The Casimir-Polder force: A manifestation of the QED vacuum · • Modification of atomic level...
Transcript of 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
Quantum Optics II 2004 Cozumel FSU Jena
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.
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
Content
• Introduction: Body-assisted QED vacuum
– Quantization scheme
– vacuum QED effects
Quantum Optics II 2004 Cozumel FSU Jena
Content
• Introduction: Body-assisted QED vacuum
– Quantization scheme
– vacuum QED effects
• Casimir-Polder force: Perturbative approach
– Ground-state atom near magnetodielectric half space
Quantum Optics II 2004 Cozumel FSU Jena
Content
• Introduction: Body-assisted QED vacuum
– Quantization scheme
– vacuum QED effects
• Casimir-Polder force: Perturbative approach
– Ground-state atom near magnetodielectric half space
• Casimir-Polder force: Beyond perturbation theory
– General theory
– Dynamics in weak-coupling limit
– Excited-state atom
Quantum Optics II 2004 Cozumel FSU Jena
Content
• Introduction: Body-assisted QED vacuum
– Quantization scheme
– vacuum QED effects
• Casimir-Polder force: Perturbative approach
– Ground-state atom near magnetodielectric half space
• Casimir-Polder force: Beyond perturbation theory
– General theory
– Dynamics in weak-coupling limit
– Excited-state atom
• Summary and outlook
Quantum Optics II 2004 Cozumel FSU Jena
The QED vacuum
vacuus [lat.]: empty
Classical electrodynamics:
E(r) = 0, B(r) = 0 (no electromagnetic field)
Quantum Optics II 2004 Cozumel FSU Jena
The QED vacuum
vacuus [lat.]: empty
Classical electrodynamics:
E(r) = 0, B(r) = 0 (no electromagnetic field)
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)
Quantum Optics II 2004 Cozumel FSU Jena
The QED vacuum
vacuus [lat.]: empty
Classical electrodynamics:
E(r) = 0, B(r) = 0 (no electromagnetic field)
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
Quantum Optics II 2004 Cozumel FSU Jena
Normal-mode quantization
Quantized electric field:
E(r) =∑k
gk(r)ak + H.c.
gk(r): normal modes
a†k, ak: creation, annihilation operators
Quantum Optics II 2004 Cozumel FSU Jena
Normal-mode quantization
Quantized electric field:
E(r) =∑k
gk(r)ak + H.c.
gk(r): normal modes
a†k, ak: creation, annihilation operators
QED vacuum: ak|0〉 = 0
⇒ 〈E(r)〉 = 0, ∆E(r) 6= 0
Quantum Optics II 2004 Cozumel FSU Jena
Normal-mode quantization
Quantized electric field:
E(r) =∑k
gk(r)ak + H.c.
gk(r): normal modes
a†k, ak: creation, annihilation operators
QED vacuum: ak|0〉 = 0
⇒ 〈E(r)〉 = 0, ∆E(r) 6= 0
Applicability:
• free space
Quantum Optics II 2004 Cozumel FSU Jena
Normal-mode quantization
Quantized electric field:
E(r) =∑k
gk(r)ak + H.c.
gk(r): normal modes
a†k, ak: creation, annihilation operators
QED vacuum: ak|0〉 = 0
⇒ 〈E(r)〉 = 0, ∆E(r) 6= 0
Applicability:
• free space• arbitrary arrangement of
– perfectly reflecting bodies
Quantum Optics II 2004 Cozumel FSU Jena
Normal-mode quantization
Quantized electric field:
E(r) =∑k
gk(r)ak + H.c.
gk(r): normal modes
a†k, ak: creation, annihilation operators
QED vacuum: ak|0〉 = 0
⇒ 〈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!
Quantum Optics II 2004 Cozumel FSU Jena
Relevance of dispersionand absorption
1. World is not perfect:
absorption always present and relevant in experiments
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
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
3. New materials:
artificial metamaterials with left-handed properties1
→ strongly dispersing and absorbing
1D. R. Smith et. al., PRL 84, 18, 4184 (2000)
Quantum Optics II 2004 Cozumel FSU Jena
Generalized quantization scheme
Normal-mode quantization of electric field:
E(r) =∑k
gk(r)ak + H.c.
Quantum Optics II 2004 Cozumel FSU Jena
Generalized quantization scheme
Normal-mode quantization of electric field:
E(r) =∑k
gk(r)ak + H.c.
⇑c-number functions
⇓
⇑creation/annihilation
operators
⇓
Quantum Optics II 2004 Cozumel FSU Jena
Generalized quantization scheme
Normal-mode quantization of electric field:
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.
Quantum Optics II 2004 Cozumel FSU Jena
Classical Green tensor
[∇× µ−1(r, ω)∇× −
ω2
c2ε(r, ω)
]G(r, r′, ω) = δ(r− r′)
Quantum Optics II 2004 Cozumel FSU Jena
Classical Green tensor
[∇× µ−1(r, ω)∇× −
ω2
c2ε(r, ω)
]G(r, r′, ω) = δ(r− r′)
Physical interpretation:
Source at r
Quantum Optics II 2004 Cozumel FSU Jena
Classical Green tensor
[∇× µ−1(r, ω)∇× −
ω2
c2ε(r, ω)
]G(r, r′, ω) = δ(r− r′)
Physical interpretation:
Source at r G(r,r′,ω)−→−→
Quantum Optics II 2004 Cozumel FSU Jena
Classical Green tensor
[∇× µ−1(r, ω)∇× −
ω2
c2ε(r, ω)
]G(r, r′, ω) = δ(r− r′)
Physical interpretation:
Source at r G(r,r′,ω)−→−→ Electric field at r′
Quantum Optics II 2004 Cozumel FSU Jena
Creation/annihilation operators
Bosonic commutation relations:[fλi(r, ω), f†
λ′j(r′, ω′)
]= δλλ′δijδ(r− r′)δ(ω − ω′) (λ = e,m)
Quantum Optics II 2004 Cozumel FSU Jena
Creation/annihilation operators
Bosonic commutation relations:[fλi(r, ω), f†
λ′j(r′, ω′)
]= δλλ′δijδ(r− r′)δ(ω − ω′) (λ = e,m)
Physical interpretation:
Noise polarization:
PN(r, ω) = i
√~ε0π
×√
Imε(r, ω) fe(r, ω)
+ +
++
++
+
+
+−−
−−
−−
−−
Quantum Optics II 2004 Cozumel FSU Jena
Creation/annihilation operators
Bosonic commutation relations:[fλi(r, ω), f†
λ′j(r′, ω′)
]= δλλ′δijδ(r− r′)δ(ω − ω′) (λ = e,m)
Physical interpretation:
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
Quantum Optics II 2004 Cozumel FSU Jena
Atom-field dynamics
H = HMF + HA + HAMF
Quantum Optics II 2004 Cozumel FSU Jena
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)
Quantum Optics II 2004 Cozumel FSU Jena
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)
Atomic Hamiltonian:
HA =∑α
pα2
2mα+
1
2ε0
∫d3rPA
2(r)
Quantum Optics II 2004 Cozumel FSU Jena
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)
Atomic Hamiltonian:
HA =∑α
pα2
2mα+
1
2ε0
∫d3rPA
2(r)
Electric dipole interaction:
HAMF = −dE(rA) +1
2mA
[pA, d× B(rA)
]+
Quantum Optics II 2004 Cozumel FSU Jena
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)]
Quantum Optics II 2004 Cozumel FSU Jena
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!
Quantum Optics II 2004 Cozumel FSU Jena
Manifestations of the QED vacuum
• Casimir force [C. Raabe et. al., PRA 68, 033810 (2003)]
Quantum Optics II 2004 Cozumel FSU Jena
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)]
Quantum Optics II 2004 Cozumel FSU Jena
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)]
• Modification of atomic level structure [as above]
Quantum Optics II 2004 Cozumel FSU Jena
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)]
• Modification of atomic level structure [as above]
• Spontaneous decay [Ho et. al., PRA 68, 043816 (2003)]
Quantum Optics II 2004 Cozumel FSU Jena
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)
Quantum Optics II 2004 Cozumel FSU Jena
Relevance of Casimir-Polder forces
• Adsorption of atoms/molecules
to surfaces1
• Atom optics2
1M. A. Chesters et. al., Surf. Sci. 35, 161 (1973);J. Darville, in Vibrations at Surfaces (Plenum, New York, 1982)
2F. Shimizu and J. Fujita, PRL 88, 123201 (2002)
Quantum Optics II 2004 Cozumel FSU Jena
Relevance of Casimir-Polder forces
• Adsorption of atoms/molecules
to surfaces1
• Atom optics2
• Atomic-force microscopes3
1M. A. Chesters et. al., Surf. Sci. 35, 161 (1973);J. Darville, in Vibrations at Surfaces (Plenum, New York, 1982)
2F. Shimizu and J. Fujita, PRL 88, 123201 (2002)3G. Binnig et. al., PRL 56, 930 (1986)
Quantum Optics II 2004 Cozumel FSU Jena
Casimir-Polder force:
Perturbative approach (ground-state atoms)
Open issues
• Role of material absorption
• Influence of magnetic properties
Quantum Optics II 2004 Cozumel FSU Jena
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)
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
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
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)
Quantum Optics II 2004 Cozumel FSU Jena
Atom near half space1
Azε (ω) µ ω( )
Quantum Optics II 2004 Cozumel FSU Jena
Atom near half space1
Azε (ω) µ ω( )
Nonretarded limit: zA c/ωt
Quantum Optics II 2004 Cozumel FSU Jena
Atom near half space1
Azε (ω)
Nonretarded limit: zA c/ωt
Purely dielectric half space: U0(zA) = −C3
z3A
C3 =~
16π2ε0
∫ ∞
0duα(0)
0 (iu)ε(iu)− 1
ε(iu) + 1≥ 0
Quantum Optics II 2004 Cozumel FSU Jena
Atom near half space1
Azµ ω( )
Nonretarded limit: zA c/ωt
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
Quantum Optics II 2004 Cozumel FSU Jena
Atom near half spaceRetarded limit: zA c/ωr, c/ωk0
U0(zA) =C4
z4AC4 = C4[α(0), ε(0), µ(0)] T 0
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
Comparison of different forces
distance nonretarded retardedobjects e↔ e e↔ m e↔ e e↔ m
U ∝ +1
z3U ∝ −
1
zU ∝ +
1
z4U ∝ −
1
z4
Quantum Optics II 2004 Cozumel FSU Jena
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
z6U ∝ −
1
z4U ∝ +
1
z7U ∝ −
1
z7
Quantum Optics II 2004 Cozumel FSU Jena
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
z6U ∝ −
1
z4U ∝ +
1
z7U ∝ −
1
z7
F ∝ +1
z3F ∝ −
1
zF ∝ +
1
z4F ∝ −
1
z4
Quantum Optics II 2004 Cozumel FSU Jena
Casimir-Polder force:
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
Quantum Optics II 2004 Cozumel FSU Jena
General theory
Lorentz force on charged particles (electric dipole app.):
fα = qαE(rA)−∇ϕA(rA)
+12
[˙rα × B(rA)− B(rA)× ˙rα
]
Quantum Optics II 2004 Cozumel FSU Jena
General theory
⇒
Lorentz force on charged particles (electric dipole app.):
fα = qαE(rA)−∇ϕA(rA)
+12
[˙rα × B(rA)− B(rA)× ˙rα
]Lorentz force on an atom:⟨
F⟩AMF
=∇
⟨dE(r)
⟩AMF
+d
dt
⟨d× B(r)
⟩AMF
r=rA
Quantum Optics II 2004 Cozumel FSU Jena
General theory
⇒
Lorentz force on charged particles (electric dipole app.):
fα = qαE(rA)−∇ϕA(rA)
+12
[˙rα × B(rA)− B(rA)× ˙rα
]Lorentz force on an atom:⟨
F⟩AMF
=∇
⟨dE(r)
⟩AMF
+d
dt
⟨d× B(r)
⟩AMF
r=rA
Applicability:
• Field state: arbitrary
• Atomic state: arbitrary
• Coupling: strong/weak
Quantum Optics II 2004 Cozumel FSU Jena
General theory
⇒
Lorentz force on charged particles (electric dipole app.):
fα = qαE(rA)−∇ϕA(rA)
+12
[˙rα × B(rA)− B(rA)× ˙rα
]Lorentz force on an atom:⟨
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
Quantum Optics II 2004 Cozumel FSU Jena
CP force: Weak-coupling limitRemaining task: Solving the dynamics (Markov approximation)
E(r) = E(r, t) =?, d = d(t) =?
Quantum Optics II 2004 Cozumel FSU Jena
CP force: Weak-coupling limitRemaining task: Solving the dynamics (Markov approximation)
E(r) = E(r, t) =?, d = d(t) =?
Casimir-Polder force:
F(rA, t) =∑m,n
σnm(t)Fmn(rA)
Quantum Optics II 2004 Cozumel FSU Jena
CP force: Weak-coupling limitRemaining task: Solving the dynamics (Markov approximation)
E(r) = E(r, t) =?, d = d(t) =?
Casimir-Polder force:
F(rA, t) =∑m,n
σnm(t)Fmn(rA)
⇑ ⇑
Atomic density
matrix elements:
σnm(t)
Associated force
components:
Fmn(rA)
Quantum Optics II 2004 Cozumel FSU Jena
CP force: Weak-coupling limitRemaining task: Solving the dynamics (Markov approximation)
E(r) = E(r, t) =?, d = d(t) =?
Casimir-Polder force:
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)
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
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
Example: Two-level atom near dielectric half space
(single-resonance medium, nonretarded limit)
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
Quantum Optics II 2004 Cozumel FSU Jena
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!
Quantum Optics II 2004 Cozumel FSU Jena
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!
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!
Quantum Optics II 2004 Cozumel FSU Jena
Excited-(eigen)state force
Example: Two-level atom near dielectric half space
(single-resonance medium, nonretarded limit)
-3
0
3
1.12 1.13 1.14
(F11r )
z
10/ ωTω
x10 9/(3C / λT4 )
z =4.5 nmA
ω10
|1>
|0>
Quantum Optics II 2004 Cozumel FSU Jena
Excited-(eigen)state force
Example: Two-level atom near dielectric half space
(single-resonance medium, nonretarded limit)
-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>
Quantum Optics II 2004 Cozumel FSU Jena
Excited-(eigen)state force
Example: Two-level atom near dielectric half space
(single-resonance medium, nonretarded limit)
-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>
Quantum Optics II 2004 Cozumel FSU Jena
Excited-(eigen)state force
Example: Two-level atom near dielectric half space
(single-resonance medium, nonretarded limit)
-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>
Quantum Optics II 2004 Cozumel FSU Jena
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
Quantum Optics II 2004 Cozumel FSU Jena
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