Chapter 11
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Bragg Diffraction and the Interference of Two Atom Lasers: An Analogy Roger A . Hegstrom Department of Chemistry, Wake Forest University, Winston-Salem, NC 27109
The interference pattern produced by the beams from two atom lasers can be understood in a relatively easy way by exploiting a formal equivalence between the expression for the quantum mechanical probability density for the atoms in the laser pair and the expression for the intensity of x-rays diffracted from a crystal lattice.
Introduction Within the past decade, an exciting new form of matter, the Bose-Einstein (BE) condensate, has been produced (7). In a typical experiment, a gaseous sample of pure sodium containing several million atoms is cooled to about a microkelvin and trapped in the F = 1, m = -1 hyperfine state by a combination of stationary magnetic fields and laser fields. Under these conditions, the sodium atoms are bosons and essentially all of them enter the lowest energy state of the trapping potential, which is similar to the familiar threedimensional particle-in-a-box system. When two such "boxes" of sodium atoms are prepared, and the atoms are subsequently released from these traps, the F
© 2002 American Chemical Society In Structures and Mechanisms; Eaton, G., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
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condensates expand, overlap, and are found to form beautiful interference fringes reminiscent of a two-slit interference pattern. Indeed, this interference is a direct confirmation of coherence, so it has now become common to describe the BE condensate (plus the associated apparatus) as an "atom laser". The usual theoretical description of a BE condensate is given in terms of a macroscopic wavefunction with a definite phase, the latter supposedly being produced by a process known as "spontaneous symmetry breaking". A definite phase for each member of a pair of condensate wavefunctions is necessary for interference to occur when the condensates overlap. Justification of this description, which is approximate for a finite number of atoms, has been the subject of much theoretical work (i). In this paper I begin with a simple, direct yet novel theoretical description of a pair of BE condensates which is exact for any number of atoms in the limit of an ideal gas (2). I emphasize a formal equivalence to the theory of Bragg diffraction of a beam of x-rays from a crystal. Once this equivalence is recognized, the location of the regions of greatest atom density in the interference pattern follows directly from the analog of the Bragg diffraction law.
The Exact Wavefunction for a Pair of Ideal Gas Bose-Einstein Condensates Consider two electromagnetic traps a and b, one centered at coordinate x and the other at x , respectively. Let the wavefunction for a single atom in the ground state of the electromagnetic trap at a or b be denoted a(x) or b(x) respectively, where χ is the coordinate of the center of mass of the atom. For Ν noninteracting identical atoms with integer total spin, and assuming for simplicity that half of the atoms are in each trap, the exact many-atom wave function is given by a symmetrized product of the orbitals a(x) and b(x): a
b
(1)
where W = N! / (N/2)!(N/2)! is the number of distinct permutations of the a's and b's in the product a(l)a(2)...a(N/2)b(N/2 + l)b(N/2 + 2)...b(N), and P. N
denotes a permutation operator. In this notation a(j) is an abbreviation for either a(Xj) or a(pj) where Xj and pj denote the spatial coordinate or the momentum, respectively, for the jth atom. The orbitals a and b and hence also
In Structures and Mechanisms; Eaton, G., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
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the total wavefunction Ψ have a time dependence which is not denoted explicitly. In obtaining Eq. (1) overlap integrals between the orbitals a and b have been set equal to zero, which does not introduce any noticeable error because the trapped condensates are typically each about 10 μιη wide and their centers separated by about 40 μπι and hence their overlap is effectively zero. The assumption of noninteracting atoms is made here for simplicity and because it is a reasonably good approximation; more accurate results taking into account the atomic interactions can be made using the Gross-Pitaevskii equation (which can be considered "the Hartree-Fock equation for bosons") but the results obtained are essentially the same as in the present treatment. It is important to realize that, although there are no dynamic interactions between them, the atoms are correlated, even when they are spatially separated by a large distance, due to the symmetrization of the many-atom wavefunction In fact it is this correlation which leads to the interference pattern which the theory predicts and which has been observed experimentally.
Analogy with Bragg Diffraction It is now possible to see how interference between the two condensates occurs by drawing an analogy with Bragg diffraction. In order to do this most simply, we consider the wavefunction in momentum space, and assume that the condensate orbitals a and b are identical except for their spatial location. It then followsfromtranslational symmetry that both condensates have the same momentum distribution, and more specifically that the condensate orbitals must have the form a(p) = e ™ /
