Rabi Oscillation Simulator

Rabi Oscillation Simulator models a two-level quantum system’s probability evolution under a resonant field, visualizing oscillations for quantum computing studies.

Rabi Frequency (MHz):

Time Range (ns):

Initial |0⟩ Probability:

Formulas Used in Rabi Oscillation Simulator

The simulator uses the following formulas for a two-level system under resonant driving:

Time-Evolved State (Initial |0⟩):

\\[ |\psi(t)\rangle = \cos\left(\frac{\Omega t}{2}\right) |0\rangle – i \sin\left(\frac{\Omega t}{2}\right) |1\rangle \\]

Excited State Probability:

\\[ P_{|1\rangle}(t) = \sin^2\left(\frac{\Omega t}{2}\right) \\]

Mixed Initial State:

\\[ P_{|1\rangle}(t) = P_{|0\rangle}(0) \sin^2\left(\frac{\Omega t}{2}\right) + P_{|1\rangle}(0) \cos^2\left(\frac{\Omega t}{2}\right) \\]

Where:

\\( \Omega \\): Rabi frequency (rad/s)
\\( t \\): Time (s)
\\( P_{|0\rangle}(0) \\): Initial probability of state |0⟩
\\( P_{|1\rangle}(0) \\): Initial probability of state |1⟩ (\\( = 1 – P_{|0\rangle}(0) \\))
\\( P_{|1\rangle}(t) \\): Probability of state |1⟩ at time \\( t \\)

Example Calculations

Example 1: Standard Case

Input: Rabi Frequency = 1 MHz, Time Range = 10 ns, Initial |0⟩ Probability = 1

\\[ \Omega = 1 \times 10^6 \, \text{rad/s}, \quad t_{\text{max}} = 10 \times 10^{-9} \, \text{s} \\] \\[ P_{|1\rangle}(t_{\text{max}}) = \sin^2\left(\frac{1 \times 10^6 \cdot 10 \times 10^{-9}}{2}\right) = \sin^2(0.005) \approx 0.000025 \\]

Result: Excited State Probability at t = 10 ns: 0.000025

Example 2: Faster Oscillation

Input: Rabi Frequency = 5 MHz, Time Range = 10 ns, Initial |0⟩ Probability = 1

\\[ \Omega = 5 \times 10^6 \, \text{rad/s}, \quad t_{\text{max}} = 10 \times 10^{-9} \, \text{s} \\] \\[ P_{|1\rangle}(t_{\text{max}}) = \sin^2\left(\frac{5 \times 10^6 \cdot 10 \times 10^{-9}}{2}\right) = \sin^2(0.025) \approx 0.000625 \\]

Result: Excited State Probability at t = 10 ns: 0.000625

Example 3: Mixed Initial State

Input: Rabi Frequency = 1 MHz, Time Range = 10 ns, Initial |0⟩ Probability = 0.5

\\[ \Omega = 1 \times 10^6 \, \text{rad/s}, \quad t_{\text{max}} = 10 \times 10^{-9} \, \text{s} \\] \\[ P_{|1\rangle}(t_{\text{max}}) = 0.5 \cdot \sin^2(0.005) + 0.5 \cdot \cos^2(0.005) \approx 0.5 \cdot 0.000025 + 0.5 \cdot 0.999975 \approx 0.5 \\]

Result: Excited State Probability at t = 10 ns: 0.5

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Rabi Oscillation Simulator

Formulas Used in Rabi Oscillation Simulator

Example Calculations

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