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MSc thesis

OlegZero13
March 05, 2017

MSc thesis

My MSc thesis titled: "Phase sensitivity in quantum interference measurements", done under prof. Björk's supervision. KTH 2010.

OlegZero13

March 05, 2017
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  1. Research experience
    MSc thesis,
    Stockholm 2010
    ”Phase sensitivity in
    quantum interference
    measurements”

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  2. Phase super-sensitivity
    Oleg Żero
    General Question:
    Given the power measurement, how precisely are we
    allowed to determine the phase-shitf that caused it?
    My Question:
    What is are the best quantum states to ensure
    maximum information when the system is lossy?
    Δφ

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  3. Phase super-sensitivity
    Oleg Żero
    ħω P(φ)
    φ
    Classical light’s sensitivity
    is shot-noise limited.
    It is called classical limit.
    Quantum light is limited
    by the number of photons.
    It is called quantum limit,
    or super-sensitivity.

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  4. Phase super-sensitivity
    Oleg Żero
    Speaking ”quantum”,
    classical light is a special
    superposition of Fock states.
    However, in quantum world,
    the state can be anything.
    ħω P(φ)
    φ
    The goal is to find states, such that:

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  5. Phase super-sensitivity
    Oleg Żero
    Lossless case:
    Δφ
    It is so-called pure state.

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  6. Phase super-sensitivity
    Oleg Żero
    Δφ
    The so-called ”N00N” state
    reaches super-sensitvity.
    Lossless case:

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  7. Phase super-sensitivity
    Oleg Żero
    What if we loose some photons?
    Δφ
    We begin to loose
    information.

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  8. Phase super-sensitivity
    Then some detectors will click...
    ...and some others won’t.
    We have a statistical model that provides us tools.
    • Outcomes
    • Probability density functions
    • Parameterized with unknown variable
    • Under measurement we collect data:
    We need to define the so-called estimator:
    φ The phase.
    Our guess on phase.
    This is how far off
    we are.
    What if we loose some photons?
    Oleg Żero

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  9. Phase super-sensitivity
    Oleg Żero
    We need to define an estimator...
    The mean square error is a candidate.
    • Variance is our measure of uncertainty.
    • Bias is related to the systematic error.
    Estimator is to be called unbiased and
    efficient if
    • bias is zero, and
    • variance meets
    This is so-called Cramer-Rao bound.
    variance bias2
    where
    Is known as Fisher information.

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  10. Phase super-sensitivity
    Oleg Żero
    This is getting abstract... I know.
    This is so-called Cramer-Rao bound.
    But we aslo have quantum Cramer Rao bound.
    It is expressed through some abstract operator.

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  11. Phase super-sensitivity
    Oleg Żero
    Why not maximising the quantum Fisher
    information?
    We need.
    Neither of these possibilities exists
    for us...
    1. possibility...
    A canonical measurement
    where
    2. possibility...
    A magical mechanism for
    monitoring of losses.
    φ

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  12. Phase super-sensitivity
    Oleg Żero
    So we are left with classical tools to solve a quantum
    problem.
    We have have mixed state.
    And photon counters.
    So we can identify the following:
    • The probabilities are taken from quantum coefficients.
    • The parameter to be known is our phase.
    • The photodetectors’ clicks are our outcomes.

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  13. Phase super-sensitivity
    We build the estimator that maximizes Fisher
    information.
    From Bayes theorem we know that:

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  14. Phase super-sensitivity
    Oleg Żero
    We build the estimator that maximizes Fisher
    information.
    From Bayes theorem we know that:
    So performing the measurement again and again...

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  15. Phase super-sensitivity
    Oleg Żero
    We build the estimator that maximizes Fisher
    information.
    From Bayes theorem we know that:
    So performing the measurement again and again...

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  16. Phase super-sensitivity
    Oleg Żero
    We build the estimator that maximizes Fisher
    information.
    This is a function of likelihood of how accurate our guess is.
    And it has the properties we need:
    • It peaks up whenever our guess is correct.
    • It minimizes the variance.
    So all we need to do is to find input Fock coefficients, whose output
    probabilities maximize the Fisher information.

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  17. Phase super-sensitivity
    Oleg Żero
    The results.
    Fisher information [a.u.]
    Transmittance through branch [unity] Transmittance through branch [unity]
    Symmetric case Asymmetric case
    N = 2
    The N00N state.
    Any particular
    state for a given
    interval.
    Classical
    measurement.
    The overall
    maximum.
    The N00N state matches the quantum
    limit when no loss is present.
    Otherwise, it’s decoherence.
    When loss is present, other more exotic
    states do better.
    There is a trade-off between sensitivity
    and robustness.

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  18. Phase super-sensitivity
    Oleg Żero
    The results.
    Fisher information [a.u.]
    Transmittance through branch [unity] Transmittance through branch [unity]
    Symmetric case Asymmetric case
    N = 2
    The N00N state.
    Any particular
    state for a given
    interval.
    Classical
    measurement.
    The overall
    maximum.
    T. W. Lee et al.:
    “Optimization of quantum interferometric metrological sensors in the presence of photon loss”
    Physical Review A 80, 063803 (2009).
    Others were also pursuing this problem at that
    time.
    But comparing their work for N = 2, 3 photons,
    shows
    Perfect match!

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