We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross-Pitaevskii (GP) limit where the trap frequency ω, the temperature T and the particle number N are related by N ∼ (T /ω) 3 → ∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap.We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose-Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.2.4. The condensate energy and the interaction between the condensate and the thermal cloud .

We investigate the dynamics of two bosons trapped in an infinite one-dimensional optical lattice potential within the framework of the Bose-Hubbard model and derive an exact expression for the wave function at finite time. As initial condition we chose localized atoms that are separated by a distance of d lattice sites and carry a center-of-mass quasimomentum. An initially localized pair (d = 0) is found to be more stable as quantified by the pair probability (probability to find two atoms at the same lattice site) when the interaction and/or the center-of-mass quasimomentum is increased. For initially separated atoms (d = 0) there exists an optimal interaction strength for pair formation. Simple expressions for the wave function, the pair probability, and the optimal interaction strength for pair formation are computed in the limit of infinite time. Whereas the time-dependent wave function differs for values of the interaction strength that differ only by the sign, important observables such as the density and the pair probability do not. With a symmetry analysis this behavior is shown to extend to the N -particle level and to fermionic systems. Our results provide a complementary understanding of the recently observed [Winkler et al., Nature (London) 441, 853 (2006)] dynamical stability of atom pairs in a repulsively interacting lattice gas.

We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross-Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa 2̺ 2 − ̺ 2 0 . Here ̺ denotes the density of the system and ̺ 0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose-Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.

Recently it was shown that molecules rotating in superfluid helium can be described in terms of the angulon quasiparticles [Phys. Rev. Lett. 118, 095301 (2017)]. Here we demonstrate that in the experimentally realized regime the angulon can be seen as a point charge on a 2-sphere interacting with a gauge field of a non-abelian magnetic monopole. Unlike in several other settings, the gauge fields of the angulon problem emerge in the real coordinate space, as opposed to the momentum space or some effective parameter space. Furthermore, we find a topological transition associated with making the monopole abelian, which takes place in the vicinity of the previously reported angulon instabilities. These results pave the way for studying topological phenomena in experiments on molecules trapped in superfluid helium nanodroplets, as well as on other realizations of orbital impurity problems.In Maxwell's unification of electricity and magnetism, there was one piece missing that would make the electric and magnetic forces perfectly symmetric with respect to each otherthe magnetic monopoles. As demonstrated by Dirac [1], the existence of a single magnetic monopole would explain quantisation of electric charge everywhere in the Universe. Since Dirac's work, the existence of magnetic monopoles -as real elementary particles or effective quasiparticles -has preoccupied physicists working in several different fields. In high-energy physics, 't Hooft [2] and Polyakov [3] demonstrated the existence of non-abelian magnetic monopoles in the context of the unification of the fundamental interactions [4]. Despite the lack of experimental evidence for elementary monopoles in nature [5], collective phenomena exhibiting the behavior of magnetic monopoles have been predicted to emerge in condensed matter systems [6][7][8][9][10], and subsequently observed in experiment [11][12][13][14].As shown by Berry [15], magnetic monopoles can also emerge in an external parameter space of a simple quantum mechanical problem [16,17]. Moreover, the parameter space can be further generalized to coordinates of a particle interacting with another particle [18][19][20]. For example, Moody et al. [18] showed that the effective Hamiltonian describing nuclear rotation of a diatomic molecule can be rewritten as that of a charged particle interacting with a gauge field of a magnetic monopole. In follow-up studies the emerging gauge fields have been studied in various contexts, from different types of atomic problems [21][22][23][24] to spinor Bose-Fermi mixtures [25], to the fractional quantum Hall effect [26], to Bose-Einstein Condensates [8][9][10]. Finally, emergence of monopole-like gauge fields represents an important tool for computing topological invariants of quantum systems using Chern numbers [27], and thereby classifying the topology of the problem [8,[28][29][30][31]. Such a topological classification of quantum states is particularly relevant in the context of current research on topological states of matter [32][33][34][35][36][37].In this Lette...