Подход эффективного гамильтониана
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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

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1 1957 2 u fano phys rev 124 1866 1961

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

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).


1 1957 2 u fano phys rev 124 1866 1961
Coupled defect mode with propagating over waveguide light гамильтонианаManolatou, et al, IEEE J. Quant. Electronics, (1999)


Coupled mode theory

Одно модовый резонатор гамильтониана

Coupled mode theory


1 1957 2 u fano phys rev 124 1866 1961

Инверсия по времени гамильтониана

Одно-модовый резонатор

CMT

  • Х. Хаус, Волны и поля в оптоэлектронике


1 1957 2 u fano phys rev 124 1866 1961
CMT гамильтониана

  • Много-модовый резонатор

IEEE J. Quantum Electronics, 40, 1511 (2004)


1 1957 2 u fano phys rev 124 1866 1961
Два порта, две моды гамильтониана

%CMT for transmission through resonator with two modes

clear all

E=-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


1 1957 2 u fano phys rev 124 1866 1961

T гамильтонианаволновод с двумя резонаторами, Булгаков, Садреев, Phys. Rev. B84, 155304 (2011)


1 1957 2 u fano phys rev 124 1866 1961

W is matrix NxM where N is the number of eigen states of closed quantum

system, M is the number of continuums (channels)


1 1957 2 u fano phys rev 124 1866 1961

S.Datta, ( closed quantumElectronic transport in mesoscopic systems) (1995).


1 1957 2 u fano phys rev 124 1866 1961

Проекционные операторы closed quantum:

Уравнение Липпмана-Швингера


S matrix
S-matrix closed quantum

Basis of closed billiard

The biorthogonal basis


1 1957 2 u fano phys rev 124 1866 1961
c closed quantum

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
2d case closed quantum

Limit to continual case


Matlab calculation

Na=input('input length along transport Na=') closed quantum

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;

%Leads

E=-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:NL

kpp=acos(-E/2+EL(np,np)/2);

kp(np,1:length(E))=kpp;

end

HR=HL;

%Dot

N=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 matrix

psiBin=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;

Matlab calculation



Effective hamiltonian for time periodic case

For stationary case closed quantum

l

Effective Hamiltonian for time-periodic case


1 1957 2 u fano phys rev 124 1866 1961

Волновая функция полубесконечного m-го провода

N=1


Numerical results n 1

m=-1, 0, 1 полубесконечного

21 quasi energies

Numerical results N=1

l=0.75, vC=0.25

H. Fukuyama, R. A. Bari, and H.C. Fogedby, PRB (1973).

BS, J. Phys. C (1999): Критерий применимости теории возмущений