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Atomic clocks Martin Fraas Avronfest, July 2013
Transcript of Atomic clocks - huji.ac.ilmath.huji.ac.il › ~avronfest › Fraas.pdfBrief history of man made...
Atomic clocks
Martin Fraas
Avronfest, July 2013
” For example, a clock, with a running time of a day and anaccuracy of 10−8 second, must weigh almost a gram—for reasonsstemming solely from uncertainty principles and similarconsiderations.”
” For example, a clock, with a running time of a day and anaccuracy of 10−8 second, must weigh almost a gram—for reasonsstemming solely from uncertainty principles and similarconsiderations.”
Accuracy of Clocks
Frequency ω(t) = ω0 + ϕ(t)
Clocktime tclock = 1ω0
∫ t0 ω(s)ds
Clocktime variance (∆t)2 = 〈(tclock − t)2〉
log(∆tt )
Random walkof frequency
Quartz
Atomic clock
1 100 104 106�12
�10
�8
�6
�4
t[s]
Accuracy of Clocks
Frequency ω(t) = ω0 + ϕ(t)
Clocktime tclock = 1ω0
∫ t0 ω(s)ds
Clocktime variance (∆t)2 = 〈(tclock − t)2〉
log(∆tt )
Random walkof frequency
Quartz
Atomic clock
1 100 104 106�12
�10
�8
�6
�4
t[s]
Accuracy of Clocks
Frequency ω(t) = ω0 + ϕ(t)
Clocktime tclock = 1ω0
∫ t0 ω(s)ds
Clocktime variance (∆t)2 = 〈(tclock − t)2〉
log(∆tt )
Random walkof frequency
Quartz
Atomic clock
1 100 104 106�12
�10
�8
�6
�4
t[s]
Accuracy of Clocks
Frequency ω(t) = ω0 + ϕ(t)
Clocktime tclock = 1ω0
∫ t0 ω(s)ds
Clocktime variance (∆t)2 = 〈(tclock − t)2〉
log(∆tt )
Random walkof frequency
Quartz
Atomic clock
1 100 104 106�12
�10
�8
�6
�4
t[s]
Accuracy of Clocks
Frequency ω(t) = ω0 + ϕ(t)
Clocktime tclock = 1ω0
∫ t0 ω(s)ds
Clocktime variance (∆t)2 = 〈(tclock − t)2〉
log(∆tt )
Random walkof frequency
Quartz
Atomic clock
1 100 104 106�12
�10
�8
�6
�4
t[s]
Brief history of man made clocks
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Brief history of man made clocks
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Brief history of man made clocks
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Brief history of man made clocks
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Brief history of man made clocks
Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )Year Clock Accuracy
1761
1930
1955
2010
Harrison’s H4
Quartz
Cs Atomic Clock
AI+ Optical Clock
0.2 s
500 µs
10 µs
10−6 µs
( ∆tday )
Theoretical Challenges
I Improvement of atomic clocksI Employment of entangled states [Bollinger et. al. 96]
I Quantum logic spectroscopy [Schmidt et. al. 05]
I Fighting noise
I Limits of the atomic clock accuracy [Itano et. al. 93]I Inclusion of decoherence [Huelga et. al. 97]
I Limits on size, mass, power, etc.
Theoretical Challenges
I Improvement of atomic clocksI Employment of entangled states [Bollinger et. al. 96]
I Quantum logic spectroscopy [Schmidt et. al. 05]
I Fighting noise
I Limits of the atomic clock accuracy [Itano et. al. 93]I Inclusion of decoherence [Huelga et. al. 97]
I Limits on size, mass, power, etc.
Theoretical Challenges
I Improvement of atomic clocksI Employment of entangled states [Bollinger et. al. 96]
I Quantum logic spectroscopy [Schmidt et. al. 05]
I Fighting noise
I Limits of the atomic clock accuracy [Itano et. al. 93]I Inclusion of decoherence [Huelga et. al. 97]
I Limits on size, mass, power, etc.
Theoretical Challenges
I Improvement of atomic clocksI Employment of entangled states [Bollinger et. al. 96]
I Quantum logic spectroscopy [Schmidt et. al. 05]
I Fighting noise
I Limits of the atomic clock accuracy [Itano et. al. 93]I Inclusion of decoherence [Huelga et. al. 97]
I Limits on size, mass, power, etc.
The operation of a Cesium Atomic Clock
Classical oscillator ω(t) = ω0 + ϕ(t)︸︷︷︸frequency error
Quantum oscillator ω0 − frequency reference
Main idea: want to adjust ω(t) to ω0 (i.e. make ϕ(t) small)
by means of repeated synchronization.
.A B
C
D
Cesiumbeam
Detector
Quartz crystal
Cavity
The operation of a Cesium Atomic Clock
Classical oscillator ω(t) = ω0 + ϕ(t)︸︷︷︸
frequency error
Quantum oscillator ω0 − frequency reference
Main idea: want to adjust ω(t) to ω0 (i.e. make ϕ(t) small)
by means of repeated synchronization.
.A B
C
D
Cesiumbeam
Detector
Quartz crystal
Cavity
The operation of a Cesium Atomic Clock
Classical oscillator ω(t) = ω0 + ϕ(t)︸︷︷︸frequency error
Quantum oscillator ω0 − frequency reference
Main idea: want to adjust ω(t) to ω0 (i.e. make ϕ(t) small)
by means of repeated synchronization.
.A B
C
D
Cesiumbeam
Detector
Quartz crystal
Cavity
The operation of a Cesium Atomic Clock
Classical oscillator ω(t) = ω0 + ϕ(t)︸︷︷︸frequency error
Quantum oscillator ω0
− frequency reference
Main idea: want to adjust ω(t) to ω0 (i.e. make ϕ(t) small)
by means of repeated synchronization.
.A B
C
D
Cesiumbeam
Detector
Quartz crystal
Cavity
The operation of a Cesium Atomic Clock
Classical oscillator ω(t) = ω0 + ϕ(t)︸︷︷︸frequency error
Quantum oscillator ω0 − frequency reference
Main idea: want to adjust ω(t) to ω0 (i.e. make ϕ(t) small)
by means of repeated synchronization.
.A B
C
D
Cesiumbeam
Detector
Quartz crystal
Cavity
The operation of a Cesium Atomic Clock
Classical oscillator ω(t) = ω0 + ϕ(t)︸︷︷︸frequency error
Quantum oscillator ω0 − frequency reference
Main idea: want to adjust ω(t) to ω0 (i.e. make ϕ(t) small)
by means of repeated synchronization.
.A B
C
D
Cesiumbeam
Detector
Quartz crystal
Cavity
The operation of a Cesium Atomic Clock
Classical oscillator ω(t) = ω0 + ϕ(t)︸︷︷︸frequency error
Quantum oscillator ω0 − frequency reference
Main idea: want to adjust ω(t) to ω0 (i.e. make ϕ(t) small)
by means of repeated synchronization.
.A B
C
D
Cesiumbeam
Detector
Quartz crystal
Cavity
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρ
depends on the accumulated frequency error ϕτ :=∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Ramsey interferometry
A BC
D
. . .
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
ρρ0
ω(t) = ω0 + ϕ(t)
A BC
D
Cesiumbeam
. . .
Quartz crystal
.
.
|g�
|e�
�τ
Relative evolution of the state for time τ : ρ0 7→ ρdepends on the accumulated frequency error ϕτ :=
∫ τ0 ϕ(s)ds
ρ0 → ρ(ϕτ ) := e−iϕτHρ0eiϕτH
Detection and feedback
C
D
.B
ω(t) = ω0 + ϕnew(t)
ϕτ
x
POVM
ϕ̂
A POVM measurement on the final state Rπ2ρ
assigns a measurement outcome x to the accumulated frequency error ϕτ .
ϕ̂ is an estimation of ϕτ , based on the measurement outcome x .
A feedback uses ϕ̂ to adjust the original frequency error ϕ.
Detection and feedback
C
D
.B
ω(t) = ω0 + ϕnew(t)
ϕτ
x
POVM
ϕ̂
A POVM measurement on the final state Rπ2ρ
assigns a measurement outcome x to the accumulated frequency error ϕτ .
ϕ̂ is an estimation of ϕτ , based on the measurement outcome x .
A feedback uses ϕ̂ to adjust the original frequency error ϕ.
Detection and feedback
C
D
.B
ω(t) = ω0 + ϕnew(t)
ϕτ
x
POVM
ϕ̂
A POVM measurement on the final state Rπ2ρ
assigns a measurement outcome x to the accumulated frequency error ϕτ .
ϕ̂ is an estimation of ϕτ , based on the measurement outcome x .
A feedback uses ϕ̂ to adjust the original frequency error ϕ.
Detection and feedback
C
D
.B
ω(t) = ω0 + ϕnew(t)
ϕτ
x
POVM
ϕ̂
A POVM measurement on the final state Rπ2ρ
assigns a measurement outcome x to the accumulated frequency error ϕτ .
ϕ̂ is an estimation of ϕτ , based on the measurement outcome x .
A feedback uses ϕ̂ to adjust the original frequency error ϕ.
Detection and feedback
C
D
.B
ω(t) = ω0 + ϕnew(t)
ϕτ
x
POVM
ϕ̂
A POVM measurement on the final state Rπ2ρ
assigns a measurement outcome x to the accumulated frequency error ϕτ .
ϕ̂ is an estimation of ϕτ , based on the measurement outcome x .
A feedback uses ϕ̂ to adjust the original frequency error ϕ.
The Mathematical Model
I The evolution of ϕ(t) in absence of synchronization:
ϕ(t + s) := ϕ(t) +√
2D Ws︸︷︷︸Wiener process
I Two fixed time scales:
Ttime between two consecutive synchronizations
≥ τinterrogation time
I A function ϕτ 7→ ρ(ϕτ )
I An estimation strategy {ρ(ϕτ ) 7→ x , x 7→ ϕ̂}.
I A linear feedback ϕ(t) 7→ ϕ(t)− ϕ̂.
The Mathematical Model
I The evolution of ϕ(t) in absence of synchronization:
ϕ(t + s) := ϕ(t) +√
2D Ws︸︷︷︸Wiener process
I Two fixed time scales:
Ttime between two consecutive synchronizations
≥ τinterrogation time
I A function ϕτ 7→ ρ(ϕτ )
I An estimation strategy {ρ(ϕτ ) 7→ x , x 7→ ϕ̂}.
I A linear feedback ϕ(t) 7→ ϕ(t)− ϕ̂.
The Mathematical Model
I The evolution of ϕ(t) in absence of synchronization:
ϕ(t + s) := ϕ(t) +√
2D Ws︸︷︷︸Wiener process
I Two fixed time scales:
Ttime between two consecutive synchronizations
≥ τinterrogation time
I A function ϕτ 7→ ρ(ϕτ )
I An estimation strategy {ρ(ϕτ ) 7→ x , x 7→ ϕ̂}.
I A linear feedback ϕ(t) 7→ ϕ(t)− ϕ̂.
The Mathematical Model
I The evolution of ϕ(t) in absence of synchronization:
ϕ(t + s) := ϕ(t) +√
2D Ws︸︷︷︸Wiener process
I Two fixed time scales:
Ttime between two consecutive synchronizations
≥ τinterrogation time
I A function ϕτ 7→ ρ(ϕτ )
I An estimation strategy {ρ(ϕτ ) 7→ x , x 7→ ϕ̂}.
I A linear feedback ϕ(t) 7→ ϕ(t)− ϕ̂.
The Mathematical Model
I The evolution of ϕ(t) in absence of synchronization:
ϕ(t + s) := ϕ(t) +√
2D Ws︸︷︷︸Wiener process
I Two fixed time scales:
Ttime between two consecutive synchronizations
≥ τinterrogation time
I A function ϕτ 7→ ρ(ϕτ )
I An estimation strategy {ρ(ϕτ ) 7→ x , x 7→ ϕ̂}.
I A linear feedback ϕ(t) 7→ ϕ(t)− ϕ̂.
The Mathematical Model
I The evolution of ϕ(t) in absence of synchronization:
ϕ(t + s) := ϕ(t) +√
2D Ws︸︷︷︸Wiener process
I Two fixed time scales:
Ttime between two consecutive synchronizations
≥ τinterrogation time
I A function ϕτ 7→ ρ(ϕτ )
I An estimation strategy {ρ(ϕτ ) 7→ x , x 7→ ϕ̂}.
I A linear feedback ϕ(t) 7→ ϕ(t)− ϕ̂.
Repeated Synchronization
←→τ
nT (n + 1)T
ϕn ϕn+1
Notation: ϕn := ϕ((n + 1)T−).
Equation for the jump:
ϕn+1 = ϕn − ϕ̂n +√
2DWT
I The equation defines a non-linear Markovian process;
I We aim to study its stationary solutions;
I ϕn provides ϕ(t), which gives the clock time;
Repeated Synchronization
←→τ
nT (n + 1)T
ϕn ϕn+1
Notation: ϕn := ϕ((n + 1)T−).
Equation for the jump:
ϕn+1 = ϕn − ϕ̂n +√
2DWT
I The equation defines a non-linear Markovian process;
I We aim to study its stationary solutions;
I ϕn provides ϕ(t), which gives the clock time;
Repeated Synchronization
←→τ
nT (n + 1)T
ϕn ϕn+1
Notation: ϕn := ϕ((n + 1)T−).
Equation for the jump:
ϕn+1 = ϕn − ϕ̂n +√
2DWT
I The equation defines a non-linear Markovian process;
I We aim to study its stationary solutions;
I ϕn provides ϕ(t), which gives the clock time;
Repeated Synchronization
←→τ
nT (n + 1)T
ϕn ϕn+1
Notation: ϕn := ϕ((n + 1)T−).
Equation for the jump:
ϕn+1 = ϕn − ϕ̂n +√
2DWT
I The equation defines a non-linear Markovian process;
I We aim to study its stationary solutions;
I ϕn provides ϕ(t), which gives the clock time;
Repeated Synchronization
←→τ
nT (n + 1)T
ϕn ϕn+1
Notation: ϕn := ϕ((n + 1)T−).
Equation for the jump:
ϕn+1 = ϕn − ϕ̂n +√
2DWT
I The equation defines a non-linear Markovian process;
I We aim to study its stationary solutions;
I ϕn provides ϕ(t), which gives the clock time;
Repeated Synchronization
←→τ
nT (n + 1)T
ϕn ϕn+1
Notation: ϕn := ϕ((n + 1)T−).
Equation for the jump:
ϕn+1 = ϕn − ϕ̂n +√
2DWT
I The equation defines a non-linear Markovian process;
I We aim to study its stationary solutions;
I ϕn provides ϕ(t), which gives the clock time;
Unbiased clock
Unbiased clock is accurate in average, E[tclock ] = t.
⇓
E[ϕ(s)] = 0 provided E[ϕ(0)] = 0.
(variational argument) ⇓ (variational argument)
For some ζ ∈ R, E[ϕ− ϕ̂|ϕ] = ζϕ.
Definition (ζ-unbiased clock)
The clock is ζ-unbiased if the estimation procedure satisfies
E[ϕ− ϕ̂|ϕ] = ζϕ, |ζ| < 1.
Unbiased clock
Unbiased clock is accurate in average, E[tclock ] = t.
⇓
E[ϕ(s)] = 0 provided E[ϕ(0)] = 0.
(variational argument) ⇓ (variational argument)
For some ζ ∈ R, E[ϕ− ϕ̂|ϕ] = ζϕ.
Definition (ζ-unbiased clock)
The clock is ζ-unbiased if the estimation procedure satisfies
E[ϕ− ϕ̂|ϕ] = ζϕ, |ζ| < 1.
Unbiased clock
Unbiased clock is accurate in average, E[tclock ] = t.
⇓
E[ϕ(s)] = 0 provided E[ϕ(0)] = 0.
(variational argument) ⇓ (variational argument)
For some ζ ∈ R, E[ϕ− ϕ̂|ϕ] = ζϕ.
Definition (ζ-unbiased clock)
The clock is ζ-unbiased if the estimation procedure satisfies
E[ϕ− ϕ̂|ϕ] = ζϕ, |ζ| < 1.
Unbiased clock
Unbiased clock is accurate in average, E[tclock ] = t.
⇓
E[ϕ(s)] = 0 provided E[ϕ(0)] = 0.
(variational argument) ⇓ (variational argument)
For some ζ ∈ R, E[ϕ− ϕ̂|ϕ] = ζϕ.
Definition (ζ-unbiased clock)
The clock is ζ-unbiased if the estimation procedure satisfies
E[ϕ− ϕ̂|ϕ] = ζϕ, |ζ| < 1.
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)Fisher information is inversely proportional to the width .
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)Fisher information is inversely proportional to the width .
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)Fisher information is inversely proportional to the width .
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)Fisher information is inversely proportional to the width .
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)Fisher information is inversely proportional to the width .
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)Fisher information is inversely proportional to the width .
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)
Fisher information is inversely proportional to the width .
Fisher information
How much information about ϕ is in ρ(ϕ)?
F (ϕ) := Tr(ρ(ϕ)L2ϕ),
1
2{Lϕ, ρ(ϕ)} = ρ̇(ϕ).
For a pure state, F (ϕ) is the Fubini-Study metric.
Scaling: ρ(ϕ)→ ρ(τϕ), F (ϕ)→ τ2F (τϕ).
Example (Coherent states)
< x |ψ(ϕ) >=F 1/4
(2π)1/4exp
(−F
4(x − ϕ)2
)Fisher information is inversely proportional to the width .
Stationary states
TheoremLet ϕn be a stationary state of an ζ-unbiased clock. Then
E[ϕ2n]
≥ 1
τ2F
1− ζ1 + ζ
+2DT
1− ζ2 g(ζ,τ
T),
E[(tclock − t)2] ≥ tT
ω20
(1
τ2F+
2DT
3(1− ζ)2f (ζ,
τ
T)
),
where
g(ζ, x) = ζ2 +1 + ζ − 2ζ2
3x ,
f (ζ, x) = 1 + ζ + ζ2 + (1 + 2ζ)(1− ζ)x + (1− ζ)2x2,
1
F= E
[1
F (τϕn)
].
Stationary states
TheoremLet ϕn be a stationary state of an ζ-unbiased clock.
Then
E[ϕ2n]
≥ 1
τ2F
1− ζ1 + ζ
+2DT
1− ζ2 g(ζ,τ
T),
E[(tclock − t)2] ≥ tT
ω20
(1
τ2F+
2DT
3(1− ζ)2f (ζ,
τ
T)
),
where
g(ζ, x) = ζ2 +1 + ζ − 2ζ2
3x ,
f (ζ, x) = 1 + ζ + ζ2 + (1 + 2ζ)(1− ζ)x + (1− ζ)2x2,
1
F= E
[1
F (τϕn)
].
Stationary states
TheoremLet ϕn be a stationary state of an ζ-unbiased clock. Then
E[ϕ2n] ≥ 1
τ2F
1− ζ1 + ζ
+
2DT
1− ζ2 g(ζ,τ
T),
E[(tclock − t)2] ≥ tT
ω20
(1
τ2F+
2DT
3(1− ζ)2f (ζ,
τ
T)
),
where
g(ζ, x) = ζ2 +1 + ζ − 2ζ2
3x ,
f (ζ, x) = 1 + ζ + ζ2 + (1 + 2ζ)(1− ζ)x + (1− ζ)2x2,
1
F= E
[1
F (τϕn)
].
Stationary states
TheoremLet ϕn be a stationary state of an ζ-unbiased clock. Then
E[ϕ2n] ≥ 1
τ2F
1− ζ1 + ζ
+2DT
1− ζ2 g(ζ,τ
T),
E[(tclock − t)2] ≥ tT
ω20
(1
τ2F+
2DT
3(1− ζ)2f (ζ,
τ
T)
),
where
g(ζ, x) = ζ2 +1 + ζ − 2ζ2
3x ,
f (ζ, x) = 1 + ζ + ζ2 + (1 + 2ζ)(1− ζ)x + (1− ζ)2x2,
1
F= E
[1
F (τϕn)
].
Stationary states
TheoremLet ϕn be a stationary state of an ζ-unbiased clock. Then
E[ϕ2n] ≥ 1
τ2F
1− ζ1 + ζ
+2DT
1− ζ2 g(ζ,τ
T),
E[(tclock − t)2] ≥ tT
ω20
(1
τ2F+
2DT
3(1− ζ)2f (ζ,
τ
T)
),
where
g(ζ, x) = ζ2 +1 + ζ − 2ζ2
3x ,
f (ζ, x) = 1 + ζ + ζ2 + (1 + 2ζ)(1− ζ)x + (1− ζ)2x2,
1
F= E
[1
F (τϕn)
].
Stationary states
TheoremLet ϕn be a stationary state of an ζ-unbiased clock. Then
E[ϕ2n] ≥ 1
τ2F
1− ζ1 + ζ
+2DT
1− ζ2 g(ζ,τ
T),
E[(tclock − t)2] ≥ tT
ω20
(1
τ2F+
2DT
3(1− ζ)2f (ζ,
τ
T)
),
where
g(ζ, x) = ζ2 +1 + ζ − 2ζ2
3x ,
f (ζ, x) = 1 + ζ + ζ2 + (1 + 2ζ)(1− ζ)x + (1− ζ)2x2,
1
F= E
[1
F (τϕn)
].
Stationary states
TheoremLet ϕn be a stationary state of an ζ-unbiased clock. Then
E[ϕ2n] ≥ 1
τ2F
1− ζ1 + ζ
+2DT
1− ζ2 g(ζ,τ
T),
E[(tclock − t)2] ≥ tT
ω20
(1
τ2F+
2DT
3(1− ζ)2f (ζ,
τ
T)
),
where
g(ζ, x) = ζ2 +1 + ζ − 2ζ2
3x ,
f (ζ, x) = 1 + ζ + ζ2 + (1 + 2ζ)(1− ζ)x + (1− ζ)2x2,
1
F= E
[1
F (τϕn)
].
Analysis of the clock operation
Qualitative analysis of the stationary state:
I The clock time diffuses;
I For D = 0 the diffusion does not depend on the correlationlength ζ;
Quantitative analysis, the case T = τ :
I The optimal interrogation time is determined by a balance ofthe dissipation and estimation precision. For fixed ζ:
4DT = (1− ζ)21
FT 2.
I For the optimal time, ζ ≈ 0.35 minimize the variance of thestationary state;
Analysis of the clock operation
Qualitative analysis of the stationary state:
I The clock time diffuses;
I For D = 0 the diffusion does not depend on the correlationlength ζ;
Quantitative analysis, the case T = τ :
I The optimal interrogation time is determined by a balance ofthe dissipation and estimation precision. For fixed ζ:
4DT = (1− ζ)21
FT 2.
I For the optimal time, ζ ≈ 0.35 minimize the variance of thestationary state;
Analysis of the clock operation
Qualitative analysis of the stationary state:
I The clock time diffuses;
I For D = 0 the diffusion does not depend on the correlationlength ζ;
Quantitative analysis, the case T = τ :
I The optimal interrogation time is determined by a balance ofthe dissipation and estimation precision. For fixed ζ:
4DT = (1− ζ)21
FT 2.
I For the optimal time, ζ ≈ 0.35 minimize the variance of thestationary state;
Pieces of the Proof
I ϕn is a supermartingale =⇒ existence of a stationary state;
I Cramer-Rao type inequality: Suppose E[ϕϕ̂] = ζE[ϕ2] then
E[(ϕ− ϕ̂)2] ≥ (1− ζ)21
F+ ζ2E[ϕ2];
Local CR, ζ = 0 Global CR, infζ = 1/(F + E[ϕ2]−1)
I Compute covariance of ϕ(t), e.g. E[ϕn+hϕn] = ζhE[ϕ2n].
Integration gives variance of the clock time.
Pieces of the Proof
I ϕn is a supermartingale =⇒ existence of a stationary state;
I Cramer-Rao type inequality: Suppose E[ϕϕ̂] = ζE[ϕ2] then
E[(ϕ− ϕ̂)2] ≥ (1− ζ)21
F+ ζ2E[ϕ2];
Local CR, ζ = 0 Global CR, infζ = 1/(F + E[ϕ2]−1)
I Compute covariance of ϕ(t), e.g. E[ϕn+hϕn] = ζhE[ϕ2n].
Integration gives variance of the clock time.
Pieces of the Proof
I ϕn is a supermartingale =⇒ existence of a stationary state;
I Cramer-Rao type inequality: Suppose E[ϕϕ̂] = ζE[ϕ2] then
E[(ϕ− ϕ̂)2] ≥ (1− ζ)21
F+ ζ2E[ϕ2];
Local CR, ζ = 0 Global CR, infζ = 1/(F + E[ϕ2]−1)
I Compute covariance of ϕ(t), e.g. E[ϕn+hϕn] = ζhE[ϕ2n].
Integration gives variance of the clock time.
Pieces of the Proof
I ϕn is a supermartingale =⇒ existence of a stationary state;
I Cramer-Rao type inequality: Suppose E[ϕϕ̂] = ζE[ϕ2] then
E[(ϕ− ϕ̂)2] ≥ (1− ζ)21
F+ ζ2E[ϕ2];
Local CR, ζ = 0
Global CR, infζ = 1/(F + E[ϕ2]−1)
I Compute covariance of ϕ(t), e.g. E[ϕn+hϕn] = ζhE[ϕ2n].
Integration gives variance of the clock time.
Pieces of the Proof
I ϕn is a supermartingale =⇒ existence of a stationary state;
I Cramer-Rao type inequality: Suppose E[ϕϕ̂] = ζE[ϕ2] then
E[(ϕ− ϕ̂)2] ≥ (1− ζ)21
F+ ζ2E[ϕ2];
Local CR, ζ = 0 Global CR, infζ = 1/(F + E[ϕ2]−1)
I Compute covariance of ϕ(t), e.g. E[ϕn+hϕn] = ζhE[ϕ2n].
Integration gives variance of the clock time.
Pieces of the Proof
I ϕn is a supermartingale =⇒ existence of a stationary state;
I Cramer-Rao type inequality: Suppose E[ϕϕ̂] = ζE[ϕ2] then
E[(ϕ− ϕ̂)2] ≥ (1− ζ)21
F+ ζ2E[ϕ2];
Local CR, ζ = 0 Global CR, infζ = 1/(F + E[ϕ2]−1)
I Compute covariance of ϕ(t), e.g. E[ϕn+hϕn] = ζhE[ϕ2n].
Integration gives variance of the clock time.
Pieces of the Proof
I ϕn is a supermartingale =⇒ existence of a stationary state;
I Cramer-Rao type inequality: Suppose E[ϕϕ̂] = ζE[ϕ2] then
E[(ϕ− ϕ̂)2] ≥ (1− ζ)21
F+ ζ2E[ϕ2];
Local CR, ζ = 0 Global CR, infζ = 1/(F + E[ϕ2]−1)
I Compute covariance of ϕ(t), e.g. E[ϕn+hϕn] = ζhE[ϕ2n].
Integration gives variance of the clock time.
Conclusions & Outlooks
Conclusions:
I Mathematically minded model of atomic clocks;
I Analysis of the stationary state;
Outlooks:
I ”Central limit theorem“;
I Beyond unbiased clock, unbiased stationary state;
I Entropy production;
Conclusions & Outlooks
Conclusions:
I Mathematically minded model of atomic clocks;
I Analysis of the stationary state;
Outlooks:
I ”Central limit theorem“;
I Beyond unbiased clock, unbiased stationary state;
I Entropy production;
Conclusions & Outlooks
Conclusions:
I Mathematically minded model of atomic clocks;
I Analysis of the stationary state;
Outlooks:
I ”Central limit theorem“;
I Beyond unbiased clock, unbiased stationary state;
I Entropy production;
Happy Birthday
Yosi!
