10.5 Dynamical Tunneling in Atom Optics Experiments
359
In these experiments, because of the very low temperature and the fact that the
atomic recoil is quantized in steps of 2 ¯
hk L , the allowed momentum states appear
to be quantized in integer multiples of 2 ¯
hk L with very small spread about those
values. This allows us to use Floquet analysis for this system rather than FloquetBloch analysis (Mouchet et al. 2001) as one might expect from the form of the
Hamiltonian in Eq. (10.38). Because of the quantization of the recoil momentum,
an atom cannot transition through the continuous range of energies in the energy
bands allowed by the Hamiltonian in Eq. (10.38), at least for the timescales of the
experiment.
Let us now write the Hamiltonian in dimensionless form. We perform a scaling
that explicitly quantizes the momentum in units of 2 ¯
hk L (Luter and Reichl 2002).
Let
φ = 2k L x,
p = 2 n ¯
hk L , ω = ω m /4ω r , ω r = ¯
hk 2
L /2m, τ = 4ω r t, and
H th =
m
H /2k 2
L ¯
h 2 to obtain
H th = n
2
−
αω 2
8π 2
cos(
φ) +
1
2
cos(
φ − ωτ ) +
1
2
cos(
φ + ωτ )
,
(10.39)
where α = 8ω r T 2 V 0 / ¯
h. All quantities are dimensionless, and ˆ
n is the dimensionless
momentum operator with eigenstates |n and integer eigenvalues −∞≤n≤∞.
Note that, in the experimental papers, the following scaling is performed:
φ =
2k L x, τ = ω m t/2π = t/T ,
ρ = 4πk L
p/mω m ,
H exp = 16π 2 k 2
L
H /mω 2
m .
With this scaling, the Hamiltonian in Eq. (10.38) takes the form
H exp =
ρ 2
2 −
2αcos 2 (π τ )cos(
φ). However, this choice of scaling does not allow the type of
analysis described below.
The system described by Eq. (10.39) has three primary resonances centered at
(n = 0, φ = 0) and (n = ±ω/2, φ = 0). For small values of α (α < 1.5),
the primary resonances have pendulum-like structure, and the resonance at n = 0
has half-width n 0 =
αω 2
4π 2 , while the resonances at n ± = ±ω/2 have half-width
n ± = n 0 /
√
2. The primary resonance at n = 0 bifurcates at α≈7.0. The two
outer primaries remain visible for 0 < α ≤ 13.0 and disappear for larger values
of α.
The Hamiltonian in Eq. (10.39) has a classical analog if we let the dimensionless
momentum ˆ
n→n, where n can take on a continuum of values, −∞≤n≤∞. The
classical motion is obtained from Hamilton’s equations ˙
n = −
∂H th
∂φ and ˙
φ =
∂H th
∂n .
In the experiment, ω r = 13,000 rad/s and T = 2π/ω m = 20 μs, so the
dimensionless radial frequency is ω = 6.0. In Fig. 10.8a, we show a strobe plot
of the classical phase space for α = 2.0. The three primary resonances are clearly
visible in this plot. A strobe plot of the classical phase space for α = 9.7 is shown
in Fig. 10.8b. The central primary resonance has bifurcated and is largely destroyed,
and the outer primary resonances have been reduced significantly in size and are
centered at momentum values n = ±4.1. Note also that the chaotic region lies in
the interval −5 ≤ n ≤ +5, indicating that 11 quantized momentum states determine
the dynamics in the chaotic region.
359
In these experiments, because of the very low temperature and the fact that the
atomic recoil is quantized in steps of 2 ¯
hk L , the allowed momentum states appear
to be quantized in integer multiples of 2 ¯
hk L with very small spread about those
values. This allows us to use Floquet analysis for this system rather than FloquetBloch analysis (Mouchet et al. 2001) as one might expect from the form of the
Hamiltonian in Eq. (10.38). Because of the quantization of the recoil momentum,
an atom cannot transition through the continuous range of energies in the energy
bands allowed by the Hamiltonian in Eq. (10.38), at least for the timescales of the
experiment.
Let us now write the Hamiltonian in dimensionless form. We perform a scaling
that explicitly quantizes the momentum in units of 2 ¯
hk L (Luter and Reichl 2002).
Let
φ = 2k L x,
p = 2 n ¯
hk L , ω = ω m /4ω r , ω r = ¯
hk 2
L /2m, τ = 4ω r t, and
H th =
m
H /2k 2
L ¯
h 2 to obtain
H th = n
2
−
αω 2
8π 2
cos(
φ) +
1
2
cos(
φ − ωτ ) +
1
2
cos(
φ + ωτ )
,
(10.39)
where α = 8ω r T 2 V 0 / ¯
h. All quantities are dimensionless, and ˆ
n is the dimensionless
momentum operator with eigenstates |n and integer eigenvalues −∞≤n≤∞.
Note that, in the experimental papers, the following scaling is performed:
φ =
2k L x, τ = ω m t/2π = t/T ,
ρ = 4πk L
p/mω m ,
H exp = 16π 2 k 2
L
H /mω 2
m .
With this scaling, the Hamiltonian in Eq. (10.38) takes the form
H exp =
ρ 2
2 −
2αcos 2 (π τ )cos(
φ). However, this choice of scaling does not allow the type of
analysis described below.
The system described by Eq. (10.39) has three primary resonances centered at
(n = 0, φ = 0) and (n = ±ω/2, φ = 0). For small values of α (α < 1.5),
the primary resonances have pendulum-like structure, and the resonance at n = 0
has half-width n 0 =
αω 2
4π 2 , while the resonances at n ± = ±ω/2 have half-width
n ± = n 0 /
√
2. The primary resonance at n = 0 bifurcates at α≈7.0. The two
outer primaries remain visible for 0 < α ≤ 13.0 and disappear for larger values
of α.
The Hamiltonian in Eq. (10.39) has a classical analog if we let the dimensionless
momentum ˆ
n→n, where n can take on a continuum of values, −∞≤n≤∞. The
classical motion is obtained from Hamilton’s equations ˙
n = −
∂H th
∂φ and ˙
φ =
∂H th
∂n .
In the experiment, ω r = 13,000 rad/s and T = 2π/ω m = 20 μs, so the
dimensionless radial frequency is ω = 6.0. In Fig. 10.8a, we show a strobe plot
of the classical phase space for α = 2.0. The three primary resonances are clearly
visible in this plot. A strobe plot of the classical phase space for α = 9.7 is shown
in Fig. 10.8b. The central primary resonance has bifurcated and is largely destroyed,
and the outer primary resonances have been reduced significantly in size and are
centered at momentum values n = ±4.1. Note also that the chaotic region lies in
the interval −5 ≤ n ≤ +5, indicating that 11 quantized momentum states determine
the dynamics in the chaotic region.