Statistical properties of spectral fluctuations ofNinteracting bosons in a harmonic trap

Abstract

Spectral fluctuations of a system of N weakly interacting bosons in an isotropic harmonic trap are studied, with the focus on the deviations from Poisson spectral statistics, typical of a quantum integrable systems. We have utilized the ideas formulated by Makino et al. [Phys. Rev. E 67, 066205 (2003)1063-651X10.1103/PhysRevE.67.066205] who have extended the approach of Berry and Robnik [J. Phys. A 17, 2413 (1984)JPHAC50305-447010.1088/0305-4470/17/12/013]. Earlier investigations of the Berry conjecture [Proc. R. Soc. London, Ser. A 356, 375 (1977)1364-502110.1098/rspa.1977.0140] of Poisson spectral statistics mainly considered quantum systems whose classical counterparts are integrable. However, the system of N weakly interacting bosons in the external trap has no classical counterpart. Also, it is a realistic and experimentally achievable system with close relation to Bose-Einstein condensation. Thus, a stringent analysis of the applicability of the Berry conjecture to this kind of systems is indeed required. We observe that for small boson number, the system is close to integrability and the nearest-neighbor level spacing distribution and the level number variance exhibit deviations from Poisson statistics similar to those of rational rectangular billiards.

Random k-Body Ensembles for Chaos and Thermalization in Isolated Systems

“…These results are consistent with the observation in Figure 6 a. Besides a varying number of levels, for a fixed number of low-lying levels by varying number of bosons, it is seen from Figure 6 b that for small boson systems, the spacing distribution shows a Shrielman peak which arises due to the large number of quasi-degenerate states (besides this, the NNSD and number variance exhibit deviations from Poisson statistics similar to those of a rectangular billiard [ 81 ]). Also, as seen from Figure 6 b, with an increasing boson number, the lower levels in the system change smoothly from Poisson to Wigner-like.…”

“…As already mentioned, experiments with ultra-cold atoms in which the interaction strength and other parameters are controlled provide the main motivation for these studies [ 61 , 62 , 63 , 64 , 65 , 80 ]. There is now a series of theoretical investigations by a Kolkata group (references [ 25 , 26 , 27 , 28 , 29 , 81 , 82 , 83 ]), and the results of these studies are briefly described in this section. These include (i) an analysis of level fluctuations using GOE and related ensembles in trapped interacting boson systems, diffuse van der Walls clusters and molecular resonances in erbium isotopes; and (ii) statistical relaxation in interaction quench dynamics of ultra-cold bosons and thermalization in isolated quantum systems.…”