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arXiv:1412.3192 · Physical Review A

Simulating the dynamical quantum Hall effect with superconducting qubits

Xu-Chen Yang, Dan-Wei Zhang, Peng Xu, Yang Yu, Shi-Liang Zhu

Published Feb 5, 2015• 8 citations
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Abstract

We propose an experimental scheme to simulate the dynamical quantum Hall effect and the related interaction-induced topological transition with a superconducting-qubit array. We show that a one-dimensional Heisenberg model with tunable parameters can be realized in an array of superconducting qubits. The quantized plateau, which is a feature of the dynamical quantum Hall effect, will emerge in the Berry curvature of the superconducting qubits as a function of the coupling strength between nearest-neighbor qubits. We numerically calculate the Berry curvatures of two-, four-, and six-qubit arrays and find that the interaction-induced topological transition can be easily observed with the simplest two-qubit array. Furthermore, we analyze some practical conditions in typical experiments for observing this dynamical quantum Hall effect.

DOI
10.1103/physreva.91.022303

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We propose an experimental scheme to simulate the dynamical quantum Hall effect and the related interaction-induced topological transition with a superconducting-qubit array. We show that a one-dimensional Heisenberg model with tunable parameters can be realized in an array of superconducting qubits. The quantized plateau, which is a feature of the dynamical quantum Hall effect, will emerge in the Berry curvature of the superconducting qubits as a function of the coupling strength between nearest-neighbor qubits. We numerically calculate the Berry curvatures of two-, four-, and six-qubit arrays and find that the interaction-induced topological transition can be easily observed with the simplest two-qubit array. Furthermore, we analyze some practical conditions in typical experiments for observing this dynamical quantum Hall effect.
Topologically ordered phase has emerged as one of most exciting concepts that not only broadens our understanding of phases of matter, but also has been found to have potential application in fault-tolerant quantum computation. The direct measurement of topological properties, however, is still a challenge, especially in interacting quantum system. Here we realize two to four spin one-dimensional Heisenberg chains using nuclear magnetic resonance simulators and observe interaction-induced topological transitions, where Berry curvature in the parameter space of Hamiltonian is probed by means of dynamical response and then the first Chern number is extracted by integrating the curvature over the closed surface. The utilized experimental method provides a powerful means to explore topological phenomena in quantum systems with many-body interactions.
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.
We investigate a square-lattice architecture of superconducting transmon qubits with interqubit interactions mediated by inductive couplers. Via periodically modulating the couplers, the Abelian gauge potential, termed effective magnetic flux, can be synthesized artificially, making the system an excellent platform for simulating two-dimensional topological physics. First, we focus on the three-leg ladder which only has three rows and investigate the chiral dynamics therein for the single-particle ground state when the effective magnetic flux varies. We find what we call the ``staggered vortex-Meissner phase transition,'' where the vortex number can typically stagger a few times between one (defined as ``vortex phase'') and larger integers (defined as ``Meissner phase'') when the effective magnetic flux changes between $\ensuremath{-}\ensuremath{\pi}$ and $\ensuremath{\pi}$. This phenomenon, actually not a phase transition by definition, is quite different from the vortex-Meissner phase transition in the two-leg ladder that, in contrast, possesses only two rows and is usually treated as the quasi-two-dimensional model. Also, we find that the chiral current relies on the effective magnetic flux according to a squeezed sinusoidal function. Both the staggering of the vortex number and squeezing of the chiral current can be controllable by the coupling ratio, which is defined by the coupling-strength ratio between the column direction and row direction. Second, we continue to increase the number of rows beyond three, and the topological band structure to be anticipated at an infinite number of rows begins to occur even for a relatively small number (10 or so for typical parameters) of rows. This heralds a small circuit scale enough to observe the topological band. The behavior of the edge state in the band gap can be interpreted by the topological Chern number, which can be calculated through integrating the Berry curvature with respect to the first Brillouin zone. Last, we present a systematic method on how to measure the topological band structure based on time- and space-domain Fourier transformation of the wave function after properly exciting the qubits, which should be helpful for comprehensively analyzing the topological physics since all the topological properties are mainly contained in the band structure. Our results offer an avenue for simulating two-dimensional topological physics on the state-of-the-art superconducting quantum chips.
Geometric phases in quantum mechanics play an extraordinary role in broadening our understanding of fundamental significance of geometry in nature. One of the best known examples is the Berry phase [M.V. Berry (1984), Proc. Royal. Soc. London A, 392:45], which naturally emerges in quantum adiabatic evolution. So far the applicability and measurements of the Berry phase were mostly limited to systems of weakly interacting quasi-particles, where interference experiments are feasible. Here we show how one can go beyond this limitation and observe the Berry curvature, and hence the Berry phase, in generic systems as a nonadiabatic response of physical observables to the rate of change of an external parameter. These results can be interpreted as a dynamical quantum Hall effect in a parameter space. The conventional quantum Hall effect is a particular example of the general relation if one views the electric field as a rate of change of the vector potential. We illustrate our findings by analyzing the response of interacting spin chains to a rotating magnetic field. We observe the quantization of this response, which we term the rotational quantum Hall effect.
Topological states of quantum matter have inspired both fascinating physics findings and exciting opportunities for applications. Due to the over-complicated structure of, as well as interactions between, real materials, a faithful quantum simulation of topological matter is very important in deepening our understanding of these states. This requirement puts the quantum superconducting circuits system as a good option for mimicking topological materials, owing to their flexible tunability and fine controllability. As a typical example herein, we realize a Z2\mathbb{Z}_2Z2​-type topological insulator featuring the quantum spin Hall effect state, using a coupled system of transmission-line resonators and transmons. The single-excitation eigenstates of each unit cell are used as a pseudo-spin 1/2 system. The boundary of the topological phase transition is fixed in the phase diagram. Topological edge states are shown, which can be experimentally verified by detecting the population at the boundary of the plane. Compared to the previous simulations, this compositional system is fairly controllable, stable and less limited. Therefore, our scheme provides a reliable platform for faithful quantum simulations of topological matter.
The recent experimental realization of a programmable two-dimensional square superconducting qubit array opens a new avenue for quantum simulation with qubits [Science 372, 948 (2021)]. Here, we present an experimentally feasible method to achieve two-dimensional superconducting qubit lattice with tunable coupling strengths. A configuration with higher-order topological phases is constructed, featuring topologically protected boundary states in lower dimension (corner states). We show that the quadrupole topological phases can be effectively characterized by the dynamics of the single-excitation quantum state. Moreover, we also explore the detection and dynamics of the corner modes in our qubit lattice system. Particularly, we propose an effective scheme to realize quantum information transfer via the corner states. Our work suggests that superconducting circuits systems are a fertile platform to study novel topological quantum phases and may shed light on the ongoing exploration of topologically protected quantum information processing.