Quantum breathing dynamics of ultracold bosons in one-dimensional harmonic traps: Unraveling the pathway from few- to many-body systems

Following a “bottom-up approach” in understanding many-particle effects and dynamics we provide a systematic ab initio study of the dependence of the breathing dynamics of ultracold bosons in a one-dimensional (1D) harmonic trap on the number of bosons ranging from few to many. To this end, we employ the multilayer multiconfiguration time-dependent Hartree method for bosons (ML-MCTDHB) which has been developed very recently [Krönke, Cao, Vendrell, and Schmelcher, New J. Phys. 15, 063018 (2013)]. The beating behavior for two bosons is found numerically and consequently explained by an analytical approach. Drawing on this, we show how to compute the complete breathing mode spectrum in this case. We examine how the two-mode breathing behavior of two bosons evolves to the single-frequency behavior of the many-particle limit when adding more particles. In the limit of many particles, we numerically study the dependence of the breathing mode frequency on both the interaction strength as well as on the particle number. We provide an estimate for the parameter region where the mean-field description provides a valid approximation.

Figure 1

Occurrence of a beating in the breathing mode as indicated by $\langle {\widehat{X}}^2\rangle$ (inset) its Fourier spectrum. The signal $S_t$ refers to this expectation value $S_t=\langle {\widehat{X}}^2\rangle$. Two orbitals have been provided, the breathing has been triggered by quenching the trap from $\Omega =1$ to $\Omega =\sqrt{0.9}$. [HO-U] refers to harmonic oscillator units with respect to the Hamiltonian before the quench.

Figure 2

Relative motion breathing mode frequency for two particles and $M=1,...,11$ orbitals as functions of $g$. The error owing to the finite spectral resolution is $\Delta \omega =0.013$.

Figure 3

Energy of the relative harmonic oscillator for two particles with an arbitrary interaction strength. The broken lines indicate the asymptotic values that are reached by the energies (solid lines) in the limit $g\to \infty$. Computed using the analytic solution in Ref. [21] (cf. figures therein).

Figure 4

Prediction for the full breathing spectrum at any interaction strength, up to $\approx$$6\Omega$. Here we have depicted all such modes from states with up to 20 quanta. The inset provides a detailed view on the lowest band. The frequencies are labeled by ${\omega }_{2i,2I,2j,2J}$ which refers to the frequency arising from $\langle {\Phi }_{2I}{\varphi }_{2i}\left|{\widehat{X}}^2\right|{\Phi }_{2J}{\varphi }_{2j}\rangle$.

Figure 5

Numerical hint on the existence of sidebands. The signal $S_t$ refers to the expectation value of $\widehat{X^2}$, $S_t=\langle {\widehat{X}}^2\rangle$. Note that in this case the trap has been quenched to $\Omega =\sqrt{0.3}$. The interaction strength has been set to $g=0.4$. The peak positions are ${\omega }_1/\Omega =1.916$, ${\omega }_2/\Omega =1.975$, ${\omega }_3/\Omega =2.000$. The position of the middle peak ${\omega }_2$ agrees with the first sideband ${\omega }_{2,2I,4,2I}$ in Fig. 4, coming from $\left.〈{\varphi }_2\right|{\widehat{r}}^2\left|{\varphi }_4\right.〉$.

Figure 6

Breathing frequency of the relative motion for 3 (red, plus markers), 4 (green, diamonds), and 5 (light blue, crosses) particles as a function of $g$. For all these cases, $M=9$ orbitals have been provided. For comparison we have also depicted the graph for two particles and $M=11$ (dark blue, squares).

Figure 7

Breathing mode frequency as a function of particle number for various interaction strengths ranging from $g=0.2$ (uppermost) to $g=0.8$ (lowermost solid line). Inset: Breathing mode frequency as a function of interaction strength for various particle numbers ranging from $N=10$ (uppermost) to $N=130$ (lowermost dashed line). Dashed dotted lines: breathing frequency within Thomas-Fermi approximation.

Figure 8

Contour plot for the frequency of the breathing mode ${\omega }_{\mathrm{br}}/\Omega$. The white dashed lines indicate lines of constant $g(N-1)$. The inset shows the dependence of the breathing frequency on the interaction strength for some characteristic particle numbers and compares these curves with the corresponding mean-field results. MF denotes mean-field, i.e., effective one-body results, whereas MB refers to many-body, i.e., converged ML-MCTDHB results.