Let photons perform the dance of the Hall effect

The quantum Hall effect is favored by scientists for its unique physical properties, prompting queries into the mysterious quantum mechanisms underlying it. In two-dimensional electron gas systems, due to electrons filling up Landau levels under the influence of a strong magnetic field, electrical resistance shows quantized plateaus proportional to the magnetic field strength, forming the theoretical foundation of the integer quantum Hall effect. This phenomenon is not only relatively intuitive to understand but also has broad application value in precision measurement and electronics.

Surprisingly, just one year after the revelation of the integer quantum Hall effect, the fractional quantum Hall effect was discovered by Tsui and Horst Störmer, presenting even more fascinating phenomena: unlike the integer plateaus of Hall resistance in the integer effect, the fractional effect shows fractional plateaus, seemingly splitting the electron charge into several parts, leading Tsui to jest that he had discovered particle-like quarks.

Nobel laureate Robert B. Laughlin, in trying to explain the fractional quantum Hall effect, proposed a microscopic wave function and envisioned an easy-to-understand physical picture: in this state, electrons bind with magnetic flux to create quasi-particles or quasi-holes carrying fractional charge. These quasi-particles have a fraction of the elementary charge, though it is worth noting that Laughlin’s model only explains cases with odd denominator fractional charges.

Meanwhile, American physicist Frank Wilczek also proposed a corresponding theory, introducing a fractionally-statistic particle called anyon, which behaves differently than traditional fermions or bosons and is confined to two-dimensional movement. Later scientific developments confirmed that the quasi-particles with fractional charges mentioned by Laughlin are indeed anyons. Excitingly, anyons are considered potential building blocks for constructing topological quantum bits and achieving fault-tolerant quantum computers.

In recent years, the quantum Hall effect continues to reveal new phenomena in experiments, such as occurring without strong magnetic fields or at room temperature, known as the quantum anomalous Hall effect. While theoretically predicted, experimental detection remains challenging. Chinese physicist Shoucheng Zhang believed that magnetic topological insulators might be the key to realizing this effect. In 2013, Chinese physicist Xue Qikun and his colleagues observed the integer quantum anomalous Hall effect for the first time, and the fractional anomalous Hall effect was subsequently observed independently by different research groups over the following decade.

Increasingly, more systems demonstrate the potential to exhibit the quantum Hall effect, not limited to solid-state systems. On this, Chinese physicist Wen Xiaogang offered insights that whether it’s the fractional quantum Hall effect under two-dimensional and strong magnetic field conditions or the high-temperature superconductivity phenomena not explainable by traditional BCS theory, they are fundamentally “phenomena,” just like the falling of apples and the rotation of celestial bodies, all aspects of a unified gravitational theory.

In the strange world of quantum, a particularly unique state of matter has attracted widespread attention, that is, topological order. Unlike the disordered particle states in materials under normal conditions, under extreme circumstances such as very low temperatures, high purity two-dimensional materials, and sufficiently strong magnetic fields, particles behave in ways that stray from traditional physical theory predictions. Especially electrons – at this time, they no longer act independently but rather interact with each other in an organized way, forming a new state of matter referred to as the fractional quantum Hall state. In this state, the rotating interactions between electrons resemble a delicate quantum waltz.

To delve deeper into this state of matter, scientists proposed a new concept: topological order. Topological order describes the special ordered phase that matter exhibits under extreme conditions, characterized by strong topological properties and correlations, based on long-range many-body quantum entanglement. This discovery provides not only the basis for understanding a multitude of mystical phenomena such as quantum spin liquids and quantum Hall states but also drives the study of important matter states related to but not containing topological order, such as topological insulators. Examining the quantum Hall state from a topological perspective not only helps us understand the phenomenon itself but also aids us in recognizing and understanding other various novel physical phenomena and exploring their interconnections. Furthermore, the study of topological order also touches on deeper issues, such as the true nature of quantum entanglement.

Scientists, in an effort to understand the fractional quantum Hall effect (FQH), have adopted methods analogous to basic physics experiments. Just as middle school students experience the force of gravity when an apple falls, researchers have constructed an FQH system using photons to explore the essence of quantum Hall states. Traditional two-dimensional solid-state materials face various limitations in studying Hall states, such as the requirements for low temperature and strong magnetic fields, as well as the materials’ own imperfections and difficult-to-control characteristics. Therefore, researchers have adopted a ‘bottom-up’ strategy, from simple to complex. For example, a Harvard University research team successfully simulated the fractional quantum Hall effect using ultracold atoms in an optical lattice. Meanwhile, Chinese scientists have established a superconducting cavity QED lattice, a 4×4 chessboard-like platform that makes the system’s Hamiltonian clearly visible and allows precise photon manipulation, thereby producing the FQH effect without the need for an external magnetic field.

The reason an external magnetic field is not required is that the system utilizes the Aharonov-Bohm effect (A-B effect). This effect demonstrates that even in the absence of a magnetic field, as long as the paths of two photons enclose a nonzero magnetic flux, their phase difference can manifest in interference. On this special platform, the magnetic flux can be changed through precision control, indicating that physicists can use photons to simulate and study the fractional quantum Hall effect, and possibly unveiling deeper secrets of the quantum world behind all this.

In the pursuit of quantum information processing, scientists have adopted a unique approach: constructing a four-by-four superconducting cavity lattice, where each lattice point is composed of a newly developed type of superconducting quantum bit called “little boxes.” These quantum bits are designed with an anharmonic characteristic, meaning that the spacings between their energy levels are uneven. Therefore, when a photon enters a little box, it prompts a quantum bit to jump from the lower level to the next level. However, due to a larger gap between the third and second energy levels, subsequent photons cannot match the existing energy difference, and thus are rejected.

With the help of couplers between quantum bits, we have successfully constructed an equivalent magnetic field on the two-dimensional chessboard, thus enabling the control of the photon’s hopping between boxes and achieving precise manipulation of the motion and rhythm of photon flow.

Using this platform, we are able to reveal some important information about the behavior of photons on a two-dimensional lattice checkerboard. First, we can assess the ability to control photon dynamics through the influence of the constructed artificial magnetic field. Experimental results show that under different strengths of magnetic fields, photons experience varying Lorentz forces and thus deflect at different angles in the lattice. Scientists further observed the “butterfly” shaped spectra of single-photon energy levels with changes in the magnetic field, which is consistent with theoretical predictions.

Delving further into the research, we attempted to formally prepare fractional quantum Hall (FQH) states in the experiment. By adjusting the system coupling and magnetic field strength, the system undergoes an adiabatic evolution, and photons transition from a naive state to a new quantum state. With increasing magnetic field strength, a topological phase transition occurs, and once the magnetic field strength reaches a certain threshold, the anticipated FQH state emerges.

Concerning this emerging FQH state, we are more interested in its inherent long-range correlation characteristics and topological properties. In the past, these properties were mainly derived from theoretical calculations, but now, through artificially constructed quantum systems, we can directly observe these properties in experiments. The research results indicate that in the system transformed to an FQH state, photons are more likely to appear between two lattice points that are a long distance apart. Additionally, there is a significant change in the movement pattern of photons—in the FQH state, density flows with opposite chiralities on the inside and edges were observed, while radial density flows were effectively suppressed, all consistent with the theoretical description of the FQH state.

Especially noteworthy is the existence of quasiparticle characteristics in the FQH state. In traditional two-dimensional materials, capturing and manipulating quasiparticles is a major challenge since charged particles have difficulty crossing magnetic field lines. In current research, by precise control over the lattice potential structure, scientists are able to track the formation process of quasiparticles. Quasiparticle “traps” built at the edges of the checkerboard affect the clustering behavior of photons in the normal state, and as the traps become sufficiently deep, the amount of photon clustering correlates linearly with the depth of the traps.

In the wonderful realm of the fractional quantum Hall effect (FQH), when observing a less deep potential well, we note that the rate of increase in photon number is limited. However, when the depth of the potential well exceeds a certain threshold, we see a rapid increase in the number of photons, forming a distinct plateau phenomenon. This phenomenon is caused by the incompressibility of quasiparticles.

In the FQH state, the Hall conductivity is a key indicator, which can be determined by the average number of photons responding to changes in magnetic flux—that is, the bulk density at the center of the lattice. In recent experiments, the measured value of the fractional quantum Hall conductivity was 0.52, which is very close to the ideal thermodynamic limit of 0.5 and theoretical simulation value of 0.6 for finite sizes.

All these findings undoubtedly point to one conclusion: the FQH state can be realized in a photon-filled lattice. Today, our scientists have successfully prepared fractional quantum Hall states with photons on a checkerboard of gauge fields formed by superconducting circuits. This two-dimensional system is programmable and photons can be precisely controlled through our manipulation. Such precise control of photons not only carries meaning far beyond the study of Hall states but also opens up possibilities for bottom-up research of other strongly-correlated topological states and provides direction for fault-tolerant operations in topological quantum computing. This ability for high-precision manipulation of microscopic particles is a core part of the “second quantum revolution.”

Let us not forget that the Laughlin description of fractional quantum Hall states can only explain fractionally charged particles with odd denominators. For situations with even denominators, in 1991, Gregory Moore and Nicholas Read proposed the Moore-Read (MR) states, which can describe these cases. In this state, quasi-particles not only possess fractional charge, but also follow non-Abelian statistical rules. The interaction among electrons results in an energy gap between the ground state and the excited state, which protects the ground state’s degeneracy from being affected by small perturbations in the Hamiltonian.

This resistance to change, similar to the concept of topology in mathematics, where a shape maintains its topological properties despite transformations, leads physicists to call substances such as FQH states topological states of matter. Because of their robustness, topological quantum computing becomes possible. In the topologically protected degenerate space, information storage is non-local, while most system-environment interactions are local. Hence, the stored information, due to its protected degeneracy, is less susceptible to decoherence.

An increasing number of experiments have shown that it is feasible to use fractional quantum Hall systems for topological quantum computing. By braiding different types of non-Abelian anyons in FQH states, various universal quantum logic gates can be implemented. The combination of topological protection and universal quantum logic gates makes fault-tolerant quantum computing an important prospective application in FQH states and other topological states of matter.

In the future, the artificially constructed chessboard will become larger, allowing for the creation of larger and more complex FQH states. The localized topological coherent movement of quasi-particles can benefit from single-site programming, and the implementation of braiding operations with multiple non-Abelian quasi-particles in larger systems has profound significance for building fault-tolerant quantum information storage and processing devices.

As an outstanding ancient Chinese thinker and scientist, Mozi left a rich legacy of thought and scientific achievements, representing an important figure in the early Chinese scientific thought. The Mozi Salon is named after him, dedicated to inheriting and developing this precious scientific tradition, and committed to promoting the spirit of science.

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University of Science and Technology of ChinaJian-Wei PanChao-Yang LuMing-Cheng ChenPlasmoniumSuperconductivityHigh Nonlinearity Optical Resonator Array.