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