We’ve long since grown used to the idea that atoms in a double potential well can go back and forth between wells, even though classically an atom with insufficient energy is precluded from surmounting the energy barrier between the two compartments. We should not be too surprised then to learn that an atom executing one type of regular motion can suddenly be found to be moving 180° out of phase with its initial motion: At a point in the motion when the atom should be moving to the right, it’s seen to be moving to the left. That’s the idea behind dynamical tunneling, or the hopping between separate, stable regions in phase space. Although there’s been some evidence for dynamical tunneling in molecular systems, it has now been seen very directly in two new experiments on ultracold atoms. 1,2
Eric Heller of Harvard University says that the two experiments highlight the dramatic quantum manipulation of ultracold gases that is now possible, and raise new issues in quantum dynamics. Ultracold atomic systems have been used for a number of years to explore the quantum behavior of nonlinear dynamical systems (see the news story in Physics Today June 1995, page 18, and the article by Mark Raizen, Christophe Salomon, and Quan Niu in Physics Today July 1997, page 30). Whereas many studies are done on systems that are totally chaotic, the two new experiments enter less well-charted territory: a mixed region where both stable motion and chaos are present. One of the new experiments 2 claims evidence that the chaos has actually facilitated the dynamical tunneling.
Dynamical tunneling was demonstrated in a Bose–Einstein condensate (BEC) by William Phillips and his coworkers at NIST in Gaitherburg, Maryland, in collaboration with Halina Rubinsztein-Dunlop, Gerard Milburn, and their colleagues at the University of Queensland in Brisbane, Australia. 1 It was also seen in a system of ultracold atoms by Raizen and his group at the University of Texas at Austin. 2 (Tunneling of a BEC through an energy barrier was already seen several years ago. 3 )
Symmetric modes
In each of the new experiments, the ultracold atoms, all having nearly identical momenta, were placed in a one-dimensional optical lattice; such a lattice essentially provides a washboardlike energy potential. The heights of the potential walls were temporally modulated in a periodic way. In the NIST–Queensland experiment, the height was varied between finite values so that atoms in the BEC sloshed back and forth within each well. In the Texas experiment, the walls were periodically driven to zero, so that the atoms could skim over them, moving like a traveling wave consistently either to the right or to the left.
The two research groups always sampled the momenta of the atoms at the same point in their periodic motion. That way, the atoms were expected to be always moving with the same velocity. The NIST–Queensland experimenters, for example, sampled the momenta when the atoms were in the lowest part of their swing. Even when the atoms started out moving, say, from left to right, the researchers found that, after a certain number of modulation periods, the atoms were found to be going from right to left, signaling a 180° shift in the motion. The atoms shifted from right-going to left-going motions at regular intervals.
Both left-moving and right-moving modes of oscillation are allowed. Once set in motion in one mode, however, the system cannot switch to the other mode, at least not classically. The two modes are separated not by an energy barrier but by a barrier in some other constant of the system. The transition from one mode to another was termed dynamical tunneling in 1981 by Heller and Michael Davis (now at Argonne National Laboratory). 4
Dynamical tunneling
One example of dynamical tunneling is a rotating molecule like formaldehdye (H2CO). This molecule can flip between two states, with the line from the carbon to the oxygen atom either parallel or antiparallel to the molecule’s angular momentum vector. One can’t directly photograph the molecule flipping between these two states, but one can infer that such flipping occurs from the splitting of the rotational energy levels: Dynamical tunneling splits the energy levels just as tunneling through an energy barrier splits the energy levels of a particle in a potential well.
To picture dynamical tunneling, consider the phase space plot shown in figure 1. The diagram represents a snapshot of the phase space for classical motion in a system driven by the same Hamiltonian as in the NIST–Queensland experiment. It is not a plot of the continuous motion of the atoms, but rather a stroboscopic picture, with images taken once every period of the motion.
Figure 1. Phase-space diagram of momentum p versus position q shows islands of stable motion (elliptical curves) in a sea of chaos (dots). Diagram is a stroboscopic sampling of atoms in a periodic, modulated system. Dynamic tunneling takes a system from the island near +p 0 (enlarged in the inset) to the bottom island near -p 0.
Figure 1. Phase-space diagram of momentum p versus position q shows islands of stable motion (elliptical curves) in a sea of chaos (dots). Diagram is a stroboscopic sampling of atoms in a periodic, modulated system. Dynamic tunneling takes a system from the island near +p 0 (enlarged in the inset) to the bottom island near -p 0.
The closed curves shown above and below the p = 0 axis are the regions of phase space occupied by the atoms that move in phase with the driving modulation. Atoms in the top island (see inset) are moving to the right when they are sampled at the bottom of the well, and those in the bottom island are moving to the left. (The central island corresponds to motions with very small amplitudes, which were not excited in the NIST–Queensland experiment.) The dots filling the area between islands indicate the phase-space positions of particles in chaotic motion.
Theorists have predicted that, in such mixed regions of stable and chaotic motion, the presence of chaos can actually facilitate the dynamical tunneling. 5 Normally the probability of tunneling is determined by the overlap of the wavefunctions on the two islands. But if a third state in the chaotic region becomes involved, the tunneling probability can be enhanced. Theorists Steven Tomsovic, Dennis Ullmo, and Oriol Bohigas termed such a process “chaos-assisted tunneling.” 6 (Tomsovic is at Washington State University and the other two are from the Laboratoire de Physique Théorique et Modeles Statistiques in Orsay, France.) One nonrigorous way to view the process is to imagine that the system moves coherently, little by little, from one island of stability to another by traveling along chaotic trajectories that closely approach the two islands. Raizen commented that chaos-assisted tunneling can have applications in other fields, including the tunneling transport of electrons, coupling of light, and coupling of molecular rotations.
The evidence
Raizen and his colleagues produced evidence for dynamical tunneling by selecting, from a sample of a million ultracold cesium atoms, only about 1% having a very narrow momentum spread about a desired value. (Only atoms whose rough positions and momenta fall within the stable regions will tunnel.) The experimenters then put the selected atoms in the lowest energy levels of a static optical lattice. To start the oscillatory motion, they suddenly shifted the potential relative to the atoms; the effect is similar to lifting a ball from the bottom to the side of a bowl and releasing it to initiate a back-and-forth motion.
Then the Texas team ramped up the amplitude of a time-dependent modulation of the lattice’s potential wells, causing the atoms to move in one direction, much like a traveling wave. To determine the direction of motion of the atoms, the researchers turned off the trapping lattice beams and measured the atomic momenta every 40 µs, twice the period of the driving modulation.
As shown in figure 2, the population of atoms shifted in time from a distribution peaked at a momentum of +8 (2ħk) to one peaked at −8 (2ħk), where k is the wave number of the laser beams that create the optical lattice. In other words, the atoms were sometimes moving to the left, sometimes to the right, and sometimes in superpositions of the two motions. The atoms tunneled from one mode to the other at intervals of about 360 µs until the oscillations are damped out.
Figure 2. Evolution of momentum distribution of ultracold atoms in a one-dimensional potential, sampled every two modulation periods. By 160 µs, a significant fraction of the atoms have tunneled from a state with positive momentum to one with negative momentum; by 360 µs, they have tunneled back again.
Figure 2. Evolution of momentum distribution of ultracold atoms in a one-dimensional potential, sampled every two modulation periods. By 160 µs, a significant fraction of the atoms have tunneled from a state with positive momentum to one with negative momentum; by 360 µs, they have tunneled back again.
Raizen and his Texas collaborators verified that their system exhibits certain characteristics of dynamical tunneling: Specifically, they found that the tunneling is suppressed when the system is not initially centered on the island of stability and when the symmetry is broken between two stable islands. Both effects confirm the prediction that dynamical tunneling is highly sensitive to the parameters of the system.
The NIST and Queensland collaborators demonstrated dynamical tunneling in a BEC, in which all atoms, by definition, are in the same momentum state. Following a similar procedure but with a different amplitude of modulation, this group also saw the atoms oscillate between modes that feature motions that are 180° out of phase with one another, as seen in figure 3. The discrete diffraction peaks in the figure result from interference between atoms from different cells of the optical lattice; most atoms are found in the peaks centered at 4 and 6 ħk. By varying the system parameters, Phillips and his colleagues observed as many as seven coherent tunneling oscillations, as well as tunneling periods that varied by as much as a factor of two.
Figure 3. Momentum distribution of a Bose–Einstein condensate oscillating within the wells of an optical lattice. The diffraction peaks result from interference between atoms from different wells. Sampled after 0.25 (front), 2.25 (middle), and 5.25 (back) modulation periods, the distributions show the evolution from a negative momentum to a positive momentum state.
Figure 3. Momentum distribution of a Bose–Einstein condensate oscillating within the wells of an optical lattice. The diffraction peaks result from interference between atoms from different wells. Sampled after 0.25 (front), 2.25 (middle), and 5.25 (back) modulation periods, the distributions show the evolution from a negative momentum to a positive momentum state.
Chaos-assisted tunneling
The Texas group believes that its experiment provides evidence of chaos-assisted tunneling. If chaos-assisted tunneling is involved, one would expect the tunneling rate to be faster when chaos is present than when it is absent. Raizen and company can remove the chaos by turning off the driving modulation of their optical lattice; the resulting motion is simply Bragg scattering. Dynamical tunneling should be manifested by above-the-barrier reflection, or reflection of atoms having a greater energy than the potential walls of the optical lattice. Such reflection is estimated to take about 1 s, much longer than the 360-µs period observed for dynamical tunneling in the driven system. Indeed, the experimenters saw no sign of tunneling oscillations in the undriven system over an interval of 1600 µs.
Raizen and his colleagues also measured the momentum distribution in time steps of 1 µs to study in detail how the tunneling proceeds. They saw a periodically enhanced probability in the chaotic region near p = 0, which they believe is evidence that a third (chaotic) state is involved in the transport between the two islands.
Dynamical and chaos-assisted tunneling have been observed in an electromagnetic analog of these quantum mechanical systems: a microwave cavity. 7 They also underlie the fundamental processes in chaos-assisted ionization and may be responsible for the decay of superdeformed nuclei.
Tomsovic thinks that both of the new experiments show clear evidence for dynamical tunneling. Although they are suggestive of chaos-assisted tunneling, he would like to see more evidence, such as determinations of the sensitivity of the tunneling time to variations in the system parameters.
