Quantum effects in curved spacetime

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. 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. 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…” • 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 Q: What happens when the vacuum fluctuations which result in the Lamb shift are modified?

6. 2. Casimir-Polder force 1. Casimir effect Lamb shift induced by spacetime curvature • Our interest If modes are modified, what would happen? How spacetime curvatureaffects the Lamb shift? Observable?

7. A neutral atom fluctuating electromagnetic fields 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. • How • Bethe’s approach, Mass Renormalization (1947) Propose “renormalization” for the first time in history! (non-relativistic approach) • Relativistic Renormalization approach (1948)

8. A neutral atom 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

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

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

11. Vacumm fluctuations Radiation reaction

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

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

14. uncertain? Symmetric operator ordering

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

16. 4. Study the atomic Lamb shift in various backgrounds … 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;

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

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

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

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

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

22. The revision caused by spacetime curvature. The Lamb shift of a static one in Minkowski spacetime with no boundaries. — The grey-body factor It is logarithmically divergent , but the divergence can be removed by exploiting a relativistic treatment or introducing a cut-off factor.

23. Vl(r) r 2M 3M The effect of backscattering of field modes off the curved geometry. Consider the geometrical approximation:

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

25. sum position 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 .

26. Target: In the asymptotic regions, the analytical formalism of the radial functions: • Numerical computation Key problem: How to solve the differential equation of the radial function?

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

29. The dashed lines represents and the solid represents .

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

31. 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,

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. Environment (Bath) System How – theory of open quantum systems The von Neumann equation (interaction picture) The interaction Hamiltonian The evolution of the reduced system The Lamb shift Hamiltonian The dissipator

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

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

43. 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 87.012101 (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.