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10 Time-Periodic Quantum Systems
The spatial distribution of the αth Floquet eigenstate is given by
α (x, y)=
1
π
∞
n x =1
∞
n y =1
¯
P α (0) n x ,n y +i ¯
Q α (0) n x ,n y
sin(n x x)sin(n y y). (10.104)
10.8.1.2 Behavior of Quantum States
We can follow the behavior of the quantum states as we turn on the amplitude V 1 of
the time-periodic modulation of the optical lattice. We shall assume that the driving
frequency is ω = 2π , in dimensionless units. For the case V 1 = 0, the Hamiltonian
is independent of time and we can write
¯ ¯
L
(0) ≡
0
¯
H (0)
− ¯
H (0) 0
.
(10.105)
The solution to the Schrödinger equation Eq. (10.90) can be written
¯ ¯
(t) = e
¯ ¯
L (0) t ¯ ¯
(0).
(10.106)
We denote the nth and (n + 1)th eigenvectors of ¯ ¯
L
(0)
as
¯
φ 2n−1 =
1
√
2
i|E n
|E n
and ¯
φ 2n =
1
√
2
−i|E n
|E n
(10.107)
Then ¯ ¯
L
(0) ¯
φ 2n−1 = −iE n ¯
φ 2n−1 and ¯ ¯
L
(0) ¯
φ 2n = iE n ¯
φ 2n . Furthermore, ¯
φ 2n−1 =
¯
φ ∗
2n . For V 1 = 0, the Floquet eigenphases occur in pairs that are equal to ±iE n
and the real and imaginary parts of the Floquet eigenstates are equal to the energy
eigenstates. Therefore, at V 1 = 0 we can identify each Floquet eigenphase and
eigenstate with an energy eigenvalue and eigenstate. However, as we turn on the
modulation, each Floquet state changes from a pure energy eigenstate to an energy
entangled state as the amplitude of the time periodic modulation increases. Energy
is no longer conserved.
In Fig. 10.18 we show the average energy E = ¯
T
α · ¯
L 0 · ¯
T
α for the Floquet
eigenstates α = 1, . . . 8. At V 1 = 0, the average energy is equal to the energy
eigenvalue E n=α and remains essentially unchanged for small non-zero values
of V 1 . However, for larger, non-zero values of V 1 the average energy undergoes
significant excursions, indicating that the Floquet eigenstates begin to consist of a
superposition of a number of energy eigenstates. In Fig. 10.18a, we show the average
energy of Floquet eigenstates α = 1, . . . 8 for b = 0.2. There is a slight increase
of the average energy of some states as V 1 increases, but not a significant increase.
Two of the states, α = 3 and α = 6, are almost completely unchanged by the
time periodic modulation. In Fig. 10.18b, we show the average energy of Floquet
10 Time-Periodic Quantum Systems
The spatial distribution of the αth Floquet eigenstate is given by
α (x, y)=
1
π
∞
n x =1
∞
n y =1
¯
P α (0) n x ,n y +i ¯
Q α (0) n x ,n y
sin(n x x)sin(n y y). (10.104)
10.8.1.2 Behavior of Quantum States
We can follow the behavior of the quantum states as we turn on the amplitude V 1 of
the time-periodic modulation of the optical lattice. We shall assume that the driving
frequency is ω = 2π , in dimensionless units. For the case V 1 = 0, the Hamiltonian
is independent of time and we can write
¯ ¯
L
(0) ≡
0
¯
H (0)
− ¯
H (0) 0
.
(10.105)
The solution to the Schrödinger equation Eq. (10.90) can be written
¯ ¯
(t) = e
¯ ¯
L (0) t ¯ ¯
(0).
(10.106)
We denote the nth and (n + 1)th eigenvectors of ¯ ¯
L
(0)
as
¯
φ 2n−1 =
1
√
2
i|E n
|E n
and ¯
φ 2n =
1
√
2
−i|E n
|E n
(10.107)
Then ¯ ¯
L
(0) ¯
φ 2n−1 = −iE n ¯
φ 2n−1 and ¯ ¯
L
(0) ¯
φ 2n = iE n ¯
φ 2n . Furthermore, ¯
φ 2n−1 =
¯
φ ∗
2n . For V 1 = 0, the Floquet eigenphases occur in pairs that are equal to ±iE n
and the real and imaginary parts of the Floquet eigenstates are equal to the energy
eigenstates. Therefore, at V 1 = 0 we can identify each Floquet eigenphase and
eigenstate with an energy eigenvalue and eigenstate. However, as we turn on the
modulation, each Floquet state changes from a pure energy eigenstate to an energy
entangled state as the amplitude of the time periodic modulation increases. Energy
is no longer conserved.
In Fig. 10.18 we show the average energy E = ¯
T
α · ¯
L 0 · ¯
T
α for the Floquet
eigenstates α = 1, . . . 8. At V 1 = 0, the average energy is equal to the energy
eigenvalue E n=α and remains essentially unchanged for small non-zero values
of V 1 . However, for larger, non-zero values of V 1 the average energy undergoes
significant excursions, indicating that the Floquet eigenstates begin to consist of a
superposition of a number of energy eigenstates. In Fig. 10.18a, we show the average
energy of Floquet eigenstates α = 1, . . . 8 for b = 0.2. There is a slight increase
of the average energy of some states as V 1 increases, but not a significant increase.
Two of the states, α = 3 and α = 6, are almost completely unchanged by the
time periodic modulation. In Fig. 10.18b, we show the average energy of Floquet