Pure-state, single-photon wave-packet generation by parametric down conversion in a distributed...

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Pure-state, single-photon wave-packet generation by parametric down conversion in a distributed

microcavityM. G. Raymer, Jaewoo Noh*

Oregon Center for Optics, University of Oregon

--------------------------------------

I.A. Walmsley, K. Banaszek, Oxford Univ.-----------------------------------------------------------------

* Inha University, Inchon, Korea-----------------------------------------------------------------

ITR - [email protected]



Single-Photon Wave-Packet

1

€

1 = dω ψ∫ (ω) 1 ω

Wave-Packet is a Superposition-state:

(like a one-exciton state)



Interference behavior of Single-Photon Wave-Packets

At a 50/50 beamsplitter a photon transmits or reflects with 50% probabilities.

1

1

€

1 = dω ψ∫ (ω) 1 ω


0

beam splitter



Interference behavior of Single-Photon Wave-Packets

At a 50/50 beamsplitter a photon transmits or reflects with 50% probabilities.

1

0

€

1 = dω ψ∫ (ω) 1 ω


1

beam splitter



Single-Photon, Pure Wave-Packet States Interfere as Boson particles

1

1beam splitter

2

0



Single-Photon, Pure Wave-Packet States Interfere as Boson particles

1

1beam splitter

0

2



Spontaneous Parametric Down Conversion in a second-order nonlinear, birefringent crystal

Phase-matching (momentum conservation):

rkS +

rkI =

rkP ±π / L

rkS

rkI

rkP

pumpSignal V-Pol

Idler H-Pol

Energy conservation:ωS +ωI =ωP

red red blue

L

kz

frequency

P

VH

H-Pol

phase-matching bandwidth



Correlated Photon-Pair Generation by Spontaneous Down Conversion (Hong and Mandel, 1986)

0 or 1

Monochromatic Blue Light

Red photon pairs

2nd-order Nonlinear optical crystal

ω+ω '=ωP

ωP

ω '

ω

0 or 1

Ψ 2P = dω C∫ (ω) 1SIGNAL ω 1IDLER ωP −ω

IDLER

SIGNAL

• Creation time is uncontrolled

• Correlation time ~ (bandwidth)-1

Perfect correlation of photon frequencies:

optional



1

1

Correlated Photon-Pair Measurement (Hong, Ou, Mandel, 1987)

Time difference

2 or 0

0 or 2

Red photons

Nonlinear optical crystal

Time difference

Coincidence Rate

Correlation time ~ (bandwidth)-1

Creation time uncontrolled

MC Blue light

boson behavior



1

1

trigger if n = 1

filter

Pulsed blue light

For Quantum Information Processing we need

pulsed, pure-state single-photon sources.

Create using Spontaneous Down Conversion and conditional detection:

shutter

nonlinear optical crystal

(Knill, LaFlamme, Milburn, Nature, 2001)



1

trigger if n = 1

filter

Pulsed blue light

For Quantum Information Processing we need

pulsed, pure-state single-photon sources.

Create using Spontaneous Down Conversion and conditional detection:

shutter

nonlinear optical crystal

SIGNAL

(Knill, LaFlamme, Milburn, Nature, 2001)



trigger

Pulsed Pump Spectrum has nonzero bandwidth

Ψ2P

= dω '∫ dω C∫ (ω,ω ') 1IDLER ω ' 1SIGNAL ω

Zero-Bandwidth Filter , 0

ωP

ω'

ω1IDLER '

1SIGNAL

→ dω∫ C(ω0 ,ω ) 1SIGNAL ωdetect signal

Pure-state creation at cost of vanishing data rate



1

1

1

trigger

trigger

1Time difference

Coincidence Counts

Do single photons from independent SpDC sources interfere well? Need good time and frequency correlation.

large data rate

filter

Pulsed blue light

filter

random delay

vanishing data rate



Goal : Generation of Pure-State Photon Pairs without using Filtering

Single-photon Wave-Packet States:

1 S0 = dω ψ 0∫ (ω) 1 Sω

1 I0 = dω φ0∫ (ω) 1 I ω

signal

idler

Ψ2P

= dω '∫ dω C∫ (ω,ω ') 1IDLER ω ' 1SIGNAL ω

= 1I 0⊗ 1

S 0

€

C(ω,ω ') = ψ 0 (ω) × φ0 (ω ')Want :

(no entanglement)



Decomposition of field into Discrete Wave-Packet Modes. (Law, Walmsley, Eberly, PRL, 2000)

Ψ = vac + dω'∫ dω C∫ (ω,ω' ) 1 Sω 1 Iω'

Ψ = vac + λ jj

∑ 1 Sj ⊗ 1 I j

1 S j = dω ψ j∫ (ω) 1 Sω

1 I j = dω φ j∫ (ω) 1 Iω

Single-photon Wave-Packet States:

(Schmidt Decomposition)



The Schmidt Wave-Packet Modes are perfectly correlated.

But typically it is difficult to measure, or separate, the Schmidt Modes.

Ψ = vac + λ jj

∑ 1 Sj ⊗ 1 I j

Mode Amplitude Functions: Mode spectra overlap.

No perfect filters exist, in time and/or frequency.

frequency

€

CS1(ω)

€

CS 2(ω)

€

CS 3(ω)

filter

1 S j = dω ψ j∫ (ω) 1 Sω 1 I j = dω φ j∫ (ω) 1 Iω



Why does the state generally NOT factor?

€

Ψ2P

= dω'∫ dω C∫ (ω,ω ') 1SIGNAL ω 1IDLER ω '

≠ 1S 0⊗ 1

I 0

€

€

'

€

C(ω,ω ')

need to engineer the state to make it factor

Energy conservation and phase matching typically lead to frequency correlation



Spontaneous Parametric Down Conversion inside a Single-Transverse-Mode

Optical Cavity

rkS

rkI

rkP

pump

Nonlinear optical crystal with wave-guide

1 mm

DOES NOT WORK

the problem:cavity FSR ~ 1/Lphase-matching BW ~ 10/L



Spontaneous Parametric Down Conversion inside a Distributed-Feedback Cavity

4 mm

0.2 mm cavity

second-order nonlinear-optical crystal

pump

H-PolH-Pol idler

V-Pol signal

4 mm

Linear-index Distributed-Bragg Reflectors (DBR)

Linear-index wave-guide

• large FSR = c /(2x0.2 mm)• small phase-matching BW: ~ 10 c /(4 mm)



0 =800 nm KG = 25206/mmn/n ~ 6x10-4 (k = 2/mm)

4 mm

DBR 99% mirror

SIMPLIFIED MODEL: Half-DBR Cavity

Reflectivity

frequency/1015

DBR band gap

0.2 mm cavity

cavity mode



Quantum Generation in a Dielectric-Structured Cavity: Phenomenological Treatment

∂x2E(x,t) −∂t

2D(x,t) = J(x,t) = χ NL Ep (x,t)E *(x, t)Signal Source Pump

Frequency Domain:

∂x2 + ε (x,ω ) ω 2⎡⎣ ⎤⎦ %E(x,ω ) = %J(x,ω )

∂x2 + ε (x,ω )ω 2⎡⎣ ⎤⎦u(x,ω ) = 0 (modes)

space and frequency dependent electric permeability:

ε(x,ω ) = ε (x)n2 (ω )



€

x

internal Signal, Idler modes

pump fieldinteraction

€

˜ E S (x,ω)

€

˜ E S (x,ω) = ˜ E VAC (x,ω) + uout(x,ω) dω'∫ C(ω,ω') ˆ a I† (ω')

0 L

€

C(ω,ω') = χ NL dx'0

L∫ ˜ E p (x ' ,ω +ω') uS *(x ' ,ω)uI * (x' ,ω')

€

˜ E p

two-photon amplitude



Heisenberg Picture Schrodinger Picture


€

C(ω,ω ') = α p (ω + ω') Φ(ω,ω ')Amplitude for Photon Pair Production:

€

Φ(ω,ω') = χ NL dx '0

L∫ uP (x ' ,ω +ω')uS *(x ' ,ω)uI * (x ' ,ω')

pump spectrum

Cavity Phase-Matching

pump mode internal Signal, Idler modes



Type-II Collinear Spontaneous Parametric Down Conversion in a second-order nonlinear, birefringent crystal

Phase-matching (momentum conservation):

rkS +

rkI =

rkP ±π / L

rkS

rkI

rkP

pump Signal

Idler

Energy conservation:

ωS +ωI =ωP

red red blue

L

k

frequency

P

VH

H-PolH-Pol

V-Pol

phase-matching bandwidth



P

I

S

0 kS kI kP

S = I = P/2

Birefringent Nonlinear Crystal, Collinear, Type-II, Bulk Phase Matched, with Double-

Period Grating:

KGS/2 KGI/2

kS + kI = kP

KTP -->



0 L

• grating index contrast • crystal length L = 4 mm, giving kG L = 8• cavity length ~ 0.2mm• signal and idler fields are phase matched at degeneracy wavelength S = I = 800 nm • pump wavelength = 400 nm• pump pulse duration 10 ps

n / n = 5 ×10−4

KTP Crystal with Double Gratings

95% mirror


are needed to see this picture.Two-Photon Amplitude C(, ’)

’ ’

No Grating, No Cavity

’ ’

Two Gratingszo

om

in

Two Gratings

with Cavity


are needed to see this picture.Two-Photon Amplitude C(, ’)

’ ’

’

’

Two Gratings with Cavity

x Pump Spectrum zoom

in

(hi res)



€

C(ω,ω ') = χ NL dx '0

L∫ ˜ E p (x ' ,ω +ω') uS * (x ' ,ω)uI * (x ',ω')

Schmidt-Mode Decomposition

Schmidt-mode eigenvalues for different values of cavity-mirror reflectivity r2r2 j=1 j=2 j=3 j=4 j=50.95 0.951 0.0196 0.0196 0.0044 0.00440.99 0.998 0.0007 0.0007 0.0002 0.0002


Ψ = vac + λ jj

∑ 1 Sj ⊗ 1 I j

1 S j = dω ψ j∫ (ω) 1 Sω 1 I j = dω φ j∫ (ω) 1 Iω


are needed to see this picture.First Four Schmidt Modesfor 95% Cavity Mirror

j=1 j=2

j=4j=3

frequency frequency

amplitude

DBR


are needed to see this picture.Unfiltered Measurement-Induced Wave-function Collapse

• For cavity-mirror reflectivity = 0.99, the central peak contains 99% of the probability for photon pair creation, without any external filtering before detection.

• If any idler photon is detected, then the signal photon will be in the first Schmidt mode with 99% probability.

• Promising for high-rate production of pure-state, controlled single-photon wave packets.



CONCLUSIONS & DIRECTIONS:• Spontaneous Down Conversion can be controlled by modifying the density of states of vacuum modes using distributed cavity structures.• One can engineer the vacuum to create single-photon pairs in well defined, pure-state wave packets, with no spectral no spectral entanglemententanglement.• In the absence of detector filtering, detection of one of the pair leaves the other in a pure single-photon state. • Waveguide development at Optoelectronics Research Center (Uni-Southampton, Peter Smith)



cavity 1

cavity 2

beamsplitter

photon pairweak single-mode squeezed

Alternative Scheme: Single-mode squeezers combined at a beam splitter

Pure-state, single-photon wave-packet generation by parametric down conversion in a distributed...

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