Nonadiabatic transitions in a Stark decelerator

In a Stark decelerator, polar molecules are slowed down and focused by an inhomogeneous electric field which switches between two configurations. For the decelerator to work, it is essential that the molecules follow the changing electric field adiabatically. When the decelerator switches from one configuration to the other, the electric field changes in magnitude and direction, and this can cause molecules to change state. In places where the field is weak, the rotation of the electric field vector during the switch may be too rapid for the molecules to maintain their orientation relative to the field. Molecules that are at these places when the field switches may be lost from the decelerator as they are transferred into states that are not focused. We calculate the probability of nonadiabatic transitions as a function of position in the periodic decelerator structure and find that for the decelerated group of molecules the loss is typically small, while for the un-decelerated group of molecules the loss can be very high. This loss can be eliminated using a bias field to ensure that the electric field magnitude is always large enough. We demonstrate our findings by comparing the results of experiments and simulations for the Stark deceleration of LiH and CaF molecules. We present a simple method for calculating the transition probabilities which can easily be applied to other molecules of interest.

Figure 1

(a) Structure of the Stark decelerator, showing three consecutive deceleration stages centered on the $n$th stage. Each stage is formed from a pair of rods whose axes are parallel to the $z$ axis for the odd-numbered stages, and to the $x$ axis for the even-numbered stages ($n$ is even). For clarity, the two rods forming the $n$th stage are shown slightly displaced, though in reality one should be exactly behind the other. (b) Electric field profile versus reduced position for the two switch configurations when $R=1.5$ mm, $r_0=1$ mm, $L=6$ mm, $V=15$ kV. (c) Longitudinal phase-space acceptance for synchronous phase angles of $0^{°}$ (outermost line) through to ${80}^{°}$ (innermost line) in steps of ${10}^{°}$. The plot is for CaF molecules in the $|4,0\rangle$ state and for the electric field profile shown in (b).

Figure 2

Stark shift of the low-lying states of a rigid rotor molecule. States are labeled by their values of $\mathcal{N}$ and $M$.

Figure 3

The rotation rate, $d\beta /\mathit{dt}$ (solid), the angular frequency separation of LiH $|1,0\rangle$ and $|1,1\rangle$ states (dotted), and the angular frequency separation of CaF $|4,0\rangle$ and $|4,1\rangle$ states (dashed), as a function of time since throwing the switch and at various reduced positions ($\theta$) within the decelerator. The switch rise and fall time was set to $\tau =500$ ns, and the electric field profile used in the calculation is the one shown in Fig. 1b. Note the logarithmic scale on both axes.

Figure 4

Evolution of the populations in the sublevels of the $\mathcal{N}=1$ manifold of LiH as a function of time since throwing the switch and at the position $\theta =-{90}^{°}$. The curves are labeled by their value of $\left|M\right|$. The initial state is $M=0$, the switch rise and fall time is $\tau =500$ ns, and the electric field profile is the one shown in Fig. 1b.

Figure 5

Evolution of the populations in the various sublevels of the $\mathcal{N}=4$ manifold of CaF as a function of time since throwing the switch, and at three different reduced positions in the decelerator. The curves are labeled by their value of $\left|M\right|$. The initial state is $M=0$, the switch rise and fall time is $\tau =500$ ns, and the electric field profile is the one shown in Fig. 1b.

Figure 6

Transition probability out of the $M=0$ state as a function of the reduced position in the decelerator. The position dependence is symmetric about $\theta =-{90}^{°}$. The switch rise and fall time is $\tau =500$ ns, and the electric field profile is the one shown in Fig. 1b. (Solid line) LiH in $\mathcal{N}=1$. (Dashed line) CaF in $\mathcal{N}=4$.

Figure 7

The Stark shift of the energy levels of CaF that correlate to $N=4$ at low field. The states are identified by their values of $J$ and $F$ at low field and by the value of $\left|M_F\right|$. At higher field the states separate into five groups that can be labeled by the projection quantum number $\left|M_N\right|$. Those that join the $M_N=0$ group are shown by bold black lines. The inset shows the same set of states up to a field of 200 kV/cm.

Figure 8

Time-of-flight profiles of LiH molecules decelerated from an initial speed of 420 m/s to a final speed of 213 m/s. The deceleration parameters are $V_{\mathrm{hi}}=9.5$ kV, $V_{\mathrm{lo}}=0$, and ${\varphi }_0={51}^{°}$. (a) Lower trace, experimental data. (b) Middle trace, simulation including nonadiabatic loss. (c) Top trace, simulation for the adiabatic case. In all cases the signal is normalized to that obtained in dc mode. The traces are offset for clarity.

Figure 9

(a) Experimental time-of-flight profiles for CaF in the $\mathcal{N}=4$ state with a synchronous phase angle of $0^{°}$ (no deceleration), a synchronous speed of 370 m/s, and an applied voltage of $V_{\mathrm{hi}}=14$ kV. (i) No switching. (ii) $V_{\mathrm{lo}}=1.45$ kV. (iii) $V_{\mathrm{lo}}=0$. (iv) $V_{\mathrm{lo}}=0.5$ kV. (b) Simulations of these experiments. All the results share a common baseline.