Atomic Waveguides for Atom Chips 

by William M. Golding 

ARL-TR-5014 October 2009 

Approved for public release; distribution unlimited. 
NOTICES 

Disclaimers 

The findings in this report are not to be construed as an official Department of the Army position 

unless so designated by other authorized documents. 

Citation of manufacturer’s or trade names does not constitute an official endorsement or 

approval of the use thereof. 

Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Adelphi, MD 20783-1197 

ARL-TR-5014 October 2009 

Atomic Waveguides for Atom Chips 

William M. Golding 

Sensors and Electron Devices Directorate, ARL 

Approved for public release; distribution unlimited. 
ii 

REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently 

valid OMB control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

October 2009 

2. REPORT TYPE 

Summary/interim 

3. DATES COVERED (From - To) 

4. TITLE AND SUBTITLE 

Atomic Waveguides for Atom Chips 

5a. CONTRACT NUMBER 

5b. GRANT NUMBER 

5c. PROGRAM ELEMENT NUMBER 

6. AUTHOR(S) 

William M. Golding 

5d. PROJECT NUMBER 

5e. TASK NUMBER 

5f. WORK UNIT NUMBER 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

U.S. Army Research Laboratory 

ATTN: RDRL-SEE-O 

2800 Powder Mill Road 

Adelphi, MD 20783-1197

8. PERFORMING ORGANIZATION 

 REPORT NUMBER 

ARL-TR-5014

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) 

11. SPONSOR/MONITOR'S REPORT 

 NUMBER(S) 

12. DISTRIBUTION/AVAILABILITY STATEMENT 

Approved for public release; distribution unlimited. 

13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

This report studies the quantum behavior of a spin one-half magnetic atom guided by a two-dimensional quadrupole magnetic 

field of infinite extent. The technique used is based on the Frobenius series technique and gives a complete understanding of 

the guide mode structure for small radial coordinates. Asymptotic solutions are used to establish boundary conditions for large

radii. Various guide modes that have not been included in earlier studies are made clear by this technique. Although, in 

principle, the series solutions are complete, convergence is slow and the present study, therefore, uses these solutions only to 

establish consistent initial conditions for a differential equation solver. In this way, radial modes and eigenvalues are 

calculated. The results can be extended to higher spin atoms and more complicated guiding fields. These extensions along 

with detailed analysis of experimental limitations will be the subject of future work. 

15. SUBJECT TERMS

Atom chips, atomic waveguides, BEC 

16. SECURITY CLASSIFICATION OF:

17. LIMITATION 

 OF 

 ABSTRACT 

UU 

18. NUMBER 

 OF 

 PAGES 

38 

19a. NAME OF RESPONSIBLE PERSON

William M. Golding 

a. REPORT 

Unclassified 

b. ABSTRACT 

Unclassified 

c. THIS PAGE 

Unclassified 

19b. TELEPHONE NUMBER (Include area code) 

(301) 394-1535 

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18 
iii 

Contents 

List of Figures iv

1. Introduction 1

2. Magnetic Waveguides—Atom Guides using Magnetic Fields 2

3. The Hamiltonian for a Straight Magnetic Waveguide 5

3.1 Hamiltonian for a Magnetic Atom ..................................................................................5 

3.2 The Alignment Operator .................................................................................................7 

3.3 Eigenstates Common to H and Lz .................................................................................10 

3.4 Numerical Work ............................................................................................................14 

4. Farfield Solutions 19

5. Numerical Solutions 20

6. Conclusions 26

7. References 28

Distirbution List 29 
iv 

List of Figures 

Figure 1. Cross-sectional view of the magnetic null above a single wire waveguide. Lowfield-seeking atoms are guided in this local magnetic minimum. .............................................3 

Figure 2. Two alternatives to the single-wire guide that could be constructed using etching 

techniques on a silicon chip. (Upper) The four-wire guide with one pair of currents going 

into and another pair coming out of the page, in which the current directions simply 

alternate from one wire to the next. The symmetry of the current distribution creates a 

null in the magnetic field at the center of the guide, which forms a transverse quadrupole 

field without the need of a large external bias coil. (Lower) A hexapole configuration 

shown as a simple generalization of the quadrupole guide; however, the magnitude of the 

potential varies quadratically from the center as opposed to the linear variation of the 

quadrupole. In addition, the alignment operator would be changed to Lz – 2Sz instead of 

Lz – Sz. ........................................................................................................................................4 

Figure 3. Cross-sectional field plot of the general quadrupole guiding field used in this work. 

Fields come inwards along x and go outwards along y, resulting in a quadrupole null at 

the center that can be used for atom guiding. ............................................................................6 

Figure 4. Illustration of the backwards rotation of the spin that must occur to keep a 

magnetic moment aligned with the magnetic field. The particular form of the alignment 

operator used here is specific to the quadrupole field. ...............................................................9 

Figure 5. These plots show a μ = 1/2 eigenstate (left) and the μ = –1/2 degenerate eigenstate 

(right). The mode on the left is spin up in the low field limit and the mode on the right is 

spin down in this limit. Notice that the two components in each mode oscillate as out of 

phase Bessel functions at large radius and that the roles of R+ and R–have been reversed 

relative to each other. The fact that these two modes are degenerate allows them to be 

combined to form pure bound or unbound states. ....................................................................21 

Figure 6. Local basis eigenstates corresponding to the global basis states shown in figure 5. 

The components of the local basis states are derived from the global states by forming the 

sum and difference of the global components. This can also be accomplished as a basis 

transformation using rotation matrices to go from the Sz to Sx basis. The significant 

difference between the local and global basis representations is the appearance of an 

exponentially decaying bound component along with an oscillatory or unbound 

component. The bound components in the two degenerate modes are equal and have the 

same sign, but the unbound components have opposite signs. For this reason the sum and 

the difference of the two degenerate modes can be used to produce pure bound and 

unbound states as shown in figure 7. .......................................................................................22 

Figure 7. Purely bound state obtained by forming a superposition of the degenerate states 

shown in figure 6. ....................................................................................................................22 

Figure 8. These two states are the degenerate pair of the first excited modes in the system of 

modes that act like 1 J when the transverse field goes to zero. Since they are degenerate, 

they can be combined to obtain pure unbound and bound modes. ..........................................23 
v 

Figure 9. These are the next two higher order modes in the system of modes that act like 1 J

when the transverse field goes to zero. As expected, the number of zero crossings 

increases as the mode energy increases. ..................................................................................24 

Figure 10. This state is the lowest order mode of the system of modes that behave like 0 J as 

the transverse field is decreased. Notice the large on axis component that distinguishes 

this mode from the 1 J type modes. This mode is the likely ground state of the system; 

however, there is some mixing of modes from the spin-up solutions in this system as 

discussed in the text of section 3. Thus, this mode might represent a superposition of 

spin-up and spin-down eigenstates. It is shown in both the global and local bases. ...............25 

Figure 11. Local behavior of the higher order modes that behave like 1 2 J and J at low 

transverse field .........................................................................................................................25 

Figure 12. These two plots for the lowest energy μ = 3/2 state show that the same basic 

behavior holds for alignment other than μ = ±1/2. The solutions can be found for higher 

values of alignment as well. .....................................................................................................26 
vi 

INTENTIONALLY LEFT BLANK. 
1 

1. Introduction 

Atom chips are miniature structures built to control and manipulate ultracold atoms. They can be 

constructed using techniques borrowed from the integrated circuit, microelectromechanical 

systems (MEMS), and integrated optics industries. The goal is to create miniature structures that 

can be used to produce optical, electrical, or magnetic fields for manipulating and transporting 

atoms in a coherent way on a general atom chip circuit. A recent review of atom chip techniques 

is available in Fortágh (2007). 

Some atom chip structures may be thought of as miniature versions of larger devices common in 

atomic physics laboratories where cold atom techniques are used. However, some devices rely 

on the miniature scale of the traps available on atom chips to produce quantum effects. Single 

mode operation of an atomic waveguide using ultracold atoms requires that the physical scale of 

the guiding potentials be on the order of the deBroglie wavelength of the atom. The deBroglie 

wavelength is inversely proportional to the atomic momentum, so to increase the wavelength the 

momentum of the atoms must be decreased. This is accomplished by cooling the atoms using 

some combination of laser cooling and magnetic evaporation. Atoms can be cooled so that their 

wavelengths are on the order of a few hundred nanometers. This is conveniently in the range of 

available lithographic and other miniature construction techniques. 

A useful device that might be constructed using atom chips is the atom interferometer. A chipbased atom interferometer could be used to sense inertial fields or magnetic fields. Successful 

development of miniature interferometers on atom chips would allow the development of very 

sensitive detectors based on the interference of matter waves in cold atom systems such as Bose 

condensates or Fermi degenerate gases. 

In one possible implementation of an atom chip-based interferometer, cold atoms would be 

created on the surface of an atom chip and then launched into an atom waveguide. In the 

waveguide, they would propagate towards a beam splitter. The splitter would be designed to 

cause the atomic wave packets to travel along two alternate paths through a sensing region. The 

atoms would then arrive at a beam combiner completing the basic interferometric structure. By 

counting the relative number of atoms exiting at the two output ports of the combiner (using light 

scattering or some other technique), a sensitive measure of the phase difference between the two 

alternate paths can be obtained. The main idea is that all of the necessary functions should take 

place on the surface of the atom chip so that a useful miniature instrument might eventually be 

constructed. 

The phase difference between the paths in the interferometer is the basic quantity that is 

measured in normal operation. Many things can disturb the phase difference causing errors in the 

measurement. For example, local optical, electric, or magnetic fields can perturb the internal and 
2 

external atomic energy levels causing the guided atoms to accumulate phase errors. In addition, 

mechanical accelerations, rotations, or vibrations of the atom chip could cause relative phase 

errors to accumulate, disturbing the measurement. The various sources of phase error in the 

interferometer must be understood, controlled, and compensated for to a high level in order that 

very sensitive measurements can be made. 

The atom waveguide in one form or another is central to the development of basic 

interferometric structures on atom chips. The quadrupole atom guide as well as the quantum 

behavior of atoms in magnetic traps has been studied by several authors (Hinds, 2000 and 2001 

Bill, 2006; Lesanovsky, 2004; Sukumar, 1997; Potvliege, 2001; Bergman, 1989); however, the 

results of these investigations are difficult to interpret because of several problems in the analysis 

techniques used. It is clear that a certain class of modes may have been missed in these other 

works. In addition, the calculations of Hinds (2000) claim logarithmically singular solutions as 

physically valid guide modes for the spin one case. This is incorrect. The missing modes, 

singularities, and other things will be discussed here and in future work based on this analysis. 

It is necessary to have a complete and correct understanding of the mode structure and the 

eigenvalues in order to predict various experimentally interesting properties of the magnetic 

guides, such as lifetime limitations or longitudinal phase shifts that may be caused by small 

perturbations in the fields. For example, small perturbations might be caused by limitations of 

the lithographic techniques currently available to construct the small wires needed. However, 

without a proper understanding of the behavior of the guide modes under the action of these 

perturbations, it is impossible to know if available construction techniques are adequate to create 

a useful atom guide. This report studies the quantum behavior of a spin one-half magnetic atom 

guided by a two-dimensional quadrupole magnetic field of infinite extent. The technique used is 

based on the Frobenius series technique (Ince, 1956; Bender, 1978) and gives a complete 

understanding of the guide mode structure for small radial coordinates. Asymptotic solutions are 

then used as boundary conditions at a large distance from the guide center. Although, in 

principle, the series solutions are complete, the present study uses these solutions only to 

establish consistent initial conditions for a differential equation solver. In this way, a set of radial 

modes and eigenvalues are calculated; however, issues pertaining to proper normalization still 

remain to be resolved before these modes can be used for detailed quantum calculations. 

2. Magnetic Waveguides—Atom Guides using Magnetic Fields 

One of the simplest types of atomic waveguides that can be constructed on an atom chip is the 

single-wire magnetic guide. This type of guide is designed as a single straight current-carrying 

wire printed lithographically on a silicon chip. The cross section of a lithographically produced 

wire is often nearly rectangular. Assuming the detailed cross-sectional shape of the wire can be 

ignored, the magnetic field created by a current flowing along the wire encircles the wire in a 
3 

right-hand sense relative to the current direction. Using large external Helmholtz coils, a uniform 

bias magnetic field is applied in a direction parallel to the chip surface and perpendicular to the 

direction of the wire. The two fields will cancel at a fixed distance above the wire creating a null 

in the total transverse magnetic field that extends the length of the wire. This long null in the 

magnetic field forms a two-dimensional quadrupole field that can be used to guide atoms along 

and just above the wire. 

A two-dimensional finite element solution of the magnetic fields just above a rectangular 

current-carrying conductor is shown in figure 1. The uniform current flowing in the rectangular 

wire produces a magnetic field that is cancelled at about 60 μm directly above the wire surface 

by the field of an external pair of coils. The contour lines represent the flux density of the 

magnetic field and there is a magnetic null within the small circular contour. This null region is 

where low-field-seeking atoms will be trapped and guided in this single-wire guide design. 

Figure 1. Cross-sectional view of the magnetic null above a single wire waveguide. Low-field-seeking 

atoms are guided in this local magnetic minimum. 

The single-wire guide is one of the simplest structures for magnetically guiding atoms on atom 

chips; however, it is not an ideal guide. There are several basic difficulties that make this design 

inconvenient for use in general atom chip circuits. The basic problem is that the wire must be 

perpendicular to the fixed external bias field so that the trapping region maintains its shape and 
4 

strength. This means that the waveguides must be straight and run in one direction only. This is a 

serious design limitation for a general waveguide. Some alternative are shown in figure 2 and 

others are discussed in Fortágh (2007). 

Quadrupole Guide

Guided Cold

Atom Cloud

Silicon Substrate

Gold Wires

Embedded in

Substrate

Hexapole Guide

Guided Cold

Atom Cloud

Silicon Substrate

Cold Atom Guide Cross Sections

Figure 2. Two alternatives to the single-wire guide that could be constructed using etching techniques on a 

silicon chip. (Upper) The four-wire guide with one pair of currents going into and another pair 

coming out of the page, in which the current directions simply alternate from one wire to the 

next. The symmetry of the current distribution creates a null in the magnetic field at the center of 

the guide, which forms a transverse quadrupole field without the need of a large external bias 

coil. (Lower) A hexapole configuration shown as a simple generalization of the quadrupole 

guide; however, the magnitude of the potential varies quadratically from the center as opposed to 

the linear variation of the quadrupole. In addition, the alignment operator would be changed to 

Lz – 2Sz instead of Lz – Sz. 

By using multiple coplanar wires or by etching into the surface of the chip, it is possible to 

construct magnetic guides that can be laid out in a flexible way on the surface of an atom chip. 

These techniques can, in principle, produce approximate quadrupole guiding fields that twist and 

turn along the surface of an atom chip and preclude the need for a system of external bias coils. 
5 

Generalized quadrupole guiding structures for atoms might be used to create coherent splitters, 

combiners, directional couplers, and tunneling devices that are analogous designs to those 

presently used to create similar microwave or optical devices. However, the detailed analysis 

begins with the simple straight magnetic waveguide for atoms. 

3. The Hamiltonian for a Straight Magnetic Waveguide 

To study the dynamics of an atom in a magnetic waveguide, the Schrödinger equation describing 

a single neutral magnetic atom trapped in the magnetic guiding field must be solved. To 

minimize the initial complexities, the initial model is chosen to be a neutral spin 1/2 atomic level 

since it has only two magnetic levels. Although this system is essentially the same as the 

magnetically trapped or guided neutron, it is not completely representative of the magnetic 

properties of the alkali atoms that are normally used in laser cooling work. Once the study of this 

model atomic system is complete, a model of a more realistic system, such as the rubidium atom 

with F = 1,2, will be completed. A full model including hyperfine interactions, important in 

experimental systems that explore atomic clock development, could include eight or more 

magnetic sublevels. Rubidium is used at the U.S. Army Research Laboratory (ARL) for 

experimental atom chip development and a complete rubidium waveguide model will be useful 

for design and analysis of various experiments. 

3.1 Hamiltonian for a Magnetic Atom 

The quantum description of an atom moving in a spatially dependent magnetic field must include 

both internal and external degrees of freedom. This means that the momentum, position, and spin 

degrees of freedom must all be treated as quantum operators. Using m for the mass of the atom 

and M



 for the magnetic moment, the Hamiltonian is written as the sum of the kinetic energy, 

2

2

p

m

 and the interaction energy of the magnetic moment and the field B



. The result is 

 

2

2

p H M Br

m      . (1) 

The magnetic field B  r   is independent of the z-direction since it is the guiding field and is 

taken to be an ideal quadrupole field equation (equation 2), extending to infinity in the x-y plane 

and uniform along z. An additional uniform bias field B0 is added in the z-direction to help 

control possible spin-dependent losses that may occur at the zero field point at the center of the 

guide. The spatial dependence of the ideal quadrupole field is given by 

B  r B xx yy B z    1 0  ˆ ˆ ˆ   . (2) 
6 

A cross-sectional plot of this field is shown in figure 3. The guiding center of this field is at the 

origin where the transverse field is zero. The quantity B1 is the magnitude of the transverse field 

gradient. It is taken as greater than or equal to zero in this work, although changing the sign of B1

simply changes the quadrupole field configuration from one with the field point inward along the 

x-axis to one with the field pointing inward along the y-axis. 

Figure 3. Cross-sectional field plot of the general quadrupole guiding field used in this 

work. Fields come inwards along x and go outwards along y, resulting in a 

quadrupole null at the center that can be used for atom guiding. 

The gradient of the transverse field is what provides the trapping force that keeps the atom 

confined in the transverse direction. The transverse field is zero at the center of the guide and the 

only nonzero field component at the very center is the longitudinal bias field B0. The magnitude 

of the transverse field is 2 2 B   Bx y B 1 1 , where  is the radial coordinate in the 

cylindrical system. Even though the field is varying in direction as one goes round the guide, the 

contours of constant field magnitude are simply circles. 

The potential is independent of the position along the guide z and therefore pz is a constant of the 

motion and is ignored here. In future problems, when variations in the potential along the zdirection are considered, the variations in the longitudinal momentum will be included. 

Variations in the transverse potential should show up as variations in the guide propagation 

constant and this will affect the accuracy of sensitive measurements based on interferometry. 
7 

In this simplified model of the atom (equation 1), the magnetic moment is just the magnetic 

moment of a single outer electron. The atomic model used here can be thought of as an alkali 

atom with a spin zero nucleus and a total mass m. The magnetic moment is M   S  , where  is 

the gyromagnetic ratio of the level considered and S



 is the spin angular momentum of the atom. 

Since a spin 1⁄2 system is being considered, the spin angular momentum is proportional to the 

Pauli matrices, 2

S      . 

The Schrödinger equation for the guided atom eigenstates E is 

     

2

2

2 E EE r SBr r E r

M

       

      . (3) 

It will be solved for the energies and eigenstates. The equation 3 is made dimensionless by 

choosing a length scale such that x  x , where the new x is dimensionless. Then 

   

2 22

2 2

2 2 E E

k

r r

M M

       and the recoil energy is defined as 

2 2

2 R

k E

M   , where 1 k  

is 

the wavenumber of the optical field used to cool the atom. This is a rather arbitrary choice at this 

stage, but the recoil energy is a convenient scale for atoms that have been laser cooled. After 

dividing through by the recoil energy, pulling out the relevant factors, and defining the 

dimensionless transverse field parameter 1

1 2 R

B b

E  

  , the longitudinal field parameter 

0

0 2 R

B b

E  

 , and the scaled energy 

R

E

E

  , the scaled Schrödinger equation becomes 

          2

1 0 xy z rb x y rb r r                    . (4) 

The next step is to convert to the cylindrical coordinate system and derive the equation for the 

radial wave functions. In cylindrical coordinates, we have x =  cos , y =  sin , and the 

Schrödinger equation takes the following form: 

          2

1 0 , cos sin , , , xy z b b                         . (5) 

Before reducing equation 5 further, an important conserved quantity in this problem will be 

introduced. 

3.2 The Alignment Operator 

The total angular momentum is not conserved in this system. Neither the orbital angular 

momentum 

1 Lz i

  nor the spin angular momentum 1

2 z z S   individually commute with 

the Hamiltonian but the difference of these two operators does commute with H as shown in the 
8 

Erratum (Hinds, 2001). To show that this is the case, one can explicitly calculate the 

commutators of Lz and Sz with H. The appropriate operator form of the Hamiltonian for this 

calculation is 

  2

1 0 cos sin Hb b           x y z . (6) 

Both Lz and Sz obviously commute with the first and third terms of H, but the middle term that 

corresponds to the quadrupole potential is more interesting. The following straightforward 

calculations (equation 7) of both commutators show that they are equal to each other and 

nonzero unless, of course, the transverse field gradient b1 is zero or if we consider the point 

 = 0. 

   

 

     

 

 

1

1

1

1

1

, cos sin

sin cos

, , cos , sin 2

2 cos 2 sin

2

cos sin

z xy

x y

z zx zy

y x

y x

b L H

i

ib

b S H

b i i

ib



   

   

     

   

   

 

 

     

 

 

. (7) 

The fact that the commutators are equal means that the difference Lz – Sz is a conserved quantity. 

For simplicity, the operator Lz – Sz is relabeled as z and called the alignment operator. The 

alignment is obviously related to the angular momentum. Instead of being the sum of the two 

types of angular momentum, it is the difference. If one considers the angular momentum as a 

generator of rotation, then the alignment generates a coordinated rotation about the axis that 

maintains the angle between the field and the magnetic moment of the atom. The eigenvalues of 

z are denoted by μ. Since the eigenvalues of Lz are integral and the eigenvalues of Sz are halfintegral in the spin 1⁄2 case, the allowed values of μ are half-integral and μ has the dimensions of 

an angular momentum. 

Consider an atom located on the positive x-axis of the waveguide with its magnetic moment 

aligned with the local magnetic field. At this position the magnetic field is pointing in the 

negative x-direction towards the origin. If the position of the atom is rotated 90° 

counterclockwise around the z-axis, it will end up on the y-axis. If the rotation is performed 

without changing the orientation of the magnetic moment, then the moment will be perpendicular 

to the y-axis at this new position. This is illustrated by the red vector in figure 4. However, the 

quadrupole magnetic field defined in equation 2 is pointing in the positive y-direction at this 

position. 
9 

Figure 4. Illustration of the backwards rotation of the spin that must occur to keep a magnetic moment aligned 

with the magnetic field. The particular form of the alignment operator used here is specific to the 

quadrupole field. 

By rotating the spin of the atom 90° clockwise, the magnetic moment will be aligned along the 

magnetic field as it was at the starting point on the x-axis. This backward 90° rotation is 

illustrated by the green vector in figure 4, which is always aligned with the quadrupole field as it 

moves around the center. The operator z is the generator of the special rotation z i

e

   needed 
10 

to keep the spin aligned with the magnetic field, keeping the magnetic potential energy constant. 

It is a special property of the quadrupole field that two equal rotations of opposite sense in 

different spaces conserve the potential energy in this way. However, similar relations hold for 

higher multipole magnetic fields, in which the rotation angle of the spin must be an integer 

multiple of the actual rotation angle. 

3.3 Eigenstates Common to H and Lz 

Using the alignment operator, the Schrödinger equation in polar coordinates (equation 5) reduces 

to a radial equation for the spin components of the wave function. The first step is to rewrite the 

Schrödinger equation using the Pauli matrices, i. The quantization axis for the spin description 

is the +z-direction and this is along the propagation axis of the guide. This basis is referred to as 

the global spin basis. The spin-up, +, and spin-down, –, components are referred to this global 

spin basis. Using the following definitions 

         

   

 

2

1 0 , cos sin , , ,

, , ,

01 0 1 0 , , 10 0 0 1

xy z

xy z

b b

i

i

  



           

 

 

 

 





        

       

                  

 (8) 

in the Schrödinger equation, we arrive at the coupled system 

 

 

 

 

 

 

 

 

2

2 1 0

, , , ,

, , , ,

i

i

e

b b

e





       

 

       

   



   

                      

. (9) 

The components are completely uncoupled if the transverse field b1 is turned off by setting 

b1 = 0. In this limit, the problem reduces to that of a free particle with a magnetic moment in a 

uniform magnetic field, b0. Also in this limit, the operators Lz, Sz, and the alignment   z zz L S

commute with H and are conserved quantities. Since these operators commute with each other as 

well as with H, simultaneous eigenstates can be found when b1 = 0. We can use this limit as a 

way to identify states when the transverse field is large by allowing b1 to reduce towards zero in 

the complete solutions. 

The common eigenstates of Lz, Sz, and z are found using the eigenstates of Lz, Sz in the 

following way 
11 

   

, ,

, ,

,, , ,

then let to define simultaneous eigenstates of ,

, ,.

z

z

z zz

z

z

L lm llm

S lm mlm

lm lm L S lm l m lm

l m

mm mm





 





    

 

  

 (10) 

In the coordinate representation, 1 Lz i

   and therefore the angular dependence of the 

component with alignment μ and spin m is given by i m   e m    . Using this angular dependence, 

the general multi-component wave function in the matrix representation is 

 

 

   

   

1/2

1/2

,

,

i

i

R e

R e

 

 

  

  



 

  

   

    

   

. (11) 

Although very similar, this is not the simple product form that is used in basic separation of 

variables techniques because the angular dependence is different for each component of the trial 

function. The result of using this trial form of the eigenstates in the Schrödinger equation 

(equation 9) is 

 

 

 

 

 

 

 

 

2 1/2 1/2 1/2 1/2

1 0 2 1/2 1/2 1/2 1/2

i i ii i

i i ii i

Re e Re Re Re

b b

Re e Re Re Re

    

    

 

  

  

   

  

                

        

. (12) 

The Laplacian operator acting on the states (equation 11) produces the following terms: 

       

2

2 2 1/2 1/2

2

i i 1 1/2

Re R R R e   

 



 

 

  

  

       

 

. (13) 

Using this expression in equation 12 results in the following system of coupled differential 

equations: 
12 

   

  

   

  

2

2 1/2

2

1/2 1/2 1/2

1 0

2

2 1/2

2

1/2 1/2 1/2

1 0

1 1/2

,

1 1/2

.

i

i i ii

i

i i ii

R R Re

b e Re bRe Re

R R Re

b e Re bRe Re

 

 

    

 

 

    



 

 



 

 



 

 

 



 

  

 

         

 

 

         

 

 

 (14) 

An interesting point to notice is the way that the angular dependence factors out exactly. This 

cancellation is due to the way the angular factor in the quadrupole field part of the potential 

compensates for the action of the spin operators and is a direct consequence of using 

simultaneous eigenstates of H and the alignment operator z . Because of this, the angular 

dependence can be completely factored out of the problem leaving only the radial equations: 

 

 

2

2

2 1 0

2

2

2 1 0

1 1/2 ,

1 1/2 .

R R R b R bR R

R R R b R bR R

 

 



 

 



 

 

    

    



      

       

 (15) 

The coupled radial equations for the eigenstates of the quadrupole waveguide are shown in 

equation 15. They must be solved for eigenvalues and eigenvectors using the allowed halfintegral values of the alignment 135 , ,

222     . 

As mentioned earlier, when the transverse field is turned off, the two equations in equation 15 

are uncoupled. In this limit, the solutions are simply Bessel functions and they may also be 

common eigenstates of H, Lz, Sz, andz . Using this fact the complete solutions at b1 = 0 can be 

written as 
13 

     

     

     

   

1/2

1/2 0

1

2

1/2

1/2 0

1/2

1/2 0

1

2

1/

1/2 0

 bounded spin up

0 ,, ,

 unbounded spin up

0

0

 bounded spin down

, , 0

i

i

i

i

J be

l mm

Y be

J be

l mm

Y be

 



 



 







 

 

 

 

 

 

















     

     

     



 

          

   2

 .

unbounded spin up 











     



  

 (16) 

The set of solutions in equation 16 represents the four free space solutions to the pair of 

uncoupled second order radial equations in the limit of a small transverse field, that is, when 

b1  0. It is seen that the limiting cases are clearly distinguished. There will be one mode that 

behaves like 1/2 J that will be referred refer to as a spin-up mode and one mode that behaves 

like 1/2 J that will be referred to as a spin-down mode. As the transverse field 1 b is turned on, 

these modes continuously evolve into modes with significant amplitudes of both components. 

This can be thought of as precession of a pure spin state around a transverse field. To keep things 

straight, the states can still be labeled by their character as 1 b  0 . 

As a specific example, the case of alignment 1⁄2 can be considered. In this case, the two 

physically acceptable solutions in the 1 b  0 limit are the spin-up 1 J state and the spin-down 0 J

state. Then, to calculate the properties of the spin-down state at larger values of 1 b , the series 

solution that reduces to 0 J spin down is used to calculate the complete behavior of the finite 1 b

state near the origin. This series solution provides all initial conditions required near the origin 

for and R  R to initialize an ordinary differential equation (ODE) solver for the calculation of 

the behavior of both components of the spin-down mode at large radii. 

The two solutions that involve the Bessel functions of the second kind Yi  are unbounded at 

the origin and should be excluded as unphysical in problems that include the origin in the 

domain. This is the case in the quadrupole waveguide problem considered here. However, the 

unbounded solutions can be useful as they may help to identify the unphysical modes when all 

that is available is a complicated series solution. In the analysis by Hinds (2000) and others 

based on this earlier work, the two bounded modes seem to be combined into one mode, 

apparently leading to the use of mixed boundary conditions at the origin. For this reason, the 

results of that work are difficult to interpret. In this study, the modes are separately identified and 
14 

kept separate throughout the calculations. Although the scheme chosen here to separate the 

modes at low transverse field seems useful, more work on the effect of the various boundary 

conditions at the guide center is needed. 

3.4 Numerical Work 

Numerical techniques are used to find the eigenvalues and eigenvectors at finite b1. The 

differential equation solvers that come with the Maple computer algebra system have been used 

in this work. The integration technique that is used is called the shooting technique and was the 

technique used in the work by Hinds (2000). Specific values of the parameters μ, , b0, and the 

transverse field b1 are chosen. An initial value is provided for both of the radial wave function 

components and their derivatives at the origin, and the coupled equations are integrated outwards 

towards large values of . If the eigenvalue  is not correct, the solution quickly grows large as 

an exponentially growing solution becomes dominant. The process is repeated after adjusting the 

eigenvalue. In this way, consistent values for the energies can be found that correspond to 

various modes of the guide. 

However, in order to get accurate results from the differential equation solvers, accurate initial 

values are needed at the origin. The equations are singular at the origin and a Taylor series 

expansion of the general solution about the origin does not, in general, exist. The singularity at 

the origin is a regular singularity and series solution techniques can be used to accurately solve 

for the behavior of the solutions near the origin. These solutions are used to provide accurate 

initial conditions to the ODE solvers. In this way, specific modes that are bounded at the origin 

can be studied. 

To obtain an equation convenient for the use of the series technique, the coupled 2nd order radial 

equations (equation 15) are transformed into a single 4th order equation for one specific 

component such as R    of the wave function, as shown in equation 17. Once the 4th order 

equation is solved, the general solution for R   contains four independent functions in series 

form. The four functions describing R   are found by substitution of each R    into 

equation 18. In this way, four consistent pairs of components solving the original coupled 

equation (equation 15) are obtained. This process can be reversed by solving first for R   

using equation 19 and then using equation 20 to find the corresponding R    solutions. 

Solving in either order produces a valid general solution to the system, but often one way 

produces a better form for the resulting solutions. This can be a useful way to identify log free 

regular solutions to the system. These types of solutions are free from singularities at the origin 

and are the physical solutions required in the waveguide problem under study. 
15 

 

 

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

  

  

      





 

















 

    



       

      

             

       

 (17) 

   

   

  

2

2

2 2 0

1

1

1 2

R R

d d R R bR

b d

d

  

   

  

 

  

                

 

 

 (18) 

 

 

4 22

3 2

0

26 2 2 4 2 2 2 4

4

4

4

2

2

1 00 0

5 2 2

2

2 54 6

1 19 35 22 9

2 2 16

0

d R

d

d R

d

d R

d

R

b

b bb b

  

  

        



















       

     

  



 

    

           

 

 

      

 (19) 

   

   

  

2

2

2 2 0

1

1

1 2

R R

d d R R bR

b d

d

  

   

  







 

                

 

 

 (20) 

Each of the 4th order equations 17 and 19 are solvable using the Frobenius technique for higher 

order equations (Ince, 1956; Bender, 1978). The basic idea is to use a modified power series 

form for the solution and to solve for the unknown coefficients. An assumed series solution of 

the form 

 

0

s R c 





 

 





  (21) 

is used in one of the 4th order equations. The coefficients cv and the index s are determined by 

equating the coefficients of like powers on either side of the equation to zero. The first equation 

will be of the form c0f0(s) = 0 and c0 is assumed equal to one. The equation f0(s) = 0 is called the 
16 

indicial equation and there are four roots si. Each root corresponds to one solution of one of the 

differential equations 17 or 19. The indicial equations and the roots are both given as follows: 

4 3 22 2 4 2 17 3 3

1/ 2, 7 / 2,

5 19 6 2 6 6 9 0, 2 2 16 2

 1/ 2, +5/2.

R s ss s

s

 

 

 







                

 



 

 (22) 

4 3 22 2 4 2 17 3 3

1/ 2, 7 / 2,

5 19 6 2 6 6 9 0, 2 2 16 2

 1/ 2, +5/2.

R s ss s

s

 

 

 







                

 



 

 (23) 

In the case of μ =1/2, the indices for the R equation are s+ = (3,3,1,–1) and for the R equation the 

indices are s– = (4,2,0,0). The largest index always produces a solution free from logarithms in 

the Frobenius method and is thus bounded at the origin. Thus, two types of log-free modes with 

leading behaviors of R+ ~ 

3

 and R– ~ 

4

 at the origin are found. 

Since μ must be half integral, the solutions to the indicial equations are all integers. This is a 

special case in the Frobenius technique that results in some of the series solutions containing 

logarithmic terms like ln() times another power series. These logarithmic terms are unbounded 

unless multiplied by a positive power of . They must be excluded from the physical solutions if 

they cause either component of the solution to become unbounded. 

In the modes studied so far, two bounded and two unbounded solutions are always found by this 

technique for any value of μ. This should be expected by examining the limiting behavior of the 

solutions demonstrated in equation 16 for small values of b1. Since b1 and  are independent, any 

logarithmic terms in  cannot simply be cancelled by allowing b1  0. 

However, there are cases when straightforward application of the series technique apparently 

results in only one solution that is bounded at the origin, the other three being unbounded. The 

four solutions together must represent the general solution to the problem. By inspecting the 

limiting forms (equation 16), a linear superposition of the three unbounded solutions can be 

constructed to create two unbounded and one bounded solution, again resulting in two 

independent bounded solutions. Essentially, a singularity present in two of the calculated series 

is cancelled by this technique producing the additional regular log free solution. 

Once the series solutions (equation 21) for various values of s are determined, they are 

individually substituted into the corresponding equation (equation 18) to obtain the related series 

solution for the other solution. In this way, four consistent pairs of series solutions are generated 

for the system (equation 15). 
17 

These four functions together represent a fundamental solution to the waveguide problem for a 

particular value of the alignment eigenvalue μ. Two modes are bounded and two are unbounded 

at the origin, as mentioned previously. The series solutions do not converge quickly for large 

distances from the guide center and the basic requirement in the search for eigenvalues is that 

solutions remain bounded at large radii as well as at the origin. A better technique for large radii 

is to use an ODE solver and integrate outwards from the origin. The series solutions are ideal 

tools to establish consistent initial conditions at some point away from the origin so that the 

singularities do not negatively affect the ODE solver. 

Samples of the four series solutions obtained via the Frobenius technique for the case μ = 1/2 are 

displayed in equations 24 through 27. The solutions are labeled Rms1, etc., indicating series one 

for R– and so on. The series labeled Rps1, etc., are derived directly from the corresponding Rm

series using equation 20. The bounded solutions in this case are Rms1 and Rms3 and these series 

are the basis of two types of waveguide modes that are distinguished by their behavior at the 

origin. 

For the case μ = 1/2 the allowed values of s– are (4,2,0,0). Rms1 is the solution with the indicial 

exponent 4. This is the largest exponent of the group and, therefore, Rms1 is finite at the origin 

according to the Frobenius theory (Ince, 1956; Bender, 1978). Both components of the Rms1 

solution are zero at the origin, which may lead to an inherent insensitivity to certain types of spin 

flips or field fluctuations that are possible at the zero field point at the guide center. As b1  0 

the component Rps1 dominates and the combination behaves like the m = 1/2 spin-up state 

identified in equation 16. This is how the states are identified for arbitrary b1. 

   

2 3 10 2 8 6 4

1

2 49 2 37 52 3

1 1

1 1 1 11

1 1 11 1

4800 23040 384 12

11 1 24 5 2400 200

1 2 16

115200 115200 12

Rms b

b b

Rps b b b bb

     

      

         

     

 (24) 

The solution Rms2 is the s = 2 solution and both components have logarithmic terms that are 

multiplied by positive powers of . These terms are actually finite at the origin, but the 1/ term 

in Rps2 is unbounded so this term must be excluded from the physical solutions. The singularity 

in Rps2 comes from substituting Rms2 into the differential equation 20. The derivative of the log 

produces the singularity. 
18 

 

 

   

83 62 4

2 3 8 62 4 2

1

47 35 23

1 1 11

4 27 2 35 2 3

1 1

1 11

1 11 2 ln

3072 96 8

1 7 83 299

1536 5760 1920 480

1 11 2 2 ln

3072 96 4

1 1 127 56 30 1920 1832

46080 46080 120

Rms

b

Rps b b bb

b b

b bb

   

    

       

     

         

        

 

        

 

 

1 1

299 4

30 b b







 (25) 

The indicial exponent s = 0 produces the solution Rms3. This solution is an acceptable solution 

that is nonzero at the origin. However, it also behaves as R J   1     for small values of b1

and could also be a m = 1/2 state. Closer inspection shows that the pair of solutions generated 

by the b1  0 limit can be created as superpositions of the Rms1 and Rms3 series. For example, 

the series RMS3 in equation 26 should be J0     . The two leading terms are correct but a 

term proportional to 4  is missing. Since the leading term of Rms1 in equation 24 is also 4 

these series can be added to fill in the missing term. This produces both a modified Rms3 and a 

modified Rps3 of precisely the correct limiting behavior. This type of correction is currently 

done by hand for the various series generated at different values of alignment; however, a 

general way of combining and creating solutions to automatically obtain the desired limiting 

behavior for various values of μ is being developed. This will enhance the usefulness of any 

results based on these modes. 

   

2 36 2

1

2 45 3 23 2

1 1

1 11

11 1 3 1

288 1152 4

11 1 4 36 144

3

1152 1152 4

Rms b

b b

Rps b bb

  

     

        

     

 (26) 

The solution Rms4 also contains logarithmic terms and is unbounded at the origin. It is not a 

physically acceptable solution as it has a term proportional to 1



. 
19 

 

     

   

2 36 2

1

2 3 6 42 2

1

4 25 3 23 2

1 1

1 11

2 45 3 2

1 1

1

1 1 36 4 ln

816 3264 34 17

75 9 2 1

3264 6528 1088 17

11 3 2 8 72 288

4 ln

6528 6528 34

1 1 14 5 150 600

6528 6528

Rms b

b

b b

Rps b bb

b b

b

   

   

      

   

               

        

          

 

   

3 2

1 11

1 3

b bb 4 17

 



 

 (27) 

4. Farfield Solutions 

The coupled equation 15 can be solved approximately in the limit as   . This is 

accomplished by ignoring terms that are small relative to the linearly increasing term 

proportional to b1. The resulting approximate equation is independent of μ, , and b0. It has the 

following solutions: 

 

 

3 3 33

2 2 22

10 1 2 0 1 3 0 1 40 1

1 1 11

3 3 33

2 2 22

10 1 2 0 1 3 0 1 40 1

1 1 11

2 2 22 () () () () 3 3 33

2 2 22 () () () () 3 3 33

R CI b CJ b CK b CY b

b b bb

R CI b CJ b CK b CY b

b b bb

   

   





      

                

      

                

. (28) 

The only unacceptable solution at  is the I0 solution. By adding and subtracting these two 

equations, the oscillatory and exponential terms are isolated. The resulting equations are 

 

 

3 3

2 2

10 1 3 0 1

1 1

3 3

2 2

2 0 1 40 1

1 1

2 2 () () 3 3

2 2 () () 3 3

R R CI b CK b

b b

R R CJ b CY b

b b

  

  

 

 

  

       

  

       

. (29) 

The physical basis of this behavior is that expressions R+()  R–() are proportional to the two 

components of the eigenstates of Sx and the magnetic field is aligned with the x-axis at  = 0. 

The components R+()  R–() form the components of what is called the local basis in the 

section 5. These two components then exhibit purely bound or purely unbound behavior. 
20 

Physically acceptable solutions are obtained only for C1 = 0, thus the sum R+() +R–() decays as 

3/2

0 1

1

2 ( ) 3

K b

b 

     

 for    . This is the basis of the procedure to find the eigenvalues, . The 

equations are integrated using an ODE routine for various values of . The correct eigenvalues 

are the ones that effectively force C1 = 0 producing the physical boundary condition 

R+() + R–()  0 for   . In the section 5, many samples of these eigenstates and 

eigenenergies are discussed. 

5. Numerical Solutions 

In this section, the results of the numerical techniques described in the text are displayed for 

various parameters. The components R+()  R–() discussed in the text form the components of 

the local basis. The components in the local basis are labeled by and Rs d  RR R RR      . 

This pair of components is what is meant by local basis components in the text of this section. 

The component Rs is proportional to the amplitude for the spin to be pointed along the positive x

direction and Rd is proportional to the amplitude for the spin to be found pointing in the negative 

x-direction. Since the magnetic field is chosen to be pointing along the negative x-direction and 

the spin is chosen to be parallel to the magnetic moment, the Rs component of the wavefunction 

is aligned opposite to the magnetic field and must act as a bound component at large radii where 

the magnitude of the magnetic field gets very large. Since the Rd component is aligned with the 

field, it must be unbound as the effective magnetic potential energy decreases for large radii. 

The solutions, therefore, are of a peculiar nature in that one component in the local basis appears 

to be a localized bound state yet the other component extends to infinity. This makes it difficult 

to identify solutions that can be considered to be purely bound states. However, the solutions for 

μ form a degenerate pair when b0 = 0. This means that the solution found for one value of μ is 

only one component of a degenerate pair of solutions. Any linear superposition of these two 

states is then also a solution with the common eigenvalue. It turns out that by simply adding and 

subtracting the two degenerate solutions, a new set of degenerate states that are either purely 

bound or purely unbound states can be formed. 

In this way, the solutions are separated into bound and unbound pairs. The bound states are 

concentrated near the center of the guide, decaying exponentially away from the guide axis and 

the unbound states oscillate and extend to infinity. The modes of interest in atom guiding are the 

bound modes, as the unbound modes will tend to interact with things far from the guide center 

and thereby be easily lost from the guide. However, the bound and unbound modes are 

degenerate and it will be difficult experimentally to keep them from mixing with each other and 

eventually causing atoms to be lost from the guide. The introduction of a nonzero bias field b0

can remove the degeneracy, leaving isolated bound and unbound modes. This is analogous to the 
21 

introduction of a bias field in magnetic traps to reduce spin flips in the adiabatic approximation. 

The details of this connection with the adiabatic picture will be a subject of future study. 

In figure 5, the degenerate low energy states of the modes with zero axial amplitudes are shown 

in the global basis. On the left of figure 5, the mode with  1/2 is shown, and on the right, the 

degenerate mode for   1/2 is shown. For zero bias field, these modes are identical except for 

the exact reversal of the roles of R and R . 

Figure 5. These plots show a μ = 1/2 eigenstate (left) and the μ = –1/2 degenerate eigenstate (right). The mode 

on the left is spin up in the low field limit and the mode on the right is spin down in this limit. Notice 

that the two components in each mode oscillate as out of phase Bessel functions at large radius and 

that the roles of R+ and R–have been reversed relative to each other. The fact that these two modes are 

degenerate allows them to be combined to form pure bound or unbound states. 

In figure 6, the same modes shown in figures 5 are displayed except they are now both in the 

local basis. The bound component of each of these modes is apparent as the red curve that decays 

at large radius. However both modes are clearly mixed with bound and unbound components that 

oscillate as    . Since these modes are degenerate, they can be cleanly separated into bound 

and unbound modes by simply superpositioning the degenerate pair. By adding and subtracting 

the modes from one another, new modes are formed that each contains a single nonzero 

component and one component that is identically zero. It is not simply the individual sum and 

difference of spin components that takes the mode from the global to the local spin basis but a 

superposition of two degenerate modes both of which are in the local spin basis. The result of 

combining the modes is shown in figure 7. Only one component is visible in each of these plots 

because the other component has been exactly cancelled out. Since this separation into bound 

and unbound modes is based on the degeneracy of the   1/2 , modes that occurs when 0 b  0 
22 

it is expected that nonzero values of 0 b will lift the degeneracy and provide a purely quantum 

description of the stabilizing effect of the bias field in magnetic traps. A complete study of this 

effect and a comparison with the adiabatic approximation will be the subject of a future study. 

Figure 6. Local basis eigenstates corresponding to the global basis states shown in figure 5. The components of 

the local basis states are derived from the global states by forming the sum and difference of the global 

components. This can also be accomplished as a basis transformation using rotation matrices to go 

from the Sz to Sx basis. The significant difference between the local and global basis representations is 

the appearance of an exponentially decaying bound component along with an oscillatory or unbound 

component. The bound components in the two degenerate modes are equal and have the same sign, but 

the unbound components have opposite signs. For this reason the sum and the difference of the two 

degenerate modes can be used to produce pure bound and unbound states as shown in figure 7. 

Figure 7. Purely bound state obtained by forming a superposition of the degenerate states shown in figure 6. 
23 

The first higher order local mode for the 1/ 2    case is displayed in figure 8. This is the 

second mode in the series of modes that behave like 1 J at low transverse field. The bound 

component, shown in red, clearly crosses the axis once and returns exponentially to zero from 

negative values. The unbound modes, shown in green, oscillate in a manner similar to a Bessel 

function, but these two components are clearly exactly out of phase for the two different values 

of  . This behavior is the same in all degenerate pairs and again allows the formation of pure 

bound and unbound modes. In figure 9, the next two higher order modes of this sequence are 

displayed. It is clear that the bound components are behaving in the same way as in other 

quantum mechanical bound state problems. The difference here is the intermingling of the bound 

and unbound behavior within one mode and is due to the extra spin degree of freedom not found 

in the more common potential well problems that involve scalar particles. This makes the 

identification of pure bound states difficult. 

Figure 8. These two states are the degenerate pair of the first excited modes in the system of modes that act like 

1 J when the transverse field goes to zero. Since they are degenerate, they can be combined to obtain 

pure unbound and bound modes. 
24 

Figure 9. These are the next two higher order modes in the system of modes that act like 1 J when the transverse 

field goes to zero. As expected, the number of zero crossings increases as the mode energy increases. 

The second sequence of modes behaves like 0 J at the origin. These lowest of these modes are 

lower in energy than the 1 J type modes and should be considered the ground state of the system. 

This apparent ground state is shown in figure 10. This mode may have some contamination in it 

as discussed in section 3. We are currently developing a way to exclude these types of 

contamination so that a complete and systematic study of all the modes can be completed. 

Another issue may be that this ground state designation is system dependent, as the differing 

behavior of these modes at the origin might be used along with various mode suppression 

strategies to eliminate some levels, possibly including the original ground state. In figure 11, a 

few higher modes of this sequence are displayed in the local basis. In the local basis, both 

components of the   1/2modes are nonzero at the origin. In figure 12, the lowest energy 

state for the   3/2 case is presented. Its bound state component also has no zero crossings and 

could be considered a ground state for atoms constrained to be in the   3/2 state of alignment. 
25 

Figure 10. This state is the lowest order mode of the system of modes that behave like 0 J as the transverse field 

is decreased. Notice the large on axis component that distinguishes this mode from the 1 J type 

modes. This mode is the likely ground state of the system; however, there is some mixing of modes 

from the spin-up solutions in this system as discussed in the text of section 3. Thus, this mode might 

represent a superposition of spin-up and spin-down eigenstates. It is shown in both the global and 

local bases. 

Figure 11. Local behavior of the higher order modes that behave like 1 2 J and J at low transverse field 
26 

Figure 12. These two plots for the lowest energy μ = 3/2 state show that the same basic behavior holds for 

alignment other than μ = ±1/2. The solutions can be found for higher values of alignment as well. 

6. Conclusions 

A technique for solving for the eigenmodes and eigenvalues of the magnetic quadrupole 

waveguide has been developed. The technique is robust, but there is still some room for 

improvement. A good understanding of the modes and mode structure has been obtained; 

however, the numerical techniques being used for the calculations can still produce some 

contamination between modes that can result in errors in the calculated energies. This is 

currently being resolved and an improved technique will be incorporated soon. 

The calculated modes are of a peculiar nature that is not completely bound or completely 

unbound, which causes a problem as they are not strictly normalizeable. Rather, the modes are of 

a hybrid nature. A possible resolution of this problem is to use degeneracy to split the hybrid 

modes into pure bound and pure unbound states. This technique appears to be useful, but we are 

still left with modes that must be normalized properly to be used in further calculations. A 

reliable technique to handle this problem for the hybrid modes has not yet been developed. It 

may be the case that some type of box normalization could be useful for the unbound states, but 

this would have to include the bound states in some natural way so that consistent expectations 

could be calculated. Another issue is the possible close coupling of the degenerate bound and 

unbound states. This coupling could cause possible lifetime limitations. This degeneracy must be 

studied and the usefulness of the bias field in lifting the degeneracy in a purely quantum 
27 

situation, reducing any related decay mechanisms, must be considered. Some details still need to 

be resolved, but as the remaining issues are cleared up these mode calculations will be useful for 

further calculations of important effects in the use of atomic waveguides in atom interferometry 

based on atom chips. 
28 

7. References 

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

McGraw-Hill: New York, NY, 1978. 

Bergman, T. H. et al. JOSA B 1989, 6 (11), 2249. 

Bill, J. Phys. Rev. A 73 2006, 053609. 

Fortágh, J.; Zimmerman, C. RMP 2007, 79, 235. 

Hinds, E. A.; Eberlein, C. Phys. Rev. A 61 2000, 033614. 

Hinds, E. A.; Eberlein, C. Phys. Rev. A 64 2001, 039902(E). 

Ince, E. L. Ordinary Differential Equations; Dover: New York, NY, 1956, p. 396. 

Lesanovsky, I.; Schmelcher, P. Phys. Rev. A 70 2004, 063604. 

Potvliege, R.M. and Zehnlé, V. Phys. Rev. A 63, 2001, 025601. 

Sukumar, C. V.; Brink, D. M. Phys. Rev. A 56 1997, 2451. 
29 

NO. OF 

COPIES ORGANIZATION 

 1 ADMNSTR 

 ELEC DEFNS TECHL INFO CTR 

 ATTN DTIC OCP 

 8725 JOHN J KINGMAN RD STE 0944 

 FT BELVOIR VA 22060-6218 

 1 DARPA 

 ATTN IXO S WELBY 

 3701 N FAIRFAX DR 

 ARLINGTON VA 22203-1714 

 1 CD OFC OF THE SECY OF DEFNS 

 ATTN ODDRE (R&AT) (1 CD) 

 THE PENTAGON 

 WASHINGTON DC 20301-3080 

 1 US ARMY RSRCH DEV AND 

 ENGRG CMND 

 ARMAMENT RSRCH DEV AND 

 ENGRG CTR 

 ARMAMENT ENGRG AND 

 TECHNLGY CTR 

 ATTN AMSRD AAR AEF T J MATTS 

 BLDG 305 

 ABERDEEN PROVING GROUND MD 

 21005-5001 

 1 PM TIMS, PROFILER (MMS-P) 

 AN/TMQ-52 

 ATTN B GRIFFIES 

 BUILDING 563 

 FT MONMOUTH NJ 07703 

 1 US ARMY INFO SYS ENGRG CMND 

 ATTN AMSEL IE TD A RIVERA 

 FT HUACHUCA AZ 85613-5300 

 1 COMMANDER 

 US ARMY RDECOM 

 ATTN AMSRD AMR 

 W C MCCORKLE 

 5400 FOWLER RD 

 REDSTONE ARSENAL AL 35898-5000 

 1 US GOVERNMENT PRINT OFF 

 DEPOSITORY RECEIVING SECTION 

 ATTN MAIL STOP IDAD J TATE 

 732 NORTH CAPITOL ST NW 

 WASHINGTON DC 20402 

NO. OF 

COPIES ORGANIZATION 

 1 US ARMY RSRCH LAB 

 ATTN RDRL CIM G T LANDFRIED 

 BLDG 4600 

 ABERDEEN PROVING GROUND MD 

 21005-5066 

 17 US ARMY RSRCH LAB 

 ATTN IMNE ALC HRR 

 MAIL & RECORDS MGMT 

 ATTN RDRL CIM L TECHL LIB 

 ATTN RDRL CIM P TECHL PUB 

 ATTN RDRL SEE O 

 W M GOLDING (10 COPIES) 

 P LEE 

 ATTN RDRL SER E F CROWNE 

 ATTN RDRL SES E H BRANDT 

 ATTN RDRL SES P A EDELSTEIN 

 ATTN RDRL SER U C FAZI 

 ADELPHI MD 20783-1197 

 1 NATIONAL INSTITUTE OF 

 STANDARDS AND TECHNOLOGY 

 ATTN DR W D PHILLIPS 

 100 BUREAU DRIVE, STOP 8424 

 GAITHERSBURG MD 20899-8424 

 1 DEPT. OF THE ARMY 

 WEAPONS SCIENCES DIRECTORATE 

 ATTN AMSRD AMR WS H EVERITT 

 REDSTONE ARSENAL AL 

 35898-5000 

 1 DEPT. OF THE ARMY 

 WEAPONS SCIENCES DIRECTORATE 

 ATTN AMSRD AMR WS T BAHDER 

 REDSTONE ARSENAL AL 

 35898-5000 

 1 DEPT. OF THE ARMY 

 WEAPONS SCIENCES DIRECTORATE 

 ATTN AMSRD AMR WS K MYNENI 

 REDSTONE ARSENAL AL 

 35898-5000 

 1 CD US ARMY RESEARCH OFFICE 

 1 HC ATTN P REYNOLDS 

 PO BOX 12211 

 RESEARCH TRIANGLE PARK NC 

 27709-2211 
30 

NO. OF 

COPIES ORGANIZATION 

 1 DARPA 

 ATTN DR. AMIT LAL 

 3701 N FAIRFAX DR 

 ARLINGTON VA 22203-1714 

 1 DARPA 

 ATTN DR. JAY LOWELL 

 3701 N FAIRFAX DR 

 ARLINGTON VA 22203-1714 

TOTAL: 34 (1, ELEC, 2 CDS, 31 HCS)