Probing Complex-Energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator

M.-M. Cao, K. Li, W.-D. Zhao, W.-X. Guo, B.-X. Qi, X.-Y. Chang, Z.-C. Zhou, Y. Xu, and L.-M. Duan

Phys. Rev. Lett. 130, 163001 – Published 21 April 2023

Abstract

Non-Hermitian systems generically have complex energies, which may host topological structures, such as links or knots. While there has been great progress in experimentally engineering non-Hermitian models in quantum simulators, it remains a significant challenge to experimentally probe complex energies in these systems, thereby making it difficult to directly diagnose complex-energy topology. Here, we experimentally realize a two-band non-Hermitian model with a single trapped ion whose complex eigenenergies exhibit the unlink, unknot, or Hopf link topological structures. Based on non-Hermitian absorption spectroscopy, we couple one system level to an auxiliary level through a laser beam and then experimentally measure the population of the ion on the auxiliary level after a long period of time. Complex eigenenergies are then extracted, illustrating the unlink, unknot, or Hopf link topological structure. Our work demonstrates that complex energies can be experimentally measured in quantum simulators via non-Hermitian absorption spectroscopy, thereby opening the door for exploring various complex-energy properties in non-Hermitian quantum systems, such as trapped ions, cold atoms, superconducting circuits, or solid-state spin systems.

Received 27 October 2022
Accepted 31 March 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.163001

Physics Subject Headings (PhySH)

Geometric & topological phases Open quantum systems Quantum simulation

Non-Hermitian systems Trapped ions

General PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

M.-M. Cao ^1,*, K. Li^1,*, W.-D. Zhao², W.-X. Guo ¹, B.-X. Qi¹, X.-Y. Chang¹, Z.-C. Zhou^1,3, Y. Xu ^1,3,†, and L.-M. Duan^1,3,‡

¹Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
²HYQ Co., Ltd., Beijing 100176, People’s Republic of China
³Hefei National Laboratory, Hefei 230088, People’s Republic of China

^*These authors contributed equally to this work.
^†yongxuphy@tsinghua.edu.cn
^‡lmduan@tsinghua.edu.cn

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Vol. 130, Iss. 16 — 21 April 2023

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Images

Figure 1
(a) Schematic of our experimental setup where a ${}^{171}{Yb}^{+}$ ion is confined in a linear Paul trap. We use two 369 nm laser beams to perform cooling, pumping, and detection of an ion and to produce a population loss, respectively. A 935 nm laser beam is applied to repump the leakage back to the Doppler cooling cycle. We also shine a focused 435 nm narrow-band laser beam to drive the transition between the $| 0 ⟩$ level and the auxiliary level $| a ⟩$ . Microwaves are used to implement the Hermitian part of the models in Eqs. (1) and (6). (b) Schematic of laser and microwave configurations. Colored and black arrows specify the transitions induced by laser beams shown in (a) and microwave fields, respectively. Note that energy levels are plotted for visual clarity rather than based on a real energy scale. (c) Experimental sequences consisting of Doppler cooling, sideband cooling, initialization, time evolution, and detection.
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Figure 2
Schematics of the (a) unlink, (b) unknot, and (c) Hopf link topological structures formed by complex energies in the $[Re (E), Im (E), k]$ space by connecting the energies at $k = 0$ and $k = 2 π$ .
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Figure 3
Experimentally measured complex eigenenergies of the modified non-Hermitian Rice-Mele model in Eq. (1). (a1),(a2) Spectral lines of $N_{a}$ with respect to the detuning $δ$ obtained by experimental measurements (diamonds with error bars), fitting the experimental data (red lines) and numerical simulations (green lines) at $k = 2 π / 5$ [the corresponding extracted energies are highlighted by red circles in (b)–(d)]. In the numerical simulation, we set $N_{0} = 1$ . (b1),(b2) Complex energies in the complex-energy plane. Real (c1),(c2) and imaginary parts (d1),(d2) of complex energies as a function of the momentum parameter $k$ . The energies are extracted by fitting spectral lines (circles with error bars) or obtained by diagonalizing the system Hamiltonian (solid lines) based on the realized system parameters. Here, we realize a topologically nontrivial Hamiltonian with $J_{1} = 0.315 MHz$ , $J_{2} = 0.098 MHz$ , $J_{3} = 0.122 MHz$ , $m_{z} = 0.035 MHz$ , $γ = 0.092 MHz$ , and $Ω = 0.019 MHz$ in (a1)–(d1) and a trivial Hamiltonian with $J_{3} = 0$ and $m_{z} = 0.038 MHz$ in (a2)–(d2) [the other parameters are the same as those in (a1)–(d1)]. In both Figs. 3 and 4, experimental data are averaged over 20 experimental repetitions (each contains 1000 shots) with error bars estimated by the standard deviation of the 20 rounds of experiments.
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Figure 4
Experimentally measured complex energies with the unknot (a),(b) and Hopf link (c),(d) topological structures. Complex energies are plotted in the complex-energy plane in (a) and (c), and in the $[Re (E), Im (E), k]$ -space in (b) and (d). We realize the model in Eq. (1) with $J_{1} = 0.195 MHz$ , $J_{2} = 0.098 MHz$ , $J_{3} = 0.100 MHz$ , $m_{z} = 0.038 MHz$ , and $γ = 0.127 MHz$ , in (a) and (b) and the model in Eq. (6) with $m_{x} = 0.13 MHz$ , $g_{1} = 0.05 MHz$ , $g_{2} = 0.08 MHz$ , $γ_{0} = 0.15 MHz$ , and $g_{3} = 0.07 MHz$ in (c) and (d). In (b), we shift the second band (in red) along the $z$ axis by $2 π$ in order to display the full periodicity of the energy band.
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