Lattice modulation spectroscopy of one-dimensional quantum gases: Universal scaling of the absorbed energy

Lattice modulation spectroscopy is a powerful tool for probing low-energy excitations of interacting many-body systems. By means of bosonization we analyze the absorbed power in a one-dimensional interacting quantum gas of bosons or fermions, subjected to a periodic drive of the optical lattice. For these Tomonaga-Luttinger liquids we find a universal ${\omega }^3$ scaling of the absorbed power, which at very low-frequency turns into an ${\omega }^2$ scaling when scattering processes at the boundary of the system are taken into account. We confirm this behavior numerically by simulations based on time-dependent matrix product states. Furthermore, in the presence of impurities, the theory predicts an ${\omega }^2$ bulk scaling. While typical response functions of Tomonaga-Luttinger liquids are characterized by exponents that depend on the interaction strength, modulation spectroscopy of cold atoms leads to a universal power-law exponent of the absorbed power. Our findings can be readily demonstrated in ultracold atoms in optical lattices with current experimental technology.

Figure 1

Response function $\chi \left(\omega \right)$ to a modulation of the lattice at the frequency $\omega$ and temperature $T$ evaluated from Eq. (20) for $F\left(K\right)=1$ and $\alpha /u=1$.

Figure 2

Absorbed power density. Using matrix product states, we have evaluated the absorbed power in a periodically driven Bose-Hubbard model on an open chain of $L=120$ sites and density $\rho =1.2$ for two values of the interaction strength (see the legend). The absorbed power density, renormalized by the drive strength $\delta J^2$, shows a universal ${\omega }^3$ scaling, as predicted from the Tomonaga-Luttinger theory. Deviations at low frequencies are decreasing with increasing system size and are expected to arise from the residual contributions of system edges which add an ${\omega }^2/L$ contribution that is leading in frequency but vanishing in the thermodynamic limit.

Figure 3

Luttinger liquid parameters. We evaluate (a) the Luttinger parameter $K$, (b) the Luttinger velocity $u$, and (c) the derivative of the Luttinger parameter with respect to the kinetic energy $dK/d{J}_0$ as a function of the inverse system size $L$ for various values of the interaction strength $U$ and fixed density ${\rho }_0=1.2$. Lines (dashed and solid) are guides to the eyes.

Figure 4

Prefactor of the absorbed power. The analytically predicted prefactor $F\left(K\right)$ obtained in Eq. (22) (closed circles) is compared to the numerically obtained prefactor from the full time evolution (stars). The data are shown for various values of the interaction strength $U$ and for a fixed density ${\rho }_0=1.2$ as a function of the inverse system size $1/L$.