A macroscopic clock model to solve the paradox of Schrödinger’s cat

We propose detecting the moment an atom emits a photon by means of a nearly classical macroscopic clock and discuss its viability. It is shown that what happens in such a measurement depends on the relation between the clock’s accuracy and the width of the energy range available to the photon. Implications of the analysis for the long standing Schrödinger’s cat problem are reported.


A meaningful question?
Does it make sense to talk about the moment the atom decayed?Not always.Decay of a metastable state is often described by a model 8,9 where a discrete state |e� , corresponding to an excited "atom" with energy E e , is connected to "reservoir" states {|E r �} , representing an "atom" in its ground state, E g = 0 , plus an emitted "photon" with energy E r .The corresponding Hamiltonian takes the form, where �(E r ) is the matrix element responsible for the transitions between the system's discrete and continuum states, i.e., for the decay of the excited atom.In the continuum limit, whenever the final probabilities are added up, one can replace the sum r by an integral ρ(E r )dE r , where ρ(E r ) is the density of the reservoir states 8 .
After preparing an atom in its excited state, and waiting for t seconds, one can find a photon with an energy E r .Expanding the transition amplitude �E r | exp(−i Ĥt)|e� in powers of V reveals a variety of scenarios where the photon, emitted for the first time at τ first is re-absorbed and re-emitted until settling down into its final state |E r � at some τ last .Thus, the emission process may not occur via a single transition to the ground state, but can have a finite duration τ last − τ first .Measuring even "the first passage time" τ first presents considerable difficulties 10 , and we do not know if τ last − τ first can be measured at all.
A helpful exception is the first order transition in the weak coupling limit, which does indeed occur via a single jump, (1)
Yet, there is a case where the transition proceeds via a single jump, and the Zeno effect does not occur.We will discuss it next.

Results and discussion
The wide band (Markovian) case In the Markovian (wide band) approximation 8 , both �(E r ) and ρ(E r ) , are taken to constant, very small and very big respectively, i.e. → 0 , ρ → ∞ , in such a manner that a product ρ� 2 remains finite, The model admits an exact solution for any Ŵ , and there is no need to limit oneself to the first order approxi- mation (5).The amplitudes of the four possible processes are given by 8 : By (4), atom's decay is exponential at all times, and by (5), the energy distribution of the emitted photons is Lorentzian Further helpful to our purpose is the fact that, according to Eqs. ( 5) and ( 6), the atom can emit a photon only once, and never re-absorbs it afterwards.The moment of transition can, therefore, be defined at least in terms of the virtual scenarios available to the system.With the purely exponential decay in Eq. ( 4) frequent checks of the atom's state do not affect the decay rate, which stays the same with or without such checks [hence, the adjective Even so, destruction of coherence between the moments of emission in Eq. ( 5) must change something akin to the interference pattern in a double slit experiment.Below we will show that it is the energy spectrum of the emitted photons (8) that is affected by the measurement's accuracy.

A quantum hourglass and its macroscopic limit
Suppose Alice the experimenter, does not wish to subject the system to frequent checks, and prefers instead to have, at the end of the experiment, a single record of the moment the atom decayed.For this purpose, she might consider a clock which stops at the moment the atom leaves its excited state.The clock could be an hourglass, in which case the number of the sand grains escaped, would tell Alice the time of the event.A quantum analogue of an hourglass is not difficult to find.Alice could use an array of identically polarised distinguishable spins precessing in a magnetic field, and estimate the elapsed time by counting the spins which have been flipped.Alternatively, Alice can employ a large number of non-interacting bosonic atoms, N >> 1 , initially in the left well of a symmetric double well potential (see Fig. 1).
The clock Hamiltonian given by where â+ R(L) creates a boson in the right (R), or left (L) well, and ω is the hopping matrix element and the ampli- tude of finding n bosons in the right well is easily found to be where C N n = N! n!(N−n)! is the binomial coefficient.Alice can choose ωt << 1 , so that the Rabi period of a single boson is very large, and have a practically irreversible flow of bosons from left to right.She can also assure, by making N very large, that the mean number of atoms in the right well is also large (except perhaps at very short times), n(t) ≡ p(t)N >> 1 .Under these conditions, binomial distribution under the root sign in (10) can be approximated by a normal distribution 12 , and after some algebra [see the "Methods" section (Derivation of Eq. ( 11))] we have Alice can now count the atoms in the right well and use t n in Eq. ( 11) as an estimate for the elapsed time.Equation (11) shows that her estimate is likely to be within an error margin t of the true value t.A good clock is the one which has a small relative error.If ωt is kept constant while N → ∞ , the error tends to zero, since �t/t n = 1/ √ n ≈ 1/(ωt √ N) ∼ 1/ √ N , and with many bosons Alice has a good clock (see Fig. 2).
(11)  We make a further remark.As N → ∞ , a large system of independent particles begins to develop certain clas- sical properties 13,14 [see also the "Methods" section (Derivation of Eq. ( 11))].For example, denoting one-partial states in the wells as |L� and |R� , and preparing the bosons in a state |� clock Bose (0 , where u LL (t) = cos(ωt) and u RL (t) = −i sin(ωt) are the matrix elements of the one-particle evolution operator.The evolved state |� clock Bose (t)� is not an eigenstate of an operator n = N i=1 |R� i �R| i = â+ R âR , which gives the number of bosons in the right well.However, expanding it in the eigenstates of n , n|n� = n|n� , n = 0, 1 . . .N , one finds 14 the coefficients localised in a range ∼ √ N around a mean value n(t) A similar localisation would occur if |�(t)� expanded in any basis, and this has important consequences.Firstly, one can accurately measure n (or any other operator 14 ) and obtain a result close to its mean value ( ∼ N ) with an error margin ∼ √ N .This is a good measurement, since its relative error tends to zero.Secondly, one can measure it inaccurately, e.g., by using a von Neumann pointer prepared in a Gaussian state of a width ∼ N 1/2+ε , where 0 < ε < 1/2 14 .This is still a good measurement since ∼ N 1/2+ε /N → 0 , but also one which in the limit N → ∞ leaves the state ( 12) almost intact, since N 1/2+ε / √ N → ∞ [see the "Methods" section (A macroscopic clock) for details].Alice can keep reading this macroscopic nearly classical clock without affecting its operation, like she would do with a classical wrist watch.

A clock which first runs and then stops
Next Alice needs to make the clock run until the moment the atom emits a photon.6][17] ).The corresponding Schrödinger equation is easily solved, and the amplitude for the composite {(a)tom+(ph)oton + clock} , starting with the right well empty, to end with n bosons there, is found to be [see the "Methods" section (Coupling the clock to a quantum system)] where A clock Bose (n ← 0, τ ) is given by Eq. ( 11), and A a+ph (j ← e, t|τ ) is the amplitude for the atom-photon system to reach a final state |e� or |E r � after remaining in |e� for exactly τ seconds, where Ûa+ph (t|τ ) is the conditional evolution operator.This is clearly the desired result.The clock runs only while the atom remains in the excited state, and the amplitudes are added for all possible durations τ , which may lie between 0 and t.The integral in Eq. ( 15) is evaluated by noting that adding πe to ĤM only shifts the energy of the discrete state E e by [see the "Methods" section (Timing the transition in the Markovian case)].The result ( 0 confirms what is already known from Eqs. ( 4) and ( 5).An atom, still found in the excited state at t, has remained in that state all the time.An atom, found in the ground state, has not returned to the excited state after making a single transition at some τ between t = 0 and t.
Alice the practitioner can now prepare the atom in its excited state, couple it with a "good" clock (11), wait until time t, and then measure the energy of the photon (if any), as well as count the bosons in the right well.She can find no photon and n bosons, with a probability where P clock Bose (n ← 0, t) = |A clock Bose (n ← 0, t)| 2 [see Eq. ( 11)].She may find n bosons, a photon with an energy E r , and conclude that the emission occurred around [see Eq. ( 11)] ( 12) The error of this result is determined by the width of the Gaussian (11) which, restricts the possible values of τ in Eq. ( 14).Alice's relative error is, therefore, �t/τ n ∼ 1/ √ n << 1 , where �t = ω −1 N −1/2 was defined in Eq. (11).The probability of this outcome is given by the absolute square of A a+ph+clock Bose (E r , n ← e, 0) in Eq. (14).Extending in Eq. ( 14) the limits of integration to ±∞ , and evaluating Gaussian integrals yields for 0 < τ n < t , and P(E r , τ n ← e, 0) = 0 otherwise.
The net probability of an outcome τ n is and replacing n → t 0 dτ n helps to verify that the overall decay rate is not affected by the presence of the clock, P decay �t (t) = n P(τ n ← e, 0) = 1 − exp(−Ŵt) .Finally, the spread of the energies of the emitted photons is no longer Lorentzian, but Gaussian, and becomes broader as Alice's accuracy improves, t → 0 .[Note that we cannot arrive at the Lorentzian distribution (8) simply by sending t → ∞ in Eq. ( 21), since Eq. ( 20) was derived under assumption that the number of bosons in the right well is large].

A clock which first waits and then runs
Alice can also consider a Markovian clock which starts running only after the transition has taken place and continues doing so until the time of observation t.(It will be clear shortly why this case is of interest).Replacing in Eq. ( 13) projector πe by 1 − πe = ∞ −∞ dE r |E r ��E r | , τ with t − τ , and acting as before yields [see the "Methods" section (Coupling the clock to a quantum system)] where δ n0 is the Kronecker delta.Now the number of the bosons in the right well is determined by the time which has elapsed since the moment of emission, and we can attend to the cat which dies as a result of the atom's decay.

Exploding powder kegs and poisoned cats
It is difficult to resist the temptation to relate the present discussion to the famous Schrödinger's Cat problem.In 1935 Einstein and Schrödinger discussed a hypothetical case in which explosion of a powered keg was caused by a photon emitted by a decaying atom.In 6 Schrödinger dramatised the narrative further by replacing the unstable powder by a now famous live cat, which dies in the event.The perceived contradiction was due to the fact that, prior to the final observation of the cat's state, the wave function of the joint system was deemed to be a superposition of the states |atom: excited� ⊗ |cat: alive� and |atom: decayed� ⊗ |cat: dead� .With wave func- tion believed to reflect on the actual condition of a system, this left a big question mark over the cat's situation prior to be found either dead or alive.The same contradiction was observed in the powder keg example, where, again, macroscopically distinguishable states |unexploded� and |exploded� were forced into superposition through entanglement with the atom.
It is worth revisiting the situation by replacing the cat (the keg) with the (nearly) classical clock of Section "A clock which first waits and then runs".So far, the cat paradox did not arise because we only required a matrix element of a unitary operator Ûa+ph+clock (t) = exp(−i Ĥa+ph+clock t) between the states |E r � ⊗ |n� and |e� ⊗ |0� in the Hilbert space of the composite a+ph+clock .The question, we recall, was "is there a photon, and how many bosons are there in the right well at t?" Although there appears to be no need for it, one can create a kind of "cat" problem by looking at the ket and object the appearance of a superposition of distinguishable macroscopic states in r.h.s. of Eq. (23).Indeed, for an accurate clock, i.e. t → 0 ( N >> 1 ), the clock's states in the r.h.s. of Eq. ( 23) are practically orthogonal [cf.Eq. ( 12)], 0 .Alternatively, one can avoid the paradox ( 19) www.nature.com/scientificreports/ of the cat being both dead and alive by considering the superposition to be a transient artefact of the calculation, needed only to establish the likelihood of finding n escaped bosons, and having no further significance.The analogy can be taken further.Neither the cat's demise, nor an explosion are purely instantaneous events.By looking at the deterioration of the cat's body (we leave outside the question of what it means to be alive) one can tell how long ago it stopped functioning.By looking at how much of the powder has been burnt, or how much dust thrown up in the air has settled, it is possible to deduce the moment when explosion started.Remarkably, the waiting clock of Section "A clock which first waits and then runs" keeps a similar record, only in a more direct way (Fig. 3).
Alice may find no bosons in the right well (cat is alive), or a certain number of them (a particular stage of decay of the dead cat's body).The more accurately Alice is able to deduce the "moment of death", the broader will be the energy distribution of the photon whose emission has killed the cat [cf.Eq. ( 21)].A valid analogy could be a very long fuse, whose burnt length (number of bosons in the right well) would let one deduce the moment when it was set on fire.

Beyond the wide band approximation
Next we revisit a more general (non-Markovian) case of Section "Results and discussion" [cf.Eq. ( 2)], where the product |�(E r )| 2 ρ(E r ) may depend on the photon's energy, ) is finite, and the transition occurs via a single jump.Only a small proportion of all atoms will be found decayed by the time t, but Alice may still want to know when this unlikely transition did occur.A simple calculation [see the "Methods" section (Timing the first order transition in a non-Markovian case)] shows that the probability of the clock's reading τ n for a system ending in a state |E r � , is still given by an expression similar to Eq. ( 19), so that measuring the moment of emission to an accuracy t broadens the range of the photon's energies, which grows as 1/�t owing to the Gaussian in r.h.s. of Eq. ( 24).Therefore, it is the availability of the final system's states that restricts the decay rate, and is responsible for the Zeno effect already mentioned in Section "Results and discussion".Indeed, acting as before (cf.Section "Methods"), for the probability to decay by the time t we find In the Markovian wide band limit Ŵ �t in Eq. (26) does reduce to Fermi's golden rule 11 , Ŵ �t = Ŵ Fermi = 2π� 2 ρ .But if the integration of an ever broader Gaussian is restricted to a finite range, the factor of t in Eq. ( 26) is no longer cancelled, and the decay rate eventually decreases as the measurement becomes more accurate.For example, consider a special case of an energy band of a width E r = E max − E min , wherein ρ(E r )� 2 (E r ) = const .Comparing the decay rates prescribed by Eq. ( 26) and by Fermi's rule, we have The accuracy with which the moment of emission can be determined without significantly altering the decay rate is, ultimately, limited by the width of the energy range, available to the emitted photon.What happens for not too small values of t depends, however, on whether the excited atom's energy lies within the allowed range, as explained in Fig. 4. If E e < E min or E e > E max unobserved atom cannot decay, and the decay rate first increases as t becomes smaller, leading to a kind of "anti-Zeno" effect 18 .It eventually begins to fall off in agreement with Eq. ( 27), when the exponential in Eq. ( 24) can be approximated by unity. (24)

Feynman's "only mystery of quantum mechanics"
All this leaves one with a question: "what can be said about the moment of emission if it has not been timed by a cat, gunpowder, or a clock?"Very little, according to Feynman 7,19 .In a double slit experiment a particle can reach a point on the screen by passing through the holes, with the probability amplitudes A 1 and A 2 , respec- tively.The probability of arriving at the screen with both slits open is With no restriction on the signs of the amplitudes, it is possible to have (e.g., near a dark fringe) , so that eliminating one of the routes increases the number of arriving particles.For this reason, it is not possible to assume that a setting of the particle's internal machinery (or any other hidden variable) predetermines the hole to be chosen by each particle on its way to the screen.The mathematics cannot be simpler, and one must conclude that "... when you have no apparatus to determine through which hole the thing goes, then you cannot say that it either goes through one hole or the other".This is an illustration of the Uncertainty Principle 7 which states that one cannot determine which of the alternatives has been taken without destroying interference between them.
The same principle, applied to the case of a decaying atom, states that with no apparatus to determine the moment of decay, one cannot say that the atom emits a photon with an energy between E r and E r + dE r at one moment or the other.Indeed, if each atom were predestined to decay at a given time, the number of decayed atoms could only increase or stay the same as the time span available for the atom's decay becomes longer.However, the corresponding probability is given by W(E r , E r + dE r ) = P(E r )dE r , and P(E r ) = ρ| t 0 A a+ph (E r ← e, t|τ )dτ | 2 , shown in Fig. 5, can decrease with t. (Note that the probability in Fig. 5 is that of a single measurement made at different times.If the decayed atoms are counted twice, the number measured at a later time is, of course, always greater.)The decrease cannot be blamed on the re-absorption of the photon, impossible in the Markovian model [cf.Eq. ( 6)].Neither can it be explained by the change in the emitted photon's energy [cf.Eq. ( 7)].
This seems even stranger than the double slit case.One could imagine the routes passing through different holes merged, like two confluent rivers, where it is impossible to say on which of the two a boat is.Merging time intervals may be even more difficult to fathom, but to conclude that an unobserved transition has occurred (28) .For E min < E e < E max better accuracy means a smaller decay rate (Zeno effect); for E e < E min there is an initial increase in the value of Ŵ (anti-Zeno effect).Both possibilities are illustrated in the inset.The anti-Zeno effect also occurs in the case E e > E max , not shown here.
at a particular moment would lead to "an error in prediction" 19 , as was discussed above.This is, according to Feynman 7 , the only mystery of quantum mechanics, which defies "any deeper explanation".

Conclusions
The story of the Schrödinger's cat, whose death is caused by the decay of an excited atom, is one of the best known illustration of a problem which one expects to arise when the classical world meets its quantum counterpart 6 .
A classical system, believed to have an unbroken continuous history, appears to loose this property if forced to interact with a quantum object, for which no continuous description is thought to be available 21 .To bridge the gap between the classical and quantum views we design a nearly classical macroscopic clock, capable of timing the moment of decay to a good, yet finite accuracy.The complete narrative is as follows.
An atom, prepared in its excited state is found decayed at time t, after having emitted a photon with energy E r .The instant of emission is unknown, and to determine it the experimenter needs a device which would measure it.One suitable choice is a clock consisting of large number N of noninteracting bosonic atoms, initially trapped in the left well of a double well potential.Finding that n bosons have made the transition to the right well, one can estimate the elapsed time t as t n ≈ ω −1 √ n/N , with an error �t ≈ ω −1 / √ N .With the transition amplitude ω small, and the number of bosons large, N >> 1 the clock is a source of irreversible current flowing from left to right.With many bosons in the right well, N >> n >> 1 , the clock is seen to acquire an important classi- cal property.Its wave function becomes localised, and one is able to measure time to a good accuracy without significantly perturbing the clock's evolution (for more details see also 14 ).
The clock can be arranged to run until the moment of emission, which would yield a good estimate of the time of emission provided �t/t n << 1 , except in the unlikely case of the decay occurring almost immediately.The effect of the measurement on the atom's decay depends on the range of energies E r , available to the emitted photon.In the wide band limit, �E r �t >> 1 the decay rate Ŵ remains the same, and destruction of interference between the moments of emission leads only to broadening of the photon's energy spectrum, whose shape is no longer Lorentzian, but Gaussian, with a width ∼ 1/�t .Having obtained a result t n , and knowing that more measurements could have been added both before and after t = t n (almost) without altering the clock's evolu- tion, the experimenter has a complete history of what has happened.The atom remained in its excited state until t n − t t n + t , and then continued in the ground state until the time when the clock is read.Note that essential for recovering such a continuous description is the classical property of the macroscopic clock reached in the limit N >> 1.
The Zeno effect sets in when the inverse clock's accuracy become comparable to the range of available photon's energies, E r t 1 .Now the notion of the moment of decay is meaningful only in the weak coupling limit, Ŵ Fermi t << 1 [cf.Eq. ( 2)].In the "narrow band" limit, �E r �t << 1 , the decay rate is proportional to t , and the unlikely atomic decay is further suppressed as t → 0.
The clock set up to run after the decay has occurred, helps provide an additional insight into the fate of the Schrödinger's feline 6 .Now one knows that there were no bosons in the right well until t n (within an error margin t ), after which their number there was steadily growing.One can leave the question of what it means to be alive outside the scope of quantum theory, and concentrate instead on the deterioration of the cat's macroscopic physical body.The waiting clock is a blueprint for a very primitive "cat", said to be alive if there are no bosons in the right well, n = 0 , and dead in some stage of decay with n >> 1 .If the analogy holds, a real cat's physical frame should be characterised by a quantum uncertainty t cat , which limits the ability of an experienced forensic

Figure 1 .
Figure 1.A classical hourglass (left), and its quantum version (right).(a) With the barrier closed (the clock is switched off) the bosons remain in the left well.(b) If the barrier is down (the clock is switched on), the number of bosons escaping into the right well allows one to estimate the elapsed time.

Figure 3 .
Figure 3.An artists's impression of a primitive cat (a) alive and well, and (b) sadly, dead for some time.Any resemblance to real cats, living or dead, is purely coincidental.

Figure 4 .
Figure 4.The rate of decay into a finite-sized band E min ≤ E r ≤ E max as a function of the clock's accuracy t [ ξ = 2(E e − E min )/�E r ].For E min < E e < E max better accuracy means a smaller decay rate (Zeno effect); for E e < E min there is an initial increase in the value of Ŵ (anti-Zeno effect).Both possibilities are illustrated in the inset.The anti-Zeno effect also occurs in the case E e > E max , not shown here.

Figure 5 .
Figure 5. Probability of finding the photon in a unit interval around energy E r in a single measurement made at time t.Note that similar results were observed in 20 .