Supersymmetric States of Quantum Droplets

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/341622507 Supersymmetric states of quantum droplets Technical Report · May 2020 DOI: 10.13140/RG.2.2.19602.68806 CITATIONS READS 0 49 2 authors: Sergio Manzetti Alexander Trunev Fjordforsk A/S Likalo LLC, Toronto, Canada 107 PUBLICATIONS 1,095 CITATIONS 99 PUBLICATIONS 107 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Development of method for in-situ rapid detection of mercury in water samples View project Reaction mechanisms of molecular mutagens in DNA-base adduct formation. View project All content following this page was uploaded by Sergio Manzetti on 25 May 2020. The user has requested enhancement of the downloaded file. Supersymmetric states of quantum droplets. Sergio Manzetti 1;2∗ and Alexander Trounev 2;3 1. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596, SE-75124 Uppsala, Sweden. 2. Fjordforsk A/S, Midtun, 6894 Vangsnes, Norway. Email: [email protected] 3. A & E Trounev IT Consulting, Toronto, Canada. May 25, 2020 Abstract 1 A dropleton, or quantum droplet made of bosonic particles interacting via the long range dipole-dipole interaction is a collective state found in rare-earth metal gases. The properties of dropleton are particularly interesting, as droplets composed of a higher number of atoms with large enough magnetic moment may assume a supersymmetric ground state. This effect results when laser electromagnetic fields affect the quantum gas, and reduce the number of available symmetries for the system of magnetic dipoles. Here, we devise a recently developed supersymmetric equation to identify the lowest energy state of dropleton composed of N atoms. We also inspect on the existence of preferred symmetries for various values of N and by modulating the strength and property of the external vector potential. We finally extend our analysis identify the fate of droplets and formed torii during the evolution of time. The results are impor- 1Please cite as: Manzetti Sergio and Trounev Alexander (2020). Supersymmetric states of quantum droplets. Technical Reports, 2024:1-21. Fjordforsk AS, www.fjordforsk.no 1 tant for advanced applications in computing and data-processing units based on dropletons, which may form the basis for CPU technologies of the future. 1 Introduction Quantum information processing and computing are dependent on a generation of processors and computational units which radically differs from the older generation. The emerging technologies in quantum computing devise the spin of electrons as ideals for generating computational signals for the execution of chains of command in the CPU [1, 2, 3, 4]. Alternative methods to generate computational signals can also be generated by using other for fermionic systems, for instance anyons [5] or quantum liquids, where the re-arrangement of quasi-/particle symmetry by an external magnetic field generates the computational signal. One particular quantum fluid of interest is represented by liquid-like quantized quasiparticle ensembles knows as dropletons, formed in metal lattices which, considered incompressible, assume often a lower energy state when composed of several quasiparticles compared to states for one single quasiparticle [6]. Dropletons are formed in rare-earth metal lattices under the influence of an imposed dipolar electromagnetic fields, as recently observed in 164Dy and erbium atoms [7, 6, 8, 9, 10, 11, 12, 13]. The appealing property of dropletons is their existence at ultra-low atom densities (108 times lower than in a helium droplet) which follows strict symmetry-reorganization by symmetry breaking [11]. Determining the symmetry of dropletons is experimentally chal- lenging, and models for the analysis have been developed [6, 14, 7] using the classical Bose-Einstein equation for Bose-Enstein condensates. Previously, we developed a Hamiltonian for the description of vorticity in quantized as well as continuum systems [15, 16, 17, 18], which of we also developed analytical solutions to [19]. Using this operator, we studied the impact of homogeneous and in- homogeneous magnetic fields on vortex and quasiparticle dynamics, and derived the preferred symmetries under the influence of magnetic fields [15, 16, 17]. As dropletons are formed by a balance of the mutual attractive and repulsive forces 2 that derive from a nuclear potential [6, 8, 9, 10, 11], theoretical studies with alternating (rotating, oscillating or translocating) magnetic fields are of fundamental importance for the study of dropletons. Furthermore, the formation of dropletons is extremely sensitive to physical parameters, such as temperature [13], and theoretical models that best describe ultrasensitive phenomena require the use of non-self adjoint operators [20, 21, 22, 23]. Non-self adjoint operators have the following advantages, a) they pertain a wide spectrum of eigenvalues and b) generate often real solutions in the lower or upper limit, by using various methods i.e. transformation or perturbation methods [20]. The non-self-adjoint supersymmetric operator we developed: 2 e2 H = − ~ r2 − A~2; (1) SUSY 2m c2 has no upper limit and yields analytical solutions in the lower limit [17, 19, 24] in the form of the supersymmetric wave-equation HSUSY Ψ = EΨ: (2) In this study, we devise this equation to study the formation of dropletons in an electron gas, where we particularly investigate the preferred symmetries assumed by the system composed of N particles, their respective energy, their wavefunction and the type of symmetry arising during phase-transitions. To describe the evolution of the wave function of a system of bosons, we use the nonlinear equation @ 1 2 @ @ 2 3 ~ 2 (3) @t = 2 r − iΩ(x @y − y @x ) − gj j − gqf j j + (A) Here A~ is a dimensionless vector potential and q; qqf ; Ω - parameters of the model describing the particle interaction and the angular velocity, respectively. Eq. (3) is different from the standard eGPE [7, 6], because we explicitly take into account the rotation of the system and describe the electromagnetic ~ interaction using the vector potential. For a system of magnetic dipoles di in an external field B~ , we have N ~ ~ X [di × ~ri] [B × ~r] A~ = + (4) r3 2 i=1 i 3 As is known, a system of magnetic dipoles in an external magnetic field acquires a magnetic moment Mi = χikBk: (5) On the other hand, if the body acquires a magnetic moment, then it begins to rotate at an angular velocity e Ω = g χ−1M : (6) l 2mc lk ik i here glk are gyromagnetic coefficients. This effect, first discovered by Einstein and de Haas [25], should also be observed in quantum systems. Consider the local model of the electromagnetic interaction of dipoles with two parameters g [M~ × ~r] [B~ × ~r] A~ = dd + : (7) 2 2 3=2 (r + r0) 2 Using (7), one can solve equation (2), find the eigenvalues and eigenfunctions. Then the solution of equation (2) is used as the initial data for equation (3). By these solutions of equation (3) we therefore aim to determine the number of quantum droplets, their stability and the physical parameters that determine their formation, existence and lifetime. 2 Numerical analysis and results 2.1 Droplet formation Consider a sphere of radius R = 4, put B~ = M~ = (0; 0; 1); r0 = 0:3; gdd = 1 and find the first 10 eigenvalues of equation (2) - see Figure 1. Figure 1 shows the eigenfunctions of (2), which yields ten symmetric fluid-like states with parallel and concentric amplitudes. The positive and negative eigenvalues give all wave-amplitudes which locate both at the radial boundaries of the system as well as in the center. Particular features are found for eigenvalues E=- 0.135926 and E=-0.29154 which show similarities to liquid droplets similar to the droplet simulated by devising Taylor expansions in water simulation models 4 [26]. When we study the other eigenstates of (2), we find a preferred C2 symmetry (belonging to the cyclic permutation group in group-theory) for eigenstates E=0.0989539 and E=0.0992089, E=0.245759 and E=0.24609, E-0.115858, E=- 0.117923 (Fig 1). The symmetry of the states of E=-0.135926, E=0.29154 and E=0.203948 are also C2 symmetries with a clear rotation by π about the z-axis. We find however no symmetry for E=0.205016. Figure 2 shows the system potential −A~2 at z = 0 and five eigenfunctions corresponding to the energy levels E = 0:0989539; 0:24609; −0:135926; −0:117923; 0:203948 (indicated above the figures). The potential is not infinite and vanishes at the origin (denoted by a red peak emerging from the center of the potential), forming a combina- tion of repulsion and attraction, as is observed in droplet-experiments [9]. The first eigenfunction (Fig 1) describes 12 droplets located on two circles parallel to plane (x; y), with an apparent icosahedral Ih-symmetry (Fig 2 top middle) similar to an atomic g-orbital. The ninth eigenfunction (Fig 1) describes 10 droplets located on a circle in plane (x; y) with a Ih symmetry as well (Fig 2, top left), also similar to an atomic orbital of type fundamental (f-). The fifth eigenfunction (Fig 1) describes 3 concentric torii (Fig 2, lower left), which has 2 similarities to the 3dy atomic orbital in the arrangement of the torii. The fourth and sixth eigenfunctions from figure 1 describes droplets in shapes similar to a cucumber (Fig 2, lower middle and right). Neither of these shapes have been observed before in neither atomic orbitals nor liquid /crystal shapes according to our knowledge, however both pertain C2 symmetries, as their parental eigenfunctions. 5 . Figure 1 The first ten eigenfunctions of (2) in a case of spherical region (with eigenvalues indicated above the figures) in plane (x; z).

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