Quantum trajectories for the
laboratory: modeling engineered
quantum systems
Andrew Doherty
University 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 oscillating
Understand how the performance depends on
feedback gain, measurement backaction means
that there is an optimum gain.
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
6
Coherently Driven 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
7
Derive equations of motion
8
Bloch Equations
9
Feedback leads to permanent Rabi oscillations
Concept of a quantum trajectory
Interact one at a time
undergo projective
measurement
Harmonic oscillators
representing input field
approach system
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
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
larger or smaller than expected?
, “innovation” was measurement
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”
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.
Why This Feedback?
Ansatz for solution
We would like
So we define
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 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
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 the
average 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 be
easy 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 oscillating
Understand how the performance depends on
feedback gain, measurement backaction means
that there is an optimum gain.
Optimal Perfomance
Efficiency of the measurement is
Total dephasing rate is
Optimal feedback gain is
Optimal performance