Comparison of atom and photon interferometers using 5D optics
Christian J. Bordé SYRTE & LPL
E(p)
ENERGY μ
μ
p = ( E / c, p x , p y , p z ) pμ = ( E / c, − p x , − p y , − p z )
p pμ = E / c − p = m c 2
2
2
2 2
G E ( p ) = m 2 c 4 + p 2 c 2 = mR c 2
1905
Mass
MOMENTUM
p
E μˆ
p = ( E / c, p x , p y , pz , mc )
μˆ
p pμˆ = 0
mc
μˆ = 0,1,2,3,4
p
E(p)
b a
mbc2 mac2
p//
μˆ
x = (ct , x, y , z, cτ )
μˆ
t2
dx dxμˆ = c dt − dx − ds = 0 2
2
2
s = cτ
μ
μ
dx p v = = dt mR μ
dcτ mc v = = dt mR 4
v4
ds = c dt − dx = 0 2
2
2
p μˆ = ( E / c, p x , p y , pz , mc ) μˆ ˆ S = − ∫ pμˆ dx
ˆˆ μν ˆ S = =φ → g ∂ μˆφ ∂νˆφ = 0
2
x
E(p) → =Ω ∂ϕ μ μ p → i= = popϕ ∂ x hν μ ENERGY
dB
1 ∂ ,≡ 2 2 − Δ c ∂t 2
1905
Mass
2 2
1923
mc p pμ = m c → ϕ + 2 ϕ = 0 =h / λ dB → =K MOMENTUM p μ
2 2
FROM 3 TO 4 SPATIAL DIMENSIONS E ∂ ϕ ∂ϕ μ μ 4 = popϕ p → i= p → i= = −mop cϕ ∂xμ ∂cτ mc
μˆ
p pμˆ = 0 →
p
1 ∂ϕ ,ˆ ϕ ≡,ϕ − 2 2 = 0 c ∂τ 2
de Broglie wavelength and de de Broglie-Compton frequency
1 ∂ϕ ,ˆ ϕ ≡,ϕ − 2 2 = 0 c ∂τ 2
t
G k
s = cτ
x
2
1 ⎛E⎞ 1 1 k = 2 =⎜ ⎟ = 2 + 2 dBC dB ⎝ =c ⎠ 2
h h λ dBC = λ dB = p No clock at mc the de Broglie-Compton frequency (possible in the future)
CHEMIN OPTIQUE & PRINCIPE DE FERMAT
1 ∂ϕ ,ˆ ϕ ≡,ϕ − 2 2 = 0 c ∂τ équation d'iconale à 5D (μˆ ,νˆ = 0,1, 2,3, 4): 2
g ∂ μˆ φ ∂νˆφ = 0 ˆˆ μν
(4) ⎛ g j0 j ⎞ E dl μˆ hφ = − ∫ pμˆ dx = − ⎜ ∫ cdt − ∫ dx ⎟ +∫ ⎟ c ⎜⎝ g 00 g 00 ⎠
dl ( 4) = − fij dxi dx j + c 2 dτ 2 fij = gij −
g 0i g 0 j g 00
(4 )
λ
h = c g 00 E
BASICS OF ATOM /PHOTON OPTICS Schroedinger-like equation for the atom /photon field:
∂ϕ mR c 1 1 j μν 4 ⎡ ⎤ i= = p p j + p p4 ⎦ ϕ + pμ h pν ϕ ϕ− ⎣ ∂t 2 2 mR 2 mR 2
p j = i=∂ j ;
p4 = i=∂ cτ ;
p0 = mR c (=ω / c for photons)
GG 2 G⇒ G 2 - gravitation field: h = −2 g.q / c − q. γ .q / c ⇒ G G - rotation field: h = − α .q / c 00
⇒
⇒
⇒
- gravitational wave: h = β − δ
G⇒ G G⇒ G G⇒ G GG GG Hext = p.α (t ).q + p. β (t ). p / 2mR − mRq. γ (t).q / 2 − mR g.q + f . p
ABCDξφ LAW OF ATOM OPTICS G⇒ G G⇒ G G⇒ G GG GG Hext = p.α (t ).q + p. β (t ). p / 2mR − mRq. γ (t ).q / 2 − mR g.q + f . p wavepacket (q, t ) = exp ( iScl / = ) exp exp ⎡⎣⎡⎣ip ipcc((tt))( qq −− qqcc((tt))) // ==⎤⎦⎤⎦ F F ( qq −− qqcc((tt),), X X ((tt),),YY((tt))) p = ( px , p y , pz , Mc); q = ( x, y, z , cτ )
qc (t ) = Aqc (t0 ) + Bpc (t0 ) / mR + ξ (t , t0 )
pc (t ) / mR = Cqc (t0 ) + Dpc (t0 ) / mR + φ (t , t0 ) X (t ) = AX (t0 ) + BY (t0 ) Y (t ) = CX (t0 ) + DY (t0 )
Framework valid for Hamiltonians of degree ≤ 2 in position and momentum
Hamilton’s equations for the external motion
G⇒ G G⇒ G G⇒ G GG GG Hext = p.α (t ).q + p. β (t ). p / 2mR − mRq. γ (t ).q / 2 − mR g.q + f . p ⎛ q ⎞ χ =⎜ ⎟ p m / R⎠ ⎝
⎛ dH ext ⎞ ⎟ ⎛ α (t ) β (t ) ⎞ dp ⎛ f (t ) ⎞ d χ ⎜⎜ ⎟=⎜ χ +⎜ = = Γ( t ) χ + Φ ( t ) ⎟ ⎟ dt ⎜ 1 dH ext ⎟ ⎝ γ (t ) α (t ) ⎠ ⎝ g (t ) ⎠ − ⎜ m dq ⎟ R ⎝ ⎠ ⎛ A ( t , t0 ) B ( t , t0 ) ⎞ ⎛ ξ ( t , t0 ) ⎞ χ (t ) = ⎜ ⎟ χ ( t0 ) + ⎜ ⎟ C t , t D t , t φ t , t ( ) ( ) ( ) 0 0 ⎠ 0 ⎠ ⎝ ⎝ ⎛ A ( t , t0 ) B ( t , t0 ) ⎞ ⎡ t ⎛ α (t ') β (t ') ⎞ ⎤ M ( t , t0 ) = ⎜ ⎟ = T exp ⎢ ∫t ⎜ ⎟dt '⎥ 0 C t , t D t , t ( 0 )⎠ ⎣ ⎝ γ (t ') α (t ') ⎠ ⎦ ⎝ ( 0)
t ⎛ ξ ( t , t0 ) ⎞ = ⎜ ⎟ ∫t M ( t , t ')Φ (t ')dt ' 0 ⎝ φ ( t , t0 ) ⎠
Atomic Gravimeter
Space coordinate z
arm II z 1' v 1'
τ4
z2' v 2' z2 v2
⎛ z2 − z ' 2 ⎞ mR c (τ 1 + τ 3 − τ 2 − τ 4 ) + ( p2 + =k + p2 ' ) ⎜ ⎟=0 ⎝ 2 ⎠ 2
τ2
v0'
z0
τ3 z1
τ1
v0
p dx = 0 j ∫ j
z1 1= Az(T=)(A z0(T− )( g z/ γ−) +g B v R + g /γ / γ(T) +) pB0 (/Tm)v 1 0 0
/ mR = C(T )( z0 − g / γ ) + D(T ) p0 / mR + =k / mR armp1 I
T
T'
Time coordinate t
δϕ = −k ( z2 − z1 − z '1 + z0 ) + k ( z2 − z '2 ) / 2
Exact phase shift for the atom gravimeter δϕ = −k ( z2 − z1 − z '1 + z0 ) + k ( z2 − z '2 ) / 2 k ⎧⎡ = ⎨ ⎣sinh γ ⎩
(
+ γ ⎡1 + cosh ⎣
(
⎛ =k ⎞ ⎤ γ T ⎜ v0 + ⎟ ⎦⎝ 2 mR ⎠
γ ( T + T ' ) − 2 cosh
⎛ g ⎞⎫ ⎤ γ T ⎜ z0 − ⎟ ⎬ ⎦⎝ γ ⎠⎭
)
γ ( T + T ' ) − 2sinh
(
)
)
(
)
which can be written to first-order in γ, with T=T’:
⎡7 ⎤ =k ⎞ ⎛ 2 T − z0 ⎥ δϕ = kgT + kγ T ⎢ gT − ⎜ v 0 + * ⎟ 2M ⎠ ⎝ ⎣12 ⎦ 2
2
Reference: Ch. J. B., Theoretical tools for atom optics and interferometry, C.R. Acad. Sci. Paris, 2, Série IV, p. 509-530, 2001
ARBITRARY 3D TIME-DEPENDENT GRAVITO-INERTIAL FIELDS
G ⇒ G G ⇒ G G⇒ G Hamiltonian: H = p. α (t ).q + p. β (t ). p / 2mR − mR q. γ (t ).q / 2 ⎛A B⎞ ⎛α β ⎞ Hamilton's equns: ⎜ = T exp ∫ dt ⎜ ⎟ ⎟ C D γ α ⎝ ⎠ ⎝ ⎠
Example: Phase shift induced by a gravitational wave ⇒ ⇒ ⇒ ⇒ ⇒ Einstein coord.: β = 1 + h cos (ξ t + φ ) , γ = 0, with h = {h ij } ⇒
⇒ ⇒
⇒
Fermi coord.: β = 1 , γ = (ξ / 2 ) h cos (ξ t + φ ) 2
A =1 ⎧ ⎪ Einstein coord.: ⎨ h ⎪ B = t + ξ ⎡⎣sin (ξ t + φ ) − sin φ ⎤⎦ ⎩ h hξ t ⎧ A = 1 − ⎡⎣cos (ξ t + φ ) − cos φ ⎤⎦ − sin φ ⎪⎪ 2 2 Fermi coord.: ⎨ ⎪ B = t + h ⎡⎣sin (ξ t + φ ) − sin φ ⎤⎦ − ht ⎡⎣cos (ξ t + φ ) + cos φ ⎤⎦ ξ 2 ⎪⎩
Atomic phase shift induced by a gravitational wave
δϕ = −khV0ξ T sin (ξ T + φ ) sinc (ξ T / 2 ) −khq0 / 2 ⎡⎣cos ( 2ξ T + φ ) − 2 cos (ξ T + φ ) + cos φ ⎤⎦ 2
2
−khV0T ⎡⎣cos ( 2ξ T + φ ) − cos (ξ T + φ ) ⎤⎦ + ϕ 0 − 2ϕ1 + ϕ * =k ⎞ ⎛ V0 = ⎜ p0 + ⎟ / M 2 ⎠ ⎝
Ch.J. Bordé, Gen. Rel. Grav. 36 (March 2004) Ch.J. Bordé, J. Sharma, Ph. Tourrenc and Th. Damour, Theoretical approaches to laser spectroscopy in the presence of gravitational fields, J. Physique Lettres 44 (1983) L983-990
Bordé-Ramsey interferometers Laser beams Atom beam 2
mR c ⎛ =k ⎞ δϕ = − T⎜ ⎟ h cos (ξ T + φ ) sinc (ξ T ) = ⎝ mR c ⎠ 2
Relativistic atom optics and interferometry
http://christian.j.borde.free.fr/st163023.pdf
