We propose to detect quadrupole interactions of neutral ultra-cold atoms via their induced mean-field shift. We consider a Mott insulator state of spin-polarized atoms in a two-dimensional optical square lattice. The quadrupole moments of the atoms are aligned by an external magnetic field. As the alignment angle is varied, the mean-field shift shows a characteristic angular dependence, which constitutes the defining signature of the quadrupole interaction. For the $^{3}P_{2}$ states of Yb and Sr atoms, we find a frequency shift of the order of tens of Hertz, which can be realistically detected in experiment with current technology. We compare our results to the mean-field shift of a spin-polarized quasi-2D Fermi gas in continuum

We investigate the occurrence of rotons in a quadrupolar Bose-Einstein condensate confined to two dimensions. Depending on the particle density, the ratio of the contact and quadrupole-quadrupole interactions, and the alignment of the quadrupole moments with respect to the confinement plane, the dispersion relation features two or four point-like roton minima or one ring-shaped minimum. We map out the entire parameter space of the roton behavior and identify the instability regions. We propose to observe the exotic rotons by monitoring the characteristic density wave dynamics resulting from a short local perturbation, and discuss the possibilities to detect the predicted effects in state-ofthe-art experiments with ultracold homonuclear molecules.

Following the recent proposal to create quadrupolar gases [Bhongale et al., Phys. Rev. Lett. 110, 155301 (2013)], we investigate what quantum phases can be created in these systems in one dimension. We consider a geometry of two coupled one-dimensional systems, and derive the quantum phase diagram of ultra-cold fermionic atoms interacting via quadrupole-quadrupole interaction within a Tomonaga-Luttinger-liquid framework. We map out the phase diagram as a function of the distance between the two tubes and the angle between the direction of the tubes and the quadrupolar moments. The latter can be controlled by an external field. We show that there are two magic angles θ c B,1 and θ c B,2 between 0 to π/2, where the intratube quadrupolar interactions vanish and change signs. Adopting a pseudo-spin language with regards to the two 1D systems, the system undergoes a spin-gap transition and displays a zig-zag density pattern, above θ c B,2 and below θ c B,1 . Between the two magic angles, we show that polarized triplet superfluidity and a planar spin-density wave order compete with each other. The latter corresponds to a bond order solid in higher dimensions. We demonstrate that this order can be further stabilized by applying a commensurate periodic potential along the tubes.

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