Amplitude Mode in the Quantum Phase Model

We derive the collective low-energy excitations of the quantum phase model of interacting lattice bosons within the superfluid state using a dynamical variational approach. We recover the well-known sound (or Goldstone) mode and derive a gapped (Higgs-type) mode that was overlooked in previous studies of the quantum phase model. This mode is relevant to ultracold atoms in a strong optical lattice potential. We predict the signature of the gapped mode in lattice modulation experiments and show how it evolves with increasing interaction strength.

Figure 1

Schematic overview of models describing the superfluid phase of lattice Bosons. For small (dimensionless) interactions $u=U/2t\overline{n}z<1/{\overline{n}}^2$, the Bose-Hubbard model is well approximated by the continuum Gross-Pitaevskii action $S_{\mathrm{GP}}(\psi )$. Close to the critical interaction $u\approx u_c=2$, the dynamics is described by the critical theory $S_{\mathrm{Mott}}(\psi )$. For $u>1/{\overline{n}}^2$, the Bose-Hubbard model is equivalent to the quantum phase model, analyzed in detail in this work.

Figure 2

(a) Spectra for the phonon (dash-dotted line) and the amplitude mode (dotted line). Black curves correspond to $u=U/Jz=0.25$, gray curves to $u=1$. (b) Gap ${\Delta }_{\sigma }$ of the amplitude mode (dotted line) and the phonon bandwidth $W_{\phi }$ (dash-dotted line) vs dimensionless coupling $u=U/Jz$ in the superfluid phase. The gray shaded area corresponds to the Mott phase. The solid line denotes the condensate fraction $|{\psi }_0{|}^2$.

Figure 3

The response function to the lattice modulation $S^{\mathrm{kin}}(\omega )$ within the quadratic theory for different values of the interaction strength $u$ in two dimensions. The dark gray peak relates to the absorption of a single amplitude mode at $\hslash \omega ={\Delta }_{\sigma }$ (for illustrational purposes we give the delta-peak a finite width). The area in faint gray corresponds to the two-phonon continuum excited at energies below twice the phonon bandwidth $\hslash \omega \le 2W_{\phi }$ (the logarithmic divergence is a density-of-states effect peculiar to two dimensions). Both absorption probabilities scale as $u$ at small $u$.