Physics with Atomic Clocks: Beyond What Time Is It?•How Atomic Clocks work•Cold Collisions - Rb•Microwave photon recoils

–Is the photon momentum ħk?•Juggling atoms in fountains•Optical Clocks

–How do you count at optical frequencies?

Support from NASA, ONR, NSF, NIST, & Penn State.

TheoryServaas Kokkelmans (now Asst. Prof.)Boudewÿn VerhaarEindhoven Technological University, the Netherlands

Penn StateRuss HartRuoxin LiDr. Lingze DuanDr. Steve GensemerDr. Paul LammertKurt Gibble

Tyler AndersonSeongsik ChangChad FertigRon LegereIrfon ReesVeronica SavuDr. Sid CahnDr. Wenko SüptitzDr. Ivan MistrikDr. Xinye XuJérémie BouttierQuentin Beaufils

3/20/2007-2

Atomic Clocks

υ=E h

State 1

State 2?

Atom X

Quantum Mechanics

If nucleus

Then or

2 Well Defined Energy States

υ ≡ 9,192,631,770Hz

1 valence e−: 6SS= 1/2

Nuclear spin: I= 7/2

Current Atom of Choice is 133Cs.

Microwave or Laser ν.

( )P #2Accurately Determine Center

F = +I S

•I

= 4 z

S

= 4F

= 3F

υΔ =E h1 26S

3/20/2007-3

= + =Ψ =

3 4

2

F F

F=4F=3

Atomic Beam Clocks

Pendulum Analogy

υ osc

F=4 Probability or pendulum amplitude

Versus voscillator

υ atom

Precession time T

Δ =1

2f

TT=0.005s ⇒100Hz

Can’t observe an atomic (quantum) pendulum.

“Ramsey Fringes”

Accuracy:δν

ν−= ± × 141 10

Cs beam

F = 3

S

N

F = 4

Oscillator 9.192 GHz

S

NF=3

F=4( )π2 9.192i GHz te

Detector

4

3or

±100 μHz

3/20/2007-4

• Microwave Spectroscopy– laser-cooling: Doppler

shifts & narrow linewidth

Laser-Cooled Fountain Clocks & the Cold Collision Frequency Shift

Tiesinga, Verhaar, Stoof, & van Bragt, PRA 45, 2671 (’92).Gibble & Chu, PRL 70, 1771 (’93)

The dominant problem for Cs fountain clocks

( )

2

43 92 citGHzie ϕπ ++=Ψ

ν≡9,192,631,770 Hz

4=F

3=F

Cs 6s ground state

ΔνCs

Density (108 cm-3)0 1 2 3 4 5

Fre

quen

cy (

mH

z)

-1.0

-0.5

0.0

0.5

1.0

• Collisions can shift the phase of the atomic coherence

phase shifts coherence by ϕcσ=−106 Å2

ν − 6.834 GHz (Hz)-4 -2 0 2 4

Tra

nsiti

on P

roba

bilit

y

0.0

0.5

1.0

9 GHz

Cs

3/20/2007-5

Cold Atom ScatteringSemi-classically:

Cs-Cs at 1μK

μ= ≈max max 0.3L vb h

v=1 cm/sμ=m/2

bmax

−1μKbmax ≈200Å

V(R)

R-6

R

( )= +1L l l h

Quantum Mechanics

No p-wave 0=∴

Cs-Cs - 99.9(1)% s-wave @ 0.89 μKKG, Chang, Legere, PRL (1995).

s-wave scattering for T → 0:

Spherical outgoing waves → isotropic scattering

( )λσ δπ

= ≈2

2 6 20 0sin 10dB Å

λμ

= ≈ >> m ax3,000dB

hÅ b

v

3/20/2007-6

87Rb Clock

Δν = 0.953 Hzσy(τ) = 2.1×10−13 τ−1/2

S/N = 200:1 (local oscillator)

34cm

6.8GHz

Rb

Δν

ν − 6,834,612,611 Hz (Hz)-4 -3 -2 -1 0 1 2 3 4

Tra

nsiti

on P

roba

bilit

y

0.0

0.5

1.0

3/20/2007-7

• -0.38(8)mHz @ n=1.0(6)109×cm−3

• 50× less than 133Cs• Consistent with calculations• GPS & SI second

87Rb Cold Collision Frequency Shift

Cs: Gibble & Chu, PRL 70, 1771 (‘93)Rb: Fertig & Gibble, PRL 85, 1622 (‘00)

Sortais Bize, Nicolas, Clairon, Salomon, & Williams, PRL 85, 3117 (‘00)Theory: Kokkelmans, Verhaar, Gibble, & Heinzen, PRA 56, RC4389 (‘97)

Kempen, Kokkelmans, Heinzen, & Verhaar, PRL 88, 093201 (‘02)

34cm

6.8GHz

Rb

ΔνCs

δ = +Γ

δ = −Γ

δ = 0

Density (109 cm-3)0 0.1 0.2 0.3 0.4 0.5

Fre

quen

cy (

mH

z)

-1.0

-0.5

0.0

0.5

1.0

• -0.38(8)mHz @ n=1.0(6)109×cm−3

• 50× less than 133Cs• Consistent with calculations• GPS & SI second

87Rb Cold Collision Frequency Shift

Cs: Gibble & Chu, PRL 70, 1771 (‘93)Rb: Fertig & Gibble, PRL 85, 1622 (‘00)

Sortais Bize, Nicolas, Clairon, Salomon, & Williams, PRL 85, 3117 (‘00)Theory: Kokkelmans, Verhaar, Gibble, & Heinzen, PRA 56, RC4389 (‘97)

Kempen, Kokkelmans, Heinzen, & Verhaar, PRL 88, 093201 (‘02)

34cm

6.8GHz

Rb

ΔνCs

δ = +Γ

δ = −Γ

δ = 0

Density (109 cm-3)0 0.1 0.2 0.3 0.4 0.5

Fre

quen

cy (

mH

z)

-1.0

-0.5

0.0

0.5

1.0

RECOMMENDATION CCTF 1 (2004):Concerning secondary representations of the secondThe Consultative Committee for Time and Frequency,…recommends that the unperturbed ground-state hyperfine quantum

transition of 87Rb may be used as a secondary representation of the second with a frequency of fRb = 6 834 682 610.904 324 Hz and an estimated relative standard uncertainty (1σ) of 3 × 10-15, … .

CCTF (2006):recommends 199Hg+ @ 282nm, 88Sr+ @ 674nm, & 171Yb+ @ 436nm.

3/20/2007-9

Atomic Clocks6 834 682 610.904 324 17(7) Hz

Doppler shift for atoms at room temperature.

Doppler shift for a slow walk.

Gravitational red-shift for 2m.

5s in the age of the universe!

Time dilation for walking.2

2

1

2

v

c

δνν

= −

3/20/2007-10

E (m)-600 0 400

N (

m)

-400

0

200Handheld $200 receiver.

Lime Rock, CT

v (m

ph)

50

100

a tr(g

's)

-1

0

1

distance (m)

0 1000 2000

Better Clocks - ApplicationsGlobal Positioning System (GPS)

– 24 satellites with atomic clocks– Accuracy of 15 feet.– Many applications

• Auto, marine, & aviation navigation• surveying & mapping• agriculture, construction, mining,

fleet management

Car racing & teaching physics

Time North (m) East (m)13:53:54 102.1 -259.413:53:55 86.8 -222.113:53:57 48.4 -143.413:53:58 28.9 -101.213:53:59 9.4 -57.613:54:01 -31.9 32.213:54:03 -75.7 127.813:54:04 -98.2 174.913:54:06 -136.0 248.213:54:07 -151.9 282.513:54:08 -167.8 305.6

E (m)-600 0 400

N (

m)

-400

0

200

Lime Rock, CT

3/20/2007-11

Better Clocks• Global Positioning System (GPS)

– earth-based - ionosphere & water vapor are current limitations

– Interplanetary - VLBI (radio)

– NASA Deep Space Network

• Academic interest: How accurately can we measure?– Fundamental constants – new techniques

• High speed Communication networks

• Tests of General Relativity– 100 to 104 times better

• Time variation of constants– string theory

– Cs & Rb clocks, optical clocks

• Unforeseen applications

Mars

d

dt

α α = ≈e

c

2 1

137

3/20/2007-12

NASA DSN (Deep Space Network)

3 sites: Goldstone, Canberra, & Madrid.

24 m to 70 m radio telescopes.

Uses the most stable & fieldableclocks.

Different atmospheric delays.

3/20/2007-13

Vapor Trapv=10 m/s

6.8 GHz

0.5 m

shutter

v=5cm/s

UHV Trap

Penn State & JPL

Atomic Clock for SpaceRubidium Atomic Clock Experiment: RACE

Much longer observation times. T=10s (Δν = 50 mHz)

ν

Tra

nsiti

on P

roba

bilit

y

• Local Oscillator• ISS vibrational noise• 2 clocks for accuracy evaluation• Clock science, red-shift, c

3/20/2007-14

High Performance Clocks in a Space-Based DSN

Very Long Baseline Interferometry

Initial range comes from 2-way.

Different than GPS - transmit only.

Geometry is very different than GPS.

Precise timing allows <10 m error.

Time differences are key.

Differential relative to Mars station.

Geo Synch orbit &10 m @ Mars

Δs

Δx

b

r

θΔ = Δx bΔ

=b s

r= 2.5mm

Δ Δ=

x f

r f−= × 141.4 10

Δ = Δ = 8.5t c x ps

3/20/2007-15

Microwave photon recoilMicrowave Lensing

SYRTE FO2

IEN CsF1

PTB CSF1

NIST F1

Clock Accuracy & Stability

• Biggest offset is Gravitational Redshift• Black-body is difficult – differential AC Stark shift• Cold collisions inhibit evaluation of other systematics

1.4

300020006000280Instability @ 1s

1693.36.5Total uncertainty (10−16)

<221.44.3Electronics, microwave leakage

<0.35<0.3<31st order Doppler - Distributed cavity phase

0.722.62.5Black-body radiation

12712.0Cold collisions

Uncertainty in Frequency (10−16)

40 days

Wynands & Weyers, Metrologia (2005)

R. Li & KG, Metrologia ’04.

3/20/2007-16

Microwave Photon Recoil?

Infinite plane wave

0.1 /xv m sμ≈

Recoil Frequency Shift:

Conserve E & p.

v

E

1F =

2F =

rv2 2

22 2

k

m mc

δυ ωυ ω

= =

161.5 10−= ± ×

kxmv k=

3/20/2007-17

Is an atom’s recoil equal to the photon’s momentum?

Finite beam:Maxwell: ( )2 2 0k E∇ + =

so 2 2 2 2x y zk k k k+ + = & xk k<

Three good choices:1. The photon momentum comes in

discrete units of k, in the x direction.2. The atom has a recoil of kx in the x

direction; vy & vz are unchanged.3. The atom has a recoil of kx in the x

direction, and also ±ky & ±kz.

k

xk

xkzk±

k

xn k

xz

Gould, Ruff, Pritchard, PRL (1986)Wicht, … Chu, PRA (2005)

Wicht…, Chu, Phys. Script (2002)Cladé…, Biraben, PRL (2006)

,0

2

2y zkw mm

≈=

8xk ppbδ⇒ = −

3/20/2007-18

ħkħkx

( )

( )

12

int

12

,2

,2

i t

i t

r t eH

r t e

ω

ω

ω

ω

−⎛ ⎞Ω⎜ ⎟= ⎜ ⎟

⎜ ⎟Ω −⎜ ⎟⎝ ⎠

Transverse (Microwave) Photon Recoils?

There is no grating in the z direction.

→ No recoil in z direction.

2

4

k

mδυ

π≠

xk

xz

zk±k

“Microwave” Stern-Gerlach regime:

Same problem for atomic clocks: The dipole force of the microwave field acts as a lens on the atomic wavefunction.

Shift: ≈ ±4nm

Δwidth: ≈ ±2nm

173 10δυυ

−≈ ×161.5 10δυυ

−≠ ×

( ) ( ) ( ) ( )2 2

2 22 1 sin cosz xP t t k k xδ ϕ⎡ ⎤= Ψ − Ψ ⎡ ⎤⎣ ⎦⎢ ⎥⎣ ⎦

( ) 1

2z R

wk

wδυ ϕ υ= Δ

1z

E

2

KG PRL (‘06)

f

2e e

( )2

22 tΨ( )2

21 tΨ

3/20/2007-19

Juggling Atomic Fountains

Siz

e (

cm2 )

10−10

Ec (μK)2 10 100 200

10−12

10−11

10−13

10−15

10−14

s-wave

T 300nK=K19

7ms

μ==Δ

E

t Launch

•State-to-state velocity-selected differential crossed-beam scattering at µK energies.

•Ramsauer-Townsend effect•Next generation of clocks will juggle.

Emg t

c =2 2

4

Δ

Legere & Gibble, PRL (‘98)

3/20/2007-20

Juggling Frequency Shift

S. Kokkelmans & KG

• Higher partial waves– other patterns

6.8 GHz

Rb

66ms22ms

• Optimal delay is 22 ms

44ms

• Avoid 44 ms

0.0 0.5 1.0 1.5 2.0

-0.05

0.00

0.05

E (mK)

Fre

quen

cy S

hift

(mH

z)

3/20/2007-21

scattered state 3 & 4

100×

n(vz)

vz (cm/s)-10 0 10

( )12

3 4ikzeψ + +⎡ ⎤= +⎣ ⎦

unscattered

Direct Measurement of s−wave Phase Shifts

Hart, Xu, Legere, & KG, quant-ph/0702146 (in press)

-2 -1 0 1 2

unsc

atte

red

atom

s (1

03ar

b.)

Microwave Detuning (Hz)

0

10

15

5

0

10

20

scat

tere

d at

oms

(arb

.)

( ) δ δψ δ δ+ +⎡ ⎤= + + +⎢ ⎥

⎣ ⎦3 41

3 423 4 sin 3 sin 4

ikr ikri iikz e e

e e ekr kr

n1 (arb.)0

20

A (

arb.

)

0 1

n1 (arb.)

−0.3

−0.2

−0.1

0

δ 3−δ

4(r

ad)

0 0.5 1

• Juggle “clock” atom in a coherent superposition of 2 states with an atom in F,m (4,4).

• Detect only scattered atoms.• Phase shift of Ramsey fringes is

independent of atomic density.• Clock-like accuracy - 104× more

accurate interactions. PPM scattering lengths.

3/20/2007-22

A Quantum Scattering Interferometer

•Mach-Zehnder

φΔ

φΔ

3 4-φ δ δΔ =

2

0,30,4 +

•Phase shift, not frequency shift•8 mrad statistical error

frequency shift

−0.2

−0.1

00 0.1 0.2 0.3 0.4 0.5

T (s)

δ 3−δ

4(r

ad.)

Det

ecte

d A

tom

s (a

rb.)

-10 -5 0 5 10Microwave Detuning (Hz)

0

10

20

3/20/2007-23

Optical Clocks

• How do you make a very short pulse of light (sound)?

A laser with a million colors!

• Fractional frequency is the key performance measure.

δυ υυ πυ

Δ=

/S Nυ ≈ 1 51 0o p t H z ×1 0 0 , 0 0 0

Count optical cycles!

Pulse rate

3/20/2007-24

2005 Nobel Prize in Physics

"for his contribution to the quantum

theory of optical coherence"

Roy GlauberHarvardUSA, 1925 -

Jan HallJILA,USA, 1934 -

Ted HänschMax Planck Germany, 1941 -

"for their contributions to …precision spectroscopy, including the optical frequency comb"

3/20/2007-25

Counting the Ticks of Light

• Measure f by measuring the difference between f & 2f!• Much better clocks - also huge impacts for chemistry.

Stable Laser

f

f

Crystal

2f – f =

2f

(Pulse Rate) × 429,228

Pulse rate

= f

2f

Diddams et al., PRL 84, 5102 (‘00); Rafac et al., PRL 85, 2462 (‘01).

3/20/2007-26

• Distortions when vωà1→can’t lock

• Frequency domain– 3.5 kHz cavity, 3 MHz laser fm “noise”

• Pound-Drever-Hall locking• Laser linewidth ffmà cavity

linewidth Δν• Gravity wave detectors - LIGO

• What if the laser sweeps through resonance faster than the build up time?

• Previous analyses in time domain.

Directly Locking Lasers with Large FM Noise to High Q Cavities

Laser

Rohde et al., JOSA B 19, 1425, ’02.

2vω

ωω

≡Δ

Lawrence et al., JOSA B 16, 523, ’99.

(1 )cavcav in

dEiv t E iE

dt ω= − − +

EOM

Servo PD

-20-10 10

2030

40

SPDH

ν/Γ

t

Drever et al., Appl. Phys. B 31, 97 ’83.

.01

20

Duan & Gibble, Opt. Lett 30, 3317, ’05.

3/20/2007-27

Laser Lock to Cavity

Tra

nsm

issi

on

0

10

20

30

40

-20 -10 0 10 20Detuning (kHz)

6.97 kHz

66,600 Finesse

δυυ

=× 14

6.97

3.8 10 /

kHz

Hz S N

99.995% reflectivity.Photon lives for 22 μs!

100 200 300

Inte

nsity

f (Hz)

22.7 Hz

0.1 0.2 0.5 1

2

4

1020

0.01

σ y(H

z)

τ (s)

4 Hz

1.6Hz τ

Laser Stability

3/20/2007-28

• Laser-cooled atoms– TAI – international atomic time

• 87Rb Cold collision shift – 50 times smaller for 87Rb– cancel collision shift– Juggling clocks– Rb clocks will be the GPS system clocks

• RACE: High Performance Rb– French clock ACES (Cs) in 2008??

• Atom’s recoil is less than ħk.– No discrete transverse recoils– Lensing frequency shift

• Juggling Cs fountain– Direct measurement of scattering phase

shifts

• Optical clocks and oscillator

Summary

Density (109 cm-3)0.0 0.1 0.2 0.3 0.4 0.5

Fre

quen

cy (

mH

z)

-1.0

-0.5

0.0

0.5

1.0

6.8 GHz

Rb

-20 -10 10 20 30 40

SPDH

ν/Γ

t