Mesoscopic aspects of strongly interacting cold atoms

Harmonically trapped lattice bosons with strong repulsive interactions exhibit a superfluid-Mott-insulator heterostructure in the form of a “wedding cake.” We discuss the mesoscopic aspects of such a system within a one-dimensional scattering matrix approach and calculate the scattering properties of quasiparticles at a superfluid-Mott-insulator interface as an elementary building block to describe transport phenomena across such a boundary. We apply the formalism to determine the heat conductivity through a Mott layer, a quantity relevant to describe thermalization processes in the optical lattice setup. We identify a critical hopping below which the heat conductivity is strongly suppressed.

Figure 1

(a) One-dimensional model of a S-MI heterostructure in a trap defined by the potential $V_i$ at position $i$. The dark areas denote Mott-insulating regions with a fixed density ${\rho }_i$, while the bright areas are superfluids of condensed particles (center, $S_p$) and condensed holes (wings, $S_h$). (b) Sketch of a scattering event at the superfluid-Mott-insulator interface, where a phonon incident from the superfluid generates back-reflected quasiparticles of phonon- and amplitude-type as well as particle- and hole-type excitations transmitted into the Mott insulator.

Figure 2

(Color online) Scattering coefficients for an incoming sound mode at an energy $ϵ=\hslash {\omega }_s\left(\pi \right)/4$. (a) For all values of $t/t_c$. The full lines show the exact values, while the dash-dotted line is the approximate value from Eq. (14). (b) Comparison between the exact and approximate values for $r_{ss}$. (c) All coefficients around the “critical” value $t\approx 0.971$ where for $ϵ=\hslash {\omega }_s\left(\pi \right)/4$ the hole and massive modes become exponentially damped. Note that values of the scattering coefficients larger than 1 are not in contradiction with particle conservation as these excitations do not have a well-defined particle character. (d) For values $t\approx t_c$ the dash-dotted results from Eq. (16) again describe the exact solutions well.

Figure 3

(Color online) Sketch of a scattering event of an incoming phonon from $S_p$. (a) The incident sound mode is transformed into a particle and a hole with amplitudes ${\tau }_{sp}$ and ${\tau }_{sh}$, respectively. Inside the Mott layer, the particle propagates until it hits the potential, where it is either reflected or tunnels under the barrier. The hole emerges from under the barrier and propagates to the other end of the layer. At the boundary to $S_h$, both modes are converted into the modes of the superfluid with the corresponding amplitudes. (b) The same situation with a higher energy of the incoming sound mode that allows for undamped propagation through the Mott region.

Figure 4

(Color online) Left: sketch of the local excitation spectra at different points $A,a,b,c,B$ (columns) in the inhomogeneous system for three different values of the hopping amplitude at $t/t_c=0.9,0.78,0.25$ (rows). Holelike spectra are dashed; particlelike spectra are solid lines. Right: local phase diagram with Mott-insulating and superfluid regions depending on the chemical potential $\delta {\mu }_i$ and the hopping $t/t_c$. The blue (light gray) shaded area of the Mott lobe corresponds to values of $t/t_c$ for which the hole at point $a$ has spectral overlap with the holelike spectra at $c$, allowing for undamped propagation in the energy window given in blue (light gray) on the left. The red (dark gray) shaded area of the Mott lobe marks values of $t$ for which the modes in the superfluid have mutual spectral overlap for a nonvanishing range of energy as shown on the left. For values $t<\tilde{t}$, the crossed terms do not contribute ($\text{phonon}\leftrightarrow \text{massive}$ mode) to the heat conductivity (see text).

Figure 5

(a) Heat conductivity $\kappa \left(T\right)$ for different values of $t$ ($t=0.9,0.82,0.78,0.74t_c$, from the back to the front) for a Mott layer of thickness $L_0=10a$ at $t=0$ ($a$ denotes the lattice constant). The structure of $\kappa \left(T\right)$ for small $T$ is dictated by the spectral gap in the intermediate Mott region. For smaller values of $t$, where only damped modes contribute to transport, the heat conductivity is exponentially suppressed and not shown here. (b) The dependence of the maximal heat conductivity ${\kappa }_{\infty }=\kappa \left(T⪢U/k_B\right)$ on the hopping amplitude $t$ displaying the qualitative change for $t<\tilde{t}$ in the plot, where the strong suppression sets in. The change at $t_{\star }$ is masked by a scattering resonance in the intermediate $\tilde{t}<t<t_{\star }$ regime.