[PPT] - Phase Measurements at Phase Measurements at the Theoretical Limit PowerPoint Presentation

SLIDE 1

Phase Measurements at Phase Measurements at the Theoretical Limit the Theoretical Limit

Dominic Berry Dominic Berry

Institute for Quantum Computing Institute for Quantum Computing Brendon Higgins, Howard Wiseman, Steve Bartlett, Brendon Higgins, Howard Wiseman, Steve Bartlett, Morgan Mitchell, Geoff Morgan Mitchell, Geoff Pryde Pryde

SLIDE 2

Applications of phase measurement Applications of phase measurement

Communication Frequency and time measurement Distance measurement

SLIDE 3

The Heisenberg Uncertainty Principle The Heisenberg Uncertainty Principle

2 p x Δ Δ ≥

position & momentum

Y X

Y Δ X Δ

quadratures

1 X Y Δ Δ ≥ 1 2

y z x

J J J Δ Δ ≥

spin

y z x

SLIDE 4

The Heisenberg limit The Heisenberg limit vs vs the standard quantum limit the standard quantum limit

The Heisenberg Limit The Heisenberg Limit

If one uncertainty is

If one uncertainty is reduced as much as reduced as much as possible. possible.

Uncertainty scaling

Uncertainty scaling

1 N φ Δ ∝

The Standard The Standard Quantum Limit Quantum Limit

If the two uncertainties

If the two uncertainties are equal. are equal.

Uncertainty scaling

Uncertainty scaling

1 N φ Δ ∝

SLIDE 5

Types of phase measurement Types of phase measurement

Single Single-

mode phase

mode phase

φest φ

input state

φest

Interferometric measurement Interferometric measurement

φ

input state

ptical
ptical

y z x

atomic atomic

y z x y z x y z x

Jz measurement z rotation y rotation y rotation

SLIDE 6

Interferometric measurement Interferometric measurement Single Single-

mode phase

mode phase

φest φ

input state

φest φ

input state

ptical
ptical

y z x

atomic atomic

y z x y z x y z x

Jz measurement z rotation y rotation y rotation

Types of phase measurement Types of phase measurement

SLIDE 7

Single Single-

mode measurements

mode measurements

Signal is beam is mixed with

Signal is beam is mixed with strong strong “ “local oscillator local oscillator” ”. .

Heterodyne

Heterodyne – – linear variation linear variation

f
f θ

θ. .

Homodyne

Homodyne – – θ θ close to close to φ φ. .

φ

input state

φest

local

scillator

θ (t) φ θ (t)

processor

−

input state local

scillator
Use an estimate of the

Use an estimate of the phase to approximate a phase to approximate a homodyne measurement. homodyne measurement.

SLIDE 8

Adaptive phase measurement Adaptive phase measurement

Task: measure an arbitrary phase Task: measure an arbitrary phase

φ

input state local oscillator

θ θ

processor

−

H. M. Wiseman, Phys. Rev. A 56, 944 (1997).
H. M. Wiseman, Phys. Rev. A 57, 2169 (1998).

final phase final phase estimate estimate

Total phase uncertainty Total phase uncertainty Δφ

Δφ 2 = (intrinsic uncertainty)2 + (uncertainty due to measurement)2

SLIDE 9

Adaptive phase measurement Adaptive phase measurement

Task: measure an arbitrary phase Task: measure an arbitrary phase

φ

input state local oscillator

θ θ

processor

−

H. M. Wiseman, Phys. Rev. A 56, 944 (1997).
H. M. Wiseman, Phys. Rev. A 57, 2169 (1998).

final phase final phase estimate estimate

Total phase uncertainty Total phase uncertainty Δφ

Δφ 2 = (intrinsic uncertainty)2 + (uncertainty due to measurement)2 heterodyne measurements

1 N ∝

SLIDE 10

Adaptive phase measurement Adaptive phase measurement

Task: measure an arbitrary phase Task: measure an arbitrary phase

φ

input state local oscillator

θ θ

processor

−

H. M. Wiseman, Phys. Rev. A 56, 944 (1997).
H. M. Wiseman, Phys. Rev. A 57, 2169 (1998).

final phase final phase estimate estimate

Total phase uncertainty Total phase uncertainty Δφ

Δφ 2 = (intrinsic uncertainty)2 + (uncertainty due to measurement)2 heterodyne measurements

1 N ∝

The standard quantum limit

SLIDE 11

Wiseman Mark I Wiseman Mark I

Task: measure an arbitrary phase Task: measure an arbitrary phase

☺ ☺ ≡ ≡ best phase estimate best phase estimate

≡

≡ poor phase estimate poor phase estimate φ

input state local oscillator

processor

−

Mark I feedback phase estimate: final phase estimate: ⇒ ideal phase measurement for N = 1 ⇒ for N >> 1

1/4

1 N φ Δ ∝

H. M. Wiseman, Phys. Rev. A 56, 944 (1997).
H. M. Wiseman, Phys. Rev. A 57, 2169 (1998).
Worse than standard

quantum limit!

SLIDE 12

Wiseman Mark II Wiseman Mark II

Task: measure an arbitrary phase Task: measure an arbitrary phase

☺ ☺ ≡ ≡ best phase estimate best phase estimate

≡

≡ poor phase estimate poor phase estimate φ

input state local oscillator

processor

−

Mark II feedback phase estimate: final phase estimate: ☺ ⇒ ideal phase measurement for N = 1 ⇒ for N >> 1

3/4

1 N φ Δ ∝

H. M. Wiseman, Phys. Rev. A 56, 944 (1997).
H. M. Wiseman, Phys. Rev. A 57, 2169 (1998).

☺ ☺

Beats the standard quantum limit

SLIDE 13

Optimal adaptive Optimal adaptive

Task: measure an arbitrary phase Task: measure an arbitrary phase

☺ ☺ ≡ ≡ best phase estimate best phase estimate

≡

≡ poor phase estimate poor phase estimate ☯ ☯ ≡ ≡ intermediate intermediate φ

input state local oscillator

☯ ☯

processor

−

Mark II feedback phase estimate: ☯ final phase estimate: ☺ ⇒ ideal phase measurement for N = 1 ⇒ for N >> 1

ln N N φ Δ ∝

D. W. Berry and H. M. Wiseman,
Phys. Rev. A 63, 013813 (2001).

☺ ☺

Almost the Heisenberg limit

SLIDE 14

Single Single-

mode phase

mode phase

φ

input state

φest

y z x

atomic atomic

y z x y z x y z x

Jz measurement z rotation y rotation y rotation

φest φ

input state

ptical
ptical

Interferometric measurement Interferometric measurement

Types of phase measurement Types of phase measurement

SLIDE 15

Optical interferometry Optical interferometry

Theoretical limit

Theoretical limit

Squeezed states

Squeezed states1

1

NOON states

NOON states2

2

Theoretical

Theoretical-

limit adaptive measurements

limit adaptive measurements3

3

Theoretical

Theoretical-

limit

limit non nonadaptive adaptive measurements measurements4

4

Hybrid measurements

Hybrid measurements4

4

3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman,

and G. J. Pryde, Nature 450, 393 (2007).

4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell,

H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

1 C. M. Caves, Phys. Rev. D 23, 1693 (1981). 2 B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

SLIDE 16

Optical interferometry Optical interferometry

Theoretical limit

Theoretical limit

Squeezed states

Squeezed states1

1

NOON states

NOON states2

2

Theoretical

Theoretical-

limit adaptive measurements

limit adaptive measurements3

3

Theoretical

Theoretical-

limit

limit non nonadaptive adaptive measurements measurements4

4

Hybrid measurements

Hybrid measurements4

4

3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman,

and G. J. Pryde, Nature 450, 393 (2007).

4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell,

H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

1 C. M. Caves, Phys. Rev. D 23, 1693 (1981). 2 B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

SLIDE 17

Optimal measurements Optimal measurements

ptimal two-mode

joint measurement3

N photons

ptimal two-mode

entangled input state1,2

N φ π Δ ≈

The theoretical limit

φ

1 A. Luis and J. Peřina, Phys. Rev. A 54, 4564 (1996). 2 D. W. Berry and H. M. Wiseman, PRL 85, 5098 (2000). 3 B. C. Sanders and G. J. Milburn, PRL 75, 2944 (1995).

φest

SLIDE 18

θ

θ( (t t) ) is adjusted to minimise the expected variance after is adjusted to minimise the expected variance after the next detection. the next detection.

Gives uncertainty

Gives uncertainty

processor

N photons

How to perform the measurement? How to perform the measurement?

θ(t) φ

D. W. Berry and Wiseman, PRL 85, 5098 (2000).

1 N φ Δ ∼

SLIDE 19

How to create the input state? How to create the input state?

Two problems: Two problems:

1. 1.

The state needs to be a special coherent superposition The state needs to be a special coherent superposition

f the form
f the form

There is no known way of producing such a state. There is no known way of producing such a state.

2. 2.

The input mode needs to be very long so that The input mode needs to be very long so that θ θ

(

(t t) ) can can be adjusted between detections. be adjusted between detections.

N n n

n N n ψ

=

−

∑

SLIDE 20

Optical interferometry Optical interferometry

Theoretical limit

Theoretical limit

Squeezed states

Squeezed states1

1

NOON states

NOON states2

2

Theoretical

Theoretical-

limit adaptive measurements

limit adaptive measurements3

3

Theoretical

Theoretical-

limit

limit non nonadaptive adaptive measurements measurements4

4

Hybrid measurements

Hybrid measurements4

4

3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman,

and G. J. Pryde, Nature 450, 393 (2007).

4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell,

H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

1 C. M. Caves, Phys. Rev. D 23, 1693 (1981). 2 B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

SLIDE 21

Mach Mach-

Zehnder

Zehnder interferometer interferometer with coherent states with coherent states

φ

coherent state α vacuum

φest

without squeezing:

1/ N φ Δ ≈

The standard quantum limit

SLIDE 22

Mach Mach-

Zehnder

Zehnder interferometer interferometer with squeezed states with squeezed states

φ

coherent state α squeezed vacuum

φest

without squeezing:

1/ N φ Δ ≈

with squeezing:

/

r

e N φ

−

Δ ≈

C. M. Caves, Phys. Rev. D 23, 1693 (1981).

Beats the standard quantum limit

SLIDE 23

1/ N φ Δ ≈

Mach Mach-

Zehnder

Zehnder interferometer interferometer with NOON states with NOON states

input state

φ

,0 0, N N + ,0 0,

iN

e N N

φ

+

φ/π p(φ)

B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

φest

SLIDE 24

Optical interferometry Optical interferometry

Theoretical limit

Theoretical limit

Squeezed states

Squeezed states1

1

NOON states

NOON states2

2

Theoretical

Theoretical-

limit adaptive measurements

limit adaptive measurements3

3

Theoretical

Theoretical-

limit

limit non nonadaptive adaptive measurements measurements4

4

Hybrid measurements

Hybrid measurements4

4

3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman,

and G. J. Pryde, Nature 450, 393 (2007).

4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell,

H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

1 C. M. Caves, Phys. Rev. D 23, 1693 (1981). 2 B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

SLIDE 25

Eliminating the fringes Eliminating the fringes

θ(t) φ

φ/π p(φ)

φest

,0 0, N N +

SLIDE 26

Equivalence of NOON states and Equivalence of NOON states and multiple passes multiple passes

φ φest

,0 0, N N +

Photons detected at times t1, t2, … tN. ⇒ Passed through phase shift at times t1−Δt, t2 −Δt, … tN −Δt.

SLIDE 27

Equivalence of NOON states and Equivalence of NOON states and multiple passes multiple passes

φ φest

1,0 0,1 +

Electro-optic switches pass single photon through phase shift at times t1−Δt, t2 −Δt, … tN −Δt.

SLIDE 28

Equivalence of NOON states and Equivalence of NOON states and multiple passes multiple passes

φ φest

1,0 0,1 +

Electro-optic switches pass single photon through phase shift at times t1−Δt, t2 −Δt, … tN −Δt.

SLIDE 29

Equivalence of NOON states and Equivalence of NOON states and multiple passes multiple passes

1,0 0,1 +

φ φ φ φ φ φ

SLIDE 30

Equivalence of NOON states and Equivalence of NOON states and multiple passes multiple passes

1,0 0,1 +

Each splitting copies the photon:

φ φ φ φ φ φ

1,0 0,1 1,0 1,0 0,1 0,1 + +

SLIDE 31

Equivalence of NOON states and Equivalence of NOON states and multiple passes multiple passes

φ φ φ φ φ φ

,0 0, N N +

Copy the photons at the beginning to get the NOON state.

SLIDE 32

Eliminating the fringes Eliminating the fringes

θ(t) φ

φ/π p(φ)

φest

,0 0, N N +

B. L. Higgins, D. W. Berry, S. D. Bartlett,
H. M. Wiseman, and G. J. Pryde, Nature

450, 393 (2007).

SLIDE 33

Eliminating the fringes Eliminating the fringes

1 photon

θ(t) φ

φ/π p(φ)

φest

B. L. Higgins, D. W. Berry, S. D. Bartlett,
H. M. Wiseman, and G. J. Pryde, Nature

450, 393 (2007).

SLIDE 34

Eliminating the fringes Eliminating the fringes

φ/π p(φ) 1 photon

φest θ(t) φ

B. L. Higgins, D. W. Berry, S. D. Bartlett,
H. M. Wiseman, and G. J. Pryde, Nature

450, 393 (2007).

SLIDE 35

θ(t)

Eliminating the fringes Eliminating the fringes

φ/π p(φ) 1 photon

φest φ

B. L. Higgins, D. W. Berry, S. D. Bartlett,
H. M. Wiseman, and G. J. Pryde, Nature

450, 393 (2007).

SLIDE 36

θ(t)

φ/π p(φ)

Eliminating the fringes Eliminating the fringes

1 photon

φest φ

B. L. Higgins, D. W. Berry, S. D. Bartlett,
H. M. Wiseman, and G. J. Pryde, Nature

450, 393 (2007).

SLIDE 37

φ

( ) p φ

The uncertainty The uncertainty

The uncertainty is

The uncertainty is

This does not beat

This does not beat the SQL! the SQL!

The distribution has

The distribution has fat tails. fat tails.

2/ N φ Δ ≈

SLIDE 38

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

H H H H . . . . H

2K

U

The phase shifts are obtained from unitary

The phase shifts are obtained from unitary U U satisfying satisfying

i

U u e u

φ

=

2

1

K

i

e

φ

+

2

1

i

e

φ

+

1

2

1

i

e

φ

+

2

1

i

e

φ

+

1

2

1

K

i

e

φ

−

+

SLIDE 39

. . . . . . . .

Inverse quantum Fourier transform Inverse quantum Fourier transform

Provided

Provided φ φ is of the form is of the form φ φ = =π πr r/2 /2K

K, the inverse

, the inverse quantum Fourier transform gives the bits of quantum Fourier transform gives the bits of r r at at the output. the output.

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H . . . . . . . . . . . .

1/2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H H

2K

U

1/2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

inverse QFT

SLIDE 40

. . . . . . . .

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H . . . . . . . . . . . .

1/2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H H

2K

U

1/2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

SLIDE 41

. . . . . . . .

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H . . . . . . . . . . . .

1/2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H H

2K

U

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

SLIDE 42

. . . . . . . .

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H . . . . . . . . . . . .

1/2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H H

2K

U

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

SLIDE 43

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H

1/2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H

2K

U

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

H

SLIDE 44

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H

1/ 2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H

2K

U

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

H

SLIDE 45

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H

1/ 2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H

2K

U

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

H

SLIDE 46

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H

1/ 2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H

2K

U

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

H

SLIDE 47

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . . . . . .

2

U

1

2

U

2

U

u

1

2K

U

−

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H

1/ 2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H

2K

U

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

H

SLIDE 48

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . .

[ ] r

1

[ ] r

2

[ ] r [ ]K r

2

K

Z

− 1

2

K

Z

−

2 K

Z

−

H H H H H H

1/ 2

Z

1/4

Z

H

1/ 2

Z

H . . . .

1

[ ]K r

−

H H

2Kφ

1/ 2

Z

2

K

Z

− 1

2

K

Z

− 3

2

K

Z

−

1

2K φ

− 2

2 φ

1

2 φ 2 φ

1. 1.

The qubits are dual The qubits are dual-

rail single photons.

rail single photons.

2. 2.

The The Hadamard Hadamard is a beam splitter. is a beam splitter.

3. 3.

The controlled unitaries are the unknown phase in the The controlled unitaries are the unknown phase in the interferometer. interferometer.

4. 4.

The controlled phase operations are feedback to the phase The controlled phase operations are feedback to the phase θ θ( (t t). ).

5. 5.

The operations may be performed in sequence to reuse the The operations may be performed in sequence to reuse the same interferometer. same interferometer.

SLIDE 49

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . . . . . . . . . . . . . .

[ ] r

1

[ ] r

2

[ ] r [ ]K r

H H H H H H H

( )

K

t θ

H . . . .

1

[ ]K r

−

H H

2Kφ

1

( ) t θ

1

2K φ

− 2

2 φ

1

2 φ 2 φ

2

( )

K

t θ

− 1

( )

K

t θ

−

1. 1.

The qubits are dual The qubits are dual-

rail single photons.

rail single photons.

2. 2.

The The Hadamard Hadamard is a beam splitter. is a beam splitter.

3. 3.

The controlled unitaries are the unknown phase in the The controlled unitaries are the unknown phase in the interferometer. interferometer.

4. 4.

The controlled phase operations are feedback to the phase The controlled phase operations are feedback to the phase θ θ( (t t). ).

5. 5.

The operations may be performed in sequence to reuse the The operations may be performed in sequence to reuse the same interferometer. same interferometer.

SLIDE 50

Inverse quantum Fourier transform Inverse quantum Fourier transform

. . . .

[ ] r

1

[ ] r

2

[ ] r [ ]K r

H H H H H H H

( )

K

t θ

H . . . .

1

[ ]K r

−

H H

2Kφ

1

( ) t θ

1. 1.

The qubits are dual The qubits are dual-

rail single photons.

rail single photons.

2. 2.

The The Hadamard Hadamard is a beam splitter. is a beam splitter.

3. 3.

The controlled unitaries are the unknown phase in the The controlled unitaries are the unknown phase in the interferometer. interferometer.

4. 4.

The controlled phase operations are feedback to the phase The controlled phase operations are feedback to the phase θ θ( (t t). ).

5. 5.

The operations may be performed in sequence to reuse the The operations may be performed in sequence to reuse the same interferometer. same interferometer.

1

2K φ

− 2

2 φ

1

2 φ 2 φ

2

( )

K

t θ

− 1

( )

K

t θ

−

. . . .

SLIDE 51

The equivalent state The equivalent state

The sequence of different numbers of passes is equivalent to a

The sequence of different numbers of passes is equivalent to a tensor product of NOON states: tensor product of NOON states:

This is equivalent to

This is equivalent to for for N N = 2 = 2K

K+1 +1−

−1. 1.

( ) ( )

( )

1 1

2 ,0 0,2 2 ,0 0,2 1,0 0,1

K K

+ ⊗ ⊗ + ⊗ + … ,

N n

n N n

=

−

∑

SLIDE 52

Two problems: Two problems:

1. 1.

The state needs to be a special coherent superposition The state needs to be a special coherent superposition

f the form
f the form

There is no known way of producing such a state. There is no known way of producing such a state.

2. 2.

The input mode needs to be very long so that The input mode needs to be very long so that θ θ

(

(t t) ) can be adjusted between detections. can be adjusted between detections.

1. 1.

Using multiple passes of single photons we obtain an Using multiple passes of single photons we obtain an effective state of the form effective state of the form even though the actual state is just single photons. even though the actual state is just single photons.

2. 2.

The input mode does not need to be long The input mode does not need to be long – – we can we can send photons through one at a time. send photons through one at a time.

How to create the input state? How to create the input state?

N n n

n N n ψ

=

−

∑

SLIDE 53

What do we need for theoretical What do we need for theoretical-

limit

limit scaling? scaling?

The squared error is approximately (for real

The squared error is approximately (for real ψ ψn

n)

) where we add the dummy state coefficients where we add the dummy state coefficients ψ ψ−

−1 1 =

= ψ ψN

N+1 +1 = 0.

= 0.

( )

2 2 1 1 N n n n

φ ψ ψ +

=−

Δ ≈ −

∑

SLIDE 54

What do we need for theoretical What do we need for theoretical-

limit

limit scaling? scaling?

The squared error is approximately (for real

The squared error is approximately (for real ψ ψn

n)

) where we add the dummy state coefficients where we add the dummy state coefficients ψ ψ−

−1 1 =

= ψ ψN

N+1 +1 = 0.

= 0.

For scaling at the theoretical limit we need

For scaling at the theoretical limit we need ψ ψn

n+1 +1−

−ψ ψn

n ∝

∝ 1/ 1/N N3/2

3/2.

.

The state coefficients just need to increase then decrease in a

The state coefficients just need to increase then decrease in a gradual way. gradual way.

n

1 N ∼ N ∼

3/ 2

1 N ∼

n

ψ

N n n

n N n ψ

=

−

∑

( )

2 2 1 1 N n n n

φ ψ ψ +

=−

Δ ≈ −

∑

SLIDE 55

The equivalent state The equivalent state

equivalent state

ψn

N n n

n N n ψ

=

−

∑

SLIDE 56

N n n

n N n ψ

=

−

∑

The equivalent state The equivalent state

ptimal state

ψn

equivalent state

SLIDE 57

The equivalent state The equivalent state

ptimal state

equivalent state for M = 2

ln N N φ Δ ∝

ψn

SLIDE 58

The equivalent state The equivalent state

ptimal state

equivalent state for M = 3

1 N φ Δ ∝

ψn

SLIDE 59

The equivalent state The equivalent state

ptimal state

equivalent state for M = 4

1 N φ Δ ∝

ψn

SLIDE 60

What about the feedback? What about the feedback?

φest

1 photon

θ(t) φ

φ/π p(φ) set theta to maximise sharpness after next detection

SLIDE 61

What about the feedback? What about the feedback?

φest

1 photon

θ(t) φ

φ/π p(φ)

SLIDE 62

Predicted variances Predicted variances

variance × (number of resources)2 M = 1 M = 2 t h e

r

e t i c a l l i m i t M = 3 M = 4 M = 5 M = 6 SQL for single passes number of resources

SLIDE 63

Experimental results Experimental results

M = 6 SQL for single passes M = 1 > 10dB variance × (number of resources)2 number of resources t h e

r

e t i c a l l i m i t

B. L. Higgins, D. W. Berry, S. D. Bartlett,
H. M. Wiseman, and G. J. Pryde, Nature

450, 393 (2007).

SLIDE 64

Optical interferometry Optical interferometry

Theoretical limit

Theoretical limit

Squeezed states

Squeezed states1

1

NOON states

NOON states2

2

Theoretical

Theoretical-

limit adaptive measurements

limit adaptive measurements3

3

Theoretical

Theoretical-

limit

limit non nonadaptive adaptive measurements measurements4

4

Hybrid measurements

Hybrid measurements4

4

3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman,

and G. J. Pryde, Nature 450, 393 (2007).

4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell,

H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

1 C. M. Caves, Phys. Rev. D 23, 1693 (1981). 2 B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

SLIDE 65

φ

Nonadaptive Nonadaptive measurements measurements

SLIDE 66

φ

Size of region is < 21−0π/3

Nonadaptive Nonadaptive measurements measurements

Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. .

SLIDE 67

φ

Nonadaptive Nonadaptive measurements measurements

Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. . 1. 1. Perform enough measurements Perform enough measurements with 2 with 21

1 = 2 passes to ensure that

= 2 passes to ensure that the system phase is in one of the the system phase is in one of the two purple regions with two purple regions with high high probability. probability.

SLIDE 68

φ

Size of region is < 21−1π/3

Nonadaptive Nonadaptive measurements measurements

Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. . 1. 1. Perform enough measurements Perform enough measurements with 2 with 21

1 = 2 passes to ensure that

= 2 passes to ensure that the system phase is in one of the the system phase is in one of the two purple regions with two purple regions with high high probability. probability.

SLIDE 69

φ

Nonadaptive Nonadaptive measurements measurements

Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. . 1. 1. Perform enough measurements Perform enough measurements with 2 with 21

1 = 2 passes to ensure that

= 2 passes to ensure that the system phase is in one of the the system phase is in one of the two purple regions with two purple regions with high high probability. probability. 2. 2. Perform enough measurements Perform enough measurements with 2 with 22

2 passes to ensure that the

passes to ensure that the system phase is in one of the four system phase is in one of the four green regions with green regions with high high probability. probability.

SLIDE 70

φ

Size of region is < 21−2π/3

Nonadaptive Nonadaptive measurements measurements

Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. . 1. 1. Perform enough measurements Perform enough measurements with 2 with 21

1 = 2 passes to ensure that

= 2 passes to ensure that the system phase is in one of the the system phase is in one of the two purple regions with two purple regions with high high probability. probability. 2. 2. Perform enough measurements Perform enough measurements with 2 with 22

2 passes to ensure that the

passes to ensure that the system phase is in one of the four system phase is in one of the four green regions with green regions with high high probability. probability.

SLIDE 71

Nonadaptive Nonadaptive measurements measurements

φ

Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. . 1. 1. Perform enough measurements Perform enough measurements with 2 with 21

1 = 2 passes to ensure that

= 2 passes to ensure that the system phase is in one of the the system phase is in one of the two purple regions with two purple regions with high high probability. probability. 2. 2. Perform enough measurements Perform enough measurements with 2 with 22

2 passes to ensure that the

passes to ensure that the system phase is in one of the four system phase is in one of the four green regions with green regions with high high probability. probability.

. . . . . .

Perform enough measurements Perform enough measurements with 2 with 2K

K passes to ensure that the

passes to ensure that the system phase is in one of 2 system phase is in one of 2K

K

regions with high probability. regions with high probability. Size of region is < 21−Kπ/3

SLIDE 72

φ

Size of region is < 21−Kπ/3

At stage k, if the system phase is not in

the region, then the maximum error is ∝ 2−k.

More measurements are needed for

small k to ensure that the contribution to the variance is not large.

The resource cost of additional

measurements is less for small k.

The best results are obtained if M

decreases linearly with k.

Nonadaptive Nonadaptive measurements measurements

Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. . 1. 1. Perform enough measurements Perform enough measurements with 2 with 21

1 = 2 passes to ensure that

= 2 passes to ensure that the system phase is in one of the the system phase is in one of the two purple regions with two purple regions with high high probability. probability. 2. 2. Perform enough measurements Perform enough measurements with 2 with 22

2 passes to ensure that the

passes to ensure that the system phase is in one of the four system phase is in one of the four green regions with green regions with high high probability. probability.

. . . . . .

Perform enough measurements Perform enough measurements with 2 with 2K

K passes to ensure that the

passes to ensure that the system phase is in one of 2 system phase is in one of 2K

K

regions with high probability. regions with high probability.

SLIDE 73

At stage k, if the system phase is not in

the region, then the maximum error is ∝ 2−k.

More measurements are needed for

small k to ensure that the contribution to the variance is not large.

The resource cost of additional

measurements is less for small k.

The best results are obtained if M

decreases linearly with k. Perform enough measurements Perform enough measurements with 2 with 20

0 = 1 pass to ensure that

= 1 pass to ensure that the system phase is in the blue the system phase is in the blue region with high probability region with high probability. . 1. 1. Perform enough measurements Perform enough measurements with 2 with 21

1 = 2 passes to ensure that

= 2 passes to ensure that the system phase is in one of the the system phase is in one of the two purple regions with two purple regions with high high probability. probability. 2. 2. Perform enough measurements Perform enough measurements with 2 with 22

2 passes to ensure that the

passes to ensure that the system phase is in one of the four system phase is in one of the four green regions with green regions with high high probability. probability.

. . . . . .

Perform enough measurements Perform enough measurements with 2 with 2K

K passes to ensure that the

passes to ensure that the system phase is in one of 2 system phase is in one of 2K

K

regions with high probability. regions with high probability.

φ

Size of region is < 21−Kπ/3

Nonadaptive Nonadaptive measurements measurements

1/ N φ Δ ∝

SLIDE 74

Optical interferometry Optical interferometry

Theoretical limit

Theoretical limit

Squeezed states

Squeezed states1

1

NOON states

NOON states2

2

Theoretical

Theoretical-

limit adaptive measurements

limit adaptive measurements3

3

Theoretical

Theoretical-

limit

limit non nonadaptive adaptive measurements measurements4

4

Hybrid measurements

Hybrid measurements4

4

3 B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman,

and G. J. Pryde, Nature 450, 393 (2007).

4 B. L. Higgins, D. W. Berry, S. D. Bartlett, M. W. Mitchell,

H. M. Wiseman, and G. J. Pryde, e-print: 0809.3308 (2008).

1 C. M. Caves, Phys. Rev. D 23, 1693 (1981). 2 B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

SLIDE 75

Hybrid measurements Hybrid measurements

φ

Supplement the

Supplement the M M = 1 measurement with additional measurements with = 1 measurement with additional measurements with single passes. single passes.

p(φ)

Estimate from M = 1 Estimate from single passes

SLIDE 76

Hybrid measurements Hybrid measurements

φ

Supplement the

Supplement the M M = 1 measurement with additional measurements with = 1 measurement with additional measurements with single passes. single passes.

p(φ)

If estimates agree, use

If estimates agree, use the the M M = 1 estimate. = 1 estimate.

Estimate from M = 1 Estimate from single passes

SLIDE 77

Hybrid measurements Hybrid measurements

If estimates agree, use

If estimates agree, use the the M M = 1 estimate. = 1 estimate.

If the estimates differ,

If the estimates differ, use estimate from use estimate from single photons. single photons.

This yields error

This yields error

p(φ) φ

3/ 4

1/ N φ Δ ∝

Supplement the

Supplement the M M = 1 measurement with additional measurements with = 1 measurement with additional measurements with single passes. single passes.

Estimate from single passes Estimate from M = 1

SLIDE 78

Hybrid measurements Hybrid measurements

The equivalent state is the (approximate) Gaussian from single p

The equivalent state is the (approximate) Gaussian from single photon measurements convoluted hoton measurements convoluted with the flat distribution from the with the flat distribution from the M M = 1 measurement: = 1 measurement:

The resulting equivalent state still has a region where the stat

The resulting equivalent state still has a region where the state coefficients rise sharply: e coefficients rise sharply:

* =

SLIDE 79

Hybrid measurements Hybrid measurements

SQL for single passes t h e

r

e t i c a l l i m i t nonadaptive hybrid uncertainty × (number of resources) number of resources

B. L. Higgins, D. W. Berry, S. D. Bartlett,
M. W. Mitchell, H. M. Wiseman, and
G. J. Pryde, e-print: 0809.3308 (2008).

SLIDE 80

Adapting the number of passes Adapting the number of passes

As well as adapting a

As well as adapting a feedback phase, the feedback phase, the number of passes can be number of passes can be adapted. adapted.

N V×N2

scheme 1 scheme 2 scheme 3

ln N N φ Δ ∼

Almost the theoretical limit

SLIDE 81

Summary Summary

Single mode phase

Feedback is needed to beat the

Feedback is needed to beat the standard quantum limit. standard quantum limit.

The best feedback is not the

The best feedback is not the best phase estimate. best phase estimate.

SLIDE 82

Summary Summary

Single mode phase

Feedback is needed to beat the

Feedback is needed to beat the standard quantum limit. standard quantum limit.

The best feedback is not the

The best feedback is not the best phase estimate. best phase estimate.

Interferometry

Special states give improved accuracy,

Special states give improved accuracy, but have problem with ambiguity. but have problem with ambiguity.

Using multiple measurements gives true

Using multiple measurements gives true scaling at the theoretical limit. scaling at the theoretical limit.

This may be achieved even without

This may be achieved even without adaptive measurements! adaptive measurements!

SLIDE 83