Velocity-selected molecular pulses produced by an electric guide
Electrostatic velocity filtering is a technique for the production of continuous guided beams of slow polar molecules from a thermal gas. We extended this technique to produce pulses of slow molecules with a narrow velocity distribution around a tunable velocity. The pulses are generated by sequentially switching the voltages on adjacent segments of an electric quadrupole guide synchronously with the molecules propagating at the desired velocity. This technique is demonstrated for deuterated ammonia (ND), delivering pulses with a velocity in the range of and a relative velocity spread of at FWHM. At velocities around , the pulses contain up to molecules each. The data are well reproduced by Monte-Carlo simulations, which provide useful insight into the mechanisms of velocity selection.
Present address: ]Universiteit Twente, Mesa+ Institute for Nanotechnology, Postbus 217, 7500AE Enschede, The Netherlands.
Beams of cold molecules offer exciting prospects for experiments in physics and chemistry Dulieu2006 (); Smith2008 (); Carr2009 (); Krems2009 (). This includes cold-collision studies and cold reaction dynamics Krems2005 (); Gilijamse2006 (); Krems2008 (); Willitsch2008 (); Sawyer2009 (); Zuchowski2009 () as well as high-resolution experiments to determine, for example, the electric dipole moment (EDM) of the electron Hinds1997 (); Meyer2008 (); Tarbutt2009 (); Vutha2009 (). For these measurements a beam of slow molecules with well-defined velocity and a high degree of internal-state purity is advantageous or even mandatory. Velocity-selected pulses of slow cold molecules have been produced so far only by deceleration techniques Bethlem1999 (); Gupta2001 (); Fulton2004 (); Nar08b (); Hogan2009 (). In this work we present an alternative method based on velocity filtering.
In its original concept, the technique of velocity filtering by an electric or magnetic guide Rangwala2003 (); Junglen2004a (); Patterson2007 () is a method for producing a continuous beam of slow molecules. The main advantage of this method lies in its simplicity and in the high flux obtainable at low average velocities. In addition, it is a very general method since it is applicable to polar (electric guide) and paramagnetic (magnetic guide) molecules Sommer2009 (); Krems2009 () as long as they have a reasonably large positive Stark shift (linear or quadratic MotschWater ()) or Zeeman shift. Besides, no modifications of the guide are required when changing molecular species. Typically, these molecules are extracted from a thermal effusive source. The extracted beam therefore contains molecules populating different internal states Motsch2007 (). By adding a buffer-gas cooling scheme prior to velocity filtering, the purity of the internal-state distribution can be strongly increased Maxwell2005 (); vanBuuren2009 (); Sommer2009 ().
To obtain monokinetic molecular pulses, we have now extended the electrostatic guiding technique by including velocity selection. The latter is reminiscent of the experiment carried out by Eldridge Eldridge1927 () in 1927 and employed in many other experiments afterwards Scoles1988 (), in which velocity selection was achieved by mechanically chopping a continuous molecular beam effusing from a slit. In our setup the selection is done electrically by switching the voltages applied to the electrode segments of the guide on and off in a specific sequence. The switching times are adjusted to the desired molecular velocity. In this way molecular pulses with a narrow velocity distribution around a desired velocity are obtained. Our velocity-selection technique is highly versatile, demonstrating the generality of the electric-guiding technique. It allows for an immediate transition from a continuous guided beam to a pulsed mode of operation along with tuning to different desired velocities.
Ii Experimental Setup
The setup used to guide polar molecules is shown schematically in Fig. 1. This is the same setup used for previous experiments and has been described in detail elsewhere MotschBoosting (); MotschWater (). Briefly, it consists of three differentially pumped vacuum chambers accommodating the electrodes of the electric guide.
Four parallel stainless-steel rods with a diameter of in a quadrupole configuration with spacing between the rods constitute the electrodes of the guide. Alternating high voltages on neighboring electrodes give rise to a quadrupole field, which has a field minimum at the center (see inset, Fig. 1). Thus polar molecules in low-field-seeking (lfs) states, experiencing a positive Stark shift, are transversely confined, provided their kinetic energy does not exceed the potential barrier (trap depth). The Stark shift at the maximum of the trapping field determines the maximum transverse velocity . A bend in the guide’s electrodes limits the molecular velocity in the longitudinal direction. The longitudinal cutoff velocity is obtained by equating the centrifugal force to the restoring force caused by the Stark shift , where is the radius of curvature of the guide electrodes (see Fig. 1) and is the free inner radius of the guide, at which the maximum trapping field is reached (see inset of Fig. 1).
The guide is split into four segments. The guide segments and made of bent electrodes are located in the first and second vacuum chamber, respectively, while the straight segments and are placed in the differential pumping sections connecting the vacuum chambers. The guide segments are separated from each other by gaps of . The molecules are injected into the guide in the first vacuum chamber through a ceramic tube with a diameter of connected to a gas reservoir. The ceramic tube is detached from the guide by a gap of . The guided molecules are detected in the third vacuum chamber by a quadrupole mass spectrometer (QMS, Pfeiffer QMG422) positioned downstream from the guiding electrodes. The guided molecules are ionized by electron impact and mass-filtered. Single-ion detection is achieved by employing a secondary electron multiplier.
Iii Velocity Selection Scheme
The molecules which are trapped in the electrostatic guiding potential span a large continuum of longitudinal velocities. In our experiment we start with a continuous flow of guided molecules whose flux is described by the relation
where is the velocity distribution of the molecules with respect to their longitudinal velocity . The integration is performed over the interval of longitudinal velocities from 0 to the maximum longitudinal velocity in the beam, . This implies that under steady-state conditions molecules with all possible longitudinal velocities are present in the guide at every instant of time. To filter only molecules with a certain longitudinal velocity out of the total flux, the velocity-dependent selection scheme described below is applied. In order to avoid modification of the velocity distribution stemming from collisions of slow molecules with fast molecules near the exit of the ceramic tube MotschBoosting (), all our experiments are performed at a low reservoir pressure ().
As described in the previous section, the guide is composed of four segments, which can be switched independently to a guiding or non-guiding configuration. By applying an appropriate switching sequence to the guide segments ideally only molecules that continuously experience a guiding field are steered to the end of the guide. Most of the other molecules are lost from the guide and do not reach the detector. This results in molecular pulses characterized by a certain velocity and velocity spread. The process is schematically presented in Fig. 2. At time segment is in guiding configuration while segments and are switched off. Segment is on all the time to avoid switching transients in the response of the mass spectrometer. Segment remains on for time (), during which molecules with all possible longitudinal velocities below are guided. They cannot, however, propagate farther through segment . At time (), shortly before segment is turned off, segment is turned on. This gives rise to an overlap time interval during which both segments are in guiding configuration. This ensures that in this time window molecules in the gap and its immediate vicinity will experience a continuous electric quadrupole field and can enter the subsequent segment without being disturbed by the switching of the electric field. Thus the overlap interval defines a molecular pulse with duration containing molecules with all longitudinal velocities below the cutoff velocity . This pulse traverses the gap and is launched into segment , while segment remains off. The time interval during which segment is in guiding configuration depends on the desired longitudinal velocity according to , where is the length of segment . Just before segment is switched off at time , segment is turned on at time , resulting in the overlap interval . This opens up a pathway for molecules to bridge the gap and enter segment . Segment remains in non-guiding configuration. It is important to point out that when moving along segment , the initially short molecular pulse spreads along the propagation line as a result of longitudinal velocity dispersion. Molecules that arrive at the gap between segments and before segment is switched on are lost. The same occurs for molecules which are too slow and are still in segment when the latter is turned off. Only the molecules whose velocity is matched to their arrival time at the gap between segments and experience a continuous guiding field and are accepted by segment .
Since the overlap intervals and are not infinitesimally short, the velocity distribution of the molecules has a finite width around the selected velocity. The center velocity as well as the minimum and the maximum velocity of the molecules comprising the output molecular pulse arriving at the detector can be graphically derived with the help of the diagram shown in Fig. 2 (c). The guiding configurations are designated by the white boxes, while the grey shading designates the non-guiding configurations. The slopes of the black lines represent the lower and the upper boundaries for the velocity of the molecules arriving at the detector at time . The slopes of the red dashed lines mark the lowest and the highest admissible velocities in the produced pulse.
From the switching times, the maximal repetition rate of the pulses can be determined. The rate is given by the time the molecules need to move from the entrance of the first segment to the exit of the second segment. This stems from the fact that the first segment can be switched on again only when the molecular pulse has entered the third guide segment. In our setup repetition rates of the order of a few hundred Hertz can be realized for molecular pulses with velocities of a few tens of . To give an example, at a velocity of a maximal repetition rate of can be employed. In principle, higher repetition rates are attainable by using a configuration with shorter segments.
To determine the properties of the molecular pulses we have performed time-of-flight measurements. The obtained information includes the shape of the pulse, its intensity, its width, and its center velocity.
Fig. 3 shows the time-of-flight signal of a molecular pulse with a center velocity of . The velocities of the detected molecules are determined from their arrival times by converting the acquired signal from the time domain to the velocity domain. This is done via the relation , where stands for the arrival time of the molecules relative to the middle of the time interval , and is the traversed length. The bin width of the histogram, , is much smaller than all relevant time scales.
To rationalize the pulse formation we analyze velocity distributions at a selected center velocity of (see Fig. 4). First, we performed time-of-flight measurements switching off all four electrodes of the respective segments to achieve the non-guiding regime. This non-guiding configuration of the electrodes is referred to as grounded off-configuration (Fig. 4 (left inset)). The resulting velocity distribution is given by the black filled-dotted curve in Fig. 4. The broad shape results from molecules that survive in the guide during the off-configuration. In our further discussion these molecules are referred to as residual molecules. These molecules are not guided by an electric field but manage to reach the next guiding segment simply by free rectilinear flight. The asymmetric shape of the broader velocity distribution results from the bend of segment after the gap (see Fig. 1). Residual molecules whose velocities are higher than the desired velocity set by the switching time can make it through the gap and enter segment C by free rectilinear flight. If these molecules reach the bend of segment at times they escape from the guide. This leads to a reduction of the broadening at the high-velocity side of the velocity distribution. On the other hand, residual molecules whose velocities are smaller than the desired velocity , can remain in segment and manage to reach segment even at times by free rectilinear flight. These molecules are guided in segment since this segment is on for times , and thus contribute to the broadening of the velocity distribution at its low-velocity side. To reduce the contribution of residual molecules to the pulse, we applied another non-guiding configuration, termed gradient off-configuration, in which three of the electrodes of the guide are switched off and one of the electrodes remains at (Fig. 4 (right inset)). This non-guiding scheme results in a deflection field of in the center of the guide pushing the residual molecules out of the guide. This leads to the narrowed velocity distribution given by the red open dots in Fig. 4. The temporal profile of the corresponding pulse is shown in Fig. 3.
The overlap intervals have been optimized to maximize the number of molecules per pulse without significantly increasing the width of the velocity distribution. The duration of the overlap intervals has been adjusted proportional to to ensure that for every the molecules travel the same distance during the overlap times. In our experiment . In this way molecules of all velocities experience the same switching transient fields. For our setup optimal overlap times around hundred microseconds are found. For example, an overlap time of has been used for and for the data shown in Fig. 4.
An important parameter of the segmented-guiding technique is the number of molecules in a pulse. To determine this value we have used the background-subtracted data of the histograms representing single-molecule detections of the QMS. Using a previous calibration of the QMS Sommer2009 (), we have estimated that the pulse at contains molecules. This number can be increased to molecules by increasing the pressure in the reservoir MotschBoosting ().
As a next step, we compare the velocity distributions at different center velocities. In Fig. 5 the experimentally determined velocity distributions of molecular pulses for several center velocities are displayed by red dots. Overall, good agreement between the heights of the velocity distributions of the velocity-selected pulses and the measured velocity distribution for a continuous beam of guided molecules is found. This demonstrates that electric switching does not lead to a reduction of the number of molecules at the desired center velocity of the pulse. We have also successfully extracted velocity-selected molecular pulses from our buffer-gas cooled source vanBuuren2009 (); Sommer2009 () to demonstrate that the same technique can be employed to create velocity-selected pulses with a high degree of internal-state purity.
To further substantiate the experimental results we compare them to the results from Monte-Carlo simulations for a bent electric guide. The simulations employed a model guide accounting for all physical aspects of the experiment and propagated molecules with the appropriate parameters and starting conditions. The input range of longitudinal velocities was from to . Simulations for pulsed operation as well as for continuous guiding (with all segments of the guide on at all times) have been carried out. As in the experiment, the velocities are obtained from their arrival time in the detector with respect to the time . Due to the relatively large statistical uncertainties the simulated peaks have been smoothed using the SavitzkyGolay smoothing filter Press1996 () employing a third-order polynomial with 6 points. The simulated continuous velocity profile has been smoothed as well using the above method with 30 points. The results of the simulations are presented in Fig. 5 (black solid curves). Good agreement is found between the experimental data and the simulations for the continuous beam as well as for the pulses at different longitudinal velocities .
The full widths at half maximum (FWHM) of the velocity distributions, , are obtained from fits with Voigt profiles Olivero1977 (). In this way the ratios can be derived. Overall, a value of is obtained for the relative velocity spread. The non-zero width of the velocity distributions is attributed to two effects, the finite duration of the overlap intervals proportional to and the presence of residual molecules. In the following we describe these two effects and their contributions to the widths of the velocity distributions in more detail.
The desired longitudinal velocity is determined by (see Section 3). By differentiating this expression with respect to time and by substituting for , we obtain the following relationship between the pulse width and the width of its velocity distribution, . The two short but non-zero overlap intervals () determine a triangular velocity distribution resulting from a convolution of two rectangular velocity distributions. Therefore we obtain the following relation for the FWHM of the velocity distribution stemming from the finite overlap intervals,
We determine that for our settings. From the measured widths we can conclude that residual molecules substantially contribute to the observed broadening. Therefore, decreasing the overlap intervals to smaller values does not lead to much narrower velocity distributions. Indeed, we have observed that for shorter overlap intervals of a few tens of microseconds the widths of the velocity distributions do not decrease significantly. They are an order of magnitude broader than expected from Eq. 2 for these short overlap times. Additionally, a substantial reduction in peak height is observed for the slowest pulse with a center velocity of . This additional reduction has not been seen with overlap times of a few hundred microseconds.
The residual molecules manage to reach the subsequent segment shortly before or shortly after the overlap intervals. The number of these molecules depends on the mean survival time they can spend in gradient off-configuration before being kicked out by the deflection field. This time depends on the transverse velocity of the molecules and on the strength of the deflection field. For a straight guide, both the transverse velocity and the strength of the deflection field are independent of the set velocity . Indeed, we have verified this by fitting the transverse velocity distributions obtained from the Monte-Carlo simulations with Gaussian profiles. The fits demonstrate that for all longitudinal velocities the average value of the transverse velocity roughly equals .
To account for the broadening resulting from residual molecules in addition to the broadening stemming from the finite overlap time, we use the following model including both effects. Let (see Fig. 2). Let us also first assume that there are no residual molecules in the vicinity of the gap between segments and , and the overlap time . This assumption implies that the broadening of the velocity distribution originates only from the finite overlap time and from residual molecules in the vicinity of the gap between segments and . The minimum velocity at which molecules in segment can still reach the subsequent segment is . The maximum velocity at which some molecules remain guided in segment is . After substituting , the above expressions are transformed into and , respectively. The difference between and defines a rectangular distribution of longitudinal velocities determined by the overlap time and the survival time of the residual molecules in the vicinity of the gap between segments and . By analogy, the same model is applicable to the gap between segments and if we assume that now there are residual molecules only in the vicinity of the gap between segments and , , and . The overall velocity distribution is thus obtained by the convolution of the two rectangular velocity distributions. The resulting velocity distribution has a triangular profile with a FWHM given by the formula
for a guide without bends. It is obvious that for infinitesimally short survival times () and for overlap times satisfying the condition , the above formula reduces to Eq. 2. In this model the survival time can be considered as an effective additional overlap time.
To be able to verify the analytical model and to determine , we have performed simulations for a segmented straight guide with experimental conditions similar to the ones in the real experiment with the bent guide. The FWHM of the velocity distributions of the molecules reaching the QMS have been fitted to , resulting in as shown in Fig. 6. This value agrees with our estimate of based on the typical field strength of our deflection field. Note that the mean longitudinal velocities tend to shift to values higher than , especially for large (See gray points in Fig. 6). This effect is caused by non-deflected molecules as well, which are more likely to reach the detector at high longitudinal velocities. For disagreement between the analytical model and the simulation appears due to the limited range of generated velocities. This shows that molecules with velocities beyond the generated range of velocities contribute to these pulses. That is why bends are required to avoid contributions from high-velocity molecules and to keep the mean velocity close to .
The FWHM of the velocity distributions for the bent guide are presented as a function of the mean velocity of the distribution for both the experimental data and the Monte-Carlo simulations in Fig. 6 (red circles and inverted triangles, respectively) as well. The results were obtained from fits of the velocity distributions with Voigt profiles. Good agreement is found between experimental data and simulations. This indicates that the main broadening effects, finite overlap times and contribution of residual molecules, are well understood. Compared to the straight guide, the widths increase less strongly at high . This is caused by the second bend, which accepts molecules only while being in guiding configuration whereas in a straight guide larger contributions arise from residual molecules (which can survive the deflection fields during time ). This clearly shows that the second bend reduces the broadening of the peaks.
An interesting aspect is the effect of the different elements of the guide on the behavior of the molecular pulses, which is important for optimization of the experimental setup. Towards this end, we looked at the results from the Monte-Carlo simulations. We recorded the instantaneous velocities of the propagated molecules at the beginning of the guide and at the detector and compared them with the velocities obtained from their time of flight. The pulses in the time domain for the velocities of and produced at the beginning of the guide are much narrower than the ones produced at the detector. This can be explained by velocity dispersion in the guide. Another reason for this effect, however, might be velocity mixing, which is most likely to occur in the bends of the molecular guide. To study this effect we compared the simulated initial (at the entrance of the guide) velocity distribution of the molecules reaching the detector with their simulated final velocity distribution at the position of the detector. The two velocity distributions are very similar, which implies that the longitudinal velocities do not alter strongly during the propagation in the guide. Therefore, we conclude that no observable velocity mixing occurs in the bends and that the only reason for the pulse broadening in the time domain under our experimental conditions is the velocity dispersion.
Our method offers prospects for further improvements. As long as the velocity pulses do not overlap, the pulse width depends only on the total length of the guide and the strength of the deflection field. Smaller pulse widths can be obtained using higher deflection fields in non-guiding configuration to eliminate more efficiently contributions of rectilinearly flying molecules. This can be achieved by supplying a higher positive voltage to the non-grounded electrode during off-configuration (see Fig. 4). From simulations for a straight-guide geometry, we estimate that increasing the voltage from kV to kV will decrease the relative pulse width from (see Fig. 6) to for ms. Another approach to minimize these undesired contributions is to use a bent-guide geometry, in which the molecules are more easily expelled from the guide during the deflection period. Ultimately the pulse width will depend only on the overlap time used. In our setup the relative contribution amounts to of , which can even be further reduced by applying shorter overlap times at the expense of a reduced number of molecules per pulse. Higher overall fluxes can be obtained by running the experiment at a higher repetition rate (up to a few hundred Hertz) without sacrificing on width. By segmenting the guide into more and smaller parts and employing faster switching times, multiple pulses can be stored in the guide at the same time, resulting in even higher rates.
The possibility to tune the width of the velocity independent of the set velocity adds a new degree of freedom to the velocity-filtering technique, which can easily be varied and optimized. In continuous mode the beam is characterized by high density and high flux with a broad velocity spread, whereas pulses with small widths in velocity are obtained at lower density and flux. This technique combined with a buffer-gas cooled source has promising applications in collision and chemistry experiments at low energies. For example, in the collision experiments of Willitsch2008 (), velocity filtered and guided molecules react with cold Coulomb-crystallized ions. Sofar the energy-dependence of the scattering rates has been studied by lowering the voltages on the guide electrodes resulting in a lower cutoff velocity and therefore an on average lower . Unfortunately, lowering the guiding field also reduces . This can be avoided by using a segmented guide, which has also the advantage to narrow the velocity around a set velocity. In addition, the rotational temperature can be independently adjusted by varying the density of the buffer gas. Cold collisions can also be studied by directing velocity-selected pulses into a dense beam as has been shown in Gilijamse2006 () or by guiding the slow molecules into a long-lived trapped sample. Velocity-selected molecules could then collide with laser-cooled atoms stored in an electric trap Rieger2007 (); Schlunk2007 (). Another realm of application of our method could be molecular interferometry with guidable molecules. A major challenge in this field is the production of molecules with well-defined and controllable velocities Brezger2003 (); Gerlich2007 (). Pulses from a segmented guide could be tuned to a setting at which highest contrast is obtained. Therefore, the segmented-guiding technique could be well-suited for such experiments with molecules having sufficiently large dipole moments.
Financial support from Deutsche Forschungsgemeinschaft (Cavity-Mediated Molecular Cooling CMMC and Munich-Centre for Advanced Photonics MAP) is gratefully acknowledged. We appreciate initial discussions on the topic with Sadiq A. Rangwala.
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