Interest Group Stream: Australian Optical Society

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

1Centre for Laser Science, Department of Physics, The University of Queensland, Brisbane QLD 4072, Australia,

2Laboratoire de Mathematique et de Physique, Tours, France

3Laboratoire Kastler-Brossel, Université P. et M. Curie, Paris, France

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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] W. H. Zurek, Nature 412 (16 Aug), 712 (2001), N. L. Balazs and A. Voros, Ann. Phys. 199, 123 (1990), M. V. Berry, Proc. R. Soc. Lond A 413, 183 (1987)

[2] W. K. Hensinger et al., Nature 412 (5 July), 52 (2001)

[3] J. Goldstone and R. L. Jaffe, Phys. Rev. B 45, 14100 (1992)