1. Quantum effects in curved spacetime
Hongwei Yu

2. Outline
Motivation
Lamb shift induced by spacetime curvature
Thermalization phenomena of an atom outside
a Schwarzschild black hole
Conclusion

3. Quantum effects unique to curved spacetime
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?

4. 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

5. 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
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?

6. 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?

7. fluctuating electromagnetic fields
How
Bethe’s approach, Mass Renormalization (1947)
A neutral atom
fluctuating electromagnetic fields
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.

8. fluctuating electromagnetic fields
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

9. C. Cohen-Tannoudji 1997 Nobel Prize Winner
DDC formalism (1980s)
J. Dalibard
J. Dupont-Roc
C. Cohen-Tannoudji Nobel Prize Winner

10. Reservoir of vacuum fluctuations
a neutral atom
Reservoir of vacuum fluctuations
Field’s variable
Free field
Source field
Atomic variable
0≤λ ≤ 1

11. Vacumm fluctuations
Radiation reaction

12. How to separate the contributions of vacuum fluctuations
and radiation reaction?
Model:
a two-level atom coupled with vacuum scalar field fluctuations.
Atomic operator

13. —— corresponding to the effect of vacuum fluctuations
Atom + field Hamiltonian
Heisenberg equations for the field
Heisenberg equations for the atom
Integration
The dynamical equation of HA
—— corresponding to the effect of vacuum fluctuations
—— corresponding to the effect of radiation reaction

14. Symmetric operator ordering
uncertain?
Symmetric operator ordering

15. For the contributions of vacuum fluctuations and radiation reaction to the atomic level ,
with

16. 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

17. Waves outside a Massive body
A complete set of modes functions satisfying the Klein-Gordon equation:
Radial functions
Spherical harmonics
outgoing
ingoing
with the effective potential
and the Regge-Wheeler Tortoise coordinate:

18. Positive frequency modes → the Schwarzschild time t.
reflection coefficient
transmission coefficient
The field operator is expanded in terms of these basic modes, then we can define the vacuum state and calculate the statistical functions.
Positive frequency modes → the Schwarzschild time t.
Boulware vacuum:
D. G. Boulware, Phys. Rev. D 11, 1404 (1975)
It describes the state of a spherical massive body.

19. Is the atomic energy mostly shifted near r=3M?
For the effective potential:
Is the atomic energy mostly shifted near r=3M?

20. For a static two-level atom fixed in the exterior region of the spacetime with a radial distance (Boulware vacuum),
with

21. Analytical results In the asymptotic regions:
P. Candelas, Phys. Rev. D 21, 2185 (1980).

22. The revision caused by spacetime curvature.
The grey-body factor
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.

23. Consider the geometrical approximation:
Vl(r)
The effect of backscattering of field modes off the curved geometry.

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

25. 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.

26. 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:

27. 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,

28. They are evaluated at large r!
For the outgoing modes,
with
They are evaluated at large r!
Similarly,

29. The dashed lines represents and the solid represents .

30. For the summation of the outgoing and ingoing modes:
4M2gs(ω|r) as function of ω and r.

31. 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.

32. 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!

33. 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.

34. 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

35. 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.

36. 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

37. 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

38. The master equation (Schrödinger picture)
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

39. The line element of a Schwarzschild black hole
The trajectory of the atom
The Wightman function
The Fourier transform

40. The summation concerning the radial functions in asymptotic regions
The spontaneous excitation rate of the detector
The proper acceleration

41. The effective temperature
The equilibrium state
The effective temperature
The grey-body factor

42. The geometrical optics approximation
Low frequency limit
High frequency limit
The geometrical optics approximation
The grey-body factor tends to zero in both the two asymptotic regions.

43. For an arbitrary position
Near the horizon
Spatial infinity
For an arbitrary position

44. A stationary environment out of thermal equilibrium
The effective temperature
Analogue spacetime?
B. Bellomo et al, PRA (2013).

45. 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.

46. 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.

48. I'd put my money on some other correction
Speculation
“The size of the Proton”, Nature, 466, 213(2010)
= (67) fm, which differs by 5.0 standard deviations from the CODATA value of (69) fm. It implies that either the Rydberg constant has to be shifted by -110 kHz/c (4.9 standard deviations), or the calculations of the QED effects in atomic hydrogen atoms are insufficient.
The first thing is to go through the existing calculations carefully ". It could be that an error was made, or that approximations made in existing quantum calculation simply aren't good enough.
I'd put my money on some other correction