Discrete time quasicrystals

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Discrete time quasicrystals

Krzysztof Giergiel, Arkadiusz Kuroś, and Krzysztof Sacha

Phys. Rev. B 99, 220303(R) – Published 11 June 2019

Abstract

Between space crystals and amorphous materials there exists a third class of aperiodic structures which lack translational symmetry but reveal long-range order. They are dubbed quasicrystals and their formation, similar to the formation of space crystals, is related to spontaneous breaking of translational symmetry of underlying Hamiltonians. Here, we investigate spontaneous emergence of quasicrystals in periodically driven systems. We consider a quantum many-body system which is driven by a harmonically oscillating force and show that interactions between particles result in spontaneous self-reorganization of the motion of a quantum many-body system and in the formation of a quasicrystal structure in time.

Received 11 July 2018
Revised 3 December 2018

DOI:https://doi.org/10.1103/PhysRevB.99.220303

Physics Subject Headings (PhySH)

Bose gases Bose-Einstein condensates Cold atoms & matter waves Crystal phenomena Quasicrystalline structure

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Krzysztof Giergiel¹, Arkadiusz Kuroś¹, and Krzysztof Sacha^1,2

¹Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland
²Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland

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Vol. 99, Iss. 22 — 1 June 2019

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Images

Figure 1
Generation of the one-dimensional Fibonacci quasicrystal. The solid black line cuts the square lattice. The tangent of the angle that the line forms with the vertical axis is equal to the golden ratio. Consecutive cuts of the line with the vertical ( $L$ ) and horizontal ( $R$ ) lines of the lattice form the Fibonacci quasicrystal sequence, $L R L L R L R L \dots$ . Dashed green and dotted-dashed orange lines correspond to rational approximation of the golden ratio, 3/2 and 13/8, respectively. In the case of ultracold atoms bouncing between two mirrors which oscillate with frequency $ω$ (see the schematic plot in the inset where ${\vec{F}}_{g}$ denotes the gravitational force), the green and orange lines are related to spontaneous formation of finite fragments of the Fibonacci quasicrystal in the time domain with $Ω_{x} / Ω_{y} = s_{y} / s_{x} = 3 / 2$ and $13 / 8$ , respectively. That is, bounces of atoms off the left $L$ and right $R$ mirrors form a sequence of events that reproduces a fragment of the Fibonacci quasicrystal of length $s_{x} + s_{y}$ . Parameters $Ω_{x}$ and $Ω_{y}$ are frequencies of bouncing of atoms off the left and right mirrors, respectively, see text. The axes of the main figure can be considered as two independent time axes [54], $t_{x, y}$ , related to periodic motion along the $x$ and $y$ directions.
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Figure 2
Density of atoms bouncing between two orthogonal oscillating mirrors at $t = 2 π / 3 ω$ . The left ( $L$ ) mirror is located at $x = 0$ and the right ( $R$ ) mirror at $y = 0$ and the gravitational force ${\vec{F}}_{g}$ points in the $- (x + y)$ direction, see inset of Fig. 1. Left panel is related to the symmetry preserving state which evolves periodically with the driving period $2 π / ω$ —the left and right mirrors are visited by atoms alternately: $L R L R L R$ . The presented density consists of $s_{x} s_{y}$ localized Wannier-like wavepackets ( $s_{x} = 2, s_{y} = 3$ ). The trajectory the Wannier wavepackets are moving along is drawn in the panels. Right panel corresponds to a symmetry broken state where interactions between atoms result in spontaneous breaking of discrete time translation symmetry of the Hamiltonian and emergence of a quasicrystal structure in time. Atoms are visiting the left and right mirrors in an order that matches the sequence $L R L L R$ i.e. a finite fragment of the Fibonacci quasicristal is reproduced because the golden ratio gradient of the line in Fig. 1 is approximated by the rational number $Ω_{x} / Ω_{y} = s_{y} / s_{x} = 3 / 2$ . The parameters of the system are: $λ_{x} = 0.094, λ_{y} = 0.030, ω = 1.1, Δ ϕ = 2 π / 3, g_{0} N = 0$ (left) and $g_{0} N = - 0.022$ (right). The latter results in $U_{i i} N / J = - 81$ , with $J = 4.8 \times 10^{- 6}$ , in the Hamiltonian (2) that describes an effective $s_{x} \times s_{y}$ lattice. The results are obtained within the quantum secular approach [79].
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Figure 3
Scaled probabilities for the detection of atoms close to the left mirror $ρ_{L} (t)$ (blue lines) and close to the right mirror $ρ_{R} (t)$ (red lines), where $ρ_{L} (t) = \int d y {| ψ (x \approx 0, y, t) |}^{2}$ and there is an analogous expression for $ρ_{R} (t)$ . Top panels are related to symmetry preserving states while bottom panels to states where the discrete time translation symmetry of the Hamiltonian is spontaneously broken. Left panels correspond to $Ω_{x} / Ω_{y} = s_{y} / s_{x} = 3 / 2$ while in the right panels such ratios are equal 13/8. Symmetry preserving states form periodic sequences of the elementary cells $L$ and $R$ associated with the alternate appearance of maxima of $ρ_{L} (t)$ and $ρ_{R} (t)$ . In the symmetry broken case, bounces of atoms off the left and right mirrors form a sequence of events that reproduces a finite fragment (of length $s_{x} + s_{y}$ ) of the Fibonacci quasicrystal which is repeated in the time evolution of the system with the period $T = s_{x} s_{y} 2 π / ω$ . The results shown in the left panels correspond to the same parameters as in Fig. 2, while in the right panels: $λ_{x} = 0.087, λ_{y} = 0.026, ω = 1.77, Δ ϕ = π / 2, s_{y} = 13, s_{x} = 8, g_{0} N = 0$ (top right panel) and $g_{0} N = - 0.029$ (bottom right panel). The latter results in $U_{i i} N / J = - 80$ and $J = 2.3 \times 10^{- 6}$ in the Bose-Hubbard Hamiltonian that describes an effective $s_{x} \times s_{y}$ lattice.
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