=(1,0)$ and $\bm{e}_{y}=(0,1)$. The
shift operators move the walker in the corresponding directions, depending on
the walker’s internal degrees of freedom in the basis of
$\\{|0\rangle,|1\rangle\\}$ (dubbed the coin states). The coin operator acts
in the subspace of coin states
$R(\theta_{1,2})=\begin{pmatrix}\cos\theta_{1,2}&i\sin\theta_{1,2}\\\
i\sin\theta_{1,2}&\cos\theta_{1,2}\end{pmatrix}\otimes\mathds{1}_{\bm{r}}$,
where $\mathds{1}_{\bm{r}}=\sum_{\bm{r}}\ket{\bm{r}}\bra{\bm{r}}$. The gain-
loss operators are given by $M_{j}=\begin{pmatrix}e^{\gamma_{j}}&0\\\
0&e^{-\gamma_{j}}\end{pmatrix}\otimes\mathds{1}_{\bm{r}}$, which make the
quantum walk non-unitary for finite $\gamma_{x}$ or $\gamma_{y}$.

In the experiment, we encode the internal coin states
$\\{|0\rangle,|1\rangle\\}$ in the photon polarizations
$\\{|H\rangle,|V\rangle\\}$, and the external spatial modes through the
discretized temporal shifts. For the latter, we build path-dependent time
delays into the loop, so that the spatial superposition of the photonic walker
is translated to the temporal superposition of multiple well-resolved pulses
within each discrete time step Silberhorn . To encode spatial modes in two
dimensions, the temporal modes are further separated into two different time
scales by the free-space Mach-Zehnder interferometer: $80$ ns in the
$x$-dimension and $4.83$ ns in the $y$-dimension. For detection, we record the
arrival time of the photons using avalanche photodiodes (APDs) with the help
of an acoustic-optical modulator (AOM) serving as an optical switch to remove
undesired pulses LuPan22 .

In the quantum-walk dynamics, the time-evolved state at the end of each
discrete time step $t$ is
$|\psi(t)\rangle=U^{t}|\psi(0)\rangle=e^{-iHt}|\psi(0)\rangle$, where we
define an effective Hamiltonian $H$. Apparently, the quantum walk implements a
stroboscopic simulation of the Hamiltonian $H$ at integer time steps. We
measure the center-of-mass position of the walker, defined through stefano3

$\displaystyle\bm{n}_{CM}(t)=\frac{\sum_{\bm{r}}\langle\psi(t)|\bm{r}|\psi(t)\rangle}{\sum_{\bm{r}}\langle\psi(t)|\psi(t)\rangle}.$
(2)

As illustrated in Fig. 1, starting from a local excitation, the motion of
$\bm{n}_{CM}=(x_{CM},y_{CM})$ is closely connected with the spectral geometry
of the effective Hamiltonian $H$ on the complex plane. More explicitly,
transforming $H$ to the momentum space, we have
$H(\bm{k})|\psi_{\pm}(\bm{k})\rangle=E_{\pm}(\bm{k})|\psi_{\pm}(\bm{k})\rangle$,
where $\bm{k}$ belongs to the first Brillouin zone, and $E_{\pm}(\bm{k})$ and
$|\psi_{\pm}(\bm{k})\rangle$ are respectively the eigenenergies and
eigenstates under the periodic boundary condition (PBC), with the subscripts
$\pm$ indicating the band index. For a local initial state that is an equal-
weight superposition of all eigenstates within a given band, for instance,
$|\psi(0)\rangle=\sum_{\bm{k}}|\psi_{+}(\bm{k})\rangle\otimes|\bm{k}\rangle$,
the short-time behavior of $\bm{n}_{CM}$ reads (see Methods)

$\displaystyle x_{CM}(t)\simeq\frac{1}{2}a_{x}t^{2}-\frac{1}{2},\quad
y_{CM}(t)\simeq\frac{1}{2}a_{y}t^{2}+\frac{1}{2},$ (3)

where

$\displaystyle a_{x}$
$\displaystyle=\frac{1}{\pi^{2}}\int_{-\pi}^{\pi}\text{d}k_{x}\text{d}k_{y}E_{I}\frac{\partial
E_{R}}{\partial
k_{x}}=\frac{2}{\pi}\int_{-\pi}^{\pi}\text{d}k_{y}\mathcal{A}_{x}(k_{y}),$
$\displaystyle a_{y}$
$\displaystyle=\frac{1}{\pi^{2}}\int_{-\pi}^{\pi}\text{d}k_{x}\text{d}k_{y}E_{I}\frac{\partial
E_{R}}{\partial
k_{y}}=\frac{2}{\pi}\int_{-\pi}^{\pi}\text{d}k_{x}\mathcal{A}_{y}(k_{x}).$

Here $E_{R}$ and $E_{I}$ are respectively the real and imaginary components of
$E_{+}$. Importantly, $a_{x}$ and $a_{y}$ suggest that the short-time self-
acceleration rate is proportional to the volume enclosed by the eigenspectrum
of the corresponding band in the complex parameter space. An alternative
understanding is that the self-acceleration rate is proportional to the
averaged area enclosed by $E_{+}(k_{x},k_{y})$ on the complex plane as $k_{y}$
traverses the Brillouin zone, as shown in Fig. 1. It should be mentioned that,
in non-Hermitian systems, self acceleration of the wave function in the
absence of external forces is a universal phenomenon observed for rather
arbitrary excitations that are initially localized (see Methods). However, it
is only when the system is initially prepared in an equal-weight superposition
of all eigenstates within a given band, that the ensuing self acceleration
relates to the spectral geometry through $a_{x}$ and $a_{y}$.

Simulating the one-dimensional dynamics. For our experimental demonstration,
we first study the simplified case of one-dimensional quantum walks, where the
self-acceleration rate is proportional to the area enclosed by the PBC
eigenspectrum on the complex plane stefano3 . Based on the general two-
dimensional setup in Fig. 1, one-dimensional quantum walks can be realized by
simply removing the free-space Mach-Zehnder interferometer within the loop.
The resulting Floquet operator is given by

$U=S_{x}M_{x}R(\theta).$ (4)

We focus on the parameter regime with $\theta$ being quite close to $\pi/2$,
where the complex eigenenergies of the effective Hamiltonian $H$ are
approximately (see Methods)

$\displaystyle
E_{\pm}(k)\approx\pm\frac{\pi}{2}\mp\cos\beta\cos(k-i\gamma_{x}),$ (5)

corresponding to those of a typical Hatano-Nelson model. We initialize the
system in the superposition state (representing a local excitation)
$|\psi(0)\rangle_{1}=e^{\gamma_{x}}|0\rangle\otimes|x=-1\rangle+|1\rangle\otimes|x=0\rangle$,
which is an equal-weight superposition of the Bloch states (in the first
Brillouin zone) of $H$ with eigenenergy $E_{+}(k)$. The experimental
implementation of such a local initial state, which is pivotal to our
measurement scheme, is discussed in the Supplemental Material. The resulting
short-time dynamics of $x_{CM}$ then follows that in Eq. (3), with the self-
acceleration rate given by

$\displaystyle
a_{x}=\frac{2}{\pi}\int_{-\pi}^{\pi}\text{d}k_{x}E_{I}\frac{\text{d}}{\text{d}k}E_{R}:=\frac{2}{\pi}\mathcal{A}.$
(6)

Here $\mathcal{A}$ corresponds to the area enclosed by the complex eigenenergy
$E_{+}(k_{x})$ in the complex plane, taken with the appropriate sign depending
on the circulation direction of the PBC energy loop. Such a sign naturally
indicates the direction of self acceleration.

In Figs. 2a-c, we show the measured spatial population evolution of the
dynamics under different gain-loss parameters $\gamma_{x}$. The wave-function
propagation becomes asymmetric when the gain-loss parameter $\gamma_{x}$
becomes finite. In Figs. 2d-f, we show the measured $x_{CM}(t)$, which are
quadratic in time when $\gamma_{x}\neq 0$, consistent with theoretical
predictions. By fitting the center-of-mass propagation of the wave functions,
we extract the quantity $\mathcal{A}$ from the self-acceleration rate (see
Fig. 2g), which agrees well with the numerically calculated area enclosed by
the eigenspectum $E_{+}({k})$ on the complex plane (see Fig. 2h). The self
acceleration in the one-dimensional model is a clear signature of the non-
Hermitian skin effect under the open-boundary condition (OBC). In fact, in
systems that do not display the non-Hermitian skin effects, the PBC energy
spectrum collapses to an open arc enclosing a vanishing area $\mathcal{A}=0$,
and thus acceleration vanishes according to Eq. (6).

Figure 2: Self acceleration in one-dimensional dynamics. a-c Dynamic
evolutions governed by the effective non-Hermitian Hamiltonian with parameters
$\gamma_{x}=0,\gamma_{x}=-0.05$ and $\gamma_{x}=0.1$, respectively. d-f
Evolution of the center of mass $n_{CM}(t)$ as a function of the discrete time
step $t$ corresponding to the dynamic evolutions in a-c, respectively. g Areas
enclosed by $E_{+}(k)$ versus the gain-loss parameter $\gamma=\gamma_{x}$. We
take a local initial state $\ket{\psi(0)}_{1}$ and the coin parameter
$\theta=0.45\pi$. h PBC energy spectra for increasing values of $\gamma$.
Figure 3: Self acceleration in two-dimensional dynamics. a Wave packet center
of mass $x_{CM}(t)$ ($y_{CM}(t)$) as a function of the discrete time step $t$.
The gain-loss parameter is taken as $\gamma=0.08$. b Probability distribution
of the $24$-time-step quantum walk with $\varphi=\pi/5$. We mark the position
occupied by the initial state as the red square. c, d The wave packet center
of mass $x_{CM}(t)$ ($y_{CM}(t)$) as a function of the discrete time step $t$
with $\varphi=\pi/5$. The other parameters are $\theta_{1}=0.12$,
$\theta_{2}=\pi/2-0.12$, and the initial state $\ket{\psi(0)}_{2}$.  Figure 4:
Measurement of the spectral volume. a PBC energy spectrum with $k_{y}$ as a
parameter and for $\gamma_{x}=\gamma_{y}$. The black, purple, red, and blue
surfaces correspond to $\gamma_{x(y)}=0$, $\gamma_{x(y)}=-0.04$,
$\gamma_{x(y)}=0.04$, and $\gamma_{x(y)}=0.08$, respectively. b, c The average
of the areas enclosed by $E_{+}(k_{x},k_{y})$ as a function of
$\gamma_{x(y)}$. The other parameters are $\theta_{1}=0.12$,
$\theta_{2}=\pi/2-0.12$, and the initial state $\ket{\psi(0)}_{2}$. The blue
dashed line represents the theoretical results, and the gray dashed lines
correspond the results fitted by experimental data.  Figure 5: Connecting the
short- and long-time dynamics. a Evolution of the center of mass $n_{CM}(t)$
as a function of the discrete time step $t$ in the one-dimensional quantum
walks with $\theta=0.41\pi$ and $\gamma_{x}=0.15$. We take a local initial
state $\ket{\psi(0)}_{1}$. b Wave packet center of mass $x_{CM}(t)$
($y_{CM}(t)$) as a function of the discrete time step $t$ in the two-
dimensional quantum walks with $\gamma_{x}=\gamma_{y}=0.26$,
$\theta_{1}=0.12$, $\theta_{2}=\pi/2-0.12$ the initial state
$\ket{\psi(0)}_{2}$.

Simulating the two-dimensional dynamics. In two dimensions, we focus on the
coin parameters close to $(\theta_{1}=0,\theta_{2}=\pi/2)$, where the
eigenenergies of the effective Hamiltonians are

$\displaystyle
E_{\pm}(k_{x},k_{y})\approx\pm\frac{\pi}{2}\pm\tilde{E}(k_{x},k_{y}).$ (7)

Here
$\tilde{E}(k_{x},k_{y})=\theta_{1}\cos(k_{y}-i\gamma_{y}-k_{x}+i\gamma_{x})-(\pi/2-\theta_{2})\cos(k_{x}-i\gamma_{x}+k_{y}-i\gamma_{y})$.
We initialize the system in the local state
$|\psi(0)\rangle_{2}=|0\rangle\otimes|x=0,y=0\rangle-e^{\gamma_{x}-\gamma_{y}}|1\rangle\otimes|x=-1,y=1\rangle$,
which is a superposition of all the Bloch states corresponding to
$E_{+}(k_{x},k_{y})$, again facilitated by the choice of the coin parameters.

In Fig. 3a, we show the time evolution of $(x_{CM},y_{CM})$ under different
gain-loss parameters $\gamma_{x}$ and $\gamma_{y}$, which are parameterized
through $\gamma_{x}=\gamma\cos\varphi$ and $\gamma_{y}=\gamma\sin\varphi$.
Consistent with a previous study ql3 , the tuning of the parameters gives rise
to directional propagation in the two-dimensional plane, which underlies the
emergence of the non-Hermitian skin effect when open boundaries are enforced.
An example of the full population evolution is illustrated in Fig. 3b.
Apparently, for finite $\gamma_{x}$ or $\gamma_{y}$, the corresponding
$x_{CM}$ or $y_{CM}$ exhibits quadratic behavior at early times, consistent
with the predicted self acceleration.

In Fig. 4, we explicitly demonstrate the correspondence between the spectral
volume and the self acceleration. For convenience, we focus on the case
$\gamma_{x}=\gamma_{y}$, where the dynamics along the $x$ and $y$ directions
are symmetric. Both the spectral volume and the fitted self acceleration
increase linearly with increasing $|\gamma_{x(y)}|$, consistent with the
theoretical analysis. Similar to one-dimensional quantum walks, the self
acceleration is a precursor of persistent drift (or current) at long times,
and thus indicates the accumulation of excitation at the edges or corners of a
finite two-dimensional domain (or equivalently, the appearance of the so-
called non-reciprocal skin effect fangchenskin2 ). It should be mentioned
that, in two-dimensional systems, the skin effect is a universal phenomenon
that appears under rather arbitrary boundary shapes fangchenskin2 , and thus
it persists even when the self acceleration vanishes (see Methods).

Crossover between short- and long-time dynamics. In the presence of a non-
trivial spectral point-gap topology, it is well-established that the long-time
dynamics of a local excitation is a directional propagation with a constant
drift velocity stefano ; ql ; ql2 , indicating that the self acceleration
ceases in the long time evolution. The asymptotic drift velocities for two-
dimensional dynamics are defined as the group velocity at the quasimomentum
$(k_{m_{x}},k_{m_{y}})$, with

$\displaystyle v_{m_{x,y}}$
$\displaystyle=\frac{\text{d}E_{R}(k_{m_{x}},k_{m_{y}})}{\text{d}k_{x,y}}$ (8)

where the eigenenergy $E_{+}(k_{m_{x}},k_{m_{y}})$ features the largest
imaginary component (whose corresponding eigenmode survives at long times). As
such, the combination of drift velocity at long times and self acceleration at
short times provides a complete correspondence between the spectral geometry
and bulk dynamics of a local excitation.

In Fig. 5, we experimentally characterize the crossover from self-
acceleration-dominated dynamics at short times (cyan dashed curves), to a flow
at the drift velocity (purple dashed curves) at long times. This is achieved
by choosing parameters such that the difference between self acceleration and
constant motion is appreciable at the experimentally accessible time steps.

Discussion. Unveiling the correspondence between dynamical and spectral
properties of classical and quantum systems is a fundamental problem and a
major challenge in different areas of physics. While such a correspondence is
quite well understood in closed systems described by Hermitian Hamiltonians,
it remains largely unexplored for open systems. Using dissipative photonic
quantum walks, here we have experimentally demonstrated a fundamental
correspondence between the spectral geometry and the dynamics of local
excitations in open systems described by effective non-Hermitian Hamiltonians,
showing that a non-trivial spectral topology generally corresponds to a
transient self acceleration of the wave function. Our results provide major
advancements in the understanding of the correspondence between spectral
geometry and dynamics beyond the Hermitian paradigm, and could stimulate
further studies on an emergent area of research.

Methods

Self acceleration for one-dimensional quantum walks. In the one-dimensional
quantum walk, the Floquet operator reads $U=S_{x}M_{x}R(\theta)$, where
$S_{x}=\sum_{x}\ket{0}\bra{0}\otimes\ket{x-1}\bra{x}+\ket{1}\bra{1}\otimes\ket{x+1}\bra{x}$
is the spatial shift operator,
$M_{x}=\sum_{x}\begin{pmatrix}e^{\gamma_{x}}&0\\\
0&e^{-\gamma_{x}}\end{pmatrix}\otimes\ket{x}\bra{x}$ is the gain-loss operator
with the gain-loss parameter $\gamma_{x}$, and
$R(\theta)=\sum_{x}\begin{pmatrix}\cos\theta&i\sin\theta\\\
i\sin\theta&\cos\theta\end{pmatrix}\otimes\ket{x}\bra{x}$ is the coin
operator. In the momentum space, the Floquet operator $U$ takes the form

$\displaystyle
U(k)=d_{0}\sigma_{0}+id_{x}\sigma_{x}+id_{y}\sigma_{y}+id_{z}\sigma_{z},$ (9)

where the expressions for $d_{x,y,z}$ are given in the Supplemental Material,
and $\sigma_{x,y,z}$ are the Pauli matrices. Defining the momentum-space
Hamiltonian $H_{k}$ through $U(k)=e^{-iH_{k}}$, its quasienergies are given by

$\displaystyle
E_{\pm}(k)=\pm\arccos\left[\cos\theta\cos(k-i\gamma_{x})\right],$ (10)

with corresponding eigenstates (in the coin-state basis)

$\displaystyle\left|\psi_{\pm}(k)\right\rangle=\begin{pmatrix}\frac{d_{z}\pm\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}}{d_{x}+id_{y}}\\\
1\end{pmatrix}.$ (11)

In the experiment we set $\theta\approx\pi/2$, so that one has

$\displaystyle E_{\pm}(k)$
$\displaystyle\simeq\pm\frac{\pi}{2}\mp\tilde{E}(k)$
$\displaystyle=\pm\frac{\pi}{2}\mp\cos\theta\cos(k-i\gamma_{x}),$ (12)

and

$\displaystyle\left|\psi_{\pm}(k)\right\rangle\simeq\begin{pmatrix}\pm
e^{\gamma_{x}+ik}\\\ 1\end{pmatrix}.$ (13)

Notice that $\tilde{E}(k)$ coincides with the dispersion of the Hatano-Nelson
model with asymmetric nearest-neighbor hopping amplitudes
$(1/2)\cos(\theta)e^{-\gamma_{x}}$ and $(1/2)\cos(\theta)e^{\gamma_{x}}$.

In our experiment, the quasi-local initial state can be expressed as

$\displaystyle|\psi(0)\rangle_{1}$
$\displaystyle=e^{\gamma_{x}}|0\rangle\otimes|x=-1\rangle+|1\rangle\otimes|x=0\rangle$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\text{d}k|\psi_{+}(k)\rangle\otimes|k\rangle.$
(14)

The time-evolved state is then

$\displaystyle|\psi(t)\rangle=\frac{(-i)^{t}}{2\pi}\int_{-\pi}^{\pi}\text{d}k\begin{pmatrix}e^{\gamma_{x}+ik}\\\
1\end{pmatrix}e^{i\tilde{E}(k)t}\otimes|k\rangle,$ (15)

and the center-of-mass of the normalized wave function is defined through
stefano3

$\displaystyle
n_{\mathrm{CM}}(t):=\frac{\langle\psi(t)|x|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle}.$
(16)

For short-time dynamics, making use of the truncated expansion
$e^{2\tilde{E}_{I}(k)t}\approx 1+2\tilde{E}_{I}(k)t$, where
$\tilde{E}(k)=\tilde{E}_{R}(k)+i\tilde{E}_{I}(k)$, after some straightforward
calculations one obtains

$\displaystyle n_{\mathrm{CM}}(t)=\frac{\mathcal{A}}{\pi}t^{2}-\frac{1}{2}.$
(17)

Here
$\mathcal{A}:=\int_{-\pi}^{\pi}\text{d}k_{x}E_{I}\frac{\text{d}}{\text{d}k}E_{R}$
is the area enclosed by the complex quasienergy dispersion $E_{+}(k)$ in the
complex plane.

In the long-time limit, the dynamics would be dominated by the Bloch mode
$k_{m}$ where $E_{I}(k)$ is the global maximum. This corresponds to the
conditions $\left(\frac{\text{d}E_{I}(k_{m})}{\text{d}k}\right)=0$ and
$\left(\frac{\text{d}^{2}E_{I}(k_{m})}{\text{d}k^{2}}\right)<0$. Defining
$\xi=k-k_{m}$, we expand the quasienergy around $k_{m}$, where the leading
orders give $E(k)\simeq
E(k_{m})+\left(\frac{\text{d}E(k_{m})}{\text{d}k}\right)\xi+\frac{1}{2}\left(\frac{\text{d}^{2}E(k_{m})}{\text{d}k^{2}}\right)\xi^{2}$.
It follows that

$\displaystyle n_{\text{CM}}(t)$ $\displaystyle\approx\frac{2\pi
t\left(\frac{\text{d}E_{R}(k_{m})}{\text{d}k}\right)e^{2E_{I}(k_{m})t}\int_{-\infty}^{\infty}\text{d}\xi\,e^{t\xi^{2}\left(\frac{\text{d}^{2}E_{I}(k_{m})}{\text{d}k^{2}}\right)}}{2\pi
e^{2E_{I}(k_{m})t}\int_{-\infty}^{\infty}\text{d}\xi\,e^{t\xi^{2}\left(\frac{\text{d}^{2}E_{I}(k_{m})}{\text{d}k^{2}}\right)}}$
$\displaystyle=v_{m}t,$ (18)

where $v_{m}=\left(\frac{\text{d}E_{R}(k_{m})}{\text{d}k}\right)$ is
identified as the drift velocity.

Self acceleration for two-dimensional quantum walks. Following a similar
procedure outlined in the previous section, we derive the center-of-mass
motion of the wave functions for two-dimensional quantum walks.

We start from the Floquet operator $U$ in Eq. (1), and focus on the parameter
regimes $\theta_{1}\approx 0$ and $\theta_{2}\approx\pi/2$. Under these
conditions, the momentum-space quasienergies are approximately
$E_{\pm}\left(k_{x},k_{y}\right)\simeq\pm\frac{\pi}{2}\pm\tilde{E}\left(k_{x},k_{y}\right)$,
where

$\displaystyle\tilde{E}\left(k_{x},k_{y}\right)=$
$\displaystyle\theta_{1}\cos\left(k_{y}-i\gamma_{y}-k_{x}+i\gamma_{x}\right)$
(19)
$\displaystyle-\left(\frac{\pi}{2}-\theta_{2}\right)\cos\left(k_{x}-i\gamma_{x}+k_{y}-i\gamma_{y}\right).$

The corresponding eigenstates are

$\displaystyle|\psi_{\pm}(\bm{k})\rangle\simeq\begin{pmatrix}1\\\ \mp
e^{i(k_{x}-i\gamma_{x}-k_{y}+i\gamma_{y})}\end{pmatrix}.$ (20)

The local initial state is the uniform superposition of Bloch states in the
upper band

$\displaystyle|\psi(0)\rangle_{2}$
$\displaystyle=|0\rangle\otimes|x=0,y=0\rangle-e^{\gamma_{x}-\gamma_{y}}|1\rangle\otimes|x=-1,y=1\rangle$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\text{d}k_{x}\text{d}k_{y}|\psi_{+}(\bm{k})\rangle\otimes|\bm{k}\rangle.$
(21)

We further write the time-evolved wave function at time $t$ as

$\displaystyle|\psi(t)\rangle=\sum_{x,y}\begin{pmatrix}(-i)^{t}\tilde{\psi}_{x,y}(t)\\\
-(-i)^{t}\tilde{\psi}_{x+1,y-1}(t)\end{pmatrix}\otimes|x,y\rangle,$ (22)

where

$\tilde{\psi}_{x,y}(t)=\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\text{d}k_{x}\text{d}k_{y}e^{ik_{x}x+ik_{y}y-i\tilde{E}\left(k_{x},k_{y}\right)}.$
(23)

Defining the center-of-mass positions of the wave function as in Eq. (2), we
have $x_{CM}(t)\simeq\frac{1}{2}a_{x}t^{2}-\frac{1}{2}$ and
$y_{CM}(t)\simeq\frac{1}{2}a_{y}t^{2}+\frac{1}{2}$ for short-time dynamics,
where the self accelerations are given by
$a_{x}=\frac{1}{\pi^{2}}\int_{-\pi}^{\pi}\text{d}k_{x}\text{d}k_{y}E_{I}\frac{\partial
E_{R}}{\partial k_{x}}$ and
$a_{y}=\frac{1}{\pi^{2}}\int_{-\pi}^{\pi}\text{d}k_{x}\text{d}k_{y}E_{I}\frac{\partial
E_{R}}{\partial k_{y}}$. Here we defined $E_{+}(k_{x},k_{y})=E_{R}+iE_{I}$.

Apparently, the self accelerations have a simple geometric interpretation in
terms of the energy spectrum $E_{+}(k_{x},k_{y})$ in the complex plane. For
instance, the acceleration along the $x$ direction can be written as

$\displaystyle
a_{x}=\frac{2}{\pi}\int_{-\pi}^{\pi}\text{d}k_{y}\mathcal{A}_{x}\left(k_{y}\right),$
(24)

where

$\mathcal{A}_{x}\left(k_{y}\right):=\frac{1}{2\pi}\int_{-\pi}^{\pi}\text{d}k_{x}E_{I}\frac{\partial
E_{R}}{\partial k_{x}}.$ (25)

For a fixed value of $k_{y}$ (taken as a parameter), the expression
$\int_{-\pi}^{\pi}\text{d}k_{x}E_{I}\frac{\partial E_{R}}{\partial k_{x}}$ is
the area enclosed by the spectrum $E\left(k_{x},k_{y}\right)$ in the complex
plane as $k_{x}$ traverses Brillouin zone. Alternatively, taking the
integration over $k_{y}$ into account, $a_{x}$ is proportional to the volume
enclosed by $2\pi\mathcal{A}_{x}\left(k_{y}\right)$ in the parameter space, as
$k_{y}$ traverses the Brillouin zone.

Finally, for the long-time dynamics, we have

$\displaystyle x_{\mathrm{CM}}(t)\sim v_{m_{x}}t,\quad y_{\mathrm{CM}}(t)\sim
v_{m_{y}}t,$ (26)

where $v_{m_{x}}$ and $v_{m_{y}}$ in Eq. (8) are the drift velocities,
corresponding to the location of the global maximum of $E_{I}(k_{x},k_{y})$.

Universality of self acceleration in non-Hermitian dynamics. In an Hermitian
system, according to the Ehrenfest theorem a wave packet cannot accelerate in
the absence of any external force. However, this is not the case for non-
Hermitian systems GongPRX ; AJP . To illustrate this point, let us consider
for example the single-particle dynamics on a one-dimensional lattice with
Hamiltonian in the physical space $\hat{H}=\hat{T}+V(x)$, where $x$ is the
lattice site position, $V(x)$ is the external potential,
$\hat{T}=T(\hat{p}_{x})$ is the kinetic energy operator,
$\hat{p}_{x}=-i\partial_{x}$ is the momentum operator, and $T(p_{x})$ is the
energy dispersion curve of a given lattice band. For the standard Hatano-
Nelson model, for instance, one has
$T(p_{x})=J\exp(ip_{x}+\gamma)+J\exp(-ip_{x}-\gamma)$, where
$J\exp(\pm\gamma)$ are the asymmetric left/right hopping amplitudes.

For a given initial excitation of the system $|\psi_{0}\rangle$ at time $t=0$,
with $\langle\psi_{0}|\psi_{0}\rangle=1$, the evolved wave function in the
early times is given by

$|\psi_{t}\rangle=\exp(-i\hat{H}t)|\psi_{0}\rangle\simeq\left(1-it\hat{H}-\frac{t^{2}}{2}\hat{H}^{2}\right)|\psi_{0}\rangle.$
(27)

From this equation, one can readily calculate the time evolution of the mean
position $\langle
x\rangle=\langle\psi_{t}|x|\psi_{t}\rangle/\langle\psi_{t}|\psi_{t}\rangle$,
up to the order $\sim t^{2}$, and the corresponding initial acceleration,
$a_{x}=(\text{d}^{2}\langle x\rangle/\text{d}t^{2})_{t=0}$, which reads
explicitly

$a_{x}=\langle
2\hat{H}^{{\dagger}}x\hat{H}-x\hat{H}^{2}-\hat{H}^{{\dagger}2}x\rangle_{0}+2\langle\hat{H}^{{\dagger}}-\hat{H}\rangle_{0}\langle\hat{H}^{{\dagger}}x-x\hat{H}\rangle_{0}.$
(28)

In the above equation,
$\langle\hat{A}\rangle_{0}\equiv\langle\psi_{0}|\hat{A}\psi_{0}\rangle$
denotes the mean value of any operator $\hat{A}$ over the initial state
$|\psi_{0}\rangle$. Let us now assume that there is not any external force,
$V(x)=0$, so that the Hamiltonian contains the kinetic energy term solely,
$\hat{H}=\hat{T}$. Using the generalized commutation relation
$[x,F(\hat{p_{x}})]=i(\text{d}F/\text{d}p_{x})$ for any function $F(p_{x})$ of
the momentum operator, one obtains

$\displaystyle a_{x}$ $\displaystyle=$ $\displaystyle\langle
x(2\hat{T}^{{\dagger}}\hat{T}-\hat{T}^{2}-\hat{T}^{{\dagger}2})\rangle_{0}+2i\langle\frac{\partial\hat{T}^{{\dagger}}}{\partial
p_{x}}(\hat{T}^{{\dagger}}-\hat{T})\rangle_{0}+$ (29) $\displaystyle+$
$\displaystyle
2\langle\hat{T}^{{\dagger}}-\hat{T}\rangle_{0}\langle\hat{T}^{{\dagger}}x-x\hat{T}\rangle_{0}.$

Clearly, in any Hermitian system $\hat{T}^{{\dagger}}=\hat{T}$, we necessarily
have $a_{x}=0$. Conversely, in a non-Hermitian system where
$\hat{T}^{{\dagger}}\neq\hat{T}$, the acceleration $a_{x}$ is non-vanishing
for rather arbitrary initial states $|\psi_{0}\rangle$, with its value
dependent on the specific initial excitation.

Self acceleration and the non-Hermitian skin effect. In one-dimensional
models, there is a close correspondence between the transient self
acceleration, observed when the lattice is initially excited by an equal-
weight superposition of all eigenstates within a given band, and the non-
Hermitian skin effect under the OBC stefano3 . In fact, the non-Hermitian skin
effect, that is, the localization of a macroscopic number of eigenstates near
the boundaries, appears rather generally whenever the Hamiltonian displays a
point-gap topology in the PBC energy spectrum kawabataskin ; fangchenskin . In
the presence of the point-gap topology, the area $\mathcal{A}$, which is
proportional to the self acceleration, is necessarily non-vanishing.

In two-dimensional systems the non-Hermitian skin effect can depend on the
geometry of the boundaries, and it is thus clear that the bulk dynamics of a
wave packet alone (including transient self acceleration and long-time drift
motion), cannot uniquely determine the behavior of these boundary-dependent
systems. Indeed, a recent work proved that, in higher dimensions, the non-
Hermitian skin effect is a universal phenomenon observed for almost every
local non-Hermitian Hamiltonian that displays a finite spectral area under the
periodic boundary condition, and when the shape of the open boundaries are
taken without any special symmetries fangchenskin2 . A distinction between
generalized reciprocal and non-reciprocal skin effect has also been introduced
fangchenskin2 , depending on whether the current in the system is vanishing or
not, respectively.

In our two-dimensional non-Hermitian quantum walk, the non-vanishing self
acceleration clearly corresponds to a non-vanishing current, and thus the skin
effect is of the latter type and is observable for arbitrary boundary shapes.
The vanishing of the self acceleration in a two-dimensional system does not
necessarily imply the absence of the non-Hermitian skin effect under arbitrary
shape of the boundaries, albeit it can disappear for a specific shape of the
boundaries. To clarify this point, let us consider for example the two-
dimensional square lattice described by the Bloch Hamiltonian

$H(k_{x},k_{y})=J_{x}\cos k_{x}+iJ_{y}\cos k_{y},$ (30)

with real (Hermitian) hopping amplitude $J_{x}$ along the $x$ direction, and
imaginary (non-Hermitian) hopping amplitude $iJ_{y}$ along the $y$ direction.
Note that in this non-Hermitian model, the hopping amplitudes are reciprocal,
and from the formulas of $a_{x}$ and $a_{y}$, it readily follows that
$a_{x}=a_{y}=0$, that is, transient self acceleration in the bulk is absent.
In this model, the non-Hermitian skin effect disappears for a square geometry
of the boundaries due to the existence of two mirror symmetries. However, skin
modes appear under different boundaries which break these mirror symmetries
fangchenskin2 .

##  A Supplemental Material for “Self acceleration from spectral geometry in
dissipative quantum-walk dynamics”

In this Supplemental Material, we provide some technical and experimental
details.

###  A.1 Derivation of the Floquet operator $U$ and the effective Hamiltonian

As mentioned in the main text, in the one-dimensional quantum walk, the
Floquet operator reads $U=S_{x}M_{x}R(\theta)$. As all the operators can be
expressed by Pauli matrices, the time-evolution operator can be also written
as

$\displaystyle
U(k)=d_{0}(k)\sigma_{0}+id_{x}\sigma_{x}+id_{y}(k)\sigma_{y}+id_{z}(k)\sigma_{z}$
(S1) $\displaystyle d_{0}(k)=\cos(\theta)\cos(-i\gamma_{x}+k),$ $\displaystyle
d_{x}(k)=\sin(\theta)\cos(-i\gamma_{x}+k),$ $\displaystyle
d_{y}(k)=-\sin(\theta)\sin(-i\gamma_{x}+k),$ $\displaystyle
d_{z}(k)=\cos(\theta)\sin(-i\gamma_{x}+k).$

The effective Hamiltonian is defined through the relation $U=e^{-iH}$, which
in the quasi-momentum space takes the form

$H=\int_{-\pi}^{\pi}\text{d}\bm{k}\left[E(\bm{k}){\mathbf{n}}(\bm{k})\cdot\sigma\right]\otimes|\bm{k}\rangle\langle\bm{k}|,$
(S2)

with

$\displaystyle\mathbf{n}(k)=\frac{1}{\sin
E(k)}\begin{pmatrix}-\sin(\theta)\cos(k-i\gamma_{x})\\\
\sin(\theta)\sin(k-i\gamma_{x})\\\
-\cos(\theta)\sin(k-i\gamma_{x})\end{pmatrix}.$ (S3)

Here the quasienergies are given by
$E_{\pm}(k)=\pm\arccos\left[\cos\theta\cos(k-i\gamma_{x})\right]$, which are
the eigenvalues of $H$.

Similarly, for the two-dimensional case, we have

$\displaystyle\mathbf{n}(\bm{k})$ $\displaystyle=\mathbf{n}(k_{x},k_{y})$ (S4)
$\displaystyle=\frac{1}{\sin
E(k_{x},k_{y})}\begin{pmatrix}-\cos(k_{x}+k_{y}-i\gamma_{x}-i\gamma_{y})\cos(\theta_{2})\sin(\theta_{1})-\cos(k_{x}+k_{y}-i\gamma_{x}+i\gamma_{y})\cos(\theta_{1})\sin(\theta_{2})\\\
\sin(k_{x}+k_{y}-i\gamma_{x}-i\gamma_{y})\cos(\theta_{2})\sin(\theta_{1})-\sin(k_{x}+k_{y}-i\gamma_{x}+i\gamma_{y})\cos(\theta_{1})\sin(\theta_{2})\\\
-\sin(k_{x}+k_{y}-i\gamma_{x}-i\gamma_{y})\cos(\theta_{2})\cos(\theta_{1})-\sin(k_{x}+k_{y}-i\gamma_{x}+i\gamma_{y})\sin(\theta_{1})\sin(\theta_{2})\end{pmatrix},$

where the quasienergy is

$\displaystyle
E(k_{x},k_{y})=\pm\arccos\left[\theta_{1}\cos\left(k_{y}-i\gamma_{y}-k_{x}+i\gamma_{x}\right)-\left(\frac{\pi}{2}-\theta_{2}\right)\cos\left(k_{x}-i\gamma_{x}+k_{y}-i\gamma_{y}\right)\right].$
(S5)

###  A.2 Experimental scheme

Figure S1: A time-multiplexed implementation of the two-dimensional photonic
quantum walk. The process involves splitting photonic wave packets using a
polarizing beam splitter (PBS) and guiding them through a pair of single-mode
fibers (SMF) to achieve a temporal step in the $x$ direction. Similarly, a
temporal step in the $y$ direction is achieved using another two-PBS loop in
free space. At each step, partial photons are coupled out and directed towards
avalanche photodiodes (APDs) for polarization resolving detection of their
arrival times. ND: neutral density filter; AOM: optical switch acousto-optic
modulator.

We implement quantum walks by employing a time-multiplexed configuration,
where the external spatial modes are encoded through discretized temporal
shifts, and the internal coin-state degrees of freedom are encoded using
photonic polarizations (Fig. S1). With this experimental arrangement, we
successfully carried out one-dimensional and two-dimensional quantum walks in
the same experimental platform under various configurations, as demonstrated
in the following.

The overall architecture is that of a fiber network, through which attenuated
single-photon pulses with a wavelength of $808$ nm and a pulse width of $88$ps
are sent, with each full cycle around the network representing a discrete time
step. Laser pulses are attenuated by neutral density filters, which
effectively reduce the energy of the laser pulses to the single-photon level
at the detection stage. This step aims to maintain an average photon number
per pulse below $2.6\times 10^{-4}$ to minimize the probability of multi-
photon events. The input intensity of the initial laser pulse can be increased
when we aim to obtain amplitude distribution of quantum walks after larger
numbers of steps.

The coin states $\\{|0\rangle,|1\rangle\\}$ are encoded in the photon
polarizations $\\{|H\rangle,|V\rangle\\}$. The single-photon pulse act as the
walker, exhibiting distribution across multiple temporal modes. This is
achieved by building path-dependent time delays into the four different paths
(labeled $x\pm 1$ and $y\pm 1$) in Fig. S1 within the network. Specifically,
for two-dimensional quantum walks, the temporal modes were separated by two
different time scales: $80$ns in the $x$-dimension and $4.83$ns in the
$y$-dimension. For the one-dimensional quantum walk, the temporal modes were
solely distributed along the $x$-dimension.

To implement the two-dimensional quantum walk, we rewrite the time-evolution
operator as $U^{t}=e^{\gamma_{x}\gamma_{y}t}U^{t}_{E}$, where

$U_{E}^{t}=M_{y}^{{}^{\prime}}S_{y}R(\theta_{2})M_{x}^{{}^{\prime}}S_{x}R(\theta_{1})$
(S6)

with $M_{i}^{{}^{\prime}}=e^{-|\gamma_{i}|}M_{i},i=x,y$.

The coin state is initialized after single-photon pulses pass through a
polarizing beam splitter (PBS) and a half-wave plate (HWP). Subsequently, the
photons are coupled in and out from a time-multiplexed configuration via a
beam splitter (BS) with a reflectivity of $3\%$. The coin operator
$R(\theta_{1(2)})$ is implemented by the sandwich-type set of wave plates
(QWP-HWP-QWP), where QWP is the abbreviation of quarter-wave plate. To
implement the shift operators, PBSs separate the photons with different
polarizations and direct them into the two-fiber loop ($S_{x}$) or the free-
space Mach-Zehnder interferometer ($S_{y}$). Specifically, the difference
between two distinct fiber lengths ($270$m and $287.03$m) are used to realize
the polarization-dependent time delay $80$ns in the $x$-dimension. The
corresponding time difference in the $y$ direction is $4.83$ns, which is
introduced by a $1.61$m free space path difference of the free-space Mach-
Zehnder interferometer. Importantly, the coherence of single-photon pulses is
inherently preserved by the interference condition required for the single-
particle quantum walk.

To implement the loss operation $M_{x}^{{}^{\prime}}$, two HWPs are introduced
into each fiber loop. The ones at the input and output ends of the fiber are
also used to keep the polarizations of the single-photon pulses unchanged. For
$\gamma_{x}>0$, we adjust the angle of the HWP in the $x+1$ path satisfying
$\cos\frac{\theta}{2}=e^{-2\gamma_{x}}$, the part of photons
$1-e^{-4\gamma_{x}}$ are transmitted by the second PBS, and leak out of the
setup. For $\gamma_{x}<0$, we set the angle of the HWP on the $x-1$ path to
satisfy $\cos\frac{\theta}{2}=e^{2\gamma_{x}}$ and the part of photons
$1-e^{4\gamma_{x}}$ subsequently leak out of the setup. The loss operator
$M_{y}^{{}^{\prime}}$ is realized with the same method.

Arrival time is recorded by avalanche photodiodes (APDs), aided by an
acoustic-optical modulator (AOM) that functions as an optical switch to
eliminate unwanted pulses. The time-resolved pulses within the window are
recorded and translated to the corresponding spatial position of the walker.
We measure the probability distribution of quantum walks

$P(x,y,t)=\frac{\left|\langle\psi(x,y,t)|\psi(x,y,t)\rangle\right|}{\sum_{x,y}\left|\langle\psi(x,y,t)|\psi(x,y,t)\rangle\right|}=\frac{N(x,y,t)}{\sum_{x,y}N(x,y,t)},$
(S7)

where $N(x,y,t)$ is the total photon number at the position $(x,y)$ after a
$t$-step evolution.

In our experimental setup, the loss of photons is primarily attributed to
losses incurred by various optical elements. Even in the case of a unitary
quantum walk, the overall efficiency of our round-trip single-loop is
approximately $0.71$. The overall efficiency is derived by multiplying the
transmission rates of each optical component employed in the round trip, which
include the transmission rate of the beam splitter ($\sim 0.97$), the
efficiency of collecting photons from free space to fiber ($\sim 0.80$), and
the transmission rates of all other optical components ($\sim 0.91$). Thus, we
have $0.71\simeq 0.97\times 0.80\times 0.91$. Besides, the measurement
duration for a specific time step is approximately one hour, limited by the
stability of our experimental setup.

###  A.3 Initial state preparation and reconstruction of the center-of-mass
motion

As discussed in the main manuscript, the initial excitation of the system
should correspond to an equally-weighted superposition of Bloch eigenstates in
a given lattice band. In our experiment, we cannot directly encode such an
initial state since it contains excitation of both odd and even lattice sites,
which is unfeasible with our setup. To overcome such a limitation, it is
important to note that at each step, wave packets characterized by odd or even
positions in the initial state do not interfere during the evolution process.
This occurs because the evolved state exclusively occupy either even or odd
positions when progressing to step $t$. We can take advantage of such a major
property to split our experiment in two steps, where the initial excitation
occupies either the even or odd lattice sites, and then reconstructing the
wave packet dynamics exploiting the linearity of the system. To illustrate our
strategy, let us consider the two-dimensional quantum walk as an example. We
divided the experiment into two distinct parts, and we rewrite the initial
state as

$\displaystyle|\psi(0)\rangle=\left|\psi^{1}(0)\right\rangle+e^{\gamma_{x}-\gamma_{y}}\left|\psi^{2}(0)\right\rangle,$
(S8)

where

$\displaystyle\left|\psi^{1}(0)\right\rangle=|1\rangle\otimes|0,0\rangle,\left|\psi^{2}(0)\right\rangle=|0\rangle\otimes|-1,1\rangle.$

The evolved state is then
$|\psi(t)\rangle=U^{t}|\psi(0)\rangle=U^{t}\left|\psi^{1}(0)\right\rangle+e^{\gamma_{x}t-\gamma_{y}t}U^{t}\left|\psi^{2}(0)\right\rangle$.
Thus, the wave packet center of mass is given by

$\displaystyle
x_{CM}(t)=\frac{\sum_{x,y}\left[x\left|\langle\psi^{1}(x,y,t)|\psi^{1}(x,y,t)\rangle\right|+e^{2t(\gamma_{x}-\gamma_{y})}x\left|\langle\psi^{2}(x,y,t)|\psi^{2}(x,y,t)\rangle\right|\right]}{\sum_{x,y}\left[\left|\langle\psi^{1}(x,y,t)|\psi^{1}(x,y,t)\rangle\right|+e^{2t(\gamma_{x}-\gamma_{y})}\left|\langle\psi^{2}(x,y,t)|\psi^{2}(x,y,t)\rangle\right|\right]}$
(S9)

and

$\displaystyle
y_{CM}(t)=\frac{\sum_{x,y}\left[y\left|\langle\psi^{1}(x,y,t)|\psi^{1}(x,y,t)\rangle\right|+e^{2t(\gamma_{x}-\gamma_{y})}y\left|\langle\psi^{2}(x,y,t)|\psi^{2}(x,y,t)\rangle\right|\right]}{\sum_{x,y}\left[\left|\langle\psi^{1}(x,y,t)|\psi^{1}(x,y,t)\rangle\right|+e^{2t(\gamma_{x}-\gamma_{y})}\left|\langle\psi^{2}(x,y,t)|\psi^{2}(x,y,t)\rangle\right|\right]}.$
(S10)

One-dimensional quantum walks can also be realized with our setup by simply
removing the free-space Mach-Zehnder interferometer. The probability
distribution is obtained

$P(x,t)=\frac{\left|\langle\psi(x,t)|\psi(x,t)\rangle\right|}{\sum_{x}\left|\langle\psi(x,t)|\psi(x,t)\rangle\right|}=\frac{N(x,t)}{\sum_{x}N(x,t)},$
(S11)

where $N(x,t)$ is the total photon number at the position $x$ after a $t$-step
evolution.

Similarly, the chosen initial state can be written as

$|\psi(0)\rangle=e^{\gamma_{x}}\left|\psi^{1}(0)\right\rangle+\left|\psi^{2}(0)\right\rangle,$
(S12)

where

$\left|\psi^{1}(0)\right\rangle=\ket{1}\otimes\ket{-1},\left|\psi^{2}(0)\right\rangle=\ket{0}\otimes\ket{0}.$
(S13)

The center of mass of the normalized involved state is

$n_{CM}(t)=\frac{\sum_{x}\left(e^{2t\gamma_{x}}x\left|\langle\psi^{1}(x,t)|\psi^{1}(x,t)\rangle\right|+x\left|\langle\psi^{2}(x,t)|\psi^{2}(x,t)\rangle\right|\right)}{\sum_{x}\left(e^{2t\gamma_{x}}\left|\langle\psi^{1}(x,t)|\psi^{1}(x,t)\rangle\right|+\left|\langle\psi^{2}(x,t)|\psi^{2}(x,t)\rangle\right|\right)}.$
(S14)

Therefore, we perform two individual evolutions with different initial states,
which finally enable us to reconstruct $x_{CM}(t)$, $y_{CM}(t)$, and
$n_{CM}(t)$.

Acknowledgments

This work has been supported by the National Natural Science Foundation of
China (Grant Nos. 92265209, 12025401 and 11974331).

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