From: gilligan@bldrdoc.gov (Jonathan M. Gilligan)
Newsgroups: sci.physics
Subject: Re: Quantum Zeno Effect, Part 1.
Date: 6 Oct 92 23:28:27 GMT
Organization: National Institute of Standards and Technology

In article <1992Oct3.134653.20562@oracorp.com> daryl@oracorp.com
(Daryl McCullough) writes about the quantum Zeno effect, but makes
some basic errors:

In fact, the quantum Zeno paradox (the inhibition of spontaneous decay
by frequent measurement) does not occur in spontaneous decay (such as
radioactive decay of nuclei) so long as the decay is exponential.
Here is a proof:

  Start with the particle in an unstable state |\psi_0> with lifetime \tau.

  The probability that after 
  time t, the particle has
  not decayed is:               P(t) = exp(-t/\tau)

  Peek N times between 0
  and t. Now the probability
  that the particle has not 
  decayed is:                   P_N(t) = (exp(-t/N \tau)^N = exp(-t/\tau)
                                independent of N.

Thus, the quantum Zeno paradox (inhibited spontaneous decay) occurs
_only_ if decay is nonexponential at short times (nonexponential decay
is predicted by theory, but has never been oberved). A good paper by
Ghirardi, Omero, Weber, and Rimini (Nuovo Cimento 52A, 421 (1979))
shows the difficulty (impossibility) of detecting this phenomenon
using practical detectors.

Ghirardi et al also introduce a conceptual problem with the Zeno paradox:
a measurement of duration t can only resolve energies with a spread

   \Delta E >= \hbar/t 

As we increase the frequency of measurements, we necessarily decrease
the duration of each measurement and hence increase the energy spread.
The nonexponential decay that leads to the Zeno paradox results from
the assumption that the energy spectrum of a decaying particle is
bounded (i.e., is not truly a Lorentzian all the way from E=-infinity
to E=+infinity). Since a measurement of vanishingly small duration
would put the particle in a state with an infinite energy spread,
would perturb the dynamics on exactly the time scale over which the
nonexponential behavior takes place, and hence prevents the Zeno
paradox from being observed.

The ``quantum Zeno effect,'' (as distinct from the paradox) consists
of the inhibition of a driven transition (rather than a spontaneous
one). The initial proposal was by R. J. Cook (Phys.  Script.  T21, 49
(1988)). The experiment was by Itano, Heinzen, Bollinger, and Wineland
(Phys. Rev. A 41, 2295 (1990)).

A resonantly driven transition between |\psi_0> and |\psi_1>
has P(t) = (1+cos(\Omega t))/2, where P(t) is the probability 
that a system initially in |\psi_0> will be found in |\psi_1> 
at time t, and \Omega is the Rabi frequency. Now,

P_N(t) = ((1 + cos(\Omega t/N))/2)^N

in the limit of large N, this becomes

P_N(t) = (1 - (t^2/4N^2))^N   ---> 1

So in this case, the transition is indeed suppressed by frequent measurement.

>In the limit as N -> infinity, keeping T fixed, we get P(b,T) = 0. In
>other words, if you measure a value often enough, you can prevent it
>from changing. An example where the quantum zeno effect applies is in
>observation of alpha decay. As I understand it, "watching" a
>radioactive atom closely can make the rate of alpha decay go to zero,
>and this has actually been observed.

To my knowledge, the nobody has observed a suppression of radioactive
decay (this would be the quantum Zeno _paradox_, not _effect_). Such a
measurement seems unfeasible because you would have to measure rapidly
on the timescale over which the decay was non-exponential, which is
vanishingly small. Since nobody has seen nonexponential decay, it must
be impractical to look for the Zeno paradox. Ghirardi et al (op cit.)
discuss some of the problems in detail.

>Now, this effect is easily explained using the collapse of the wave
>function; every time you observe the state of the particle, you
>collapse its state back into an eigenstate of the observable. However,
>it seems that most physicists these days don't really believe in the
>collapse of the wave function as a physical effect.

My experience is that most physicists I know believe in wavefunction
collapse and a few very vocal ones do not believe. John Bell was a
famous example of someone who was very uncomfortable with collapse. 
L. E. Ballentine, is one of the more articulate advocates of alternative,
statistical, interpretations of quantum mechanics. See, for example,
his ``The Statistical Interpretation of Quantum Mechanics,'' Rev. Mod.
Phys. 42, 358 (1970).

>Some view it as simply a matter of a change of knowledge; that is,
>collapsing the wave function simply represents a collapse in the set
>of possibilities, and has no more physical meaning than the
>corresponding collapse that occurs when you peak at a coin after you
>have flipped it. However, it is hard to reconcile a change in
>knowledge with an actual physical effect such as the slowing of the
>decay rates of radioactive particles.

E. Block and P.R. Berman (Phys. Rev. A 44, 1466 (1991)) and,
independently, V. Frerichs and A. Schenzle (Phys. Rev. A 44, 1962
(1991)) have demonstrated that the quantum Zeno effect is predicted by
a density-matrix formalism with no wavefunction collapse. To my
knowledge, the statistical versus collapse debate has more to do with
ontology than physics, as the two make identical predictions for all
experiments that have been done to date.

---Jon
-- 

Disclaimer --- The government probably disagrees with my opinions.