quantum effects in curved spacetime hongwei yu. outline motivation lamb shift induced by spacetime...
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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
Quantum effects unique to curved spacetime
• Hawking radiation
• Gibbons-Hawking effect
• Unruh effect
Challenge: Experimental test.
• Particle creation by GR field
Q: How about curvature induced corrections to
those already existing in flat spacetimes?
Motivation
What is Lamb shift?
• Theoretical result:
• 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 Dirac theory in Quantum Mechanics shows: the states, 2s1/2
and 2p1/2 of hydrogen atom are degenerate.
The Lamb shift
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?
Our interest
How spacetime curvature affects the Lamb shift? Observable?
If modes are modified, what would happen?
2. Casimir-Polder force1. Casimir effect
Lamb shift induced by spacetime curvature
How
• Bethe’s approach, Mass Renormalization (1947)
A neutral atom
fluctuating electromagnetic fieldsPAH 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 fieldsEdH 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
)(IH
)(N)()1()()(N ttAtAt
)()(N tAt
Atomic variable
Field’s variable
)(N)( ttA
0≤λ ≤ 1
)()()( tAtAtA sf
Free field Source field
Vacumm fluctuations
Radiation reaction
Model:
a two-level atom coupled with vacuum scalar field fluctuations.
Atomic operator)()( 30 RH A
))(()()( 2 xRH I
d
dtaakdHkkkF
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 HA
Integrationsf EEE
Atom + field Hamiltonian
IFAsystem HHHH
—— corresponding to the effect of vacuum fluctuationsfE—— corresponding to the effect of radiation reaction
sE
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
22222122 )/21()/21( dSindrdrrMdtrMds
A complete set of modes functions satisfying the Klein-Gordon equation:
outgoing
ingoing
Spherical harmonics Radial functions
,0)|()(22
2
rRrVdr
dl
),12/ln(2* MrMrr
and the Regge-Wheeler Tortoise coordinate:
with the effective potential
.2)1(2
1)(32
r
M
r
ll
r
MrV
)()( ll AA�
222)()(1)(1 lll BAA
�
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 coefficienttransmission coefficient
0)(
dr
rdV Mr 3
0)(
3
2
2
Mr
dr
rVd
2
2
max 27
2/1)(
M
lrV
Is the atomic energy mostly shifted near r=3M?
For the effective potential:
32
2)1(21)(
r
M
r
ll
r
MrV
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
rrvf
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:
3Mr
2M
Vl(r)
,max2 V ;1~lB
,max2 V .0~lB
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!
Mr 2 r
Discussion:
The spacetime curvature amplifies the Lamb shift!
Problematic!
Mr 2
r
2
0
)()12(
l
l rRl 2
0
)()12(
l
l rRl �
rM /21
4 2
2
02
)()12(4
1
l
lBlM
2
02
)()12(1
l
lBlr
rM /21
4 2
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
02
)()12(1
l
lBlr
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:
Mrs 2
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,
with
Similarly,
They are evaluated at large r!
For the outgoing modes, r
The dashed lines represents and the solid represents .2
)(lA 2
)(lB
4M2gs(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,
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
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 a static two-level atom evolve outside a Schwarzschild black hole?
How – theory of open quantum systems
The von Neumann equation (interaction picture)
The interaction Hamiltonian
Environment (Bath)
System
The evolution of the reduced system
The Lamb shift Hamiltonian
The dissipator
For a two-level atom
The master equation (Schrödinger picture)
The spontaneous excitation rate
The spontaneous emission rate
The time-dependent reduced density matrix
The coefficients
The line element of a Schwarzschild black hole
The Wightman function
The Fourier transform
The trajectory of the atom
The summation concerning the radial functions in asymptotic regions
The spontaneous excitation rate of the detector
The proper acceleration
The effective temperature
The grey-body factor
The equilibrium state
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
B. Bellomo et al, PRA 87.012101 (2013).
The effective temperature
Analogue spacetime?
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.