the casimir-polder force: a manifestation of the qed vacuum · • modiﬁcation of atomic level...

Quantum Optics II 2004 Cozumel FSU Jena

The Casimir-Polder force:

A manifestation of the QED

vacuum

Stefan Yoshi Buhmann and Dirk-Gunnar Welsch

Friedrich-Schiller-Universitat Jena, Germany


Puzzle

What is greater than God,

more evil than the devil?

The poor have it,

the happy need it,

and if you eat it, you will die.


Puzzle

What is greater than God,

more evil than the devil?

The poor have it,

the happy need it,

and if you eat it, you will die.

Answer: Nothing


Content

• Introduction: Body-assisted QED vacuum

– Quantization scheme

– vacuum QED effects


Content




• Casimir-Polder force: Perturbative approach

– Ground-state atom near magnetodielectric half space


Content






• Casimir-Polder force: Beyond perturbation theory

– General theory

– Dynamics in weak-coupling limit

– Excited-state atom


Content






• Casimir-Polder force: Beyond perturbation theory

– General theory

– Dynamics in weak-coupling limit

– Excited-state atom

• Summary and outlook


The QED vacuum

vacuus [lat.]: empty

Classical electrodynamics:

E(r) = 0, B(r) = 0 (no electromagnetic field)


The QED vacuum




QED: [ε0Ei(r), Bj(r′)] = −i~εijk∂kδ(r− r′)

∆A∆B ≥ 12|〈[A, B]〉| (Heisenberg uncertainty relation)

⇓〈E(r)〉 = 0, 〈B(r)〉 = 0 (no e.m. field on average)

but: ∆E(r) 6= 0, ∆B(r) 6= 0 (field fluctuations)


The QED vacuum




QED: [ε0Ei(r), Bj(r′)] = −i~εijk∂kδ(r− r′)

∆A∆B ≥ 12|〈[A, B]〉| (Heisenberg uncertainty relation)

⇓〈E(r)〉 = 0, 〈B(r)〉 = 0 (no e.m. field on average)

but: ∆E(r) 6= 0, ∆B(r) 6= 0 (field fluctuations)

QED vacuum = vanishing of average electromagnetic

fields


Normal-mode quantization

Quantized electric field:

E(r) =∑k

gk(r)ak + H.c.

gk(r): normal modes

a†k, ak: creation, annihilation operators




E(r) =∑k

gk(r)ak + H.c.

gk(r): normal modes


QED vacuum: ak|0〉 = 0

⇒ 〈E(r)〉 = 0, ∆E(r) 6= 0




E(r) =∑k

gk(r)ak + H.c.

gk(r): normal modes



⇒ 〈E(r)〉 = 0, ∆E(r) 6= 0

Applicability:

• free space




E(r) =∑k

gk(r)ak + H.c.

gk(r): normal modes



⇒ 〈E(r)〉 = 0, ∆E(r) 6= 0

Applicability:

• free space• arbitrary arrangement of

– perfectly reflecting bodies




E(r) =∑k

gk(r)ak + H.c.

gk(r): normal modes



⇒ 〈E(r)〉 = 0, ∆E(r) 6= 0

Applicability:

• free space• arbitrary arrangement of

– perfectly reflecting bodies

– nondispersive, nonabsorbing bodies

Not applicable to dispersing and absorbing bodies!


Relevance of dispersionand absorption

1. World is not perfect:

absorption always present and relevant in experiments





2. Fluctuations associated with absorption:

additional fluctuations which contribute to the net fluctuations





2. Fluctuations associated with absorption:

additional fluctuations which contribute to the net fluctuations

3. New materials:

artificial metamaterials with left-handed properties1

→ strongly dispersing and absorbing

1D. R. Smith et. al., PRL 84, 18, 4184 (2000)


Generalized quantization scheme

Normal-mode quantization of electric field:

E(r) =∑k

gk(r)ak + H.c.




E(r) =∑k

gk(r)ak + H.c.

⇑c-number functions

⇓

⇑creation/annihilation

operators

⇓




E(r) =∑k

gk(r)ak + H.c.

⇑c-number functions

⇓

⇑creation/annihilation

operators

⇓

Quantized electric field in linear, causal media:

E(r) =∫ ∞

0dω

∫d3r′ i

√~πε0

ω2

c2

√Im ε(r′, ω)G(r, r′, ω)fe(r

′, ω)

+ω

c

√−Imµ−1(r′, ω)

[∇′ ×G(r, r′, ω)

]fm(r′, ω)

+H.c.


Classical Green tensor

[∇× µ−1(r, ω)∇× −

ω2

c2ε(r, ω)

]G(r, r′, ω) = δ(r− r′)



[∇× µ−1(r, ω)∇× −

ω2

c2ε(r, ω)

]G(r, r′, ω) = δ(r− r′)

Physical interpretation:

Source at r



[∇× µ−1(r, ω)∇× −

ω2

c2ε(r, ω)

]G(r, r′, ω) = δ(r− r′)


Source at r G(r,r′,ω)−→−→



[∇× µ−1(r, ω)∇× −

ω2

c2ε(r, ω)

]G(r, r′, ω) = δ(r− r′)


Source at r G(r,r′,ω)−→−→ Electric field at r′


Creation/annihilation operators

Bosonic commutation relations:[fλi(r, ω), f†

λ′j(r′, ω′)

]= δλλ′δijδ(r− r′)δ(ω − ω′) (λ = e,m)




λ′j(r′, ω′)



Noise polarization:

PN(r, ω) = i

√~ε0π

×√

Imε(r, ω) fe(r, ω)

+ +

++

++

+

+

+−−

−−

−−

−−




λ′j(r′, ω′)



Noise polarization:

PN(r, ω) = i

√~ε0π

×√

Imε(r, ω) fe(r, ω)

+ +

++

++

+

+

+−−

−−

−−

−−

Noise magnetization:

MN(r, ω) =

√−

~κ0

π

×√

Imµ−1(r, ω) fm(r, ω)

NN

N

N

NN

N

NN

SS

S

S

S

SS

S

S


Atom-field dynamics

H = HMF + HA + HAMF


Atom-field dynamics

H = HMF + HA + HAMF

Medium-assisted field Hamiltonian:

HMF =∑

λ=e,m

∫d3r

∫ ∞

0dω ~ω f†λ(r, ω)fλ(r, ω)


Atom-field dynamics

H = HMF + HA + HAMF


HMF =∑

λ=e,m

∫d3r

∫ ∞


Atomic Hamiltonian:

HA =∑α

pα2

2mα+

1

2ε0

∫d3rPA

2(r)


Atom-field dynamics

H = HMF + HA + HAMF


HMF =∑

λ=e,m

∫d3r

∫ ∞


Atomic Hamiltonian:

HA =∑α

pα2

2mα+

1

2ε0

∫d3rPA

2(r)

Electric dipole interaction:

HAMF = −dE(rA) +1

2mA

[pA, d× B(rA)

]+


QED vacuum in presence oflinear, causal media

QED vacuum: fλ(r, ω)|0〉 = 0 (λ = e,m)

⇒ 〈E(r)〉 = 0, [∆E(r)]2 =~πε0

∫ ∞

0dω

ω2

c2Im[TrG(r, r, ω)]


QED vacuum in presence oflinear, causal media

QED vacuum: fλ(r, ω)|0〉 = 0 (λ = e,m)

⇒ 〈E(r)〉 = 0, [∆E(r)]2 =~πε0

∫ ∞

0dω

ω2

c2Im[TrG(r, r, ω)]

Highly structured fluctuations of the electromagnetic field!


Manifestations of the QED vacuum

• Casimir force [C. Raabe et. al., PRA 68, 033810 (2003)]




• Casimir-Polder force [S. Y. Buhmann et. al., PRA 70, 052117

(2004)]





(2004)]

• Modification of atomic level structure [as above]





(2004)]

• Modification of atomic level structure [as above]

• Spontaneous decay [Ho et. al., PRA 68, 043816 (2003)]


Relevance of Casimir-Polder forces

• Adsorption of atoms/molecules

to surfaces1

1M. A. Chesters et. al., Surf. Sci. 35, 161 (1973);J. Darville, in Vibrations at Surfaces (Plenum, New York, 1982)




to surfaces1

• Atom optics2


2F. Shimizu and J. Fujita, PRL 88, 123201 (2002)




to surfaces1

• Atom optics2

• Atomic-force microscopes3


2F. Shimizu and J. Fujita, PRL 88, 123201 (2002)3G. Binnig et. al., PRL 56, 930 (1986)


Casimir-Polder force:

Perturbative approach (ground-state atoms)

Open issues

• Role of material absorption

• Influence of magnetic properties


Van der Waals potential

Idea: system in state |0〉 ⊗ |0〉

interaction HAMF ⇒ energy shift ∆E0

⇒ van-der-Waals potential ∆E0 = ∆E(0)0 + U0(rA)

⇒ van-der-Waals force F0(rA) = −∇AU0(rA)







2nd-order perturbation theory: ∆2E0 =∑ψ

|〈0|〈0|HAMF|ψ〉|2

E0 − Eψ







2nd-order perturbation theory: ∆2E0 =∑ψ

|〈0|〈0|HAMF|ψ〉|2

E0 − Eψ

Result:

U0(rA) =~µ0

2π

∫ ∞

0duu2α

(0)0 (iu)TrG(1)(rA, rA, iu)

atomic polarizability:

α(0)0 (ω) = lim

ε→0

2

3~∑k

ωk0|d0k|2

ω2k0 − ω2 − iωε

Scattering Green

tensor:

G(1)(rA, rA, iu)


Atom near half space1

Azε (ω) µ ω( )



Azε (ω) µ ω( )

Nonretarded limit: zA c/ωt



Azε (ω)


Purely dielectric half space: U0(zA) = −C3

z3A

C3 =~

16π2ε0

∫ ∞

0duα(0)

0 (iu)ε(iu)− 1

ε(iu) + 1≥ 0



Azµ ω( )


Purely dielectric half space: U0(zA) = −C3

z3A

C3 =~

16π2ε0

∫ ∞

0duα(0)

0 (iu)ε(iu)− 1

ε(iu) + 1≥ 0

Purely magnetic half space: U0(zA) = +C1

z3A

C1 =~

16π2ε0

∫ ∞

0du

(u

c

)2α(0)0 (iu)

µ(iu)− 1

µ(iu) + 1+

[µ(iu)− 1]

2

≥ 0

1S. Y. Buhmann, T. Kampf, and D.-G. Welsch, in preparation


Atom near half spaceRetarded limit: zA c/ωr, c/ωk0

U0(zA) =C4

z4AC4 = C4[α(0), ε(0), µ(0)] T 0


Atom near half spaceRetarded limit: zA c/ωr, c/ωk0

U0(zA) =C4

z4AC4 = C4[α(0), ε(0), µ(0)] T 0

0

2

4

6

8

10

12

1 1.5 2 2.5 3

µ(0)

(0)εattractive

repulsive


Atom near half space

Numerical results: two-level atom, Drude-Lorentz model

-0.0004

0

0.0004

0.0008

0 2 4 6 8 10

’bar200.txt’ u 3:8’bar180.txt’ u 3:8’bar160.txt’ u 3:8’bar120.txt’ u 3:8’bar043.txt’ u 3:8

PSfrag replacements

zAωT,m/c

U0(z

A)1

2π

2c/

(µ0ω

3 T,m

d2 10) ωP,m = 2.00

ωP,m = 1.80ωP,m = 1.60ωP,m = 1.20ωP,m = 0.43

C1

C3

C4


Comparison of different forces

distance nonretarded retardedobjects e↔ e e↔ m e↔ e e↔ m

U ∝ +1

z3U ∝ −

1

zU ∝ +

1

z4U ∝ −

1

z4




U ∝ +1

z3U ∝ −

1

zU ∝ +

1

z4U ∝ −

1

z4

U ∝ +1

z6U ∝ −

1

z4U ∝ +

1

z7U ∝ −

1

z7




U ∝ +1

z3U ∝ −

1

zU ∝ +

1

z4U ∝ −

1

z4

U ∝ +1

z6U ∝ −

1

z4U ∝ +

1

z7U ∝ −

1

z7

F ∝ +1

z3F ∝ −

1

zF ∝ +

1

z4F ∝ −

1

z4



Beyond perturbation theory

Open issues

• Temporal evolution of the force

• Influence of body-induced shifting and broadening of atomic

transition lines

• Force for arbitrary atomic states

• Force in case of strong atom-field coupling

• Force for arbitrary field states


General theory

Lorentz force on charged particles (electric dipole app.):

fα = qαE(rA)−∇ϕA(rA)

+12

[˙rα × B(rA)− B(rA)× ˙rα

]


General theory

⇒



+12


]Lorentz force on an atom:⟨

F⟩AMF

=∇

⟨dE(r)

⟩AMF

+d

dt

⟨d× B(r)

⟩AMF

r=rA


General theory

⇒



+12



F⟩AMF

=∇

⟨dE(r)

⟩AMF

+d

dt

⟨d× B(r)

⟩AMF

r=rA

Applicability:

• Field state: arbitrary

• Atomic state: arbitrary

• Coupling: strong/weak


General theory

⇒



+12



F⟩AMF

=∇

⟨dE(r)

⟩AMF

+d

dt

⟨d× B(r)

⟩AMF

r=rA

Applicability:

• Field state: arbitrary

• Atomic state: arbitrary

• Coupling: strong/weak

In the following:

→ vacuum: ρMF = |0〉〈0|

→ weak coupling


CP force: Weak-coupling limitRemaining task: Solving the dynamics (Markov approximation)

E(r) = E(r, t) =?, d = d(t) =?



E(r) = E(r, t) =?, d = d(t) =?


F(rA, t) =∑m,n

σnm(t)Fmn(rA)



E(r) = E(r, t) =?, d = d(t) =?


F(rA, t) =∑m,n

σnm(t)Fmn(rA)

⇑ ⇑

Atomic density

matrix elements:

σnm(t)

Associated force

components:

Fmn(rA)



E(r) = E(r, t) =?, d = d(t) =?


F(rA, t) =∑m,n

σnm(t)Fmn(rA)

⇑ ⇑

Atomic density

matrix elements:

σnm(t)

Associated force

components:

Fmn(rA)

⇑ ⇑

Body-induced change of atomic level structure:

ωnm → ωnm(rA), Γn(rA)


Change of atomic level structure

Shift of atomic transition frequencies:

ωnm(rA) = ωnm + δωn(rA)− δωm(rA)

δωn(rA) =∑k

µ0

π~P

∫ ∞

0dω ω2dnkImG(1)(rA, rA, ω)dkn

ωnk − ω


Change of atomic level structure

Shift of atomic transition frequencies:

ωnm(rA) = ωnm + δωn(rA)− δωm(rA)

δωn(rA) =∑k

µ0

π~P

∫ ∞

0dω ω2dnkImG(1)(rA, rA, ω)dkn

ωnk − ω

Example: Two-level atom near dielectric half space

(single-resonance medium, nonretarded limit)

-0.002

0

0.002

1.12 1.13 1.14

δω/ ωT

/ ωTω10-0.002

0

0.002

1.12 1.13 1.14

zA = 4.5 nm

zA = 5.4 nm


Decay-induced broadening of atomic levels:

Γn(rA) =∑k

Θ[ωnk(rA)]Γnk(rA)

=2µ0

~Θ[ωnk(rA)]ω2

nk(rA)dnkImG [rA, rA, ωnk(rA)]dkn


Decay-induced broadening of atomic levels:

Γn(rA) =∑k

Θ[ωnk(rA)]Γnk(rA)

=2µ0

~Θ[ωnk(rA)]ω2

nk(rA)dnkImG [rA, rA, ωnk(rA)]dkn



0

0.004

0.008

1.12 1.13 1.14

T/ ωω10

Γ/ ωT

0

0.004

0.008

1.12 1.13 1.14

zA = 4.5 nm

zA = 5.4 nm


Excited-(eigen)state force

Dominant contribution to the force:

Frnn(rA)

=µ0

2

∑k

Θ[ωnk(rA)]Ω2nk(rA)

∇ ⊗ dnkG

(1)[r, r,Ωnk(rA)]dknr=rA

+H.c.

Ωnk(rA) = ωnk(rA) + i[Γn(rA) + Γk(rA)]/2

→ Influenced by level shifting and broadening!



Dominant contribution to the force:

Frnn(rA)

=µ0

2

∑k

Θ[ωnk(rA)]Ω2nk(rA)

∇ ⊗ dnkG

(1)[r, r,Ωnk(rA)]dknr=rA

+H.c.

Ωnk(rA) = ωnk(rA) + i[Γn(rA) + Γk(rA)]/2

→ Influenced by level shifting and broadening!

Example: Two-level atom near dielectric half space(single-resonance medium, nonretarded limit)

F11(zA) = −3(d2x + d2y + 2d2z)

32πε0z4A

|ε[Ω10(zA)]|2 − 1

|ε[Ω10(zA)] + 1|2

ε[Ω10(zA)] = 1 +ω2P

ω2T − ω2

10(zA)− i[Γ(zA) + γ]ω10(zA)

→ Γ(zA) + γ plays the role of total absorption parameter!





-3

0

3

1.12 1.13 1.14

(F11r )

z

10/ ωTω

x10 9/(3C / λT4 )

z =4.5 nmA

ω10

|1>

|0>





-3

0

3

1.12 1.13 1.14

(F11r )

z

10/ ωTω

x10 9/(3C / λT4 )

z =4.5 nmA

-3

0

3

1.12 1.13 1.14

ω10

|1>

|0>

|1>

ω10

|0>





-3

0

3

1.12 1.13 1.14

(F11r )

z

10/ ωTω

x10 9/(3C / λT4 )

z =4.5 nmA

-3

0

3

1.12 1.13 1.14

-3

0

3

1.12 1.13 1.14

ω10

|1>

|0>

|1>

|0>

ω10

|1>

ω10

|0>





-3

0

3

1.12 1.13 1.14

(F11r )

z

10/ ωTω

x10 9/(3C / λT4 )

z =4.5 nmA

-3

0

3

1.12 1.13 1.14

-3

0

3

1.12 1.13 1.14

-3

0

3

1.12 1.13 1.14

ω10

|1>

|0>

|1>

|0>

ω10

|1>

ω10

|0>

|1>

ω10

|0>


Casimir-Polder force: Summary

• Absorption: included in Green tensor formalism

• Magnetic bodies → attractive vs repulsive van der Waals po-

tentials

• Spontaneous decay → temporal evolution of the force

• Shifting and broadening of atomic transition lines

→ noticeable influence in nonretarded limit


Casimir-Polder force: Summary

• Absorption: included in Green tensor formalism

• Magnetic bodies → attractive vs repulsive van der Waals po-

tentials

• Spontaneous decay → temporal evolution of the force

• Shifting and broadening of atomic transition lines

→ noticeable influence in nonretarded limit

Outlook

• Closer investigation of off-diagonal force components

• Force in case of strong atom-field coupling

• Force for arbitrary field states

the casimir-polder force: a manifestation of the qed vacuum · • modiﬁcation of atomic level...

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