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Physical Review Letters

Atomic Landau-Zener Tunneling and Wannier-Stark Ladders in Optical Potentials

Qian Niu, Xian-Geng Zhao, G. A. Georgakis, Mark G. Raizen

Published Jun 10, 1996• 190 citations
Publisher

Abstract

We calculate the quantum motion of ultracold atoms in an accelerating optical potential, and show how they may be used to observe Landau-Zener tunneling and Wannier-Stark ladders, two fundamental quantum effects in solid state physics. The optical potential is spatially periodic, yielding an energy spectrum of Bloch bands for the atoms. The acceleration provides an inertial force in the moving frame, emulating an electric force on Bloch electrons.

DOI
10.1103/physrevlett.76.4504

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Quantum Electronics and Laser Science Conference•Jan 1, 1997•0 citations
We calculate the quantum motion of ultracold atoms in an accelerating optical potential, and show how they may be used to observe Landau-Zener tunneling and Wannier-Stark ladders, two fundamental quantum effects in solid state physics. The optical potential is spatially periodic, yielding an energy spectrum of Bloch bands for the atoms. The acceleration provides an inertial force in the moving frame, emulating an electric force on Bloch electrons.
We report the first observation of atomic Wannier-Stark ladders in a potential formed by an accelerating standing wave of light.1 This is a fundamental effect of quantum transport in optical lattices. Wannier-Stark ladder resonances were predicted to occur in the band structure of an electron moving in a periodic lattice with a dc electric field, and have been recently observed in superlattices. Our work is the first spectroscopic study of Wannier-Stark ladders free from complications such as the impurities, lattice vibrations, and multi-particle interactions present in solids.
The early quantum theory of electrical conductivity in crystal lattices by Bloch and Zener [1,2] led to the striking prediction that a homogeneous static electric field induces an oscillatory rather than uniform motion of the electrons. These so-called Bloch oscillations have never been observed in natural crystals because the scattering time of the electrons by the lattice defects is much shorter than the Bloch period. In semiconductor superlattices the larger spatial period leads to a much shorter Bloch period (,600 fs) and Bloch oscillations have recently been observed through the emission of THz radiation by the electrons [3]. Here we present Bloch oscillations of atoms in the fundamental energy band of a periodic optical potential. We directly measure the atomic momentum distribution evolving in time under the influence of a constant inertial force for various potential depths. We experimentally observe oscillation periods in the millisecond range as well as positive and negative effective masses. Bloch oscillations are a pure quantum effect which can be explained in a simple one-dimensional model. The periodicity of the lattice (period d) leads to a band structure (Fig. 1) of the energy spectrum of the particle and the corresponding eigenenergies Ensqd and eigenstates jn, ql (Bloch states) are labeled by the discrete band index n and the continuous quasimomentum q; Ensqd and jn, ql are periodic functions of q with period 2pyd and q is conventionally restricted to the first Brillouin zone g2pyd, pydg [4]. Under the influence of a constant external force F, weak enough not to induce interband transitions, a given Bloch state jn, qs0dl evolves (up to a phase factor) into the state jn, qstdl according to qstd ­ qs0d 1 Ftyh . (1) This evolution is periodic with a period tB ­ hyjFjd corresponding to the time required for the quasimomentum to scan a full Brillouin zone. The mean velocity in jn, qstdl
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~Received 30 October 1996! We observe quantum tunneling of cold sodium atoms from an accelerating one-dimensional standing wave of light. Atoms are trapped in a far-detuned standing wave that is accelerated for a controlled duration. For sufficiently large values of the acceleration, we observe an exponential decay in the number of atoms that remain trapped as a function of the interaction time. We show that this loss is due to quantum tunneling, and compare the decay rates with Landau-Zener theory. We also observe oscillations in the tunneling rate as a function of the acceleration which are due to quantum interference effects. @S1050-2947~97!50702-5# PACS number~s!: 03.75.Be, 32.80.Pj, 42.50.Vk Tunneling in the center-of mass motion of atoms should become an important process at the low temperatures that are now achieved with laser cooling. This can appear, for example, as a loss mechanism for atoms confined in an optical lattice constructed from interfering beams of light. While atomic motion in these structures has become an active area of research @1,2#, tunneling has been obscured by spontaneous scattering @3#. An alternative atom optics system where tunneling can be significant has appeared in our efforts to develop a cold atomic beam for interferometry. Our approach has been to launch laser-cooled atoms in the potential created by an accelerating, far-detuned, standing wave of light. This system was previously used to study Wannier-Stark ladders @4# and Bloch oscillations @5#. The results reported in this Rapid Communications provide an observation of tunneling in atom optics, and the high degree of experimental control has enabled a quantitative comparison with theory. We consider atomic motion in an accelerating standing wave of light. For sufficiently large detuning from atomic resonance, the atoms remain in their internal ground state and experience an effective one-dimensional potential given by V0cos@2kL(x2at 2 /2)#, where k L52p/l L is the wave number, and a is the acceleration of the standing wave @6#. The well depth V0 is the amplitude of the optical dipole potential. It is proportional to the laser intensity and inversely proportional to detuning from atomic resonance. We neglect variations of the potential in the two transverse directions, which is justified for beams that are sufficiently