Tight-binding representation for rf floquet states

Volume 140, number 7,8 PHYSICS LETTERS A 9 October 1989

T IGHT-BINDING REPRESENTATION FOR rf FLOQUET STATES

Bala SUNDARAM and Avinash SINGH Department of Phystcs and Astronomy, The Johns Hopkins Umverstty, Baltimore, MD 21218, USA

Received 18 April 1989; rexased manuscript received 13 June 1989; accepted for pubhcation 16 June 1989 Commumcated by A.R. Bxshop

The interaction of highly excited hydrogen atoms with rf fields is described in terms of a tight-binding Hamiltoman with long- range energy transfer terms. Modifications to localization characteristics are summarized and applied to the question of quantal suppression of classical diffusion seen for this system.

The search for quantum mechanical manifesta- tions of classical chaos has led to a great deal of the- oretical activity over the past few years [ 1 ]. This test of the correspondence principle has produced some interesting new results as well as some novel asso- ciations. In particular, the question of quantum suppression of the classically predicted, diffusive energy growth has provoked special interest. This suppression is a direct consequence of quantal interference effects and is analogous to localization in disordered electronic systems [ 1,2]. Two systems that have been studied extensively and exhibit suppression effects in their dynamics are the kicked rotor (KR) and the driven surface-state electron DSSE. The DSSE Hamiltonian is of special interest as it has been shown to describe, approximately, the behavior of highly-excited hydrogen atoms interact- ing with a microwave field [ 1 ].

The KR system was analytically mapped onto a tight binding Hamiltonian in terms of the quasienergy or Floquet states [2 ]. Within a basis, @,, of unperturbed hydrogenic states, the DSSE can also be mapped onto a real, symmetric tight-binding Ham- iltonian of the form

H = ~ Z ~ l ~ , > < ¢ , l + ~ Vm, lCm><0,1, (1) n n #~ r r l

o t ~ o t with 1". = Z ~ E,~ I a.~ 12 and Vm. - Z ~ E,~ (a , . ) a . . In terms of ~., the quasienergy states are given by q,:' = Z . a~'O. with corresponding quasienergies E~ such that

U(T) ~'~ = ear~'~ ~ . (2)

Here z is the period of the external field and U(T) is the single-cycle time evolution operator. It is easily seen that the choice for T, and Vm, ensures the same eigensolutions, E,~, ~u,, as obtained for U(z) . Our construction of H amounts to taking the logarithm of U:

U(T) = e ~m . (3)

Therefore the tight-binding mapping provides an effective time-independent Hamiltonian which describes the one-cycle and, hence, the long-time dynamics. It should be noted that this is a general procedure valid for any temporally periodic Hamiltonian.

In the tight-binding representation, Vr~, are the effective transition matrix elements coupling unperturbed states m and n. The T, are like the on-site energies and have a pseudorandom distribution as a consequence of the quasienergies E,~ being defined modulo the external (microwave) frequency, 12. The off-diagonal matrix elements Vm. correspond to the hopping terms in a tight-binding representation. In the limit of very weak fields, F- ,0 , the projection coefficients a~ are well approximated by 6,~, where m runs over the basis index. This simply means that the quasienergy states and quasienergies are very nearly the unperturbed basis states and energies, The V,,, all vanish in this limit but the pseudorandom distribution of T, is retained. Therefore, studying the

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behavior of the off-diagonal terms is sufficient to characterize this system.

Fig. 1 shows the nature of Vm,, for fixed n ( = no) and all m, for typical values of the DSSE parameters. The external field parameters are expressed in scaled variables [ 1 ], 12o=I2no 3 and Fo=Fng, where f2 and F are the frequency and amplitude of the rf field (in atomic units) and no is the quantum number of the initial state. In our mapping to a tight-binding Ham- iltonian the hopping term exhibits three distinct features - (a) long range effects; (b) oscillations in sign of hopping and (c) preferential coupling of particular sites (resonances). As seen from fig. 1, both the range of the hopping and the positions of resonances are dependent on I2o. In particular, in the limit I20 >> 1, a single resonance dominates V,~, whereas for f2o~ 1.0 the long range effects are most pro- nounced. Increasing F0 simply leads to a larger range in the hopping term.

To understand the modifications due to long range hopping we first discuss localization characteristics in a simple I-D model with a hopping term of the form V/rE It was suggested in ref. [2] that ~,= 1 de-

- 1

.05

- 0 5

4

3

.2

1

0

- 1

• i ' ' ' i ' ' ' i ' ' ' i ' ' '

(a) -

i ' ' • P ' ' '

(b)

. : ~ : : ,, , • : ', ', ',

i (c)

- 2 0 20 40

(m-n)

Fig. I. EffeeUve transition matrix elements between Floquet states of the DSSE Hamiltonian. Parameter values are defined in terms of scaled variables: field Fo=Fn~ and frequency ~o=g2no 3. For all cases Fo=0 .04 and no= n = 66. (a) 12o=0.8, (b) Do= 1.2 and (c) ao=2.8.

marcated cases of localized and extended solutions. However, we find that y=2 is the critical value be- low which departures from exponential behavior appear, as a consequence of a reduction in effective disorder. This crossover is displayed in quantities appropriate to the DSSE system and implications to recent experiments [3,4] are discussed. It is suggested that the varying range of hopping, as a function of field parameters, is largely responsible for the wide range of behavior seen in the DSSE problem.

We consider the following tight-binding Hamiltonian,

H= ~'~,at, a, - ~ -~(a~a~+a~a,), (4) t I t - - J [ = r

where t, are the random on-site energies chosen in- dependently from a uniform distribution on - W/2 to W/2. The limit y~oo gives the usual nearest- neighbor (NN) case. The critical value of y can be related to changes in the energy relation for the pure system, W= 0. Fourier transforming to momentum space, the energy dispersion is simply

Ev(k) = - 2 V c o s ( k r )

r=l r r (5)

This expression is easily evaluated for integer ~, with the energy for even Y being expressed in terms of Ber- noulli polynomials [5]. Odd values of y lead to a more complicated k-dependence. To illustrate, the dispersion relations for 7 of 1 through 4 are given by

EI(k)/V= log2 ( 1 - c o s k ) ,

E2(k)/V= - ~ x E + n k - ½k2,

Ea(k)/V= - 2 ( ( 3 ) + 3 k E - k 2 log k (for small k) ,

E4(k)/V= 1 4 2 2 | 3 W 4 W , - ~ n +re k / 3 . -nk /3.+k /4. (6)

where ~(n) is the Riemann zeta function. It is clear that the qualitative nature of the spectrum is mod- ified by the long range hopping. A detailed analysis is presented elsewhere [6 ] but the features relevant to the DSSE problem are emphasized here.

The coherence effect arising from long range hopping modifies the spectrum near the lower band edge and this alters the localization characteristics signif- icantly. In the range 1 ~<?< 3, dF, v(k)/dk~kv-2, for small k. Therefore, for ~< 2 the density of states,

401


N(E) , which is proportional to the inverse of the slope, actually vanishes at the lower band edge. This ~s in sharp contrast to the NN case where N ( E ) diverges at both the upper and lower band edges. W~thin the Boltzmann approximation, weak disorder leads to a finite mean free path, l, and an elastic scattering rate r - I . Since the probability of scattering depends on availabdity of other momentum states, the effective disorder strength as measured by r - i depends on the density of states and is given by r - ~ (E) ~ N(E) W 2. In view of the vanishing of the density of states as k--,0 when y< 2, it is clear that the relaxation time diverges as k- ,0. Since the mean free path,/ , is equal to the group velocity (dEr (k ) / dk) times the relaxation rate, both of which diverge (for y < 2 ) as k--,0, lk diverges as 1/k 4-2r, In a one dimensional system we expect the localization length, which characterizes the exponentially localized wavefunct~ons, to be roughly equal to the mean free path.

The effects of disorder are studied using the exact elgenstates obtained by diagonalizing the Hamilto- nian (4) with a periodic boundary condition [7]. These eigenstates are equivalent to the quasienergy states for the DSSE system. The crossover from ex- ponentml to power law behavior is clearly seen in the asymptotic t~me-averaged probability d~stnbution, P(r), and its second moment ( r 2 ) . We chose to study P(r) because it gives the time-averaged probability (as t--,oo) of finding a particle at distance r from the ongin where it was placed at t=0. P(r) thus relates to a physically meaningful quantity in the context of the DSSE problem where it corresponds to the transition probabihties. In terms of the site representation, P ( r ) = El ( ~ ) 2 ( ~ + r ) 2 , where t, the site occupied at t=0 , was chosen to be the central lattice point. Fig. 2 shows that P(r) decays expo- nentmlly with r for y= 10 whereas it decays via a power law for ~= 1. This can easily understood in terms of the earlier discussion of the pure spectrum. W~thin the weak disorder limit, it was shown that for y< 2, the mean free path and hence the localization length, ~, diverges as k--,0. It can then be shown that P(r), which consists of a sum over states of exponentially decaying functions with ~ ranging to infin- ity, exhibits a power-law dependence.

A s~mple signature of the crossover is seen in the second moment, ( r 2), which scales with system size

' ' I ' ' ' I • ' ' I ' ' • I ' ' ' I ' '

ol

ool

000 !

l 0 - 6

1 0 - 6

1 0 - 7 , . . . . . . i i i

- 4 0 - 2 0 0 2 0 4 0

~2~ r 1 I . . . . 1 . . . . . . . . I ' ' ' ' I

ol

OOl

1oo 10 0 lO 100

trl

Fig. 2. Decay of the asymptotic time-averaged probability distribution as a function of r for IV~ V= 5.0 and (a) 7= 10 and (b) 7=1

L given the power law dependence o fP ( r ) . For P(r) proportional to 1/r '~, it can be shown that ( r 2) is constant, for or> 3; ~ L 3-'~, for 1 < a < 3 and ~ L 2,

for a < 1. Note that the normalization for P(r) has to be taken into account. Fig. 3 shows a log-log plot of ( r E) as a function of the lattice size, L, for IV/ V= 5.0 and for several values of y. The straight lines are drawn as least square fits to the points. Changing ~, from 20 (which is essentially the NN case) to 3 results in very little absolute change in ( r 2 ) and vir- tually no dependence on size. This implies that the wavefunctions must all be exponentially localized with localization length smaller than the lattice size. However, when 7 is decreased to 2 there is a sharp increase in < r 2 >. This, together with the system-size dependence for ? between 1 and 2 is a clear indi- cation of a sharp crossover at y=2. The slopes ob- served are consistent with the simple analytic esti- mates suggested earlier.

The implications to the DSSE system are seen in analogous measures constructed in terms of the Flo- quet or quasienergy states. The asymptotic time-averaged probabdity distribution over hydrogenic states

402


A

3

. . . . i . . . . i , .

a o

!

2 1 1 1 1 1 1 1 1 1 [ i i i i 4 5 5 5 5

In L

Fig. 3. The second moment of the asymptotic time-averaged probability distribuUon as a function of system size for IT'/V= 5.0 and ~,=20 (triangles), 3 (circles), 2 (squares) and 1 (crosses).

i i i i

6

3

2

I I I , I I

3 8 4 4 2 4 4 4 6

L n L

Fig. 4 Variation of the second moment of the asymptotic time- averaged distribution, for the DSSE, with bas~s size. The parameters are ~=2.8, no--66 and Fo=0.03 (triangles), 0.07 (crosses) and 0.1 (circles).

m

e"

V

_=

n, with initial condition no, is given by P(n) = E~ (a~o)2 (a~) 2, where the sum is over quasi-

energy states. The second moment of this distribution ( ( n - n o ) 2) is equivalent to ( r2) . A localiza- Uon theory based on NN coupling predicts a transition from exponentially localized to delocalized behavior [ 1 ] at a critical scaled field value which, for fixed #2 o and no, is given by --0~cr =~-o~7/6/(6-6no) 1 / 2 . / This means that the variation of the second moment with system size, on a log-log scale, should show an abrupt change in slope from 0 to 2 on increasing the field strength past F~ r. Fig. 4 shows the size dependence of ( ( n - n o ) 2) for different field values with fixed #20 and no. The basts size was varied to provide from 30 to 80 states for upward transitions and 20 for transitions downward. The delocalization thresh- old predicts a critical field value of ~ 0.16. The size dependence for Fo of 0.07 and 0.1 clearly points to power-law behavior in the distribution over unperturbed states. This is in contrast to the case of Fo=0.03 where the power-law fall off is (at least) faster than 1 / ( n - no) 3 and, hence, indistinguishable (in the second moment) from exponential behavior. The highest field value, F0=0.1 has a slope of ~ 1.6. Thus, the transition from exponentially localized to

extended distributions moves smoothly through a re- gion of power-law behavior.

Our results are particularly relevant to recent experiments directed to studying quantal suppression effects [3,4]. In these experiments the presence of additional fields leads to a high-n cutoff, no, in the space of bound states. The equwalent pseudo-lattice (eq. (1) ) is, therefore, always finite [4]. In ionization experiments, our results would be manifest in the "experimental ionization", the sum of probability in states with n> nc plus true ionization [4], as a function of n~. I f n-changing distributions are measured [ 3 ], a more direct verification is possible. For instance, we reanalysed the data shown in figs. l a and l b of ref. [3] by plotting In [P(n) ] as a function of In (n - no), where no is the imtlal state and n > no. The P(n) are found to satisfy powerlaws with slopes of ~ - 1.2 and ~ - 2 . 0 respectively. The corresponding second moments would scale, approximately, with basis size as L 1 s and L. Both cases would be in the power-law regime rather than the exponentially localized regime. This analysis is included to indi- cate ways of confirming experimentally the predic- tions m this paper. Of course, specific aspects of the

403


experiment, like finite interaction time and field profile, should also be considered.

Considerations of long range energy transfer terms provide information on the gross features of the probability distribution. The consequences of additional features, in the effective transition matrix elements for the DSSE problem, to localization characteristics can also be easily understood. The oscillations in the sign of the hopping make the energy spectrum bounded for all 7 but also reduce the coherence effects. Preferential couplings of specific sites show up as peaks in the probability distribution (resonances) which compete with the "background" distnbutmn. Inclusion of resonances in a tight-binding scheme also leads to Fano-like profiles within the band. These arise from the interference of resonant and nonresonant pathways, leading to the same point in the continuum band. For the DSSE, this would appear to conform with the idea [ 8 ] that the nonresonant background provides a quasicontinuum that couples coherently with the resonant process. The "effective" power-law satisfied by the distribution is altered with the inclusion of these processes but the overall trends remain the same. Further, the effective matrix elements Vm,, for ~o >> 1, can be used to test approximations involving "essential" states, where, for example, only states that are successively one photon resonant are considered [9 ]. A detailed analysis of these additional processes will be presented elsewhere.

We have mapped the DSSE problem to a tight- binding Hamiltoman which effectively describes the one-cycle and hence the long-time dynamics. This motivates the consideration of long-range energy transfer terms in a simple 1-D system. The modifi-

cations in localization behavior lead to added insight into quantal suppression effects reported for the DSSE problem. In particular, the power law behavior of the probability distribution function reported here is related to experiments in progress to test localization effects [ 3,4 ]. Further, it would appear that understanding the effects of long range hopping may help extend the analogy with disordered electronic systems to quantized versions of other nonlinear maps [2 ].

This work was supported in part by the National Science Foundation grant No. PHY-8518368. We are grateful to R.V. Jensen, Lloyd Armstrong and M.O. Robblns for helpful suggestions and to J.E. Bayfield for providing us with the experimental data.

References

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[3] J.E. Bayfield and D.W. Sokol, Phys. Rev. Lett. 61 (1988) 2007.

[4] E.J Galvez, B.E. Sauer, L Moorman, P.M. Koch and D. Richards, Phys. Rev. Lett. 61 ( 1988 ) 2011

[5] I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, senes and products (Academic Press, New York, 1980).

[ 6 ] Avmash Smgh and Bala Sundaram, to be published. [7] A. Singh and W.L. McMillan, J. Phys. C 17 (1984) 2097 [8] J.N. Bardsley, B. Sundaram, L.A. Pinnaduwage and J.E.

Bayfield, Phys. Rev. Lett. 56 (1986) 1007. [9] R.V. Jensen, S.M. Susskind and M.M. Sanders, Phys Rev.

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Tight-binding representation for rf floquet states