One, Two, Three,…, Infinity

Barry R. Parker

Miguel Ángel García-March, Bruno Juliá-Díaz, G. E. Astrakharchik +2 more

We study a system of two bosons of one species and a third atom of a second species in a one-dimensional parabolic trap at zero temperature. We assume contact repulsive inter- and intraspecies interactions. By means of an exact diagonalization method we calculate the ground and excited states for the whole range of interactions. We use discrete group theory to classify the eigenstates according to the symmetry of the interaction potential. We also propose and validate analytical Ans\"atze gaining physical insight over the numerically obtained wave functions. We show that, for both approaches, it is crucial to take into account that the distinguishability of the third atom implies the absence of any restriction over the wave function when interchanging this boson with any of the other two. We find that there are degeneracies in the spectra in some limiting regimes, that is, when the interspecies and/or the intraspecies interactions tend to infinity. This is in contrast with the three-identical boson system, where no degeneracy occurs in these limits. We show that, when tuning both types of interactions through a protocol that keeps them equal while they are increased towards infinity, the systems's ground state resembles that of three indistinguishable bosons. Contrarily, the systems's ground state is different from that of three-identical bosons when both types of interactions are increased towards infinity through protocols that do not restrict them to be equal. We study the coherence and correlations of the system as the interactions are tuned through different protocols, which permit us to build up different correlations in the system and lead to different spatial distributions of the three atoms.

Anne Boutet de Monvel, Lech Zieliński

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences ( E m + ) m = 0 ∞ , ( E m - ) m = 0 ∞ satisfying a three-term asymptotic formula with the remainder estimate O ( m - 1 ln m ) when m tends to infinity. By analogy to the one-photon quantum Rabi model, the leading three terms of this asymptotic formula, describe a generalized rotating-wave approximation for large eigenvalues of the two-photon quantum Rabi model.

Philippe Grangier

BOYANG LIU, Jiangping Hu

A weakly interacting Bose–Fermi mixture model was investigated using Wisonian renormalization group (RG). This model includes one boson–boson interaction term and one boson–fermion interaction term. The scaling dimensions of the two interaction coupling constants were calculated as 2-D at tree level and the flow equations were derived at one-loop level. We find that in the flow equations the contributions from the fermion loops go to zero as the length scale approaches infinity. In three-dimensional case two fixed points are calculated. One is the Gaussian fixed point and the other one is Wilson–Fisher fixed point. We observe that the boson–fermion interaction decouples at the Wilson–Fisher fixed point. We also find that under RG transformation the boson–fermion interaction coupling constant runs to negative infinity with a small negative initial value, which indicates a boson–fermion pairing instability. Furthermore, the possibility of emergent supersymmetry in this model was discussed.

D. J. Reilly