Interferometry using adiabatic passage in dilute-gas Bose-Einstein condensates

We theoretically examine three-well interferometry in Bose-Einstein condensates (BECs) using adiabatic passage. Specifically, we demonstrate that a fractional coherent transport adiabatic passage protocol enables stable spatial splitting in the presence of nonlinear interactions. A reversal of this protocol produces a coherent recombination of the BEC with a phase-dependent population of the three wells. The effect of nonlinear interactions on the interferometric measurement is quantified and is found to lead to an enhancement in sensitivity for moderate interaction strengths.

Figure 1

Schematic of a three-well system: at (a) $t=0$, at (b) $t_p/2<t<t_p/2+\tau$ where $\tau$ is the hold time required for phase accumulation, and at (c) $t=t_p+\tau$. The system consists of two parallel repulsive Gaussian barriers embedded in an ambient harmonic trap, dividing the system into three wells. At $t=0$—(a)—the BEC resides in a well. At $t=t_p/2$—(b)—the BEC is split into two equal components residing in wells 1 and 3. To perform the interferometric measurement, the tunneling rates are then kept constant for some hold time $\tau$ during which a relative phase difference may be accumulated. At $t=t_p+\tau$—(c)—the two BECs are recombined, which leads to a phase-dependent population in well 1.

Figure 2

Ideal three-well system with $\Delta =0$ and $g=0$: (a) Proposed pulsing scheme: Eqs. (5) and (6). (b) Energies of the eigenmodes: $D_0$ and $D_{\pm }$. (c) Evolution of the occupation of the three wells: solid curve: $N_1$ and dashed curve: $N_3$. (d) Adiabaticity parameter $\alpha t_p$. In these figures, we have assumed that $\tau =0$.

Figure 3

Eigenenergies for a range of interaction strengths with $\Delta =0.1$ and $t_p\to \infty$. As the interaction strength is increased, new eigenstates appear. When $\left|g\right|\approx 1.0$, the $D_0^{\prime }$ state at $t=0$ is no longer ${\psi }_1$, precluding the possibility of a 50:50 split.

Figure 4

Wave-function components for the $D_0^{\prime }$ state with nonzero $g,\Delta =0.1$, and $t_p\to \infty$. Upper (blue): ${\psi }_1$; middle (green): ${\psi }_2$; and lower (red): ${\psi }_3$. Solid curves: $g=0.2$; long dashed curves: $g=-0.6$; short dashed curves: $g=1.4$; and dot-dashed curves: $g=2.0$. Interactions lead to occupation of the middle well during transport as can be seen for $g=0.7$ and $g=1.0$. Note: ${\psi }_i$ is real.

Figure 5

Real part of the eigenvalue of the Jacobian at the $D_0^{\prime }$ state. (a) $\Delta =0.0$. (b) $\Delta =0.1$. Transport through regions with $\mathrm{Re}\left(\lambda \right)=0$ allows complete fidelity for the protocol. For $\Delta =0$, the $D_0^{\prime }$ state is unstable for all $g\ne 0$, albeit with small $\mathrm{Re}\left(\lambda \right)$.

Figure 6

Fidelity $\epsilon$ of split as defined in Eq. (23) as a function of pulse time $t_p$ and interaction strength $g$ at $t=t_p/2$ with $\Delta =0.1$. Regions of good fidelity $\epsilon \approx 1$ are denoted by dark red. The short dashed lines denote $g=g_c$ where the bifurcation appears, and the long dashed lines denote $\left|g\right|=0.2$.

Figure 7

Occupation of well 1 after ideal BEC splitting and recombination using reversal of f-CTAP protocol for (a) $t_p=200{\Omega }^{-1}$, (b) $t_p=500{\Omega }^{-1}$, (c) $t_p=1000{\Omega }^{-1}$, and (d) $t_p=2000{\Omega }^{-1}$ with $\Delta =0.1$ as a function of the phase difference ($\varphi$) between wells 1 and 3 at $t=t_p/2$ and the strength of the interatomic interactions $\left|g\right|\le 1$.

Figure 8

Deviation in the population of well 1 from the noninteracting expected value $N_1\left(\right|g|\le 0.1)-N_1(g=0)$ as a function of the phase difference ($\varphi$) between wells 1 and 3 at $t=t_p/2$ for (a) $t_p=100{\Omega }^{-1}$ and (b) $t_p=2000{\Omega }^{-1}$ where $N_1(g=0)={\cos }^2(\varphi /2)$. The classical first-order perturbation, Eq. (24), is plotted for dashed curves: $\left|g\right|=0.05$ and solid black curves: $\left|g\right|=0.1$.

Figure 9

Phase-sensitive measurement of the occupation of well 1 $N_1\left(t_p\right)$ for interatomic interaction strengths $\left|g\right|\le 0.35$ after recombination times of (a) and (b) $t_p=100{\Omega }^{-1}$ and (c) and (d) $t_p=3000{\Omega }^{-1}$. Ideal splitting of a BEC as defined in Eq. (8) is assumed.

Figure 10

Sensitivity of the density measurements for interaction strengths of (a) and (b) $g=-0.06$ and (c) and (d) $g=0.3$ with respect to changes in phase and total pulse time. The dashed lines correspond to the phase sensitivity $\partial N/\partial \varphi$, and the solid lines correspond to the pulse-time sensitivity $\partial N/\partial t_p$ between $90{\Omega }^{-1}\le t_p\le 110{\Omega }^{-1}$ for (a) and (c) and $1800{\Omega }^{-1}\le t_p\le 2200{\Omega }^{-1}$ for (b) and (d).