investigation of high-precision phase estimation in the ... · 1 motivation 2 the quantum...

Investigation of high-precision phase estimationwith trapped ions in the presence of noise

Sanah Altenburg, S. Wolk and O. Guhne

Department Physik, Universitat Siegen, Siegen, Germany

Bilbao, 23. April 2015

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 1

Siegen

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 2

Motivation: Why Quantum Metrology?

Classical setup:

N particles in a classical state,The Standard Quantum Limit(SQL):

(∆ϕ)2 ∝ 1/N

Quantum setup:

N particles in a quantumstate.The Heisenberg Limit (HL):

(∆ϕ)2 ∝ 1/N2

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 3

Motivation: Why Quantum Metrology?

The Cramer-Rao bound

(∆ϕ)2 ≥ 1/FQ [%,Λϕ]

With the tool;Quantum Fisher Information (QFI) FQ [%,Λϕ]

But noise destroys the advantage of using quantum states.

The Question:

For a given set-up, with a given noise model;Which quantum state should be used?

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 4

1 Motivation

2 The Quantum Fisher-Information

3 Quantum Metrology with Ions

4 Metrology with different states

5 Conclusions and next steps

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 5

1 Motivation

2 The Quantum Fisher-Information

3 Quantum Metrology with Ions

4 Metrology with different states

5 Conclusions and next steps

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 6

The Quantum Fisher-Information

The QFI is propor-tional to the statisticalvelocity of the evolu-tion of a given state

(∂tDH)2 ∝ F

with the Hellinger Dis-tance DH (distance inprobability space).

Picture from G. Toth

and I. Apellaniz, J. Phys.

A 42, 424006 (2004)

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 7

The Quantum Fisher-Information

Initial state in its eigendecomposition % =∑i

λi |i〉〈i |

Unitary time evolution U = exp [−iθM]

Definition of the Quantum Fisher-Information

F (%,M) = 2∑α,β

(λα − λβ)2

λα + λβ|〈α|M |β〉|2

Example:

The fully mixed state % = 1/d is insensitive to any kind of unitarytime evolution, so that F (%,M) = 0.

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 8

1 Motivation

2 The Quantum Fisher-Information

3 Quantum Metrology with Ions

4 Metrology with different states

5 Conclusions and next steps

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 9

Quantum Metrology with IonsSetup:

N trapped ions in achain.

Magnetic field~B0 = B0~ez

Dynamics:

Ideal dynamicsH = γB0t︸ ︷︷ ︸

ϕ

Sz

Noise: Magnetic field fluctuations B = B0 + ∆B(t)

η = ϕ+ γ

∫ t

0dτ∆B(τ)︸ ︷︷ ︸δϕ(t)

Estimate ϕ for a given time t = T

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 10

The noise model1

Dynamics

U = exp[−iϕSz ]︸ ︷︷ ︸Uϕ

exp[−iγ∫ T

0dτ∆B(τ)Sz ]︸ ︷︷ ︸

Unoise

The state %0 evolves in a noisy state %T = Unoise%0U†noise

Time correlation function for the magnetic field fluctuations ∆B(τ)

〈∆B(τ)∆B(0)〉 = exp[− ττc

] 〈∆B2〉

Assume Gaussian phase fluctuations with 〈δϕ(T )〉 = 0

Average 〈%T 〉 and calculate the QFI FQ [〈%T 〉 ,Sz ]

For frequency estimation with ϕ = ωT :

FωQ = T 2FϕQ

1T. Monz et al.: 14-Qubit Entanglement: Creation and Coherence, PRL 106 (2011)

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 11

1 Motivation

2 The Quantum Fisher-Information

3 Quantum Metrology with Ions

4 Metrology with different states

5 Conclusions and next steps

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 12

The product-state

The classical state....

...is a Product-state:

|PN〉 =

(1√2

(|0〉+ |1〉))⊗N

Observations:

For T = 0 we find FQ = N, thisis the SQL.

For T > 0 the QFI decreases.

The more ions the faster theQFI decreases. 0 10 20 30 40 50

T in a.u.

0.0

0.2

0.4

0.6

0.8

1.0

FQ

[%,Sz]/N

N=2N=4N=6

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 13

The GHZ-state

The GHZ-state...

...is defined as

|Ψ〉GHZ =1√2

(|000...0〉+ |111...1〉).

The QFI is

FQ = N2e−N2C(T ).

Solving the master equationleads to

FQ = N2e−N2C(T ).

[F.Frowis et al., New Journal of Phys.8 (2014)]

0 10 20 30 40 50T in a.u.

0

1

2

3

4

5

6

FQ

[%,Sz]/N

N=2N=4N=6

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 14

The sym. Dicke-state

Definition: Dicke-state

|DNsym.〉 ∝

∑j

Pj|0〉⊗N/2 ⊗ |1〉⊗N/2

Rotation around the X -axis: UNx (α) |Dsym.

N 〉

α = 0 α = π/4 α = π/2

x

y

|0〉⊗N

|1〉⊗N

x

y

|0〉⊗N

|1〉⊗N

x

y

|0〉⊗N

|1〉⊗N

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 15

The sym. Dicke-state

For N = 4:

For α = 0, FQ = 0

For α = π/2 and T = 0,FQ = N(N + 2)/2

For α = π/4, somewherein between

For a given T and N, we areable to tell which state givesthe best precision for phaseand frequency estimation.

|0〉⊗N

|1〉⊗N

|0〉⊗N

|1〉⊗N

|0〉⊗N

|1〉⊗N0 5 10 15 20 25 30

T in a.u.

0

1

2

3

4

FQ

[%,Sz]/N

GHZ stateProduct state

9.5 13.5

Details

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 16

Dicke state under any rotation angle α

N = 4

Rotation around the X -axis: UNx (α) |Dsym.

N 〉

xy

|0〉⊗N

|1〉⊗N

xy

|0〉⊗N

|1〉⊗N

xy

|0〉⊗N

|1〉⊗N

0 π4

π2

3π4

α

0.0

0.5

1.0

1.5

2.0

2.5

3.0FQ

[%,Sz]/N

T=0

T=4

T=8

T=12

T=16

T=20

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 17

1 Motivation

2 The Quantum Fisher-Information

3 Quantum Metrology with Ions

4 Metrology with different states

5 Conclusions and next steps

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 18

Conclusions

Trapped ions in the presence ofnoise

Sym. Dicke state under noise

0 π4

π2

3π4

α

0.0

0.5

1.0

1.5

2.0

2.5

3.0

FQ

[%,Sz]/N

T=0

T=4

T=8

T=12

T=16

T=20

Optimization over states

|0〉⊗N

|1〉⊗N

|0〉⊗N

|1〉⊗N

|0〉⊗N

|1〉⊗N0 5 10 15 20 25 30

T in a.u.

0

1

2

3

4

FQ

[%,Sz]/N

GHZ stateProduct state

9.5 13.5

Details

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 19

Next steps

Differential interferometry withtrapped Ions

Trapped ions withHamiltonian

H = γ

∫ t

0dτ∆B(τ)(Sz ⊗ Sz)

+γφ(1⊗ Sz).

For photons and atoms seeM. Landini et al., New J. ofPhys. 11, 113074 (2014)

Other states

Investigation of the usefulness ofother Dicke-states, e.g. theW-state.

Collaboration with Ch.Wunderlich (Siegen)

I. Baumgart et al., arXiv:1411.7893(2014)

Trapped ions with Hamiltonian

H = γ

∫ t

0

dτ∆B(τ)Sz

+γ(Ω + ε)tSx .

Optimization for the estimationof ε.

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 20

Acknowledgements

This work was founded by the Friedrich-Ebert-Foundation.

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 21

Thank you for your Attention! Questions?

Trapped ions in the presence ofnoise

Sym. Dicke state under noise

0 π4

π2

3π4

α

0.0

0.5

1.0

1.5

2.0

2.5

3.0

FQ

[%,Sz]/N

T=0

T=4

T=8

T=12

T=16

T=20

Optimization over states

|0〉⊗N

|1〉⊗N

|0〉⊗N

|1〉⊗N

|0〉⊗N

|1〉⊗N0 5 10 15 20 25 30

T in a.u.

0

1

2

3

4

FQ

[%,Sz]/N

GHZ stateProduct state

9.5 13.5

Details

Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 22