Quantum trajectories for the laboratory : modeling engineered quantum systems

1. Andrew Doherty University of Sydney Quantum trajectories for the laboratory: modeling engineered quantum systems

2. Goal of this lecture will be to develop a model of the most important aspects of this experiment using the theory of quantum trajectories I hope the discussion will be somewhat tutorial and interactive.

3. Goal of this lecture will be to develop a model of the most important aspects of this experiment using the theory of quantum trajectories I hope the discussion will be somewhat tutorial and interactive.

4. Feedback leads to permanent Rabi oscillations

5. Check that the qubit state is really oscillating Understand how the performance depends on feedback gain, measurement backaction means that there is an optimum gain.

6. CoherentlyDriven Atom • Atom in free space spontaneously emits • Laser leads to stimulated emission and absorption • Photodetector makes it possible to see statistics of emission events • Stimulated absorption and emission can become much faster than spontaneous emission

7. CoherentlyDriven Atom • Master equation method treats coupling to bath in perturbation theory Coherent driving Emission into bath Absorption from bath Dephasing due to bath Interpretation of terms in master equation

8. Derive equations of motion

9. Bloch Equations

10. Feedback leads to permanent Rabi oscillations

11. Concept of a quantum trajectory Interact one at a time undergo projective measurement Harmonic oscillators representing input field approach system

12. Toy Model of QND Measurement Detector reads out qubit in white noise background Measurement outcome Can obtain this equation phenomenologically using the picture on the previous slide Or as the limit of a realistic model of the device

13. Toy Model of QND Measurement Detector reads out qubit in white noise background Measurement outcome Is a normally distributed random variable with mean zero and variance Update of quantum state, depending on: quality of measurement, uncertainty about , “innovation” was measurement larger or smaller than expected?

14. Toy Model of QND Measurement Detector reads out qubit in white noise background Measurement outcome Is a normally distributed random variable with mean zero and variance Update of x depends on correlations between x and y Dephasing damps x, is a reflection of “measurement backaction”

15. Measurement and Feedback We need to add measurement and feedback to our Rabi flopping system Measurement modelled as we have discussed Feedback described by feedback Hamiltonian Modulate amplitude of coherent drive depending on measurement result to speed up or slow down oscillations as necessary.

16. Why This Feedback? Ansatz for solution We would like So we define Consider

17. Why This Feedback? So on average for the feedback we have If the qubit is rotating too fast, then we reduce the rotation rate, if it is lagging we speed it up. We need an equation to describe how successful the feedback is, how close to Rabi perfect oscillation we are, something like

18. Toy Model of Feedback Detector reads out qubit in white noise background Measurement outcome After that detection, the feedback acts

19. Toy Model of Feedback Expanding out we find the following

20. Toy Model of Feedback We can then simplify and average over measurement results to find the average performance

21. Complete Model (T=0) Then we add back all the rest of the stuff This model is a little difficult to solve analytically still, although it should be easy to code. We can do an approximate analysis, similar to the one in the paper where we average over a Rabi cycle.

22. Transform into rotating frame We can consider the following rotating wave state Rotate our Bloch sphere as follows. Note that

23. Rotating Frame Master Equation With all these definitions we can find the master equation in the rotating frame Then the rotating wave approximation amounts to ignoring all time dependent coefficients of this equation

24. Rotating Frame Bloch Equation After all this we get the following simple equation And the steady state

25. Rotating Frame Bloch Equation After all this we get the following simple equation And the steady state Ideal performance would be Back in the real world with no rotating frame this is an infinite Rabi oscillation

26. Check that the qubit state is really oscillating Understand how the performance depends on feedback gain, measurement backaction means that there is an optimum gain.

27. Optimal Perfomance Efficiency of the measurement is Total dephasing rate is Optimal feedback gain is Optimal performance