Eugene PolzikNiels Bohr Institute

Copenhagen University

Entanglement assisted metrology and sensing

Entangled atom clock sub-femtoTesla magnetometry

Canonical quantum variables for light X,P – can be well measured by homodyne detection or by polarization rotation

[ ] iPXaaPaaX i =−=+= ++ ˆ,ˆ)ˆˆ(ˆ),ˆˆ(ˆ22

1

1/ 2X Pδ δ ≥

X X or P Pδ δ± ±measure

X̂

P̂2 2 1/ 2X Pδ δ= =

Coherent state (or vacuum)

All states inthis talk have Gaussian

Wigner functions

-5

0

5

-5

0

5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

2 1/ 2Xδ <Squeezed state

2 21 2 1 2( ) ( ) 2X X P Pδ δ− + + <

Two-mode squeezed (EPR)

-450 450

PolarizingBeamsplitter 450/-450

Measurement of canonical variables via Stokes operators - polarization

ˆ ˆ,X P

12 2

ˆ ˆS nX=

S2 measurement

Strongclassical

light

Quantum field

Ensemble of N polarized atoms = a giant spin“spin up”

“spin down”Jx

2ˆ ˆ ˆ ˆ, ,

ˆ ˆ ˆ ˆ/ , /

iy z x

y x z x

J J iJ N X P i

X J J P J J

⎡ ⎤ ⎡ ⎤= = =⎣ ⎦ ⎣ ⎦

= =

Jy~P

Jz~X

Uncorrelated atoms:

( ) ( ) 14z yVar J Var J N= =Projection noise

Quantum noise of the initial state of atoms

Quantum measurement changesthe state: back action noise of the meter (light)

The meter (light) has its own quantum noisewhich adds to the measurement error

arXiv:0907.2453W. Wasilewski, K. Jensen, H. Krauter, J.J. Renema, M. V. Balabas, E.S. Po

http://xxx.lanl.gov/abs/0907.2453http://xxx.lanl.gov/find/quant-ph/1/au:+Wasilewski_W/0/1/0/all/0/1http://xxx.lanl.gov/find/quant-ph/1/au:+Jensen_K/0/1/0/all/0/1http://xxx.lanl.gov/find/quant-ph/1/au:+Krauter_H/0/1/0/all/0/1http://xxx.lanl.gov/find/quant-ph/1/au:+Renema_J/0/1/0/all/0/1http://xxx.lanl.gov/find/quant-ph/1/au:+Balabas_M/0/1/0/all/0/1http://xxx.lanl.gov/find/quant-ph/1/au:+Polzik_E/0/1/0/all/0/1

BBias magnetic fieldLarmor frequency Ω

cos( )RFB b t= Ω

Detection of tiny oscillating magnetic fieldsSpin dynamics

top view

2RF AJ B N Tγ⊥ ≈

2Tγ −−

Gyromagnetic constant

Transverse spin coherence time

Minimal quantum noiseof uncorrelated spins

AJ Nδ ⊥ =

Atomic levels and geometry of experiment

LΩ

B

cosRF RF LB b t= Ω

BRF

2RFb Tϕ γ=

Cesium ground state

mF =4mF =3

Room Temperature Spins1012 atomsCesium

2/36P

2/16S432

99.5% opticalpumping

Harmonic oscillatorin the ground state

320kHzΩ =

1GHzΔ ≈

Alkene-basedcoating. T2 = 40msecPerspectively up to

many seconds

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4 Atomic Quantum Noise

Ato

mic

noi

se p

ower

[arb

. uni

ts]

3

3

ˆ cos

ˆ sin

inz x

iny x

J J S t

J J S t

α

α

= Ω

= Ω

2 2ˆ ˆ ˆo u t in L a b

zS S Jα= +

Ĵ yz2

ˆ ( )S t1S

Atomic magnetometer – simplified quantum theory

BRF=B0 cos(Ωt) 2/16S 320kHzΩ =

Quantum back action of probe light on atoms:calcellation via entanglement of two ensembles

LΩmF =4mF =3

L−ΩmF =3mF =-4

Calcellation of measurement back action with two cells

1 2 1 2 1 2ˆ ˆ ˆ ˆ ˆ ˆ, (z z y y x xJ J J J i J J⎡ ⎤ ) 0+ + = + =⎣ ⎦

measurement doesnot change relativespin orientation

RF

Faraday rotation signal doubles

RF

† † †ˆ ˆˆ ˆ . .H a b ab h cχ χ= + +

a

bmF = F F-1 F-2 F-3

1 2

Double Λ”butterfly”

2 2

1

14aa

ξ =Tensor polarizability

Vector polarizability

Off-resonant interactionfor realistic atoms and polarized light

2ˆ ˆ ˆ ˆ( )L A L Ak P P X Xξ= +

Quantum Nondemolition Interaction limit ↔ tensor term→0:1. For spin ½2. For alkali atoms, if Δ >> HF of excited state and the

interaction time is not too long

Atom

s IN

Ideal read out of the atomic state:atom-light state swap plus squeezing of the probe light

1

1

out in out inA L A Lout in out inL A L A

X P P X

X P P X

ξ ξ

ξ ξ

−

−

= =

= =

W. Wasilewski et al, Optics Express 17, 14444-14457 (2009)

Photon

s IN

AtomsOUT

PhotonsOUT

† † †ˆ ˆˆ ˆ . .H a b ab h cχ χ= + +1 2

2ˆ ˆ ˆ ˆ( )L A L Ak P P X Xξ= +

Cs probe detuning 850nmDepends only on detuning for

a given transition

Optimized temporal modes for the swap operation

Swap of state from atoms to light providesthe best quantum measurement of atomic state

21 1 1 12cos 2cos 1 2

ˆ ˆ ˆ ˆ1 ( )swap swapt tout in y yNFS S e e J Jγ γ

ξ− −

Φ Φ= + − +

Projection noiselimited

Entanglementassisted

6ξ =

Magnetic field sensitivity with 1.5*1012 atoms

150.42 10 /T Hz−⋅

State-of-the-art cell magnetometer with 1016 K atomsLee at al, Appl. Phys.Lett. 2006

150.24 10 /T Hz−⋅

100-fold improvementin sensitivity per atom

15 1036 10 3.6 10RFB Tesla G− −= ⋅ = ⋅

1012 spins in each ensemble

y z

x

y z

xSpins which are “more parallel” than that

are entangled

Entanglement of two

macroscopic objects.

21

~ −N

B. Julsgaard, A. Kozhekin and EP, Nature, 413, 400 (2001)

( ) ( )1 2 1 2ˆ ˆ ˆ ˆ/ 2 / 2 1z z x y y xVar J J J Var J J J+ + + <Can be created by a measurement

Magnetometry beyond the projection noise limit

Entanglingpulse

RF field Measuringpulse

( ) ( )2 21 2 1 2ˆ ˆ ˆ ˆ 1.3 2z z y y x xJ J J J J Jδ δ+ + + = ⋅ < ⋅

Entanglement by QND measurement

2ms

( ) ( )2 21 2 1 2ˆ ˆ ˆ ˆ 1.3 2X X P Pδ δ+ + + =

EPR entanglementSignal/noise times

bandwidthτ

Thomas Fernholz

Hanna Krauter

Kasper Jensen

Lars Madsen

WojtekWasilewski

Frequency of atomictransition as

• standard of time

2

1

ω12

Appel et al, PNAS – Proceedings of the National Academy of Science (2009)106:10960-10965

Two level atom of a clock as a quasi-spin

F=4mF =0

( )( )

14 ,4 3,32

14 32

ˆ ˆzJ N

N N

ρ ρ= −

= −

Measurable atomic operator

F=3mF =0

Jx

2ˆ ˆ, iy z xJ J iJ N⎡ ⎤ = =⎣ ⎦

Jy~P

Jz~X

Uncorrelated atoms:

( ) ( ) 14z yVar J Var J N= =Projection noise

ω12 = 9,192,631,770 Hz

Ramsey method – a way to determine ω12

020

Ti td E e tωπ δ= ∫

2

1

ω12

π/2 pulseπ/2 pulse

12( )tω ω θ− =0

02

Ti td E e tωπ δ= ∫

θRamsey fringes

2N

N2

N1

Jx

δJ2z = ½N <

0Jz

Measurement of the populationdifference

2 12( )zJ Nδ =

Coherent spin state:PROJECTION NOISE

( )12

2 1NN

NΨ = + N independent atoms

Spin squeezed state

( )2 1 NΨ ≠ +

Metrologically relevant spin squeezing = entanglement

Angular uncertainty ofthe spin defines

metrological significanceWineland et al 1992

Uncorrelated atoms(coherent spin state)

Entangled atoms

1/ 2

1 12 4( )z x

NVar J J Nδϕ −=

= =( ) ( )

1

2 214

( )

( ) / /z x x x

Var N

Var J J N N J J

ϕ −<

′ ′< =

Entangled state cannot be writtenas a product of individual atom states

( )...2 ...1 ... ...1 ...2 ...i k i k+∼

Z

Atomic clock with spin squeezed atoms

Quantum Nondemolition Measurement (QND) and Spin Squeezing

An ideal QND measurement

1. Conserves one quantum variable (operator P)

2. Channels the backaction of the measurementinto the conjugate variable X

3. Yields information about the P

ˆ ˆ,X P i⎡ ⎤ =⎣ ⎦2

1ω12

Our goal – measure the population difference in a QND way togenerate a spin squeezed state

Proposal by Kuzmich, Bigelow, Mandel in 1999.

A measurement changes the measured statestandard QM textbook

ˆ ˆ

ˆ ˆ ˆ ˆ,

A L

iL L A

H P P

X H X P⎡ ⎤= ⎣ ⎦

∼

∼

QND of atomic population differenceBalanced photocurrent:

ˆ ˆ( )ph phi n a a n φ+

− − +∼Probe shot noise

Atomic signal

4,0 3,0 AN N Pφ − ∝∼

out in inL L AX X Pκ= +

In canonical variables:

†2

( )iLX a a= − CESIUM LEVEL SCHEMECESIUM LEVEL SCHEME6P3/2

6P1/2

9.192 GHz6S1/2

1.17 GHz852nm

F=4

F=4

F=3

F=5

F=3

F=2

251 MHz

152 MHz201 MHz F=4F=3

Ideally no spontaneousemission

nondemolitionmeasurement of

population difference

Interferometry on a pencil-shaped atomic sample

(dipole trapped Cs atomsT= 100 microK, 200.000 atoms)

QND measurement of atomic population difference

2 1( ),n N N

n Aδ γ σφ β β= + − =

Δ

Interferometer measured phase shift around balanced position:

Probe shotnoise

Atomic signal1/ 2n−

Atoms

Monochromatic versus bichromatic QND measurementD. Oblak, J. Appel, M. Saffman, EP

PRA 2009

CESIUM LEVEL SCHEMECESIUM LEVEL SCHEME6P3/2

6P1/2

9.192 GHz

6S1/2

1.17 GHz852nm

F=4

F=4

F=3

F=5

F=3

F=2

251 MHz

152 MHz201 MHz

F=4F=3

Maximal spin squeezingscales as 1/ d

20

1 11 (1 )d

ξ ηη η

⎛ ⎞= +⎜ ⎟+ −⎝ ⎠

Maximum spin squeezingscales as 1/d

20

1 11 (1 )d

ξη η

=+ −

Daniel Oblak

NielsKjaergaard

Ulrich Hoff

Patrick Windpassinger

Juergen Appel

Step 1:coherent spin state preparation

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

Linear Pol

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

Linear Pol

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

Linear Pol

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

Linear Pol

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

Linear Pol

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

Linear Pol

π-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

Linear Pol

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F=4

F=3

-4 -3 -2 -1 0 +1 +2 +3 +4

-3 -2 -1 0 +1 +2 +3

mF

mF

F'=4

F'=3

-4 -3 -2 -1 0 +1 +2 +3 +4 F'=5

-2 -1 0 +1 +2mF F'=2

+5-5mF

6 P2 3/2

6 S2 1/2

STEPS 2 and 3: generation and verification of spin squeezed statevia QND measurement

Best guess forsecond

measurement

(1) (1) (1)2

ˆ1A L L L

P x x kxκκ

= =+

(2) (1)L Lx kx−

(1)Lx

First probe pulse:(1) (1)ˆ ˆ ˆinL L AX X Pκ= +

Outcome: (1)Lx

Experimental parameters

80 microns

Up to 2*105 atoms probed

55 microns

Resonant optical depth up to 30Optimal decoherence parameter η=0.2Optimal photon number per QND 2*107

Expected projection noise reduction about 6 dBState lifetime ~ 0.2 msec

Determination of QND-induced decoherence η

Two probe pulses

Visibility reductionis (1-η)

Spin squeezing and entanglement

Shot noise of QND probe

J. Appel, P. Windpassinger, D. Oblak, U. Hoff, N. Kjærgaard, and E.S. Polzik. PNAS – Proceedings of the National Academy of Science (2009) 106:10960-10965

Multiparticle entanglement of an atom clock

1

4

5

3

2

2010

Sørensen, Mølmer, PRL 2001Sanders et al

Multiparticleentanglementof a spin squeezedstate

Our result

Entanglement assistedcold atom clock

Quantum state engineeringand measurement allows for unprecedented precision of sensing

Sub-femtotesla magnetometry with1012 entangled atoms

Entanglement assisted metrology andsensing is a part of Quantum Information

Science

Step 1:�coherent spin state preparation