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arXiv:physics/0607141 · Physical Review Letters

Time-Frequency Domain Analogues of Phase Space Sub-Planck Structures

Ludmiła Praxmeyer, Piotr Wasylczyk, Czesław Radzewicz, Krzysztof Wódkiewicz

Published Feb 7, 2007• 37 citations
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Abstract

We present experimental data of the frequency-resolved optical gating measurements of light pulses revealing interference features which correspond to sub-Planck structures in phase space. For superpositions of pulses, a small, sub-Fourier shift in the carrier frequency leads to a state orthogonal to the initial one, although in the representation of standard time-frequency distributions these states seem to have a nonvanishing overlap.

DOI
10.1103/physrevlett.98.063901

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We present experimental data of the frequency-resolved optical gating measurements of light pulses revealing interference features which correspond to sub-Planck structures in phase space. For superpositions of pulses, a small, sub-Fourier shift in the carrier frequency leads to a state orthogonal to the initial one, although in the representation of standard time-frequency distributions these states seem to have a nonvanishing overlap.
Exploiting the correspondence between the Wigner distribution function and the frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate the existence of chessboardlike interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase-space representation of compass states. Using superpositions of four electric pulses as the realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exactly chronocyclic correspondence to the sub-Planck structure in the interference pattern appearing for the superposition of two Schr\"odinger-cat-like states in a position-momentum phase space.
A sub-Planck structure from Compass states, composited by four coherent states in superposition, is reported experimentally through the correspondence between Wigner distribution function and frequency-resolved optical gating (FROG) measurement. Demonstration on the chessboard-like patterns with a time-bandwidth product smaller than that of a transform-limited pulse is realized through simultaneously sufficient time and frequency separations among optical pulses. The sub-Planck structure in the interference pattern from Compass states not only manifests itself as a generalized Schödinger's cat state, but also provides the platform to exploit quantum metrology and quantum information processing through ultrafast optics.
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