Energy-level splitting for weakly interacting bosons in a harmonic trap

We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap. When the interactions are turned off, the energy levels are equidistant and highly degenerate. At linear order in the coupling parameter, these degenerate levels split, and we study the patterns of this splitting. It turns out that the problem is mathematically identical to diagonalizing the quantum resonant system of the two-dimensional Gross-Pitaevskii equation, whose classical counterpart has been studied previously in the mathematical literature on turbulence. Our purpose is to explore the implications of the symmetries and energy bounds of this resonant system, previously studied for the classical case, for the quantum level splitting. Simplifications in computing the splitting spectrum numerically result from exploiting the symmetries. The highest energy state emanating from each unperturbed level is explicitly described by our analytics. We furthermore discuss the energy-level spacing distributions in the spirit of quantum chaos theory. After separating the eigenvalues into blocks with respect to the known conservation laws, we observe the Wigner-Dyson statistics within specific large blocks, which leaves little room for further integrable structures in the problem beyond the symmetries that are already explicitly known.

Figure 1

Action of raising operators $Z_{+}, Z_{-}$, and $W$ in the $(M,E)$ plane. The patterns of consecutive moves generated by $Z_{+}, Z_{-}$, and $W$ are closely reminiscent of (though not exactly identical to) the so-called Motzkin walks. (Definitions and some recent applications to quantum spin chains can be found in [32, 33].)

Figure 2

Wignerian distribution of the unfolded level spacings in the $Z^{†}$ kernel of the (27,27) block of the LLL Hamiltonian. The green curve is the Wigner surmise (53), while the red curve is the Poisson distribution (52).

Figure 3

Wignerian distribution of the unfolded level spacings in the joint $(Z_{+}^{†},Z_{-}^{†},W^{†})$ kernel of the (7,22,16) block of the resonant Hamiltonian. The green curve is the Wigner surmise (53), while the red curve is the Poisson distribution (52).

Figure 4

(a) Crossover and (b) Poissonian level spacing distributions for two different blocks in the LLL sector: (a) $N=55$ and $E=30$ and (b) $N=7$ and $E=43$.

Figure 5

Wignerian distributions in two different blocks away from the LLL sector: (a) $N=10, E=16$, and $M=2$ and (b) $N=10, E=17$, and $M=3$.