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Interest group stream: Australian Optical Society
Quantum magnification of classical sub-Planck phase space features
W. K. HENSINGER1, A. MOUCHET2, D. DELANDE3, N. R. HECKENBERG1 AND H.
RUBINSZTEIN-DUNLOP1
1
Centre for Laser Science, Department of Physics, The University of Queensland, Brisbane QLD 4072,
Australia,
2
Laboratoire de Mathematique et de Physique, Tours, France
3
Laboratoire Kastler-Brossel, Université P. et M. Curie, Paris, France
email: [email protected]
To understand the relationship between quantum mechanics and classical physics a crucial question to be
answered is how distinct classical dynamical phase space features translate into the quantum picture. This
problem becomes even more interesting if these phase space features occupy a much smaller volume than
in a phase space spanned by two non-commuting variables such as position and momentum. The question
whether phase space structures in quantum mechanics associated with sub-Planck scales have physical
signatures has recently evoked a lot of discussion [1]. Here we will show that sub-Planck classical
dynamical phase space structures, for example regions of regular motion, can give rise to states whose
phase space representation is of size  or larger. This is illustrated using period-1 regions of regular motion
(modes of oscillatory motion of a particle in a modulated well) whose volume is distinctly smaller than
Planck’s constant. They are magnified in the quantum picture and appear as states whose phase space
representation is of size  or larger. Cold atoms provide an ideal test bed to probe such fundamental aspects
of quantum and classical dynamics. In the experiment a Bose-Einstein condensate is loaded into a far
detuned optical lattice. The lattice depth is modulated resulting in the emergence of regions of regular
motion surrounded by chaotic motion in the phase space spanned by position and momentum of the atoms
along the standing wave [2]. Sub-Planck scaled phase space features in the classical phase space are
magnified and appear as distinct broad peaks in the atomic momentum distribution. The corresponding
quantum analysis shows states of size which can be associated with much smaller classical dynamical
phase space features. This effect may considered as the dynamical equivalent of the Goldstone and Jaffe
theorem [3] which predicts the existence of at least one bound state at a bend in a two or three dimensional
spatial potential.
[1]
[2]
[3]
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R. Soc. Lond A 413, 183 (1987)
W. K. Hensinger et al., Nature 412 (5 July), 52 (2001)
J. Goldstone and R. L. Jaffe, Phys. Rev. B 45, 14100 (1992)