dynamics of reﬂection of ultracold atoms from a periodic

Dynamics of reflection of ultracold atoms from a periodic one-dimensional magnetic latticepotential

Mandip Singh, Russell McLean, Andrei Sidorov, and Peter Hannaford*Centre for Atom Optics and Ultrafast Spectroscopy and ARC Centre of Excellence for Quantum-Atom Optics,

Swinburne University of Technology, Melbourne 3122, Australia�Received 23 January 2009; published 12 May 2009�

We report on an experimental study of the dynamics of the reflection of ultracold atoms from a periodicone-dimensional magnetic lattice potential. The magnetic lattice potential of period 10 �m is generated byapplying a uniform bias magnetic field to a microfabricated periodic structure on a silicon wafer coated with amultilayered TbGdFeCo/Cr magneto-optical film. The effective thickness of the magnetic film is about 960 nm.A detailed study of the profile of the reflected atoms as a function of externally induced periodic corrugationin the potential is described. The effect of angle of incidence is investigated in detail. The experimentalobservations are supported by numerical simulations.

DOI: 10.1103/PhysRevA.79.053407 PACS number�s�: 37.10.Gh, 03.75.�b

I. INTRODUCTION

The control offered by optical lattices provides a rich plat-form to explore the physics of ultracold gases in periodicpotentials. Optical lattices, which are based on the opticaldipole force, have been used to realize coherent spin-dependent transport of atoms �1� and multiparticle entangle-ment �2�. Many interesting phenomena which were earliertheoretically studied in condensed-matter physics have nowbeen experimentally explored using ultracold atoms in opti-cal lattices �3�. Another approach to realize periodic poten-tials utilizes a magnetic lattice which is based on the mag-netic dipole force. A magnetic lattice is a periodic array ofmagnetic traps which can be created either by structures ofcurrent-carrying wires �4,5� or permanent magnetic filmsfabricated on a substrate �6–8�. Since magnetic lattices arebased on miniaturized structures they are well suited to inte-gration on a microchip. Permanent magnetic films offer rela-tively high magnetic field gradients and curvatures withoutany resistive heating on the chip. In addition, the thin struc-ture and high electrical resistance of magnetic films can sup-press thermal fluctuations, thereby offering higher trap sta-bility. One-dimensional periodic magnetic structures producea magnetic field that decays exponentially from the surfaceand which can be used to reflect atoms in weak-field-seekingstates �9�. Such structures have been used to realize magneticmirrors for atoms in weak-field-seeking states �10,11� and tomanipulate atoms from a corrugated reflector �12�. In recentexperiments on a one-dimensional permanent magnetic lat-tice, radial trap frequencies of up to 90 kHz have been mea-sured for 87Rb atoms trapped in the lattice at a distance closeto 5 �m from the surface �8�. In another experiment a min-iaturized structure of current-carrying wires on a microchiphas been used to diffract a Bose Einstein condensate �BEC��13,14�. In addition, interesting experiments on the manipu-lation of ultracold atoms in a two-dimensional array of mag-netic microtraps have recently been reported �15�. In thispaper we present a detailed study of the dynamics of the

reflection of ultracold atoms from a 1D permanent magneticlattice potential with an externally controlled periodic corru-gation.

II. MAGNETIC FIELD FROM A MAGNETIC LATTICESTRUCTURE

In order to produce the required potential a grooved sili-con microstructure of period 10 �m was coated with multi-layered Tb6Gd10Fe80Co4 /Cr magneto-optical film �16�. Sucha structure when perpendicularly magnetized generates peri-odically varying magnetic field components and by applyinga uniform bias magnetic field to the structure an array ofmagnetic traps can be produced �7�. A schematic of a peri-odic magnetic structure of period a, thickness t, and perpen-dicular magnetization Mz is shown in Fig. 1.

For an infinite array the components of the magnetic fieldalong the y and z directions are given by �7�

By = Bs��1 − e−kt�e−k�z−t� sin�ky�

− 13 �1 − e−3kt�e−3k�z−t� sin�3ky� + ¯� + Bby , �1�

Bz = Bs��1 − e−kt�e−k�z−t� cos�ky�

− 13 �1 − e−3kt�e−3k�z−t� cos�3ky� + ¯� + Bbz, �2�

where k=2� /a, Bs=4Mz �Gaussian units�, Mz is the magne-tization �assumed to be in the perpendicular direction�, andBbi for i=x ,y ,z are the bias field components along x, y, andz directions, respectively.

For distances from the surface large compared to a /4�the effect of higher spatial harmonics is negligible and thefield components can be written as

Bx = Bbx, �3�

By = B0 sin�ky�e−kz + Bby , �4�

Bz = B0 cos�ky�e−kz + Bbz, �5�

where B0=Bs�ekt−1�. In the absence of bias fields the mag-nitude of the magnetic field is given by*[email protected]

PHYSICAL REVIEW A 79, 053407 �2009�

1050-2947/2009/79�5�/053407�7� ©2009 The American Physical Society053407-1

http://dx.doi.org/10.1103/PhysRevA.79.053407

B�x,y� = B0e−kz, �6�

i.e., the magnitude of the field decreases exponentially withdistance z from the surface. Thus the field gradient repelsatoms in weak-field-seeking states and such a structure be-haves as a magnetic mirror �9�. In the absence of bias mag-netic fields Bby and Bbz the intensity of the magnetic fieldgenerated by the magnetic structure is uniform in the x-yplane. However, by applying a bias field in the y or z direc-tion one can induce a periodic corrugation in the field mag-nitude along the y direction. Such an exponentially increas-ing corrugated potential can significantly affect the spatialprofile of an ultracold cloud of atoms projected toward it.The main objective of this paper is to study the detaileddynamics of an ultracold cloud reflected from the periodiccorrugated potential.

III. ATOMS IN THE CORRUGATED MAGNETICPOTENTIAL

In order to study the profile of the ultracold cloud re-flected or released from the potential we start by evaluatingthe forces acting on an atom in the potential. In the casewhere the spatial size of an atomic wave packet moving in amagnetic field B�x ,y ,z� is much smaller than the corrugationperiod, the atom can be treated as a classical point object.The components of the force acting on the atom in the mag-netic lattice potential with gravity �g� along the z directioncan then be expressed as

md2x

dt2 = − mFgF�B� �B�x,y,z��x

� , �7�

md2y

dt2 = − mFgF�B� �B�x,y,z��y

� �8�

md2z

dt2 = − mFgF�B� �B�x,y,z��z

� + mg , �9�

where m is the mass of the atom, mF is the magnetic quan-tum number, gF is the Landé g factor, and �B is the Bohr

magneton. From Eqs. �3�, �4�, and �5� the magnitude of themagnetic field in the presence of uniform bias fields is givenby

B�x,y,z� = �Bbx2 + Bby

2 + Bbz2 + B0

2e−2kz

+ 2B0e−kz�Bbz cos�ky� + Bby sin�ky��1/2

�10�

and the partial derivatives are

�B�x,y,z��x

= 0, �11�

�B�x,y,z��y

=kB0e−kz�Bby cos ky − Bbz sin ky�

B�x,y,z�, �12�

�B�x,y,z��z

= −kB0e−2kz�B0 + ekz�Bbz cos ky + Bby sin ky��

B�x,y,z�.

�13�

Inserting Eqs. �11�–�13� into Eqs. �7�–�9� and solving, thetrajectory of an atom in the magnetic potential can be calcu-lated. Profiles of �B�x ,y ,z� /�y and �B�x ,y ,z� /�z in thex=c plane �where c is a constant� are shown in Figs. 2�a� and2�b�, respectively, for Bbx=45 G, Bby =8 G, Bbz=0 G,t=1 �m, a=10 �m, and 4�Mz=3 kG. Figure 2�a� indi-cates that the y component of the force, which is proportionalto �B�x ,y ,z� /�y, is an oscillatory function of y and provideshorizontal momentum to the atoms while the negative valueof �B�x ,y ,z� /�z causes repulsion from the lattice �for atomsin weak-field-seeking states�. For the one-dimensional poten-tial considered, �B�x ,y ,z� /�x=0, which results in a zeroforce along the x axis.

IV. EXPERIMENT: EFFECT OF CORRUGATION

The experimental setup is based on a hybrid atom chip�8,17� which consists of a permanent magnetic structure andcurrent-carrying wires. The permanent magnetic structurewas produced by coating a multilayered magneto-optical film�Tb6Gd10Fe80Co4 /Cr� with effective thickness of 960 nm ona grooved Si structure. The period of the �10�10 mm2�grooved structure is 10 �m. The coated structure was per-pendicularly magnetized with 4�Mz 3 kG. Measure-ments indicate that the coercivity of the film was about 6kOe and the Curie temperature is about 300 °C. The groovedSi wafer was then positioned on the current-carrying wirestructure. The position of the permanent magnetic structurewith respect to the wires is shown schematically in Fig. 3,where gravity is acting along the z direction �the chip ismounted face down�.

The periodic corrugation in the magnetic potential wasintroduced externally by applying a bias field Bby and keep-ing Bbz=0. The experimental protocol is as follows. About2�105 87Rb atoms in the weak-field-seeking stateF=2,mF=2� were cooled close to the BEC transition byevaporative cooling in a Z-shaped wire �pin numbers 1 and 2in Fig. 3� magnetic trap located 150–250 �m from the

Y

X

a/2 a/2

X- bias field

Y- bias field

t

Mz

Z

FIG. 1. �Color online� Schematic showing an array of perpen-dicularly magnetized parallel slabs with period a. A one-dimensional magnetic lattice of elongated microtraps with nonzeropotential minima is formed by applying bias fields along the x andy directions.

SINGH et al. PHYSICAL REVIEW A 79, 053407 �2009�

053407-2

surface of the structure. At this distance the exponentiallydecaying magnetic field from the 10 �m period permanentmagnetic structure is negligibly small. After the evaporativecooling stage the externally applied magnetic field along they axis �which allows adjustment of the trap offset in theZ-wire trap� was linearly reduced to zero in 10 ms keepingIz=35 A �pin numbers 1 and 2 in Fig. 3� and Bbx=45 G.After waiting another 10 ms the value of Iz was linearlyreduced from 35 to 22 A in 1.4 ms. Immediately after this 1.4ms interval Iz was quickly reduced to zero and the corruga-tion inducing bias field Bby was simultaneously switched tothe desired value. Both Iz and Bby were able to be switchedon �off� in less than 1 ms. The process of decreasing Iz in 1.4ms moves the trap center toward the periodic magnetic lat-tice potential and therefore the trapped ultracold cloud ispropelled upward toward the lattice potential with a velocityvz in the range of −100 to −130 mm /s. When Iz wasswitched off the Z trap disappeared but the ultracold cloudcontinued to move in the presence of Bbx=45 G and Bby.The moving cloud enters the exponentially increasing mag-netic field of the magnetic lattice and experiences an expo-nentially increasing repulsive force in the z direction. In thecase of an external applied bias field Bby the y component ofthe force on the atoms due to the nonzero value of�B�x ,y ,z� /�y also spreads them horizontally �in the y direc-tion�.

Absorption images were taken after the atom cloud hadinteracted with the periodic corrugated potential. The ampli-tude of the corrugation is varied by changing the value ofBby. A set of images is shown in Fig. 4 for a 9 ms time offlight for a range of values of Bby where the atomic profilesclosely resemble a circular arc. It is evident from Fig. 4 thatas the magnitude of Bby and hence the corrugation is reducedthe angular spread of the reflected atoms decreases. The an-gular spread is given by the angle subtended by the circulararc at the center of the circle and is equal to2 arctan��y /2�z0−�z�� for the horizontal ��y� and the ver-tical ��z� widths of the profile shown in Fig. 4. The radius ofthe circular arc z0 1.4 mm. A plot showing the angular

spread of the released cloud as a function of Bby is presentedin Fig. 5. It is evident from Fig. 5 that the minimum angularspread appears at Bby =2.2 G instead of Bby =0. This offset isdue to the presence of a stray field of the same order whichcanceled the applied field Bby. This experiment establishesthat the angular spread originates from the periodic corruga-tion in the magnetic field pattern that can be controlled ex-ternally. In the case of an evanescent light wave mirror it hasbeen shown that the effect of roughness can also producecurvature in the profile of the reflected atomic cloud �18�.

V. SIMULATION: EFFECT OF CORRUGATION

In order to verify the interpretation of the above experi-mental observations the differential equations �Eqs. �7�–�9��were solved numerically for different values of Bby. In thecalculations the magnetic structure parameters a=10 �m,

FIG. 2. �Color online� Plots showing the partial derivatives of the magnetic field magnitude in the lattice potential �a� with respect to yand �b� with respect to z. The parameters used in the calculations are given in the text.

X

Z

Y

Z - Bias Field

X - Bias Field

Y - Bias Field

DC Current

Lattice Structure

1

2

3

4

FIG. 3. �Color online� Periodic magnetic structure and current-carrying wires.

DYNAMICS OF REFLECTION OF ULTRACOLD ATOMS… PHYSICAL REVIEW A 79, 053407 �2009�

053407-3

4�Mz=3 kG, and t=1 �m were used. The calculationswere performed by taking the initial velocity �at t=0�vz=−125 mm /s, the initial position of the cloud �at t=0� zi=60 �m, Bbx =45 G, Bbz=0 G, and zero temperature. Aninitial cloud width of 30 �m along the y direction was as-sumed which is three times the lattice period. A set of calcu-lated profiles of the cloud as a function of Bby is shown inFig. 6 which indicates that the angular spread gradually re-duces as the corrugation is reduced to zero, as observed inthe experiment.

VI. VARYING THE ANGLE OF INCIDENCE

In the previous experiment the ultracold cloud was pro-jected perpendicular to the plane of the magnetic structure,i.e., vx=0, vy =0, and only vz was nonzero. For an ultracoldcloud moving toward the lattice in the y-z plane the angle ofincidence with respect to the z axis is �=arctan�vy /vz� whichwas zero in the previous experiment. In this experiment the

effect of angle of incidence is studied. The angle of incidencewas varied by varying vy and keeping vz constant.

In order to provide a velocity component vy, the ultracoldcloud was made to oscillate in the y direction in the Z trapbefore moving toward the lattice. The protocol used toproject the cloud at an angle with respect to the z axis is asfollows. After completion of the evaporative cooling stage anultracold cloud of 2.5�105 atoms close to the critical tem-perature was obtained in the Z-wire trap. Immediately afterevaporation Bby was decreased linearly to zero in 10 ms. Inthe next 50 ms Iz=35 A was decreased to 10 A and thecurrent �Izs� in the semilong Z wire �pin numbers 1 and 4 inFig. 3� was increased from 0 to 25 A synchronously such thatIz+ Izs=35 A. In the next 10 ms both Iz and Izs were restoredlinearly to their initial values of 35 and 0 A, respectively,where Bbx=45 G was constant throughout. This process pro-vides a kick to the cloud along the y direction in the Z trapand the cloud starts to oscillate. Therefore, vy also oscillatesback and forth. This is the first part of the protocol. In thesecond part the cloud was moved toward the lattice afterholding for a variable time thold while it was oscillating. Thecloud was moved toward the lattice by linearly decreasing Izfrom 35 to 17 A in 15 ms. After holding Iz at 17 A for 0.3 msit was quickly reduced to zero. The atoms reflected from thecorrugated potential were detected through absorption imag-ing after 6 ms time of flight. The corrugation in this experi-ment originates from the magnetic field generated by thecurrent through the Z wire. Measurements were taken fordifferent values of thold to vary vy of the cloud �at the timewhen it interacts with the magnetic potential�.

The axial position �y0� and axial velocity �vy� of the os-cillating cloud at the time when it interacts with the latticewere evaluated by fitting a harmonic function to the axialoscillations. The axial position and axial velocity are givenby y0=A0 cos��athold+�� and vy =−�aA0 sin��athold+��, re-spectively. Thus, in this experiment the ultracold cloud canbe made to interact with the lattice potential with different vyby varying thold. This also results in the interaction occurringat different locations.

In Fig. 7 a series of absorption images of the reflectedatoms is shown for different values of thold over a full cycleof the axial oscillation where the time step in successiveimages is 4 ms. It is apparent from these data that the profile

(1) (2) (3) (4) (5)

(6) (7) (8) (9) (10)

(11) (12) (13) (14) (15)

-15.4 G -13.2 G -11 G -8.8 G -6.6 G

(3)-4.4 G 2.2 G 4.4 G-2.2 G 0 G

6.6 G 8.8 G 13.2 G 15.4 G11 G

4.2 mm x 2.5 mm

�y

�z

z0

FIG. 4. Absorption images showing experimentally observed spatial profiles of atoms reflected from a corrugated potential for differentvalues of applied bias field Bby �as indicated in the bottom right corner of each image� and hence different amplitudes of corrugation �for 9ms time of flight�. The horizontal and vertical directions correspond to the y and z axes, respectively. Gravity is in the downward direction,and the chip surface is above the top of each image.

FIG. 5. �Color online� Angular spread of atoms as a function ofapplied bias field Bby. The minimum of the angular spread occurs atBby �2.2 G rather than at Bby =0 G due the presence of a strayfield.


053407-4

of the reflected atoms is tilted at an angle which depends onvy but is independent of the axial position y0. At the turningpoint of the axial oscillation the axial velocity vy is zero andtherefore the angle of incidence �� is also zero �normal in-cidence�. The horizontal distance between the two turningpoints �located at the center of each profile� when �=0 rep-resents 2A0. These two turning points correspond approxi-mately to image numbers 1 and 6 in Fig. 7 and the distancebetween them is 2A0=2.32 mm. These profiles show thatuniformity of the corrugation persists over many lattice pe-riods.

The experimental results were verified by calculating theprofile of reflected atoms for different values of y0 and vy bysolving Eqs. �7�–�9�. The values of y0 and vy for each simu-lation were calculated from y0=A0 cos��athold+�� andvy =−�aA0 sin��athold+�� for �=� and �a=2��24 rad /sin successive time steps of 4 ms. A series of calculatedatomic profiles for the experimental parameters isshown in Fig. 8, where the time of flight is 6 ms andvz=−102 mm /s for each plot.

VII. SUMMARY AND CONCLUSIONS

A detailed study and analysis of the reflection of an ultra-cold cloud of atoms from a one-dimensional magnetic latticepotential of period 10 �m has been presented. The magnetic

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�a�

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�Z�m

m�

�b�

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�c�

��

��

��

��

��

��

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�d�

��

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�e�

��

��

��

��

��

��

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�Z�m

m�

�f�

��

��

��

��

��

��

��

��

��

��

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�g�

��

��

��

��

��

��

��

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�h�

��

��

��

��

��

��

��

��

��

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.53.

2.5

2.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�i�

FIG. 6. �Color online� Calcu-lated spatial profiles of the atomsas a function of corrugation. Bby

= �a� −6.6, �b� −4.4, �c� −2.2, �d�−1, �e� 0, �f� 1, �g� 2.2, �h� 4.4,and �i� 6.6 G.

�1�4.0 mm � 1.3 mm �7�

�2� �8�

�3� �9�

�4� �10�

�5� �11�

�6� �12�

FIG. 7. �Color online� Series of absorption images showing theatomic density profile of reflected atoms for different values of thold.The time step in successive measurements is 4 ms. The horizontaland vertical directions correspond to the y and z axes, respectively.


053407-5

lattice was constructed on a grooved Si structure coated withperpendicularly magnetized multilayered TbGdFeCo/Crmagneto-optical film and mounted on a hybrid atom chip. Ithas been shown experimentally and theoretically that thefield induced corrugation can significantly change the profileof the cloud to a curved shape. In addition a detailed study of

the effect of angle of incidence has been presented; in thiscase the atomic cloud interacts with different regions of thelattice where the extreme points are separated by up to about2.3 mm. The observed atomic profiles have been explainedusing classical equations of motion applied to atoms in themagnetic lattice potential. The results provide insights into

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�1�

Y0 � �1.16 mm Vy � 0.0 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�7�

Y0 � 1.03 mm Vy � �80.2 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�2�

Y0 � �0.95 mm Vy � 99.0 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�Z�m

m�

�8�

Y0 � 0.55 mm Vy � �154.0 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�3�

Y0 � �0.41 mm Vy � 163.1 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�9�

Y0 � �0.13 mm Vy � �173.5 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�4�

Y0 � 0.27 mm Vy � 169.6 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�10�

Y0 � �0.76 mm Vy � �131.7 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�5�

Y0 � 0.86 mm Vy � 116.3 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�11�

Y0 � �1.12 mm Vy � �43.4 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�6�

Y0 � 1.15 mm Vy � 21.9 mm�s

�2 �1 0 1 22.

1.5

1.

0.5

0.

Y �mm�

Z�m

m�

�12�

Y0 � �1.09 mm Vy � 60.2 mm�s

FIG. 8. �Color online� Calcu-lated spatial profiles of the re-flected atoms as a function ofaxial position and axial velocity.


053407-6

the manipulation of ultracold atoms reflecting from a peri-odic corrugated potential. In future we would like to explorepotential applications such as how the reflection from theperiodic corrugated potential can be used to characterize themagnetic lattice potential and how the effect of periodic cor-rugation differs from a random corrugation.

ACKNOWLEDGMENTS

We would like to thank Brenton Hall for useful discus-sions. This work was supported by the Australian ResearchCouncil Centre of Excellence for Quantum-Atom Optics anda Swinburne University Strategic Initiative Grant.

�1� O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, andI. Bloch, Phys. Rev. Lett. 91, 010407 �2003�.

�2� O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, andI. Bloch, Nature �London� 425, 937 �2003�.

�3� M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I.Bloch, Nature �London� 415, 39 �2002�.

�4� J. Yin, W. Gao, J. Hu, and Y. Wang, Opt. Commun. 206, 99�2002�.

�5� A. Grabowski and T. Pfau, Eur. Phys. J. D 22, 347 �2003�.�6� C. D. J. Sinclair, J. A. Retter, E. A. Curtis, B. V. Hall, I. L.

Garcia, S. Eriksson, B. E. Sauer, and E. A. Hinds, Eur. Phys. J.D 35, 105 �2005�.

�7� S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, J.Phys. B 39, 847 �2006�.

�8� M. Singh, M. Volk, A. Akulshin, R. McLean, A. Sidorov, andP. Hannaford, J. Phys. B 41, 065301 �2008�.

�9� G. I. Opat, S. J. Wark, and A. Cimmino, Appl. Phys. B: LasersOpt. 54, 396 �1992�.

�10� T. M. Roach, H. Abele, M. G. Boshier, H. L. Grossman, K. P.Zetie, and E. A. Hinds, Phys. Rev. Lett. 75, 629 �1995�.

�11� A. I. Sidorov, R. J. McLean, W. J. Rowlands, D. C. Lau, J. E.Murphy, M. Walkiewicz, G. I. Opat, and P. Hannaford, Quan-tum Semiclassic. Opt. 8, 713 �1996�.

�12� P. Rosenbusch, B. V. Hall, I. G. Hughes, C. V. Saba, and E. A.Hinds, Phys. Rev. A 61, 031404�R� �2000�.

�13� A. Gunther, S. Kraft, M. Kemmler, D. Koelle, R. Kleiner, C.Zimmermann, and J. Fortagh, Phys. Rev. Lett. 95, 170405�2005�.

�14� A. Gunther, S. Kraft, C. Zimmermann, and J. Fortagh, Phys.Rev. Lett. 98, 140403 �2007�.

�15� R. Gerritsma, S. Whitlock, T. Fernholz, H. Schlatter, J. A.Luigjes, J.-U. Thiele, J. B. Goedkoop, and R. J. C. Spreeuw,Phys. Rev. A 76, 033408 �2007�.

�16� J. Y. Wang, S. Whitlock, F. Scharnberg, D. S. Gough, A. I.Sidorov, R. J. McLean, and P. Hannaford, J. Phys. D 38, 4015�2005�.

�17� B. V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford, and A. I.Sidorov, J. Phys. B 39, 27 �2006�.

�18� H. Perrin, Y. Colombe, B. Mercier, V. Lorent, and C. Henkel,J. Phys. B 39, 4649 �2006�.


053407-7

dynamics of reﬂection of ultracold atoms from a periodic

Documents