### Focus

## BIT gains significant progress in Bloch oscillations dominated by effective anyonic particle

News Resource: School of Physics

Editor: News Agency of BIT

Translator: Hou Bingqing, News Agency of BIT

Beijing Institute of Technology, May 5nd, 2022: Recently, the two research groups of BIT, one led by Professor Zhang Xiangdong from School of Physics, the other led by Professor Sun Houjun from School of Integrated Circuits and Electronics, have cooperated and gained significant progress in the research on simulating anyonic Bloch oscillations in electric circuit networks. The related research results were published in the recent Nature Communications, and the research has been founded by National Nature Science Foundation of China and National Key R&D Program of China.

Bloch oscillations (BOs) are exotic phenomena describing the periodic motion of a wave packet subjected to an external force. Its theoretical paradigms were originally proposed for electrons in crystals. The first experimental observation of Bloch oscillations was based on semiconductor superlattice. In fact, BO is a universal wave phenomenon. Hence, it has also been observed in various classical wave systems. Recent researches show that, not only single-body systems, but also few-body strongly correlated quantum systems can exhibit the exotic phenomena of Bloch oscillations. For example, the two strongly correlated Bosons can exhibit fractional BOs. Such novel effect can also be achieved in optical waveguide lattice array.

We know that boson follows Bose-Einstein statistics and fermion follows Fermi-Dirac statistics. Except for boson and fermion, anyon are quantum quasiparticles with statistics intermediate between them, which were proposed 40 years ago. To be specific, when exchanging two anyons’ position in space, the systematic wave function will gain a phase between 0 and π. Scientists found that two anyons can demonstrate statistically dependent Bloch oscillations. Although theoretical results indicate that the oscillation frequency of two pseudofermion becomes half of that for two bosons, these Bloch oscillations with an anyonic intermediate statistical angle (0<statistical angle<π) have never been observed in experiments. More importantly, because of the complexity of the problem, experimental observations of Bloch oscillations about anyon have never been achieved.

Recently, the joint research group from BIT achieved experimental observation of novel Bloch oscillations simulated by anyonic quantum statistics by using classical electric circuits. Firstly, the researchers made a 2D single-particle model equivalent to the 1D two-anyon model through stimulating the two-anyons in the 1D lattice by a single particle in the mapped 2D lattice. Further, by designing 2D circuit networks (as shown in Fig1a), the corresponding eigenstate equation perfectly accords with the stationary Schrodinger equation of 2D tight binding model derived from mapping. Through ingenious design, the eigenstate frequency of circuit networks can exhibit equal-spaced energy spectrum determined by quantum statistics (Wannier-stark ladder), as shown in Fig1b and d. It can be found that just as the eigen-spectra of two anyons, equal-spaced energy distribution appears when the statistic angle is 0 or π, and the energy spacing of the former is twice of the latter. This demonstrates that circuit simulator can exhibit two anyonic energy paradigm characteristics determined by effective quantum statistics. It is worth mentioning that the energy paradigm with Wannier-stark ladder characteristics is the essential reason of anyonic BOs.

Fig. 1. Schematic diagram and eigen-energies of designed circuit simulators for the Bloch oscillation of a pair of anyons. a,c: The mapped 2D lattice of the single particle for simulating the 1D two-anyon effect in the absence of on-site interaction under an external forcing and the schematic diagram. b, d: Calculated eigen-energies of two anyons and eigen-frequencies of the circuit simulator.

The theoretical simulation and experimental measure in Fig. 2 and 3 evidenced the effectiveness of the circuit networks further. Fig. 3a is the sample diagram of circuit network. Firstly, the sum of impedance of frequency-spectra can clearly show the equal-spaced energy spectrum, and it satisfies the requirement that the frequency-spacing of two adjacent impedance peaks for the bosonic simulator is two times that for two pseudofermions (as shown in Fig. 2a,2b and Fig. 3b,3c). The results of time-domain simulations and measure further showed the statistics-dependent BO, which is consistent with the two anyons lattice model. Fig. 2c and 2g show the evolution behavior of the voltage signal over time for all circuit lattice points in simulation results. It is clear that the voltage signals also oscillate periodically in classical circuit network, and the BO frequency in the two-boson circuit simulator is almost twice that in the two-pseudofermion simulator. To show the oscillation of voltage signals more intuitively, Fig. 2b(2e) and 2h(2i) respectively show the measured amplitude (phase distributions) of the voltages at excited circuit nodes at different times. We can see that the symmetric voltage distribution in Boson circuit simulator and the asymmetric distribution in pseudofermion simulator are consistent with the wave function properties dominated by quantum statistics. Relevant experimental results are shown in Fig. 3d(3e) and 3g(3h), which accord with the simulated results. Finally, the numerically calculated oscillation of the single-lattice voltage signal in the boson circuit simulator (pseudofermions circuit simulator). It can be seen further that oscillation frequency of two-pseudofermion circuit simulators is half of that for two-boson. The significant decay of the voltage signal results from the large lossy effect. Finally, the researchers also theoretically show that by balancing the competition between the applied potential field and statistical angles, the two anyons system with an intermediate statistical angle between 0 and π can exhibit the oscillation three times of two bosons. Such novel effect can also be achieved in circuit networks.

The related results have been published in Nature communications [Nat Commun 13, 2392 (2022)]. Zhang Weixuan, doctor at School of Physics (now postdoctor at XUTELI School of Integrated Circuit and Electronics), and Yuan Hao, doctor of 2020, are co-first authors of the paper.

Fig. 2 Numerical results for simulating anyonic Bloch oscillations in electric circuit networks.

Fig. 3 Experimental results for observing anyonic Bloch oscillations in electric circuit networks.

Paper Link: https://doi.org/10.1038/s41467-022-29895-0