EXCITON ENTANGLED-TWO-PHOTON ABSORPTION BY SEMICONDUCTOR QUANTUM WELLS L. J. Salazar 1 ., D. Guzmán 1,2 , F.J. Rodriguez 1 y L. Quiroga 1 Condensed Matter Physics Group. http://fimaco.uniandes.edu.co Quantum Optics Lab. http://opticacuantica.uniandes.edu.co International Workshop on Quantum Coherence and Decoherence (IWQCD-Cali 24-28 Sept. 2012) 1 Departamento de Física Universidad de los Andes. 2 Laboratorio de Óptica Cuántica Universidad de los Andes
Transcript
EXCITON ENTANGLED-TWO-PHOTON ABSORPTION BY SEMICONDUCTOR QUANTUM
WELLS L. J. Salazar1., D. Guzmán1,2, F.J. Rodriguez1 y L. Quiroga1
International Workshop on Quantum Coherence and Decoherence (IWQCD-Cali 24-28 Sept. 2012)
1 Departamento de Física Universidad de los Andes. 2 Laboratorio de Óptica Cuántica Universidad de los Andes
arXiv:1109.6895
Outline
• Motivation. • Theoretical model to calculate absorption of Entangled photons -Biphoton wave function. - Continuos transitions. - Continuos and excitonic transitions • Results. • Conclusions.
SPDC(Spontaneous Parametric Down Conversion) Birth of entangled photons
- You shine with a blue laser on a BBO (Beta Barium Borate) crystal. In the crystal, one blue photon (about 400nm wavelength) is split up into two red photons (800nm wavelength). This process is called "parametric down conversion" -Orthogonally polarized photons leave two wavelengths which satisf ies conservation of momentum and energy. -Birefringent nonlinear crystal to produce specific type SPDC type II.
MOTIVATION
Type I
In SPDC type-I the pump photon has an orthogonal polarization with respect to idler and signal photons which are equally polarized (non-collinear case). Due to the fact that the two photons have the same polarization they will have the same refractive index and thus lie on the same cone.
Nonlinear crystal
Signal
Idler
Type II
In SPDC type-II the idler and signal have orthogonal polarizations. Due to the fact that the refractive indices of the two created photons are different the cones are not coaxial.
Nonlinear crystal
Signal
Idler
Degenerate case: The wavelength of both idler and signal are double the pump wavelength. By changing the angle of the pump with respect to the crystal the emission cone changes its size.
Again the size of the cones depends on the angle between pump beam and the optical axis of the crystal. Here the collinear case is, when the two emission cones are tangential. Thus idler and signal can be emitted in pump beam direction
Spontaneous Parametric Down Conversion
Entangled Photons Type-I and Type II, produced in Quantum Optics Lab. at Uniandes
Entangled Photons In the intersection between the two rings the photons cannot be distiguished, althought each photon pair must be orthogonal, also cannot be distinguished by momentum, therefore we get entangled photons in polarization
Blue-Laser
Photon pairs
no-linear crystal. Tipo II
MOTIVATION
Entangled Two Photon Absorption
13 de abril de 2011
1. Desarrollo teorico Nakanishi extendido a varios niveles
El campo electromagnetico propagandose en una dimension, con polarizacion !, se define en modoscontinuos mediante la expresion:
"E(z, t) =! !
"!d#
"h#
4$!0cAa(#)e"i!(t"z/c)! + H.C. (1)
En donde el operador de aniquilacion satisface la siguiente relacion de conmutacion:
[a(#), a†(##)] = %(# ! ##) (2)
El campo electrico se relaciona con el potencial vector "A:
"E = !& "A
&t(3)
"A(z, t) = i
! !
"!d#
"h
4$!0cA#a(#)e"i!(t"z/c)! + H.C. (4)
Si se asume que el ancho de banda del campo es menor a la frecuencia de la portadora #, el termino conla raız cuadrada puede tomarse como constante. De este modo el potencial vector puede escribirse:
"A(z, t) = i
"h
4$!0cA#
! !
"!d# a(#)e"i!(t"z/c)! + H.C. (5)
Definiendo el operador a(t) como:
a(t) " 1#2$
! !
"!d# a(#)e"i!t, (6)
se puede reescribir el potencial vector una manera mas sencilla:
"A(z, t) = i
"h
2!0cA#a(t! z/c)! + H.C (7)
1.1. Transicion de dos fotones en un sistema de varios niveles
El Hamiltoniano del sistema en imagen de interaccion esta dado por la expresion:
HI(t) = !#
"
q"
m"
"P"(t) · "A(t) (8)
Usando teorıa de perturbacion a segundo orden:
1
Entangled Two Photon Absorption
13 de abril de 2011
1. Desarrollo teorico Nakanishi extendido a varios niveles
El campo electromagnetico propagandose en una dimension, con polarizacion !, se define en modoscontinuos mediante la expresion:
"E(z, t) =! !
"!d#
"h#
4$!0cAa(#)e"i!(t"z/c)! + H.C. (1)
En donde el operador de aniquilacion satisface la siguiente relacion de conmutacion:
[a(#), a†(##)] = %(# ! ##) (2)
El campo electrico se relaciona con el potencial vector "A:
"E = !& "A
&t(3)
"A(z, t) = i
! !
"!d#
"h
4$!0cA#a(#)e"i!(t"z/c)! + H.C. (4)
Si se asume que el ancho de banda del campo es menor a la frecuencia de la portadora #, el termino conla raız cuadrada puede tomarse como constante. De este modo el potencial vector puede escribirse:
"A(z, t) = i
"h
4$!0cA#
! !
"!d# a(#)e"i!(t"z/c)! + H.C. (5)
Definiendo el operador a(t) como:
a(t) " 1#2$
! !
"!d# a(#)e"i!t, (6)
se puede reescribir el potencial vector una manera mas sencilla:
"A(z, t) = i
"h
2!0cA#a(t! z/c)! + H.C (7)
1.1. Transicion de dos fotones en un sistema de varios niveles
El Hamiltoniano del sistema en imagen de interaccion esta dado por la expresion:
HI(t) = !#
"
q"
m"
"P"(t) · "A(t) (8)
Usando teorıa de perturbacion a segundo orden:
1U (2)
I (t) = ! 1h2
! !
"!
! !
"!dt2 dt1 HI(t2)HI(t1)!(t2 ! t1) (9)
en donde !(t) corresponde a la funcion escalon de Heaviside definida por:
!(t) ="
1 si t " 00 si t < 0
El estado del sistema en un instante t esta dado por:
|"(t)# = U (2)I (t)|Ri# $ |Mg# (10)
en donde la materia se encuentra inicialmente en el estado |Mg# y el campo en el estado |Ri#.La probabilidad de transicion de un estado inicial |Ri#$ |Mg# a uno final |Rf #$ |Mf # esta dada en general
en donde {#n, |n#} son valores y vectores propios del Hamiltoniano de la materia H0 (se sobreentientesuma entre ındices repetidos segun la convencion de Einstein).
Reemplazando los operadores $A en terminos de los operadores de aniquilacion a(t), se obtiene:
%Rf |A!(t1)Aµ(t2)|Ri# =$ h
2%0A#
%%Rf |a1(t1)a2(t2)|Ri# (12)
El campo |"# contiene un par de fotones enredados. El par es absorbido (|"# ' |0#) por la materia(|Ri# ' |Rf #). Se define entonces la funcion del bifoton:
%0|a1(t1)a2(t2)|"# = "(t1, t2) (13)
Por otro lado, los operadores correspondientes a la materia se denotan por:
%f |P ! |n# = P !f,n (14)
%n|Pµ|g# = Pµn,g (15)
Reemplazando las contribuciones de radiacion y materia se obtiene:
La funcion escalon de Heaviside !(t2 ! t1) define un orden: se absorbe primero el foton que correspondea la transicion interbanda y posteriormente el foton que genera la transicion intrabanda (figura 1).
La expresion anterior puede escribirse de forma mas simple si se define la siguiente transformada deFourier (Nakanishi):
2
U (2)I (t) = ! 1
h2
! !
"!
! !
"!dt2 dt1 HI(t2)HI(t1)!(t2 ! t1) (9)
en donde !(t) corresponde a la funcion escalon de Heaviside definida por:
!(t) ="
1 si t " 00 si t < 0
El estado del sistema en un instante t esta dado por:
|"(t)# = U (2)I (t)|Ri# $ |Mg# (10)
en donde la materia se encuentra inicialmente en el estado |Mg# y el campo en el estado |Ri#.La probabilidad de transicion de un estado inicial |Ri#$ |Mg# a uno final |Rf #$ |Mf # esta dada en general
en donde {#n, |n#} son valores y vectores propios del Hamiltoniano de la materia H0 (se sobreentientesuma entre ındices repetidos segun la convencion de Einstein).
Reemplazando los operadores $A en terminos de los operadores de aniquilacion a(t), se obtiene:
%Rf |A!(t1)Aµ(t2)|Ri# =$ h
2%0A#
%%Rf |a1(t1)a2(t2)|Ri# (12)
El campo |"# contiene un par de fotones enredados. El par es absorbido (|"# ' |0#) por la materia(|Ri# ' |Rf #). Se define entonces la funcion del bifoton:
%0|a1(t1)a2(t2)|"# = "(t1, t2) (13)
Por otro lado, los operadores correspondientes a la materia se denotan por:
%f |P ! |n# = P !f,n (14)
%n|Pµ|g# = Pµn,g (15)
Reemplazando las contribuciones de radiacion y materia se obtiene:
La funcion escalon de Heaviside !(t2 ! t1) define un orden: se absorbe primero el foton que correspondea la transicion interbanda y posteriormente el foton que genera la transicion intrabanda (figura 1).
La expresion anterior puede escribirse de forma mas simple si se define la siguiente transformada deFourier (Nakanishi):
2
U (2)I (t) = ! 1
h2
! !
"!
! !
"!dt2 dt1 HI(t2)HI(t1)!(t2 ! t1) (9)
en donde !(t) corresponde a la funcion escalon de Heaviside definida por:
!(t) ="
1 si t " 00 si t < 0
El estado del sistema en un instante t esta dado por:
|"(t)# = U (2)I (t)|Ri# $ |Mg# (10)
en donde la materia se encuentra inicialmente en el estado |Mg# y el campo en el estado |Ri#.La probabilidad de transicion de un estado inicial |Ri#$ |Mg# a uno final |Rf #$ |Mf # esta dada en general
en donde {#n, |n#} son valores y vectores propios del Hamiltoniano de la materia H0 (se sobreentientesuma entre ındices repetidos segun la convencion de Einstein).
Reemplazando los operadores $A en terminos de los operadores de aniquilacion a(t), se obtiene:
%Rf |A!(t1)Aµ(t2)|Ri# =$ h
2%0A#
%%Rf |a1(t1)a2(t2)|Ri# (12)
El campo |"# contiene un par de fotones enredados. El par es absorbido (|"# ' |0#) por la materia(|Ri# ' |Rf #). Se define entonces la funcion del bifoton:
%0|a1(t1)a2(t2)|"# = "(t1, t2) (13)
Por otro lado, los operadores correspondientes a la materia se denotan por:
%f |P ! |n# = P !f,n (14)
%n|Pµ|g# = Pµn,g (15)
Reemplazando las contribuciones de radiacion y materia se obtiene:
La funcion escalon de Heaviside !(t2 ! t1) define un orden: se absorbe primero el foton que correspondea la transicion interbanda y posteriormente el foton que genera la transicion intrabanda (figura 1).
La expresion anterior puede escribirse de forma mas simple si se define la siguiente transformada deFourier (Nakanishi):
2
U (2)I (t) = ! 1
h2
! !
"!
! !
"!dt2 dt1 HI(t2)HI(t1)!(t2 ! t1) (9)
en donde !(t) corresponde a la funcion escalon de Heaviside definida por:
!(t) ="
1 si t " 00 si t < 0
El estado del sistema en un instante t esta dado por:
|"(t)# = U (2)I (t)|Ri# $ |Mg# (10)
en donde la materia se encuentra inicialmente en el estado |Mg# y el campo en el estado |Ri#.La probabilidad de transicion de un estado inicial |Ri#$ |Mg# a uno final |Rf #$ |Mf # esta dada en general
en donde {#n, |n#} son valores y vectores propios del Hamiltoniano de la materia H0 (se sobreentientesuma entre ındices repetidos segun la convencion de Einstein).
Reemplazando los operadores $A en terminos de los operadores de aniquilacion a(t), se obtiene:
%Rf |A!(t1)Aµ(t2)|Ri# =$ h
2%0A#
%%Rf |a1(t1)a2(t2)|Ri# (12)
El campo |"# contiene un par de fotones enredados. El par es absorbido (|"# ' |0#) por la materia(|Ri# ' |Rf #). Se define entonces la funcion del bifoton:
%0|a1(t1)a2(t2)|"# = "(t1, t2) (13)
Por otro lado, los operadores correspondientes a la materia se denotan por:
%f |P ! |n# = P !f,n (14)
%n|Pµ|g# = Pµn,g (15)
Reemplazando las contribuciones de radiacion y materia se obtiene:
La funcion escalon de Heaviside !(t2 ! t1) define un orden: se absorbe primero el foton que correspondea la transicion interbanda y posteriormente el foton que genera la transicion intrabanda (figura 1).
La expresion anterior puede escribirse de forma mas simple si se define la siguiente transformada deFourier (Nakanishi):
2
U (2)I (t) = ! 1
h2
! !
"!
! !
"!dt2 dt1 HI(t2)HI(t1)!(t2 ! t1) (9)
en donde !(t) corresponde a la funcion escalon de Heaviside definida por:
!(t) ="
1 si t " 00 si t < 0
El estado del sistema en un instante t esta dado por:
|"(t)# = U (2)I (t)|Ri# $ |Mg# (10)
en donde la materia se encuentra inicialmente en el estado |Mg# y el campo en el estado |Ri#.La probabilidad de transicion de un estado inicial |Ri#$ |Mg# a uno final |Rf #$ |Mf # esta dada en general
en donde {#n, |n#} son valores y vectores propios del Hamiltoniano de la materia H0 (se sobreentientesuma entre ındices repetidos segun la convencion de Einstein).
Reemplazando los operadores $A en terminos de los operadores de aniquilacion a(t), se obtiene:
%Rf |A!(t1)Aµ(t2)|Ri# =$ h
2%0A#
%%Rf |a1(t1)a2(t2)|Ri# (12)
El campo |"# contiene un par de fotones enredados. El par es absorbido (|"# ' |0#) por la materia(|Ri# ' |Rf #). Se define entonces la funcion del bifoton:
%0|a1(t1)a2(t2)|"# = "(t1, t2) (13)
Por otro lado, los operadores correspondientes a la materia se denotan por:
%f |P ! |n# = P !f,n (14)
%n|Pµ|g# = Pµn,g (15)
Reemplazando las contribuciones de radiacion y materia se obtiene:
La funcion escalon de Heaviside !(t2 ! t1) define un orden: se absorbe primero el foton que correspondea la transicion interbanda y posteriormente el foton que genera la transicion intrabanda (figura 1).
La expresion anterior puede escribirse de forma mas simple si se define la siguiente transformada deFourier (Nakanishi):
2
2
ture showing new features associated to a transformeddensity of available states as well as to inter-particle inter-action e!ects. As additional facts, it should be mentionedthat since TPA process provides information about cor-relation functions of the radiation field, QWs nanostruc-tures may be useful for the in-situ detection of photonentanglement produced by other on-chip semiconductorsources.
Spectral filters introduced in the paths of the signaland idler photons allow for the production of narrow-band temporally entangled photons. These kind of en-tangled photons are the ones to be considered in thiswork. Additional control mechanisms on the biphotonwavefunction shape, including the engineering of the spa-tial profile of the pump laser and selecting the geometryof the nonlinear crystal, provide useful tools to manipu-late the two-photon absorption profile of semiconductornanostructures18–20.
In the present work we perform a comparative studyof TPA by semiconductor nanostructures, for light withwave-like classical properties (laser light) and a new kindof quantum entangled light. We limit ourselves to thecase when one-photon absorption is excluded. The pa-per is organized as follows: In Section II we briefly re-view the general theory for the TPA for laser light andextend it to the case of entangled photons. In Sec-tion III we study the entangled-photon absorption fornon-interacting electron-hole pairs QWs, and finally weconsider the Coulomb interaction between electrons andholes (exciton e!ects). In Section IV, the main conclu-sions of the present work are summarized.
II. THEORETICAL BACKGROUND
The interaction of light with matter systems has beenstudied since the first days of the modern quantum the-ory. A semiclassical (or semi-quantum) approach inwhich the light is treated classically but the matter isquantum-mechanically described, has been mostly used.With the development of new sources of radiation, someof them producing light which cannot be considered interms of classical wave models, new theories for the un-derstanding of the interaction of these new forms of lightwith matter are under development. We begin by brieflyreviewing the standard Born approximation for light-matter interaction.
The radiation-matter Hamiltonian is given by
H =!
!
"!P! ! q!
!A(!r!, t)#2
2m!+
!
"k
!""k(a†"ka"k +12) (1)
where # denotes particle indexes, a†"k and a"k describe cre-ation and annihilation photon operators in mode !k, re-spectively. In the dipolar approximation the potentialvector becomes
!A(!r!, t) = !A(t) = !A(+)(t) + !A(!)(t) (2)
with
!A(+)(t) =$
!2V
!
"k
!$"k""k
a"ke!i#!kt (3)
where V is the radiation quantization volume and !$"k theunitary vector indicating the polarization of mode !k. Ad-
ditionally, !A(!)(t) =%!A(+)(t)
&†. Neglecting quadratic
terms in the potential vector, the Hamiltonian in Eq.(1)can be written as
H = H0 + HI (4)
where H0 denotes the matter Hamiltonian and theradiation-matter interaction term, in the interaction pic-ture, is given by
HI(t) = !!
!
q!
m!c!P!(t) · !A(t) (5)
By invoking the standard second-order time-dependentperturbation theory, the evolution operator turns out tobe
U (2)I (t) = ! 1
!2
' t
0dt1
' t
0dt2"(t1 ! t2)HI(t1)HI(t2) (6)
with "(t) representing the Heaviside step function. Westart by considering the evolution from an initial sepa-rate state where the radiation is in state |Ri > whilethe matter system is initially in its ground state |Mg >.Therefore, the whole system starts in state |#(0) >=|Ri > "|Mg >. At time t the system has evolved to thestate
|#(t) >= U (2)I (t)|Ri > "|Mg > (7)
We are interested in the transition probability from thegiven initial state to a final state denoted by |Rf >"|Mf >
Pf,i(t) =(((< Mf |" < Rf |U (2)
I (t)|Ri > "|Mg >(((2
(8)
which can also be written as7
P (2)f,i (t) =
((((' t
0dt1
' t
0dt2 < Rf |A†
$(t1)A†µ(t2)|Ri > M$,µ(t1, t2)
((((2
(9)
Where α denotes particle indexes a+k (ak) describe creation and (annihilation)
photon operators in mode k
Theoretical approach
Matter-Radiation interaction
!!"!#$
!!$!#"
%&'&()*+,-
%('(.,/0-
%1'1231-
45*!,(*/+06!!"#6708&9+,6
"$&
6
The role played by the entanglement time is to mod-ulate the oscillator strength for a given transition. Insome cases this e!ect can be as drastic as to eliminatean allowed two-photon transition. In other words, selec-
tion rules for transitions between electronic states can bemanipulated as a consequence of the entanglement time.
Now we calculate the entangled two photon absoprtionfor SPDC-I and II. Starting from the general formula (9)
P (2)f,i (t) =
!!!!" t
0dt1
" t
0dt2 < Rf |A†
!(t1)A†µ(t2)|Ri > M!,µ(t1, t2)
!!!!2
=!!!!!
" t
0dt1
" t
0dt2 < Rf |A†
!(t1)A†µ(t2)|Ri > "(t1 ! t2)
# e
!mc
$2 %
n
P !f,nPµ
n,ge!i("n!"f )t1!i("g!"n)t2
!!!!!
2
(27)
The terms in the integral can be arranged as a new vari-able Sf,i(t), with P (2)
f,i (t) = |Sf,i(t)|2
S(2)f,i (t) =
# e
!mc
$2 %
n
P !f,nPµ
n,g
" t
0dt1
" t1
0dt2 < Rf |A†
!(t1)A†µ(t2)|Ri > e!i("n!"f )t1!i("g!"n)t2 (28)
Acording to (12), the vector potential operator con-tibutes with the a(!2,"#2)a(!1,"#1)|# >, where |# > is thebiphoton wave function. This function can be changedif Type I or Type II are considered. For SPDC-II thetime wave function is well known2 and it contributes to
the the terms corresponding to light in expresion (28)< Rf |A†
!(t1)A†µ(t2)|Ri >. If we consider now a biphoton
instead a coherent state, the matrix element for light arewritten as
In the last equation we take into account for the CWbiphoton wave function2. It is important to remark that
for any biphoton function the wave plane terms have been
Theoretical model for entangled photpn absorption: Biphoton wave function. Continuos transitions. excitons and continuous transitions
E(k)
k
Matter
Incident entangled photons
2.3. Reglas de seleccion
Por convencion se dice que el campo tiene polarizacion paralela cuando esta es paralela al planoen donde no hay confinamiento. Por otro lado, el campo tiene polarizacion perpendicular cuando lapolarizacion es perpendicular al plano en donde no hay confinamiento. En la figura (2) se presenta comoejemplo un esquema para el caso de la polarizacion paralela.
Es importante hacer tal diferenciacion porque los elementos P !f,n y Pµ
n,g tienen diferentes expresionessegun la polarizacion del campo.
Figura 2: Convencion polarizacion haz de luz incidente. Se muestra el caso de polarizacion paralela.
2.3.1. CASO: polarizacion perpendicular
Se supone polarizacion en la direccion z.
P zn,g = !"k,"k!!n,n!P (34)
P zf,n = !n,n!f
memc
2ihL [1! (!1)n+nf ] n nf
n2"n2f[1! !n,nf ] (35)
2.3.2. CASO: polarizacion paralela
P zn,g = !"k,"k!!n,n!P (36)
P xf,n = m0
mch|"k| (37)
3. Algoritmo en MATLAB
El algoritmo debe evaluar la expresion (33) con las reglas de seleccion correspondientes a la polarizacionque se seleccione.
El programa calcula la probabilidad de absorcion de dos fotones P2 para diferentes valores de las frecuen-cias de los fotones signal y idler. El resultado se desplega en una matriz de 10" 10 con ejes #1 (signal) y #2
(idler). Un ejemplo de tal resultado se presenta en la figura (3).
5
Entangled light interacting with semiconductor nanostructures: quantum wells
• Interaction between biphotons and semiconductor quantum well
• Probability of two-photon absorption – Does the absorption depend on quantum correlations? – Can it be controlled?
Radiation wavefunction
Biphoton wave function
Fourier transform for SPDC-II
where
– – – T : Pump coherence time (inverse of pump bandwidth) – τ : Quantum correlation time (depends on the crystal lenght and propagation velocities)
Tailoring Quantum-Correlated Two-Photon
Matter and light parameters (experimental)
• Light Type-II SPDC bifoton _
– – Degenerate case:
• Matter GaAs semiconductor quantum well – Exciton Bohr Radius 3D. – Energy Gap: – Quantum well width:
SPDC biphotons type II
• Schmidt Number (amount of entanglement)
Biphoton joint intensity
ωs /ωp
ωi /ωp
Frequency anticorrelated (FAC) Non-entangled (K=1)
Frequency correlated(FC)
Two photon transition in quantum wells
Non-excitonic 2 photon absorption
Excitonic 2 photon absorption
Ci Conduc'onsubband
HHi Heavyholesubband
Xi Discreteexcitonicstate
Continuos transitions
Discrete and Continuos transitions
(a)
0 2 4 6 8 100
1
2
3
4
5
!!T
K
6
The role played by the entanglement time is to mod-ulate the oscillator strength for a given transition. Insome cases this e!ect can be as drastic as to eliminatean allowed two-photon transition. In other words, selec-
tion rules for transitions between electronic states can bemanipulated as a consequence of the entanglement time.
Now we calculate the entangled two photon absoprtionfor SPDC-I and II. Starting from the general formula (9)
P (2)f,i (t) =
!!!!" t
0dt1
" t
0dt2 < Rf |A†
!(t1)A†µ(t2)|Ri > M!,µ(t1, t2)
!!!!2
=!!!!!
" t
0dt1
" t
0dt2 < Rf |A†
!(t1)A†µ(t2)|Ri > "(t1 ! t2)
# e
!mc
$2 %
n
P !f,nPµ
n,ge!i("n!"f )t1!i("g!"n)t2
!!!!!
2
(27)
The terms in the integral can be arranged as a new vari-able Sf,i(t), with P (2)
f,i (t) = |Sf,i(t)|2
S(2)f,i (t) =
# e
!mc
$2 %
n
P !f,nPµ
n,g
" t
0dt1
" t1
0dt2 < Rf |A†
!(t1)A†µ(t2)|Ri > e!i("n!"f )t1!i("g!"n)t2 (28)
Acording to (12), the vector potential operator con-tibutes with the a(!2,"#2)a(!1,"#1)|# >, where |# > is thebiphoton wave function. This function can be changedif Type I or Type II are considered. For SPDC-II thetime wave function is well known2 and it contributes to
the the terms corresponding to light in expresion (28)< Rf |A†
!(t1)A†µ(t2)|Ri >. If we consider now a biphoton
instead a coherent state, the matrix element for light arewritten as
In the last equation we take into account for the CWbiphoton wave function2. It is important to remark that
for any biphoton function the wave plane terms have been
€
Ψ(ω1,ω2) = 2 τTπe−(ω1−ω2−ω p
0 )T 2 e−(ω1−ω2−δω p )
τ 2
4 +2iπD(ω1 −ω2 −δω p )
τ2)
δω p =ωsignal −ω idler ; ω p0 =ωsignal +ω idler
€
K =12τT
+Tτ
K: Schmidt number T : Pump coherence time (inverse of pump bandwidth) t : Quantum correlation time
€
Ψ(ω1,ω2)
RESULTS
Biphoton joint density
(a)
0 2 4 6 8 100
1
2
3
4
5
!!TK
€
Ψ(ω1,ω2)2 RESULTS
6x105
3x105
3x105
1x105
1.5x105
1x104
Non-excitonic effects
!"# !"$ !"% !"& !"' (
!!!)
!
*+(!!!
#+(!!!
,-./
!"# !"$ !"% !"& !"'
!#+(!!0
#+(!!"
1!,-./
!"$ !"% !"& !"' (!
0+(!!"
!g"2#!
33(!45
335!4(
33*!45
!g"2#!!
(a)
0 2 4 6 8 100
1
2
3
4
5
!!T
K
€
K =12τT
+Tτ
RESULTS
0
3x10!!
6x10!!
0
7.5x10!"
1.5x10!#
0
2x10!#
4x10!#
ETPA
0
5x10!#
0.48 0.6 0.72
!!!g
0
3x10!"
6x10!"
0.48 0.6 0.72
!!!g
0
3.5x10!!
L=aB/2
L=aB
L=2aB
WITH excitonic effects
(a)
0 2 4 6 8 100
1
2
3
4
5
!!T
K
RESULTS
0
0.05
0
0.05
ETPA
0.5 0.6 0.7
!!!g
0
0.05
0.5 0.6 0.7
!!!g
K=1 K=1.1
K=1.2 K=1.3
K=1.4 K=1.5
(a)
0 2 4 6 8 100
1
2
3
4
5
!!T
K
With excitonic effects: Complete spectra
RESULTS
CONCLUSIONS
• Two photon transitions to excitons in semiconductors QWs by absorbing an entangled biphoton have been addressed
• Analytical expresions for the ETPA rate including proper second-order correlation functions.
• We have demonstrated that optical non-linear effects at the level of single couple of photons can be observed by playing with the correlation time window of cross polarized photons emitted by a type-II SPDC process.
• We found that a high ETPA rate occurs when the sum of frequencies of the signal/idler photon pair is on resonance with the discrete QW transitions.
• Semiconductor nanostructures like QWs provide new large-bandwidth multiple-photon detectors where both discrete and continuous carrier states play important roles in discriminating the degree of photon entanglement.
• Is possible to explore new transitions that ususally are violated in traditional optical experiments.
• Solid state system can be used as a controllable single photon sources.
![Exciton diffusion, end quenching, and exciton …exciton binding energy can be larger than a third of the band gap energy [4–6], making them stable even at room temperature. The](https://static.documents.pub/doc/80x56/5f9e847eddf44d4ccd689439/exciton-diffusion-end-quenching-and-exciton-exciton-binding-energy-can-be-larger.jpg)
