Accessing finite-momentum excitations of the one-dimensional Bose-Hubbard model using superlattice-modulation spectroscopy

We investigate the response to superlattice modulation of a bosonic quantum gas confined to arrays of tubes emulating the one-dimensional Bose-Hubbard model. We demonstrate, using both time-dependent density-matrix renormalization-group and linear response theory, that such a superlattice modulation gives access to the excitation spectrum of the Bose-Hubbard model at finite momenta. Deep in the Mott insulator, the response is characterized by a narrow energy-absorption peak at a frequency approximately corresponding to the on-site interaction strength between bosons. This spectroscopic technique thus allows for an accurate measurement of the effective value of the interaction strength. On the superfluid side, we show that the response depends on the lattice filling. The system can either respond at infinitely small values of the modulation frequency or only above a frequency threshold. We discuss our numerical findings in light of analytical results obtained for the Lieb-Liniger model. In particular, for this continuum model, bosonization predicts power-law onsets for both responses.

Figure 1

Sketch of the superlattice-modulation spectroscopy. The amplitude of the equilibrium optical lattice $V_0\left(x\right)=V_0{sin}^2\left(k_{L}x\right)$ (gray solid line) is time-periodically modulated in a dimerized fashion, i.e., the perturbing potential is given by $\delta V(x,t)\approx Asin\left(\omega t\right)sin\left(k_{L}x\right)$ with small amplitude $A$. The lattice amplitude is modulated between the two configurations indicated by (orange) dashed and (purple) dash-dotted lines, illustrating that while one potential barrier is increased the neighboring one is decreased, and vice versa.

Figure 2

Time evolution of the absorbed energy $E\left(t\right)-E_0$ at $U=4J$ in a system of $L=64$ sites and filling $\overline{n}=1$ per site for a modulation amplitude $A=0.01J$ and two different modulation frequencies. If this frequency is chosen within the resonant region, $\hslash \omega =7.5J$ (orange), energy is absorbed, whereas when the modulation is off-resonant, $\hslash \omega =3J$ (blue), very little energy absorption takes place.

Figure 3

Energy-absorption rates deep in the Mott insulator for a system of size $L=40$ and a modulation amplitude $A=0.01J$. Symbols are t-DMRG results and solid lines show the analytical result within perturbation theory [see Eq. (5)]. For $U=60J$, a comparison to the normal lattice modulation is shown (open symbols). The dash-dotted line is the response to normal lattice modulation within perturbation theory [24]. The inset shows a comparison to a system of size $L=96$ at $U=60J$.

Figure 4

Energy-absorption rates for intermediate interaction strengths on the Mott-insulating side of the phase transition for a system of size $L=64$ and a modulation amplitude $A=0.01J$. Symbols are t-DMRG results and solid lines are guide to the eyes. For $U=4J$ and $U=10J$, dotted lines show that the analytical predictions of perturbation theory deviate more and more from numerical results as $U$ is approaching the phase transition to the superfluid state. The inset shows a comparison to a system size $L=32$ at $U=6J$. Plateaus in the absorption rate appear to wash out with increasing system size.

Figure 5

The square markers indicate the frequency at which the maximum of the energy-absorption rate occurs as a function of interaction strength $U$ within t-DMRG using the same parameters as in Figs. 3, 4, and 7. Error bars indicate the observed bandwidth. We define the bounds as the mean between the frequency for which $\overline{dE/dt}/\left(A^{2}L\right)<0.1/\hslash$ and the neighboring frequency for which $\overline{dE/dt}/\left(A^{2}L\right)>0.1/\hslash$. The dashed blue line indicates the expected frequency within perturbation theory, $\hslash {\omega }_{\text{peak}}\approx U\left[1+{(2J/U)}^2\right]$, and the gray shaded region is the corresponding bandwidth ($=4J$) within perturbation theory. The dash-dotted orange line indicates the naive expectation that $\hslash {\omega }_{\text{peak}}\approx U$. The inset shows a zoom into the small-$U$ region.

Figure 6

Sketch of the excitation spectrum of the Lieb-Liniger model for a given interaction strength $\gamma$. The Lieb I mode is soundlike at small momenta and becomes particlelike at larger momenta. The Lieb II mode exhibits the same soundlike behavior at small momenta, but it becomes maximal at $k=\pi n$ and vanishes again at $k=2\pi n$ and reopens at $k>2\pi n$, where $n$ is the density. The shaded region represents the continuum of excitations bounded between the two modes. The inset shows a sketch of the onset of the corresponding energy-absorption rates within linear response for two different densities (using $K=3/2$). The corresponding momentum transfer $\mathrm{\Delta }k=\pi /a$ (marked by vertical lines in the main plot) either corresponds to a momentum at which the Lieb II mode is finite (dashed green line), which leads to a finite onset for the response, or to a momentum at which the the energy of the Lieb II mode vanishes (dotted purple line), which leads to a finite response at all frequencies.

Figure 7

The energy-absorption rate in the superfluid region for a system of size $L=64$ and a modulation amplitude $A=0.01J$ at different interaction strengths. The dotted vertical lines indicate the corresponding energy of the Lieb II mode at momentum $ka=\pi$. The fillings and the corresponding continuum densities $n\gtrsim 1/a$ are chosen such that the momentum $ka=\pi$ appears to the left of the maximum of the Lieb II branch (see Fig. 6). Solid lines are guides to the eye.