I Geiger counters and measurement

[WernerQH]

Suppose we have a Schrodinger's cat setup in a transparent box. I use $A(t)$ as the coefficient of the "undecayed" atom state, and $B(t)$ as the coefficient of the "decayed" atom state.

You are oversimplifying. You need more than two coefficients to describe the situation using the Schrödinger equation. In the description of the decay you need to include also the decay products (typically an escaping electron and anti-neutrino). Whether or not a Geiger counter is present, in the derivation of Fermi's Golden Rule we take the squared modulus of the coefficients of the final states (Born rule) and add them up. This would lead to a decay probability increasing with time like $t^2$, were it not for the density of final states. As time progresses, the contributing final states decrease, their distribution in energy becoming ever sharper (the width decreasing proportional to $t^{-1}$). In this way we arrive at a constant decay rate. And for all we know, radioactive decay happens even without Geiger counters present.

What really happens, is that a radioactive nucleus just sits there for a long time, and suddenly decays. A slowly (over the course of minutes, days, or years) evolving wave function exists only in the minds of theoreticians. You should not think that your two coefficients $A(t), B(t)$ have some direct physical meaning.

 

[jeeves]

Note that this posting (and imho the entire thread) does not belong to the quantum mechanics forum but to the interpretation subforum ;-)).

I object to this characterization. I do not believe I am asking anything about interpretation. I am simply asking: What is the state of the atom at various times? This is a question with empirically testable consequences. In principle, the atom could be measured, and its state determined. Interpretation has nothing to do with it. (Also, I did not mention collapse in that most recent post.)

What really happens, is that a radioactive nucleus just sits there for a long time, and suddenly decays. A slowly (over the course of minutes, days, or years) evolving wave function exists only in the minds of theoreticians. You should not think that your two coefficients $A(t), B(t)$ have some direct physical meaning.

For similar reasons, I hold that am not imbuing my coefficients with any physical meaning. I just want to know what the coefficients are at $t=1,2,3$ in the situation I described, and why they are that way. By "why" I mean I am asking for strictly empirical explanations (such as decoherence, lack of interactions, etc.), not anything to do with interpretation.

 

[Morbert]

I do not believe I am asking anything about interpretation. I am simply asking: What is the state of the atom at various times?

The problem is the answer to this question is interpretation-dependent (e.g. Some interpretations ascribe no objective character to the state).

An interpretation-independent question that might still serve your purposes might be something like: "Given preparation of the particle X and sequence of observations Y that occur at times {t1, t2, t3, ...} what are the likelihoods of the possible outcomes"

 

[jeeves]

Sure, we can gloss "What is the state?" as asking about statistics of identical ensembles. And my question "Why is the state that way?" can be glossed as "How does one derive the empirically observed statistics a priori by reasoning about Schrodinger evolution and entanglement?" .

The key point is: We all know what the empirically observed statistics are going to be. I would like to know how to correctly reason about the formalism of quantum mechanics to derive those statistics. (It's easy to reason incorrectly and predict a quantum Zeno effect if, for example, one views watching the cat as a "measurement" of the atom.) I presented some reasoning on the last page and am curious if it is right.

 

[Morbert]

Sure, we can gloss "What is the state?" as asking about statistics of identical ensembles. And my question "Why is the state that way?" can be glossed as "How does one derive the empirically observed statistics a priori by reasoning about Schrodinger evolution and entanglement?" .

The key point is: We all know what the empirically observed statistics are going to be. I would like to know how to correctly reason about the formalism of quantum mechanics to derive those statistics. (It's easy to reason incorrectly and predict a quantum Zeno effect if, for example, one views watching the cat as a "measurement" of the atom.) I presented some reasoning on the last page and am curious if it right.

So here his how I would model a decaying particle $\phi$ and detector $\psi$:

The particle and detector at some initial time $t_i$ are prepared as $$|\Psi_0\rangle = |\phi_s\rangle|\psi_\text{ready}\rangle$$ with unitary evolution $$U(t)|\Psi_0\rangle = a(t)|\phi_s\rangle|\psi_\text{not clicked}\rangle + b(t)|\phi_d\rangle|\psi_\text{clicked}\rangle$$ where $s$ and $d$ are for "survived" and "decayed" respectively. We can model the behaviour of the detector with a projective decomposition of the identity operator on the detector's state space $I = E_\text{clicked} + E_\text{not clicked}$ . We construct a chain of projectors for whatever temporal resolution we demand to represent the various possible outcomes. A simple example: we can model an outcome like "the detector clicks at time $T$" as $$E_\text{not clicked}(T-\delta t)\otimes E_\text{clicked}(T+\delta t)$$ which has a probability $$p_a = \mathrm{Tr}\left[E_\text{clicked}(T+\delta t) \otimes E_\text{not clicked}(T-\delta t) |\Psi_0\rangle\langle\Psi_0|E_\text{not clicked}(T-\delta t)\otimes E_\text{clicked}(T+\delta t) \right]$$

 

[PeroK]

(It's easy to reason incorrectly and predict a quantum Zeno effect if, for example, one views watching the cat as a "measurement" of the atom.) I presented some reasoning on the last page and am curious if it is right.

IMO, that's why it's not a good idea to try to relate QM directly to macroscopic behaviour. One thing you must do is deconstruct your macroscopic ideas before you can assume they are directly related to underlying QM phenomena. In other words, an "observation" at the macroscopic level may not map very well to measurements of a QM system, as generally understood. For example:

When you are watching a cat, what is actually happening? If the cat is poisoned, how long does it take to react? How long before you know it's dying rather than yawning or whatever?

These are not silly questions once you've taken the step of assuming that watching a cat is equivalent to a continuous measurement of a microscopic system on whose state its live depends!

In answer to my own question, watching a cat is an enormously fuzzy and imprecise set of measurements compared to the sort of precise measurement required in a QM experiment.

We could compare this with the recent experiments regarding the anomalous dipole moment of the muon. The mean lifetime of a muon is about $2.2$ microseconds. You have no ability in watching a cat to pinpoint changes in its macroscopic state over those timescales.

If you rigged up a system where the muon's decay was linked to the death of a cat by some mechanism, then you simply cannot claim that by "continuously" watching the cat you are continuously aware of whether the muon has decayed or not. The muon will have been created and have decayed in a timescale shorter than the timescales of any macroscopic phenomenon. Even if the cat reacted to poison in $1$ second, that is still 500,000 times longer than the muon's lifetime. And your observations of the cat and its reactions do not pin down the lifetime of the muon in any meaningful way.

 

[PeterDonis]

I am simply asking: What is the state of the atom at various times?

And the best answer that basic QM, with no interpretation involved, can give you is that the QM math is not telling you "the state of the atom" in the sense you appear to be using the term. It's just telling you how to predict probabilities for possible observations. There is a mathematical thingie in the machinery used to predict probabilities that is called "the state of the atom" (more precisely the "wave function" or "state vector" of the atom), but basic QM, without adopting any particular interpretation, does not make any claim whatsoever about that mathematical thingie's physical meaning. All basic QM says is that you can use the mathematical thingie to predict probabilities.

This is a question with empirically testable consequences.

No, it isn't. You can empirically test the predictions for probabilities of possible observations without bringing in any particular QM interpretation, but you cannot empirically test what "the state of the atom" is without bringing in some particular QM interpretation, because different interpretations make different claims about what "the state of the atom" is in any particular situation, even though they all agree on the predictions of probabilities.

 

[PeroK]

PS watching a cat is very different from continually subjecting an atom to a series of ultraviolet pulses to suppress its evolution to an excited state - as described in a quantum Zeno experiment:

https://en.wikipedia.org/wiki/Quantum_Zeno_effect#Experiments_and_discussion

Fuzzy, imprecise macroscopic measurements cannot be used to interrogate microscopic QM systems in the way you are imagining.

 

[jeeves]

And the best answer that basic QM, with no interpretation involved, can give you is that the QM math is not telling you "the state of the atom" in the sense you appear to be using the term. It's just telling you how to predict probabilities for possible observations.

Sure. As I said in post #64, I just want to know how to apply the math appropriately to correctly predict the empirically observed outcomes. We make predictions in QM by constructing a wave function and reading off the desired probabilities. I am asking only how to construct the wave function appropriately. I do not intend to give it any special ontological status, or interpret it metaphysically in any way. When I ask "What is the state of the atom?" I am only asking about the appropriate wave function to assign. I will endeavor to be more precise about this distinction.

PS watching a cat is very different from continually subjecting an atom to a series of ultraviolet pulses to suppress its evolution to an excited state - as described in a quantum Zeno experiment:

https://en.wikipedia.org/wiki/Quantum_Zeno_effect#Experiments_and_discussion

Fuzzy, imprecise macroscopic measurements cannot be used to interrogate microscopic QM systems in the way you are imagining.

I completely agree. I would just like to know how to correctly account for this (for all practical purposes, etc.) when writing down wave functions to predict things.

What you say in post #66 seems consistent with the description sketched in my post #59. (In particular, "You have no ability in watching a cat to pinpoint changes in its macroscopic state over those timescales.") Do you find the description I give there reasonable? (Again, please read "state" in an ontologically minimal way.)

 

[PeterDonis]

I just want to know how to apply the math appropriately to correctly predict the empirically observed outcomes.

You use the known probability of decay per unit time for whatever atom you have in your apparatus, and integrate that over the time since you prepared the atom to get a cumulative probability. In other words, you use the familiar math of radioactive decay that was already known even before QM was discovered (unless you are using a setup that involves QM corrections to the exponential decay law, in which case you have to use the decay probability math that includes those corrections).

You can, of course, write things in terms of a QM wave function with coefficients $A(t)$ and $B(t)$ in front of the "non-decayed" and "decayed" terms, where $B(t)$ is determined by the radioactive decay law and $A(t)$ is determined by normalization (the squared norm of the wave function as a whole must be 1). But this doesn't tell you anything new as far as probabilities go that the radioactive decay law doesn't already tell you. And if you're not picking any particular QM interpretation, you have no reason to even bother writing down the wave function in the first place since you're not assigning it any physical meaning and it doesn't add any ability to predict probabilities that you don't already have.

 

[jeeves]

You can, of course, write things in terms of a QM wave function with coefficients $A(t)$ and $B(t)$ in front of the "non-decayed" and "decayed" terms, where $B(t)$ is determined by the radioactive decay law and $A(t)$ is determined by normalization (the squared norm of the wave function as a whole must be 1). But this doesn't tell you anything new as far as probabilities go that the radioactive decay law doesn't already tell you. And if you're not picking any particular QM interpretation, you have no reason to even bother writing down the wave function in the first place since you're not assigning it any physical meaning and it doesn't add any ability to predict probabilities that you don't already have.

Yes, I would like to write things in terms of a wave function with coefficients $A(t)$ and $B(t)$. In particular, I would like to know whether what I wrote in post #59 is correct (regarding both the coefficients themselves and the reasoning used to reach them).

I disagree that if I am not picking a particular interpretation, there is no reason to do this. The reason is to see if my understanding of the QM formalism is correct. This knowledge will be useful for when I consider more complicated situations, where I do not have prior knowledge of the probabilities or a suitable classical approximation.

 

[PeroK]

Sure. As I said in post #64, I just want to know how to apply the math appropriately to correctly predict the empirically observed outcomes. We make predictions in QM by constructing a wave function and reading off the desired probabilities. I am asking only how to construct the wave function appropriately.

Usually you would be dealing with some specific microscopic phenomenon. Most introductory texts deal with the Hydrogen atom and the prediction of its spectrum. That's a classic solution to the SDE for the energy eigenstates (wavefunctions).

But, QM is not just about wavefunctions. If you want to try some genuine QM, there's an accessible and insightful treatment at undergraduate level here:

http://physics.mq.edu.au/~jcresser/Phys304/Handouts/QuantumPhysicsNotes.pdf

 

[jeeves]

I found the following quote in the paper

Peres, Asher (1989). Quantum limited detectors for weak classical signals. Physical Review D, 39(10), 2943–2950.

He writes of quantum Zeno and continuous measurements that:

This problem is peculiar to quantum systems. It disappears
in the semiclassical limit, where eigenvalues become extremely dense.
From the quantum point of view,
classical measurements are always fuzzy. This is why a
watched pot may boil, after all: the observer watching it
is unable to resolve the energy levels of the pot. Any hy-
pothetical device which could resolve these energy levels
would also radically alter the behavior of the pot. Like-
wise, the mere presence of a Geiger counter does not
prevent a radioactive nucleus from decaying. The
Geiger counter does not probe the energy levels of the
nucleus (it interacts with decay products whose Hamiltonian
has a continuous spectrum). As the preceding calculations show,
peculiar quantum effects, such as the
Zeno "paradox" occur only when individual levels are
resolved (or almost resolved).

This explanation is consistent with Nugatory's (and others) given earlier.

What does the parenthetical "it interacts with decay products whose Hamiltonian has a continuous spectrum" contribute to the explanation? Why is the fact that the Hamiltonian has continuous spectrum relevant to the argument that a Zeno effect cannot occur? The Hamiltonian of the decay product is only relevant after the decay happens, and once decay happens, we no longer have to worry about a Zeno effect.

 

[WernerQH]

Yes, I would like to write things in terms of a wave function with coefficients $A(t)$ and $B(t)$. In particular, I would like to know whether what I wrote in post #59 is correct (regarding both the coefficients themselves and the reasoning used to reach them).

With just two coefficients, you are dealing with a two-level system. There is no way that the solution can have any semblance to the decay of a radioactive atom. You cannot just assume coefficients. How would you specify the Hamiltonian? Have you ever done some simple exercises in QM?

What does the parenthetical "it interacts with decay products whose Hamiltonian has a continuous spectrum" contribute to the explanation? Why is the fact that the Hamiltonian has continuous spectrum relevant to the argument that a Zeno effect cannot occur? The Hamiltonian of the decay product is only relevant after the decay happens, and once decay happens, we no longer have to worry about a Zeno effect.

Do you know Fermi's Golden Rule? Have you looked at its derivation? Although the Born rule is usually stated as relating to "measurements", Fermi's Golden Rule is routinely applied in situations where it doesn't make sense to talk about measurements or observations. For example in the calculation of nuclear reaction rates in the interior of stars. And for the derivation of Fermi's Golden Rule it is irrelevant when (or if at all) a measurement is made. "Measurement" and "wave function collapse" still linger prominently in textbooks, but IMHO they are not actually a crucial part of the formalism. John Bell, in his essay "Against Measurement" railed against treating "measurement" as a fundamental process. I agree with him that measurements should be explained in terms of something more fundamental.

 

[gentzen]

Why is the fact that the Hamiltonian has continuous spectrum relevant to the argument that a Zeno effect cannot occur?

Because no classical measurement can ever resolve all energy levels of a continuous measurement. At least that is the argument Asher Peres gives in your quote:

From the quantum point of view, classical measurements are always fuzzy. This is why a watched pot may boil, after all: the observer watching it is unable to resolve the energy levels of the pot.
...
As the preceding calculations show, peculiar quantum effects, such as the Zeno "paradox" occur only when individual levels are resolved (or almost resolved).

For the more detailed explanation, he refers to "the preceding calculations".

 

[jeeves]

With just two coefficients, you are dealing with a two-level system. There is no way that the solution can have any semblance to the decay of a radioactive atom. You cannot just assume coefficients. How would you specify the Hamiltonian? Have you ever done some simple exercises in QM?

I am merely repeating the analysis in post #12 of this thread. Do you disagree with that analysis?

For the more detailed explanation, he refers to "the preceding calculations".

The preceding calculations do not seem to have a direct bearing on this particular example. The only relevant comment is:

One can even consider a passive detector,
such as a Geiger counter waiting
for the decay of a nucleus, but this situation does not fit
at all with our definition of a measurement. This setup is
best described as a single metastable system with several
decay channels.

 

[PeterDonis]

I would like to write things in terms of a wave function with coefficients $A(t)$ and $B(t)$.

I described how to do that in what you quoted, including how to determine $B(t)$ and $A(t)$.

In particular, I would like to know whether what I wrote in post #59 is correct (regarding both the coefficients themselves and the reasoning used to reach them).

As far as I can tell, your post #59 is saying that $A(t)$ and $B(t)$ get determined by the method I described in what you quoted (the radioactive decay law for $B(t)$ and normalization for $A(t)$) until the cat is observed to die; at that point we deduce that the atom has decayed and the state collapses to $A = 0$, $B = 1$ (the "atom decayed, cat dead" state). This is correct (as long as you are careful not to assign any physical meaning to "the state collapses" and to use it as a mathematical method only, to tell you what state you will now use for future predictions).

The reason is to see if my understanding of the QM formalism is correct.

The QM formalism in this case is so simple that, as I said before, it doesn't really add anything to your knowledge--including to your knowledge of how to work with the QM formalism for more complicated cases.

The best simple experiment I know of to exercise your QM formalism-fu is an experiment on a pair of entangled particles to test for violations of the Bell inequalities.

 

[jeeves]

As far as I can tell, your post #59 is saying that $A(t)$ and $B(t)$ get determined by the method I described in what you quoted (the radioactive decay law for $B(t)$ and normalization for $A(t)$) until the cat is observed to die; at that point we deduce that the atom has decayed and the state collapses to $A = 0$, $B = 1$ (the "atom decayed, cat dead" state). This is correct (as long as you are careful not to assign any physical meaning to "the state collapses" and to use it as a mathematical method only, to tell you what state you will now use for future predictions).

Great, thank you. This completely answers my original question. I will look into the Bell inequality violations; thank you for that also.

My only remaining worry is about that comment of Peres. As far as I can tell, he seems to be arguing that:

1. The Geiger counter does not probe the energy levels of the nucleus, only the energy of the decay product after the nucleus has decayed.
2. The decay product has continuous spectrum, and Zeno effects only apply in the presence of a discrete spectrum.
3. Therefore, there can't be a quantum Zeno effect.

But I don't see why we need step two. It seems valid to say that:

1. The Geiger counter does not probe the energy levels of the nucleus, only the energy of the decay product after the nucleus has decayed.
2. Since (for all practical purposes) the Geiger counter does not entangle with the nucleus before decay, the counter cannot prevent decay, and there can be no quantum Zeno effect.

So the continuity of the spectrum seems irrelevant.

 

[PeterDonis]

My only remaining worry is about that comment of Peres.

I don't think the Peres comment that was quoted really applies to the Geiger counter case. Or, for that matter, the cat case we have been discussing.

In the case of the counter and the cat, observation of those systems is only an indirect proxy for what we are interested in, namely, whether or not some particular atom has decayed. And because of the way those scenarios are set up, there is no meaningful "quantum Zeno" interaction between the counter, or the cat, and the atom we are interested in, until the atom decays. And that is true regardless of what the atom's energy levels are, whether they are continuous or discrete, how closely spaced they are, etc.

The kind of thing Peres is talking about is something different; he is talking about something like comparing measuring an atom in some unstable/metastable state directly vs. observing whether a cat is alive or dead. In both cases, you get something like a "binary" result (not decayed/decayed, alive/dead), but in the case of the atom, if you are measuring it directly (rather than relying on an indirect proxy like a Geiger counter), your measurement can involve an interaction (say probing the atom with a laser) that does have a "quantum Zeno" effect on the atom. But that is possible only if the states of the atom you are trying to distinguish are "spaced far enough apart", so to speak, that your measurement can reliably tell one from the other (or more precisely can reliably collapse the atom into one or the other).

In the case of observing a cat to see whether it is alive or dead, no such "quantum Zeno" effect is possible--you can't keep a cat alive just by constantly watching it. And that is because, unlike a single atom, a cat has an enormous number of possible states, which are "spaced very close together", so to speak, and what we refer to as "alive" and "dead" are not single states of the cat but huge subspaces of the cat's state space, and our observations cannot reliably force the cat into just one single state; we can't "collapse" the cat into some single desired state in its "alive" state space (which is what we would have to do to have a "quantum Zeno" effect on the cat) by observing it. In fact we can't do that by any means we have at our disposal now or in the foreseeable future.

To put this another way: probing a single atom with a probe laser can have a significant effect on its dynamics, but looking at a cat and seeing that it's alive does not have any significant effect on its dynamics. That's why we can do quantum Zeno experiments with atoms but not with cats.

 

[Morbert]

How does Peres's account for the disappearance of the zeno effect in "detection of decay product" experiments (see post #73) inform the zeno paradox given by Home and Whitaker?

They establish the paradox in a thought experiment involving a hollow sphere of inward-facing detectors surrounding the atom in question. They set the radius of the sphere to be very large, and define a measurement as a rapid contraction of the sphere, such that if the particle has decayed (survived), the products will be detected (not be detected) during the contraction, and hence one of the two possible energy levels of the atom will be registered. A correlation is therefore established between detector and energy of the atom even if no decay product was detected, and without any ambiguity of continuous measurement. They claim that, using only orthodox quantum theory, the choice of sequences of measurements would affect the decay statistics even though no direct interaction takes place.

 

[gentzen]

How does Peres's account for the disappearance of the zeno effect in "detection of decay product" experiments (see post #73) inform the zeno paradox given by Home and Whitaker?

Even so I disagree with vanhess71 that this entire thread should have been in the Quantum Interpretations and Foundations subforum, the abstract of that paper feels suspiciously like a typical interpretation paradox:

... Gedanken experiments are outlined which illustrate the key features of the paradox, and its implications for the realist interpretation are discussed.

I know that the abstract also says: "It is demonstrated that collapse of state-vector is not a requirement for the paradox, which is independent of interpretation of quantum theory." But that statement too has more to do with interpretation than with plain "calculate and be happy" quantum mechanics, from my POV.

 

[DrChinese]

How does Peres's account for the disappearance of the zeno effect in "detection of decay product" experiments (see post #73) inform the zeno paradox given by Home and Whitaker?

They establish the paradox in a thought experiment involving a hollow sphere of inward-facing detectors surrounding the atom in question. They set the radius of the sphere to be very large, and define a measurement as a rapid contraction of the sphere, such that if the particle has decayed (survived), the products will be detected (not be detected) during the contraction, and hence one of the two possible energy levels of the atom will be registered. A correlation is therefore established between detector and energy of the atom even if no decay product was detected, and without any ambiguity of continuous measurement. They claim that, using only orthodox quantum theory, the choice of sequences of measurements would affect the decay statistics even though no direct interaction takes place.

I couldn't find a free version of the reference. I did find a similar paper by the same 2 authors:

https://www.fi.muni.cz/usr/buzek/zaujimave/home.pdf
A Conceptual Analysis of Quantum Zeno; Paradox, Measurement, and Experiment

I don't see where they use QM to make a testable prediction that decay rate is in any way affected by the presence (or lack thereof) of Geiger counters (measurement devices). Given these papers were written 25+ years ago, I would think we would have heard about that if this novel prediction had made a splash. On the other hand, a more recent analysis of the Zeno effect via indirect measurement finds there is no such effect (in contradiction).

https://arxiv.org/abs/quant-ph/0406191
"We study the quantum Zeno effect in the case of indirect measurement, where the detector does not interact directly with the unstable system. Expanding on the model of Koshino and Shimizu [Phys. Rev. Lett., 92, 030401, (2004)] we consider a realistic Hamiltonian for the detector with a finite bandwidth. We also take explicitly into account the position, the dimensions and the uncertainty in the measurement of the detector. Our results show that the quantum Zeno effect is not expected to occur, except for the unphysical case where the detector and the unstable system overlap."

A quantum system (such as a radioactive isotope) does not change (its statistical decay rate) in any way due to the presence or absence of an indirect measurement system designed to detect a decay product. Interestingly, it does, however, react to gravitational effects (time dilation).

 

[Morbert]

The Home and Whitaker paper references a 1980 paper by Peres which gives a time-evolution that is "valid for small times" $$|\left(\phi,e^{-iHt}\phi\right)|^2 \approx 1-(\Delta H)^2t^2+\cdots$$They use this in their derivation of the Zeno's paradox, which is peculiar to me. When I use the more general time evolution of an exponential decay, everything works out fine.

 

[jeeves]

The Home and Whitaker paper references a 1980 paper by Peres which gives a time-evolution that is "valid for small times" $$|\left(\phi,e^{-iHt}\phi\right)|^2 \approx 1-(\Delta H)^2t^2+\cdots$$They use this in their derivation of the Zeno's paradox, which is peculiar to me. When I use the more general time evolution of an exponential decay, everything works out fine.

Yes, the exponential distribution is memoryless and cannot produce a Zeno effect. The effect relies on the deviations from exponential decay at short times. Quoting Wikipedia:

Unstable quantum systems are predicted to exhibit a short-time deviation from the exponential decay law. This universal phenomenon has led to the prediction that frequent measurements during this nonexponential period could inhibit decay of the system, one form of the quantum Zeno effect.

References are available in the article, and the calculation is done in equations (11) through (15) of the Home–Whitaker article you posted. I think it's also discussed somewhere in Sakurai, but my memory may be unreliable.

 

[DrChinese]

1. Yes, the exponential distribution is memoryless and cannot produce a Zeno effect.

2. The effect relies on the deviations from exponential decay at short times. Quoting Wikipedia: ...

1. Agreed.

2. I would not agree that there is such a deviation for radioactive decay. As far as I know, there isn't any indirect quantum measurement effect that would inhibit decay even in that program ("short" times).

In fact, I am not sure that there is a hypothetical short time program (regardless of what Wikipedia says). I'm also not sure what *direct* quantum Zeno effects have been studied at any level regarding radioactive decay.

 

[Morbert]

I am wondering if the model for time-evolution used in the [Home + Whitaker] papers above just isn't correct with these negative-measurement thought experiments. Assuming like before we have a system ($\phi$) and passive detector ($\psi$) that evolves like so $$U(t)|\phi_0, \psi_0\rangle = \alpha(t)|\phi_s, \psi_s\rangle + \beta(t)|\phi_d, \psi_d\rangle$$ The probability that the atom has survived until some time $T$ is $$\langle\phi_0,\psi_0|U^\dagger(T)|\phi_s,\psi_s\rangle\langle\phi_s,\psi_s|U(T)|\phi_0,\psi_0\rangle = |\alpha(T)|^2$$I can add an identity operator, evolved to some intermediate time $t$ without changing the result like so $$\langle\phi_0,\psi_0|I^\dagger(t)U^\dagger(T)|\phi_s,\psi_s\rangle\langle\phi_s,\psi_s|U(T)I(t)|\phi_0,\psi_0\rangle = |\alpha(T)|^2$$ But the identity operator can be projectively decomposed $I = |\phi_s,\psi_s\rangle\langle\phi_s,\psi_s| + |\phi_d,\psi_d\rangle\langle\phi_d,\psi_d|$ which is exactly what we would do to compute probabilities for an intermediate measurement. This intermediate measurement therefore can't change the probability computed for survival of the atom at $T$, so long as we have a valid $U$.

 

[Morbert]

https://arxiv.org/abs/quant-ph/0406191
"We study the quantum Zeno effect in the case of indirect measurement, where the detector does not interact directly with the unstable system. Expanding on the model of Koshino and Shimizu [Phys. Rev. Lett., 92, 030401, (2004)] we consider a realistic Hamiltonian for the detector with a finite bandwidth. We also take explicitly into account the position, the dimensions and the uncertainty in the measurement of the detector. Our results show that the quantum Zeno effect is not expected to occur, except for the unphysical case where the detector and the unstable system overlap."

A quantum system (such as a radioactive isotope) does not change (its statistical decay rate) in any way due to the presence or absence of an indirect measurement system designed to detect a decay product. Interestingly, it does, however, react to gravitational effects (time dilation).

Interesting paper. Thanks!

 

[jeeves]

I would not agree that there is such a deviation for radioactive decay. As far as I know, there isn't any indirect quantum measurement effect that would inhibit decay even in that program ("short" times).

In fact, I am not sure that there is a hypothetical short time program (regardless of what Wikipedia says). I'm also not sure what *direct* quantum Zeno effects have been studied at any level regarding radioactive decay.

There must be. An exact exponential decay law at all times is inconsistent with the laws of quantum mechanics. Refer to: Greenland, P. T. Seeking non-exponential decay. Nature 335, 298 (1988).

Further, there is indeed a program of theoretical predictions and attempted experimental tests of deviations from exponential decay in various contexts. For experimental evidence in quantum tunneling, see: Experimental evidence for non-exponential decay in quantum tunnelling, Nature 1997. I don't think experimental evidence has been found for radioactive decay specifically yet, but following the references in those papers (and citations to those paper) will lead you to theoretical work on the issue.

Interesting paper. Thanks!

My inclination is to say that Home and Whitaker are simply wrong about their thought experiment. We have come to the conclusion in this thread that when the detector is not moving (e.g. it is a cat in a box), non-observation does not count as a "measurement" and will not produce a quantum Zeno effect, because the cat/detector does not directly probe the nucleus, it only interacts with the decay product after the decay.

If you accept that explanation, I am not sure how moving the detector around in space changes anything. Non-detection still won't count as a measurement because non-detection still doesn't produce an interaction with the nucleus (or anything else in that thought experiment).

 

[DrChinese]

1. An exact exponential decay law at all times is inconsistent with the laws of quantum mechanics. Refer to: Greenland, P. T. Seeking non-exponential decay. Nature 335, 298 (1988).

1. Here is a link to your reference:

https://www.nature.com/articles/335298a0

However, it's not really a good reference for your point, as I question whether Greenland's analysis is generally accepted (not well cited). When Greenland compared his prediction to actual relevant experiment, he found no support:

"Neither isotope revealed any deviation from exponential decay, and nor has any other test."

His actual prediction was: "In fact the decay of an isolated quantum state can never be exponential." So, there's that.

My point in all this is that deviation from exponential decay really has nothing to do with whether or not a Geiger counter type measurement can induce a quantum Zeno effect in a radioactive sample (as the OP seems to suggest). It cannot.2. I could not find a free link to your second reference. But I found another from the similar author group that might be of interest:

https://arxiv.org/abs/quant-ph/0104035
Observation of the Quantum Zeno and Anti-Zeno effects in an unstable system
"We report the first observation of the Quantum Zeno and Anti-Zeno effects in an unstable system. Cold sodium atoms are trapped in a far-detuned standing wave of light that is accelerated for a controlled duration. For a large acceleration the atoms can escape the trapping potential via tunneling. Initially the number of trapped atoms shows strong non-exponential decay features, evolving into the characteristic exponential decay behavior. We repeatedly measure the number of atoms remaining trapped during the initial period of non-exponential decay. Depending on the frequency of measurements we observe a decay that is suppressed or enhanced as compared to the unperturbed system."

This effect results from direct "measurement", rather than something indirect (like a Geiger counter).

 

[jeeves]

1. Here is a link to your reference:

https://www.nature.com/articles/335298a0

However, it's not really a good reference for your point, as I question whether Greenland's analysis is generally accepted (not well cited). When Greenland compared his prediction to actual relevant experiment, he found no support:

"Neither isotope revealed any deviation from exponential decay, and nor has any other test."

His actual prediction was: "In fact the decay of an isolated quantum state can never be exponential." So, there's that.

Here is a link to the second reference: https://web.archive.org/web/2010033...chargement/Optique_Quantique/Raizen_decay.pdf

The Greenland article is a short expository article, and I would not expect it to be highly cited.

Note that the Khalfin paper cited in the paper I just linked (reference 1) predicting corrections to exponential decay has 635 citations (according to Google scholar). This is also discussed in reference 3 (708 citations), and some others. I believe the observation that the decay cannot be exactly exponential is even made in standard graduate textbooks; the Nature article cites Ballentine's book. I conclude that deviations from exponential decay are generally accepted by the physics community.

I agree that direct experimental evidence of these deviations does not seem to exist (to my knowledge). This appears to be a limitation of our experimental abilities (specifically probing short enough time scales), not of our theoretical understanding. I would reconsider my view if there were a paper that claims to access the time scales where quantum Zeno-permitting deviations are predicted for radioactive decay and finds no effect. However, my understanding is that no such paper exists, and my response is then simply that absence of evidence is not evidence of absence.

I agree these deviations are irrelevant to the Geiger counter example.