University Advanced Quantum Mechanics · topic UAQM-14.08
Quantum trajectories: unravelling the master equation into quantum jumps
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Unravelling the master equation into quantum jumps: predict a finite change
A quantum trajectory is the pure state of a monitored system conditioned on its measurement record: it evolves smoothly under a non-Hermitian Hamiltonian between detections, jumps when a detector clicks, and averages over all records to the master-equation ρ. Use the supplied topic model Y_out = 2+1/om^2, where Y_out is MEAN WAIT Γ⟨τ⟩ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. om is Rabi frequency Ω/Γ, dimensionless. Keep t (Time since last click Γτ) = 2 fixed. For this exercise use only 1.0675 ≤ om ≤ 4.5325 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at om = 1.8298 and 2.3149. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δom across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
MEAN WAIT Γ⟨τ⟩ against om
Predictions from the supplied Unravelling the master equation into quantum jumps model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): MEAN WAIT Γ⟨τ⟩
Horizontal axis (x): om: Rabi frequency Ω/Γ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| om: Rabi frequency Ω/Γ | MEAN WAIT Γ⟨τ⟩ |
|---|---|
| 1.0675 | 2.877534566635039 |
| 1.9337499999999999 | 2.26742336335857 |
| 2.8 | 2.127551020408163 |
| 3.66625 | 2.0743970727541763 |
| 4.5325 | 2.0486770638846914 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of om separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At om = 1.8298, substitute into Y_out = 2+1/om^2: Y₀ = 2.29867.
- At om = 2.3149, the same fixed-parameter model gives Y₁ = 2.18661.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.112061; the input change is Δom = 0.4851.
- Average sensitivity is ΔY_out/Δom = -0.231005 1 per unit input. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 2.29867, Y₁ = 2.18661; ΔY_out = -0.112061; average sensitivity = -0.231005 1 per unit input.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Unravelling the master equation into quantum jumps: infer an input with uncertainty
A quantum trajectory is the pure state of a monitored system conditioned on its measurement record: it evolves smoothly under a non-Hermitian Hamiltonian between detections, jumps when a detector clicks, and averages over all records to the master-equation ρ. Use the supplied topic model Y_out = 2+1/om^2, where Y_out is MEAN WAIT Γ⟨τ⟩ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. om is Rabi frequency Ω/Γ, dimensionless. Keep t (Time since last click Γτ) = 2 fixed. For this exercise use only 1.0675 ≤ om ≤ 4.5325 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 2.098817, with a bounded reading uncertainty ±0.007521. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown om by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0003465, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
MEAN WAIT Γ⟨τ⟩ against om
Predictions from the supplied Unravelling the master equation into quantum jumps model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): MEAN WAIT Γ⟨τ⟩
Horizontal axis (x): om: Rabi frequency Ω/Γ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| om: Rabi frequency Ω/Γ | MEAN WAIT Γ⟨τ⟩ |
|---|---|
| 1.0675 | 2.877534566635039 |
| 1.9337499999999999 | 2.26742336335857 |
| 2.8 | 2.127551020408163 |
| 3.66625 | 2.0743970727541763 |
| 4.5325 | 2.0486770638846914 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(om) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(1.0675) = 2.87753 and Y_out(4.5325) = 2.04868. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives om_line = 1.0675 + (2.09882 − 2.87753)(4.5325 − 1.0675)/(2.04868 − 2.87753) = 4.32289. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives om = 3.18115; substituting back gives Y_out = 2.098817.
- The interpolation error in the input is |om_line − om| = 1.14174. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.091296 and 2.106338. Sort the roots even if the response decreases: 3.06659 ≤ om ≤ 3.30959.
- The lower and upper input deviations are 0.11456 and 0.128439. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer om ≈ 3.18115; compatible interval [3.06659, 3.30959]. Endpoint interpolation gives 4.32289.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.