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Figure 3.2: Simulation of LC Oscillator – a) Eigenstates: Lowest 30 eigenstates of
harmonic oscillator potential: V (x) = x 2 . b) Transition matrix: Absolute value
of transition matrix elements |T nm | showing the expected √ n behavior.
can only drive transitions between neighboring levels (only elements on the first
off-diagonals are non-zero). The fact that the diagonal elements of T nm are zero
implies that it is not possible to change the phase of a level population with this
kind of a drive, making Z-rotations impossible (see Section 3.3.1). These results
match the analytic solution:
⎛
√
T nm =
2µω
⎜
⎝
√
1
√
1
√
2
√
2
√
3
√
3
. . .
⎞
⎟
⎠
.. .
(3.30)
Figure 3.3b shows |T nm | for the qubit-like potential shown with its eigenstates
in Figure 3.3a. This plot consists of three distinct regions:
• The bottom-left corner corresponds to transitions between states confined
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