Quantum breathing mode of trapped bosons and fermions at arbitrary coupling

Interacting particles in a harmonic trap are known to possess a radial collective oscillation—the breathing mode (BM). We show that a quantum system has two BMs and analyze their properties by exactly solving the time-dependent Schrödinger equation. We report that the frequency of one BM changes with system dimensionality, the particle spin and the strength of the pair interaction and propose a scheme that gives direct access to key properties of trapped particles, including their many-body effects.

Figure 1

(Color online) Time evolution of the one-particle potential energy $\Delta U_{\text{pot}}=⟨U_{\text{pot}}⟩\left(t\right)-U_{\text{pot},t=0}$ and $\Delta \left|x\right|=⟨\left|x\right|⟩\left(t\right)-\left|x_{t=0}\right|$ for $N=2$ particles in a 1D trap at $\lambda =1.0$, obtained by solving Eq. (1), symbols. Lines: fit, Eq. (4), with frequencies ${\omega }_r=1.901$ and ${\omega }_R=2.0$.

Figure 2

Resonance excitation (II) for $\lambda =1.0$ and three frequencies, ${\omega }_{\text{ext}}=\alpha {\omega }_r$ with $\alpha =0.95,0.995,1.0$. Left: antisymmetric initial state (top) and particle density (gray area). Arrows indicate the two breathing motions. Right: time evolution of the total energy for the three external frequencies (top) and the exciting pulse (bottom).

Figure 3

QBM frequencies ${\omega }_r$ and ${\omega }_R$ for $N=2$ particles in a 1D trap versus $\lambda$ from solution of Eq. (5) with a basis expansion method (gray line) and a cn solution of Eq. (1), $(+)$ using excitation method (I). Grey resonance spectra $\left(E_{\text{tot}}^{\infty }\right)$ are the result of method (II). Dashed lines: classical $\left(\lambda >30\right)$ and quantum $\left(\lambda <1\right)$ mean-field models, symbols: TDHF results for $N=2,3,4$, which coincide for $\lambda \le 1$.

Figure 4

(Color online) Frequency ${\omega }_r$ for two particles with a symmetric initial state in a 1D trap from numerical solution of the TDSE using a softened Coulomb potential. Crosses correspond to solution method (I) and lines to method (II), see Sec. 2B. For $\kappa =0$, ${\omega }_r$ is the same for symmetric and antisymmetric states. The result for the antisymmetric case is shown by the monotonic full (black) line. A finite $\kappa$ drastically influences ${\omega }_r$ for the symmetric case which is explained by the behavior of the relative wave function at $x=0$ (see inset).

Figure 5

(Color online) QBM frequency ${\omega }_r$, for $N=2$ in a 2D trap for symmetric and antisymmetric states from solution of the TDSE (Eq. (5)) with $\kappa =0$. For comparison, also the 1D result and the two fits $w_r^{fit}\left(\lambda \right)$, Eq. (9), are shown by the (red) dotted lines.