Quantum trajectories for the laboratory: modeling engineered quantum systems

Andrew DohertyUniversity of Sydney

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.

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.

Feedback leads to permanent Rabi oscillations

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

Feedback leads to permanent Rabi oscillations

Concept of a quantum trajectory

Harmonic oscillators representing input field approach system

Interact one at a time

undergo projective measurement

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

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?

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”

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

Why This Feedback?

Ansatz for solution

So we define

We would like

Consider

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

Toy Model of Feedback

Detector reads out qubit in white noise background

Measurement outcome

After that detection, the feedback acts

Toy Model of Feedback

Expanding out we find the following

Toy Model of Feedback

We can then simplify and average over measurement results to find theaverage performance

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.

Transform into rotating frame

We can consider the following rotating wave state

Rotate our Bloch sphere as follows.

Note that

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

Rotating Frame Bloch Equation

After all this we get the following simple equation

And the steady state

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

Check that the qubit state is really oscillatingUnderstand how the performance depends on feedback gain, measurement backaction meansthat there is an optimum gain.

Optimal Perfomance

Efficiency of the measurement is

Total dephasing rate is

Optimal performance

Optimal feedback gain is