A Two Wire Waveguide and Interferometer for Cold Atoms. E. A. Hinds, C. J. Vale, and M. G. Boshier

Transcription

1 A Two Wire Waveguide and Interferometer for Cold Atoms E. A. Hinds, C. J. Vale, and M. G. Boshier Susse Centre for Optical and Atomic Phsics, Universit of Susse, Brighton, BN 9QH, U.K. A versatile miniature de Broglie waveguide is formed b two parallel currentcarring wires in the presence of a uniform bias field. We derive a variet of analtical epressions to describe the guide and present a quantum theor to show that it offers a remarkable range of possibilities for atom manipulation on the submicron scale. These include controlled and coherent splitting of the wavefunction as well as cooling, trapping and guiding. In particular we discuss a novel microscopic atom interferometer with the potential to be eceedingl sensitive. An atom whose magnetic moment has projection along an eternal magnetic field of magnitude B eperiences a Zeeman interaction potential U B. The associated force is able to guide weak-field-seeking atoms along a minimum of B. This principle underlies the Stern-Gerlach effect and the magnetic heapole lens, which have plaed such important roles in the histor of atomic beams. Magnetic forces are now a central feature of atom optics the subject of manipulating, confining, and guiding cold neutral atom clouds and Bose-condensates [,, 3].

2 Recentl there has been great interest in building miniature magnetic structures where small features make a strong field gradient, while a superimposed uniform bias field makes a zero whose position is adjustable [4]. This idea has been realized in several laboratories, using either supported wires [5, 6, 7], printed circuits [8, 9,,, ] or microscopic patterns of permanent magnetization [3]. Miniature guides are attractive because the offer the possibilit of propagating de Broglie waves in a single transverse mode in D [4] or D [5]. This is a central goal of man eperimental groups because it is required for achieving atom interferometr with guided de Broglie waves. Miniature structures are also promising for studing the phsics of quantum gases confined to less than 3D [6]. In this letter we discuss the guide formed b two wires carring parallel currents I spaced A apart in the presence of a bias field, as illustrated in Fig.. We point out that the guide is far more adaptable than previousl realized [9,], deriving simple formulae for the various configurations it can produce. We present a quantum theor to show how the guide can be used to manipulate atoms on the sub-micron scale, to make a controlled and coherent splitting of the wavefunction, and to realize a novel atom interferometer. Let us adopt the natural units of A for lengths, B I / FA for magnetic fields, and B for energies. Dimensionless quantities based on these units are indicated b lower case letters, e.g. means X A. With this scaling the magnetic field produced b the current-carring wires in Fig. has Cartesian components

3 b b ( ) ( ) ( ) ( ) () On the -ais, b and the field produced b the wires is b. This has a single maimum of b at height as shown in Fig.. The addition of a bias field > (in normalized units) along the positive -direction cancels b at two positions, indicated for > / b the circles in Fig.. For weak-field-seeking atoms, these zeros form guides parallel to the z-ais (a small field ma be added along the z- direction to suppress non-adiabatic spin flips, although states without angular momentum around the guide ais can be stable without it [4]). The inset in Fig. shows the dimensionless guiding potential u. The barrier between the two guides is >, while the barrier above the upper guide is >. As the strength of the bias field is increased, these zeros approach one another until the coalesce to form a single guide at height when >. Further increase of the bias splits the guide horizontall and the zeros follow the circle (Fig. ). In the first two rows of Table we present simple formulae giving the guide centers (,) and trap depths u in each of the three bias field regimes. Fig. 3a shows the interaction potential u and the field lines for >. 8. The trap centered on / is smaller and 4 times stronger than the upper one at (the ratio of gradients is alwas the inverse of the ratio of heights). Each guide is clindricall smmetric over a limited region around its center, and has a constant 3

4 potential gradient, a characteristic of the quadrupole smmetr evident in the field lines. We note that the fields are oppositel directed in the two guides. At the two bottom corners, the field can be seen circulating around the current-carring wires, which are taken in this diagram to be of negligible radius. Figure 3b shows the potential and field lines for the single guide, formed at the critical bias. Here the linear gradient vanishes: the guide is harmonic with curvature u / H and the corresponding heapole smmetr can be seen in the field lines. At higher bias field the potential splits laterall into two quadrupole guides as illustrated in Fig. 3c for >. 5. Simple formulae for the gradients are given in the last row of Table. If we allow the bias field to have a component in the -direction, man other possibilities open up, one of which is particularl relevant, as we will see below. Let us appl both the critical bias > along, and an etra bias of magnitude, > at angle to the -ais. When,>, the potential splits into two guides separated b a distance,>, and the line joining their centers makes an angle / with the -ais. Thus, when the etra bias field is rotated the two guides orbit around the coalescence point at half the frequenc. Each of the three regimes of bias field has interesting features to offer for atom optics eperiments. To take a concrete eample, consider a guide with a 3 m spacing and A flowing in the wires, for which the characteristic field and gradient are B =.7 mt and B /A = 8 T/m (such a structure alread eists in our laborator). With a weak bias field of.3 mt ( >. ), the upper guide is centered 3 mm ( A / > ) above the wires and has a field gradient of. T/m ( > B / A ). This is ver well suited to operate as a 4

5 magneto-optical trap (MOT) with two pairs of suitabl polarized light beams in the - plane, propagating along aes rotated b 45º from the - and -aes. For the purpose of initiall collecting atoms in the MOT, a pair of auiliar coils can create a field gradient along the z-direction, allowing a third pair of light beams to produce MOT confinement along the z-ais. The MOT can be lowered and compressed to a maimum gradient of 4.4 T/m ( B / A ) b increasing the bias field to.3 mt ( B / ). At this point the 4 light can be turned off and the auiliar coils producing the field gradient along the z-ais can be switched to produce parallel fields. This makes a purel magnetic Ioffe-Pritchard trap, where weak-field-seeking atoms can be left in the ground state b evaporation. Alternativel, if a cold dense sample of atoms is alread available in a macroscopic magnetic trap, as for eample with a Bose-Einstein condensate, a single mode of the upper guide can be loaded in a mode-matched wa b suddenl switching off the trap and turning on the guide with I, > and a z-bias field chosen to duplicate the original trap spring constants. In either case, a subsequent adiabatic variation of the field can bring atoms to the coalescence point of the guide, where the potential is harmonic and the 3 transverse frequenc M is given b mm A B. This has the value.8 3 s for the particular guide we are considering here. The ground state size I D / mm is 5 nm. Suppose now that the atoms have all been prepared in the transverse ground state of the harmonic guide. An increase, > of the bias field deforms the guide potential into a smmetric double well, providing a highl controlled and reproducible 5:5 coherent splitter for the de Broglie wave. If the two halves of the atom cloud are held apart for some length of time the ma act as the arms of an interferometer. In order to understand 5

6 the operation of this interferometer in more detail, we have calculated the eigenmodes of the guide numericall b solving the two-dimensional time-dependent Schrödinger equation with the spin degree of freedom adiabaticall eliminated. We start with the eigenstates {( n, n )} of a harmonic potential, then slowl deform the potential to obtain the corresponding eigenstates of the Hamiltonian for the -wire guide at the critical bias >. Net we slowl var the bias field to find how the three lowest eigenstates and their energies evolve as a function of >. We assume that the atom densit is low, although of course the mean field interactions at higher densit should produce interesting non-linearities and phsics beond the Gross-Pitaevskii equation. Figures 4 and 5 show the eigenstates and their energies, labeled b the quantum numbers n, n ) which count the number of nodes along and. At the critical bias, ( the spectrum is (almost) harmonic with the (,) state ling D M below the nearl degenerate pair of first ecited states (,) and (,) (a small anisotrop of the guide prevents eact degenerac). An increase, > of the bias splits the guide and the singlepeaked (,) state deforms adiabaticall into a double-peaked wavefunction with even reflection smmetr in the z-plane. As long as it is not perturbed, this of course returns to the harmonic ground state when > is restored to. If, however, a differential phase of F is introduced between its two peaks while >, the wavefunction becomes a zantismmetric function, which we recognize in Fig. 4 as the (,) state. When > this evolves adiabaticall into the first z-antismmetric ecited state of the harmonic guide. For arbitrar phase shifts the final state of the harmonic guide is a superposition of (,) and (,). Thus smmetr dictates that the two output ports of the 6

7 interferometer are two different vibrational states of the harmonic guide. The third state (,) in Fig. 4 is even under reflection in the z-plane, like the ground state, but it has a vibrational ecitation in the -direction. This ecitation is preserved as the bias increases and consequentl the energ lies approimatel D M above (,). It is also interesting to consider decreasing the bias to >. The interferometer states (,) and (,) both emerge in the upper guide, with (,) having one quantum of ecitation along the -ais. The (,) state however, goes into the ground state of the lower guide (Fig. 4). Now we turn to the practical aspects of the interferometer. One can estimate the minimum, > needed to achieve splitting,, > min, b setting the displacement of the wells equal to the ground state diameter I. To lowest order the result can be epressed in the elegant form, > min I / A DM / B, which for the guide in our eample amounts to a change of 64 nt in the bias field. Achieving this level of control over the field would require some care but it is not a major technical challenge. When, > is increased further, the splitting between the (,) and (,) levels becomes eponentiall small (Fig. 5), being equal to the tunneling frequenc between the left and right potential wells. As the bias changes, one wants to avoid non-adiabatic transitions induced b / t. This operator, being smmetric under reflections in the z-plane, cannot ecite the z-antismmetric state (,), but it does connect the ground state to (,). Since this and the other coupled states are at least D M awa in energ, the adiabatic condition is that, > must change slowl in comparison with the period of harmonic oscillation / M (over several milliseconds for the guide in our eample). Once the atoms are split, the 7

8 antismmetric perturbation to be measured is turned on. As is usual in interferometr, this interaction has to be non-adiabatic to mi the states (,) and (,) (i.e. to introduce a phase shift between the left and right wavepackets) and must therefore be turned on and off in times much less than the tunneling period. This is not a stringent requirement since the tunneling can be made arbitraril slow b a modest increase in, >. Finall, reducing, > to zero, it remains onl to read out the fringe pattern through the population in state (,) [ (,) ], which is proportional to cos [ sin ]. In principle the readout can be done b absorption or fluorescence imaging to determine the atom distribution in the guide, although this method requires high spatial resolution. Alternativel, there are at least two methods for separating the (,) and (,) populations. First, if the guide is operated without an aial bias field and > is reduced below to etract both states in the upper quadrupole guide, the ecited state (,) will be lost due to spin flips [4], leaving the (,) population to be measured b fluorescence. Second, the additional bias field, > can be rotated adiabaticall from the ˆ direction to ˆ before reducing its amplitude to zero, thereb transforming state (,) into (,). This leaves the output of the interferometer as a superposition of (,) and (,) in the harmonic guide, which can be read out b reducing > so that the (,) component moves into the upper guide while the (,) part is transported downward. In an interferometer the statistical signal:noise ratio is proportional to the interaction time J and to the square root of the number of atoms N. We can easil imagine 6 atoms trapped in the waveguide with a measurement time of ~ s, giving 4 J N. In comparison a macroscopic cold atom interferometer of the Kasevich-Chu tpe [7] has 8

9 8 atoms falling through the apparatus per second with an interaction time of order 3 ms, giving 3 J N over the same s. Both interferometers have a separation of ~ m between arms. It therefore seems clear that this method of splitting the atoms in time, rather than splitting atoms propagating through space, can significantl advance some kinds of measurement. For eample, the new interferometer would be etremel sensitive to electric field gradients and to gravit. Also, it would not suffer from phase shifts due to unwanted rotations because the Sagnac phase is zero. In conclusion, we have shown that two currents and a bias field form an eceedingl versatile structure, producing a waveguide that can be split in a highl controlled wa and manipulated on the sub-micron scale. We have shown eplicitl how a novel and sensitive atom interferometer could be realized with such a guide. This structure is also ideal for a variet of topical applications including a miniature magneto-optical trap and the stud of -dimensional quantum gases. Finall, the quantum model of guided atom interferometr presented here will appl to man eisting eperiments once the reach the level of single-mode operation. We are indebted to David Lau, Stephen Hopkins, and Mark Kasevich for valuable discussions and to Matthew Jones for eperimental work on the Susse microscopic guide. This work was supported b the UK EPSRC and the EU. 9

10 REFERENCES J. P. Dowling and J. Gea-Banacloche, Adv. At. Mol. Opt. Phs. 37, (996). E. A. Hinds and I. G. Hughes, J. Phs. D 3, R9 (998). 3 W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, Enrico Fermi Course CXL, M. Inguscio et al. (Eds) IOS Press, Amsterdam (999). 4 J. D. Weinstein and K. G. Libbrecht, Phs. Rev. A 5, 44 (995). 5 J. Fortagh et al., Phs. Rev. Lett. 8, 53 (998). 6 J. Denschlag, D. Cassettari and J. Schmiedmaer, Phs. Rev. Lett. 8, 4 (999). 7 M. Ke et al., Phs. Rev. Lett. 84, 37 (). 8 J. Reichel, W. Hänsel and T. W. Hänsch, Phs. Rev. Lett. 83, 3398 (999). 9 D. Müller et al., Phs. Rev. Lett. 83, 594 (999). D. Müller et al., arxiv:phsics/39 (). N. H. Dekker et al., Phs. Rev. Lett. 84, 4 (). R. Folman et al., Phs. Rev. Lett. 84, 4749 (). 3 P. Rosenbusch et al., Phs Rev. A 6, 344(R) (); Appl. Phs B 6, 79 (). 4 E. A. Hinds and Claudia Eberlein, Phs Rev. A () 5 E. A. Hinds, M. G. Boshier and I. G. Hughes, Phs. Rev. Lett. 8, 645 (998). 6 M. Olshanii, Phs. Rev. Lett. 8, 938 (998): H. Monien, M. Linn, and N. Elstner, Phs. Rev. A 58, R3395 (998). 7 M. J. Snadden et al., Phs. Rev. Lett. 8, 97 (998).

11 Figure Captions Figure. Atom guide using two current-carring wires and a bias field. With increasing bias, two guiding regions move towards each other along Y (dotted line) until the coalesce at Y = A. The then separate along the dashed circle. Figure. Field b on the -ais versus height above the wires. A bias > / along makes two zeros (circles). Inset: interaction potential u showing the two linear guides. Figure 3. Interaction potentials and field lines for (a) >. 8 (b) > (c) >. 5. Figure 4. Wavefunctions of the lowest three states in the guide for etra bias fields, > /, > min,,. The ranges for and are. 5 and.5. 7 respectivel. Figure 5. Energ spectrum for the three lowest eigenstates in the guide versus etra bias field.

12 Table. Center, depth and gradient of the -wire guides for each regime of the bias field. Bias > < > = > > (, ), > (,) >, > u >, > > / H u > > > > >

13 Bias field Y Coalescence point (,A) X A A Figure : Hinds et al.

14 Dimensionless field b Dimensionless height.5 u 4 6 Figure : Hinds et al.

15 3 3 - u.5 - u.6 - u - - (a) (b) (c) - Figure 3: Hinds et al.

16 Figure 4. Hinds et al

17 Energ / hw 4 3 (,) (,) (,) - Dimensionless etra bias field Db/Db min Figure 5. Hinds et al.