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Physical review. B./Physical review. B

Simulation of a topological phase transition in a Kitaev chain with long-range coupling using a superconducting circuit

Ziyu Tao, Tongxing Yan, Weiyang Liu, Jingjing Niu, Yuxuan Zhou, Libo Zhang, Hao Jia, Wei-Qiang Chen, Song Liu, Yuanzhen Chen, Dapeng Yu

Published Jan 8, 2020• 22 citations
Publisher

Abstract

We report an experimental work on simulating the Kitaev chain of one-dimensional $p$-wave superconductivity with long-range coupling using superconducting quantum circuits. The effective Hamiltonian of this model in the momentum space is mapped onto that of a qubit driven by a tunable microwave control field. By monitoring the dynamics of the qubit, we generate intuitive yet precise visualization of topological characteristics of different quantum phases of the system. Topological invariants can be directly deduced from the experimental results, without relying on indirect extraction from measurements such as microwave spectroscopy and temporal averaging. Therefore a much enhanced efficiency of measurement can be realized compared to other methods that are often used to simulate topological phase transitions. As a consequence, a comprehensive phase diagram covering a wide range of the parameter space is obtained. Topological phase transitions and quantum multicritical points are clearly demonstrated. In particular, new topological phases emerge as a consequence of long-range coupling. The method used here can be readily generalized to simulate more complex systems on different experimental platforms.

DOI
10.1103/physrevb.101.035109

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We report an experimental work on simulating the Kitaev chain of one-dimensional ppp-wave superconductivity with long-range coupling using superconducting quantum circuits. The effective Hamiltonian of this model in the momentum space is mapped onto that of a qubit driven by a tunable microwave control field. By monitoring the dynamics of the qubit, we generate intuitive yet precise visualization of topological characteristics of different quantum phases of the system. Topological invariants can be directly deduced from the experimental results, without relying on indirect extraction from measurements such as microwave spectroscopy and temporal averaging. Therefore a much enhanced efficiency of measurement can be realized compared to other methods that are often used to simulate topological phase transitions. As a consequence, a comprehensive phase diagram covering a wide range of the parameter space is obtained. Topological phase transitions and quantum multicritical points are clearly demonstrated. In particular, new topological phases emerge as a consequence of long-range coupling. The method used here can be readily generalized to simulate more complex systems on different experimental platforms.
Kitaev chain is one of the important physical models for studying topological quantum states and quantum computing. We here propose an approach to simulate the one-dimensional Kitaev chain via a circuit of superconducting qubits. In our approach, all coupling parameters can be controlled independently, and a nontrivial gauge phase is constructed. We also study the environmental effects on the topological states of the Kitaev chain. In addition to the independent environment surrounding each qubit, we consider the common environment shared by neighboring qubits. Such an environment can generate effective non-Hermitian dissipative coupling between qubits. Through analysis and numerical calculation, we demonstrate that the common environment can significantly affect the topological properties of the qubit chain. Moreover, we observe that dissipative couplings at the edges of the Kitaev chain affect the topological states more strongly than those located elsewhere. Our work may provide a new way to explore topological phase transitions and environmental effects on topological physics using superconducting qubit circuits.
In this paper, we have considered the extended version of the Kitaev model in one dimension, i.e., a long-range p-wave superconducting wire. In the long-range Kitaev chain, the superconducting hopping and pairing terms in the Hamiltonian decay, independently, in a power-law fashion , where l is the distance between the two sites and x is some positive constant. We have studied the appearance of Majorana zero-energy edge modes and also, massive Dirac edge modes by exact diagonalization, as well as analytical computations. Exact diagonalization indicates the existence of both kinds of massless and massive edge modes in the energy spectrum. Furthermore, we obtain the phase diagram and the topological phase transitions by calculating the winding number, which is the topological invariant.
Hyperbolic geometry hosts a plethora of exotic quantum phenomena ranging from holographic duality to novel topological phases. However, investigating these effects experimentally remains challenging due to the Euclidean constraints of natural crystalline materials. While recent synthetic lattices have achieved static realizations, they often lack the dynamical tunability required to explore the full geometric landscape. Here, we overcome these limitations by implementing a quantum simulation of hyperbolic geometry within the parameter space of a superconducting quantum circuit. We construct a Hyperbolically Driven Quantum System (HDQS) by mapping the hyperbolic properties onto the dynamically controllable Hamiltonian parameters. By probing the systems' response under geometric quantum driving, we elucidate how the intrinsic geometry of the parameter space dictates observable physical quantities. Furthermore, we extend the theoretical framework to realize ensemble simulation. This generalization enables us to harness the fast coherent control of superconducting qubits to capture essential geometric characteristics, establishing a versatile platform for exploring quantum dynamics on hyperbolic manifolds.
Abstract A number of recent works have argued that quantum complexity, a well-known concept in computer science that has re-emerged recently in the context of the physics of black holes, may be used as an efficient probe of novel phenomena such as quantum chaos and even quantum phase transitions. In this article, we provide further support for the latter, using a 1-dimensional p-wave superconductor — the Kitaev chain — as a prototype of a system displaying a topological phase transition. The Hamiltonian of the Kitaev chain manifests two gapped phases of matter with fermion parity symmetry; a trivial strongly-coupled phase and a topologically non-trivial, weakly-coupled phase with Majorana zero-modes. We show that Krylov-complexity (or, more precisely, the associated spread-complexity) is able to distinguish between the two and provides a diagnostic of the quantum critical point that separates them. We also comment on some possible ambiguity in the existing literature on the sensitivity of different measures of complexity to topological phase transitions.
Abstract Quantum simulation enables study of many-body systems in non-equilibrium by mapping to a controllable quantum system, providing a powerful tool for computational intractable problems. Here, using a programmable quantum processor with a chain of 10 superconducting qubits interacted through tunable couplers, we simulate the one-dimensional generalized Aubry-André-Harper model for three different phases, i.e., extended, localized and critical phases. The properties of phase transitions and many-body dynamics are studied in the presence of quasi-periodic modulations for both off-diagonal hopping coefficients and on-site potentials of the model controlled respectively by adjusting strength of couplings and qubit frequencies. We observe the spin transport for initial single- and multi-excitation states in different phases, and characterize phase transitions by experimentally measuring dynamics of participation entropies. Our experimental results demonstrate that the recently developed tunable coupling architecture of superconducting processor extends greatly the simulation realms for a wide variety of Hamiltonians, and can be used to study various quantum and topological phenomena.
We investigate an inter-leg coupled hybrid Kitaev ladder composed of two parallel superconducting chains with distinct pairing interactions. The upper chain of the ladder hosts conventional ppp-wave pairing, while the lower chain exhibits long-range pairing that decays algebraically with distance. We demonstrate that the mutual influence of long-range pairing exponent, chemical potential, and inter-leg coupling strength gives rise to a rich topological phase diagram characterized by multiple Majorana zero modes and massive Dirac modes. In particular, we show that the inter-leg coupling renormalizes the effective energy scales, leading to a systematic shift of the topological phase boundaries and enabling controlled tuning of the Majorana modes. Furthermore, we identify a transition from a two Majorana zero mode phase to a phase encapsulating four Majorana zero modes, as the long-range pairing exponent is varied. This transition is accompanied by a crossover regime in which Majorana zero modes coexist with massive Dirac modes, reflecting hybridization between edge and bulk excitations. This ladder thus provides a minimal and attractive platform for realizing the impact of a long-range pairing on topological phases. Our results highlight the potential of long-range hybrid systems for engineering tunable topological states relevant for quantum information applications.