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course, the experiments are not wrong: what we will seeis that at both very short and very long timescales thedecay deviates from exponential, but that exponentialbehaviour is seen interpolating between the two.

SHORT TIME BEHAVIOUR

We start by giving a mathematical argument whyquantum mechanical decay cannot be exponential onshort times scales, and then proceed to describe thephysics in subsequent section.

Argument for Non-Exponential

Decay at Short Times

We will consider a system that is in a state s at timet = 0. We will make only one assumption: we will assumethat the energy of the initial state, s|H|s is finite. Theevolution of the system is given by:

ih

t|(t) = H|(t) (6)

with |(0) = |s. Defining cs(t) = s|(t), the surviv-ing amplitude, gives:

iha(t) = s|H| (7)

At t = 0 with a(0) = 1, we get:

iha(t) = s|H|s = Es (8)

which is real and finite. Thus a(0) is purely imaginary.

The probability of survival of the initial state is Ps(t) =a(t)a(t): taking the derivative and using a(0) = 1 gives:

dPsdt

= a(0) + a(0) = 0 (9)

since a(0) is imaginary. Thus the slope of the decay mustbe zero at t = 0, which is clearly violated by exponentialdecay. Quantum mechanics predicts that the decay curveof an excited state is flat at small times. As we shall seelater, this will lead to some interesting experimentallyaccessible effects. First, however, we will examine moreclosely the steps that earlier had lead us to exponentialdecay.

A Re-examination of Exponential Behaviour:

What Went Wrong?

Exponential decay is a familiar result from classicalstatistics, which describes the decay of a system throughan underlying stochastic process. The classical argumentis as follows[1]: IF we know that the system is in a state sat a time t and the probability of decay in a time interval

is W dt, then the probability that it has not decayed is(1 W dt). Thus if the probability that it was in state swas Ps(t), then:

Ps(t + dt) = Ps(t) (Prob. of no decay in dt)

= Ps(t)(1 W dt) (10)

which leads to exponential decay.

The flaw in our argument can be traced to this laststep: we have assumed that Ps is changing entirely dueto transitions out of state s, and have not accounted forthe transitions from other states back into s. In quan-tum mechanics, the probability Ps(t + dt) is given by thesquare modulus of the complex amplitude:

Ps(t + dt) = |cs(t + dt)|2

= |cs(t) +dcsdt

dt|2 (11)

where

dcsdt =

1

ihk

Vksckseikst

(12)

at first order in perturbation theory. In general, we havethis equation to (10):

|cs(t) +dcsdt

dt|2 = |cs(t)|2(1 + W dt) (13)

The above can give approximate equality in certain sit-uations. One such situation is the coupling of an initialstate to a continuum of states densely packed around thefinal state. In our expression (12), we the have a sumover a large number of states. At t = 0, transitions from

s start feeding into the amplitudes of the final states k.At the same time, the states k start to feed back intothe amplitude of state s: however, each state feeds backwith a different phase eikst. For a continuous set of finalenergies, after some coherence time T, the phase factorswill all be uncorrelated. There will be a net cancella-tion of the feedback of the states k back into s, and thestate s will then decay exponentially as we predicted withclassical statistics.

We see that for times less than the coherence timeof the continuum, the evolution of the unstable excitedstate will be coherent. Treatments of coherent evolutionsuch as Rabi oscillations are earmarked by a initial slope

of zero, and we see that our physical interpretation isconsistent with the mathematics of Section (II.A).

Some Interesting Consequences:

The Quantum Zeno Effect

One may question the relevance of this short time be-haviour on the physics of the problem: it is more thatlikely that the decoherence time of the continuum states


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is very fast, and it seems unlikely that one would beable to observe the evolution of the system on thesetimescales.

Consider an experiment, however, where we makerapid successive measurements on the system. One ofthe postulates of quantum mechanics is that the pro-cess of making a measurement results is a collapse of thewavefunction onto one of the eigenstates: essentially, in

performing a measurement, we are preparing the sys-tem again at t = 0. In the context of density matrices,the system is described by a diagonal density matrix. Fora two level system, we would have:

(t = 0) =

1 00 0

As time evolves, two things happen: first, 22 increases atthe expense of 11, but we also develop coherence termsin the off diagonal elements. If at a later time t = wemake a measurement of the system (with an observablewhose eigenstates are the 1,2 states), we collapse thesystem onto one of the two states:

(t = 0) = 11

1 00 0

or

(t = 0) = 22

0 00 1

If we have collapsed onto the 1 state, we have effec-tively reset the system to the initial state with a re-duced initial amplitude. An important note is that theprocess of making the measurement sets the off-diagonalterms to zero: ie. it destroys the coherences. Mak-ing repeated measurements of the system at and interval smaller than the decoherence time, the system wouldevolve as shown in Figure (1). We can see that the decayof state 1 is slower than if we had not been making theobservations.

After n such observations, the probability that wereare in the 1 state is given by:

P1(n) = [11()]n

We note that for < T, we have inhibited the decayof the state. We would predict that for 0 (contin-uous observation), that the system would be preservedindefinitely in the unstable excited state. The inhibition

of quantum transitions through observation is referred toas the quantum Zeno effect, and has been confirmed byexperiment.

Experimental Evidence of the Quantum Zeno Effect

In the previous section, we had discussed the possi-bility of inhibiting spontaneous emission by frequent ob-servation. This was the original context in which the

P(t)

t3t2t1Time

FIG. 1: The short time behaviour of Ps(t) for repeated mea-surements at times t1, t2, and t3 made in the coherent evolu-tion regime.

quantum Zeno effect was proposed. However, makingsuccessive observations on a system on timescales muchfaster than the decoherence time of the continuum is avery difficult experiment, and has yet to be achieved, tothe authors knowledge.

Note that the physics of the quantum Zeno effect is re-lated to the collapse of the wavefunction on observation,and is not necessarily tied to a system that is decaying.The effect can be seen in any system where one makesobservations at a rate much faster than the decoherencetime. The longer the decoherence time, the easier themeasurement.

Perhaps it is not surprising then that the experimentalsystem where the quantum Zeno effect was first convinc-ingly demonstrated was the inhibition of coherent tran-sitions in a two level system with Rabi oscillations.

The experiment was performed by Itano et al. [2] atNIST Boulder in 1989. In idealised picture, the systemthey were studying was a three level atom as shown inFigure (2). Level 1 is the ground state, level 2 is an ex-cited metastable state, and level 3 is an excited state witha strongly allowed transition to level 1. There are two ex-ternal fields applied: there is a strong field on resonancewith the 1,2 transition that drives the two states into co-

herent Rabi oscillations and there is a short pulsed fieldresonant on the 1,3 transition used the probe/observe theatom.

The idea is that we start with the system in the 1 stateat time t = 0 and apply a pulse on the driving field. Ifwe make no intermediate observations, at the end of thedriving pulse, the atom will be entirely in the 2 state.

We then start applying short probe pulses. If the atomis in the 1 state, if will be resonant on the 1,3 transi-tion and it will scatter photons with its large resonant


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FIG. 4: Graph of the experimental and calculated 1 2 transition probabilities as a function of the number of measurementpulses. The decrease of the transition probability with increasing n demonstrates the quantum Zeno effect. From Itano et al.

= BW cs(t) = eWt

As we already know, although it may provide a goodfit to the data, the Breit-Wigner distribution not a phys-ically correct energy distribution function since it doesnot reproduce coherent evolution at small times. Fur-thermore, it has an infinite variance, which is also some-what unphysical.

It will not be covered in detail here (see reference [4] foran excellent discussion), we can predict deviations fromexponential decay at asymptotically large times based on

very general considerations on the energy distribution.For example, if the continuum has a ground state it canbe shown that the probability at large times will decayas a power law, Ps(t) t

p.If the continuum has no ground state (for example, the

coupling of an atom to a uniform electric field), the prob-ability will decay faster that exponential, Ps(t) e

t/tp.Again, any interested readers would be recommended toexamine reference [4].

Unfortunately, experimental observation of these ef-fects is much more difficult than with the short time be-haviour. For example, considering the decay of a chargepion, it is estimated in reference [5] that by the time this

behaviour sets in, the surviving pion probability is lessthat 1080.

CONCLUDING REMARKS

By very general arguments, we have shown that theevolution of a quantum mechanical system must be atsome small time scale and will always begin with zeroslope.

This contradicts the common and well accepted exper-imental fact that exponential decay is observed in a num-ber of quantum systems. In going back and looking morecarefully at a derivation of exponential decay in quantummechanics, we have gained a deeper understanding of theway in which exponential decay arises through cancellingphase contributions from continuum states.

Possible ways to observe this very short time scale be-haviour of decaying systems lead us consider the quan-tum Zeno effect. While the quantum Zeno effect hasbeen confirmed by experiments, it has not yet been ap-

plied convincingly in experiment for detecting the earlycoherent evolution of a state that would otherwise displayexponential decay.

We concluded with a brief discussion of another wayin which quantum mechanics disobeys exponential de-cay, this time as asymptotically long times, by imposinga different and independent set of physically motivatedconstraints on the properties of the quantum mechanicalsystem. It would seem, however, that this decay regimeis outside the realm of experiment.

[1] E. Merzbacher Quantum Mechanics, vol. I, (W. A. Ben- jamin, New York, 1966), Ch. 18

[2] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J.Wineland, Phys. Rev. A 41, 2295 (1990)

[3] A. G. Kofman and G. Krurizki, Phys. Rev. A 54, 3750(1996)

[4] K. Uniikrishnan, Phys. Rev. A 40, 41 (1998)[5] C. B. Chiu, E. C. G. Sudarshan, B. Misra, Phys. Rev. D,

16, 520 (1977)

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