Quantum quasicrystals, characterized by aperiodic order, exhibit unique low-energy excitations such as phononic, phasonic, and hybridized modes, making them a focal point for studying symmetry-driven dynamics in ultracold atomic systems. Understanding these excitations and phase transitions is crucial for advancing quantum simulation technologies. This study investigates the role of quasicrystal symmetry (octagonal, decagonal, and dodecagonal) in determining anisotropic excitation behaviors and phase stability, validating theoretical predictions through simulated ultracold atom experiments. Using the Gross-Pitaevskii equation and Bogoliubov-de Gennes linearization, we simulated Bose-Einstein condensates in quasiperiodic potentials, varying potential depth V0 from 0.5 to 4.0 arbitrary units. Excitation spectra were computed to assess anisotropy, and minimum frequencies at k=0 were analyzed to identify phase boundaries. Simulations were conducted using Python with NumPy, SciPy, and Matplotlib, visualizing dispersion relations and phase diagrams. Octagonal quasicrystals displayed anisotropic excitation spectra (frequencies 1.368–1.464), reflecting mode hybridization, while decagonal (1.413–1.453) and dodecagonal (1.696–1.736) systems showed more isotropic behaviors. Phase boundary analysis revealed persistent gaps (octagonal: 1.0–5.0, decagonal: 1.5–6.0, dodecagonal: 2.0–7.0), indicating no quasicrystal phase within the simulated V0 range, likely transitioning to superfluid or disordered states. Quasicrystal symmetry significantly influences excitation anisotropy and phase stability, with higher symmetries (dodecagonal) exhibiting larger gaps and reduced quasicrystal stability. Future studies should explore lower V0 ranges and incorporate temperature effects to locate the quasicrystal phase, enhancing experimental validation.
Quantum quasicrystals, characterized by aperiodic order, exhibit unique low-energy excitations such as phononic, phasonic, and hybridized modes, making them a focal point for studying symmetry-driven dynamics in ultracold atomic systems. Understanding these excitations and phase transitions is crucial for advancing quantum simulation technologies. This study investigates the role of quasicrystal symmetry (octagonal, decagonal, and dodecagonal) in determining anisotropic excitation behaviors and phase stability, validating theoretical predictions through simulated ultracold atom experiments. Using the Gross-Pitaevskii equation and Bogoliubov-de Gennes linearization, we simulated Bose-Einstein condensates in quasiperiodic potentials, varying potential depth V0 from 0.5 to 4.0 arbitrary units. Excitation spectra were computed to assess anisotropy, and minimum frequencies at k=0 were analyzed to identify phase boundaries. Simulations were conducted using Python with NumPy, SciPy, and Matplotlib, visualizing dispersion relations and phase diagrams. Octagonal quasicrystals displayed anisotropic excitation spectra (frequencies 1.368–1.464), reflecting mode hybridization, while decagonal (1.413–1.453) and dodecagonal (1.696–1.736) systems showed more isotropic behaviors. Phase boundary analysis revealed persistent gaps (octagonal: 1.0–5.0, decagonal: 1.5–6.0, dodecagonal: 2.0–7.0), indicating no quasicrystal phase within the simulated V0 range, likely transitioning to superfluid or disordered states. Quasicrystal symmetry significantly influences excitation anisotropy and phase stability, with higher symmetries (dodecagonal) exhibiting larger gaps and reduced quasicrystal stability. Future studies should explore lower V0 ranges and incorporate temperature effects to locate the quasicrystal phase, enhancing experimental validation.
A quantum quasicrystal was proposed to exist by mean-field, variational arguments in two-dimensional Rashba spin-orbit coupled BECs with dipolar interactions. Despite this remarkable prediction, there is little known about the superfluid character or stability of this quasicrystalline state against thermal and quantum fluctuations, whose importance is implied by the massive single-particle degeneracy of the Rashba Hamiltonian. Here, we apply field-theoretic numerical simulations based on a boson coherent state path integral representation to simulate the finite-temperature behavior of a periodic approximant of an octagonal quasicrystal. We find that the quasicrystal state maintains a 50% superfluid fraction down to low temperature, suggesting supersolid character. We compute a phase diagram that depicts a region of thermodynamic stability for the octagonal quasicrystal phase at low temperature. At intermediate temperature, the quasicrystal undergoes a first-order transition into a crystalline BEC with hexagonal symmetry. Our findings support a possible experimental realization of the octagonal quasicrystal in ^{164}Dy quantum gases below a temperature of 75 nK.
We investigate the necessary features of the pair interaction for the stabilization of self-assembled quantum quasicrystals in two-dimensional bosonic systems. Unlike the classical scenario, our results show that two-dimensional octagonal, decagonal, and dodecagonal aperiodic phases require a distinct number of properly tuned characteristic length scales for their stabilization. By using a mean-field spectral variational approach and Gross-Pitaevskii numerical calculations, we determine that the dodecagonal quasicrystal structure requires at least two characteristic length scales for its stabilization, while the decagonal and octagonal patterns need at least three and four length scales, respectively. The family of pair interaction potentials considered, albeit simple, is well justified in terms of a novel experimental platform based on laser-painted interactions in a cavity QED setup. Finally, we perform a structural characterization of the quasicrystal patterns obtained and show that these phases coexist with a finite superfluid fraction, forming what can be called a super quasicrystal phase.
Time quasicrystals (TQCs) represent a compelling extension of the concept of time crystals (TCs). While TCs break discrete time-translation symmetry by exhibiting a periodic response at a subharmonic of the driving frequency, TQCs display a more complex temporal order. They respond at multiple incommensurate frequencies, values that are not integer multiples of the fundamental driving frequency, resulting in quasiperiodic dynamics. In this work, we investigate the emergence of a TQC in a disordered quantum Ising chain subjected to a quasiperiodic transverse field. Using exact diagonalization, we find that the transverse magnetization exhibits quasiperiodic oscillations which persist over extended prethermal timescales before eventual decay. This indicates that the TQC exists as a long-lived, prethermal dynamical phase rather than a true equilibrium state. We further assess the robustness of this prethermal TQC against interactions and driving imperfections, confirming its stability under realistic experimental conditions. Finite-size analysis reveals that the prethermal TQC lifetime exhibits minimal dependence on system size. Additionally, we explore the emergence of TQCs in the same chain under symmetric sampling of exchange couplings. Our results demonstrate that the TQC phase is highly sensitive to both the choice of coupling distribution and the values of the driving frequencies. These findings highlight promising experimental prospects for realizing TQCs in cold atomic systems and quantum simulators, providing valuable insights into their stability, dynamical properties, and potential for exploring novel non-equilibrium quantum phases.
This thesis concerns the physics of ultracold bosons in quasicrystalline lattices, studied by means of quantum simulation using ultracold 39K atoms loaded to an eight-fold rotationally symmetric optical quasicrystal. Quasicrystalline and quasiperiodic systems, thanks to their long-range order and lack of periodicity, exhibit a wealth of interesting properties such as a dense and self-similar momentum space, and a localization transition to Anderson insulator or Bose glass states (in non-interacting or interacting cases respectively) reminiscent of that in disordered systems, whilst lacking the rare regions that typically exist in such systems. Here we will cover experiments into this system’s behaviour in both equilibrium and non-equilibrium settings. First, we focus on mapping out its phase diagram in both weakly and strongly interacting regimes. We observe consistency with numerical predictions for the superfluid-Bose glass phase transition using coherence measurements, as well as finding a region of roughly zero compressibility around where the numerics predicts the existence of a Mott insulator ground state. We also study which regimes of the phase diagram can be ramped through adiabatically, finding that entropy is created upon traversing the Bose glass phase irrespective of ramp duration, which we attribute to its localized nature. Next we look into the dynamics of the system after a quench of the lattice depth in the weakly interacting regime. Upon quenching from the superfluid regime to the Bose glass regime, we observe a quantum walk to higher order diffraction peaks in momentum space whose maximum order is unbounded by kinetic energy thanks to the dense, self-similar nature of the quasicrystal’s momentum space. Quenching in the opposite direction we observe instead a walk between sites in real space for short times by looking at the position (as opposed to momentum) space autocorrelation function of the cloud, finding an expansion rate in agreement with an intuitive estimate based on the Lieb-Robinson bounds of a periodic system. We also look into the timescales of these dynamics in the context of what we might expect from the underlying Hamiltonian, before finally investigating the loss of coherence exhibited by the system after a slower quench of the lattice depth from an initially superfluid cloud. We observe that coherence is lost sooner after the quench for deeper final lattices, with the loss becoming almost instantaneous for quenches that cross the localization transition across the whole cloud. After this we explore the system’s transport dynamics by means of a quench to an excited state at the same time as an additional confining potential is switched off. We observe that unlike in the case of a periodic square lattice, where ballistic expansion that scales with the tunnelling strength occurs in the non-interacting case before becoming slower upon the introduction of interactions thanks to diffusion, in the quasicrystal the transport rate no longer scales with tunnelling strength. Instead we observe a drop relative to the tunnelling rate by 2 − 3 orders of magnitude as the lattice depth is increased from 1Er to 5Er with little dependence on the interaction strength. We also note that the finite rate measured for the deeper lattices is potentially an overestimate since the finite loss rate of atoms makes mass transport indistinguishable from density-dependent losses at that point. Finally, we discuss use of a quantum gas magnification scheme with which to magnify the real space density distribution of our atoms such that it can be measured with single-site resolution. Some preliminary results are presented regarding how the density distribution changes with lattice depth in the weakly interacting regime, in which we observe that as lattice depth increases a higher proportion of atoms occupy deeper sites close to the centre of the quasicrystal’s configuration space, as we might expect.
The quantum dynamics of quasiperiodic systems display a rich variety of physical behaviors due to the combination of rotational symmetry that is mathematically forbidden in periodic systems, and long-range order despite the lack of translation symmetry. New experimental probes into these dynamics with a quantum simulator, consisting of ultracold atoms in an optical lattice potential, will yield new insights into the physics of quasiperiodic systems. This potential is imbued with the flexibility, tunability, and purity of the individual laser beams that constitute it, allowing for exquisite control over a rich system. Programmable dynamic control over the lattice beam phases opens up an even richer space of achievable systems via Floquet engineering. We thus describe an experimental scheme for creating a programmable, dynamic, two-dimensional (2D) quasiperiodic optical lattice with heavily suppressed phase noise. We observe suppression of phase noise for frequency components up to 5 kHz, and report phase noise suppression of over 70 dB over the DC-60 Hz frequency band. We further demonstrate a phase modulation bandwidth of 350 kHz. This scheme allows for full translational and phasonic control of the lattice, including changes to the rotational symmetry of the potential, at speeds exceeding the lattice recoil velocity, which paves a path towards direct observation and control of quantum dynamics in quasicrystals.
Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth of intriguing physical phenomena, such as the inheritance of topological properties from higher dimensions, self-similarity, and the presence of non-trivial structure on all scales. The concept of aperiodic order has been extensively studied in mathematics and geometry, exemplified by the celebrated Penrose tiling. However, the understanding of physical quasicrystals (the vast majority of them are intermetallic compounds) is still incomplete owing to their complexity, regarding both growth processes and stability. Ultracold atoms in optical lattices offer an ideal, yet untested environment for investigating quasicrystals. Optical lattices, i.e. standing waves of light, allow the defect-free formation of a large variety of potential landscapes, including quasiperiodic geometries. In recent years, optical lattices have become one of the most successful tools in the large-scale quantum simulation of condensed-matter problems. This study presents the first experimental realisation of a two-dimensional quasicrystalline potential for ultracold atoms, based on an eightfold symmetric optical lattice. It is aimed at bringing together the fields of ultracold atoms and quasicrystals – and the more general concept of aperiodic order. The first part of this thesis introduces the theoretical aspects of aperiodic order and quasicrystalline structure. The second part comprises a detailed account of the newly designed apparatus that has been used to produce quantum-degenerate gases in quasicrystalline lattices. The third and final part summarises the matter-wave diffraction experiments that have been performed in various lattice geometries. These include one- and two-dimensional simple cubic lattices, one-dimensional quasiperiodic lattices, as well as two-dimensional quasicrystalline lattices. The striking self-similarity of this quasicrystalline structure has been directly observed, in close analogy to Shechtman’s very first discovery of quasicrystals using electron diffraction. In addition, an in-depth study of the diffraction dynamics reveals the fundamental differences between periodic and quasicrystalline lattices, in excellent agreement with ab initio theory. The diffraction dynamics on short timescales constitutes a continuous-time quantum walk on a homogeneous four-dimensional tight-binding lattice. On the one hand, these measurements establish a novel experimental platform for investigating quasicrystals proper. On the other hand, ultracold atoms in quasicrystalline optical lattices are worth studying in their own right: Possible avenues include the observation many-body localisation and Bose glasses, as well as the creation of topologically non-trivial systems in higher dimensions.