Driven dipole oscillations and the lowest-energy excitations of strongly interacting lattice bosons in a harmonic trap

We show that the analysis of the time evolution of the occupation of site and momentum modes of harmonically trapped lattice hard-core bosons, under driven dipole oscillations, allows one to determine the energy of the lowest one-particle excitations of the system in equilibrium. The analytic solution of a single particle in the absence of a lattice is used to identify which function of those time-dependent observables is best fit for the analysis, as well as to relate the dynamic response of the system to its single-particle spectrum. In the presence of the lattice and of multiple particles, a much richer and informative dynamical response is observed under the drive.

Figure 1

(a) Lowest one-particle energy excitations (see text) as a function of the number of particles in the ground state. The inset shows the site occupations for $N_p=50$. (b) Expected frequencies for the largest response in the Fourier transform of $ln\left[n_{i=51}\left(t\right)\right]$ and $ln\left[m_{k=0}\left(t\right)\right]$, which follows from the prediction for one particle in the continuum while taking into account that the spectrum changes because of the presence of a lattice.

Figure 2

Fourier transform of the time evolution of the occupation of (a) the site at the center of the trap ($i=51$) and (b) the zero-momentum mode in a system with $N_p=5$ HCBs. The inset in each panel depicts the time evolution of the respective observable ($t$ is given in units of $J^{-1}$). Vertical dashed lines in the main panels indicate the most prominent frequencies highlighted in the Fourier transform of both observables. They correspond to ${\omega }_1=E_1-{\omega }^{\prime }$, ${\omega }_2=2{\omega }^{\prime }$, ${\omega }_3=E_2-2{\omega }^{\prime }$, ${\omega }_4=E_1+{\omega }^{\prime }$, ${\omega }_5=E_2$, ${\omega }_6=E_1+3{\omega }^{\prime }$, ${\omega }_7=E_3-{\omega }^{\prime }$, ${\omega }_8=E_2+2{\omega }^{\prime }$, and ${\omega }_9=E_3+\omega$.

Figure 3

Density plots of (a) ${\left|N\left(\omega \right)\right|}^2$ and (b) ${\left|M\left(\omega \right)\right|}^2$ as a function of $\omega /{\omega }_0$ and $N_p$. We also report, as open symbols, results for ${\omega }_1$ through ${\omega }_9$ (see caption in Fig. 2) ordered from bottom to top on the left side of each panel. Note that, following the convention in Fig. 1, results involving $E_1$ (${\omega }_{1,4,6}$) are depicted by circles, $E_2$ (${\omega }_{3,5,8}$) by squares, $E_3$ (${\omega }_{7,9}$) by diamonds, and $2{\omega }^{\prime }$ (${\omega }_2$) by crosses.