Chapter 15: Quantum Optics of Optical Parametric Oscillators

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Chapter 15: Quantum Optics of Optical Parametric Oscillators

In this lecture you will learn:

• Descriptions of OPOs• Classical equations of OPOs• Phase transition in OPOs• Quadrature noise in OPOs below and above rhreshold• Squeezing in OPOs


Introduction to OPOs (Second Order Optical Nonlinearity)

Fundamental Characteristics of Oscillators

i) Loss

OPO: Photons can be lost from the cavity

i) Gain

OPO: Parametric gain due to stimulated down-conversion of pump photons

ii) Gain Saturation

OPO: ?

o o2

22

12

122211 ˆ)ˆ()ˆ(ˆ2

ˆˆ2ˆˆ aaaaiaaaaH oo

op 2

oo


Introduction to OPOs (Second Order Optical Nonlinearity)

a o2

2222 ˆ)ˆˆ()ˆˆ(ˆ2

ˆˆ2ˆˆˆ aaaaaaiaaaaaaH bbbaobbbaaa

b

A non-degenerate optical parametric oscillator

op 2

ob a o

o oo


Introduction to OPOs (Third Order Optical Nonlinearity)

o2

22

2222 )ˆ)(ˆˆ()ˆˆ()ˆ(

2ˆˆˆˆˆ aaaaaaiaaaaaaH bbbaobbbaaa

A non-degenerate optical parametric oscillator

op

ob a o

o oo

op

a

b 3


22

12

122211 ˆ)ˆ()ˆ(ˆ2

ˆˆ2ˆˆ aaaaiaaaaH oo

Hamiltonian for OPO:

Degenerate OPO: Equations for Mode 1

11 1 1 2

11 1 2 11

1 0 ˆˆ ˆ2 ˆ ˆ( ) ( ) ( )1ˆ1ˆ ˆ ˆ ˆ( ) ( ) ( )0

2

ˆ ˆ( ) ( ) 0

ˆ ˆ( ) ( ) ( )

op in

p inop

in in

in in

ia t a t S ta ad

dt a t a t S ta ai

S t S t

S t S t t t

Equations for Mode 1:

ˆ ˆ 0,ˆ ˆ 0,

o

o

i tin g L

i tin g L

S t v b z t e

S t v b z t e

o o2


)()(ˆ)(ˆ

0)(ˆ)(ˆ

tttStS

tStS

inin

inin

ˆ ˆ 0,ˆ ˆ 0,

o

o

i tin g L

i tin g L

S t v b z t e

S t v b z t e

11

11

11

11

1ˆ ˆ ˆˆ0, 0,

1 ˆˆ

1ˆ ˆ ˆˆ0, 0,

1 ˆˆ

o o

o o

i t i tout g R g L

p

inp

i t i tout g R g L

p

inp

S t v b z t e a t v b z t e

a t S t

S t v b z t e a t v b z t e

a t S t

Equations for Mode 1


Equations for Mode 2

222 2 12 22 2 1

2

12 0 ˆˆ ˆ2 ˆ( ) ( ) ( )( ) 1ˆ1 2ˆ ˆ ˆ( ) ( ) ( )( ( ))0 2

2

op in

p inop

ia t a t F ta td

dt a t a t F ta ti

Equations for Mode 2 (Pump):

2

2

ˆ ˆ 0,

ˆ ˆ 0,

o

o

i tin g L

i tin g L

F t v d z t e

F t v d z t e

2 2ˆ ˆ( ) ( )p po oi ii t i tin p in pF t r e e F t r e e

Assume input for mode 2 is a CW coherent state:

pr Pump photons per sec


Semi-Classical Analysis

Assume:

ti oetta 11ˆ

ti oetta 222ˆ

2

*11

11

21

pdttd

pip

pper

dttd

2

212

22 1

221

Take the average of the quantum equations:


The Semi-Classical Solution: Gain and Loss for Mode 1

ti oetta 222ˆ

ti oetta 11ˆSuppose:

ti

ti

ett

ett2

1

22

11

Equation for mode 1:

2

*11

1

12

1

pdttd

21

1122

21 12cos2 ttttdttd

p

Parametric gain Loss

For maximum gain: 2

21

tt 2

21

ttor

For gain/loss balance in steady state:

21

12 0p


ti oetta 222ˆ


ti

ti

ett

ett2

1

22

11

The Semi-Classical Solution: Gain and Loss for Mode 2


pip

pper

dttd

2

212

22 1

221

2

2 221 2 2 1 2 2 2

2 2

1cos 2 2 cospp

p p

rd tt t t t

dt

Parametric loss Loss External pumping

22 2 1 22p p pr

For gain/loss balance in steady state:



Assume:

ti oetta 11ˆ

ti oetta 222ˆ

2

*11

1

12

1

pdttd

pip

pper

dttd

2

212

22 1

221

Take the average of the quantum equations:

021 dttd

dttd

In steady state:



Obtain for Mode 1: *

1212

12

211

1 22

1

pippp

per

dttd

Write the mode 1 field as:x1

x2

ytxxtxtr ˆ)(ˆ)()( 21

Define:

21

22

21

2 )()()( ttxtxtrtrtr

trVdttrd

)(

Mode 1 equation becomes:

ppppp

rrrrrV

2cos244

22

42

21

2

1 1 1 2( ) ( ) ( ) ( )i tt t e x t i x t

1( )t

( )t


The Semi-Classical Field Potential

trVdttrd

)(

Mode 1 equation becomes:

ppppp

rrrrrV

2cos244

22

42

21

2

x1x2

)(rV

rp > rpth

x1 x2

)(rV rp < rpth

pthpp

p rr 22

21161

pthpp

p rr 22

21161

2,

2

22

2

pp

pthpp

rrr0r


The Semi-Classical Solution

2 for2

for 0

122

1 pspthp

ipthp

p

pthp

s rrerr

rrp

Steady State Solution for Mode 1:

Steady State Solution for Mode 2:pi

ppsps er 22122 2

pspthpp

ipspthp

ipp

srre

rrerp

p

21

22

2 for

2

for2

Use:


The Semi-Classical Solution

ti oetta 222ˆ


ti

ti

ett

ett2

1

22

11


2

*11

11

21

pdttd

21

1122

21 12cos2 ttttdttd

p

Parametric gain Loss

For maximum gain: 2

21

tt 2

21

ttor


The Semi-Classical Solution: Gain SaturationWe have:

21

1122

21 12cos2 ttttdttd

p

pippsps er 2

2122 2

pspthpp

ipspthp

ipp

srre

rrerp

p

21

22

2 for

2

for2

The solution for mode 2 was:

Above threshold gain must equal loss:1

212p

s

22

21

12

12

161

14

12

pppth

ppthp

ps

r

r

Threshold pumping rate


OPOs: Quantum Fluctuations, Squeezing, and Noise

po

po

itis

itis

etbta

etbta

2

222

111

)(ˆ)(ˆ

)(ˆ)(ˆ 2

Consider an OPO operating in steady state:

1)(ˆ),(ˆ)(ˆ),(ˆ 2211 tbtbtbtb

Where:

)(ˆ)(ˆ)(ˆ 2221 txitxtb pp

And the quadrature fluctuations in mode 1 are:

p

2p

Pump

Mode 1

p

2p

Pump

Mode 1

Below Threshold

Above Threshold

x1

x2

x1

x2


OPO Below Threshold

11 1 1 2

11 1 2 11

1 0 ˆˆ ˆ2 ˆ ˆ( ) ( ) ( )1ˆ1ˆ ˆ ˆ ˆ( ) ( ) ( )0

2

op in

p inop

ia t a t S ta ad

dt a t a t S ta ai

The equations for mode 1 were:

ppss

pthp

r

rr

221 20

211 2 1

1 1

ˆ ( ) 1 1 ˆˆ ˆ( ) ( ) ( )2

po ii t

s inp p

d b t b t b t S t e edt

2 2

2 2

22 2

1 1

2 22 2 2

1 1

ˆ ( ) ˆ ˆ1 1 ( ) ( )ˆ ( )2 2

ˆ ( ) ˆ ˆ1 1 ( ) ( )ˆ ( )2 2

p po op

p

p po op

p

i ii t i tin in

sp p

i ii t i tin in

sp p

d x t S t e e S t e ex tdt

d x t S t e e S t e ex tdt i


OPO Below Threshold

ppss

pthp

r

rr

221 20

pth

p

pth

p

rr

tx

rr

tx

p

p

1

141)(ˆ

1

141)(ˆ

222

22

p

2p

Pump

Mode 1Below Threshold

x1

x2

Noise increases as threshold is approached

Noise decreases as threshold is approached

161

1

1161)(ˆ)(ˆ 2

222

2

pth

prr

txtxpp


OPO Below Threshold

ppss

pthp

r

rr

221 20

2

11

11

1ˆ ˆˆ ˆ0,

1 ˆˆ ( )

o

po

i tout g R in

p

i t iin

p

S t v b z t e a t S t

b t e S t

22 )(ˆ)(ˆ)(ˆ 1111p

op

o itiitis etbetbta

0

i

etzbetzbtzx

etzbetzbtzx

pp

p

pp

p

iR

iRout

iR

iRout

2,ˆ,ˆ

),(ˆ

2,ˆ,ˆ

),(ˆ

2222

222

Recall the definitions of quadratures for propagating states:


OPO Below Threshold

ppss

pthp

r

rr

221 20

i

etzbetzbtxv

ietzbetzbtzx

etzbetzbtxv

etzbetzbtzx

pp

p

pp

p

pp

p

pp

p

iL

iL

pg

iR

iRout

iL

iL

pg

iR

iRout

2,0ˆ,0ˆ

)(ˆ1

2,0ˆ,0ˆ

),0(ˆ

2,0ˆ,0ˆ

)(ˆ1

2,0ˆ,0ˆ

),0(ˆ

2222

1

2222

222

1

222


OPO Below Threshold

ppss

pthp

r

rr

221 20

2,0ˆ,0ˆ

12

1

1

)(ˆ22

1

12

pp

p

iL

iL

g

pth

p

p

p ezbezbv

rr

ix

iezbezbv

rr

ix

pp

p

iL

iL

g

pth

p

p

p

2,0ˆ,0ˆ

12

1

1

)(ˆ22

1

122

Solve the quadrature equations in frequency domain inside the cavity:


OPO Below Threshold

ppss

pthp

r

rr

221 20

2,0ˆ,0ˆ

12

1

12

1

),0(ˆ22

1

12

pp

p

iL

iL

pth

p

p

pth

p

pout ezbezb

rr

i

rr

izx

iezbezb

rr

i

rr

izx

pp

p

iL

iL

pth

p

p

pth

p

pout2

,0ˆ,0ˆ

12

1

12

1

),0(ˆ22

1

122


OPO Below Threshold: Spectral Density of the In-Phase Quadrature

ppss

pthp

r

rr

221 20

22

1

22

1

22

12

1

12

1

41

),0(ˆ),0(ˆ2

pth

p

p

pth

p

p

g

outout

rr

rr

v

zxzxdSpp


OPO Below Threshold: Spectral Density of the Out-of-Phase Quadrature

ppss

pthp

r

rr

221 20

22

1

22

1

2222

12

1

12

1

41

),0(ˆ),0(ˆ2

pth

p

p

pth

p

p

g

outout

rr

rr

v

zxzxdSpp


Measurements of Quadrature Squeezing in an OPO

Laser

Nonlinear crystal(BBO)

o

o

o2

OPO

o

Pump

o

o

Phase delay

o

_

Balanced homodyne detectionLO

Wu and Kimble (1987)


OPO Above Threshold

p

p

ipthps

ipthp

ps

pthp

er

err

rr

22

22

1

2

2

2 221

1

2 21 2

2 2

ˆ ( ) ˆ ˆ1 ( ) ( )ˆ2

ˆ ˆ ˆ1 1 ( ) ( )ˆ ˆ ( )2 2

p po op

p

p po opp p

i ii t i tin in

sp

i ii t i tin in

sp p

d x t S t e e S t e ey tdt

d y t G t e e G t e ey t x tdt

)(ˆ)(ˆ)(ˆ

)(ˆ)(ˆ)(ˆ

22

2221

tyitytb

txitxtb

pp

pp

Mode 1

Mode 2 (pump)

Quadratures of mode 1 and pump are now coupled! pipinin ertFtG ˆˆ

Plug in the OPO equations and linearize:


OPO Above Threshold

p

p

ipthps

ipthp

ps

pthp

er

err

rr

22

22

1

2

2

2 2

2 22 2 1 2

1

1

22 1 2 2

2

ˆ 1 ˆ ˆ ( )

ˆ ˆ1 ( ) ( ) 2

ˆ ( ) 1 ˆ ˆ( )2

pp p

p po o

pp p

sp

i ii t i tin in

p

sp

d x tx t y t

dt

S t e e S t e ei

d y ty t x t

dt

2 2

2

ˆ ˆ1 ( ) ( ) 2

p po oi ii t i tin in

p

G t e e G t e ei

pipinin ertFtG

ˆˆ


OPO Above Threshold: In-Phase Quadrature

2 22 2

1

11 2

22 21

2

ˆ ˆ( ) ( )1 1ˆ ˆ( 0, ) ( )2

1ˆ ˆ( 2 ) ( 2 ) 1 2

2

p p

p p

p p

i iout in o in o

g p g

s i ig p p in o in o

sp

S e S ex z xv v

v G e G e

i

2 22 122 2

12

1 1ˆ ˆ2 ( ) ( )1 11 2

2

p pi ip p in o in o

gsp

iS e S e

vi

p

p

ipthps

ipthp

ps

pthp

er

err

rr

22

22

1

2

2


OPO Above Threshold: Out-of-Phase Quadrature

p

p

ipthps

ipthp

ps

pthp

er

err

rr

22

22

1

2

2

11 2

2 222 2

11 2 1 2

2 1

22 21

1 2 1 2

1ˆ ˆ( 2 ) ( 2 )ˆ ( 0, )

21 1 12 2

1 12

11 1 1

2 2

p p

p

s i ig p pout in o in o

sp p p p

p p

sp p p p

v G e G ex zi

i

i

i

2 2ˆ ˆ( ) ( )12

p pi iin o in o

g

S e S ev i


OPO Above Threshold: Quadrature Noise Spectral Density

p

p

ipthps

ipthp

ps

pthp

er

err

rr

22

22

1

2

2

2

12

21

2

12

21

221

2

21

1

410,02222

pps

ppss

gxx v

zSpp

2

21

212211

410,022

sppgxx v

zSpp

Both approach 1/4vg as 1s becomes large (much above threshold)

Maximum squeezing obtained in the out-of-phase quadrature at threshold


OPO Above Threshold: Quadrature Noise Spectral Density

p

p

ipthps

ipthp

ps

pthp

er

err

rr

22

22

1

2

2

Above threshold

Above threshold


Miniature OPOs: 2 Nonlinearity

Lithium Niobate wafer polished into a disk


Chip Based OPOs: 2 Nonlinearity


Chip Based OPOs and Frequency Combs: 3 Nonlinearity

OPOs based on four-wave mixing (Kerr nonlinearity) in Si3N4 ring resonators

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Chapter 15: Quantum Optics of Optical Parametric Oscillators

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