Transcript Lamb shift

Quantum effects in curved spacetime
Hongwei Yu
Outline
 Motivation
 Lamb shift induced by spacetime curvature
 Thermalization phenomena of an atom outside
a Schwarzschild black hole
 Conclusion
Motivation

Quantum effects unique to curved spacetime
•
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?
Lamb shift
 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?
Lamb shift induced by spacetime curvature
 Our interest
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
Vacumm
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 a k a k
d
3
Atomic operator
Atom + field Hamiltonian
H system  H A  H F  H I
Heisenberg equations
for the field
Heisenberg equations
for the atom
Integration
The dynamical
equation of HA
E E E
f
s
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?
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
r  2M
r 
2

 (2l  1) Rl ( r )
2

 (2l  1) Rl ( r )


l 0
4 2
1  2M / r
1
r2
?

 (2l  1) B ( )
l 0
l
2
l 0


1
4M 2

4 2
1  2M / r
 (2l  1) B ( )
2
l
l 0
?
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
2
negative with their absolute values larger than 1  (2l  1) B ( ) .
r2

l 0
l
 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.
 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%.
 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.
 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.
Thermalization of an atom outside a Schwarzschild black hole
 How a static two-level atom evolve outside a Schwarzschild black hole?
 Model:
A radially polarized two-level atom coupled to a bath of fluctuating
quantized electromagnetic fields outside a Schwarzschild black hole
in the Unruh vacuum.
 The Hamiltonian
How – theory of open quantum systems
The von Neumann equation (interaction picture)
Environment (Bath)
System
The interaction Hamiltonian
The evolution of the reduced system
The Lamb shift Hamiltonian
The dissipator
For a two-level atom
The master equation (Schrödinger picture)
The spontaneous emission rate
The spontaneous excitation rate
The time-dependent reduced density matrix
The coefficients
The line element of a Schwarzschild black hole
The trajectory of the atom
The Wightman function
The Fourier transform
The summation concerning the radial functions in asymptotic regions
The spontaneous excitation rate of the detector
The proper acceleration
The equilibrium state
The effective temperature
The grey-body factor
Low frequency limit
High frequency limit
The geometrical optics approximation
The grey-body factor tends to zero in both the two asymptotic regions.
Near the horizon
Spatial infinity
For an arbitrary position
A stationary environment out of thermal equilibrium
The effective temperature
Analogue spacetime?
B. Bellomo et al, PRA 87.012101 (2013).
Summary
 In the Unruh vacuum, the spontaneous excitation rate of the
detector is nonzero, and the detector will be asymptotically
driven to a thermal state at an effective temperature,
regardless of its initial state.
 The dynamics of the atom in the Unruh vacuum is closely
related to that in an environment out of thermal equilibrium
in a flat spacetime.
Conclusion
 The spacetime curvature may cause corrections to quantum
effects already existing in flat spacetime, e.g., the Lamb shift.
 The Lamb shift is weakened by the spacetime curvature, and
the corrections may be found by looking at the spectra from a
distant astrophysical body.
 The close relationship between the dynamics of an atom in the
Unruh vacuum and that in an environment out of thermal
equilibrium in a flat spacetime may provides an analogue
system to study the Hawking radiation.