VOLUME 81, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 10 AUGUST 1998

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Javanainen and Wilkens Reply: The disagreement be-tween our Letter [1] and the preceding Comment [2]about the state of the atomic system after a trapped BoEinstein condensate has been split by erecting a potenbarrier in the middle. Our view is that, while both the Letter and the Comment seem correct within their premisat this point there is no conclusive evidence to decidefavor of either set of premises.

We elaborate in terms of the Hartree-Fock-type meafield theory (MFT), which seems to have been quite sucessful at describing the state of an alkali vapor condensAccording to MFT, in the ground state of the condensain a single trap, all atoms are in the lowest-energy sotion ci of the Gross-Pitaevskii equation (GPE). The samshould apply as long as the notch in the potential betwethe halves of the trap is low enough. This is the single-trregime.

On the other hand, when the trap is split (symmetcally) and the halves are completely separated, the loweenergy state is obviously the number state in which tatoms are divided evenly between the one-particle stacl andcr obtained by solving the GPE separately for thground states of the left and right halves of the trap.a somewhat smaller separation, one adds a coupling mtrix element to describe tunneling between the two potetial wells. The Hamiltonian underlying the Comment [3is coached in different terms, but may be derived easfrom this prescription by canonical transformation. This the tunneling regime.

When the barrier height in the GPE is increaseddescribe splitting of the trap, the ground state of thsingle trapci evolves continuously into a superposition othe left- and right-trap ground states,cf ­ 1p

2scl 1 cr d.

However, while the state in which all atoms are in thstateci is the ground state of the single trap, by virtuof the preceding paragraph, the same does not applycf in the limit of the split trap. The single-trap andtunneling limits are both individually discussed in termof a MFT, but a different MFT. Somewhere in betweenthe qualitative nature of the description changes.

So far we have not been able to build a bridge betwethe two regimes. For instance, the superposition principdoes not hold for the GPE, so that the relation betwethe wave function that evolves with a rising barrier fromthe ground state of the single trap and the wave functiothat are predominantly confined to one or the other trapnot trivial. In this view, the interesting and important buobviously also difficult key issue is, how should one thinabout superpositions of Hartree-Fock one-particle wafunctions.

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Noting that “at the moment we have no proof to thieffect, but at least in the limit of weak interactions theassumption is clearly valid,” in our Letter we simplyassume that the split condensate is represented bywave functioncf . We are extrapolating from the single-trap regime to the tunneling regime, a step that is opeto questions. Ironically, though, the Comment is subjeto the mirror image of precisely the same criticism. ThHamiltonian underlying the Comment is not valid in thelimit of a single trap. The assumption in [2] that the trapemerges from the splitting in the ground state of thHamiltonian is anad hoc extrapolationbackward fromthe tunneling regime to the single trap. For a conclusivanalysis, one needs an approach that explicitly connethe two regimes.

When the trap is split too rapidly, the condensate halvare evidently left in a macroscopic sloshing motionThe condition of adiabaticity in our Letter was meanas a qualitative safeguard against such a case. This no a priori guarantee that our condition governsadiabaticity when the system negotiates its way frothe single trap to the tunneling regime. The time scaof adiabaticity is a combination of the characteristitime scale for elementary excitations and dimensionleparameters (atom numberN , parameterQ as in theComment, . . .), and we know of no explicit analysis othis issue.

Whether the quantitative results of our Letter or of thComment are more appropriate depends on adiabaticitythe transition between single-trap and tunneling regimeAt this time there is insufficient evidence on the table twarrant a firm conclusion.

Juha JavanainenDepartment of Physics, University of ConnecticutStorrs, Connecticut 06269-3046

Martin WilkensInstitut für Physik, Universität PotsdamD-14469 Potsdam, Germany

Received 24 March 1998 [S0031-9007(98)06784-2PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 74.50.+r

[1] J. Javanainen and M. Wilkens, Phys. Rev. Lett.78, 4675(1997).

[2] A. J. Leggett and F. Sols, preceding Comment, Phys. ReLett. 81, 1344 (1998).

[3] See A. J. Leggett and F. Sols, Found. Phys.21, 353(1991).

© 1998 The American Physical Society 1345