Observation of topological transitions in interacting quantum circuits

Energy dynamics, heat production and heat-work conversion with qubits: toward the development of quantum machines

We present an overview of recent advances in the study of energy dynamics and mechanisms for energy conversion in qubit systems with special focus on realizations in superconducting quantum circuits. We briefly introduce the relevant theoretical framework to analyze heat generation, energy transport and energy conversion in these systems with and without time-dependent driving considering the effect of equilibrium and non-equilibrium environments. We analyze specific problems and mechanisms under current investigation in the context of qubit systems. These include the problem of energy dissipation and possible routes for its control, energy pumping between driving sources and heat pumping between reservoirs, implementation of thermal machines and mechanisms for energy storage. We highlight the underlying fundamental phenomena related to geometrical and topological properties, as well as many-body correlations. We also present an overview of recent experimental activity in this field.

Observation of Spin-Tensor Induced Topological Phase Transitions of Triply Degenerate Points with a Trapped Ion

Triply degenerate points (TDPs), which correspond to new types of topological semimetals, can support novel quasiparticles possessing effective integer spins while preserving Fermi statistics. Here by mapping the momentum space to the parameter space of a three-level system in a trapped ion, we experimentally explore the transitions between different types of TDPs driven by spin-tensor-momentum couplings. We observe the phase transitions between TDPs with different topological charges by measuring the Berry flux on a loop surrounding the gap-closing lines, and the jump of the Berry flux gives the jump of the topological charge (up to a 2π factor) across the transitions. For the Berry flux measurement, we employ a new method by examining the geometric rotations of both spin vectors and tensors, which lead to a generalized solid angle equal to the Berry flux. The controllability of a multilevel ion offers a versatile platform to study high-spin physics, and our Letter paves the way to explore novel topological phenomena therein.

Observing the quantum topology of light

Topological photonics provides a powerful platform to explore topological physics beyond traditional electronic materials and shows promising applications in light transport and lasers. Classical degrees of freedom are routinely used to construct topological light modes in real or synthetic dimensions. Beyond the classical topology, the inherent quantum nature of light provides a wealth of fundamentally distinct topological states. Here we implement experiments on topological states of quantized light in a superconducting circuit, with which one- and two-dimensional Fock-state lattices are constructed. We realize rich topological physics including topological zero-energy states of the Su-Schrieffer-Heeger model, strain-induced pseudo-Landau levels, valley Hall effect, and Haldane chiral edge currents. Our study extends the topological states of light to the quantum regime, bridging topological phases of condensed-matter physics with circuit quantum electrodynamics, and offers a freedom in controlling the quantum states of multiple resonators.

A synthetic monopole source of Kalb-Ramond field in diamond

Magnetic monopoles play a central role in areas of physics that range from electromagnetism to topological matter. String theory promotes conventional vector gauge fields of electrodynamics to tensor gauge fields and predicts the existence of more exotic tensor monopoles. Here, we report the synthesis of a tensor monopole in a four-dimensional parameter space defined by the spin degrees of freedom of a single solid-state defect in diamond. Using two complementary methods, we characterized the tensor monopole by measuring its quantized topological charge and its emanating Kalb-Ramond field. By introducing a fictitious external field that breaks chiral symmetry, we further observed an intriguing spectral transition, characterized by spectral rings protected by mirror symmetries. Our work demonstrates the possibility of emulating exotic topological structures inspired by string theory.

Measurement of Spin Chern Numbers in Quantum Simulated Topological Insulators

The topology of quantum systems has become a topic of great interest since the discovery of topological insulators. However, as a hallmark of the topological insulators, the spin Chern number has not yet been experimentally detected. The challenge to directly measure this topological invariant lies in the fact that this spin Chern number is defined based on artificially constructed wave functions. Here we experimentally mimic the celebrated Bernevig-Hughes-Zhang model with cold atoms, and then measure the spin Chern number with the linear response theory. We observe that, although the Chern number for each spin component is ill defined, the spin Chern number measured by their difference is still well defined when both energy and spin gaps are nonvanished.

Quantum control of bosonic modes with superconducting circuits

Bosonic modes have wide applications in various quantum technologies, such as optical photons for quantum communication, magnons in spin ensembles for quantum information storage and mechanical modes for reversible microwave-to-optical quantum transduction. There is emerging interest in utilizing bosonic modes for quantum information processing, with circuit quantum electrodynamics (circuit QED) as one of the leading architectures. Quantum information can be encoded into subspaces of a bosonic superconducting cavity mode with long coherence time. However, standard Gaussian operations (e.g., beam splitting and two-mode squeezing) are insufficient for universal quantum computing. The major challenge is to introduce additional nonlinear control beyond Gaussian operations without adding significant bosonic loss or decoherence. Here we review recent advances in universal control of a single bosonic code with superconducting circuits, including unitary control, quantum feedback control, driven-dissipative control and holonomic dissipative control. Various approaches to entangling different bosonic modes are also discussed.

Simulation of higher-order topological phases and related topological phase transitions in a superconducting qubit

Higher-order topological phases give rise to new bulk and boundary physics, as well as new classes of topological phase transitions. While the realization of higher-order topological phases has been confirmed in many platforms by detecting the existence of gapless boundary modes, a direct determination of the higher-order topology and related topological phase transitions through the bulk in experiments has still been lacking. To bridge the gap, in this work we carry out the simulation of a two-dimensional second-order topological phase in a superconducting qubit. Owing to the great flexibility and controllability of the quantum simulator, we observe the realization of higher-order topology directly through the measurement of the pseudo-spin texture in momentum space of the bulk for the first time, in sharp contrast to previous experiments based on the detection of gapless boundary modes in real space. Also through the measurement of the evolution of pseudo-spin texture with parameters, we further observe novel topological phase transitions from the second-order topological phase to the trivial phase, as well as to the first-order topological phase with nonzero Chern number. Our work sheds new light on the study of higher-order topological phases and topological phase transitions.

Experimental Observation of Tensor Monopoles with a Superconducting Qudit

Monopoles play a center role in gauge theories and topological matter. There are two fundamental types of monopoles in physics: vector monopoles and tensor monopoles. Examples of vector monopoles include the Dirac monopole in three dimensions and Yang monopole in five dimensions, which have been extensively studied and observed in condensed matter or artificial systems. However, tensor monopoles are less studied, and their observation has not been reported. Here we experimentally construct a tunable spin-1 Hamiltonian to generate a tensor monopole and then measure its unique features with superconducting quantum circuits. The energy structure of a 4D Weyl-like Hamiltonian with threefold degenerate points acting as tensor monopoles is imaged. Through quantum-metric measurements, we report the first experiment that measures the Dixmier-Douady invariant, the topological charge of the tensor monopole. Moreover, we observe topological phase transitions characterized by the topological Dixmier-Douady invariant, rather than the Chern numbers as used for conventional monopoles in odd-dimensional spaces.

Extended harmonic mapping connects the equations in classical, statistical, fluid, quantum physics and general relativity

One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that  of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.

Non-Abelian generalizations of the Hofstadter model: spin-orbit-coupled butterfly pairs

The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin-orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

Digital Simulation of Topological Matter on Programmable Quantum Processors

Simulating the topological phases of matter in synthetic quantum simulators is a topic of considerable interest. Given the universality of digital quantum simulators, the prospect of digitally simulating exotic topological phases is greatly enhanced. However, it is still an open question how to realize the digital quantum simulation of topological phases of matter. Here, using common single- and two-qubit elementary quantum gates, we propose and demonstrate an approach to design topologically protected quantum circuits on the current generation of noisy quantum processors where spin-orbital coupling and related topological matter can be digitally simulated. In particular, a low-depth topological quantum circuit is performed on both the IBM and Rigetti quantum processors. In the experiments, we not only observe but also distinguish the 0 and π energy topological edge states by measuring the qubit excitation distribution at the output of the circuits.

Topological edge states of interacting photon pairs emulated in a topolectrical circuit

Topological physics opens up a plethora of exciting phenomena allowing to engineer disorder-robust unidirectional flows of light. Recent advances in topological protection of electromagnetic waves suggest that even richer functionalities can be achieved by realizing topological states of quantum light. This area, however, remains largely uncharted due to the number of experimental challenges. Here, we take an alternative route and design a classical structure based on topolectrical circuits which serves as a simulator of a quantum-optical one-dimensional system featuring the topological state of two photons induced by the effective photon-photon interaction. Employing the correspondence between the eigenstates of the original problem and circuit modes, we use the designed simulator to extract the frequencies of bulk and edge two-photon bound states and evaluate the topological invariant directly from the measurements. Furthermore, we perform a reconstruction of the two-photon probability distribution for the topological state associated with one of the circuit eigenmodes.

Experimental measurement of the quantum geometric tensor using coupled qubits in diamond

Geometry and topology are fundamental concepts, which underlie a wide range of fascinating physical phenomena such as topological states of matter and topological defects. In quantum mechanics, the geometry of quantum states is fully captured by the quantum geometric tensor. Using a qubit formed by an NV center in diamond, we perform the first experimental measurement of the complete quantum geometric tensor. Our approach builds on a strong connection between coherent Rabi oscillations upon parametric modulations and the quantum geometry of the underlying states. We then apply our method to a system of two interacting qubits, by exploiting the coupling between the NV center spin and a neighboring ¹³C nuclear spin. Our results establish coherent dynamical responses as a versatile probe for quantum geometry, and they pave the way for the detection of novel topological phenomena in solid state.

An experimental test of the geodesic rule proposition for the noncyclic geometric phase

The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics and may become advantageous for quantum technology. In noncyclic evolutions, a proposition relates the geometric phase to the area bounded by the phase-space trajectory and the shortest geodesic connecting its end points. The experimental demonstration of this geodesic rule proposition in different systems is of great interest, especially due to the potential use in quantum technology. Here, we report a previously unshown experimental confirmation of the geodesic rule for a noncyclic geometric phase by means of a spatial SU(2) matter-wave interferometer, demonstrating, with high precision, the predicted phase sign change and π jumps. We show the connection between our results and the Pancharatnam phase. Last, we point out that the geodesic rule may be applied to obtain the red shift in general relativity, enabling a new quantum tool to measure gravity.

Optically Controlled Orbitronics on a Triangular Lattice

The propagation of electrons in an orbital multiplet dispersing on a lattice can support anomalous transport phenomena deriving from an orbitally induced Berry curvature. In striking contrast to the related situation in graphene, we find that anomalous transport for an L=1 multiplet on the primitive 2D triangular lattice is activated by easily implemented on site and optically tunable potentials. We demonstrate this for dynamics in a Bloch band where point degeneracies carrying opposite winding numbers are generically offset in energy, allowing both an anomalous charge Hall conductance with the sign selected by off-resonance coupling to circularly polarized light and a related anomalous orbital Hall conductance activated by layer buckling.

Distinct Bound of the Quantum Speed Limit via the Gauge Invariant Distance

We derive a distinct bound of the quantum speed limit for a non-Hermitian quantum system by employing the gauge invariant and geometric natures of quantum mechanics. The bound is of geometric properties since it relates to the geometric phase of the quantum system, and it is tighter than the Mandelstam-Tamm and Margolus-Levitin bounds in some cases. Also, by making the geodesic assumption, the analog of the Margolus-Levitin bound is derived for the time-dependent (non-)Hermitian quantum system. These two bounds reflect the impacts of the transmission modes of the state vectors on the evolution path in the manifold.

Observation of Topological Magnon Insulator States in a Superconducting Circuit

Searching topological states in artificial systems has recently become a rapidly growing field of research. Meanwhile, significant experimental progress on observing topological phenomena has been made in superconducting circuits. However, topological insulator states have not yet been reported in this system. Here, for the first time, we experimentally realize a tunable dimerized spin chain model and observe the topological magnon insulator states in a superconducting qubit chain. Via parametric modulations of the qubit frequencies, we show that the qubit chain can be flexibly tuned into topologically trivial or nontrivial magnon insulator states. Based on monitoring the quantum dynamics of a single-qubit excitation in the chain, we not only measure the topological winding numbers, but also observe the topological magnon edge and defect states. Our experiment exhibits the great potential of tunable superconducting qubit chain as a versatile platform for exploring noninteracting and interacting symmetry-protected topological states.

Topological Quantum Transition Driven by Charge-Phonon Coupling in the Haldane Chern Insulator

In condensed matter physics many features can be understood in terms of their topological properties. Here we report evidence of a topological quantum transition driven by the charge-phonon coupling in the spinless Haldane model on a honeycomb lattice, a well-known prototypical model of the Chern insulator. Starting from parameters describing the topological phase in the bare Haldane model, we show that increasing the strength of the charge lattice coupling drives the system towards a trivial insulator. The average number of fermions in the Dirac point, characterized by the lowest gap, exhibits a finite discontinuity at the transition point and can be used as a direct indicator of the topological quantum transition. Numerical simulations show, also, that the renormalized phonon propagator exhibits a two peak structure across the quantum transition, whereas, in the absence of the mass term in the bare Haldane model, there is indication of a complete softening of the effective vibrational mode, signaling a charge density wave instability.

Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit

A Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and is well studied in many branches of modern physics; however, the quantum metric as a real component of the QGT is less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure the quantum metric tensor for characterizing the geometry and topology of underlying quantum states in parameter space. The first method is to probe the transition probability after a sudden quench, and the second one is to detect the excitation rate under weak periodic driving. Furthermore, based on quantum metric and Berry-curvature measurements, we explore a topological phase transition in a simulated time-reversal-symmetric system. The work opens up a unique approach to explore the topology of quantum states with the QGT.

Machine Learning Topological Phases with a Solid-State Quantum Simulator

We report an experimental demonstration of a machine learning approach to identify exotic topological phases, with a focus on the three-dimensional chiral topological insulators. We show that the convolutional neural networks-a class of deep feed-forward artificial neural networks with widespread applications in machine learning-can be trained to successfully identify different topological phases protected by chiral symmetry from experimental raw data generated with a solid-state quantum simulator. Our results explicitly showcase the exceptional power of machine learning in the experimental detection of topological phases, which paves a way to study rich topological phenomena with the machine learning toolbox.

Probing Weyl Physics with One-Dimensional Sonic Crystals

Recently, intense efforts have been devoted to realizing classical analogues of various topological phases of matter. In this Letter, we explore the intriguing Weyl physics by a simple one-dimensional sonic crystal, in which two extra structural parameters are combined to construct a synthetic three-dimensional space. Based on our ultrasonic experiments, we have not only observed the synthetic Weyl points, but also probed the novel reflection phase singularity that connects inherently with the topological robustness of Weyl points. The presence of topologically nontrivial interface modes has been demonstrated further. As the first realization of topological acoustics in synthetic space, our study exhibits great potential of probing high-dimensional topological phenomena by such easily fabricated and detected low-dimension acoustic systems.

Two classes of singularities and novel topology in a specially designed synthetic photonic crystals

Zak phase and topological protected edge state are usually studied in one-dimensional (1D) photonic systems with spatial inversion symmetry (SIS). In this work, we find that specific classes of 1D structure without SIS can be mapped to a system with SIS and also exhibit novel topology, which manifest as phase cut lines (PCLs) in our specially designed synthetic photonic crystals (SPCs). Zak phase defined in SIS is extended to depict the topology of PCLs after redefinition, and a topological protected edge state is also achieved in our 1D structure without SIS. In our SPCs, the relationship between Chern numbers in two dimensions (2D) and the extended Zak phases of 1D PCLs is given, which are bound by the first type singularities. Higher Chern numbers and multi-chiral edge states are achieved utilizing the concept of synthetic dimensions. The effective Hamiltonian is given, based on which we find that the band edges of each PCL play a role analogous to the valley pseudospin, and our SPC is actually a new type of valley photonic crystal that is usually studied in graphene-like honeycomb lattice. The chiral valley edge transport is also demonstrated. In higher dimensions, the shift of the first type singularity in expanded parameter space will lead to the Weyl point topological transition, which we proposed in our previous work. In this paper, we find a second type of singularity that manifests as a singular surface in our expanded parameter space. The shift of the singular surface will lead to the nodal line topological transition. We find the states on the singular surface possess extremely high robustness against certain randomness, based on which a topological wave filter is constructed.

Tunable THz generalized Weyl points

Weyl points, as linearly double degenerated point of band structures, have been extensively researched in electronic and classical wave systems. However, Weyl points' realization is always accompanied with delicate "lattice structures". In this work, frequency-tunable terahertz (THz) generalized Weyl points inside the parameter space have been investigated and displayed by a specially designed photonic crystal with polydimethylsiloxane (PDMS) immersed in 4-cyano'-pentylbipenyl (5CB) liquid crystals (LCs). The reflective phase vortices as a signature of the generalized Weyl points are observed through our numerically simulations. Besides, interface states between photonic crystals and any reflective substrates are fulfilled too. Meanwhile, we could also change the orientation of LC molecule by the external magnetic field so as to tune the frequency of the first two bands' Weyl point from 0.27698THz to 0.30013THz. This band lies in the short-range wireless communication. Thus, our proposal may be beneficial to the investigation and application of Weyl points' properties and strongly localized states.

Simulation and Manipulation of Tunable Weyl-Semimetal Bands Using Superconducting Quantum Circuits

We simulated highly tunable Weyl-semimetal bands using superconducting quantum circuits. Driving the superconducting quantum circuits with microwave fields, we mapped the momentum space of a lattice to the parameter space, realizing the Hamiltonian of a Weyl semimetal. By measuring the energy spectrum, we directly imaged the Weyl points, whose topological winding numbers were further determined from the Berry curvature measurement. In addition, we manipulated the band structure with an additional pump microwave field, producing a momentum-dependent Weyl-point energy together with an artificial magnetic field, which are indispensable for generating chiral magnetic topological currents in some special Weyl semimetals and may have significant impact on topological physics.

Second Chern number of a quantum-simulated non-Abelian Yang monopole

Topological order is often quantified in terms of Chern numbers, each of which classifies a topological singularity. Here, inspired by concepts from high-energy physics, we use quantum simulation based on the spin degrees of freedom of atomic Bose-Einstein condensates to characterize a singularity present in five-dimensional non-Abelian gauge theories-a Yang monopole. We quantify the monopole in terms of Chern numbers measured on enclosing manifolds: Whereas the well-known first Chern number vanishes, the second Chern number does not. By displacing the manifold, we induce and observe a topological transition, where the topology of the manifold changes to a trivial state.

Yang Monopoles and Emergent Three-Dimensional Topological Defects in Interacting Bosons

The Yang monopole as a zero-dimensional topological defect has been well established in multiple fields in physics. However, it remains an intriguing question to understand the interaction effects on Yang monopoles. Here, we show that the collective motion of many interacting bosons gives rise to exotic topological defects that are distinct from Yang monopoles seen by a single particle. Whereas interactions may distribute Yang monopoles in the parameter space or glue them to a single giant one of multiple charges, three-dimensional topological defects also arise from continuous manifolds of degenerate many-body eigenstates. Their projections in lower dimensions lead to knotted nodal lines and nodal rings. Our results suggest that ultracold bosonic atoms can be used to create emergent topological defects and directly measure topological invariants that are not easy to access in solids.

Topological Floquet-Thouless Energy Pump

We explore adiabatic pumping in the presence of a periodic drive, finding a new phase in which the topologically quantized pumped quantity is energy rather than charge. The topological invariant is given by the winding number of the micromotion with respect to time within each cycle, momentum, and adiabatic tuning parameter. We show numerically that this pump is highly robust against both disorder and interactions, breaking down at large values of either in a manner identical to the Thouless charge pump. Finally, we suggest experimental protocols for measuring this phenomenon.

Topological Maxwell Metal Bands in a Superconducting Qutrit

We experimentally explore the topological Maxwell metal bands by mapping the momentum space of condensed-matter models to the tunable parameter space of superconducting quantum circuits. An exotic band structure that is effectively described by the spin-1 Maxwell equations is imaged. Threefold degenerate points dubbed Maxwell points are observed in the Maxwell metal bands. Moreover, we engineer and observe the topological phase transition from the topological Maxwell metal to a trivial insulator, and report the first experiment to measure the Chern numbers that are higher than one.

Quantum information processing with superconducting circuits: a review

During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of quantum supremacy with fifty qubits is anticipated in just a few years. Quantum supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces. Therefore, the time is ripe for describing some of the recent development of superconducting devices, systems and applications. As such, the discussion of superconducting qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry.

Local probe of single phonon dynamics in warm ion crystals

The detailed characterization of non-trivial coherence properties of composite quantum systems of increasing size is an indispensable prerequisite for scalable quantum computation, as well as for understanding non-equilibrium many-body physics. Here, we show how autocorrelation functions in an interacting system of phonons as well as the quantum discord between distinct degrees of freedoms can be extracted from a small controllable part of the system. As a benchmark, we show this in chains of up to 42 trapped ions, by tracing a single phonon excitation through interferometric measurements of only a single ion in the chain. We observe the spreading and partial refocusing of the excitation in the chain, even on a background of thermal excitations. We further show how this local observable reflects the dynamical evolution of quantum discord between the electronic state and the vibrational degrees of freedom of the probe ion.

An exhausting characterization of the coherence properties of quantum system becomes challenging with increasing system size. Here the authors demonstrate that phonon autocorrelation functions and quantum discord can be measured with local control, and validate it in a string of 42 trapped ions.

Quantum systems under frequency modulation

We review the physical phenomena that arise when quantum mechanical energy levels are modulated in time. The dynamics resulting from changes in the transition frequency is a problem studied since the early days of quantum mechanics. It has been of constant interest both experimentally and theoretically since, with the simple two-state model providing an inexhaustible source of novel concepts. When the transition frequency of a quantum system is modulated, several phenomena can be observed, such as Landau-Zener-Stückelberg-Majorana interference, motional averaging and narrowing, and the formation of dressed states with the appearance of sidebands in the spectrum. Adiabatic changes result in the accumulation of geometric phases, which can be used to create topological states. In recent years, an exquisite experimental control in the time domain was gained through the parameters entering the Hamiltonian, and high-fidelity readout schemes allowed the state of the system to be monitored non-destructively. These developments were made in the field of quantum devices, especially in superconducting qubits, as a well as in atomic physics, in particular in ultracold gases. As a result of these advances, it became possible to demonstrate many of the fundamental effects that arise in a quantum system when its transition frequencies are modulated. The purpose of this review is to present some of these developments, from two-state atoms and harmonic oscillators to multilevel and many-particle systems.

Direct Measurement of Topological Numbers with Spins in Diamond

Topological numbers can characterize the transition between different topological phases, which are not described by Landau's paradigm of symmetry breaking. Since the discovery of the quantum Hall effect, more topological phases have been theoretically predicted and experimentally verified. However, it is still an experimental challenge to directly measure the topological numbers of various predicted topological phases. In this Letter, we demonstrate quantum simulation of topological phase transition of a quantum wire (QW), by precisely modulating the Hamiltonian of a single nitrogen-vacancy (NV) center in diamond. Deploying a quantum algorithm of finding eigenvalues, we reliably extract both the dispersion relations and topological numbers. This method can be further generalized to simulate more complicated topological systems.

Measuring the Second Chern Number from Nonadiabatic Effects

The geometry and topology of quantum systems have deep connections to quantum dynamics. In this Letter, I show how to measure the non-Abelian Berry curvature and its related topological invariant, the second Chern number, using dynamical techniques. The second Chern number is the defining topological characteristic of the four-dimensional generalization of the quantum Hall effect and has relevance in systems from three-dimensional topological insulators to Yang-Mills field theory. I illustrate its measurement using the simple example of a spin-3/2 particle in an electric quadrupole field. I show how one can dynamically measure diagonal components of the Berry curvature in an overcomplete basis of the degenerate ground state space and use this to extract the full non-Abelian Berry curvature. I also show that one can accomplish the same ideas by stochastically averaging over random initial states in the degenerate ground state manifold. Finally, I show how this system can be manufactured and the topological invariant measured in a variety of realistic systems, from superconducting qubits to trapped ions and cold atoms.

Measurement of a vacuum-induced geometric phase

Berry's geometric phase naturally appears when a quantum system is driven by an external field whose parameters are slowly and cyclically changed. A variation in the coupling between the system and the external field can also give rise to a geometric phase, even when the field is in the vacuum state or any other Fock state. We demonstrate the appearance of a vacuum-induced Berry phase in an artificial atom, a superconducting transmon, interacting with a single mode of a microwave cavity. As we vary the phase of the interaction, the artificial atom acquires a geometric phase determined by the path traced out in the combined Hilbert space of the atom and the quantum field. Our ability to control this phase opens new possibilities for the geometric manipulation of atom-cavity systems also in the context of quantum information processing.

Effective geometric phases and topological transitions in SO(3) and SU(2) rotations

We address the development of geometric phases in classical and quantum magnetic moments (spin-1/2) precessing in an external magnetic field. We show that nonadiabatic dynamics lead to a topological phase transition determined by a change in the driving field topology. The transition is associated with an effective geometric phase which is identified from the paths of the magnetic moments in a spherical geometry. The topological transition presents close similarities between SO(3) and SU(2) cases but features differences in, e.g. the adiabatic limits of the geometric phases, being 2π and π in the classical and the quantum case, respectively. We discuss possible experiments where the effective geometric phase would be observable.

Singularity of the time-energy uncertainty in adiabatic perturbation and cycloids on a Bloch sphere

Adiabatic perturbation is shown to be singular from the exact solution of a spin-1/2 particle in a uniformly rotating magnetic field. Due to a non-adiabatic effect, its quantum trajectory on a Bloch sphere is a cycloid traced by a circle rolling along an adiabatic path. As the magnetic field rotates more and more slowly, the time-energy uncertainty, proportional to the length of the quantum trajectory, calculated by the exact solution is entirely different from the one obtained by the adiabatic path traced by the instantaneous eigenstate. However, the non-adiabatic Aharonov- Anandan geometric phase, measured by the area enclosed by the exact path, approaches smoothly the adiabatic Berry phase, proportional to the area enclosed by the adiabatic path. The singular limit of the time-energy uncertainty and the regular limit of the geometric phase are associated with the arc length and arc area of the cycloid on a Bloch sphere, respectively. Prolate and curtate cycloids are also traced by different initial states outside and inside of the rolling circle, respectively. The axis trajectory of the rolling circle, parallel to the adiabatic path, is shown to be an example of transitionless driving. The non-adiabatic resonance is visualized by the number of cycloid arcs.

Stimulated Raman adiabatic passage in a three-level superconducting circuit

The adiabatic manipulation of quantum states is a powerful technique that opened up new directions in quantum engineering—enabling tests of fundamental concepts such as geometrical phases and topological transitions, and holding the promise of alternative models of quantum computation. Here we benchmark the stimulated Raman adiabatic passage for circuit quantum electrodynamics by employing the first three levels of a transmon qubit. In this ladder configuration, we demonstrate a population transfer efficiency >80% between the ground state and the second excited state using two adiabatic Gaussian-shaped control microwave pulses. By doing quantum tomography at successive moments during the Raman pulses, we investigate the transfer of the population in time domain. Furthermore, we show that this protocol can be reversed by applying a third adiabatic pulse, we study a hybrid nondiabatic–adiabatic sequence, and we present experimental results for a quasi-degenerate intermediate level.

The precise control and manipulation of the states of a multi-level quantum system are fundamental for quantum information processing. Here, the authors demonstrate the robust adiabatic manipulation of the quantum states of a superconducting circuit via stimulated Raman adiabatic passage.

Quantum Simulator for Transport Phenomena in Fluid Flows

Transport phenomena still stand as one of the most challenging problems in computational physics. By exploiting the analogies between Dirac and lattice Boltzmann equations, we develop a quantum simulator based on pseudospin-boson quantum systems, which is suitable for encoding fluid dynamics transport phenomena within a lattice kinetic formalism. It is shown that both the streaming and collision processes of lattice Boltzmann dynamics can be implemented with controlled quantum operations, using a heralded quantum protocol to encode non-unitary scattering processes. The proposed simulator is amenable to realization in controlled quantum platforms, such as ion-trap quantum computers or circuit quantum electrodynamics processors.

Qubit Architecture with High Coherence and Fast Tunable Coupling

We introduce a superconducting qubit architecture that combines high-coherence qubits and tunable qubit-qubit coupling. With the ability to set the coupling to zero, we demonstrate that this architecture is protected from the frequency crowding problems that arise from fixed coupling. More importantly, the coupling can be tuned dynamically with nanosecond resolution, making this architecture a versatile platform with applications ranging from quantum logic gates to quantum simulation. We illustrate the advantages of dynamical coupling by implementing a novel adiabatic controlled-z gate, with a speed approaching that of single-qubit gates. Integrating coherence and scalable control, the introduced qubit architecture provides a promising path towards large-scale quantum computation and simulation.