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Physical Review A

Reply to “Comment on ‘Quasicontinuum modeling of photoassociation’ ”

Juha Javanainen, Matt Mackie

Published Jan 4, 2002• 13 citations
Publisher

Abstract

In the preceding Comment [Phys. Rev. A 65, 027401 (2002)], the authors maintain that we have used incorrect initial conditions in our calculations on photoassociation of a thermal, nondegenerate gas [Phys. Rev. A 58, R789 (1998)], and that our conclusion that stimulated Raman adiabatic passage (STIRAP) is absent in two-color photoassociation is therefore invalid. We refute the technical argument of the Comment. Nonetheless, corresponding to the fact that in quantum mechanics the decomposition of a mixed state into pure states is not unique, the question of the presence or absence of STIRAP turns out to be a matter of semantics.

DOI
10.1103/physreva.65.027402

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In the preceding Comment [Phys. Rev. A 65, 027401 (2002)], the authors maintain that we have used incorrect initial conditions in our calculations on photoassociation of a thermal, nondegenerate gas [Phys. Rev. A 58, R789 (1998)], and that our conclusion that stimulated Raman adiabatic passage (STIRAP) is absent in two-color photoassociation is therefore invalid. We refute the technical argument of the Comment. Nonetheless, corresponding to the fact that in quantum mechanics the decomposition of a mixed state into pure states is not unique, the question of the presence or absence of STIRAP turns out to be a matter of semantics.
We maintain that the reservations against the use of stimulated Raman adiabatic photoassociation in a nondegenerate gas, expressed by Mackie and Javanainen in a recent manuscript [Phys. Rev. A 60, 3174 (1999)], do not account for the coherence induced by the photoassociating laser. The authors treat the problem of two-color stimulated photoassociation using a set of decoupled three-mode problems, each containing a box eigenstate and the two bound states. We argue that such decomposition is not justified, since in the limit of a very large box all three-mode problems are, in fact, coherently coupled. The three-mode condition can only be met with a set of wave packets whose bandwidth matches the spectral width of the photoassociating pulse. Using this basis set, we show that by the authors' own formalism; stimulated Raman photoassociation is a viable process in the thermal ensemble.
We analyze coherent two-color photoassociation of a Bose-Einstein condensate, focusing on stimulated Raman adiabatic passage (STIRAP) in free-bound-bound transitions from atoms to molecules. This problem raises an interest because STIRAP has been predicted to be absent in the nondegenerate case [Javanainen and Mackie, Phys. Rev. A 58, R789 (1998)]. Nevertheless, we find that Bose stimulation enhances the free-bound dipole matrix element for an atomic condensate, and photoassociative STIRAP turns out to be a viable mechanism for converting an atomic condensate to a molecular condensate with near-unit efficiency.
Collective two-color photoassociation of a freely interacting 87Rb^{87}\mathrm{Rb}87Rb Bose-Einstein condensate is theoretically examined, focusing on stimulated Raman adiabatic passage (STIRAP) from an atomic to a stable molecular condensate. In particular, Drummond et al. [Phys. Rev. A. 65, 063619 (2002)] have predicted that particle-particle interactions can limit the efficiency of collective atom-molecule STIRAP, and that optimizing the laser parameters can partially overcome this limitation. We suggest that the molecular conversion efficiency can be further improved by treating the initial condensate density as an optimization parameter.
We examine photoassociation by using a quasicontinuum to describe the colliding atoms. The quasicontinuum system is analyzed using methods adapted from the theory of laser spectroscopy and quantum optics, and a continuum limit is then taken. In a degenerate gas the equilibrium probability of photoassociation may be close to unity. In the continuum limit, for a thermal atomic sample, the stimulated Raman adiabatic passage (STIRAP) mechanism cannot be employed to eliminate unwanted spontaneous transitions.
In the preceding Comment, the authors suggest that the molecular conversion efficiency in atom-molecule stimulated Raman adiabatic passage can be improved by lowering the initial atomic density, which in turn requires longer pulse durations to maintain adiabaticity. Apart from the fact that the mean-field approximation becomes questionable at low densities, we point out that a low-density strategy with longer pulses has several problems. It generally requires higher pulse energies and increases radiative losses. We also show that even within the approximations used in the Comment, their example leads to no efficiency improvement compared to our high-density case. In a more careful analysis including radiative losses neglected in the Comment, the proposed strategy gives almost no conversion owing to the longer pulse durations required.
This paper begins with the premise that stimulated Raman adiabatic passage (STIRAP) from one state to another by coupling to an intermediate state necessarily requires some population of that intermediate state. The usual view of the process depends on having no population in the intermediate state, but a simple argument from the time-dependent Schr\"odinger equation shows that such adiabatic transfer cannot occur. From another perspective, removing the time dependence of the Hamiltonian by making the rotating frame transformation cancels the optical frequency oscillations but does not account for the envelopes of the pulses commonly used for STIRAP. By assuming a small but finite intermediate state amplitude ${\ensuremath{\varepsilon}}_{1}$ in the wave function, the propagator for the evolution between initial and final states is derived. Applying the propagator to the initial state produces a single compact formula for the final intermediate-state population that is readily evaluated for an arbitrary choice of laser pulse parameters. Several plots for various experimental conditions show that under the conditions normally used for STIRAP experiments the effects are typically a few percent or less, but in the end this effect puts an ultimate limit on total efficiency.