Measuring sub-Planck structural analogues in chronocyclic phase space

Phase space quasi-probability distributions of certain quantum states reveal structure on a scale that is small compared to the Planck area. Using an analog between the wavefunction of a single photon and the electric field of a classical ultrashort optical pulse we show that spectral shearing interferometry enables measurement of such structure directly, thereby extending an idea of Krzysztof Wódkiewicz and others. In particular, we use multiple-shear spectral interferometry to fully characterize a pulse consisting of two subpulses which are temporally and spectrally disjoint, without a relative-phase ambiguity. This enables us to compute the Wigner distribution of the pulse. This spectrographic representation of the pulse field features fringes that are tilted with respect to both the timeand frequency axes, showing that in general the shortest sub-Planck distances may not be in the directions of the canonical (and easily experimentally accessible) directions. Further, independent of this orientation, evidence of the sub-Planck scale of the structure may be extracted directly from the measured signal. 1. Classical and quantum wave fields The electromagnetic field is a proxy for the wavefunction of a single photon [1]. That is, a spatiotemporally localized optical pulse field plays the role of a probability amplitude (with suitable normalization) for the measurement of a single quantum of light [2]. Interference phenomena that are observed with classical fields therefore bear a close relationship to quantum interference phenomena for single particles, and this provides a route to exploring quantum phase space structures in a simple way. It is not perhaps surprising that there is a similarly close relationship between the characterization of a classical ultrafast optical pulse (i.e. the measurement of its electric field) and the quantum state tomography of a single photon, and this analog may be used to apply recently developed measurement techniques in ultrafast optics to the study of quantum phase space. Krzysztof Wódkiewicz exploited this analog to effect in a number of directions, most recently in a study of so-called subPlanck phase-space structures discovered by Zurek [3]. The key elements of state tomography for localized photons are the same as those required to fully characterize the electric field of a short optical pulse. The central idea for this is related to the Wódkiewicz-Eberly notion of the “physical spectrum”[4]. This is simply the time-resolved intensity of a spectrally filtered pulse,