Energy-level statistics and time relaxation in quantum systems

We study a quantum-mechanical system, prepared, at t (cid:53) 0, in a model state, that subsequently decays into a sea of other states whose energy levels form a discrete spectrum with given statistical properties. An important quantity is the survival probability P ( t ), deﬁned as the probability, at time t , to ﬁnd the system in the original model state. Our main purpose is to analyze the inﬂuence of the discreteness and statistical properties of the spectrum on the behavior of P ( t ). Since P ( t ) itself is a statistical quantity, we restrict our attention to its ensemble average (cid:94) P ( t ) (cid:38) , which is calculated analytically using random-matrix techniques, within certain approximations discussed in the text. We ﬁnd, for (cid:94) P ( t ) (cid:38) , an exponential decay, followed by a revival, governed by the two-point structure function of the statistical spectrum, thus giving a nonzero asymptotic value for large t ’s. The analytic result compares well with a number of computer simulations, over a time range discussed in the text. (cid:64) S1063-651X (cid:126) 97 (cid:33) 03603-9 (cid:35) PACS


I. INTRODUCTION
The problem of finding the time relaxation of a quantummechanical system after it has been prepared in some model state is of great interest in many fields of physics. Perhaps one of the oldest examples goes back to the analysis by Weisskopf and Wigner ͓1͔ of a decaying state. The main features of the derivation can be obtained in the following schematic model. The state ͉͘ is coupled at tϭ0 to an infinite discrete spectrum of equally spaced levels ͉n͘, ⌬ being the level spacing, via the matrix elements v n ϭ͗i͉v͉n͘, assumed to be independent of n: v n ϭv. Then, taking the continuum limit ⌬→0 (͉n͘→͉E͘), subject to the condition 2v 2 /⌬ϭ⌫ ͑where ⌫ is a constant), the survival probability, i.e., the probability of finding the system in state ͉͘ at time t, is found to have the familiar exponential behavior P(t)ϭe Ϫ⌫t ͑in units in which បϵ1).
The above schematic model can be taken as the paradigm for a number of physical applications. Consider, for instance, the quasiparticle concept in many-body theory ͓2͔. In this case, the model state ͉͘ can be taken to represent a singleparticle state, which is coupled to the remaining states via the two-body interaction. For a bound system, the resulting width is sometimes known as the ''spreading width.'' Actually, various Green function techniques have been developed to treat this problem ͓2͔. Still in the framework of manybody theory, one can put in the above language the important problem of describing the resonances generated by the decay of ''bound states embedded in the continuum'' via the twobody interaction ͓3͔. Actually, the general problem of resonance decay in scattering theory and the deviations from the pure exponential decay have been of interest to several authors over the years: see, e.g., Ref. ͓4͔, and the references cited therein. Another problem that fits the above scheme and was of interest a few decades ago in the realm of nuclear physics is the decay, via the Coulomb interaction, of a state of good isospin embedded in a sea of states of a different isospin ͓5͔, and the related spreading width.
In the case of a bound system, the spectrum the model state decays into is discrete; in the approaches indicated above, the influence, in the relaxation process, of the discreteness of the spectrum or its statistical properties was not contemplated. In contrast, Ref. ͓6͔ and some of the references cited there consider the relaxation of a harmonic oscillator coupled to a bath of N harmonic oscillators with equally spaced frequencies (⌬ being the spacing͒, without taking the continuum limit. In the N→ϱ limit an exponential decay is found, with a revival at r Ϸ2/⌬ and multiples thereof. Reference ͓7͔ discusses the dynamics of an electron inside a chaotic quantum dot ͑for a review of quantum dots see Ref. ͓8͔͒: the analysis considers the time evolution of the probability to find the particle at a point r 2 if it was located at r 1 at tϭ0. Using supersymmetry methods, it is found that quantum interference, the discreteness of the energy spectrum, and the corresponding level statistics lead to a nontrivial time-dependent behavior governed by the Fourier transform of the two-level correlation function ͓9,10͔ ͑see also Refs. ͓11,12͔͒. A similar statement is made, without proof, in Ref. ͓13͔: there, a wave packet is constructed at tϭ0 as a linear combination of N states-contained in a stretch of energy ␦E, the mean spacing being ⌬-obeying one of the random-matrix statistics ͓9,10͔; it is stated that the ensemble-averaged survival probability is governed by the Fourier transform of the two-level correlation function, in agreement with Ref. ͓7͔. The purpose of the present paper is to analyze the influence of the discreteness and statistical properties of the spectrum in the survival probability P(t) of a model state. The analysis is performed in a language entirely similar to that described at the beginning of this Introduction in connection with the Weisskopf-Wigner model, but using a randommatrix spectrum instead of an equally spaced one and without ever taking the continuum limit. The ensemble average ͗P(t)͘ is calculated using random-matrix techniques: although the calculation is performed within certain approximations to be defined below, we found the simplicity of the derivation appealing.
In the next section we define the model and solve the corresponding time-independent Schrödinger equation. The survival probability and its ensemble average are calculated in Sec. III, where a comparison with numerical simulations is also presented.

II. THE MODEL. THE TIME-INDEPENDENT SOLUTION
In the Introduction we described a number of physical systems in which the relaxation of a model state ͉͘ into a ''sea'' of levels ͉͘ via a residual interaction is of great interest. In this section we propose a model for this problem and analyze the solution of the corresponding timeindependent Schrödinger equation. The time-dependent solution-the topic of main interest in this paper-will be studied in the next section.
The states ͉͘ and ͉͘ are assumed to be solutions of an unperturbed Hamiltonian H 0 , E being embedded in the sea of levels E . The total Hamiltonian is where the residual interaction v only couples the state ͉͘ with the states ͉͘; i.e., its only nonzero matrix elements are ͉͗v͉͘ϭv .

͑3͒
Some of the properties of the solutions of the complete Schrödinger equation can be obtained in an elementary way, as discussed in Refs. ͓5,14͔ and briefly indicated in what follows. Expanding the eigenstates ͉i͘ in terms of ͉͘ and ͉͘ as we can write Eq. ͑4͒ as the system of coupled equations whose solutions are the eigenvalues E i and the eigenvector components c (i) . We find, for c , which can thus be eliminated from Eqs. ͑6͒ to give, for E, the dispersionlike relation A plot of the two sides of Eq. ͑8͒ as a function of E gives the solutions E i as the abscissas of the intersections. We can see that there is a solution between every two successive E 's, and one smaller and one larger than all the E 's. The solutions E i approach E as we go ever farther away from E . The ''strength function,'' i.e., the probability of finding the model state ͉͘ in the exact eigenstate ͉i͘, is given, from Eqs. ͑5͒ and ͑7͒, by We shall see in the next section ͓see Eq. ͑30͔͒ that the strength function plays an important role in the calculation of the time-dependent survival probability; we thus devote the rest of this section to studying some of its properties.
Since the state ͉͘ is normalized, the w i 's obey the sum rule Far away from E , where E i ϷE , Eq. ͑8͒ allows writing w i as in agreement with perturbation theory. It is interesting that for a ''picket-fence'' ͑PF͒ model, in which the E 's are equally spaced-their distance being ⌬-and infinite in number, and all the ͉v ͉ 2 ϭv 2 , the strength function can be found exactly as i.e., a Lorentzian of width which in turn goes over, in the continuum limit, to the result mentioned in the Introduction. In the model with which are we concerned, the energy levels E are assumed to obey some statistical distribution with constant average spacing ⌬ and to be infinite in number; we suppose, for simplicity, that all the ͉v ͉ 2 ϭv 2 . The strength function w i will then depend not only on E i , as for the PF, Eq. ͑12͒, but also on all the other E j 's. This last dependence is negligible far from E , as Eq. ͑11͒ shows, but this may not be the case in other regions. We represent w i as Here, w(E i ) is a continuous function of its argument that represents, for fixed E i , an average over all the other levels; ␦w i represents the fluctuations around that average and thus averages to zero for fixed E i ͑its full ensemble average thus vanishes too: ͗␦w i ͘ϭ0). The sum rule ͑10͒ reads

͑15͒
The first term, i.e., is a statistical quantity, because the E i 's fluctuate from sample to sample: it is actually a linear statistic, in the language of Dyson and Mehta ͓15͔. Thus W fluctuates around ͗W͘ϭ1 ͑17͒ ͓the magnitude of the fluctuations will be found below, in Eq. ͑25͔͒; but W, added to the second term in Eq. ͑15͒, always gives 1. Since the E i 's occur symmetrically in the distribution function, the probability density p(E i ) for a single eigenvalue is identical for all i's. For N levels in an interval L, p(E i )ϭ1/L and Eq. ͑17͒ gives Letting N, L→ϱ, with L/Nϭ⌬, we find 1 ⌬ ͵ w͑E͒dEϭ1.

͑19͒
Inspired by the behavior of W i far from E ͓see Eq. ͑11͔͒ and the result for the PF Eq. ͑12͒, we propose, a simple model for w(E i ), the Lorentzian that the fluctuating quantities ␦w i must fulfill.

III. THE RELAXATION OF THE MODEL STATE
Suppose that at tϭ0 we prepare the system described in the preceding section in the model state ͉͘. As a function of time, the system will be found in the state ͑we use units in which បϵ1).
The survival probability amplitude for finding the system, at time t, in the original state ͉͘ is which is thus expressed in terms of the strength function of the preceding section, Eq. ͑9͒. For an ensemble of spectra, as we considered in the preceding section, ␣(t) is a random quantity; its ensemble average is given by in the same limit as in the preceding section, right below Eq.

͑32͒
We now turn to the main topic of the present paper: the study of the survival probability which, from Eqs. ͑30͒ and ͑14͒, can be written as Of course, the full statistical distribution of the random quantity P(t) for fixed t is of interest; here we shall confine ourselves to the calculation of its first moment. From Eq. ͑34͒ we can write and The first term in Eq. ͑35͒, i.e., ͗P(t)͘ (0) , can be calculated if we realize that the quantity ␣ (t) is again a linear statistic: this we do below ͓Eq. ͑41͔͒. We have not been able to evaluate the term Q(t); however, we show that this term is quite small for not too large t and verify this statement numerically in cases where ⌫ӷ⌬. For large t we also show that its average over a long time interval is positive and verify this statement numerically as well ͑see Sec. III A below͒.
where b(k) is again the two-level form factor and which reduces to () of Eq. ͑22͒ when tϭ0. Equations ͑31͒, ͑36͒, and ͑39͒ then yield

͑41͒
For the Lorentzian model ͑20͒ for w(E), we have is the main result of this paper: it gives ͗P(t)͘ (0) as a quadrature of known functions. For ⌫ӷ⌬, ͗P(t)͘ (0) describes an exponential decay, which is then followed by a ''revival,'' governed by the two-level form factor. Equation ͑41͒ can be approximated by for ⌫ӷ⌬. We first apply Eq. ͑41͒ for tϭ0. Using Eqs. ͑19͒, ͑21͒, ͑25͒, and ͑40͒, we find

͑45͒
Here we used Eq. ͑28͒. Results ͑44͒ and ͑45͒ add up to 1, as they should. If ⌫ӷ⌬, Eq. ͑45͒ shows that Q(0) is second order in the small quantity ⌬/⌫; we thus expect, for not too large times, ͗P(t)͘ (0) to dominate over Q(t).
Consider now the average of ͗P(t)͘ over an infinite time interval, that we denote with a bar. From Eq. ͑35͒ we find ͗P͑t͒͘ϭ͗P͑t͒͘ ͑ 0 ͒ ϩQ͑t ͒.

͑46͒
Using Eqs. ͑36͒ and ͑37͒, we can express the first term as which, evaluated with the Lorentzian model ͑20͒, gives which should then also be the asymptotic value of ͗P(t)͘ (0) for large times. This result has a very appealing interpretation. The probability of finding the eigenstate ͉i͘ of H in the vector state ͉(t)͘ is w i , independent of time ͓see Eq. ͑29͔͒; w i is concentrated in a region of width ⌫ ͓Eq.
͑20͔͒, where there are ϳnϭ⌫/⌬ levels, so that ͗w i ͘ϳ1/n inside that region. From the standpoint of the eigenstates ͉͘, ͉͘ of H 0 , the probability is concentrated, at tϭ0, in the model state ͉͘ and in none of the states ͉͘; as t→ϱ, according to Eq. ͑48͒, it is as if the probability had been spread among ϳn neighboring levels.
We can estimate the second term in Eq. ͑46͒ by time averaging Eq. ͑38͒ and approximating ͗w(E i )(␦w i )͘Ϸ0, so that Q(t)Ϸ ͚ i ͗(␦w i ) 2 ͘. This indicates that the correction on ͗P(t)͘ (0) provided by Q(t) is positive. This correction can be further estimated as follows. Call x i ϭ͉͗i͘, so that w i ϭx i 2 . As in the above paragraph, we recall that the energy span over which w i is appreciable is ϳ⌫ and contains ϳnϭ⌫/⌬ levels. Suppose that the x i are Gaussian variables with zero centroid and the same variance V. Then ͗x i 2 ͘ϭ͗w i ͘ϭV and ͗x i 4 ͘ϭ͗w i 2 ͘ϭ3V 2 , so that var(w i )ϭ ͗w i 2 ͘Ϫ͗w i ͘ 2 ϭ2V 2 . From Eq. ͑10͒, Vϭ1/n, so that Q(t)ϳn var(w i )ϳ1/nϳ⌬/⌫. This estimate indicates that for large times Q(t) is no longer negligible compared with ͗P(t)͘ (0) and that the two quantities could be of the same order of magnitude.

A. Numerical simulations
We present a scheme that is useful for the numerical calculation of the survival probability P(t) of Eq. ͑33͒. We can expand the wave function of the system at time t ͓Eq. ͑29͔͒ in terms of the model state ͉͘ and the basis ͉͘ as The coefficients ␣(t) and ␣ (t) satisfy the set of ordinary differential equations Using Eqs. ͑50͒ we find the set of (Nϩ1) 2 ordinary differential equations from which we can find the time evolution of P(t). In Eqs. ͑51͒ we have defined with , ϭ1, . . . ,N. The system of equations ͑51͒ is formally equivalent to the one obtained in Ref. ͓6͔ and the magnitudes defined by Eqs. ͑52͒ can be thought of as a particular case of the ''relevant'' operators introduced also in that reference. A number of numerical simulations were performed, in order to verify the approximations under which the above analytical results were obtained. For the E 's an unfolded ͑i.e., with uniform density͒ GOE spectrum with Nϭ800 levels centered at E was used. The coupling matrix element, in units of the mean spacing ⌬, was chosen as v/⌬ϭ2.08, giving, for the width of the strength function, in the same units, ⌫/⌬ϭ2v 2 /⌬ 2 ϭ27.27. The results of the computer simulations are given in Figure 1, which shows P(t) averaged over an ensemble of 100 members; the error bars indicate the fluctuations arising from the finite size of the ensemble. The dash-dotted line is a plot of the theoretical expression ͗P(t)͘ (1) , Eq. ͑43͒; no significant difference was found with ͗P(t)͘ (0) , obtained carrying out numerically the integration indicated in Eq. ͑41͒. In the abscissa, the time is indicated in units of the ''Heisenberg time'' 1/⌬. We observe that, up to times tϳ2.4/⌬ϳ65/⌫ ͑see inset͒, ͗P(t)͘ (1) gives a good quantitative description of the data.
The reason for the difference, or discrepancy, in the vicinity of the minimum is not very clear; numerically we have not found a clear indication of saturation in this difference with the number of levels, which was increased up to Nϭ800. There is some evidence ͓17͔ that such a discrepancy might be connected with the difference in statistical properties between the spectrum of the E i 's and that of the E 's; if this is the case, using, in Eq. ͑43͒, the b(⌬t/2) appropriate for the levels E should improve the agreement. Also, should there be any evidence for a deviation with respect to the Lorentzian model of the strength function, we could modify the t () of Eq. ͑42͒ accordingly. For larger times the numerical results are larger than the analytical ones, in agreement with the discussion given above. The qualitative trend, i.e., an exponential decay followed by a nonzero asymptotic value, is similar to the one found in Ref. ͓12͔; this asymptotic behavior is similar to that indicated in Ref. ͓11͔ and to the ''echo'' observed in Ref. ͓7͔.

IV. SUMMARY
In this paper we consider a quantum-mechanical system, prepared, at tϭ0, in a model state, that subsequently decays into a sea of other states whose energy levels form a discrete spectrum with an average level spacing ⌬. The main purpose of the paper is to analyze the influence of the discreteness and statistical properties of the spectrum in the time dependence of the survival probability P(t). The ensemble average ͗P(t)͘ of the survival probability is given in Eq. ͑35͒. The first term, ͗P(t)͘ (0) , is calculated explicitly in Eq. ͑41͒, under the assumption that the stationary ͑i.e., energy-independent͒ statistical properties of the original spectrum are not significantly altered in the perturbed spectrum. For ⌫ӷ⌬, ͗P(t)͘ (0) of Eq. ͑41͒ does not differ significantly from its approximation ͗P(t)͘ (1) of Eq. ͑43͒. Its general trend, for ⌫ӷ⌬, is an exponential decay e Ϫ⌫t , followed by a revival, governed by the two-point structure function. We have not been able to evaluate the second term, Q(t); however, for not too large times we found it to be quadratic in the small quantity ⌬/⌫, so that ͗P(t)͘ (1) alone describes well ͑except, to a certain extent, for the vicinity of the minimum, as discussed above͒ the results of computer simulations up to times on the order of 2.4 times the Heisenberg time 1/⌬, or ϳ65 decay times 1/⌫.
FIG. 1. The time evolution of the average survival probability obtained from a numerical simulation is represented as the solid line: an unfolded GOE spectrum with 800 levels centered at E was used, with v/⌬ϭ2.08 and ⌫/⌬ϭ27.27; an ensemble of 100 elements was constructed. The dash-dotted line represents the result of the analytical study ͗P(t)͘ (1) . The inset shows the evolution for shorter times, where the error bars due to the finite sample size were included.