10 september 20016th symposium on frequency standards & metrology relativistic quantum theory of...
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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
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th 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 20016th Symposium on Frequency
Standards & Metrology
MOMENTUM
E(p)
p
atomslope=v
photonslope=c
rest mass
ENERGY
Mc2
h
h / h dB/
h dB
K
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES
ba ba a b
z y
xa
b
10 September 20016th Symposium on Frequency
Standards & Metrology
E(p)
pTMkB2
n(p)
h
h /
Recoil energy 22 2/ Mh
10 September 20016th Symposium on Frequency
Standards & Metrology
E(p)
p
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES
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
10 September 20016th Symposium on Frequency
Standards & Metrology
E(p)
p
Recoil energy 22 2/ Mh
10 September 20016th Symposium on Frequency
Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES
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
10 September 20016th Symposium on Frequency
Standards & Metrology
E(p)
p
10 September 20016th Symposium on Frequency
Standards & Metrology
RAMSEY FRINGES WITH TWO SPATIALLY SEPARATED FIELD ZONES
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology FOUNTAIN CLOCK
2
xz
2
1
v/)v(
gT
kk ba
a
a
b
b k
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
Atom InterferometerLaser beams
Atom
beam
10 September 20016th Symposium on Frequency
Standards & Metrology Interféromètres atomiques
Jets
atomiquesFaisceaux
laser
10 September 20016th Symposium on Frequency
Standards & Metrology
E(p)
p
E(p)
p
SATURATION SPECTROSCOPY
22 / Mchrecoil doublet
10 September 20016th 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 20016th Symposium on Frequency
Standards & Metrology
Time-domain Ramsey-Bordé interferences with cold Ca atoms
Time-domain Ramsey-Bordé interferences with cold Ca atoms
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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°
10 September 20016th Symposium on Frequency
Standards & Metrology
theoremStokes D-4
2
1with
2
1
2
2
hpAAAdxdx
dxhpdtphpE
c
Quite generally, the spin-independent part of the phase shift is:
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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:
10 September 20016th Symposium on Frequency
Standards & Metrology
DOPPLER-FREE TWO-PHOTON SPECTROSCOPY
E(p)
p
10 September 20016th Symposium on Frequency
Standards & Metrology
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
10 September 20016th Symposium on Frequency
Standards & Metrology
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)
10 September 20016th Symposium on Frequency
Standards & Metrology
RECOIL SHIFT IN DOPPLER-FREE TWO-PHOTON SPECTROSCOPY
E(p)
p
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