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Correlations in anticrossing spectra and scattering theory: Numerical simulations
Chemical Physics 150 ( 199 1) 3 1 I-320 North-Holland
Correlations in anticrossing spectra and scattering theory: numerical simulations U. Hartmann, H.A. Weidenmiiller and T. Guhr Max-Planck-Institutfiir Kernphysik, W-6900 Heidelberg, Germany Received 26 September 1990
For the interpretation of experimental results of anticrossing spectroscopy in molecules, the results of numerical simulations are presented. We have focussed our attention on the visibility of the correlation hole. In the formulation of the model, we combine scattering theory and a random-matrix model.
1. Introduction The identification of experimental indicators of chaotic motion in quantum systems is a problem of considerable interest. The standard procedure consists in the measurement of the energies of a sequence of levels having the same quantum numbers, with the aim of determining the eigenvalue statistics. The procedure is based on the fact that for regular (or integrable) systems, the eigenvalue distribution is Poissonian, while for time-reversal invariant systems, which are chaotic in the classical limit, the distribution is conjectured to coincide locally with the distribution of eigenvalues of the Gaussian orthogonal ensemble of random matrices (the GOE). Historically, the first evidence for GOE level statistics came from data on low-energy neutron scattering on heavy nuclei, cf. refs. [ 1,2 ] and earlier work cited therein. Later, evidence for chaos was also found in atomic [ 3,4] and molecular [ 5-71 spectra. Our present investigation was triggered by the work of the Grenoble group [ 8 ] on the organic molecule methylglyoxal. This group observed the resonance fluorescence yield of laser-excited molecules as a function of an external magnetic field. The laser light excites a singlet state at fixed excitation energy, independent of the strength of the magnetic field. As the strength of the magnetic field is changed, the m = f 1 components of the nearby triplet states are shifted past the singlet state, giving rise to Landau-Zener anticross-
ings. Each anticrossing causes a resonance-like reduction of the resonance-fluorescence intensity with a Lorentzian line shape. We became interested in methylglyoxal because here several thousand anticrossings were observed and, more importantly, the Lorentzians of neighbouring anticrossings overlap fairly strong, making it impossible to perform a detailed analysis of each anticrossing and to establish thereby the eigenvalue statistics as a signal for chaotic motion. We were intrigued by the idea proposed by the Grenoble group [ 9 ] : in the autocorrelation function of the Fourier transform of the fluorescence yield, they found a “correlation hole” at small values of the displacement parameter t. This correlation hole was interpreted as being due to long-range GGE-type level correlations and, thereby, as a fingerprint of chaos. The possible existence of such a fingerprint, observable without the need to perform a painstaking levelby-level analysis of the spectrum (which, in the case of methylglyoxal, would not even be feasible) is obviously of general interest and far exceeds the special case of methylglyoxal. The question is: can we show theoretically that the correlation hole is indeed a manifestation of chaos? This question was addressed in a preceding paper by two of the present authors [ 10 1. The investigation was carried out analytically in the small coupling limit which is based on two assumptions: (i) The rms matrix element VSldescribing the dy-
0301-0104/91/S 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
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U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
namical coupling between the singlet state and the triplet states was assumed to be small compared to the mean spacing D of the triplet states. This assumption permits an expansion in powers of V,,/D which can be terminated after a few terms. The assumption is realistic for methylglyoxal because the mean level spacing D that is relevant is the mean level spacing between triplet states of the same symmetry [ lo]. (ii) With ys the width for photoemission of the singlet state and yt the width for photoemission from the triplet states, it was assumed [ lo] that YJ,> 4V$. This assumption, however, does not apply to methylglyoxal, severely limiting the significance of the results of ref. [ lo] in the case of this molecule. To investigate the domain YJ, 5 Vz,, we had to resort to Monte Carlo simulations. At the same time, we considered it useful to investigate the domain of overlapping resonances, which also is not accessible analytically. The results of these simulations are reported in the present paper. We point out that the terms non-overlapping resonances (T/D +c 1) and overlapping resonances (r/ D >, 1)) where r is the total width, refer to pure spectra where all levels have the same symmetry. Spectra of molecules like methylglyoxal are usually superpositions of several pure spectra. Hence those spectra may look like consisting of overlapping resonances although they are superpositions of spectra of nonoverlapping resonances, cf. ref. [ 10 1. The model, the anticrossing spectroscopy, the scattering matrix formalism, and the experimental results were presented in detail in ref. [ 10 1. To make the present paper sufficiently self-contained, we confine ourselves to a rather compact presentation of these topics ( section 2 ) . In section 3, we describe how the Fourier transforms of anticrossing spectra were simulated, and in section 4 we present simulations for the regime of overlapping resonances ( V,,? D). In section 5, we investigate the influence of the distribution of widths and depths of the anticrossings on the Fourier transform of non-overlapping resonances, and we compare our results with those of ref. [ lo]. The conclusions are formulated in section 6.
was used to excite selectively a rotationless singlet state Is) with excitation energy E,x 25000 cm-‘. Via the singlet-triplet interaction Vst, this level (which we refer to as a doorway state) is mixed with the triplet states ltp), PL= 1, 2, .... N where Nz+ 1 (fig. 1). As the external magnetic field is changed, the energy EG of the m,= + 1 component of a triplet state may get close to the singlet state (so that IEs-E; I S V,,), and an anticrossing occurs. The anticrossing mechanism is quite simple: Because of the mixing of the two states, some transition strength y is shifted into the non-recorded transition from the triplet state, and is thereby lost for the resonance fluorescence yield, causing a resonance-like dip in the latter. We note that for purposes of anticrossing spectroscopy, the magnetic field plays the role of an excitation energy scanning the triple states. In the case of isolated anticrossings, the line shape isgivenby [11] Z(B) =I0
’
1-
2V:&,lY, (E, ~B-Es)2+2V~,ys,(y;’
+y;‘)+yzt
(1) where Yst is the total width of the mixed state. We measure the magnetic field B in energy units. The background intensity I0 is given by np /yr with 1: the singlet excitation rate. For experiments performed at sufficiently low pressure, we have Ystz 4 ( y. + Yt); this relation will be used in the following. Eq. ( 1) was confirmed by very precise measure-
Itp>
2. The model In the experiment
of the Grenoble
group, a laser
Fig. I. Scattering model for the laser experiment on methylglyoxal. Dynamic couplings are drawn as wavy lines.
U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
ments [ 121. Inspection shows that an anticrossing causes a dip in the fluorescence yield of Lorentzian shape with width rand depth h given by r=2J4Vf~y:~/(y,y,)+y:~, h=2A, VZt(~/2)-2~stl~s.
(2)
For non-overlapping resonances and in the limit fi Z+2 I Vs, 1 considered in ref. [ lo], the expressions simplify to
(3)
k, = 210Et/ (LYE1 .
In the case of the experiment of Lombardi and coworkers ySz y,x 0.3 MHz e 1V,, 1 and thus rexp = 4 I Kt I YstlJrsut h,x,=~ortl(Ys+Yt)
3
.
(4)
Please note that in the last case the depth is independent of Vs,. In the glyoxal measurement (resolved anticrossings) all observed anticrossings had the same depth [ 131. This leads to the conclusion that all triplet states have the same radiation width y, which was confirmed by life-time rf experiments [ 14 1. Eq. ( 1) is easily derived using the density matrix formalism but can also be obtained from the scattering matrix formalism [ lo]. In the latter case the measured intensity in case of N triplet states is given by [lOI
I(E)=&c j: IS(E,B)-1]2dE,
(5)
.”
-03
where S(E,B)=l-iy,[G-‘1,
(6)
and G=EIN+,
_
Es--iys
H-
VT
tir, +BIN > .
(7)
Here V is an N-component column vector describing the coupling of the doorway state with any of the triplet states I tp); lN is the unit matrix in Ndimensions, and r, is an NX N dyadic matrix given by y$‘y,‘/‘. The ytllare the y decay widths of the triplet states Itp) , and B is the strength of the magnetic field in appropriate units. Finally, His the NX N Hamiltonian matrix which describes the mutual interaction of the N
313
triplet states. In line with the arguments presented in section 1, we assume below that H is an element of the GOE of dimension N. In eq. (7), we have taken account of the only one spin projection (m, = + 1 or m,= - 1) although both spin projections contribute to the experimental result. We do so because the Monte Carlo simulation becomes forbiddingly formidable if both projections are included. The analytical results of ref. [ lo] justify our procedure: the fluctuations due to the mixing of different spin projections are not generic in the sense that the mod square of their Fourier transform is proportional to the delta function. For non-overlapping resonances, the origin of the correlation hole mentioned in the introduction is easily seen in a phenomenological model [ 9,13 1. We suppose that an intensity spectrum consists of an infinite sequence of isolated anticrossings, each with shape given by eq. (1 ), with constant width r and constant depth h, the spacings being those of the eigenvalues of the Hamiltonian H. (The finiteness of an actual spectrum does not affect the argument, see ref. [ lo].) Then the Fourier transform of the intensity autocorrelation function, henceforth denoted by Ic(t)12,hastheform Ic(t)~2~[d(t)+1-b2(t)]exp(-2ntI’/D).
(8)
Here, b,(t) is the Fourier transform of the two-level cluster function Y2(r) first introduced by Dyson and Metha [ 15 1. The result in eq. ( 8 ) is the product of two terms, one being the Fourier transform of the shape of the anticrossing [ exp ( - Zxtr/D) 1, the other accounting for correlations between triplet levels. For a Poissonian triplet spectrum, both Y2(r) and b2( t) vanish, while for a spectrum with GOE level statistics both functions behave as shown analytically in ref. [ 15 I.. For the GOE case the analytical result for l-b2(t) is plotted in fig. 2. In particular l-b,(t) displays a dip at the origin which becomes the correlation hole observed in refs. [ 9,131. We note that this phenomenological way of reasoning is independent of the value of the ratio y,y,/4V$.
3. Numerical simulation In this section the method used to generate the mod squares of the Fourier transforms of anticrossing
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U. Hartmann et al. /Correlations in anticrossingspectra and scattering theory
(ensemble) average over the mod square of the Fourier transform, i.e. the quantity 1c(t) 12, or smooth the mod square of the Fourier transform (cf. section 3.4). 3.2. Generation of the spectrum
00 0
1
2
t
3
Fig. 2. Plot of 1-b,(t) for the GOE, b2 (1) is the Fourier transform of the two-level cluster function Y,(r).
spectra is described. At the end of this section we describe the checks of the programs. 3.1. Overview The random matrix H (see eq. (7) ) used for the simulations was drawn from the GOE. Forced by computer-time requirements we had to restrict the simulation to the case of one pure sequence. This simplification goes beyond the restriction to a single value of the spin projection which we introduced and discussed in and below eq. (7). Indeed, it is known that the sequence of triplet states with fixed spin projection consists of states belonging to several symmetry classes. A realistic simulation would have to use in eq. (7) a Hamiltonian H having a block structure, each block simulating triplet states with fixed symmetry and represented by a separate GOE. We return to this point in the discussion (section 6). The procedure to obtain the simulated Fourier transforms is very similar to the way the experimentalists obtained their Fourier transforms. The main steps are as follows: ( 1) Generate a GOE matrix H for the N triplet states. (2) Calculate for every desired value of the magnetic field B the intensity ( fluorescense yield) Z(B) from eqs. (5)-(7). ( 3 ) Calculate the mod square of the Fourier transform of the intensity Z(B) generated in step 2. (4) Repeat steps ( 1 )-( 3), thereby generating the
In an arbitrarily chosen energy unit, we took ES= 0, yS:=y,= 1, and all y$’ = 1. The last choice was motivated by the observation by Lambardi and co-workers [ 14 ] that in glyoxal (non-overlapping anticrossings), all anticrossings had the same widths for the triplet states. The restrictions inherent to this choice are removed in section 5. We also took all matrix elements VP of the singlet-triplet coupling equal to a fixed value V,,; again, this restriction is lifted in section 5. Finally, our choice yS= yt was motivated by the experimental situation in methylglyoxal [ 141. With the choices just made, the results of the calculation depend on two parameters, i.e., the singlet-triplet coupling strength VStand the mean level spacing D. To generate efficiently spectra of constant level density, we diagonalized H and unfolded the eigenvalue spectrum, using the semicircle law for the mean level density. This diagonalization has to be performed only once for every choice of H, and every spectrum. After the diagonalization, spectra having different distributions of widths and depths can be generated from the same matrix H by using different distributions for V,, and y:/‘. The diagonalization of H affects both the triplet-singlet coupling matrix elements VP, and the width amplitudes r$‘. To see this, we write H in the form OAOT, where 2 is the diagonal matrix of eigenvalues, and 0 the diagonalizing orthogonal matrix. In the eigenvalue representation, VP and y$’ are transformed into (OV), and ( OY:‘~),, respectively. For NB 1, the elements of 0 are uncorrelated Gaussian distributed random variables, and the quantities (OV), and (Oy:‘2), therefore have probability distributions which differ from those of the original variables VP and r$‘. 3.3. Energy integration To obtain the resonance fluorescence yield Z(B), it is necessary to integrate over the energy E of the laser beam; this integration has to be carried out for every value B of the magnetic field (cf. eq. ( 5 ) ) and con-
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U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
stitutes the main difficulty of the numerical simulation, since the evaluation of the integrand involves the inversion of a matrix of dimension N+ 1. The energy integration was performed analytically by diagonalizing [ G-El N+, ] (which is independent of the energy, cf. eq. (7) ) and by using the residue theorem. The numerical effort (number of floating point operations) for the diagonalization and inversion of the matrix [G-El,,,, , ] --I scales as N’ and inhibits the generation of very long sequences. This problem was circumvented by calculation of several short spectra. A test of the validity of this technique is described below. 3.4. Fourier transformation We used the fast Fourier transform routine from the NAG-library [ 18 1. Furthermore, we had to process the “raw” Fourier transform. Lombardi and coworkers [ 131 have already pointed out that the Fourier transform of a spectrum shows very strong fluctuations which they compare with the well-known speckle phenomenon of laser light shining on a rough surface. For the Fourier transforms shown in this paper we employed the following smoothing techniques: ( 1) Ensemble averaging of the mod squares. This is the best method. Due to the enormous amount of CPU time required for this method we could apply it only for simulations with small matrices. (2) Averaging of the mod square over some finite interval At. To continue the smoothed curves also to values oft < f At we exploited the time reflection symmetry of the mod square of the Fourier transform. To discuss method (2 ), we write the mod square of the Fourier transform in the form I
dbexp(ibt)
I
dBI(B+ fb) I(B- fb) .
(9)
The averaging necessary to obtain smooth fumtions is contained in eq. (9) by the integration over B. If this integration extends over the entire axis, then an ergodicity argument can be used to show the equality of eq. (9) with the ensemble average, and to infer the smoothness of Ic( t) 1’. Therefore, the fluctuations actually found derive from the finite dimension N of the matrices used in the simulation. The smoothing procedure described above is not well-founded theoretically. We have checked its validity by comparing
10 -
c-4 -
:5-u
I
0
1
I
2 t
Fig. 3. Comparison of differently smoothed mod squares of the Fourier transforms: the solid zigzagged Fourier transform results from averaging 10 independent Fourier transforms, after smoothing over 20 points the smooth solid line was obtained. The dashed line shows the result after smoothing a single mod square of the Fourier transform over 100 points.
the result with a combination of smoothing plus ensemble averaging for a small ensemble. The result is shown in fig. 3. This and similar checks have convinced us of the validity of the procedure. Prior to the Fourier transformation, we normalized the spectrum by a constant shift of the intensity in such a way that the mean intensity vanishes. This procedure affects only the value of the Fourier transform at t=O, which becomes zero. We believe that this procedure is very reasonable because it avoids the identical correlation contribution at the origin mentioned in ref. [ 13 1. 3.5. Consistency checks To check the basic routines, we performed the following tests. We calculated the ensemble-averaged intensity 1S( E= 0, B) - 11’ and compared it with the analytical result obtained by using supersymmetry methods [ 161. The generation of the random matrix and the normalization of the level density were checked by calculating, after unfolding, the DysonMetha two-level cluster function 1 - Y,(r) for a GOE Hamiltonian N with finite dimension N. To check our procedure for the calculation of the S-matrix in eq. ( 6 ) , we generated 1 - Y,(r) by calculating the product of the traces of [E, -I-Z+ ic ] --I and of [ I$ - H- ie ] - ’ , for small values of e and large N. Fig.
il. Hartmann et al. /Correlations in anticrossing spectra and seatiering theory
316
4 shows the result for N= 50 versus r= ( E2 - El ) /D.
The agreement with the analytical result is consistent with the finite value of c=O.OlD used in the calculation. Fig. 5 shows tests of the concatination technique. The solid (dashed) line shows the Fourier transform obtained from 6 spectra with N=48 anticrossings ( 13 spectra with N= 18 anticrossings, respectiveiy ). No significant difference is seen.
I
I
I
I
I
-4
-2
0
2
4
f
Fig. 4. Test of the two-level correlation function 1 - Y,(r) on the unfolded scale r= (S, -E2)/Lk The zigzagged line is the numerical result for an ima~na~ part rdl.0113, The smooth line is the
8
1
I
1
I
I
I
Fig.5. Test of the ~n~teMtio~ technique. The solid (dashed) line is the mod square of the Fourier transform obtained from 6 spectra with 48 ( 13 spectra with 18) anticrossings calculated using a GOE Hamiltonian H. To show the significance of the correlation hole the curve generated by using a diagonal H with Poissonian level statistics is also shown as dotted line.
4. Overlapping anticmsings In the present section, we investigate the transition from the regime of isolated anticrossings (r* D) to that of strongly overlapping (Q L)) anticrossin~. We are especially interested in the fate of the correlation hole, the existence of which was demonstrated for isolated anticrossings by a phenomenologi~l argument in section 2. Our choice of parameters is that of section 3 with V, w ys= y,=: 1 and V, and y$” independent of p. Hence, all anticrossings have the same depth fZoand the same width I’= 4 Vst,at least for parameter values where they are isolated. Overlapping anticrossings are generated by increasing Vst,keeping D fixed. Without analytical justification we use r=4 V,, also in the domain of overlapping anticrossings. The upper half of fig. 6 shows a typical intensity spectrum in the regime of strongly overlapping anticrossings. Only six dips are seen, although the spectrum is generated by about 30 anticrossings. The dips can, therefore, no be assigned to single anticrossings, but are generated by the supe~sition of several neighbouring anticrossings and should be labeled jlwtuations. (They correspond to the Ericson tluctuations seen in low-energy nuclear reactions [ 17 1. ) We note that the mean intensity reduction is 0.14& although an isolated anticrossing has a depth of 0.510. The low intensity reduction of 0.141, is caused by the high density of anticrossings. It is obviously impossible to deduce an eigenvalue statistic from this kind of fluctuation pattern. But the correlation hole in 1c( t ) I* survives: it is visible with width s 0.04 in the curve corresponding to T/D=: 8 in the lower part of fig. 6. Two more curves are shown in the lower part of fig. 6, corresponding to r/D= I.6 and 0.1, respectively. We observe that the width of the correlation hole increases with decreasing r/D. This is consistent with eq. (8), although this equation holds only for isoiated resonances: The factor 1-b2( t) increases monotonically with t. Nitoreover, bz ( t ) is independent of T/D. The local extremum of the product of 1-b2( t > and the exponential is shifted to smaller values oft when r/D is increased. The curves shown in fig. 6 were obtained by smoothing a single Fourier transform. This is why they do not pass through zero for t=O. Use of the ensemble average would have resulted in more signifi-
317
U. Hartmann et al. /Correlations in anticrossingspectraand scatteringtheory
0.0
Fig. 6. In the upperpart a smallsectionof an anticrossingspectrum from the strongly overlapping regime ( V,=200, D= 100, T/D=8) is shown. The lower part contains three mod squares of Fourier transforms ranging from the regime of well resolved to strongly overlapping anticrossings: The dashed line corresponds to V,,=25, LklOOO, T/D=O.l, the dotted line to V,,=400, D= 1000, r/D= 1.6, and the solid line to V,,=ZOO,D= 100, r/ D=8. Every spectrum contains 288 anticrossings. For better comparison, the curves are scaled to the same height.
cant correlation holes, particularly for the large ratios of r/D. However, in the regime of strongly overlapping resonances long sequences of anticrossings had to be used, making ensemble averaging numerically infeasible. To test the significance of the correlation hole for r/D= 8, we repeated the calculation with a diagonal Hamiltonian H for the triplet states with eigenvalues obeying Poisson statistics. Fig. 7 shows the two curves, for GOE and Poisson statistics, cf. also fig. 5. The curve generated with the Poisson distribution does not show any indication of a correlation hole. We conclude that even for V:, >,y.y, and for strongly overlapping anticrossings, the correlation hole remains a significant fingerprint for GOE level statistics.
5. Non-overlapping anticrossing with variable widths and depths
The mod squares of the Fourier transforms shown
0.1
0.2
,
0.3
Fig. 7. To test the significance of the correlation hole in the regime of strongly overlapping anticrossings, the mod square of the Fourier transform for T/D=8 shown in fig. 6 (dotted line) is compared with corresponding quantity generated by using a diagonal H with Poissonian level spacings (solid line).
in the previous section were calculated for anticrossings with constant matrix elements VPand r$’ (both sets independent of the index p) and, therefore, with constant widths and depths. This assumption is not realistic. This is why in the present section we investigate the fate of the correlation hole when the abovementioned restriction is lifted. We confine ourselves to non-overlapping anticrossings because this is the situation of experimental interest. We focus attention on the case ySy,5 4Vz,. From a naive point of view, we expect that the less uniform the line shapes are, the more the correlation hole is smeared out. This expectation is supported by arguments given in ref. [ 9 1, and by the analytical results of ref. [ lo]. It was shown that for constant line width rbut varying depth h the correlation hole is filled in. More precisely, in eq. (8 ) one has to substitute 1-b,(t)~(h2)-(h)2bz(t).
(10)
At t = 0, the function (c( 1) 12 in appropriate units no longer assumes the value zero but rather the value l~(0)1~=1-(h)~/(h~), which obeys 0<(c(0)12 dl. When we started the simulations, predictions on the influence of variable widths on the fate of the correlation hole were not available. But in the experiments by Lombardi and co-workers [ 81 which had stimulated our investigations all anticrossings had the
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U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
same depths but different widths. For this reason, we extended the line of reasoning from ref. [ 91. We state the results (which were derived in the framework of a rather crude model) without going into details. For both Lorentzian and Gaussian line shapes, we found the substitution rule 1-b,(t)+(
(hr)2)-(hT)*b,(t).
(11)
This rule suggests that the suppression of the correlation hole is determined by the first two moments of the area of the anticrossing line shapes. Result ( 11) is intuitively appealing; we believe that it applies whenever the line shape f( E, r, h ) obeys the scaling rulef(E, h,r)=hf(E/r, 1, 1). To test the validity of rule ( 11), we have performed a numerical simulation for the following three cases. (An intensity spectrum for each case is shown in fig. 8.) ( 1) V,= 25, D= 1000, and 7::’ and VPGaussian distributed with zero mean value and variances given by yt and Vi, respectively. This is the generic case: both the widths and the depths of the anticrossings fluctuate. (2) Same as ( 1) with the exception of y$’ which is now constant. Most anticrossings have now a depth of about f&,. This corresponds more or less to the experiments of Lombardi and co-workers. However, in their experiments all anticrossings have the same depth and the widths do not seem to follow a Gaussian distribution law [ 121. (3) V,,=O.2, D=50, yi/* constant and equal to unity, and V,, Gaussian distributed with variance V:, . This is close to the limit ysy,3 4 V:, which was treated analytically in ref. [ 10 1. Please note the suppression of the origin at the ordinate in the lower part of fig. 8. All anticrossings have the same width of y, + y, but have different depths. To allow for a direct comparison with the calculation in ref. [ lo] we replaced in eq. ( 7 ) the dyadic product by yt1N In ref. [ 10 ] this was done for technical reasons. The mod squares of the Fourier transforms corresponding to these three cases are shown in fig. 9. These values were obtained by averaging over an ensemble of 10 Fourier transforms, and by smoothing over 20 data points. To facilitate the analysis, we have plotted log 1c(t) I *. The Lorentzian form of the anticrossings leads via eq. ( 1) to a straight line with slope (--2x(r/D)tloge). The correlation hole manifests
1.0-
,” ; 0.5 l-i case 2 01
l.O-
I_
0.8 -
I
casa 3 B/U
Fig. 8. Three spectra obeying different uniformness. In cases ( 1) and(2) V,=25,D=lOOOischosen,incase(3) VS=0.2,D=50. In case ( 1) the V,, and the y$* are random, in case (2) the V,, are random but the yi/* arefixed,incase(3)r,issety,l,proportional to the unit matrix. Note that the intensity scale is different in case ( 3 ) .
itself as a deviation from the straight line which is expected to occur for 0 < t < 1. In the generic case (upper part of fig. 9), the correlation hole is barely visible; without ensemble averaging, it disappears altogether. The implication is that in experiments on non-overlapping anticrossings where both depth and width vary, the detection of the correlation hole becomes very difficult, and requires very long sequences of anticrossings (corresponding to our ensemble average). The correlation hole is very well developed for case (2 ), which is close to the situation met in the experiments by Lombardi and co-workers. In the latter case, all anticrossings have exactly the same depth, and the
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U. Hartmann et al. /Cowelations in anticrossingspectra and scattering theory
Table 1 Comparison of the Monte Carlo simulated and the theoretical values for the suppression of the correlation hole for the mod square Fourier transforms shown in fig, 9. The error of the simulation values (labeled mc ) is about 10%. Case
((hO’/<
1 2 3
38% 51% 41%
(hW> ),
(<~r>2/((~r)2>)Sk. 40.5% 63.5% 39.4%
pressing r,, and h, as functions of V,, and y$’ via eqs. (2). Given the large statistical uncertainty of the Monte Carlo simulations, the agreement is perfectly acceptable. To improve our statistics significantly, we estimate that an ensemble of about 100 members would be needed. For the comparison in case (3) (where recourse to eq. (2) is not needed and the theoretical results of ref. [ lo] can be used directly), we comment that our value of 41% does not contradict the analytical result of 33.3% of ref. [ 10 ] because our simulation, with ySyt= 1 and 2 V,= 0.4, does not fully meet the condition fi =B2 1V, I used in ref. [ lo]. -20
I 1
I 2
I 3
I 4,5
c_
Fig. 9. The mod squares of the Fourier transforms of the three cases of fig. 8.
correlation hole is even better visible than in our simulations. Because of our different choice of scale the correlation hole in case (3 ) looks larger than it is; this choice is also responsible for the fact that the statistics of the data points looks worse than in cases ( 1) and (2). To make a comparison with theoretical expectations, we have deduced quantitative information from the three simulations just described. The depth of the correlation hole was determined by an extrapolation of the straight-line sections of the curves in fig. 9 to the point t=O; this value was used to determine the ratio (hr) ‘/( (hr)‘), using the substitution rule ( 11) . The results are shown in the second column of table 1. They must be compared with the “theoretical” values shown in the last column, obtained by ex-
6. Discussion We the help of our simulations, we extended the investigations on the correlation hole performed in ref. [ 10 ] into several domains that have not been analytically accessible so far. On the one hand, we consider the regime y.y, ;5 V: mainly for overlapping but also for non-overlapping anticrossings with constant widths and depths (section 4). In the latter case (nonoverlapping anticrossings), the existence of the correlation hole confirmed the phenomenological model discussed in section 2. It is satisfactory that we found the correlation hole to survive intact as a fingerprint of GGE level statistics even in the domain of overlap ping anticrossings. A second extension was that into the regime YJ, 6 V: and non-overlapping anticrossings with variable depths and/or widths. This extension demonstrated the importance of the uniformity of the spectrum for the visibility of the correlation hole. Our results show that the depth of the correlation hole is given by the ratio ( hr) 2/ ( ( hQ2 ) , while
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the width of the correlation hole depends on the line shape of the anticrossings and on the ratio T/D. For isolated anticrossings, this width is almost unity and independent of the line shape. Our experience indicates that correlation holes with a depth less than 0.4 I or so are very difficult to identify unambiguously: we had to average over a set of 10 independent calculations. Using a single (and short) Fourier transform does not work because the statistical fluctuations are too large, and the correlation hole is washed out. In conclusion we find that in all cases investigated the existence of the correlation hole is a safe fingerprint of GOE statistics, while its absence does not imply regular motion. Our investigations in section 5 show that the correlation hole is indeed mainly a pure spectral GOE effect: although the three spectra in fig. 8 look very different, the corresponding correlation holes in fig. 9 are rather similar. The differences due to the distributions of the V, and y{,/’ are important but in most cases they do not change the picture entirely. Hence our investigations support and illustrate the observation made in ref. [ lo] that the (unrealistic) assumption ySyt>>4 I’:* made in ref. [ lo] does not have too negative an impact on the results and leads to a reasonable description of the correlation hole. From the methodical point of view, the concatenation technique has proved very useful for generating enough data in a reasonable time. In the regime of isolated anticrossings, even short spectra containing only about 20 anticrossings can be used. This technique might be of interest also for experimentalists who can measure a large number of short spectra. All spectra must, of course, be normalized to a uniform level density prior to concatenation. Our investigation leaves open a few questions, some unfortunately related to the actual experiments. A better test of rule ( 11) in the regime of isolated anticrossings, and a check of its extension to overlapping resonances, is desirable. Likewise, it is necessary to discuss triplet states with more than one symmetry class. Refs. [ 9, lo] suggests that only the width of the correlation hole is decreased by superposing several classes. But this problem requires considerably more
CPU time than used in the present study. For this problem the density matrix formalism does not offer a preferable alternative because in the S-matrix formalism the dimension of the matrix is linear in N while the number of independent elements of the density matrix is proportional to N *.
Acknowledgement
We thank R. Jost, M. Lombardi, and J.P. Pique for stimulating discussions and W. Brllckner, H.-W. Heyng, and T. Kihm for providing us with lots of CPU time on their computers.
References [ 1] G. Hackett, R. Werbin and J. Rainwater, Phys. Rev. C I7 (1978) 43. [2] R.U. Haq, A. Pandey and 0. Bohigas, Phys. Rev. Letters 48 (1982) 1086. [3] N. Rosenzweig and C.E. Porter, Phys. Rev. 126 ( 1960) 1698. [ 4 ] H.S. Camarda and P.D. Geotgopulos, Phys. Rev. Letters 50 (1983) 492. [ 51 E. Haller, H. Kiippel and L.S. Cederbaum, Chem. Phys. Letters 101 (1983) 215. [ 61 S. Mukamel, J. Sue and A. Pandey, Chem. Phys. Letters 105 (1984) 134. [ 71 T. Zimmennann, H. Kijppel, L.S. Cederbaum, G. Persch and W. Demtrdder, Phys. Rev. Letters 61 (1988) 3. (81 R. Jost and M. Lombardi, Lecture Notes in Physics, Vol. 263 (Springer, Berlin) p. 72, and references therein. [ 91 L. Leviandier, M. Lombardi, R. Jost, and J.P. Pique, Phys. Rev. Letters 56 (1986) 2449. [ 10 ] T. Guhr and H.A. Weidenmiiller, Chem. Phys. 146 ( 1990) 21. [ 111 H. Wieder and T.G. Eck, Phys. Rev. 153 (1967) 103. [ 121 E. Pebay-Peyroula, R. Jost, M. Lombardi and J.P. Pique, Chem. Phys. 106 ( 1986) 243, and references therein. [ 13 ] R. Jost and M. Lombardi, ET. Analysis, preprint. [ 141 M. Lombardi, Grenoble, private communication. [ 15 ] M.L. Metha, Random Matrices (Academic Press New York, 1967). [ 161 J.J.M. Verbaarschot, H.A. Weidenmttller and M.R. Zimbauer, Phys. Rep. 129 (1985) 367. [ 171 A. Bohr and B.R. Mottelson, Nuclear Structure (W.A. Benjamin, New York, 1969). [ 18 ] The NAG Fortran Library ManuaJ-mark 13 (The Numerical Algorithms Group Limited, Oxford, 1988).
Correlations in anticrossing spectra and scattering theory: numerical simulations U. Hartmann, H.A. Weidenmiiller and T. Guhr Max-Planck-Institutfiir Kernphysik, W-6900 Heidelberg, Germany Received 26 September 1990
For the interpretation of experimental results of anticrossing spectroscopy in molecules, the results of numerical simulations are presented. We have focussed our attention on the visibility of the correlation hole. In the formulation of the model, we combine scattering theory and a random-matrix model.
1. Introduction The identification of experimental indicators of chaotic motion in quantum systems is a problem of considerable interest. The standard procedure consists in the measurement of the energies of a sequence of levels having the same quantum numbers, with the aim of determining the eigenvalue statistics. The procedure is based on the fact that for regular (or integrable) systems, the eigenvalue distribution is Poissonian, while for time-reversal invariant systems, which are chaotic in the classical limit, the distribution is conjectured to coincide locally with the distribution of eigenvalues of the Gaussian orthogonal ensemble of random matrices (the GOE). Historically, the first evidence for GOE level statistics came from data on low-energy neutron scattering on heavy nuclei, cf. refs. [ 1,2 ] and earlier work cited therein. Later, evidence for chaos was also found in atomic [ 3,4] and molecular [ 5-71 spectra. Our present investigation was triggered by the work of the Grenoble group [ 8 ] on the organic molecule methylglyoxal. This group observed the resonance fluorescence yield of laser-excited molecules as a function of an external magnetic field. The laser light excites a singlet state at fixed excitation energy, independent of the strength of the magnetic field. As the strength of the magnetic field is changed, the m = f 1 components of the nearby triplet states are shifted past the singlet state, giving rise to Landau-Zener anticross-
ings. Each anticrossing causes a resonance-like reduction of the resonance-fluorescence intensity with a Lorentzian line shape. We became interested in methylglyoxal because here several thousand anticrossings were observed and, more importantly, the Lorentzians of neighbouring anticrossings overlap fairly strong, making it impossible to perform a detailed analysis of each anticrossing and to establish thereby the eigenvalue statistics as a signal for chaotic motion. We were intrigued by the idea proposed by the Grenoble group [ 9 ] : in the autocorrelation function of the Fourier transform of the fluorescence yield, they found a “correlation hole” at small values of the displacement parameter t. This correlation hole was interpreted as being due to long-range GGE-type level correlations and, thereby, as a fingerprint of chaos. The possible existence of such a fingerprint, observable without the need to perform a painstaking levelby-level analysis of the spectrum (which, in the case of methylglyoxal, would not even be feasible) is obviously of general interest and far exceeds the special case of methylglyoxal. The question is: can we show theoretically that the correlation hole is indeed a manifestation of chaos? This question was addressed in a preceding paper by two of the present authors [ 10 1. The investigation was carried out analytically in the small coupling limit which is based on two assumptions: (i) The rms matrix element VSldescribing the dy-
0301-0104/91/S 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
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U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
namical coupling between the singlet state and the triplet states was assumed to be small compared to the mean spacing D of the triplet states. This assumption permits an expansion in powers of V,,/D which can be terminated after a few terms. The assumption is realistic for methylglyoxal because the mean level spacing D that is relevant is the mean level spacing between triplet states of the same symmetry [ lo]. (ii) With ys the width for photoemission of the singlet state and yt the width for photoemission from the triplet states, it was assumed [ lo] that YJ,> 4V$. This assumption, however, does not apply to methylglyoxal, severely limiting the significance of the results of ref. [ lo] in the case of this molecule. To investigate the domain YJ, 5 Vz,, we had to resort to Monte Carlo simulations. At the same time, we considered it useful to investigate the domain of overlapping resonances, which also is not accessible analytically. The results of these simulations are reported in the present paper. We point out that the terms non-overlapping resonances (T/D +c 1) and overlapping resonances (r/ D >, 1)) where r is the total width, refer to pure spectra where all levels have the same symmetry. Spectra of molecules like methylglyoxal are usually superpositions of several pure spectra. Hence those spectra may look like consisting of overlapping resonances although they are superpositions of spectra of nonoverlapping resonances, cf. ref. [ 10 1. The model, the anticrossing spectroscopy, the scattering matrix formalism, and the experimental results were presented in detail in ref. [ 10 1. To make the present paper sufficiently self-contained, we confine ourselves to a rather compact presentation of these topics ( section 2 ) . In section 3, we describe how the Fourier transforms of anticrossing spectra were simulated, and in section 4 we present simulations for the regime of overlapping resonances ( V,,? D). In section 5, we investigate the influence of the distribution of widths and depths of the anticrossings on the Fourier transform of non-overlapping resonances, and we compare our results with those of ref. [ lo]. The conclusions are formulated in section 6.
was used to excite selectively a rotationless singlet state Is) with excitation energy E,x 25000 cm-‘. Via the singlet-triplet interaction Vst, this level (which we refer to as a doorway state) is mixed with the triplet states ltp), PL= 1, 2, .... N where Nz+ 1 (fig. 1). As the external magnetic field is changed, the energy EG of the m,= + 1 component of a triplet state may get close to the singlet state (so that IEs-E; I S V,,), and an anticrossing occurs. The anticrossing mechanism is quite simple: Because of the mixing of the two states, some transition strength y is shifted into the non-recorded transition from the triplet state, and is thereby lost for the resonance fluorescence yield, causing a resonance-like dip in the latter. We note that for purposes of anticrossing spectroscopy, the magnetic field plays the role of an excitation energy scanning the triple states. In the case of isolated anticrossings, the line shape isgivenby [11] Z(B) =I0
’
1-
2V:&,lY, (E, ~B-Es)2+2V~,ys,(y;’
+y;‘)+yzt
(1) where Yst is the total width of the mixed state. We measure the magnetic field B in energy units. The background intensity I0 is given by np /yr with 1: the singlet excitation rate. For experiments performed at sufficiently low pressure, we have Ystz 4 ( y. + Yt); this relation will be used in the following. Eq. ( 1) was confirmed by very precise measure-
Itp>
2. The model In the experiment
of the Grenoble
group, a laser
Fig. I. Scattering model for the laser experiment on methylglyoxal. Dynamic couplings are drawn as wavy lines.
U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
ments [ 121. Inspection shows that an anticrossing causes a dip in the fluorescence yield of Lorentzian shape with width rand depth h given by r=2J4Vf~y:~/(y,y,)+y:~, h=2A, VZt(~/2)-2~stl~s.
(2)
For non-overlapping resonances and in the limit fi Z+2 I Vs, 1 considered in ref. [ lo], the expressions simplify to
(3)
k, = 210Et/ (LYE1 .
In the case of the experiment of Lombardi and coworkers ySz y,x 0.3 MHz e 1V,, 1 and thus rexp = 4 I Kt I YstlJrsut h,x,=~ortl(Ys+Yt)
3
.
(4)
Please note that in the last case the depth is independent of Vs,. In the glyoxal measurement (resolved anticrossings) all observed anticrossings had the same depth [ 131. This leads to the conclusion that all triplet states have the same radiation width y, which was confirmed by life-time rf experiments [ 14 1. Eq. ( 1) is easily derived using the density matrix formalism but can also be obtained from the scattering matrix formalism [ lo]. In the latter case the measured intensity in case of N triplet states is given by [lOI
I(E)=&c j: IS(E,B)-1]2dE,
(5)
.”
-03
where S(E,B)=l-iy,[G-‘1,
(6)
and G=EIN+,
_
Es--iys
H-
VT
tir, +BIN > .
(7)
Here V is an N-component column vector describing the coupling of the doorway state with any of the triplet states I tp); lN is the unit matrix in Ndimensions, and r, is an NX N dyadic matrix given by y$‘y,‘/‘. The ytllare the y decay widths of the triplet states Itp) , and B is the strength of the magnetic field in appropriate units. Finally, His the NX N Hamiltonian matrix which describes the mutual interaction of the N
313
triplet states. In line with the arguments presented in section 1, we assume below that H is an element of the GOE of dimension N. In eq. (7), we have taken account of the only one spin projection (m, = + 1 or m,= - 1) although both spin projections contribute to the experimental result. We do so because the Monte Carlo simulation becomes forbiddingly formidable if both projections are included. The analytical results of ref. [ lo] justify our procedure: the fluctuations due to the mixing of different spin projections are not generic in the sense that the mod square of their Fourier transform is proportional to the delta function. For non-overlapping resonances, the origin of the correlation hole mentioned in the introduction is easily seen in a phenomenological model [ 9,13 1. We suppose that an intensity spectrum consists of an infinite sequence of isolated anticrossings, each with shape given by eq. (1 ), with constant width r and constant depth h, the spacings being those of the eigenvalues of the Hamiltonian H. (The finiteness of an actual spectrum does not affect the argument, see ref. [ lo].) Then the Fourier transform of the intensity autocorrelation function, henceforth denoted by Ic(t)12,hastheform Ic(t)~2~[d(t)+1-b2(t)]exp(-2ntI’/D).
(8)
Here, b,(t) is the Fourier transform of the two-level cluster function Y2(r) first introduced by Dyson and Metha [ 15 1. The result in eq. ( 8 ) is the product of two terms, one being the Fourier transform of the shape of the anticrossing [ exp ( - Zxtr/D) 1, the other accounting for correlations between triplet levels. For a Poissonian triplet spectrum, both Y2(r) and b2( t) vanish, while for a spectrum with GOE level statistics both functions behave as shown analytically in ref. [ 15 I.. For the GOE case the analytical result for l-b2(t) is plotted in fig. 2. In particular l-b,(t) displays a dip at the origin which becomes the correlation hole observed in refs. [ 9,131. We note that this phenomenological way of reasoning is independent of the value of the ratio y,y,/4V$.
3. Numerical simulation In this section the method used to generate the mod squares of the Fourier transforms of anticrossing
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U. Hartmann et al. /Correlations in anticrossingspectra and scattering theory
(ensemble) average over the mod square of the Fourier transform, i.e. the quantity 1c(t) 12, or smooth the mod square of the Fourier transform (cf. section 3.4). 3.2. Generation of the spectrum
00 0
1
2
t
3
Fig. 2. Plot of 1-b,(t) for the GOE, b2 (1) is the Fourier transform of the two-level cluster function Y,(r).
spectra is described. At the end of this section we describe the checks of the programs. 3.1. Overview The random matrix H (see eq. (7) ) used for the simulations was drawn from the GOE. Forced by computer-time requirements we had to restrict the simulation to the case of one pure sequence. This simplification goes beyond the restriction to a single value of the spin projection which we introduced and discussed in and below eq. (7). Indeed, it is known that the sequence of triplet states with fixed spin projection consists of states belonging to several symmetry classes. A realistic simulation would have to use in eq. (7) a Hamiltonian H having a block structure, each block simulating triplet states with fixed symmetry and represented by a separate GOE. We return to this point in the discussion (section 6). The procedure to obtain the simulated Fourier transforms is very similar to the way the experimentalists obtained their Fourier transforms. The main steps are as follows: ( 1) Generate a GOE matrix H for the N triplet states. (2) Calculate for every desired value of the magnetic field B the intensity ( fluorescense yield) Z(B) from eqs. (5)-(7). ( 3 ) Calculate the mod square of the Fourier transform of the intensity Z(B) generated in step 2. (4) Repeat steps ( 1 )-( 3), thereby generating the
In an arbitrarily chosen energy unit, we took ES= 0, yS:=y,= 1, and all y$’ = 1. The last choice was motivated by the observation by Lambardi and co-workers [ 14 ] that in glyoxal (non-overlapping anticrossings), all anticrossings had the same widths for the triplet states. The restrictions inherent to this choice are removed in section 5. We also took all matrix elements VP of the singlet-triplet coupling equal to a fixed value V,,; again, this restriction is lifted in section 5. Finally, our choice yS= yt was motivated by the experimental situation in methylglyoxal [ 141. With the choices just made, the results of the calculation depend on two parameters, i.e., the singlet-triplet coupling strength VStand the mean level spacing D. To generate efficiently spectra of constant level density, we diagonalized H and unfolded the eigenvalue spectrum, using the semicircle law for the mean level density. This diagonalization has to be performed only once for every choice of H, and every spectrum. After the diagonalization, spectra having different distributions of widths and depths can be generated from the same matrix H by using different distributions for V,, and y:/‘. The diagonalization of H affects both the triplet-singlet coupling matrix elements VP, and the width amplitudes r$‘. To see this, we write H in the form OAOT, where 2 is the diagonal matrix of eigenvalues, and 0 the diagonalizing orthogonal matrix. In the eigenvalue representation, VP and y$’ are transformed into (OV), and ( OY:‘~),, respectively. For NB 1, the elements of 0 are uncorrelated Gaussian distributed random variables, and the quantities (OV), and (Oy:‘2), therefore have probability distributions which differ from those of the original variables VP and r$‘. 3.3. Energy integration To obtain the resonance fluorescence yield Z(B), it is necessary to integrate over the energy E of the laser beam; this integration has to be carried out for every value B of the magnetic field (cf. eq. ( 5 ) ) and con-
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U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
stitutes the main difficulty of the numerical simulation, since the evaluation of the integrand involves the inversion of a matrix of dimension N+ 1. The energy integration was performed analytically by diagonalizing [ G-El N+, ] (which is independent of the energy, cf. eq. (7) ) and by using the residue theorem. The numerical effort (number of floating point operations) for the diagonalization and inversion of the matrix [G-El,,,, , ] --I scales as N’ and inhibits the generation of very long sequences. This problem was circumvented by calculation of several short spectra. A test of the validity of this technique is described below. 3.4. Fourier transformation We used the fast Fourier transform routine from the NAG-library [ 18 1. Furthermore, we had to process the “raw” Fourier transform. Lombardi and coworkers [ 131 have already pointed out that the Fourier transform of a spectrum shows very strong fluctuations which they compare with the well-known speckle phenomenon of laser light shining on a rough surface. For the Fourier transforms shown in this paper we employed the following smoothing techniques: ( 1) Ensemble averaging of the mod squares. This is the best method. Due to the enormous amount of CPU time required for this method we could apply it only for simulations with small matrices. (2) Averaging of the mod square over some finite interval At. To continue the smoothed curves also to values oft < f At we exploited the time reflection symmetry of the mod square of the Fourier transform. To discuss method (2 ), we write the mod square of the Fourier transform in the form I
dbexp(ibt)
I
dBI(B+ fb) I(B- fb) .
(9)
The averaging necessary to obtain smooth fumtions is contained in eq. (9) by the integration over B. If this integration extends over the entire axis, then an ergodicity argument can be used to show the equality of eq. (9) with the ensemble average, and to infer the smoothness of Ic( t) 1’. Therefore, the fluctuations actually found derive from the finite dimension N of the matrices used in the simulation. The smoothing procedure described above is not well-founded theoretically. We have checked its validity by comparing
10 -
c-4 -
:5-u
I
0
1
I
2 t
Fig. 3. Comparison of differently smoothed mod squares of the Fourier transforms: the solid zigzagged Fourier transform results from averaging 10 independent Fourier transforms, after smoothing over 20 points the smooth solid line was obtained. The dashed line shows the result after smoothing a single mod square of the Fourier transform over 100 points.
the result with a combination of smoothing plus ensemble averaging for a small ensemble. The result is shown in fig. 3. This and similar checks have convinced us of the validity of the procedure. Prior to the Fourier transformation, we normalized the spectrum by a constant shift of the intensity in such a way that the mean intensity vanishes. This procedure affects only the value of the Fourier transform at t=O, which becomes zero. We believe that this procedure is very reasonable because it avoids the identical correlation contribution at the origin mentioned in ref. [ 13 1. 3.5. Consistency checks To check the basic routines, we performed the following tests. We calculated the ensemble-averaged intensity 1S( E= 0, B) - 11’ and compared it with the analytical result obtained by using supersymmetry methods [ 161. The generation of the random matrix and the normalization of the level density were checked by calculating, after unfolding, the DysonMetha two-level cluster function 1 - Y,(r) for a GOE Hamiltonian N with finite dimension N. To check our procedure for the calculation of the S-matrix in eq. ( 6 ) , we generated 1 - Y,(r) by calculating the product of the traces of [E, -I-Z+ ic ] --I and of [ I$ - H- ie ] - ’ , for small values of e and large N. Fig.
il. Hartmann et al. /Correlations in anticrossing spectra and seatiering theory
316
4 shows the result for N= 50 versus r= ( E2 - El ) /D.
The agreement with the analytical result is consistent with the finite value of c=O.OlD used in the calculation. Fig. 5 shows tests of the concatination technique. The solid (dashed) line shows the Fourier transform obtained from 6 spectra with N=48 anticrossings ( 13 spectra with N= 18 anticrossings, respectiveiy ). No significant difference is seen.
I
I
I
I
I
-4
-2
0
2
4
f
Fig. 4. Test of the two-level correlation function 1 - Y,(r) on the unfolded scale r= (S, -E2)/Lk The zigzagged line is the numerical result for an ima~na~ part rdl.0113, The smooth line is the
8
1
I
1
I
I
I
Fig.5. Test of the ~n~teMtio~ technique. The solid (dashed) line is the mod square of the Fourier transform obtained from 6 spectra with 48 ( 13 spectra with 18) anticrossings calculated using a GOE Hamiltonian H. To show the significance of the correlation hole the curve generated by using a diagonal H with Poissonian level statistics is also shown as dotted line.
4. Overlapping anticmsings In the present section, we investigate the transition from the regime of isolated anticrossings (r* D) to that of strongly overlapping (Q L)) anticrossin~. We are especially interested in the fate of the correlation hole, the existence of which was demonstrated for isolated anticrossings by a phenomenologi~l argument in section 2. Our choice of parameters is that of section 3 with V, w ys= y,=: 1 and V, and y$” independent of p. Hence, all anticrossings have the same depth fZoand the same width I’= 4 Vst,at least for parameter values where they are isolated. Overlapping anticrossings are generated by increasing Vst,keeping D fixed. Without analytical justification we use r=4 V,, also in the domain of overlapping anticrossings. The upper half of fig. 6 shows a typical intensity spectrum in the regime of strongly overlapping anticrossings. Only six dips are seen, although the spectrum is generated by about 30 anticrossings. The dips can, therefore, no be assigned to single anticrossings, but are generated by the supe~sition of several neighbouring anticrossings and should be labeled jlwtuations. (They correspond to the Ericson tluctuations seen in low-energy nuclear reactions [ 17 1. ) We note that the mean intensity reduction is 0.14& although an isolated anticrossing has a depth of 0.510. The low intensity reduction of 0.141, is caused by the high density of anticrossings. It is obviously impossible to deduce an eigenvalue statistic from this kind of fluctuation pattern. But the correlation hole in 1c( t ) I* survives: it is visible with width s 0.04 in the curve corresponding to T/D=: 8 in the lower part of fig. 6. Two more curves are shown in the lower part of fig. 6, corresponding to r/D= I.6 and 0.1, respectively. We observe that the width of the correlation hole increases with decreasing r/D. This is consistent with eq. (8), although this equation holds only for isoiated resonances: The factor 1-b2( t) increases monotonically with t. Nitoreover, bz ( t ) is independent of T/D. The local extremum of the product of 1-b2( t > and the exponential is shifted to smaller values oft when r/D is increased. The curves shown in fig. 6 were obtained by smoothing a single Fourier transform. This is why they do not pass through zero for t=O. Use of the ensemble average would have resulted in more signifi-
317
U. Hartmann et al. /Correlations in anticrossingspectraand scatteringtheory
0.0
Fig. 6. In the upperpart a smallsectionof an anticrossingspectrum from the strongly overlapping regime ( V,=200, D= 100, T/D=8) is shown. The lower part contains three mod squares of Fourier transforms ranging from the regime of well resolved to strongly overlapping anticrossings: The dashed line corresponds to V,,=25, LklOOO, T/D=O.l, the dotted line to V,,=400, D= 1000, r/D= 1.6, and the solid line to V,,=ZOO,D= 100, r/ D=8. Every spectrum contains 288 anticrossings. For better comparison, the curves are scaled to the same height.
cant correlation holes, particularly for the large ratios of r/D. However, in the regime of strongly overlapping resonances long sequences of anticrossings had to be used, making ensemble averaging numerically infeasible. To test the significance of the correlation hole for r/D= 8, we repeated the calculation with a diagonal Hamiltonian H for the triplet states with eigenvalues obeying Poisson statistics. Fig. 7 shows the two curves, for GOE and Poisson statistics, cf. also fig. 5. The curve generated with the Poisson distribution does not show any indication of a correlation hole. We conclude that even for V:, >,y.y, and for strongly overlapping anticrossings, the correlation hole remains a significant fingerprint for GOE level statistics.
5. Non-overlapping anticrossing with variable widths and depths
The mod squares of the Fourier transforms shown
0.1
0.2
,
0.3
Fig. 7. To test the significance of the correlation hole in the regime of strongly overlapping anticrossings, the mod square of the Fourier transform for T/D=8 shown in fig. 6 (dotted line) is compared with corresponding quantity generated by using a diagonal H with Poissonian level spacings (solid line).
in the previous section were calculated for anticrossings with constant matrix elements VPand r$’ (both sets independent of the index p) and, therefore, with constant widths and depths. This assumption is not realistic. This is why in the present section we investigate the fate of the correlation hole when the abovementioned restriction is lifted. We confine ourselves to non-overlapping anticrossings because this is the situation of experimental interest. We focus attention on the case ySy,5 4Vz,. From a naive point of view, we expect that the less uniform the line shapes are, the more the correlation hole is smeared out. This expectation is supported by arguments given in ref. [ 9 1, and by the analytical results of ref. [ lo]. It was shown that for constant line width rbut varying depth h the correlation hole is filled in. More precisely, in eq. (8 ) one has to substitute 1-b,(t)~(h2)-(h)2bz(t).
(10)
At t = 0, the function (c( 1) 12 in appropriate units no longer assumes the value zero but rather the value l~(0)1~=1-(h)~/(h~), which obeys 0<(c(0)12 dl. When we started the simulations, predictions on the influence of variable widths on the fate of the correlation hole were not available. But in the experiments by Lombardi and co-workers [ 81 which had stimulated our investigations all anticrossings had the
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U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
same depths but different widths. For this reason, we extended the line of reasoning from ref. [ 91. We state the results (which were derived in the framework of a rather crude model) without going into details. For both Lorentzian and Gaussian line shapes, we found the substitution rule 1-b,(t)+(
(hr)2)-(hT)*b,(t).
(11)
This rule suggests that the suppression of the correlation hole is determined by the first two moments of the area of the anticrossing line shapes. Result ( 11) is intuitively appealing; we believe that it applies whenever the line shape f( E, r, h ) obeys the scaling rulef(E, h,r)=hf(E/r, 1, 1). To test the validity of rule ( 11), we have performed a numerical simulation for the following three cases. (An intensity spectrum for each case is shown in fig. 8.) ( 1) V,= 25, D= 1000, and 7::’ and VPGaussian distributed with zero mean value and variances given by yt and Vi, respectively. This is the generic case: both the widths and the depths of the anticrossings fluctuate. (2) Same as ( 1) with the exception of y$’ which is now constant. Most anticrossings have now a depth of about f&,. This corresponds more or less to the experiments of Lombardi and co-workers. However, in their experiments all anticrossings have the same depth and the widths do not seem to follow a Gaussian distribution law [ 121. (3) V,,=O.2, D=50, yi/* constant and equal to unity, and V,, Gaussian distributed with variance V:, . This is close to the limit ysy,3 4 V:, which was treated analytically in ref. [ 10 1. Please note the suppression of the origin at the ordinate in the lower part of fig. 8. All anticrossings have the same width of y, + y, but have different depths. To allow for a direct comparison with the calculation in ref. [ lo] we replaced in eq. ( 7 ) the dyadic product by yt1N In ref. [ 10 ] this was done for technical reasons. The mod squares of the Fourier transforms corresponding to these three cases are shown in fig. 9. These values were obtained by averaging over an ensemble of 10 Fourier transforms, and by smoothing over 20 data points. To facilitate the analysis, we have plotted log 1c(t) I *. The Lorentzian form of the anticrossings leads via eq. ( 1) to a straight line with slope (--2x(r/D)tloge). The correlation hole manifests
1.0-
,” ; 0.5 l-i case 2 01
l.O-
I_
0.8 -
I
casa 3 B/U
Fig. 8. Three spectra obeying different uniformness. In cases ( 1) and(2) V,=25,D=lOOOischosen,incase(3) VS=0.2,D=50. In case ( 1) the V,, and the y$* are random, in case (2) the V,, are random but the yi/* arefixed,incase(3)r,issety,l,proportional to the unit matrix. Note that the intensity scale is different in case ( 3 ) .
itself as a deviation from the straight line which is expected to occur for 0 < t < 1. In the generic case (upper part of fig. 9), the correlation hole is barely visible; without ensemble averaging, it disappears altogether. The implication is that in experiments on non-overlapping anticrossings where both depth and width vary, the detection of the correlation hole becomes very difficult, and requires very long sequences of anticrossings (corresponding to our ensemble average). The correlation hole is very well developed for case (2 ), which is close to the situation met in the experiments by Lombardi and co-workers. In the latter case, all anticrossings have exactly the same depth, and the
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U. Hartmann et al. /Cowelations in anticrossingspectra and scattering theory
Table 1 Comparison of the Monte Carlo simulated and the theoretical values for the suppression of the correlation hole for the mod square Fourier transforms shown in fig, 9. The error of the simulation values (labeled mc ) is about 10%. Case
((hO’/<
1 2 3
38% 51% 41%
(hW> ),
(<~r>2/((~r)2>)Sk. 40.5% 63.5% 39.4%
pressing r,, and h, as functions of V,, and y$’ via eqs. (2). Given the large statistical uncertainty of the Monte Carlo simulations, the agreement is perfectly acceptable. To improve our statistics significantly, we estimate that an ensemble of about 100 members would be needed. For the comparison in case (3) (where recourse to eq. (2) is not needed and the theoretical results of ref. [ lo] can be used directly), we comment that our value of 41% does not contradict the analytical result of 33.3% of ref. [ 10 ] because our simulation, with ySyt= 1 and 2 V,= 0.4, does not fully meet the condition fi =B2 1V, I used in ref. [ lo]. -20
I 1
I 2
I 3
I 4,5
c_
Fig. 9. The mod squares of the Fourier transforms of the three cases of fig. 8.
correlation hole is even better visible than in our simulations. Because of our different choice of scale the correlation hole in case (3 ) looks larger than it is; this choice is also responsible for the fact that the statistics of the data points looks worse than in cases ( 1) and (2). To make a comparison with theoretical expectations, we have deduced quantitative information from the three simulations just described. The depth of the correlation hole was determined by an extrapolation of the straight-line sections of the curves in fig. 9 to the point t=O; this value was used to determine the ratio (hr) ‘/( (hr)‘), using the substitution rule ( 11) . The results are shown in the second column of table 1. They must be compared with the “theoretical” values shown in the last column, obtained by ex-
6. Discussion We the help of our simulations, we extended the investigations on the correlation hole performed in ref. [ 10 ] into several domains that have not been analytically accessible so far. On the one hand, we consider the regime y.y, ;5 V: mainly for overlapping but also for non-overlapping anticrossings with constant widths and depths (section 4). In the latter case (nonoverlapping anticrossings), the existence of the correlation hole confirmed the phenomenological model discussed in section 2. It is satisfactory that we found the correlation hole to survive intact as a fingerprint of GGE level statistics even in the domain of overlap ping anticrossings. A second extension was that into the regime YJ, 6 V: and non-overlapping anticrossings with variable depths and/or widths. This extension demonstrated the importance of the uniformity of the spectrum for the visibility of the correlation hole. Our results show that the depth of the correlation hole is given by the ratio ( hr) 2/ ( ( hQ2 ) , while
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U. Hartmann et al. /Correlations in anticrossing spectra and scattering theory
the width of the correlation hole depends on the line shape of the anticrossings and on the ratio T/D. For isolated anticrossings, this width is almost unity and independent of the line shape. Our experience indicates that correlation holes with a depth less than 0.4 I or so are very difficult to identify unambiguously: we had to average over a set of 10 independent calculations. Using a single (and short) Fourier transform does not work because the statistical fluctuations are too large, and the correlation hole is washed out. In conclusion we find that in all cases investigated the existence of the correlation hole is a safe fingerprint of GOE statistics, while its absence does not imply regular motion. Our investigations in section 5 show that the correlation hole is indeed mainly a pure spectral GOE effect: although the three spectra in fig. 8 look very different, the corresponding correlation holes in fig. 9 are rather similar. The differences due to the distributions of the V, and y{,/’ are important but in most cases they do not change the picture entirely. Hence our investigations support and illustrate the observation made in ref. [ lo] that the (unrealistic) assumption ySyt>>4 I’:* made in ref. [ lo] does not have too negative an impact on the results and leads to a reasonable description of the correlation hole. From the methodical point of view, the concatenation technique has proved very useful for generating enough data in a reasonable time. In the regime of isolated anticrossings, even short spectra containing only about 20 anticrossings can be used. This technique might be of interest also for experimentalists who can measure a large number of short spectra. All spectra must, of course, be normalized to a uniform level density prior to concatenation. Our investigation leaves open a few questions, some unfortunately related to the actual experiments. A better test of rule ( 11) in the regime of isolated anticrossings, and a check of its extension to overlapping resonances, is desirable. Likewise, it is necessary to discuss triplet states with more than one symmetry class. Refs. [ 9, lo] suggests that only the width of the correlation hole is decreased by superposing several classes. But this problem requires considerably more
CPU time than used in the present study. For this problem the density matrix formalism does not offer a preferable alternative because in the S-matrix formalism the dimension of the matrix is linear in N while the number of independent elements of the density matrix is proportional to N *.
Acknowledgement
We thank R. Jost, M. Lombardi, and J.P. Pique for stimulating discussions and W. Brllckner, H.-W. Heyng, and T. Kihm for providing us with lots of CPU time on their computers.
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