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