Topological molecules and topological localization of a Rydberg electron on a classical orbit

Letter

Topological molecules and topological localization of a Rydberg electron on a classical orbit

Ali Emami Kopaei, Xuedong Tian, Krzysztof Giergiel, and Krzysztof Sacha

Phys. Rev. A 106, L031301 – Published 1 September 2022

Abstract

It is common knowledge that atoms can form molecules if they attract each other. Here, we show that it is possible to create molecules where bound states of the atoms are not the result of attractive interactions but have the topological origin. That is, the bound states of the atoms correspond to the topologically protected edge states of a topological model. Such topological molecules can be realized if the interaction strength between ultracold atoms is properly modulated in time. A similar mechanism allows one to realize topologically protected localization of an electron on a classical orbit if a Rydberg atom is perturbed by a properly modulated microwave field.

Received 31 January 2022
Accepted 16 August 2022

DOI:https://doi.org/10.1103/PhysRevA.106.L031301

Physics Subject Headings (PhySH)

Atomic & molecular processes in external fields Bose-Einstein condensates Topological insulators

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Ali Emami Kopaei^1,*, Xuedong Tian^1,2,*, Krzysztof Giergiel^1,3, and Krzysztof Sacha ¹

¹Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland
²College of Physics Science and Technology, Guangxi Normal University, 541004 Guilin, China
³Optical Sciences Centre, Swinburne University of Technology - Hawthorn, Victoria 3122, Australia

^*These authors contributed equally to this work.

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Issue

Vol. 106, Iss. 3 — September 2022

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Images

Figure 1
Topological localization of a Rydberg electron. Panel (a) shows the spectrum of the quantized version of the classical Hamiltonian (2) (red dashed lines) and the corresponding energy levels of the quantum effective Hamiltonian (3) (solid blue lines) for $s = 4$ . For $V_{1} / V_{2} > 0$ , two edge states of the Hamiltonian (2) form with nearly degenerate energies visible in the middle of the spectrum. Energy values are presented in the atomic units multiplied by $n_{s}^{2}$ and the constant term $- 3 / 2$ is subtracted. Panel (b) presents probability densities of these two edge states for $V_{1} / V_{2} = 2$ —the edge is located around $Θ = 0$ (or equivalently $Θ = 2 π$ ) and the edge states localize close to it. Other eigenstates (bulk states) are delocalized along the entire range of $Θ$ [in (c) the fifth excited eigenstate of (2) is shown]. The parameters of the system are the following: the resonant principal quantum number $n_{s} = I_{s} = 800, n_{s}^{3} ω = 4, n_{s}^{4} F = 1.5 \times 10^{- 4}, n_{s}^{4} F_{1} = 1.258 \times 10^{- 3}, n_{s}^{4} λ = 1.172 \times 10^{- 5}$ , and $f_{k} = e^{i k ε} cos (k π / 21) {sinc}^{2} (k π / 14) k / J_{k}^{'} (k)$ with $ε = 5 \times 10^{- 3}$ for $| k | \leq 20$ and $f_{k} = 0$ for higher $| k |$ [29]. In (b) and (c), $n_{s}^{4} F_{2} = 1.93 \times 10^{- 3}$ . Quantities are in atomic units.
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Figure 2
Topological localization of a Rydberg electron. Probability densities of an eigenstate of (3) corresponding to one of the edge states identified with the help of (2). The densities are plotted in the laboratory frame and in the cylindrical coordinate for different moments of time, i.e., $ω t / s = 7 π / 4 + j π / 4$ where integer $j$ goes from 0 to 4 from left to right panels, respectively. The presented edge state reveals the wavepacket evolving on the elongated Kepler orbit with the period $s 2 π / ω$ and its localization is protected by topology. The sequence of the panels illustrates time evolution of the wavepacket during one half of the period. During the other half, the time evolution can be illustrated by the same panels but ordered from right to left. The parameters are the same as in Fig. 1. Quantities are in atomic units.
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Figure 3
Topological molecules. Panel (a) shows probability densities of two edge states in the moving frame. The densities are localized around the certain position $x$ , indicating that two atoms form bound states. Panel (b) presents signatures of the formation of the topological molecule in the laboratory frame. If the atoms are prepared in, e.g., the edge state drawn with the blue line in (a), the probability density for the measurement of the atoms at $x_{1} + x_{2} = π$ reveals a periodic appearance of the localized edge state. The results are obtained within the exact diagonalization of the Floquet Hamiltonian, but they are indistinguishable from the densities obtained with the help of the effective Hamiltonian (5). The parameters of the system are the following: $F_{1} / F_{2} = 2$ with $F_{2} = 10, ω = 10^{5}, λ = 35$ and the Fourier components of $f (t)$ are $f_{k} = e^{i k ε} cos (k π / 21) {sinc}^{2} (k π / 14)$ with $ε = 10^{- 4}$ for $| k | \leq 40$ and $f_{k} = 0$ for greater $| k |$ . Unit are specified following Eq. (4).
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