the floquet theory of the time-periodic hamiltonian systems. i. the … · 2015. 10. 22. · 5 the...

Romanian Reports in Physics, Vol. 52, Nos. 3–4, P. 233–263, 2000

THE FLOQUET THEORY OF THE TIME-PERIODIC HAMILTONIAN SYSTEMS.

I. THE PSEUDO-EVOLUTION METHOD

RADU PAUL LUNGU

Physics Department, University of Bucharest

(Received January 11, 1998)

Abstract. A method that describes the evolution of a time-periodic Hamiltonian system is presented. This method is founded on the properties of the Floquet Hamiltonian in the extended Hilbert space, on the use of an auxiliary parameter which is analogous to the time, and on the defi-nition of the vectors associated to the state of the system, such that the matrix elements in the ex-tended Hilbert space have physical correspondence. The formalism presents similarities to the standard formalism of the quantum mechanics of the conservative systems and also, it enables to develop a perturbation technique analogous to those used in the theory of the conservative many-particle systems.

Key words: Floquet Hamiltonian, pseudo-evolution method.

1. INTRODUCTION

The theoretical study of the time-periodic Hamiltonian systems has become important in the last years because of many experimental results performed on quantum systems interacting with the laser field of high intensities. It is well known that the use of the perturbation theory is not suitable to describe the inter-action between a quantum system and an intense electromagnetic field; therefore it is most convenient to use non-perturbative methods, and considerable progress is achieved adapting the Floquet theorem to the evolution equation of a quantum system having a time-periodic Hamiltonian [1, 2]; the resulting non-perturbative method describing the evolution of such system is usually named the quantum Floquet formalism. H. Sambe [3] has shown that the quantum Floquet formalism is expressed more naturally if an extended Hilbert space is introduced, including time as a supplementary coordinate and J. S. Howland has studied the mathe-matical properties of this extended space [11]. This formalism was used in many papers for the study of atomic systems [4, 5] or of semiconductor superlattices [6, 7], which interact with intense laser fields.

In this paper we shall present a method that uses systematically the ex-tended Hilbert space and introduces an auxiliary parameter simulating the physi-

234 Radu Paul Lungu 2

cal time in order to describe the evolution of a quantum system with time-peri-odic Hamiltonian; this method has many formal similarities to the standard method used in the quantum mechanics of the time-independent Hamiltonian systems and in addition it has the advantage that it may be adapted for the sec-ond quantization formalism, the natural way to manipulate the many-particle systems. We mention that our method is derived from the (t, t′) formalism pro-posed by U. Peskin and N. Moiseyev [12], but our point of view is different in many respects from that of Peskin and Moiseyev: a) we define more naturally the state pseudo-vectors in the Schrödinger picture and the physical signifiance of the quantities from the Floquet space; b) we extend the formalism to the Heis-enberg and the interaction pictures; c) our metod is used as a general treatement of the quantum systems having time-periodic Hamiltonians, but not as a simple mathematical expedient for numerical computations.

In Section 2 we define the extended Hilbert space (named the Floquet space) as the direct product between the Hilbert space of the quantum states and the time Hilbert space associated with the time-periodic functions; using a time basis, we define the vectors and the operators of this space, instead of using the Fourier components, as is frequently done.

In Section 3 we present the most important properties of the Floquet Ham-iltonian, and we derive from them a relationship between the Floquet Hamilto-nian and the evolution operator of the system.

Using the above relationship, in Section 4 we construct a simulation method of the system evolution defining adequately in the Floquet space the vec-tors associated to the states and the operators associated to the observable quanti-ties, and introducing an auxiliary parameter which simulates the physical time; then we show that it is possible to define, in the Floquet space, the Schrödinger, the Heisenberg and the interaction pictures in a similar manner as the corre-sponding pictures of the standard quantum mechanics, and in each of the above pictures the matrix elements of the observables between state vectors (both de-fined in the Floquet space) have a direct physical correspondence.

The simulation method of the system evolution (presented in Section 4) is used in Section 5 to obtain a variant of the Gell-Mann and Low theorem, adapted to the Floquet space; from this theorem we deduce a perturbation series, which is similar to the Feynman-Dyson series used in the quantum field theory. We note that our results are not in contradiction with those of D. Hone, R. Kutzmerick and W. Kohn [13], as we have considered a pseudo-evolution of the system and not a physical time evolution.

In Section 6 we present some concluding remarks about the results and the possibility of the application of the above method to the study of the many-par-ticle systems in the second quantization formalism.

In the Appendix, we present the relation between the Floquet Hamiltonian and the evolution operator of the system, and in addition we show that the Flo-

3 The Floquet theory of the time-periodic Hamiltonian systems (I) 235

quet decomposition of the evolution operator can be obtained directly, from the general properties of the Floquet Hamiltonian.

2. THE FLOQUET SPACE

We consider the manifold of time-periodic complex valued functions ( ) ,tψ with period T and square integrable on a period; this function set can be

organized as a Hilbert space (named the time Hilbert space T) by the following procedure:

i. we define the vector )ψ from the T-space as being represented by the continuous column matrix ( )[ ] 2 ;t Tt ≤ψ i.e. the set of the components of this vector is the same as the set of the values of the function ( )tψ during one period;

ii. the scalar product between two vectors )1ψ and )2ψ of the T-space is defined by the equation:

( ) ( ) ( )1 2 1 22d ;

t Tt t tT

∗≤

ψ ψ = ψ ψ∫ (2.1)

iii. we define the operators in the T-space, denoted as ,a by the correspond-ence between the vectors: ) ).aa ψ = ψ

In the following it will be convenient to introduce a time basis in the T-space, denoted as ){ };t then, the basis vectors satisfy the orthonormality and completeness relations: ( ) ( )t t T t t′ ′= δ − (2.2a)

)(2

dt T

t t t IT≤=∫ (2.2b)

where I is the unit operator in the T-space; consequently, the vector )ψ has the expansion: ) ( ) )

2d

t Tt t tT≤ψ = ψ∫ (2.3)

The former construction of the time Hilbert space will be used for the construc-tion of the Floquet space, as we shall present in the following.

a) We consider the manifold of the vectors ( ) ,tχ defined in the Hilbert space of the system physical states (H) which are time-periodic dependent, the period being T.


b) The set of all vectors obtained by the time evolution during a period, of the vector ( ) ,tχ defines the vector ,χ that is represented by the continuous column matrix ( )

2;

t Tt

≤ χ then χ is defined in the Hilbert space F

(named the Floquet space), and this space is represented by the continuous sum of all Hilbert spaces associated to all the times during one period: 2 .tt T≤⊕ H

c) We notice that the extension from ( )tχ to χ is similar to the defini-tion of the vector )ψ from the time periodic function ( );tψ it results that we can consider the time t as a supplementary parameter of the extended space F, and this space is the direct product between the Hilbert space H and the time Hilbert space T: .= ⊗F H T Consequently, the vector χ ∈F has a decompo-sition in the time basis similar to Eq. (2.3):

( ) )2

dt T

t t tT≤χ = χ∫ (2.4)

We note some consequences of Eqs. (2.4) and (2.2) that will be important in the subsequent Sections:

i. the scalar product between the vectors ′χ and ′′χ is

( ) ( )2

d ;t T

t t tT≤′ ′′ ′ ′′χ χ = χ χ∫ (2.5)

ii. ( )tχ is the t-time component of the vector ,χ i.e. ( ) ( ;t tχ = χ iii. as ( ) ( ) ,t T tχ + = χ we can extend by periodicity the time basis, with

the equation

) ) , ;t nT t n+ = ∈Z

in this way the vector χ may be equivalently decomposed in every time period

( ) )( )( )1 2

1 2d ;

n T

n Tt t tT

+

−χ = χ∫

iv. the state vectors of the quantum system (in the Schrödinger picture) ( ) ,tΞ do not have a periodic time dependence; therefore they cannot be ex-

tended directly in the Floquet space (in Sect.3 we will show that it is possible to define vectors from the Floquet space associated to the states of the system, us-ing the pseudo-evolution method).


In the physical Hilbert space H, the system is characterized by operators ( )ˆ ,A t that can have a parametric dependence on time or they can contain time

derivation operations; a characteristic example is the Floquet Hamiltonian ( ) ( ) ˆˆ ˆ 1i ,L t H t t≡ − ∂ ∂ which was used in the study of systems with a

time-periodic Hamiltonian [3, 6]. We remark that the operators containing time derivation operators [in other words, they are functions of the formal operator ( ) ˆˆ 1iZ t t≡ ∂ ∂ ], are not well defined operators in the H-space, in the sense that

the action of the operator ( ) ( )( )ˆ ˆA t f Z t= on the vector ( )tχ (having a well defined value of the t-parameter) is the formal vector ( )A tχ which does not correspond to the same value of t, therefore this kind of operator is a nonsyn-chronous one; in contradistinction to the last operators, the operators having only a parametric dependence on time are synchronous operators. In the following we shall treat formally both kind of operators (the synchronous and the nonsynchro-nous ones), written ( )ˆ ,A t as integral operators in respect to the time variable and acting on the time dependent vectors ( ) :tχ

( ) ( ) ( ) ( ) ( )2

dˆ ˆ ,A t Ttt A t t A t t tT′ ≤′ ′ ′χ ≡ χ = χ∫ (2.6)

where the kernel of the integral operator is the formal operator ( )ˆ ,A t t′ = ( ) ( )ˆ .A t T t t′= δ − If we perform the extension of the vectors in the Floquet

space with Eq.(2.4), then the formal operators become well defined operators in the F-space; using Eqs. (2.4) and (2.6) we obtain the operator A (from the Flo-quet space) associated to the formal operator ( )ˆ :A t

) ( ) ) ( )(

) ( )(2 2

2 2

d d ˆ

d d ˆ ,

A At T t T

t T t T

t tt t t A t tT Tt t t A t t tT T

≤ ≤

′≤ ≤

χ ≡ χ = χ = χ

′ ′ ′= χ

∫ ∫

∫ ∫

A

that is:

) ( )( ) ( )(2 2 2

d d dˆ ˆ ,t T t T t T

t t tt A t t t A t t tT T T′≤ ≤ ≤′ ′ ′= =∫ ∫ ∫A (2.7)

We note some consequences of Eq. (2.7) that we shall use in the next Section.

i. The operators ( )ˆ ,A t which have a time parametric dependence, must be periodic in time, in order to ensure the condition that ( )A tχ is a time periodic vector, if ( )tχ is a time periodic vector.


ii. A time independent operator Â has a trivial extension in the F-space:

)(2

dˆ ˆt T

tA t t A IT≤= ⊗ = ⊗∫A

Particularly, in the F-space, the unit operator is 1̂ ,I= ⊗I and the null operator is 0̂ .I= ⊗O iii. The time matrix of the operator A is identical to the kernel of the inte-

gral operator ( )ˆ :A t ( ) ( )ˆ ,t t A t t′ ′=A

iv. The operator algebra is conserved by the Floquet extension:

( ) ( ) ( )ˆˆ ˆA t B t C t⋅ = ⇒ ⋅A B=C

v. There are two remarkable operators, having pure time actions: the time operator T and the pseudo-energy operator Z:

a) the time operator is

) (2

d ˆ ˆ1 1t T

t t t t tT≤≡ = ⊗∫T

and the corresponding time operator (from the T-space) t defines the time basis, accordingly to the eigenvalue equation: ) );t t t t=

b) the pseudo-energy operator is

) ( )(2

d ˆˆ 1t T

t t Z t t zT≤≡ = ⊗∫Z

where ( ) ˆˆ 1iZ t t= ∂ ∂ was discussed above, and z is the corresponding operator in the T-space ) (

2d ˆi 1

t Ttz t t zT t≤

∂= = ⊗∂∫

We note some important properties of the pseudo-energy operator:

1. the eigenvalue equation in the T-space: ) ) ,z n n n= ω where the time component of the eigenvectors are ( ) ie n tt n − ω= (the Fourier functions),

2. Z is a hermitian operator in the F-space, as is seen from the relation: ( ) ( )†′ ′′ ′ ′′χ χ = χ χZ Z [this is easily verified using the time components ( )t′χ and ( )t′′χ of the vectors ′χ and ,′′χ which are time-periodic];


c) T and Z are canonical conjugate operators, by the commutation relation: [ ], i .=Z T I

We can obtain a vector basis in the F-space, using the following procedure: i. we choose the set ( ){ }tα αρ of time periodic vectors, being a basis of

the tH -space; then, these vectors satisfy the relations:

1. ( ) ( )t T tα αρ + = ρ 2. ( ) ( ) ,t t′ ′α α α αρ ρ = δ 3. ( ) ( ) 1̂t tα αα ρ ρ =∑ ii. we define the vectors: ( ) ( ) ( )ie ,n n tt t− ωα αρ = ρ where ;n∈Z then we

obtain:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

, ,2d

1̂

n nn nt T

n n

n

t t tT

t t T t t

′′ ′α α α′α≤

∞

α α=−∞ α

ρ ρ = δ δ

′ ′ρ ρ = δ −

∫

∑ ∑

iii. we make the extension of the vectors ( ) ( )n tαρ in the F-space, using Eq. (2.4):

( ) ) ( ) ( )2

dn nt T

t t tTα α≤ρ = ρ∫

Then, the set ( ){ },

n

nα

αρ satisfies the orthonormality and the completeness re-

lations:

( ) ( )

( ) ( )

, ,n n

n n

n n

n

′′ ′α α α′α

∞

α α=−∞ α

ρ ρ = δ δ

ρ ρ =∑ ∑ I (2.8)

therefore, it is a basis of the F-space. We note that the Floquet space may be constructed equivalently, using the

Fourier series expansions of the time-periodic vectors ( )tχ and of the opera-tors ( )ˆ ;A t in this case the vector basis in the time Hilbert space T is the Fourier basis ){ },n which are the eigenvectors of the pseudo-energy operator z (see above). Although the Fourier method has some theoretical and practical advan-tages and it is used in many papers [4, 5], however we shall employ exclusively


the time method (which is a complementary point of view), because it assures an easier physical interpretation, may be used for the direct construction of the the pseudo-evolution method (see Sect. 4) and may be adapted directly to the second quantization formalism.

3. THE FLOQUET HAMILTONIAN AND ITS RELATION TO THE EVOLUTION OPERATOR OF THE QUANTUM SYSTEM

In Sect. 2 we have introduced the extended Hilbert space (the Floquet space) and we have shown that the operators in this space have the time compo-nents with possible time derivative operations, but if they have a parametric time dependence, then this time dependence is necessarily a periodical one; conse-quently, in the following we shall consider only a quantum system having the Hamiltonian to be time periodic.

The Floquet Hamiltonian of this system is, by the definition, the formal op-erator ( ) ( ) ( )ˆ ˆ ˆL t H t Z t= − (this operator is in the H-space a nonsynchronous and a nonhermitian operator); we define the corresponding operator in the extended space, using Eq. (2.7):

) ( )(2

d ˆt T

t t L t tT≤= ∫L (3.1)

We note some properties of the L operator that we shall use in the following.

i. L is a well defined and hermitian operator in the F-space. ii. L has the translation property:

, p pp = − ω L M M (3.2)

where pM is the Fourier translation operator, defined as the pure time-operator which makes a uniform translation of the Fourier components of the vectors:

) (

( ) ) ( ) )

i2

d 1̂e p tp t T

pn n

t t tT

n n n p n

− ω≤

∞ ∞

=−∞ =−∞

=

χ = χ ⇒ χ = χ −

∫

∑ ∑

M

M

iii. If we consider { },α αε Φ a solution of the eigenvalue equation α α αΦ = ε ΦL (3.3)

then, according to the translation property [Eq. (3.2)], it is easy to show that


( ) ( ){ },p pp

α α∈

ε ΦZ

are also solutions of Eq. (3.3), where

( )

( )

p

pp

pα α

α α

ε = ε − ωΦ = Φ

M (3.4)

iv. Because L is a hermitian operator in the Hilbert space F, it results that the eigenvalues ( )pαε are real and the eigenvectors satisfy the orthonormality relation:

( ) ( ) , ,p p

p p′

′ ′α α α′αΦ Φ = δ δ (3.5)

We suppose that the set ( ){ },

p

pα

αΦ is a complete vector set in the

F-space, satisfying the completeness relation:

( ) ( )p p

p

∞

α α=−∞ α

Φ Φ =∑ ∑ I

(from the above construction of the basis in the F-space, a sufficient condition of completeness is that the set { }α αΦ should be a complete one in the tH -space).

In the Appendix we shall prove that the operator ( )iexp − τL can be ex-pressed in terms of the evolution operator of the system ( )0ˆ ,U t t by the equation:

( ) ) ( )(2 di ˆexp ,t T t t U t t tT≤− τ = − τ − τ∫L (3.6) Using Eq. (3.6) together with the time decomposition of a vector χ and

of an operator A [see Eqs. (2.4) and (2.7)] we obtain:

( ) ) ( ) ( )2 di ˆexp ,t T t t U t t tT≤− τ χ = − τ χ − τ∫L (3.7)

( ) ( )

) ( ) ( ) ( )(2

i iexp exp

d ˆ ˆ ˆ, ,t T

t t U t t A t U t t tT≤

τ ⋅ ⋅ − τ =

= + τ ⋅ + τ ⋅ + τ∫

L A L (3.8)

In the next Section we shall use Eqs. (3.7) and (3.8) to construct a method which simulates the evolution of the system without the direct use of the evolution op-erator; this method is formally similar to that used for the conservative systems and we call it the pseudo-evolution method.


4. THE PSEUDO-EVOLUTION METHOD

We consider a quantum system having a time-periodic Hamiltonian and L is its Floquet Hamiltonian in the extended Hilbert space F, defined by Eq. (3.1). We define the pseudo-evolution operator ( ), ,′τ τU as the operator in the F-space, dependent on the real parameters τ and τ′ (we refer to these as the pseudo-times) and satisfying the following differential equation together with the boundary condition: ( ) ( )0 0i , ,∂ τ τ = ⋅ τ τ∂τU L U (4.1)

( )0 0,τ τ =U I (4.2)

The solution of Eqs.(4.1) and (4.2) is:

( ) ( ){ }0 0i, expτ τ = − τ − τU L (4.3) and it shows that ( )0,τ τU has similar properties as the evolution operator of a conservative system:

1. it is invariant with respect to the pseudo-times translations, 2. it is a unitary operator, 3. it satisfies group properties.

In addition, using Eqs. (4.3) and (3.6), we obtain the relation between the time matrix of the pseudo-evolution operator and the physical evolution operator of the system: ( ( ) ) ( ) ( )ˆ, 0 ,t t U t t T t t′ ′′ ′ ′′ ′ ′′τ = δ − − τU (4.4)

We shall use the operator ( )0,τ τU to define vectors and operators in the F-space which depend on the τ parameter and which simulate the time evolution of the system in similar pictures to those used for the conservative systems [8].

4.1. THE SCHRÖDINGER PICTURE

We define the Schrödinger picture in the Floquet space by the following conditions:

a) the state of the system is characterized by the state pseudo-vector ( ) ,Ξ τ whose dependence on the τ parameter is generated by the pseudo- evo-

lution operator in accordance with the equation


( ) ( ) ( )0 0, ;Ξ τ = τ τ Ξ τU (4.5)

b) the operators associated to the observables and the special operators (as is the Floquet Hamiltonian) are obtained using the standard extension, repre-sented by Eq. (2.7), from the usual Schrödinger operators (defined in the H-space);

c) the matrix element of an operator between two state pseudo-vectors is equal with the corresponding physical matrix element at the time ,t = τ accord-ingly to the condition

( ) ( ) ( ) ( ) ( )ˆt

t A t t=τ

′ ′′ ′ ′′Ξ τ Ξ τ = Ξ ΞA (4.6)

Equation (4.6) assures the physical correspondence of the results obtained after reasonings in the F-space; on the other hand, we observe that the time integral appearing in the scalar product of the left side of Eq. (4.6) can be equal to a sin-gle value of the integrand only when the state pseudo-vector has a singular be-havior. In order to realize the condition contained in Eq. (4.6) we shall define the state pseudo-vector ( )0Ξ associated to the physical vector of the initial state

( ) 00Ξ ≡ Ξ (at 0t = ), by the following procedure:

i. we consider the time-periodic vector ( )0 tΞ from the H-space, with the period T, which satisfies the boundary condition ( )0 00 ;Ξ = Ξ

ii. we choose a sequence of time-periodic functions with the period T, which converges to the Dirac function ( ){ } ,M Mt ∈δ N when ;M →∞

iii. we define the initial state pseudo-vector ( )0Ξ as

( ) ( ) ( ) )

( ) )

02

2

d0 w lim

d 0;

Mt TM

t T

t T t t tTt t tT

≤→∞

≤

Ξ = − δ Ξ

≡ Ξ

∫

∫ (4.7)

where, the weak convergence implies that the limit M →∞ is performed only when we make the time integration of the scalar product in the F-space. We present some consequences of the definition Eq. (4.7).

a) Using Eqs.(4.5), (4.7), (4.3) and (3.7), the state pseudo-vector ( )0Ξ

evolves at the value τ of the pseudo-time to the vector (in the F-space):


( ) ( ) ( ) ) ( )( ( )

( ) ( ) ( ) )

2

02

d ˆ, 0 0 , 0

d ˆw lim ,

t T

Mt TM

t t U t t tTt T t U t t t tT

≤

≤→∞

Ξ τ = τ Ξ = − τ − τ Ξ

= − δ − τ − τ Ξ − τ

∫

∫

U (4.8)

According to Eq. (2.4), the time component of this vector is a singular vector in the H-space

( ) ( ( ) ( ) ( ); w lim ;MM

t t T t t→∞

Ξ τ ≡ Ξ τ = − δ − τ Ξ τ

and its nonsingular part is the vector

( ) ( ) ( )0ˆ; ,t U t t tΞ τ ≡ − τ Ξ − τ

We can also define the nonsingular vector in the F-space:

( ) ) ( ) ( ) 02d ; , 0

t T

t t tT≤Ξ τ ≡ Ξ τ = τ Ξ∫ U

This vector has physical correspondence only for the time component :t = τ

( ) ( ) ( ) ( ) ( )0ˆ ˆ; , 0 0 , 0 ttt U U t =τ=τΞ τ = τ Ξ = τ Ξ = Ξ (4.9)

that is, this time component is equal to the physical state vector at the time .t = τ b) The state pseudo-vector can be expressed in a more compact form intro-

ducing the operator

( ) ) ( ) (2

d ˆw lim 1 Mt TMt t T t tT≤→∞

τ ≡ − δ − τ∫D (4.10)

which satisfies the following commutation relation with the pseudo-evolution operator:

( ) ( ) ( ) ( ), 0 , 0′ ′τ ⋅ τ = τ + τ ⋅ τU D D U

[as it is obtained using Eqs. (3.8) and (4.3)]; then, Eq. (4.8) becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( )0 0, 0 0 , 0Ξ τ = τ ⋅ Ξ = τ ⋅ τ Ξ = τ Ξ τU D D U D (4.11)

c) Using an arbitrary operator A, defined by Eq. (2.7), and having the time matrix of the form ( ) ( ) ( )ˆt t A t T t t′ ′= δ −A together with state pseudo-vectors, of the form given by Eq. (4.8), we shall verify the condition of the physical cor-respondence for the matrix elements, expressed by Eq. (4.6):


( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 2

2

d dlim

; ;

d ˆlim ; ;

ˆ ˆ; ;

Mt T t TM

M

Mt TM

t t

t t T tT T

t t t t T t

t T t t A t tT

t A t t t A t t

′≤ ≤→∞

≤→∞

=τ =τ

′′ ′′Ξ τ Ξ τ = δ − τ

′ ′ ′′ ′ ′× Ξ τ Ξ τ δ − τ

′ ′′= δ − τ Ξ τ Ξ τ

′ ′′ ′ ′′= Ξ τ Ξ τ = Ξ Ξ

∫ ∫

∫

A

A

where the last equality was obtained using Eq. (4.9). The restriction 2Tτ ≤ (apparently needed from the time integration) is unnecessary, because we may change conveniently the time variable 1 2t t t nT→ = + and using the periodic-ity property of the integrand, we obtain a similar integral defined on a time inter-val containing the value of the τ-parameter.

d) The quantum Floquet theorem [1, 2] asserts that the evolution operator of a system having the Hamiltonian periodic in time can be expressed in the form:

( ) ( ) ( )iˆ ˆ ˆ, 0 expU t W t t K= ⋅ − where ( )Ŵ t is a unitary, time periodic operator and K̂ is a hermitian, time in-dependent operator. If at 0 0t = the system is in the eigen-state αΥ of K̂ (cor-responding to the eigenvalue αε ), then the state vector of the system at the time t is:

( ) ( )i

e tt tα− εα αΞ = Φ

where ( ) ( )ˆ ;t W tα αΦ = Υ in addition ( )tαΦ is an eigenvector of the Floquet Hamiltonian corresponding to the eigenvalue αε :

( ) ( ) ( )L̂ t t tα α αΦ = ε Φ

In the Appendix we present a derivation of these results, using properties of the F-space.

The pseudo-vectors associated to the state ( )tαΞ can be constructed by the procedure exposed before:

i. Using the fact that ( )tαΦ is a time periodic vector and it satisfies the condition ( )0 ,α αΦ = Υ we can choose ( ) ( )0 ;t tαΞ = Φ in accordance with Eqs. (3.1) and (3.3), the extension in the F-space of this vector is an eigen-vector of the operator L, i.e. 0 .αΞ = Φ


ii. The nonsingular part of the state vector in the F-space, is then obtained using Eqs. (4.8) and (4.3):

( ) ( ) iiexp e α− ε τα α αΞ τ = − τ Φ = ΦL and its time component

( ) ( ( ) ( )i

; et t tα− ε τα α αΞ τ ≡ Ξ τ = Φ

satisfies the physical correspondence condition, expressed by Eq. (4.9). For an arbitrary state of the system, we can decompose the vector of the

initial state 0Ξ in the basis { }α αΥ and then we can apply the above proce-dure for each eigen-state; it results that ( )Ξ τ is expressed as a superposition

of the vectors ( )αΞ τ with the constant coefficients and the singular vector

( ) ,Ξ τ which is used for the matrix elements, is obtained with Eq. (4.8). The method presented above is able to study completely the evolution of a

quantum system having a time-periodic Hamiltonian, using the extended Hilbert space; because L is the system pseudo-evolution generator (in respect to the τ parameter) and it behaves like a time independent Hamiltonian, this method uses similar expressions to those of the standard theory of the conservative systems (in the so-called Schrödinger picture), and this property is very convenient al-lowing us to transpose many of standard methods of the quantum mecanics to these non-conservative systems (particularly, we shall derive the perturbation theory adapted to these systems in the interaction picture).

4.2. THE HEISENBERG PICTURE

The Heisenberg picture is obtained performing the unitary transformation ( ) ( )† , 0τ = τR U on the state pseudo-vectors and on the operators of the

Schrödinger picture. We shall present some important properties of the objects of the Heisenberg picture.

According to Eq. (4.11), a state pseudo-vector HΞ is of the form:

( ) ( ) ( ) ( )† 0,0 0 0HΞ = τ Ξ τ = Ξ = ΞU D (4.12)

Analogously to the Schrödinger picture, the physical correspondence for the pseudo-state vectors is contained in the equation: ( )

0,H Htt =Ξ = Ξ where

HΞ is the physical state vector of the system.


An operator of the Schrödinger picture A, becomes in the Heisenberg pic-ture the operator:

( ) ( ) ( )† , 0 , 0H τ = τ ⋅ ⋅ τA U A U (4.13)

The Heisenberg operators have the properties: a) ( )H τA satisfies the differential equation and the boundary condition:

( ) ( )[ ]

( )

i ,

0

H H

H

∂ τ = τ∂τ

=

A A L

A A (4.14)

which are obtained from Eqs. (4.12), (4.1) and (4.2); b) using Eqs. (4.3) and (3.8), we obtain the time decomposition of ( )H τA

( ) ) ( )(

( ) ( ) ( ) ( )2

†

d ˆ ;

ˆ ˆ ˆ ˆ; , ,

H Ht T

H

t t A t tT

A t U t t A t U t t

≤τ = τ

τ = + τ ⋅ + τ ⋅ + τ

∫A (4.15)

hence, only the time component 0t = of ( )H τA has physical correspondence, as is seen from Eq. (4.15)

( ) ( ) ( ) ( ) ( )†0

ˆ ˆ ˆ ˆ ˆ; , 0 , 0H H Ht tA t U A U A t= =ττ = τ ⋅ τ ⋅ τ =

where ( )ˆHA t is the physical Heisenberg operator (in the H-space); c) the matrix elements of an operator product between two state pseudo-

vectors are equal to the corresponding physical quantities, in accordance with the equation

( ) ( ) ( ) ( )ˆ ˆH H a H y H H H a H y HA Y′ ′′ ′ ′′Ξ τ τ Ξ = Ξ τ τ ΞA Y (4.16) which is proved in a similar manner to the former demonstration of Eq. (4.6).

4.3. THE INTERACTION PICTURE

We consider from now on that the Hamiltonian of the system ( )Ĥ t can be decomposed in the free part ( )0ˆ ,H t which is a time-periodic operator having the period T, and the perturbation part ( )ˆ ,V t which can be a time-periodic operator having the same period or it can be a time-independent operator; hence, the Flo-quet Hamiltonian is decomposed in the form:


( ) ( ) ( )0ˆ ˆ ˆL t L t V t= +

where ( ) ( ) ( )0 0ˆ ˆ ˆL t H t Z t= − is the Floquet Hamiltonian of the free system, and ( ) ˆˆ 1iZ t t= ∂ ∂ was discussed before. We perform the extension in the F-space,

using Eq. (2.7), for the preceding decomposition of the Floquet Hamiltonian:

0= +L L V (4.17)

We define the pseudo-evolution operator for the free system ( )0 0,τ τU with similar equations to Eqs. (4.1) and (4.2):

( ) ( )

( )

0 0

0 0

i , ,

,

∂ τ τ = ⋅ τ τ∂τ

τ τ =

0 0 0

0

U L U

U I (4.18)

resulting the solution

( ) ( ){ }0 0i, expτ τ = − τ − τ0 0U L (4.19) which shows that ( )0,τ τ0U has similar properties to the operator ( )0, .τ τU

The interaction picture, in the F-space, is obtained performing the unitary transformation ( ) ( )†0 , 0τ = τ0R U to the state pseudo-vectors and to the operators of the Schrödinger picture.

The operators of the interaction picture are defined by the relation:

( ) ( ) ( )† 00 , 0 , 0I τ = τ ⋅ ⋅ τA U A U (4.20)

and they have similar properties to the Heisenberg operators (the differential equation, the boundary condition, the time decomposition and the physical cor-respondence).

The state pseudo-vector ( )IΞ τ is obtained from the correspondent

Schrödinger state pseudo-vector ( ) ,Ξ τ by the equation:

( ) ( ) ( )†0 , 0IΞ τ = τ Ξ τU (4.21)

and its pseudo-evolution is generated by the action of the pseudo-evolution op-erator of the interaction picture ( )0, ,I τ τU according to the equation:

( ) ( ) ( )0 0,I I IΞ τ = τ τ Ξ τU (4.22)


The operator ( )0,I τ τU has similar properties to the evolution operator of the interaction picture used in the standard theory of the conservative systems [8] (but the latter is an operator in the H-space); therefore, we present its principal properties without their proof:

a) ( )0,I τ τU is a unitary operator and it has group properties; b) ( )0,I τ τU is related to the Schrödinger evolution operator ( )0,τ τU by

the equation ( ) ( ) ( ) ( )†0 0 0 00, , ;I R Rτ τ = τ ⋅ τ τ ⋅ τU U (4.23)

c) ( )0,I τ τU satisfies the boundary condition

( )0 0, ;I τ τ =U I (4.24)

d) ( )0,I τ τU satisfies the differential equation

( ) ( ) ( )0 0i , ,I I I∂ τ τ = τ ⋅ τ τ∂τU V U (4.25)

The differential equation Eq. (4.25), together with the boundary condition Eq. (4.24), are equivalent to the integral equation:

( ) ( ) ( )0

0 0i, d ,I I I

τ

τ′ ′ ′τ τ = − τ τ ⋅ τ τ∫U I V U (4.26)

which has a perturbation solution (the von Neumann series):

( ) ( ) ( ) ( ){ }0 0

0 1 10

1 i, d d!n

I n I I nn

Tn

∞ τ τττ τ

=

τ τ = − τ τ τ τ∑ ∫ ∫U V V (4.27)

where Tτ is the Dyson chronological ordering operation in respect to the values of the pseudo-time τ. We have considered V to be a bounded operator in the F-space (for assuring the convergence of the von Neumann series), and in addi-tion we remark that Eq. (4.27) is valid in more general conditions, when the Schrödinger operator V has a parametric dependence of the pseudo-time.

Using Eqs. (4.11), (4.12), (4.20), (4.21) and (4.23) we obtain the relations connecting the state pseudo-vectors and the operators of the Heisenberg and the interaction pictures: ( ) ( )0,H I IΞ = τ Ξ τU (4.28)

( ) ( ) ( ) ( )0, , 0H I I Iτ = τ ⋅ τ ⋅ τA U A U (4.29) which are similar to the correspondent results of the standard theory [8, 10]. Us-ing Eqs. (4.23) and (4.27) we obtain that ( )′τD commutes with ( ), ,I ′τ τU there-


fore ( )IΞ τ has the same singularity as ( ) ,HΞ τ i.e.

( ) ( ) ( )

( ) ( ) ( )0 0

0

,

I I

I I I

Ξ τ = Ξ τΞ τ = τ τ Ξ τ

D

U (4.30)

We conclude this section observing that the pseudo-evolution method (pre-sented above in the Schrödinger, Heisenberg and interaction pictures) performs the description of the evolution of the system in the F-space, which is deter-mined by the parameter τ as a causal evolution parameter, as the standard theory of the conservative systems. Since we use an extended Hilbert space, we can manipulate the operators, which in the usual Hilbert space depend periodically on time (or they contain time derivative operations), analogously to the time-independent operators of the H-space; on the other side, in this extended space, the vectors associated to the states and the operators associated to the ob-servable quantities of the system have not direct physical signification, but, in-cluding the singular factor in the state pseudo-vectors only the matrix elements (defined in the F-space) are equal to the physical matrix elements. The pseudo- evolution method gives a unique procedure to construct the quantities in the F-space (from the corresponding ones in the usual H-space) and it has a physical interpretation of the final results (obtained after considerations in the F-space).

5. THE PSEUDO-TIME PERTURBATION THEORY

We consider a quantum system for which we can make the decomposition of the Hamiltonian in the free part and the perturbation part, in the manner de-scribed when we defined the interaction picture (Sect. 4); then, the Floquet Ham-iltonian of the system is given by Eq. (4.17). We shall use a method similar to the method of adiabatic switching on the perturbation, of the standard quantum mechanics [10], to obtain states of the perturbed system from the states of the free system, by the continuous variation of an auxiliary parameter. For this, we introduce an auxiliary quantum system (named the adiabatic system), that in the Schrödinger picture has the free Hamiltonian identical to that of the physical sys-tem and it has the perturbation Hamiltonian parametric dependent on the pseudo-time: ( ) ( ) e ,−η τη τ =V V where 0η> is the adiabatic parameter. The Floquet Hamiltonian of the adiabatic system is ( ) ( ) ( ) ( )0 ;η ητ = + τL L V there-fore, by the variation of the pseudo-time parameter from 0τ = ±∞ to 0,′τ = the adiabatic system evolves from the free system to the perturbed system and the adiabatic system becomes the physical system at the limit 0.η→ We shall em-ploy the interaction picture because this picture is the most adequate one to the


present problem. It is easy to verify that the results presented above for the physical system (see Sect. 4) can be transposed for the adiabatic system:

a) the operators which in the Schrödinger picture remain unchanged when we pass to the adiabatic system, in the interaction picture also they remain the same with the physical operators;

b) the pseudo-evolution operator ( ) ( )0,Iη τ τU for the state pseudo-vectors of the adiabatic system in the interaction picture, has a perturbation series in the form of Eq. (4.27), where ( )I τV is substituted with ( ) ( ) ( )e :II

η −η ττ = τV V

( ) ( ) ( )

( ) ( ) ( ){ }0 0

1

0 10

1

1 i, d d!

e ;n

nnI

n

I I n

n

T

∞ τ τη

τ τ=

−η τ + + ττ

τ τ = − τ τ

× τ τ

∑ ∫ ∫U

V V

(5.1)

c) the state pseudo-vectors are dependent on the adiabatic parameter and they differ from the state pseudo-vectors of the physical system:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )0 0

0

,

I I

I I I

η η

η η η

Ξ τ = Ξ τ Ξ τ = τ τ Ξ τ

D

U (5.2)

When 0 ,τ = ±∞ the adiabatic system reproduces the free physical system

[because ( ) ( )0I η τ =V O ] and the interaction picture becomes the free Heisenberg picture; then, it is possible to choose the state of the adiabatic system to be a free Floquet eigen-state:

( ) ( )0

0 0Iη

ατ =±∞

Ξ τ = Φ (5.3)

where 0αΦ is an eigenvector of the free Floquet Hamiltonian 0.L After the

adiabatic pseudo-evolution, when 0τ = the interaction picture coincides with the Heisenberg picture and the nonsingular part of the Heisenberg state pseudo-vector is:

( ) ( ) ( ) ( ) ( ) 0, , 0 0,H I Iη η η α±α ±αΞ = Ξ = ±∞ ΦU (5.4)

then, the Heisenberg state pseudo-vector of the physical system is obtained from Eq. (5.4) in the limit 0.η→ This situation is similar to those studied in the Gell-Mann and Low theorem [9]; therefore we can justify the following result:


a) the vector ( ),Hη±αΞ contains at 0η ≈ a divergent phase factor of the

type 1−η and this is the only singularity; b) the vector

( )

( )

,

00 ,

limH

H

η±α

±α ηη→α ±α

ΞΦ ≡

Φ Ξ (5.5)

is well defined in the F-space and it is an eigenvector of the Floquet Hamiltonian of the perturbed system, i.e.

±α ±α ±αΦ = ε ΦL (5.6)

The above result implies that the Heisenberg state pseudo-vector of the perturbed system, corresponding to the vector ±αΦ is:

( )H±α ±αΞ = ΦD 0 (5.7)

The proof of the Gell-Mann and Low theorem adapted to the time periodic sys-tems can be made similarly as the proof of the standard theorem, with the differ-ence that these quantities are objects defined in the extended Hilbert space F. We shall present briefly the proof based on the validity of the perturbation theory, which is analogous to the standard proof [10].

i. We consider an eigenvector ( )0pαΦ of the free Floquet Hamiltonian 0 ,L

corresponding to the eigenvalue ( )0 ;pαε then we have the identity:

( )( ) ( ) ( ) ( ) ( ) ( ) ( )0 00 0 00, , 0,p p pI Iη ηα α α − ε ± ∞ Φ = ±∞ Φ L U L U (5.8) ii. We consider the perturbation Hamiltonian in the form ( ) ( )ˆ ˆ ,V t gv t=

where g is the coupling constant; using the perturbation series Eq. (5.1) and the differential equation Eq. (4.13) applied for ( ) ,I τV we obtain the identity:

( ) ( ) ( ) ( ) ( ) ( )0 , 0, 0, i 0,I I Ig gη η η∂ ± ∞ = − ⋅ ± ∞ η ±∞∂ ∓L U V U U (5.9)

iii. Combining Eqs. (5.8) and (5.9), we obtain

( )( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 00, i 0,p p pI Ig gη ηα α α∂− ε ± ∞ Φ = η ±∞ Φ∂∓L U U (5.10) iv. We perform the scalar product between the vector ( )0

pαΦ and the both


sides of Eq. (5.10), and we obtain:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0 0

0 0

0 0

i ln 0,

0,

0,

p pI

p pI

p pI

g gη

α α

ηα α

ηα α

∂+ η Φ ±∞ Φ∂

Φ ⋅ ± ∞ Φ=

Φ ±∞ Φ

U

V U

U

(5.11)

Equation (5.11) shows that the vector ( ) ( ) ( )00, pIη α± ∞ ΦU behaves at 0η ≈ in the form:

( ) ( ) ( ) ( ) ( ){ }0 i0, expp p pIη ±α ±αα± ∞ Φ ≈ χ ϕηU (5.12) where ( )p±αχ and

( )p±αϕ are quantities which are independent of the adiabatic

parameter η, that is the vector in the question has in the limit 0η→ as the only singularity a divergent phase factor; then, we can perform a multiplicative re-normalization (in order to cancel this phase factor), resulting that the vector

( )( ) ( ) ( )

( ) ( ) ( ) ( )0

00 0

0,lim

0,

pIp

p pI

ηα

±α ηη→α α

± ∞ ΦΦ ≡

Φ ±∞ Φ

U

U (5.13)

is well defined in the F-space and its norm is finite. v. We apply the operator ( )0 i

p g gα− ε ± η ∂ ∂L to the vector

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )0 0

0 0

0,

0,

p pI

p pI

ηα α

ηα α

Φ ⋅ ± ∞ Φ

Φ ±∞ Φ

V U

U

and we use Eq. (5.10); it results the equation:

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

0 00

0 0

0

0 0

0

0 0

0,

0,

0,

0,

0,i

0,

p pIp

p pI

pI

p pI

pI

p pI

g g

ηα α

α ηα α

ηα

ηα α

ηα

ηα α

Φ ⋅ ± ∞ Φ − ε −

Φ ± ∞ Φ

± ∞ Φ× =

Φ ± ∞ Φ

± ∞ Φ∂= + η∂ Φ ± ∞ Φ

V UL

U

U

U

U

U

(5.14)


In Eq. (5.14), we are now in a position to let η go to zero and using Eq. (5.13) we find:

( ) ( ) ( )p p p±α ±α ±αΦ = ε ΦL (5.15)

where ( ) ( ) ( ) ( )0 0 .p p p p±α ±αα αε ≡ ε + Φ ΦV Equation (5.15) shows that

( )p±αΦ is an

eigenvector of L; finally, putting 0p = in Eqs. (5.13) and (5.15), we obtain Eqs. (5.5) and (5.6), that is the assertions of the theorem are proved.

In the following we shall consider the conjuncture when the vectors

+αΦ and −αΦ are equivalent; then, employing the above theorem we can

express the Floquet eigen-state expectation value of an operator product corre-sponding to the perturbed physical system in terms of Floquet eigen-state expec-tation values corresponding to the free physical system:

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

, ,

, ,

†0 , 0 ,

00 , 0 ,

0, 0,lim

,

H H a H y Ha y

H H

I H a H y II I

I II

+ −α α

+ −αα α

η ηα α

ηη→α α

Ξ τ τ Ξτ τ ≡

Ξ Ξ

Ξ ∞ ⋅ τ τ ⋅ − ∞ Ξ=

Ξ ∞ −∞ Ξ

A YA Y

U A Y U

U

(5.16)

where ( )0 , 00Iα αΞ ≡ ΦD is the state pseudo-vector of the free physical sys-tem (in the Heisenberg, or equivalently in the interaction picture) and this state pseudo-vector is associated to the Floquet eigen-state which evolves adiabati-cally in the state , .H±αΞ Equation (5.16) is formally similar to the analogous

expressions used in the theory of the conservative systems. We shall apply the last result to the situation when the product of the opera-

tors is chronological ordered; using the relation between the Heisenberg opera-tors and the interaction operators of the adiabatic system [i.e. we use Eq. (4.29) where IU is substituted with

( )IηU ], we obtain from Eq. (5.16) the result:

( ) ( ){ }( ) ( ) ( ) ( ){ }

( ) ( )0 , 0 ,

00 , 0 ,

,lim

,

a y

I I a I y II

I II

T

T

τα

ηα τ α

ηη→α α

τ τ

Ξ ∞ −∞ ⋅ τ τ Ξ=

Ξ ∞ −∞ Ξ

A Y

U A Y

U

(5.17)


Equation (5.17) contains in a condensed form the perturbation expansion of an eigen-state expectation value for a chronological product of Heisenberg opera-tors; indeed, using the perturbation series Eq. (5.1) and performing the limit

0,η→ we obtain from Eq. (5.17) the expression:

( ) ( ){ } ( )

( ) ( ) ( ) ( ){ }0 0

10

01

1 1 i d d!n

a y nn

I I n I a I y

T S n

T

∞ τ ττ τ τα α =

τα

τ τ = − τ τ

× τ τ ⋅ τ τ

∑ ∫ ∫A Y

V V A Y

(5.18)

( ) ( ) ( ){ }0 0

01 1

0

1 i d d!n

n I I nn

S Tn

∞ τ τα ττ τ α

=

= − τ τ τ τ∑ ∫ ∫ V V (5.19)

In the preceding expressions Eqs. (5.18) and (5.19) we used the notation

0 0 , 0 ,I Iα αα ≡ Ξ Ξ

for the expectation value on the free system ground state described by the state pseudo-vector 0 , .IαΞ

The above perturbation expansion can be expressed in terms of physical quantities (defined in the H-space); therefore, we perform the time integrations like in Eqs. (4.6) and (4.16) for the matrix elements presented in Eqs. (5.16), (5.18) and (5.19). For the perturbed system we use the time decomposition of the state vector, according to Eqs. (5.7) and (4.10):

( ) ( ) ( ) ), 2d0 w limH Mt TM

t T t t tTα α α≤→∞Ξ = Φ = − δ Φ∫D

and the property: ( )0α αΦ = Υ (which is the eigenvector of the quasi-ham-iltonian K̂ ); finally we find:

( ) ( ){ } ( ) ( ){ }ˆ ˆ

H a H ya y

T A YT α τ ατ

α α α

Υ τ τ Υτ τ =

Υ ΥA Y (5.20)

For the expectation value of the free system we perform similar operations and we find:

( ) ( ){ } ( ) ( ){ }0

0 0ˆ ˆ

I a I y I a I yT T A Yτ α τ αα

τ τ = Υ τ τ ΥA Y (5.21)

where 0αΥ is a normalized eigenvector of the free quasi-hamiltonian 0ˆ .K


The preceding results show that we obtained a perturbation expansion for the expectation values of the chronological product of Heisenberg operators on the quasi-hamiltonian eigen-states of the physical system and this perturbation expansion is similar to the perturbation expansion on the energy eigen-states used in the standard theory of the conservative systems (the Feynman-Dyson perturbation series) [10].

6. CONCLUSIONS

The pseudo-evolution method is a formalism which, using quantities de-fined in the extended Hilbert space and an auxiliary parameter, simulates the evolution of a system having time-periodic Hamiltonian. The obtained expres-sions are formally similar to the corresponding expressions of the standard for-malism of the conservative systems: instead of the Hamiltonian we use the Floquet Hamiltonian and instead of time we use the pseudo-time (the auxiliary evolution parameter). This method gives a procedure for the construction of the quantities from the extended Hilbert space and it assures the physical correspon-dence of the matrix elements, with singular state pseudo-vectors.

The important consequence is the possibility of the adiabatic switching on of the perturbation, from which we have obtained a theorem of Gell-Mann and Low type and then we have deduced a perturbation expansion of Feynman and Dyson type. The perturbation theory presented in this paper is different from that proposed by H.Sambe [3] in two respects:

a) we have included in the free Hamiltonian the periodic in time coupling between the quantum system and the external field, and we have considered the perturbation as a supplementary interaction (e.g. when we deal with a many- par-ticle system the perturbation is the self-interaction);

b) we have formulated the unperturbed problem for the Floquet Hamiltonian, in contrast to the preceding methods where the physical time-independent Hamil-tonian was used and the time periodic term was considered as the perturbation.

The formalism presented above, although more complicated than the stan-dard formalism of the quantum mechanics, can nevertheless be adapted to the second quantization method, so that it can be used to the study of the many- par-ticle systems which interact with time-periodic external fields; in this situation it will be possible to use the methods of the quantum field theory adapted to the extended space and it will be possible to define the Green functions and to com-pute them using perturbation methods.


APPENDIX

THE QUANTUM FLOQUET THEOREM

We consider the eigenvector system of the Floquet Hamiltonian in the

F-space, ( ){ },

,pp

αα

Φ and we suppose that it is a basis of the F-space, in ac-

cordance with Eqs. (3.5). We choose an arbitrary set of integer numbers { }pα α and we define the operators:

{ }( ) ( ) †p p

pp pα α

α αα α α αα α

≡ Φ Φ = Φ Φ∑ ∑P M M (A.1)

( ) ( ) †p pp p pα α α αα α

≡ Φ Φ = Φ Φ∑ ∑P M M (A.2)

where pM is {the Fourier translation operator} and we used Eq. (3.4). We remark the following properties of the above operators:

i. pP satisfies the completeness relation

;pp

∞

=−∞

=∑ P I (A.3)

ii. { }pP is a projector operator

{ } { } { }†2 ;pp p= =P P P (A.4)

iii. the time matrix diagonal element of the operator { }pP is obtained from the identity

) ( )( ( )

) ( ) (

†0

i02 2

02

d d e

d

p pp

p t tt T t T

p

t T

t t t t t tT T

t t t t tT

∞

=−∞

∞′− ω −

′≤ ≤=−∞

≤

= ⋅ ⋅

′′ ′=

=

∑

∑∫ ∫

∫

I M P M

P

P

resulting

( { } ) ( )0 1̂;pt t t t= =P P (A.5)


iv. from the definition of the Fourier translation operator it results

( ) )(ie p tp pp p

p t t t∞ ∞

ω

=−∞ =−∞

| = =∑ ∑M M

and, using this result together with Eq. (3.5), we obtain

( ) ) ( )( , ;p pt t ′α α ′α α α′αΦ Φ = δ (A.6) v. { }pP has the projection properties

{ } )(( ) ( )p p

p t tα α

α αΦ = ΦP (A.7)

{ } ) ( { } { };p p pt t =P P P (A.8) vi. { }pP commutes with the Floquet Hamiltonian

( ) { } { } ( )( ) ( )( ) ( )p p pp pf f fα α αα α α

α

⋅ = ⋅ = Φ ε Φ∑L P P L (A.9)

where ( )f x is an arbitrary function. On the other side, if a, b are any complex numbers and X, Y are any opera-

tors defined in a Hilbert space and satisfying the commutation relation [ ], ,X Y aY= it is easy to prove the identity:

e e eb X ba b XY Y=

then, using this identity together with Eqs. (A2) and (A3) we obtain

( ) ( )

( )

†0

i †0

i iexp exp

ie exp

p pp

pp p

p

∞

=−∞

∞ωτ

=−∞

− τ = − τ ⋅ ⋅ ⋅

= − τ ⋅ ⋅

∑

∑

L L M P M

M L P M (A.10)

In the right side of Eq. (A10) we perform the time decomposition of the Fourier translation operators and then we sum over the index p:

( )

) ( ( ) )( ( )i02 2

iexp

d d iexp e p t tt T t T

p

t t t t t tT T

∞′− ω − −τ

′≤ ≤=−∞

− τ =

′′ ′− τ ⋅ ∑∫ ∫

L

= L P (A.11)


) ( ( ) )() ( )(

02

2

d iexp

d ˆ ,

t T

t T

t t t t tTt t U t t tT

≤

≤

= − τ ⋅ − τ − τ

= − τ − τ

∫

∫

L P

where the operator ( )ˆ ,U t t′ is defined by the equation:

( ) ( ( ){ } )0iˆ , expU t t t t t t′ ′ ′≡ − − ⋅L P (A.12) resulting that it is an operator in the H-space. Using Eqs. (A8) and (A9) we de-compose the operator ( )ˆ ,U t t′ in the following form:

( ) ( ( ) ( ) )( ( ) )( ( ) )( ) ( )

0

0 0

†

i iˆ , exp exp

i iexp 0 0 exp

ˆ ˆ, 0 , 0

U t t t t t t

t t t t

U t U t

′ ′ ′= − ⋅ ⋅

′ ′= − ⋅ ⋅

′= =

L P L

L P P L (A13)

We remark some properties of the operator ( ) ( ( ) )i 0ˆ , 0 exp 0 .U t t t≡ − ⋅L P i. the boundary condition

( ) ( )00ˆˆ , 0 0 0 1

tU t

== =P (A.14)

according to Eq. (A5). ii. using Eqs. (A4), (A5) and (A9) it results

( ) ( ) ( ( ) )( ( ) )( ( ) ( ) )( )

†0 0

0

0

i iˆ ˆ, 0 , 0 exp 0 0 exp

i iexp exp

1̂

U t U t t t t t

t t t t

t t

⋅ = − ⋅ ⋅

= − ⋅ ⋅

= =

L P P L

L P L

P

(A.15)

and in a similar manner we obtain: ( ) ( )† ˆˆ ˆ, 0 , 0 1U t U t⋅ = [consequently, from Eq. (A15) we see that ( )ˆ , 0U t is a unitary operator];

iii. the spectral decomposition

( ) ( ) ( ) ( )i iˆ , 0 e 0 e 0t tU t t tα α− ε − εα α α α

α α

= Φ Φ = Φ Φ∑ ∑ (A.16)

where we used Eq. (A9) and then we performed the time decomposition of the eigenvectors of the Floquet Hamiltonian ,αΦ according to Eq. (2.4);


iv. we perform the time projection of the eigenvalue equation of L [Eq. (3.3)], obtaining the corresponding equation in the H-space

( ) ( ) ( ) ( ) ( )ˆ ˆ iL t H t t t ttα α α α α∂Φ ≡ Φ − Φ = ε Φ∂

(A.17)

Then, using Eqs. (A16) and (A17), we obtain the differential equation for ( )ˆ , 0U t

( ) ( ) ( )i ∂ ∂

,,

U tt

H t U t0

0= ⋅ (A18)

Equations (A18) and (A13) show that the operator ( )ˆ , 0U t satisfies the same differential equation and boundary condition as the physical evolution op-erator of the system (from the initial time 0 0t = to the time t); since this is a

Cauchy problem with a unique solution, then ( )ˆ , 0U t is the physical evolution operator of the system and thus Eq. (3.6) is proved.

We make in addition the following remarks:

a) Equations (A13), (A14), (A16) and (A17) can be proved in a similar manner for the operator ( )0ˆ , ,U t t which is then identified as the evolution op-erator of the system from 0t to t.

b) Using Eq. (A16), we can express ( )ˆ , 0U t in a more general form:

( ) ( ( ) )

( ) ( ) ( )

( ( ) { } )

ii

i

ˆ , 0 e e 0

e

iexp 0

p

p tp t

tp p

p

U t t

t t

t t

α αα

αα α

− ε − ω− ωα α

α

− εα α

α

= Φ Φ

= Φ Φ

= − ⋅

∑

∑

L P

(A.19)

which shows that ( )ˆ , 0U t is independent from the set { } .pα α

c) We consider the vector ( ) ( ) ( ) ( )iep p p tt t tα α α− ωα α αΦ ≡ Φ = Φ and

we define the set ( ) ( ){ } ,p tααα

Φ where to each α we associate an integer num-

ber ;pα then, using Eqs. (A5) and (A6), we obtain:


( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ( ) ( ) )

( { } )

,

1̂

p p p p

p p p p

p

t t t t

t t t t

t t

α α α α

α α α α

′ ′′α α α α′ ′α α

α α α αα α

Φ Φ = Φ Φ = δ Φ Φ = Φ Φ

= =

∑ ∑P

(A.20)

Equations (A20) are the orthonormality and the completeness relations for

the vector set ( ) ( ){ } ,p tααα

Φ thus this is a basis of the Hilbert space H. We de-

fine in addition the vector ( ) ( )0

;pt

tαα α=

Υ = Φ from the above relations we

obtain that αΥ is independent from pα is also a basis in the H-space. d) Using Eqs.(A19), (A8) and (A9), we can express the evolution operator

( )ˆ , 0U t in the generalized Floquet decomposition:

( ) ( ( ) { } ) ( { } ( ) )( { } )( { } ( ) ){ } ( ) { }( )

i iˆ , 0 exp 0 exp 0

i0 0 exp 0

iˆ ˆexp

p p

p p

p p

U t t t t t

t t

W t t K

= − ⋅ = ⋅

= ⋅

≡ ⋅ −

L P P L

P P L (A.21)

where we have introduced the following operators:

{ } ( ) ( { } )( ) ( )ˆ 0 pp pW t t tαα α

α

= = Φ Υ∑P (A.22)

{ } ( { } ) ( )ˆ 0 0 pp pK αα α αα

≡ ⋅ = Υ ε Υ∑L P (A.23)

We note the following properties of { } ( )ˆ :pW t

1. { } ( ) { } ( )ˆ ˆ ,p pW t T W t+ = because ( ) ( )p tααΦ is time-periodic;

2. { } ( ) ( { } )0ˆˆ 0 0 1p pt

W t=

= =P [according to Eq. (A5)];

3. using Eq. (A7) we obtain

{ } ( ) ( { } )( ) ( ) ( ) ( ) ( ) ( )

{ } ( )( ) ( ) ( { } )

( ) ( ) ( ) ( )†

ˆ 0 0

ˆ 0 0

p p pp p

p p ppp

W t t t t t t

W t t t t t t

α α α

α α α

α α α α

α α α α

Υ = Φ = Φ = Φ

Φ = Φ Φ = Υ

P

P


The latter equations show that { } ( )ˆ pW t is an operator which performs the change of the basis in the H-space, thus it is a unitary operator. We remark how-ever that this property can be deduced directly in the operator form, using Eqs. (A5) and (A8):

{ } ( ) { } ( ) ( { } )

{ } ( ) { } ( ) ( { } )

†

†

ˆˆ ˆ 0 0 1

ˆˆ ˆ 1

p pp

p pp

W t W t

W t W t t t

⋅ = =

⋅ = =

P

P

The other operator of the Floquet decomposition, { }ˆ pK has the following proper-

ties [as is seen directly from its definition]: 1. it is a time-independent and hermitian operator in the H-space, 2. it satisfies the eigenvalue equation:

{ }( )ˆ p

pKα

α α αΥ = ε Υ

and its eigenvector system is an orthonormalized basis in the H-space, 3. it is related to the Floquet Hamiltonian by following the equation:

{ }( ) ( { } ( ) )ˆ 0 0p pf K f= ⋅P L according to Eqs. (A4) and (A9), where ( )f x is an arbitrary function having Taylor expansion about the origin; therefore, follows the equation:

{ }( ) ( { } ( ) )i iˆexp 0 exp 0p pt K t− = ⋅ −P L that proves the Floquet decomposition expressed by Eq. (A21).

The above results are a proof of the quantum Floquet theorem grounded exclusively on the assertion that the Floquet Hamiltonian of a time-periodic Hamiltonian system has a complete eigenvector system in the F-space; in addi-tion, the present considerations led directly to the general form of the Floquet decomposition for the evolution operator, where we used a set of arbitrary satel-lites, expressed by any integer number set { } .pα α

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