Entangled photon pair generation using guided wave SPDCiwqi12/iwqi12_talks/K_Thyagarajan.pdfRef:...
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Entangled photon pair generation
using guided wave SPDC
K. Thyagarajan
Physics Department, IIT Delhi
IWQI12, HRI, Allahabad, February 20-26, 2012 1
Thanks to:
Ms. Jasleen Lugani, IIT Delhi
Dr. Sankalpa Ghosh, IIT Delhi
Dr. Ritwick Das, NISER, Bhubaneswar
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Outline
• Modes in waveguides
• Second order nonlinear optical effects
• Spontaneous parametric down conversion
– Generation of photon pairs
• Domain engineering for generation of
– Polarization entangled photon pairs
– Modal and path entangled photon pairs
• Bragg reflection waveguides (BRW) for
– Increasing pump acceptance bandwidth and
– Narrow signal bandwidth
• Conclusions 2
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Optical waveguides
nc
nf
ns
Planar
waveguide
Channel
waveguide
n2
n1
Optical fiber
• Optical waveguides: •High index region surrounded by lower index regions
• Wave guidance by total internal reflection
• Materials
• Glass, Lithium niobate, GaAs, Silicon etc.
3
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Modes of propagation
Mode:
• Certain electric field patterns that propagate unchanged
• Solution to Maxwell’s equations satisfying appropriate
boundary conditions
• Characteristics:
• Definite transverse electric field pattern
• Definite phase and group velocity
• Discrete set of guided modes
Similar to
• Modes of oscillation of a string fixed at two ends
• Eigenstates of a potential well in quantum physics
4
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Modal field
• Discrete number of guided modes
• Single mode waveguide: only a single guided mode is
possible
• Each guided mode characterized by a different modal
field pattern and propagation constant
5
cczti
eyxzAtzyxE
),()(21),,,(
A: Amplitude of the mode
(x,y): Transverse modal field distribution
: Propagation constant of the mode
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Modes of propagation
in a waveguide
6
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Modes of coupled waveguides
Waveguide # 1
Waveguide # 2
Symmetric mode Anti-symmetric mode
Modes have
different
velocities
Modes have
different
oscillation
frequencies
7
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Directional coupler
Input P1 Output P3
Output P4
Coupling region
Used to split or combine optical signals
8
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Power coupling in a directional coupler
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10
Normalized distance (kz)
Couple
d p
ow
er
Ref: Introduction to fiber optics, A Ghatak and K Thyagarajan, Cambridge Univ Press, 1998
9
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Y-branch with single mode guides
Input 3 dB splitter
Inputs
In phase
Inputs
p out of phase
No output
10
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Nonlinear polarization
For light waves with high intensity, the electric fields
are high and then polarization is a nonlinear function of
electric field
...)3(
02
0203EEE d
E
P
t
t
Not a sine wave
11
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Second order nonlinearity
20
2 EdNL
If we consider a plane em wave incident in the medium
)cos( kztA E
)](2cos[20
20
)(2cos20
2
kztAdAd
kztAdNL
• The 2 term responsible for the generation of second
harmonic electromagnetic field
Then
12
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Sum and difference frequency
generation
)22
cos(2
)11
cos(1
zktAzktA E
Input electric field:
Nonlinear polarization gets generated at the following
frequencies:
21,
21,
22,
12,0
SHG SFG DFG
SHG: Second harmonic generation
SFG: Sum frequency generation
DFG: Difference frequency generation 13
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Phase matching
In general the velocity of nonlinear polarization is not
equal to the velocity of the electromagnetic wave at the same
frequency that it is trying to generate
For efficient generation, these two velocities have to be equal
PHASE MATCHING CONDITION
)()2(or 122 nnkk
• Due to dispersion this is normally not possible
• Use birefringence of the crystal
• Use periodic interaction to compensate for mismatch 14
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Photon picture
SHG can be considered as a fusion of two photons at
frequency to form one photon at frequency 2
For efficient interaction, we need to conserve momentum
hk1 hk1
hk2
k2 = 2 k1
Phase matching
condition Vector diagram
h
h
h 2
15
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Sub harmonic generation
Is the following possible?
Second harmonic generation
2
SHG
Red Blue
2
Nonlinear crystal
Red Blue
CLASSICALLY PROHIBITED PROCESS
16
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17
Parametric fluorescence • Incident photon at one frequency spontaneously
generates a pair of photons at lower frequencies
Ref: Martin et al., Opt Exp 17 (2009) 1033
655 nm 1255 nm
1370 nm
17
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One photon at p splits spontaneously into one photon at
s and another at i
Explanation for the process is quantum mechanical
For efficient down conversion
Energy conservation
Momentum conservation
Spontaneous parametric down conversion
(SPDC)
p s
i
isp
isp
kkk
18
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SPDC using Quasi Phase Matching
• Periodic variation in the nonlinear coefficient
• Spatial frequency K chosen to compensate for phase mismatch
• Most widely used technique for SPDC
• Can be applied to any pair of signal and idler ls
• Use highest nonlinear coefficient tensor element
Kkkk isp
19
z
x Signal (s)
Idler (i)
p
k k K
k2
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Bulk vs. waveguide configuration
• Lightwaves get guided through the device
• Photons generated in well defined spatial modes
• Ease of collection and further processing
• Due to restricted modal structure, much higher probability of emission into distinct modes
• Effective decoupling of spectral and spatial degrees of freedom
• Novel configurations and integrated geometry
Waveguide
t ~ 2 mm
410~)(
)( bulkA
waveguideA
20
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Entangled photons via SPDC
• Entanglement in different degrees of freedom
– Polarization
– Mode or path
• Generation SPDC using (2) in waveguides
• Many existing schemes for polarization
entanglement
– need extra experimental steps to entangle signal and
idler photon pairs
• Direct generation of non degenerate entangled
signal and idler photons interesting
21
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Type II quasi phase matching in LiNbO3
• Using different QPM periods one can downconvert an o-polarized pump to either
– An o-polarized signal and an e-polarized idler or
– An e-polarized signal and an o-polarized idler
• In both cases the polarization states of output are well defined
o-signal
e-idler
e-signal
o-idler
QPM period L1 QPM period L2
L i
ie
s
so
p
po nnn
lll1
1
L i
io
s
se
p
po nnn
lll2
1
22
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Doubly periodic poling
• It is possible to satisfy both QPM conditions simultaneously
• Variation of nonlinear coefficient d along propagation direction is doubly periodic
• The grating contains two spatial frequencies required to phase match both the processes simultaneously
Pump
Signal
Idler
Domain engineered crystal
Ref: Thyagarajan, Lugani, Ghosh, Martin, Ostrowsky, Alibart, Tanzilli,
PHYSICAL REVIEW A 80, 052321 (2009) 23
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Doubly periodic poling
• x-variation of nonlinear
coefficient )()( 2124 xfxfdd
d
f1(x) f2(x)
L0 Lp
x
24
( .....4
2211224
xiKxiKxiKxiKeeee
dd
p
p
p
KKK
KKK
02
01
By choosing appropriate values of K0 and Kp, it is possible to achieve
phase matching for both o->o+e and o->e+o processes
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25
Polarization state of output photons
• Classically we would say that the output is
either of the following:
– Signal is H polarized and idler is V polarized or
– Signal is V polarized and idler is H polarized
• According to quantum mechanics – Polarization state of individual signal and idler
photons are undefined but are orthogonal to each other
– The output signal and idler photons are entangled in polarization
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Fields at pump, signal and idler
• Pump is taken to be a classical wave and signal and
idler are treated quantum mechanical.
( ( ( yeeEreEtxiktxik
popopopppp ˆ)(
2
1
( yeaea
LrediE
xik
so
xik
so
so
ssosso
soso ˆˆˆ2
)(ˆ
int
( zeaea
LrediE
xik
se
xik
se
se
ssesse
sese ˆˆˆ2
)(ˆ
int
( yeaea
LrediE
xik
io
xik
io
io
iioiio
ioio ˆˆˆ2
)(ˆ
int
( zeaea
LrediE
xik
ie
xik
ie
ie
iieiie
ieie ˆˆˆ2
)(ˆ
int
Pump(o)
Signal(o)
Signal(e)
Idler(o)
Idler(e)
26
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( ( ti
iose
ti
ioseeo
ti
ieso
ti
iesooespppp eaaeaaCeaaeaaCdH
ˆˆˆˆˆˆˆˆˆ )1()1(
int
2sin
2exp
4intint
2
024)1( Lkc
Lki
nn
IEdC oe
oe
ieso
oeisp
oep
2sin
2exp
4intint
2
024)1( Lkc
Lki
nn
IEdC eo
eo
iose
eoisp
eop
,22
1
L
i
ie
s
so
p
po
oe
nnnk
lllp
p
L
i
io
s
se
p
po
eo
nnnk
lllp
p2
2
1
Interaction Hamiltonian
27
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Output state
28
( oeeoeooes isCisCdi ,,
Entangled in polarization
• The relative magnitudes of the C coefficients would
determine if the output is maximally entangled or not.
• We have defined the degree of entanglement as
( ( eooe
eooe
CC
CC
,max
,min 10
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Ti:LiNbO3 waveguide
• Refractive index profile:
0;
0;2),(
2
22 2222
zn
zenennzyn
c
hzwy
bb
• Use standard variational analysis to calculate
– Effective indices at pump, signal and idler wavelengths
– Mode field profiles of the interacting modes
z
y
x
Pump
Signal
Idler
29
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Transverse mode distributions Signal Idler
o-signal
e-signal
e-idler
o-idler
30 Good modal overlap for both the processes
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Spectrum of the two processes
E-signal + O-idler
O-signal + E-idler
5 nm
sinc2
(kL
/2)
Signal wavelength (nm)
770 775 780 785 790 795
0
0.2
0.4
0.6
0.8
1.0
31
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Entanglement vs. width Depth (mm)
12
10
8
6.5
7 8 9 10 11 12
Enta
ngle
me
nt
measure
6 0.92
0.94
0.96
0.98
1.0
Width (mm) 32
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Experimental verification
33 Ref: Thomas, Herrmann and Sohler, ECIO, Cambridge (2010), ThC4
SPDC
Interlaced domain structures
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Modal and path entangled
photons
• Modal and path entangled photons can find
applications in quantum information processing,
lithography etc.
• Waveguide device supporting two spatial modes
– Incident pump generates signal and idler photons
entangled in modal degree of freedom
– Modal entanglement can be converted to path
entanglement using waveguide device
34
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Waveguide device geometry
Ref: Jasleen Lugani, Sankalpa Ghosh, and K. Thyagarajan, Phys. Rev. A 83
(2011) 062333
I II
III IV
Pump
y
x z
d
Doubly periodic
domain reversed
region
Single mode
waveguide Y-splitter Symmetric
directional
coupler
Y-splitter to two non-
identical waveguides
35
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Mode Entangled photons Generating non-degenerate, co-polarised mode
entangled photons.
Satisfying QPM conditions of two different SPDC
processes, simultaneously, leading to mode entangled
pairs of photons.
All waves have e-polarization and use d33 coefficient
a. pump (0) -> signal(0) + idler(0)
b. pump(0) -> signal(1) + idler(1)
L
i
i
s
s
p
p nnnK
lllp
p 000
1
1 22
L
i
i
s
s
p
p nnnK
lllp
p 110
2
2 22
36
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Domain engineering
37
I II III IV
• Incident pump photon down converts
– Either into symmetric signal and symmetric idler modes
– Or into antisymmetric signal and antisymmetric idler modes
• Both processes almost equally efficient
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Quantum mechanical analysis
• Pump is assumed to be classical and signal and idler
treated quantum mechanically.
• The interaction Hamiltonian in the interaction picture under
energy conservation and RWA is
( (
1111100000intˆˆˆˆˆˆˆˆˆ
isisisiss aaaaCaaaaCdH
2sinc
4
2sinc
4
12/
112
10331
02/
002
0330
1
0
Lke
nn
ItEdC
Lke
nn
ItEdC
Lki
is
isp
Lki
is
oisp
p
p
dydzzyezyezyeI isp ),(),(),( 0000
dydzzyezyezyeI isp ),(),(),( 1101
38
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Output state
• Output state at the end of Region III:
39
( 111000 ,, isCisCdi s
• Output represents a mode entangled state
• 0: symmetric mode
• 1: antisymmetric mode
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Results • We have carried out simulations for the planar domain
engineered LN waveguide.
• The propagation constants vary with core separation
Wavelength
(nm)
∆ne
750 .0033
1452 .0026
1550 .0025
0 1 2 3 4
x 10-5
9.265
9.2665
9.268
9.2695
9.271x 10
6
d (m)
(
m -
1)
ANTISYMMETRIC
SIGNAL
SYMMETRIC
0 1 2 3 4
x 10-5
8.667
8.6688
8.6706
8.672x 10
6
d (m)
(
m-1
)
ANTISYMMETRIC
SYMMETRICIDLER
40
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Modal patterns • Titanium indiffused waveguides in lithium niobate
• The modes have very good modal overlap
• This leads to maximal modal entanglement
41
0 2 4 -2 -4 0 2 4 -2 -4
y (mm) y (mm)
Ez(y
)
pump (0) -> signal(0) + idler(0) pump(0) -> signal(1) + idler(1)
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Bandwidth of the two processes
42
Bandwidths for both the processes are almost identical (16 nm)
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Modal entanglement to path entanglement
43
• Fundamental symmetric modes exit from the upper waveguide
having higher propagation constant
• First order antisymmetric modes exit from the lower waveguide
with smaller propagation constant
• Output state is path entangled
I II III IV
( lluus isCisCdi ,, 10
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SPDC with increased pump bandwidth
• SPDC source with
– Signal photon at the telecom wavelength of 1550
nm
• Essential requirements: – High efficiency generation
– Increased pump acceptance bandwidth so that
femtosecond pump could be used
– Narrow signal bandwidth so that signal photons
can be used as flying qubits
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Idea
• Efficiency of the down conversion varies as
2
2csin L
0 Kisp
L: Length of interaction
• Now
ip
pdd
pdsd
ll
• If waveguide design is such that
(p-i) exhibits a minimum at a specific lp, then
• As lp changes, ls will remain fixed
• This will lead to
• large pump acceptance bandwidth and
• narrow signal bandwidth
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46
Bragg reflection waveguides
• Variation of the propagation constants at different
wavelengths depends on
– Material dispersion
– Waveguide dispersion
• We need to counter the strong material dispersion at
pump wavelength by a strong waveguide dispersion
at the idler wavelength
• Bragg reflection waveguides (BRW) can provide such
a possibility
Ref: Thyagarajan, Das, Alibart, de Micheli, Ostrowsky, Tanzilli,
Optics Express, 16 (2008) 3577.
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Total internal reflection and Bragg
reflection
• Total internal
reflection
– Reflection total
– nc > ncl
• Bragg reflection
– Reflection partial
– No restrictions on nc
nc
ncl
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TIR and Bragg modes
n(x)
x
n1
n2
nc
O
(neff)TIR
(neff)BRW
TIR modes: n1 < neff < nc
Bragg modes: neff < n2
• Dispersion of TIR and Bragg modes can be very different • Large design space for dispersion of Bragg modes
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Modal dispersion
Bragg modes exhibit very strong dispersion
nc: GaN
ns: AlGaN
nc: GaN
n1: AlxGa1-xN
n2: AlyGa1-yN
TIR guided
Bragg mode
1.50 1.54 1.58 1.62
ne
ff
2.0
2.1
2.2
Wavelength (mm)
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SPDC with BRW
• Pump (~0.8 mm): TIR mode
• Signal (~1.55 mm): TIR mode
• Idler (~ 1.653 mm): BRW mode
PPGaN core
Al0.45Ga0.55N (n2)
Al0.02Ga0.98N (n1)
s and i
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Phase matching
• Appearance of a minimum implies that as lp changes
– Phase matching condition will continue to be satisfied
lp (nm)
p-
i (m
m-1
)
L = 2.77 mm
11.2814
11.2816
11.2818
11.2820
800 805 810 795 790
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Wavelength variation of signal and idler
• Modal dispersion designed so that as lp changes, ls remains the same
lp (nm)
800 804 796
1540
1580
1620
1660
1700
Idler
Signal
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Bandwidth
• Pump acceptance bandwidth increased 30 times compared to conventional geometry
• Very small signal bandwidth; useful in quantum communication systems
BRW:
BW ~ 12 nm
TIRW
BW ~ 0.4 nm
BRW
BW ~ 1 nm TIRW
BW ~ 21 nm
Pump bandwidth Signal bandwidth
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Separable states • The idea proposed also leads to separable
signal-idler pair states
– Useful for various applications like heralding identical
single photon states etc.
54
0
ippd
d l
• implies
id
id
pd
pd
• This condition leads to the generation of separable state
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Conclusions
• Using waveguide geometries
– Provides with additional degrees of freedom
– Output photons in well defined discrete spatial modes
– Higher efficiency
• Domain engineering
– Direct generation of polarization, mode or path entangled photon pair
• BRWs have very interesting dispersion behaviour
– Designs to achieve large pump acceptance bandwidth and small
signal bandwidth or separable states
• Integrated quantum optical circuits should play a very
important role in the future
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Acknowledgement
• Work partially supported by an Indo-French project
sponsored by Department of Science and
Technology (DST), India and and Centre Nationale
de la Recherche Scientifique (CNRS), France
56