Quasi-energies, loss-energies and stochasticity

Vol. 29 (1991)

REPORTS

ON

MATHEMATICAL

No. 2

PHYSICS

QUASI-ENERGIES, LOSS-ENERGIES AND STOCHASTICITY G. FakultPt

fiir Physik,

KARNER

Universitat

Bielefeld, D-4800 Bielefeld I, FRG

V. I. MAN’KO P. N. Lebedev

Institute

of Physics,

Leninsky

Prospect

53, 117 333, Moscow,

USSR

L. STREIT Bibos, Universitat

Bielefeld, D-4800 Bielefeld 1, FRG

(Received

October

12, IY89)

We review the concepts of quasi-energy and loss-energy by discussing examples of simple time-dependent quantum systems such as a free particle with time-varying mass, a one-dimensional harmonic oscillator with time-dependent frequency and a damped harmonic oscillator. Based on these findings we derive conditions for chaotizing, resp. stabilizing multidimensional parametric oscillators in interaction (in the sense of different quasi-energy, resp. loss-energy spectra).

1. Quasi-energy

spectra,

general

considerations

Interest in simple time-dependent quantum systehs has grown rapidly during the last decade. Inspired by exciting discoveries in the classical mechanics of nearintegrable systems, namely existence of the so-called deterministic chaos (e.g. [I]), researchers began to look for related phenomena in quantum mechanics (for a review and references see [2] for instance). Up to now an important part of work was concentrated on quantum systems under time-periodic external perturbations. Such systems are described by time-periodic Hamiltonians, a fact which (at least in principle) allows us to apply the notions of quasi-energy in case of real-time periodicity (e.g. [3]) and 1oss;energy if the .time periodicity is purely imaginary [4]. Let us remind what we understand as quasi-energy of a quantum system with Hermitean Hamiltonian H(t) such that H (t + T) = H(t),

where T is the real period. Schrhdinger equation

In that case the wave vector

Cl771

(1.1)

1$(t) > which solves the

G. KARNER,

178

V. I. MAN’KO

and L,. STREIT

(l-2) gives rise to an evolution its state space, namely

operator

U (t, s) for the quantum

= U(t,

= H(t) U,

U(0) = E. H -

of the Hamiltonian U(t+T,

see (1.1) -

(1.4) we have

s) = U(T, O).U(t, s).

Thus the Floquet (monodromy) operator the discrete group of time shifts

(1.5)

U (T, 0) realizes a unitary representation

U(PrT, 0) = [U(T, O)]“. Unitarity of the discrete Abelian Decomposition one-dimensional states. In fact,

(1.3)

O)W (0) > >

the equation ih:

Due to the periodicity

of the system in

. Iti@) ’

with U (t, s) obeying

trajectory

*

of (1.6)

representation follows from the hermiticity of the Hamiltonian. The group has only one-dimensional unitary irreducible representations. of the states of our quantum system into a sum over these irreducible representations leads to the notion of quasi-energy wave functions IC/,(t) with the property

$,(t+T)

’1

= exp - fT [

$,(t)

(1.7)

are called quasi-energy states with quasi-energy ‘E. This quantum number is determined modulo 27&/T. In the case of arbitrary T the states (1.7) coincide with the stationary states and the quasi-energies with the energy eigenvalues of the system. Using property (1.6), we have U(nT, 0)$,(t) = exp

The

factor

2. Quasi-energy

- YT

T is the character L. of the discrete time translation

exp

representation

-y 1

[.

1

$,(t).

of a one-dimensional

(1.8)

irreducible

group.

spectra, some examples

Let us first discuss the case when the Hamiltonian is a time-periodic linear form of the Hermitean generators of the Lie algebra 8 of some group G: H = zci(t)Si,

(2.1)

QUASI-ENERGIES,

LOSS-ENERGIES

AND STOCHASTICITY

179

where [ei,

~j]

=

U~j

C

ok.

(2.2)

k

Here !& are generators in a representation of the Lie algebra 6 and the Ufj are structure constants. We consider the case where the real coefficients in the linear form (2.1) are time-periodic functions, i.e. ci (t + T) = ci (t).

(2.3)

’ For a quantum

system with Hamiltonian (2.1) the evolution operator U(t, s) exists and is the operator of finite group “rotations” in the space of quantum states. The space of quantum states is a representation space (reducible or irreducible) of the Lie algebra in question. Thus the evolution operator in this situation may be written as U (t, 0) = exp (i (K (t)) and the operator follows:

(2.4)

K(t) belongs to the same Lie algebra and may be parametrized K(t) = CSi(t)&.

as

(2.5)

*

The parameters s,(t) of this operator are determined through the parameters in the Hamiltonian. In fact we need to obtain an explicit expression for the time-ordered - exponential type solution of the Schrijdinger equation with Hamiltonian (2.1) in terms of group parameters. With U(t, 0) = 2exp(-ijH(t’)&t’)

(2.6)

0 (Z

denotes

time-ordering)

the Floquet exp(iK(T))

(monodromy)

operator

= Zexp(-iyH(t’)dt’).

has the form (2.7)

0

There exists an element g of the group G for which the operator of the group representation z(g): r(g) = exp(iK(T)).

(2.7) is the operator

(2.8)

By definition, the quasi-energy spectrum is determined by the eigenvalues of the Floquet operator. This spectrum is the same for all operators of the form r(go).r(g)*r(&)

= r(go*gXC).

(2.9)

This means the quasi-energy spectrum is the same for all group elements of any one conjugacy class. The adjoint representation is created by the transformation g -+ go*g.g; ’ and reads

180

G. KARNER,

V. I. MAN’KO

and L. STRElT

(2.10) As a consequence, Floquet operators have the same quasi-energy spectrum whenever their generators are related through an equation of the form s:(T) = +(gO)sj(T).

(2.11)

Thus it is sufficient to determine the quasi-energy spectrum just for one representative of the Floquet operators in each class (choosing of course the simplest one possible). Such a representative will be an element of the Lie algebra corresponding to one group element from the given conjugacy class. We can count and classify all of these elements. If the Hamiltonian (2.1) has non-vanishing coefficients ci(t) only for a subalgebra of the Lie algebra 6, then the Floquet operator (2.4) is determined by an operator K (T) which also belongs to this subalgebra. If the subalgebra corresponds to a one-dimensional subgroup then the quasi-energy spectrum coincides with the spectrum of the Lie generator of this subgroup. In the general case where all the parameters c,(t) in (2.4) differ from zero we can formulate a general statement regarding possible quasi-energy spectra of non-stationary quantum systems with the “group” Hamiltonian (2.1) which is a linear time-periodic form with respect to the Lie algebra generators: Let us consider all the conjugacy classes of the group; then the possible quasi-energy spectra arise from the spectra of all those Lie algebra generators which correspond to the different conjugacy classes. In this way we have reduced the problem of stochasticity for quantum systems with time-periodic group Hamiltonians (2.1) to the spectrum of Lie algebra generators in a given representation. Next we consider one of the simplest examples of quantum systems with time-periodic Hamiltonians. At first we discuss the unbounded one-dimensional free motion of a particle with varying mass, then we study the motion of such a particle in a square well. The Hamiltonian for these models is H(t)

=

p’

2m@I

(2.12)

with m(t+ T) = m(t).

(2.13)

We can think of various physical situations where such a time dependence arises. For example, electrons moving through a material have an effective mass determined by properties of the material. If we change these properties through external influences on the material this may accordingly produce a time dependent effective mass for the electron. If the material is located in an external plane wave field we would have a periodic time dependence. In any case (2.12) offers a model for the construction of quasi-energy spectra. The Schrodinger equation

QUASI-ENERGIES,

LOSS-ENERGIES

i/r&$(x, may be reduced to the Schriidinger the change of variable

AND

STOCHASTICITY

t) = - &“:i(x, problem

181

(2.14)

t)

with time-independent

Hamiltonian

by

(2.15) This ansatz works for any Hamiltonian we have the equation

with time-dependent

mass. Instead of (2.14)

(2.16) This is the equation that we must consider in both cases: for the unlimited motion with the particle coordinate ranging between -- 00 and + a, and likewise for the particle confined to an interval by an infinitely deep square well potential. The latter case is realized by imposing Dirichlet boundary condition at both ends of the interval which we shall put at x = 0 and x = L. For the unbounded free motion the Hamiltonian (2.12) may in fact be considered as the generator of an infinite dimensional representation of the group SL(2, R) which in turn is a subgroup of the inhomogeneous symplectic group to be used further down. Traditional variable separation methods allow us to solve the equation (2.16) in the form $(x,

If we have no potential

1

(2.17) -T $E(~). c well the function $E(x) may be found from the relation -

z) = exp

&,“:‘L” =E$,(x)

(2.18)

which gives the usual plane wave function eE (x) = (2rcr2)) ‘I2 exp ( f ipx/h), where

the quantum

number

E reads in terms of the momentum

E=PZ_

(2.19) PER (2.20)

2nz (0)'

We stress that the quantum number E labelling our wave functions is not the energy. Energy is not conserved for non-stationary quantum systems. The time-dependent wave function has the form $E(x,t)=exp

-y [

1w $E

(2.21)

182

G. KARNER,

The wave function

V. I. MAN’KO

and L. STREIT

(2.21) has the property tiE(x, t+T)

(2.22)

= exp

which follows from (2.23)

z(t+T) = T(t)-tz(T). We have used the abbreviation t’=- ET(T) T

(2.24)

’

The quantum number E is called the quasi-energy. Consequently, the quasi-energy spectrum is continuous - it differs from that of particles with constant mass only by a factor. Explicitly, we have for the quasi-energy of a free particle whose mass is varying with period T

p2 T dt’

s-&= 2T,m(t’) At the same time E is the eigenvalue Iexp{-i[&dt]I$,>

(2.25)

of the Floquet

operator

of our system since

=exP(--F)We>

(2.26)

if we express the evolution operator as a time-.ordered exponential and relabel the wave function. Thus we have obtained for the free periodic mass motion a continuous quasi-energy spectrum and a complete set of eigenfunctions with b-function normalization. Now let us turn towards the potential well. Essentially all the former considerations carry over to this model, the only necessary changes being due to the boundary conditions at both ends of the interval I: (2.27)

$(x, t)lxea1 = 0.

The spectrum of the Laplacean -8,” is discrete in this case and we have for the eigenvalues in (2.18) the usual discrete spectrum k, = x.n/L,

The quasi-energy

spectrum

n = 0, 1, 2, . . .

in this case is discrete

(2.28)

as well and given by (2.29)

We want to point out that this formula applies to any Hamiltonian

that is stationary

QUA!%ENERGIES.

LOSS-ENERGIES

AND

except for a periodically varying mass. A complete set of orthonormal for the potential well is given by lj, = (2/L)l”

sin . (rcnx/L) exp

183

STOCHASTICITY

eigenfunctions

7‘ dt’

-i (Fi27r2n2/L2)1~ 0 m(t’) *

(2.30)

This expression shows that everything, and in particular the quasi-energy spectrum, coincides with the stationary case: except for the appearance of an average inverse mass

The multidimensional generalizations of these models are straightforward. The only difference is that the quasi-energy is now given for the free motion by sp = fm-’ (p is the momentum

vector)

. . . . P,)

(2.32)

and

E(“~....,~~)- 19 for the N-dimensional Our final example

1~~1, P = (pl,

-‘(fiX/L)Q:+

well. is the harmonic

. . . +n;-J

oscillator

with time-dependent

H = (m -1pZ+mo2(t)qz)/2.

(2.33) frequency: (2.34)

If the mass of the oscillator were time dependent too and equal to m(t) the change of the time variable would reduce the problem to one with constant mass and new time-dependent frequency Q (t) given by

m(t) m(0)

Q(t) = -o(t).

(2.35)

Now we shall consider the case of constant mass and periodically varying frequency w(t) with o (0) = 1 as well as FI= 1. The frequency obeys the periodicity condition (2.36)

0 (t + T) = 0 (t).

We shall use the method of linear integrals of motion [S, 61 for the solution of this problem. It can be checked that the parametric oscillator has two integrals of motion [S] which are given as the two Hermitean linear combinations of the non-Hermitean invariant a(t) = i {e (t)p - c’:(t)x}/JZ and its adjoint

a+, where the function a(t)+U2(t)E(t)

c(t) solves the classical equation =

0.

(2.37)

of motion (2.38)

184

G. KARNER.

(Dot means time-derivative.)

V. 1. MAN’KO

There

and L. STREIT

are two Hermitean

p0 = (a-a+)/ifi

= (Res)p--

invariants:

(Re8)x

(2.39)

(1mE)x.

(2.40)

and x0 = (a+a+)/JZ

= (Ims)p-

(These operators should not be confused with the time-dependent operators occurring in the Heisenberg or interaction picture, they are defined by the Schrodinger picture operators on the right-hand sides.) The formulas (2.39) and (2.40) can be rewritten in matrix form (2.41) where the 2 x 2 matrix n(t) with matrix elements n,(t), i = 1, . . ., 4, belongs to the group SL(2, R) or SP(2, R). The real matrix elements &(t) are expressed in terms of the classical trajectory c(t) of the parametric oscillator >_,(t) = Res, The Wronskian

n,(t) = -ReE,

2, (t) = Imr,

for the classical equation G--E

The initial conditions

of motion

for the classical trajectory

solution

(2.42) to be (2.43)

are

Rei(O) = Im(-:(O) = 0,

(2.44)

c(t) obeys the conditions 6(O) = 1

Alternatively

(2.38) is chosen

= 2i.

Ree(0) = Imi:(O) = 1, i.e. the complex

n,(t) = -1mi:.

we introduce

and

i(O) = i.

(2.45)

the two real solutions

cl(t) = Rea In terms of them the Wronskian

and

(2.46)

= ImE.

takes the form d,E, -&a2

and the matrix n(t) determining

I

= 1

the linear invariants

(2.47) p0 and x0 looks as follows: (2.48)

The real solutions

cl.2 obey the initial conditions

El(O) = 1,

E2(0) = 0,

t, (0) = 0,

8, (0) = 1.

Due to these conditions the invariants pa(t) and x,(t)‘coincide momentum and position operators p and x.

(2.49)

at t = 0 with the

QUASI-ENERGIES,

LOSS-ENERGIES

AND STOCHASTICITY

185

The Green’s function in coordinate representation G(x,, x1; t) of the oscillator with time-dependent frequency has been found by Husimi [7]. We shall write down the kernel of the evolution operator in the Weyl-Wigner representation [5,6] G(p, q; t) = 2(2+E1 +kJ’12exp

(2+,2,1+&.,)c-%P2+

(~2-~1)Pq+i:1q21

.

(2.50)

Note that, in view of the initial conditions (2.49), the Weyl-Wigner representation of the propagator is equal to unity for t = 0, in the same way as its coordinate representation G (x2, x 1; t) = 0 equals Dirac’s b-function. This is verified easily from the following explicit expression: G(x,,

x1; t) = [2rcis2(t)]-1’2exp

.

(2.51)

The Green’s function (2.51) turns out to be an eigenfunction for the integral of the motion x0(t); applying the operator (2.40) to G as a function of x = x2 one finds xo(t)G(.,

x1; t) = x1 G(-, x1; t).

(2.52)

At the same time the other invariant pa(t) gives rise to a first order differential equation for the configuration space kernel of the evolution operator G PO@) G t.7 x 1; ~1 = 8x1 G(., ~1; 0.

(2.53)

Thus the integrals of motion x0 and p. are closely related to the Floquet operator in coordinate representation. Let us study properties of these operators, particularly with a view towards their invariance under time shifts by the period T. We know that the classical equation of motion (2.38) in the stable region, has two complex solutions E and E with the following properties: T) = exp(ixT)e(t)

(2.54)

E(t+ T) = exp(-ixT)E(t)

(2.55)

c(t+

and

with real-valued number following properties:

X. Furthermore,

a (t + T) = exp (ixT) a (t), Thus the Hermitean integral shifts by the period T:

of motion

u+ (t+ T).u(t+

the invariants u+(t+T)

a and a+ in (2.37) have the

= exp(-ixT)a’(t).

a+ (t).u(t) is invariant T) = a+ (t).u(t).

(2.56)

with respect to time (2.57)

The construction of solutions for the Schrddinger equation proceeds with use of the operators a and u+ in close analogy to the stationary case. The first step is to find a solution of the Schrodinger equation which is annihilated by u(t)

5 - Renorts29.2

G. KARNER,

186

V. I. MAN’KO

and L. STREIT

a(O$o(x, 0 = 0. Equation

(2.58)

(2.58) can be solved easily and the normalized $o(X,

The modulus

t) = x-114E-112(t)exp

of the function

E is of course

[

solution

is of the form

i$x2 .

(2.59)

1

also a periodic

function,

i.e.

(2.60)

IE(t + T)I = IE@)I and obeys the equation

~lE(t)l+w2(t)lc(t)l-,E(t)l-3 =0. The function

E can then be expressed

in terms of its modulus

as follows:

c(t) = IE(t)l exp which implies in the stability E(t+

The “ground property

(2.61)

(2.62)

regime

T) = s(t)exp

state” wave function

(2.63)

1 ((&gJT).

of the parametric

oscillator

( -1 .(27s ’ :"").T)$o(x,

$o(x, t+ T) = exp

u+ @).a@) = &{IE(t)12p2+

t).

,E(u)12

All the other solutions are obtained a+ (t).a(t), which is of the form

as eigenfunctions Iti(t)l 2x2-

has the periodicity

of the invariant

[E(t)z(t)xp+E(t)~(t)pX]},

(2.64) operator (2.65)

and are given by ljn(x, t) = y$o(x,

t).

(2.66)

From this expression and from the periodicity of the wave function $. it follows that

properties

(2.56) and (2.63) of a+ and

rjn(x, t+T) = exp(-i($+n)x.T)IC/,(x,

t),

(2.67)

where x=-

1r s-

dO

T o IE(@I 2 ’

(2.68)

QUASI-ENERGIES,

LOSS-ENERGIES

187

AND STOCHASTICITY

Comparing this formula with the definition of the quasi-energy spectrum E, we see that the quasi-energy spectrum of the quantum parametric oscillator in the case of stable classical motion is discrete, equidistant and of a form similar to the usual spectrum of the quantum harmonic oscillator yet with a modified frequency x

lT dtI En=-_S ---((n+$), The matrix

n(t)

(2.69)

n = 1, 2, . . .

T o 1~(0”

in the stable case has the simple form

if T is measured in units of x-r be represented as

sin T

cos T

A(T) =

COST

-sinT

and the Floquet

1

operator

(2.70) for the stable regime may

U(T, 0) = exp(-i(P2+x2)T/2}. Let us now consider the non-stable and s2 with the properties

situation.

s1 (t + T) = eXTcl (t)

and

(2.7 1)

Here we have two real solutions

s2 (t+ T) = eeXTs2 (t),

s1

(2.72)

where x is a real parameter. In this case the integral of the motion p. (t) of (2.39) has a “periodicity” of the form (2.73a)

p. (t + T) = eXTPO(t) and the invariant

x0 of (2.40) obeys x,(t+

The quadratic

Hermitean

integral

T) = eKXTxo(t).

(2.73b)

of the motion

d(t) = L-PO (Ox,

(2.74)

(t) + xo WP~ @)3/2

is constant under shifts by T: d(t+ T) = d(t). At the initial time t = 0 it coincides with the dilation

operator

d(t = 0) = ;la,x+xi3J.

(2.75)

Since e-Txdxxe+Txa,

the Floquet

operator

=

e-TX

and

U (T, 0) of the unstable VT,

0) = exp

e-

Txa,

axei

TX?,

=

eT2_

(2.76)

case has the form

1,

(2.77)

G. KARNER,

188

V. I. MAN’KO and L. STREIT

where x0 and p. are the integrals of the motion introduced above, taken at time T. Clearly the operator d of (2.54) is unitarily equivalent to the operator K = (pf) - xi)/2. The rotation

matrix

U which connects (1(l)

=

(2.78)

the matrices eT

0

1

(2.79)

[ 0 ewT

and (2.80) in the sense that /i(l) = U /it’) U- ‘, is given by

Jto which there corresponds

1

1 1

U=1

the unitary

2 [ -11

operator

exp [.$+ i-

(2.81)

;)I.

This unitary equivalence gives us the possibility to reduce the quasi-energy spectrum to that of an oscillator with inverted parametric resonance effectively transforms the attractive linear force one. The classical behaviour of the inverted oscillator is described by motion z-($x

= 0

problem of the potential. The into a repulsive the equation of (2.82)

with the solution x(t) = x0 ch ot + (&/w) shot,

(2.83)

where x0 and &, are the initial position and velocity of the inverted oscillator. Infinitesimal variations of these initial conditions produce exponentially growing variations of the trajectories as one sees explicitly in 6x(t) = 6x, ch ot + (&Z,/w) shot,

(2.84a)

da(t) = o6x,shot+&&,chot.

(2.84b)

Enclosed within reflecting walls (billiard on the line) such a system chaotizes quickly and trajectories diverge exponentially fast. Thus the monodromy operator (2.77) is unitarily equivalent to the operators described by the matrices (2.79) and (2.80). Hence the continuous spectrum may correspond either to the diagonal Abelian subgroup of the real group SL(2, R) or to a subgroup in the same conjugacy class described by the matrix (2.80).

QUASI-ENERGIES,

LOSS-ENERGIES

A matrix n (7) which also belongs to an Abelian subgroup a different conjugacy class is A(T)=

;; [

which corresponds

to a quasi-energy

operator

189

AND STOCHASTICITY

of SL (2, R) but to

1

(2.85)

proportional

to the integral of motion (2.86)

K (r) = Pzl(Q/2. In this case the quasi-energy Hamiltonian.

spectrum

is equivalent

to that

one of the free

3. Loss energy and loss energy states In Section 2 we considered Hamiltonians which have the periodicity now focus on quantum systems which have the symmetry H(t+iz)

(1.1). Let us

= H(t).

(3.1)

Systems of this kind are obtained from (real-) time-perodic ones by Wick rotation of t. In [4] the. notion of loss-energy was introduced in complete analogy to quasi-energy yet up to now no connection was made between loss-energies and chaoticity. A possible application of this idea is the theory of quantum system with dissipation such as the damped harmonic oscillator. We present a short review of the results obtained in [4]. A damped harmonic oscillator may be described by

H(t)= Cp2exp (- 2yt) + 02x2 exp (2yt)]/2 (withm=r?= l).Ob viously, H(t)obeys the periodicity condition There exist two integrals of motion of the form (2.41):

(3.2) (3.1) with z = rc/y.

(3.3) The n (t)-matrix

reads A(t) =

where a:=

exp (iQ2t- rt)

(y - iQ) exp (iS2t+ at)

exp(-iQ2t--yt)

(y+iSZ)exp(-iQt+yt)

[02--y 2] ‘I2 . Following

(i) Weak damping

1

[4] we list up three different

o > y: Here the loss-energy E, = (Q+iy)(n+1/2),

(3.4)

’

situations:

spectrum

is discrete and given by

II = 0, 1, . . .

(3.5)

We note that for y +O the usual oscillator spectrum (ii) Strong damping w < y: In this case the character is different: Using 52”:= [y2 -0~1~‘~ we obtain

emerges. of the loss-energy

spectrum

190

G. KARNER,

V. I. MAN’KO

and L. STREIT

ef = yWfiy/2 with continuous y E R. (iii) Critical damping

(3.6)

c.~= y: Here we arrive at E: = (0+iy)/2,

(3.7)

where rr > 0 is continuous. Hence in complete analogy to the quasi-energy spectra of the time-periodic harmonic oscillators discussed in Section 2 we can distinguish three types of motion. We remark that a combination of quasi-energy (i.e. real time-periodicity) and loss-energy is possible if one considers Hamiltonians obeying H(t + T+ iz) = H(t).

(3.8)

This can be arranged for instance by use of Jacobi’s O-function instead of sin t (or cost) and sh t (or ch t) to model the time-dependence of the system in question. 4. Multidimensional In this section

quadratic systems we shall concentrate

on Hamiltonians

of the type

H 0) = 2 B (t) 2 /2 + ,c 0) 2 with the 2N-vectors Consequently

x = (pl, . . ., pN; x1, . . ., xJ,

(4.1)

,C (t) and the 2N x 2N-matrix

B(t).

(4.2)

z B(r) g = i;” 4A&)q,. a=1

The coefficients B(t) and G (t) are supposed (4.1) possesses the 2N integrals of motion

to be T-periodic (in Schrodinger

functions. picture)

L(t) = A (t) 2 + g (t), where A(t)

is the solution

to the classical equations A(t) = A(t)CB(t),

The 2N x 2N-matrix

(4.3) of motion

/l(O) = 12N.

(4.4)

C reads in block form C=

The matrix

The system

A(t) is a real symplectic A(t+T)

0 1,

L I -z,o.

matrix

(4.5)

and obeys

= A(T).A(t),

(4.6)

which implies that the integrals of motion are T-periodic too. The spectrum of the Floquet operator U (T, 0), which takes the system over one period of oscillation,

QUASI-ENERGIES,

LOSS-ENERGIES

AND STOCHASTICITY

191

is completely determined by the properties of the “monodromy matrix” /i (T), i.e. by properties of the classical solution for the quadratic system. The nature of the quasi-energy spectrum is connected with the conjugacy class the matrix /1(T) belongs to. (A classification of possible conjugate classes of the symplectic group is found in [S].) For instance, if /1(T) looks like

c?

cos h, cos h,.

e A(T) =

sin h, (4.7)

cos h,

?

-sin h,.

cos h,.

. . 2

c?

2

cos h,

-sin h,

sin h,.

- sin h,

2

C ..

cos 12,

all the eigenvalues have the form lj = exp’(ihj), j = 1, . . ., N and all h, are real-valued. Then the quasi-energy spectrum is discrete and of the form [9]:

&=~,~hj(nj+l,~)+f;~,(I)S,(t)dl 0

(43)

J-1

with LJ (t) = (i l(t), i(t)) and nj&. We remark that the linear term ,C (t) in (4.1) does not severly change the discrete oscillator-type spectrum. Concluding this section, we remark that a stable multidimensional classical oscillator corresponds to discrete quasi-energy spectrum of the quantum system. On the other hand, if n(T) has eigenvalues of the type exp( + hj) with real-valued hj we observe unstable classical motion together with continuous quasi-energy spectrum t: = f .; J-1

hjvj+ ;;

~2(t)dl(t)dt

(4.9)

0

with \lj E R. Of course the monodromy matrix A(T) can have a mixed structure which corresponds to stable motion in certain directions and irregular behaviour in others. Finally we remark that with the help of /1(T) and J (t) the kernel of the time-evolution operator U (t, s) can be written down explicitly. 5. Two interacting time-dependent quantum oscillators Based on our findings in the preceeding sections we present the final theme of this article. We shall answer questions of the following type: Given two harmonic oscillators with time-dependent frequencies acting in x- and y-directions respectively and which are coupled via a potential of xy-type. Is it

192

G. KARNER,

V. I. MAN’KO and L. STREIT

possible to organize the interaction in such a way that for instance chaotic x-oscillator is stabilized by the influence of the y-oscillator? Expressed in mathematical terms we have two Hamiltonians

the initially

H, (t) = (& + o2 (t) x2)/2

(5.1)

Y2)/2

(5.2)

m2 (t) = W; (1 + h cos cot).

(5.3)

and H2

(0

=

(Py’

+

co2

@I

with the specific frequency

Then the classical oscillators have regions of (non-)stability as shown in Fig. 1 (which follows from [lo], shaded areas correspond to unstable regime): Ihl

t

0

200 Fig. I

Let us couple both oscillators = xyog [~+xcosot] to arrive at

(5.1)

and

(5.2) with

HI,2 0) = H,(t)+H,(t)+F’, where i and 3t are positive constants.

the

help

of

V(x, y) (5.4)

A certain change of variables, namely the use of

x = (x +y)/2l’2, Y = (x-y)/2r’2 yields a new Hamiltonian

without

XY-interaction:

(5.5)

QUASI-ENERGIES,

LOSS-ENERGIES

H(t) = (p: +py” + [w’(t)+o;(;1+xcos

AND STOCHASTICITY

wt)] X2 + [w2 (t)-e&L+xcos

19.1 or)] r2}/2. (5.6)

It can be seen from Fig. 1 that for this new Hamiltonian the two normal modes may be shifted from one region to the other. In fact, the interaction changes the coupling constants of the individual oscillators:

(9 h+h, = for the x-oscillator

G,* G&+g =Efl0 +ijl12

and correspondingly (ii) h--+/i, = s,

for the second $&+z

(5.7)

system:

= $(,_:)1,2.

(5.8)

0

A glance at Fig. 1 assures that these alterations may shift points into or out of the shaded area, i.e. in that way we may destabilize or stabilize the individual oscillators. Of course the same mechanism takes place in higher dimensional systems and in particular for kicked quantum systems considered in [ 11, 121. REFERENCES [l] A. J. Lichtenberg and M. A. Lieberman: Regular and Stochastic Motion, Springer, New York (1983). [2] G. Casati and L. Molinari: “Quantum chaos with time-periodic hamiltonians”, in Supplement to Progress in Theoretical Physics on NeM: Trends in Chaotic Dynamics of Hamiltonian Systems (1988). [3] a) Ya. B. Zeldovich: Sov. Phys. JETP 24 (1967) 1006. b) V. I. Ritus: ibid, 1041. [4] V. V. Dodonov and V. I. Man’ko: Nuovo Cimento 44 (1978), 265. [S] I. A. Malkin and V. 1. Man’ko: Dynamical Symmetries and Coherent States of Quantum Systems, Nauka Pub., Moscow (1979) (in Russian). [6] I. A. Malkin and V. I. Man’ko: Phys. Lett. A32 (1970) 243. [7] K. Husimi: Prog. Theor. Phys. 9 (1953), 381. [S] M. Sugiura: J. Math. Sot. Japan 11 (1959) 374. [9] V. V. Dodonov and V. I. Man’ko: Invariants and Evohttion of Nonstationary Quantum Systems, ed. M. A. Mackow, Proc. Lebedev Physics Institute, Vol. 183, Nova Science, New York (1989). [lo] A. M. Perelomov and V. S. Popov: Teor. i. Mat. Physica 1 (1969) 360 (in Russian). [l 11 B. V. Chirikov: Phys. Rep. 52 (1979) 263. [12] F. Haake and D. L. Shepelyansky: Europhys. Lett. 5 (1988), 671.