electromagnetically induced transparency with rydberg...

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Electromagnetically Induced Transparency

with Rydberg Atoms

David Petrosyan

OPTICS11, 8/09/11 – p. 1/22

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FORTHIESLOutline

Background:

Electromagnetically induced transparency (EIT)

Rydberg atoms: dipole-dipole (DD) & van der Waals (VdW) interactions

OPTICS11, 8/09/11 – p. 2/22

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FORTHIESLOutline

Background:



Cross-phase modulation of single photons via static DDI

OPTICS11, 8/09/11 – p. 2/22

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FORTHIESLOutline

Background:




Strong-field EIT with VdW interacting Rydberg atoms:

Experiment

Theoretical model

Numerical simulations

OPTICS11, 8/09/11 – p. 2/22

FORTHIESL

FORTHIESLOutline

Background:




Strong-field EIT with VdW interacting Rydberg atoms:

Experiment

Theoretical model

Numerical simulations

Conclusions

OPTICS11, 8/09/11 – p. 2/22

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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

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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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ωrb

ωar ωar

ωrb

b

r

aatomj

a

b

iatom

rωar = ωrb

OPTICS11, 8/09/11 – p. 6/22

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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

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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

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Dij

Dij

Dji

Dji

bb

r

aatomj

aiatom

r

δ δ


(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

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ij∆r

aatomj

ai

r

bb

atom


(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

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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

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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


Ei → Ψi = cos θEi − sin θ√Nσgri (i = 1, 2) propagate with ±vg = c cos2 θ

OPTICS11, 8/09/11 – p. 9/22

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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


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

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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

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0 L/wτ

0

1

φ(τ)

−L/w 0 L/wζ

−2

−1

0

D(z

)


D(z − z′) = 1πw2

∫ 2π0 dϕ′∫∞


′)


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

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0 L/wτ

0

1

φ(τ)

−L/w 0 L/wζ

−2

−1

0

D(z

)


D(z − z′) = 1πw2

∫ 2π0 dϕ′∫∞


′)


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

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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

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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

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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

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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

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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

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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.

OPTICS11, 8/09/11 – p. 15/22

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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

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Ωcpε r


S(r) ≈ ∑Nsa


= ∆(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

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Ωcpε r


S(r) ≈ ∑Nsa


= ∆(0)ΣRR(r) + s(r)


V \V(r)sa


′)d3r′ ≃ w8 ΣRR(r)

Blockade ∆(0) = 1Vsa

∫

Vsa∆(r′)d3r′ = 3C6

R3sa

∫ Rsa

0dr′

r′4→ ∞ (≫ γe)

OPTICS11, 8/09/11 – p. 16/22

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Ωcpε r


S(r) ≈ ∑Nsa


= ∆(0)ΣRR(r) + s(r)


V \V(r)sa


′)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

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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

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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

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γe − i∆p︸︷︷︸

αTLA



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

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γe − i∆p︸︷︷︸

αTLA



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

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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

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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

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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

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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

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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




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






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






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

electromagnetically induced transparency with rydberg...

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