Download Diapositiva 1

Can we use atom interferometers in searching
for gravitational waves?
• C.J. Bordé, University of Paris N.
• G. Tino, University of Firenze
• F. Vetrano, University of Urbino
F.Vetrano - Aspen Winter Conference, FEB 2004
Intuitive concept:
δΦ ~ 1/λ
lower λ as much as you can !!
 MW  OW mc 2


 10
 OW  MW 
10
 10 12
But only if any other thing is the same. Is it so? Is it easy?
• The Physics (e.g. : mp > 0, vp < c; mL = 0, vL = c
dispersion equations E = E(p))
different
• The Technology: very different state of evolution for OW and MW
“optical components”
F.Vetrano - Aspen Winter Conference, FEB 2004
What kind of interference are we speaking about ? - 1
A coherent description of quantum particles exists, both for Dirac and K.G. scalar particles,
from which the main contribution from gravity fields to the phase can be deduced. Neglecting
coupling between spin and curvature we have:
t
c2

2
E
• scalar (newtonian)
2
(G.W.)

t0
dt
p  h p 
E(p)
t1
h
oo
dt
t0
• vectorial (e.g. Sagnac, Gale…..)
• tensorial
1
c2
2E
c

t1
i
h
p
dt
io

i = 1,2,3
t0
t1
i
j
p
h
p
 ij dt
i,j = 1,2,3
t0
[C.J.Bordé et al, in “Gyros,Clocks,Interferometers…”, C.Lammerzahl, C.W.F.Everitt, F.W.Hehl (Eds), Springer, Berlin (2001)
C.J.Bordé, Gen.Rel.Grav., 36, (2004) in press]
F.Vetrano - Aspen Winter Conference, FEB 2004
What kind of interference are we speaking about ? - 2
Ramsey Interference
Only internal degrees
of freedom are involved
in the processes
• Two level atoms (ground, excited)
• Two successive electromagnetic interactions
• No external momentum exchange (for the moment)
e.m.1
e.m.2
decay
[w.p.] g
0
g, e
t1
Overlap (g, e): e
t2
g
t
No possibility to know which are g and which are e along the (time) path: in the finale state there is
interference between the two states: this is the Ramsey Interference.
It involves only internal D.O.F. : we look at the number of decays/s
|e>
|g> , which shows interference fringes in the “time” space.
F.Vetrano - Aspen Winter Conference, FEB 2004
M.W. Evolution - 1
Standard quantum approach (in “rotating” system): first order perturbation theory for a
dipole e.m./w.p. interaction. If for simplicity we assume zero detuning δ, the population of
the e state after the interaction time τ with the e.m. field (by a laser of frequency ωL) is
1
c e ()  (1 - cos  eg )
2
2
where Ω eg is the Rabi frequency (
(♦)
 
 eg  -  e d  E g  ), being d the dipole momentum
of the atom and E the electric oscillating field. Putting ωeg = ωe – ωg , the detuning is defined
by δ = ωL – ωeg. Assumiming δ = 0, the equation (♦) describes the so called resonant Rabi
oscillations .
Assume: t = 0
|cg(0)|² = 1; τ such that Ω eg τ = π/2 (π/2 pulse). From (♦) we have :
1
c e () 
2
2
F.Vetrano - Aspen Winter Conference, FEB 2004
1
c g ( ) 
2
2
M.W. Evolution - 2
 0
1 1
  
 
2 i
1
π/2 pulse
This is a perfect Beam Splitter
for internal states
1
1/2
g
g
e
0
e
τ
F.Vetrano - Aspen Winter Conference, FEB 2004
M.W. Evolution - 3
Use two π/2 pulses in order to have 2Ωeg τ = π (π pulse): we have
c e (2)  1
2
 0
1
    
1
0
π pulse
This is a perfect Mirror
for internal states
1
g
e
e
g
0
2τ
Note:
If δ ≠ 0, with two π/2 pulses (time interval T between them) same considerations as before with
F.Vetrano - Aspen Winter Conference, FEB 2004
c e ( 2  T ) 
2
1
1  cosT 
2
M.W. Evolution - 4
Take into account external D.O.F. considering the momentum exchange :
e, p e   e   p e 
g, p g   g   p g 
 
 
 
V  -dE
E  E o cos ( k  r - L t  )
• kinetic term in the Hamiltonian (and in the phase too)
• recoil term in the detuning δ:
 
p  k  k2
  L  (eg 

)
M
2M
Even if the interference is on internal d.o.f., the recoil opens the path: if we look for interference,
we must close it spatially. Anyway, we are always looking at the fringes in the number of decays/s.
F.Vetrano - Aspen Winter Conference, FEB 2004
The simplest “closed” atom interferometer
Counter (decays/s)
g
e
e
g
Source
π/2
g
π
π/2
1


(2T  2cos  (2T  2)  c (1
)


2
2

2
c
 
e , p  k
 
e , p  k
ΔΦ takes into account all that happens to the atoms (GW too)
• the absorption (emission) of momenta modifies both internal and external states
simultaneously
• this Mach-Zender atom interferometer is too symmmetric: no hope for effects
from GWs on the interference fringes on the number of decays/s
F.Vetrano - Aspen Winter Conference, FEB 2004
The Ramsey-Bordé Interferometer
2
a,2
b,1
b,1
Atoms source
a,2
b = excited s.
a,0
b,1
b,1
a,0
a,0
a = ground s.
a,0
b,-1
a,0
a,0
1
b,-1
b,-1
L1
L2
Atoms source
L3
L4
Cat eyes
Laser
F.Vetrano - Aspen Winter Conference, FEB 2004
Towards the optics of M.W. - 1
• Atom Interferometers exist (and do work)
• Coherent approaches to quantum particle in curved space (weak field)
has been developed
• Find general tools for phase calculation in A.I. in presence of G.W.
• Design a “good” interferometer: optimal configuration, source, detection…..
• Find all possible noise sources and lower them to the best you can
We discuss only the first point with a simple look at the second one
F.Vetrano - Aspen Winter Conference, FEB 2004
Towards the optics of M.W. - 2
• In a limited interval of velocity the dispersion function E = E(p) can be “simplified” through
a power series expansion
a Schroedinger type equation is obtained
• If we look at the equation of a (laser) beam in a cavity with some (possibly) non-linear susceptivity,
indicating by U the shape of the mode and z the propagation axis, we have:
k o2
U 1 2
n
i

t U 

r
U

n n
z 2k
2k
which is a Schroedinger-like equation, provided the exchanges t
z; M/ћ
k
• Gaussian modes are a good basis for light; but gaussian wave packets are a basis too for
particle “beams”
[C.J.Bordé in “Fundamental systems in quantum optics”, LXIII Les Houches Session, J.Dalibard,J.M.Raimond,J.Zinn-Justin (Eds),
Elsevier, Amsterdam (1992)
C.J.Bordé, Gen.Rel.Grav, 36 (2004) in press]
F.Vetrano - Aspen Winter Conference, FEB 2004
Towards the optics of M.W. - 3
Look at the gaussian (lowest) modes for light and for a particle:
 Y k o w o2 2 
wo
U i
exp - i
r 
2
X
 X

w o2
k o w o2
i Y
Y
1
2
kowo 

i
;

w2
2X
2R
X Q
1
 (q , t ) 
2  q 
1
4
1
2


it
1


2 


2
M

q






q2


exp 



i

t
 4q 2 1 



2

2M q   



Note : M/ћ
k; t
F.Vetrano - Aspen Winter Conference, FEB 2004
z; Δq
wo/2 ;
1
2
Towards the optics of M.W. - 4
• similar equations
• similar basis
Can we use the tools of gaussian (light) optics in gaussian (m.w.) optics?
And especially the ABCD approach?
ABCD matrices approach for light
Gaussian mode(P1)
medium [ABCD matrices]
AQ1  B
Q2 
CQ1  D
gaussian mode(P 2)
ABCD law for gaussian
(light) optics
Yes, we can; but:
• remember the correspondances
• most important: how can we write the ABCD matrices for M.W.s propagation?
F.Vetrano - Aspen Winter Conference, FEB 2004
The path to reach ABCD matrices - 1
w.p.(P )
?
w.p.(P )
1
2
[ABCD]
Suppose the Hamiltonian quadratic at most:
  
   p p M   
H pq
- qq
2M
2
• write the classical action Scl(1,2)
• write the quantum propagator for w.p.(1)
w.p.(2) (e.g. applying the principle of
correspondence to Scl)
• Apply the quantum propagator to a basis of gaussian w.p.s from (1) to (2)
• Write formally the ABCD law and require it be identically satisfied
F.Vetrano - Aspen Winter Conference, FEB 2004
The path to reach ABCD matrices - 2
We obtain:
iS cl ( t1,t 2 ) 

w.p.( t 2 , p 2 , q 2 )  exp 
w.p.(t 1 , p1 , q1 )




where:
 q2   A
 p    21 B 21 

 2 
C 21 D 21 


M
   
 -1 -   0
 X 2   A 21

Y 
 

 2   C 21
 q1    
p 


 1 
M 


M
B 21   X1 

Y 

D 21 
 1
and:
 A 21 B21 

 
 C 21 D 21 
τ
being a time ordering operator.
F.Vetrano - Aspen Winter Conference, FEB 2004

t 2     
 dt 
exp   
 t1     
The Beam Splitter influence
Standard 1st order perturbation approach for weak dipole interaction
ttt theorem
The B.S. introduces a multiplicative amplitude and phase factor simply related to the laser
beam quantities (indicated by a star):

 
 
M bs exp  t  k q  
[C.Antoine, C.J.Bordé, Phys.Lett.A, 306 , 277-284 (2003) and references therein]
F.Vetrano - Aspen Winter Conference, FEB 2004


Phase shift for a sequence of pairs of
homologous paths (an interferometer geometry) - 1
q
kβN
kβ1
β1
Mβ1
β2
kα1
Mβ2
Mβ3
Mα2
kα2
kβi
β3
Mα1
α1
kβ3
kβ2
Mβi
βi
Mα3
α2
kαi
αi
βD
MβN
Mαi
α3
kα3
βN
MαN
αD
αN
kαN
t
t1
t2
t3
F.Vetrano - Aspen Winter Conference, FEB 2004
ti
tN
tD
Phase shift for a sequence of pairs of
homologous paths (an interferometer geometry) - 2
From ABCD law, ttt theorem, and properties of classical action Scl(P1,P2)
for N beam-splitters :
  -
p 1  p1
2
q i  q  i


q1  q 1    k i  k i 
- i  i  t i  i  i  
2
i 1 

N
M i  M i  Si p i 1

p i  k i
qi1  q i1  qi  q i 



2
2M i
i 1

 M i 2M i
N
 Si pi 1
 
pi  k i
qi  q i  q i  qi  


2M i
 M i 2M i
 
[C.Antoine, C.Bordé, J.Opt. B: Quantum Semiclass.Opt., 5, 199-207 (2003)]
F.Vetrano - Aspen Winter Conference, FEB 2004
Build the simplest A.I. for G.W. - 1
BS1
BS2
BS3
BS4
a,0
T’
T1
T2
b,k
b, -k
Atom source
a,0
a,0
Put T1 = T2 , T’
F.Vetrano - Aspen Winter Conference, FEB 2004
a,0
0
a,0
the simplest unsymmetrical A.I.
Counter
Build the simplest A.I. for G.W. - 2
The machine
• Choose FNC (It’s better!!!)
• Calculate ABCD matrices in presence of GW at the 1st order in the strain
amplitude h (e.g.: a wave with polarization “+” impinging perpendicularly
on the plane of the interferometer)
• Apply ΔΦ expression (slide 19) to the interferometer
• Use ABCD law to substitute all qj coordinates
• Fully simplify
• Print ΔΦ
• End
Note : the job should be worked in the frequency space: use Fourier transform, please!
F.Vetrano - Aspen Winter Conference, FEB 2004
Build the simplest A.I. for G.W. -3
The phase shift:
where:
k
  - p1 h  2 T 2 f 
m
2/m = 1/Ma + 1/Mb ; kћ = transversal momentum
(by BS);
p1 = initial (longitudinal) momentum;
f(ω) = complex frequency response of the interferometer

T

sin



2 
2
f  
sin T  T
 



2



2



cos2T - cosT 
 

T







T

 sin
 sin2 T - sin T
2
i
- cosT 

T
T




2


F.Vetrano - Aspen Winter Conference, FEB 2004






2






Build the simplest A.I. for G.W. - 4
The sensitivity (1):
Suppose the A.I. “shot noise” limited, with η a kind of efficiency of the decay process we use:
shot noise 


N
At the level of S.N.R. = 1 we have:
h  
2
1
 p L2 T f 
N
T
T
sin
2
2
f  
 T
2
Note: |f(ω)| ~ T
(ω→0);
F.Vetrano - Aspen Winter Conference, FEB 2004
T

sin

sin T
2
1- 2

T
 T

2







2
|f(ω)| ~ (2/ω) |sin (ωT/2)/ (ωT/2)| → 0
(ω→∞)
Build the simplest A.I. for G.W. - 5
The sensitivity (2):
  1018 atoms (H)/s;
v L  10 m/s; N
6
h (1/Hz)
10  20
T  10 -3 s; L  10 3 m; vT  10 m/s
10  22
v L  10 7 m/s; L  10 5 m
T  10 - 2 s
10  24
v L  10 m/s ; vT  5 m/s ; L  50 m;
  1018 atoms (Cs) / s
N
10  26
0.1
F.Vetrano - Aspen Winter Conference, FEB 2004
1
10
100
1000
Hz
Final Remarks
• comprehensive approach to the problem (A.I. + G.W.)
• “automatic” tool for solving atom interferometers
• realistic values of physical parameters (someone on the borderline…)
• interesting value of the sensitivity even for a “minimal” atom interferometer
• don’t forget BEC
• noise budget ?
• from the idea to the experiment
Looking at a future (not too far from now, hopefully) and at a very hard work, bearing
in mind the title of this talk, in my opinion we can say…………….
F.Vetrano - Aspen Winter Conference, FEB 2004
Conclusion
….be optimistic: we can!!!
F.Vetrano - Aspen Winter Conference, FEB 2004
Some general references:
• P.Berman (Ed), Atom Interferometry, Ac.Press, N.Y. (1997)
• S.Chu, in “Coherent atomic matter waves”, 72° Les Houches Session, R.Kaiser, C.Westbrook,
F.David (Eds), Springer Verlag, N.Y. (2001)
• C.J. Bordé, CR Acad.Sc.Paris, t2-SIV, 509-530 (2001)
• C.J.Bordé, Metrologia, 39, 435-463 (2002)
Please Note:
Not all of the Authors have had the possibility of revising the final form of this talk; they
share obviously the scientific content, but mistakes, misunderstandings, misprints are
totally under my own responsibility alone (F.V.).
F.Vetrano - Aspen Winter Conference, FEB 2004