Long Range Colloquium

The Long Range Colloquium is a bi-weekly seminar series covering the latest developments in condensed matter physics and quantum information. We alternate experimental and theory talks to keep the schedule exciting for a varied audience: have a look at our 2021 schedule below! You can also find all the information and recordings for our 2020 talks here.


Speaker Title
Corentin Coulais The Non-Hermitian physics of odd Robotic Matter
David Goldhaber-Gordon Can a solid-state quantum simulator help us understand materials?
Nicole Yunger Halpern MBL-mobile: Many-body-localized engine
Atac Imamoglu Strongly correlated electrons in atomically thin semiconductors
Cui-Zu Chang Quantum Anomalous Hall Effect in the Magnetic Topological Insulator Thin Films
Andrea Caviglia Ultrafast control of magnetic interactions via light-driven phonons
Richard Kueng Provably efficient machine learning for quantum many-body problems
Sebastian Huber Topological Mechanics
Prineha Narang Predicting and Controlling Scalable Quantum Systems
Giuseppe Carleo Many-body wave functions in the era of machine learning and quantum computing
Jonathan Simon When Photons Self-Organize: Making Matter from Light
Michael Sentef Cavity quantum materials
Andreas Wallraff Microwave Networks for Solid State Quantum Information Processors
Andrew Mackenzie Benefits of good old-fashioned crystalline perfection - new physics in ultra pure delafossite metals
Monika Aidelsburger Ultracold atoms in optical lattices out-of-equilibrium
Xie Chen Fracton and Chern-Simons Theory
Andrea Young Orbital magnetism in graphene heterostructures
Leonid Levitov ‌Electrons Bloch-waltzing in Moire superlattices

The Non-Hermitian physics of odd Robotic Matter

By Corentin Coulais (University of Amsterdam),

Controlling how waves propagate, attenuate and amplify in simple, scalable geometric structures is a daunting challenge for science and technology. In this talk, I will discuss how odd matter—in which microscopic interactions are asymmetric and non-conservative—can be used to steer mechanical waves in unprecedented ways. Combining experiments on mechanical lattices of distributed robots with wave physics and continuum and topological mechanics, I will discuss the emergence of the non-Hermitian skin effect, of non-Hermitian topological waves and of one-way solitons. I will further show how these odd waves can be used to induce locomotion and unusual responses to impacts and hence pave the way towards a novel generation of materials with animate properties.


  1. Active impact and locomotion in robotic matter with nonlinear work cycles
    Brandenbourger M., Scheibner C., Veenstra J., Vitelli V., Coulais C., arXiv:2108.08837

  2. Topology and broken hermiticity
    Coulais C., Fleury R. and van Wezel J., Nat. Phys. 17, 9–13 (2021)

  3. Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial
    Ghatak A., Brandenbourger M., van Wezel J. and Coulais C., Proc. Natl. Ac. Sc. U.S.A. 117 (47) 29561 (2020)

  4. Non-reciprocal robotic metamaterials
    Brandenbourger M., Locsin X., Lerner E. and Coulais C., Nat. Commun., 10, 4608 (2019)

Can a solid-state quantum simulator help us understand materials?

By David Goldhaber-Gordon (Stanford University),

When we speak of a quantum computer today, we usually mean one that can run any algorithm. But the idea of using a quantum-based computer to simulate aspects of our quantum world goes back at least to Feynman in the early 1980s. Effective model Hamiltonians are central to condensed matter physics, as they can capture the essence of material properties while avoiding the full complexity of actual materials. But even those model Hamiltonians can be intractable to treat on classical computers. Very perfect realizations of model Hamiltonians relevant to materials can now be built with ensembles of cold atoms or with nanopatterned solid state systems similar to those used to make qubits. I will compare new experimental results to cutting-edge classical computation, and then ask: What happens when we (soon!) get our wish and achieve a "quantum simulator" that can describe certain interesting Hamiltonians with power beyond the best classical computers and algorithms?

MBL-mobile: Many-body-localized engine

By Nicole Yunger Halpern (National Institute of Standards and Technology (NIST), Joint Institute for Quantum Information and Computer Science (QuICS) University of Maryland),

Many-body-localized (MBL) systems do not thermalize under their intrinsic dynamics. The athermality of MBL, we propose, can be harnessed for thermodynamic tasks. We illustrate this ability by formulating an Otto engine cycle for a quantum many-body system. The system is ramped between a strongly localized MBL regime and a thermal (or weakly localized) regime. The difference between the energy-level correlations of MBL systems and of thermal systems enables mesoscale engines to run in parallel in the thermodynamic limit, enhances the engine’s reliability, and suppresses worst-case trials. We estimate analytically and calculate numerically the engine’s efficiency and per-cycle power. The efficiency mirrors the efficiency of the conventional thermodynamic Otto engine. This work introduces a thermodynamic lens onto MBL, which, having been studied much recently, can now be leveraged in thermodynamic tasks.

Serious reference: NYH, White, Gopalakrishnan, and Refael, Phys. Rev. B 99, 024203 (2019) https://dx.doi.org/10.1103/PhysRevB.99.024203.

Fun reference: NYH, “Quantum Steampunk,” Sci. Am. (May 2020) https://www.scientificamerican.com/article/quantum-steampunk-19th-century-science-meets-technology-of-today/

Strongly correlated electrons in atomically thin semiconductors

By Atac Imamoglu (ETH Zurich),

In this talk, I will describe recent experiments in atomically-thin transition metal dichalcogenides (TMDs) where Coulomb interactions between electrons dominate over their kinetic energy. Our measurements provide a direct evidence that the electrons at densities < 3 · 10^11 cm-2 in a pristine MoSe2 monolayer form a Wigner crystal even at B = 0. This is revealed by our low-temperature (T = 80 mK) magneto-optical spectroscopy experiments that utilize a newly developed technique allowing to unequivocally detect charge order in an electronic Mott-insulator state. This method relies on the modification of excitonic band structure arising due to the periodic potential experienced by the excitons interacting with a crystalline electronic lattice. Under such conditions, optically-inactive exciton states with finite momentum matching the reciprocal Wigner lattice vector k = k­W get Bragg scattered back to the light cone, where they hybridize with the zero-momentum bright exciton states. This leads to emergence of a new, umklapp peak in the optical spectrum heralding the presence of periodically-ordered electronic lattice.

Quantum Anomalous Hall Effect in the Magnetic Topological Insulator Thin Films

By Cui-Zu Chang (Pennsylvania State University),

The quantum anomalous Hall (QAH) effect can be considered as the quantum Hall (QH) effect without an external magnetic field, which can be realized by time-reversal symmetry breaking in a topologically non-trivial system [1, 2]. A QAH system carries spin-polarized dissipationless chiral edge transport channels without the need for external energy input, hence may have a huge impact on future electronic and spintronic device applications for ultralow-power consumption. The many decades' quest for the experimental realization of the QAH phenomenon became a possibility in 2006 with the discovery of topological insulators (TIs). In 2013, the QAH effect was observed in thin films of Cr-doped TI for the first time [3]. Two years later in a near-ideal system, V-doped TI, contrary to the negative prediction from first principle calculations [2], a high-precision QAH quantization with more robust magnetization and a perfectly dissipationless chiral current flow was demonstrated [4]. In this talk, I will introduce the route to the experimental observation of the QAH effect in the aforementioned two systems [3, 4], and also talk about our recent progress on the high Chern number QAH effect in magnetic TI multilayers [5].
[1] Haldane, Phys. Rev. Lett. 61, 2015 (1988).
[2] Yu et al, Science 329, 61 (2010).
[3] Chang et al, Science 340, 167(2013).
[4] Chang et al, Nat. Mater. 14, 473(2015).
[5] Zhao et al, Nature 588, 419 (2020).

Ultrafast control of magnetic interactions via light-driven phonons

By Andrea Caviglia (Delft University of Technology),

Resonant ultrafast excitation of infrared-active phonons is a powerful technique with which to control the electronic properties of materials that leads to remarkable phenomena such as light-induced superconductivity, switching of ferroelectric polarization and ultrafast insulator-to-metal transitions. We will discuss how light-driven phonons can be utilized to coherently manipulate macroscopic magnetic states. Intense mid-infrared electric field pulses tuned to resonance with a phonon mode of the archetypical antiferromagnet DyFeO3 induce ultrafast and long-living changes of the fundamental exchange interaction between rare-earth orbitals and transition metal spins. Non-thermal lattice control of the magnetic exchange, which defines the stability of the macroscopic magnetic state, allows us to perform picosecond coherent switching between competing antiferromagnetic and weakly ferromagnetic spin orders. In this talk, the potential of resonant phonon excitation for the manipulation of ferroic order on ultrafast timescales will be emphasized.

Provably efficient machine learning for quantum many-body problems

By Richard Kueng (Johannes Kepler University, Linz, Austria),

Classical machine learning (ML) provides a potentially powerful approach to solving challenging problems in quantum physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. We prove that classical ML algorithms can efficiently predict ground state properties of a physical system, after learning from data obtained by measuring related systems. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state.

This is joint work with Robert Huang (Caltech), Giacomo Torlai (AWS), Victor Albert (University of Maryland) and John Preskill (Caltech+AWS).

Topological Mechanics

By Sebastian Huber (ETH Zurich),

Topological band theory provides us with a powerful design-principle for metamaterials that control how waves propagate in space. These metamaterials are ideal to investigate novel phenomena difficult to observe in other platforms. In particular, I will present how artificial gauge fields lead to the appearance of chiral Landau levels in neutral Weyl systems and crystalline symmetries to novel fragile topological states in an acoustic super-structure.

Predicting and Controlling Scalable Quantum Systems

By Prineha Narang (Harvard University),

Quantum matter hosts spectacular excited-state and nonequilibrium effects, but many of these phenomena remain challenging to control and, consequently, technologically underexplored. My group’s research, therefore, focuses on how quantum systems behave, particularly away from equilibrium, and how we can harness these effects. By creating predictive theoretical and computational approaches to study dynamics, decoherence and correlations in quantum matter, our work could enable technologies that are inherently more powerful than their classical counterparts ranging from scalable quantum information processing and networks, to ultra-high efficiency optoelectronic and energy conversion systems (1). In this talk, I will present work from my research group on describing, from first principles, the microscopic dynamics, decoherence and optically-excited collective phenomena in quantum matter at finite temperature to quantitatively link predictions with 3D atomic-scale imaging and quantum spectroscopy. Capturing these dynamics poses unique theoretical and computational challenges. The simultaneous contribution of processes that occur on many time and length-scales have remained elusive for state-of-the-art calculations and model Hamiltonian approaches alike, necessitating the development of new methods in computational physics (2–4). I will show selected examples of our approach in ab initio design of active defects in quantum materials (5–7), and control of collective phenomena to link these active defects (8–10). Finally, I will discuss ideas in directly emulating quantum information systems, particularly addressing the issues of model abstraction and scalability, and present an outlook on various co-design strategies with algorithms efforts underway.

  1. Head-Marsden, K., Flick, J., Ciccarino, C. J. & Narang, P. Quantum Information and Algorithms for Correlated Quantum Matter. Chem. Rev. (2020) doi:10.1021/acs.chemrev.0c00620.

  2. Rivera, N., Flick, J. & Narang, P. Variational Theory of Nonrelativistic Quantum Electrodynamics. Phys. Rev. Lett. 122, 193603 (2019).

  3. Flick, J., Rivera, N. & Narang, P. Strong light-matter coupling in quantum chemistry and quantum photonics. Nanophotonics 7, 1479–1501 (2018).
  4. Flick, J. & Narang, P. Cavity-Correlated Electron-Nuclear Dynamics from First Principles. Physical Review Letters vol. 121 (2018).
  5. Narang, P., Ciccarino, C. J., Flick, J. & Englund, D. Quantum Materials with Atomic Precision: Artificial Atoms in Solids: Ab Initio Design, Control, and Integration of Single Photon Emitters in Artificial Quantum Materials. Adv. Funct. Mater. 29, 1904557 (2019).
  6. Hayee, F. et al. Revealing multiple classes of stable quantum emitters in hexagonal boron nitride with correlated optical and electron microscopy. Nat. Mater. 19, 534–539 (2020).

  7. Ciccarino, C. J. et al. Strong spin–orbit quenching via the product Jahn–Teller effect in neutral group IV qubits in diamond. npj Quantum Materials 5, 75 (2020).

  8. Neuman, T., Wang, D. S. & Narang, P. Nanomagnonic Cavities for Strong Spin-Magnon Coupling and Magnon-Mediated Spin-Spin Interactions. Phys. Rev. Lett. 125, 247702 (2020).

  9. Wang, D. S., Neuman, T. & Narang, P. Dipole-coupled emitters as deterministic entangled photon-pair sources. Phys. Rev. Research 2, 043328 (2020).

  10. Neuman, T. et al. A Phononic Bus for Coherent Interfaces Between a Superconducting Quantum Processor, Spin Memory, and Photonic Quantum Networks. arXiv [quant-ph] (2020).

Many-body wave functions in the era of machine learning and quantum computing

By Giuseppe Carleo (EPFL Lausanne),

The theoretical description of several complex quantum phenomena fundamentally relies on many-particle wave functions and our ability to represent and manipulate them. Variational methods aim at compact descriptions of many-body wave functions in terms of parameterised ansatz states, and are at present living exciting transformative developments, thanks to advances both in machine learning and in quantum computing. In this talk I will first present variational representations of quantum states based on artificial neural networks [1]. Highlighting the connection with other known representations based on tensor networks [2], I will also discuss the crucial role of physical symmetries in these representations. Then, I will discuss parameterised quantum circuits, and hybrid classical-quantum variational schemes with an emphasis on near-term quantum hardware and limited-depth circuits. In this context, I will discuss strategies to efficiently target ground-state problems [3] and unitary dynamics [4]. For both classical and quantum variational parameterisations, I will focus on physics-oriented applications of these approaches, discussing their limitations and possible paths for improvement. ——— [1] Carleo and Troyer, Science 365, 602 (2017) [2] Sharir, Shashua, and Carleo, arXiv:2103.10293 (2021)

[3] Stokes, Izaac, Killoran, and Carleo, Quantum 4, 269 (2020) [4] Barison, Vicentini, and Carleo, arXiv:2101.04579 (2021)

When Photons Self-Organize: Making Matter from Light

By Jonathan Simon (University of Chicago),

In this talk I will discuss ongoing efforts at University of Chicago to explore matter made of light. I will begin with a broad introduction to the challenges associated with making matter from photons, focusing specifically on (1) how to trap photons and imbue them with synthetic mass and charge; (2) how to induce photons to collide with one another; and (3) how to drive photons to order, by cooling or otherwise. I will then provide as examples two state-of-the-art photonic quantum matter platforms: microwave photons coupled to superconducting resonators and transmon qubits, and optical photons trapped in multimode optical cavities and made to interact through Rydberg-dressing. In each case I will describe a synthetic material created in that platform: a Mott insulator of microwave photons, stabilized by coupling to an engineered, non-Markovian reservoir, and a Laughlin molecule of optical photons prepared by scattering photons through the optical cavity. I will conclude with an outlook on new experimental platforms we are developing to marry these techniques and ideas!

Cavity quantum materials

By Michael Sentef (Max Planck Institute for the Structure and Dynamics of Matter),

Recent years have seen tremendous progress in utilizing ultrafast light-matter interaction to control the macroscopic properties of quantum materials [1]. Many of the most intriguing effects are based on nonthermal pathways, with the material (quantum many-body system) being driven away from its thermal equilibrium by strong laser pulses. While this has deepened our understanding of quantum matter far from equilibrium and enabled us to build bridges to other fields (quantum simulators, Floquet states of matter, (pre)thermalization, …), there are a number of challenges: (i) the need for strong lasers, (ii) heating, (iii) short lifetime of light-induced states.

This has motivated an emergent community of researchers to search for new directions that draw inspiration from the discoveries in ultrafast materials science and combine them with expertise gleaned from quantum optics, cavity QED, polaritonic chemistry, and nanoplasmonics, creating the new field of "cavity quantum materials“.

In this Colloquium, I will provide a personal perspective on this new field and highlight a few of our recent works. Specifically, I will discuss the quantum-to-classical crossover of Floquet engineering in correlated systems and show how a many-photon classical coherent state can be replaced by a few-photon number state, provided that one can reach the regime of sufficiently strong light-matter coupling in a cavity [2]. I will then show how the quantum geometry of wavefunctions impacts their light-matter coupling, and how we envision this to be a key ingredient for future light-matter-based engineering of flat-band (Moiré) materials [3].

[1] A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W. McIver, M. A. Sentef, Nonthermal pathways to ultrafast control in quantum materials, https://arxiv.org/abs/2103.14888 [2] M. A. Sentef, J. Li, F. Künzel, M. Eckstein, Quantum to classical crossover of Floquet engineering in correlated quantum systems, Phys. Rev. Research 2, 033033 (2020) [3] G. E. Topp, C. J. Eckhardt, D. M. Kennes, M. A. Sentef, P. Törmä, Light-matter coupling and quantum geometry in moiré materials, https://arxiv.org/abs/2103.04967

Microwave Networks for Solid State Quantum Information Processors

By Andreas Wallraff (ETH Zurich),

Quantum computing is a radically new approach to processing information. It is one of the approaches which has the potential to address the ever-growing need of society, industry and research for computing power. At ETH Zurich, we have designed, realized and tested a first data link which allows superconducting-circuit-based quantum processors located in different systems to directly exchange quantum information [1]. This link, for a quantum computer, takes the role of a network transferring data between computing nodes located in a high-performance computing data center. However, unlike its conventional counterparts, our data link is operated at ultra-low temperatures, close to the absolute zero. This allows our quantum data link to directly connect to quantum processors operating at the same temperature [2]. The system we have constructed is a first of its kind in the world and could play an important role in growing the power of quantum computers in the future.

[1] P. Magnard et al., Phys. Rev. Lett. 125, 260502 (2020) [2] P. Kurpiers et al., Nature 558, 264-267 (2018)

Benefits of good old-fashioned crystalline perfection - new physics in ultra pure delafossite metals

By Andrew Mackenzie (Max Planck Institute for Chemical Physics of Solids),

The delafossites are a series of layered compounds with triangular lattices similar to that of NaCoO2 but with a different stacking sequence along the c axis. They are host to intriguing magnetic insulators and semimetals, as well as metals such as PdCoO2, PtCoO2, PdCrO2 and PdRhO2. The properties of these metals are remarkable. Although they are strongly two-dimensional, their room temperature electrical conductivity is higher per carrier than that of any elemental metal, and PdCoO2 crystals can have a low temperature resistivity of only a few nΩcm, corresponding to mean free paths of tens of microns. I will describe recent experiments from my group in which we have established that the huge mean free paths result from extraordinarily low defect densities in the conducting planes of as-grown crystals, and give examples of new mesoscopic physics that can be observed as a result. If time permits I will also discuss intriguing bulk and surface states that can be observed with extremely high resolution in angle resolved photoemission experiments.

Ultracold atoms in optical lattices out-of-equilibrium

By Monika Aidelsburger (Ludwig-Maximilians-University Munich),

Well-controlled synthetic quantum systems, such as ultracold atoms in optical lattices, offer intriguing possibilities to study complex many-body problems relevant to a variety of research areas, ranging from condensed matter to high-energy physics. In particular, out-of-equilibrium phenomena constitute natural applications of quantum simulators, which have already successfully demonstrated simulations in regimes that are beyond reach using state-of-the-art numerical techniques.

This enables us to shed new light on fundamental questions about the thermalization of isolated quantum many-body systems. While generic models are expected to thermalize according to the eigenstate thermalization hypothesis (ETH), violation of ETH is believed to occur mainly in two types of systems: integrable models and many-body localized systems. In between these two extreme limits there is, however, a whole range of models that exhibit more complex dynamics, for instance, due to an emergent fragmentation of the Hilbert space into many dynamically disconnected subspaces. Here, we realize such a model by implementing the 1D Fermi-Hubbard model with a strong linear potential [1] and observe strong initial-state dependent thermalization - a smoking-gun signature of Hilbert-space fragmentation. Engineering quantum systems out-of-equilibrium, on the other hand, further can be used as a tool to engineer novel quantum phases of matter, which cannot be accessed in static realizations. To this end, the system’s parameters are varied periodically, a method commonly known as Floquet engineering [2]. This facilitated the realization of paradigmatic topological lattice models and recently inspired ideas for implementing Z2 lattice gauge theories [3]. The rich properties of topological Floquet systems, however, transcend those of their static counterparts, resulting in a generalized bulk-edge correspondence. As a consequence, topological edge modes can exist even in situations where the bulk bands have zero Chern numbers. The novel properties of such anomalous Floquet systems open the door to exciting new non- equilibrium many-body phases without any static analogue [4].

[1] S. Scherg et al., arXiv:2010.12965 (2020) [2] A. Eckardt, Phys. Mod. Phys. 89, 311 (2017) [3] C. Schweizer et al., Nat. Phys. 15, 1168-1173 (2019) [4] K. Wintersperger et al., Nature Physics 16, 1058-1063 (2020)

Fracton and Chern-Simons Theory

By Xie Chen (Caltech),

Fracton order describes the peculiar phenomena that point excitations in certain strongly interacting systems either cannot move at all or are only allowed to move in a lower dimensional sub-manifold. It has recently been discovered in various lattice models, tensor gauge theories, etc. In this talk, we discuss how another powerful field theory framework -- the 2+1D Chern-Simons (CS) gauge theory -- can be used to provide new insight and explore new possibilities in 3+1D fracton order. 2+1D U(1) gauge theories with a CS term provide a simple and complete characterization of 2+1D Abelian topological orders. To study 3+1D fracton order, we extend the theory by taking the number of component gauge fields to be infinity. In the simplest case of infinite-component CS gauge theory, different components do not couple to each other and the theory describes a decoupled stack of 2+1D fractional Quantum Hall systems with quasi-particles moving only in 2D planes -- hence a fractonic system. More interestingly, we find that when the component gauge fields do couple through the CS term, more varieties of fractonic orders are possible. For example, they may describe foliated fractonic systems which extends the framework found in exactly solvable models. Moreover, we find examples which lie beyond the foliation framework, characterized by 2D excitations of infinite order and braiding statistics that are not strictly local.

Orbital magnetism in graphene heterostructures

By Andrea Young (UCSB),

The earliest reports of ferromagnetism date to Thales of Miletus who lived and wrote around 600 BC. Thales noted the ability of natural magnetite to attract iron, and is said to have taken this as proof that matter itself was alive. Our theories of magnetism have evolved considerably since then: we now know that ferromagnetism arises from the interplay of the Coulomb repulsion between electrons and their fermionic statistics. However, in one sense our science has advanced only little: the vast majority of magnets, like magnetite, consist of ordered arrangements of the electron spins stabilized by the spin orbit interaction. In my talk, I will describe a new class of magnets based on the spontaneous alignment of electron orbitals. Such orbital ferromagnetism may be a generic phenomena, but has, to date, found its fullest expression in graphene heterostructures in which the two dimensional orbits of electrons in distinct momentum space valleys provide the underlying degree of freedom. Because orbital degrees of freedom arise directly from the band wavefunctions, they are uniquely susceptible to experimental control via materials design. Orbital magnets also enable new forms of magnetic control using in situ knobs. For instance, orbital magnets in moire superlattice systems, where the band structure features nontrivial topology, allow for field-effect switching of magnetic moments and the resulting quantized anomalous Hall effects. I will conclude with an outlook for realizing more exotic topological phases of matter based on orbital magnetism.

‌Electrons Bloch-waltzing in Moire superlattices

By Leonid Levitov (MIT),

A striking prediction of the quantum theory of solids is that Bloch electrons in the presence of a constant electric field oscillate at a frequency tunable by the field and taking identical values for all carriers in the band. As a paradigmatic example of collective electron dynamics, Bloch oscillations received much attention, and yet, achieving this regime in solids proved to be challenging. This talk will discuss the new ideas and a new hope triggered by the advent of van der Waals heterostructures, in particular the flat bands in Moire graphene. There are several interesting not-yet-explored regimes harbored by these systems. One is synchronous oscillations of “enlightened” electrons coupled to “the big power in the sky” (the EM field of a proximal resonator)—a collective many-body dynamics that holds promise for THz electronics. Another is magnetic Bloch oscillations achieved by combining constant electric and magnetic fields. The unique dynamical regimes arising in this case are described by the Ashcroft-Mermin (AM) Hamiltonian. We will demonstrate that the AM dynamics is integrable and identify two distinct phases, electric and magnetic, exhibiting position-momentum duality—the orbits extended in the \(k\) space and confined in the \(x\) space, and vice versa. The boundary between the electric and magnetic phases hosts an interesting and presently poorly understood chaotic dynamical phase. The two phases—electric and magnetic—are of keen interest both as examples of complex quantum dynamics arising in a simple setting and because of their relevance for THz research.