Self acceleration from spectral geometry in dissipative quantum-walk dynamics

The dynamic behavior of a physical system often originates from its spectral properties. In open systems, where the effective non-Hermitian description enables a wealth of spectral structures in the complex plane, the concomitant dynamics are 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 a 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 [1][2][3][4].Just as the energy of a celestial body impacts its trajectory [5], so the energy quantization accounts for the spontaneous collapse and revival in quantum models [2].In solid materials, transport of electrons depends on the lattice dispersion [6][7][8], 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 [9].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 [10][11][12].This enables a rich variety of spectral geometries on the complex plane, with non-trivial consequences in the system behavior [13][14][15][16][17].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 [18], leading to phase-space dynamics with a changing metric structure [18][19][20].Beyond the semiclassical models, recent studies of non-Hermitian Hamiltonians with the non-Hermitian skin ef-fect [17,21] unravelled that, spectral features such as the closing of the imaginary gap on the complex plane [21][22][23][24][25][26][27][28], or the overall spectral topology [29][30][31][32][33][34][35][36][37], can have detectable dynamic consequences, including anomalous relaxation dynamics [38][39][40], boundary accumulations of loss in the dynamics (known as the edge burst) [26][27][28], and the persistent directional flow which has served as an experimental signature for the non-Hermitian skin effect [41][42][43][44][45][46][47][48][49][50].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 [51] 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 dy- namics in non-Hermitian systems, complementing existing experiments on the long-time dynamics [52][53][54].As the spectral topology is intimately connected with the non-Hermtian skin effect [14][15][16], self acceleration offers a practical and sensitive dynamic signal for its detection, particularly in quantum systems where decoherence dominates at long times.
Results 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.[53][54][55], 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.
In the experiment, we encode the internal coin states {|0⟩, |1⟩} in the photon polarizations {|H⟩, |V ⟩}, 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 [56].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 [57].
In the quantum-walk dynamics, the time-evolved state at the end of each discrete time step t is |ψ(t)⟩ = U t |ψ(0)⟩ = e −iHt |ψ(0)⟩, 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 [51] n As illustrated in Fig. 1, starting from a local excitation, the motion of 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 where k belongs to the first Brillouin zone, and E ± (k) and |ψ ± (k)⟩ are respectively the eigenenergies and eigenstates under the periodic boundary condition (PBC), with the subscripts ± indicating the band index.For a local initial state that is an equal-weight superposition of all eigenstates within a given band, for instance, |ψ(0)⟩ = k |ψ + (k)⟩ ⊗ |k⟩, the short-time behavior of n CM reads (see Methods) where 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 [51].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 We focus on the parameter regime with θ being quite close to π/2, where the complex eigenenergies of the effective Hamiltonian H are approximately (see Methods) corresponding to those of a typical Hatano-Nelson model.We initialize the system in the superposition state (representing a local excitation) |ψ(0 , 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 Here 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 γ x .The wave-function propagation becomes asymmetric when the gain-loss parameter γ x becomes finite.In Figs.2d-f, we show the measured x CM (t), which are quadratic in time when γ x ̸ = 0, consistent with theoretical predictions.By fitting the center-of-mass propagation of the wave functions, we extract the quantity 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 A = 0, and thus acceleration vanishes according to Eq. ( 6).
Simulating the two-dimensional dynamics.In two dimensions, we focus on the coin parameters close to (θ 1 = 0, θ 2 = π/2), where the eigenenergies of the effective Hamiltonians are Here . We initialize the system in the local state |ψ(0)⟩ 2 = |0⟩⊗|x = 0, y = 0⟩−e γx−γy |1⟩⊗|x = −1, y = 1⟩, 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 γ x and γ y , which are parameterized through γ x = γ cos φ and γ y = γ sin φ.Consistent with a previous study [55], 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 γ x or γ 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 γ x = γ 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 |γ x(y) |, consistent with the theoretical analysis.Similar to one-dimensional quantum walks, the self acceleration is a precursor of per- sistent 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 [16]).
It should be mentioned that, in two-dimensional systems, the skin effect is a universal phenomenon that appears under rather arbitrary boundary shapes [16], 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 pointgap topology, it is well-established that the long-time dynamics of a local excitation is a directional propagation with a constant drift velocity [52][53][54], 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 mx , k my ), with where the eigenenergy E + (k mx , k my ) 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 sys- In the momentum space, the Floquet operator U takes the form where the expressions for d x,y,z are given in the Supplemental Material, and σ 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 with corresponding eigenstates (in the coin-state basis) In the experiment we set θ ≈ π/2, so that one has and Notice that Ẽ(k) coincides with the dispersion of the Hatano-Nelson model with asymmetric nearest-neighbor hopping amplitudes (1/2) cos(θ)e −γx and (1/2) cos(θ)e γx .In our experiment, the quasi-local initial state can be expressed as The time-evolved state is then and the center-of-mass of the normalized wave function is defined through [51] n CM (t) := ⟨ψ(t)|x|ψ(t)⟩ ⟨ψ(t)|ψ(t)⟩ .
For short-time dynamics, making use of the truncated expansion e 2 ẼI (k)t ≈ 1 + 2 ẼI (k)t, where Ẽ(k) = ẼR (k) + i ẼI (k), after some straightforward calculations one obtains Here 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 where v m = dE R (km) dk 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 θ 1 ≈ 0 and θ 2 ≈ π/2.Under these conditions, the momentum-space quasienergies are approximately The corresponding eigenstates are The local initial state is the uniform superposition of Bloch states in the upper band We further write the time-evolved wave function at time t as where ψx,y (t) = 1 (2π) 2 π −π π −π dk x dk y e ikxx+ikyy−i Ẽ(kx,ky) .

(23)
Defining the center-of-mass positions of the wave function as in Eq. ( 2), we have x CM (t) ≃ 1 2 a x t 2 − 1 2 and y CM (t) ≃ 1 2 a y t 2 + 1 2 for short-time dynamics, where the self accelerations are given by a x = 1 ∂ky .
Here we defined 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 where For a fixed value of k y (taken as a parameter), the expression ∂kx is the area enclosed by the spectrum E (k x , k y ) 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 2πA x (k y ) in the parameter space, as k y traverses the Brillouin zone.
Finally, for the long-time dynamics, we have where v mx and v my in Eq. ( 8) are the drift velocities, corresponding to the location of the global maximum of 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 [13,18].To illustrate this point, let us consider for example the single-particle dynamics on a one-dimensional lattice with Hamiltonian in the physical space Ĥ = T + V (x), where x is the lattice site position, V (x) is the external potential, T = T (p x ) is the kinetic energy operator, px = −i∂ 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 + γ) + J exp(−ip x − γ), where J exp(±γ) are the asymmetric left/right hopping amplitudes.
For a given initial excitation of the system |ψ 0 ⟩ at time t = 0, with ⟨ψ 0 |ψ 0 ⟩ = 1, the evolved wave function in the early times is given by From this equation, one can readily calculate the time evolution of the mean position ⟨x⟩ = ⟨ψ t |x|ψ t ⟩/⟨ψ t |ψ t ⟩, up to the order ∼ t 2 , and the corresponding initial acceleration, a x = (d 2 ⟨x⟩/dt 2 ) t=0 , which reads explicitly In the above equation, ⟨ Â⟩ 0 ≡ ⟨ψ 0 | Âψ 0 ⟩ denotes the mean value of any operator Â over the initial state |ψ 0 ⟩.Let us now assume that there is not any external force, V (x) = 0, so that the Hamiltonian contains the kinetic energy term solely, Ĥ = T .Using the generalized commutation relation [x, F ( px )] = i(dF/dp x ) for any function F (p x ) of the momentum operator, one obtains Clearly, in any Hermitian system T † = T , we necessarily have a x = 0. Conversely, in a non-Hermitian system where T † ̸ = T , the acceleration a x is non-vanishing for rather arbitrary initial states |ψ 0 ⟩, 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 equalweight superposition of all eigenstates within a given band, and the non-Hermitian skin effect under the OBC [51].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 [14,15].In the presence of the pointgap topology, the area 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 [16].A distinction between generalized reciprocal and non-reciprocal skin effect has also been introduced [16], 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 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 [16].

SUPPLEMENTAL MATERIAL FOR "SELF ACCELERATION FROM SPECTRAL GEOMETRY IN DISSIPATIVE QUANTUM-WALK DYNAMICS"
In this Supplemental Material, we provide some technical and experimental details.
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(θ).As all the operators can be expressed by Pauli matrices, the time-evolution operator can be also written as The effective Hamiltonian is defined through the relation U = e −iH , which in the quasi-momentum space takes the form with Here the quasienergies are given by E ± (k) = ± arccos [cos θ cos(k − iγ x )], which are the eigenvalues of H.
Similarly, for the two-dimensional case, we have where the quasienergy is Experimental scheme 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 88ps 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 × 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⟩, |1⟩} are encoded in the photon polarizations {|H⟩, |V ⟩}.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 ± 1 and y ± 1) in Fig. S1 within the network.Specifically, for twodimensional quantum walks, the temporal modes were separated by two different time scales: 80ns in the x-dimension and 4.83ns 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 γxγyt U t E , where

FIG. S1.
A time-multiplexed implementation of the two-dimensional photonic quantum walk.The 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.
with M ′ i = e −|γi| 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(θ 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 (270m and 287.03m) are used to realize the polarization-dependent time delay 80ns in the x-dimension.The corresponding time difference in the y direction is 4.83ns, which is introduced by a 1.61m 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 , 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 γ x > 0, we adjust the angle of the HWP in the x + 1 path satisfying cos θ 2 = e −2γx , the part of photons 1 − e −4γx are transmitted by the second PBS, and leak out of the setup.For γ x < 0, we set the angle of the HWP on the x − 1 path to satisfy cos θ 2 = e 2γx and the part of photons 1 − e 4γx subsequently leak out of the setup.The loss operator M ′ y 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) = |⟨ψ(x, y, t)|ψ(x, y, t)⟩| x,y |⟨ψ(x, y, t)|ψ(x, y, t)⟩| = N (x, y, t) 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 (∼ 0.97), the efficiency of collecting photons

FIG. 1 .
FIG.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 ky (upper panel) and kx (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.

FIG. 2 .
FIG.2.Self acceleration in one-dimensional dynamics.a-c Dynamic evolutions governed by the effective non-Hermitian Hamiltonian with parameters γx = 0, γx = −0.05 and γx = 0.1, respectively.d-f Evolution of the center of mass nCM (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 γ = γx.We take a local initial state |ψ(0)⟩ 1 and the coin parameter θ = 0.45π.h PBC energy spectra for increasing values of γ.

FIG. 3 .
FIG. 3. Self acceleration in two-dimensional dynamics.a Wave packet center of mass xCM (t) (yCM (t)) as a function of the discrete time step t.The gain-loss parameter is taken as γ = 0.08.b Probability distribution of the 24-time-step quantum walk with φ = π/5.We mark the position occupied by the initial state as the red square.c, d The wave packet center of mass xCM (t) (yCM (t)) as a function of the discrete time step t with φ = π/5.The other parameters are θ1 = 0.12, θ2 = π/2 − 0.12, and the initial state |ψ(0)⟩ 2 .

2 FIG. 4 .FIG. 5 .
FIG. 4. Measurement of the spectral volume.a PBC energy spectrum with ky as a parameter and for γx = γy.The black, purple, red, and blue surfaces correspond to γ x(y) = 0, γ x(y) = −0.04,γ x(y) = 0.04, and γ x(y) = 0.08, respectively.b, c The average of the areas enclosed by E+(kx, ky) as a function of γ x(y) .The other parameters are θ1 = 0.12, θ2 = π/2 − 0.12, and the initial state |ψ(0)⟩ 2 .The blue dashed line represents the theoretical results, and the gray dashed lines correspond the results fitted by experimental data.a b

2 < 0 .
dE I (km) dk = 0 and d 2 E I (km) dk Defining ξ = k−k m , we expand the quasienergy around k m , where the leading orders give E