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Transcript of Deterministic quantum computation with photonic · PDF fileIESL FORTH Deterministic quantum...
IESLFORTH
Deterministic quantum computationwith photonic qubits
Single-Photon Nonlinearitiesvia Electromagnetically Induced Transparency
David Petrosyan
IESL-FORTH, Greece
QUDAL, 1/3/06 – p. 1/16
IESLFORTHOutline
Motivation: Quantum Information Processing with Single Photons
Background: EIT and Photon Storage in Atomic Media
Photonic Memory
Single Photon Sources
Cross-Phase ModulationSingle Photon Detection
Summary
QUDAL, 1/3/06 – p. 2/16
IESLFORTHOutline
Motivation: Quantum Information Processing with Single Photons
Background: EIT and Photon Storage in Atomic Media
Photonic Memory
Single Photon Sources
Cross-Phase ModulationSingle Photon Detection
Summary
QUDAL, 1/3/06 – p. 2/16
IESLFORTHOutline
Motivation: Quantum Information Processing with Single Photons
Background: EIT and Photon Storage in Atomic Media
Photonic Memory
Single Photon Sources
Cross-Phase ModulationSingle Photon Detection
Summary
QUDAL, 1/3/06 – p. 2/16
IESLFORTHOutline
Motivation: Quantum Information Processing with Single Photons
Background: EIT and Photon Storage in Atomic Media
Photonic Memory
Single Photon Sources
Cross-Phase ModulationSingle Photon Detection
Summary
QUDAL, 1/3/06 – p. 2/16
IESLFORTHOutline
Motivation: Quantum Information Processing with Single Photons
Background: EIT and Photon Storage in Atomic Media
Photonic Memory
Single Photon Sources
Cross-Phase Modulation
Single Photon Detection
Summary
QUDAL, 1/3/06 – p. 2/16
IESLFORTHOutline
Motivation: Quantum Information Processing with Single Photons
Background: EIT and Photon Storage in Atomic Media
Photonic Memory
Single Photon Sources
Cross-Phase ModulationSingle Photon Detection
Summary
QUDAL, 1/3/06 – p. 2/16
IESLFORTHOutline
Motivation: Quantum Information Processing with Single Photons
Background: EIT and Photon Storage in Atomic Media
Photonic Memory
Single Photon Sources
Cross-Phase ModulationSingle Photon Detection
Summary
QUDAL, 1/3/06 – p. 2/16
IESLFORTHPhotonic Qubit
Qubit: Single-photon wavepacket in the polarization state
|ψ〉 = α |V 〉 + β |H〉 with |α|2 + |β|2 = 1
|V 〉 ≡ |0〉 & |H〉 ≡ |1〉 form the computational basis |0〉, |1〉
General single-qubit unitary operation U = eiαT (φ1)R(ϑ)T (φ2) can bedecomposed into the product of
eiα , R(ϑ) =[
cosϑ − sinϑsinϑ cosϑ
]
, T (φ) =
[
1 00 eiφ
]
I = T (0), X = R(π/2)T (π), Y = eiπ/2R(π/2), Z = T (π), H = R(π/4)T (π).
R(ϑ) – Photon polarization rotation, andT (φ) – Relative phase-shift of |V 〉 & |H〉 componentscan be implemented with linear-optics operations, e.g.
ψ ψ
PBS
T( )φθR( ) UFR
QUDAL, 1/3/06 – p. 3/16
IESLFORTHPhotonic Qubit
Qubit: Single-photon wavepacket in the polarization state
|ψ〉 = α |V 〉 + β |H〉 with |α|2 + |β|2 = 1
|V 〉 ≡ |0〉 & |H〉 ≡ |1〉 form the computational basis |0〉, |1〉
General single-qubit unitary operation U = eiαT (φ1)R(ϑ)T (φ2) can bedecomposed into the product of
eiα , R(ϑ) =[
cosϑ − sinϑsinϑ cosϑ
]
, T (φ) =
[
1 00 eiφ
]
I = T (0), X = R(π/2)T (π), Y = eiπ/2R(π/2), Z = T (π), H = R(π/4)T (π).
R(ϑ) – Photon polarization rotation, andT (φ) – Relative phase-shift of |V 〉 & |H〉 componentscan be implemented with linear-optics operations, e.g.
ψ ψ
PBS
T( )φθR( ) UFR
QUDAL, 1/3/06 – p. 3/16
IESLFORTHTwo-Photon Logic Gates
Controlled-NOT gate Controlled-Z (PHASE) gate
WCNOT |a〉 |b〉 −→ |a〉 |a⊕ b〉 WCZ |a〉 |b〉 −→ (−1)ab |a〉 |b〉a, b ∈ 0, 1 a, b ∈ 0, 1
WCZ requires nonlinear (Kerr) photon-photon interaction – XPM
H
2ψ
1ψ
V
V
H
=outΦCZ 1ψ 2ψπ
XPM
PBS
PBS
W
Any multiqubit transformation can be decomposed intosingle-qubit U and two-qubit WCNOT or WCZ transformations⇒ U and W are Universal
QUDAL, 1/3/06 – p. 4/16
IESLFORTHTwo-Photon Logic Gates
Controlled-NOT gate Controlled-Z (PHASE) gate
WCNOT |a〉 |b〉 −→ |a〉 |a⊕ b〉 WCZ |a〉 |b〉 −→ (−1)ab |a〉 |b〉a, b ∈ 0, 1 a, b ∈ 0, 1
WCZ requires nonlinear (Kerr) photon-photon interaction – XPM
H
2ψ
1ψ
V
V
H
=outΦCZ 1ψ 2ψπ
XPM
PBS
PBS
W
Any multiqubit transformation can be decomposed intosingle-qubit U and two-qubit WCNOT or WCZ transformations⇒ U and W are Universal
QUDAL, 1/3/06 – p. 4/16
IESLFORTHOptical Quantum Computer
V
V
V
V
V
V
H
H
. . .
. . .
. . . . . .
. . .
SPhS
SPhS
SPhS
SPhS
SPhD
SPhD
SPhD
SPhD
U
U
U
U
U
W
Optical Quantum ProcessorInitialization Read−out
W
W
SPhS: Single Photon Sources
W=XPM: Cross Phase ModulationSPhD: Single Photon Detectors
Petrosyan, J. Opt. B. 7, S141 (2005) QUDAL, 1/3/06 – p. 5/16
IESLFORTHElectromagnetically Induced Transparency
Γ
∆
δRs
g
e
dΩE
H = ~
N∑
j=1
0 gE†(zj)e−ikzj 0
gE(zj)eikzj ∆ Ωd(t)e
ikdzj
0 Ω∗d(t)e
−ikdzj δR
j
with E(z, t) =P
q aq(t)eiqz and g =℘ge
~
q
~ω2ε0V
δR = 0 ⇒ Dark states (H |Dq1〉 = 0 |Dq
1〉) for single-photons |1q〉 = aq† |0〉|Dq
1〉 = cos θ |1q, s(0)〉 − sin θ |0q, s(1)〉 tan2 θ(t) = g2N|Ωd(t)|2
with collective atomic states|s(0)〉 ≡ |g1, g2, . . . , gN 〉|s(1)〉 ≡ 1√
N
∑Nj=1 e
i(k+q−kd)zj |g1, . . . , sj , . . . , gN 〉
Fleischhauer, Lukin, PRL 84, 5094 (2000); PRA 65, 022314 (2002)
θ = 0 (|Ωd|2 g2N ) ⇒ |Dq1〉 = |1q〉 |s(0)〉 purely photonic excitation
θ = π/2 (|Ωd|2 g2N ) ⇒ |Dq1〉 = |0q〉 |s(1)〉 purely atomic excitation
QUDAL, 1/3/06 – p. 6/16
IESLFORTHElectromagnetically Induced Transparency
Γ
∆
δRs
g
e
dΩE
H = ~
N∑
j=1
0 gE†(zj)e−ikzj 0
gE(zj)eikzj ∆ Ωd(t)e
ikdzj
0 Ω∗d(t)e
−ikdzj δR
j
with E(z, t) =P
q aq(t)eiqz and g =℘ge
~
q
~ω2ε0V
δR = 0 ⇒ Dark states (H |Dq1〉 = 0 |Dq
1〉) for single-photons |1q〉 = aq† |0〉|Dq
1〉 = cos θ |1q, s(0)〉 − sin θ |0q, s(1)〉 tan2 θ(t) = g2N|Ωd(t)|2
with collective atomic states|s(0)〉 ≡ |g1, g2, . . . , gN 〉|s(1)〉 ≡ 1√
N
∑Nj=1 e
i(k+q−kd)zj |g1, . . . , sj , . . . , gN 〉
Fleischhauer, Lukin, PRL 84, 5094 (2000); PRA 65, 022314 (2002)
θ = 0 (|Ωd|2 g2N ) ⇒ |Dq1〉 = |1q〉 |s(0)〉 purely photonic excitation
θ = π/2 (|Ωd|2 g2N ) ⇒ |Dq1〉 = |0q〉 |s(1)〉 purely atomic excitation
QUDAL, 1/3/06 – p. 6/16
IESLFORTHElectromagnetically Induced Transparency
Γ
∆
δRs
g
e
dΩE
H = ~
N∑
j=1
0 gE†(zj)e−ikzj 0
gE(zj)eikzj ∆ Ωd(t)e
ikdzj
0 Ω∗d(t)e
−ikdzj δR
j
with E(z, t) =P
q aq(t)eiqz and g =℘ge
~
q
~ω2ε0V
δR = 0 ⇒ Dark states (H |Dq1〉 = 0 |Dq
1〉) for single-photons |1q〉 = aq† |0〉|Dq
1〉 = cos θ |1q, s(0)〉 − sin θ |0q, s(1)〉 tan2 θ(t) = g2N|Ωd(t)|2
with collective atomic states|s(0)〉 ≡ |g1, g2, . . . , gN 〉|s(1)〉 ≡ 1√
N
∑Nj=1 e
i(k+q−kd)zj |g1, . . . , sj , . . . , gN 〉
Fleischhauer, Lukin, PRL 84, 5094 (2000); PRA 65, 022314 (2002)
θ = 0 (|Ωd|2 g2N ) ⇒ |Dq1〉 = |1q〉 |s(0)〉 purely photonic excitation
θ = π/2 (|Ωd|2 g2N ) ⇒ |Dq1〉 = |0q〉 |s(1)〉 purely atomic excitation
QUDAL, 1/3/06 – p. 6/16
IESLFORTHDark-State Polariton
Define operator Ψ(z, t) = cos θ(t)E(z, t) − sin θ(t)√Nσgs(z, t)
with σgs(z, t) = 1Nz
PNzj=1 |gj〉〈sj | , Nz = N
Ldz 1
Ψ(z, t) =∑
q ψq(t)eiqz ⇒ |Dq
1〉 = ψq† |0q〉 |s(0)〉
Fleischhauer, Lukin, PRL 84, 5094 (2000); PRA 65, 022314 (2002)
Equation of motion(
∂
∂t+ vg(t)
∂
∂z
)
Ψ(z, t) = 0 ⇒ Ψ(z, t) = Ψ
(
z −∫ t
0
vg(t′)dt′, 0
)
vg(t) = c cos2 θ(t) = c |Ωd(t)|2g2N+|Ωd(t)|2 group velocity (time-dependent)
⇒ One can decelerate/accelerate the propagation of Ψ(z, t)
QUDAL, 1/3/06 – p. 7/16
IESLFORTHDark-State Polariton
Define operator Ψ(z, t) = cos θ(t)E(z, t) − sin θ(t)√Nσgs(z, t)
with σgs(z, t) = 1Nz
PNzj=1 |gj〉〈sj | , Nz = N
Ldz 1
Ψ(z, t) =∑
q ψq(t)eiqz ⇒ |Dq
1〉 = ψq† |0q〉 |s(0)〉
Fleischhauer, Lukin, PRL 84, 5094 (2000); PRA 65, 022314 (2002)
Equation of motion(
∂
∂t+ vg(t)
∂
∂z
)
Ψ(z, t) = 0 ⇒ Ψ(z, t) = Ψ
(
z −∫ t
0
vg(t′)dt′, 0
)
vg(t) = c cos2 θ(t) = c |Ωd(t)|2g2N+|Ωd(t)|2 group velocity (time-dependent)
⇒ One can decelerate/accelerate the propagation of Ψ(z, t)
QUDAL, 1/3/06 – p. 7/16
IESLFORTHStopping of Light
Ωd
gvE
E
c Ψ
Single-photon WP: |1〉 =∑
q ξq |1q〉 = 1
L
∫
dzf(z)E†(z) |0〉
i) At the entrance 0 < θ(0) . π/2 (0 < Ωd(t) g2N )
⇒ Pulse is spatially compressed by vg(0)c
= cos2 θ(0) 1
ii) Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Pulse is stopped vg(t) = 0
and stored as atomic excitation |Dq1〉 = |0q〉 |s(1)〉
iii) Rotate θ(t′) < π/2 (Ωd(t′) > 0)
⇒ Pulse is released vg(t′) > 0
Requires Tvg(0) < L & T−1 < δωtw ⇒ large optical depth 2κ0L 1
QUDAL, 1/3/06 – p. 8/16
IESLFORTHStopping of Light
Ωd
gvE
E
c Ψ
Single-photon WP: |1〉 =∑
q ξq |1q〉 = 1
L
∫
dzf(z)E†(z) |0〉
i) At the entrance 0 < θ(0) . π/2 (0 < Ωd(t) g2N )
⇒ Pulse is spatially compressed by vg(0)c
= cos2 θ(0) 1
ii) Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Pulse is stopped vg(t) = 0
and stored as atomic excitation |Dq1〉 = |0q〉 |s(1)〉
iii) Rotate θ(t′) < π/2 (Ωd(t′) > 0)
⇒ Pulse is released vg(t′) > 0
Requires Tvg(0) < L & T−1 < δωtw ⇒ large optical depth 2κ0L 1
QUDAL, 1/3/06 – p. 8/16
IESLFORTHStopping of Light
Ωd
gvE
E
c Ψ
Single-photon WP: |1〉 =∑
q ξq |1q〉 = 1
L
∫
dzf(z)E†(z) |0〉
i) At the entrance 0 < θ(0) . π/2 (0 < Ωd(t) g2N )
⇒ Pulse is spatially compressed by vg(0)c
= cos2 θ(0) 1
ii) Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Pulse is stopped vg(t) = 0
and stored as atomic excitation |Dq1〉 = |0q〉 |s(1)〉
iii) Rotate θ(t′) < π/2 (Ωd(t′) > 0)
⇒ Pulse is released vg(t′) > 0
Requires Tvg(0) < L & T−1 < δωtw ⇒ large optical depth 2κ0L 1
QUDAL, 1/3/06 – p. 8/16
IESLFORTHStopping of Light
Ωd
gvE
E
c Ψ
Single-photon WP: |1〉 =∑
q ξq |1q〉 = 1
L
∫
dzf(z)E†(z) |0〉
i) At the entrance 0 < θ(0) . π/2 (0 < Ωd(t) g2N )
⇒ Pulse is spatially compressed by vg(0)c
= cos2 θ(0) 1
ii) Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Pulse is stopped vg(t) = 0
and stored as atomic excitation |Dq1〉 = |0q〉 |s(1)〉
iii) Rotate θ(t′) < π/2 (Ωd(t′) > 0)
⇒ Pulse is released vg(t′) > 0
Requires Tvg(0) < L & T−1 < δωtw ⇒ large optical depth 2κ0L 1
QUDAL, 1/3/06 – p. 8/16
IESLFORTHPhotonic Memory
Ωd
dΩ
45
λ/4
ER
EL
E
g
Ωd
1
1
2
2s
e e
s
REL E
λ/4 plate oriented at 45
|V 〉 → |R〉 = 1√2( |V 〉 + i |H〉) |H〉 → |L〉 = 1√
2( |V 〉 − i |H〉)
⇒ |ψ(0)〉 → α |R〉 + β |L〉
Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Photon is stopped and stored as superposition of |s(1)1 〉 & |s(1)1 〉
|ψ(t)〉 = α |s(1)1 〉 + β |s(1)2 〉 With low decoherence!
Rotate θ(t′) < π/2 (Ωd(t′) > 0)
⇒ Photon is released |ψ(t′)〉 → α |R〉 + β |L〉
Fleischhauer, Mewes (2002); Mewes, Fleischhauer, PRA 72, 022327 (2005) QUDAL, 1/3/06 – p. 9/16
IESLFORTHPhotonic Memory
Ωd
dΩ
45
λ/4
ER
EL
E
g
Ωd
1
1
2
2s
e e
s
REL E
λ/4 plate oriented at 45
|V 〉 → |R〉 = 1√2( |V 〉 + i |H〉) |H〉 → |L〉 = 1√
2( |V 〉 − i |H〉)
⇒ |ψ(0)〉 → α |R〉 + β |L〉Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Photon is stopped and stored as superposition of |s(1)1 〉 & |s(1)1 〉
|ψ(t)〉 = α |s(1)1 〉 + β |s(1)2 〉 With low decoherence!
Rotate θ(t′) < π/2 (Ωd(t′) > 0)
⇒ Photon is released |ψ(t′)〉 → α |R〉 + β |L〉
Fleischhauer, Mewes (2002); Mewes, Fleischhauer, PRA 72, 022327 (2005) QUDAL, 1/3/06 – p. 9/16
IESLFORTHPhotonic Memory
Ωd
dΩ
45
λ/4
ER
EL
E
g
Ωd
1
1
2
2s
e e
s
REL E
λ/4 plate oriented at 45
|V 〉 → |R〉 = 1√2( |V 〉 + i |H〉) |H〉 → |L〉 = 1√
2( |V 〉 − i |H〉)
⇒ |ψ(0)〉 → α |R〉 + β |L〉Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Photon is stopped and stored as superposition of |s(1)1 〉 & |s(1)1 〉
|ψ(t)〉 = α |s(1)1 〉 + β |s(1)2 〉 With low decoherence!
Rotate θ(t′) < π/2 (Ωd(t′) > 0)
⇒ Photon is released |ψ(t′)〉 → α |R〉 + β |L〉
Fleischhauer, Mewes (2002); Mewes, Fleischhauer, PRA 72, 022327 (2005) QUDAL, 1/3/06 – p. 9/16
IESLFORTHSingle-Photon Sources
Parametric Down-Conversions
i
pump NLC V
HD
Pairs of P&M entangled photons|Φ〉 = 1√
2( |V 〉s |H〉i + |H〉s |V 〉i)
Detect idler in |H〉i ⇒ project signal onto |V 〉i
Cavity QED
pΩκ
1Vg
Ωpg
Γe
g
Two-Level System (atom, QD ...) with Γ < g < κ
Apply ΩpT = π pulse ⇒ single photon is emitted(Purcell effect)
Khitrova et al, Nature Physics 2, 81 (2006)
g
e
Ωp
s
gΓ
Three-Level System with g > κ,Γ
|D〉 = cos θ |g, 0〉 − sin θ |s, 1〉 tan θ =Ωp
g
Apply Ωp adiabatic pulse ⇒ single photon is emitted(intracavity STIRAP)
Kuhn, Hennrich, Rempe, PRL 89, 067901 (2002); McKeever et al, Science 303, 1992 (2004)
Requires high-Q cavities
QUDAL, 1/3/06 – p. 10/16
IESLFORTHSingle-Photon Sources
Cavity QED
pΩκ
1Vg
Ωpg
Γe
g
Two-Level System (atom, QD ...) with Γ < g < κ
Apply ΩpT = π pulse ⇒ single photon is emitted(Purcell effect)
Khitrova et al, Nature Physics 2, 81 (2006)
g
e
Ωp
s
gΓ
Three-Level System with g > κ,Γ
|D〉 = cos θ |g, 0〉 − sin θ |s, 1〉 tan θ =Ωp
g
Apply Ωp adiabatic pulse ⇒ single photon is emitted(intracavity STIRAP)
Kuhn, Hennrich, Rempe, PRL 89, 067901 (2002); McKeever et al, Science 303, 1992 (2004)
Requires high-Q cavities
QUDAL, 1/3/06 – p. 10/16
IESLFORTHSingle-Photon Sources
Cavity QED
pΩκ
1Vg
Ωpg
Γe
g
Two-Level System (atom, QD ...) with Γ < g < κ
Apply ΩpT = π pulse ⇒ single photon is emitted(Purcell effect)
Khitrova et al, Nature Physics 2, 81 (2006)
g
e
Ωp
s
gΓ
Three-Level System with g > κ,Γ
|D〉 = cos θ |g, 0〉 − sin θ |s, 1〉 tan θ =Ωp
g
Apply Ωp adiabatic pulse ⇒ single photon is emitted(intracavity STIRAP)
Kuhn, Hennrich, Rempe, PRL 89, 067901 (2002); McKeever et al, Science 303, 1992 (2004)
Requires high-Q cavitiesQUDAL, 1/3/06 – p. 10/16
IESLFORTHEIT Based Single-Photon Sources
s
e
g
?
i) Create symmetric spin (Raman) excitation state
|s(1)〉 = 1√N
∑Nj=1 e
iδkzj |g1, . . . , sj , . . . , gN 〉
⇒ |D1(0)〉 = |0〉 |s(1)〉
QUDAL, 1/3/06 – p. 11/16
IESLFORTHEIT Based Single-Photon Sources
E
s
e
g
dΩ
i) Create symmetric spin (Raman) excitation state
|s(1)〉 = 1√N
∑Nj=1 e
iδkzj |g1, . . . , sj , . . . , gN 〉
⇒ |D1(0)〉 = |0〉 |s(1)〉ii) Apply Ωd 6= 0 (θ < π/2) & Release SPh WP⇒ |D1(t)〉 → |1〉 |s(0)〉
QUDAL, 1/3/06 – p. 11/16
IESLFORTHEIT Based Single-Photon Sources
E
s
e
g
dΩ
i) Create symmetric spin (Raman) excitation state
|s(1)〉 = 1√N
∑Nj=1 e
iδkzj |g1, . . . , sj , . . . , gN 〉
⇒ |D1(0)〉 = |0〉 |s(1)〉ii) Apply Ωd 6= 0 (θ < π/2) & Release SPh WP⇒ |D1(t)〉 → |1〉 |s(0)〉
wΩ
s
e
g
D
Spontaneous Raman Scatteringi) Apply write pulse Ωw & detectforward scattered Stokes photon
⇒ Click of D corresponds to |s(1)〉(with δk = kw − ks)
Go to ii)
Duan et al, Nature 414, 413 (2001); Kuzmich et al, Nature 423, 731 (2003);van der Wal et al, Science 301, 196 (2003); Chou et al, PRL 92, 213601 (2004);Eisaman et al, PRL 93, 233602 (2004); Eisaman et al, Nature 438, 837 (2005) QUDAL, 1/3/06 – p. 11/16
IESLFORTHEIT Based Single-Photon Sources
E
s
e
g
dΩ
i) Create symmetric spin (Raman) excitation state
|s(1)〉 = 1√N
∑Nj=1 e
iδkzj |g1, . . . , sj , . . . , gN 〉
⇒ |D1(0)〉 = |0〉 |s(1)〉ii) Apply Ωd 6= 0 (θ < π/2) & Release SPh WP⇒ |D1(t)〉 → |1〉 |s(0)〉
Ω rΩ r
i jd
g gi j
VDDd
Dipole BlockadePair of atoms i, j in Rydberg states |d〉⇒ Dipole-Dipole Interaction (anisotropic)
VDD = ~∆(ri − rj) |di dj〉〈di dj |Resonant DDI (Föster process) + Static DDI (in dc E-field)
∆(ri − rj) ≈ − n4e2a20
π~ε0|ri−rj |3
Double-excitation is nonresonantPdouble ∼ |Ωr|2
∆2 1 if Ωr < ∆
QUDAL, 1/3/06 – p. 11/16
IESLFORTHEIT Based Single-Photon Sources
E
s
e
g
dΩ
i) Create symmetric spin (Raman) excitation state
|s(1)〉 = 1√N
∑Nj=1 e
iδkzj |g1, . . . , sj , . . . , gN 〉
⇒ |D1(0)〉 = |0〉 |s(1)〉ii) Apply Ωd 6= 0 (θ < π/2) & Release SPh WP⇒ |D1(t)〉 → |1〉 |s(0)〉
sg
e
Ωr Ωr(1)
(2)
d
Dipole Blockade
Atomic ensemble VDD = ~∑N
ij ∆(ri − rj) |ri rj〉〈ri rj |i) Apply Ω
(1)r for
√NΩ
(1)r T1 = π (collective π pulse)
⇒ |s(0)〉 ≡ |g1, g2, . . . , gN 〉 →→ 1√
N
∑
j eik(1)
r zj |g1, . . . , dj , . . . , gN 〉 ≡ |d(1)〉Single collective Rydberg excitation (
√NΩ
(1)r < ∆)
i’) Apply Ω(2)r T2 = π ⇒ |d(1)〉 → |s(1)〉 (with δk = k
(1)r k
(2)r )
Go to ii)
Lukin et al, PRL 87, 037901 (2001) QUDAL, 1/3/06 – p. 11/16
IESLFORTHEIT: Spectral Properties
Ωd
E1
∆e
δRs
Γ E
g
Ωd
1
1
−4 −2 0 2 4Detuning δR/γge
−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)
E1(z, t) = E1
(
0, t− zvg
)
eiαz = E1(0, τ)e−κz+iφ vg = c
1+cκ0γge
|Ωd|2
' |Ωd|2κ0γge
c
Medium Polarizability α(∆1) = κ0iγge
γge−i∆1+|Ωd|2
γR−iδR
⇒ Absorption κ = 1vg
[
γR +δ2R
|Ωd|2]
κ0 Phase shift φ(z) = δR
vgz
EIT conditions: |Ωd|2 (γge + |∆1|)(γR + δR)
QUDAL, 1/3/06 – p. 12/16
IESLFORTHEIT: Spectral Properties
∆e
1
E1E2
δRs
Γ
g
Ωd
∆f
2
Ωd
E2
E1
−4 −2 0 2 4Detuning δR/γgs
−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)
E1(z, t) = E1
(
0, t− zvg
)
eiαz = E1(0, τ)e−κz+iφ
α(∆1) = κ0iγge
γge−i∆1+|Ωd|2
γR−iδR+iS
with S ' |Ω2|2∆2
– ac Stark shift of |s〉
δR = 0 ⇒ φ(z) = Svgz = |Ω2|2
∆2vgz – XPM; κz
φ(z) ' γfs
∆2 1 – Small XA
(requires |∆2 ± k2v| γfs ⇒ cold atoms)
Schmidt, Imamoglu, Opt. Lett. 21, 1936 (1996) QUDAL, 1/3/06 – p. 12/16
IESLFORTHCross-Phase Modulation
g(2)v
1
2E
Evg
(1)
v(1)g = |Ωd|2
κ0γge c
but
v(2)g =
[
1c− κ0
|Ω1|2|Ωd|2
γfs
∆22
]−1
' c
⇒ Group velocity mismatch limits interaction time/length (maxφ ∼ 0.1π)
Harris, Hau, PRL 82, 4611 (1999)
Local interaction
⇒ Strong focusing (w ∼ σ0 ∼ λ2) of E1,2 for 0 ≤ z ≤ L
⇒ NL phase-shift is not uniform (spectral broadening)
QUDAL, 1/3/06 – p. 13/16
IESLFORTHCross-Phase Modulation
∆b
dΩ
2
(B)
s
g
e
E
Atoms B
g1 g2
E
1
2
E
g(2)v
vg(1)
dΩ
E1
(A)
g
s
e
f
E2
e
dΩ
∆d
E12E
s
Atoms A
Lukin, Imamoglu, PRL 84, 1419 (2000) Petrosyan, JOB 7, S141 (2005)
Group velocities can be matched v(1)g ' v
(2)g
⇒ φ = π of SPh pulses possible
|1〉 |1〉 CZ−→ − |1〉 |1〉
Harris, Yamamoto, PRL 81 3611 (1998); Petrosyan, Kurizki, PRA 65, 033833 (2002);Ottaviani et al., PRL 90 197902 (2003); Masalas, Fleischhauer, PRA 69, 061801 (2004);Friedler et al., PRA 71, 023803 (2005); Andre et al., PRL 94, 063902 (2005)
Local interaction
⇒ Strong focusing (w ∼ σ0 ∼ λ2) of E1,2 for 0 ≤ z ≤ L
⇒ NL phase-shift is not uniform (spectral broadening)
QUDAL, 1/3/06 – p. 13/16
IESLFORTHCross-Phase Modulation
∆b
dΩ
2
(B)
s
g
e
E
Atoms B
g1 g2
E
1
2
E
g(2)v
vg(1)
dΩ
E1
(A)
g
s
e
f
E2
e
dΩ
∆d
E12E
s
Atoms A
Lukin, Imamoglu, PRL 84, 1419 (2000) Petrosyan, JOB 7, S141 (2005)
Group velocities can be matched v(1)g ' v
(2)g
⇒ φ = π of SPh pulses possible
|1〉 |1〉 CZ−→ − |1〉 |1〉
Harris, Yamamoto, PRL 81 3611 (1998); Petrosyan, Kurizki, PRA 65, 033833 (2002);Ottaviani et al., PRL 90 197902 (2003); Masalas, Fleischhauer, PRA 69, 061801 (2004);Friedler et al., PRA 71, 023803 (2005); Andre et al., PRL 94, 063902 (2005)
Local interaction
⇒ Strong focusing (w ∼ σ0 ∼ λ2) of E1,2 for 0 ≤ z ≤ L
⇒ NL phase-shift is not uniform (spectral broadening)
QUDAL, 1/3/06 – p. 13/16
IESLFORTHCross-Phase Modulation via Static DDI
2E
|E | 12 |E
|2 22EE1
E1
g
dd1 2
e e1 2
Ω 2
V
Ω1
DD Est
w
w
Ψ Ψ1 2v vg g
Static Estez ⇒ Rydberg states |di〉 have large permanent dipole moments℘dez = 3
2nqea0ez (Stark eigenstates)
Ei → Ψi = cos θEi − sin θ√Nσgdi
(i = 1, 2) propagate with ±vg = c cos2 θ
Atomic components of Ψi interact via Static DDI ⇒ induces XPM
VDD = ~ρ2
∫∫
d3r d3r′σdd(r)∆(r− r′)σdd(r
′)
∆(r − r′) = C
1 − 3 cos2 ϑ
|r− r′|3 with C =
℘dl℘dl′
4πε0~
Resonant DDI (state mixing) is suppressed for q = n − 1, m = 0
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 θv
R L0 dz′∆(2z′ − L) = 2C
vw2
⇒ φ = 2Cvgw
= π of SPh pulses possible |1〉 |1〉 CZ−→ − |1〉 |1〉
Advantages
Weak focusing (w ∼ 30µm) of E1,2 for 0 ≤ z ≤ L
NL (Collisional) phase-shift is uniform
Friedler, Petrosyan, Fleischhauer, Kurizki, PRA 72, 043803 (2005)
QUDAL, 1/3/06 – p. 14/16
IESLFORTHCross-Phase Modulation via Static DDI
0 L/wτ
0
1
φ(τ)
−L/w 0 L/wζ
−2
−1
0
∆(ζ)
• DD level shift∆(z − z′) = 1
πw2
R 2π0 dϕ′R ∞
0 dr′⊥r′⊥e−r′2⊥/w2
∆(zez − r′)
• Phase shiftφ(z1, z2, t) = − sin4 θ
R t0dt′∆(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 θv
R L0 dz′∆(2z′ − L) = 2C
vw2
⇒ φ = 2Cvgw
= π of SPh pulses possible |1〉 |1〉 CZ−→ − |1〉 |1〉
Advantages
Weak focusing (w ∼ 30µm) of E1,2 for 0 ≤ z ≤ L
NL (Collisional) phase-shift is uniform
Friedler, Petrosyan, Fleischhauer, Kurizki, PRA 72, 043803 (2005)
QUDAL, 1/3/06 – p. 14/16
IESLFORTHCross-Phase Modulation via Static DDI
0 L/wτ
0
1
φ(τ)
−L/w 0 L/wζ
−2
−1
0
∆(ζ)
• DD level shift∆(z − z′) = 1
πw2
R 2π0 dϕ′R ∞
0 dr′⊥r′⊥e−r′2⊥/w2
∆(zez − r′)
• Phase shiftφ(z1, z2, t) = − sin4 θ
R t0dt′∆(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 θv
R L0 dz′∆(2z′ − L) = 2C
vw2
⇒ φ = 2Cvgw
= π of SPh pulses possible |1〉 |1〉 CZ−→ − |1〉 |1〉
Advantages
Weak focusing (w ∼ 30µm) of E1,2 for 0 ≤ z ≤ L
NL (Collisional) phase-shift is uniform
Friedler, Petrosyan, Fleischhauer, Kurizki, PRA 72, 043803 (2005)
QUDAL, 1/3/06 – p. 14/16
IESLFORTHCross-Phase Modulation via Static DDI
0 L/wτ
0
1
φ(τ)
−L/w 0 L/wζ
−2
−1
0
∆(ζ)
• DD level shift∆(z − z′) = 1
πw2
R 2π0 dϕ′R ∞
0 dr′⊥r′⊥e−r′2⊥/w2
∆(zez − r′)
• Phase shiftφ(z1, z2, t) = − sin4 θ
R t0dt′∆(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 θv
R L0 dz′∆(2z′ − L) = 2C
vw2
⇒ φ = 2Cvgw
= π of SPh pulses possible |1〉 |1〉 CZ−→ − |1〉 |1〉
Advantages
Weak focusing (w ∼ 30µm) of E1,2 for 0 ≤ z ≤ L
NL (Collisional) phase-shift is uniform
Friedler, Petrosyan, Fleischhauer, Kurizki, PRA 72, 043803 (2005) QUDAL, 1/3/06 – p. 14/16
IESLFORTHSingle Photon Detection
V
Hψ
DPBS
D
Single photon WP |ψ〉 = α |V 〉 + β |H〉passes through PBS⇒ |V 〉 and |H〉 polarization components
are directed into Photodetectors D
⇓Qubit Measurement Requires High-Efficiency Photodetectors
Avalanche Photodetectors — quantum efficiency η . 70%
EIT based Photodetection — quantum efficiency η → 100%f
s
ΩpdΩ
e
g
ΓfE
Ωd
Ωp
E
DDi) Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Pulse is stopped|1q〉 |s(0)〉 → |0q〉 |s(1)〉
ii) Apply pump Ωp tocycling transition |s〉 → |f〉 ⇒ Rf ' 1
2Γf
⇒ Collect fluor. with D (η < 1) for time T : Sf ' 12ηΓfT 1
Imamoglu, PRL 89, 163602 (2002); James, Kwiat, PRL 89, 183601 (2002)
QUDAL, 1/3/06 – p. 15/16
IESLFORTHSingle Photon Detection
V
Hψ
DPBS
D
Single photon WP |ψ〉 = α |V 〉 + β |H〉passes through PBS⇒ |V 〉 and |H〉 polarization components
are directed into Photodetectors D
⇓Qubit Measurement Requires High-Efficiency Photodetectors
Avalanche Photodetectors — quantum efficiency η . 70%
EIT based Photodetection — quantum efficiency η → 100%f
s
ΩpdΩ
e
g
ΓfE
Ωd
Ωp
E
DDi) Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Pulse is stopped|1q〉 |s(0)〉 → |0q〉 |s(1)〉
ii) Apply pump Ωp tocycling transition |s〉 → |f〉 ⇒ Rf ' 1
2Γf
⇒ Collect fluor. with D (η < 1) for time T : Sf ' 12ηΓfT 1
Imamoglu, PRL 89, 163602 (2002); James, Kwiat, PRL 89, 183601 (2002)
QUDAL, 1/3/06 – p. 15/16
IESLFORTHSingle Photon Detection
V
Hψ
DPBS
D
Single photon WP |ψ〉 = α |V 〉 + β |H〉passes through PBS⇒ |V 〉 and |H〉 polarization components
are directed into Photodetectors D
⇓Qubit Measurement Requires High-Efficiency Photodetectors
Avalanche Photodetectors — quantum efficiency η . 70%
EIT based Photodetection — quantum efficiency η → 100%f
s
ΩpdΩ
e
g
ΓfE
Ωd
Ωp
E
DD
i) Rotate θ(t) → π/2 (Ωd(t) → 0)
⇒ Pulse is stopped|1q〉 |s(0)〉 → |0q〉 |s(1)〉
ii) Apply pump Ωp tocycling transition |s〉 → |f〉 ⇒ Rf ' 1
2Γf
⇒ Collect fluor. with D (η < 1) for time T : Sf ' 12ηΓfT 1
Imamoglu, PRL 89, 163602 (2002); James, Kwiat, PRL 89, 183601 (2002) QUDAL, 1/3/06 – p. 15/16
IESLFORTHSummary
Scalable and efficient quantum computation with photonic qubits requires:
Deterministic sources of single-photons
Reversible photon storage devise
Giant nonlinearities (XPM) to entangle pairs of photons
Reliable single-photon detectors.
EIT based (or related) techniques can implement these requirements⇒ Deterministic all-optical quantum computation & communicationmay become possible
Various sources of decoherence, such as Doppler, time-of-flight &collisional broadening, etc, have to be carefully studied and eliminated.
QUDAL, 1/3/06 – p. 16/16
IESLFORTHSummary
Scalable and efficient quantum computation with photonic qubits requires:
Deterministic sources of single-photons
Reversible photon storage devise
Giant nonlinearities (XPM) to entangle pairs of photons
Reliable single-photon detectors.
EIT based (or related) techniques can implement these requirements⇒ Deterministic all-optical quantum computation & communicationmay become possible
Various sources of decoherence, such as Doppler, time-of-flight &collisional broadening, etc, have to be carefully studied and eliminated.
QUDAL, 1/3/06 – p. 16/16
IESLFORTHSummary
Scalable and efficient quantum computation with photonic qubits requires:
Deterministic sources of single-photons
Reversible photon storage devise
Giant nonlinearities (XPM) to entangle pairs of photons
Reliable single-photon detectors.
EIT based (or related) techniques can implement these requirements⇒ Deterministic all-optical quantum computation & communicationmay become possible
Various sources of decoherence, such as Doppler, time-of-flight &collisional broadening, etc, have to be carefully studied and eliminated.
QUDAL, 1/3/06 – p. 16/16