Two-dimensional Thouless pumping in time-space crystalline structures

Y. Braver, C.-h. Fan, G. Žlabys, E. Anisimovas, and K. Sacha

Phys. Rev. B 106, 144301 – Published 3 October 2022

Abstract

Dynamics of a particle in a resonantly driven quantum well can be interpreted as that of a particle in a crystal-like structure, with the time playing the role of the coordinate. By introducing an adiabatically varied phase in the driving protocol, we demonstrate a realization of the Thouless pumping in such a time crystalline structure. Next, we extend the analysis beyond a single quantum well by considering a driven one-dimensional optical lattice, thereby engineering a 2D time-space crystalline structure. Such a setup allows us to explore adiabatic pumping in the spatial and the temporal dimensions separately, as well as to simulate simultaneous time-space pumping.

Received 5 July 2022
Revised 12 September 2022
Accepted 24 September 2022

DOI:https://doi.org/10.1103/PhysRevB.106.144301

Physics Subject Headings (PhySH)

Cold gases in optical lattices Crystal phenomena

Bose-Einstein condensates Ultracold gases

Wannier function methods

General PhysicsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Y. Braver ^1,*, C.-h. Fan ^2,3,*, G. Žlabys ^1,4,*, E. Anisimovas ¹, and K. Sacha²

¹Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, LT-10257 Vilnius, Lithuania
²Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland
³School of Physics and Optoelectronics, Xiangtan University, Hunan 411105, China
⁴Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan

^*These authors contributed equally to this work.

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Vol. 106, Iss. 14 — 1 October 2022

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Images

Figure 1
Temporal adiabatic pumping in a 2D time-space crystal with $s = 2$ temporal cells and only one ( $N = 1$ ) spatial cell that consists of two spatial sites. The following values of parameters were used: $V_{S} = 7640$ , $V_{L} = 2$ , $ω = 410$ , $s = 2$ , $λ_{S} = 100$ , $λ_{L} = 40$ , and $φ_{x} = 0$ . (a) Quasienergy levels $ɛ_{β}$ (see text for level numbering convention) of the Floquet Hamiltonian $\hat{H}$ vs the adiabatic phase $φ_{t}$ . (b) The Wannier functions $| w_{α} {(x, t) |}^{2}$ at $φ_{t} = 0$ , represented by black regions. The shaded areas (green and pink) indicate the extent of the temporal cells ( $γ = 1, 2$ ). (c) Wannier functions $| w_{α} (x_{0}, t) |^{2}$ (where $x_{0} = 0.3 π$ ) at $φ_{t} = 0$ and $φ_{t} = π / 2$ . (d) Change of the Wannier functions $| w_{α} (x_{0}, t) |^{2}$ as $φ_{t}$ scans across a complete cycle of length $2 π$ . The green- and pink-shaded areas indicate the extent of the two temporal lattice cells ( $γ = 1, 2$ ).
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Figure 2
Spatial adiabatic pumping in a 2D time-space crystal with $s = 2$ temporal cells and $N = 2$ spatial cells.. The same values of parameters were used as in Fig. 1 except that $φ_{t} = 0$ , while $φ_{x}$ is varied. (a) Quasienergy levels $ɛ_{j, β}$ of the Floquet Hamiltonian $\hat{H}$ vs the adiabatic phase $φ_{x}$ . (b) Wannier functions $| w_{i, 1} (x, t_{0}) |^{2}$ (where $t_{0} = π / 2 Ω$ ) at $φ_{x} = 0$ and $φ_{x} = π / 4$ . (c) Changes of the Wannier functions $| w_{i, 1} (x, t_{0}) |^{2}$ as $φ_{x}$ scans across a complete cycle of length $π$ . The cyan- and green-shaded areas indicate the extent of the two spatial lattice cells ( $k = 1, 2$ ), each consisting of two sites separated by unshaded regions.
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Figure 3
Simultaneous temporal and spatial adiabatic pumping in a 2D time-space crystal. The same values of parameters were used as in Fig. 1 except that both $φ_{t}$ and $φ_{x}$ are varied. (a) Quasienergy levels $ɛ_{j, β}$ of the Floquet Hamiltonian $\hat{H}$ vs the adiabatic phase $φ_{t} = 2 φ_{x}$ . [(b)–(f)] The Wannier functions at $φ_{t} = 2 φ_{x} = 0, π / 2, π, 3 π / 2, 2 π$ . The probability densities $| w_{i, α} {(x, t) |}^{2}$ are represented by black regions, while the shaded areas indicate the extent of the spatial ( $k = 1, 2$ ) and temporal ( $γ = 1, 2$ ) cells. In top-left panels in [(b)–(f)], $x_{0}^{(1)}$ and $x_{0}^{(2)}$ indicate the locations of two detectors, see text.
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Figure 4
Phase-space maps of the particle motion governed by the secular effective Hamiltonian (B10) (left panel) and the exact Hamiltonian (B3) (right panel). The yellow line in the left panel indicates the working point $I_{s} = 24.3$ . The map in the right panel was generated by numerically integrating the classical equations of motion resulting from the exact Hamiltonian (B3) and registering particle's position $(θ, I)$ stroboscopically at the intervals of the lattice driving period. The same values of parameters were used as in Fig. 1.
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Figure 5
Adiabatic pumping in a two-cell time crystal simulated based on the quasiclassical approach. The same values of parameters were used as in Fig. 1. (a) Energy levels $E_{β}$ of quantized effective Hamiltonian (B11) vs the adiabatic phase $φ_{t}$ . (b) Wannier functions $| w_{α} {(Θ) |}^{2}$ superimposed on the effective potential (blue curves) $U_{eff} = H_{eff} - P^{2} / 2 M$ at $φ_{t} = 0$ and $φ_{t} = π / 2$ . The scale of the $y$ axis relates to the potential. The Wannier functions are depicted in arbitrary units; they are positioned such that the flat tails mark the corresponding mean energies $〈 w_{α} | {\hat{H}}_{eff} | w_{α} 〉$ . (c) Changes of the Wannier functions $| w_{α} {(Θ) |}^{2}$ as $φ_{t}$ scans across a complete cycle of length $2 π$ . The shaded areas indicate the extent of the two temporal lattice cells ( $γ = 1, 2$ ).
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