Dissipative dynamics of a parabolic confined particle in the presence of magnetic field

Physica A 292 (2001) 238–254

www.elsevier.com/locate/physa

Dissipative dynamics of a parabolic con$ned particle in the presence of magnetic $eld S. Baskoutasa; b; ∗ , C. Politisa; b , M. Rietha; b , W. Schommersa; b

a Department

of Engineering Science, School of Engineering, University of Patras, 26110 Patras, Greece Karlsruhe, Institut f"ur Nanotechnologie, D - 76021 Karsruhe, Germany

b Forschungszentrum

Received 27 July 2000; received in revised form 5 October 2000

Abstract Quantum mechanical properties of a parabolic con$ned electron in a small quantum dot system with the presence of magnetic $eld and dissipation were studied. Considering the dissipation as being due to the coupling of the system with a phonon bath at low temperature, we ignore the Brownian motion according to the Yu et al. approach [Phys. Rev. A 49 (1994) 592]. Such an approximation is valid for small quantum dots due to the apparent very strong energy quantization. Therefore assuming that the e
1. Introduction It is well known that the inclusion of dissipation in quantum mechanics has followed two main lines: (i) one starts from a classical equation of motion for a dissipative system, $nds the Langrange function leading to this equation and then applies the canonical quantization method [1], and (ii) one considers the dissipation as being ∗

Corresponding author. E-mail address: [email protected] (S. Baskoutas).

c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter  PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 5 8 8 - 4

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due to the coupling between two systems: a dissipative system S and an absorptive system R – the so-called loss reservoir. The quantization is applied to the whole system S + R, which is conservative, and the dissipation in the system S being obtained by eliminating the variables of the system R in a set of coupled equations of motion [2]. Apart from the controversy about approach (i), concerning its ability [3], or not [4], to consistently describe a dissipative system, dissipation in this category has been discussed essentially by using an e
(1.1)

where x1 ; x2 are the coordinates of the electron in two dimensions, m∗ is the bare-band mass of the electron and !0 is the oscillation frequency of the well. Kumar et al. [18] have also shown that even if the de$ning cap layer is square shaped, the con$ning potential seen by the electrons in a quantum dot at a $xed-gate voltage and the evolution of energy levels with increasing magnetic $eld is similar to that of a parabolic potential.

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In this work we represent a study of the quantum mechanical properties of an electron which is con$ned by a parabolic potential of form (1.1) in a quantum dot, in the presence of a magnetic $eld perpendicular to the dot plane. Assuming also the existence of dissipation of category (ii) that comes from the presence of a phonon bath which is coupled to the system at low temperature and considering the formulation of Refs. [7,8], we ignore the Brownian motion. Actually such an approximation is valid for small quantum dots due to the large separation of neighboring energy levels ( ≈ 45 meV for small self-assembled quantum dots [17, p. 61]) and therefore the condition !0
is generally satis$ed. Then the corresponding Lagrangian and (C–K) Hamiltonian [19] which describe the dissipative quantum-dot system at low temperature have the following forms:
 2 1  e 2 1 ∗ 2  2 L(t) = p − A − m !0 xi e t ; (1.2a) 2m∗ c 2 i=1

2 1  e 2 − t 1 ∗ 2  2 t H (t) = p − A e + m !0 xi e ; 2m∗ c 2

(1.2b)

i=1

where A is the electromagnetic potential, c is the velocity of light and xˆ i ; i = 1; 2, are the unit vectors of the Cartesian basis (r = x1 xˆ 1 + x2 xˆ 2 ). Given that the potential A has the form A = 12 B(x2 xˆ 1 − x1 xˆ 2 ) ;

(1.3)

where B is the magnetic $eld, then the corresponding classical equations of motion which correspond to the Lagrangian (1.2a) and to the Hamiltonian (1.2b), have the forms  !c !c  xO 1 − x˙ 2 + x˙ 1 − x2 + !12 x1 = 0 ; (1.4a) 2 2  !c !c  xO 2 + (1.4b) x˙ 1 + x˙ 2 + x1 + !22 x2 = 0 ; 2 2 where  !1 = !2 = !02 + !c2 =4 and !c = |e|B=m∗ c is the cyclotron frequency. As we can see the above equations of motion for (i) !c = 0 they take the form of the classical equations of motion of the usual damped harmonic oscillator in two-dimensions [11], corresponding to the Lagrangian (1.2a) for B = 0 (ii) = 0 correspond to the Lagrangian (1.2a) for = 0. It is well known that Feynman path-integral method is a very powerful tool for obtaining the propagator and the wave function of (C–K) Hamiltonians [11]. On the other hand, an alternative procedure, which will be used in this work, is based on the ordering technique of operators [20]. As is known this method exploits the algebraic structure of the Hamiltonian, which can be written as linear combination of the generators of the underlying algebra, and gives an analytical form of the system’s evolution operator [21–24].

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Thus the organization of the paper is arranged as follows. In Section 2 we present the mathematical formalism which is needed to obtain the exact evolution operator as well as the exact wave function of the system using the Wei–Norman ordering method. In Section 3 we give the analytical forms of the energy and Hamilton operator expectation values and we present numerical results of their behavior as a function of time for the case of weak dissipation. Furthermore, we present also a graphic illustration of the energy operator expectation values in terms of the e
2. Mathematical formalism 2.1. The exact evolution operator of the system Taking into account Eq. (1.3) the Hamiltonian (1.2b) takes the form  2   pi2 − t 1 ∗ 2 2 t !c − t H (t) = + e + m !i xi e e (x1 p2 − x2 p1 ) : ∗ 2m 2 2

(2.1)

i=1

The Hamiltonian (2.1) can be separated into two terms H (t) = H  (t) + H  (t) ;

(2.2)

 2   pi2 − t 1 ∗ 2 2 t H (t) = e + m !i xi e 2m∗ 2

(2.3)

!c − t e (x1 p2 − x2 p1 ) : 2

(2.4)

where 

i=1

and H  (t) =

As it is well known H  which describes the quantum dissipative harmonic oscillator, can be written in terms of the SU(1,1) algebra generators and the evolution operator which corresponds to this Hamiltonian has the form [22–24]


2

U i (xi ; t) = e(t) ea(t)xi eb(t)xi @=@xi ec(t)@

2

=@xi2




(U i (xi ; 0) = 1) ;

(2.5)

where the Wei–Norman characteristic functions (t); a(t); b(t); c(t) for weak ( 2 =4¡ !02 + !c2 =4) and strong ( 2 =4¿!02 + !c2 =4) dissipations are given in Appendix A.

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Then the corresponding evolution operator which corresponds to the Hamiltonian H  has the following form: U  (xi ; t) =

2




U i (xi ; t) :

(2.6)

i=1

Since U  (xi ; t) is known, the evolution operator U (xi ; t) corresponding to the Hamiltonian H will be given by U (xi ; t) = U  (xi ; t)U  (xi ; t) ;

(2.7)

where U  (t; 0) satis$es the evolution equation @U  (xi ; t) = HI (t)U  (xi ; t) ; @t with HI (t) being de$ned by i˝

(2.8)




HI (t) = U + (xi ; t)H  (t)U  (xi ; t) :

(2.9)

By straightforward evaluation of (2.9) with the help of Eq. (2.4) we obtain !c HI (t) = e− t (x1 p2 − x2 p1 ) : (2.10) 2 Exploiting the SU(2) algebraic structure of the Hamiltonian (2.10) we can obtain the corresponding evolution operator in an ordered form according to the Wei–Norman theorem U  (xi ; t) = ef1 (t)x2 @=@x1 ef3 (t)x2 @=@x2 e−f3 (t)x1 @=@x1 ef2 (t)x1 @=@x2 ;



(2.11)





with U  (xi ; 0) = 1 and U + U  = U  U + = 1. In order to obtain the exact forms of the Wei–Norman characteristic functions f1 ; f2 ; f3 we solve the following system of di
(2.12b)

f˙ 3 + f1 f˙ 2 e−2f3 = 0 :

(2.12c)

Then we take f1 (t) = tan



 !c t (e − 1) ; 2

(2.13a)



 !c t f2 (t) = − tan (e − 1) ; 2

(2.13b)

  !c t (e − 1) f3 (t) = − log cos 2

(2.13c)

and f1 (0) = f2 (0) = f3 (0) = 0 with the convergence condition (!c =2 )(e t − 1)¡=2.

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2.2. Analytical form of the wave function Assuming that our system is initially prepared in the form of a Gaussian wavepacket  el (x1 ; x2 ; 0) =

m∗ ! 0 ˝

1=2

e−m

∗

!0 =2˝(x12 +x22 )

(2.14)

and acting the evolution operator (2.7) on (2.14), e.g. el (x1 ; x2 ; t) = U (xi ; t)el (x1 ; x2 ; 0) ;

(2.15a)

we obtain the following analytical form of the wave function: 2

2

el (x1 ; x2 ; t) = A(t)eA1 (t)x1 eA2 (t)x2 e−A3 (t)x1 x2 ;

(2.15b)

where the analytical forms of the quantities involved in (2.15b) are given in Appendix B. After some algebra and numerical calculations wavefunction (2.15b) satis$es the condition +∞ +∞ ∗ el (x1 ; x2 ; t)el (x1 ; x2 ; t) d x1 d x2 = 1 : (2.16) −∞

−∞

3. Energy and Hamilton operators 3.1. Energy expectation values Following a similar procedure to that presented in Refs. [11,25], and using the Lagrangian (1.2a) the energy operator Eop (t) has the form  2   pi2 −2 t 1 ∗ 2 2 |e|B −2 t Eop (t) = e + m !i xi + (x1 p2 − x2 p1 ) : e 2m∗ 2 2m∗ c

(3.1)

i=1

We emphasize that the energy operator is not identical to the Hamiltonian Hˆ (t) itself. Hˆ (t) does not represent the total energy of the system, but rather is the generator of the motion of an energy-dissipative open system. It is straightforward to calculate the expectation values of Eop (t): Eop (t) = el |Eop (t)|el 

(3.2)

and after some algebra we obtain the following analytical expression, where the quantities involved in (3.3) are given in analytical form in Appendix B: Eop (t) = E1 (t) + E2 (t) + E3 (t) + E4 (t) ;

(3.3)

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with E1 (t) =

2 1 |A(t)|

× −

 2

−2 Re(A1 (t))(Re (A3 (t))=2 Re(A1 (t)) − 2 Re(A2 (t)))

1 Re2 (A3 (t)) + 4 Re(A1 (t)) 4 Re2 (A1 (t))

1 × 2 2(Re (A3 (t))=2 Re(A1 (t)) − 2 Re(A2 (t))) E2 (t) =

2 2 |A(t)|

× − × E3 (t) =

× E4 (t) =

:

(3.4a)

 2

−2 Re(A2 (t))(Re (A3 (t))=2 Re(A2 (t)) − 2 Re(A1 (t)))

1 Re2 (A3 (t)) + 4 Re(A2 (t)) 4 Re2 (A2 (t))

1 2 2(Re (A3 (t))=2 Re(A2 (t)) − 2 Re(A1 (t)))

2 3 |A(t)|



 :

(3.4b)

 2

(Re (A3 (t))=2 Re(A1 (t)) − 2 Re(A2 (t)))

1 Re(A3 (t)) : 4 Re(A1 (t)) (Re2 (A3 (t))=2 Re(A1 (t)) − 2 Re(A2 (t)))

2 4 |A(t)|

 2

−2 Re(A1 (t))(Re (A3 (t))=2 Re(A1 (t)) − 2 Re(A2 (t)))

(3.4c) : (3.4d)

Calculation of the Hamiltonian H (t) expectation values lead to the same forms as above but with di
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245

◦

Fig. 1. The energy operator expectation values as a function of time for ( ) = 0=s; (∇) = 5 × 1010 =s; ◦ (•) = 1011 =s and B = 0:2 T; l0 = 5 nm. The curve ( ) represents also the behavior of the Hamilton operator expectation values with time.

increase of the con$nement frequency !0 or with the decrease of the e
1=2 (4.1)

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S. Baskoutas et al. / Physica A 292 (2001) 238–254

Fig. 2. Energy operator expectation values as a function of the e
and  @p1; 2 =

−

˝2 |A(t)|2 (4A21 (t) 2m∗

×

 2

−2Re(A1 (t))(Re (A3 (t))=2Re(A1 (t)) − 2Re(A2 (t)))

× −

Re2 (A3 (t)) 1 + 4Re(A1 (t)) 4Re2 (A1 (t))

1 × 2 2(Re (A3 (t))=2Re(A1 (t)) − 2Re(A2 (t)))





+ A23 (t)

−2Re(A2 (t))(Re2 (A3 (t))=2Re(A2 (t)) − 2Re(A1 (t)))

× − ×

Re2 (A3 (t)) 1 + 4Re(A2 (t)) 4Re2 (A2 (t))

1 2 2(Re (A3 (t))=2Re(A2 (t)) − 2Re(A1 (t)))



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Fig. 3. (Weak dissipation) Behavior of the variances @ x1; 2 for = 1011 =s and B = 0:2 T; l0 = 5 nm.

− 4A1 (t)A3 (t) × ×

Re(A3 (t)) 2Re(A1 (t))  2

−2Re(A1 (t))(Re (A3 (t))=2Re(A1 (t)) − 2Re(A2 (t)))

1 + 2A1 (t) 2(Re (A3 (t))=2Re(A2 (t)) − 2Re(A1 (t)))

×

2

 −2Re(A1 (t))(Re2 (A3 (t))=2Re(A1 (t)) − 2Re(A2 (t)))

1=2 )

: (4.2)

A graphic illustration of the above formulas is clearly depicted in Figs. 3 and 4 for weak friction = 1011 =s; B = 0:2 T; l0 = 5 nm and in Figs. 5 and 6 for strong friction = 1012 =s; B = 0:2 T; l0 = 5 nm. As is understood for weak dissipation the uncertainty principle is satis$ed, obtaining also squeezing in the Kuctuation @ x1; 2 at the expense in the Kuctuation @p1; 2 . On the other hand, for strong dissipation the uncertainty principle is not valid. Therefore when the inKuence of Brownian motion is neglected, i.e., for the case !1 or !2 ¿ =2 and low temperature [27], the uncertainty principle is valid and the

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Fig. 4. (Weak dissipation) Behavior of the variances @p1; 2 for = 1011 =s and B = 0:2 T; l0 = 5 nm.

Fig. 5. (Strong dissipation) Behavior of the variances @x1; 2 for = 1012 =s and B = 0:2 T; l0 = 5 nm.

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Fig. 6. (Strong dissipation) Behavior of the variances @p1; 2 for = 1012 =s and B = 0:2 T; l0 = 5 nm.

Table 1 The behavior of variances @ x1; 2 as a function of the dot e
variances x1; 2 (102 )(nm)

5 10 20 30 40 50 60

0.555 1.112 2.273 3.348 4.456 5.565 6.683

existence of dissipation is responsible for the production of squeezed coherent states, in agreement with the results of Ref. [27] for the case of the damped oscillator without magnetic $eld. Studying also the behavior of the variances @ x1; 2 with the dot e
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5. Conclusions In the present work we have investigated the quantum mechanical properties of a parabolic con$ned electron in a small quantum dot system with the presence of a magnetic $eld perpendicular to the dot, assuming also the existence of dissipation which comes through the coupling of the electron with a phonon bath. Due to the very strong energy quantization which is evident for small quantum dot systems, the condition of weak dissipation, i.e., !1 or !2 ¿ =2 characterizes generally the small quantum dot system and therefore permit us to ignore the Brownian motion at low temperature and to work with the e
Acknowledgements The authors gratefully acknowledge $nancial support from the International BOuro of the Bundesministerium fOur Bildung und Forschung, Bonn, Germany and from the Research Committee of the University of Patras, Patras, Greece.

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Appendix A Weak friction !1 or !2 ¿ =2: a(t) = − i

m∗



!02 +

 cos t b(t) = − log   2

(t) =

4

2˝ 

c(t) = i

!c2

 2m∗

+

!c2 4

!02

!c2 4

2 4t

−

cos !

!c2 4

2 4t



sin + −

  e t ; 2 2 ! cos !02 + 4c − 4 t − !

 !02 +

˝ !02



 −!   ; 

(A.2)

  2 !2 sin !02 + 4c − 4 t

 ; !c2 2 2 cos !0 + 4 − 4 t − !

b(t) ; 2 

(A.3)

(A.4) 

 ; ! = arctan   2 !02 + !c2 =4 − 2 =4 Strong friction !1 or !2 ¡ =2: a(t) = − i

m∗



!c2

!02 +

4

2˝ 

(t) =

˝

 2m∗ b(t) ; 2

!02 +

!c2 4



(A.5)

2 4

(!02

!c2 4 )t



sinh − +

  e t ; !c2 2 2 sinh 4 − (!0 + 4 )t + ’



 sinh t b(t) = − log   2

c(t) = i

(A.1)

2 4

 − !02 + sinh ’

!c2 4



 t+’   ; 

   2 2 2 + !c t sinh − ! 0 4 4 ;

   2 2 2 + !c t + ’ sinh − ! 0 4 4

(A.6)

(A.7)

(A.8)

(A.9)

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S. Baskoutas et al. / Physica A 292 (2001) 238–254

   2 2 − !02 + 4   ’ = arctan 

!c2 4

   : 

(A.10)

Appendix B The quantities involved in expressions (2.15b), (3.4a) – (3.4d) have the forms #1 (t) = c(t) e−2f3 (t) [1 + f12 (t)] ;  2 #2 (t) = c(t) ef3 (t) + f1 (t)f2 (t) e−f3 (t) + f22 (t) e−2f3 (t) ; #3 (t) = 2c(t)

# $ ef3 (t) + f1 (t)f2 (t) e−f3 (t) e−f3 (t) f1 (t) + f2 (t) e−f3 (t) ;

!"

˝ + #1 (t) ; 2m∗ !0   ˝ #23 (t) ; R(t) = 4 + #2 (t) − 2m∗ !0 4(˝=2m∗ !0 + #1 (t)) M (t) =

(B.1) (B.2) (B.3) (B.4) (B.5)

g1 (t) = e2b(t) e−2f3 (t) ;

(B.6)

g2 (t) = e−2f3 (t) e2b(t) f12 (t) ;

(B.7)

g3 (t) = 2e−2f3 (t) e2b(t) f1 (t) ;

(B.8)

g4 (t) = e−2f3 (t) e2b(t) f22 (t) ;

(B.9)

" #2 g5 (t) = e2b(t) ef3 (t) + f1 (t)f2 (t) e−f3 (t) ;

(B.10)

" # g6 (t) = 2e2b(t) f2 (t) e−f3 (t) ef3 (t) + f1 (t)f2 (t) e−f3 (t) ;

(B.11)

g7 (t) = e−2f3 (t) e2b(t) f2 (t) ;

(B.12)

# " g8 (t) = e2b(t) e−f3 (t) f1 (t) ef3 (t) + f1 (t)f2 (t) e−f3 (t) ;

(B.13)

 " # g9 (t) = e2b(t) e−f3 (t) ef3 (t) + f1 (t)f2 (t) e−f3 (t) + f1 (t)f2 (t) e−2f3 (t) ; (B.14) g10 (t) =

g7 (t) g1 (t) 2 # (t) + g4 (t) − #3 (t) ; 4M 2 (t) 3 M (t)

(B.15)

g11 (t) =

g2 (t) 2 g8 (t) #3 (t) + g5 (t) − #3 (t) ; 2 4M (t) M (t)

(B.16)

S. Baskoutas et al. / Physica A 292 (2001) 238–254

g12 (t) =

g3 (t) 2 g9 (t) #3 (t) + g6 (t) − #3 (t) ; 2 4M (t) M (t) 

A1 (t) = −  A2 (t) = −  A3 (t) =

253

(B.17)



g1 (t) g10 + − a(t) 4M (t) R(t)

;

(B.18)

;

(B.19)



g11 g2 (t) + − a(t) 4M (t) R(t)

g12 g2 (t) + 4M (t) R(t)

 ;

(B.20)

w1 (t) = Re(4A21 (t) + A23 (t)) ;

(B.21)

w2 (t) = Re(4A22 (t) + A23 (t)) ;

(B.22)

w3 (t) = 4A3 (t)(A1 (t) + A2 (t)) ;

(B.23)

  !c ˝2 − t 1 ∗ !c2 2 + i˝ e− t A3 (t) ; e w1 (t) + m !0 + 1= − ∗ 2m 2 4 2

(B.24)

  !c ˝2 − t 1 ∗ !c2 2 − i˝ e− t A3 (t) ; ! e w (t) + + m 2 0 2m∗ 2 4 2

(B.25)

2= −

˝2 − t !c e w3 (t) + i˝ e− t (A1 (t) − A2 (t)) ; ∗ 2m 2

3

=

4

=−

˝2 − t e (A1 (t) + A2 (t)) ; m∗

(B.26)

(B.27)

Appendix C The factors 1 ; expectation values

2

which are involved in the de$nition of the Hamilton operator

  !c ˝2 − t 1 ∗ t !c2 2 + i˝ e− t A3 (t) ; e w1 (t) + m e !0 + 1= − 2m∗ 2 4 2

(C.1)

  !c ˝2 − t 1 ∗ t !c2 2 m − i˝ e− t A3 (t) : ! e w (t) + e + 2 0 2m∗ 2 4 2

(C.2)

2= −

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

M. Razavy, Phys. Rev. A 36 (1987) 482. A.O. Caldeira, A.J. Leggett, Ann. Phys. NY 149 (1983) 374. P. Croxson, Phys. Rev. A 49 (1994) 588. D.M. Greenberger, J. Math. Phys. 20 (1979) 672. P. Caldirola, Nuovo Cimento 18 (1941) 393. E. Kanai, Prog. Theor. Phys. 3 (1948) 440. L.H. Yu, C.P. Sun, Phys. Rev. A 49 (1994) 592. C.P. Sun, L.H. Yu, Phys. Rev. A 51 (1995) 1845. A.O. Caldeira, A.J. Leggett, Ann. Phys. NY 149 (1987) 374. Q. Niu, J. Stat. Phys. 65 (1991) 317. H.G. Oh, H.R. Lee, T.F. George, C.I. Um, Y.M. Choi, W.H. Kahng, Phys. Rev. A 40 (1989) 45. G.S. Agarwal, S.A. Kumar, Phys. Rev. Lett. 67 (1991) 3665. M. Fang, K. Wang, Phys. Lett. A 197 (1995) 135. B. Baseia, V.S. Bagnato, M.A. Marchiolli, M.C. de Oliveira, Quant. Semiclass. Opt. 8 (1996) 1147. C.F. Lo, D. Kiang, Int. J. Mod. Phys. B 14 (2000) 993. U. Woggon, Optical Properties of Semiconductor Quantum Dots, Springer Tracts in Modern Physics, Springer, Berlin, 1997. L. Jacak, P. Hawrylak, A. Wojs, Quantum Dots, Springer, Berlin, Heidelberg, 1998, p. 31. A. Kumar, S.E. Laux, F. Stern, Phys. Rev. B 42 (1990) 5166. C.F. Lo, Phys. Rev. A 45 (1992) 5262. W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. G. Dattoli, S. Solimeno, A. Torre, Phys. Rev. A 34 (1986) 2646. S. Baskoutas, A. Jannussis, R. Mignani, Phys. Lett. A 164 (1992) 17. S. Baskoutas, Quant. Semiclass. Opt. 8 (1996) 989. S. Baskoutas, A. Jannussis, R. Mignani, Nuovo Cimento B 95 (1993) 953. J.R. Brinati, S.S. Mizrahi, J. Math. Phys. 21 (1980) 2154. K.D. Zhu, T. Kobayashi, Phys. Lett. A 190 (1990) 337. Y.H. Jin, S.P. Kou, J.Q. Liang, B.Z. Li, Mod. Phys. Lett. B 14 (2000) 267.