cooling effects in the stark deceleration of rydberg atoms

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Cooling effects in the Stark deceleration ofRydberg atomsmolecules with time-dependentelectric fieldsTo cite this article Y Yamakita et al 2007 J Phys Conf Ser 80 012045

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Cooling effects in the Stark deceleration of Rydberg atomsmolecules with time-dependent electric fields

Y Yamakita12 R Takahashi1 K Ohno1 S R Procter2 G Maguire2 and

T P Softley2

1 Department of Chemistry Graduate School of Science Tohoku University Aramaki Aoba-ku Sendai 980-8578 Japan 2 Department of Chemistry University of Oxford Chemistry Research Laboratory Mansfield Road Oxford OX1 3TA United Kingdom

E-mail yyqpcrkkchemtohokuacjp

Abstract This paper presents calculations for realizing the deceleration of H2 Rydberg molecules with n = 16 where n is the principal quantum number A double-dipole decelerator (Softley T P Procter S R Yamakita Y Maguire G and Merkt F 2005 J Elec Spectrosc Relat Phenom 144-147 113) operated with an optimum time-dependent electric field allows in principle complete deceleration of the Rydberg molecules to zero mean velocity A bunch of molecules in a supersonic beam is decelerated from the initial velocity centered at ~900 ms-1 and translational temperature 1 K to the final velocity 0 ms-1 and temperature 13 mK The calculations are performed using the 4th-order symplectic integrator based on representations in phase space q p and show that an ensemble with narrow q0 and broad p0 distribution is converted to one at standstill with broad q and narrow p Cooling effects are reinforced by field ionization in which the fast components that move to regions of high electric field are effectively filtered out

1 Introduction Controlling the translational motion of gas-phase molecules particularly through the use of multipolar devices has played an important role in reaction dynamics experiments [1] The potential-energy gradient in inhomogeneous electric fields (or magnetic fields) gives rise to forces acting on a molecule and trajectory simulations make it possible to design optics for neutral molecules When the field is kept static the sum of the kinetic energy and the potential energy is conserved However if the field is time-dependent the energy of the molecule is not determined by the position but by the integrated work exerted in the field

)1()d()( 2

1

r

rrrr tFtW

At the position where the force F(rt) is zero ramping updown the electric potential does not exert any work on the molecule Such a property allows cooling of the translational energy of molecules to which conventional laser cooling techniques cannot be applied Indeed the fast-switched multi-stage decelerator developed by Meijer and co-workers has demonstrated the deceleration of neutral

7th Asian International Seminar on Atomic and Molecular Physics IOP PublishingJournal of Physics Conference Series 80 (2007) 012045 doi1010881742-6596801012045

ccopy 2007 IOP Publishing Ltd 1

molecules such as ND3 [2] The deceleration of molecules with respect to the laboratory frame and the cooling down to low temperatures (lt1 K) are expected to lead to advances in ultra-high resolution spectroscopy and physics of ultracold collisions [3] Time-dependent electric fields can also be utilized for trapping neutral molecules in high-field-seeking states The electric trapping of neutral molecules has been demonstrated for low-velocity components of an effusive molecular beam after filtering out with static inhomogeneous fields [4] Other approaches such as He buffer gas cooling [5] photoassociation in a magneto-optical trap [6] and Stark control by optical dipole forces [7] have also been reported Stark deceleration has been demonstrated for Rydberg atomsmolecules including H2 molecules [8-10] Ar and H atoms [1112] and Na atoms [13] However a complete halting has only been demonstrated for H atoms

In this paper we investigate the problem of slowing down the Rydberg atomsmolecules by assuming the two-dipole setup proposed previously [14] In the high-n Rydberg states of any atommolecule the Stark effect is significantly larger than in ground-state molecules The magnitude of the Stark shift for the extreme levels of given principal quantum number n is approximately proportional to n2 The wavefunctions of the hydrogenic Rydberg states are represented by the product of wavefunctions u1 and u2 in parabolic coordinates with quantum numbers n1 and n2 [15]

)2()()()( 21

imeuu The fact that the first-order Stark effect is proportional to k = n1 - n2 is equivalent to the existence of

an electric dipole moment proportional to k located on the atom or molecule of interest Within first-order perturbation theory the effective dipole moment is estimated to be 915 D for the hydrogenic state |nn1n2m = |161500 (1 D 0393 ea0 e elementary charge a0 Bohr radius) which is greater by nearly three orders in magnitude than those for ground-state dipolar molecules It follows that Rydberg atomsmolecules have the potential to be controlled in moderate-strength electric fields (say 1 kVcm) [2] which are significantly weaker than those required for ground-state counterparts (typically 01 - 02 MVcm) On the other hand the drawbacks of this approach include the short lifetimes of Rydberg states typically about a few s for n 16 or less due to spontaneous or blackbody-induced emissions for atomicmolecular systems and due to predissociation and autoionization for molecular systems Thus deceleration must be completed in a period less than ~ 10 s unless special stabilization methods are applied A steep electric-field gradient is required over a short length of flight but exposing the Rydberg atommolecule to a large field-strength would cause field ionization and level crossing In order to avoid this complication time-dependent alternation of the spatial distribution of the electric field is an efficient approach [111314]

2 Calculations An atom or molecule with energy U in an inhomogeneous electric field E(rt) experiences a force which is given by

)3()()(

)(rr

rrF tE

tEUt

The first part of the product is a slope for the Stark effect as a function of electric field strengths E =

(Ex2 + Ey

2 + Ez2) U is calculated by diagonalisation of the effective Hamiltonian

)4(0 eEzHH


2

with a basis set of the form N+ n l J MJ where J is in this case the vector sum of the core rotation N+ and the Rydberg electron angular momentum l The perturbation term in Eq (5) is evaluated using the conventional vector-coupling algebra [16]

)5(10

1)12)(12()1(

1 lrll

NJJl

MJ

MJ

JJ

MJlnNznlJMN

JJ

MlNJJ

JJ

J

The second part of the product in Eq 3 represents the gradients of electric fields which are obtainable using well-known formula for electric field gradients applied with thin wires [1]

Now let us consider a decelerator of the same geometry as in Ref [14] except for the position for photoexcitation (figure 1) Briefly two dipoles comprising thin wires of 05 mm diameter with a space gap 35 mm are placed with a separation of 8 mm along the molecular beam A quadrupole field created by the two dipoles changes its minimum position synchronously with the motion of a bunch of molecules We assume a cold beam containing H2 molecules for which the initial velocity and translational temperature are v0 = 900 ms-1 and 1 K respectively The molecules are populated in the extreme low-field-seeking level of the n = 16 Rydberg state by two-colour laser excitation at a position 2 mm away from the centre of the dipoles Gaussian beam waists of 100 m define the volume of excitation Monte Carlo calculations are performed to yield 10000 trajectories using a 4th order symplectic integrator DH

)6()exp(0

0

qp

Dqp

Ht

where the operator DH is derived from the Lagrange equations In principle the infinitesimal integrator DH can be decomposed to arbitrary higher orders asymptotically in the forms of kinetic and potential parts [17] The forces are calculated for the Stark sublevel n = 16 N+ = 2 MJ = 0 based on the experimentally-confirmed Stark map which shows energy level variations as a function of electric field [8]

3 Results and Discussion Figure 1 presents the time-dependent voltages applied to dipoles A and B which increase or decrease exponentially respectively A local minimum of electric field strength |E| created in the quadrupole field shown in figure 2 can be optimized to follow the translational motion of the molecules An electric-field gradient is applied in the direction opposing the molecular beam whilst the absolute field strength experienced by the molecules is kept below the limit of field ionization or level crossing In fields greater than 1540 Vcm the Stark sublevels of the bluest state of n = 16 and the reddest state of n = 17 will undergo avoided crossings with each other [9] such that on increasing fields beyond that limit the energy levels that were originally going up with field will go down and vice versa Therefore the field experienced by the molecule must be kept below the Inglis-Teller limit 1540 Vcm otherwise the system will lose the linear response to the electric field At the avoided-crossing points the probability of diabatic transition between the neighbouring sublevels is negligible since the energy separation of the crossing ~1 cm-1 is far too large for such transitions A population in the extreme low-field-seeking state should be prevented from undergoing population dispersion by the Majorana transition at zero fields In optimizing fields we therefore impose the condition that the electric fields are kept in a range 100 Vcm - 1540 Vcm


3

Figure 1 (a) Schematic illustration of a two-dipole decelerator and (b) time dependence of the voltages applied on dipoles A and B

Figure 2 (Colour online) Contour maps showing initial and final electric field strengths and the positions of Rydberg H2 molecules (red dots) at (a) t = 0 s (b) t = 5 s and (c) t = 985 s respectively The molecules are excited to Rydberg states n = 16 N+ = 2 MJ = 0 at the position (y z) = (00) at t = 0 s by laser beams with a Gaussian beam waist of 100 m

Figure 2 shows the initial and final electric fields at time t = 0 s and 985 s respectively The

minima of the potential valleys deviate from the centre of the two dipoles due to the different magnitudes of the voltages VA and VB These voltages are displayed more quantitatively in figure 1(b) The separation of the contours in figure 2 indicates that the gradient become steeper on going away from the local minimum towards the right-hand and left-hand sides respectively The H2 molecules represented as the red dots experience opposing forces on the right-hand side walls throughout the process The temporal and spatial shapes of the field gradients play an important role in narrowing the velocity distributions of the ensemble The bottom of the potential forms a negative cusp peaking at |E| = 0 leading to possible Majorana transitions One might think of creating a minimum of non-zero


4

electric field strength but this is impossible in principle [18] and neither can a field maximum be formed in space without charges

Figure 3 (a) Representative points in phase space showing Rydberg H2 molecules in the time-dependent electric fields shown in Figure 1 (b) corresponding velocity distributions and (c) position distributions

Figure 3 presents the calculated snapshots in phase space for the representative points of the H2

molecules in the linear bluest state of the n = 16 N+ = 2 MJ = 0 Rydberg state and provides a comprehensive description of the trajectories The image at t = 0 represents the ensemble defined by a laser beam waist z 100 m and a velocity distribution defined for a translational temperature of 1 K with respect to the mean velocity vector of the molecular beam The H2 molecules are completely decelerated to mean zero velocity in a time interval 985 s and the translational temperature is calculated to be cooled down to 13 mK In the deceleration process the molecules travel about 46 mm on average The ensemble of points becomes inclined which indicates that the faster components (ie the right-hand portion of the ensemble in figure 3) move more rapidly in the z-direction than the slower components (represented as the left-hand portion) It is noted that the phase-space volume of the distribution become smaller in the deceleration process This might seem to be strange since Liouvillersquos theorem requires that the volume in the phase space should remain unchanged in the absence of dissipative processes The loss of the trajectories is due to field ionization level crossings and Majorana transitions The fast components of the trajectories exceed the Inglis-Teller limit (1995 Vcm) on the wall of the potential gradient Correspondingly the upper part of the distribution is lsquoshearedrsquo in phase space at t = 4 ndash 7 s [see figure 3(a)] and predominantly contributes to ~30 loss as de-tracking Nonetheless the faster the velocity the steeper the gradient experienced by the Rydberg molecules giving rise to cooling effects The translational energy distribution is compressed to give a narrow distribution at t 4 s as shown in figure 3(b) and the cooling process continues further to yield the final temperature of T = 13 mK at t = 985 s The slow components are calculated to be caught up by the local minimum of the electric field at the range of positions z = 42 mm - 5 mm


5

at t = 985 s The off-axis velocity distribution along the y-axis becomes slightly wider than the initial distribution and is defined by the separation between the nozzle and the laser beam (l = 100 mm) The focusing effect for the low-field seeking states is known to be in particularly important to obtain a large beam flux at a distant position from the source

Figure 4 (Colour online) (a) The initial velocities and (b) positions along the z-axis for the trajectories which result in loss of control due to high fields exceeding the Inglis-Teller limit in the range t = 35 ndash 65 s (in blue) and population dispersion by the Majorana transition in the ranges t = 1 ndash 2 s and t = 9 ndash 10 s (in red) and (c) the ratio of successfully decelerated trajectories to total trajectories (yield)

To present the performance more analytically figure 4 shows the initial velocities and initial

positions for the de-tracked trajectories as a function of the time at which they are lost by the above two mechanisms The de-tracked trajectories shown in the range t = 35 ndash 65 s are lost as a result of the high fields beyond the Inglis-Teller limit (1995 Vcm) Those with initial velocities faster than the average (900 ms-1) result in the loss in figure 4(a) The fact that the initial positions have Gaussian distributions with a full width ~100 m in figure 4(b) indicates that the initial velocity is a determining factor Field ionization takes place at fields greater than 4800 Vcm for the extreme low-field-seeking state of the n = 16 Rydberg state after being de-tracked due to the level crossing The other loss mechanism is the Majorana transition in extremely low fields The component with slow initial velocities decays at t = 1ndash2 s when it is caught up by the field minimum The initial positions show that the late-coming part of a pulsed beam will become out-of-control as soon as the field minimum begins to move At the final stage of deceleration the slower components with slow initial velocities decay at t = 6ndash10 s regardless of the initial positions In this case the loss is predominantly caused by the high field at t = 35 ndash 65 s [figure 4(c)]

4 Conclusions


6

The present calculations have demonstrated the possibility of translational cooling down to 13 mK in time-varying inhomogeneous fields for Rydberg H2 molecules If complete deceleration were realized for Rydberg atomsmolecules ultracold collisions which have not been studied with merged molecular beams could become possible Furthermore the reduced velocity with respect to the laboratory frame could present opportunities to develop scattering experiments at surfaces Such interesting phenomena would open up interesting new research fields for study in the future

Acknowledgements We gratefully acknowledge a research grant from the Matsuo Foundation and a Grant-in-Aid for Scientific Research (C) from the Ministry of Education Science Sports and Culture Japan (No 17550004 2005) to one of the authors (YY)

References [1] Reuss J 1988 State selection by nonoptical methods Atomic and Molecular Beam Methods vol 1

ed by G Scoles (New York Oxford University Press) pp 276ndash292 [2] Bethlem H L and Meijer G 2003 Production and Application of Translationally Cold Molecules

Int Rev Phys Chem 22 73 [3] 2004 Special Issue Ultracold Polar Molecules Formation and Collisions Eur Phys J D

31(2) [4] Junglen T Rieger T Rangwala S A Pinkse P W H and Rempe G 2004 Two-Dimensional

Trapping of Dipolar Molecules in Time-Varying Electric Fields Phys Rev Lett 92 223001 [5] Weinstein J D Decarvalho R Guillet T Friedrich B and Doyle J M 1998 Magnetic Trapping of

Calcium Monohydride Molecules at Millikelvin Temperatures Nature 395 148 [6] Zwierlein M W Stan C A Schunck C H Raupach S M F Gupta S Hadzibabic Z and Ketterle

W 2003 Observation of Bose-Einstein Condensation of Molecules Phys Rev Lett 91 250401

[7] Fulton R Bishop A L and Barker P F 2004 Optical Stark Decelerator for Molecules Phys Rev Lett 93 243004

[8] Procter S R Yamakita Y Merkt F and Softley T P 2004 Controlling the Motion of Hydrogen Molecules Chem Phys Lett 374 667

[9] Yamakita Y Procter S R Goodgame A L Softley T P and Merkt F 2005 Deflection and Deceleration of Hydrogen Rydberg Molecules in Inhomogeneous Electric Fields J Chem Phys 121 1419

[10] Softley T P 2004 Applications of Molecular Rydberg States in Chemical Dynamics and Spectroscopy Int Rev Phys Chem 23 1

[11] Vliegen E and Merkt F 2005 On the Electrostatic Deceleration of Argon Atoms in High Rydberg States by Time-Dependent Inhomogeneous Electric Fields J Phys B 38 1623

[12] Vliegen E and Merkt F 2007 Stark Deceleration of Hydrogen Atoms J Phys B 39 L241 [13] Vanhaecke N Comparat D and Pillet P 2005 Rydberg Decelerator Using a Travelling Electric-

Field Gradient J Phys B 38 S409 [14] Softley T P Procter S R Yamakita Y Maguire G and Merkt F 2005 Controlling the Motion of

Hydrogen Molecules Design of a Two-Dipole Decelerator J Elec Spectrosc Relat Phenom 144-147 113

[15] Quantum Mechanics of One- and Two-Electron Atoms Bethe H A and Salpeter E E 1957 (Springer Berlin) sect51

[16] Vrakking M J J 1996 Lifetimes of Rydberg states in ZEKE experiments III Calculations of the dc electric field dependence of predissociation lifetimes of NO J Chem Phys 105 7336

[17] Suzuki M 1992 General Theory of Higher-Order Decomposition of Exponential Operators and Symplectic Integrators Phys Lett A 165 387

[18] Meek S A Abraham E R I and Shafer-Ray N E 2005 Impossibility of a Biased Stark Trap in Two Dimensions Phys Rev A 71 065402


7

Cooling effects in the Stark deceleration of Rydberg atomsmolecules with time-dependent electric fields

Y Yamakita12 R Takahashi1 K Ohno1 S R Procter2 G Maguire2 and

T P Softley2

1 Department of Chemistry Graduate School of Science Tohoku University Aramaki Aoba-ku Sendai 980-8578 Japan 2 Department of Chemistry University of Oxford Chemistry Research Laboratory Mansfield Road Oxford OX1 3TA United Kingdom

E-mail yyqpcrkkchemtohokuacjp

Abstract This paper presents calculations for realizing the deceleration of H2 Rydberg molecules with n = 16 where n is the principal quantum number A double-dipole decelerator (Softley T P Procter S R Yamakita Y Maguire G and Merkt F 2005 J Elec Spectrosc Relat Phenom 144-147 113) operated with an optimum time-dependent electric field allows in principle complete deceleration of the Rydberg molecules to zero mean velocity A bunch of molecules in a supersonic beam is decelerated from the initial velocity centered at ~900 ms-1 and translational temperature 1 K to the final velocity 0 ms-1 and temperature 13 mK The calculations are performed using the 4th-order symplectic integrator based on representations in phase space q p and show that an ensemble with narrow q0 and broad p0 distribution is converted to one at standstill with broad q and narrow p Cooling effects are reinforced by field ionization in which the fast components that move to regions of high electric field are effectively filtered out

1 Introduction Controlling the translational motion of gas-phase molecules particularly through the use of multipolar devices has played an important role in reaction dynamics experiments [1] The potential-energy gradient in inhomogeneous electric fields (or magnetic fields) gives rise to forces acting on a molecule and trajectory simulations make it possible to design optics for neutral molecules When the field is kept static the sum of the kinetic energy and the potential energy is conserved However if the field is time-dependent the energy of the molecule is not determined by the position but by the integrated work exerted in the field

)1()d()( 2

1

r

rrrr tFtW

At the position where the force F(rt) is zero ramping updown the electric potential does not exert any work on the molecule Such a property allows cooling of the translational energy of molecules to which conventional laser cooling techniques cannot be applied Indeed the fast-switched multi-stage decelerator developed by Meijer and co-workers has demonstrated the deceleration of neutral


ccopy 2007 IOP Publishing Ltd 1



)2()()()( 21




)3()()(

)(rr

rrF tE

tEUt


(Ex2 + Ey


)4(0 eEzHH


2


)5(10

1)12)(12()1(

1 lrll

NJJl

MJ

MJ

JJ

MJlnNznlJMN

JJ

MlNJJ

JJ

J



)6()exp(0

0

qp

Dqp

Ht




3






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)(rr

rrF tE

tEUt


(Ex2 + Ey


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1 lrll

NJJl

MJ

MJ

JJ

MJlnNznlJMN

JJ

MlNJJ

JJ

J



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0

qp

Dqp

Ht




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NJJl

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JJ

MJlnNznlJMN

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MlNJJ

JJ

J



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0

qp

Dqp

Ht




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cooling effects in the stark deceleration of rydberg atoms

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