Analytic wave model of Stark deceleration dynamics

Stark deceleration relies on time-dependent inhomogeneous electric fields which repetitively exert a decelerating force on polar molecules. Fourier analysis reveals that such fields, generated by an array of field stages, consist of a superposition of partial waves with well-defined phase velocities. Molecules whose velocities come close to the phase velocity of a given wave get a ride from that wave. For a square-wave temporal dependence of the Stark field, the phase velocities of the waves are found to be odd-fraction multiples of a fundamental phase velocity $\lambda ∕\tau$, with $\lambda$ and $\tau$ the spatial and temporal periods of the field. Here we study explicitly the dynamics due to any of the waves as well as due to their mutual perturbations. We first solve the equations of motion for the case of single-wave interactions and exploit their isomorphism with those for the biased pendulum. Next we analyze the perturbations of the single-wave dynamics by other waves and find that these have no net effect on the phase stability of the acceleration or deceleration process. Finally, we find that a packet of molecules can also ride a wave which results from an interference of adjacent waves. In this case, small phase stability areas form around phase velocities that are even-fraction multiples of the fundamental velocity. A detailed comparison with classical trajectory simulations and with experiment demonstrates that the analytic “wave model” encompasses all the longitudinal physics encountered in a Stark decelerator.

Figure 1

(Color online) A prototypical switchable field array that generates fields suited for accelerating or decelerating polar molecules. The field stages are longitudinally separated by a distance $\lambda ∕2$. Every other field stage is energized and every other grounded. There are two possible field configurations. (a) Electric fields generated by the two field configurations (for the case of four field stages). The electric fields that pertain to the upper and lower field configurations are shown by the red and blue curves and are referred to here as the red, ${\epsilon }_r$, and blue, ${\epsilon }_b$, fields, respectively. Also shown is the longitudinal coordinate $z$; (b) Alternation between the red and blue fields as a function of time, $t$. A given field stage is energized or grounded during a time $\tau ∕2$, after which the fields are switched, i.e., the field stages that were energized become grounded and vice versa. An energized field stage becomes grounded or vice versa during a transient time, $\Delta \tau$. The figure pertains to the case of guiding, for which the period $\tau$ is constant. The case of a varying period is shown in Fig. 2.

Figure 2

The time dependence of the field pertaining to the case of deceleration, for which the period $\tau$ is a function of time, $\tau =\tau \left(t\right)$. The case of a constant period is shown in Fig. 1. The timing sequence, generated by Eq. (33), is suitable for decelerating OH radicals on the $(+,1,1)$ wave with ${\varphi }_s=53°$ from an initial velocity of $370\mathrm{m}∕\mathrm{s}$ to a final velocity of $25\mathrm{m}∕\mathrm{s}$ in the decelerator presented in Refs. 17, 16. The upper panel shows the corresponding dependence of the switching half period, $\tau \left(t\right)∕2$. The points mark the time difference between two subsequent switching times. See Sec. 4C.

Figure 3

(Color online) A synchronous and a nonsynchronous molecule subjected to the field $\epsilon (z,t)$ of a $(+,n,\mathcal{l})$ wave moving at a phase velocity $V_{n,\mathcal{l}}$ (all motion is from left to right). The change of the velocity, $v_s$, of a synchronous molecule is such that its phase ${\varphi }_s$ with respect to the traveling field remains constant. This is the case when $v_s=V_{n,\mathcal{l}}$. The velocity, $v$, of a nonsynchronous molecule and its phase, $\varphi$, change with time. Also shown are the spatial coordinates of the synchronous and nonsynchronous molecule, $z_s$ and $z$, respectively.

Figure 4

(Color online) Realizations of a plane biased pendulum: a bob of mass $m$ is fixed to a rigid suspension of length $r$ which is attached to an axle of diameter $R$; wound around the axle is a string that carries a bias of mass $M$. A plane biased pendulum is a one-dimensional system, whose only coordinate is the angle $\varphi$ between the vertical axis $z$ and the bob suspension $r$. The stable and unstable equilibrium points are located symmetrically with respect to a plane perpendicular to the direction of the $z$ axis at angles ${\varphi }_s$ and $\pi -{\varphi }_s$, respectively. The stable-equilibrium angle ${\varphi }_s$ of the biased pendulum corresponds to the synchronous phase of the Stark accelerator/decelerator.

Figure 5

(Color online) The potential $V\left(\varphi \right)$ of a biased pendulum or, interchangeably, of an accelerator/decelerator, Eq. (47) (red curve), along with the pure pendulum potential $V_P\left(\varphi \right)$ (blue curve) and the potential of the bias $V_B\left(\varphi \right)$ (black curve). Also shown are the minimum (stable) and maximum (unstable) equilibrium points. One can see that the unstable equilibrium point coincides with the outermost turning point, ${\varphi }_{\mathit{out}}$, in the biased-pendulum potential. Angles in excess of ${\varphi }_{\mathit{out}}$ result in a nonuniform accelerating rotation of the pendulum about the axle, propelled by the falling bias. On the other hand, the inner turning point, ${\varphi }_{\mathit{in}}$, cannot be exceeded, since the potential at $\varphi <{\varphi }_{\mathit{in}}$ is repulsive. The cases of acceleration and deceleration for ${\alpha }_{n,\mathcal{l}}<0$ and ${\alpha }_{n,\mathcal{l}}>0$ are shown in panels (a)–(d), as labeled. The potential is expressed in terms of its amplitude $\frac{2W_n}{\pi \mathcal{l}}$, see Eqs. (13, 14).

Figure 6

(Color online) Biased pendulum or, interchangeably, Stark accelerator/decelerator potential $V\left(\varphi \right)$ for a range of values of the stable equilibrium point or, interchangeably, of the synchronous phase, ${\varphi }_s$. For ${\varphi }_s=\pm \pi ∕2$, the stable and unstable equilibrium points coincide and the potential cannot support any bound states. The potential is expressed in terms of its amplitude $\frac{2W_n}{\pi \mathcal{l}}$, see Eqs. (13, 14).

Figure 7

The separatrices, determined from Eqs. (53, 55, 57, 59), as functions of the synchronous phase, ${\varphi }_s$. Contours demarcate domains in phase space $({\dot{\varphi }}_{n,\mathcal{l}},{\varphi }_{n,\mathcal{l}})$ where stable oscillations take place. Note that ${\dot{\varphi }}_{n,\mathcal{l}}∕{\left(2{\alpha }_{n,\mathcal{l}}\right)}^{1∕2}$ plays the role of a (dimensionless) momentum and the angle $\varphi$ of its conjugate coordinate. Depending on the sign of ${\alpha }_{n,\mathcal{l}}$ and on the sign of $a_s$ (acceleration or deceleration), four cases are distinguished and shown in panels (a)–(d).

Figure 8

Phase-space distribution of a molecular beam as it enters a Stark accelerator/decelerator. The best overlap between a phase stable area (black fishes) and the molecular beam pulse (swarm of dots) is obtained when the synchronous molecule matches the mean position and velocity of the beam molecules. Cf. Fig. 2.

Figure 9

(Color online) Dynamics of a nonsynchronous molecule riding the (3, 5) wave and perturbed by the (1, 1) wave. The dynamics was determined by numerically integrating the differential equation of the molecule interacting with the two waves, for initial conditions that make the (3, 5) wave resonant and the (1, 1) wave perturbing. Both the longitudinal velocity, $v\left(t\right)$, panel (a), and the longitudinal position, $z\left(t\right)-v_{s}t$, relative to the synchronous molecule moving at a velocity $v_s$, panel (b), exhibit slow oscillations superposed by fast oscillations. The slow oscillations arise from the single-wave interaction of the molecule with the resonant (3, 5) wave and are described by Eqs. (72, 73). The fast oscillations are due to the perturbing (1, 1) wave and are described by Eqs. (77, 78). While the influence of the nonresonant wave on the velocity is significant, its effect on the position $z\left(t\right)$ is strongly suppressed. The time scale is given in terms of the slow-oscillation period ${\mathcal{T}}_{n,\mathcal{l}}$

Figure 10

(Color online) (a) Relative magnitude of the electric fields due to the (1, 1) wave and the (3, 3) wave (black dotted curves), which are traveling at the same velocity. These relative magnitudes are typical for two waves with successive $n$ and the same velocity. We see that the net field (red curve) is determined predominantly by the (1, 1) wave, i.e., the one with the smaller value of $n$. The conclusions about phase stability can be reached by considering solely this wave. (b) Typical relative magnitudes of the force due to the (1, 1), (3, 3), and (5, 5) waves (black dotted curves), which are all traveling at the same velocity. Since a small deviation in the force can accumulate when calculating a switching sequence comprising many stages, one should rely on the net force (red curve) rather than on the dominant term, cf. Eq. (80).

Figure 11

(Color online) Global phase portrait showing the phase stable areas due to the various waves (case of guiding). The contours pertain to average velocities of OH molecules plotted as a function of their initial velocity $v$ and initial spatial phase $kz$; the link between the average velocity and phase stability is given by Eq. (69). The contour plot is obtained by numerically integrating the full equation of motion (81) for 80 waves with a temporal phase and switching sequence given, respectively, by Eqs. (32, 33) corresponding to guiding $(\sin {\varphi }_s=0)$ at $V_0=150\mathrm{m}∕\mathrm{s}$. The phase portrait is in perfect agreement with Monte Carlo trajectory simulations which, in turn, are in perfect agreement with experiment.

Figure 12

(Color online) Detailed view of the phase stable areas of Fig. 11 around (a) $V_0$ and (b) $\frac{5}{3}{V}_0$ (the case of guiding). The contours pertain to average velocities of OH molecules plotted as a function of their initial velocity $v$ and initial spatial phase $kz$. Zooming in at the global phase portrait allows for an accurate comparison of the full-fledged numerical result with the analytic treatment of the dynamics. The white curves show the separatrices calculated from Eqs. (86, 87) for a resonant, single-wave interaction. The green curves are obtained from Eq. (89) and comprise the perturbations due to all the other, nonresonant waves. The yellow curves combine the two and are seen to render a perfect agreement with the full-fledged calculation.

Figure 13

(Color online) Detailed view of the phase stable areas around (a) $V_0$ and (b) $\frac{5}{3}{V}_0$ for the case of deceleration. The contours pertain to average velocities of OH molecules plotted as a function of their initial velocity $v$ and initial spatial phase $kz$. The contours were obtained by numerically integrating the equation of motion (81) for 80 waves, with a temporal phase and switching sequence given, respectively, by Eqs. (32, 33) corresponding to deceleration $(A_{n,\mathcal{l}}\sin {\varphi }_s<0)$. Panel (a) shows the full-fledged numerical calculation for deceleration on the first-harmonic (1, 1) wave with ${\varphi }_s=20°$. Panel (b) shows the full-fledged numerical calculation for deceleration on the $(+,3,5)$ wave with ${\varphi }_s=-170°$. The white curves pertain to the separatrices obtained for a resonant, single-wave interaction, as given by Eqs. (53, 57). The green curves are obtained from Eq. (89) and comprise the perturbations due to all the other, nonresonant waves. The yellow curves combine the two and are seen to be in perfect agreement with the full-fledged calculation.

Figure 14

Interference dynamics of a nonsynchronous molecule interacting with the (1, 1) wave and the (1, 3) wave which jointly create phase stability at $2V_0$. The dynamics was obtained by numerically integrating the equation of motion (92) for a molecule interacting with the two waves, with the initial condition $v\left(0\right)=2V_0$. Both the longitudinal velocity, $v\left(t\right)$, panel (a), and the relative position, $\Delta z=z\left(t\right)-2V_{0}t$, panel (b), exhibit slow oscillations superposed by fast oscillations. The two interfering waves (1, 1) and (1, 3) act jointly as a single (2, 4) wave, giving rise to slow oscillations of period ${\mathcal{T}}_{2,4}$. Note the similarity with Fig. 9.

Figure 15

(Color online) Detailed view of the phase stable area around $2V_0$ for the case of guiding (with ${\varphi }_s=0°$), panel (a), and of deceleration (with ${\varphi }_s=20°$), panel (b). The contours pertain to average velocities of OH molecules plotted as a function of their initial velocity $v$ and initial spatial phase $kz$. The full-fledged numerical calculations are compared with the analytic result for the case of the interference of the (1, 1) wave with the (1, 3) wave, which give jointly rise to a (2, 4) wave. The white curves show the separatrices (at $t=0$) obtained from Eq. (53) for $n\to n+r$ and $\mathcal{l}\to \mathcal{l}+s$, and with ${\varphi }_{n+r,\mathcal{l}+s}$ and ${\alpha }_{n+r,\mathcal{l}+s}$ as given by Eqs. (107, 112). The green curves were obtained from Eq. (114) and comprise the perturbations due to all nonresonant waves. The yellow curves combine the two and are seen to be in excellent agreement with the full-fledged numerical calculation. The temporal phase and switching sequence used for deceleration, panel (b), is given, respectively, by Eqs. (121, 122). We note that the $A_2$ coefficient is negative for the case of low-field seekers, considered in this calculation.