Quantum trajectories for the laboratory : modeling engineered quantum systems
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Quantum trajectories for the laboratory: modeling engineered quantum systems
Andrew DohertyUniversity of Sydney
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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.
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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.
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Feedback leads to permanent Rabi oscillations
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Check that the qubit state is really oscillatingUnderstand how the performance depends on feedback gain, measurement backaction meansthat there is an optimum gain.
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Coherently Driven 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
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Coherently Driven Atom
- Master equation method treats coupling to bath in perturbation theory
Coherent drivingEmission into bathAbsorption from bathDephasing due to bath
Interpretation of terms in master equation
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Derive equations of motion
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Bloch Equations
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Feedback leads to permanent Rabi oscillations
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Concept of a quantum trajectory
Harmonic oscillators representing input field approach system
Interact one at a time
undergo projective measurement
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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 slideOr as the limit of a realistic model of the device
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Toy Model of QND Measurement
Detector reads out qubit in white noise background
Is a normally distributed random variable with mean zero and variance
Update of quantum state, depending on:
Measurement outcome
quality of measurement, uncertainty about , “innovation” was measurementlarger or smaller than expected?
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Toy Model of QND Measurement
Detector reads out qubit in white noise background
Is a normally distributed random variable with mean zero and variance
Update of x depends on correlations between x and y
Measurement outcome
Dephasing damps x, is a reflection of “measurement backaction”
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Measurement and Feedback
We need to add measurement and feedback to our Rabi flopping system
Modulate amplitude of coherent drive depending on measurement result tospeed up or slow down oscillations as necessary.
Measurement modelled as we have discussed
Feedback described by feedback Hamiltonian
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Why This Feedback?
Ansatz for solution
So we define
We would like
Consider
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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 laggingwe speed it up.
We need an equation to describe how successful the feedback is, how close to Rabi perfect oscillation we are, something like
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Toy Model of Feedback
Detector reads out qubit in white noise background
Measurement outcome
After that detection, the feedback acts
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Toy Model of Feedback
Expanding out we find the following
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Toy Model of Feedback
We can then simplify and average over measurement results to find theaverage performance
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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 beeasy to code.
We can do an approximate analysis, similar to the one in the paper where we average over a Rabi cycle.
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Transform into rotating frame
We can consider the following rotating wave state
Rotate our Bloch sphere as follows.
Note that
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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
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Rotating Frame Bloch Equation
After all this we get the following simple equation
And the steady state
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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
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Check that the qubit state is really oscillatingUnderstand how the performance depends on feedback gain, measurement backaction meansthat there is an optimum gain.
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Optimal Perfomance
Efficiency of the measurement is
Total dephasing rate is
Optimal performance
Optimal feedback gain is