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arXiv:quant-ph/0412138 · Physical Review Letters

Observation of Nonspreading Wave Packets in an Imaginary Potential

Ralf Stützle, M. C. Goebel, Th. Hörner, E. Kierig, I. Mourachko, Markus K Oberthaler, Maxim A. Efremov, Mikhail Vladimirovich Fedorov, Valery P. Yakovlev, K. A. H. van Leeuwen, W. P. Schleich

Published Sep 8, 2005• 61 citations
arXivHTMLPDF

Abstract

We propose and experimentally demonstrate a method to prepare a nonspreading atomic wave packet. Our technique relies on a spatially modulated absorption constantly chiseling away from an initially broad de Broglie wave. The resulting contraction is balanced by dispersion due to Heisenberg's uncertainty principle. This quantum evolution results in the formation of a nonspreading wave packet of Gaussian form with a spatially quadratic phase. Experimentally, we confirm these predictions by observing the evolution of the momentum distribution. Moreover, by employing interferometric techniques, we measure the predicted quadratic phase across the wave packet. Nonspreading wave packets of this kind also exist in two space dimensions and we can control their amplitude and phase using optical elements.

DOI
10.1103/physrevlett.95.110405

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We propose and experimentally demonstrate a method to prepare a nonspreading atomic wave packet. Our technique relies on a spatially modulated absorption constantly chiseling away from an initially broad de Broglie wave. The resulting contraction is balanced by dispersion due to Heisenberg's uncertainty principle. This quantum evolution results in the formation of a nonspreading wave packet of Gaussian form with a spatially quadratic phase. Experimentally, we confirm these predictions by observing the evolution of the momentum distribution. Moreover, by employing interferometric techniques, we measure the predicted quadratic phase across the wave packet. Nonspreading wave packets of this kind also exist in two space dimensions and we can control their amplitude and phase using optical elements.
We propose and experimentally demonstrate a method to prepare a non-spreading atomic wave packet. Our technique relies on a spatially modulated absorption constantly chiselling away from an initially broad de-Broglie wave. Heisenberg's uncertainty principle puts a limit to this contraction and determines the final complex Gaussian form of the packet. Experimentally we confirm the predicted formation process of the wave packet by observing the evolution of the momentum distribution. Moreover by employing interferometric techniques we measure the predicted quadratic phase across the wave packet.
The spreading or nonspreading of wave packets in one-dimensional linear and quadratic potentials can readily be determined by solving Ehrenfest’s equations. The results for quadratic potentials can be understood in simple intuitive terms by using quantum distribution functions in phase space which, like classical distributions, rotate clocklike in time once per motional cycle. For more complicated situations, such as potentials with quantum energy spacings which are integral multiples of a basic unit, much can be learned instead by studying the quantum recurrence of their time-dependent wave functions or probability densities.
The formation of two-dimensional nonspreading atomic wave packets produced in the interaction of a beam of two-level atoms with two standing light waves polarised in the same plane is considered. The mechanism providing a dispersionless particle dynamics is the balance of two processes: a rapid decay of the atomic wave function away from the field nodes due to spontaneous transitions to nonresonance states and the quantum broadening of the wave packets formed in the close vicinity of field nodes. Coordinate-dependent amplitudes and phases of the two-dimensional wave packets were found for the j g =0 ↔ j e =1 transition.
We show the method for constructing nonspreading wave packets whose shape and motion can be general. We analyze the time evolution of nonspreading wave packets by decomposing the Hamiltonian into two parts. Of the two, one changes the instantaneous state, the other does not. Through this decomposition, the time evolution operator is shown to be effectively a spatial shifting operator. This explains why nonspreading wave packets can be nonspreading. And we show that the part of the Hamiltonian which changes the instantaneous state governs the motion of the nonspreading wave packets.
We report on the imaging of the in situ spatial distribution of deterministically prepared single-atom wave packets as they expand in a plane, finding excellent agreement with the scaling dynamics predicted by the Schrödinger equation. Our measurement provides a direct and quantitative observation of the textbook free expansion of a one-particle Gaussian wave packet, which we believe has no equivalent in the existing literature. Second, we utilize these expanding wave packets as a benchmark to develop a protocol for the controlled projection of a spatially extended wave function from continuous space onto the sites of a deep optical lattice and subsequent single-atom imaging using quantum gas microscopy techniques. By probing the square modulus of the wave function for various lattice ramp-up times, we show how to obtain a near-perfect projection onto lattice sites. Establishing this protocol represents a crucial prerequisite to the realization of a quantum gas microscope for continuum physics. The method demonstrated here for imaging a wave packet whose initial extent greatly exceeds the pinning lattice spacing, is designed to be applicable to the many-body wave function of interacting systems in continuous space, promising a direct access to their microscopic properties, including spatial correlation functions up to high order and large distances.