Experimental observation of topological transitions in interacting multispin systems

Zhihuang Luo, Chao Lei, Jun Li, Xinfang Nie, Zhaokai Li, Xinhua Peng, and Jiangfeng Du

Phys. Rev. A 93, 052116 – Published 26 May 2016

Abstract

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.

Received 15 October 2015
Revised 1 March 2016

DOI:https://doi.org/10.1103/PhysRevA.93.052116

Physics Subject Headings (PhySH)

Geometric & topological phases Topological quantum computing

Atomic, Molecular & Optical

Authors & Affiliations

Zhihuang Luo¹, Chao Lei¹, Jun Li², Xinfang Nie¹, Zhaokai Li¹, Xinhua Peng^1,3,*, and Jiangfeng Du^1,3,†

¹Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
²Beijing Computational Science Research Center, Beijing 100094, China
³Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

^*xhpeng@ustc.edu.cn
^†djf@ustc.edu.cn

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Issue

Vol. 93, Iss. 5 — May 2016

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Images

Figure 1
(a) The quasiadiabatic evolution path (blue) in the spherical parameter space of external magnetic field $\vec{h}$ ( $| \vec{h} | = 1$ ). (b) The magnetization as a function of $v_{θ}$ for the two-qubit case. In the linear zone (i.e., the left part of the dashed black line), $M_{ϕ} = F_{ϕ θ} v_{θ}$ and $v_{θ}^{max} = 1.53$ . The high-order terms of $v_{θ}$ dominate at the right of the dashed black line.
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Figure 2
The pulse sequences for (a) simulating the quasiadiabatic evolution of $\hat{H} (θ_{n}, 0)$ and effectively creating the nearest near $z z$ interactions with a uniform coupling strength $J$ from the natural interactions of NMR, for three qubits (b) and four qubits (c), respectively. The gray regions represent the evolution of ${\hat{H}}_{z z} = - J \sum_{j = 1}^{N - 1} {\hat{σ}}_{j}^{z} {\hat{σ}}_{j + 1}^{z}$ . (a) $θ_{n} = v_{θ}^{2} t_{n}^{2} / 2 π$ for $n = 1, 2, \dots, 300$ . The loop is used to approximate the quasiadiabatic evolution path. The red rectangles represent $π$ pulses. (b) $τ_{1} = - J τ / π J_{23}$ and $τ_{2} = J τ / π J_{12}$ . (c) $τ_{1} = - J τ (J_{12} + J_{34}) / 2 π J_{12} J_{34}$ , $τ_{2} = - J τ (J_{12} - J_{34}) / 2 π J_{12} J_{34}$ , and $τ_{3} = J τ / 2 π J_{23}$ .
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Figure 3
(a) The molecular structure of chloroform. The RF pulses act on ${}^{13}C$ and ${}^{1}H$ nuclei independently to fulfill the desired control tasks. The coupling constant between two nuclei is $J_{CH} = 214.6$ Hz. (b) The energy-level diagram of a two-spin Heisenberg chain. (c) The sum experimental ${}^{13}C$ spectra obtained by decoupling the other spin ${}^{1}H$ and swapping ${}^{1}H$ to the observed nucleus ${}^{13}C$ . The integration of the resonant peak of the experimental spectra stands for the total magnetization along $y$ direction. (d) The Berry curvature as a function of interaction strength $J$ . The blue circles and red diamonds represent the theoretical and experimental values, respectively. $C_{1} (= 2 F_{ϕ θ})$ count the degeneracies (small and red spheres) emerging in $\vec{h}$ parameter space (big and yellow spheres). The experimental average values of different quantized plateaus are $F_{θ ϕ} = - 0.0034 \pm 0.019$ and $F_{θ ϕ} = 0.99 \pm 0.015$ .
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Figure 4
The molecular structures and parameters of (a) diethyl-fuoromalonate, where three qubits are labeled as ${}^{1}H, {}^{13}C, and {}^{19}F$ and (b) iodotrifluoroethylene, where four qubits are labeled ${}^{13}C, {}^{19}{F_{1}}, {}^{19}{F_{2}}, and {}^{19}{F_{3}}$ . The chemical shifts and scalar coupling constants (in Hz) are given as the diagonal and off-diagonal elements in two tables, respectively. The last column shows the transversal relaxation time $T_{2}$ of each nucleus. Due to the interactions, their corresponding ${}^{13}C$ equilibrium spectra were split into $2^{N - 1}$ (i.e., four and eight) peaks, respectively.
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Figure 5
(a, b) The sum experimental ${}^{13}C$ spectra obtained by decoupling and swapping all other nuclei to the observable nucleus ${}^{13}C$ , respectively for three- and four-qubit cases.
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Figure 6
(a, b) The energy-level diagrams of the Heisenberg spin model for $N = 3$ and $N = 4$ , respectively. (c, d) The Berry curvatures as a function of interaction strength $J$ in the three- and four-qubit experiments, respectively. The blue circles and red diamonds represent the theoretical and experimental values, respectively. The average values of different plateaus are $F_{θ ϕ} = 0.48 \pm 0.029$ and $F_{θ ϕ} = 1.49 \pm 0.027$ (c) and $F_{θ ϕ} = - 0.011 \pm 0.034$ , $F_{θ ϕ} = 1.033 \pm 0.032$ , and $F_{θ ϕ} = 1.99 \pm 0.024$ (d).
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