Controllable double waveguide for atoms

1 May 2000

Optics Communications 178 Ž2000. 93–101 www.elsevier.comrlocateroptcom

Controllable double waveguide for atoms O. Zobay ) , B.M. Garraway Sussex Centre for Optical and Atomic Physics, UniÕersity of Sussex, Brighton BN1 9QH, UK Received 30 December 1999; accepted 2 March 2000

Abstract A method is proposed to create linear atom guides in the field of a periodically magnetized surface. By applying an additional homogeneous magnetic field the characteristics of the waveguides can be drastically modified; in particular, pairs of waveguides can be created that may be split or combined depending on the external field. Applications of this device to the construction of Y-switches and the implementation of an atom interferometer are discussed. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.75.Be; 39.20.q q; 32.80.Pj; 32.60.q i Keywords: Atom optics; Atom traps and waveguides; Atom interferometry

1. Introduction The rapid advances in the fields of atom cooling and atom optics have stimulated significant interest in the manufacture of miniaturized atomic traps and waveguides. These devices are considered basic building blocks for a new generation of experiments and technical applications such as the study of lowdimensional quantum gases or the construction of integrated atom-optical circuits. Among the various approaches to create potential structures the use of magnetic fields appears to be particularly attractive for a number of reasons. For example, they offer excellent isolation from the environment as the atoms

) Corresponding author. Fax: q44-1273-677196; e-mail: [email protected]

do not interact with any exterior walls etc. In particular, in contrast to optical traps, detrimental effects caused by the atom-light interaction such as heating due to spontaneous emission cannot occur. Furthermore, the atoms may be spatially confined on length scales comparable to, or below, optical wave lengths. This offers the prospect of, e.g., reaching the Lamb–Dicke regime or achieving very high atomic densities. Finally, magnetic fields are easily manipulated in the laboratory thus allowing changes in the trap configuration during the experiment. Various methods to construct miniaturized magnetic traps and waveguides have been investigated theoretically and experimentally in the past few years. In Ref. w1x, for example, different trap configurations were described that can be produced by micron-scale superconducting circuits confined to a plane. Drawing on this proposal, in Ref. w2x a wire trap was constructed that could be loaded from a magneto-

0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 0 9 - X

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O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

optical trap by slowly deforming the trapping potentials. The extreme field gradient in the vicinity of a sharp magnetized steel pin was used in Ref. w3x to create a quadrupole trap with which the density of the atomic cloud was increased by a factor of more than 100 during an adiabatic compression process. The magnetic fields of a single current-carrying wire w4x and of combinations of several microfabricated wires w5–7x were recently shown to yield effective guiding potentials into which atoms from external sources were fed successfully. Theoretical studies of the construction and properties of magnetic waveguides were reported in w8,9x. The purpose of this article is to propose a new method of constructing a two-dimensional magnetic trap or linear waveguide. The proposal is based on the ‘magnetic surface technology’ demonstrated, e.g., in Refs. w10–12x. A suitable material, such as videotape, is magnetized so that a desired field configuration is produced. The field configuration that we discuss in this paper has the interesting property that the trap characteristics can be changed drastically by simply applying an additional homogeneous magnetic field Bh parallel to the trap surface. In particular, it is found that, when increasing Bh from zero, one first enters a regime in which two traps are formed at different distances from the magnetic surface. The two traps then approach each other and combine at a certain field strength. If the field is increased even further the combined trap is split again and two separate, symmetrical potential wells are formed at equal distances from the magnetic surface. This scenario, albeit conceptually quite simple, may have interesting practical applications some of which are examined in the following. In Section 2, after giving a detailed account of the properties of the field configuration introduced above, its use to implement Y-switches for atomic waveguides is outlined. Section 3 discusses in more detail a further application, i.e., an interferometer to detect weak forces. A short conclusion is given in Section 4.

2. The atomic double waveguide Our proposal for designing a linear waveguide makes use of a recently demonstrated technique to produce magnetic mirrors for atoms w10–12x. These

experiments exploit the fact that a surface lying in the plane z s 0 and having a magnetization M s M1cosŽ kx . xˆ creates a magnetic field B s B1eyk z wycosŽ kx . xˆ q sinŽ kx . zˆ x. If an atom moves in this field slowly enough its magnetic moment will adiabatically follow the instantaneous field direction, i.e., it is effectively exposed to the potential UŽ r . s g L m B m F < B Ž r .< with g L the atomic Lande´ factor, m B the Bohr magneton, and m F the magnetic hyperfine quantum number. In the above example U A eyk z so that a weak-field seeking atom will be repelled from the magnetic surface which thus acts as a mirror. Consider now the case where the periodic surface magnetization also contains a contribution from the second harmonic, i.e., M s w M 1 cos Ž kx . q M2 cosŽ2 kx .x x. ˆ This situation gives rise to a magnetic field B s y B1eyk z cos Ž kx . q B2 ey2 k z cos Ž 2 kx . xˆ q B1eyk z sin Ž kx . q B2 ey2 k z sin Ž 2 kx . zˆ ' B x xˆ q Bz zˆ

Ž 1.

so that the ensuing potential has the form U Ž r . s g L m B m F eyk z = B12 q B22 ey2 k z q B1 B2 eyk z cos Ž kx .

1r2

.

Ž 2. Evidently, UŽ r . is translationally invariant in the y direction and periodic in the x direction with a period of 2prk so that the discussion can be restricted to the region yprkF x - prk. An alternative method of creating a potential such as Eq. Ž2. is offered by the magnetic field of arrays of parallel wires that carry electric currents in alternating directions. At a sufficiently large distance from the plane containing the wires the field of a single such array is almost identical to the one of a sinusoidally magnetized surface w13x. Superposing two such arrays with appropriate wire spacings gives a field similar to Eq. Ž1. in the far zone. An advantage of this approach in comparison to the magnetized surface is the fact that the field strength B1 and B2 can be varied during the experiment which gives

O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

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additional possibilities to manipulate the atomic sample. The field configuration Ž1. already yields a simple linear atomic waveguide which can be seen as follows. In the plane x s 0 the field component Bz vanishes. On the other hand, if B1 B2 - 0, the z dependence of B x is analogous to the behavior of familiar Morse potential. In particular, when expŽykz . s yB1rB2 , B x changes sign so that the absolute field strength < B Ž r .< and thus the potential Ž2. has a minimum at this point wsee Fig. 1Ža.x. As the potential does not depend on the y coordinate a waveguide is formed. Consider now the more general field distribution B s y B1eyk z cos Ž kx . q B2 ey2 k z cos Ž 2 kx . y Bh xˆ q B1eyk z sin Ž kx . q B2 ey2 k z sin Ž 2 kx . zˆ q B0 yˆ ,

Ž 3.

which differs from Ž1. in two ways. First of all, a holding field B0 is introduced in the y direction w8x. It prevents the field from vanishing exactly at the minima and thus suppresses unwanted spin flips of trapped atoms and their ejection from the trap. Note that the holding field does not influence the existence and the location of the field minima. Second, an additional homogeneous field Bh is applied along the x axis. Its purpose becomes apparent by again considering the field in the plane x s 0: it simply shifts the Morse ‘potential’ curve for B x Ž z . upwards or downwards. In particular, as shown in Fig. 1Žb., Žc., for B1 , Bh ) 0, the curve B x Ž z . is moved upwards so that a second zero is created which approaches the first one and coalesces with it when the minimum of B x Ž z . moves through zero. More precisely, the first zero of B x Ž z . is located at

Fig. 1. Magnetic potential UŽ z . Žfull curve. and field strength B x Ž z . Ždashed. at x s 0 as determined by Eq. Ž3.. Ža. One potential minimum appears for B0 s 0, Bh s 0 wwhen Eq. Ž3. reduces to Eq. Ž1.x. Žb. Increasing Bh yields a second minimum. Žc. The holding field B0 eliminates points of vanishing magnetic field, but does not change the location of the minima.

Žwhen it collides with the outer zero.. The outer atom trap is found at eyk z s y

e

1 sy

q 2 b2

(

1 4 b 22

q

bh b2

,

Ž 4.

where b 2 s B2rB1 - 0 and b h s BhrB1. Because x s 0, Bz s 0, and so this zero in B x results in a minimum in < B < and thus gives rise to an ‘inner’ atom trap. It exists for values of b h between 1 q b 2 Žwhere it hits the magnetic surface. and y1r4b 2

y 2 b2

(

1 4 b 22

q

bh

Ž 5.

b2

and appears as long as 0 - b h - y1r4b 2 . These two potential minima are separated by a saddle point which is fixed at eyk z s y

yk z

1

1 2 b2

.

Ž 6.

After merging into one at b h s y1r4b 2 the two traps separate again for larger fields. This time, however, they are no longer situated in the plane x s 0 but are found at the positions eyk z s

(

y

bh b2

,

cos Ž kx . s

(

1

y 4 b 2 bh

,

Ž 7.

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O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

i.e., they are symmetrically separated from x s 0. They finally hit the magnetic surface and disappear when b h s yb 2 . In Fig. 2Ža. – Žc. the deformation of the trapping potential with increasing b h is illustrated: the two waveguides are first placed on top of each other, then they combine and finally move apart sideways. Having introduced the main features of the proposed linear atom trap we now want to discuss some of its properties as a waveguide for atomic de Broglie waves. Of particular interest in this context are the extension and the energies of the lowest-lying eigenstates of the two-dimensional traps. If the potential

wells are sufficiently separated from each other, i.e., b h is not too close to y1r4b 2 , and the magnetic fields B1 , B2 , and B0 are strong enough, these quantities are immediately obtained from the harmonic approximation of the potential around the minima. Thereby, it is found that 2

U Ž r . s E1 b 0 q 12 k 2 uXXm Ž x y x m . q Ž z y z m .

½

2

5, Ž 8.

where E1 s g L m b m F B1 , b 0 s B0rB1 , and Ž x m , z m . the location of a minimum. The curvature uXXm is the same in both the x and z direction and is given by uXXm ,i s Ž 1 q 4 b h b 2 . 1 q 2 b h b 2 q Ž 1 q 4 b h b 2 .

1r2

r2 b 0 b 22

Ž 9.

for the inner well, uXXm , o s Ž 1 q 4 b h b 2 . 1 q 2 b h b 2 y Ž 1 q 4 b h b 2 . r2 b 0 b 22

1r2

Ž 10 .

for the outer well, and by uXXm , s s b h Ž 1 q 4 b h b 2 . rb 0 b 2

Ž 11 .

for the two symmetric wells Ž b h ) y1r4b 2 .. With M the mass of the trapped atom one can define the oscillator frequency v s Ž E1 k 2 uXXm rM .1r2 which yields the ground state extension Ž "rM v .1r2 and the energy spacing " v between the lower excited states. Note that v is proportional to Ž B1 k 2rb 0 .1r2 . When the two traps coincide, i.e., b h s y1r4b 2 , the potential around the minimum is approximated by the fourth-order expansion

½

U Ž r . s E1 b 0 q

k4 16b 0 b 22 2

q2 Ž z y z m . x 2 Fig. 2. Contour plots of waveguide potentials Us g L m B m F < B < in the x – z plane with B given by Eq. Ž3.. The parameter values are b 2 sy1.3, b 0 s 0.1, and Ža. bh s 0.08, Žb. bh sy1r4b 2 s 0.192, and Žc. bh s 0.30. The equipotential lines are shown with an energy difference of 0.01 g L m B B1 , their sequence is cut off at a limiting value.

4 Ž z y zm . q x 4

5

.

Ž 12 .

Defining s s E1 k 4r16 b 0 b 22 the characteristic length scale of the ground state is now given by Ž " 2rMs .1r6 , whereas the energy scale is set by Ž " 4srM 2 .1r3. Note that the ground state extension is

O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

less sensitive to changes in B1 and b 0 than in the case of the separated minima, but more sensitive to variation of k, in correspondence to intuitive expectation. If b h lies in the vicinity of y1r4b 2 the extension of the ground state depends also on the position of the potential minima besides the curvature, the ground state is elongated in either the z or x direction. To give a numerical example, for the parameter values k s 10 3 my1 , B1 s 100 G, b 2 s y1.3, b 0 s 0.1, and b h s 0 Ži.e., with only the inner well remaining. we expect a ground state size of about 6 = 10y7 m for a Rb atom with gm F s 1. Of course, the traps may also hold incoherent samples of thermally excited atoms which may be of far larger diameter. Another figure characterizing the waveguide is the number of quantum mechanically bound states it can support. This quantity may be estimated from a simple phase-space argument to be given by Apm2 rŽ2p " . 2 with A the trap cross section and pm the maximum momentum of trapped atoms. For the coalesced traps we have roughly A f z m2 and pm f w2 ME1Ž b h2 q b 02 y b 0 .x1r2 . For the above numerical example Žbut with b h s 0.192 for a combined trap. one thus expects about 6 = 10 9 bound eigenstates. A feature of this configuration of magnetic fields is the possibility to increase the density of a trapped atomic sample by manipulating the transverse field Bh . To this end one could first load the outer well, which can be made arbitrarily shallow by choosing Bh small. Then, by increasing Bh to the point of coalescence of the two wells, and subsequently decreasing it again, the steeper inner well can be filled Žsee Fig. 3.. Numerical simulations show that, for a

(

Fig. 3. The outer trap is initially loaded Ža. and the inner trap is filled by ramping the homogeneous field Bh up to the point of coalescence of the two wells Žb. and then down again Žc.. Thereby, an effective density compression for the atoms ending up in the inner well is achieved.

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system initially in the quantum-mechanical ground state of the outer trap, population may be transferred to the inner well ground state with an efficiency f of at least 35% without significantly populating any of the inner excited states. The density compression is given by f w uXXm,i Ž b h,final .ruXXm, o Ž b h,initial .x1r2 and can achieve values of the order of 10. Similar results are to be expected for the transport to the symmetric wells. Further compression may obtained by reducing the holding field B0 Žas long as spin-flips remain suppressed.. The present proposal for an atomic waveguide might be considered interesting for a number of reasons. First of all, owing to the flexibility of the ‘magnetic surface technology’, it could easily be realized over a range of length scales and for vastly varying magnetic field strengths. Due to the periodicity of the surface magnetization one automatically obtains a whole array of potential tubes that can be manipulated in parallel. Most interesting, however, seems to be the possibility to drastically vary the trap properties both in space and in time by appropriately changing the transverse field Bh . An obvious application would be the construction of beam splitters and combiners ŽY-switches.. Beam splitting or combining in the time domain is evidently achieved by changing b hŽ t . away from, or towards, the value y1r4b 2 . The corresponding effect in the space domain is achieved if the additional transverse field in the x direction depends on y. Here, it might even be feasible to construct a ‘microcircuit’, i.e., a serial combination of splitter and combiner. To this end, b h of Eq. Ž3. would be held at the value of y1r4b 2 so that only one tube is present. A small perturbation, e.g. from a permanent magnet aligned along the x direction, would be sufficient to change locally the value of the transverse field so that the original tube is split into two and the desired microcircuit is produced. To illustrate this point, Fig. 4 shows the potential created by superposing the unperturbed waveguide field and the field of a magnetic dipole. The dipole is located at rdip s Ž0,0, z dip . with z dip - 0, i.e., below the magnetic surface, and points into the negative x direction so that the transverse field is increased around y s 0. The dipole field is given by Bdip Ž r . s Bd,0 Ž3qˆ x qˆ y xˆ .rŽ kq . 3 with q s r y rdip and qˆ the unit vector in the q direction. The figure

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O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

shows that the desired microcircuit configuration is indeed obtained. However, in this simple arrange-

ment one observes an asymmetry between the two arms of the circuit so that the flux may not be the same in both channels. Another issue which would need further study in a practical implementation of the device concerns its overall transmission and reflectivity. As the shape of the microcircuit may be varied over a wide range by changing the perturbing magnetic field its properties may be systematically studied and, possibly, optimized. Another potential application of the double waveguide, namely atom interferometry to detect weak forces, is discussed in more detail in Section 3. The loading of atomic waveguides has recently been demonstrated in Refs. w5,6,14x where the atoms are ‘dropped’ or ‘funneled’ into the waveguides. Analogous schemes might also be applied in the present case. An alternative approach would consist of moving a trapped atomic sample Žfor example, an optically trapped Bose–Einstein condensate. to the location of the waveguide and then switching the initial trap off. Such a scheme would be similar in spirit to the loading of an optical trap with a condensate w15x. If the process is performed adiabatically there may indeed be a possibility to populate the quantum mechanical ground state of the guide.

3. An atom interferometer In atom interferometers the splitting and recombination of the atomic samples is usually achieved with the help of optical pulses or microfabricated structures w16x. The measured signal is then extracted from interference effects between different planewave eigenstates of the center-of-mass motion and, possibly, internal states. The magnetic device described in Section 2 suggests an alternative implementation of an atom interferometer. In this context, the magnetic potential would not be used as a wave-

Fig. 4. Potential of a microcircuit created by perturbing the waveguide field by a magnetic dipole. The parameter values are b 2 sy1.3, b 0 s 0.1, and bh s 0.192 for the waveguide; Bd,0 r B1 sy100 and kz dip sy11 for the magnetic dipole. The equipotential lines are drawn with an energy difference of 0.005 g L m B B1 .

O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

guide Žto transport atoms in the y direction. but simply as a time-evolving trap that moves atoms to predetermined positions in the x–z plane and holds them there. The necessary confinement in the y direction could be achieved, e.g., with the help of sheets of blue-detuned laser light. This interferometer is then sensitive to a weak, external, x dependent potential V Ž x . that acts on the atoms in addition to the magnetic potential UŽ r . s g L m B m F < B Ž r .<. The interferometric process would then proceed through the following steps Žsee Fig. 5.: 1. The atoms are prepared in the quantum mechanical ground state c 0Ž i. of the combined traps at b h s y1r4b 2 . 2. The transverse field b h is increased in such a way that the two emerging traps, which are equidistant from the magnetic surface wFig. 2Žc.x, become well separated from each other. 3. The system is then held at the final field strength for a time t . 4. Subsequently, the field b h is ramped down to the value y1r4b 2 .

99

5. Finally, b h is increased again which allows one to read out the interferometric signal. As indicated in Fig. 5, this sequence serves to prepare, after step 4, the wave packet in a superposition of the trap ground state c 0Ž i. and the first excited state c 1Ž i.. The relative weights of these states, which are measured after step 5, are periodic in the holding time t . This period, in turn, depends on the change of V Ž x . along the x direction. Let us now discuss the various steps of this sequence in more detail. After the second step the two trap centers are located at x 1 and x 2 so that the two well bottoms experience a potential difference of DV s V Ž x 2 . y V Ž x 1 .. If this difference is large compared to the Žsmall. quantum mechanical tunneling rate between the wells the ground state c 0Ž f . in the new configuration is exclusively localized in the lower trap. If Bh is increased sufficiently slowly the atomic population will completely go into this state. In a more rapid process, however, one can also populate the lowest-lying state c 1Ž f . of the upper trap due to nonadiabatic excitation. This state c 1Ž f . would evolve by adiabatic following from the combined-trap eigenstate c 1Ž i. that has one vibrational excitation in the x direction. To describe the nonadiabatic time development we introduce the 2 = 2 transfer matrix U1 that contains the transition matrix elements between the initial and final states. If the final state is given by aŽ0f .c 0Ž f . q a1Ž f .c 1Ž f . we thus have Ž aŽ0f .,a1Ž f . .T s U1Ž1,0.T. Of course, U1 depends on the functional form of BhŽ t .. In a first approximation it is obtained from the solution of the set of differential equations i a˙ 0 Ž t . s E0 Ž t . a0 Ž t . r" y i V eff Ž t . a1 Ž t . , i a˙1 Ž t . s E1 Ž t . a1 Ž t . r" q i V eff Ž t . a0 Ž t . ,

Fig. 5. Schematic of the interferometric process.

Ž 13 .

where E0r1 denote the energies of the instantaneous eigenstates c 0r1 at field strength BhŽ t . and V eff s Hd 3 r Ž c 0 Ec 1rE t . is the nonadiabatic coupling. These quantities was well as the solution of Eqs. Ž13.x have to be determined numerically. Their knowledge allows us to estimate the necessary rate of change of Bh , as the nonadiabatic excitation is only efficient if V eff is comparable to E1 y E0 . Note that Eqs. Ž13. only yield an approximate determination of U1 as

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O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

they do not include coupling to other excited states. These additional couplings also render the 2 = 2 matrix U1 non-unitary. During the holding process of step 3 the trap states c 0Ž f . and c 1Ž f . acquire a phase difference of expŽyiD E Ž f .tr" . with D E Ž f . s E1Ž f . y E0Ž f . f DV. By analogy with step 2, the transfer of the wave packets during step 4 is characterized by the transfer Žf. Ž i. matrix U2 that connects the states c 0,1 to c 0,1 . The Ž i. fractional population of c 0 at the end of this sequence is given by P Žt . 2

s U2Ž0 ,0. exp Ž yiD E Ž f .tr" . U1Ž0 ,0. q U2Ž0 ,1.U1Ž1 ,0. . Ž 14 . It is thus periodic in t with a period of p "rD E Ž f ., i.e., one determines D E Ž f . from interferometric information stored in the t dependence of the final population of the vibrational ground state. In Fig. 6 P is shown as a function of t for an example with Rb atoms in which V Ž x . was chosen as the gravitational potential Mgx. The parameters characterizing the trap are B1 s 10G, k s 10 6 my1 , b 2 s y1.3, b 0 s 0.1; the transverse field is linearly ramped up and down between b h s 0.2 and 0.3 with a rise and fall time of 1.6ms each. The distance between the trap centers at b h s 0.3 is 1.28 mm. The full circles show the final ground state population as obtained from the simulation of the atomic time evolution in the two-dimensional time-dependent trapping potential. These results are well approximated by Eq. Ž14. in which the transition matrix elements and the energy difference D E Ž f . have been determined numerically. Note that in Fig. 6 the visibility of the signal is Õ s Ž Pmax y Pmin .rPmax f 0.9. To read out the fractional population P the transverse field b h is increased again, but this time adiabatically. In this way the ground state population is completely transferred to the lower trap whereas the atoms in the excited state are transported to the higher-lying trap. The number of atoms in each of the two traps is then determined by, e.g., an optical measurement. The dashed line in Fig. 6 shows the population of the lower trap after the field b h is ramped up again to the value of 0.3 during a rise time of 8ms.

Fig. 6. Ground state population P as a function of the holding time t after step 4 of the interferometric process described in the text. Full curve: approximation according to Eq. Ž14., full circles: numerically determined values. Parameter values as given in the text. The dashed curve shows the population of the lower-lying trap after step 5.

A practical implementation of this scheme is certainly faced with a number of difficulties. First of all, the trap ground state has to be populated very efficiently. Given the recent progress in atom cooling and especially in the creation of coherent Bose–Einstein condensates, however, this problem may not be unsurmountable as discussed at the end of Section 2. Another technical issue is the accurate repetition of the various changes of the field b h . A more fundamental problem concerns the fact that the quantity which is actually measured is not the potential difference DV but the energy difference D E of the eigenstates c 0Ž f . and c 1Ž f .. These quantities differ from each other due to the finite extension of the traps and the deformation of the original trapping potential by V Ž x .. In the above example this difference is about 0.1%, under other circumstances this value may be significantly larger. It may be possible to extrapolate DV from the measured D E by calibration with a known potential or additional numerical calculations. A high-precision measurement, however, does not seem to be feasible. Applications of the scheme might thus be more likely in situations where the interferometric process has to go on over a very long time t due to the weakness of the perturbation and has to be confined to a small region. Under such circumstances the excellent isolation from the environment that is provided by the magnetic trap, and the ensuing long decoherence time, present an important advantage w17x.

O. Zobay, B.M. Garrawayr Optics Communications 178 (2000) 93–101

With an efficient loading process one could also imagine using the whole array of interferometers provided by the periodically magnetized surface in parallel. In this way one might directly measure, e.g., the intensity gradient along a weakly focused laser beam sampled by the interferometer array. In this case the external potential results from the intensitydependent ac Stark shift experienced by the atoms in the laser field.

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potentials generated by more complicated magnetizations. One could imagine structures with three or more traps that could be manipulated in a similar same way as described above. Acknowledgements Stimulating discussions with Prof. E.A. Hinds are gratefully acknowledged. This work was supported by the United Kingdom Engineering and Physical Sciences Research Council.

4. Conclusion In this paper a linear double waveguide for atoms was proposed and theoretically investigated. Its manufacture relies on imprinting a specific periodic magnetization onto an appropriate recordable medium such as a videotape. Since the corresponding techniques have already been demonstrated experimentally the practical realization should be currently feasible. A particular feature of the proposed setup is its sensitivity to the application of an additional homogeneous magnetic field. It can be used to drastically alter the separation of the two waveguides and their mutual orientation with respect to the surface. Furthermore, owing to the periodicity of the surface magnetization a whole array of such double waveguides could be manipulated in parallel. Recently demonstrated methods to load atom guides with incoherent atomic samples might also be applicable in the present case. In this context the proposed device could be used to implement Yswitches and microcircuits whose properties are variable over a wide range. Even more interesting applications are conceivable if the atoms are loaded into single quantum mechanical eigenstates of the system. Under these circumstances the device could be used, for example, as an atom interferometer which is capable of detecting weak forces. The present double waveguide is constructed from a periodic surface magnetization containing only the two lowest Fourier components. Its interesting properties might make it worthwhile to examine the

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