electromagnetically induced transparency with rydberg...
TRANSCRIPT
FORTHIESL
FORTHIESL
Electromagnetically Induced Transparency
with Rydberg Atoms
David Petrosyan
OPTICS11, 8/09/11 – p. 1/22
FORTHIESL
FORTHIESLOutline
Background:
Electromagnetically induced transparency (EIT)
Rydberg atoms: dipole-dipole (DD) & van der Waals (VdW) interactions
OPTICS11, 8/09/11 – p. 2/22
FORTHIESL
FORTHIESLOutline
Background:
Electromagnetically induced transparency (EIT)
Rydberg atoms: dipole-dipole (DD) & van der Waals (VdW) interactions
Cross-phase modulation of single photons via static DDI
OPTICS11, 8/09/11 – p. 2/22
FORTHIESL
FORTHIESLOutline
Background:
Electromagnetically induced transparency (EIT)
Rydberg atoms: dipole-dipole (DD) & van der Waals (VdW) interactions
Cross-phase modulation of single photons via static DDI
Strong-field EIT with VdW interacting Rydberg atoms:
Experiment
Theoretical model
Numerical simulations
OPTICS11, 8/09/11 – p. 2/22
FORTHIESL
FORTHIESLOutline
Background:
Electromagnetically induced transparency (EIT)
Rydberg atoms: dipole-dipole (DD) & van der Waals (VdW) interactions
Cross-phase modulation of single photons via static DDI
Strong-field EIT with VdW interacting Rydberg atoms:
Experiment
Theoretical model
Numerical simulations
Conclusions
OPTICS11, 8/09/11 – p. 2/22
FORTHIESL
FORTHIESL
Electromagnetically Induced Transparency
εp
εp
−4 −2 0 2 4Detuning ∆p/γe
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Dis
pers
ion
Re(α
)
0.0
0.2
0.4
0.6
0.8
1.0
Abs
orpt
ion
Im(α)
Γe
∆p
e
g
Stationary propagation ∂zEp = iκ2αEp with κ = ς0ρ [ρ ≫ ρphot]
2LA Polarizability α = iγe
γe−i∆p≡ αTLA
OPTICS11, 8/09/11 – p. 3/22
FORTHIESL
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Electromagnetically Induced Transparency
εp
Ωc
εp Ωc
−4 −2 0 2 4Detuning ∆p/γe
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Dis
pers
ion
Re(α
)
0.0
0.2
0.4
0.6
0.8
1.0
Abs
orpt
ion
Im(α)
Γe
δ
∆
Γ
p
c
e
g
r
r
Stationary propagation ∂zEp = iκ2αEp with κ = ς0ρ [ρ ≫ ρphot]
3LA (EIT) Polarizability α = iγe
γe−i∆p+|Ωc|2
γr−i(∆p+δc)
≡ αEIT
Fleischhauer, Imamoglu, Marangos, RMP 77, 633 (2005) OPTICS11, 8/09/11 – p. 3/22
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Rydberg Atoms
R 0a2n≅+ −
High principal quantum number
n ≫ 1 (H-like)
OPTICS11, 8/09/11 – p. 4/22
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Rydberg Atoms
q
−q
Est
−q+1
q−1
−
+
High principal quantum number
n ≫ 1 (H-like)
Static electric field Est
⇒ Stark eigenstates with
permanent dipole moments
℘r = 32nqea0 q ∈ [−n+ 1, n− 1]
Gallagher, Rydberg Atoms (Cambridge 1994)OPTICS11, 8/09/11 – p. 4/22
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Static Dipole-Dipole Interaction
atomjiatom
pi pj
r i −r jθ
Atoms i, j in state |r〉 possess permanent dipole moments ez℘i,j
OPTICS11, 8/09/11 – p. 5/22
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Static Dipole-Dipole Interaction
atomjiatom
pi pj
r i −r jθ
Atoms i, j in state |r〉 possess permanent dipole moments ez℘i,j
⇒ Static DDI
VSDD = ~σirrDijσ
jrr
Dij ≡ D(ri − rj) ∝ ℘i℘j(1−3 cos2 θ)
|ri−rj |3∝ n4 — SDDI strength
OPTICS11, 8/09/11 – p. 5/22
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Resonant Dipole-Dipole Interaction
r
g
Energy Er = − Ryn∗2
effective PQN n∗ = n− δl (δl quantum defect)
OPTICS11, 8/09/11 – p. 6/22
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Resonant Dipole-Dipole Interaction
ωrb
ωar ωar
ωrb
b
r
aatomj
a
b
iatom
rωar = ωrb
OPTICS11, 8/09/11 – p. 6/22
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Resonant Dipole-Dipole Interaction
Dij
Dij
Dji
Dji
b
r
aatomj
a
b
iatom
rωar = ωrb
⇒ |ri〉 |rj〉 → |ai,j〉 |bj,i〉: Resonant exchange (Förster process)
VRDD = ~(σibrDijσ
jar + σj
brDjiσiar
)+H.c
Dij ≡ D(ri − rj) ∝ ℘br℘ar
|ri−rj |3∝ n4 — RDDI strength
OPTICS11, 8/09/11 – p. 6/22
FORTHIESL
FORTHIESLVan der Waals Interaction
Dij
Dij
Dji
Dji
bb
r
aatomj
aiatom
r
δ δ
ωrb − ωar = δ ≫ D
(δ ∝ n−3)
OPTICS11, 8/09/11 – p. 7/22
FORTHIESL
FORTHIESLVan der Waals Interaction
Dij
Dij
Dji
Dji
bb
r
aatomj
aiatom
r
δ δ
ωrb − ωar = δ ≫ D
(δ ∝ n−3)
⇒ |ri〉 |rj〉 9 |ai,j〉 |bj,i〉: Non-Resonant DDI (Adiabatic elim. |ai,j〉 |bj,i〉)
OPTICS11, 8/09/11 – p. 7/22
FORTHIESL
FORTHIESLVan der Waals Interaction
ij∆r
aatomj
ai
r
bb
atom
ωrb − ωar = δ ≫ D
(δ ∝ n−3)
⇒ |ri〉 |rj〉 9 |ai,j〉 |bj,i〉: Non-Resonant DDI (Adiabatic elim. |ai,j〉 |bj,i〉)
⇒ Energy shift of |ri〉 |rj〉 (2nd-order in D/δ)
VVdW = ~σirr∆ijσ
jrr
∆ij ≡ ∆(ri − rj) = 2|D(ri−rj)|2
δ= C6
|ri−rj |6∝ n11 — VdWI strength
Saffman, Walker, Mølmer, RMP 82, 2313 (2010) OPTICS11, 8/09/11 – p. 7/22
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Photonic phase gate with SDDI
Friedler, Petrosyan, Fleischhauer, Kurizki, PRA 72, 043803 (2005)Shahmoon, Kurizki, Fleischhauer, Petrosyan, PRA 83, 033806 (2011) OPTICS11, 8/09/11 – p. 8/22
FORTHIESL
FORTHIESLPhoton-Photon Interaction
2E
E1
2
2EE1
E1
g
rr 1 2
e e1 2
Ω 2
V
Ω1
DD Est
w
wE
22 |
|
| |
Ψ Ψ1 2v vg g
Static Estez ⇒ Stark eigenstates |ri〉 with SDMs ℘rez = 32nqea0ez
OPTICS11, 8/09/11 – p. 9/22
FORTHIESL
FORTHIESLPhoton-Photon Interaction
2E
E1
2
2EE1
E1
g
rr 1 2
e e1 2
Ω 2
V
Ω1
DD Est
w
wE
22 |
|
| |
Ψ Ψ1 2v vg g
Static Estez ⇒ Stark eigenstates |ri〉 with SDMs ℘rez = 32nqea0ez
Ei → Ψi = cos θEi − sin θ√Nσgri (i = 1, 2) propagate with ±vg = c cos2 θ
OPTICS11, 8/09/11 – p. 9/22
FORTHIESL
FORTHIESLPhoton-Photon Interaction
2E
E1
2
2EE1
E1
g
rr 1 2
e e1 2
Ω 2
V
Ω1
DD Est
w
wE
22 |
|
| |
Ψ Ψ1 2v vg g
Static Estez ⇒ Stark eigenstates |ri〉 with SDMs ℘rez = 32nqea0ez
Ei → Ψi = cos θEi − sin θ√Nσgri (i = 1, 2) propagate with ±vg = c cos2 θ
Atomic components of Ψi interact via VSDD ⇒ induces XPM
Resonant DDI (state mixing) is suppressed for q = n− 1, m = 0
OPTICS11, 8/09/11 – p. 9/22
FORTHIESL
FORTHIESLCross-Phase Modulation
0 L/wτ
0
1
φ(τ)
−L/w 0 L/wζ
−2
−1
0
D(z
)
• DD level shift [vs. ζ = (z − z′)/w]
D(z − z′) = 1πw2
∫ 2π0 dϕ′∫∞
0 dr′⊥r′⊥e−r′2⊥/w2D(zez − r
′)
• Phase shift [vs. τ = vgt/w]
φ(z1, z2, t) = − sin4 θ∫ t0dt
′D(z1 − z2 − 2vg(t− t′))
OPTICS11, 8/09/11 – p. 10/22
FORTHIESL
FORTHIESLCross-Phase Modulation
0 L/wτ
0
1
φ(τ)
−L/w 0 L/wζ
−2
−1
0
D(z
)
• DD level shift [vs. ζ = (z − z′)/w]
D(z − z′) = 1πw2
∫ 2π0 dϕ′∫∞
0 dr′⊥r′⊥e−r′2⊥/w2D(zez − r
′)
• Phase shift [vs. τ = vgt/w]
φ(z1, z2, t) = − sin4 θ∫ t0dt
′D(z1 − z2 − 2vg(t− t′))
Initially t = 0, z1 = 0 & z2 = L ⇒ φ(0, L, 0) = 0
After the interaction t = L/vg, z1 = L & z2 = 0
φ(L, 0, L/v) = − sin4 θvg
∫ L0 dz′D(2z′ − L) = 2C
vgw2
OPTICS11, 8/09/11 – p. 10/22
FORTHIESL
FORTHIESLCross-Phase Modulation
0 L/wτ
0
1
φ(τ)
−L/w 0 L/wζ
−2
−1
0
D(z
)
• DD level shift [vs. ζ = (z − z′)/w]
D(z − z′) = 1πw2
∫ 2π0 dϕ′∫∞
0 dr′⊥r′⊥e−r′2⊥/w2D(zez − r
′)
• Phase shift [vs. τ = vgt/w]
φ(z1, z2, t) = − sin4 θ∫ t0dt
′D(z1 − z2 − 2vg(t− t′))
Initially t = 0, z1 = 0 & z2 = L ⇒ φ(0, L, 0) = 0
After the interaction t = L/vg, z1 = L & z2 = 0
φ(L, 0, L/v) = − sin4 θvg
∫ L0 dz′D(2z′ − L) = 2C
vgw2
• Phase shift φ = π [spatially uniform!]
⇒ Universal CPHASE gate between SPh pulses E1 & E2|x〉1 |y〉2 → (−1)xy |x〉1 |y〉2 (x, y ∈ [0, 1])
OPTICS11, 8/09/11 – p. 10/22
FORTHIESL
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EIT with strong VdWI
Petrosyan, Otterbach, Fleischhauer, arXiv:1106.1360 [quant-ph] OPTICS11, 8/09/11 – p. 11/22
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Experiment
OPTICS11, 8/09/11 – p. 12/22
FORTHIESL
FORTHIESLTheoretical Model
VVdW
ΩΓe
Ω
δ
∆
Γ
p
p
c
ec
g
r
r
Hamiltonian H = Ha + Vaf + VVdW
Ha = −~∑N
j [∆pσee(rj) + (∆p + δc)σrr(rj)]
Vaf = −~∑N
j [Ωp(rj)σeg(rj) + Ωcσre(rj) + H.c.]
VVdW = ~∑N
i<j σrr(ri)∆(ri − rj)σrr(rj)
⇓
OPTICS11, 8/09/11 – p. 13/22
FORTHIESL
FORTHIESLTheoretical Model
VVdW
ΩΓe
Ω
δ
∆
Γ
p
p
c
ec
g
r
r
Hamiltonian H = Ha + Vaf + VVdW
Ha = −~∑N
j [∆pσee(rj) + (∆p + δc)σrr(rj)]
Vaf = −~∑N
j [Ωp(rj)σeg(rj) + Ωcσre(rj) + H.c.]
VVdW = ~∑N
i<j σrr(ri)∆(ri − rj)σrr(rj)
⇓Stationary probe-field propagation [Ωp ≡ ηEp]
∂z〈E†p(r)Ep(r)〉 = −κ(r)〈E†
p(r)Im[α(r)]Ep(r)〉
Polarizability α(r) =iγe
γe − i∆p +|Ωc|2
γr−i[∆p+δc−S(r)]
with S(r) ≡ ∑Nj ∆(r− rj)σrr(rj) total VdW shift of |r〉 at position r
OPTICS11, 8/09/11 – p. 13/22
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Rydberg Excitation Blockade (3LA)
Population of |r〉: 〈σrr(∆2)〉 ≈〈Ω†
pΩp〉
|Ωc|2+∆22
γ2e
|Ωc|2
[∆2 = ∆p + δc]
⇒ 〈σrr(0)〉 =〈Ω†
pΩp〉
|Ωc|2& 〈σrr(w)〉 = 1
2 〈σrr(0)〉 w ≡ |Ωc|2
γe
OPTICS11, 8/09/11 – p. 14/22
FORTHIESL
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Rydberg Excitation Blockade (3LA)
Population of |r〉: 〈σrr(∆2)〉 ≈〈Ω†
pΩp〉
|Ωc|2+∆22
γ2e
|Ωc|2
[∆2 = ∆p + δc]
⇒ 〈σrr(0)〉 =〈Ω†
pΩp〉
|Ωc|2& 〈σrr(w)〉 = 1
2 〈σrr(0)〉 w ≡ |Ωc|2
γe
An atom in |r〉 blocks Rydberg excitations for ∆(R) & w [∆2 → ∆2 −∆(R)]
⇒ Blockade radius Rsa ≃ 6
√|C6|w
& (superatom) volume Vsa = 4π3 R3
sa
OPTICS11, 8/09/11 – p. 14/22
FORTHIESL
FORTHIESL
Rydberg Excitation Blockade (3LA)
Population of |r〉: 〈σrr(∆2)〉 ≈〈Ω†
pΩp〉
|Ωc|2+∆22
γ2e
|Ωc|2
[∆2 = ∆p + δc]
⇒ 〈σrr(0)〉 =〈Ω†
pΩp〉
|Ωc|2& 〈σrr(w)〉 = 1
2 〈σrr(0)〉 w ≡ |Ωc|2
γe
An atom in |r〉 blocks Rydberg excitations for ∆(R) & w [∆2 → ∆2 −∆(R)]
⇒ Blockade radius Rsa ≃ 6
√|C6|w
& (superatom) volume Vsa = 4π3 R3
sa
0 2 4 6 8R/Rsa
0
1
2
3g r(2
) (R)
Ωp/2π (MHz)0.20.51.02.0
g(2)r (R) ≡ 〈σrr(0)σrr(R)〉
〈σrr(0)〉〈σrr(R)〉
σrr(R) =
|Ωc|2Ω†
pΩp
|Ωc|2Ω†pΩp+[|Ωc|2−∆p∆2(R)]2+∆2
2(R)γ2e
[∆2(R) ≡ ∆2 − S(R)]OPTICS11, 8/09/11 – p. 14/22
FORTHIESL
FORTHIESL
Superatom
(1)E
(1)R
(1)R (2)E
Ωpnsa
Ωc
(2)E
(3)E2Ωc
3Ωc(1)R (1)E
2Ph
3Ph
4Ph
Ωp
Ωp
G
san
san −1)2
−2)3
(
(
nsa = ρVsa
|G〉 = |g1, g2, . . . , gnsa 〉
|E(1)〉 = 1√nsa
∑nsaj |g1, g2, . . . , ej , . . . , gnsa 〉
|R(1)〉 = 1√nsa
∑nsaj |g1, g2, . . . , rj , . . . , gnsa 〉
|E(2)〉 = 1√nsa(nsa−1)
∑nsai<j |g1, . . . , ei, . . . , ej , . . . , gnsa 〉
|R(1)E(1)〉 = 1√nsa(nsa−1)2
∑nsai,j |g1, . . . , ri, . . . , ej , . . . , gnsa 〉
etc.
OPTICS11, 8/09/11 – p. 15/22
FORTHIESL
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Superatom
(1)E
(1)R
(1)R (2)E
Ωpnsa
Ωc
(2)E
(3)E2Ωc
3Ωc(1)R (1)E
2Ph
3Ph
4Ph
Ωp
Ωp
G
san
san −1)2
−2)3
(
(
nsa = ρVsa
|G〉 = |g1, g2, . . . , gnsa 〉
|E(1)〉 = 1√nsa
∑nsaj |g1, g2, . . . , ej , . . . , gnsa 〉
|R(1)〉 = 1√nsa
∑nsaj |g1, g2, . . . , rj , . . . , gnsa 〉
|E(2)〉 = 1√nsa(nsa−1)
∑nsai<j |g1, . . . , ei, . . . , ej , . . . , gnsa 〉
|R(1)E(1)〉 = 1√nsa(nsa−1)2
∑nsai,j |g1, . . . , ri, . . . , ej , . . . , gnsa 〉
etc.
• Adiabatic elimination of |E(k)〉 [γe = 12Γe ≫ Ωp,c]
• while ΣGG +ΣRR = 1 [saturation]
⇒ ΣRR =|Ωc|
2nsaΩ†pΩp
|Ωc|2nsaΩ†pΩp+[|Ωc|2−∆p∆2]2+∆2
2γ2e
OPTICS11, 8/09/11 – p. 15/22
FORTHIESL
FORTHIESL
Total VdW shift at position r
Ωcpε r
Nsa = V/Vsa superatoms
S(r) ≈ ∑Nsa
j ∆(r− rj)ΣRR(rj)
= ∆(0)ΣRR(r) + s(r)
OPTICS11, 8/09/11 – p. 16/22
FORTHIESL
FORTHIESL
Total VdW shift at position r
Ωcpε r
Nsa = V/Vsa superatoms
S(r) ≈ ∑Nsa
j ∆(r− rj)ΣRR(rj)
= ∆(0)ΣRR(r) + s(r)
MF shift s(r) ≡ ρsa∫
V \V(r)sa
∆(r− r′)ΣRR(r
′)d3r′ ≃ w8 ΣRR(r)
OPTICS11, 8/09/11 – p. 16/22
FORTHIESL
FORTHIESL
Total VdW shift at position r
Ωcpε r
Nsa = V/Vsa superatoms
S(r) ≈ ∑Nsa
j ∆(r− rj)ΣRR(rj)
= ∆(0)ΣRR(r) + s(r)
MF shift s(r) ≡ ρsa∫
V \V(r)sa
∆(r− r′)ΣRR(r
′)d3r′ ≃ w8 ΣRR(r)
Blockade ∆(0) = 1Vsa
∫
Vsa∆(r′)d3r′ = 3C6
R3sa
∫ Rsa
0dr′
r′4→ ∞ (≫ γe)
OPTICS11, 8/09/11 – p. 16/22
FORTHIESL
FORTHIESL
Total VdW shift at position r
Ωcpε r
Nsa = V/Vsa superatoms
S(r) ≈ ∑Nsa
j ∆(r− rj)ΣRR(rj)
= ∆(0)ΣRR(r) + s(r)
MF shift s(r) ≡ ρsa∫
V \V(r)sa
∆(r− r′)ΣRR(r
′)d3r′ ≃ w8 ΣRR(r)
Blockade ∆(0) = 1Vsa
∫
Vsa∆(r′)d3r′ = 3C6
R3sa
∫ Rsa
0dr′
r′4→ ∞ (≫ γe)
⇒ S(r) = ∆(0)ΣRR(r) + 〈s(r)〉
ΣRR(r) — Projector
〈ΣRR(r)〉 ∈ [0, 1] — Blockade probability : ∆(0) ≫ γe
OPTICS11, 8/09/11 – p. 16/22
FORTHIESL
FORTHIESL
Probe field intensity evolution
∂z〈Ω†p(r)Ωp(r)〉=−κ(r)〈Ω†
p(r)Im[α(r)]Ωp(r)〉⇒−κ(r)Im[〈α(r)〉r]〈Ω†p(r)Ωp(r)〉
OPTICS11, 8/09/11 – p. 17/22
FORTHIESL
FORTHIESL
Probe field intensity evolution
∂z〈Ω†p(r)Ωp(r)〉=−κ(r)〈Ω†
p(r)Im[α(r)]Ωp(r)〉⇒−κ(r)Im[〈α(r)〉r]〈Ω†p(r)Ωp(r)〉
Conditional polarizability:
〈α(r)〉r = 〈ΣRR(r)〉riγe
γe − i∆p︸ ︷︷ ︸
αTLA
+[1−〈ΣRR(r)〉r]iγe
γe −−i∆p +|Ωc|2
γr−i[∆p+δc−〈s(r)〉]︸ ︷︷ ︸
αEIT
OPTICS11, 8/09/11 – p. 17/22
FORTHIESL
FORTHIESL
Probe field intensity evolution
∂z〈Ω†p(r)Ωp(r)〉=−κ(r)〈Ω†
p(r)Im[α(r)]Ωp(r)〉⇒−κ(r)Im[〈α(r)〉r]〈Ω†p(r)Ωp(r)〉
Conditional polarizability:
〈α(r)〉r = 〈ΣRR(r)〉riγe
γe − i∆p︸ ︷︷ ︸
αTLA
+[1−〈ΣRR(r)〉r]iγe
γe −−i∆p +|Ωc|2
γr−i[∆p+δc−〈s(r)〉]︸ ︷︷ ︸
αEIT
Superatom operator: ΣRR(˜r) =|Ωc|
2nsaΩ†p(˜r)Ωp(˜r)
|Ωc|2nsaΩ†p(˜r)Ωp(˜r)+[|Ωc|2−∆p∆2]2+∆2
2γ2e
〈Ω†p(r)Ωp(r)〉r → 〈Ω†
p(r)Ωp(r)〉g(2)p (r, r) 〈Ω†p(r)Ωp(r)〉 → 〈Ω†
p(r)Ωp(r)〉
OPTICS11, 8/09/11 – p. 17/22
FORTHIESL
FORTHIESL
Probe field intensity evolution
∂z〈Ω†p(r)Ωp(r)〉=−κ(r)〈Ω†
p(r)Im[α(r)]Ωp(r)〉⇒−κ(r)Im[〈α(r)〉r]〈Ω†p(r)Ωp(r)〉
Conditional polarizability:
〈α(r)〉r = 〈ΣRR(r)〉riγe
γe − i∆p︸ ︷︷ ︸
αTLA
+[1−〈ΣRR(r)〉r]iγe
γe −−i∆p +|Ωc|2
γr−i[∆p+δc−〈s(r)〉]︸ ︷︷ ︸
αEIT
Superatom operator: ΣRR(˜r) =|Ωc|
2nsaΩ†p(˜r)Ωp(˜r)
|Ωc|2nsaΩ†p(˜r)Ωp(˜r)+[|Ωc|2−∆p∆2]2+∆2
2γ2e
〈Ω†p(r)Ωp(r)〉r → 〈Ω†
p(r)Ωp(r)〉g(2)p (r, r) 〈Ω†p(r)Ωp(r)〉 → 〈Ω†
p(r)Ωp(r)〉
Intensity correlation within V(r)sa :
g(2)p (r, r) ≡ 〈E†
p(r)E†p(r)Ep(r)Ep(r)〉
〈E†p(r)Ep(r)〉〈E
†p(r)Ep(r)〉
≡ g(2)p (r)
∂zg(2)p (r) = −κ(r)Im[〈α(r)〉 − αEIT]g
(2)p (r)
OPTICS11, 8/09/11 – p. 17/22
FORTHIESL
FORTHIESL
Numerical Simulations: comput. procedure
Stochastic (Monte-Carlo)
Divide the propagation distance L into L2Rsa
intervals (superatoms)
For each z ∈ SAj generate uniform random number pz ∈ [0, 1]
if pz ≤ 〈ΣRR(r)〉r ⇒ 〈α(r)〉r = αTLA
if pz > 〈ΣRR(r)〉r ⇒ 〈α(r)〉r = αEIT
Continue to z ∈ SAj+1, etc.
Average over several independent realizations
OPTICS11, 8/09/11 – p. 18/22
FORTHIESL
FORTHIESL
Numerical Simulations: comput. procedure
Stochastic (Monte-Carlo)
Divide the propagation distance L into L2Rsa
intervals (superatoms)
For each z ∈ SAj generate uniform random number pz ∈ [0, 1]
if pz ≤ 〈ΣRR(r)〉r ⇒ 〈α(r)〉r = αTLA
if pz > 〈ΣRR(r)〉r ⇒ 〈α(r)〉r = αEIT
Continue to z ∈ SAj+1, etc.
Average over several independent realizations
Continuous limit
Infinitely many realizations ⇒ ∂zIp(r) = −κ(r)Im[〈α(r)〉r]Ip(r)
with 〈α(r)〉r = 〈ΣRR(r)〉rαTLA + [1− 〈ΣRR(r)〉r]αEIT
and ∂zg(2)p (r) ≃ −κ(r)〈ΣRR(r)〉Im[αTLA − αEIT]g
(2)p (r)
OPTICS11, 8/09/11 – p. 18/22
FORTHIESL
FORTHIESL
Numerical Simulations: exper. parameters
Atoms : 87Rb at T = 20 µK
|g〉 ≡ 5S1/2 |F = 2,mF = 2〉 |e〉 ≡ 5P3/2 |F = 3,mF = 3〉 |r〉 ≡ 60S1/2
Γe = 3.8× 107 s−1, δω1 ≃ 2π · 5.7× 104 s−1 γe = 12Γe + δω1
Γr = 5× 103 s−1, δω2 ≃ 2π · 1.1× 105 s−1 γr = 12Γr + δω2
C6/2π = 1.4× 1011 s−1µm6
ρ(z) = ρ0 exp[−(z − z0)2/2σ2ρ]; ρ0 = 1.32× 107 mm−3 σρ = 0.7 mm
[ρ = 1.2× 107 mm−3, L = 1.3 mm ⇒ κL = 4.524]
Ωc = 2π · 2.25× 106 s−1 ( δc2π
= −105 s−1) ⇒ Rsa ≃ 6.6 µm & nsa ≃ 14.7
vg(∆2 ≃ 0) =2|Ωc|2κγe
≃ 6000 m/s
Pritchard et al., PRL 105, 193603 (2010); Singer et al., JPB 38, S295 (2005) OPTICS11, 8/09/11 – p. 19/22
FORTHIESL
FORTHIESL
Numerical Simulations [stochastic MC]
−10 −5 0 5 10∆p/2π (MHz)
0
0.5
1
1.5
g p(2) (L
)
0
0.2
0.4
0.6
0.8
Tra
nsm
issi
on I p(
L)/
I p(0)
0.010.150.51.0
Ωp(0)/2π
averaged over
10 realizations
Experiment ⇔ Theory : negligible shift & broadening of EIT line
OPTICS11, 8/09/11 – p. 20/22
FORTHIESL
FORTHIESL
Numerical Simulations [continuous]
−10 −5 0 5 10∆p/2π
0
0.5
1
1.5
g p(2) (L
)
0
0.2
0.4
0.6
0.8I p(
L)/
I p(0)
−10 −5 0 5 10∆p/2π
0 1 2 3Ωp(0)/2π (MHz)0
0.10.20.3
∆ pmax/2
π
00.20.40.60.8
1
Tm
ax
11.5
22.5
δωE
IT/2
πΩp(0)/2π
0.010.250.51.0
3.02.0
(a) (b)
(c)
(d)
(e)
averaged over
inf. realizations
OPTICS11, 8/09/11 – p. 21/22
FORTHIESL
FORTHIESLConclusions
Each Rydberg excitation |r〉 is delocalized over blockade volume Vsa
The medium is effectively composed of Nsa superatoms (only!)
OPTICS11, 8/09/11 – p. 22/22
FORTHIESL
FORTHIESLConclusions
Each Rydberg excitation |r〉 is delocalized over blockade volume Vsa
The medium is effectively composed of Nsa superatoms (only!)
The field Ep(r) is affected by an atom at r via α(r):An excited SA surrounding r blocks the excitation |r〉 of the atom: α → αTLA
All the other superatoms induce a small mean-field shift of |r〉: δc → δc + 〈s〉
OPTICS11, 8/09/11 – p. 22/22
FORTHIESL
FORTHIESLConclusions
Each Rydberg excitation |r〉 is delocalized over blockade volume Vsa
The medium is effectively composed of Nsa superatoms (only!)
The field Ep(r) is affected by an atom at r via α(r):An excited SA surrounding r blocks the excitation |r〉 of the atom: α → αTLA
All the other superatoms induce a small mean-field shift of |r〉: δc → δc + 〈s〉
Conditional (nonlinear) absorption changes photon statistics g(2)p (r, r)
When g(2)p (r, r) ≃ 0, absorption saturates: ρphot . ρsa ⇒ 〈Ω†
pΩp〉 ≃ 4ρsaρ
|Ωc|2
Photons are antibunched within the temporal window of δt ≃ 2Rsavg
(≃ 1.6 ns)
OPTICS11, 8/09/11 – p. 22/22
FORTHIESL
FORTHIESLConclusions
Each Rydberg excitation |r〉 is delocalized over blockade volume Vsa
The medium is effectively composed of Nsa superatoms (only!)
The field Ep(r) is affected by an atom at r via α(r):An excited SA surrounding r blocks the excitation |r〉 of the atom: α → αTLA
All the other superatoms induce a small mean-field shift of |r〉: δc → δc + 〈s〉
Conditional (nonlinear) absorption changes photon statistics g(2)p (r, r)
When g(2)p (r, r) ≃ 0, absorption saturates: ρphot . ρsa ⇒ 〈Ω†
pΩp〉 ≃ 4ρsaρ
|Ωc|2
Photons are antibunched within the temporal window of δt ≃ 2Rsavg
(≃ 1.6 ns)
Thanks to
for collaboration for hospitality for fin. support
OPTICS11, 8/09/11 – p. 22/22