Effects of higher-order energy bands and temperature on the bosonic Mott insulator in a periodically modulated lattice

We show that a certain class of higher-order excitations in ultracold atoms experiments can be described by straightforward extension of the standard strong coupling approach in the coherent state path integral formalism. It is achieved by theoretical analysis of energy absorption spectroscopy in the three-dimensional system of strongly correlated bosons described by the Bose-Hubbard model. In particular, for unit filling, an explicit form of the single-particle Mott insulator Green function at finite temperatures is derived which goes beyond the standard Hubbard bands description. Moreover, for relevant densities, we calculated the energy absorption rate and performed thermometry on rubidium atomic cloud gas by using previously obtained experimental data. Within the local density approximation, we explain that in such systems the nature of absorption spectrum depends significantly on local chemical potential: (a) the crossover region between lobes is characterized by different types of particle-hole excitations from neighboring Mott lobes and (b) origin of higher-order energy excitations changes from hole type to particle type for higher bosonic densities.

Figure 1

(a) Phase diagram of the Bose-Hubbard model at $T/U=0.1$ for the three-state approximation (TSA) and higher-state approximation (HSA). For comparison, we plot the zero-temperature phase diagram at mean-field level (TSA/HSA). In the $\mu /U-J/U$ plane the red line corresponds to TSA at $T/U=0.1$, the black solid line to HSA at $T/U=0.1$, and the black dashed line to TSA/HSA at $T/U=0$. (b) and (c) correspond to the first and second lobes respectively, in which quasiparticle excitations in HSA and TSA are plotted. The remaining parameters are $J/U=0.025, \mu /U\approx 0.41$ for $n_0=1$ and $J/U=0.015, \mu /U\approx 1.41$ for $n_0=2$. In (b) and (c), we set also $k_y=k_z=0$.

Figure 2

HSA calculation of frequency $\omega /U$ dependent energy absorption rate in terms of $\omega S\left(\omega \right)/U$ for different values of chemical potential: (a) $\mu /U=0.1$, (b) $\mu /U=0.5$, (c) $\mu /U=0.9$, (d) $\mu /U=1.1$, (e) $\mu /U=1.5$, (f) $\mu /U=1.9$. Gray dashed line denotes the resultant effect of all transition types summed. Calculations are made at $T/U=0.2$ with the interaction strength $U/J=90$.

Figure 3

$\omega S\left(\omega \right)$ in $U$ units versus frequency $\omega /U$ for $\mu /U=0.5$ where $n_0=1$. (a) Present HSA (black circles) and TSA (red line) results in the zero-temperature limit. In (b) the data points from HSA are plotted for $T/U=0.1$ (red line) and $T/U=0.2$ (black line).

Figure 4

Schematic explanation of (a) $|1,1\rangle \to |0,2\rangle$ and (b) $|2,2\rangle \to |1,3\rangle$ excitations.

Figure 5

The full width at half maximum (FWHM) as a measure of the introduced energy [3] compared with the frequency spectrum of $\omega S\left(\omega \right)/U$, which is proportional to the energy absorption rate. Black squares denote the data obtained within HSA and red circles denote the experimental data from Ref. [3]. To compare qualitatively the experimental data from Ref. [3] with HSA, we have normalized in vertical and horizontal axes the $U$ peak from HSA to the $U$ peak from Ref. [3], which enables analysis of the $2U$ peak behavior (see Secs. 3c and 3d). The inset shows the cross section of the average density distribution of the trapped gas. The simulation is made for $T/U=0.11$.

Figure 6

HSA calculation of frequency $\omega /U$ dependent energy absorption rate in terms of $\omega S\left(\omega \right)/U$ for different values of chemical potential: (a) $\mu /U=0.7$, (b) $\mu /U=1.3$, (c) $\mu /U=1.7$. Gray dashed lines denote the resultant effect of all transition types summed together. Calculations are made at $T/U=0$ with interaction strength $U/J=90$.