PDF (double-sided) - Physics Department, UCSB - University of ...
PDF (double-sided) - Physics Department, UCSB - University of ... PDF (double-sided) - Physics Department, UCSB - University of ...
Figure 3.1: Examples of Numerical Simulations: Potential shown with eigenstates offset by their energy. – a) Eigenstates of coarse harmonic oscillator potential: V (x) = x 2 , −3.0 ≤ x ≤ 3.0, dx = 1.0. b) Lowest 7 eigenstates of fine harmonic oscillator potential: V (x) = x 2 , −5.0 ≤ x ≤ 5.0, dx = 0.1 c) Lowest 17 eigenstates of qubit-like potential: V (δ) = δ 2 − 5 cos δ + 5δ, −8.0 ≤ δ ≤ 3.0, dδ = 0.05. States localized in the shallow (deep) minimum are shown in green (red), while states that span both minima are shown in gray. Plotting the eigenvectors offset by their corresponding energies gives a plot like Figure 3.1a. For the lower energy states this plot clearly shows the usual oscillating behavior of the wave functions and for the ground state even the exponential decay outside the potential. If the x-step-size is decreased from 1 to 0.1, i.e. the resolution of the approximation is increased by 10×, the first 7 eigenvectors look like Figure 3.1b. Their corresponding energies are: 1.58, 4.74, 7.89, 11.06, 14.23, 17.45, 20.80 These numbers show the expected equal spacing of the levels fairly well. The energies of the levels are quite different in the two approximations. Es- 44
pecially for the higher levels, for which the wave functions should extend significantly past the chosen x-range, the approximation becomes fairly poor in the low-resolution case. But as the resolution of the approximation increases and the x-range is expanded, the energy levels and wave-functions get closer and closer to their true values. Unfortunately this also increases the size of the matrix M and therefore the time to diagonalize it. The latter increases exponentially with the size of M. This makes finding the eigenstates of a quantum system from its potential a computationally hard problem. To ensure accurate conclusions, the approximation should be repeated with several different step sizes and ranges to verify that the energy levels have indeed converged to their true values. 3.2.1 The Eigenstates of the Qubit Potential Applying this technique to the qubit potential is as straightforward and yields a plot similar to Figure 3.1c. The important pieces of information to take away from this analysis are: • The number of states localized in the operating minimum as a function of flux bias: Even though the potential might have two or more minima, they might not be deep enough compared to the “mass” of the particle to support the required number (≥ 2) of localized quantum states. 45
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- Page 55 and 56: Figure 2.4: Josephson Qubits: Sligh
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- Page 59 and 60: Figure 2.5: Example Qubit Coupling
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- Page 63: states in the qubit’s inductor, t
- Page 66 and 67: at time t. r is not restricted to b
- Page 68 and 69: 3.1.2 Effects of a Time Dependent P
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- Page 76 and 77: like this: V = ( V (−1, −1), V
- Page 78 and 79: Figure 3.2: Simulation of LC Oscill
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- Page 82 and 83: with ω mn = Em−En . Multiplying
- Page 84 and 85: α, it can be ignored. Thus, the in
- Page 86 and 87: e solved exactly: A(t + ∆t) = e
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- Page 97 and 98: Chapter 4 Designing the Phase Qubit
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- Page 107 and 108: Figure 4.3: Squid I/V Traces - a) L
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pecially for the higher levels, for which the wave functions should extend significantly<br />
past the chosen x-range, the approximation becomes fairly poor in the<br />
low-resolution case. But as the resolution <strong>of</strong> the approximation increases and the<br />
x-range is expanded, the energy levels and wave-functions get closer and closer<br />
to their true values. Unfortunately this also increases the size <strong>of</strong> the matrix M<br />
and therefore the time to diagonalize it. The latter increases exponentially with<br />
the size <strong>of</strong> M. This makes finding the eigenstates <strong>of</strong> a quantum system from its<br />
potential a computationally hard problem. To ensure accurate conclusions, the<br />
approximation should be repeated with several different step sizes and ranges to<br />
verify that the energy levels have indeed converged to their true values.<br />
3.2.1 The Eigenstates <strong>of</strong> the Qubit Potential<br />
Applying this technique to the qubit potential is as straightforward and yields<br />
a plot similar to Figure 3.1c. The important pieces <strong>of</strong> information to take away<br />
from this analysis are:<br />
• The number <strong>of</strong> states localized in the operating minimum as a function <strong>of</strong><br />
flux bias: Even though the potential might have two or more minima, they<br />
might not be deep enough compared to the “mass” <strong>of</strong> the particle to support<br />
the required number (≥ 2) <strong>of</strong> localized quantum states.<br />
45
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