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�

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�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�

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�τ

A BC

D

Cesiumbeam

. . .

Quartz crystal

.

.

|g�

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�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

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.

.

|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

. . .

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.

.

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�τ

A BC

D

Cesiumbeam

. . .

Quartz crystal

.

.

|g�

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�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!