Amplification paramétrique et réduction du bruit quantique dans des microcavités semiconductrices

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Amplification paramétrique et réduction du bruit

quantique dans des microcavités semiconductrices

Augustin Baas

To cite this version:

Augustin Baas. Amplification paramétrique et réduction du bruit quantique dans des microcavités

semiconductrices. Physique Atomique [physics.atom-ph]. Université Pierre et Marie Curie - Paris VI,

2003. Français. �tel-00007954�


   

    
    
 

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)

†



(c

h

k

h

)

†

('90&(1&4( &0) 491) (&0) &01  4( +((1& 490)4(

k

e



k

h

4)' ( ) 4(' 01@('n #' 90&(1&(,0) )' 02()1 )( ''/

'  &(0) 4( 0uu10) 4(' 20'0)'(  @0&( 4(' (&1&20)')( (1

1' &( -1(n 9(,&(''0) 4(  +(1& u04())( 41 0uu1(1& 4( (

0&(1& 1 &(u(& 0&4&( () 4()' (,0)-1( (' w13@ 53 

h[bb

k

,bb

†

k

]i ≈ 1 − O(n.a

2

B

)

 ! r $  0

n

(' ( )0u2&( 49(,0) & 1) 4( '1&/( (

a

B

49(&0)' ( 4( &01' 4)' 1)( '1&/( 4( 90&4&( 4( 9(,()'0) 4( 9(,0)n

0&'-1( ( &(01+&(u()'()&( 4(1, /0)0)'490)4( 49(,0) 4(+() )0)

)33(2(0) (' '()'2(   )1&( /(&u0)-1( 4(' 0)'1)' 4( 9(,0)n

=01& 01+0& &(& 9(,0) 0uu( 1) 20'0) 4)' 1) '4'Au( 0)'1

491) 3&)4 )0u2&( 49(,0)' () )(&0)0) ((1( 1)( &)'/0&u0) 01&

''(& 491) ('( /(&u0)-1( +(&' 1) ('( 20'0)-1(n 7(( &)'/0&u0)

1)&( ('  &)'/0&u0) 498'1 0)
1( 0uu(  1)( u@04( '4'u-1(

-1 (&u( 4( 4&&( 1) 3o 49(&0)' () (&u('4( 20'0)' 8'1> 9

n )1&(

/(&u0)-1( 4(' (&0)' ( 4(' &01' (' &('0)'2( 4( (&u(' '1u()&('

 01'(' 0&4&('n8) @u0)() 49)(&0) ()&( (,0)' &()4 40)  /0&u(

491) @u0)() 49)(&0) ()&( 20'0)' 1-1( 0) 40 &?01(& 4(' (&u('

0&&(/'n 5

)' 1) &(u() u 1 '(0)4 0&4&( () 4()' (&0)-1(

0uu( 0) ( +(&& 1 &3&@( 7

n !

4( ( @&( (' (&u('  -1&( 0r

&(1&' 4( 9)(&0) 4( 7010u2 '0) &)'/0&u' () 1)( '0uu( 4( &041'

490&(1&' 20'0)-1('  01' (' 0&4&(' !0@ 99

n

7(( u@04( (&u( 4902()& 4(' -10)' 4( p0@ (,0)-1('n 7(' -1r

0)' &(() (' -10)' 4( p0@  4(1, )+(1, 4(  @4'-1( 0u-1(

u' ((' 0)())() 4( /
0) 0)'')( (' )0)r)&' /(&u0)-1(' 41(' 

 )1&( 0u0'(  1'(1&' 0&' 4( 9(,0) 4)' 4(' (&u(' 0&&(/' 

&()0&u'0) 4( 9)(&3( )-1( '1&0) 4(  /0&( 490'(1& ( 20r 3( 4( =1 ( 9)(&0) 7010u2())( (,0)r(,0) 4()4)( 41 ')n 7(( 4('&0) !0@ 99 (' 40( u(u()4)'-1'u() 01(' ('

4('&0)' 4(' (,&()(' ((1(' 4)' ('u&0+'() &3u( 4( 013(

/0&  /2( 4()'0 9(,0) (' '4'u-1(u()0)'4& 0uu( 1) 20'0)n

7(( &0@( &()4 01( ' 4u()'0) 4)'  4('&0) 4(' (,&()(' 4(

1' () 1' )0u2&(1'(' 0&) '1& 1)( +()1(( 0)4()'0) 4(' 0&0)'

&1( u,( (,0)r@00)n

B NTKLILHQ RQ HIUFJN

=01& 02()& ( &3u( 4( 013( /0&0) ( ( 1' -1)-1( 4)' 1)(

+ 0-1( 2&4r=&0n 1'( 4(  &)'u''0) 1)( 4(' u&0&'(' u04('

@00)' 01& 1) u04( 0&0) 4( +((1& 490)4( 4)' ( ) 40)) /0&u()

()0&( 1) 0))11u uu( ' ( 0))11u (' '&11& () ' &0' (1, 4(

 + 2&4r=&0n(' (,0)' 0)40) 1)( &022 4( &(0u2)'0) &

1) 4( (u' 1)( 5(+14 5!

 -1 u(  41&( 4( 9@)3( 49)(&3( ()&(

(' 4(1, '4'Au(' () &3u( 4( 013( /0&n

6) '( &00'( 4)' (( '(0) 4( 4&&( 2&A+(u() (' &0&' 4( 

+n (' u&0&' 4(  + '0) 4(' u&0&' 4(&-1(' u101@('n 8)(

'1(''0) 4( &(' 4( 01@(' 4( 4(1, '(ur0)41(1&' 49)4(' 4&()' 4(

uu( ''(1& 0-1(

λ

0

/4

 0

λ

0

('  0)31(1& 490)4( 4(  &'0))( 4( 

+0)'1() & 1)?(1 49)(&/&()(' 0)'&1+(' () &(,0) ( 4('&1 r

2'0&0) '0) )33(2(' 0)&&(u() 1 ' 4(' u&0&' u-1(' 01&

('-1('  &(+ )( 4''( ' ('

95 %

4)' ( &0@( )/&r&013(n9(r  4(' u&0&' 4( p&33 13u()( +( ( )0u2&( 4( 01@(' (u(' ( ()

4-1(u() 4(' +(1&' +0')(' 4(

99, 95 %

!0'()@(& 53 +0) 5$ +0) 55

n 7(' u&0&' '0) &A' 0&&((u() 4&' &  u@04( 4(' u&(' 4(

&)'/(& +0) 55

 n $' (3)() (1& &(+ u,u( '1& 1)( 3( ()

)(&3( (( 2)4( )(&4(n ) 4(@0&' 4( ( )(&+(  &(+ 0'(

( &'()( 4( )0u2&(1, u)u -1 0&&('0)4() 1, u04(' 4( /1( +0)

55 n

8)( + 2&4r=&0 )r)  u&0&' 4( p&33 (' 0)'1( 4( 4(1,

u&0&' 4( p&33'&' &  0)31(1& 0-1(

L

c

= pλ

0

/2

+(  ()(&n =01&

1)( (,0)  9)4()( )0&u(  &'0))( 4( (( + &q 4)'

 2)4( )(&4( 4(' u&0&' 4( p&33 ( 1' '01+() 1 ()&(n 9''(1&

4-1( 4(  0)31(1& 4( + (( 1'' ('(1&4( 90&4&( 4(  0)31(1&

λ

0

/ -1( 9)(&+( '(& 2&( 4(  + (' '1&(1&   2)4( )(&4(

4(' u&0&'n 8) '(1 u04( 0-1( (1 40) &q&( 4)'  2)4( )(&4(n

) 4(@0&' 4(  2)4(0) &(&01+( (' ' '(0)4&('n7(' u04('4( /1( 0) 1)

0(() 4( &(,0) 1' /2( -1( ( u04( &) ( '0) 40) 1' &3('

'(&(u()n 6

) (' &(&01+( 01& )9u0&( -1( +((1& 490)4( 4)' ( )

( (1& 4'(&'0) (' 'u&(  ((' 4(' u04(' 4( +n$' '0) 1) ) &A'

(( 49u''0) 4)' ( '12'& 0  1uA&( (' 2'0&2( ( & 0)'-1()

(&41(n 5

( u)A&( 3)&( ' ?01() 1) & ( &1 4)' (' 4'0'/' '(ur

0)41(1&' u((1&' 4( 1uA&( +0) 55

n

 + 2&4r=&0 &'()( 1)( 4'(&'0) /0)0) 4(  0)')( 4r

(&-1( 41 '(ur0)41(1&n 6

) 4&+( 1)( u''( ((+( 01& ( @00) 

&& 4(  4'(&'0) '10'( &20-1( 4(' )(&3('n 7(&)(' 4')0)'

'0)  0&(&  1'( 4(  )1&( 4(' u&0&'n  )&0) 41 @u 4)'

(' u&0&' ()&q)( 1)( 4()4)( () )(&3( 4(  @'(   &(,0)n  0)r

31(1& 4( )&0) 4(' u&0&' ( 0)31(1& 4( p&33(' &('0)'2( 491)

(&u( 0&&(/   0)31(1& 4( + 4)'  41)0) 491)( 0)31(1& 4( +

((+( +0) 55

n

B
NFCQ RQ HKPDEIQ KMJ

&
&% /- "*#-."/

8)( u&0+ '(ur0)41&( (' 1)( + 0-1(  u&0&' 4( p&33

&'0))( +( 1)( &)'0) 1)-1( 491) 1' -1)-1(n  0'0) 41 1'

4)'  + 4()4 4(  0)31(1&4( +n( 0'0))(u()41 1' 0

1 4('

1' (' @0' 01& u,u'(& 9)(&0)n  + 0-1( (&u( 4( u0r

                           

Energie

Désaccord (meV)

0

4

6

-4

-6

-2

2

41(& (' &0&' 4(  &0)'( 1u)(1'( 41 u(1 u1(1&n ( (&u(

+) 01  4( '( (& () &3u( 4( 013( /0&n  0)40) 4( &3u( 4(

013( /0& u0'( 1)( 0)')( 4( 013( (,0)r@00)

g

3&)4( 4(+) (' &3(1&' 41 @00) 4( +

γ

a

( 4( 9(,0)

γ

b

n5( /
0) 3)&( ( &3u( 4( 013( /0&(' () '0 () 13u()) ( 013( ()&( ( u04( 4( +

(  &'0))( u&(('0 () 4u)1) (' 1, 4( &(,0)n ) @4'-1(

0u-1( (' 4(1, '010)' '0) (,0(' 4)' ( @0, 4(' 0u('491)( &

+( (' 0u(' 4( !442(&3 ( 491&( & +( (' 0u(' /&04'n  1)(''( 4(

 + (1 4)' ( &(u(& ' ()4&( 4(' +(1&' 0)'4&2(' r1' 4(

10

7

r 01& 4(' u&0&' '1&0)41(1&' !u0)4 5$

n 5)' (' u&0+' 

1' -1)-1(' 0) (1 1'' 2() 13u()(&  &(+ 4(' u&0&' -1( (

)0u2&( 4( 1'n ) (( /0&( 490'(1&-1 40))( 9)()' 41 013(

13u()( +(  &)( &&( 41 )0u2&( 4( 1' w014& 5

>n

&
&6

.-"/./  #"*8+ ./'. 

(' u04(' (,0) ( @00) '0) &' 0uu( 4(' 0'(1&' @&u0)-1('

20'0)-1(' 01' )&(u()n 7( &(u() '10'( 01& 9(,0) 91'r

0) 4(  &)'/0&u0) 498'1n 6

) ((1( 40) 1) 4+(0(u() (&1&2/

41 @u0)() 0 4)' 9('( 20'0)-1( &&  90&4&( 4(1, () 4()'

(,0)-1(n ( @u0)() 49)(&0) (' 4+(0 4)' 9&0,u0) 4r

0&(( -1( ?1'1( 1) &A' ( &40) 4( p0@& () 0u&'0) 4(  0)31(1&

490)4(n

6) )( 0)'4A&( -1( (' ' (,0)-1( ( @00)-1( 4) 1)( 0&'0)

&1&( 2() 41)(-1 '0) (''(1' ?01(&1) & ( 4)' ( '491)( (,0)

0

20

40

-20

-40

Angle (deg)

Energie

K

K = (k, k

z

θ

k

|k| = |K|sinθ

490)4( 4)' ( ) 4(' 01@(' (' )0

k

n 70uu( (' ' 4( 4&()'

k

)( '0) ' 01' 1'( 4( 9)+&)( & &)'0) 4)' ( ) 4(' 01@('

( @u0)() &q 0uu( 1)( '0uu( 4( (&u(' '1& ('

k



H =

X

k

{E

exc

(k) b

†

k

b

k

+ E

cav

(k) a

†

k

a

k

+

Ω

R

2

³

a

†

_k

b

k

+ b

†

_k

a

k

´

}

 ! r> 0

a

k

(

b

k

'0) (' 0&(1&' 49))@0) 491) @00) 4)' ( u04(

k

( 491) (,0) 4)' ( u04(

k

n

a

†

_k

(

b

†

_k

'0) (' 0&(1&' 4( &0) 0&&('0)4)'n (' )(&3(' 4(' u04(' @00)-1(' ( (,0)-1(' '0) )0(' &('(+(u()

E

cav

(k)

(

E

exc

(k)

n( 4(&)(& (&u( 4& ( 013( )&( (,0)r@00) 4(

4( s4)('r71uu)3'n(' u04(' &0&(' '9(,&u() 0uu( 4(' 0u2)'0)'

)&(' 4(' 4(1, 0'(1&' )1,

µ

p

k

q

k

¶

=

µ

X

k

−C

k

C

k

X

k

¶

.

µ

b

k

a

k

¶

 ! r 0  (' 0(()' 4( w01(4

C

k

(

X

k

4()4() 41 4'0&4 ()&(  &'0))(

4( + (  &'0))( (,0)-1(

δ

k

= E

cav

(k) − E

exc

(k)

X

k

=

Ã

δ

k

+

p

δ

2

k

+ Ω

2

R

2

p

δ

2

k

+ Ω

2

R

!

1/2

 ! r 3 

C

k

=





Ω

2

R

2

p

δ

2

k

+ Ω

2

R

³

δ

k

+

p

δ

2

k

+ Ω

2

R

´





1/2

 ! r 5  $' +&1()  0)40) 4( )0&u'0)

C

2

k

+ X

k

2

= 1

n ( @u0)() 4)'  2'( 4(' u04(' 01' &()4 1)( /0&u( 430)(

H =

X

k

{E

LP

(k)p

†

k

p

k

+ E

U P

(k)q

k

†

q

k

}

 ! r !9 

0 (' )(&3(' 4(' u04(' &0&(' )4(,(' = ( 8= 01& 0 (& (  1(&

0&0)' '9(,&u()

E

LP

(k) = E

exc

(k) +

δ

k

2

−

1

2

q

δ

2

k

+ Ω

2

R

 ! r !! 

E

U P

(k) = E

exc

(k) +

δ

k

2

+

1

2

q

δ

2

k

+ Ω

2

R

 ! r ! .

7( 013( 0@&() ()&( (' 4(1, 0'(1&' )1, ()&q)( 1) )&0'(r

u() &&'-1( 4)' ( 43&uu( 4(' )(&3(' 4(' u04(' &0&(' 41 '4'r

Au( +0 & 131&( ! n ! n 9 )&0'(u() (' 1'' +'2( 4)' 9('( 4('

k

n  4'(&'0) 4( 9(,0) &q 0uu( ( & &0&  (( 41 @00) (

-1( 90) (1 &(,&u(& () 4') -1(  u''( ((+( 41 @00) (' 2(101

1' (( -1( (( 4( 9(,0)n  131&( !

n .   &( () /0)0) 4( 9)3(

49)4()( & &0&  9)4()( )0&u( 01& 1) 4'0&4 )1 

k

= 0

n  0'0) &(+( 4(  4'(&'0) 4(' u04(' 401' ( & 0)'-1() 

4'(&'0) 4(' u04(' &0&(' 4()4 41 4'0&4  9)4()( )0&u(n 8)(

4('&0) 4(' &0&' )&(' 4(' 0&0)' 4( + (' 40))( 1 &r

3&@( p 41 '(0)4 @&(n

( 013( /0& 40))( 40) (1  4( )01+(1, 02?(' -1 (1+() &( (,'

)4+41((u() & 1) /'(1 '(& +( 9)(&3( ( ( +((1& 490)4( 4'n

 )1&( u,( 4( (' 02?(' (1 &( u041( 'u(u()() /0)0) 41 4'r

0&4 (,0)r+n(' (,&()(' 0)'4&(' &  '1( )( /0) )(&+()&-1(

 2&)@( 2''( 4( 0&0)'n  /0&u( () 1+(( (,-1( -19(( '0 &/0'

0)'4&( 0uu( 1) A3( 01& (' 0&0)'n ( u04(

&A' +0& &( 9(,&(''0) 491) @u -1)-1( u0)0u04(0) u0)&(

-1( ( &)( 49)(&14( 49w('()2(&3 0)41  4(' 110)' 4)' ('

u('1&(' 40) 90&4&( 4( 3&)4(1& (' 40)) &  u( -1)-1( ')4&4n

6) (,-1( ()'1( 0uu() 0)'&1&( 4(' ' 41 @u -1 &'()() 4('

110)' )/&(1&('   u( -1)-1( ')4&4 (' ' 0u&u'n

7(( 0)'&10) )&041 )1&((u() (' &0(''1' &u&-1(' 0uu(

1)( u@04( 4( 3)&0) 49' )0)r''-1(' 41 @un  4(&)A&( '(0)

(' 0)'&( 1 &(u() '(ur''-1( 4(' 110)'n

 QT NJIJT HKCDMFCNT

&%&% 8 '#- .(* "/""

6) '( ( 4)' ( 4&( 4(  '(0)4( -1)10)n6) 0)'4A&( 1) @u

u0)0u04( 4( /&-1()( 4( 4&(0) ( 4( 0&'0) 1,('

b

E(t) b

= −

1

_c

_∂t

∂

A(t) = i

b

E

0

2

(bae

−iωt

_{− ba}

†

_e

iωt

₎

 ! r !* 

ba

(

ba

†

'0) &('(+(u() (' 0&(1&' 49))@0) ( 4( &0) 491) @00) 4)' ( u04( 41 @u 4( /&-1()(

ω

n $' +&1()  &(0) 4( )0)r 0uu10)

[ba, ba

†

_{] = 1}

n  0)')( 4( )0&u'0)

E

₀

+1

q

~

_ω

2²

0

V

0

²

0

('  0)')( 4(&-1( 41 +4( (

V

( +01u( 4( -1)10)n7(( 0)')( 40))( 90&4&( 4( 3&)4(1& 41 @u (&-1( 0&&('0)4)  1) @00) 4)'

( +01u( 4( -1)10)n

6) )&041 (' 0&(1&' 4( -14&1&(-1+()' 1, 3&)4(1&' 1'(' ()

0-1( ''-1( 01&  &(&'()0) 491) @u 4)' ( &(A&( 4( &(')(

b

X = E

0

(ba + ba

†

)

 ! r ! 2

b

Y

= iE

0

(ba − ba

†

)

 ! r !$ 

-1( 90) &(u( 4)' 9(,&(''0) 41 @u (&-1(

b

E(t) = b

Xsin(ωt) + b

Y cos(ωt)



! r ! > (' 0&(1&'

b

X

(

b

Y

)( 0uu1() 'n$' )90) ' 49' &0&(' 0uu1)' ( ( @u (&-1(0u2)'0) )&( 4( (' 0&(1&')9  40) ' 49

2 Y

D

_q

X

Y

j

q

2 X

D

q

φ

θ

=1' &'u() 0) u0)&( -1( (' +&)(' 4(' 110)' 4(' 0&(1&' 4(

-14&1&( +&1() 1)( )3 49w('()2(&3

∆ b

X∆ b

_{Y ≥ E}

2

₀

 ! r !0 

E

2

0

0&&('0)4   u( -1)-1( ')4&4 40))( & ( 0uu1(1& 4('

4(1, -1)' 0) ?131(' 01& ('-1((' 9)3 (' &(n 7(( )3

(' +&1( 01& 1) 01( -1(0)-1( 4( -14&1&(' 0) ?131('n 8) 01( 4(

-14&1&(' (' &&' & 9 )3(

θ

+0 & 131&( ! n * 

b

X

θ

= E

0

(ba

w

e

iθ

+ ba

†

w

e

−iθ

)

 ! r !3 

b

Y

θ

= iE

0

(ba

w

e

iθ

− ba

†

w

e

−iθ

)

 ! r !5  &%&6 --, #"' /-,

9 41 +4( ?01( ( & ( 49 4( &/&()( 41 0) 4( +1( 4(' 110)'

-1)-1('n +(1& u04())( 41 @u 4)' (  (' )1(n (' 110)'

41 @u 4)' (  '0) (' uu(' 01& 01(' (' -14&1&(' ( &())() 

+(1& u)u( 10&'( & (')3'4( w

('()2(&3n7(' &0&' 4(' 1r

10)' 41 @u 41 +4( 41)''()  ''( 4(' ' 0@&()' 01& ('-1('

(' 110)' '0)   u( -1)-1( ')4&4 -1(-1( '0  -14&1&(n

6

θ

sq

4(  1uA&( 4(' '01&(' ''-1('n 7(( u( (' ()( & (,(u( & 1)

@u '(& &A' 1r4(''1' 41 '(1n

(' 110)' 491) 01( -1(0)-1( 4( -14&1&(' 41 @u 4)' 1)

 0@&() )0

|αi

+&1() )' (' &(0)' '1+)('

∆ b

X

_|αi,θ

= ∆ b

Y

_|αi,θ

= E

2

0

 ! r . 9   131&( ! n * u0)&( (' 110)' 491) @u 4)' 1)  -1(0)-1( -1( 90) (1 0u&(& 1, 110)' 491) @u 4)' 1)  0@&() 4( uu( u14( ( 4( uu( @'(n 9 &( 4(' 110)' 491)  0@&() ('  1'

(( 0''2( 10&'( & (' )3' 4( w('()2(&3n

&%&
--, #"8 .',

8) @u('44)'1) 0u&u1 '()'&3('1)( 4( '('-14&1&('

 4(' 110)' )/&(1&('   u( -1)-1( ')4&4n 6) + 0)'&1&( 

4(' ' 0u&u' -1 0) 4( 1'  &0& 49&( u)u1, n(n 9&( 4('

110)'41 @u 4)'( ('3  9&( u)u( 49)(&14(n9

0u&u (1 '( 0)'&1&( /0&u((u()  && 4( 90) 4( 90&(1& 4(

0u&(''0) '1& 1)  0@&()

|α, ξi = b

S(ξ)|αi



! r.

! 

0 0) 41) 90&(1& 4( 0u&(''0)

b

S(ξ) = exp [

1

2

(ξ

∗

_ba

2

_{− ξba}

†2

_)]

 ! r..

ξ = re

2ıθ

sq

(' 1) )0u2&( 0u(,( &2&&(n 6) &(&'()( '1&  131&( !

n2

9 1) 02()1 &  &)'/0&u0) 1)&(n  0u&(''0) u,u( (

40) 90)30) u,u('0) 02()1(' 01& ( 01( 4( -14&1&(' 49 )3(

θ

sq



∆X

_θ

2

_sq

= E

2

₀

e

−2r

et ∆Y

_θ

2

_sq

= E

2

₀

e

2r

 ! r. * 

r

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