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Подход эффективного гамильтониана. 1 . М. С. Лифшиц, ЖЭТФ ( 1957 ). 2 . U.Fano, Phys. Rev. 124, 1866 (1961). 3 . H. Feshbach,, Ann. Phys. (New York) 5 (1958) 357 ; 19 (1962) 287. 4 . C. Mahaux, H.A. Weidenmuller, ( Shell-Model Approach to Nuclear - PowerPoint PPT Presentation
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1. М. С. Лифшиц, ЖЭТФ (1957).2. U.Fano, Phys. Rev. 124, 1866 (1961).3. H. Feshbach,, Ann. Phys. (New York) 5 (1958) 357; 19 (1962) 287.4. C. Mahaux, H.A. Weidenmuller, (Shell-Model Approach to Nuclear Reactions), North-Holland, Amsterdam, 1969.5. I.Rotter, Rep. Prog. Phys., 54, 635 (1991).6. S.Datta, (Electronic transport in mesoscopic systems) (1995).
7. S. Albeverio, et al J.Math. Phys. 37, 4888 (1996).8. Y.V. Fyodorov and H.-J. Sommers, J. Math. Phys. 38, 1918 (1997)
9. F. Dittes, Phys. Rep. (2002).10. Sadreev and I. Rotter, J.Phys.A (2003).11. J. Okolowicz, M. Ploszajczak, and I. Rotter, Phys. Rep. 374, 271(2003).12. D.V. Savin, V.V. Sokolov V.V., and H.-J. Sommers, PRE (2003). 13. Sadreev, J.Phys.A (2012).
• Coupled mode theory (оптика)H.A.Haus, (Waves and Fields in Optoelectronics) (1984).C. Manolatou, et al, IEEE J. Quantum Electron. (1999).S. Fan, et al, J. Opt. Soc. Am. A20, 569 (2003).S. Fan, et al, Phys. Rev. B59, 15882 (1999).W. Suh, et al, IEEE J. of Quantum Electronics, 40, 1511 (2004).
Bulgakov and Sadreev, Phys. Rev. B78, 075105 (2008).
Подход эффективного гамильтонианаПодход эффективного гамильтониана
Coupled defect mode with propagating over waveguide light
Manolatou, et al, IEEE J. Quant. Electronics, (1999)
Coupled mode theory
Одно модовый резонатор
CMT• Х. Хаус, Волны и поля в оптоэлектронике
0
2
( )
, W=|a| .
i tin
out in
dai a kS e
dt
S CS a
Одно-модовый резонатор
22 2 2 2| |
2 | | | | | |out
dW d aa S a
dt dt
= 2 0
0
( ) exp( ), ( ) ,
, 2in
in
a t a i t i i a kS
i a kS
Инверсия по времени2 2
2 2| |2 | | | |
2 in
d a ka S
dt
2k
22 2 2 * *
2 2 2 2 * *
* * * * 2
2 2 2 * *
| || | | | 2 | | 2 ( ).
2 ,
2 | | | | | | ( ),
2 ( ) 2 | | ,
| | | | ( 1)( ) 2 |
in out in in
out in
out in in out in out
in in in out in out in
out in in out in out in
d aS S a a S aS
dt
a S CS
a S C S C S S S S
a S aS S S S S C S
S C S C S S S S C S
2 2 2| | | | |in outS S
1C
0( ) 2
2
i tin
out in
dai a S e
dt
S S a
CMT• Много-модовый резонатор
IEEE J. Quantum Electronics, 40, 1511 (2004)40, 1511 (2004)
Два порта, две моды
1 2 1 1 1 2
1 2 2 2
1 1; D= ; 0.5 ;
1 12 K D D
%CMT for transmission through resonator with two modesclear allE=-2:0.01:2;D=[sqrt(0.1) sqrt(0.25) sqrt(0.1) sqrt(0.25)];G=0.5*D'*D;H0=diag([-0.25 0.25]);H=H0-1i*G;for j=1:length(E)Q=E(j)*diag([1 1])-H;in=[1; 0];IN=1i*D'*in;A=Q\IN;;A1(j)=A(1); A2(j)=A(2);t(:,j)=-in+D*A;end
T волновод с двумя резонаторами, Булгаков, Садреев, Phys. Rev. B84, 155304 (2011)
'
1| | ' 2 | | '
0CCeff
eff B
C S C i C W W CE i H
H H i W W
W is matrix NxM where N is the number of eigen states of closed quantum system, M is the number of continuums (channels)
1
1 1 1 1
1 1
1 1 1
1 1
1 11
1 11
1,
0
0 ;
( 0 ) 0,
( ) 1.
1,
0
1[ ] 1.
0
1
0
B CC
B B
B B B B
B B B
B B B B
B B B
B B B B
Beff
G H H HE i H
G G E i H V
G G V E H
E i H G V G
E H G V G
G V GE i H
G E H V VE i H
GE i H
,
,
21D box: ( ) sin ;
1 1
1Leads: the left: ( ) sin (1 );
2 | sin |
1 the right: ( ) sin ( );
2 | sin |
n
E L
E R
njj
N N
j k jk
j k j Nk
1 2 3 4
2 2
1 11
1 1 1 1 11
1 1 1 1 2 2 1 1 3 3 1 4 41
2 2 2
1 1 12 21 12 2
2
2 exp
1
0
1
0
1
0
1 ( / 2)1(1) (1) sin (1) (1)
0 0
exp( )
( )
B B
B B
B Bj j j j
t tm n m n
eff B C CC
W V VE i H
m W n dE m V E E V nE i E
dE m j j V j j E E j j V j j nE i E
EdE k dE
E i E E i E
t ik
H H t ik
S.Datta, (Electronic transport in mesoscopic systems) (1995).
0
'
2
2
2
2
2
, 2
| |, | '
| , , |, < , |E',L>= ( ')
| , , |, < , |E',R>= ( ')
( , ) | , | . .
B L R
B n nnn
L
R
nC L R n
H H H H
H E n n n n
H dEE E L E L E L E E
H dEE E R E R E R E E
V dE V E C E C n C C
0
0 0 0
( ) ,
, ,
E H V
H E H E
0B CC leads
H H H V H V
1 10 0 0| ( 0 ) | |E i H V G V
1 10 0 0 [1 ( 0 ) ] | |или E i H V F
Уравнение Липпмана-Швингера
Проекционные операторы:2
2
| , , |; | |;C Bn
P dE E C E C P n n
0
| ,
| 0 ;
| ,
E L
E R
| |
| |
| |
0 0
0 ;
0 0
L L
B B
R R
L L L B L R LB
B L B B B R BL BR
R L R B R R RB
P
P
P
P VP P VP P VP V
V P VP P VP P VP V V
P VP P VP P VP V
0
1 1 - 0
0
1 1 11 - 1 -
0 0 0
1 0 - 1
0
LBL
BL BRB B
RBR
VE i H
F V V VE i H E i H E i H
VE i H
1 10
|| ,
| | 0 | .
| , |
L
B
R
E L
F F
E R
1
1 1 1 11
1 G G ;
D1 1 1 1
1+
1;
0
LB BL LB LB BRL L L
BL BR
RB BL RB RB BRR R R
eff
V GV V V GVE H E H D E H
F V V
V GV V V GVE H E H D E H
GE i H
1 1 1 1[1 ] , , ;
1 1 1 1, [1 ] , ;
1 1, , ;
L LB BL LB BRL eff L eff
R LB BL LB BRL eff L eff
B BL BReff eff
V V E L V V E RE H E H E H E H
V V E L V V E RE H E H E H E H
V E L V E RE H E H
S-matrix
' ' '
12 , | | ', ;
'0CC CC CB BCeff
r tS i E C V V C E
t rE i H
Basis of closed billiard
*
| |
12 ( , ) | | ( , )
0
B m
m nmn eff
H m E m
t i V E L m n V E RE i H
The biorthogonal basis
*'| ) | ), | )=| >, ( |=< | , ( | ')= ;
| )( |
, | | )( | ,2
0
eff
eff
H z
P
E L V V E Rt i
E i z
c H.-W.Lee, Generic Transmission Zeros and In-Phase Resonances in Time-Reversal Symmetric Single Channel Transport, Phys. Rev. Lett. 82, 2358 (1999)
2d case
Limit to continual case
L
R
2
1
2
1
mm' nn'p
p
, | |m',n'>= - ( , , ) ( ' ', ) exp( )
- ( , , ) ( ', ', ) exp( ) ;
( , , ) (1) ( ) ( ).
( , , ) ( ) ( ) ( ).
L
L
R
R
eff mn L L L L p
R R R R p
N
L L m n pj N
N
R R m x n pj N
m n H E W m n p W m n p ik
W m n p W m n p ik
W m n p v j j
W m n p v N j j
2 2
1 1
2
21, 2cos 2cos( ) 2 2cos( ) 2 ;1
cos( ) 1; /2, exp( ) cos( ) sin( ) ;
( , , ) (1) ( ) ( ) ( (1) (0))
C p pC C
p p p p p
eff B L L R R
N N
L L m n p m mj N j N
p pN E k k k
N N
k k ik k i k i
H H iW W iW W
W m n p v j j
0
( ) ( )
(0)( ) ( );
n p
L
n p
j j
dy y yx
Matlab calculationNa=input('input length along transport Na=')Nb=input('input length cross to transport Nb=')Nin=input('input numerical position of the input lead Nin=')Nout=input('input numerical position of the output lead Nout=')NL=length(Nin); NR=length(Nout);vL=1; vR=vL; tb=1;%LeadsE=-2.9:0.011:1;HL=zeros(NL,NL); HL=HL-diag(ones(1,NL-1),1);HL=HL+HL';HL=HL-diag(sum(HL),0);for np=1:NLkpp=acos(-E/2+EL(np,np)/2);kp(np,1:length(E))=kpp;endHR=HL;%DotN=Na*Nb;HB=zeros(N,N); HB=HB-diag(ones(1,N-1),1)-diag(ones(1,N-Na),Na);HB(Na:Na:N-Na,Na+1:Na:N-Na+1)=0;HB=tb*(HB+HB');%Coupling matrixpsiBin=psiB(Nin,:); psiBout=psiB(Nout,:);WL=vL*psiBin'*psiL'; WR=vR*psiBout'*psiL';DB=diag(ones(Na*Nb,1));for j=1:length(E) g=diag(exp(i*kp(:,j)));gg=diag(sin(real(kp(:,j))).^0.5);WW=WL*g*WL'+WR*g*WR';Heff=diag(EB)-WW;QQ=DB*E(j)-Heff;PP=QQ^(-1);SS=2*i*(WL*gg)'*PP*WR*gg;t(n,j)=SS(1,1);psS=psiB*PP*WL;
Datta’s site representation
Cv
21,, exp( )C N Cv z v ik
1, 0,
, 0,jC
j Nv
v j N
Effective Hamiltonian for time-periodic case
1, 0,
, 0,jC
j Nv
v j N
For stationary case
Волновая функция полубесконечного m-го провода
N=1
Numerical results N=1
m=-1, 0, 121 quasi energies
BS, J. Phys. C (1999): Критерий применимости теории возмущений 1M
H. Fukuyama, R. A. Bari, and H.C. Fogedby, PRB (1973).
vC=0.25
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