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Quantum trajectories for the laboratory: modeling engineered quantum systems Andrew Doherty University 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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Single photon source Modes of the light approach atom and are scattered off Every so often a photon is emitted into the light field, spontaneous emission

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Single photon source Modes of the light approach atom and are scattered off Every so often a photon is emitted into the light field, spontaneous emission Detector counts photons in scattered field

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Single photon source Modes of the light approach atom and are scattered off Every so often a photon is emitted into the light field, spontaneous emission Count rate

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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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Single photon source Evolution for no photon emission This happens most of the time Photon emission Happens some of the time

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No photon emission Probability of no photon emission up to time t

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Photon emission at t Photon emission at time t Probability of photon emission between t and t+dt Total probability of photon emission

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Distant Detector Photon emission at time t Probability of photon emission between t and t+dt Total probability of photon emission

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Detector Efficiency Photon detection at time t+t p Probability of photon detection between t+t p and t+t p +dt Total probability of photon detection

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Beam Splitter

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Quantum Interference

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Two single photon sources Evolution for no photon emission Photon detection at c Photon detection at d

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Two single photon sources Evolution for no photon emission Photon detection at c Photon detection at d

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Detection at c Photon detection at c Free evolution with no photon emission

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Detection at d Photon detection at d Free evolution with no photon emission

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Detection at d Photon detection at d Free evolution with no photon emission

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Coincidence Probability Hong-Ou-Mandel Dip

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State Preparation Free evolution Microwave pi/2-pulses v v State preparation (shelving) v v Optical pi-pulses v v

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Detection at c

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Wait Wait for any subsequent photon emission

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Microwave pi-pulses Microwave pi-pulses swap spin states

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Optical pi-pulses Optical pi-pulses prepare for second single photon emission

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Free evolution Free evolution up to second emission

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Detection at c Detection at c and wait

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Remote Entanglement We have a state roughly equal to With the probability we would have guessed And output state given by: (could be improved by tightening co-incidence window)

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Coincidence Probability Hong-Ou-Mandel Dip

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Notes on this calculation Fidelity of entangled state has same dependence on physical parameters as for HOM experiment. Need propagation phases to be constant over interval between photons Easy to see that other factors like detector efficiency and path length difference after the beam splitter factor out also Timing error hasn’t been included here, could be done easily by modeling the pulse that prepares the state e. This is another kind of mode-match error and would look much like the dependence on the decay rates. Note that difference in the delay time for the two paths factors out. This is because we have modeled the propagation of the single photon pulse as if the beam were perfectly monochromatic. This approximation is valid because we are in the Markov regime but if the delays are too large, this needs to be corrected

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