Andreev-Like Reflections with Cold Atoms

We propose a setup in which Andreev-like reflections predicted for 1D transport systems could be observed time dependently using cold atoms in a 1D optical lattice. Using time-dependent density matrix renormalization group methods we analyze the wave packet dynamics as a density excitation propagates across a boundary in the interaction strength. These phenomena exhibit good correspondence with predictions from Luttinger liquid models and could be observed in current experiments in the context of the Bose-Hubbard model.

Figure 1

(a) A propagating excitation exhibits normal reflections (top) or Andreev reflections (bottom) at an interaction boundary depending on the relative interaction strengths on the two sides. (b) Observation via bosons in an optical lattice in three steps: Preparation of the initial excitation using a superimposed trap (left); propagation of the excitation towards the interaction boundary, formed by coupling the atoms off resonantly to an additional internal state (right); and detection via measurement of the atom density in a region between the initial excitation and the interaction boundary.

Figure 2

Numerical simulation results for propagation of an initial density excitation (${\epsilon }_0=2J$, $\sigma =3$, $x_0=45$) across an interaction boundary from $V_L=0$ to varying $V_R$ at $x_b=90$ ($M_b=1$), with open boundary conditions. (a) Shaded density plot for $V_R=J$, showing a normal (positive density) reflection at the boundary. (b) Shaded density plot for $V_R=-J$, showing an Andreev (negative density, or hole) reflection at the boundary. (c) Difference from the initial value of the integrated density over sites 60–75, showing an initial peak due to propagation of the initial density excitation, and a secondary maximum ($V_R=J$, dashed line) or minimum ($V_R=-J$, solid line) due to the reflected excitation. (d) Reflection coefficients as a function of $V_R$, showing the comparison between simulation values (solid line) and estimated parameters from LL theory using numerically computed Luttinger parameters [from Eq. (4), dashed line] and the analytical form $g=1/\sqrt{1+a{V}_i/v_F}$ (dotted line). The inset shows the dependence on the depth ${ϵ}_0$ with $\sigma =2$ (solid lines) and $\sigma =4$ (dashed lines) for $V_R=-0.5J$ (upper curves) and $V_R=-J$ (lower curves), computed from measurement sites 50–70.

Figure 3

Reflections of a density excitation (${\epsilon }_0=2J$, $x_0=45$, $x_b=90-M_b$) from a thick boundary with $V_i$ varying from $V_L=0$ to $V_R=-J$ linearly over $M_b$ lattice sites. (a) Shaded density plot showing reflection of an initial density excitation with $\sigma =3$ from a boundary with $M_b=10$, showing the considerably broader reflected wave produced by the thick boundary. (b) Reflection coefficient as a function of barrier thickness $M_b$ (measurement sites 55–85), showing an increase in the amplitude of the negative density reflection as the barrier is increased, for initial wave packets with $\sigma =3$ (solid line) and $\sigma =2$ (dashed line). The amplitude is also slightly larger for the narrower initial wave packet. ${\epsilon }_0(t=0)=2J$.

Figure 4

Realization of Andreev reflections within the Bose-Hubbard model. (a) Shaded density plot showing an Andreev-like reflection from a boundary at $X_b=90$, with $U_L=10J$ and $U_R=J$. (b) Reflection coefficients for different $U_R$ as a function of $V_{R,\mathrm{eff}}$ for the Bose-Hubbard model (solid line). These are compared with results from the extended Hubbard model (dashed line) and estimated parameters from LL theory (dotted line) computed as in Fig. 2. Parameters used: ${\epsilon }_0=2J$, $\sigma =3$, $x_0=45$, $X_b=90$, $M_b=1$, $U_L=10J$, measurement sites 55–85.