The cold atoms are subjected to an intensity modulated optical standing wave (Animation 1).
The Hamiltonian describing the dynamics of the atoms is given by
Where q is scaled position, p is scaled momentum, e is depth of modulation, t is the time, and k is scaled well depth.
The atoms exhibit 2 distinct
types of motion:
1. A regular type motion in which the atoms oscillate in each of the
potential wells of the standing wave in phase with the intensity modulation
(Animation 2).
2. A chaotic type motion in which the atoms bounce inside the wells in
a random manner.
Poincare sections provide
an easy way to understand the dynamics of the system. They are a
stroboscopic plot of the position and momentum of the atoms (where the “strobe”
is applied at a certain time in the modulation of the standing wave).
Islands of regular motion in a sea of chaos are observed. The islands are formed
due to the periodic return of atoms oscillating in phase with the modulation
(Figure 1) and are termed resonances.
Figure 1: Poincare section
The dynamic tunneling occurs between these two groups of atoms (or resonances).
Figure
2: A schematic of the experimental setup
Experimental observation
of resonances is achieved by allowing the resonances to reach the bottom of
the wells of the standing wave ie. when they have maximum velocity, at which
point the standing wave is turned off. After a period of free evolution the
momentum distribution can be resolved experimentally by taking a picture of
the atomic spatial distribution (Animation 3).
Animation 3: Ballistic expansion
Resonances can be represented by a superposition of Floquet states of opposite parity. A superposition of Floquet states can be prepared such that all the atoms are localised in one resonance using phase space preparation methods. Time evolution can lead to a coherent tunneling process into the other resonance. This tunneling process occurs in both momentum and position coordinates in which the atoms tunnel into a motion, which is 180 degrees out of phase with their initial motion. In fact the momentum is completely reversed. Phase space tunneling cannot occur in classical systems of one dimension as atoms cannot cross Kolmogorov, Arnold and Moser (KAM) surfaces.
The first step in the observation of tunneling is preparing the atoms such that they are in a superposition of Floquet states but initially localised in one resonance. This has been achieved by using a Sagnac interferometer configuration for the standing wave. With this type of experimental setup, the optical standing wave can be given a certain velocity relative to the cold atoms. This enables one to localise all the atoms in one resonance. Animation 4 illustrates this tunneling process.
Animation 4: One group of atoms tunneling into the the other group.
Quantum effects usually occur on relatively long time scales. Therefore it is important that major sources of noise, which could destroy coherent quantum processes, have been minimized. Spontaneous emission caused by the optical standing wave is the main cause of concern. To overcome this obstacle, the standing wave must be detuned as far as possible from atomic resonance.
The experimental observation of dynamical tunneling was performed at NIST with W.Phillips' atom optics group.
[1] W.K. Hensinger, A.G. Truscott, B. Upcroft, N.R. Heckenberg and H. Rubinsztein-Dunlop, Atoms in an amplitude-modulated standing wave – dynamics and pathways to quantum chaos, J. Opt. B: Quantum Semiclass. Opt. 2, 659-667, 2000
[2] W.K. Hensinger, H. Häffner, A. Browaeys, N.R. Heckenberg, K. Helmerson, C. McKenzie, G.J. Miburn, W.D. Phillips, S.L. Rolston, H. Rubinsztein-Dunlop, and B. Upcroft, Dynamical Tunneling of Ultracold Atoms, Nature, p. 52, Vol. 412, 2001