Entangled photon pair generation using guided wave SPDCiwqi12/iwqi12_talks/K_Thyagarajan.pdfRef:...

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

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

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

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

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

Modes of propagation

in a waveguide

6

Modes of coupled waveguides

Waveguide # 1

Waveguide # 2

Symmetric mode Anti-symmetric mode

Modes have

different

velocities

Modes have

different

oscillation

frequencies

7

Directional coupler

Input P1 Output P3

Output P4

Coupling region

Used to split or combine optical signals

8

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

Y-branch with single mode guides

Input 3 dB splitter

Inputs

In phase

Inputs

p out of phase

No output

10

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

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

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

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

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

Sub harmonic generation

Is the following possible?

Second harmonic generation

2

SHG

Red Blue

2

Nonlinear crystal

Red Blue

CLASSICALLY PROHIBITED PROCESS

16

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

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

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

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

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

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

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

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

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

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

( ( 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

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

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

Transverse mode distributions Signal Idler

o-signal

e-signal

e-idler

o-idler

30 Good modal overlap for both the processes

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

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

Experimental verification

33 Ref: Thomas, Herrmann and Sohler, ECIO, Cambridge (2010), ThC4

SPDC

Interlaced domain structures

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

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

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

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

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

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

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

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)

Bandwidth of the two processes

42

Bandwidths for both the processes are almost identical (16 nm)

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

44

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

45

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

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.

47

Total internal reflection and Bragg

reflection

• Total internal

reflection

– Reflection total

– nc > ncl

• Bragg reflection

– Reflection partial

– No restrictions on nc

nc

ncl

48

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

49

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)

50

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

51

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

52

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

53

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

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

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

55

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