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Generating Forbidden 10-Fold Symmetry Quasicrystals using an Optical System

Mittal, Jahnavee; Corcovilos, Theodore A.Department of Physics, Duquesne University; Pittsburgh Quantum Institute (pqi.org)

INTRODUCTION

ACKNOWLEDGEMENTS

Funding for this project and support for Jahnavee Mittal was provided by the DuquesnePhysics Department.

REFERENCES

Quasicrystals are non-periodic arrangements of atoms, first discovered in the 1980s,that possess no translational symmetry but still maintain long-range order. Theempirical mechanical, thermal, and electronic properties of quasicrystals matchneither those of true crystals nor non amorphous materials, making these propertiesdifficult to predict theoretically. We will study the quantum mechanical properties ofquasicrystals by building analogs to them using atoms at nanokelvin temperatures inan optical potential equal to that of a quasicrystal. We numerically simulated aquasicrystal with 10-fold symmetry built by the interference of five nearly co-propagating laser beams using Fresnel propagation software written in Python andconstructed a simple optical setup to demonstrate the potential. The quasicrystalstructure of the laser interference pattern is experimentally verified using the Fouriertransform of a photograph of the pattern, which clearly shows “forbidden” 10-foldrotational symmetry.

In order to physically generate a quasicrystal with 10-fold symmetry, a simpleoptical system was built using a Helium-Neon laser with a wavelength ofapproximately 633 nm. Utilizing kinematic mirrors allows for the beam of the laser tobe directed into a telescope, used to collimate the beam. Once the beam has beencollimated, another kinematic mirror will reflect the beam perpendicular to theincoming beam, which will then propagate 0.5 m to an optical mask in front of afocusing lens. The optical mask will then dissect the beam into five individual beamswhich will then be focused by a lens (focal length of 0.5 m). This focused beam willthen be collected by an optical camera (Fig. 3).

Python was used to extract the k- vectors from the experimental image. By takingthe Fast Fourier Transform (FFT) of the k- vectors from the image, symmetry, orstructural integrity of the crystal, can be verified. After taking the FFT of the originalimage, two rings containing 10 high intensity spots (FFT peaks), were recovered.Using the distance formula, the separation of the twenty points from the center of theFFT was calculated, then compared to the predicted FFT peaks.

Conclusion

CONTACT INFORMATION

Ted Corcovilos, Department of Physics, Duquesne University, 600 Forbes Ave. 317 Fisher Hall,Pittsburgh, PA 15282. Email: corcovilost@duq.edu Ph: (412) 396-5973

Quasicrystals have existed for many centuries in art and architecture due to theiraesthetic appearances, but have recently been discovered in the world of materialsystems. First detected in 1982, D. Shechtman’s notable work with synthesized alloys(Fig. 1) displayed the non-periodic and translational periodicity presented inquasicrystals. These unique structures are solids, usually ternary metallic alloys thatcan lead to noteworthy geometrical properties like 5-, 8-, 10-, or 12- fold rotationalsymmetry.

This chimera of materials has unusual, but useful mechanical properties such as,high temperature ductility, extreme hardness, and low coefficients of friction, makingthe quasicrystals much more advantageous than previously thought. Our goal for thisproject is to implement an uncommon method of generating non-periodic quasicrystalpotentials with 5-fold rotational symmetry, through the use of five interfering, nearlyco-propagating laser beams.

The interference patters are generated by the principle of superposition- theaddition of two waves with the same wavelength and polarization, to form a standingwave. In other words, the electric field amplitudes of the beams add. These waves,consisting of constant frequency and amplitudes, propagate along an infinitely longstraight line and are normal to the phase of the velocity vector. Therefore, we can usethe plane-wave approximation when determining the optical potential of the system(Fig. 2).

A pair of interfering beams generates a one-dimensional lattice, three beamsgenerates triangular/ hexagonal lattices, four generates square/rhombic lattices, butonce we have five interfering beams, we venture into the world of quasicrystals.Unlike the superposition of two plane waves, the superposition of 5 plane wavesrequires the use of Poynting vectors (time-averaged squared) to recover an image ofthe quasicrystal.

BACKGROUND

Obtained from our simple optical setup, the generated quasicrystal image(Fig. 4) and its obviously symmetric Fourier Transform (Fig. 5) provide a “proof-of-concept demonstration,” previously thought to be uncommon. Although there is aslight shift in the experimental Fourier Transform (caused by the tilt in the opticalcamera), overall symmetry is maintained, and two distinct circular rings are shown.

If an interference pattern can be generated using the principle of superposition, then a quasicrystal lattice can be generated from the interference of 5 nearly co-propagating laser beams.

METHODS

HYPOTHESIS

Optical Setup

Results: Setup

The image obtained from the optical setup, clearly presents a 10-foldrotational symmetry, previously understood to be “forbidden.” Various calculationswere completed in order to confirm the validity of the experiment. To verify that thepredicted model and the experimental model were compatible, the locations of the k-vectors had to be retrieved from the predicted model. In the experimental image, theshifted intensity peaks were corrected. After the corrections were made, geometricalmethods were employed to extract the predicted k- vectors. In doing so, the locationsof the predicted k-vectors were extremely similar to the positions of theexperimentally obtained image. Therefore, confirming our initial hypothesis to betrue.

Results: Ultracold Atoms

Figure 1. D. Schechtman’s synthesized 𝐴𝑙14𝑀𝑛86alloy

Summer Undergraduate Research Symposium 2018

Figure 3. The optical layout used to achieve the

image of a quasicrystal.

Figure 5. Fast Fourier Transform of generated

quasicrystal image

Figure 4. The image of the optical lattice, or

quasicrystal, generated through 5 beam interference

Ultracold Atoms

Ultracold atoms located in optical lattices can be used to tune and customize aquantum system. Using confined optical potentials in quasicrystals, we will be able toconstruct materials from the ground up by utilizing ultracold atoms. Although wehave been successful in creating quasicrystals, the laser used to develop these imagesis too inefficient to be used in the ultracold experiment. Thus, a drawing (Fig. 6) of theoptical system will be used to construct the “real” setup.

The images that will be generated from this setup were numerically simulated inpython (poppy simulation) using the Fresnel approximation and diffraction effects.

Figure 6. The “real” layout of the optical system used

for generating the quasicrystal potential. Optical fibers (blue) will transport the beams to an integrated 6-way beam-splitter. 5 output fiber optics will lead towards a collimated lens. The intersecting beams (red), form the interference pattern which will be de-magnified 50x by a microscope.

The numerical simulation developed two images, the first being an opticallattice (Fig. 7), and the second being the FFT of the optical lattice (Fig. 8). Comparingthe results from the poppy simulation to those from the optical layout, we can verifythat both versions of the experiment yield similar results.

Figure 7. The image of the optical

lattice acquired from the poppy simulation

Figure 8. The FFT of the optical

lattice shown in Figure 7.

Future Work

Once the optical system utilizing the optical fibers has been completed, thenext phase in our exploration of quasicrystals is to cause the phenomenology ofphason modes. Phasons are a type of excitation present only in quasicrystals and canbe seen through the modulation of the phases of the 5 interfering beams. Adjustingthe phases of each beam will show a translation in the potential pattern. Unlikeperiodic lattices, this translation does not occur in physical space, but rather in ahigher-dimensional configuration space. Phason modes are therefore equivalent totranslating a slice, or nonlocal correlations between lattice sites as they slide in andout of the cut plane, through a five-dimensional hypercube of the configurationspace. These “phason modes” are not typically found in solid materials, thus makingtheir measurement pivotal.

1. D. Shechtman, I. Blech, D. Gratias, and J. Cahn. “Metallic phase with long-range orientational order and no translational symmetry”. Physical Review Letters 53.20 (1984), pp. 1951-1953. DOI: 10.1103/PhysRevLett.53.1951

Figure 2. Equation used to represent the

potential of a Poynting vector