We apply the Bose-Hubbard Hamiltonian to a three-well system and show analytically that coherent transport via adiabatic passage (CTAP) of N non-interacting particles across the chain is possible. We investigate the effect of detuning the middle well to recover CTAP when on-site interparticle interactions would otherwise disrupt the transport. The case of small interactions is restated using first-order perturbation theory to develop criteria for adibaticity that define the regime where CTAP is possible. Within this regime we investigate restricting the Hilbert space to the minimum necessary basis needed to demonstrate CTAP, which dramatically increases the number of particles that can be efficiently considered. Finally, we compare the results of the Bose-Hubbard model to a mean-field three-mode Gross-Pitaevskii analysis for the equivalent system.

Recently, it has been proposed that the adsorption transition for a single polymer in dilute solution, modeled by lattice walks in three dimensions, is not universal with respect to intermonomer interactions. Moreover, it has been conjectured that key critical exponents ϕ, measuring the growth of the contacts with the surface at the adsorption point, and 1/δ, which measures the finite-size shift of the critical temperature, are not the same. However, applying standard scaling arguments the two key critical exponents should rather be identical, hence pointing to a potential breakdown of these standard scaling arguments. Both of these conjectures are in contrast to the well-studied situation in two dimensions, where there are exact results from conformal field theory: these exponents are both accepted to be 1/2 and universal. We use the flatPERM algorithm to simulate self-avoiding walks and trails on the hexagonal, square, and simple cubic lattices up to length 1024 to investigate these claims. Walks can be seen as a repulsive limit of intermonomer interaction for trails, allowing us to probe the universality of adsorption. For each lattice model we analyze several thermodynamic properties to produce different methods of estimating the critical temperature and the key exponents. We test our methodology on the two-dimensional cases, and the resulting spread in values for ϕ and 1/δ indicates that there is a systematic error which can far exceed the statistical error usually reported. We further suggest a methodology for consistent estimation of the key adsorption exponents which gives ϕ=1/δ=0.484(4) in three dimensions. Hence, we conclude that in three dimensions these critical exponents indeed differ from the mean-field value of 1/2, as had previously been calculated, but cannot find evidence that they differ from each other. Importantly, we also find no substantive evidence of any nonuniversality in the polymer adsorption transition.

We present a coupled pair approach for studying few-body physics in harmonically trapped ultracold gases. The method is applied to a two-component Fermi system of N particles. A stochastically variational gaussian expansion method is applied, focusing on optimization of the two-body correlations present in the strongly interacting, or unitary, limit. The groundstate energy of the four-, six-and eight-body problem with equal spin populations is calculated with high accuracy and minimal computational effort. We also calculate the structural properties of these systems and discuss their implication for the many-body ultracold gas and other few-body calculations.

We study uniform 3-star polymers with one branch tethered to an attractive surface and another branch pulled by a force away from the surface. Each branch of the 3-star lattice is modelled as a self-avoiding walk on the simple cubic lattice with one endpoint of each branch joined at a common node. Recent theoretical work [1] found four phases for this system: free, fully adsorbed, ballistic and mixed. The mixed phase occurs between the ballistic and fully adsorbed phase. We investigate this system by using the flatPERM Monte Carlo algorithm with special restrictions on the endpoint moves to simulate 3-stars up to branch length 128. We provide numerical evidence of the four phases and in particular that the ballistic-mixed and adsorbed-mixed phase boundaries are first-order transitions. The position of the ballistic-mixed and adsorbed-mixed boundaries are found at the expected location in the asymptotic regime of large force and large surface-monomer interaction energy. These results indicate that the flatPERM algorithm is suitable for simulating star lattice polymers and opens up new avenues for numerical study of non-linear lattice polymers.

We analytically determine the properties of two interacting particles in a harmonic trap subject to a rotation or a uniform synthetic magnetic field, where the spherical symmetry of the relative Hamiltonian is preserved. Thermodynamic quantities such as the entropy and energy are calculated via the second order quantum cluster expansion. We find that in the strongly interacting regime the energy is universal, however the entropy changes as a function of the rotation or synthetic magnetic field strength.PACS numbers: 03.75.Hh, 03.75.Ss, Over the last few years ultracold degenerate gases have attracted much interest due to their controllability and stability. Advances in tight confining harmonic traps and the use of magnetic fields and Feshbach resonances in controlling atomic collisions have made it possible to explore the BCS-BEC crossover [1][2][3]. Difficulties with developing a many-body theory for these systems in the strongly interacting regime using mean-field approximations have motivated the study of few-body problems as a means to gain insight into the many-body problem. Few-body systems with contact interactions are exactly solvable or numericaly tractable [4][5][6][7], particularly in the strongly interacting regime and have been experimentally studied in their own right [8]. The virial expansion of fewbody physics can be used to calculate the thermodynamics of many-body systems [9][10][11] and has been verified experimentally [12].In this work we address the problem of unitary gases subject to a rotation or synthetic magnetic field by solving the two-body problem and finding the virial expansion to second order. This enables us to show that entropy in the presence of a rotation or synthetic magnetic field is not universal, in contrast to the universal character of the total energy.A system subject to a rotation and one subject to a synthetic magnetic field have several similarities. In both systems angular momentum states and time-reversal symmetry are broken. Furthermore, both problems can be described by gauge-dependent Hamiltonians, making it convenient to consider the systems together and to draw comparisons between the two. In ultracold trapped gases the dominant contribution to the low energy behavior is from the two-particle s-wave interactions.To begin the analysis the rotating system is considered first. Specifically, the motion of two particles of mass m in a harmonic trap potential V trap (r) subject to a rotation Ω and a contact interaction potential V int (r 1 − r 2 ) are described by the Hamiltonianwhere r i and p i are the positions and momenta of each particle. Equation (1) can be decoupled in center of mass and relative coordinates, yieldingwhere R = (r 1 + r 2 )/ √ 2 and r = (r 1 − r 2 )/ √ 2 are the center of mass and relative coordinates. We consider the case where the rotation is about the z-axis with frequency Ω z so that Ω = (0, 0, Ω z ). The harmonic trapping potential is chosen to be axially symmetric with transverse and axial frequencies ω ⊥ and ω z , respectively. Using the ...