Exquisite control of quantum systems has allowed researchers to connect reality to ideas of how an exotic form of particle transport known as the quantum Hall effect can occur in the absence of a magnetic field. See Letters p.237 & p.241
A magnetic field produces a force on a charged particle that is perpendicular to both the field and the particle's velocity. This force, called the Lorentz force, induces circular orbits, the chirality (handedness) of which breaks timereversal symmetry, giving rise to unusual particletransport phenomena such as 'edge states' protected by topological properties and the related quantum Hall effect (Fig. 1a). In two papers published in this issue of Nature, researchers have experimentally demonstrated that the quantum Hall effect can persist in the absence of a magnetic field (Jotzu et al.^{1}, page 237), and have developed a new method to quantify topological protection in arbitrary dynamical systems (Roushan et al.^{2}, page 241).
In 1988, the physicist F. Duncan Haldane introduced a model that exhibited the quantum Hall effect without a net magnetic field, instead using a field that alternated within each unit cell of a honeycomb lattice^{3}. Although his model has never been observed in the laboratory, it forced researchers to ask what ingredients are absolutely necessary for the quantum Hall effect and how they could predict whether its signatures will be present in a particular material. At its heart, the quantum Hall effect is a voltage induced across a piece of material (the 'Hall bar') when a current flows along the material; the voltage is proportional to small magnetic fields, and quantized to discrete values for large magnetic fields. That such a quantized voltage could persist in the absence of a magnetic field was quite unexpected. We now understand that the key ingredient is not a magnetic field, but rather Berry curvature^{4}, a mathematical entity that smoothly connects quantum states of different momenta and acts like a generalized magnetic field. These deep ideas are at last connected to experimental reality: Jotzu et al. implemented and studied Haldane's model using cold atoms (Fig. 1b), whereas Roushan et al. used a superconducting circuit to measure the Berry curvature in systems of one and two quantum bits (qubits; Fig. 1c).
The first step to realizing Haldane's model is creating a honeycomb lattice, which Jotzu and colleagues previously demonstrated for potassium atoms trapped by laser beams^{5}. Their new breakthrough is a synthetic magnetic field for the atoms, which, being charge neutral, do not experience a Lorentz force from an actual magnetic field. The key is to generate a large synthetic magnetic flux, which is the number of field lines piercing an area. Although small amounts of flux had previously been generated^{6}, the large flux necessary for Haldane's model had not. Jotzu et al. show that circular shaking of individual sites of a honeycomb lattice results in potassium atoms hopping to nonadjacent lattice sites (Fig. 1b) in a way that is equivalent to a large, alternating flux. The idea of generating chiral motion of quantum particles by shaking is a modern development^{7,8}, and, before Jotzu and colleagues' work, had been definitively observed only in bismuth selenide (Bi_{2}Se_{3}) samples subjected to shortduration laser pulses^{9}.
Jotzu et al. measured the presence of their synthetic magnetic flux by accelerating the potassium atoms and searching for a force perpendicular to their motion, akin to the Lorentz force for electrons in magnetic fields. They observed this perpendicular force as motion of the atoms perpendicular to their velocities. Crucially, they found that the direction of the force changed depending on the direction of atomic velocity, indicating a timereversalbreaking force, because a timereversalsymmetric force would be in the same direction regardless of the particle's velocity. Finally, the authors mapped out the sensitivity of Haldane's model to disorder in the latticesite energies, demonstrating that such perpendicular motion persists only when the shaking produces enough flux to overcome the disorder imposed for testing purposes.
Meanwhile, Roushan et al. demonstrated a technique for measuring Berry curvature in arbitrary quantum systems, generalizing the perpendicularmotion signature used by Jotzu and coworkers: for slow variations of physical parameters such as momenta or magnetic fields, a quantum system can remain in its ground state up to a Lorentzforcelike correction to its wavefunction that is proportional to both the rate of change of parameters and the Berry curvature. Imagine transporting a brimming bowl of soup: move it slowly and it will not spill, but sudden movements make a mess. Now, if stumbling forward made the soup spill to the right, the apparent force that the soup experienced could be understood in terms of Berry curvature. By tuning the parameters of their quantum systems and measuring deviations of the quantum state from the ground state, Roushan et al. extracted the Berry curvature at each point in the mathematical space formed by the parameters of the system^{10}.
They first implemented the simplest possible quantum system — a single qubit with an adjustable energy difference between its two states, along with a tunable microwave field driving the qubit from one state to the other. Formally, this system is equivalent to a particle of known momentum in a honeycomb lattice, where the qubit state reflects the probability of the particle being on each lattice site within a unit cell. By varying the energy difference and the microwave field, Roushan et al. dragged the qubit through momentum space and watched for deviations, not in position as in Jotzu and colleagues' study, but in qubit state. Exquisite control of the system let them precisely measure the quantum state at any instant, and determine the Berry curvature.
But Roushan and colleagues went a step further: they studied a system of two interacting qubits and mapped out its Berry curvature. This is a stimulating and wonderful idea because the twoqubit system is not analogous to particles on a lattice, but the Berry curvature remains well defined and nonzero. We must then reconsider what the Berry curvature means — it is not just about particles moving in circular orbits, but more subtly quantifies global topological properties of arbitrary quantum systems that are insensitive to local disorder. Roushan and coworkers' technique is potentially scalable to systems containing many qubits, suggesting that quantum magnets and even quantum computers could be understood in terms of Berry curvature and topological properties. Nonetheless, the number of measurements required to scale up is daunting, and the connection between Berry curvature and dynamics in the highdimensional parameter spaces associated with such systems remains an active field of research.
Meanwhile, Jotzu and colleagues' work is a crucial step towards coldatom realizations of exotic phenomena such as fractional quantum Hall phases and fractional Chern insulators. These are both ordered arrangements of electrons in which imperfections in the ordering, known as anyons, behave as particles with a bizarre property — they remember their past locations and can be braided around one another, becoming entangled like shoelaces. These anyons would be fascinating to observe in their own right, and could be a powerful paradigm for quantum computing.
All that remains for Jotzu and coworkers is to add interactions between their potassium atoms and to harness the extraordinary control of the coldatom toolbox to search for these phenomena. Nonetheless, these phenomena exist near the ground state of ultracold atoms interacting with one another in the presence of a magnetic field, and reaching that ground state requires cooling to onebillionth of a degree above absolute zero, something we do not know how to do for general quantum systems. The next challenge is thus to find a way to smoothly convert the lowesttemperature state of matter yet created, the Bose–Einstein condensate, into a fractional quantum Hall phase, or to make a stateofthe art refrigerant^{11}.
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Simon, J. Magnetic fields without magnetic fields. Nature 515, 202–203 (2014). https://doi.org/10.1038/515202a
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