Bloch oscillations, Landau–Zener transition, and topological phase evolution in an array of coupled pendula

Significance

We demonstrate in a classical system of coupled pendula, phenomena that are usually attributed to quantum dynamics of electrons or atoms in ultra-clean lattices and to topological insulators. In dynamics that lasted hundreds of seconds, we identified Bloch oscillations, in which an initial wave-packet drifts in Fourier space and therefore moves back and forth in real space. We also measured the Landau–Zener transition probability between nearby energy bands and the analog of quantum superposition of states. Finally, we directly measured the Zak topological phase by comparing the phase accumulated in a topologically nontrivial experiment to a topologically trivial one. Our work opens numerous possibilities of classical mechanical realizations of other solid-state effects, with and without topological signatures.

Abstract

We experimentally and theoretically study the dynamics of a one-dimensional array of pendula with a mild spatial gradient in their self-frequency and where neighboring pendula are connected with weak and alternating coupling. We map their dynamics to the topological Su–Schrieffer–Heeger model of charged quantum particles on a lattice with alternating hopping rates in an external electric field. By directly tracking the dynamics of a wave-packet in the bulk of the lattice, we observe Bloch oscillations, Landau–Zener transitions, and coupling between the isospin (i.e., the inner wave function distribution within the unit cell) and the spatial degrees of freedom (the distribution between unit cells). We then use Bloch oscillations in the bulk to directly measure the nontrivial global topological phase winding and local geometric phase of the band. We measure an overall evolution of 3.1 $\pm$ 0.2 radians for the geometrical phase during the Bloch period, consistent with the expected Zak phase of $\pi$. Our results demonstrate the power of classical analogs of quantum models to directly observe the topological properties of the band structure and shed light on the similarities and the differences between quantum and classical topological effects.

Classical analogues of condensed and topological states of matter proved to be an invaluable tool in visualizing fundamental concepts and exploring the relations between quantum and classical dynamics. The topological classifications of states of systems, which originate in quantum lattice models, quantum Hall effect, quantum spin-Hall effect, topological insulators, and superconducting qubits (1–10) were adapted to various classical systems such as photonic crystals (11–14), phononic crystals (2, 15–18), classical mechanical oscillators (19–24), and electrical resonator circuits (25–28). Many of these works focused on probing stable edge channels, which provide indirect evidence for the nontriviality of the band topology. Some measured the topology more directly via the internal structure of the edge mode. The classification of topological invariants via direct bulk measurement, however, is limited (29, 30) and remains a challenge in both quantum and classical systems.

Here, we report a direct measurement of a topological and geometric phase by tracking the dynamics in the bulk of a classical system: a one-dimensional array of coupled pendula. The analysis of the measurement is based on an approximate mapping between the time evolution of the classical coupled oscillators and the quantum tight-binding or discrete Schrödinger equation of electrons on lattice potentials, as was recently demonstrated for the nonlinear case (26, 31, 32). The mapping is in the spirit of the mapping in ref. 33, but ours is strictly local (i.e., the local wave function is associated with a single pendulum’s motion). In particular, our system is mapped onto the canonical Su–Schrieffer–Heeger (SSH) model (34, 35), a prototypical model for topological phase transitions in one dimension. This mapping is significant, as the SSH model has two phases: one with a trivial Zak phase (36, 37) that is adiabatically connected to an atomic limit and the other with a Zak phase of $\pi$, with an obstruction to the atomic limit. In addition, a mild monotonic change in the pendulum self-frequency along the array is mapped to an external electric Field.

We experimentally realized such a system of $\sim$50 pendula. Using the mapping to the SSH model, we theoretically, computationally, and experimentally show that our system exhibits phenomena that are usually discussed in the context of the quantum dynamics of electrons on ultra-clean lattices. This includes Bloch oscillations (38), where, due to the electric field and the periodicity of the lattice, an electron’s wave function oscillates in the presence of a uniform electric field instead of accelerating, and Landau–Zener (LZ) tunneling (39, 40), in which the electron’s wave-packet can leak to a higher energy band, or be in a superposition of the two bands, due to the time-dependent energy difference between the bands. Finally, these observations enable us to extract the nontrivial topological phase winding of the SSH bands from bulk measurements in our system. Specifically, we use the fact that, due to Bloch oscillations, a wave-packet that is initially localized in reciprocal space samples the entire Brillouin zone during its dynamics, thereby possessing information on the geometrical phase and the Zak phase after a full Bloch period (8, 41).

The extraction of the geometrical phase is enabled due to the classical and macroscopic nature of our system, which allows direct measurement of wave-packet dynamics and the amplitude and phase evolution of each oscillator. Specifically, we extract the geometrical phase by comparing the wave-packet phase evolution in two experiments—one with trivial and the other with topological band structure. We accurately extracted the difference of the geometrical phase evolution between these two experiments over 400 periods of the fundamental oscillation of an individual pendulum, which lasted about 700 s, and which indeed ended in a Zak phase difference of $\pi$ after one Bloch period, as theoretically expected.

Classical Realization of the SSH Model

We constructed a one-dimensional array, schematically shown in Fig. 1A, of $N=51$ pendula hanging using v-shaped strings from a common horizontal fixed beam. All pendula had identical mass $m=0.35$ kg yet varying string lengths $r_j$, ($j=1,\cdots ,N$), carefully tuned to satisfy

$$\begin{array}{c}\hfill \frac{1}{r_j}=\frac{1}{r}\left[1+\alpha \left(j-\frac{N}{2}\right)\right].\end{array}$$ [1]

Fig. 1.

(A) Sketch of the mechanical system: pendula coupled to each other by knots (depicted by red and blue dots) connecting adjacent strings at varying and alternating heights ${\delta }_{j,j+1}$. The pendulum lengths $r_j$ have a mild gradient according to Eq. 1. (B) The experiment started with a wave pattern implemented using a board that was cut according to the desired wave-packet. Then the board was abruptly removed and the pendula started to evolve freely. (C) The solution to Eq. 7 in Fourier space, exhibiting Bloch oscillations due to the external field: The initial wave packet (black) travels toward negative $k$ values and follows the SSH lower energy band (blue). Then at $k=-\pi$, depending on the parameters of Eq. 7, the wave continues its travel either adiabatically following the lower band, or by jumping through LZ diabatic transition and following the upper band (red), or by following in a superposition of the two bands.

The parameter $r=0.76m$ is the length of the central pendulum, and $\alpha =2.4\times {10}^{-3}$ controls the spatial gradient in string length. The accuracy in tuning the length of each pendulum was $<1$ mm (SI Appendix).

Each pendulum is coupled to its two nearest neighbors by connecting adjacent strings by knots at alternating heights, close to the beam, as depicted in Fig. 1A, (see SI Appendix). This results in weak and alternating coupling between neighboring pendula, i.e., the coupling constitutes a relatively weak perturbation to the basic pendulum oscillations. The pendulum oscillations are kept at small angles during the dynamics, such that we can model the system as a set of simple harmonic oscillators coupled to each other with harmonic springs of small and alternating effective stiffnesses, ${\kappa }_{j,j+1}$. Each coupling ${\kappa }_{j,j+1}$ is controlled by adjusting the height of the knots that couple adjacent pendula according to

$$\begin{array}{c}\hfill {\kappa }_{j,j+1}=\left\{\begin{array}{cc}\kappa \hfill & j\mathrm{even}\hfill \\ {\kappa }^{\prime }\hfill & j\mathrm{odd}\hfill \end{array}\right..\end{array}$$ [2]

Details on how the knot heights determine the couplings $\kappa$ and ${\kappa }^{\prime }$ appear in SI Appendix. We designed and performed three experiments, with different values of $\kappa$ and ${\kappa }^{\prime }$. Experiments 1 and 2 were designed to observe Bloch oscillations and the change in topological phase in the adiabatic LZ regime. In Experiment 1, the couplings were $\kappa =0.07$ N/m and ${\kappa }^{\prime }=0.035$ N/m. In Experiment 2, $\kappa$ and ${\kappa }^{\prime }$ were switched to $\kappa =0.035$ N/m and ${\kappa }^{\prime }=0.07$ N/m. Experiment 3 was designed to observe LZ tunneling and wave-packets that are in a superposition of the upper and lower bands. There, the couplings were set to $\kappa =0.082$ N/m and ${\kappa }^{\prime }=0.064$ N/m, in order to enhance the LZ transition.

The three experiments were all performed in a similar fashion. Initially, all pendula rested in their minimum energy point, except ten adjacent pendula in the bulk of the lattice, that were given an initial offset from their rest position. As shown in Fig. 1B, these translations were given by a board, cut in advance to produce a particular initial wave pattern (SI Appendix). At time $\tau =0$, the board was abruptly removed, and the system was allowed to evolve freely. The full evolution of the pendula in each experiment was video recorded from below [Video (42)]. Finally, the position of each of the pendula as a function of time was extracted from the video.

Fig. 2 presents the results of Experiments 1 and 3, while Experiment 2 (not shown) produced results almost identical to those of Experiment 1. In Experiment 1, the energy stored in each pendulum performed slow oscillations: The wave-packet moved to lower pendulum numbers and then moved back, and returned to the very same initial wave-packet at $\tau =700$ s, only with smaller amplitude due to $\sim$70% energy loss to friction. Apart from friction, the fraction of energy that leaked and appeared as motion of other pendula and did not return in the original wave-packet after 700 s, was $5\%$ or less. Moreover, the inset clearly shows that in the middle of the oscillation, at $\tau =350$ s the wave was checkerboard-like: Adjacent pairs of pendula were swinging out of phase, namely, the wave-packet was centered around $k=\pi$ when the unit cell is a dimer. In Experiment 3, at that middle point, the wave split into two branches, which moved toward two opposite directions at later times. However, the two partial waves reached a maximal translation at $\tau =750$ s, and at that point, a zoom-in shows that at one branch the pendula all swing in phase, while in the other branch, neighboring pendula swing with alternating phases. The two branches then turned back and merged again at about $\tau =1125$ s.

Fig. 2.

(A and B) The pendulum displacements in Experiments 1 (A) and 3 (B). Each line shows a pendulum displacement as a function of time and is shifted vertically according to the pendulum index. Bloch oscillations are clearly seen in both experiments. In Experiment 1, the wave follows adiabatically the lower band, while in Experiment 3 after one Bloch period, the system is in a superposition of waves in the Lower and Upper bands. The Insets show zoom-ins of the pendulum displacements at certain times, in which the state of the partial wave (its mean $k$ value, and the band) can be identified (see text); (C and D) Discrete spatial Fourier transform of the pendulum motion at each time, projected onto the two theoretical eigenstates of the SSH problem, Eq. 11. These represent the partial wave in the Lower (Bottom) and Upper (Top) bands.

As we now show, these observations can be understood via a mapping to the quantum SSH model, and are direct manifestations of Bloch oscillations and LZ transitions.

Mapping to the Quantum SSH Model

The dynamics of the pendulum system is described by $2N$ linearized Hamilton equations for the translation of each pendulum in the transverse direction, $y_j(\tau )=r_j{\theta }_j(\tau )$, where ${\theta }_j(\tau )$ is the pendulum’s angle from the vertical axis, and the momentum $p_j(\tau )$ of each of the pendula, as a function of time $\tau$,

$${\dot{y}}_j=\frac{1}{m}{p}_j,$$ [3]

$${\stackrel{\cdot }{p}}_j=-\frac{mg}{r_j}{y}_j-{\kappa }_{j-1,j}\left(y_j-y_{j-1}\right)+{\kappa }_{j,j+1}\left(y_{j+1}-y_j\right).$$ [4]

Experimentally, after hundreds of oscillations of the pendula, we observed significant energy dissipation of more than 50% due to friction. However, it appeared as an overall global decay of the amplitude of all pendula and did not affect any of the effects or the measurements of the topological phases as discussed below. We confirmed this by simulations of Eqs. 3 and 4 in which effective dissipation terms $-{\mathrm{\Gamma }}_j{p}_j$, were added to the r.h.s of Eq. 4 with ${\mathrm{\Gamma }}_j$ randomly taken from a uniform distribution with a large range of values, $[0,1/350{\mathrm{s}}^{-1}]$. We found that, while the overall decay rate was similar to that observed in the experiments, the extracted topological phase and the LZ transition rate were not affected. Therefore, for simplicity, in our theoretical analysis hereafter energy dissipation is not considered. We discuss the discrepancies between our model and the actual setup in the discussion ahead as well as in SI Appendix.

Crucial to the accuracy of the mapping to the SSH model, we design the system such that the natural frequency of each pendulum, $\sqrt{\frac{g}{r}}$, fulfills $\frac{g}{r}\gg \frac{\kappa +{\kappa }^{\prime }}{m}$, and the gradient is small, $\alpha \ll 1$. Namely, that $\sqrt{\frac{g}{r}}$ is significantly larger than any other frequency scale in the problem. Under these conditions, we find that if one introduces a complex dynamic variable

$$\begin{array}{cc}\hfill {\psi }_j& \equiv e^{i{\omega }_0\tau }{u}_j,\hfill \end{array}$$ [5]

such that

$$\begin{array}{cc}\hfill u_j& \equiv \left(\sqrt{\frac{\mathit{gm}}{2r}}{y}_j+i\sqrt{\frac{1}{2m}}{p}_j\right),\hfill \end{array}$$ [6]

and where ${\omega }_0\equiv \sqrt{\frac{g}{r}}+\sqrt{\frac{r}{g}}\frac{1}{m}\left(\kappa +{\kappa }^{\prime }\right)$, then to a very good approximation, ${\psi }_j$ evolves according to

$$\begin{array}{cc}\hfill i\dot{{\psi }_j}& \approxeq \frac{\mathit{Ea}}{2}\left(j-\frac{N}{2}\right){\psi }_j-t_{j-1,j}{\psi }_{j-1}-t_{j,j+1}{\psi }_{j+1}.\hfill \end{array}$$ [7]

The approximation used in deriving Eq. 7 is similar to the rotating wave approximation used, e.g., in NMR analysis (see full derivation in SI Appendix). This equation is identical to the Schrödinger equation for the SSH model with lattice constant $a$ in the presence of an electric field $E$. The alternating hopping terms are related to the system’s parameters by

$$\begin{array}{cc}\hfill t_{j,j+1}& =\left\{\begin{array}{cc}\sqrt{\frac{r}{g}}\frac{1}{2m}\kappa \equiv t\hfill & j\mathrm{even},\hfill \\ \sqrt{\frac{r}{g}}\frac{1}{2m}{\kappa }^{\prime }\equiv t^{\prime }\hfill & j\mathrm{odd}.\hfill \end{array}\right.\hfill \end{array}$$ [8]

Given Eq. 7, the combination

$$\begin{array}{cc}\hfill \mathit{Ea}& =\sqrt{\frac{g}{r}}\alpha \hfill \end{array}$$ [9]

is related to the Bloch frequency by $Ea=\frac{\mathit{dk}}{d\tau }$, where $-\pi <k<\pi$ is the dimensionless wave vector that corresponds to the discrete Fourier transform with respect to unit cell numbers (note that the unit cell is a dimer). The accuracy of Eq. 7 depends on the original system’s parameters; Specifically, Eqs. 3 and 4 have two small parameters. The first is $\alpha$, and the second can be defined as $ϵ\equiv \sqrt{\frac{g\left(\kappa +{\kappa }^{\prime }\right)}{\mathit{rm}}}$, which is the ratio between the pendulum’s and the coupling’s self-frequencies. Both should be made as small as possible for a faithful mapping to the SSH model.

Experimental Measurement of Bloch Oscillations and LZ Transition

In order to compare the phenomena seen in Fig. 2 A and B to the prediction of Eq. 7 we analyze them in Fourier space by performing two Fourier transforms over the odd-$j$ “a” sites and the even-$j$ “b” sites. On the theoretical level, Eq. 7 with zero electric field leads to two energy bands with eigenvalues ${\omega }_k=\pm v_k$ and eigenstates in the $(a,b)$ dimer inner space ${\xi }_{k,1/2}\equiv \frac{1}{\sqrt{2}}{(1,\pm e^{i{\phi }_k})}^T$ where $v_k{e}^{i\phi }=t+t^{\prime }{e}^{-ik}$ (see SI Appendix for details). Due to the small electric field, during the Bloch oscillation, an initial wave-packet changes the value of its central wavenumber $k$ in momentum space at a constant rate, periodically sampling the whole Brillouin zone $-\pi <k<\pi$. At $k=\pi$ the gap is minimal, and depending on the rate $\frac{\mathit{dk}}{d\tau }=Ea$, the wave packet evolution can vary, from following adiabatically the lower band, to jumping to the upper band through the LZ transition, to being in a superposition of the two bands; see illustration in Fig. 1C.

The Bloch oscillations observation is more convincing in $k$-space with respect to the unit cell (the dimers) number. Thus, each of the three experiments was further analyzed by taking the complex values of $u_j$ obtained from Eq. 5 using the measured values of $y_j$ and $p_j$, and performing, at each point in time, two discrete Fourier transforms of the wavefunctions in the odd “a” sites $u_{2l+1}$ and of the even “b” sites $u_{2l+2}$ and defining

$$\begin{array}{c}\hfill u_{k,a}\equiv {\displaystyle \sum _{l=0}^{N/2-1}}{u}_{2l+1}{e}^{-ilk},u_{k,b}\equiv {\displaystyle \sum _{l=0}^{N/2-1}}{u}_{2l+2}{e}^{-ilk}.\end{array}$$ [10]

We project the resulting vector ${(u_{k,a},u_{k,b})}^T$ onto the analytically derived lower- and upper-energy eigenstates of the SSH model ${\xi }_{k,1/2}$

$$\begin{array}{cc}\hfill u_{k,1}& \equiv \frac{1}{\sqrt{2}}\left(u_{k,a}+e^{-i{\phi }_k}{u}_{k,b}\right),\hfill \\ \hfill u_{k,2}& \equiv \frac{1}{\sqrt{2}}\left(u_{k,a}-e^{-i{\phi }_k}{u}_{k,b}\right),\hfill \end{array}$$ [11]

where ${\phi }_k$ is estimated from the experimental $\kappa$ and ${\kappa }^{\prime }$, and their mapping to $t$ and $t^{\prime }$. The result is shown in Fig. 2C for Experiment 1 and in Fig. 2D for Experiment 3. One can see that the wave-packet started its evolution almost entirely in the lower band. Indeed, the initial condition $u_j(\tau =0)$ implemented by the cut board was designed to be a Gaussian in the lower band, centered at $k=0$ with $\mathrm{\Delta }k\approx 0.35$ small compared to the Brillouin zone dimensionless size of $2\pi$.

At early times, the Gaussian in $k$-space stayed in the lower band and shifted to lower values of $k$ at a constant rate, until reaching the middle of the Bloch cycle, corresponds to $k=\pm \pi$, as can be clearly observed from the zoom-in in the insets of the corresponding figures in $j$-space. The wave-packet continued to evolve along the lower band in Experiments 1 and 2, whereas in Experiment 3, the wave split and some of its amplitude was transferred to the upper-band via a LZ transition. This could be easily spotted in $j$-space, because the continuing Bloch oscillation at later times made the two parts of the wave-packet move in different directions, due to the opposite signs of the group velocities $\frac{d{\omega }_k}{\mathit{dk}}$ of the two bands. Furthermore, the zoom-ins in $j$-space clearly show the two different inner dimer states of the two bands at $k=0$— the state (1,$-$1) for the upper band and the state (1,1) for the lower band. In this respect, the real space evolution of the wave-packets was coupled to the pseudo-spin degree of freedom defined by the sublattice (i.e., the upper and lower bands) —similar to the situation in the Stern-Gerlach experiment.

Validation of the Mapping with Numerical Simulations

Simulating the system can further demonstrate the strength and robustness of the mapping for displaying quantum-analog effects in classical mechanical systems, over a wide range of parameter values. We simulated the system of coupled pendula by solving numerically Eqs. 3 and 4 for various sets of parameters $N$, $\alpha$, $r$, $\kappa$, and ${\kappa }^{\prime }$. We used the 4–5 Runge–Kutta method with both absolute and relative tolerances of ${10}^{-9}$.

We checked the accuracy of the mapping by quantitatively predicting the details of Bloch oscillations and the LZ transition. Fig. 3A verifies the validity of the theoretical expression for the Bloch frequency $\frac{\mathit{dk}}{d\tau }=\sqrt{\frac{g}{r}}\alpha$, by showing excellent agreement as the parameters $\alpha ,g,r$ were varied over considerable ranges. Other features of the oscillations, such as their amplitude, are also well predicted by the mapping to the SSH model; see SI Appendix for details.

Fig. 3.

(A) The simulated rate of Bloch oscillations compared with the theoretical prediction $\frac{\mathit{dk}}{d\tau }=\sqrt{\frac{g}{r}}\alpha$ for a significant range of the values of the parameters $g$, $r$, and $\alpha$. The solid line marks the identity. The experimentally measured value (red) of the Bloch period agrees with the theoretical prediction, with the error-bar marking the spread between Experiments 1, 2, and 3. (B) The LZ diabatic transition probability as extracted from experiments and simulations, Eq. 13, vs. the theoretical prediction, Eq. 12. The simulations covered variations in all parameters, $g$, $r$, $\kappa$, ${\kappa }^{\prime }$, and $\alpha$. The solid line marks the identity. Inset: relative deviation of the simulated values from the prediction. For reasonable values of $ϵ=\sqrt{r(\kappa +{\kappa }^{\prime })/gm}$, which control the accuracy of the mapping of the array of pendula to the SSH model, the deviation is 5% or less.

The LZ diabatic transition probability is given by $P_D= exp\left(-2\pi V^2/d\right)$, where $V$ is the off-diagonal element and $d$ is the rate of change of the diagonal part of the Hamiltonian in the LZ model (39, 40). Given the energy bands in the SSH model and the mapping in Eqs. 8 and 9, we can express $P_D$ using the pendulum array’s parameters (see derivation in SI Appendix),

$$\begin{array}{c}\hfill P_D= exp\left[-\frac{\pi r{\left(\kappa -{\kappa }^{\prime }\right)}^2}{2mg\alpha \sqrt{\kappa {\kappa }^{\prime }}}\right].\end{array}$$ [12]

Fig. 3B shows results of simulations of Eqs. 3 and 4 using $N=200$ pendula, using an initial Gaussian wave-packet in the lower band, for various values of $g,r,\kappa$ and ${\kappa }^{\prime }$. The relative energy transfer to the upper band after one Bloch oscillation, at $\tau =T_B=2\pi /\frac{\mathit{dk}}{d\tau }$, observed in the simulations,

$$\begin{array}{c}\hfill P_{D,\mathrm{sim}}\approx \frac{{\sum }_k{\left|u_{k,2}(\tau = T_B)\right|}^2}{{\sum }_k{\left|u_{k,1}(\tau = 0)\right|}^2}\end{array}$$ [13]

was compared to the theoretical expression in Eq. 12. The agreement between the simulation and the analytical formula extends over almost three orders of magnitude with no fitting parameters and over a considerable range in $g,r,\kappa ,{\kappa }^{\prime }$, and $\alpha$, thus demonstrating the robustness and insensitivity of the mapping to changing parameters. When $ϵ$ is less than 0.2, the error in predicting the LZ transition probability is a few percent or less. As a side note, this accuracy is why both the analytical calculations and the simulations were proven to be useful in designing the experiments in this work. In Fig. 3, we also present the measured values of $\mathit{Ea}$ and of $P_D$ from our experiments, again in good agreement with the theoretical results. Error-bars were estimated from the spread of the results in the three experiments and from the accuracy of determining the parameters in the experiment; see SI Appendix for more details.

Geometrical Phase Evolution and Band Topology

When the values of $\kappa$ and ${\kappa }^{\prime }$ are switched (such was the situation between Experiments 1 and 2), the topology of the bands of the SSH Hamiltonian switches from trivial to nontrivial (where the triviality/nontriviality here is subjected to the choice of the unit cell with the “a” cell on the left). The topology of the band is imprinted in the evolution of the wave-packet in the form of a global $\pi$ phase difference after one Bloch period. Contrary to the situation in the quantum case, the global phase of the classical pendulum motion complex variables $u_j$ (and of ${\psi }_j$) was easily measurable. Therefore, in our classical system, it is possible to fix a gauge and extract the whole evolution of the geometrical phase as a function of the quasi-momentum $k$ during the Bloch oscillation period that leads to this $\pi$ phase shift. This is achieved by extracting the phase of the complex wave function component in Fourier space, $u_{k(t),1}$ in Fig. 2C, at the maximum of the wave-packet as a function of $\tau$.

To understand the relation between this phase of the wave-packet maximum and the topological phase, we solve the dynamic equation of the SSH model in the limit of adiabatic evolution. In SI Appendix, we show that in the case of an adiabatic evolution in the lower band, the solution of Eq. 7 is such that the maximum of the wave packet moves in momentum space along the line $k_c(\tau )=(-Ea\tau +\pi )mod2\pi -\pi$ (the wave-packet maximum in Fig. 2C). In addition, we show that the wave function $u_{k(\tau ),1}$ has an additional phase term which is added to the basic oscillations, $e^{-i\left[{\omega }_0\tau +\varphi (k,\tau )\right]}$, and along this line, $k=k_c(\tau )$ it is given by

$$\begin{array}{c}\hfill \varphi \left[k_c(\tau ),\tau \right]={\varphi }_0-\frac{N}{4}Ea\tau -{\int }_0^{\tau }{v}_{k^{\prime }=k_c({\tau }^{\prime })}d{\tau }^{\prime }+\frac{1}{2}{\left.{\phi }_k\right|}_0^{k_c(\tau )}.\end{array}$$ [14]

Only the last term, which is the geometrical phase evolution picked along the line $k=k_c(\tau )$ between the times $0$ and $\tau$, changes between Experiments 1 and 2, namely when switching $\kappa \leftrightarrow {\kappa }^{\prime }$. This is because in the first term, ${\varphi }_0$ depends only on the initial condition, the second term is related to the global potential being centered around the $l=N/4$ unit cell, and the third term is the dynamic phase evolution, according to the lower band eigenvalues, along the line $k^{\prime }=k_c({\tau }^{\prime })$, ${\tau }^{\prime }\in [0,\tau ]$. The last term depends on the specific choice of gauge, which is fixed by the choice of the global phase for the eigenstate ${\xi }_{k,1}\equiv \frac{1}{\sqrt{2}}{(1,e^{i{\phi }_k})}^T$ at every $k$, as manifested in Eq. 11 above. We conclude that within our model, the only difference in the phase of $u_{k(\tau ),1}$ between Experiments 1 and 2 originates from the changes in the geometrical phase.

Fig. 4B shows this phase difference of $u_{k(\tau ),1}\left(\tau \right)$ along the line $k(\tau )=k_c(\tau )$ (the maximum of the wave packet) in the two experiments. The complex values of $u_{a,k_c(\tau )}$ and $u_{b,k_c(\tau )}$ in each experiment were extracted from the Fourier transform of the measured pendulum displacements, followed by 2D interpolation in Fourier space. While the phase in each experiment alone completed about 2,500 radians during the first Bloch period (Inset), the evolution of the phase difference between the two experiments followed to a very good approximation the theoretically predicted difference of the geometrical phase evolution, the fourth term in Eq. 14, $\frac{1}{2}arg(\kappa +{\kappa }^{\prime }{e}^{-iEat})$, and completed a $\pi$ phase shift when completing a full Bloch oscillation, within a measurement accuracy of 0.2 rad. Note that the phase difference is gauge independent, as long as the same gauge is used in both experiments.

Fig. 4.

(A) Phase difference between the dimer components, $arg\left(u_{b,k(\tau )}\right)-arg\left(u_{a,k(\tau )}\right)$ at $k(\tau )=-Ea\tau$, as extracted from the displacements of the pendula in Experiments 1 (dark green line) and 2 (blue line). The dashed red lines show the theoretical curves $arg(\kappa +{\kappa }^{\prime }{e}^{iEa\tau })$, where in Experiment 1, $\kappa =0.0035$ N/m,${\kappa }^{\prime }=0.07$ N/m and in Experiment 2, $\kappa =0.07$ N/m, ${\kappa }^{\prime }=0.035$ N/m. The accuracy of the couplings is about $5\%$ (1 mm in the knots’ heights). The purple dotted line marks the Bloch oscillation period $T_B$. The winding number of the phase after a full Bloch oscillation clearly shows that in Experiment 1 the set-up was in the non-trivial topological phase, while in Experiment 2 it was in the trivial topological phase. (B) Difference in global phase evolution of $u_{a,-eE\tau }$ between Experiments 2 and 1, where the full phase evolution from the two experiments are shown in the Inset. After 700 s, and about 2,500 radians of phase evolution of the swings of the pendula, the difference between the phases reached $\pi$ to very good accuracy. The dashed red line is the theoretical prediction $\frac{1}{2}arg({\kappa }^{\prime }+\kappa e^{-iEa\tau })-\frac{1}{2}arg(\kappa +{\kappa }^{\prime }{e}^{-iEa\tau })$, with $\kappa =0.035$ N/m, ${\kappa }^{\prime }=0.07$ N/m.

Information about the local geometry and global topology of the band can also be extracted from a single experiment by comparing the phase evolution of $u_{k,a}\left(\tau \right)$ and $u_{k,b}\left(\tau \right)$. The phase difference of the two entries of ${\xi }_{k,1}$ equals ${\phi }_k$, independent of the choice of gauge. In Fig. 4A, we plot the phase difference between $u_{k,a}\left(\tau \right)$ and $u_{k,b}\left(\tau \right)$ in the two experiments. This phase difference is shown to follow the theoretical curve $arg(\kappa +{\kappa }^{\prime }{e}^{-iEat})$, which results in a different winding number for switching $\kappa$ and ${\kappa }^{\prime }$. The small ripples of the experimental curves, which appear to dominate the accuracy in measuring the geometrical phase and the winding are attributed to a small but finite LZ diabatic transition (1 to 2$\%$). Similar ripples were seen in simulations that we performed with similar conditions and can, in principle, be improved by future experiment with, e.g., a larger band gap.

Discussion

We have shown that a system of coupled pendula with alternating couplings and a mild gradient in the lengths of the pendula obeys remarkably accurately the Schrödinger equation of the topological SSH model in an external electric field. As such, its dynamics exhibits Bloch oscillations, LZ transitions that are followed by entanglement between the band and spatial degrees of freedom, and the appearance of the nontrivial topological phase of the bands. These features that are usually attributed to microscopic quantum systems now appear on a macroscopic classical system, which is easily observed in full detail through direct measurement of the evolution of its “wave function”. This enabled us to quantify the Bloch oscillations rate and the LZ diabatic transition probability that both perfectly agree with theoretical predictions. Finally, we directly extract the topology of the bands from an experimental measurement in the bulk of a macroscopic system—both within one evolution of the system by measuring the relative phase of the component of the dimer, and by comparing the phase evolution in two experiments performed on the topological and the trivial states of the system. Our measurement of the Zak phase was remarkably accurate—an accuracy of $0.2$ rad after $800\pi$ radians evolution—a relative error of less than ${10}^{-4}$. This enabled us to easily observe the accumulated Zak phase of $\pi$. It appears that pendulum arrays with high Q-factors such as our system hold advantages in bulk topological measurements compared to passive electrical resonators, although active ones (28) could potentially reach similar results.

The fact that these phenomena could be accurately measured in a simple, macroscopic, mechanical system is not trivial in several aspects; first, the mapping we used is strictly local, in the sense that it uses one pendulum motion per wave function value, but it is approximate. This, in contrast to the one used in ref. 33 for the classification of the topological phases, which was exact, but for our system, it would use a linear combination of the motion of at least three pendula; moreover a different linear combination for each of the three experiments. Here we used an approximate mapping that is simple, and common to all experiments, in order to have a simple observation of the wave function evolution, and in order to rule out the possibility that the observed change in topology between experiments is somehow connected to the mapping (for example, through some indirect choice of the unit cell). Our choice of a simple and common mapping ensures that the measurement of the Zak phase is directly connected to the evolution itself.

Second, we have shown that, unlike the quantum case, the significant dissipation of energy to heat in our experiment did not affect the coherence of the wave. Moreover, we note that there were additional degrees of freedom in the motion of each pendulum—such as motion along the axis of the beam (on top of the oscillations in the perpendicular direction that we focused on), and rotation of each pendulum due to the unbalanced torque acting on each pendulum from its coupled neighbors. Apparently those did not affect the coherence either, even to the above-mentioned degree of accuracy. In SI Appendix, we made a list of the known possible discrepancies between the model and the experimental setup, and the means taken to reduce or mitigate their influence.

These results open the way to quantitatively explore further quantum-like phenomena in classical systems. Future theoretical and experimental works can explore the nonlinear nature of physical pendula. Another possible direction of study is to examine the effect of point defects or imperfect lattices on the topological effects. In addition, note that the SSH one-dimensional topological insulator is only one example among many quantum lattice models with nontrivial features; our work can be generalized quite naturally to other systems by considering different higher-dimensional lattices of oscillators. This includes single and double-layer honeycomb lattices, and 2D or even 3D topological insulators. The advantage of our system, in which every quantity can be easily measured by video recording, would be useful in investigating those systems as well.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.