Regularizing a Singular Special Lagrangian Variety

Suppose M1 and M2 are two special Lagrangian submanifolds of Rtn with boundary that intersect transversally at one point p. The set M1 cup M2 is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle condition (which always holds in dimension n=3). Then, M1 cup M2 is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds M(alpha) with boundary that converges to M1cup M2 in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of M1 cap M2 and then by perturbing this approximate solution until it becomes minimal and Lagrangia

Exploring the Conformal Constraint Equations

One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity. This problem can be avoided by means of a technique of H. Friedrich, which replaces the Einstein equations in the unphysical spacetime by an equivalent system of equations which is regular at the boundary. The initial value problem for these equations produces a system of constraint equations known as the conformal constraint equations. This work describes some of the properties of the conformal constraint equations and develops a perturbative method of generating solutions near flat space under certain simplifying assumption

Deformations of Minimal Lagrangian Submanifolds with Boundary

Let Lbe a special Lagrangian submanifold of a compact, Calabi-Yau manifold Mwith boundary lying on the symplectic, codimension 2 submanifold W. It is shown how deformations of L which keep the boundary of L confined to W can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near L with boundary on W is found to be finite dimensional and is parametrised over the space of harmonic 1-forms of L satisfying Neumann boundary conditions. The second is that if W is a symplectic, codimension 2 submanifold sufficiently near W then under suitable conditions, there exists a minimal Lagrangian submanifold L near L with boundary on

Exploring Interacting Quantum Many-Body Systems by Experimentally Creating Continuous Matrix Product States in Superconducting Circuits

Improving the understanding of strongly correlated quantum many body systems such as gases of interacting atoms or electrons is one of the most important challenges in modern condensed matter physics, materials research and chemistry. Enormous progress has been made in the past decades in developing both classical and quantum approaches to calculate, simulate and experimentally probe the properties of such systems. In this work we use a combination of classical and quantum methods to experimentally explore the properties of an interacting quantum gas by creating experimental realizations of continuous matrix product states - a class of states which has proven extremely powerful as a variational ansatz for numerical simulations. By systematically preparing and probing these states using a circuit quantum electrodynamics (cQED) system we experimentally determine a good approximation to the ground-state wave function of the Lieb-Liniger Hamiltonian, which describes an interacting Bose gas in one dimension. Since the simulated Hamiltonian is encoded in the measurement observable rather than the controlled quantum system, this approach has the potential to apply to exotic models involving multicomponent interacting fields. Our findings also hint at the possibility of experimentally exploring general properties of matrix product states and entanglement theory. The scheme presented here is applicable to a broad range of systems exploiting strong and tunable light-matter interactions.Comment: 11 pages, 9 figure

Lifetimes of ultralong-range Rydberg molecules in vibrational ground and excited state

Since their first experimental observation, ultralong-range Rydberg molecules consisting of a highly excited Rydberg atom and a ground state atom have attracted the interest in the field of ultracold chemistry. Especially the intriguing properties like size, polarizability and type of binding they inherit from the Rydberg atom are of interest. An open question in the field is the reduced lifetime of the molecules compared to the corresponding atomic Rydberg states. In this letter we present an experimental study on the lifetimes of the ^3\Sigma (5s-35s) molecule in its vibrational ground state and in an excited state. We show that the lifetimes depends on the density of ground state atoms and that this can be described in the frame of a classical scattering between the molecules and ground state atoms. We also find that the excited molecular state has an even more reduced lifetime compared to the ground state which can be attributed to an inward penetration of the bound atomic pair due to imperfect quantum reflection that takes place in the special shape of the molecular potential