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10 Time-Periodic Quantum Systems
10.5 Dynamical Tunneling in Atom Optics Experiments
Floquet theory can be used to compute the tunneling frequencies observed in cold
atom optics experiments for which the underlying classical phase space has regions
dominated by chaos. In the experiments of Steck et al. (2001, 2002), which we
consider here, a cluster of cesium atoms in the gas phase, is cooled to a temperature
T ≈ 10 −7 K. The density is low enough that interactions between the atoms
can be neglected. These ultracold cesium atoms are allowed to interact with two
counterpropagating laser beams that create a periodically modulated standing wave
of light. The atoms can be treated as two-level systems with energy spacing ¯
hω 0 ,
and the laser beams are detuned away from resonance with these two energy levels.
Interaction of the atoms with the standing wave of light stimulates absorption
and then emission of a photon in one of two dominant modes. In the first mode, the
photon is absorbed and then emitted in the incident direction (transmitted through
the atom), causing no net recoil of the atom. In. the second mode, the photon is
absorbed and then emitted in a direction 180 o from the incident direction (reflected
from the atom), resulting in a net atomic recoil of 2 ¯
hk L , where k L is the wave vector
of the standing wave of light and ¯
h is Planck’s constant. It is this second process that
dominates the dynamics observed in the experiments.
When the laser detuning δ L = ω 0 − ω L is large, resonant absorption can be
neglected and the atoms have a large probability of being in their ground state on
time-scales of importance for the experiment. Under these conditions, the dynamics
is determined by the center-of-mass motion of the atoms. It was first shown by
Graham et al. (1992) that simple time-periodic Hamiltonians describe the dynamics
of such systems. In the subsections below, we show that Floquet analysis gives
excellent agreement with experiment (Luter and Reichl 2002).
10.5.1 Hamiltonian for Atomic Center-of-Mass
In the experiments reported in Steck et al. (2001, 2002), the dynamical evolution
of noninteracting cold cesium atoms in a periodically modulated standing wave of
light was measured. The Hamiltonian used to model the center-of-mass motion of
the cesium atoms (in S.I. units) is
H =
p 2
2m
− 2V o cos
2
ω m t
2
cos (2k L x) ,
(10.38)
where ˆ
p, ˆ
x, and m are the momentum, position, and mass, respectively, of a cesium
atom, ω m =
2π
T is the modulation frequency of the standing wave of light, and
V o =
¯
hh 2
max
8δ L
. Here max = 2E 0 d/ ¯
h is the Rabi frequency, E 0 is the electric field
strength, and d is the dipole moment of cesium. As we will see below, this system
has three primary resonances.
10 Time-Periodic Quantum Systems
10.5 Dynamical Tunneling in Atom Optics Experiments
Floquet theory can be used to compute the tunneling frequencies observed in cold
atom optics experiments for which the underlying classical phase space has regions
dominated by chaos. In the experiments of Steck et al. (2001, 2002), which we
consider here, a cluster of cesium atoms in the gas phase, is cooled to a temperature
T ≈ 10 −7 K. The density is low enough that interactions between the atoms
can be neglected. These ultracold cesium atoms are allowed to interact with two
counterpropagating laser beams that create a periodically modulated standing wave
of light. The atoms can be treated as two-level systems with energy spacing ¯
hω 0 ,
and the laser beams are detuned away from resonance with these two energy levels.
Interaction of the atoms with the standing wave of light stimulates absorption
and then emission of a photon in one of two dominant modes. In the first mode, the
photon is absorbed and then emitted in the incident direction (transmitted through
the atom), causing no net recoil of the atom. In. the second mode, the photon is
absorbed and then emitted in a direction 180 o from the incident direction (reflected
from the atom), resulting in a net atomic recoil of 2 ¯
hk L , where k L is the wave vector
of the standing wave of light and ¯
h is Planck’s constant. It is this second process that
dominates the dynamics observed in the experiments.
When the laser detuning δ L = ω 0 − ω L is large, resonant absorption can be
neglected and the atoms have a large probability of being in their ground state on
time-scales of importance for the experiment. Under these conditions, the dynamics
is determined by the center-of-mass motion of the atoms. It was first shown by
Graham et al. (1992) that simple time-periodic Hamiltonians describe the dynamics
of such systems. In the subsections below, we show that Floquet analysis gives
excellent agreement with experiment (Luter and Reichl 2002).
10.5.1 Hamiltonian for Atomic Center-of-Mass
In the experiments reported in Steck et al. (2001, 2002), the dynamical evolution
of noninteracting cold cesium atoms in a periodically modulated standing wave of
light was measured. The Hamiltonian used to model the center-of-mass motion of
the cesium atoms (in S.I. units) is
H =
p 2
2m
− 2V o cos
2
ω m t
2
cos (2k L x) ,
(10.38)
where ˆ
p, ˆ
x, and m are the momentum, position, and mass, respectively, of a cesium
atom, ω m =
2π
T is the modulation frequency of the standing wave of light, and
V o =
¯
hh 2
max
8δ L
. Here max = 2E 0 d/ ¯
h is the Rabi frequency, E 0 is the electric field
strength, and d is the dipole moment of cesium. As we will see below, this system
has three primary resonances.