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Two-Dimensional Trapping of Dipolar Molecules in Time-Varying Electric Fields

T. Junglen, T. Rieger, S. A. Rangwala, P.W. H. Pinkse, and G. RempeMax-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

(Received 9 October 2003; published 1 June 2004)

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Simultaneous two-dimensional trapping of neutral dipolar molecules in low- and high-field seekingstates is analyzed. A trapping potential of the order of 20 mK can be produced for molecules such asND3 with time-dependent electric fields. The analysis is in agreement with an experiment where slowmolecules with longitudinal velocities of the order of 20 m=s are guided between four 50 cm long rodsdriven by an alternating electric potential at a frequency of a few kHz.

DOI: 10.1103/PhysRevLett.92.223001 PACS numbers: 33.80.Ps, 33.55.Be, 39.10.+j

FIG. 1. Schematic of the four-wire setup for typical operat-ing voltages of �5 kV , where E0 � 35 kV=cm and � � 2:3�103 kV=cm3 . The voltages on a pair of opposite electrodes areswitched, as shown in (a), in order to produce a time-varyingfield that alternates between configurations 1 and 2, shownin (b), with a repetition rate in the kHz range. The radius

ping can be realized in many different ways. Our trapping rmax used in simulations is indicated by the dashed circles.

Cold molecules offer new perspectives, e.g., for highprecision measurements [1] and collisional physics stud-ies [2]. Pioneering work on cold molecules has been doneusing cryogenic buffer gas cooling [3]. Another promis-ing technique for the production of cold molecules isbased on the interaction of dipolar molecules with in-homogeneous electric fields. For example, low-field seek-ing molecules (LFS) have been slowed down in suitablytailored time-varying electric fields [4] and have beentrapped in inhomogeneous electrostatic fields [5,6].Furthermore, efficient filtering [7] of slow LFS from aneffusive thermal source using a bent electrostatic quadru-pole guide has been demonstrated [8].

Compared to LFS, the manipulation of high-field seek-ing molecules (HFS) is much more difficult. This is duemainly to the fact that electrostatic maxima are not al-lowed in free space, and hence HFS are quickly lost onthe electrodes. Nevertheless, guiding of HFS in Kepplerorbits [9] and deceleration as well as acceleration of HFS[10] is possible. Despite this progress in manipulating di-polar molecules, all techniques realized so far are suitedonly for either LFS or HFS, not both simultaneously.However, in future experiments with trapped samples ofcold molecules, collisions or the interaction with lightfields are likely to change HFS into LFS and vice versa.Therefore, a technique to trap both species simultane-ously is vital. In this Letter, we investigate both theoreti-cally and experimentally a new technique that can trapboth HFS and LFS. In particular, we report on the firstexperimental demonstration of two-dimensional trappingof slow ND3 molecules from an effusive source in a bentfour-wire guide driven by an alternating electric field.

Trapping neutral particles in oscillating electric fieldsworks as follows [11]. The force on a molecule in anelectric field, E, is given by ~FF � � ~rrW�E�, with W�E�the Stark energy of the molecule. Polar molecules such asND3 and H2CO predominantly experience a linear Starkshift, W � sE, where s is the slope of the Stark shift.Other molecules, however, and atoms usually experiencea quadratic Stark effect, W � � 1

2�E2, with the polar-

izability �.The inhomogeneous electric fields required for trap-

0031-9007=04=92(22)=223001(4)$22.50

configuration is sketched in Fig. 1. It consists of fourparallel rods with voltages as indicated. The field israpidly switched between two dipolelike configurationswith angular frequency !. Close to the center, the fieldcan be expanded harmonically:

E � E0 �H�t���x2 � y2�; (1)

where E0 is the field in the center and � is (half) thecurvature of the field. The step function H�t� � 1 if nT <t < �n� 1

2�T and H�t� � �1 otherwise, with T � 2�=!the period of the driving field and n an integer.

Independent of the slope and the sign of the Stark shift,the saddlelike dipole potential derived from Eq. (1) con-fines the particle in one direction and repels it in theperpendicular direction, depending on time. Therefore,the time average of the force is small at every position andidentical to zero for a linear Stark shift. However, theparticle performs a micromotion, which is locked to theexternal driving field so that the time-averaged force doesnot cancel, and the particle experiences a net attractiveforce towards the center. This conclusion is independentof the exact form of H�t�. Therefore, a sinusoidal change

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FIG. 2 (color online). Area of guided molecules intransverse-velocity space obtained from a two-dimensionalMonte-Carlo simulation. Curves for LFS [down-pointing blackand grey (red) triangles] and HFS [up-pointing black and grey(red) triangles] ND3 molecules with the given Stark shifts aredisplayed.

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between two configurations would also work. However,instantaneous switching is more convenient to realize inthe laboratory.

It is not difficult to solve the equation of motion formolecules in this dynamic field. As the x and y degrees offreedom separate, the equation of motion is reduced toone dimension. It reads �xx � �H�t��2x, where we haveintroduced �2 � 2s�=m for a linear Stark shift and�2 � 2�E0�=m for a quadratic Stark shift of a particlewith mass m. Following [12], we search for orbits of theform �x�t�; _xx�t�� � U�t��x�0�; _xx�0��, with U�nT � t� �U�t�U�T�n. Periodic solutions have the form U�T� �U2�T=2�U1�T=2�, with

U1�t� ��

cosh�t�� 1� sinh�t��

�sinh�t�� cosh�t��

�;

U2�t� ��

cos�t�� 1� sin�t��

��sin�t�� cos�t��

�

the evolution operators for the first and the second halfcycle. From this, the main stable region can be found forj�Tj � 3:75. Narrow higher-order stable regions alsoexist but will further be neglected. Hence there exists asharp drive-frequency threshold for every Stark shift,above which trapping is possible, irrespective of thesign of the Stark shift and independent of the initialconditions.

Note that the finite extension of our trap, jxj � xmax,limits the trap depth. This can be estimated by averagingover the micromotion, which is assumed to be smaller andfaster than the macromotion. This results in a trap depthm�2x2max�

4=�24!2�, typically of the order of 10 mK [13].For a more realistic estimate, a detailed analysis is madebelow.

It is interesting to compare molecule trapping withion trapping in a Paul trap [13]. An ion experiencesthe Coulomb force ~FFC, which has zero divergence,~rr ~FFC � 0. As a consequence, the magnitude of theforce is the same for positive and negative ions. Formolecules with a linear Stark shift and in the harmonicapproximation, ~rr ~FF � 0, and there is an accidental cor-respondence with the equations of motion of an ion in aPaul trap. It should nevertheless be emphasized that trap-ping neutral molecules differs from ion trapping in manyways. First, whereas ions are monopolar particles whichcan be trapped in an oscillating quadrupole field, neutralshave permanent or induced dipoles which can be trappedin an oscillating dipole field. Second, in contrast to ions,the force on a molecule in an electric field dependsdramatically on the internal state of the molecule.Third, as is obvious from Fig. 1, the harmonic approxi-mation is valid only in a small region around the center. Incontrast to ions, which have vanishingly little kineticenergy compared to the 105 K trap depth, the temperatureof the molecules is likely to be comparable to the trapdepth. Hence, molecules explore the full volume of the

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shallow trap, including the outer regions. These outerregions are of paramount importance in our experiment.It is here that the harmonic approximation breaks downand the molecule-field interaction allows ~rr ~FF � 0 . Thisleads to different behavior for HFS and LFS away fromthe center, where the time average of the force inconfigurations 1 and 2 is not zero. In particular, theregions near the electrodes are net repulsive for LFSand net attractive for HFS.

To take all these effects into account, we have per-formed a two-dimensional Monte Carlo simulation. Inthe simulation, pointlike particles are injected on axisin a random phase of the driving field. The input veloci-ties vx and vy are varied. The particles are propagatedunder the influence of the periodically poled Stark forcefor 10 ms, a typical time in the experiment. They areconsidered lost if they leave a radius rmax � 1:25 mm.The amount of trapped trajectories can be expressed as anarea A in the two-dimensional vxvy-velocity space quad-rant. A is a good measure for the flux, if the guidablevelocities are equally present in the source. Results forfour different molecular states of ND3 exhibiting linearStark shifts with a slope, s, of �0:6 cm�1=�100 kV=cm�and �1:2 cm�1=�100 kV=cm� are plotted in Fig. 2. Fora molecular state with a Stark shift of 1:2 cm�1=�100 kV=cm� and at 9 kHz for our experimental parame-ters, the occupied vxvy space has almost a quarter-discshape, with which a trap depth of approximately 20 mKcan be associated.

As expected from the analytic analysis, there is asharp turn-on frequency above which guiding is possibleboth for LFS and HFS. For increasing frequencies, thearea A reaches a maximum before decreasing againfor higher frequencies. Note that the maximum of A isapproximately 4 times larger for LFS than for HFS,indicating that LFS can be guided more efficiently. Thisis due to the anharmonicity of the trapping potential away

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from the center, where stable trajectories exist for LFSwith comparatively high initial velocities. This effect alsocauses the substructure in the LFS curves in Fig. 2. Foreven higher frequencies, the amplitude of the micromo-tion becomes very small, and the force on the moleculesapproaches the time average of the force in configu-rations 1 and 2. This time-averaged force is small nearthe center but substantial and repulsive near the elec-trodes. Stable orbits therefore exist in the high-frequencylimit for LFS. The stable orbits occupy disjunct areas invelocity space, of which the area A saturates. The notionof a trap depth becomes difficult in this limit. The behav-ior of HFS, however, is completely different: With thedisappearing micromotion, there are no stable trajectoriesand A vanishes in the limit of high drive frequency.

Obviously, the motion in the full anharmonic potentialcan be very complicated. In fact, we have found numeri-cal evidence for chaotic motion for some initial condi-tions. Yet despite of the complicated dynamics, LFS andHFS are simultaneously trapped and there is a broadoverlap between the trapping regions for different Starkshifts at the same frequencies, as evident from Fig. 2.

Our experimental demonstration of ac guiding uses aneffusive source of thermal polar ND3 molecules; see thesetup shown in Fig. 3 [8]. The double-bent guide consistsof four wires which pass through two vacuum chamberswith two differential pumping sections before ending in aultrahigh vacuum (UHV) detection chamber. The guidehas a length of 500 mm and is made of 2 mm diameterstainless steel rods, with a 1 mm gap between neighbor-ing rods. The radii of curvature of the two bends are25 mm and the rods are built around the 0.8 mm innerdiameter ceramic nozzle, constituting the effusive source.Typical operation pressures in the nozzle are around0.05 mbar in order to maintain molecular-flow condi-tions. Most of the molecules are not guided and escapeinto the first vacuum chamber, where an operational pres-

FIG. 3. Schematic of the experimental setup. ND3 moleculesemerging from the nozzle are injected into the guide. Theslowest molecules are kept within the guide, and after passingtwo differential pumping stages, they enter a UHV chamberwhere they are detected with a mass spectrometer. The fastmolecules are pumped away.

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sure of a few times 10�7 mbar is maintained. In thedetection chamber, where a pressure below 10�10 mbaris achieved, the guided molecules are detected with anefficiency of about 10�4 counts=molecule by a quadrupolemass spectrometer (QMS) [Hiden Analytical, HAL 301/3F] with a pulse-counting channeltron. The pulses fromthe channeltron are processed in a multichannel scaler.The ionization volume of the QMS begins 22 mm behindthe guide. To protect the QMS from the high electricfields, a metal shield with a 5 mm diameter hole is placed1 cm behind the exit of the guide. The time-varyingelectric fields are generated by switching high voltageswith fast push-pull switches. Switching frequencies aretypically in the range of a few kHz. In each phase of thealternating field, opposite rods carry voltages of up to�7 kV. Because of the inversion splitting of the vibra-tional ground state, ND3 shows in good approximation alinear Stark shift in the relevant electric field range�0–100� kV=cm [14].

For detecting ND3 molecules, the QMS is set to mass20. The influence of other gases at this mass is negligible.The QMS does not provide state-selective detection,which is very difficult to achieve if one considers thevariety of molecular states involved in a thermal (room-temperature) ensemble. Our simulation, however, showsthat the detection of HFS is suppressed, because under theinfluence of the high electric fields most of the HFS turnaround at the exit of the guide, moving backwards on itsoutside, or they are reflected back into the guide.

To discriminate the guided molecules from backgroundgas and dark counts, the alternating electric field is peri-odically applied for 200 ms, followed by 200 ms whereno field is applied. This on-off sequence is repeated about40 000 times to obtain a good signal-to-noise ratio. Fromthe instant when the alternating field is switched on, themolecules can propagate along the guide and the signalrises. After about 25 ms, the signal has reached 50%of the maximum increase, corresponding to velocitiesaround 20 m=s or about 0.5 K. When the alternating fieldis switched off, the flux instantaneously falls off. The fluxof guided molecules is determined by subtracting thebackground count rate from that during the last 50 msof the 200 ms-long interval where the field is applied. Thesame measurement without injecting gas yields no signal.

We now investigate the frequency dependence of thesystem. For electrode voltages of �5 kV, the switchingfrequency is varied between 4 and 15 kHz, and the fluxof guided molecules is recorded in frequency steps of500 Hz. The experimental data in Fig. 4(a) show thatthe signal amplitude reaches a pronounced maximum ata frequency of �8 kHz. No discernible signal is ob-served for frequencies below 4 kHz. This behavior canbe understood from the simple analytic theory above,acknowledging the thermal ensemble of molecular states.For every molecular state there exists a cutoff frequencycorresponding to the associated Stark shift, below whichno guiding is possible. This frequency increases with the

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FIG. 4. (a) Molecular flux as a function of the applied fre-quency for rod voltages of �5 kV . The graph in the inset showsthe frequency of maximum flux, !0, as a function of theapplied voltage. The dashed line indicates the !0 /

����V

pdepen-

dence. (b) Flux as a function of the applied voltage. For eachvoltage, the frequency is adjusted for maximum flux accordingto (a). The line is a quadratic fit.

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Stark shift. For example, the cutoff frequencies for mole-cules with Stark shifts of 0.6 and 1:2 cm�1 at an electricfield strength of 100 kV=cm in our guide are 3.3 and4.7 kHz, respectively. For small frequencies stable trajec-tories exist only for molecules with small Stark shifts.These molecules are unlikely to be guided as the potentialdepth is too small compared to their kinetic energy.Higher frequencies allow stable trajectories for moleculeswith higher Stark shifts, and so the number of guidablestates increases. Furthermore the guiding efficiency formolecules with higher Stark shifts is much higher. Botheffects cause the rise in the count rate for frequenciesbelow 8 kHz. For higher frequencies the count rate de-creases because the guided flux for every molecular statedepends on the depth of the potential, which drops offwith !�2.

Our conclusions are supported by two additional mea-surements. First, the dependence between the appliedvoltage and the frequency !0 for which optimum guidingexists is analyzed. In the harmonic approximation, thestable region is given by j�Tj � 3:75, with � /

�����

p. As

the curvature � scales linearly with the applied voltage V,we obtain !0 /

����V

p. Experimental data are shown to-

gether with the expected square root dependence in theinset of Fig. 4(a). Unfortunately, the measurements coveronly a small voltage interval, because for small voltagesno guiding signal could be obtained, whereas sparks

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occurred for higher voltages. Nevertheless, the measureddata agree with the calculated scaling. This allows one tochoose the optimum switching frequency for a givenvoltage. In a second measurement, the flux is measuredas a function of the applied voltage, using the optimumfrequency for each voltage. The experimental data inFig. 4(b) show a quadratic rise in flux with increasingvoltage. This is expected for ND3, where most states havea linear Stark shift [8].

To summarize, two-dimensional trapping of neutralmolecules with alternating electric fields is demonstratedexperimentally. Our result proves that molecules withtransverse and longitudinal temperatures of 20 mKand 0.5 K, respectively, can be filtered out of a room-temperature effusive source efficiently. Our simulationshows that both HFS and LFS are guided, whereas thelow-field seekers are guided and detected with a higherefficiency. The guiding efficiency for both classes willfurther increase if the injected molecules are precooledwith cryogenic methods. An important aspect of ourguide is that it is suited to trap laser-cooled atoms aswell [11]. Two-dimensional trapping of both atoms andmolecules should therefore be possible. Finally, an exten-sion of our technique to trap sufficiently slow moleculesin three dimensions [11] should be feasible.

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