Experimental verification of the Heisenberg uncertainty principle for fullerene molecules

Abstract

The Heisenberg uncertainty principle for material objects is an essential corner stone of quantum mechanics and clearly visualizes the wave nature of matter. Here we report a demonstration of the Heisenberg uncertainty principle for the most massive, complex and hottest single object so far, the fullerene molecule C70 at a temperature of 900 K. We find a good quantitative agreement with the theoretical expectation: ∆x × ∆p = h, where ∆x is the width of the restricting slit, ∆p is the momentum transfer required to deflect the fullerene to the first interference minimum and h is Planck's quantum of action.Complementarity is one of the essential paradigms of quantum mechanics [1]. Two quantities are mutually complementary in that complete (or partial) knowledge of one implies the complete (or partial) uncertainty about the other and vice versa. The most generally known case is the complementarity between position and momentum, as expressed quantitatively in the Heisenberg uncertainty principle ∆x × ∆p ≥h/2. For neutrons the uncertainty relation has been demonstrated already back in 1966 by Shull [3]. Following the growing experimental efforts in atom optics during the last decade, the uncertainty principle has shown up implicitly in several experiments and has also been explicitly investigated in both the spatial [4] and in the time domain [5].While being a physical phenomenon of interest in its own right, the complementarity between momentum and position is also an important factor for practical purposes: for example it is applied for the preparation of transverse coherence in all experiments using collimated beams, a fact that can be mathematically phrased using the van Cittert-Zernike theorem [6][7][8].There are good reasons to believe that complementarity and the uncertainty relation will hold for all sufficiently well isolated objects of the physical world and that these quantum properties are generally only hidden by technical noise for larger objects. It is therefore interesting to see how far this quantum mechanical phenomenon can be experimentally extended to the macroscopic domain.Here we report on an experiment investigating for the first time in a quantitative way the uncertainty relation upon diffraction at a single slit for a molecule as complex, massive and hot as the fullerene C 70 (m = 840 amu) at an internal and translational temperature of 900 K.It is well known that the limith/2 of the uncertainty relation ∆x × ∆p ≥h/2 is only reached for particular wave packets, for example of the Gaussian type. Evidently, the wave packet after passage through a rectangular slit is very different from this minimal uncertainty shape. This is also reflected in the far field distribution which is described by the well known sinc-function rather than a Gaussian. It is therefore a matter of definition and convenience which quantities to take as a measure of the position and momentum uncertainty in our case. Obviously, for a wave traversing a slit, one can take the slit width to be the measure of the spatial uncertai...

Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement

“…Our work addresses a cornerstone relation in quantum mechanics and, to the best of our knowledge, is the first to test one of its entropic versions. In the past, experiments have come close to the original uncertainty limit [13][14][15][16], but did not involve entangled quantum systems. We also illustrate the practical usefulness of the new inequality by applying it as an effective entanglement witness.…”

An atomically thin matter-wave beamsplitter

“…In diffraction experiments at a single slit, the relationship ∆x · ∆p diff ≥ 0.89h holds [30], when ∆x = s designates the width of the single slit and ∆p diff the full width at half maximum (FWHM) of the diffraction curve. The momentum uncertainty ∆p diff ≥ 0.89h/s is the FWHM of the envelope to all diffraction orders of the grating.…”

Quasi-optical Systems Based on Periodic Structures

“…This famous phase is directly related to the covariance between momentum and position and since for the "free particles" we are considering xx pp 2 xp constant we see that Gouy phase can be indirectly measured from the coordinate and momentum variances, quantities a lot easier to measure than covariance between x and p. On the other hand, as far as free atomic particles are concerned, experiments elaborated to test the uncertainty relation (Nairz et al, 2002) will reveal to us the matter wave equivalence of Gouy phase. Unfortunately the above quoted experiment was not designed to determine the phase and that is the reason why, so far, we have only an indirect evidence of the compatibility of theory and experiment.…”