Observation of anyonic Bloch oscillations

Bloch oscillations are exotic phenomena describing the periodic motion of a wave packet subjected to the external force in a lattice, where the system possessing single- or multipleparticles could exhibit distinct oscillation behaviors. In particular, it has been pointed out that quantum statistics could dramatically affected the Bloch oscillation even in the absence of particle interactions, where the oscillation frequency of two pseudofermions with the anyonic statistical angle being pi becomes half of that for two bosons. However, these statisticdependent Bloch oscillations have never been observed in experiments up to now. Here, we report the first experimental simulation of anyonic Bloch oscillations using electric circuits. By mapping eigenstates of two anyons to modes of designed circuit simulators, the Bloch oscillation of two bosons and two pseudofermions are verified by measuring the voltage dynamics. It is found that the oscillation period in the two-boson simulator is almost twice of that in the two-pseudofermion simulator, which is consistent with the theoretical prediction. Our proposal provides a flexible platform to investigate and visualize many interesting phenomena related to particle statistics, and could have potential applications in the field of the novelty signal control.

On the other hand, in the few-body quantum systems described by the Bose-Hubbard or Fermi-Hubbard model, many previous investigations have shown that BOs could be significantly modified by the strong particle interaction [20][21][22][23][24][25][26][27][28][29]. In particular, the frequency doubling of BOs for two strongly correlated Bosons, that is called fractional BOs, is experimentally observed based on a photonic lattice simulator [29], where the two-boson dynamics is directly mapped to the propagation of light fields in the designed two-dimensional waveguide array.
Except for bosons and fermions, anyons are quantum quasi-particles with the statistics intermediate between them [30][31][32][33]. Anyons play important roles in several areas of modern physics research, such as fractional quantum Hall systems [34][35][36][37] and spin liquids [38][39][40]. Another potential application of non-Abelian anyons is to realize the topological quantum computation [41]. Interestingly, a previous theoretical work has shown that two non-interacting anyons could exhibit exotic BOs, where the frequency halving of BOs for two pseudofermions exists if the ratio of the applied external force to the hopping rate is less than or equal to 0.5 [42]. While, the experimental observation of anyonic BOs is still a great challenging in condensed-matter systems, ultracold quantum gases and other classical wave systems. In this case, a newly accessible and fully controllable platform is expected to be constructed to simulate the anyonic BOs with novel behaviors.
In this work, we demonstrate both in theory and experiment that the anyonic BOs can be simulated by designed electric circuits. Using the exact mapping of two anyons in the external forcing to modes of designed circuit lattices, the periodic breathing dynamics of voltages have been observed by time-domain measurements in both two-boson and two-pseudofermion circuit simulators. In particular, we find that the oscillation frequency in the two-boson simulator is almost twice of that in the two-pseudofermion simulator, that is consistent with the theoretical prediction. Our work provides a flexible platform to implement many interesting phenomena depended on particle statistics, and could have potential applications in the field of the intergraded circuit design and the electronic signal control.

Results
The theory of simulating anyonic Bloch oscillations by electric circuits. We consider a pair of correlated anyons hopping on a one-dimensional (1D) chain subjected to an external force F. In this case, the system can be described by the extended version of the anyon-Hubbard model as: where + ( ) and = + are the creation (annihilation) and particle number operators of the anyon at the lth lattice site, respectively. N is the number of lattice sites. J is the singleparticle hopping rate between adjacent sites, and U defines the on-site interaction energy. The anyonic creation and annihilation operators obey the generalized commutation relations as: where is the anyonic statistical angle, and sgn(x) equals to -1, 0 and 1 for x<0, x=0 and x>0, respectively. It is worthy to note that anyons with = are "pseudofermions" as they behave like fermions off site, while being bosons on site. The two-anyon solution can be expanded in the Fock space as: where |0 > is the vacuum state and is the probability amplitude with one anyon at the We note that Eq. (4) can also be regarded as the eigen-equation describing a single particle hopping on the 2D lattice with the spatially modulated on-site potential and hopping configuration, as shown in Fig. 1a with U=0. In this case, the probability amplitude for the 1D two-anyon model with one anyon at the site m and the other at the site n is directly mapped to the probability amplitude for the single particle locating at the site (m, n) of the 2D lattice. The position-dependent on-site potential could simulate the effect of external force. Moreover, the hopping of the single particle along a certain direction in the 2D lattice represents the hopping of one anyon in the 1D lattice. In this case, the behavior of two anyons in the 1D lattice can be effectively simulated by a single particle in the mapped 2D lattice, that inspires the design of classical simulators to study the statistic dependent anyonic physics.  One of the fascinating phenomena dominated by the quantum statistics in Eq. (4) is that the BO frequency of two pseudofermions ( = ) becomes half of that for two bosons ( = 0), when the on-site particle interaction is zero (U=0) and the ratio of the external forcing to the hopping rate satisfies / ≤ 0.5 [42]. To clarify this effect, the evolution of two-anyon eigenenergies as a function of is displayed in Fig. 1b  Based on the similarity between circuit Laplacian and lattice Hamiltonian [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], electric circuits can be used as an extremely flexible platform to fulfill the above mapped 2D lattice with different statistical angles. Fig. 1c illustrates the schematic diagram for a part of designed circuit simulator with = , that corresponds to four lattice sites enclosed by the blue dash block in Fig. 1a. Here, a pair of circuit nodes connected by the inductor L are considered to form an effective site in the 2D lattice model. The voltages at these two nodes are marked by ( , ),1 and ( , ),2 , which are suitably formulated to form a pair of pseudospins ↑( , ),↓( , ) = ( ( , ),1 ± ( , ),2 )/√2 . To simulate the real-valued hopping rate, two capacitors (the capacitance equals to C) are used to directly link adjacent nodes without a cross.
As for the realization of hopping rate with a phase ( ± ), two pairs of adjacent nodes are cross connected via C. And, the position-dependent capacitors ( + ) are used for grounding to simulate the spatially modulated on-site potential induced by the external forcing. Moreover, the extra capacitor , that is crucial for the achievement of anyonic BOs in circuits demonstrated below, is also added to connect each circuit node to the ground. Through the appropriate setting of grounding and connecting, the circuit eigen-equation can be derived as: , where the complex hopping rate ± could be realized by suitably braiding the connection pattern of o adjacent circuits nodes in a single lattice site [43,44]. In this case, the relationship between and f with different values of stays the same.
To analyze the behavior of BOs in the circuit simulator with respect to , eigenfrequencies of the designed circuit as a function of the statistical angle are calculated, as shown in Fig. 1d. Here, only the eigenfrequencies related to the eigen-energies in the range of (4F, 16F) are plotted. The parameters are set as C=10pF, Ce=2nF, L=10uH and CF=5pF. It is noted that, due to the nonlinear relationship between the eigen-frequency of circuit simulator and the eigen-energy of two anyons ( = 0 /( + 4 + / ) 1/2 ), the distribution of eigenspectrum for the circuit simulator should deviate from that of two anyons with equal-spacings.
At first, we focus on the circuit simulator of two non-interacting bosons with = 0. In this case, the excitation frequency is set as 1.56MHz, which locates in the region with equally spaced eigen-values at = 0, as shown in Fig. 1d. The voltage-pseudospin could be suitably excited by setting the input signal with (12,12),1 = 0 and (12,12),2 = − 0 ( 0 = 1 ). Next, we simulate the BO of two non-interacting pseudofermions by the designed electric circuit with = . The excitation frequency is also set as 1.56MHz, locating in the equally spaced region of eigen-frequencies at = . Similar to the two-boson case, the input signal is set as [ 1, (12,12) = 0 , 2,(12,12) = − 0 ] to excite the voltage-pseudospin. The time-dependent evolution of | ↓,[ , ] ( )| 2 at each node in the 2D circuit simulator (with = ) is shown in Fig.   2d. Moreover, the calculated voltage signals of | ↓, [12,12] ( )| and | ↓, [6,5] ( )| are presented in Figs. 2e and 2f. We can see that the periodic breathing dynamics of voltage could also appear in the circuit simulator for two pseudofermions. And, the oscillation period is about 1.085ms, which is in a good consistence with = 1 ∆ = 1.074 predicted by the eigen-spectrum.
Moreover, comparing with the result of bosonic circuit simulators, we find that the oscillation period in the two-boson simulator is almost twice of that in the two-pseudofermion simulator.
This phenomenon is consistent with the theoretical prediction based on the 1D anyon-Hubbard model (See S3 of Supplementary Materials for details).
It is worthy to note that above results only focus on the two-anyon model at a fixed excitation frequency 1.56MHz and constant values of = 5 and = 2 . In S4 of Supplementary Materials, we also simulate the anyonic BOs by our designed electric circuits with different excitation frequencies, external forces and grounding capacitor . It is shown that the smaller the external force is, the larger the oscillation period and amplitude become.
Moreover, we find that more ideal BOs could be realized with a larger value of , that could make the eigen-spectrum of circuit simulators become equally spaced. Then, we turn to the fabricated circuit with = (close switches), and the excitation signal is the same to that used for bosonic circuit simulators. As shown in Fig. 3d, we measure the time-dependent evolution of | ↓, [6,5] ( )|. It is shown that the damped oscillation period is about 1.08ms, which is also in a good consistence with the simulated result. Comparing to the measured period of circuit simulator with = 0, we note that the Bloch oscillation frequency related to a pair of pseudofermions ( = ) is half of that for two bosons ( = 0), being consistent with the theoretical prediction. Similar to the bosonic results, the significant decay of the voltage signal is also resulting from the large lossy effect in the fabricated circuit.

Discussion and conclusion.
With the advantage of diversity and flexibility for circuit elements, except for the above designed LC circuit with matched stationary eigen-equations to the 1D two-anyon system, we can design another kind of electric circuits, which is based on resistances and capacitances, to

Methods.
Sample fabrications and circuit signal measurements. We exploit the electric circuits by using PADs program software, where the PCB composition, stack up layout, internal layer and grounding design are suitably engineered. Here, the well-designed PCB possesses totally eight layers to arrange the intralayer and interlayer site-couplings. It is worthy to note that the ground layer should be placed in the gap between any two layers to avoid their coupling. Moreover, all PCB traces have a relatively large width (0.5mm) to reduce the parasitic inductance and the spacing between electronic devices is also large enough (1.0mm) to avert spurious inductive coupling. The SMP connectors are welded on the PCB nodes for the signal input and detection.
To ensure the realization of BOs in electric circuits, both the tolerance of circuit elements and series resistance of inductors should be as low as possible. For this purpose, we use WK6500B impedance analyzer to select circuit elements with high accuracy (the disorder strength is only 1%) and low losses.
As for the measurement of BOs, we use the signal generator (NI PXI-5404) with eight output ports to act as the current source for exciting two circuit nodes related to a single lattice site with a constant amplitude and node-dependent initial phases. One output of the signal generator (the initial phase is set to 0) is directly connected to one end of the oscilloscope

S2. The influence of the value of on the correspondence between eigen-spectra of 2D circuit simulators and 1D two-anyon models.
It is known that the appearance of Bloch oscillations depends on the equally spaced eigen-spectrum of two bosons and two pseudofermions, and the periods are determined by the associated energy level spacings. While, due to the nonlinear relationship between the eigen-frequency of circuit simulator and the eigen-energy of two anyons = 0 /( + 4 + / ) 1/2 , the distribution of eigen-spectrum for the circuit simulator should not be equally spaced.    3.01MHz, 1.56MHz, and 1.117MHz. Other parameters are the same to that used in Fig. 2. We can see that the larger the value of is, the more ideal BOs appear. This is due to the fact that the nearly perfect eigen-spectrum with equal spacings could only be realized with an extremely large value of , as shown in Fig. S1.
Then, we will simulate anyonic Bloch oscillations with a different external force by our designed electric circuits, that is = 3 . Before circuit simulations, we calculate the evolution of two-anyon eigen-energies as a function of with J=1 and F=0.3, as shown in Fig. S5a.    3 ), we find that the more symmetric BO could be realized with a higher excitation frequency.

S5. The precise correspondence between time-dependent Schrödinger equation of two bosons and two pseudofermions and designed RC circuit simulators.
It is worthy to note that the stationary eigen-equation of our designed LC circuit is consistent with the stationary Schrödinger equation of the 1D anyon-Hubbard model with two anyons. As for the timedependent evolution equation, the voltage of LC circuit follows second-order time differential, which is different from the first-order time differential of quantum wave functions. In this part, we will design another kind of electric circuit based on resistances and capacitances to precisely match the timedependent Schrödinger equation of two anyons with = 0 and = .
The designed RC circuit simulator with = 0 is plotted in Fig. S8a. Here, the associated 1D lattice length is N. We note that the designed circuit simulator contain 2N 2 nodes, where the row (column) with Π being a × matrix. In this case, when the nodes connecting and grounding resistances are suitably applied, the form of × matrix Π can be the same to the Hamiltonian of the 1D two-boson model. In this case, the voltage evolution in the designed RC circuit could be the same to the probability amplitude of two bosons.
Based on the similar method, the RC circuit related to two pseudofermions = could also be designed. Fig. S8b presents the corresponding connection pattern at different circuit nodes. Comparing to the circuit for two bosons, the only difference is that there are a few of effective hopping rates sustaining a phase . This could be easily fulfilled by reversing the biased voltage of the associated grounding and connecting INICs.  For comparations, we also calculate the evolution of | ( )| 2 of two bosons and two pseudofermions in the 1D anyon-Hubbard model, as shown in Figs. S10a and S10b. The associated parameters are set as J=1, F=0.5 and ( = 0) = 12,12 . And, the corresponding time-dependent evolutions of | 12,12 | 2 are presented in Figs. S10c and S10d. We note that a good agreement for the time-dependent evolution of voltages and probability amplitude is obtained. In particular, it is clearly shown that the oscillation period in the two-boson simulator is twice of that in the two-pseudofermion simulator, that is consistent with the theoretical prediction.