to Lamb shift be observable?

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16 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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



Q
uantum effects unique to curved space




Hawking radiation



Gibbons
-
Hawking effect



Why



Unruh effect

Challenge: Experimental test.

Q: How about curvature induced corrections to
those already existing in flat spacetimes?



Particle creation by GR field



What is Lamb shift?



Theoretical result:




Experimental discovery:


In 1947, Lamb and Rutherford show that the level 2s
1/2

lies about
1000MHz, or 0.030cm
-
1

above the level 2p
1/2.

Then a more accurate
value 1058MHz.


The Dirac theory in Quantum Mechanics shows: the states, 2s1/2
and 2p1/2 of hydrogen atom are degenerate.

The Lamb shift



Important meanings




Physical interpretation



The Lamb shift results from the coupling of the atomic electron to
the vacuum electromagnetic field which was ignored in Dirac 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




The Lamb shift and its explanation marked the beginning of modern
quantum electromagnetic field theory.

Q: What happens when the vacuum fluctuations which result in the Lamb shift


are modified?



What happens if vacuum fluctuations are modified?


How spacetime curvature

affects the Lamb shift? Observable?

If modes are modified, what would happen?

2. Casimir
-
Polder force

1.
Casimir effect



How



Bethe’s approach, Mass Renormalization (1947)

A neutral atom

fluctuating electromagnetic fields

P
A
H
I






Relativistic Renormalization approach (1948)

Propose “renormalization” for the first time in history!
(non
-
relativistic approach)

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.



Interpret the Lamb shift as a Stark shift

A neutral atom

fluctuating electromagnetic fields

E
d
H
I






Feynman’s interpretation (1961)

It

is

the

result

of

emission

and

re
-
absorption

from

the

vacuum

of

virtual

photons
.



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
.

J. Dalibard

J. Dupont
-
Roc

C. Cohen
-
Tannoudji
1997 Nobel Prize Winner



DDC formalism (1980s)

a neutral atom

Reservoir of vacuum fluctuations

)
(

I
H
)
(
N
)
(
)
1
(
)
(
)
(
N

t
t
A
t
A
t











)
(
)
(
N
t
A
t



Atomic
variable

Field’s
variable

)
(
N
)
(
t
t
A




0
≤λ ≤

1

)
(
)
(
)
(
t
A
t
A
t
A
s
f





Free field

Source field

Vacuum
fluctuations

Radiation
reaction

Vacuum
fluctuations

Radiation
reaction

Model: a two
-
level atom coupled with vacuum scalar

field



fluctuations.

Atomic operator

)
(
)
(
3
0



R
H
A

))
(
(
)
(
)
(
2





x
R
H
I




d
dt
a
a
k
d
H
k
k
k
F







3
)
(
How to separate the contributions of vacuum fluctuations and
radiation reaction?

Heisenberg equations
for the field

Heisenberg equations
for the atom

The dynamical
equation of
H
A

Integration

s
f
E
E
E


Atom + field Hamiltonian

I
F
A
system
H
H
H
H



——

corresponding to the effect of vacuum fluctuations

f
E
——

corresponding to the effect of radiation reaction

s
E
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



2
2
2
2
2
1
2
2
)
/
2
1
(
)
/
2
1
(



d
Sin
d
r
dr
r
M
dt
r
M
ds







A complete set of modes functions satisfying the Klein
-
Gordon equation:

outgoing


ingoing

Spherical
harmonics


Radial

functions

,
0
)
|
(
)
(
2
2
2










r
R
r
V
dr
d
l


),
1
2
/
ln(
2
*



M
r
M
r
r
and the Regge
-
Wheeler Tortoise coordinate:

with the effective potential

.
2
)
1
(
2
1
)
(
3
2
















r
M
r
l
l
r
M
r
V
)
(
)
(


l
l
A
A



2
2
2
)
(
)
(
1
)
(
1



l
l
l
B
A
A






The field operator is expanded in terms of these basic modes, then we can
define the vacuum state and calculate the statistical functions.

It describes the state of a spherical massive body.


Positive frequency modes → the Schwarzschild time
t.

Boulware vacuum:

D. G. Boulware, Phys. Rev. D
11
, 1404 (1975)

reflection coefficient

transmission coefficient

0
)
(

dr
r
dV
M
r
3

0
)
(
3
2
2


M
r
dr
r
V
d


2
2
max
27
2
/
1
)
(
M
l
r
V


Is the atomic energy
mostly shifted near r=3M
?

For the effective potential:

















3
2
2
)
1
(
2
1
)
(
r
M
r
l
l
r
M
r
V
For a static two
-
level atom fixed in the exterior region of the spacetime with a
radial distance (Boulware vacuum),








B
2
2
64




with



Lamb shift induced by spacetime curvature





rr
vf








In the asymptotic regions:

P. Candelas, Phys. Rev. D 21, 2185 (1980).



Analytical results

The Lamb shift of a static one in Minkowski spacetime with no boundaries.

M



It is logarithmically divergent , but the divergence can be removed by exploiting
a relativistic treatment or introducing a cut
-
off factor.

M

The revision caused by
spacetime curvature.

The grey
-
body factor

Consider the geometrical approximation:


3M

r

2M

V
l
(r)

,
max
2
V


;
1
~
l
B
,
max
2
V


.
0
~
l
B
The effect of backscattering of field modes off the curved geometry.

2.
Near r~3M, f(r)~1/4, the revision is positive and is about 25%
!
It is
potentially observable.


1. In the asymptotic regions, i.e., and , f(r)~0, the revision
is negligible!

M
r
2



r
Discussion:

The spacetime curvature
amplifies

the Lamb shift!

Problematic!

M
r
2



r
position

sum

Candelas’s result keeps only the leading order for both the outgoing and
ingoing modes in the asymptotic regions.

1.

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!

2.

?

?

Numerical computation reveals that near the horizon, the revisions are
negative

with their absolute values larger than .

3.

(
2
l

1
)
R
l
(

r
)
l

0


2
4

2
1

2
M
/
r
(
2
l

1
)
R
l
(

r
)
l

0


2
1
4
M
2
(
2
l

1
)
B
l
(

)
l

0


2
1
r
2
(
2
l

1
)
B
l
(

)
l

0


2
4

2
1

2
M
/
r


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:

M
r
s
2

Set:

with

The recursion relation of a
k
(l,
ω
)

is determined by the differential of

the radial functions and a
0
(l,
ω
)=1, a
k
(l,
ω
)=0 for k<0
,

with

Similarly,

They are evaluated
at large r
!

For the outgoing modes,



r
The dashed lines represents and the solid represents .

2
)
(

l
A

2
)
(

l
B
4M
2
g
s
(
ω
|r
) as function of
ω

and r.

For the summation of the outgoing and ingoing modes:

The relative Lamb shift F(r) for the static atom at different position.

For the relative Lamb shift of a static atom at position r,

Conclusion:

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!

2.

The

relative

Lamb

shift

decreases

from

near

the

horizon

until

the


position

r~
4
M

where

the

correction

is

about

25
%
,

then

it

grows


very

fast

but

flattens

up

at

about

40
M

where

the

correction

is

still


about 4.8%.


1.



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.