Filtering of matter-wave vibrational states via spatial adiabatic passage

We discuss the filtering of the vibrational states of a cold atom in an optical trap by chaining this trap with two empty ones and adiabatically controlling the tunneling. Matter-wave filtering is performed by selectively transferring the population of the highest populated vibrational state to the most distant trap while the population of the rest of the states remains in the initial trap. Analytical conditions for two-state filtering are derived and then applied to an arbitrary number of populated bound states. Realistic numerical simulations close to state-of-the-art experimental arrangements are performed by modeling the triple well with time-dependent Pöschl-Teller potentials. In addition to filtering of vibrational states, we discuss applications for quantum tomography of the initial population distribution and engineering of atomic Fock states that, eventually, could be used for tunneling-assisted evaporative cooling.

Figure 1

(a) Temporal variation of the positions of the PT trap centers, where $d_0$ is the minimum trap separation and $T$ is the time delay between the two approaching sequences. (b) Illustration of the spatial profile of the PT potentials at the three different times corresponding to the vertical lines in (a). Each isolated PT trap has depth $V_0=-20\hslash {\omega }_x$ and supports only four vibrational states. (c) Temporal variation of the tunneling rate ${\Omega }_n^{\mathit{LM}}$ (${\Omega }_n^{\mathit{MR}}$) between left and middle (middle and right) traps for the ground ($n=0$) and first ($n=1$) excited states; ${\theta }_1$ is the mixing angle for the first excited state. Temporal variation of the population of the ground and first excited states of the (d) left and (e) right traps assuming the following initial distribution $P_0^L(t_{\mathrm{ini}})=P_1^L(t_{\mathrm{ini}})=1/2$, respectively.

Figure 2

Curves ${\Omega }_1(d_0)T=10$ (solid blue) and ${\Omega }_0(d_0)T=1$ (dashed red) in the parameter plane ($d_0/\alpha$, ${\omega }_{x}T$) for three different values of the potential depth (a) $s=2$, (b) $s=3,$ and (c) $s=4$. In all three cases, the dotted green curve corresponds to ${\Omega }_1(2d_0)T=1$. The gray region defines the parameter domain for which the filtering conditions (15) and (16) are fulfilled. (d) Contour plot of the fidelity (see text) of the filtering process for $s=4$ obtained numerically by integration of the Schrödinger equation with the temporal variation of the trap centers given by Eq. (13). The cross in (d) marks the parameter setting used in Fig. 1.

Figure 3

Curves ${\Omega }_{n,s}\left(d_0\right)T=10$ (solid blue) and ${\Omega }_{n-1,s}\left(d_0\right)T=1$ (dashed red) in the parameter plane ($d_0/\alpha$, ${\omega }_{x}T$) for PT potentials with depth $s=6$. Filtering for the vibrational level $n$ can be achieved in the corresponding gray region.

Figure 4

Quantum tomography protocol via a sequence of adiabatic passage processes in three PT potentials. Population distribution for the lowest eight energy levels in the (a) left and (b) right traps after each step $k$ of the protocol. The minimum distance between the traps at each step, $d_0\left[k\right]$, is shown in (c). Mean $\langle n_i\rangle$ (closed squares) and variance $\langle \Delta n_i^2\rangle$ (closed circles) of the population of each state for the (d) left $(i=L)$ and the (e) right $(i=R)$ traps along the protocol. The trap-approaching sequence at each step of the protocol is given by Eq. (13), with ${\omega }_{x}T/2=40$ and $v_0/\alpha {\omega }_x=3\times {10}^{-3}$ and the corresponding $d_0[k]$.

Figure 5

Ground-state filtering by applying a single adiabatic transport process. Population distribution for the lowest eight energy levels in the (a) left and (b) right traps before and after the single step-filtering protocol. The minimum distance $d_0$ is chosen to fulfill the filtering conditions for the first excited vibrational state and coincides with the value used in the last step shown in Fig. 4. The fidelity of the process is above $99\%$. Other parameters are the same as in Fig. 4.

Figure 6

Gaussian potential (black solid curve) approximated with harmonic (red dotted curve) and Pöschl-Teller (blue dashed curve) potentials. The corresponding eigenenergies are represented by horizontal black solid (Gaussian potential), red dotted (harmonic potential), and blue dashed (PT potential) lines. $N_G$, $N_H$, and $N_{\mathit{PT}}$ give the number of vibrational states for the given potential depth for the Gaussian, harmonic, and Pöschl-Teller potentials, respectively.