OlegZero13
March 05, 2017
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# MSc thesis

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

March 05, 2017

## Transcript

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

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

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.

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:

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

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

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

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

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.

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.

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

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.

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

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...

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...

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.

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.

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!