Homework 9 Solution
Prob. 9.1
For a twolevel system with static couplingin Eq. (9.18),
(a)
Find
A
and
B
in terms of
θ
if
c
1
(0)
= 0 and
c
2
(0)
=1.
(5 pts)
A = sin
θ
; B = cos
θ
.
(b)
Find
c
1
(t)
,
c
2
(t)
,
p
1
(t)
and
p
2
(t)
following the initial condition in (a).
(5 pts)
(
29
(
29
(
29
(
29


=

=

+

=



=

+

+
+

+

t
E
E
t
p
t
E
E
t
p
t
E
i
t
E
i
t
c
t
E
i
t
E
i
t
c
B
B
B
B
B
B
2
sin
2
sin
1
2
sin
2
sin
exp
cos
exp
sin
exp
exp
cos
sin
2
2
2
2
2
1
2
2
1
1
θ
θ
θ
θ
θ
θ
(c)
Find
A
and
B
in terms of
θ
if
c
1
(0)
=
c
2
(0)
=
2
/
1
.
(5 pts)
(
29
(
29
θ
θ
θ
θ
sin
cos
2
1
;
cos
sin
2
1

=
+
=
B
A
(d)
When is the first time
p
1
(t) = 0
following the initial condition in (c).
(10 pts)
(
29
(
29
0
2
sin
4
sin
2
1
2
1
2
2
1
1
=

+
=
=

+
t
E
E
t
c
t
p
B
θ
In general,
θ
4
sin
1
2
sin
2

=


+
t
E
E
B
.
Notice that this equality can ONLY be true for a
specific
θ
that gives sin
4
θ
= 1 where
8
3
1
1
cos
2
2
12
1
π
θ
=

+
=


H
E
V
.
If this condition is
not met then
p
1
(t)
will never be zero (the oscillation is not full swing).
If the condition is
met, then the first time p1(t) is zero at

+

=
E
E
t
π
.
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 '05
 TANG
 2m, 4m, 2 2m, 1Nm, Rabi cycle, 2 4m

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