Asymmetric double-well potential for single-atom interferometry

We consider the evolution of a single-atom wave function in a time-dependent double-well interferometer in the presence of a spatially asymmetric potential. We examine a case where a single trapping potential is split into an asymmetric double well and then recombined again. The interferometer involves a measurement of the first excited state population as a sensitive measure of the asymmetric potential. Based on a two-mode approximation a Bloch vector model provides a simple and satisfactory description of the dynamical evolution. We discuss the roles of adiabaticity and asymmetry in the double-well interferometer. The Bloch model allows us to account for the effects of asymmetry on the excited state population throughout the interferometric process and to choose the appropriate splitting, holding, and recombination periods in order to maximize the output signal. We also compare the outcomes of the Bloch vector model with the results of numerical simulations of the multistate time-dependent Schrödinger equation.

Figure 1

Energy difference $\Delta$ between ground and first excited states (solid line) for $\widehat{V_{as}}=0.02\widehat{x}$ as a function of the splitting parameter $\beta$. Dotted line—energy difference ${\Delta }_0$ for symmetric Hamiltonian, dashed line—asymmetry quantity $V_{as}$.

Figure 2

Stationary eigenfunctions of the ground state (dashed-dotted line) and the first excited state (dotted line) for different values of the splitting parameter $\beta$ = 1 (a), 2.5 (b), and 5 (c). The potential $V\left(x\right)$ is shown as the solid line.

Figure 3

Evolution of the Bloch vector $\vec{\sigma }$ and the torque vector $\vec{\Omega }$ at different moments of the splitting stage: (a) at the beginning when ${\Delta }_0\gg V_{as}$ and $\vec{\Omega }\approx (-{\Delta }_0,0,0)$, (b) when ${\Delta }_0=V_{as}$, and (c), when $V_{as}\gg {\Delta }_0$ and $\vec{\Omega }\approx (0,0,V_{as})$.

Figure 4

(a) Time evolution of the Bloch vector components ${\sigma }_x$ (solid line), ${\sigma }_y$ (dashed line), and ${\sigma }_z$ (dotted line) for $T_s=T_h=T_r=20$ and ${\beta }_{\max }=12.5$; (b) time evolution of the first excited state population $P_1$ (dotted line), the asymmetry parameter $V$ (dashed line), and the splitting parameter $\beta ∕{\beta }_{max}$ (solid line).

Figure 5

Dependence of the first excited state population $P_1$ at the end of the interferometric process on the duration of the holding stage $T_h$ for various durations of the splitting and recombining stages $T_s=T_r=5$ (a), 20 (b), and 200 (c). Results of the Bloch vector model are represented by solid lines and outcomes of full numerical simulations are presented by circles.

Figure 6

Dependence of the filling factor $F$ on the duration of the splitting stage $T_s$ for ${\beta }_{\max }=12.5$ and various values of asymmetry $\widehat{V_{as}}=0.01\widehat{x}$ (a), $0.02\widehat{x}$ (b), $0.05\widehat{x}$ (c), and $0.1\widehat{x}$ (d).