New perturbation algorithms for time-dependent quantum systems

&5 . __ * __ IIB 18 August 1997

PHYSICS LETTERS A

ELSEW~R Physics Letters A 233 (1997) l-6

New perturbation algorithms for time-dependent quantum systems

Wolfgang Scherer * Institut fiir Theoretische Physik A, TU Clausthal, LeibnizstraJe 10, 38678 Clausihal-Zellefleld, Germany

Received 26 March 1997; accepted for publication 10 June 1997 Communicated by J.P. Vigier

Abstract

The averaging method for time-dependent quantum systems is used to construct quantum mechanical analogues of the perturbation expansions by Poincare and by Kolmogorov for time-dependent operators. An example shows that the resulting two perturbation algorithms are much better than the usual time-dependent perturbation theory. @ 1997 Elsevier Science B.V.

PAC.9 03.65.-w; 3 1.15.+q; 02.30.M~; 02.9O.+p

1. Introduction 2. Poincarb-von Zeipel algorithm

Formulating quantum mechanics as a Hamiltonian

system one can construct analogues of Hamiltonian methods used in classical mechanics. For time-

independent perturbation theories this has been done

in Refs. [ l-31. Here we shall construct quantum mechanical analogues of classical perturbation algorithms for time-dependent systems using the method

of averaging. This way we obtain a quantum mechanical version of the time-dependent Poincarbvon Zeipel [4,5] as well as Kolmogorov’s (KAM-type) algorithm [ 6,7] which both differ from standard time- dependent quantum mechanical perturbation theory.

Suppose that we are given a time-dependent self-

adjoint operator which also depends on the perturbation parameter E,

H=H(t,c) =Ho(t) +FcH,(t), p=l p!

where the unitary evolution U,, (t, to) generated by the unperturbed (but possibly time-dependent) part HO is assumed known. The aim of the quantum mechanical version of the time-dependent perturbation algorithms presented here is to find or approximate

the time evolution generated by H. The idea of these methods is to transform (once in the PoincarB-von Zeipel and iteratively in Kolmogorov’s. method) H

with the help of a unitary transformation Ur(e) de- pending on t as well as E such that the problem of finding the time evolution of the transformed Hamil- ’ E-mail: [email protected]

(1)

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2 W Scherer/Physics Letters A 233 (1997) l-6

tonian is reduced to a time-independent (autonomous) initial value problem.

Let V be an E and time-dependent self-adjoint op-

erator with an E-expansion

v(t,E) = g ;wp+,w p=o .

(2)

in terms of e-independent time-dependent operators

W,,. V generates U,(E) which is the solution of

$w = +p)l/(t,t),

U,(O) = 1, vt E lit, (3)

and which by construction is a unitary operator. With

U, (E) we define the following transformation U, (6) *

on time-dependent operators

u,(E)*F(t) := Ut(E)+F(t)Ut(E). (4)

Here t denotes the adjoint operator and the notation * is taken from Hamiltonian mechanics because Eq. (4) is akin to a “pullback.” This transformation is ex-

panded in terms of time-dependent linear operators T, acting now on operators

Ut(e>* = 2 ST,, p=o .

(5)

where the T, are then recursively defined through TO = 1 and

T p+l = (6)

where dDF( G) := (i/n) [ 6 G] . With the help of this transformation we define the new Hamiltonian K as

K(t,c) := UJE)*

x (H(t,r)idlil,(*)~(t.*lil,j*)i), (7)

which is assumed to have the e-expansion

K(t,c) = 2 $K,(t) (8)

in terms of time-dependent operators Kp. From this we derive the following recursive formulae for the

operators Kp,

Ko = Ho, K, = Fp - ;[Ho, W,] - %, (9)

where Fl := HI and for p 3 2

The idea is now to choose the W, such that K( t, E) in Eq. (7) is somehow “better” than the original H( t, E) in Eq. ( 1). For this we require that the W, and the Kp satisfy

(10)

Below we shall show that we can always construct W, with the averaging method such that Eq. (IO) holds

true. Before we do that, however, let us see how the validity of Eq. (10) leads to an approximation to the time evolution. The choice of W, satisfying Eq. ( 10)

has as a consequence

K(t,e) = rr,,(to,t)+K(to,E)UHo(to,t) (11)

for any t, to E R. This invariance of K implies that

$ (U”&O, t>o;c(t, to>)

i O” cp =-- 6

c $$(to) (U&o, t)UK(t, to)) 7

p=l

(12)

where UH,,( t, to) and lJ,( t, to) are the quantum mechanical time evolutions generated by the time- dependent Hamiltonian operators HO and K. On the

other hand C,“=, ( l /p!) Kp (to) does not depend on t and Eq. ( 12) constitutes a time-independent problem.

The time evolution generated by K is thus given by

UK(t,tO) = UHo(t'fO)

x exp

U! Scherer/Physics Letters A 233 (1997) I-6

This way our requirement (10) has indeed resulted in a “better” K, since we can now compute VK( t, to). Using Eqs. (3) and (7) it can be shown that the desired non-autonomous flow V” ( t, to) is then given by

Ufl(t, to> = Ut(E)UK(t, tO)&&)+. ( 14)

The right hand side of Eq. (14) certainly satisfies the

initial data for t = to. Taking its time derivative one finds that the claim is true provided

c

+ ; ‘S

dM&(a)$(t, A)V,(A)+ = 0, (15)

0

which in turn is proven by noting its validity for E = 0

and showing that it is a constant as a function of E.

Alternatively, the claim ( 14) can also be proven using

the concept of extended phase space [ 8,9]. Combining Eq. (13) with Eq. (14) yields the de-

sired flow

vfftt> to) = Ut(E)

( 16)

Eq. ( 16) describes how VH (t, to) can be obtained

from knowledge of the W, and the K,, which are computed according to the averaging method given be-

low. If we can compute all Kp in Eq. (8) and V,(E) then Eq. ( 16) gives the exact solution. This happens, for example in cases where Fp = 0 from some p on.

However, in general it may not be possible to compute W,, Fp, Kp for all p E N but only up to some finite N E N. In this case we have to use the following (non-unitary) approximation Vy (15) for V,(E),

Thus, in general it will be the approximation

V,(t,ro) = V;N(WH,(~* to)

+ WNL ( 19)

which can be computed. As some fairly straightfor-

ward examples show this approximation is already dif- ferent from standard time-dependent perturbation the-

ory.

3. Time-dependent quantum averaging

For the execution of this algorithm it is necessary

to find W, such that Eq. (10) holds. The existence of

such a W, is due to a quite general construction using averaging as follows. Let F(t) , G(t) be two (possi-

bly) time-dependent operators such that the time evolution VF( t, to) generated by F is known and such that

Jim, VF(t-T,t)+G(t-T)&(t-T,t) -G(T)

T =o

for all t E R and

T

ZcF) (t) := :im, f - J do-VF(t-a,t)+

0

x G(t-a)VF(t--,t),

T + 1

ScF)(G)(t) :=!irnaT - JJ dr da

0 0

x V~(t-,,t)+G(t-(+)VF(t--(T,t) (

- P(t) > (17)

where each VP is given by the recursive relation

i v ptl = --

li &,=lH. (18)

,

exist. Then it follows that

G- i[FS(“(G), - dScF) (G)

h ’ at = p

(20)

(21)

(22)

(23)

(24)

4 W Scherer/Physics Letters A 233 (1997) 1-6

which is proved by a straightforward computation [ 91 using the assumed property (20) and assuming that taking commutator as well as the time derivative “com-

mutes” with the limits and integrals involved (see Ref. [ I] for a similar proof in the time-independent case).

Eq. (IO) can thus be satisfied by the choice W, = StHo) ( Fp) and from Eqs. (23) and (9) it follows that

KP = q’““‘. For this it is necessary that Eq. (20) holds which is the case if G is uniformly bounded in t. General conditions (e.g. on the HP(t) ) which assure that the Fp ( t) are such that Eq. (20) holds for G = Fp are not known at the moment and are the subject of

current investigations.

4. Time-dependent quantum KAM

The quantum mechanical algorithm constructed

above is modeled on a classical perturbation algorithm called the Poincare-von Zeipel method. An

improvement (in classical mechanics) of this method was suggested by Kolmogorov [6,7]. It consists of an iteration of the above method where each step uses

a new (and “improved”) unperturbed Hamiltonian. Its quantum mechanical analogue for time-dependent systems runs as follows. For each n = 1,2,. . . let

Vn(t,c), W,(~),U:(E),U:(E)*, and Ti be defined similarly to K W,, U,, U,?, and Tp. For each IZ we define

K”(t,c) := U;(E)* H”-‘(t,~)

05

= c cK”(t),

P=o p! I’

(25)

where I@ = H = Cra(&‘/p!)q( t) denotes now the original perturbed Hamiltonian whose time evolution is to be approximated. This results in expres- sions for KL which are of the form (9) and ( 10). The

generator V’ of the first transformation U/ (E) is now chosen such that

W; = S@(F;), w:, = 0, vp 2 2, (26)

where S(g) ( F,’ ) is defined as in Eq. (22). This implies

(d&v; )p-‘(f$+,).

(27)

(28)

After this first transformation we introduce the Hamil- tonian Hi = H’( t, E) which is considered as the

perturbed Hamiltonian of the unperturbed part # +

(29)

(30)

such that

-_( f& HA=@++ , H;=O, H;=K;,

VP b 2, (31)

and H’ (t, E) has no perturbation of first order in E. As in Eq. ( 11) we have then

H;(U) = ~~(to,r)+H~(ro,t)~~(ro,r)

for any t, tn E R and thus

(32)

The property (33) helps us to find the flow of HA since it again constitutes an autonomous initial value problem and implies

xexp - (

i(t -firo)e$@) (rO) .

) (34)

For the second transformation Uf ( E) we choose the generating operator V2 ( E) as follows,

W. Scherer/Physics Letters A 233 (1997) 1-6 5

wf = 0, w; = P’(H;), P =2,3, (35)

w; =o, vp 34, (36)

which implies

K;=H;, Kf =O, (37)

K; = gcH”, K; = @(M’), (38)

-_(+I Here Kt = HA = @ + EH: IS the unperturbed

Hamiltonian after the first step whose flow is found with the help of Eq. (34).

After the second transformation we introduce the

Hamiltonian H2 = H2( t, E), which is considered as the perturbed Hamiltonian of the unperturbed part

q-_(G) H’ + “HI y(H;)

0 2! 2 +$& 7

H2 := K2 = H; + $‘““’ \ v

=H;+-+$ p=2 p!

such that Hi = Kf,, Vp 3 22, and H2 has no perturbation of second and third order in E. Again, by con-

struction

x exp (

i(t - to) - p(H;(ro,e) -H&o+))

n I (40)

Now transform H2 with a third transformation U:(E) and so on and so forth [ 8,9]. Similar to Eq. ( 14) we have thus after n transformations

UH(t, LO)

d;(E).. .v:(E)UH"(t,rO)U::(E)...U:,(E).

(41)

The U;(E) are generated by the V”(t,c) (see Eq. (3) ) and may have to be approximated similarly

to Eqs. ( 17) and ( 18). The time evolution generated by H” can be approximated by the one generated by its unperturbed part H;,

Uf,“(t,tO) =U”$,tO) +o(c2”“),

where now as in Eqs. (34) and (40)

(42)

uH;;(t,tO)

=UH,O(f,tO)U~H~-H~)(,o)(t)...U~Hbl_Hb'-')(,o)(t)'

(43)

where the U~Hp_up-~~~lo~ (t) denote the solutions

of autonomous \ro&lems of the type (33). Putting

Eqs. (41) -(43) together yields an approximation of the time evolution generated by H which differs from

the standard time-dependent perturbation algorithm as well as from the one outlined in the first part of

the paper (Poincare-von Zeipel) .

5. An example

We illustrate the two new methods with the follow-

ing simple example taken from the dynamics of spin-i

particles in oscillating magnetic fields. Using the dimensionless variable t and the dimensionless param-

eters a and E the system to be investigated is

i$$(t) = (~+r~COS(t))x(t), (44)

=:Ho =:H;(r)

where Sj := ( n/2) CTj and the Uj are the Pauli matrices. Suppose that at the initial time to = 0 the system is

in a spin-up eigenstate

/y(O) := ; . 0 We can then compute the state at a later time t,

to a desired order in E with the various approximations to the time evolution. Note that via resealing t in Eq. (44) the coefficient a takes the role of an inverse frequency for which the value a = 1 leads to highly resonant behaviour in all perturbation algorithms. Here

K! Scherer/Physics Letters A 233 (1997) 1-6

6. Conclusion

I- 0 5 10 15 20

Time (dimensionless)

>

Fig. 1. Transition probability Ix_ (t) I2 for n = 0.5 and E = 1 com-

puted by standard perturbation theory (long dashes), Poincar&von

Zeipel (short dashes), Kolmogorov’s algorithm (dotted), and nu-

merically (solid).

we shall compare the results for the transition proba-

bility

P+--(r) = Ix-W2 (47)

from the initial spin-up state to the spin-down state

computed up to fourth order in E with the standard, Poincar&von Zeipel, and Kolmogorov’s method and

compare it furthermore with a numerical solution of

Eq. (44) using a numerical algorithm contained in the Mathematics package.

Fig. 1 shows (x_(t) I* for the case a = 0.5 and E = 1 and it is evident that the standard perturbation result is quite good for short times r < 3~7 but then increas-

ingly deviates from the numerical result and violates unitarity due to the presence of secular terms, whereas Poincar&von Zeipel as well as the superconvergent

method reproduce the numerical result. They do so with great accuracy and for arbitrarily long times in the case of smaller perturbations not shown here. Compu- tations for other a and E show that as the perturbation increases in strength standard perturbation theory be-

comes highly inaccurate and PoincarC-von Zeipel de-

viates more strongly from the numerical result while the Kolmogorov’s method remains still close to it.

Eqs. ( 16) and (41) are exact results for the time evolution which may have to be approximated by

Eq. ( 19) or Eqs. (41)-(43) respectively. If one can-

not calculate all W, these approximations introduce via Eq. (17) violations of unitarity which, however, do not grow with t as is the case in standard time-

dependent perturbation theory which contains so called secular terms.

Finally, it should be stressed that the methods presented here are quite universal in the sense that, provided the condition (20) is satisfied and Eqs. (2 1) and

(22) exist, they are applicable to any time-dependent

perturbation problem involving self-adjoint operators

independent of the particular form of these operators.

Acknowledgement

Helpful conversations

gratefully acknowledged.

References

with H.-D. Doebner are

[I ] W. Scherer, Phys. Rev. Len. 74 ( 1995) 1495.

[2] W. Scherer, J. Phys. A 27 ( 1994) 8231.

131 W. Scherer, J. Phys. A 30 (1997) 2825.

[4] H. Poincare, Les mCthodes nouvelles de la mCcanique cCli?ste

I, II, III, reprint (Dover, New York, 1957).

[5] H. von Zeipel, Ark. Astron. Mater. Phys. 11, 12, 13 (1916-

17).

[6] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 98 ( 1954) 527.

[7] A.J. Lichtenberg and M.A. Liebermann, Regular and

stochastic motion (Springer, Berlin, 1983).

[8] W. Scherer, Habilitationsschrift, TU Clausthal (1996); TU-

Clausthal preprint ASI-TPA/ 12/96.

[ 91 W. Scherer. TU-Clausthal preprint ASI-TPAI 16/96.

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New perturbation algorithms for time-dependent quantum systems