Download Can Spacetime Curvature Induced Corrections to Lamb Shift Be

Can Spacetime curvature induced corrections
to Lamb shift be observable?
Hongwei Yu
Ningbo University and Hunan Normal University
Collaborator: Wenting Zhou (Hunan Normal)
OUTLINE

Why-- Test of Quantum effects

How -- DDC formalism

Curvature induced correction to Lamb shift

Conclusion

Why

Quantum effects unique to curved space

Hawking radiation

Gibbons-Hawking effect

Particle creation by GR field

Unruh effect
Challenge: Experimental test.
Q: How about curvature induced corrections to
those already existing in flat spacetimes?

What is Lamb shift?

Theoretical result:
The Dirac theory in Quantum Mechanics shows: the states, 2s1/2
and 2p1/2 of hydrogen atom are degenerate.

Experimental discovery:
In 1947, Lamb and Rutherford show that the level 2s1/2 lies about
1000MHz, or 0.030cm-1 above the level 2p1/2. Then a more accurate
value 1058MHz.
The Lamb shift
 Physical interpretation
The Lamb shift results from the coupling of the atomic electron to
the vacuum electromagnetic field which was ignored in Dirac theory.
 Important meanings
The Lamb shift and its explanation marked the beginning of modern
quantum electromagnetic field theory.
In the words of Dirac (1984), “ No progress was made for 20 years.
Then a development came, initiated by Lamb’s discovery and
explanation of the Lamb shift, which fundamentally changed the
character of theoretical physics. It involved setting up rules for
discarding … infinities…”
Q: What happens when the vacuum fluctuations which result in the Lamb shift
are modified?
 What happens if vacuum fluctuations are modified?
If modes are modified, what would happen?
1. Casimir effect
2. Casimir-Polder force
How spacetime curvature affects the Lamb shift? Observable?
 How
 Bethe’s approach, Mass Renormalization (1947)
A neutral atom
fluctuating electromagnetic fields
 
HI  A P
Propose “renormalization” for the first time in history!
(non-relativistic approach)
 Relativistic Renormalization approach (1948)
The work is done by N. M. Kroll and W. E. Lamb;
Their result is in close agreement with the non-relativistic
calculation by Bethe.
 Welton’s interpretation (1948)
The electron is bounded by the Coulomb force and driven by the fluctuating
vacuum electromagnetic fields — a type of constrained Brownian motion.
 Feynman’s interpretation (1961)
It is the result of emission and re-absorption from the vacuum of virtual
photons.
 Interpret the Lamb shift as a Stark shift
A neutral atom
fluctuating electromagnetic fields
 
HI  d  E
 DDC formalism (1980s)
J. Dalibard
J. Dupont-Roc
C. Cohen-Tannoudji
1997 Nobel Prize Winner
a neutral atom
H I ( )
Reservoir of vacuum fluctuations

f
s
A(t )  A (t )  A (t )
Field’s
variable


N(t )  A(t )
Atomic
variable
Free field Source field


 A(t )  N(t )




  N(t )  A(t )  (1   ) A(t )  N(t )
0≤λ ≤ 1
Vacuum
fluctuations
Radiation
reaction
How to separate the contributions of vacuum fluctuations and
radiation reaction?
Model:
a two-level atom coupled with vacuum scalar field
fluctuations.
H A ( )  0 R3 ( )
H I ( )  R2 ( ) ( x( ))

dt
H F ( )   d kk ak ak
d
3
Atomic operator
Atom + field Hamiltonian
H system  H A  H F  H I
Heisenberg equations
for the field
Integration
E  E f  Es
Heisenberg equations
for the atom
The dynamical
equation of HA
E sf —— corresponding to the effect of vacuum fluctuations
E —— corresponding to the effect of radiation reaction
uncertain?
Symmetric operator ordering
For the contributions of vacuum fluctuations and radiation reaction
to the atomic level b ,
with
Application:
1. Explain the stability of the ground state of the atom;
2. Explain the phenomenon of spontaneous excitation;
3. Provide underlying mechanism for the Unruh effect;
4. Study the atomic Lamb shift in various backgrounds
…
 Waves outside a Massive body

ds 2  (1  2 M / r )dt 2  (1  2 M / r ) 1 dr 2  r 2 d 2  Sin 2d 2

A complete set of modes functions satisfying the Klein-Gordon equation:
outgoing
ingoing
Radial functions
 d2

2



V
(
r
)
 2
 Rl ( | r )  0,
dr
 

with the effective potential
 2 M   l (l  1) 2 M 
V ( r )  1 
  2  3 .
r
r 

 r
and the Regge-Wheeler Tortoise coordinate:
r*  r  2M ln( r / 2M 1),
Spherical
harmonics
transmission coefficient
reflection coefficient


Al ( )  Al ( )


2
2
2
1  Al ( )  1  Al ( )  B l ( )
The field operator is expanded in terms of these basic modes, then we can
define the vacuum state and calculate the statistical functions.
Boulware vacuum:
Positive frequency modes → the Schwarzschild time t.
D. G. Boulware, Phys. Rev. D 11, 1404 (1975)
It describes the state of a spherical massive body.
For the effective potential:
 2 M   l (l  1) 2 M 
V ( r )  1 
 2  3 
r  r
r 

dV ( r )
0
dr
r  3M
d 2V (r )
0
2
dr
r 3 M
V (r ) max
2

l  1 / 2

27 M 2
Is the atomic energy
mostly shifted near r=3M?
 Lamb shift induced by spacetime curvature
For a static two-level atom fixed in the exterior region of the spacetime with a
radial distance (Boulware vacuum),
     vf    rr
 B    
2

64 2
with

Analytical results
In the asymptotic regions:
P. Candelas, Phys. Rev. D 21, 2185 (1980).
M
The revision caused by
spacetime curvature.
The grey-body factor
M
—
The Lamb shift of a static one in Minkowski spacetime with no boundaries.
It is logarithmically divergent , but the divergence can be removed by exploiting
a relativistic treatment or introducing a cut-off factor.
Consider the geometrical approximation:
Vl(r)
r
2M
3M
 2  Vmax , Bl ~ 1;
 2  Vmax , Bl ~ 0.
The effect of backscattering of field modes off the curved geometry.
Discussion:
1. In the asymptotic regions, i.e., r  2M and r  , f(r)~0, the revision
is negligible!
2.
Near r~3M, f(r)~1/4, the revision is positive and is about 25%! It is
potentially observable.
The spacetime curvature amplifies the Lamb shift!
Problematic!
sum
position
¥
å(2l +1) R (w r)
¥
å(2l +1) R (w r)
2
l
l
l=0
l=0
r  2M
r 
4w 2
1- 2M / r
1 ¥
å(2l +1) Bl (w )
4M 2 l=0
?
1 ¥
(2l +1) Bl (w )
2 å
r l=0
2
2
4w 2
1- 2M / r
2
?
1. Candelas’s result keeps only the leading order for both the outgoing and
ingoing modes in the asymptotic regions.
2. The summations of the outgoing and ingoing modes are not of the same
order in the asymptotic regions. So, problem arises when we add the
two. We need approximations which are of the same order!
3. Numerical computation reveals that near the horizon, the revisions are
negative with their absolute values larger than
.
 Numerical computation
Target:
Key problem:
How to solve the differential equation of the radial function?
In the asymptotic regions, the analytical formalism of the radial functions:
rs  2M
Set:
with
The recursion relation of ak(l,ω) is determined by the differential of
the radial functions and a0(l,ω)=1, ak(l, ω)=0 for k<0,
For the outgoing modes, r  
with
Similarly,
They are evaluated
at large r!

2
The dashed lines represents Al () and the solid represents Bl ( ) .
2
For the summation of the outgoing and ingoing modes:
4M2gs(ω|r) as function of ω and r.
For the relative Lamb shift of a static atom at position r,
The relative Lamb shift F(r) for the static atom at different position.
Conclusion:
1. The relative Lamb shift decreases from near the horizon until the
position r~4M where the correction is about 25%, then it grows
very fast but flattens up at about 40M where the correction is still
about 4.8%.
2. F(r) is usually smaller than 1, i.e., the Lamb shift of the atom at an
arbitrary r is usually smaller than that in a flat spacetime. The
spacetime curvature weakens the atomic Lamb shift as opposed to
that in Minkowski spacetime!
 What about the relationship between the signal emitted from the
static atom and that observed by a remote observer?
It is red-shifted by gravity.
F(r): observed by a static observer at the position of the atom
F′(r): observed by a distant observer at the spatial infinity
 Who is holding the atom at a fixed radial distance?
circular geodesic motion
bound circular orbits for massive particles
stable orbits
 How does the circular Unruh effect contributes to the Lamb shift?
 Numerical estimation
Summary
 Spacetime curvature affects the atomic Lamb shift.
It weakens the Lamb shift!
 The curvature induced Lamb shift can be remarkably significant
outside a compact massive astrophysical body, e.g., the
correction is ~25% at r~4M, ~16% at r~10M, ~1.6% at r~100M.
 The results suggest a possible way of detecting fundamental
quantum effects in astronomical observations.