EI.SEVIER

16January1995

Physics Letters A 197 (1995) 135-138

PHYSICS LETTERS A

Exact solution for the motion of a particle in a Paul trap

Mang Feng, Kelin Wang Center for Fundamental Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received 4 April 1994; revised manuscript received 15 September 1994; accepted for publication 14 October 1994 Communicated by A.R. Bishop

Abstract

The motion of a particle in a time-dependent potential V(x, t) = ½ [ U+ Vcos(~ot) ]x 2 in a Paul trap can be described exactly by using a function series expansion to solve the corresponding Schr/Sdinger equation. The validity of our approach is discussed and some physically meaningful results are shown.

1. Introduction

The Paul trap [ 1 ] is an important instrument to trap charged and neutral particles for observation and measurement. Since it was reviewed by Paul in his Nobel Prize lecture, there has been an increasing in- terest in its application and mechanism [ 2-6 ]. In fact, the potential V ( x ) = v ( x ) cos(Ot) which is very sim- ilar to the Paul trap potential had been discussed the- oretically in the past in Ref. [7]. But in that paper, the time-dependent potential was replaced by a time- independent effective potential before solving the Schrtidinger equation and the motion of a particle with mass m was discussed. With the development of the theory of time-dependent quantum systems, re- cently methods of deriving the explicit time-depen- dent wave functions of the Paul trap have been pro- posed [2,3]. Particularly in Ref. [3], the author provided a good method to achieve his goal. But un- fortunately, the related Mathieu equation with a time- dependent coefficient cannot be generally solved. Therefore, the problem was only solved in principle. The purpose o f this Letter is to try to deal with the time-dependent Hamiltonian system of the Paul trap in the SchriSdinger representation and we will find

that the problem is finally concentrated on how to solve a typical nonlinear differential equation. In or- der to solve this nonlinear differential equation, we present an interesting method named the function se- ries expansion which we proposed before [8 ]. Then the quantum mechanism of a particle motion in a Paul trap is studied. Meanwhile for testing our approach, some numerical calculations were made. Finally, the results of direct calculations of the radius values and peak values of the probability wave packet are listed.

2. System of the Paul trap

The potential of a Paul trap depends on the dc volt- ages applied to it. In this paper we make the general assumption that the situation of any one dimension of this system can be described independently and analogically. Therefore it is possible for us to take ~(x, t) as the representative for a solution. ~(y, t) and ~'(z, t) can be obtained analogically. According to Ref. [ 1 ], the one-dimensional Paul trap system can be described as follows,

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136 M. Feng, K. Wang / Physics Let ters A 197(1995) 135-138

O 1 02 i ~ U ( x , t ) - 2 0 x 2 ~'(x,t)+½£22x(t)x27J(x,t) •

(1)

Let us first try to perform a transformation

~(x, t) =q~' (x, t) exp[iot( t )x 2 ] . (2)

Then Eq. ( 1 ) can be written as

. 0 1 02 O 1~ q~ (x, t )= -- ~ Ox--- 5 qb~(x, t ) - - 2 i a ( t ) x ~ ~b~(x, t)

+[2o~2( t )+&(t)+½t22( t ) lx2~, (X, t ) , (3)

where

t x

q,[ (x, t)= q~' (x, t ) e x p / J o~(t' ) d r ) . (4)

Assuming

a(t )=(o( t ) /4~o( t ) , y=~o-~/z(t)x,

S = i (p--l(o') do', (5)

and

q~(y, s) =q~ (x, t) , (6)

it is easy to obtain

. 0 I ~ S S ~ . . . . .

1 02 20y 2 ~ + ( ¼(~0- ~(02 + ½g22( t)~o2)y2qb .

(7)

Assuming

~0~- I ~ 2 + ' 2 2 ~t2x(t)~o =½c, (8)

where c is a constant, Eq. (7) is the Schr6dinger equation with a time-independent harmonic oscilla- tor potential. The solution is

qb(y, s) =IV. exp( - ½z2y2)H.(zy) e x p ( - i E . s ) , (9)

where

N.=(z /2 .n !v /~ ) l /2 1/4 , Z~--C ,

E . = ( n + ½ ) x / c , (10)

and H. is the Hermitian polynomial. So we can write the original wave function

~'.(x, t)=M.~o-I/4H.(c'/4~o-l/2x)

×exp[ioL( t )x 2 - ~cl/2x2/q~] t

× e x p ( _ i ( n + l ) x / ~ f d ~ ) ) , ( l l ,

to

where

M.=(c~ /4 /2"n!x /~AT) 1/2 , A T = t - t o . (12)

From the above deduction, it is found that only ( 8 ) is left to be solved and the final results of this system cannot be obtained unless the ~o(t) is obtained first.

3. Solution of the auxiliary equation

In the experiment of the Paul trap, ~2~(t) is

£22(t) = U+ Vcos(~ot). 13)

Before solving Eq. ( 8 ), we suppose

q~(t) =B2( t ) , 14)

so (8) can be written as

B3(t)B'(t)+f22~(t)B4(t) =c. 15)

Assuming

B(t ) = ~ Bk cosk(ogt) , 16) k = 0

it is clear that

B o= (c/U) 1/4. 17)

We set

F7 = ~ ' mZBiBjBIB., , i+y+l+m=n

F~= ~, rn (rn-1)B , BjBtBm, i + j + l + m = n

F~ = Z BiB~BIB,,,, i + j + l + m = n

F'~= Z ' BiBjB, B,. , (18) i + j + l + m = n

where ~' presents the situation in which the B3B. term is not included.

Finally a recursion formula of Bk is obtained,

M. Feng, K. Wang / Physics Letters A 197 (1995) 135-138 137

1 Bk= k2o)2B3o _ 4 U B 3

×(--o)2Fkl +to2Fkz+2+UF~4+VFk3-' ) . (19)

It is hard to prove the convergence of the function series expansion (16) strictly mathematically. But we can provide two circumstantial evidences. One is that the max imum of the function cos(cot) is 1. So the convergence of B( t ) should depend on that Of Bk. Ac- cording to Table 1, Bk decreases quickly when V is less than U and when V is much larger than U, the convergence Of Bk will still exist if ~o increases corre- spondingly. Provided we make a calculation very close to the divergence region, much more Bk terms have to be involved because in this case Bk decreases much slower. But fortunately, the value of¢o is very large in the Paul trap experiment, so a calculation of 30 to 50 B~ terms is sufficient to obtain a satisfactory, answer. That is to say, our approach can be used practically. Another is that our calculation can be compared with

the result of perturbation theory in order to test our approach and results. Transi t ion rates P00 at differ- ent times were calculated in detail for this compari- son and the values at the lowest points of these two curves are listed in Table 2 where V varies from 0.1

Table 1 B(t) for inclusion of a different number of Bk terms

30 Bk terms 50 Bk terms

U= 5, V= 1.0, m= 5 0.1896553 0.1896553 U= 5, V=2.6, m= 5 0.4654625 0.4657045 U= 5, V=2.6,~= 10 0 .032640997 0.032640997 U= 5, V=26, m= 10 0.3408110 0.3408110 U= 5, V=66, m= 10 0.9534281 0.9531162

Table 2 Comparison of the bottom values of Po0 curves between exact solution and the perturbation approach for U= 10 and o)= 1.0

V Exact solution Perturbation

0.1 0.9999999 0.9999568 0.3 0.9999801 0.9996115 0.5 0.9999309 0.9989208 0.7 0.9998599 0.9978848 1.0 0.9997145 0.9956831 1.5 0.9993826 0.9912772 2.0 0.9989830 0.9827325

to 2.0, t is from - ½ ~ to ½~z, U a n d oJ are assumed respectively to be 10 and 1.0. It can be found from Table 2 that the relative error between our calcula- tion and the perturbation approach is about 10 -5

when V= 0.1 and this relative error increases with in- crease of V. When Vis 2.0, the relative error is about

l T6~. This comparison shows not only the credibility of the method introduced in this paper but also the superiority of the exact solution.

4. Calculations for the probability

After the exact form of the wave function has been obtained, the corresponding probabilities at dif- ferent x and t are easily calculated. As shown in Fig. 1, it is a typical curve of the probability p(x , t) at a certain time. In order to make clear the motion of a particle in the Paul trap, we have calculated a series ofp(x, t) in different situations and have listed them in Table 3 where [rl is the max imum radius of the

probability and Pmax is the peak values of the proba- bility. It is shown in Table 3 that when t = 0, I r l in- creases with increase of V and decrease of U and o). But Pmax increases with decrease of V and increase of U and ~o. When t = 7r, the variations are opposite to the situation at t=0 . When t= in , I rl and Pmax vary only with U. It is clear that the variations of I r l and Pmax are periodic with time because of the periodic form of the potential 's "spring constant" in the Paul

trap. In conclusion, with the method introduced in this

?(×,t> 0,80

0.50

0.40

0 ,20

0 ,00 . . . . . . . . . , ,'>', . . . . . . . . . ,

- 4 , 0 0 - 2 , 0 0 0 . 0 0 2 ,00 4 ,00 X

Fig. 1. Typical curve ofp(x, t) at a certain time.

138 M. Feng, K. Wang / Physics Letters A 197(1995) 135-138

Table 3 Maximum radius values and peak values of the probability for different values of U, V, ~o and t

U V ~ t=O t = ~ / 2 t=~Z

Irl Pmax Irl Pmax Irl pmax

1 4 10 2.274 0.5895 2.193 0.6142 2.112 0.6406 30 2.202 0.6115 2.193 0.6142 2.184 0.6170 50 2.196 0.6133 2.193 0.6142 2.190 0.6152

1 1 10 2.2ll 0.6079 2.193 0.6142 2.172 0.6207 30 2.193 0.6136 2.193 0.6142 2.190 0.6149 50 2.193 0.6140 2.193 0.6142 2.190 0.6145

1 0.3 10 2.199 0.6123 2.193 0.6142 2.187 0.6162 30 2.193 0.6140 2.193 0.6142 2.190 0.6145 50 2.193 0.6142 2.193 0.6142 2.193 0.6143

2 8 10 2.025 0.6708 1.875 0.7305 1.731 0.7984 30 1.890 0.7240 1.875 0.7305 1.860 0.7371 50 1.881 0.7281 1.875 0.7305 1.869 0.7328

2 2 10 1.914 0.7148 1.875 0.7305 1.839 0.7466 30 1.881 0.7288 1.875 0.7305 1.872 0.7321 50 1.878 0.7299 1.875 0.7305 1.875 0.7311

2 0.6 10 1.887 0.7257 1.875 0.7305 1.866 0.7353 30 1.878 0.7300 1.875 0.7305 1.875 0.7310 50 1.875 0.7303 1.875 0.7305 1.875 0.7306

Let ter , we can desc r ibe t he m o t i o n o f a pa r t i c l e in a

Pau l t r ap en t i r e ly by a d j u s t i n g U a n d V. N o app rox i -

m a t i o n such as Ref. [ 3 ] u sed is needed . M e a n w h i l e ,

t he va l i d i t y o f th i s a p p r o a c h ha s b e e n d i s c u s s e d in

deta i l . T h e speci f ic c a l c u l a t i o n also shows t h a t th i s

a p p r o a c h is p rac t i ca l for dea l ing w i th the P a u l t r ap

p r o b l e m .

Acknowledgement

T h e a u t h o r s t h a n k P ro fe s so r Z h u D o n g p e i a n d Mr .

Shi M i n g j u n for p r o v i d i n g the i r p r e l i m i n a r y work a n d

one o f t he a u t h o r s ( F e n g M a n g ) wou ld l ike to ex-

press his t h a n k s to Mr . W u J u h a o for h is e n t h u s i a s t i c

help .

References

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Proc. 124 (1993) 89. [7] R.J. Cook et al., Phys. Rev. 31 (1985) 564. [8] H. Lan and K. Wang, Phys. Lett. A 137 (1989) 369; 139

(1989) 61; 144 (1990) 465.