† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11434011, 11522436, and 11774425), the National Key R&D Program of China (Grants No. 2018YFA0306501), the Beijing Natural Science Foundation, China (Grant No. Z180013), and the Research Funds of Renmin University of China (Grants Nos. 16XNLQ03 and 18XNLQ15).
We investigate the properties of Bose–Einstein condensates (BECs) in a two-dimensional quasi-periodic optical lattice (OL) with eightfold rotational symmetry by numerically solving the Gross–Pitaevskii equation. In a stationary external harmonic trapping potential, we first analyze the evolution of matter-wave interference pattern from periodic to quasi-periodic as the OL is changed continuously from four-fold periodic to eight-fold quasi-periodic. We also investigate the transport properties during this evolution for different interatomic interaction and lattice depth, and find that the BEC crosses over from ballistic diffusion to localization. Finally, we focus on the case of eightfold symmetric lattice and consider a global rotation imposed by the external trapping potential. The BEC shows vortex pattern with eightfold symmetry for slow rotation, becomes unstable for intermediate rotation, and exhibits annular solitons with approximate axial symmetry for fast rotation. These results can be readily demonstrated in experiments using the same configuration as in Phys. Rev. Lett. 122 110404 (2019).
Quasicrystals, which can exhibit long-range order without translational symmetry, have been used to study quantum states between the limits of periodic order and disorder.[1–4] According to the theorems of crystallography, the rotational symmetries of periodic lattices are highly restricted to a few possibilities, namely, two-, three-, four-, and six-fold symmetries. However, quasicrystals can show all the rotational symmetries forbidden to crystals, including five-, seven-, eight-, and higher-fold symmetries,[5–7] as can be retrieved from the experimental data of electron microscopy, spectroscopy, surface imaging methods, and diffraction pattern.[6]
The remarkable tunability and dynamical control of optical lattices (OLs) offer an opportunity to investigate quantum many-body states in periodic systems.[8,9] Ultracold atomic gases trapped in OLs can work as promising candidates to simulate versatile quantum phenomena in various fields of physics, such as condensed matter physics, high energy physics, and astrophysics. For example, the novel quantum transition from extended to localized states has been investigated in numerical simulation and observed in experimental exploration in the presence of OLs.[10–13] In addition, a carefully arranged configuration of laser beams can generate a quasi-periodic optical lattice with fascinating spatial patterns which are neither periodic as crystals (i.e., lack of translational symmetry) nor totally disordered (i.e., possession of long-range order).[14–17] Recently, a Bose–Einstein condensate (BEC) has been realized experimentally in a two-dimensional (2D) quasi-periodic OL with eightfold rotational symmetry.[18] This achievement paves the route to study the quasi-periodic systems which are expected to present hybrid features of crystals and amorphous matters.
Stimulated by the experimental achievement of BECs in eightfold symmetric OL,[18] we study in this work the effects of quasi-periodicity on various static and dynamic properties of the BECs. By numerically solving the Gross–Pitaevskii (GP) equation, we first discuss the matter-wave interference pattern of the BECs obtained from time-of-flight images, which reveals the eightfold symmetry and the self-similarity of the quasicrystal lattice. Next, we investigate the diffusion of particles upon released from the harmonic trap while the background lattice potential is present. A crossover from the ballistic diffusion to spatial localization is observed by continuously tuning the lattice configuration from a periodic square lattice to an eightfold symmetric quasi-periodic lattice. The effects of the interaction and lattice depth on the particle diffusion are also discussed. In addition, by applying a tilt of the lattice potential, we study the quasi-periodic Bloch oscillation and compare to the conventional Bloch oscillation induced by static force in a regular OL. Finally, we impose an external rotation on the system and numerically solve for the ground state of the BECs in the combined potential of OL and harmonic trap. When the rotation is slow in comparison to the harmonic trapping frequency, vortices can be generated to form a lattice structure with eightfold symmetry. When the rotating frequency is much greater than the trapping frequency, annular solitons can be formed along the radial direction with an approximate axial symmetry. For intermediate rotating speed, the system is dynamically unstable with no steady solution. These results extend our understanding on BECs through the crossover from ordered to disordered lattices, and can be readily implemented in experiments.
The remainder of this manuscript is organized as follows. In Section
The structure of a two-dimensional eightfold rotationally symmetric lattice can be created by four mutually incoherent one-dimensional (1D) optical lattices, which are yielded by retro-reflection of four single frequency laser beams, lying in the x–y plane and separated by an angle of π/4. The four lasers are all linearly polarized to be perpendicular to the plane as illustrated in Fig.
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![]() | Fig. 1. (a) Schematic illustration of laser arrangement to create an eightfold symmetric optical lattice. (b) A false-color plot of the quasi-periodic lattice potential with V0 = −2ER. |
We consider interacting BECs trapped in a combined potential of the lattice potential (
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The dynamical evolution of the BEC wave function ψ(r, t) in such a potential can be described by the time-dependent GP equation,[19–22] which takes the following dimensionless form:
In this section, we first analyze the case without global rotation, i.e., Ω = 0. One typical experiment to reveal the quasi-periodic lattice structure is the matter-wave interference pattern in the momentum space extracted from time-of-flight images. Figure
Another characteristic feature of quasicrystal is the transport properties subjected to an external potential gradient. In a periodic lattice, the coherent diffusion of BEC is frictionless as a result of the off-diagonal long-range order. On the other hand, a strong enough disorder would hinder the transport of the particles and lead to an insulating phase. Thus, one would expect an evolution from diffusive to localized states by continuously evolving the OL from periodic to quasi-periodic, although a quantitative description of the process could be very complicated in the presence of interaction.[31–34] To understand the transport properties, we assume that the BECs are initially prepared in a ground state ψ0 in the combined potential of OL and harmonic trap. At time t = 0, the harmonic trap is suddenly switched off while the OL is maintained as the background. The diffusion of the BECs is then described by applying the real TSSP method to numerically solve the GP Eq. (
Upon released from the harmonic trap, the BEC tends to expand in space as the potential and interaction energies are transformed into kinetic energy. In short period of time, the mean-square position of the particles along the x-direction can be approximated by a diffusive behavior
In the diffusive regime Γ ≲ 0.5, the interaction strength g and overall lattice depth
Before concluding this section, we investigate the transport of the BECs under a static and homogeneous force. In periodic OLs, a constant driving force induces oscillatory motion of the particles known as Bloch oscillation. In cold atoms, Bloch oscillation can be observed experimentally by applying a constant force via tilt of the optical lattice or gravity.[35–38] In the context of quasicrystals, numerical stimulations of 1D tilted Fibonacci lattice and 2D tilted Penrose tiling predict a quasi-periodic Bloch oscillation.[15,39] Here, we consider a tilting along the x-axis of the eightfold symmetric quasi-periodic OL with Γ = 1, and analyze the displacement of the particles upon time evolution. In order to eliminate the effect of the interaction which can induce damping and suppression of the Bloch oscillation, we assume that the BEC is non-interacting with g = 0, and the initial state is the wave function ψ0 of ground state for BECs in the combined potential of harmonic trap and OL. At t = 0, the harmonic trap is switched off and the tilting potential is applied, so that the system is subjected to a constant force along the x-axis. As shown in the inset of Fig.
Similar method can be implemented to create a rotating quasi-periodic OL to investigate the dynamical steady states of BECs therein. Here we obtain the stationary solutions of BECs in the combined potential of rotating OL and static harmonic trap by applying the backward Euler pseudospectral (BESP) method to compute the GP Eq. (
By imposing a relatively slow rotation with Ω = 0.05 and 0.1 as in Figs.
Next, we impose an external rotation along the z-axis and analyze the steady state of BECs in the presence of harmonic trap and quasi-periodic OL. This configuration attracts great attention as the underlying lattice potential can have nontrivial effect on the structure and stability of the BEC.[40,41] For example, a structural crossover from a triangular to a square lattice of vortices with increased potential of square OL has been observed in a rotating frame.[42] It was also predicted that fractional quantum Hall effect can present in a dilute gas trapped in a rotating lattice.[43]
When the external rotation becomes much faster (
We discuss the properties of Bose–Einstein condensates in an eightfold symmetric quasicrystal optical lattice by numerically solving the Gross–Pitaevskii equation. Starting from a ground state within a combined potential of optical lattice and harmonic trap, we investigate the matter-wave interference pattern, coherent diffusion of particles, and Bloch oscillation subjected to a constant force, focusing on the characteristic features induced by the quasi-periodic lattice potential. In particular, we find that the matter-wave interference pattern can reveal the underlying eightfold symmetry of the system, a crossover from diffusive to localized behavior can be experienced by continuously tuning the lattice configuration from periodic to quasi-periodic, and the Bloch oscillation also acquires a quasi-periodic nature. In addition, we also study the emergence and structure of vortices and solitons by imposing an overall rotation of the system. For slow rotation, vortices can be generated and form a lattice structure with eightfold symmetry. For fast rotation, however, annular solitons would emerge along the radial direction with approximate axial symmetry. In the intermediate regime, we find that the system is unstable with no steady state solution. Our results provide useful information to the understanding of the crossover regime between ordered and disordered geometries, and can be readily implemented in experiments.
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