Classical and quantum Liouville integrability of nonlinear Heisenberg equations

Nonlinear Analysis 71 (2009) e744–e762

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Classical and quantum Liouville integrability of nonlinear Heisenberg equations V. Peřinová ∗ , A. Lukš Laboratory of Quantum Optics, Faculty of Natural Sciences, Palacký University, Třída Svobody 26, 771 46 Olomouc, Czech Republic

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MSC: 78A97 81V80 Keywords: Liouville integrability Elliptic functions Hamilton canonical equations Nonlinear Heisenberg equations Invariant subspace method Invariant operators

abstract In classical theory a surprising result has long been derived that for the integrability of 2M Hamilton canonical equations M invariants suffice. In quantum theory a similar situation is rather transparent due to the essential linearity of the theory. But in the most interesting problems the vanishing of commutators of invariants does not suffice for separation of new degrees of freedom unfortunately. We demonstrate the connection between the expansions of solutions in Fock states and those in normally ordered products of creation and annihilation operators for simple problems of quantum optics. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Nonlinear properties of Maxwellian constitutive relations have been known for a long time. Engineers have respected the nonlinear permeability in the construction of electric machines. Magnetic field at a zero or relatively low frequency is assumed. Also nonlinear properties of plasma are well known, which have been observed in the propagation of electromagnetic waves in the ionosphere. Nonlinear properties of matter in the optical region were demonstrated in experiments on generation of light harmonics. For instance, red light is converted to ultraviolet radiation by transmission through a crystal of quartz [1]. An exposition in the form of a lecture note was published soon thereafter [2]. Great effort has been devoted to the introduction of quantum description of nonlinear optical phenomena. This quantum description is based on quantum mechanics. In quantum mechanics it is derived that quantum description is linear. In fact, quantum description of a finite number, M, of harmonic oscillators utilizes annihilation operators, aˆ j , j = 1, . . . , M, of Ď

ˆ consists of these operators and of the creation ones, aˆ j , j = 1, . . . , M, energy quanta of these oscillators. A Hamiltonian H ˆ does not where Ď means the Hermitian conjugation. It is assumed that the harmonic oscillators are autonomous, i. e., H ˆ may be constructed. depend on time explicitly. This way any other operator, M, Statistical properties of oscillators are summarized in the statistical operator ρˆ . The statistical approach to quantum ˆ which does not depend on the time explicitly, mechanics is oriented to the Schrödinger picture, in which no operator M, changes. In this picture the statistical operator ρˆ changes. ˆ depends on The truth that quantum description is linear is shown in the Heisenberg picture, in which any operator M the time and the differential equation for such an operator reads d dt

∗

ˆ = {M ˆ , Hˆ } = − i [M ˆ , Hˆ ]. M h¯

Corresponding author. Tel.: +420 585634263; fax: +420 585634253. E-mail address: [email protected] (V. Peřinová).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.11.031

(1)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

e745

Here h¯ is the reduced Planck constant, {•, •} may be called a quantum Poisson bracket, [•, •] is a commutator, [Aˆ , Bˆ ] = ˆ Aˆ Bˆ − Bˆ A. Using the commutators

[ˆaj , aˆ Ďk ] = δjk 1ˆ ,

[ˆaj , aˆ k ] = 0ˆ ,

[ˆaĎj , aˆ Ďk ] = 0ˆ ,

j, k = 1, . . . , M ,

(2)

where 1ˆ is the identity operator and 0ˆ is the null operator, the equations of motion can be derived, d dt

ˆ }, aˆ j = {ˆaj , H

(3)

which are often very similar to the equations of classical description. By appearance they can be classified as linear and nonlinear. If they seem to be linear, then they can also be solved easily. If they seem to be nonlinear, analogy is fruitless in most cases. ˆ = Hˆ it follows that From relation (1) for M

ˆ = Hˆ (0). H

(4)

Hence d dt

ˆ = {M ˆ , Hˆ (0)}. M

(5)

Particularly, d dt

ˆ (0)}. aˆ j = {ˆaj , H

(6)

These equations must be classified as linear equations for aˆ j , j = 1, . . . , M. To illustrate what was said about the equations of motion for the annihilation operators in this form, let us recall the classical description. A linear first-order evolution differential equation is solved by the method of characteristics. The characteristics are described by nonlinear ordinary differential equations. Especially, they may be related to the equations of motion. Conversely, the solutions of the equations of motion depend on the initial conditions and these functions of many variables obey the mentioned linear evolution differential equation. There exist mappings, which assign to any, or at least to some kind of statistical operator ρˆ functions of complex amplitudes αj , j = 1, . . . , M. Simultaneously appropriate mappings assign to each annihilation operator aˆ j (0) a complex amplitude αj , j = 1, . . . , M. They assign to the evolved annihilation operator aˆ j (t ) a function of complex amplitudes αj . The achieved analogy with the classical description is so close that a warning must follow it. Usually one says that there does not exist any ‘‘trajectory’’, although we are adapted to this notion from the classical description. Classical descriptions of the second-harmonic generation and three-wave mixing are provided by ordinary differential equations, which have solutions in terms of the Jacobian elliptic functions. In this connection we recall the general framework characterizing the cases where the equations of motions are solvable ‘‘by quadratures’’, the idea provided by Liouville. In fact, the equations of motion even in the cases of these two nonlinear optical processes can be written using Hamiltonian functions. The numbers of degrees of freedom are M = 2, 3 respectively. Besides the Hamiltonian function in any case there exist additional M − 1 constants of motion. The corresponding quantum descriptions are not solvable in the sense introduced in the classical theory. But it still holds that the Hamiltonian and the constants of motion represented by operators facilitate numerical solution, or, which is also interesting, logical analysis. Namely it holds that, in an appropriate basis, linear equation (6) has the form of an infinite number of finite systems of equations. On the other hand finite systems are not prospective from the viewpoint of calculus and functional analysis. An exception is presented by paper [3], devoted to the three-wave mixing essentially. But in quantum optics the formulation of the linear equation (6) itself has been accepted gratefully as evidenced by paper [4]. 2. Exact solutions of two nonlinear optical processes The second-harmonic and second-subharmonic generation is described by the equations d dt d dt

α1 = 2ig ∗ α1∗ α2 , α2 = ig α12 .

(7)

Here α1 is a complex amplitude of the fundamental frequency, α2 is a complex amplitude of the second harmonic, and g is a coupling constant. We will not use the notion of the second subharmonic.

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Similarly the three-wave mixing is described by the equations d dt d dt d dt

α1 = ig ∗ α2∗ α3 , α2 = ig ∗ α1∗ α3 , α3 = ig α1 α2 .

(8)

Here αj is a complex amplitude of the frequency ωj , j = 1, 2, and α3 is a complex amplitude of the frequency ω3 , ω3 = ω1 + ω2 . On a polar decomposition of the complex amplitudes in terms of real amplitudes, ρj , and phases, ϕj , we obtain the equations d dt d dt d dt

ρ1 eiϕ1 + iρ1 eiϕ1 ρ2 eiϕ2 + iρ2 e ρ3 eiϕ3 + iρ3 e

d

dt iϕ2 d dt iϕ3 d dt

ϕ1 = ig ∗ ρ2 ρ3 ei(−ϕ2 +ϕ3 ) , ϕ2 = ig ∗ ρ1 ρ3 ei(−ϕ1 +ϕ3 ) , ϕ3 = ig ρ1 ρ2 ei(ϕ1 +ϕ2 ) ,

(9)

or d dt d dt d dt

ρ1 + iρ1 ρ2 + iρ2 ρ3 + iρ3

d dt d dt d dt

ϕ1 = i|g |ρ2 ρ3 ei(− arg g −ϕ1 −ϕ2 +ϕ3 ) , ϕ2 = i|g |ρ1 ρ3 ei(− arg g −ϕ1 −ϕ2 +ϕ3 ) , ϕ3 = i|g |ρ1 ρ2 ei(arg g +ϕ1 +ϕ2 −ϕ3 ) ,

(10)

or d dt d dt d dt

ρ1 + iρ1 ρ2 + iρ2 ρ3 + iρ3

d dt d dt d dt

ϕ1 = i|g |ρ2 ρ3 e−iθ , ϕ2 = i|g |ρ1 ρ3 e−iθ , ϕ3 = i|g |ρ1 ρ2 eiθ ,

(11)

where θ = arg g + ϕ1 + ϕ2 − ϕ3 , or d dt d dt d dt

ρ1 + iρ1 ρ2 + iρ2 ρ3 + iρ3

d dt d dt d dt

ϕ1 = |g |ρ2 ρ3 (sin θ + i cos θ ), ϕ2 = |g |ρ1 ρ3 (sin θ + i cos θ ), ϕ3 = |g |ρ1 ρ2 (−sinθ + i cos θ ).

(12)

On passing over to real and imaginary components, we obtain that d dt d dt d dt d

ρ1 = |g |ρ2 ρ3 sin θ , ρ2 = |g |ρ1 ρ3 sin θ , ρ3 = −|g |ρ1 ρ2 sin θ ,

ρ2 ρ3 cos θ , dt ρ1 d ρ1 ρ3 ϕ2 = |g | cos θ , dt ρ2 d ρ1 ρ2 ϕ3 = |g | cos θ . dt ρ3

(13)

ϕ1 = |g |

(14)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

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From this also the equation

ρ2 ρ3 ρ1 ρ3 ρ1 ρ2 θ = |g | + − dt ρ1 ρ2 ρ3 

d



cos θ

(15)

may be derived. Two constants of motion, J1 = ρ12 + ρ32 ,

J2 = ρ22 + ρ32 ,

(16)

are obvious. 2.1. Real amplitudes On introducing u1 = −ρ1 ,

u2 = −ρ2 ,

u3 = −ρ3 ,

ζ = |g |t

(17)

in (13), we obtain the equations [5] du1 dζ du2 dζ du3 dζ dθ dζ

= −u2 u3 sin θ ,

(18)

= −u3 u1 sin θ ,

(19)

= u1 u2 sin θ,

(20)

 =−

u2 u3 u1

+

u3 u1

−

u2

u1 u2



u3

cos θ .

(21)

The constants of motion are essentially the same, u22 + u23 = const. ≡ m1 , u21 + u23 = const. ≡ m2 .

(22)

Let us follow the derivation a little. Eq. (21) may be written in the form dθ dζ

=

cos θ d ln(u1 u2 u3 ) sin θ

dζ

,

(23)

.

(24)

or sin θ dθ

=

cos θ dζ

d ln(u1 u2 u3 ) dζ

On integrating both sides, we obtain that cos θ =

Γ u1 u2 u3

,

(25)

where we have introduced

Γ = u1 (0)u2 (0)u3 (0) cos (θ (0)) .

(26)

The authors consider a cubic equation for

u23 ,

u23 (m2 − u23 )(m1 − u23 ) − Γ 2 = 0.

(27)

≤ ≤ denote its roots. According to our simplification it is a cubic equation for ρ = ρ is obvious. Then

let u23a 2 2 3b , 3c

They 2 ρ3a ,ρ

γ2 =

u23b

u23c

2 3

2 2 ρ3b − ρ3a . 2 2 ρ3c − ρ3a

u23 .

The meaning of

(28)

The solution [5] u23 (ζ ) = u23a + (u23b − u23a )sn2 u22

(ζ ) =

u22

(0) +

u23

(0) −

u23a

q −(

u23c − u23a (ζ + ζ0 ), γ

u23b

−

u23a

)sn

2

u21 (ζ ) = u21 (0) + u23 (0) − u23a − (u23b − u23a )sn2

q q

u23c



,

−

u23a



(ζ + ζ0 ), γ ,

u23c − u23a (ζ + ζ0 ), γ



,

(29)

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is interpreted in the form

 2 2 ρ3c − ρ3a |g |(t + t0 ), γ , q  2 2 2 2 2 2 2 2 2 ρ2 (t ) = ρ2 (0) + ρ3 (0) − ρ3a − (ρ3b − ρ3a )sn ρ3c − ρ3a |g |(t + t0 ), γ , 2 2 2 ρ32 (t ) = ρ3a + (ρ3b − ρ3a )sn2

q

2 2 2 ρ12 (t ) = ρ12 (0) + ρ32 (0) − ρ3a − (ρ3b − ρ3a )sn2

q  2 2 ρ3c − ρ3a |g |(t + t0 ), γ ,

(30)

where the constant t0 is determined from the relation for ρ32 (t ) at t = 0. Quoting paper [5], we do not care for the fact that, in the study of light propagation in that paper, propagation distance, z, has been appropriately used instead of time t. We pay attention to the fact that the idea of their solution can be realized, even though the time t is utilized. 2.2. Phases In mathematics the discussion of nullity of solutions of differential equations is very much taken into consideration. We have not included it. We could distinguish the discussion of cases of nullity on a ‘‘whole’’ interval and the discussion of cases of the roots. As an example we would present the first, easier discussion. Can ρ3 = 0 be a solution on some interval? Then the relation d d ρ1 = ρ2 = 0, ρ1 ρ2 = 0, dt dt would hold or ρ1 = const., ρ2 = const. and one of them would be zero on that interval. Can for instance ρ1 = 0 be a solution on some interval? Then the relation

(31)

d d ρ2 = ρ3 = 0, ρ2 ρ3 = 0, (32) dt dt would hold or ρ2 = const., ρ3 = const. and one of them would be zero on that interval. It would be sufficient to remark that it is similar as in the case ρ3 = 0. The equations for phases may be given the form d 1 ϕ1 = −|g |Γ 2 , dt ρ1 d dt d dt

ϕ2 = −|g |Γ ϕ3 = −|g |Γ

1

ρ22 1

ρ32

, .

(33)

By integration these equations are solved,

ϕ1 (t ) = ϕ1 (0) − |g |Γ

t

Z

ρ (t 0 )

0

ϕ2 (t ) = ϕ2 (0) − |g |Γ

t

Z

ϕ3 (t ) = ϕ3 (0) − |g |Γ

1

ρ22 (t 0 )

0 t

Z

1 2 1

1

ρ32 (t 0 )

0

dt 0 , dt 0 , dt 0 ,

(34)

or

ϕ1 (t ) = ϕ1 (0) − |g |Γ

t

Z

ρ (0) + ρ (0) − ρ − (ρ − ρ )sn2

0

ϕ2 (t ) = ϕ2 (0) − |g |Γ

2 1

2 3a

2 3

2 3b

2 3a

t

Z

t

Z 0

q

ρ − ρ |g |(t 0 + t0 ), γ 2 3c

2 3a

1 2 2 2 ρ22 (0) + ρ32 (0) − ρ3a − (ρ3b − ρ3a )sn2

0

ϕ3 (t ) = ϕ3 (0) − |g |Γ

1

q

2 2 ρ3c − ρ3a |g |(t 0 + t0 ), γ

1

ρ + (ρ − ρ ) 2 3a

2 3b

2 3a

sn2

q

ρ −ρ 2 3c

2 3a

|g |(t 0

+ t0 ), γ

 dt 0 .

 dt 0 ,

 dt 0 ,

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V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

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2.3. Nonnegative real roots Let us note that Armstrong, Bloembergen, Docuing, and Pershan [5] evidently states that Eq. (27) possesses three nonnegative real roots. Let us show it. If Γ = 0, then u23a = 0 and obviously roots u23b and u23c are also nonnegative. If Γ 6= 0, none of the roots may be zero. Eq. (27) is equivalent to the equation

Γ2

(x − m2 )(x − m1 ) =

x

,

(36)

where we write x instead of u23 for simplicity. As for x < 0 the left-hand side is positive and the right-hand side negative, none of the roots may be (zero or) negative. But there exist at least two positive roots. 2

As on the interval [max(m1 , m2 ), ∞) the function Γx decreases and the function (x − m2 )(x − m1 ) increases from zero beyond all bounds and these functions are continuous within this interval there is a solution of the considered equation. Let us consider this equation for Γ = Γ0 = u1 (0)u2 (0)u3 (0). Then we obtain three real roots. We assert that two of them will be less than min(m1 , m2 ). To show it, let us consider these real roots, which fulfill the equation for x, x2 − u21 (0) + u22 (0) + u23 (0) x + u21 (0)u22 (0) = 0.




(37)

Writing this equation in the form x2 − u21 (0) + u22 (0) x + u21 (0)u22 (0) = u23 (0)x




(38)

and using a plot, we see that the lesser root is less than min(u21 (0), u22 (0)). In other words, there exists a solution of the equation

Γ02

(x − m2 )(x − m1 ) =

x

,

(39)

which belongs to the interval (0, min(m1 , m2 )). But then on this interval there exist two solutions. In fact, the left-hand side of this equation is less than the right-hand one for x = min(m1 , m2 ), and this holds also as x → +0. The number of solutions is at least one and must be even. The left-hand and right-hand sides are continuous functions. For generic Γ , 0 < Γ 2 ≤ Γ02 , both solutions with this property survive, but xa will already be smaller in general (as a Γ2

decreasing function of Γ02 ) and xb will already be larger in general (as an increasing function of third positive solution it has already been derived that xc > max(m1 , m2 ).

Γ02 Γ2

), xa ≤ xb ≤ xc . About the

2.4. Degenerate case In the case of second-harmonic and second-subharmonic generation it is possible to utilize the results derived for the three-wave mixing. It is possible to illustrate the relationship of the degenerate nonlinear optical process with the nondegenerate one. With respect to a symmetry of the differential equations from the condition α1 (0) = α2 (0) it follows that α1 (t ) = α2 (t ) for all t. We will introduce the quantity 1

α+ (t ) = √ [α1 (t ) + α2 (t )] ,

(40)

2

which is such that its Poisson brackets and the Poisson brackets between it and the amplitude α3 (t ) have the Bose property, i

∗ {α+ (t ), α+ (t )} = − , {α+ (t ), α3∗ (t )} = 0, h¯ ∗ {α+ (t ), α3 (t )} = 0, {α+ (t ), α3∗ (t )} = 0.

(41)

From the condition α1 (0) = α2 (0) we obtain that

α+ (t ) =

√

2α1 (t ) =

√

2α2 (t ),

1

α1 (t ) = α2 (t ) = √ α+ (t ). 2

(42)

The equations of motion then become d dt d dt

∗ α+ = ig ∗ α+ α3 ,

1

2 α3 = i g α+ .

2

(43)

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Finally we replace 1

g → g. 2 In this manner we obtain not only the differential equations

α+ → α1 , d dt d

α3 → α 2 ,

(44)

α1 = 2ig ∗ α1∗ α2 ,

α2 = ig α12 , (45) dt but also the corresponding solutions. Certainly the opinion prevails that we should go through the derivation from the ground up. In this derivation we would encounter familiar notions. For instance, m1 = |α1 |2 + |α3 |2 = m2 = |α2 |2 + |α3 |2 =

1 2 1 2

|α+ |2 + |α3 |2 , |α+ |2 + |α3 |2 ,

m3 = 0.

(46)

But traditionally one considers 2m1 = 2m2 = |α+ |2 + 2|α3 |2 ,

(47)

or, also in the traditional notation on the right-hand side, 2m1 = 2m2 = |α1 |2 + 2|α2 |2 .

(48)

3. Theoretical quantum optics We will return to the linear operators, which we have mentioned at the beginning. The quantum description of nonlinear optical phenomena remains a substantial part of quantum optics. We will pay attention to the three-wave mixing. 3.1. Blind alley of analogy In quantum optics the three-wave mixing is described by a Hamiltonian. Let us issue from the Hamiltonian Ď

Ď Ď

ˆ int = −h¯ g ∗ aˆ 1 aˆ 2 aˆ 3 − h¯ g aˆ 1 aˆ 2 aˆ 3 , H

(49)

where g is a coupling constant. This Hamiltonian may be written in terms of numbers and phases, even though in more than one way,

q

q

p p q nˆ 1 nˆ 2 nˆ 3 + 1ˆ eˆ i(−ϕ3 +ϕ1 +ϕ2 ) q q p p q p = −h¯ g ∗ eˆ i(ϕ3 −ϕ1 −ϕ2 ) nˆ 1 nˆ 2 nˆ 3 + 1ˆ − h¯ g eˆ i(−ϕ3 +ϕ1 +ϕ2 ) nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 = . . ..

ˆ int = −h¯ g ∗ H

nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 eˆ i(ϕ3 −ϕ1 −ϕ2 ) − h¯ g

p

(50) (51)

Here we have utilized polar decompositions, which are indicated by the polar decomposition aˆ = eˆ iϕ

√

q

nˆ =

nˆ + 1ˆ eˆ iϕ .

(52)

Here nˆ is a number operator, nˆ = aˆ Ď aˆ , and eˆ iϕ is an exponential phase operator [6]. The manipulations, which have been performed in the classical case, can be replaced by a use of a Hamiltonian function. But it is not so usual. In the quantum ˆ = nˆ 3 . Also Hˆ is replaced by Hˆ int not in a mere substitution, but on a ground we will case we may utilize relation (1) for M not yet expound for brevity. From this d



1 ˆ int nˆ 3 = i nˆ 3 , − H dt h¯



q q p p q p = g ∗ nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 [ˆn3 , eˆ i(ϕ3 −ϕ1 −ϕ2 ) ] + g nˆ 1 nˆ 2 nˆ 3 + 1ˆ [ˆn3 , eˆ i(−ϕ3 +ϕ1 +ϕ2 ) ] q q p p q p = −g ∗ nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 eˆ i(ϕ3 −ϕ1 −ϕ2 ) + g nˆ 1 nˆ 2 nˆ 3 + 1ˆ eˆ i(−ϕ3 +ϕ1 +ϕ2 ) ,

(53)

(54)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

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where we have used commutation relations, which are indicated by the commutation relations

[ˆn, eˆ iϕ ] = −ˆeiϕ ,

[ˆn, eˆ −iϕ ] = eˆ −iϕ .

(55)

The right-hand side of relation (54) may be written in more than one way. We may interpret it or rederive, after first substituting the form (51) into (53),

q q p p q p nˆ 2 nˆ 3 + 1ˆ + g [ˆn3 , eˆ i(−ϕ3 +ϕ1 +ϕ2 ) ] nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 q q p p q p = −g ∗ eˆ i(ϕ3 −ϕ1 −ϕ2 ) nˆ 1 nˆ 2 nˆ 3 + 1ˆ + g eˆ i(−ϕ3 +ϕ1 +ϕ2 ) nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 .

d

nˆ 3 = g ∗ [ˆn3 , eˆ i(ϕ3 −ϕ1 −ϕ2 ) ] nˆ 1

dt

(56)

Similarly we could verify that Jˆ1 (t ) = nˆ 1 (t ) + nˆ 3 (t ), Jˆ2 (t ) = nˆ 2 (t ) + nˆ 3 (t )

(57)

are constants of motion. We are presuming that an analogy between classical and quantum solutions will take place, when the expression



1 h¯

ˆ int H

2

 +

d dt

2 nˆ 3

,

(58)

d where we replace dt nˆ 3 by relation (53), will be simplified. The first and second terms of expression (58) are

2  ˆ  q  q Hint   p p q p h¯ i(ϕ3 −ϕ1 −ϕ2 ) i(−ϕ3 +ϕ1 +ϕ2 ) ∗ ˆ ˆ ˆ 2 = g nˆ 1 + 1 nˆ 2 + 1 nˆ 3 eˆ  ± g nˆ 1 nˆ 2 nˆ 3 + 1eˆ  d  nˆ 3 



1

dt



∗ i(ϕ3 −ϕ1 −ϕ2 )

× g eˆ

q q p p q p  i(−ϕ3 +ϕ1 +ϕ2 ) ˆ ˆ ˆ nˆ 1 nˆ 2 nˆ 3 + 1 ± g eˆ nˆ 1 + 1 nˆ 2 + 1 nˆ 3

= |g |2 (ˆn1 + 1ˆ )(ˆn2 + 1ˆ )ˆn3 + |g |2 nˆ 1 nˆ 2 (ˆn3 + 1ˆ ) q q p p q p ± g ∗2 nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 eˆ 2i(ϕ3 −ϕ1 −ϕ2 ) nˆ 1 nˆ 2 nˆ 3 + 1ˆ q q p p q p ± g 2 nˆ 1 nˆ 2 nˆ 3 + 1ˆ eˆ 2i(−ϕ3 +ϕ1 +ϕ2 ) nˆ 1 + 1ˆ nˆ 2 + 1ˆ nˆ 3 .

(59)

Even though we do not present grounds of all the steps of the calculation, we are quite sure that an error does not interfere. From relation (59) it is obvious that



1 h¯

ˆ int H

2

 +

d dt

2 nˆ 3

h i = 2|g |2 (ˆn1 + 1ˆ )(ˆn2 + 1ˆ )ˆn3 + |g |2 nˆ 1 nˆ 2 (ˆn3 + 1ˆ ) .

(60)

As expression (58) has been simplified, we will attempt to represent the analogy with an analog. It may still be interesting that we obtain d dt

s h

i

nˆ 3 = ± 2|g |2 (ˆn1 + 1ˆ )(ˆn2 + 1ˆ )ˆn3 + |g |2 nˆ 1 nˆ 2 (ˆn3 + 1ˆ ) −



1 h¯

2

ˆ int (0) H

,

(61)

where we have already used the conservation law

ˆ int (t ) = Hˆ int (0). H

(62)

Another manipulation, which we will indicate only, consists in the substitutions nˆ 1 = Jˆ1 − nˆ 3 = Jˆ1 (0) − nˆ 3 ,

(63)

nˆ 2 = Jˆ2 − nˆ 3 = Jˆ2 (0) − nˆ 3 ,

(64)

where we have used the conservation laws Jˆ1 (t ) = Jˆ1 (0),

Jˆ2 (t ) = Jˆ2 (0).

(65)



It may be commented on that in Eq. (61) with explanations (63) and (64) the operator nˆ 3 t =0 does not enjoy a diagonal expression in the basis, which will be denoted by |ψJ1 ,J2 ,k i, cf. relation (64) below. This however must happen in any similar problem, where we look for a time dependence of an operator. The corresponding quantity of the classical description commutes with the Hamiltonian function in usual calculations of course. The operator may not commute with the Hamiltonian, since, from Eq. (1) it would follow otherwise that the operator was time independent.

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V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

3.2. Invariant operators

ˆ of the process. We We define Jˆ to be an invariant operator for a process, when it commutes with the Hamiltonian H will consider the process of three-wave mixing, which can be modelled as three harmonic oscillators which oscillate at Ď frequencies ω1 , ω2 , and ω3 = ω1 +ω2 . In this model, the annihilation (creation) operators aˆ j (aˆ j ), j = 1, 2, 3, are introduced obeying the standard boson commutation relations [ˆaj , aˆ k ] = [ˆaĎj , aˆ Ďk ] = 0ˆ ,

[ˆaj , aˆ Ďk ] = δjk 1ˆ ,

j, k = 1, 2, 3,

(66)

with 1ˆ and 0ˆ the identity operator and the null operator, respectively. The process is described by the Hamiltonian

ˆ = Hˆ free + Hˆ int , H

(67)

with

h

Ď

Ď

Ď

i

ˆ free = h¯ ω1 aˆ 1 aˆ 1 + ω2 aˆ 2 aˆ 2 + ω3 aˆ 3 aˆ 3 + H

1 2

h¯ (ω1 + ω2 + ω3 ) 1ˆ ,

(68)

ˆ int given in (50). Using the commutation rules (66), we identify the following invariant operators and H Jˆ1 = nˆ 1 + nˆ 3 ,

Jˆ2 = nˆ 2 + nˆ 3 ,

(69) Ď aˆ j aˆ j ,

ˆ where the photon number operators nˆ j = j = 1, 2, 3. The chosen invariant operators have the property [Jˆ1 , Jˆ2 ] = 0. The free field energy can be written in the form 



ˆ free = h¯ ω1 Jˆ1 + ω2 Jˆ2 + H

1 2

h¯ (ω1 + ω2 + ω3 ) 1ˆ .

(70)

The eigenvalues of the operators Jˆ1 , Jˆ2 are nonnegative integers and to the pair of eigenvalues (J1 , J2 ) there corresponds an eigenspace of finite dimension, namely the dimension min(J1 , J2 ) + 1. We denote this eigenspace as H (J1 ,J2 ) . A basis of this eigenspace is formed by the Fock states

|J1 − n3 , J2 − n3 , n3 i,

n3 = 0, 1, . . . , min(J1 , J2 ),

J1 = 0, 1, . . . , ∞, J2 = 0, 1, . . . , ∞.

(71)

Due to the invariance of the operators Jˆ1 , Jˆ2 , these eigenspaces are invariant subspaces of the Hilbert space of the three harmonic oscillators. This Hilbert space can be expressed as a direct sum

H≡

∞ M ∞ M

H (J1 ,J2 )

(72)

J1 =0 J2 =0

of the invariant subspaces H (J ) . Each H (J ) is spanned on the basis (71). ˆ can be decomposed with respect to these invariant subspaces, Any operator M

ˆ = M

∞ X ∞ X ∞ X ∞ X

ˆ )J1 ,J2 ,K1 ,K2 , (M

(73)

J1 =0 J2 =0 K1 =0 K2 =0

where

ˆ )J1 ,J2 ,K1 ,K2 = Π ˆΠ ˆ J1 ,J2 M ˆ K1 ,K2 , (M

(74)

with min(J1 ,J2 )

ˆ J 1 ,J 2 = Π

X

|J1 − n3 , J2 − n3 , n3 ihJ1 − n3 , J2 − n3 , n3 |.

(75)

n 3 =0

In Eq. (74), we have used simple subscript notation, which does not mean matrix elements. Particularly,

ˆ J1 ,J2 = (1ˆ )J1 ,J2 ,J1 ,J2 . Π

(76)

The interaction Hamiltonian and the operators Jˆ1 , Jˆ2 have a specific (diagonal) form

ˆ int = H

∞ X ∞ X

(Hˆ int )J1 ,J2 ,J1 ,J2 ,

(77)

J1 =0 J2 =0

Jˆj =

∞ X ∞ X J1 =0 J2 =0

(Jˆj )J1 ,J2 ,J1 ,J2 ,

j = 1, 2.

(78)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

e753

The annihilation (creation) operators of the harmonic oscillator have the form aˆ 1 =

∞ X ∞ X

(ˆa1 )J1 ,J2 ,J1 +1,J2 ,

aˆ 2 =

∞ ∞ X X

(ˆa2 )J1 ,J2 ,J1 ,J2 +1 ,

J1 =0 J2 =0

J1 =0 J2 =0

aˆ 3 =

∞ X ∞ X

(ˆa3 )J1 ,J2 ,J1 +1,J2 +1 ,

(79)

J1 =0 J2 =0

Ď

aˆ 1 =

∞ X ∞ X

(ˆaĎ1 )J1 +1,J2 ,J1 ,J2 ,

Ď

aˆ 2 =

Ď

aˆ 3 =

(ˆaĎ2 )J1 ,J2 +1,J1 ,J2 ,

J1 =0 J2 =0

J1 =0 J2 =0

∞ X ∞ X

∞ X ∞ X

(ˆaĎ3 )J1 +1,J2 +1,J1 ,J2 .

(80)

J1 =0 J2 =0

3.3. Schrödinger picture The original Schrödinger picture is related to the state vector |ψ(t )i, which is decomposed as expected in the form

|ψ(t )i =

∞ X ∞ X

(|ψ(t )i)J1 ,J2 ,

(81)

J1 =0 J2 =0

where

ˆ J1 ,J2 |ψ(t )i. (|ψ(t )i)J1 ,J2 = Π

(82)

In the Schrödinger picture, the state vector evolves according to ih¯

∂ |ψ(t )i = Hˆ int |ψ(t )i. ∂t

(83)

ˆ J1 ,J2 from the left, we Substituting (77) and (81) into the right-hand side of (83) and acting on both sides by the projector Π obtain an infinite number of linear differential equations ih¯

∂ (|ψ(t )i)J1 ,J2 = (Hˆ int )J1 ,J2 ,J1 ,J2 (|ψ(t )i)J1 ,J2 . ∂t

(84)

Each of these operator equations represents a finite number of equations. It has been known for long that these equations can be solved numerically also. For analytical expressions we refer to [7,8]. As usual, the formal solution of Eq. (83) is written in the form

|ψ(t )i = Uˆ (t )|ψ(0)i,

(85)

where

  i ˆ ˆ U (t ) = exp − Hint . h¯

(86)

Also this operator has the ‘‘diagonal’’ form Uˆ (t ) =

∞ X ∞ X

(Uˆ (t ))J1 ,J2 ,J1 ,J2 .

(87)

J1 =0 J2 =0

Formal solutions similar to the solution (85) and (86) exist also on invariant subspaces and then we can easily imagine the standard matrix algebra. Let us assume that the initial state is from the finite dimensional subspace

HS1 ,S2 =

S1 M S2 M

H (J1 ,J2 ) .

(88)

J1 =0 J2 =0

This implies that the total number of equations for the determination of coefficients CJ1 ,J2 ,n3 (t ) = hJ1 − n3 , J2 − n3 , n3 |ψ(t )i

PS

PS

is finite, i. e., J11=0 J22=0 (min(J1 , J2 ) + 1). The term Schrödinger equation is used also with respect to that for the statistical operator ρ( ˆ t ), ih¯

∂ ρ( ˆ t ) = [Hˆ int , ρ( ˆ t )]. ∂t

(89)

e754

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

A formal solution corresponding to solution (85) is

ρ( ˆ t ) = Uˆ (t )ρ( ˆ 0)Uˆ Ď (t ).

(90)

We can derive that ih¯

∂ ˆ t ))K1 ,K2 ,J1 ,J2 (Hˆ int )J1 ,J2 ,J1 ,J2 . ˆ t ))K1 ,K2 ,J1 ,J2 − (ρ( (ρ( ˆ t ))K1 ,K2 ,J1 ,J2 = (Hˆ int )K1 ,K2 ,K1 ,K2 (ρ( ∂t

(91)

A formal solution of these equations reads

ˆ 0))K1 ,K2 ,J1 ,J2 (Uˆ Ď (t ))J1 ,J2 ,J1 ,J2 . (ρ( ˆ t ))K1 ,K2 ,J1 ,J2 = (Uˆ (t ))K1 ,K2 ,K1 ,K2 (ρ(

(92)

In application, Eq. (91) need not be solved numerically if the operator (87) is known and it can be substituted into (92). Let us consider S1 = 1, S2 = 1. The decomposition (88) becomes

H1,1 ≡ H (0,0) ⊕ H (0,1) ⊕ H (1,0) ⊕ H (1,1) ,

(93)

with the dimensionalities 5 = 1 + 1 + 1 + 2. It can be seen that



Uˆ (t )

 0,0,0,0

  = 1ˆ

0,0,0,0

,





Uˆ (t )

0,1,0,1

  = 1ˆ

0,1,0,1

,





Uˆ (t )

1,0,1,0

  = 1ˆ

1,0,1,0

.

(94)

It can be derived that



Uˆ (t )

 1,1,1,1

= cos(|g |t )(|1, 1, 0ih1, 1, 0| + |0, 0, 1ih0, 0, 1|)  ∗  g g + i sin(|g |t ) |1, 1, 0ih0, 0, 1| + |0, 0, 1ih1, 1, 0| . |g | |g |

(95)

To shorten the exposition we have formulated above the time dependence only for slowly varying state vectors and statistical operators. Actually we should have used the notation |ψ I (t )i, ρˆ I (t ), where I means the ‘‘interaction picture’’. In particular,

|ψ I (t )i| = exp



i h¯

ˆ free t H



|ψ(t )i,

(96)

where the state vector |ψ(t )i evolves according to ih¯

∂ |ψ(t )i| = Hˆ |ψ(t )i. ∂t

(97)

Now

  ∂ I i ∂ I ˆ ˆ ih¯ |ψ (t )i = −Hfree |ψ (t )i + ih¯ exp Hfree t |ψ(t )i h¯ ∂t ∂t = −Hˆ free |ψ I (t )i + Hˆ |ψ I (t )i = Hˆ int |ψ I (t )i,

(98)

(99)

which is the above considered meaning of relation (83). Restricting ourselves to work with slowly varying state vectors and statistical operators, the superscript I is being omitted. The statistical properties of the oscillators remain unchanged when one uses operators such as aˆ Ij (t ) = exp(−iωj t )ˆaj (0)

(100)

for their calculation. 3.4. Heisenberg picture In the Heisenberg picture, the operators aˆ j , j = 1, 2, 3, evolve according to the Heisenberg equations of motion, ih¯

daˆ j dt

= [ˆaj , Hˆ int ],

j = 1, 2, 3.

(101)

The well-known formal solution has the form, aˆ j (t ) = Uˆ Ď (t )ˆaj (0)Uˆ (t ).

(102)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

e755

We can state the following equations, ih¯ ih¯ ih¯

d dt d dt d dt

aˆ 1




aˆ 2




aˆ 3




J1 ,J2 ,J1 +1,J2

J1 ,J2 ,J1 ,J2 +1

= aˆ 1




= aˆ 2




J1 ,J2 ,J1 +1,J2 +1



ˆ int H

J1 ,J2 ,J1 +1,J2



ˆ int H

J1 ,J2 ,J1 ,J2 +1

= aˆ 3

 




,

  − Hˆ int

aˆ 2




,

J1 ,J2 ,J1 ,J2

J1 ,J2 +1,J1 ,J2 +1

ˆ int H

J1 ,J2 ,J1 +1,J2 +1

aˆ 1

J1 ,J2 ,J1 ,J2

J1 +1,J2 ,J1 +1,J2






  − Hˆ int

 J1 +1,J2 +1,J1 +1,J2 +1

  − Hˆ int

J1 ,J2 ,J1 +1,J2

J1 ,J2 ,J1 ,J2 +1

J1 ,J2 ,J1 ,J2

aˆ 3




J1 ,J2 ,J1 +1,J2 +1

.

(103)

A formal solution of these equations reads

   
, = Uˆ Ď (t ) aˆ 1 (0) J ,J ,J +1,J Uˆ (t ) 1 2 1 2 J1 +1,J2 ,J1 +1,J2 J1 ,J2 ,J1 ,J2    

aˆ 2 (t ) J ,J ,J ,J +1 = Uˆ Ď (t ) aˆ 2 (0) J ,J ,J ,J +1 Uˆ (t ) , 1 2 1 2 1 2 1 2 J1 ,J2 +1,J1 ,J2 +1 J1 ,J2 ,J1 ,J2    

aˆ 3 (0) J ,J ,J +1,J +1 Uˆ (t ) aˆ 3 (t ) J ,J ,J +1,J +1 = Uˆ Ď (t ) aˆ 1 (t )




J1 ,J2 ,J1 +1,J2

1 2 1

J1 ,J2 ,J1 ,J2

2

1 2 1

2

J1 +1,J2 +1,J1 +1,J2 +1

.

(104)

Let us put S1 = 1, S2 = 1. For brevity, we shall introduce the notation (1,1)

(1,1)

u0,0 (t ) = cos(|g |t ), (1,1)

(1,1)∗

u1,0 (t ) = −u0,1

u0,1 (t ) =

(t ),

ig ∗

|g |

(1,1)

sin(|g |t ),

(1,1)∗

u1,1 (t ) = u0,0

(t ).

(105)

The above results, more complicated ones can be obtained using formula manipulating software, suffice only for finite expansions as aˆ 1 (t ) ≈ aˆ 1 (t ) 0,0,1,0 + aˆ 1 (t ) 0,1,1,1 ,







aˆ 2 (t ) ≈ aˆ 2 (t ) 0,0,0,1 + aˆ 2 (t ) 1,0,1,1 ,







aˆ 3 (t ) ≈ aˆ 3 (t ) 0,0,1,1 .




(106)


Let us begin with aˆ 3 (t ). Using aˆ 3 (0) 0,0,1,1 = |0, 0, 0ih0, 0, 1| in (104), we obtain that (1,1)

(1,1)

aˆ 3 (t ) 0,0,1,1 = u1,1 (t )|0, 0, 0ih0, 0, 1| + u1,0 (t )|0, 0, 0ih1, 1, 0|.




(107)

The short-time or perturbative solutions provide expansions in powers of creation and annihilation operators. We will try to emulate this approach. We will rewrite (107) as (1,1)

(1,1)

aˆ 3 (t ) 0,0,1,1 = u1,1 (t )|0, 0, 0ih0, 0, 0|ˆa3 (0) + u1,0 (t )|0, 0, 0ih0, 0, 0|ˆa1 (0)ˆa2 (0).




(108)

Let us note that

  ˆ = |0, 0, 0ih0, 0, 0|M ˆ |0, 0, 0ih0, 0, 0|M

0,0,1,1

  ˆ = (|0, 0, 0ih0, 0, 0|)0,0,0,0 M 0,0,1,1         ˆ ˆ ˆ = 1ˆ M = 1ˆ M = M 0,0,0,0

0,0,1,1

0,0,1,1

0,0,1,1

,

(109)

ˆ = aˆ 3 (0), aˆ 1 (0)ˆa2 (0). From this for M aˆ 3 (t ) 0,0,1,1 = g1 (t ) aˆ 3 (0) 0,0,1,1 + g2 (t ) aˆ 1 (0)ˆa2 (0) 0,0,1,1 ,










(110)

where (1,1)

(1,1)

g1 (t ) ≡ u1,1 (t ),

g2 (t ) ≡ u1,0 (t ).

(111)

Let us continue with aˆ 1 (t ). It can be seen that aˆ 1 (t ) 0,0,1,0 = aˆ 1 (0) 0,0,1,0 = |0, 0, 0ih1, 0, 0|.







(112)


Substituting aˆ 1 (0) 0,1,1,1 = |0, 1, 0ih1, 1, 0| into (104), we obtain that (1,1)

(1,1)

aˆ 1 (t ) 0,1,1,1 = u0,1 (t )|0, 1, 0ih0, 0, 1| + u0,0 (t )|0, 1, 0ih1, 1, 0|.




(113)

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V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

Using a possible definition of number states, we obtain that (1,1)

Ď

aˆ 1 (t ) 0,0,1,0 + aˆ 1 (t ) 0,1,1,1 = |0, 0, 0ih0, 0, 0|ˆa1 (0) + u0,1 (t )ˆa2 (0)|0, 0, 0ih0, 0, 0|ˆa3 (0)







+ u(01,0,1) (t )ˆaĎ2 (0)|0, 0, 0ih0, 0, 0|ˆa1 (0)ˆa2 (0).

(114)

Now we have to use the finite expansions of the vacuum projector

|0, 0, 0ih0, 0, 0| = (1ˆ )0,0,0,0 + (1ˆ )0,1,0,1 − (ˆaĎ2 (0)ˆa2 (0))0,1,0,1 = (1ˆ )0,0,0,0 + (1ˆ − aˆ Ď2 (0)ˆa2 (0))0,1,0,1

(115)

in the first term and

|0, 0, 0ih0, 0, 0| = (1ˆ )0,0,0,0

(116)

elsewhere. So aˆ 1 (t ) 0,0,1,0 + aˆ 1 (t ) 0,1,1,1 = aˆ 1 (0) 0,0,1,0 + aˆ 1 (0) 0,1,1,1













  + f1 (t ) aˆ Ď2 (0)ˆa3 (0)

0,1,1,1

  + f2 (t ) aˆ Ď2 (0)ˆa1 (0)ˆa2 (0)

0,1,1,1

,

(117)

where (1,1)

(1,1)

f1 (t ) ≡ u0,1 (t ),

f2 (t ) ≡ u0,0 (t ) − 1.

(118)

The calculation of aˆ 2 (t ) is quite similar. We see that aˆ 2 (t ) 0,0,0,1 + aˆ 2 (t ) 1,0,1,1 = aˆ 2 (0) 0,0,0,1 + aˆ 2 (0) 1,0,1,1













  + f1 (t ) aˆ Ď1 (0)ˆa3 (0)

1,0,1,1

  + f2 (t ) aˆ Ď1 (0)ˆa1 (0)ˆa2 (0)

1,0,1,1

.

(119)

3.5. General formulation of the method In general, we introduce (J ,J )





un 1,n20 (t ) = hJ1 − n3 , J2 − n3 , n3 | Uˆ (t ) 3

3

J1 ,J2 ,J1 ,J2

|J1 − n03 , J2 − n03 , n03 i.

(120)

We consider finite expansions S1 −1 S2

aˆ 1 (t ) ≈

XX

aˆ 1 (t )




,

aˆ 2 (t )




,

J1 ,J2 ,J1 +1,J2

J1 =0 J2 =0

aˆ 2 (t ) ≈

S1 SX 2 −1 X

J1 ,J2 ,J1 ,J2 +1

J1 =0 J2 =0 S1 −1 S2 −1

aˆ 3 (t ) ≈

XX

aˆ 3 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2 +1

.

(121)

Now we substitute the expansions



Uˆ (t )

min(J1 ,J2 ) min(J1 ,J2 )

 J1 ,J2 ,J1 ,J2

X

X

n3 =0

n03 =0

=

( J ,J )

un 1,n20 (t )|J1 − n3 , J2 − n3 , n3 ihJ1 − n03 , J2 − n03 , n03 |, 3

(122)

3

min(J1 ,J2 )

aˆ 1 (0)




J1 ,J2 ,J1 +1,J2

X p

=

J1 − n3 + 1|J1 − n3 , J2 − n3 , n3 ihJ1 − n3 + 1, J2 − n3 , n3 |,

n 3 =0 min(J1 ,J2 )

aˆ 2 (0)




J1 ,J2 ,J1 ,J2 +1

X p

=

J2 − n3 + 1|J1 − n3 , J2 − n3 , n3 ihJ1 − n3 , J2 − n3 + 1, n3 |,

n 3 =0 min(J1 ,J2 )

aˆ 3 (0)




J1 ,J2 ,J1 +1,J2 +1

=

X p

n3 + 1|J1 − n3 , J2 − n3 , n3 ihJ1 − n3 , J2 − n3 , n3 + 1|,

n3 =0

(123)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

e757

and the respective ones for





Uˆ (t )

J1 +1,J2 ,J1 +1,J2



,



Uˆ (t )

J1 ,J2 +1,J1 ,J2 +1



,

Uˆ (t )



(124)

J1 +1,J2 +1,J1 +1,J2 +1

into the formal solution (104). We state that



Uˆ Ď (t )



aˆ 1 (0)






J1 ,J2 ,J1 ,J2

J1 ,J2 ,J1 +1,J2

min(J1 ,J2 ) min(J1 +1,J2 ) min(J1 ,J2 )

X

X

X

n 3 =0

n03 =0

=0 n00 3

= 

Uˆ Ď (t )



aˆ 2 (0)

J1 ,J2 ,J1 ,J2

min(J1 ,J2 ) min(J1 ,J2 +1) min(J1 ,J2 )

n 3 =0

n03 =0

n00 =0 3

= 



Uˆ Ď (t )

aˆ 3 (0)

3

min(J1 ,J2 ) min(J1 ,J2 )+1 min(J1 ,J2 )

=

X

n 3 =0

n03 =0

n00 =0 3

J1 ,J2 +1,J1 ,J2 +1

(J ,J +1)

(J ,J )∗ 3



X

3

q (t ) J1 − n003 + 1|J1 − n3 , J2 − n3 , n3 ihJ1 + 1 − n03 , J2 − n03 , n03 |,



Uˆ (t )

J1 ,J2 ,J1 +1,J2 +1

X

3

un001,n2 (t )un001,n20




J1 ,J2 ,J1 ,J2

3

3



X

(J +1,J2 )

(J ,J )∗

J1 ,J2 ,J1 ,J2 +1

X

J1 +1,J2 ,J1 +1,J2

un001,n2 (t )un001,n0




X



Uˆ (t )

3

Uˆ (t )

(J ,J )∗

3

q (t ) J2 − n003 + 1|J1 − n3 , J2 − n3 , n3 ihJ1 − n03 , J2 + 1 − n03 , n03 |,

 J1 +1,J2 +1,J1 +1,J2 +1

(J +1,J +1)

un001,n2 (t )un001+1,n20 3

3

3

3

q (t ) n003 + 1|J1 − n3 , J2 − n3 , n3 ihJ1 + 1 − n03 , J2 + 1 − n03 , n03 |. (125)

Introducing the notation, aˆ 1 (t ) aˆ 2 (t )


J1 ,J2 ,J1 +1,J2 n3 ,n03


J1 ,J2 ,J1 ,J2 +1 n3 ,n03


= hJ1 − n3 , J2 − n3 , n3 | aˆ 1 (t ) J ,J ,J +1,J |J1 + 1 − n03 , J2 − n03 , n03 i, 1 2 1 2
= hJ1 − n3 , J2 − n3 , n3 | aˆ 2 (t ) J ,J ,J ,J +1 |J1 − n03 , J2 + 1 − n03 , n03 i, 1 2 1 2


J1 ,J2 ,J1 +1,J2 +1
aˆ 3 (t ) n ,n0 = hJ1 − n3 , J2 − n3 , n3 | aˆ 3 (t ) J 3

|J1 + 1 − n03 , J2 + 1 − n03 , n03 i,

1 ,J2 ,J1 +1,J2 +1

3

(126)

we can write that

XX

(J1 ,J2 ) min(X J1 +1,J2 ) X X minX

S1 −1 S2

S1 −1 S2

aˆ 1 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2

=

J1 =0 J2 =0

n3 =0


J1 ,J2 ,J1 +1,J2

aˆ 1 (t )

n03 =0

n3 ,n03

×|J1 − n3 , J2 − n3 , n3 ihJ1 + 1 − n03 , J2 − n03 , n03 |, S1 SX 2 −1 X

aˆ 2 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 ,J2 +1

=

S1 SX (J1 ,J2 ) min(X J1 ,J2 +1) 2 −1 min X X J1 =0 J2 =0

n3 =0


J1 ,J2 ,J1 ,J2 +1

aˆ 2 (t )

n03 =0

n3 ,n03

×|J1 − n3 , J2 − n3 , n3 ihJ1 − n03 , J2 + 1 − n03 , n03 |, S1 −1 S2 −1 min(J1 ,J2 ) min(J1 ,J2 )+1

S1 −1 S2 −1

XX

aˆ 3 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2 +1

=

XX X

X

J1 =0 J2 =0

n03 =0

n3 =0


J1 ,J2 ,J1 +1,J2 +1

aˆ 3 (t )

n3 ,n03

× |J1 − n3 , J2 − n3 , n3 ihJ1 + 1 − n03 , J2 + 1 − n03 , n03 |. Using the annihilation and creation operators, we can write it as S1 −1 S2

XX

(J1 ,J2 ) min(X J1 +1,J2 ) X X minX

S1 −1 S2

aˆ 1 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2

=

J1 =0 J2 =0

n3 =0

n03 =0

1

p

(J1 − n3 )!(J2 − n3 )!n3 !(J1 + 1 − n03 )!(J2 − n03 )!n03 !


J ,J ,J +1,J Ď J −n Ď J −n Ď n J1 +1−n03 J2 −n03 n03 × aˆ 1 (t ) n1 ,n2 0 1 2 aˆ 1 1 3 aˆ 2 2 3 aˆ 3 3 |0, 0, 0ih0, 0, 0|ˆa1 aˆ 2 aˆ 3 , 3

S1 SX 2 −1 X J1 =0 J2 =0

aˆ 2 (t )




J1 ,J2 ,J1 ,J2 +1

=

3

S1 SX (J1 ,J2 ) min(X J1 ,J2 +1) 2 −1 min X X J1 =0 J2 =0

n3 =0

n03 =0

1

p

(J1 − n3 )!(J2 − n3 )!n3 !(J1 − n03 )!(J2 + 1 − n03 )!n03 !


J ,J ,J ,J + 1 Ď J − n Ď J − n Ď n J1 −n0 J2 +1−n03 n03 aˆ 3 , × aˆ 2 (t ) n1 ,n2 0 1 2 aˆ 1 1 3 aˆ 2 2 3 aˆ 3 3 |0, 0, 0ih0, 0, 0|ˆa1 3 aˆ 2 3

3

(127)

e758

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762 S1 −1 S2 −1 min(J1 ,J2 ) min(J1 ,J2 )+1

S1 −1 S2 −1

XX

aˆ 3 (t )




J1 ,J2 ,J1 +1,J2 +1

J1 =0 J2 =0

XX X

=

J 1 =0 J 2 =0

1

X

n03 =0

n 3 =0

p (J1 − n3 )!(J2 − n3 )!n3 !(J1 + 1 − n03 )!(J2 + 1 − n03 )!n03 !


J ,J ,J + 1 ,J + 1 Ď J − n Ď J − n Ď n J1 +1−n03 J2 +1−n03 n03 aˆ 3 . aˆ 2 × aˆ 3 (t ) n1 ,n2 0 1 2 aˆ 1 1 3 aˆ 2 2 3 aˆ 3 3 |0, 0, 0ih0, 0, 0|ˆa1 3

(128)

3

Now in the (J1 , J2 )th term of the first two sums, we use an infinite expansion (J1 ,J2 ) ∞ X ∞ minX X 00 00

|0, 0, 0ih0, 0, 0| =

=0 n00 3

J100 =0 J200 =0

00

00

00

(−1)J1 +J2 −n3 Ď J 00 −n00 Ď J 00 −n00 Ď n00 J 00 −n00 J 00 −n00 n00 ˆ 1 3 aˆ 2 2 3 aˆ 3 3 aˆ 11 3 aˆ 22 3 aˆ 33 . 00 00 00 00 00 a1 (J1 − n3 )!(J2 − n3 )!n3 !

(129)

Substituting the triple sum into (128), we obtain that S 1 −1 S 2

XX

aˆ 1 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2 min(J10 ,J20 ) min(J10 +1,J20 ) min(J100 ,J200 )

S1 −1 S1 −1 S1 −1

=

S2 S2 X S2 X XXX X

X

D1 =0 J 0 =0 J 00 =0 D2 =0 J 0 =0 J 00 =0 2 2 1 1

| {z } J10 +J100 =D1

n3 =0

X

1

X

n03 =0

n00 =0 3

p 0 (J1 − n3 )!(J20 − n3 )!n3 !(J10 + 1 − n03 )!(J20 − n03 )!n03 !

| {z } J20 +J200 =D2

J 00 +J 00 −n00

×


J10 ,J20 ,J10 +1,J20 Ď D1 −n3 −n003 Ď D2 −n3 −n003 Ď n3 +n003 D1 +1−n03 −n003 D2 −n03 −n003 n03 +n003 (−1) 1 2 3 ˆ aˆ 1 aˆ 2 aˆ 3 aˆ 1 aˆ 2 aˆ 3 , 00 00 00 00 a1 (t ) n3 ,n03 (J1 − n3 )!(J2 − n3 )!n3 ! 00

S1 SX 2 −1 X

aˆ 2 (t )




J1 ,J2 ,J1 ,J2 +1

J1 =0 J2 =0

=

min(J10 ,J20 ) min(J10 ,J20 +1) min(J100 ,J200 )

S1 X S1 X S1 SX 2 −1 SX 2 −1 SX 2 −1 X

X

D1 =0 J 0 =0 J 00 =0 D2 =0 J 0 =0 J 00 =0 1 1 2 2

| {z } J10 +J100 =D1

n3 =0

X

1

X

n03 =0

n00 =0 3

p 0 (J1 − n3 )!(J20 − n3 )!n3 !(J10 − n03 )!(J20 + 1 − n03 )!n03 !

| {z } J20 +J200 =D2

J 00 +J 00 −n00

×


J10 ,J20 ,J10 ,J20 +1 Ď D1 −n3 −n003 Ď D2 −n3 −n003 Ď n3 +n003 D1 −n03 −n003 D2 +1−n03 −n003 n03 +n003 (−1) 1 2 3 ˆ aˆ 1 aˆ 2 aˆ 3 aˆ 1 aˆ 2 aˆ 3 , 00 00 00 00 a2 (t ) n3 ,n03 (J1 − n3 )!(J2 − n3 )!n3 ! 00

S1 −1 S2 −1

XX

aˆ 3 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2 +1

0 0 0 0 00 00 S1 −1 S1 −1 S1 −1 S2 −1 S2 −1 S2 −1 min(J1 ,J2 ) min(J1 ,J2 )+1 min(J1 ,J2 )

=

XXX XXX

D1 =0 J 0 =0 J 00 =0 D2 =0 J 0 =0 J 00 =0 1 1 2 2

| {z } J10 +J100 =D1

X

X

X

n3 =0

n03 =0

n00 =0 3

1

p 0 (J1 − n3 )!(J20 − n3 )!n3 !(J10 + 1 − n03 )!(J20 + 1 − n03 )!n03 !

| {z } J20 +J200 =D2

J 00 +J 00 −n00

×


J10 ,J20 ,J10 +1,J20 +1 Ď D1 −n3 −n003 Ď D2 −n3 −n003 Ď n3 +n003 D1 +1−n03 −n003 D2 +1−n03 −n003 n03 +n003 (−1) 1 2 3 ˆ aˆ 1 aˆ 2 aˆ 3 aˆ 1 aˆ 2 aˆ 3 . 00 00 00 00 a3 (t ) n3 ,n03 (J1 − n3 )!(J2 − n3 )!n3 ! 00

So we have obtained the series of the form, S1 −1 S2

XX

S2 minX (D1 ,D2 ) min(DX 1 +1,D2 ) X X

S1 −1

aˆ 1 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2

=

D 1 =0 D 2 =0

m=0

S1 −1

×

S2  X X

m0 =0

(D ,D ,D1 +1,D2 )

hm,1m0 2

(t )

Ď D1 −m Ď D2 −m Ď m D1 +1−m0 D2 −m0 m0 aˆ 2 aˆ 3 aˆ 1 aˆ 2 aˆ 3

aˆ 1

 J1 ,J2 ,J1 +1,J2

J1 =D1 J2 =D2 S1 SX 2 −1 X J1 =0 J2 =0

aˆ 2 (t )




J1 ,J2 ,J1 ,J2 +1

=

S1 SX (D1 ,D2 ) min(DX 2 −1 minX 1 ,D2 +1) X D 1 =0 D 2 =0

×

m=0

S1 SX 2 −1  X J1 =D1 J2 =D2

m0 =0

(D ,D ,D1 ,D2 +1)

hm,1m0 2

(t )

Ď D1 −m Ď D2 −m Ď m D1 −m0 D2 +1−m0 m0 aˆ 2 aˆ 3 aˆ 1 aˆ 2 aˆ 3

aˆ 1

,

 J1 ,J2 ,J1 ,J2 +1

,

(130)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762 S1 −1 S2 −1 min(D1 ,D2 ) min(D1 ,D2 )+1

S1 −1 S2 −1

XX

aˆ 3 (t )




J1 =0 J2 =0

J1 ,J2 ,J1 +1,J2 +1

=

X X

X

X

D1 =0 D2 =0

m=0

m0 =0

S1 −1 S2 −1

×

X X

(D ,D ,D1 +1,D2 +1)

hm,1m0 2

(t )

Ď D1 −m Ď D2 −m Ď m D1 +1−m0 D2 +1−m0 m0 aˆ 2 aˆ 3 aˆ 1 aˆ 2 aˆ 3

aˆ 1

J1 =D1 J2 =D2

e759

 J1 ,J2 ,J1 +1,J2 +1

,

(131)

where (D ,D ,D1 +1,D2 )

hm,1m0 2

D1 X D1 X D2 X D2 X

(t ) =

J10 =0 J100 =0 J20 =0 J200 =0

{z

|

}

J10 +J100 =D1 ,J20 +J200 =D2 ,min(J10 ,J20 )+min(J100 ,J200 )≥m,min(J10 +1,J20 )+min(J100 ,J200 )≥m0 min(m,m0 )

1

X

×

p 0 (J1 − m + l)!(J20 − m + l)!(m − l)!(J10 + 1 − m0 + l)!(J20 − m0 + l)!(m0 − l)! l=0 J100 +J200 −l
J 0 ,J 0 ,J 0 +1,J 0 (−1) aˆ 1 (t ) m1 −2l,m10 −l 2 for mode 1, × 00 (J1 − l)!(J200 − l)!l! (D ,D ,D1 ,D2 +1)

hm,1m0 2

D2 D2 X D1 X D1 X X

(t ) =

J10 =0 J100 =0 J20 =0 J200 =0

|

{z

}

J10 +J100 =D1 ,J20 +J200 =D2 ,min(J10 ,J20 )+min(J100 ,J200 )≥m,min(J10 ,J20 +1)+min(J100 ,J200 )≥m0 min(m,m0 )

1

X

×

p 0 (J1 − m + l)!(J20 − m + l)!(m − l)!(J10 − m0 + l)!(J20 + 1 − m0 + l)!(m0 − l)! l=0 J100 +J200 −l
J 0 ,J 0 ,J 0 ,J 0 +1 (−1) × 00 aˆ 2 (t ) m1 −2l,m10 −2l for mode 2, 00 (J1 − l)!(J2 − l)!l! (D ,D ,D1 +1,D2 +1)

hm,1m0 2

D1 X D1 X D2 X D2 X

(t ) =

J10 =0 J100 =0 J20 =0 J200 =0

|

{z

}

J10 +J100 =D1 ,J20 +J200 =D2 ,min(J10 ,J20 )+min(J100 ,J200 )≥m,min(J10 ,J20 )+1+min(J100 ,J200 )≥m0 min(m,m0 )

X

×

1

p 0 (J1 − m + l)!(J20 − m + l)!(m − l)!(J10 + 1 − m0 + l)!(J20 + 1 − m0 + l)!(m0 − l)! l=0 J100 +J200 −l
J 0 ,J 0 ,J 0 +1,J 0 +1 (−1) × 00 aˆ 3 (t ) m1 −2l,m10 −l 2 for mode 3. (J1 − l)!(J200 − l)!l!

(132)

3.6. Connection with spectral decomposition Let us note that the Hamiltonian also can be decomposed into terms acting on the subspaces H (J1 ,J2 ) ,

ˆ = H

∞ X ∞   X ˆ H

J1 ,J2 ,J1 ,J2

J1 =0 J2 =0

.

(133)

ˆ int and the invariant operators Jˆ1 , Jˆ2 as solutions One determines the eigenvalues and joint eigenvectors of the Hamiltonian H of equations for h¯ ω, J1 , J2 , and |ψi ˆ int |ψi = h¯ ω|ψi, H Jˆ1 |ψi = J1 |ψi, Jˆ2 |ψi = J2 |ψi,

(134)

where |ψi = (|ψi)J1 ,J2 . We turn to an infinite set of equations



ˆ int H

 J1 ,J2 ,J1 ,J2

|ψi = h¯ ω|ψi.

(135)

ˆ in place of Hˆ int , h¯ (ω1 J1 + ω2 J2 ) + h¯ (ω1 + ω2 ) + h¯ ω in place of h¯ ω, and the same eigenvector. This equation holds also for H Introducing the notation |ψJ1 ,J2 ,k i,

k = 1, . . . , min(J1 , J2 ) + 1,

(136)

e760

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

for min(J1 , J2 ) + 1 linearly independent eigenvectors, we can write the spectral decomposition

ˆ int = H

J1 ,J2 )+1 ∞ X ∞ min(X X J1 =0 J2 =0

h¯ ωJ1 ,J2 ,k |ψJ1 ,J2 ,k ihψJ1 ,J2 ,k |,

(137)

k=1

where 1

ωJ1 ,J2 ,k =

h¯

hψJ1 ,J2 ,k |Hˆ int |ψJ1 ,J2 ,k i.

(138)

On substituting into relation (86), one obtains that Uˆ (t ) =

J1 ,J2 )+1 ∞ X ∞ min(X X J1 =0 J2 =0

exp(−iωJ1 ,J2 ,k t )|ψJ1 ,J2 ,k ihψJ1 ,J2 ,k |.

(139)

k=1

Now aˆ j1 ,j2 (t ) =

∞ X ∞ X

aˆ j1 ,j2 (t )




J1 =0 J2 =0

=

J1 ,J2 ,J1 +j1 ,J2 +j2

∞ X ∞ X

min(J1 ,J2 )+1

J1 =0 J2 =0

k =1

X

! aˆ j1 ,j2 (0)

exp(iωJ1 ,J2 ,k t )|ψJ1 ,J2 ,k ihψJ1 ,J2 ,k |




J1 ,J2 ,J1 +j1 ,J2 +j2

min(J1 +j1 ,J2 +j2 )+1

X

×

! exp(−iωJ1 +j1 ,J2 +j2 ,k0 t )|ψJ1 +j1 ,J2 +j2 ,k0 ihψJ1 +j1 ,J2 +j2 ,k0 | ,

k0 =1

=

j1 ,J2 +j2 )+1 min(J1 +X j1 ,J2 +j2 )+1 ∞ X ∞ min(J1 +X X J1 =0 J2 =0

exp i(ωJ1 ,J2 ,k − ωJ1 +j1 ,J2 +j2 ,k0 )t





k0 =1

k=1

×hψJ1 ,J2 ,k | aˆ j1 ,j2 (0)




J1 ,J2 ,J1 +j1 ,J2 +j2

|ψJ1 +j1 ,J2 +j2 ,k0 i|ψJ1 ,J2 ,k ihψJ1 +j1 ,J2 +j2 ,k0 |.

(140)

To avoid a threefold repetition of similar formulas, we have used the subscripts j1 , j2 , (j1 , j2 ) = (1, 0), (0, 1), (1, 1). In (140) aˆ 1,0 , aˆ 0,1 , aˆ 1,1 mean aˆ 1 , aˆ 2 , aˆ 3 respectively. Let us also examine how the spectral decomposition may be respected in formulas which are related to the Fock states and the normally ordered monomials. According to definition (120) it holds that min(J1 ,J2 )+1

(J ,J )

X

un 1n02 (t ) = hJ1 − n3 , J2 − n3 , n3 | 3 3

exp(−iωJ1 ,J2 ,k t )|ψJ1 ,J2 ,k ihψJ1 ,J2 ,k |J1 − n03 , J2 − n03 , n03 i

k=1 min(J1 ,J2 )+1

X

=

exp(−iωJ1 ,J2 ,k t )hJ1 − n3 , J2 − n3 , n3 |ψJ1 ,J2 ,k ihJ1 − n03 , J2 − n03 , n03 |ψJ1 ,J2 ,k i∗ .

(141)

k=1

Additionally we may express the complicated piece of note (126) as


(J1 ,J2 ,J1 +j1 ,J2 +j2 )

aˆ j1 ,j2 (t )

n3 ,n03

min(J1 ,J2 )

=

(J ,J )∗

X


(J1 ,J2 ,J1 +j1 ,J2 +j2 )

(J +j ,J +j )

un001n02 (t )un001+δ1 2δ 2,n0 (t ) aˆ j1 ,j2 (0) j1 1 j2 1 3 3 3 3

n00 =0 3

n00 ,n00 +δj1 1 δj2 1 3 3

,

(142)

where, of course,


(J1 ,J2 ,J1 +j1 ,J2 +j2 ) aˆ j1 ,j2 (0) n00 ,n00 +δ δ = 3

3

j1 1 j2 1

q   J − n003 + 1  q 1

for j1 = 1, j2 = 0, for j1 = 0, j2 = 1,

J − n003 + 1  q2    n00 + 1

(143)

for j1 = 1, j2 = 1.

3

Instead of transforming relation (142) let us observe that aˆ j1 ,j2 (t )




J1 ,J2 ,J1 +j1 ,J2 +j2

  = Uˆ Ď (t )

J ,J

aˆ j1 ,j2 (0)




 J1 ,J2 ,J1 +j1 ,J2 +j2



Uˆ (t )

J1 +j1 ,J2 +j2 ,J1 +j1 ,J2 +j2

min(J1 ,J2 )+1 min(J1 +j1 ,J2 +j2 )+1

=

X

X

k=1

k0 =1

exp i(ωJ1 ,J2 ,k − ωJ1 +j1 ,J2 +j2 ,k0 )t




×hψJ1 ,J2 ,k | aˆ j1 ,j2 (0) J

1 ,J2 ,J1 +j1 ,J2 +j2



|ψJ1 +j1 ,J2 +j2 ,k0 i|ψJ1 ,J2 ,k ihψJ1 +j1 ,J2 +j2 ,k0 |,

(144)

V. Peřinová, A. Lukš / Nonlinear Analysis 71 (2009) e744–e762

e761

whence


(J1 ,J2 ,J1 +j1 ,J2 +j2 )

aˆ j1 ,j2 (t )

n3 ,n03

min(J1 ,J2 )+1 min(J1 +j1 ,J2 +j2 )+1

=

X

X

k=1

k0 =1


×hψJ1 ,J2 ,k | aˆ j1 ,j2 (0) J

exp i(ωJ1 ,J2 ,k − ωJ1 +j1 ,J2 +j2 ,k0 )t



1 ,J2 ,J1 +j1 ,J2 +j2



|ψJ1 +j1 ,J2 +j2 ,k0 i

×hJ1 − n3 , J2 − n3 , n3 |ψJ1 ,J2 ,k ihJ1 + j1 − n03 , J2 + j2 − n03 , n03 |ψJ1 +j1 ,J2 +j2 ,k0 i∗ ,

(145)

where


hψJ1 ,J2 ,k | aˆ j1 ,j2 (0) J

1 ,J2 ,J1 +j1 ,J2 +j2

|ψJ1 +j1 ,J2 +j2 ,k0 i

min(J1 ,J2 )

X

= hψJ1 ,J2 ,k |


|J1 − n003 , J2 − n003 , n003 ihJ1 − n003 , J2 − n003 , n003 | aˆ j1 ,j2 (0) J

1 ,J2 ,J1 +j1 ,J2 +j2

n00 =0 3 min(J1 ,J2 )

×

X

|J1 − n003 + δj1 1 δj2 0 , J2 − n003 + δj1 0 δj2 1 , n003 + δj1 1 δj2 1 i

n00 =0 3

×hJ1 − n003 + δj1 1 δj2 0 , J2 − n003 + δj1 0 δj2 1 , n003 + δj1 1 δj2 1 |ψJ1 +j1 ,J2 +j2 ,k0 i min(J1 ,J2 )

=

X

hJ1 − n003 , J2 − n003 , n003 |ψJ1 ,J2 ,k i∗ aˆ j1 ,j2 (0)


(J1 ,J2 ,J1 +j1 ,J2 +j2 )

n00 =0 3

n00 ,n00 +δj1 1 δj2 1 3 3

×hJ1 − n003 + δj1 1 δj2 0 , J2 − n003 + δj1 0 δj2 1 , n003 + δj1 1 δj2 1 |ψJ1 +j1 ,J2 +j2 ,k0 i.

(146)

3.7. Slowly varying operator picture To shorten the exposition we have formulated above the time dependence only for slowly varying operators. Actually we should have used the notation aˆ I1 (t ), aˆ I2 (t ), aˆ I3 (t ), where I means the ‘‘interaction picture’’. The operators of the three oscillators aˆ 1 (t ), aˆ 2 (t ), aˆ 3 (t ) can be expressed in terms of these operators and vice versa, aˆ Ij (t ) = exp(iωj t )ˆaj (t ).

(147)

With respect to the direct dependence on the time it holds that ih¯

d dt

∂ I ˆ (t )] aˆ (t ) + [ˆaIj (t ), H ∂t j = −[ˆaIj (t ), Hˆ free (t )] + [ˆaIj (t ), Hˆ (t )]

aˆ Ij (t ) = ih¯

= [ˆaIj (t ), Hˆ int (t )],

(148)

(149)

which is the above considered meaning of relation (101). The statistical properties of the oscillators remain unchanged when one uses the operator



   i ˆ ˆ ρˆ (t ) = exp − Hfree (0)t ρ( ˆ 0) exp Hfree (0)t h¯ h¯ I

i

(150)

for their calculation. 4. Conclusions We have dealt with the fate of two systems of ordinary differential equations for the classical description of a nonlinear optical process after quantization. The quantum description is linear in principle. Quantum statistics are calculated using expansions in annihilation and creation operators of photons, which, the so called Heisenberg picture being considered, still resemble asymptotical approximate solutions of ordinary differential equations of the classical description. We have concentrated ourselves to equations, which are soluble in terms of Jacobian elliptic functions and by quadratures in the classical framework. We have mentioned that the integrals of motion, which suffice for the solubility ‘‘by quadratures’’ according to the Liouvillian characteristics, reappear in the quantum description. They allow us to determine the aspect by which the expansions in annihilation and creation operators are finite. This finiteness is a dim reflection of the classical solubility. Acknowledgment This work under the Research project: Measurement and Information in Optics MSM 6198959213 was supported by the Ministry of Education of the Czech Republic.

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