Abstract
The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the Nlevel Friedrichs model. The longtime asymptoticforms of both S(t) and P(t) are obtained for an arbitrary initialstate extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t4decay and a faster decay are realized depending on the initial states, where the latter is estimated as t8.
Initial states and the various longtimebehaviors of the unstable multilevel systems
Manabu Miyamoto
Department of Physics, Waseda University, 341 Okubo, Shinjukuku, Tokyo 1698555, Japan
ABSTRACT
The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the Nlevel Friedrichs model. The longtime asymptoticforms of both S(t) and P(t) are obtained for an arbitrary initialstate extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t^{4}decay and a faster decay are realized depending on the initial states, where the latter is estimated as t^{8}.
1 Introduction
The study on the unstable systems was initiated by the works of Gamow [1], who attempts to explain the exponentialdecay law of radioactivity. This law is also observed in the atomic systems coupled with the electromagnetic (EM) field, and its theoretical description is now understood by the poles on the second Riemann sheet of the complexenergy plane [2]. However, in the middle of the last century, the deviation from the exponentialdecay law was predicted by Khalfin [3] both for short times and for long times. Around the turn of the century, a shorttime deviation was successfully observed [4]. On the other hand, the longtime deviation has still not been detected [5], even though expected for arbitrary unstable systems with the continuum of the lowerbounded energy spectrum. The main cause behind the matter could be ascribed to too small survival probability S(t) at such long times, that is the component of the initial state remaining in the state at a time t.
Some of the unstable systems can be reduced into the Friedrichs model [6,7], which allows us to investigate the decaying behavior concerning such processes as the spontaneous emission of photons from the atoms [8,9], the photodetachment of electrons from the negative ions [912], and so forth. The traditional study of the model often resorts to the single lowestlevel approximation (SLA) of the atoms or negative ions, and it could be actually verified as long as such a level is quite separate from the higher ones. However, the multilevel treatment of the model has a possibility of another advantage: the choice of coherently superposed initialstates extending over various levels. In fact, it can yield a variety of temporal behavior that is never found in the SLA [1315]. Such multilevel effects on temporal behavior are still not well studied except for Refs. [1316], and much less examined with respect to nonexponential decay at long times. however examined such a longtime behavior of the survival probability S(t), incorporating the initialstate dependence, based on the Nlevel Friedrichs model [17]. discussion, obtained were proved to be quite general and general class Restricting ourselves to the weak coupling cases, we clarified how the asymptotic form of S(t), that follows a powerdecay law, depends on the initial states. In particular, we disclosed the existence of a special initial state that maximizes the asymptotic form of S(t) at long times, which could be desirable for an experimental verification of the powerdecay law, and also the initial states that eliminate the first term of the asymptotic expansion of S(t). The latter implies that S(t) for such initial states can exhibit another powerdecay law, which is faster than the usual one. These results mean that the longtime behavior is determined by not only the smallenergy behavior of the form factors but also the initial unstablestates. Such relations between the initial states and the power decay law were already studied with respect to the asymptotic behavior of wave packets, both for the freeparticle system [18] and for finiterange potential systems [19].
In the present study, we derive the longtime asymptoticform of the total probability P(t) in the basis of the Nlevel Friedrichs model, and show its dependence on the initial unstablestates explicitly. P(t) is the probability of the system to remain in the subspace spanned by the unstable states, and is useful as a candidate for experimentally measurable quantities other than S(t) (see, e.g., [13,20]). We fist consider the initial state localized at the lowest level, to look over the SLA from the multilevel approach. Then, the difference from the result based on the SLA is found unlike that for S(t). It is also proved that there exist the initial state maximizing the asymptotic form of P(t) at long times, and also the initial states eliminating the first term of the asymptotic expansion of P(t). One can then understand that these initial states have the same roles for S(t). Moreover, we numerically confirm the results for S(t) [17] by considering the spontaneous emission process for the hydrogen atom interacting with the EM field. We demonstrate the t^{4}decay of S(t), which was theoretically obtained by [8], and a faster decay predicted by [17]. The latter is estimated like t^{8} as a powerdecay law.
2 Friedrichs model
The Nlevel Friedrichs model describes the couplings between the discrete spectrum and the continuous spectrum. The model Hamiltonian is defined by
where H_{0} denotes the free Hamiltonian
and lV being the interaction Hamiltonian
with the coupling constant l. The eigenvalues w_{n} of H_{0} were supposed not to be degenerate, i.e., w_{n} < w_{n'} for n < n'. Both nñ and wñ are the bound and scattering eigenstates of H_{0}, respectively, and satisfy the orthonormality condition: ánn'ñ = d_{nn'}, áw w 'ñ = d(w w'), and án wñ = 0 , where d_{nn'} is Kronecker's delta and d(w w') is Dirac's delta function. They also compose the completely orthonormal system (CONS) with the resolution of identity. In Eq. (3), v_{n} (w) denotes the form factor characterizing the transition between nñ and wñ . In the latter discussion, we analyze the model with the assumption that the form factor v_{n} (w) is analytic in a complex domain including the cut (0, ¥), square integrable, i.e.,
and behaves like
as w ® 0 , where p_{n} is a positive constant while q_{n} is an appropriate one. The conditions (4) and (5) ensure that the integral d wv_{n} (w) (w)/w is definite. The largeenergy condition (4) ensures that d wv_{n} (w) (w)/(z w) is definite for all complex numbers z Ï [0, ¥). Both of the conditions are satisfied by several systems involving the spontaneous emission process of photons [8,9] and the photodetachment process of electrons [912]. Note that this smallenergy condition (5) excludes the photoionization processes associated with the Coulomb interaction [21]; however, the formulation developed below could be applied to those cases.
The initial unstablestate yñ of our interest is an arbitrary superposition of the discrete states nñ,
where c_{n}'s are complex numbers satisfying the normalization condition c_{n} ^{2} = 1. Then, the survival probability S(t) of the initial state yñ , that is, the probability of finding the initial state in the state at a later time t, is defined by S(t) = A(t)^{2} . The A(t) denotes the survival amplitude of yñ , i.e., A(t) = áye^{itH} yñ. The total probability P(t) that the state at the later time t remains in the subspace spanned by nñ is also defined by P(t) = áye^{itH}(nñ án)e^{itH}yñ . It is worth noting that P(t) > S(t) strictly holds for all times t, because the projection nñ án can be decomposed into the two parts of the projections yñ áy and P_{^} so that they are nonnegative and satisfy y ñ áyP_{^} = P_{^}yñ á y = 0. One actually obtains
In particular, in the SLA, i.e., N = 1, P(t) = S(t). The Hamiltonian (1) in general has the possibility of possessing not only the scattering eigenstates ñ, but also the bound eigenstates [12,15,2224]. However, the emitted particles detected in the decaying process are only brought from the initial component associated with the scattering eigenstates. We shall here confine ourselves to studying the decaying parts of A(t) and P(t), denoted by the same symbols as
and
respectively.
In order to estimate the longtime behaviors of both A(t) and P(t), let us evaluate the scattering eigenstates ñ by solving the LippmannSchwinger equation,
We here mention this procedure somewhat in detail to make an explanation in the selfconsistent way. We first use the CONS to represent ñ in the following form,
where the coefficients (w) and f^{(±)}(w' , w) are given by (w) = ánñ and f^{(±)}(w' , w) = áw'ñ, respectively. Then, the scattering component áyñ that appears in Eqs. (8) and (9) turns out to be represented simply by
Acting áw from the left to Eqs. (10) and (11) and eliminating áwñ from both equations, one obtains,
By the same way, acting án from the left to Eqs. (10) and (11), and eliminating ánñ from both equations again, at this time, one has,
Then, substitution of Eq. (13) into (11) leads to
To determine (w), we can eliminate f^{(±)}(w' , w) from Eqs. (13) and (14), and then have an algebraic equation for (w),
where
which is the (n, n')th component of the N × N matrix G^{1} (z), and s_{nn'} (z) is defined by
for all z = re^{i}^{j} (r > 0, 0 < j < 2p).
3 Smallenergy behavior of the resolvent
Under the smallenergy condition of Eq. (5), s_{nn'} (z) is guaranteed to be analytic in the whole complex plane except the cut [0, ¥). For the later convenience, G^{1}(z) is defined as an inverse of G(z), where G(z) is assumed to be regular. Note that G(z) is nothing more than the reduced (or partial) resolvent G_{nn}_{'}(z) = án(H z)^{1}n'ñ. One can confirm this fact by following the discussion in section 3.2 of Ref. [7]. Since the behavior of A(t) and P(t) at long times are characterized by that of (w) at small energies, detrmined by Eq. (16), we need to estimate the smallenergy behavior of G(z). Note that under the condition (5) we have
as w ® +0, where s_{nn}_{'} (w±i0) = I_{nn}_{'} (w)ipv_{n} (w) (w) and I_{nn}_{'} (w) º P dw' , where P denotes the principle value of the integral. The existence of I_{nn}_{'} (0) may be just guaranteed by the smallenergy condition of Eq. (5) [25]. Supposing that G_{nn}_{'} is of the form
as w ® +0, one obtains that
which leads to
We solve this equation by assuming that g_{nn}_{'} can be expanded for small l as
By substituting Eq. (23) into (22), it follows that
and for j > 1
where we have assumed that all w_{n} does not vanish. Note that and derived here accord with at least those for solvable cases, where G(z) is explicitly obtained [15,16]. We can then obtain
with
where
where p = min{p_{n}}. With use of the _{n} instead of q_{n}, we extracted only the dominant part of (w) at small w.
4 Longtime behavior of the survival and total probabilities
In this section, we shall examine the longtime behavior of both S(t) and P(t) with the various initialstates. The formula for S(t) (or A(t)) was obtained by Ref. [17] as follows,
as t ® ¥, where i^{2p+1} = e^{i }^{(2p+1)p/2}, and G(z+1) = dxx^{z} e^{x}. Eq. (29) is derived by inserting Eqs. (12) and (26) into Eq. (8) by using the asymptotic method for the Fourier integral [26]. One can clearly perceive A(t) ~ t^{2p1} , the powerdecay law.
Let us first consider the higherlevel effects on the longtime behavior of A(t) that starts from the localized initial state at the lowest level. For such an initial state, i.e., c_{n} = d_{n1}, Eq. (29) becomes
where we supposed that
_{1}¹ 0. Since there are no factors related to the higher levels in Eq. (30), it implies that the longtime asymptotic behavior of A(t) could agree with that in the SLA for a sufficiently small l. We can also find a special superposition of discrete states nñ that maximizes the asymptotic form of A(t) at long times [17]. It is worth noting that its dependence on the initial states only appears in Eq. (29) through the factor c_{n}, which can be rewritten by an inner product as
We have here introduced an auxiliary vector defined by
from which Eq. (29) reads
as t ® ¥. With resort to the Schwarz inequality, we see that the maximum of the factor (31) is just attained by if and only if  yñ µcñ, i.e.,
where c is an arbitrary complex number with c = 1. Therefore, preparing the initial state yñ according to the above weights (34), we can maximize the asymptotic form of A(t) at long times. Substituting Eq. (34) into Eq. (33), one obtains that
On the other hand, there are another kind of initial states that are coherently superposed to eliminate the factor (31) [17]. This is indeed realized by the initial states orthogonal to cñ, i.e.,
In this case, the first term in the righthand side (rhs) of Eq. (33) becomes zero. This fact may imply that A(t) for such an orthogonal state asymptotically decays faster than t^{2p1}.
As same as for S(t), we can derive the asymptotic form of P(t) at long times, which is determined by the smallenergy behavior of the integrand. By substituting Eqs. (12) and (26) into Eq. (9), and by applying the asymptotic method for the Fourier integral again, P(t) reads
It follows from Eqs. (33) and (40) that P(t) S(t) holds at long times too, in the sense of the comparison between the first terms of the asymptotic expansion of P(t) and S(t).
We next examine the initialstate dependence of the longtime behavior of P(t). Let us first consider the higherlevel effects on P(t) with the initial state localized at the lowest level. In this case, Eq. (40) reads
with assumption that
_{1}¹ 0. One clearly sees in Eq. (41) that the higherlevel effect still remains through the factor c, unlike the case for A(t). We note that
which implies that as the number of levels N increases, the SLA with N = 1 may result in an underestimation of P(t) at long times. However, the resultant gain with a large N strictly depends on the system under the consideration: for instance, in the spontaneous emission process of photon from the hydrogen atom, c ^{2}/f_{1}^{2}~ 1.29 was obtained even for N = 50 [17]. Furthermore, Eq. (40) tells us that the initial state maximizing the asymptotic form of P(t) at long times is the same as it for S(t), i.e., cñ/ c. However, for such an initial state, we merely obtain that P(t) ~ S(t) at long times. One also finds from Eq. (40) such special initialstates that eliminate the first term of the rhs of the asymptotic expansion (40). This is in fact realized by the initial states orthogonal to cñ as same as for the S(t). Therefore, P(t) for such an initial state could be expected to decay asymptotically faster than t^{4p2} too.
Befor concluding this section, we point out that the initial state extended over discrete states nñ has the possibility of increasing the intensity of A(t) (P(t)) more than a localized one would. This possibility may be recognized as follows. The coupling lV makes a transition of the initial state, that is composed of the discrete states nñ, into the continuum state wñ, whereas the hermiticity of the Hamiltonian also allows an inverse process of making the state at a later time t get back into the subspace spanned by nñ. In the latter process, there are various candidates for the reviving discretestate. Repopulation of each discrete state can make the intensity of A(t) (P(t)) grow, provided that the initial state is composed of those discrete states. However, if the initial state only consists of a specific discrete state, the other discrete states composing the state at a later time t are thrown away without any contribution to A(t) (P(t)) [27]. This is the reason why the decay of the A(t) (P(t)) for extended states can be relieved more than that for localized states.
5 Application to the excited states of the hydrogen atom
In order to illustrate our analysis developed in the foregoing sections, we consider the spontaneous emission process for the hydrogen atom interacting with the EM field [8,9]. In this case, we suppose that nñ = (n+1)pñÄ0ñ, where (n+1)pñ and 0ñ denote the (n+1)pstate of the atom and the vacuum state of the field respectively, and also wñ = 1sñ Ä1_{w}ñ, where 1sñ and 1_{w}ñ denote the 1sstate of the atom and the onephoton state respectively. Thus, an initially excited atom makes a transition to the ground state with the emission of a photon. We here choose only three excited levels: the 2p state, 3p state, and 4p state. Then, the form factors corresponding to the 2p1s, 3p1s, and 4p1s transitions become as follows [28,29],
where L_{1} = 8.498 ×10^{18}s^{1}, L_{2} = (8/9)L_{1}s^{1}, and L_{3} = (10/12)L_{1}s^{1} are the cutoff constants. Note that these form factors have different forms, however they behave in the same way at small energy like w^{1/2}. The coupling constant is also given by l^{2} = 6.435 ×10^{9}. The embedded eigenvalues of H_{0} are defined by w_{n} = W[1 (n+1)^{2}] with W = 1.55 ×10^{16}s^{1}. To derive these formfactors, we represented 1_{w}ñ in the energyangular momentum basis for photon [30] as in Ref. [8], incorporating the conservation of angular momentum and parity in each transition [31]. As was emphasized in Ref. [8], these formfactors are surely analytic results without any approximation. The Hamiltonian (1) is then derived within the fourlevel approximation and the rotatingwave approximation.
In the following, we shall compare the longtime asymptoticform of A(t) predicted by Eq. (33) and that of A_{cut}(t), which we evaluate numerically. A_{cut}(t) is defined by
The contour C runs clockwise around the half line {re^{7pi/4}0 < r < ¥} in the complex energy plane. This contour lies on the first Riemann sheet when it goes below the half line, and gets into the second Riemann sheet when it above the half line. A_{cut}(t) is related to A(t) through the equation,
where z_{p} is in general the complex pole of áy(H z)^{1}yñ located in the region between the half lines [0, ¥) and {re^{pi/4}0 < r < ¥} in the second Riemann sheet. In the weakcoupling case considered here, each of z_{p} is in the neighborhood of w_{n}, and thus the asymptotic form of A_{cut}(t) and that of A(t) are expected to exhibit the same behavior at long times, where the power decay dominates over the exponential decay [32].
Let us first restrict ourselves to the two initial states: the localized state at the 2p level 1ñ and the maximizing state cñ/c . Figure 1 shows the time evolution of A_{cut}(t)^{2} and the asymptote of A(t)^{2} for these initialstates. It is clearly seen that A_{cut}(t)^{2}'s for these initial states approach to the corresponding asymptotes of A(t)^{2} parallel to t^{4}, however the difference between them is very small [17]. At t = 10^{5} , we obtain A_{cut}(t)/A_{asp}(t)^{2}~ 0.999 for these initial states. Is is worth stressing that this time is very earlier than 1/g_{1}~ 1.36 ×10^{10} the lifetime of the 2p state [33], where g_{1} = 2pl^{2}v_{1}(w_{1})^{2} + O(l^{4}) ~ 6.268 ×10^{8}s^{1} [8].
We next choose the two special states, ñ and ñ, as the initial state yñ . The former is defined by
and the latter is
so that they eliminate the factor (31) and satisfy the relations that ácñ = 0, ácñ = 0 , and áñ = 0 . Figure 2 shows that the time evolution of A_{cut}(t)^{2} for these initial states and for the maximizing initial state. We clearly find that, as was seen in Fig. 1, A_{cut}(t)^{2} for the maximizing initial state (solid curve) asymptotically decays like t^{4}, whereas A_{cut}(t)^{2} for other initial state (longdashed and shortdashed curves) follow another decaylaw faster than t^{4}. They seem to be fitted with the power law t^{8}. For the comparison, we also depict in Fig. 2 the two straight lines 150.0 ×(L_{1}t)^{8} and 30.0 ×(L_{1}t)^{8} (solid lines), to which A_{cut}(t)^{2} for the initial state ñ and ñ approach respectively in this time region.
6 Concluding remarks
We have considered the longtime behavior of the unstable multilevel systems and examined the asymptotic behavior of not only the survival S(t) but also the total P(t) probabilities for an arbitrary initial state in the longtime region, where both S(t) and P(t) obey a power decay law. For the initial state localized at the lowest level, we have found that the SLA results in an underestimation of the asymptotic form of P(t) unlike the result for S(t). We have also discovered two kinds of special initialstate. One of them maximizes the asymptotic form of P(t) at long times, and actually the same state that maximizes S(t). However, for such an initial state, we only obtain that P(t) ~ S(t) at long times, even though P(t) > S(t) holds for all times t. The other initial state eliminates the first term of the asymptotic expansion of P(t), and also plays the same role for S(t). In addition, we numerically confirm the previous results for S(t) [17] in consideration of the spontaneous emission process for the hydrogen atom. Then, we show not only the t^{4}decay of S(t) but also a more faster decay, the latter of which is naively fitted by a powerdecay law t^{8}. However, we still do not accomplish the derivation of the asymptotic forms for such faster decays. To this end, it will be needed to take into account the zeroenergy resonance and the zeroenergy eigenstate with an appropriate state space [34,35,36]. We hope to deal with this issue in the future.
7 Acknowledgments
The author would like to express his gratitude to the organizers of the Second International Workshop DICE2004 Castello di Piombino (Tuscany), September 14, 2004, From Decoherence and Emergent Classicality to Emergent Quantum Mechanics. He also would like to thank Professor I. Ohba and Professor H. Nakazato for useful and helpful discussions. This work is partly supported by a Grant for The 21st Century COE Program at Waseda University from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Received on 8 January, 2005
 [1] G. Gamow, Z. Phys. 51, 204 (1928).
 [2] For a review, see, for example, H. Nakazato, M. Namiki, and S. Pascazio, Int. J. Mod. Phys. B 10, 247 (1996).
 [3] L. A. Khalfin, Zh. Eksp. Theor. Fiz. 33, 1371 (1957) [
 Sov. Phys. JETP 6, 1053 (1958)].
 [4] S. R. Wilkinson, et al, Nature (London) 387, 575 (1997).
 [5] P. T. Greenland, Nature (London) 335, 298 (1988).
 [6] K. O. Friedrichs, Commun. Pure Appl. Math. 1, 361 (1948).
 [7] P. Exner, Open Quantum Systems and Feynman Integrals (Reidel, Doredrecht, 1985).
 [8] P. Facchi and S. Pascazio, Phys. Lett. A 241, 139 (1998).
 [9] I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, Phys. Rev. A 63, 062110 (2001).
 [10] K. Rza\. zewski, M. Lewenstein, and J. H. Eberly, J. Phys. B 15, L661 (1982).
 [11] S. L. Haan and J. Cooper, J. Phys. B 17, 3481 (1984).
 [12] H. Nakazato, in ''Fundamental Aspects of Quantum Physics'', edited by L. Accardi and S. Tasaki (World Scientific, New Jersey, 2003).
 [13] E. Frishman and M. Shapiro, Phys. Rev. Lett. 87, 253001 (2001);
 Phys. Rev. A 68, 032717 (2003).
 [14] I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, Int. J. Theor. Phys. 42, 2403 (2003).
 [16] E. B. Davies, J. Math. Phys. 15, 2036 (1974).
 [17] M. Miyamoto, Phys. Rev. A 70, 032108 (2004).
 [18] K. Unnikrishnan, Am. J. Phys. 65, 526 (1997);
 66, 632 (1998); F. Lillo and R. N. Mantegna, Phys. Rev. Lett. 84, 1061 (2000);
 84, 4516 (2000); J. A. Damborenea, I. L. Egusquiza, and J. G. Muga, Am. J. Phys. 70, 738 (2002);
 M. Miyamoto, J. Phys. A 35, 7159 (2002);
 Phys. Rev. A 68, 022702 (2003).
 [19] M. Miyamoto, Phys. Rev. A 69, 042704 (2004).
 [20] F. Remacle and R. D. Levine, J. Chem. Phys. 104, 1399 (1996).
 [21] E. P. Wigner, Phys. Rev. 73, 1002 (1948).
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Publication Dates

Publication in this collection
06 Sept 2005 
Date of issue
June 2005
History

Received
08 Jan 2005