10 September 20016th Symposium on Frequency Standards & Metrology Relativistic Quantum Theory of...

10 September 20016th Symposium on Frequency

Standards & Metrology

Relativistic Quantum Theory of

Microwave and Optical Atomic Clocks

by

Christian J. Bordé

Laboratoire de Physique des Lasers, Villetaneuse and

Bureau National de Métrologie, Paris



ATOMS ARE WAVES !

vdeBroglie

The recoil energy is not negligible any more in Cesium clocksM

k

2

2

Atomic clocks are fully quantum devices, in which both the internal and external degrees of freedom of the atoms must be quantized

Different from small clocks carried by classical point particlesAtom sources may be coherent sources of matter-wave

Atomic frame of reference may not be well defined

Gravitation and inertia play an important role:Atomic clocks are relativistic devices



Atom laser

Rubidium atoms are extracted from a cold rubidium gas (left) and from a Bose-Einstein condensate(right). An intense low divergence atomic beam falls under the effect of gravity.

courtesy of the university of Munich



MOMENTUM

E(p)

p

atomslope=v

photonslope=c

rest mass

ENERGY

Mc2

h

h / h dB/

h dB

K



ATOMIC WAVESz

x

y

pEEe

papd

tra

ttpErrpi

/))(()(

2/3

3

00

2),(

ME

ppMEp

paraxialppMEp

wavetravellingM

pMcEicrelativistnoncpcME

zyxzyx 4

122

2

2222

222242

00

0

22

2exp

22exp,),(

ttEi

xxMEi

zpypi

xxME

ppippadpdptra zy

zyzyzy

2

2

2

2

2exp

2exp, zy

zy

ppppa TEM00



0x 1x

0

0

1

1

y

x

DC

BA

y

x

0

0

1

1

vv

x

DC

BAx

0y1y

for light rays

for massive particles

In Gaussian optics, the matrix ABCD also gives the transformation law for the waves:

2

211

kwi

Rq transforms

as DCq

BAqq

0

01

ABCD matrices for light and matter-wave optics

SpaceorTime

OpticalSystem



ABCD PROPAGATOR

200

0000002

02

0

0020

0

0

0

222/1

v2

exp

vvexpv2v2

exp1

'exp'

2exp

1''2

2exp'

2

BAzzX

YiM

BAzzDCziM

BCzDBACziM

X

zzpizz

X

YiM

XAzzzDz

B

iMdz

Bi

M

222/1

''22

exp'2

AzzzDzB

iMdz

Bi

M

For a wave packet moving with the initial velocity Mp /v 00

)(),(),(/)()(exp/exp tYtXtzzFtzztipiS clclcl

002

00

0

0

'exp'

2exp

1 zzpizz

X

YiM

X

0000

0000

,vv

,v

DYCXYDCz

BYAXXBAzz

cl

cl


Standards & Metrology RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES

ba ba a b

z y

xa

b



E(p)

pTMkB2

n(p)

h

h /

Recoil energy 22 2/ Mh



E(p)

p



RAMSEY FRINGES : FIRST-ORDER TRANSITION AMPLITUDE AFTER A SINGLE FIELD ZONE

a

b

b

z y

x

a

ATOMS

EM WAVE

packet waveinitial),(

envelope Rabi

v2),(

)0(/))(()(

v4/v

2/3

3)()1(

00

222

tpae

e

wpdeitrb

ttpErrpi

kw

x

batkzi

a

xzba

momentum additionalv/)(v 1 xzba xxkie

packet waveinitial),()0(/))(()( 00 tpae ttpErrpi a

),()1( trb



RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES

ba b

a a b

),(

v),(

)0(v/)(

v4/

)()1(

1

222

trae

e

ew

itrb

xba

xba

xxi

w

tkziba

x

..),(),( v/)()*1()1( 12 ccetrbtrb xba xxi

EM WAVE 1 EM WAVE 2




b

a,pz

b

a,pz

ab

z

x

y

a

bATOMS

EM WAVE 1

zzzba

xzbaz

papaxxki

kwdpbbdz

)*0()0(x12

222)*1(2

)1(1

v/vexp

v2/vexp

tzatxxM

kzadz ,,v/ )*0(

x12)0(

EM WAVE 2



E(p)

p

Recoil energy 22 2/ Mh




b

b

ab

z

x

ya

b

ATOMS

EM WAVE 1

kpppapa

zppidpdpedzbbdz

zzzz

zzzzikz

2''

..../'exp'

)*0()0(

2)*1(2

)1(1

a,p'z

a,pz

EM WAVE 2



E(p)

p




b

b

ab

z

x

ya

b

ATOMS

EM WAVE

kpapa

xxxkidp

xxibbdz

zz

zz

ba

2

v/22vexp

v/exp

)*0()0(

x120

x12)*1(

2)1(

1

a,pz

a,pz±2k



Microwave resonator

Microwave

Auxiliary

Height1 m

Detection of F=1,m=0

Magnetic shield

- Flux 107 atoms/s (gain of 10/ present fountains)

- Average density 109 atoms/cm3 for x=50 m

- Continuous operation

- No losses between rise and fall: vx=15 m/s

Rubidium clock with a monomode continuous coherent beam

Courtesy of Jean Dalibardand David Guéry-Odelin


Standards & Metrology 1 Non-relativistic approach

We shall consider quite generally the non-relativisticSchroedinger equation as thenon-relativistic limit of ageneral relativistic equation described in the last partof this course:

i¹h@jª (t)i

@t= H0 +

12M

¡!p op¢)g (t) ¢¡!p op

¡¡! (t) ¢(

¡!L op +

¡!S op)

¡ M ~g(t) ¢~rop ¡M2

~rop¢)° (t) ¢~rop

+V (~rop; t)]jª (t)i (1)

whereH0 isan internal atomicHamiltonianandV (~rop; t)some general interaction Hamiltonian with an exter-nal ¯eld. Gravito-inertial ¯elds are represented by thetensors

)g (t) and

)° (t) and by the vectors

¡! (t)

and ~g(t). The same terms can also be used to rep-resent the e®ect of various external electromagnetic¯elds. The operators

¡!L op = ~rop£ ¡!p op and

¡!S op are

respectively the orbital and spin angular momentum



ABCD PROPAGATOR

200

0000002

02

0

1'

2200

0020

0

0

0

v2

exp

vvexpv2v2

exp1

)(2

exp)v(exp

'exp'

2exp

1

BAzzX

YiM

BAzzDCziM

BCzDBACziM

X

iMdt

iMBAz

iM

zzpizz

X

YiM

X

t

t

222/1

'')(2)(2

exp'2

AzzzzDB

iMdz

Bi

M

)(),(),(/)()(exp/exp tYtXtzzFtzztipiS clclcl

1'

22 )2/2/(exp)(exp dtgiM

ziM t

t

0000

0000

,vv

,v

DYCXYDCz

BYAXXBAzz

cl

cl

002

00

0

0

'exp'

2exp

1 zzpizz

X

YiM

X

0 g



12

10

0

0))((

2

1/),( dtMtp

MzpttS

t

t

clt

tclcl

Quite generally, the phase shift along each arm is:

i.e. minus the time integral of the kinetic energy


Standards & Metrology FOUNTAIN CLOCK

2

xz

2

1

v/)v(

gT

kk ba

a

a

b

b k



Gravitational/Relativistic Doppler shift for fountain clocksA quantum mechanical calculation

~ Langevin twin paradox

a

a

b

b

101,

00,

12

,0,

02

,,

0

0

)(

)(2

1/

dtttM

prgM

dtMpM

ttcMS

t

tba

ba

t

t baba

baba

/)()(exp/exp tzztipiS clcl

2ba,,

2,2

2ba,2

, v2

1v1/ bababa McM

ccM

g

v2

c

v

6

11/ 0

2

20

ab

ab

EESS



Atom InterferometerLaser beams

Atom

beam


Standards & Metrology Interféromètres atomiques

Jets

atomiquesFaisceaux

laser



E(p)

p

E(p)

p

SATURATION SPECTROSCOPY

22 / Mchrecoil doublet



Optical clocks with cold atoms

use the “working horse” of laser cooling: Magneto-optical trap (MOT)

In the future new atom sources such as atom lasers



Time-domain Ramsey-Bordé interferences with cold Ca atoms

Time-domain Ramsey-Bordé interferences with cold Ca atoms



THEORY OF OPTICAL CLOCKS: SUCCESSIVE STEPS, RELEVANT STUDIES AND DIRECTIONS OF PROGRESS

To-day we combine all these elements in a new sophisticated and realistic quantum description of optical clocks.

This effort is also underway for atomic inertial sensors.Strategies to eliminate inertial field sensitivity of optical clocks

• 1977: Naive, perturbative and numerical approaches• 1982: 2x2 ABCD matrices for field pulses/zones and free propagation between pulses/zones : still used• 1991: ABCD formalism for atom wave propagation in a gravitational field• 1994: Strong field S-matrix treatment of the e.m. field zones• 1995: Rabi oscillations in a gravitational field (analogous to frequency chirp in curved wave-fronts)• 1996: Dispersive properties of the group velocity of atom waves in strong e.m. fields



RELATIVISTIC PHASE SHIFTS

±' = ¡1¹h

Z t

t0dt0

(c2

2E (~p)p¹ h¹ º(~x0 + ~vt0; t0)pº

+°

m(° + 1)

"c2p¹ ~r h¹ º (~x0 + ~vt0; t0)pº

2E 2(~p)£ ~p

#

¢~s

¡c2

"~r £

Ã~h(~x0 + ~vt0; t0)¡

)h (~x0 + ~vt0; t0) ¢

~pcE (~p)

! #

¢~s

)

where ~s is the mean spin vector

~s =X

r;r0¯¤

r;i¯ r0;i¹hw(r)y~aw(r0)=2°



theoremStokes D-4

2

1with

2

1

2

2

hpAAAdxdx

dxhpdtphpE

c

Quite generally, the spin-independent part of the phase shift is:



Atom Interferometers as Gravito-Inertial Sensors: I - Gravitoelectric field case

Laser beams

Atoms

g

2/1

002hMcdt

Gravitational phase shift:

k

T T ’ T

with light: Einstein red shiftwith neutrons: COW experiment (1975)with atoms: Kasevich and Chu (1991)

Phaseshift

Circulation of potential

Ratio of gravitoelectric flux to quantum of flux

Mass independent (time)2

)'(. TTTgk

2/./ 00

2

hxdtdM

c



Laser beams

Atoms

dtphc

.

1

with light: Sagnac (1913)with neutrons: Werner et al.(1979)with atoms: Riehle et al. (1991)

Atom Interferometers as Gravito-Inertial Sensors: II - Gravitomagnetic field case

Phaseshift

Circulation of potential

Ratio of gravitomagnetic flux to quantum of flux

Mc

AchcSd

Mc /

.2curl.

/

1 2

Sagnac phase shift:



DOPPLER-FREE TWO-PHOTON SPECTROSCOPY

E(p)

p



Supersonic beam(seeded with He)

4-mirrorFabry-Perotcavity

ultra-high resolutionspectrometer

Cavity lock

D1

AOM 2

+84.757 MHz

D2

+160 MHz

Tunable ultra-stable laser

Reference laser

RF synthesizer

phase lock

FM1

FM2

AOM 1

2-photon Ramsey fringes experiment



Hyperfine structure of the P(4)E0 23 line of SF6

33% SF6, periodicity 600 HzS/N1Hz 5

FM 465 Hz,depth 300 Hz,28 mW inside the cavity4.5x105 Pa, 4s/point

20% SF6, S/N1Hz 14 periodicity 690 Hz

FM 465 Hz, depth 300 Hz,28 mW inside the cavity,4.5x105 Pa, 2 s/point.

690 Hz

600 HzInterzone : 50 cm

a)

b)



RECOIL SHIFT IN DOPPLER-FREE TWO-PHOTON SPECTROSCOPY

E(p)

p

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