Model for the overall phase-space acceptance in a Zeeman decelerator

We present a formalism to calculate phase-space acceptance in a Zeeman decelerator. Using parameters closely mimicking previous Zeeman deceleration experiments, this approach reveals a velocity dependence of the phase stability which we ascribe to the finite rise and fall times of the current pulses that generate the magnetic fields inside the deceleration coils. It is shown that changing the current switch-off times (characterized by the reduced position of the synchronous particle ${\kappa }_0$) as the sequence progresses, so as to maintain a constant mean acceleration per pulse, can lead to a constant phase stability and hence a beam with well-defined characteristics. We also find that the time overlap between fields of adjacent coils has an influence on the phase-space acceptance. Previous theoretical and experimental results [A. W. Wiederkehr et al., Phys. Rev. A 82, 043428 (2010); J. Chem. Phys. 135, 214202 (2011).] suggested unfilled regions in phase space that influence particle transmission through the decelerator. Our model provides a means to directly identify the origin of these effects due to coupling between longitudinal and transverse dynamics. Since optimum phase stability is restricted to a rather small parameter range in terms of the reduced position of the synchronous particle ${\kappa }_0$, only a limited range of final velocities can be attained using a given number of coils. We evaluate phase stability for different Zeeman deceleration sequences, and by comparison with numerical three-dimensional particle-trajectory simulations, we demonstrate that our model provides a valuable tool to find optimum parameter sets for improved Zeeman deceleration schemes. An acceleration-deceleration scheme is shown to be a useful approach to generating beams with well-defined properties for variable-energy collision experiments. More generally, the model provides significant physical insights that are applicable to other types of particle decelerators with finite rise and fall time fields.

Figure 1

Three-dimensional magnetic field of a solenoid coil for Zeeman deceleration at a current of 300 A (shaded surface). Mesh plots indicate the magnetic fields of neighboring coils. The magnetic fields of the individual coils are not added up vectorially. The coil specifications and positions are the same as in Dulitz et al. [30].

Figure 2

Temporal profiles of the current pulses for five consecutive deceleration coils as used in the model. The pulse shape for one coil is shown with a dashed line for clarity. Time is scaled to the period $T$, i.e., the time required for the passage of one coil distance. The rise and fall times are denoted with $t_{\mathrm{r}}$, and the time overlap between adjacent pulses, defined as shown in the figure, is referred to as $t_{\mathrm{o}}$.

Figure 3

Separatrices in (a) longitudinal and (b) transverse phase space for the Zeeman deceleration of nitrogen atoms in the ${}^2{D}_{5/2},M_J=5/2$ state at different ${\kappa }_0$ and at a longitudinal velocity of 500 m/s. Particles inside the separatrix revolve in stable orbits around the synchronous particle. Transverse trajectories at ${\kappa }_0<-0.1$ are not phase stable. The volume encompassed by the longitudinal separatrices decreases as ${\kappa }_0$ is increased from $-0.3$ to 0.2 (solid arrow). There are longitudinal separatrices whose extent increases for ${\kappa }_0>0.1$ [dashed lines and dashed arrow in (a)] because the deceleration pulse sequence, applied to achieve a given final velocity, addresses particles in the same phase space as that for a lower value of ${\kappa }_0$; for example, an identical separatrix centered at $\kappa =-0.1$ exists both for ${\kappa }_0=-0.1$ and for ${\kappa }_0=0.4$.

Figure 4

A schematic illustration of the Monte Carlo numerical integration algorithm used for the calculation of the four-dimensional transverse acceptance. The separatrix (curved green line) is enclosed by a rectangular box [in blue (dark gray)]. The side lengths of the rectangle, $r_m$ and $v_{r,m}$, are determined by the point of origin and the amplitudes of the transverse position and the transverse velocity, respectively.

Figure 5

Density plot: longitudinal phase-space acceptance (in ${\mathrm{m}}^2$/s) for Zeeman deceleration or acceleration of N(${}^2{D}_{5/2},M_J=5/2$). Green contour lines (near-vertical lines labeled with white boxes): normalized mean longitudinal acceleration ${\overline{a}}_z/{\overline{a}}_{z,m}$ along the beam axis, where ${\overline{a}}_{z,m}$ is the maximum mean longitudinal acceleration. White boxes give values of ${\overline{a}}_z/{\overline{a}}_{z,m}$ for selected contour curves. In the calculation, a cutoff was implemented for acceptances $>2{\mathrm{m}}^2$/s, as stable longitudinal orbits can extend beyond $-4\le \Delta \kappa \le 4$. Blue hatches (near-horizontal lines in central region of ${\kappa }_0$) mark regions in which longitudinal phase stability is observed only because the deceleration-acceleration pulse pattern addresses the same particles as in another phase-stable region (no hatches) with the same ${\overline{a}}_z$ and thus effectively corresponds to a different adaptive ${\kappa }_0$.

Figure 6

Density plot: transverse phase-space acceptance (in ${10}^{-2}$ ${\mathrm{m}}^4/{\mathrm{s}}^2$) for Zeeman deceleration or acceleration of N(${}^2{D}_{5/2},M_J=5/2$). As in Fig., 5, the normalized mean longitudinal acceleration ${\overline{a}}_z/{\overline{a}}_{z,m}$ along the beam axis is marked with green contour lines (near-vertical lines labeled with white boxes).

Figure 7

Longitudinal phase-space distributions of N(${}^2{D}_{5/2},M_J=5/2$) atoms inside the decelerator at different mean longitudinal accelerations ${\overline{a}}_z$ that result from 3D trajectory simulations and from the phase-space model. To account for the different numbers of initial particles in the trajectory simulations, the results for deceleration and acceleration are normalized to the number of particles in the phase-space window at ${\overline{a}}_z/{\overline{a}}_{z,m}=-0.6$ and 0.6, respectively. Under these conditions, the number of unstable particles remaining in the phase-stable region is expected to be small (see text). The separatrices in longitudinal phase space, as obtained from the model, are shown in the results from trajectory simulations [green (light gray) solid curves] for comparison. The color scales are referenced to the (scaled) number of particles from the simulation and the transverse acceptance (in ${10}^{-2}$ ${\mathrm{m}}^4/{\mathrm{s}}^2$) from the model, respectively.

Figure 8

(a) Inverse time needed for one orbit in longitudinal phase space vs the maximum relative position $\Delta \kappa$ of nonsynchronous particles at different switch-off positions ${\kappa }_0$. (b) Inverse time for one revolution in transverse phase space as a function of ${\kappa }_0$ for a particle moving close to the beam axis ($\Delta r\to 0$ mm). The data are calculated for the Zeeman deceleration of nitrogen atoms in the ${}^2{D}_{5/2},M_J=5/2$ state at $v_z=500$ m/s.

Figure 9

Density plot: overall phase-space acceptance (in ${10}^{-2}$ ${\mathrm{m}}^6/{\mathrm{s}}^3$) for Zeeman deceleration or acceleration of N(${}^2{D}_{5/2},M_J=5/2$). As in Figs. 5 and 6, the normalized mean longitudinal acceleration ${\overline{a}}_z/{\overline{a}}_{z,m}$ along the beam axis is marked with green contour lines (near-vertical lines labeled with white boxes). Orange hatches (light near-horizontal lines at left- and right-hand sides): regions in which only particles with very high longitudinal velocities or large displacements with respect to the synchronous particle are phase stable. White crossed hatches: particle motion close to the synchronous particle is phase stable. Blue hatches (dark near-horizontal lines in the central region): regions in which phase stability is observed owing to a deceleration or acceleration pulse sequence which addresses particles in the same phase space as that for the same value of ${\overline{a}}_z$ in a region marked with orange or white crossed hatches, respectively.

Figure 10

(a) Overall phase-space acceptance $V$ obtained from the phase-space model and (b) the number of transmitted particles $N$ in the trajectory simulation (dots) vs the normalized mean longitudinal acceleration ${\overline{a}}_z/{\overline{a}}_{z,m}$. $V$ is proportional to $N$ since the trajectory simulation was carried out with uniform initial position and velocity distributions. Green (medium gray) traces and black traces in (a) correspond to the overall acceptance at $v_z=200$ and 800 m/s, respectively. The number of decelerated (accelerated) particles in (b) is derived from the number of particles in the phase-space windows shown in Fig. 9; the solid lines are a guide to the eye only. The number of accelerated particles [green (medium gray) dots] is upscaled for visibility.

Figure 11

Longitudinal phase-space distributions for Zeeman (top) deceleration and (bottom) acceleration of N(${}^2{D}_{5/2},M_J=5/2$) atoms. Results are obtained from particle-trajectory simulations using (left) an adaptive and (right) a constant switch-off position ${\kappa }_0$. In the case of an adaptive ${\kappa }_0$, a mean longitudinal acceleration of $|{\overline{a}}_z|/{\overline{a}}_{z,m}=0.7$ is used. The values for ${\kappa }_0$ are chosen such that the final velocities are the same in both modes of operation, i.e., ${\kappa }_0=-0.1$ for deceleration (from 800 to 410 m/s) and ${\kappa }_0=1.8$ for acceleration (from 200 to 710 m/s). Normalization is the same as in Fig. 7, but the color scale is adjusted to increase the contrast. For comparison, longitudinal separatrices from the model are shown as solid green (light gray) curves.

Figure 12

Overall phase-space acceptance $V$ as a function of normalized mean longitudinal acceleration $|{\overline{a}}_z|/{\overline{a}}_{z,m}$ for (bottom) Zeeman deceleration and (top) Zeeman acceleration of N(${}^2{D}_{5/2},M_J=5/2$) atoms at different overlap times between adjacent coils, as indicated in the legend. Overlap times are linked to the time period $T$. Infinitely fast rise and fall times are assumed.

Figure 13

Zeeman deceleration sequences imitating the (left) $s=1$ and (right) $s=3$ operating modes of a Stark decelerator. The lower traces schematically indicate the switching times of four consecutive deceleration coils, and the red lines in the top panel illustrate the magnetic field experienced by the synchronous particle (initial velocity of 500 m/s). A guiding sequence (${\overline{a}}_z=0$) is chosen for the $s=1$ mode. In the $s=3$ sequence, ${\overline{a}}_z/{\overline{a}}_{z,m}=-0.14$, and a current of 150 A is applied to every focusing coil (dashed blue curves). All other coils are operated at 300 A. The horizontal bars indicate the time periods $T$ and $2T$ that are used to obtain the mean longitudinal acceleration ${\overline{a}}_z$.

Figure 14

Longitudinal phase-space distributions for Zeeman guiding and deceleration of N(${}^2{D}_{5/2},M_J=5/2$) atoms in the (top) $s=1$ and (middle and bottom) $s=3$ modes of operation, respectively. The focusing coils in the middle panel are operated at 150 A. Otherwise, a current of 300 A is applied to the coils. The mean longitudinal acceleration is ${\overline{a}}_z/{\overline{a}}_{z,m}=0$ ($s=1$, top), $-0.21$ ($s=3$, 150 A, middle), and $-0.23$ ($s=3$, 300 A, bottom). Trajectory results and longitudinal separatrices from the model [green (light gray) solid curves] are shown in the left column. The output from the model is displayed in the right column. The color scales are referenced to the number of particles from the simulation and the transverse acceptance (in ${10}^{-3}$ ${\mathrm{m}}^4/{\mathrm{s}}^2$) from the model, respectively. For the sake of clarity, the color code is matched to the maximum value among the three plots in each column.

Figure 15

(a) Overall phase-space acceptance $V$ for the $s=1$ and $s=3$ modes as a function of the normalized mean longitudinal acceleration ${\overline{a}}_z/{\overline{a}}_{z,m}$, calculated using the model for phase stability; see legend for assignments. (b) Number of transmitted particles $N$ for the same modes of operation [color code as in (a)] vs ${\overline{a}}_z/{\overline{a}}_{z,m}$ that is obtained from the trajectory simulation (initial velocity of 500 m/s) by counting the number of particles in the phase-space window (see Fig. 14). The solid lines are a guide to the eye only. The scaling of the trajectory data is not the same as in Fig. 10. Here, a value of $N=10$ corresponds to a transmission of 100 phase-stable particles per 1 million initial particles. The final velocities are within a range of 240–690 m/s for the $s=3$, 300 A mode.

Figure 16

(a) Trajectory simulation results for Zeeman deceleration of N(${}^2{D}_{5/2},M_J=5/2$) atoms in the acceleration-deceleration mode using $|{\overline{a}}_z|/{\overline{a}}_{z,m}=0.4$, 0.7, 0.8, and 0.9, as indicated in the legend. The particle numbers within the phase-space window (see Fig. 7), $N$, are plotted against the normalized effective mean acceleration ${\overline{a}}_{z,e}/{\overline{a}}_{z,m}$, where ${\overline{a}}_{z,e}/{\overline{a}}_{z,m}=0$ corresponds to a velocity of 500 m/s. The solid lines are a guide to the eye only. The scaling of the trajectory data is the same as in Fig. 15. (b) and (c) Velocity distributions $\Delta v_z$ for $|{\overline{a}}_z|/{\overline{a}}_{z,m}=0.4$ and 0.9 at different ${\overline{a}}_{z,e}/{\overline{a}}_{z,m}$, respectively. The line colors (shades) in (b) and (c) correspond to the colors (shades) of the circles in (a).