Hydrodynamic theory of Rotating Ultracold Bose Einstein Condensates in Supersolid Phase

Within mean field Gross-Pitaevskii framework, ultra cold atomic condensates with long range interaction is predicted to have a supersolid like ground state beyond a critical interaction strength. Such mean field supersolid like ground state has periodically modulated superfluid density which implies the coexistence of superfluid and crystalline order. Ultra cold atomic system in such mean field ground state can be subjected to artificial gauge field created either through rotation or by introducing space dependent coupling among hyperfine states of the atoms using Raman lasers. Starting from this Gross-Pitaevskii energy functional that describes such systems at zero temperature, we construct hydrodynamic theory to describe the low energy long wavelength excitations of such rotating supersolid of weakly interacting ultra cold atoms in two spatial dimensions for generic type of long range interaction. We treat the supersolidity in such system within the framework of well known two fluid approximation. Considering such system in the fast rotation limit where a vortex lattice in superfluid coexists with the supersolid lattice, we analytically obtain the dispersion relations of collective excitations around this equilibrium state. The dispersion relation gives the modes of the rotating supersolid which can be experimentally measured within the current technology. We point out that this can clearly identify such a ultra cold atomic supersolid phase in an unambiguous way.Comment: Revised Latex file with .pdf figures, Accepted for publication in Journal of Physics B : At. Mol. Op

Creating superfluid vortex rings in artificial magnetic fields

Artificial gauge fields are versatile tools that allow to influence the dynamics of ultracold atoms in Bose-Einstein condensates. Here we discuss a method of artificial gauge field generation stemming from the evanescent fields of the curved surface of an optical nanofibre. The exponential decay of the evanescent fields leads to large gradients in the generalized Rabi frequency and therefore to the presence of geometric vector and scalar potentials. By solving the Gross-Pitaevskii equation in the presence of the artificial gauge fields originating from the fundamental HE$_{11}$ mode of the fibre, we show that vortex rings can be created in a controlled manner. We also calculate the magnetic fields resulting from the higher order HE$_{21}$, TE$_{01}$, and TM$_{01}$ modes and compare them to the fundamental HE$_{11}$ mode.Comment: 8 pages, 6 figure

Constructing Truth in The Waste Land

This paper won a third place writing flag award in the critical/persuasive category. Prerna Sachdeva, writing for David Kornhaber’s E 343L class, “Modernism and Literature”.Kornhaber, DavidUndergraduate Studie

Proof of a Limited Version of Mao's Partition Rank Inequality using a Theta Function Identity

Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \cite{atkin}, who proved Dyson's conjecture. More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \cite{mao10} and for $M_2$ rank differences for partitions with no repeated odd parts modulo $6$ and $10$ \cite{maom2}. Alwaise et. al. proved four of Mao's conjectured inequalities \cite{swisher}, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.Comment: First draft. Comments are welcom

Cold atoms in rotating optical lattice with nearest neighbour interaction

Extended Bose Hubbard models with nearest neighbour interaction describe minimally the effect of long range interaction on ultra cold atoms in deep optical lattices. Rotation of such optical lattices subject such neutral cold atoms to the effect of an artificial magnetic field. The modification of the phase boundaries of the density wave and Mott Insulator phases due to this rotation are shown to be related to the edge spectrum of spinorial and scalar Harper equation. Corresponding profiles of the checkerboard vortex states with sublattice modulated superfluid order parameter near density wave phase boundary are calculated.Comment: Slightly revised, Accepted in Physical Review A for publicatio