Enhanced many-body effects in the excitation spectrum of a weakly interacting rotating Bose-Einstein condensate

The excitation spectrum of a highly condensed two-dimensional trapped Bose-Einstein condensate (BEC) is investigated within the rotating frame of reference. The rotation is used to transfer high-lying excited states to the low-energy spectrum of the BEC. We employ many-body linear-response theory and show that, once the rotation leads to a quantized vortex in the ground state, already the low-energy part of the excitation spectrum shows substantial many-body effects beyond the realm of mean-field theory. We demonstrate numerically that the many-body effects grow with the vorticity of the ground state, meaning that the rotation enhances them even for very weak repulsion. Furthermore, we explore the impact of the number of bosons $N$ in the condensate on a low-lying single-particle excitation, which is describable within mean-field theory. Our analysis shows deviations between the many-body and mean-field results which clearly persist when $N$ is increased up to the experimentally relevant regime, typically ranging from several thousand up to a million bosons in size. Implications are briefly discussed.

Figure 1

(a)–(e) Many-body ground-state densities in a rotating BEC of $N=10$ bosons with interaction parameter $\mathrm{\Lambda }=0.5$ for vorticities $l=0$ to $l=4$ [in respective panels (a)–(e)]. For $l>0$ the ground state is a single vortex whose size grows with $l$. The mean-field densities are alike (not shown). (f) Corresponding low-energy excitation spectra for the ground states in the upper panels. Thick black lines indicate mean-field results from Eq. (2), i.e., with $M=1$ (BdG), and red squares denote the many-body results from Eq. (3) with $M=7$ orbitals (LR-MCTDHB). The many-body spectra show a very rich structure which cannot be accounted for within the mean-field picture. All quantities are dimensionless.

Figure 2

Enhanced many-body effects by rotation. The excitation energies ${\omega }_{(+1)}$ are shown as a function of the rotation frequency $\mathrm{\Omega }$ for interaction parameter $\mathrm{\Lambda }=0.5$. Many-body results are calculated for different particle numbers $N$. Vertical dotted lines indicate the transition from ground-state vorticity $l$ to $l+1$ between two adjacent analyzed rotation frequencies (mean-field in black and many-body in red). It is seen that the many-body effects grow with growing vorticity. Even the assignment of the vorticity to the mean-field and many-body ground states does not match as the vorticity grows, at least for systems up to $N=1000$ bosons. The inset shows a magnified view for $\mathrm{\Omega }=3.8$. All quantities are dimensionless.

Figure 3

Gap size ${\mathrm{\Delta }}_N$ for particle numbers from $N=20$ to $N=1000$. Results are given relative to ${\mathrm{\Delta }}_{10}$, the dashed lines indicate exponential-fit curves. For $l=0,{\mathrm{\Delta }}_N$ is essentially zero (not shown). The gap varies with $N$ weaker as the vorticity $l$ grows. Small oscillations in the tails of the curves ${\mathrm{\Delta }}_N/{\mathrm{\Delta }}_{10}$ are due to the numerical accuracy of order $O\left({10}^{-5}\right)$. All quantities are dimensionless.

Figure 4

Excitation energies of the lowest excited states for a BEC with $N=100$ bosons, rotation frequency $\mathrm{\Omega }=3.8$ (i.e., ground-state vorticity $l=4)$, and interaction strength $\mathrm{\Lambda }=0.5$ for $64\times 64$ and $128\times 128$ grid points. Results are computed with LR-MCTDHB (3). The energies obtained for the two different grid sizes lie atop each other, showing numerical convergence for the smaller grid size. All quantities are dimensionless.

Figure 5

Low-energy spectrum for system parameters $N=10,\mathrm{\Omega }=3.8$, and $\mathrm{\Lambda }=0.5$. On the entire energy range, $M=5$ orbitals yield accurate results. For the very lowest excitation energies, in particular for ${\omega }_{(+1)}$ which is discussed in detail in the main text, already $M=3$ self-consistent orbitals lead to numerically converged values. All quantities are dimensionless.

Figure 6

Convergence of the position variance per particle in the $x$ direction for the ground states of a BEC with $N=10$ bosons and interaction strength $\mathrm{\Lambda }=0.5$ for different vorticities $l$. Results are computed for $M=5$, 7, and 9 self-consistent orbitals. For all values of $l$, the points for different numbers of orbitals lie atop each other, demonstrating numerical convergence for $M=5$. Inset: Magnified view for $l=4$. Already for $M=5$, the variance is converged to $O\left({10}^{-3}\right)$ of the exact value. All quantities are dimensionless.

Figure 7

Same as for the $l=4$ vortex in Fig. 3 but for different widths $\sigma$ of the Gaussian repulsion. As in the example of the main text $\left(\sigma =0.25\right)$, the relative gap ${\mathrm{\Delta }}_N/{\mathrm{\Delta }}_{10}$ for the smaller and larger values of $\sigma$ stays far away from zero, even for large BECs consisting of $N=1000$ bosons. All quantities are dimensionless.

Figure 8

Several states from the first two Landau levels in the case of the anharmonic trap utilized in this work, computed with the BdG equation. The angular momentum is denoted by $l_z$. Blue lines (for $l_z\ge 0$; to the right) denote the single-particle states (plus the ground state at $l_z=0)$ of the first level, red lines (for $l_z\le 0$; to the left) denote the single-particle states of the second level. The rotation frequency $\mathrm{\Omega }=1.9$ is close to the transition of the ground state to a vortex with vorticity $l=1$. The two Landau levels greatly overlap, showing that the LLL is not applicable here. All quantities are dimensionless.

Figure 9

Low-energy excitation spectrum for a rotating BEC in a harmonic trap. The number of bosons is $N=100$ and the ground-state vorticity is $l=1$. The repulsion is of the same Gaussian type as in the main text but with a width of $\sigma =0.01$. The interaction parameter is $\mathrm{\Lambda }=0.5$. Compared are the results for the BdG spectrum $\left(M=1\right)$ and the many-body spectrum with $M=3$ orbitals, leading to a depletion of $f\approx 1.65\%$. For $0\le \omega \lesssim 0.7$, the mean-field and many-body spectra deviate only slightly, whereas for higher excitation energies the differences become more pronounced. All quantities are dimensionless.