Topologically Protected Quantum Coherence in a Superatom

Wei Nie, Z. H. Peng, Franco Nori, and Yu-xi Liu

Phys. Rev. Lett. 124, 023603 – Published 16 January 2020

Abstract

Exploring the properties and applications of topological quantum states is essential to better understand topological matter. Here, we theoretically study a quasi-one-dimensional topological atom array. In the low-energy regime, the atom array is equivalent to a topological superatom. Driving the superatom in a cavity, we study the interaction between light and topological quantum states. We find that the edge states exhibit topology-protected quantum coherence, which can be characterized from the photon transmission. This quantum coherence helps us to find a superradiance-subradiance transition, and we also study its finite-size scaling behavior. The superradiance-subradiance transition also exists in symmetry-breaking systems. More importantly, it is shown that the quantum coherence of the subradiant edge state is robust to random noises, allowing the superatom to work as a topologically protected quantum memory. We suggest a relevant experiment with three-dimensional circuit QED. Our study may have applications in quantum computation and quantum optics based on topological edge states.

Received 24 January 2019
Revised 1 October 2019

DOI:https://doi.org/10.1103/PhysRevLett.124.023603

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Quantum control Superradiance & subradiance Symmetry protected topological states Topological phase transition

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Wei Nie^1,2, Z. H. Peng³, Franco Nori ^4,5, and Yu-xi Liu^1,2,*

¹Institute of Microelectronics, Tsinghua University, Beijing 100084, China
²Frontier Science Center for Quantum Information, Beijing, China
³Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China
⁴Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
⁵Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*yuxiliu@mail.tsinghua.edu.cn

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Issue

Vol. 124, Iss. 2 — 17 January 2020

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Images

Figure 1
(a) Schematic of a 3D circuit QED. The top layer contains a microwave transmission line resonator, which plays the role of cavity, coupled with an array of superconducting artificial atoms. On the bottom layer, superconducting coplanar waveguides are fabricated and coupled to the atoms on the top panel via interconnects in the middle dielectric layer (see Ref. [64] for details). (b) The atom array in (a) has internal interactions between neighboring unit cells. The atoms are coupled by resonators represented by LC circuits. Blue and orange dots denote atoms $A$ and $B$ in unit cells. (c) Wiring of the coupling circuits, so a 1D atom array can be obtained and coupled to the transmission line resonator, as shown in (a). (d) Optically addressing edge states of the topological atom array.
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Figure 2
(a) Lattice in the single-excitation subspace. Solid and dashed lines represent parallel and cross-couplings, respectively. (b) Topology of the lattice in the auxiliary space $[d_{y} (k), d_{z} (k)]$ . The winding number for the topological phase is nontrivial. (c) Energy spectrum of the tight-binding lattice in (a). There are large gaps between edge states and bulk states. As $δ$ changes across the critical point $δ_{c}$ , edge states undergo a transition to bulk states. (d) Wave functions of edge states at $δ = 0.1 δ_{c}$ . Here $n$ labels the positions of atoms in the array, and odd (even) number of $n$ corresponds to $| A_{(n + 1 / 2)} ⟩$ ( $| B_{(n / 2)} ⟩$ ). The parameters in (c) and (d) are $t_{c} = t_{p}$ and the number of unit cells $N = 20$ .
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Figure 3
(a) Transmission of light through the left (green-dashed) and right (red-dotted) edge states with $γ_{A B} = 0.99 γ$ . The black curve represents the transmission for both left and right edge states with $γ_{A B} = 0$ . Here we consider $δ = 0.1 δ_{c}$ . (b) Real (solid) and imaginary (dashed) parts of the rescaled susceptibility $χ$ by $κ$ for the left (green) and right (red) edge states with $γ_{A B} = 0.99 γ$ . $Im [χ]$ shows the edge-bulk transition in a finite lattice. (c,d) Variations of coherence when the system is changed from the topological to the non-topological regime. The effective decays of (c) and (d) in the unhybridized regime $δ < 0.15 δ_{c}$ correspond to the left and right edge states, respectively. The $γ_{A B}$ used in (c) is the same as that in (d). Other parameters for these figures: $t_{c} = t_{p}$ , $γ = 10 κ$ , $N = 20$ .
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Figure 4
(a) Superradiance-subradiance transition with different lattice lengths. The inset shows the finite-size scaling of SST for $γ_{A B} = 0.99 γ$ , where $δ_{m}$ is defined by $γ_{eff} (δ_{m}) = γ$ . (b) Effect of symmetry breaking resulting from waveguide-induced interactions between atoms, with $γ_{A B} = 0.97 γ$ and the interactions between atoms in the same unit cells $g_{A B} = 0.1 γ$ . The inset shows the finite-size scaling behavior of $\ln (δ_{m}^{'} - δ_{m})$ , where $δ_{m}^{'}$ indicates the SST in the symmetry-breaking case. (c) The effect of disorder in atomic frequencies, with $γ_{A B} = 0.99 γ$ and $N = 50$ . The arrow indicates the position $δ = δ_{m}$ , where SST takes place. (d) The difference between averaged $γ_{eff}$ with disorder ( $ε / t_{p} = 0.5$ ) and $γ_{eff}$ without disorder. Other parameters for these figures: $t_{c} = t_{p}$ , $γ = 10 κ$ .
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