Implementing supersymmetric dynamics in ultracold-atom systems

Abstract

Supersymmetric systems derive their properties from conserved supercharges which form a supersymmetric algebra. These systems naturally factorize into two subsystems, which, when considered as individual systems, have essentially the same eigenenergies, and their eigenstates can be mapped onto each other. We first propose a one-dimensional ultracold atom setup to realize such a pair of supersymmetric systems. We propose a Mach-Zehnder interference experiment which we demonstrate for this system, and which can be realized with current technology. In this interferometer, a single atom wave packet that evolves in a superposition of the subsystems, gives an interference contrast that is sharply peaked if the subsystems form a supersymmetric pair. Secondly, we propose a two-dimensional setup that implements supersymmetric dynamics in a synthetic gauge field.Supersymmetry (SUSY) was originally introduced in particle physics beyond the Standard Model but its conceptual structure can be applied outside of high energy theory, giving rise to supersymmetric quantum mechanics [1][2][3]. Here, the algebraic structure of SUSY relates two Hamiltonians to be SUSY partner Hamiltonians. These share the same eigenspectrum, except for the ground state possibly, and the corresponding eigenstates can be mapped onto each other. By mapping a seemingly complicated Hamiltonian on its SUSY partner for which its diagonalization is known, an exact diagonalization can be constructed. This concept has been applied to a wide range of physical problems, e.g. to the hydrogen problem [4,5], solving the Fokker-Planck equation using imaginary time propagation [6,7], and for multisoliton solutions of the Korteweg-de Vries equation [8,9]. Recent applications have been reported in [10,12].The simplest supersymmetric algebra consists of a Hamiltonian H, and a supercharge operator Q. We assume that these operate on a 2-spinor ψ = (ψ (1) , ψ (2) ), note [29]. The supercharge Q is a conserved quantity of H, and fulfills the defining equation {Q, Q † } = H. We choose Q to have the form Q = Bσ + , where σ + is the Pauli spin raising matrix, and B is a scalar operator. With this, H takes the form H =Ĥ

Linear optical approach to supersymmetric dynamics*

“…Here, for the sake of better understanding, we consider the same scenario as reported in Ref. [17], where the superpotential reads…”

“…Recently, Lahrz et al proposed a scheme for implementing the supersymmetric dynamics in ultracold atom systems [17]. In their scheme, they considered verifying the translated version of the supersymmetric relationship between two partner Hamiltonianŝ B †Û (1) (t) =Û (2)…”

“…1. Interferometric approach to implement the supersysmmetric dynamics [17]. The system, which is initialized in ψi, is split into two parts: one evolves the time evolution Û (1) (t) firstly and thereafter an operator B † (green line); the other is diametrically opposite (the orange line).…”

Continuous supersymmetric transformations in optical waveguides

“…However, more complex structures could be designed by increasing the number of waveguides, using waveguides with different widths, using optical fibers [17] or combining continuous with discrete SUSY transformations. Finally, we would like to remark that such transformations are not restricted to optical systems and could be extended to the general formalism of SUSY quantum mechanics and applied to, for instance, trapping potentials modifying its shape in time instead of space [28].…”

Boundary Supersymmetry of (1+1)D Fermionic Symmetry-Protected Topological Phases

“…Originally introduced as a resolution to certain issues in fundamental and particle physics [18], supersymmetry (SUSY) has served as a powerful theoretical tool to uncover aspects of quantum field theories [19] as well as a way to elegantly prove mathematical theorems using techniques from physics [20]. Although signatures of space-time SUSY in particle interactions has not yet been detected in particle colliders [21], proposals exist for their emergence and detection in condensed matter [22][23][24][25][26][27][28][29] and cold-atomic systems [30][31][32][33]. To the best of our knowledge, all these proposals require some kind of fine tuning of parameters.…”