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... interest in experiments of classical analogues, which accurately model the salient features of some quantum systems or other fundamental undulatory phenomena, was first raised by the seminal acoustic quasicrystal experiment of He and Maynard [1], where the analogy between both the sound propagation and Schro ̈ dinger equations [2–4] was invoked. Furthermore, the correspondence between the shallow water equation and the acoustic wave equation has also been demonstrated [5, 6] and experiments with liquid surface waves have been reported and have demonstrated abstract concepts, such as Bloch states, domain walls and band gaps in periodic systems [6, 7] and Bloch-like states in quasiperiodic systems [8], in a clear visual way. Such correspondences could be exploited to investigate and address formally similar quantum e ff ects as those observed in quantum corrals [9, 10] and in grain boundaries or simple surface steps [11]. Our main goal is to build up the hydrodynamic analogy of a double square quantum well. So we assembled an experimental set up consisting of a square-box vessel with a single square drilled at its bottom; both squares are concentric and the well diagonals are parallel to the box sides. When the vessel vibrated vertically with such amplitudes that the Faraday instability was prevented, the geometry produced three kinds of linear, or weakly nonlinear, patterns on the free surface of the liquid. The first pattern is a sort of bound state restricted to the surface area occupied by the immersed well that acts as a weak potential which bound standing waves. The second pattern is produced by the meniscus at the vessel walls [12] and it can come into the region of the immersed well depending on the liquid depth and the vibration frequency of the vessel. Finally, the last pattern arises from the interference between the meniscus wave of the vessel and the bound states of the well. The observed pattern depends on the liquid depth, h 1 , which plays the role of an order parameter by controlling the amplitudes of the bound standing states inside the boundary of the immersed well. Our main observation is summarized in figure 1a, which clearly shows the binding of the surface wave produced by the drilled well when the vibration amplitude is 60 m m. This pattern will be detailed below but the physics of its origin can be explained as follows. The bound states arise from an inertial hydrodynamic instability, balanced by the liquid surface tension [8] that grows over the region of the square well. The amplitudes of the bound states increase on increasing 1/ a 2 , where a 2 1⁄4 T/ g and a is the capillary length, T is the liquid surface tension, is the liquid density and g 1⁄4 g 0 Æ ! 2 is the e ff ective gravity; is the vibration amplitude of the vessel and ! is the vibration angular frequency of the vessel. On the contrary, the amplitude of the meniscus wave depends on the variation of the meniscus volume for each vessel oscillation and it grows accordingly with a 2 [12]. In our experiment is about 60 m m and the amplitude of the meniscus wave reaches a maximum at a vibration frequency of the vessel of 64 Hz. The frequency and the wavelength of the wave patterns are related by the well known dispersion relation of gravity-capillary waves [6–8]. To show that figure 1a is not a Faraday pattern, we present a snapshot of the system vibrating at about 70 m m in figure 1b, when the Faraday instability is actually triggered in the square well. Figure 1b shows a higher wavelength that matches with the corresponding period doubling related to the Faraday wave pattern. We chose a square vessel and the configuration of the orientated square well to verify that the immersed well confined wave states. Then we used a square methacrylate box with side L of 8 cm where a single square well with depth d of 2 mm and side l of 3.5 cm was drilled at its bottom. The bottom of the vessel was covered with a shallow ethanol layer of depth h 1 . The liquid depth over the well was then h 2 1⁄4 h 1 þ d . As already mentioned, the vibration amplitude of the vessel was 60 m m, below the threshold of Faraday instability at the frequencies of the experiment. The vessel vibrated vertically at a single frequency lying within the range 35–60 Hz. An opti- mum frequency was 50 Hz. Our experimental results can be separated into three cases according to the depth h 1 . Case I. For h 1 lower than 1 mm, the experiment shows two square lattices with a shift of 45 between them, namely the lattice of the immersed square well is separated from the square lattice of the vessel, as seen in figure 1a. The external- wave amplitudes are lower than the wave amplitudes within the well and, furthermore, the external-wave reflection at the well step [13–15] is strong. For very shallow liquid layers, waves are only present within the well at the vibration amplitudes of the experiment. Case II. At a vibration frequency of the vessel of 50 Hz and when h 1 is 1.2 mm, a quasicrystalline standing wave pattern appears inside the region of the immersed square well whereas the outer-wave pattern is a square network, as seen in figure 2a. The immersed square well acts as a weak potential and binds standing plane waves with eigenvectors parallel to the well sides. Nevertheless, it is trans- parent for the standing waves of the square box which tunnel the frame of the immersed well under the experimental conditions. Inside the region of the square box, the vessel eigenstates have eigenvectors parallel to the outer box sides. For the mentioned liquid depth and vibration frequency, A A 0 , where A and A 2 are the amplitudes of the meniscus and bound waves inside the immersed well, respectively. The interference of both standing patterns increases the symmetry in the well from square crystalline to octagonal quasicrystalline. According to the dispersion relation for gravity-capillary waves described elsewhere [6–8] the di ff erence between the external and internal wave numbers k 1 and k 2 is about 2% and the refraction bending of about 1.1 at the boundary of the central window is negligible. Furthermore, the reflection of the external wave at the well step [13–15] is also negligible with such parameters. On the other hand, slender outgoing evanescent waves are emitted at the boundary of the well and they play a role in the matching between patterns. Case III. When h 1 is increased, the ‘‘potential’’ of the immersed well is seen increasingly weaker by the system, and A 0 2 decreases accordingly. Under such conditions, transitional patterns appear gradually on the hydrodynamic window making the transition from a quasicrystalline form to a crystalline one. Figure 2b shows a transitional pattern corresponding to a liquid depth h 1 of 1.5 mm and an excitation frequency of 50 Hz. The Fourier transforms of the experimental patterns are calculated, as seen in figure 2a and b, according to crystallographic techniques of image processing described elsewhere [8], and they are used to depict figure 3, where the fast decay of j A 0 2 / A 2 j 2 is shown on increasing h 1 . Figure 4 shows four frames of the numerical simulation of the quasicrystal–crystal transition on the hydrodynamic central square window. We have also performed the numerical and experimental complete movies of the corresponding phase transition. They mimic the experiment when some drops of liquid are added to the vessel in order to generate the phase transition in the central well from the octagonal pattern to the square one. To test the feasibility of a quantum scenario analogous to our experimental results, we numerically studied the quantum confinement of a double square well by using a tight-binding Hamiltonian in a L Â L cluster of the square lattice with a single atomic orbital per lattice site ...

Context 2

... Fourier transforms of the experimental patterns are calculated, as seen in figure 2a and b, according to crystallographic techniques of image processing described elsewhere [8], and they are used to depict figure 3, where the fast decay of jA 0 2 /A 2 j 2 is shown on increasing h 1 . Figure 4 shows four frames of the numerical simulation of the quasicrystal-crystal transition on the hydrodynamic central square window. We have also performed the numerical and experimental complete movies of the corresponding phase transition. ...