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Pure-state, single-photon wave-packet
generation by parametric down
conversion in a distributed microcavity
M. G. Raymer, Jaewoo Noh*
Oregon Center for Optics, University of Oregon
--------------------------------------
I.A. Walmsley, K. Banaszek, Oxford Univ.
-----------------------------------------------------------------
* Inha University, Inchon, Korea
----------------------------------------------------------------ITR - NSF
[email protected]
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Single-Photon Wave-Packet
1
Wave-Packet is a
Superposition-state:
1 

d  ( ) 1 
(like a one-exciton state)
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Interference behavior of Single-Photon Wave-Packets
At a 50/50 beamsplitter a photon transmits or reflects with
50% probabilities.
0
1
beam splitter
Wave-Packet is a
Superposition-state:
1 

d  ( ) 1 
1
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Interference behavior of Single-Photon Wave-Packets
At a 50/50 beamsplitter a photon transmits or reflects with
50% probabilities.
1
1
beam splitter
Wave-Packet is a
Superposition-state:
1 

d  ( ) 1 
0
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Single-Photon, Pure Wave-Packet States Interfere as
Boson particles
2
1
1
beam splitter
0
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Single-Photon, Pure Wave-Packet States Interfere as
Boson particles
0
1
1
beam splitter
2
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Spontaneous Parametric Down Conversion in a second-order
nonlinear, birefringent crystal
kS
Signal V-Pol
H-Pol
pump
kP
L
kI
Idler
H-Pol
Energy conservation:
S  I  P
red red blue
Phase-matching
(momentum conservation):
r
r
kS  kI  kP   / L
phase-matching bandwidth
kz
V
H
P
frequency
optional
Correlated Photon-Pair Generation by Spontaneous Down
Conversion (Hong and Mandel, 1986)
IDLER
Monochromatic
Blue Light
0 or 1
Red photon
pairs
2nd-order
Nonlinear
optical crystal

2P

 d
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Perfect
correlation of
photon
frequencies:
   ' P
SIGNAL
0 or 1
C( ) 1SIGNAL  1IDLER  
P
• Creation time is uncontrolled
• Correlation time ~ (bandwidth)-1

P
'
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Correlated Photon-Pair Measurement (Hong, Ou, Mandel, 1987)
1
Red
photons
2 or 0
MC Blue light
1
Time difference
0 or 2
Nonlinear
optical crystal
Coincidence
Rate
Correlation time ~ (bandwidth)-1
Creation time uncontrolled
boson behavior
Time difference
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For Quantum Information Processing we need
pulsed, pure-state single-photon sources.
Create using Spontaneous Down Conversion
and conditional detection:
(Knill, LaFlamme,
Milburn, Nature,
2001)
Pulsed
blue light
1
trigger if n = 1
filter
1
nonlinear optical
crystal
shutter
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For Quantum Information Processing we need
pulsed, pure-state single-photon sources.
Create using Spontaneous Down Conversion
and conditional detection:
(Knill, LaFlamme,
Milburn, Nature,
2001)
Pulsed
blue light
trigger if n = 1
filter
shutter
1
nonlinear optical
crystal
SIGNAL
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Pulsed Pump Spectrum has nonzero bandwidth
1IDLER  '

trigger
ZeroBandwidth
Filter , 0
P
'
1SIGNAL 

2P


d '

d C( , ') 1IDLER  ' 1SIGNAL 
detect signal 

d C( 0 ,  )
1SIGNAL 
Pure-state creation at cost of vanishing data rate
Pulsed
blue light
trigger
1
filter
Do single photons from
independent SpDC sources
interfere well? Need good
time and frequency
correlation.
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1
random
delay
Coincidence
Counts
1
large data
rate
filter
1
trigger
Time difference
vanishing
data rate
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Goal : Generation of Pure-State Photon Pairs
without using Filtering
Want : C(, ')   0 ( )  0 ( ')

2P




d '
1
I0
d C ( ,  ') 1IDLER  ' 1SIGNAL 
 1
S0
(no entanglement)
Single-photon Wave-Packet States:
1
S0


d  0 ( ) 1 S
signal
1
I0


d  0 ( ) 1 I 
idler
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Decomposition of field into Discrete Wave-Packet Modes.
(Law, Walmsley, Eberly, PRL, 2000)
  vac   d'  d C( ,  ' ) 1 S 1 I  '
  vac 

j
1
Sj
 1
j
Single-photon Wave-Packet States:
1
Sj

 d
 j ( ) 1 S
1
Ij

 d
 j () 1 I 
Ij
(Schmidt
Decomposition)
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The Schmidt Wave-Packet Modes are perfectly correlated.
  vac 

j
1
Sj
 1
Ij
 d
 j () 1 I 
j
1
Sj

 d
 j ( ) 1 S
1
Ij

But typically it is difficult to measure, or separate, the
Schmidt Modes.
CS1 ( )
Mode Amplitude Functions:
Mode spectra overlap.
No perfect filters exist, in
time and/or frequency.
CS 2 ( )
filter
CS 3 ( )
frequency
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Why does the state generally NOT factor?

2P



d '
1
S0

d C ( , ') 1SIGNAL  1IDLER  '
 1
I0
C(, ')
'

Energy conservation
and phase matching
typically lead to
frequency correlation
need to engineer the
state to make it factor
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Spontaneous Parametric Down Conversion inside
a Single-Transverse-Mode Optical Cavity
1 mm
kI
pump
kP
Nonlinear
optical crystal
with wave-guide
kS
the problem:
DOES NOT
WORK
cavity FSR ~ 1/L
phase-matching
BW ~ 10/L
Spontaneous Parametric Down Conversion inside a
Distributed-Feedback
Cavity
• large
FSR = c /(2x0.2 mm)
• small phase-matching BW:
0.2 mm
cavity
4 mm
~ 10 c /(4 mm)
4 mm
H-Pol
pump
Linear-index
wave-guide
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H-Pol
idler
V-Pol
signal
Linear-index Distributed-Bragg
Reflectors (DBR)
second-order nonlinear-optical crystal
SIMPLIFIED MODEL: Half-DBR Cavity
4 mm
DBR
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0.2 mm cavity
99% mirror
Reflectivity
0 =800 nm
KG = 25206/mm
Dn/n ~ 6x10-4
(k = 2/mm)
DBR band gap
cavity mode
frequency/1015
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Quantum Generation in a Dielectric-Structured Cavity:
Phenomenological Treatment
 E(x,t)   D(x,t)  J(x,t)   Ep (x,t)E *(x,t)
2
x
2
t
Signal
NL
Source
Pump
Frequency Domain:
space and frequency dependent electric permeability:
 (x, )   (x)n2 ( )
%  )  J(x,
% )
 x2   (x, )  2  E(x,
   (x,  )  u(x,  )  0 (modes)
2
x
2
E˜ p
E˜ S (x, )
0
L
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x
†
˜
˜
ˆ
E S (x, )  EVAC (x, )  uout (x, )  d ' C( ,') aI ( ')
two-photon
amplitude
C(, ')  
NL

L
0
dx' E˜ p (x',   ') uS *(x', )uI * (x', ')
interaction
pump field
internal
Signal, Idler
modes
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Heisenberg Picture
Schrodinger Picture
  vac   d'  d C( ,  ' ) 1 S 1 I  '
Amplitude for Photon Pair Production:
C(, ')   p (  ') (, ')
pump spectrum
( , ')  
NL

L
0
Cavity Phase-Matching
dx' uP (x',   ')uS *(x', )uI * (x', ')
pump mode
internal Signal, Idler modes
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Type-II Collinear Spontaneous Parametric Down Conversion
in a second-order nonlinear, birefringent crystal
H-Pol
pump
kP
L
kS
Signal
V-Pol
kI
Idler
H-Pol
Phase-matching (momentum conservation):
r
r
kS  kI  kP   / L
k
V
H
Energy conservation:
S  I  P
P
red red blue
phase-matching bandwidth
frequency
Birefringent Nonlinear Crystal, Collinear, Type-II,
Bulk Phase Matched, with Double-Period Grating:
S = I = P/2
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kS + kI = kP
P
I,S
0
kS
kI
KGS/2 KGI/2
kP
KTP -->
KTP Crystal with Double Gratings
95% mirror
0
L
4
• grating index contrastDn / n  5  10
• 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
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Two-Photon Amplitude C(, ’)
No Grating, No Cavity
’
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Two Gratings
’
Two Gratings
’
with Cavity
’


Two-Photon Amplitude C(, ’)
Two Gratings with Cavity
’
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’
x Pump Spectrum
’
(hi res)

’

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Schmidt-Mode Decomposition
C(, ')  

NL
L
0
dx' E˜ p (x',   ') uS * (x', )uI * (x', ')
  vac   d'  d C( ,  ' ) 1 S 1 I  '
  vac 

j
1
Sj
Ij

 1
Ij
j
1
Sj

 d
 j ( ) 1 S
1
 d
 j () 1 I 
Schmidt-mode eigenvalues for different values of cavity-mirror reflectivity  2
j=1
j=2
j=3
j=4
j=5
2
0.95
0.99
0.951
0.998
0.0196
0.0007
0.0196
0.0007
0.0044
0.0002
0.0044
0.0002
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First Four Schmidt Modes
for 95% Cavity Mirror
amplitude
j=1
j=2
DBR
j=3
frequency
j=4
frequency
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.
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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 singlephoton pairs in well defined, pure-state wave
packets, with no spectral entanglement.
• In the absence of detector filtering, detection of
one of the pair leaves the other in a pure singlephoton state.
• Waveguide development at Optoelectronics
Research Center (Uni-Southampton, Peter Smith)
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Alternative Scheme:
Single-mode squeezers combined at a beam splitter
cavity 1
photon pair
weak single-mode squeezed
beam
splitter
cavity 2