Geometrical Pumping with a Bose-Einstein Condensate

We realized a quantum geometric “charge” pump for a Bose-Einstein condensate (BEC) in the lowest Bloch band of a novel bipartite magnetic lattice. Topological charge pumps in filled bands yield quantized pumping set by the global—topological—properties of the bands. In contrast, our geometric charge pump for a BEC occupying just a single crystal momentum state exhibits nonquantized charge pumping set by local—geometrical—properties of the band structure. Like topological charge pumps, for each pump cycle we observed an overall displacement (here, not quantized) and a temporal modulation of the atomic wave packet’s position in each unit cell, i.e., the polarization.

Figure 1

Bipartite magnetic lattice. [(a) and (b)] Dipole trapped ${}^{87}\mathrm{Rb}$ BECs subject to a bias magnetic field $B_0{\mathbf{e}}_z$ had a Zeeman splitting ${\omega }_Z/2\pi =0.817\mathrm{MHz}$ and a quadratic shift $\hslash \epsilon =0.03E_R$. These BECs were illuminated by four Raman beams and an rf magnetic field. Each of the two Raman couplings (strengths ${\mathrm{\Omega }}_{\pm }$) was derived from two cross-polarized Raman laser beams with frequency components $\omega$ and $\omega +\delta \omega$. (c) Adiabatic potentials colored according to $⟨m_x⟩$ computed for $\hslash (\overline{\mathrm{\Omega }},{\mathrm{\Omega }}_{\text{rf}},\delta )=(6,2.2,0)E_R$, $\delta \mathrm{\Omega }/\overline{\mathrm{\Omega }}=-0.1$, and $\varphi =\pi /4$. The dashed curves plot the $\pm \hslash {\mathrm{\Omega }}_x$ contributions to the potential experienced by states $|m_x=\pm 1⟩$. (d) Lowest two energy band energies plotted as a function of $\varphi$, otherwise with the same parameters as (c).

Figure 2

Ground state spin projections. (a) Ground state spin projections at various $\varphi$ along with the predicted populations for $\hslash (\overline{\mathrm{\Omega }},\delta \mathrm{\Omega },{\mathrm{\Omega }}_{\text{rf}},\delta )=(4.4,0,2.2,0)E_R$. The associated adiabatic potentials [insets in (b)] have minima with spin projection following the observed population’s trends. (b) Magnetization derived from data in (a).

Figure 3

Band geometry and topology computed for $\hslash (\overline{\mathrm{\Omega }},{\mathrm{\Omega }}_{\text{rf}},\delta )=(6,2.2,0)E_R$. [(a) and (b)] Zak phase and $q=0$ Berry curvature showing the dependence on both $\delta \mathrm{\Omega }/\overline{\mathrm{\Omega }}$ and $\varphi$. In (b), the arrows show experimental charge pump trajectories in Fig. 4. (c) Adiabatic potentials (displaced vertically for clarity) computed for a range of $\varphi$ constituting a complete pump cycle at $\delta \mathrm{\Omega }/\overline{\mathrm{\Omega }}=-0.4$ (left panel) and $-0.8$ (right panel). Filled circles mark the local energy minima.

Figure 4

Geometric charge pumping. (a) Magnetization measured while linearly ramping $\varphi$ with period $T=2\mathrm{ms}$, along with the prediction for $\hslash \overline{\mathrm{\Omega }}=6.38(2)E_R$, $\hslash \delta \mathrm{\Omega }=4.50(2)E_R$, and $\hslash {\mathrm{\Omega }}_{\text{rf}}=2.20(3)E_R$. (b) Displacement plotted versus $\varphi /2\pi$ (number of pump cycles). Trajectories i–iii are taken at $\delta \mathrm{\Omega }/\overline{\mathrm{\Omega }}=0.7$, 0, and $-0.7$, respectively; in each case $\hslash \overline{\mathrm{\Omega }}\approx 6E_R$ and $\hslash {\mathrm{\Omega }}_{\text{rf}}=2.20(3)E_R$. Solid curves: simulation of a charge pump in the trap. The small displacement near $\varphi =0$ is introduced by our loading procedure. (c) Measured displacement $\mathrm{\Delta }x$ per pump cycle (symbols), along with the prediction obtained by integrating the Berry curvature over our pumping trajectory (solid curve). The uncertainty bars represent the 95% confidence interval.