10.3 Quantum Nonlinear Resonances
347
i
∂ψ(θ, t)
∂t
= −
∂ 2 ψ(θ, t)
∂θ 2 + [U 0 cos(θ − ωt) + V 0 cos(3θ − ωt)]ψ(θ, t).
(10.23)
We can write the Schrödinger equation in the angular momentum basis, |n, if we
expand ψ(θ, τ ) =
1
√
2π
∞
−∞ ψ n (τ )e inθ , where ψ n (τ ) = =n|ψ(τ ). Then we obtain
i
∂ψ n (τ )
∂τ
= n
2 ψ n (τ ) +
1
2
U 0
e
−iω 0 τ ψ n−1 (τ ) + e
−iω 0 τ ψ n+1 (τ )
+
1
2
V 0
e
−iω 0 τ ψ n−3 (τ ) + e
−iω 0 τ ψ n+3 (τ )
.
(10.24)
In order to construct the Floquet eigenstates for this system, we first construct the
Floquet matrix, U n,n (T ), defined in Eq. (10.13).
For the double-resonance system, the Floquet matrix truncates itself naturally.
Outside the region of influence of the two primary resonances, the angular momentum is approximately a good quantum number, and the Floquet matrix becomes
approximately diagonal. We can truncate the Floquet matrix to include only those
angular momentum states, N − ≤n≤N + , strongly affected by the resonance region.
The Floquet matrix can be constructed numerically as follows. Choose the initial
state to be the angular momentum eigenstate, |m, so n|ψ(0) = =n|m = δ n,m . If
we evolve this state for one period, T = 2π/ω, we obtain n|ψ(T ) = U n,m (T ) for
N − ≤n≤N + , which gives the mth column of the Floquet matrix. By repeating this
process for each value of the angular momentum, m, in the interval, N − ≤m≤N + ,
we obtain the full Floquet matrix, U n,m (T ).
We can write the Floquet matrix in abstract form. Let | α = | α (0) =
| α (T ) denote the αth eigenstate of the Floquet matrix. The spectral decomposition of the Floquet evolution operator is
ˆ
U(T ) =
α
exp
−i
α
¯
h
T
| α α |.
(10.25)
The αth eigenvalue of ˆ
U(T ) is exp
−i
α
¯
h T
.
In order to visualize how the Floquet eigenstates are distributed in phase space,
we use Husimi functions. The Husimi function (Husimi 1940) for a Floquet
eigenstate, | α , evaluated at phase space point (n , θ ), is defined as
W α (n
, θ
) = ||n
, θ
| α |
2 ,
(10.26)
where |n , θ is a minimum uncertainty Gaussian wave packet centered at n =
n , θ = θ . In the angular momentum picture, the minimum uncertainty wave packet
is given by
347
i
∂ψ(θ, t)
∂t
= −
∂ 2 ψ(θ, t)
∂θ 2 + [U 0 cos(θ − ωt) + V 0 cos(3θ − ωt)]ψ(θ, t).
(10.23)
We can write the Schrödinger equation in the angular momentum basis, |n, if we
expand ψ(θ, τ ) =
1
√
2π
∞
−∞ ψ n (τ )e inθ , where ψ n (τ ) = =n|ψ(τ ). Then we obtain
i
∂ψ n (τ )
∂τ
= n
2 ψ n (τ ) +
1
2
U 0
e
−iω 0 τ ψ n−1 (τ ) + e
−iω 0 τ ψ n+1 (τ )
+
1
2
V 0
e
−iω 0 τ ψ n−3 (τ ) + e
−iω 0 τ ψ n+3 (τ )
.
(10.24)
In order to construct the Floquet eigenstates for this system, we first construct the
Floquet matrix, U n,n (T ), defined in Eq. (10.13).
For the double-resonance system, the Floquet matrix truncates itself naturally.
Outside the region of influence of the two primary resonances, the angular momentum is approximately a good quantum number, and the Floquet matrix becomes
approximately diagonal. We can truncate the Floquet matrix to include only those
angular momentum states, N − ≤n≤N + , strongly affected by the resonance region.
The Floquet matrix can be constructed numerically as follows. Choose the initial
state to be the angular momentum eigenstate, |m, so n|ψ(0) = =n|m = δ n,m . If
we evolve this state for one period, T = 2π/ω, we obtain n|ψ(T ) = U n,m (T ) for
N − ≤n≤N + , which gives the mth column of the Floquet matrix. By repeating this
process for each value of the angular momentum, m, in the interval, N − ≤m≤N + ,
we obtain the full Floquet matrix, U n,m (T ).
We can write the Floquet matrix in abstract form. Let | α = | α (0) =
| α (T ) denote the αth eigenstate of the Floquet matrix. The spectral decomposition of the Floquet evolution operator is
ˆ
U(T ) =
α
exp
−i
α
¯
h
T
| α α |.
(10.25)
The αth eigenvalue of ˆ
U(T ) is exp
−i
α
¯
h T
.
In order to visualize how the Floquet eigenstates are distributed in phase space,
we use Husimi functions. The Husimi function (Husimi 1940) for a Floquet
eigenstate, | α , evaluated at phase space point (n , θ ), is defined as
W α (n
, θ
) = ||n
, θ
| α |
2 ,
(10.26)
where |n , θ is a minimum uncertainty Gaussian wave packet centered at n =
n , θ = θ . In the angular momentum picture, the minimum uncertainty wave packet
is given by