Asymptotic Techniques for Atomic Waveguide Calculations 

by William M. Golding 

ARL-TR-5343 September 2010 

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Army Research Laboratory 

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ARL-TR-5343 September 2010 

Asymptotic Techniques for Atomic Waveguide Calculations 

William M. Golding 

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William M. Golding 

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14. ABSTRACT 

Asymptotic expansions describing the behavior of the radial wavefunctions of a magnetic atomic waveguide are developed. In 

this system, some components of the spinor wavefunctions do not die off exponentially and are therefore significant at large 

distances. This is related to the quasibound nature of the system. A good representation of the nondecaying components of the 

eigenstates at large distances from the guide center is required so that the calculated eigenstates can be used reliably in further 

numerical calculations. The asymptotic expansions presented here provide this representation and are readily related to the 

power series solutions developed in ARL-TR-5335. By connecting those power series solutions to the asymptotic expansions 

developed here, an efficient representation of the exact radial wavefunctions can be obtained. These wavefunctions are needed 

for detailed studies of important properties of magnetic guides such as; sensitivity to noise driven spin flips, importance of 

quantum Majorana transitions, energy level dependence on magnetic field, the effects of guiding field imperfections as well as 

the onset and departure of adiabatic behavior. 

15. SUBJECT TERMS

Atomic waveguide, eigenstate calculation, asymptotic expansions, phase integral technique 

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William M. Golding 

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iii

Contents 

1. Introduction 1

2. Development of the Radial Equations 2

3. Uncoupled Equations 3

4. Calculating the Controlling Factor and Leading Behavior 4

5. Stokes Phenomenon and the Asymptotic Basis Functions 6

6. Results of Phase Integral Calculations 10

7. Asymptotic Expansion of the Mode Functions 11

8. Conclusions 15

9. References 17

Distribution List 20
iv

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

In this work, radial wavefunctions describing the quantum behavior of a spin one-half particle far 

from the center of a uniform atomic waveguide are formulated. The model waveguide potential 

is an ideal transverse quadrupole magnetic field that extends to infinite distances and is 

unbounded in magnitude. The spin of the particle can be considered in either a global basis or in 

a local basis. The quantization axis for the global basis is directed in the z-direction along the 

axis of the guide while the quantization axis for the local basis is taken to be directed everywhere 

along the local direction of the quadrupole magnetic field. In either basis, the atomic 

wavefunction has two components one of which might seem more “trapped” than the other. 

However, in each of these bases there is a significant component of the spinor eigenstates that 

extends to infinity and represents the amplitude for an atom to be found outside of the trapping 

region of the waveguide. 

The asymptotic solutions developed here describe the behavior of the radial wavefunctions at 

distances far from the guide center and thus provide a means of extending the series solutions 

derived in Golding (2010) to these regions. The combination of the two series can be combined 

into an efficient and complete solution to the waveguide eigenstate problem. 

In the global spin basis, both spinor components of the radial solution appear to be unbound. 

They oscillate and only gradually decay along the radial direction outward towards infinity. 

However, in the local spin basis one component of the spinor appears to be bound and the other 

component appears unbound. This results from a nearly exact cancellation of the global 

components far from the center when viewed in the local basis. This behavior is clearly related to 

the adiabatic approaches often used to describe these waveguides when the axial bias field is 

made large (Sukumar, 1997; Brink 2007). In the adiabatic approach, the unbound component is 

effectively considered to be uncoupled from the bound component and ignored in a first 

approximation. 

A goal of our research is to precisely understand the behavior of atoms guided on atom chips and 

moving through atom chip devices. A complete description of the solution to the waveguide 

problem is a required starting point in the pursuit of this goal. 

This report begins by laying out the radial equations that must be solved to describe the 

transverse modes of an atom guide. The leading behavior of the solutions to these equations at 

large distances is then developed. To properly connect the four solutions that represent the 

leading behavior the phase integral technique is used. The phase integral technique essentially 

produces the relative phase and amplitudes needed to create an asymptotic expansion that 

correctly represents the known analytical properties that the series solution has at large distances, 
2

where the series cannot simply be summed. In the last section, the techniques used to find the 

full asymptotic expansion are described. 

2. Development of the Radial Equations 

The equations for the radial wave functions of the quadrupole guide are derived by making use 

of the angular symmetry of the quadrupole magnetic field. This symmetry is expressed as the 

conservation of alignment, Λ௭  ؠ ܮ௭ െ ܵ௭ (Golding, 2009; Hinds, 2001; Lesanovsky, 2004). The 

approach taken here is to find eigenfunctions that are common to both the Hamiltonian, ܪ and 

the alignment, Λ௭. This method effectively separates the angular and radial dependence of the 

problem leaving only a coupled system of radial equations to solve. 

By removing the angular dependence, the following pair of coupled radial equations for the 

specific alignment,  ; energy,  ; bias field, 0 b ; and field gradient, 1 b , can be derived: 

 

 

2

2

2 0 1

2

2

2 0 1

1 1/2

() ,

1 1/2

() .

R R R bR b R

R R R bR b R

 

 



 

 



 

 

   

   



    

     

 (1) 

The parameters and physical relevance of these equations have been described previously by 

Golding (2009). In the case of a spin half particle with integral orbital angular momentum, the 

pair must be solved for the allowed half-integral values of the alignment, 135 , ,

222     , 

although the solutions for negative alignment are simply derived using symmetry when the bias 

field, 0 b is zero. 

An important aspect of these equations is that they are coupled. In system 1, R and R occur in 

both equations. In other studies of this system it is assumed that the boundary conditions at the 

origin can be determined by using the fact that as  0 the terms on the right hand side of 

system 1 can be ignored (Hinds, 2000; Lesanovsky,2004; Potvliege,2001). This is equivalent to 

assuming that 1 b can be set to zero and limiting the results to small radii. Using this assumption 

it would appear that the two equations reduce to an uncoupled pair of second order differential 

equations for a small region near the origin. This is a type of singular perturbation since the form 

of the equations changes drastically when the small parameter 1 b is set to zero. 

Physically, setting 1 b to zero is like turning off the trapping fields. The boundary conditions for 

the singular case of 1 b  0 must be those of a free particle in which the spin components are 

completely uncoupled. Free particle states are eigenstates of the spin operator, z S but they are 
3

not eigenstates of the full Hamiltonian. In order to get the proper boundary conditions when the 

trapping fields are on, it is important to first uncouple the equations and perform a power series 

analysis on the resulting fourth order differential equation using the Frobenius method. The 

resulting series solution is evaluated at the origin and the behavior near the regular singularity is 

then handled correctly without making the assumption that 1 b  0 . 

3. Uncoupled Equations 

The system of radial equations displayed in system 1 contains an irregular singularity at infinity. 

In order to develop an asymptotic series for large values of the radius, the same decoupled fourth 

order equations that are developed to study the behavior near the origin are used. By solving the 

top equation of system 1 for R and substituting this result into the bottom equation, the system is 

uncoupled to produce the following fourth order differential equation for R , 

 

 

4 22

3 2

0

26 2 2 4 2 2 2 4

1 00 0

4

4

4

2

2

5 2 2

2

2 54 6

1 19 35 22 9

2 6

0

2 1

b

b

d R

d

d R

d

d R

b R b b

d

  

  

      





 

















 

    



       

      

             

       

. (2) 

An equivalent equation for the R component can be obtained from equation 2 by simply 

changing the sign of both  and 0 b as well as replacing R with R . This can be seen by 

inspection of system 1. 

Equation 2 is a fourth order Hamburger equation (Ince, 1956) that can be solved explicitly using 

the Frobenius series technique as shown by Golding (2009, 2010). It has a regular singularity at 

the origin and an irregular singularity at  . The series approach is useful as an expansion around 

the regular singular point at the origin. However, the Frobenius technique cannot generally be 

used to find an expansion around the irregular singular point. 

In equation 2, the equation for R has been developed by eliminating R from the coupled 

equations in system 1. The behavior of the other spinor component is obtained from the 

following equation: 
4

   

   

  

2

2

2 2 0

1

1

1 2

R R

d d R R bR

b d

d

  

   

  

 

  

                

 

 

 (3) 

that is easily obtained from the first equation in system 1. 

In order to obtain useful information about the behavior of the solution as    , an asymptotic 

expansion must be developed. This requires first the determination of the controlling factor and 

the leading behavior of the uncoupled fourth order equations. 

4. Calculating the Controlling Factor and Leading Behavior 

The first step in a solution of the asymptotic behavior of R in equation 2 is to calculate the 

controlling factor of the expansion. This is done by assuming a trial solution of the form, 

R S   exp    (Bender,1978). A differential equation for S   is derived that must be 

approximately solved for large  . The full equation is 

 

   

   

   

 

4

2 2 2

2 2

3 2

0

3 3

2 2 22

2 22

4

22 2 2

4 1 0

2

0 0 4 2

2

2 5/2 6 2

4 65 4 2

2 5/2 3 2

2 2 1/2 35 9 19 / 2

16

d S

d

d d S S

d d

d d b S S

d d

d d S S

d d

d Sbb

d

b b





  

 

   

  

  

 

 



   







 

 

     

              

        

 

        

  

  

    



     

4  0  

 (4) 

By using the several approximations that 

n n

n

d d S S

d d  

 

    

as  gets large, the following 

simplified equation is obtained for S   , 

   

4 2

22 2

1 2 0 d d S Sb

d d     

  

         

.

 (5) 
5

By treating equation 5 as a quadratic equation in   d S

d 



, the following four differential 

equations are derived: 

1

1

, dS i b

d

dS b

d

 







  

 

. (6) 

The solutions are 

    

    

3/2

1

1

3/2

1

1

2

3

2

3

S bC

b

S ib C

b

  

  

  

  

 (7) 

where C   is an integration “constant” to be determined. The idea in the asymptotic analysis is 

that C   should be much smaller at infinity than the first term in S   . Using the techniques 

described by Bender (1978), C  is determined by substituting the solutions in equation 7 back 

into equation 4 and using the approximations that 

n n

n

d d C C

d d  

 

    

to obtain the leading 

behavior. By this technique, the factor C  is found to be 1

3 ln

4

 b  and the leading behavior is 

determined by the four functions 

 

 

3/2

1

1

3/4

1

2

exp 3b

b

b



 



    

  (8) 

in which the factor  represent the four roots of unity, i, 1 . To be consistent with the 

asymptotic nature of this calculation, the exponent is expanded and the terms that become small 

at large  are neglected. The factor S   is then given by 

      3/2

1

3/2

1

1 2 /3 2

3 1

S b

b

b

b

   



      . (9) 

This expansion of the exponent is consistent with the normal solutions at irregular singularities 

discussed by Ince (1956). However, this choice is not unique and other possible choices may or 

may not result in a consistent asymptotic series. 
6

Using equation 9, the leading behavior of the radial wave function is given by the four functions 

that we refer to as the asymptotic basis functions 

 

 

3/2 1

1

3/4

1

2

exp 3

b

b

b

   



    

  . (10) 

The behavior of either spinor component, R or R at large  must be asymptotic to some 

superposition of these four asymptotic basis functions. This is true even though the functions in 

equation 10 were derived from the equation for R , equation 2. The reason for this is that the 

equivalent fourth order equation for R only differs in the signs of  and 0 b , neither of which 

appear in equation 10. The leading factors for the two components are therefore identical. The 

actual dependence on  and 0 b will reappear as more terms in the asymptotic expansion are 

included. 

5. Stokes Phenomenon and the Asymptotic Basis Functions 

In the global basis, all the physically allowed solutions found using the Frobenius technique in 

this problem are of the form (Golding, 2010) 

2

2

0

n

n

n

a   





 , (11) 

where  is either an even or odd integer that depends only on the mode and component of 

interest. In the local basis, the components are superpositions of similar series of different 

indicial exponents  and the analytic properties of the combined series are not as simple. For 

this reason, the global basis is used and the series form in equation 11 is taken as the general 

form for the asymptotic analysis below. 

Once this series is extended to the complex plane by letting i

e     , the coefficients needed to 

combine the asymptotic basis functions of equation 10 into a proper approximation of the power 

series in equation 11 can be determined. This is accomplished by forcing both representations to 

have the same analytic properties for large  . The extension of the asymptotic basis functions to 

the complex plane are also obtained by the substitution i

e     and are given by 

 

 

 

3/2

1

1

3

3/4 4

1

2 33 exp cos sin 3 22

, , where , 1

i

b i

b

i

b e

 

 



  



                      . (12)
7

The expression in equation 11 represents a single-valued function of  in the complex plane. A 

reasonable approximation to this complex function at large values of  can be formed by using 

various linear combinations of the four asymptotic basis functions in equation 12 to create a 

single-valued function. However, the behavior of the individual basis functions in equation 12 is 

dramatically different in different sectors of the complex plane. Different linear combinations of 

the basis functions are required in each sector to approximate the series solutions. The different 

representations in each sector must in the end be matched to the adjoining sectors so that an 

effectively continuous function is formed. This matching process is handled using the phase 

integral techniques designed to deal with Stokes Phenomenon (Bender, 1978; Heading, 1962; 

White, 2005). The phase integral calculation takes place on the complex z-plane shown in figure 

1. The basic technique is described in Heading (1962) and White (2005) for second order 

differential equations. In figure 1 the lines described as anti-Stokes lines radiate outwards along 

the lines 2 0, 3

    . These lines cut the plane up into three sectors where different forms of the 

solution are used. 

Figure 1. Complex plane showing the positions of the anti-Stokes lines 

needed to perform the phase integral calculation that properly 

combines the asymptotic basis functions. 

The anti-Stokes lines are defined so that the pair of exponential solutions with   i in equation 

12 have pure imaginary phase and therefore do not decay towards infinity. Along another set of 

lines called the Stokes lines (the red lines in figure 1) these same oscillatory solutions have 

purely real exponents and one solution decays towards infinity while the other grows 

exponentially. The real component of the phase changes sign on crossing an anti-Stokes line. 

Thus, an exponentially growing solution in one sector abruptly becomes an exponentially 
8

decaying solution in the adjacent sector. Phase integral techniques are intended to smooth out 

this apparent discontinuity. 

An important consideration in this problem is that there are four functions in equation 12 that 

must be considered. These functions form two pairs and the Stokes lines of one pair are the antiStokes lines of the other pair. For example, along the real axis where   0,

 the solutions with 

  i have purely imaginary phase and thus do not decay towards infinity while the solutions 

with   1have purely real phase and break up into decaying and growing exponentials. 

The calculation proceeds by starting in sector 7 of figure 1 with a superposition of the form 

1 1

dsd s A   i i  BC D ( A,,, B C and D are assumed to be complex constants). The 

superscripts s and d refer to the subdominant or dominant nature of the basis function in the 

sector. Subdominant components decay towards infinity and dominant components grow. 

The first step is to determine a superposition in region 1 that is consistent with the assumed form 

in region 7. This is accomplished using arbitrary Stokes constants and by adjusting the 

dominancy of the solutions as described in both Heading (1962) and White (2005). Since the 

phase factor in the denominator of the basis functions is independent of  , it can be ignored 

except when crossing the branch cut. Once this process is complete the solution in region 2 is 

developed from that in region using the Stokes constant, T1 . This process proceeds around the 

complex plane until after dealing with the branch cut, region 7 is reached again. Referring to the 

diagram in figure 1 the detailed calculation goes as follows: 

1. Starting in region 7 with the assumed form 1 1

ds d s A  i i  BCD  

2. Cross into region 1 obtaining 1 11   s dd s A B C D TC i i         

3. Cross into region 2 obtaining  2 1 11    s ds d A T B B C D TC i i          

4. Cross into region 3 obtaining  2 3 1 1 11       ds s d A T B B C T D TC D TC i i           

5. Cross into region 4 obtaining 

  2 4 2 3 1 1 11        d s ds A T B B T A T B C T D TC D TC i i             

6. Cross into region 5 obtaining 

   

         

2 42

3 11 15 3 1 1

s d

i i

d s

A TB B T A TB

C T D TC D TC T C T D TC

 

 

 

 

   

     

7. Cross into region 6 obtaining 

        

         

26 4 2 4 2

3 11 15 3 1 1

s d

i i

s d

A TB T B T A TB B T A TB

C T D TC D TC T C T D TC

 

 

 

 

      

     

8. Cross the branch cut to get back to region 7 obtaining 

        

         

26 4 2 4 2

3 11 15 3 1 1

s d

i i

s d

i A TB T B T A TB i B T A TB

i C T D TC i D TC T C T D TC

 

 

 

 

       

     
9

At this point there are two different descriptions of the function in region 7 that can be made 

equal by calculating consistent values of the Stokes constants. By this process there is essentially 

a different description of the desired function in each region of the plane and these functions 

must all be connected together so that the patched representation of the function in equation 11 is 

as continuous and singled valued as possible. By comparing the final expression obtained in 

region 7 with the expression that began the calculation in region 7, four equations are obtained 

that are sufficient to determine that all of the Stokes constants, Ti are equal to i . 

Once the Stokes constants are evaluated then the behavior of the total solution in the various 

regions can be used to completely calculate the required relations between the amplitudes, 

A,,, BCD , of the asymptotic basis functions needed to properly approximate the asymptotic 

behavior of the series solutions of equation 11. 

As noted before, the series in equation 11 implies that the extended function along the negative 

real axis must be equal to i

e  times the function along the positive real axis. Setting all of the 

Stokes constants equal to i and ignoring appropriate subdominant terms that decay 

exponentially along the real axis, the expression along the negative real axis in region 1 becomes 

   1 1

ds d A iB iD D iC i         and the similar expression along the positive real axis is 

1

s d d A  i i   B C . Taking all of this into account and including the phase shift, 

3

4

i

e

 

 common to 

all of the basis functions evaluated along the negative real axis, the following equation is 

obtained: 

     

        

3/2 3/2 3/2

1 11

1 11

3

3/2 3/2 3/2 4

11 1

11 1

2 22

exp exp exp 3 33

22 2

exp exp exp 33 3

i

i

i i eA b B b C b

b bb

i i e A iB b iD b D iC b

bb b





 

  

              

     

                        

. (13) 

Equating the coefficients of common basis functions in this expression the relations 

 

 

3

4

3

4

3

4

i i

i i

i i

A iB e e C

iDe e B

D iC e e A



















 

 

 

, (14) 

are found. The eigenfunctions described by the series in equation 11 must be finite and real 

valued along the positive real axis. This is enough information to determine that the coefficients 

are given by 
10

4

4

0

i

i

i

A A e

B A e

C

D A e









 

 



 

. (15) 

The two possible sets of solutions are necessary to account for the sign of the exponentially 

decaying components of the radial eigenfunctions as the radial mode number is increased. That 

is, if the lowest energy eigenfunction of a particular mode has no zero crossings then D is taken 

to be positive allowing the bound component of the wavefunction to decay from positive values 

towards zero. If  is odd the upper sign is taken and if  is even the lower sign must be taken. 

The next higher radial mode would require D to be negative. 

 

6. Results of Phase Integral Calculations 

Using the constants determined by the phase integral technique the functional forms of the 

leading behavior for each component and each mode are displayed here: 

 

     

     

     

3/2 1 1

3/2 1 1

3/2 1 1

3/2 2 / 1 1 3 5/2

1 3/4 3/4

1 1

3/2 2 / 1 1 3 5/2

1 3/4 3/4

1 1

3/2 2 / 1 1 3 7/2

2 3/4

1 1

2 cos /

3 4 e 2 1

2 cos /

3 4 e 2 1

2 cos /

3 4 e 2 1

b b

p

b b

m

b b

m

b b

R

b b

b b

R

b b

b b

R

b b

 



 



 





 

 



 

 



 

 

 



 



 



         

          

          

     

3/2 1 1

3/4

3/2 2 / 1 1 3 7/2

2 3/4 3/4

1 1

2 cos /

3 4 e 2 1

b b

p

b b

R

b b

 





 

 

 



         

. (16) 

These four functions represent the leading behavior of the spinor components and are valid along 

the positive real axis as    . The proportionality factor A has been set equal to one as these 

solutions need to be matched to series solutions describing the behavior in the area of the origin. 
11

The functions, 1 2 and R R

p p , refer to the spin-up components of type 1 modes and type 2 modes 

(Golding, 2009) and 1 2 and Rm m R similarly refer to the associated spin-down components. The 

phase choice in these expressions produces a positive decaying exponential for the tail of the 

bound component of the ground state for   1/2 . The overall phase is unimportant but the 

relative phase between the cosine and exponential terms in each function is the mathematical 

cause of the exponential decay normally seen in the bound state components, R R

p

 m . A quick 

check shows that the sinusoidal terms cancel out in the bound components and the exponentials 

cancel out when the difference components, R R

p  m are calculated. Thus each of the global 

spinor components, , R R

p m shown in equation 12 contains a mixture of an unbound and a bound 

component. 

Within a given series of radial eigenfunctions the boundary conditions are identical for each 

mode at the center of the guide. As the eigenenergies increase from one radial mode to the next, 

the sign of the exponential factor must change to account for the increasing number of zero 

crossing at higher energies. This can be handled by multiplying each of the functions in equation 

12 by a factor, 1 1

n  or 2 1n  , the new parameters, 1 n and 2 n , are then the radial mode indices for 

the radial wavefunctions. These factors are not included above but can be easily included when 

working with a full set of modes. 

7. Asymptotic Expansion of the Mode Functions 

In order to obtain a more accurate form of the mode functions than just the leading behavior 

shown in equation 16, a full asymptotic expansion must be calculated. This is accomplished by 

solving equation 2 using an assumed form that is made up of the product of the leading behavior 

of the mode times a series consisting of inverse powers of  . The coefficients of the series 

expansion are then determined recursively by equating the factors multiplying each power of 

in the resulting series zero. 

Substituting the following expansion in equation 2, 

       

3/2 3/2

11 11

1 2 3/4 3/4

1 1

2 2 cos / sin /

3 3

bb bb

Rw w

b b

   

 

  

             (17) 

results in a set of coupled differential equations for 1 2 w w and that can be solved by assuming 

that these functions have the following asymptotic expansions: 

/2 /2

1 2

0 0

 and n n

i i

n n

wa wc  

   

 

    . (18) 
12

Solving the resulting coupled series expansions for the first few coefficients results in the 

following approximate form for w1 when 0 b  0 : 

2 2

0 0

1 0 3/2 3/2

1 1

4 22 4

0 0 00

333

11 1 1 1

1/4 1/2

1/ 4 1/ 2 1/ 32 1/ 8 1/ 8

b c

w a

b b

b b ba

bb b b b



 







 

      

       

 



 (19)

 

6 3 42 4

0 00 2

9/2 9/2 5/2 5/2 9/2

111 1 1 0

2 36 1

0 00 2

5/2 5/2 9/2 1

1

/2

1 1

3

11 5 1/ 8 1/ 32

384 64 48

5 5 1/ 3 1/ 4 1/ 48

24 48

b bb

bbb b b c

b bb b

b bb

  





  

              

           



8 5 62 4

0 00 4

6 6 4 46

1 1 1 11

32 3 6

0 00 22 2

4 46 1

1 11

4 58

00 0 0 2 22

24 4 6 0 1

11 1 1

1 1 7 11

6144 768 384 64 256

7 35 1 1/12 1/16

96 192 192

7 29 1 1/ 8 1/ 6 1/16

96 96 384

b bb

b b b bb

b bb b

b bb

bb b b b b

bb b b

  





 

 





 

        

               

                

0

2

a



   

 



and the corresponding form for w2 

 
13

 

2 2

0 0

2 0 3/2 3/2

1 1

4 22 4

0 0 00

333

11 1 1 1

1/4 1/2

1/ 4 1/ 2 1/ 32 1/ 8 1/ 8

b

w c

b b

b bb

bb b b b

a

  c









 

    

        

 



 

 

6 3 42 4

0 00 2

9/2 9/2 5/2 5/2 9/2

111 1 1 0

2 36 1

0 00 2

5/2 5/2 9/2 1

3 2

1 11

/

11 5 1/ 8 1/ 32

384 64 48

5 5 1/ 3 1/ 4 1/ 48

24 48

b bb

bbb b b

b bb b

b bb

a

  

 



 

              

           



 

8 5 62 4

0 00 4

6 6 4 46

1 1 1 11

32 3 6

0 00 22 2

4 46 1

1 11

4 58

00 0 0 2 22

24 4 6 0 1

11 1 1

1 1 7 11

6144 768 384 64 256

7 35 1 1/12 1/16

96 192 192

7 29 1 1/ 8 1/ 6 1/16

96 96 384

b bb

b b b bb

b bb b

b bb

bb b b b b

bb b b

  





 

 





 

        

               

                

0

2

c



   

 



 (20) 

In this section, some of longer expressions are written in stacked forms within parentheses to try 

and clarify the expressions. Do not confuse these expressions with matrices as they are to be read 

as a single series of terms. In order to maintain the leading behavior form found by the phase 

integral technique in equation 16, the coefficients, 0 a and 0 c , must be chosen to produce the 4



phase shift in the cosine terms shown in equation 16. This is easily accomplished by setting the 

first coefficient, 0

1

2

a  , and the second coefficient, 0

1

2

c   . 

The first few terms of the asymptotic expansion of R are then 

 

 

3/2

1 1 2 2

0

3/4 3/2 3/2

1 1 1

3/2

1 1 2 2

0

3/4 3/2 3/2

1 1 1

2 cos /

3 1 1/4 1/2 1

2 2

2 sin /

3 1/4 2

2

1 1 1/

2

b b b R

b b b

b b b

b b b







 

 

 

 



                     

               

   



. (21) 

Ignoring the second term in each bracket as    , the expression reduces to the form found in 

equation 16 along with the 4

 phase factor that comes from combining the sin and the cos

terms. Similar expansions are found for the other components and the other modes of the guide. 
14

An expansion for the purely exponential term shown in equation 16 can also be included by 

straightforward extension of the trial function shown in equation 17 to include an exponential 

times another asymptotic series, w3    . Since the exponential dies off so quickly this is only 

important for an asymptotic representation of the atomic wavefunction relatively near to the 

center of the guide yet outside the classical turning point of the mode of interest. 

To attain good accuracy, a significant number of terms are needed in the asymptotic expansions. 

However, the expressions are long and difficult to display so only a few terms are shown here. 

The unbound component is the component that is pointed in the direction of the local magnetic 

field. This component sees the magnetic potential energy decreasing towards infinity and is 

repelled from the center of the guide. The unbound component is given by R R       . It 

decays fairly slowly, is not extremely sensitive to the bias field, 0 b and the first few terms of its 

expansion are displayed in equation 22, 

 

2 2

0

3/4 3/4 9/4 5/4

1 1 3/2

1 1 15

4 2 2 4 7/4 4

7/4 00 1

1

2 2

0

3/4 3/4 9/4 5/4

1 1 3/2

1 1

2 1/2 2 1/4 2

2 cos /

3 2 1/ 4 1/ 8 2 1/ 8 2 1/ 32 2

2 1/2 2 1/4 2

2 sin /

3 2 1/4

b

b b

b b

bb b

b

b

b b

b b

R R



 

 

 



 

 











  

        

 

 

 

 





   

 

  

   



  



  15

4 2 2 4 7/4 4

7/4 00 1

1

1/ 8 2 1/ 8 2 1/ 32 2 bb b

b          



 

 

  

      

(22)

The other interesting component for any mode is the bound component. This is given by

R R       , which decays exponentially except for a small oscillatory component in the tail 

given by equation 23 when the bias field, 0 b is zero, 

15 13 17

3/2 7/4 2 4 44

11 1 1

15 13 17

3/2 7/4 2 4 44

1 11 1

2 cos / 1/ 2 2 1/ 8 2

3

2 sin / 1/ 2 2 1/ 8 2

3

bbb b

b bb

R R

b

      

      

  

 





 

        

      

  

 





    

(23)  

Notice that this oscillating term drops off like 15/4

1



and is proportional to the alignment. 
15

However, when the bias field is increased there is a pronounced effect on the behavior of the 

oscillatory terms that make up the bound component as can be seen in equation 24, 

  37

0 6 3 6 2 9/4 4

7/4 7/4 10 10 1

1

13/2 9/2 5 37

3/2 1 0 10 4 11/4

1 1 1 9/2 2 3 9/2 4

1 0 10

5 3

1 0

2 1 1/ 2 2 49152 24576

196608

2 1 24576 12288

cos / 2

3 196608 12288 3072

20480 2

1 2

196608

b bb bb b

b

b b bb

bb b

b b bb

b b

R R





 







 



 

 

 

  

     

          



 





36 7

1 0 10 37 13

7 2 32 5 3 7 4 4 1 0 1 0 10 1

53 34 3

10 10

0 6 3

7/4 7/4 1 0

1

3/2

1 1

56 108544

32768 3072 2048

10240 1536

2 1 1/ 2 2 49152 2457

196608

2 sin /

3

b b bb

b b b b bb b

bb bb

b b b

b

b b

 





 



 

 

 

 

     

 





   

 

      



  37

6 2 9/4 4 10 1

9/2 2 3 9/2 4 37

1 0 10 4 11/4

1 13/2 9/2 5

1 0 10

5 3 36 7

1 0 1 0 10

7 2 32 5 3 7

1 0 1 0 10

5 3

1

6

1 12288 3072

2

196608 24576 12288

20480 256 108544

1 2 32768 3072 2048

196608

10240

bb b

b b bb

b

b b bb

b b b b bb

b b b b bb

b



 



 











 

 



     

  

  

 



37 13

4 4 1

34 3

0 10 1536

b

b bb





 

 

 

   

     .

 (24) 

All of the terms displayed in equation 24 are proportional to the bias field, 0 b and vanish at zero 

bias. Note also that the radial dependence is much stronger in these bias dependent components. 

These effects are small at low bias fields but are expected to become important when studying 

the adiabatic problem when it is normal to include large bias fields to eliminate the possibility of 

some types of spin flips. 

8. Conclusions 

A method for calculating complete asymptotic expansions for the modes of atomic waveguides 

has been demonstrated. The technique can be extended to high orders using computer algebra 

techniques. This method complements the series solution technique presented in Golding (2010) 

and produces an accurate representation of the waveguide solutions far away from the guide 

center. 

In this system, some components of the spinor wavefunctions do not die off exponentially and 

are therefore significant at large distances. This is related to the quasibound nature of the system. 
16

A good representation of the nondecaying components of the eigenstates at large distances from 

the guide center is required so that the calculated eigenstates can be used reliably in further 

numerical calculations. The asymptotic expansions presented here provide this representation 

and are readily related to the power series solutions developed in Golding (2010). By connecting 

those power series solutions to the asymptotic expansions developed here an efficient 

representation of the exact radial wavefunctions can be obtained. These wavefunctions are 

needed for detailed studies of important properties of magnetic guides such as sensitivity to noise 

driven spin flips, importance of quantum Majorana transitions, energy level dependence on 

magnetic field, and the effects of guiding field imperfections as well as the onset and departure 

of adiabatic behavior. 

 
17

9. References 

Bender, C. M.; Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers; 

McGraw-Hill: New York, NY, 1978. 

Brink, D. M.; Sukumar, C. V. Phys. Rev. A 75 2006, 035401. 

Golding, W. M. Atomic Waveguides for Atom Chips; ARL-TR-5014; U.S. Army Research 

Laboratory: Adelphi, MD, Oct. 2009. 

Golding, W. M. Series Solutions for Atom Guides; ARL-TR-5335; U.S. Army Research 

Laboratory: Adelphi, MD, Sept. 2010. 

Heading, J. An Introduction to Phase-Integral Methods; Metheun: New York, NY, 1962. 

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