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



Photonic Memory


Cross-Phase Modulation

Single Photon Detection

Summary

QUDAL, 1/3/06 – p. 2/16

IESLFORTHOutline



Photonic Memory


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

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

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

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


Cavity QED

pΩκ

1Vg

Ωpg

Γe

g




g

e

Ωp

s

gΓ



g



Requires high-Q cavities

QUDAL, 1/3/06 – p. 10/16


Cavity QED

pΩκ

1Vg

Ωpg

Γe

g




g

e

Ωp

s

gΓ



g



Requires high-Q cavitiesQUDAL, 1/3/06 – p. 10/16


E

s

e

g

dΩ


|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


E

s

e

g

dΩ


|s(1)〉 = 1√N

∑Nj=1 e

iδkzj |g1, . . . , sj , . . . , gN 〉


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


E

s

e

g

dΩ


|s(1)〉 = 1√N

∑Nj=1 e

iδkzj |g1, . . . , sj , . . . , gN 〉


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 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


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′))



φ(L, 0, L/v) = − sin4 θv

R L0 dz′∆(2z′ − L) = 2C

vw2

⇒ φ = 2Cvgw


Advantages



Friedler, Petrosyan, Fleischhauer, Kurizki, PRA 72, 043803 (2005)

QUDAL, 1/3/06 – p. 14/16


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′))



φ(L, 0, L/v) = − sin4 θv

R L0 dz′∆(2z′ − L) = 2C

vw2

⇒ φ = 2Cvgw


Advantages



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

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

deterministic quantum computation with photonic · pdf fileiesl forth deterministic quantum...

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