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10 September 2001
6th 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
10 September 2001
6th Symposium on Frequency Standards & Metrology
ATOMS ARE WAVES !
v
ldeBroglie
k 2
The recoil energy
is not negligible any more in Cesium clocks
2M
Atom sources may be coherent sources of matter-wave
Different from small clocks carried by classical point particles
Atomic frame of reference may not be well defined
Atomic clocks are fully quantum devices, in which both the internal
and external degrees of freedom of the atoms must be quantized
Gravitation and inertia play an important role:
Atomic clocks are relativistic devices
10 September 2001
6th Symposium on Frequency Standards & Metrology
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
10 September 2001
6th Symposium on Frequency Standards & Metrology
ENERGY
E(p)  
atom
slope=v
hdB
rest mass
h
Mc
2
h/l
MOMENTUM
photon
slope=c
h / ldB
p  K
10 September 2001
z
6th Symposium on Frequency Standards & Metrology
ATOMIC WAVES
y

a(r , t ) 
x

e
 E M c  p c

2 4
2 2


d3p
a p 
3/ 2
2 
  

i  p( r  r0 )  E ( p )( t  t 0 ) / 

 p x  2ME  p y2  p z2

 E  E  p 

p2
non

relativist
ic
2
     
   E  Mc 
2M




travelling  wave
      


 p y2  p z2  
paraxial

     p x  2ME 1 



4
ME



 i p y2  p z2


i
x  x0    p y y  p z z 
a (r , t )   dp y dp z a  p y , p z exp 

  2 2 ME

i
i

exp 
2 ME  x  x0   E t  t0 



 p y2 
 p z2 
a  p y , p z   exp  2  exp  2  TEM00
 2 
 2 
10 September 2001
6th Symposium on Frequency Standards & Metrology
ABCD matrices for light and matter-wave optics
y0
x0
x1
Optical
System
y1
Space
or
Time
 x1   A B  x0 
   
 
 y1   C D  y0 
for light rays
 x1   A B  x0 
   
 
 v1   C D  v 0 
for massive particles
In Gaussian optics, the matrix ABCD also gives the transformation law for the waves:
1 1
2
 i 2
q R kw
transforms as
Aq0  B
q1 
Cq0  D
10 September 2001
6th Symposium on Frequency Standards & Metrology
ABCD PROPAGATOR
For a wave packet moving with the initial velocity
v 0  p0 / M
1/ 2
 iM Y0
1
 p0 z ' z0  
 M  
 iM
2
2
2 
2
2

exp
z
'

z

dz''exp
exp 
Dz

2
zz
'

Az
'
exp
i

  dz
Dz  2 zz ' Az' 

0 



iiB
B  
B

 22
 22B
 X 0


 2 X 0

1
 iM

 iM
2
2
Cz0  Dv 0 z  Az0  Bv 0 

exp 
ACz0  DBv 0  2 BCz 0 v 0  exp 
X
 2

 

1/ 2





 iM Y
z  Az0  Bv 0 2 
exp 
 2 X

 exp iS /  exp ipcl (t )z  zcl (t ) / F z  zcl (t ), X (t ), Y (t )
zcl  Az0  Bv 0 , X  AX 0  BY0
v cl  Cz0  Dv 0 , Y  CX 0  DY0
10 September 2001
6th Symposium on Frequency Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY
SEPARATED FIELD ZONES
y
z
x
a
a
b
a
b
a
b
b
10 September 2001
6th Symposium on Frequency Standards & Metrology
E(p)
n(p)
Recoil energy h 2 / 2Ml2
h
2Mk BT
h/l
p
10 September 2001
6th Symposium on Frequency Standards & Metrology
E(p)
p
10 September 2001
6th Symposium on Frequency Standards & Metrology
RAMSEY FRINGES :
FIRST-ORDER TRANSITION AMPLITUDE AFTER A SINGLE FIELD ZONE
z
y
b
x
a
b
ATOMS
a
EM WAVE
((11)) 
bb ((rr,,tt))  i  ei (  kz  t  )
e
 w 2   ba  kv z  2 / 4 v 2x

d 3 p w ba
2 3 / 2 v x
Rabi envelope
ei  ba  kv z  ( x  x1 ) / v x
additional momentum
  

iipp((rrrr00))EEaa((pp)()(tttt00))// ((00)) 
e
a ( p, t )
initial wave packet
10 September 2001
6th Symposium on Frequency Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY
SEPARATED FIELD ZONES
EM WAVE 1
EM WAVE 2
a
b
b
a
a
b

w
b (r , t )  i
baei ( kz  t  )
vx
(1)
e
e
 w 2   ba  2 / 4 v 2x
i   ba  ( x  x1 ) / v x

a (r , t )
(0)

(1)* 
b (r , t )b (r , t )  ei  ba  ( x2  x1 ) / v x  c.c.
(1)
10 September 2001
6th Symposium on Frequency Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY
SEPARATED FIELD ZONES
y
z
b
x
a
a,pz
b
b
ATOMS
b
a,pz
EM WAVE 1

EM WAVE 2
(1) (1)*
2
2


dz
b
b

dp
exp

w




k
v


/
2
v
ba
z
x
 1 2  z
2
exp i   ba  kv z   x2  x1  / v x a ( 0)  p z a ( 0)*  p z 
k


  dz a  z  x2  x1  / v x , t  a ( 0)* z, t 
M


(0)

a
10 September 2001
6th Symposium on Frequency Standards & Metrology
E(p)
Recoil energy h 2 / 2Ml2
p
10 September 2001
6th Symposium on Frequency Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY
SEPARATED FIELD ZONES
y
z
a
a,pz
b
b
a,p'z
EM WAVE 1
 dz b
a
b
x
ATOMS
b
(1) (1)*
1 2 
b
  dze
2 ikz
EM WAVE 2
 dp dp'
z
z
exp i p z  p' z z /  ....
a ( 0)  p z a ( 0)*  p' z     p z  p' z 2k 
10 September 2001
6th Symposium on Frequency Standards & Metrology
E(p)
p
10 September 2001
6th Symposium on Frequency Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY
SEPARATED FIELD ZONES
y
z
a
b
x
a
a,pz
b
ATOMS
EM WAVE
a,pz±2k
(1) (1)*
dz
b
 1 b2  expi  ba   x2  x1  / v x 
 dp
b
z
exp  i kv z  2 2 x0  x2  x1  / v x 
a ( 0)  p z a ( 0)*  p z  2k 
b
10 September 2001
6th Symposium on Frequency Standards & Metrology
Rubidium clock with a monomode continuous coherent beam
Auxiliary
Magnetic shield
Microwave
Height
1m
Microwave
resonator
Detection of F=1,m=0
- Flux 107 atoms/s (gain of 10/ present fountains)
- Average density 109 atoms/cm3 for x=50 mm
- Continuous operation
- No losses between rise and fall: vx=15 mm/s
Courtesy of Jean Dalibard
and David Guéry-Odelin
10 September 2001
1
6th Symposium on Frequency Standards & Metrology
Non-relat ivist ic approach
We shall consider quit e generally t he non-relat ivist ic
Schroedinger equat ion as t he non-relat ivist ic limit of a
general relat ivist ic equat ion described in t he last part
of t his course:
i ¹h
@jª (t )i
@t
=
)
1 ¡!
H0 +
p op¢ g (t ) ¢¡!p op
2M
¡!
¡!
¡!
¡ - (t ) ¢( L op + S op)
M
)
¡ M~
g(t ) ¢~
r op ¡
~
r op¢ ° (t ) ¢~
r op
2
+ V (~
r op; t )] jª (t )i
(1)
where H 0 is an int ernal at omic Hamilt onian and V (~
r op; t )
some general int eract ion Hamilt onian wit h an ext ernal ¯eld. Gravit o-inert ial ¯elds are represent ed by t he
)
¡!
)
t ensors g (t ) and ° (t ) and by t he vect ors - (t )
and ~
g(t ). T he same t erms can also be used t o represent t he e®ect of various ext ernal elect romagnet ic
¡!
¡!
¯elds. T he operat ors L op = ~
r op £ ¡!p op and S op are
respect ively t he orbit al and spin angular moment um
10 September 2001
6th Symposium on Frequency Standards & Metrology
ABCDx PROPAGATOR
 iM 

 iM
exp  x ( z  x ) exp 
 

 
1/ 2
 M 


 2iB 



x  x  g  0
 2 / 2  x 2 / 2  gx )dt 
(
x
1
t '

t


 iM

dz 'exp 
D( z  x ) 2  2( z  x ) z ' Az'2 
 2B

iM
11
iM YY00
 pp zz''zz00
22
zz''zz00 exp
exp
exp
expii 00


XX00

 22 XX00
 
iM  
 iM 

 iM t  2
2
 exp  x ( Az0  Bv 0 ) exp 
(
x

x
)
dt

xx 
1

t
'

 

 2

1
 iM

 iM

2
2
exp 
ACz0  DBv 0  2 BCz 0 v 0  exp 
Cz0  Dv 0  x  z  x  Az0  Bv 0 
X
 2

 




 iM Y
z  Az0  Bv 0  x 2 
exp 
 2 X


 exp iS /  exp ipcl (t )z  zcl (t ) / F z  zcl (t ), X (t ), Y (t )
zcl  Az0  Bv0  x , X  AX 0  BY0
vcl  Cz0  Dv0  x, Y  CX 0  DY0
10 September 2001
6th Symposium on Frequency Standards & Metrology
Quite generally, the phase shift along each arm is:
S (t, t )   p
0

t
cl cl t 0
z

 2

1
/  
(
p
(
t
)

M
x
)
dt
cl
1
1

2M t0
t
i.e. minus the time integral of the kinetic energy
10 September 2001
6th Symposium on Frequency Standards & Metrology
FOUNTAIN CLOCK
b
   k x
a
 k  (   ba  kv z   ) / v x
1 2
x   gT
2
a
b
10 September 2001
6th Symposium on Frequency Standards & Metrology
Gravitational/Relativistic Doppler shift for fountain clocks
A quantum mechanical calculation
~ Langevin twin paradox
exp iS /  expipcl (t )z  zcl (t ) / 
1
2
2

M
c

M
v
a ,b
a ,b a, b
c2
2
 2
t 
1
2
S a ,b /    M a ,b c t  t0  
( p0  M a ,bx ) dt1

t
2 M a ,b 0


t
 
p0
t1  t0   x )dt1
  M a ,b g  (r0 
t0
M a ,b
M a ,b c 2 / 1 
b
a
Sb  Sa
a
v a,2 b
b

Eb  Ea   1 v 02  2v 0 


/   
1




6 c 2  g 
10 September 2001
6th Symposium on Frequency Standards & Metrology
Atom Interferometer
Laser beams
Atom
beam
Interféromètres atomiques
10 September 2001
6th Symposium on Frequency Standards & Metrology
Jets
atomiques
Faisceaux
laser
10 September 2001
6th Symposium on Frequency Standards & Metrology
SATURATION SPECTROSCOPY
E(p)
E(p
)
p
recoil doublet
p
h 2 / Mc 2
10 September 2001
6th Symposium on Frequency Standards & Metrology
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
10 September 2001
6th Symposium on Frequency Standards & Metrology
Time-domain Ramsey-Bordé interferences
with cold Ca atoms
10 September 2001
6th Symposium on Frequency Standards & Metrology
THEORY OF OPTICAL CLOCKS:
SUCCESSIVE STEPS, RELEVANT STUDIES AND DIRECTIONS OF PROGRESS
• 1977: Naive, perturbative and numerical approaches
• 1982: 2x2 ABCD matrices for field pulses/zones
and free propagation between pulses/zones : still used
• 1991: ABCDx 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
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
10 September 2001
6th Symposium on Frequency Standards & Metrology
RELATIVISTIC PHASE SHIFTS
±'
=
Z
1 t
(
dt 0
c2 ¹
¡
p h ¹ º (~
x0 + ~
v t 0; t 0)pº
¹h t 0
2E (~
p)
"
#
2
¹
0
0
º
~
°
c p r h ¹ º (~
x0 + ~
v t ; t )p
+
£ p
~ ¢~
s
2
m (° + 1)
2E (~
p)
"
Ã
! # )
)
c ~
p
~c
¡
r £ ~
h(~
x0 + ~
v t 0; t 0)¡ h (~
x0 + ~
v t 0; t 0) ¢
¢~
s
2
E (~
p)
where ~
s is t he mean spin vect or
~
s=
X
r ;r 0
0
¯ ¤r;i ¯ r 0;i ¹h w (r )y~
aw (r ) =2°
10 September 2001
6th Symposium on Frequency Standards & Metrology
Quite generally, the spin-independent part
of the phase shift is:
2
c
1 m
m


   
p hm p dt   p hm dx
2E
2
1 m
m

  dx dx  m A   Am  with A  p hm
2
4 - D Stokes theorem
10 September 2001
6th Symposium on Frequency Standards & Metrology
Atom Interferometers as Gravito-Inertial
Sensors: I - Gravitoelectric field case
with light: Einstein red shift
with neutrons: COW experiment (1975)
with atoms: Kasevich and Chu (1991)

g

k
Laser
beams
Atoms
T
T’
T
Gravitational phase shift:
1
c2
2
   dt Mc h00 / 2 

/M
Phase Circulation of
potential
shift

 dtd x.h00 / 2

 k . g T (T  T ' )
Ratio of
Mass independent
 (time)2
gravitoelectric flux
to quantum of flux
10 September 2001
6th Symposium on Frequency Standards & Metrology
Atom Interferometers as Gravito-Inertial
Sensors: II - Gravitomagnetic field case
with light: Sagnac (1913)
with neutrons: Werner et al.(1979)
with atoms: Riehle et al. (1991)


Laser
beams
Atoms
Sagnac phase shift:
1 
1
   ch . p dt 

c / M
Phase Circulation of
potential
shift
 
 2c. A
 2
 dS .c curl h  c / M
Ratio of
gravitomagnetic flux
to quantum of flux
10 September 2001
6th Symposium on Frequency Standards & Metrology
DOPPLER-FREE TWO-PHOTON
SPECTROSCOPY
E(p)
p
10 September 2001
6th Symposium on Frequency Standards & Metrology
2-photon Ramsey fringes experiment
+84.757 MHz
D2
AOM 2
Tunable ultra-stable laser
AOM 1
+160 MHz
FM2
Reference laser
4-mirror
Fabry-Perot
cavity
D1
FM1
RF synthesizer
Cavity lock
phase lock
ultra-high resolution
spectrometer
Supersonic beam
(seeded with He)
10 September 2001
6th Symposium on Frequency Standards & Metrology
Hyperfine structure of the P(4)E0 23 line of SF6
Interzone : 50 cm
600 Hz
33% SF6,
periodicity 600 Hz
S/N1Hz 5
a)
FM 465 Hz,
depth 300 Hz,
28 mW inside the cavity
4.5x105 Pa, 4s/point
690 Hz
b)
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.
10 September 2001
6th Symposium on Frequency Standards & Metrology
RECOIL SHIFT IN DOPPLER-FREE
TWO-PHOTON SPECTROSCOPY
E(p)
p