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264 J. Martorell et al.

state have been recently revisited by Martorell et al. with the analytically solvable

model of Sect. 9.5.3 [81].

9.5.1.4 Ways to Enhance Post-exponential Decay

Most of the proposals to enhance post-exponential decay to make it visible, or to

advance the time at which it dominates, are based on the idea of placing a reso-

nance close to threshold [98, 63, 42, 123, 143, 73]. In this respect, systems with

controllable parameters are preferred, for example, photodetachment of electrons

from negative ions, spontaneous emission in a cavity close to cutoff frequency or

in a photonic band-gap material, see the chapter by Alonso and de Vega in this

volume. Lewenstein and Rzazewski [73] proposed to manipulate the decay making

the long-time decay dominant even at early stages so that repeated measurements

could lead to an acceleration of decay (anti-Zeno effect). A precedent for this idea

was included in the well-known article on the Zeno effect by Chiu, Sudarshan and

Misra [11]. They pointed out that if the decay is measured periodically at very long-

time intervals T (in the post-exponential range), the decay law at time t = NT will

be F(t) = P(T )

∝ (t/N )

−κ N

if the survival probability for the isolated system is

P(t) ∝ 1/t

Another idea is to control the initial state so as to maximize the post-exponential

term. This has been explored by Miyamoto [85] with an N -level Friedrichs model.

One further option would be to use scattering singularities, such as a zero-energy

resonance, naturally or artiﬁcially, to provide slower long-time decay [89, 87, 80].

The escape of interacting cold atoms from a trap, in the strongly interacting Tonks–

Girardeau regime, has been also proposed to improve the visibility of the long-time

decay, because the signal would be enhanced proportionally to the number of atoms

[17].

Finally, Torrontegui et al. have recently found that the probability density at the

exponential to post-exponential transition time, and thus its observability, increases

with the distance of the detector from the source, up to a critical distance beyond

which exponential decay is no longer observed [133]. This result is inspired by

an earlier classical model by Newton [94, 95]. Quantum solvable models provide

explicit expressions for the dependence of the transition on resonance and observa-

tional parameters, facilitating the choice of optimal conditions.

9.5.1.5 Complex Potentials

The detector model proposed in [92, 18] is formally Winter’s model complemented

by an imaginary absorbing potential,

H =−

∂

∂x

+ηδ(x −1) −iV

Θ(x − X

) , x ≥ 0 , (9.62)

and the initial state is as usual the ground state of an inﬁnite well between 0 and

1. This is an “optical model” where the absorbing potential represents the effect of

9 Quantum Post-exponential Decay 265

detection of the escaping atom by laser excitation and photon emission in the outer

region; see Chap. 4.

This Hamiltonian may hold N

loc

discrete, localized eigenstates of complex

energy with purely exponential decay. The detector is placed at the barrier edge at

= 1, or further out. The survival amplitude may be written in terms of eigenstates

forming a biorthogonal basis [91]:

A(t) =

loc



l=1

−iE



∞

f (q)e

−i(q

−iV

dq , (9.63)

where C

=ψ

,

=

|ψ

, f (q) =ψ

|φ



|ψ

, and |u

 and |

 are,

respectively, right and left localized eigenstates. The continuum eigenstates appear-

ing above, |φ

 and |

, satisfy

H|φ

=E

|φ

=(q

−iV

)|φ

 , (9.64)



|H = E



|=(q

−iV

)

| , (9.65)

φ



=δ(q − q



) . (9.66)

Note that |φ

 and its corresponding biorthogonal partner are not usual scattering

states because the exterior region is not free from interaction (V (x) = 0 when x →

∞). However, the potential is constant there and this enables us to write the solution

in the external region in terms of an S matrix,

(x) =

(2π)

1/2

⎧

⎪

⎨

⎪

⎩

sin kx, 0 ≤ x ≤ 1

ikx

+ Be

−ikx

, 1 ≤ x ≤ X

−iqx

−S(q)e

iqx

, x ≥ X

, (9.67)

where k = (q

−iV

)

1/2

is the wavenumber inside, q the wavenumber outside, and

, A, B, and S are obtained from the matching conditions at x = 1 and x = X

For scattering-like solutions, q is positive. There are two branch points of k in the

complex q-plane; we take the branch cut to connect these points. Similarly, the root

in q = (k

+ iV

)

1/2

is deﬁned with a branch cut joining the two branch points in

the k-plane. In contrast to scattering-like states of the continuum, localized states

are characterized by a complex q with positive imaginary part.

The main effect of increasing V

, at weak intensities, is the progressive sup-

pression of long-time deviations, as illustrated in Fig. 9.5. Beyond some threshold

strength, even the exponential decay is affected. A quantitative approximation to

the survival probability helps to understand these effects: Let q

be the resonance

with the longest lifetime and let us assume that it is narrow and isolated. If all

other resonances have already decayed, for weak enough absorption, i.e., N

loc

= 0,

the integral of Eq. (9.63) can be approximated, using contour deformation in the

266 J. Martorell et al.

Fig. 9.5 ln[P(t)] for different absorptive step potentials, see Eq. (9.62). V

= 0(thick solid line),

0.1(dashed line), 0.3(thin solid line), and 0.5(dots). X

= 1andη = 5

complex q-plane, by the residue corresponding to the ﬁrst resonance plus a saddle

contribution,

A(t) ≈−2πiRes

[

f (q)

]

q=q

−iE

−Γ

t/2

−

√

πi

f (0)e

−V

3/2

, (9.68)

where E

represents the energy of the decaying particle, Γ

the corresponding decay-

ing width, and

f (0) = [d

f (q)/dq

]

q=0

. The second term is responsible for the

deviation from the exponential decay in the survival P(t) =|A(t)|

. The novelty

with respect to the nonabsorption case, V

= 0, is that the deviation is not given by

a purely algebraic term: The usual algebraic dependence is multiplied by the expo-

nentially decaying factor exp(−V

t). By increasing V

, the deviation term decays

more and more rapidly until, at threshold, i.e., Γ

= V

, the deviation decays faster

than the residue term. This threshold value corresponds exactly to the passage from

a resonance to a localized, normalizable state with purely exponential decay. While

for V

< V

, the dominant term at long times is the saddle contribution (pro-

portional to exp(−V

t) t

−3/2

), in the opposite case, V

> V

, the decay is purely

exponential, and the dominant contribution comes from the discrete part of the

spectrum.

9.5.1.6 Effect of a Distant Detector

A still controversial and rather crucial question is how is the decay affected by the

distance between detector and system in indirect measurements [13–15, 59, 60, 79,

68, 137, 138, 110, 122]? In their conceptual analysis of the Zeno effect, Home and

Whitaker [59] stated that the only real paradox is that the system is predicted to

have its decay affected by a detector at a macroscopic distance. Indeed, a common

9 Quantum Post-exponential Decay 267

sense expectation is that a greater separation of the detector from the initial location

of the system ought to reduce the perturbing effects of measurement, but theories

conﬁrming this expectation have been disputed [13–15, 137]. The need for more

work to arrive at a deﬁnite conclusion is clear [68].

One way to proceed is by study of explicit models. In [18], with the Hamiltonian

model of Eq. (9.62), it was shown that the disturbance of the measurement (rep-

resented by the complex absorbing potential) on the survival amplitude disappears

with increasing distance X

between the initial state and the detector, as well as by

improving its efﬁciency.

9.5.1.7 Observation of Power Laws in Organic Molecules, Quantum Dots,

or Nanocrystals

In view of the above arguments favoring persistence of exponential decay at long

times and of the distorting environmental effects on the power law decay, it came

as a surprise that the experiments of Rothe et al. [121] showed a clean power law

decrease after the cross-over from the exponential regime. In those experiments, the

luminescence decays of several species of dissolved organic molecules were mea-

sured, most with lifetimes of the order of nanoseconds, but some up to a tenth of a

millisecond. Power law decays with algebraic exponents between  2 and  4were

deduced. The strategy followed was to look for resonant decays with large Γ and not

too small E

. As shown in Eq. (9.14), larger Γ increases the asymptotic component

and not too small E

avoids small Q nonexponential decay. Having the molecules

in solution increases the widths, and for large molecules, intramolecular structure

also favors increased broadening. In this case, environmental effects increase rather

than decrease the asymptotic terms.

Unfortunately, the theoretical description of luminescence decay in such organic

molecules is difﬁcult [6, 4, 5], and therefore a detailed comparison with theory is

missing. The possibility of other origins for the observed power law decrease in

ﬂuorescence has been ruled out by Rothe et al. [121]. Still, very recently Sher et al.

[129] have also measured power law decay in the ﬂuorescence blinking of various

semiconductor nanocrystals. They reproduced those with Monte Carlo simulations

on a three-level model. These authors claim that their approach can be also useful

in the analysis of nonexponential ﬂuorescence decays of molecular systems. In fact

colloidal semiconductor quantum dots, nanorods, nanowires, and some organic dyes

exhibit power law distributions of on- and off-times of emission intermittency [39].

They can be explained with models in which an electron jumps into one of the

multiple traps and returns, but not all facets of the experiment are well understood.

In condensed molecular solids composed of conjugated polymers [100], the power

law luminescence found experimentally is attributed to electron–hole pair recombi-

nation and explained using inhomogeneous, statistical theories. Therefore, the ﬁeld

is still open, both theoretically and experimentally, and more conclusive work on

these systems would be highly desirable.

268 J. Martorell et al.

9.5.2 Indirect Measurement

Rather recently, Kelkar et al. [64] made an interesting proposal for extracting the

long-time survival probability from the low-energy  = 0 phase shift of α–α scat-

tering, where it is affected by a 2

resonant state (virtual formation of

Be). By

ﬁtting the experimental phase shift to a parameterized form, they extracted a power

law behavior 1/t

6.36

for P(t). The explanation for this value is murky, since the

long-range part of the α–α potential is dominated by Coulomb repulsion, and the

post-exponential decay law should be modiﬁed by that long-range potential tail.

Applying the recipe for well-behaved potential tails with  = 2, the exponent would

be 7. The result is nevertheless interesting, as it is a different way to approach the

problem, which may have application to other systems. Their result corresponds to

an estimated changeover time of 30 lifetimes, which makes a direct observation not

feasible.

9.5.3 A Classical Analog of a Decaying Quantum System

Recently Longhi [74, 75] proposed to use a system of identical parallel wave guides

as an optical analog of a decaying quantum system. A laser beam injected in the ﬁrst

guide leaks amplitude to the second as it travels down the guide [77]. The second

guide similarly leaks to the ﬁrst and third, and so on. Distance along the guide plays

the role of time. Let c

(t) be the amplitude in the guide labeled n at distance t from

the point of injection. If the guides have identical cross section and are equidistant,

the transverse motion is in a periodic system with nearest neighbor interactions. The

system is equivalent to a particle in a tight-binding model with Schr

odinger equation

i

d|ψ

d(gt)

|ψ=

|ψ with

⎛

⎜

⎝

/g −Δ 0 ···00

−Δε

/g −1 ···00

. ···

000···ε

/g −1

000···−1 ε

⎞

⎟

⎠

. (9.69)

Here g is the coupling parameter between adjacent wave guides and ε

is the com-

mon site energy for all sites in the periodic portion of the system. Site 1 is special; its

site energy ε

may differ from ε

and its coupling parameter is gΔ, with 0 ≤ Δ ≤ 1.

For example, if the ﬁrst wave guide is placed farther apart from its neighbor, we

should expect |Δ| < 1, and if its proﬁle is wider, ε

<ε

From here on we will measure distance in units of /g and suppose that ε

= ε

by making the guides to have identical cross-sections. Then

is a dimensionless

Hamiltonian of a “particle” in a semi-inﬁnite discrete space. Initially, only c

(t =

0) = 1 is nonzero. If Δ is signiﬁcantly less than 1, the beam is trapped behind a

barrier and will only slowly leak out to site 2. The exact time-dependent solution is

9 Quantum Post-exponential Decay 269

(t) =



s=0

2s +1

2s+1

(2t) . (9.70)

Here, α

≡ 1 − Δ

is the “decoupling” parameter. If Δ = 0 (by placing the ﬁrst

guide far from the rest of the system), a well-known sum rule satisﬁed by the Bessel

functions gives c

(t) = 1 at all times. At the other extreme, for Δ ∼ 1, small α

makes the series converge rapidly. c

(t) plays the role of the survival amplitude of a

decaying quantum analog.

For the other sites, the solution is

(t) = i

n−1



s=0

2s +n

2s+n

(2t) . (9.71)

Because J

(2t) ∼ t

/N!atsmallt, it is clear that the initial conditions at t = 0

have been satisﬁed.

Longhi showed that there are three distinct regions of t-dependence. At small t,

from the properties of J

(2t),

(t) ≈ 1 −t

/2 . (9.72)

Next there is an intermediate region where exponential decay holds, with

(t) ≈

1 + α

2α

exp [−γ

t/2] , where

= Δ

/α (9.73)

is the decay constant of the Gamow state. Weak coupling gives very slow decay and

strong coupling Δ ∼ 1 very rapid decay.

At still longer t, the exponential has fallen so far that it is smaller than the asymp-

totic value of the sum of Bessel functions. In this limit we can approximate each

term and perform the sum to obtain the asymptotic value for c

(t).

Use of the asymptotic form

n+2s

(2t) ∼

(−)

√

πt

cos Φ

−

4(n + 2s)

−1

16t

sin Φ

n+2s

≡ 2t − (n +

)

−sπ (9.74)

leads, in lowest order, to

(t) ∼ Δ

n−1

√

πt

n + (n −2)α

(1 + α

)

cos(2t −(2n + 1)

) ,

(t) ∼

√

πt

1 − α

(1 + α

)

cos(2t −3π/4) . (9.75)

270 J. Martorell et al.

These can be further embellished by adding the next to leading order term in the

asymptotic form. In any case it is evident that the post-exponential decay is a power

law with exponent −3/2.

It turns out to be an excellent approximation to write c

(t) as the sum of the

exponential plus the asymptotic approximation, for all but the very shortest t.Inthe

cross-over region their phases are correct so that the interference oscillations are

nicely reproduced. For example,

(t) ≈

1 + α

2α

−γ

t/2

2t −

, (9.76)

for t >> a

, where

1 − α

(1 + α

)

, a

6α

(1 + α

)

. (9.77)

When α is small ( Δ → 1), there is almost no exponential region, whereas when

α → 1, the exponential region extends to very large times. This is illustrated in

Fig. 9.6: It can be seen that when Δ = 0.4 in the time range shown the decay

is exponential. When Δ = 0.95 the post-exponential decay is reached very soon,

and the survival probability is still quite sizeable. The ﬁgure also shows the effect

of disorder on the post-exponential decay: Instead of assuming a constant g for

the tunneling matrix elements, we have added a random ﬂuctuation of ±1%to

1e-05

1e-04

1e-03

1e-02

1e-01

5 10152025303540

P(t)

time

Fig. 9.6 Survival probability, P(t) =|c

(t)|

. Continuous line: Δ = 0.95; long dashed line:

Δ = 0.4. The dotted lines deﬁning a band of values correspond to the simulations with a random

ﬂuctuation of ±1 % in the tunneling matrix elements

The long-time regime has been studied also for ﬁnite-time switching in [81] and to examine its

observability as a function of distance in [133].

9 Quantum Post-exponential Decay 271

each of the tunneling matrix elements between consecutive sites. By running a large

number of cases we determine an average and a mean square deviation. The band

corresponds to the average ±3 standard deviations. As can be seen the effect on

the post-exponential decay is signiﬁcant and suggests that localization effects are

taking place at long times. Note that if that random ﬂuctuation could be attributed

to “environmental” effects, our model does not predict an extension of the range of

exponential decay.

We have recently investigated initial switching effects in this model. It was

assumed that the coupling Δ(t) was increased from 0 to its ﬁnal value Δ over a

time T rather than instantaneously at time 0. The result is an additional phase shift

T added to a

in Eq. (9.77), and S

is multiplied by sin T/T . Together these changes

alter the interference between exponential and asymptotic decay contributions in a

way which ought to be observable [81].

The group in Milan have carried out two experiments [19, 8] which verify

Longhi’s theory. In the ﬁrst, they arranged Δ = 1(α = 0) and found that the

measured amplitude in guide n followed the expected form of the Bessel function

(t) = J

(2t)/t [117].

In the second paper [8], Longhi and collaborators veriﬁed the optical analog of

the Zeno effect. To do this, the wave guides were broken into longitudinal segments

of length t = L. The beam in guide n = 1 entering a new segment effectively

resets the initial condition (albeit with a reduced amplitude less than unity). The

segment length L was chosen to be short, to correspond to the short-time behav-

ior, Eq. (9.72). What they found was that the decay of c

(t) in each short seg-

ment was reset to the quadratic behavior, so that it never reached the exponential

decay regime. On the contrary, without segmentation they did see a switch-over to

Eq. (9.73). It will be interesting to see whether their experiment can be extended to

longer length scales to verify the changeover to post-exponential decay, described in

Eq. (9.76).

9.6 Final Comments

We have provided an overview of quantum post-exponential decay, from the early

works to the most recent analyses and experiments. Several simple models were

discussed, which help to understand the fundamental properties of the long-time

deviation from exponential decay, and provide clues for its control and for setting

new experiments, which is indeed one of the major challenges. Post-exponential

decay is a subtle phenomenon, involving weak and low-energy signals, easy to per-

turb and hard to detect. However, recent progress in artiﬁcial mesoscopic semicon-

ductor structures or trap design and detection of ultra-cold atoms may offer access

to this regime for simple quantum systems. Further challenges are to gain a full

understanding of post-exponential decay in complex systems, such as organic

molecules in solution, as well as a more intuitive, physical interpretation of the

effect.

272 J. Martorell et al.

Acknowledgments We thank A. del Campo, F. Delgado, W. Van Dijk, L. Schulman, and E. Tor-

rontegui for useful discussions. We are grateful to DGES-Spain for support through grant FIS2006-

10268-C03-01; to UPV-EHU (GIU07/40); to NSERC-Canada for Discovery grant RGPIN-3198;

and to the Max Planck Institute for the Physics of Complex Systems in Dresden.

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