Phonon resonances in atomic currents through Bose-Fermi mixtures in optical lattices

We present an analysis of Bose-Fermi mixtures in optical lattices for the case where the lattice potential of the fermions is tilted and the bosons (in the superfluid phase) are described by Bogoliubov phonons. It is shown that the Bogoliubov phonons enable hopping transitions between fermionic Wannier-Stark states; these transitions are accompanied by energy dissipation into the superfluid and result in a net atomic current along the lattice. We derive a general expression for the drift velocity of the fermions and find that the dependence of the atomic current on the lattice tilt exhibits negative differential conductance and phonon resonances. Numerical simulations of the full dynamics of the system based on the time-evolving block decimation algorithm reveal that the phonon resonances should be observable under the conditions of a realistic measuring procedure.

Figure 1

Fermions confined to a tilted potential $V(x)$ form a Wannier-Stark ladder with a constant energy separation $\Delta$ between adjacent lattice sites. The fermion-boson interaction enables phonon-assisted transitions from site $i$ to $j$ at the rate $W_{i\to j}$. The released energy $(i-j)\Delta$ is dissipated in the form of a phonon emitted into the superfluid.

Figure 2

The dispersion relation of acoustic phonons in a solid with $\hslash {\omega }_q~|\sin (\mathit{qa}/2)|$ (dashed line) and of Bogoliubov phonons in a superfluid confined to an optical lattice (full line) in units of the bandwidth $w=4J_b\sqrt{1+U_b{n}_b/2J_b}$ for the parameters $U_b{n}_b/J_b=0.5$. (b) The corresponding density of states $D_q=|\partial \hslash {\omega }_q/\partial (\mathit{qa})|{}^{-1}$ (full line) of the Bogoliubov phonons is approximately constant for small momenta and exhibits a Van Hove singularity near the edge of the first Brillouin zone (dashed line).

Figure 3

The total drift velocity ${\mathrm{v̄}}_d$ (full line) as a function of the lattice tilt $\delta =\Delta /4J_b$. The contributions from different jump distances $\ell$ (dashed lines) result in a distinct NDC peak and a series of phonon resonances at the positions $\delta \approx 1/\ell$. The system parameters are $U_b{n}_b/J_b=0.5$, $J_f/J_b=0.4$ yielding for the phonon bandwidth $1.12\times 4J_b$ and $\zeta =0.5$. The drift velocity is given in units of ${\mathrm{v̄}}_0=a{n}_b{U}_{\mathit{bf}}^2/4J_b\hslash$.

Figure 4

The displacement of the fermion as a function of the lattice tilt $\delta$ determined numerically (symbols $+$, $○$, $\times$) and corresponding analytical fits (solid lines). The dominant NDC peak and the $\ell =1$ phonon resonance at the boundary of the phonon band can be clearly recognized. The ratio $V$ between the height of the $\ell =1$ resonance and the NDC peak is $V=\{0.09,0.14,0.18\}$ for increasing $J_f$. The $\ell =2$ resonance is visible for sufficiently large fermionic hopping $J_f$. The system parameters are $J_f=\{0.2,0.4,0.8\}$, $U_b=0.1$, $U_{\mathit{bf}}=0.5$ and the fitting parameters are $\varepsilon =0.1$, ${\mathrm{J̃}}_f=0.7J_f$, ${\ell }_{\max }=2$, with energies in units of $J_b$.

Figure 5

The displacement of the fermion as a function of the lattice tilt $\delta$ for different evolution times $\tau$ determined numerically (connected data points). The NDC peak and the $\ell =1$ phonon resonance are clearly visible also in presence of strong boson-fermion and boson-boson interactions. The additional peaks between the NDC peak and the phonon resonance are due to finite-time effects. The displacement increases approximately linearly with the evolution time $\tau$. The system parameters are $J_f=0.5$, $U_b=U_{\mathit{bf}}=1$, with energies in units of $J_b$.