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.