1 Dec 2009 H Agaeo Mltdsi Sdt Ute Utf the Justify Further to Used Afterwards, Is GSI

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GSI-experiment, so-called the the to Furthermore, applied are considerations es omte ywihwy( way which by matter no cess, F-ramn ycrflycnieigsvrlcases, several considering where carefully by QFT-treatment h trigpitfrtedsuso stesuperposi- the is discussion the for point starting The sapriua nemdaesaeo rcs.E.g., process. a of state intermediate particular a is a .TEQATMFEDTHEORY FIELD QUANTUM THE 2. π OMLTO FTEPROBLEM THE OF FORMULATION ece httedcyrt measured rate decay the that reached s ory + h eodqatzdpcuea well as picture quantized second the f tos hsi,hwvr o possible not however, is, This ations. ea evsa diinleapebfr the before example additional as serves decay ns, eecsbtenteGIanomaly GSI the between fferences le n h nta hydrogen-like initial the and olved etpeoeo ilb given, be will phenomenon Beat m eteei onw e unknown, yet new, no is there se l.Uigtodffrn formula- different two Using aly. e + e − → µ e + + µ e − − unu Beats Quantum → cteig and scattering, γ µ -, + llations µ Z − Process 0 ,or -, soesnl pro- single one is H 4]wl be will [40] en re- a means 0 -exchange add 2

− − Category Double Slit e+e → µ+µ GSI experiment 1 No slit-monitoring at all e+e−-collider N/A 2A Monitoring & read out N/A GSI-like experiment with more kinematical accuracy 2B Monitoring without read out N/A Actual GSI-experiment

Table I: The classification of the three examples. at tree-level in the Standard Model (SM) of elementary Thomas Young [38] and has later been the major exam- 0 0 particles) it is mediated. Z νeνe and Z νµνµ are, ple to illustrate the laws of QM. Its basic procedure is the however, two distinct processes.→ → following: Light emitted coherently by some source (e.g. Using this terminology, the superposition principle can a laser) hits a wall with two slits, both with widths com- be formulated in the following way: parable to the wavelength of the light. If it hits a screen behind the wall, one will observe an interference pattern, 1. If different ways lead from the same initial to the as characteristic for wave-like objects (category 1). There same final state in one particular process, then one is, however, the interpretation of light as photons, i.e., has to add the respective partial amplitudes to ob- quanta of a well-defined energy. Naturally, one could ask tain the total amplitude. The absolute square of which path such a photon has taken, meaning through this total amplitude is then proportional to the which of the two slits it has travelled. The amazing probability of the process to happen (coherent sum- observation is that, as soon as one can resolve this by mation). monitoring the slits accurately enough, the interference 2. If a reaction leads to physically distinct final states, pattern will vanish, no matter if one actually reads out then one has to add the probabilities for the differ- the information of the monitoring (2A), or not (2B). The ent processes (incoherent summation). reason is that, regardless of using the information or not, the measurement itself has disturbed the QM process in If a certain situation belongs to category 1, an interfer- a way that the interference pattern is destroyed [42]. ence pattern will be visible (or oscillations, in case the The key point is that one cannot even say that the photon interfering terms have different phases as functions of takes only one way: In the QM-formulation, amplitudes time), while if it belongs to category 2, there will be no are added (and not probabilities), and hence the photon interference. The remaining question is at which point does not take one way or the other (and we simply sum the measurement comes in. This can be trivially said for over the results), but it rather has a total amplitude that point 2: Either the experimental apparatus is sufficiently includes a partial amplitude to take way 1 as well as an- good to distinguish between the different final states (2A) other partial amplitude way 2. By taking the absolute – then no summation whatsoever is necessary simply be- square of this sum of amplitudes, interference terms ap- cause one can divide the data set into two (or more), one pear. for each of the different final states. If this is not the case A QFT-formulation involving elementary fields only (2B), the experiment will be able to lead to either of the would be much more complicated: One would sum over final states, but one would not know which one had been the amplitudes for the photon to interact with each elec- the actual result – then one would simply have to add tron and each quark in the matter the slits are made the probabilities for the different final states to occur in of, after having propagated to this particular particle order to obtain the total probability. and before further propagating to a certain point on the What if we do such a measurement for category 1? If we screen. Of course, by using an effective formulation of the can indeed distinguish several ways that a process can theory, one can find a much more economical description happen, then this has to be done by some measurement. and the easiest one is to simply comprise all possible in- Since this measurement then has selected one particular teractions into two amplitudes, one for going through the way, we have actually transformed a situation belonging first and one for going through the second slit. to category 1 into a situation of category 2. However, Let us go back to this effective formulation: If there is then there would be no terms to interfere with – the in- monitoring, one actually “kills” one of these two ampli- terference would have been “killed”. tudes, the other one remains, and the interference is de- Let us now turn to Table I, that illustrates how our three stroyed. Whenever there has been such a measurement, examples fit into the categories 1, 2A, and 2B. These the interference will vanishe. As we will see, the ques- three cases will be discussed one by one in the following. tion is if in a certain situation a measurement has been performed (or is implicitly included in the process considered), no matter if the corresponding information is 2.1. The Double Slit experiment with photons read out, or not.

This is the “classical” situation of an experiment that reveals the nature of QM. It has ﬁrst been performed by 3

+ − + − 2 −5 2 2.2. e e → µ µ scattering at a collider in the limit mν = 0, and (∆m )⊙ =7.67 10 eV [44] is the solar neutrino mass square difference· as known Let us now consider the scattering of e+e− to form a from neutrino oscillation experiments. The question is pair of muons. This is, differently from the Double Slit how to obtain an estimate for σx: If the nucleus was experiment, a fundamental process where only a small inside a lattice, one could estimate a width like the typ- number of elementary particles is involved. If one wants ical interatomic distance, σx 1A,˚ which would lead 8 ∼ to Lcoh 2 10 m. Of course, this precision cannot to calculate the scattering probability, the amplitude for ∼ · the process is again decisive. In the SM, there are only be reached in the GSI experiment. However, at least three possibilities for this process to happen at tree-level during the electron cooling [45], the nucleus will be lo- and in all three of them the e+e− pair annihilates to calized to some precision. Since the velocity of the nu- some intermediate (virtual) boson which in the end de- cleus is known, this information could in principle be cays again, but this time into a µ+µ− pair. The inter- extrapolated for each run. A fair estimate would then be mediate particle can either be a photon, a Z0-boson, or the average distance between two electrons in the cooling process, which is roughly given by 1/√3 n 0.1mm, a Higgs scalar, see Fig. 1. ∼ Here, we have three different ways to form the process. where n is the electron density [46]. This leads to a more 14 realistic coherence length of Lcoh 2 10 m. The pes- The difference to the Double Slit experiment, however, ∼ · is that these three ways cannot be separated easily. In a simistic case, where σx is taken to be the approximate diameter 108.36 m/π [47] of the Experimental Storage real collider-experiment we are not able to say that the 19 reaction e+e− µ+µ− has taken place by the exchange Ring (ESR) produces Lcoh 6 10 m. The mean free → path of a neutrino in our∼ galaxy,· however, is roughly of, e.g., a photon only, but it will always be the sum of 40 the three diagrams (and a lot more, in case we include 1 10 m (for an assumed matter density in the Milky · 3 higher orders). Hence, this process will always fall into Way of 1 10−23 g/cm ), so the assumption that the neu- · category 1 and interference terms will appear. trino does not interact before losing its coherence is com- This is an easy and familiar example for the appearance pletely safe. of interference terms in a real experiment. In the next Even if we do not know in which of the three mass eigen- paragraph it will be shown exactly what is different in states the neutrino actually is, we know that it has to the case of the GSI-experiment. be in one of them. This knowledge is somehow obtained “a posteriori”, since the mass eigenstate only reveals its identity after some propagation. But, by conservation of 2.3. The GSI experiment energy and momentum, one could treat the process as if the kinematical selection had already been present at the The remaining question is what the situation looks production point of the neutrino. This “measurement” like for the GSI-experiment. Even though the QFT- is enforced by the physical conservation laws. calculation of what happens is pretty straightforward, An analogous reasoning is given by Feynman and fitting everything in the language used above might be a Hibbs [42], using the example of neutron scattering: Neu- bit more subtle. We will, however, see in Sec. 3.2 that the trons prepared to have all spin up scatter on a crystal. formulation in terms of amplitudes additionally justifies If one of the scattered neutrons turns out to have spin the result obtained here. down, one knows by angular momentum conservation Let us at first consider the Feynman diagram of the pro- that it must have been scattered by a certain nucleus. cess involved in the GSI-experiment in Fig. 2 [17]: Here, In principle, by noting down the spin state of every nu- in the absence of extreme kinematics (meaning that the cleus in the crystal before and after the measurement, Q-value of the reaction is large enough so that all neu- one could find the corresponding scattering partner of the trino mass eigenstates can be emitted), the neutrino is neutron without disturbing it. No matter if this would produced as electron neutrino. What happens to this be difficult practically, by a physical conservation law neutrino? Since it is not detected, it escapes to infinity in one knows that a particular scattering must have been the view of QFT (in the picture of second quantization). present, even if the corresponding nucleus is not “read Physically, it loses its coherence after some propagation out”. Accordingly, the corresponding interference van- distance and travels as a unique mass eigenstate. ishes and the neutrons that have spin down after the The key point is the following: Since the neutrino will not scattering come out diffusely in all directions. interact before it loses its coherence, it must be asymp- This can also be formulated in the language of wave totically a mass eigenstate. This can be shown easily: packets: We have complete 4-momentum conservation The coherence length of a (relativistic) neutrino is given for each single component (which is a plane wave!) of by [43] the wave packets, but if we consider the whole wave packet, its central momentum does not have to be con- 2p2 √ served [48, 49]. However, all the different components can Lcoh =2 2σx 2 , (1) · (∆m )⊙ produce both possible neutrino mass eigenstates, but for a certain kinematical configuration of parent and daugh- where σx is the size of the neutrino wave packet, p is the momentum of the mean value neutrino momentum ter components only one of the mass eigenstates will ac- 4

e+ Μ+ e+ Μ+ e+ Μ+ Γ Z 0 H 0

e- Μ- e- Μ- e- Μ-

Figure 1: The diagrams contributing to e+e− → µ+µ− in the SM.

undetected neutrino 3.1. Charged pion decay mass eigenstate + parent ion νi It is well-known that a charged pion (e.g. π ) can decay into either a positron in combination with an electron neutrino, or into the corresponding pair of µ-like parti- daughter ion cles. Let us consider the case of a pure (and normalized) initial state pion π+ . As this state evolves with time (and is not monitored),| i it will become a coherent superposition of the parent-state, as well as all possible Figure 2: The Feynman diagram for the GSI experiment. daughter states:

+ + + + π (t) = π(t) π + µ(t) µ νµ + e(t) e νe , (2) | i A | i A | i A | i tually be produced. The rest is easy: If the GSI experiment had infinite kine- where all time-dependence is inside the partial amplitudes i. Of course, this state has to be normalized matical precision, one could read out which of the mass A eigenstates has been produced and it would clearly fall correctly: into category 2A. Since, however, this information is not 2 2 2 π(t) + µ(t) + e(t) =1, (3) read out but could in principle have been obtained (e.g. |A | |A | |A | by detecting the escaping neutrino), the GSI experiment with (0) = 1 and (0) = (0) = 0. One can falls into category 2B and one has to sum over proba- π µ e understandA Eq. (2) inA the followingA way: The state at bilities. This logic works because we know that the neu- time t is a coherent superposition of the basis states trino is, after some propagation, no superposition of mass π+ , µ+ν , e+ν with time-dependent coefficients. eigenstates anymore, but just one particular eigenstate µ e Note{| i that| thei | basisi} states are orthogonal. The outcome with a completely fixed mass. of a certain measurement is some state Ψ : All that a A viewpoint closer to the amplitude formulation would detector does is projecting on just this| statei Ψ . Of be: If the neutrino finally interacts, it has to “decide” course, different detectors will in general be described| i which mass eigenstate it has, even if it was a superpo- by projections on different Ψ ’s, which is a reflection of sition of several mass eigenstates before. This is then the influence of the process| ofi measurement on the mea- equivalent to the image of having produced one particu- surement itself. If one wants to know the probability for lar mass eigenstate from the beginning on. measuring that particular state, one has to calculate it according to the standard formula,

+ 2 3. AMPLITUDES - PROBABLY THE EASIEST P (Ψ) = Ψ π (t) . (4) LANGUAGE TO USE |h | i| The question is what Ψ looks like. To make that clear, let us discuss several cases:| i In this section, I use time-dependent amplitudes for the different basis states to describe another example, The (trivial) case is that there has been no detec- namely charged pion decay, which I compare then to neu- • tion at all: Then we have gained no information. trino oscillations (with referring to the actual situation This means that the projected state is just the time- in the GSI-experiment). The logical steps needed to un- evolved state itself (we do not know anything ex- derstand the familiar example of pion decay are exactly cept for the time passed since the experiment has the same as the ones needed to understand what is going started), and we get on at GSI. This description is clear enough to account for very different situations and allows for an easy and Ψ π+(t) 2 = π+(t) π+(t) 2 =1. (5) nearly intuitive understanding of the various cases. Fur- |h | i| |h | i| thermore, it yields an a posteriori justification of the view This result is trivial, since the probability for any- used in the preceding section. thing to happen must be equal to 1. 5

The next situation is when our experimental appa- 3.2. Neutrino oscillations and the GSI-experiment • ratus can give us the information that the pion has decayed, but we do not know the final state. Then, + + Let us now turn to neutrino oscillations. Here, as we it can be either µ νµ or e νe and we remain | i | i will see, a state like Ψ in Eq. (10) can indeed be phys- with a superposition of these two states. The only ical in some situations.| i To draw a clean analogy to the information that we have gained is that the ampli- experiment done at GSI, we consider a hydrogen-like ion tude for the initial pion to be still there is now zero, as initial state M that can decay to the state Dνe π = 0 in Eq. (2). Then, the properly normalized via electron capture.| i Since there was an electron in| thei Astate Ψ is | i initial state, we know that the amplitude for producing + + the mass eigenstate νi is just Uei. If there is no relative µ(t) µ νµ + e(t) e νe | i Ψ = . (6) phase between the two mass eigenstates, the neutrino A | i2 A | 2 i | i µ(t) + e(t) |A | |A | produced in the decay is exactly the particle that we p call electron neutrino. In any case, due to different kine- The absolute value square of the corresponding pro- matics, the two mass eigenstates will in general develop jection is different phases in the time-evolution. This means that, + 2 2 2 Ψ π (t) = µ(t) + e(t) , (7) in spite of the mixing matrix elements Uei being time- |h | i| |A | |A | independent, there will be a phase between the two neu- and if there is any oscillatory phase in the ampli- trino mass eigenstates. Completely analogous to Eq. (2), iωk t tudes, k(t)= ˜k(t)e , it will have no effect due the time-evolution of the initial state will be given by: to theA absoluteA values. M(t) = M (t) M +Ue1 1(t) Dν1 +Ue2 2(t) Dν2 , What if we know that the initial pion is still | i A | i A | i A | i • present? This sets µ(t) = e(t) = 0, and Ψ (11) + A 2 A | i is just π(t) π / π(t) . The projection gives 2 2 2 with M (t) + Ue1 1(t) + Ue2 2(t) = 1 and A | i p|A | |A | | A | | A | + 2 2 M (0) = 1. We can immediately look at different cases: Ψ π (t) = π(t) , (8) |h | i| |A | A The parent ion is seen in the experiment: This which again does not oscillate. • kills all daughter amplitudes, 1,2(t) = 0. The + A If one particular final state, let us say e ν , is de- only remaining amplitude is M (t), very similar to e A • tected, then (t)= (t) = 0 and we| get anotheri Eq. (8). With the proper normalization for Ψ one π µ | i term free ofA oscillations:A gets no oscillation again:

+ 2 2 2 2 Ψ π (t) = e(t) . (9) Ψ M(t) = M (t) (12) |h | i| |A | |h | i| |A | The question remains when we do get oscillations at all. The next case corresponds to the GSI-experiment: The answer is: It depends on what our detector mea- • One sees only the decay, but cannot tell which of sures. If, e.g., the detector measures not exactly the state the two neutrino mass eigenstates has been pro- + + µ νµ or e νe , but instead some (hypothetical) super- duced. This leads to M (t) = 0 and one has to |positioni (e.g.,| somei quantum number which is not yet perform a projection onA the state known, under which neither µ+ nor e+ is an eigenstate, U 1 1(t) Dν1 + U 2 2(t) Dν2 but some superposition of them), then one could measure Ψ = e e . (13) A | i2 A | 2 i the following (correctly normalized!) state: | i Ue1 1(t) + Ue2 2(t) p| A | | A | 1 + + Doing this with M(t) from Eq. (11) yields Ψ = µ νµ + e νe . (10) | i √2 | i | i | i 2 2 2 The squared overlap is 2 Ue1 1(t) 1+ Ue2 2(t) 1 Ψ M(t) = = | A | · 2 | A | 2· |h | i| Ue1 1(t) + Ue2 2(t) + 2 1 2 2 ∗ | A | | A | Ψ π (t) = µ(t) + e(t) +2 µ(t) e(t) , 2 p 2 |h | i| 2 |A | |A | ℜ A A = Ue1 1(t) + Ue2 2(t) , (14) | A | | A | ∗ where the 2 (t) e(t) -piece will, in general, lead to ℜ Aµ A which exhibits no oscillations, but is rather an inco- oscillatory terms. What has been done differently than herent sum over probabilities. This result is the jus- before? This time, we have done more than simply killing tification of the intuitive treatment in Sec. 2.3: The one or more amplitudes in Eq. (2), and this is the cause elementary QM-discussion using probability ampli- of oscillations: Whenever we are in a situation, in which tudes gives us just the correct prescription for how the state playing the role of Ψ in Eq. (10) is physical, to sum up the amplitudes for the final states. the corresponding projection| willi yield oscillatory terms. As we will see in a moment, this is exactly what happens The GSI-experiment with infinite kinematical pre- in neutrino oscillations. • cision: In this case, one could actually distinguish 6

the states Dν1 and Dν2 . If one knows that Dν1 If one consideres the state from Eq. (13) as being the is produced| (e.g.,i by| havingi very precise informa-| i one emitted but then projects onto an electron neutrino tion about the kinematics), one will again have no state, oscillations will appear:

oscillation, 2 ∗ ∗ D, νe M(t) = (Ue1 Dν1 + Ue2 Dν2 ) 2 2 |h | i| | h | h | · 2 Ψ M(t) = 1(t)Ue1 , (15) ( M (t) M + Ue1 1(t) Dν1 + Ue2 2(t) Dν2 ) = |h | i| |A | · A | 2 i A2 | i ∗A | i | = 1(t) + 2(t) +2 ( 1(t) (t)). (17) just as in Eq. (9). |A | |A | ℜ A A2 This is, however, wrong: One has not used all the infor- These are in principle all cases that can appear. One mation that could in principle have been obtained! But can, however, have a closer look at the realistic situation Nature does not care about if one uses information or not, in the GSI-experiment. Let us re-consider Eq. (11): In so this treatment does simply not correspond to what reality, the parent ion will be described by a wave packet has happened in the actual experiment. The oscillations, with a finite size or, equivalently, a finite spreading in however, only arise due to the incorrect projection, and momentum space, due to the Heisenberg uncertainty re- have no physical meaning. lation. If this wave-packet is broad enough that each The remaining question to obtain a complete understand- component can equivalently decay into Dν1 or Dν2 , | i | i ing of the situation is if the neutrino that is emitted in the then both of the corresponding amplitudes will actually GSI-experiment oscillates. The answer is yes, of course. have the same phase ( 1(t) = 2(t)), since they have A A But to see that, we will have to modify our formalism a the same energy, and one can write Eq. (11) as bit. Knowing that an electron neutrino has been emitted corresponds to M (t) = 0 in Eq. (16), and the remaining M(t) = M (t) M + (t) [Ue1 Dν1 + Ue2 Dν2 ] . (16) A | i A | i A | i | i (normalized) state is: =|Dνei | {z } (t) Ψ = A [Ue1 Dν1 + Ue2 Dν2 ] . (18) Since the knowledge of the momentum of the parent ion | i (t) | i | i is not accurate enough at the GSI-experiment to make a |A | distinction between both final states Dνk , this is a real- Re-phasing this state and measuring the time from t on istic situation. Of course, this does not| ati all change the gives as initial state: above argumentation, since the final state Ψ will expe- | i Ψ = Ue1 Dν1 + Ue2 Dν2 . (19) rience the same modification. The neutrino produced is | i | i | i an electron-neutrino, as to be expected. The question remains, why some authors come to the This is the state which will undergo some evolution in conclusion that there should be oscillations? The answer time according to is simple: If the correspondence between time-evolved ′ ′ ′ ′ ′ Ψ(t ) = (t )Ue1 Dν1 + (t )Ue2 Dν2 , (20) initial state and detected state is wrong, then oscilla- | i A1 | i A2 | i ′ ′ 2 ′ ′ 2 ′ tions may appear. As example, we will consider the sit- with 1(t )Ue1 + 2(t )Ue2 = 1 and 1(0) = ′ |A | |A | A uation that the kinematics of the parent and daughter 2(0) = 1. If we ask what happens to this neutrino are fixed so tightly, that indeed the production ampli- ifA it is detected after some macroscopic distance, it is tudes for Dν1 and Dν2 are not equal. This would | i | i necessary to take into account what has happend to the correspond to an extremely narrow wave packet in mo- daughter nucleus that has been produced together with mentum space. Let us, e.g, have in mind the extreme case the neutrino, due to entanglement. The daughter nu- when by kinematics only the production of ν1 is possible. cleus, which is accurately described by a wave packet, This is no problem in principle and we would be used to is detected, but not with sufficient kinematical accuracy it if neutrinos had higher masses, so that the Q-value of to distinguish the different components D of the wave the capture was only sufficient to produce the lightest packet. The effect of such a non-measurement| i is studied neutrino mass eigenstates. If only the disappearance of most easily in the density matrix formalism. The density the parent is seen, the corresponding state Ψ , which is ′ | i matrix ρ corresponding to Eq. (20) is given by detected, is given by Eq. (13) (with 2(t)=0 in the ex- A ′ ′ ′ 2 treme case, but anyway with 1(t) = 2(t)). The corre- Ψ(t ) Ψ(t ) = 1(t ) D ν1 ν1 D + A 6 A | ih′ 2 | |B | | i| ih |h | sponding neutrino is, however, no electron-neutrino any- + 2(t ) D ν2 ν2 D + more (which would be Ue1 ν1 + Ue2 ν2 , with the same |B ′ | ∗| i|′ ih |h | +[ 1(t ) (t ) D ν1 ν2 D + h.c.], (21) phase for both states)! Indeed| i this is| noi surprise at all, B B2 | i| ih |h | ′ ′ ′ since the kinematics in the situation considered is so tight where k(t )= k(t )Uek. If the exact kinematics of the that it changes the neutrino state which is emitted. This daughterB is notA measured, then one has to calculate the is a clear consequence of quantum mechanics, since for trace over the corresponding states. It gives obtaining the necessary pre-knowledge (namely the very ′ ′ 2 accurate information about the kinematics), one has to ρ dD D ρ D = 1(t ) ν1 ν1 + (22) do a measurement that is precise enough to have an im- ≡ Z h | | i |B | | ih | ′ 2 ′ ∗ ′ pact on the QM state. + 2(t ) ν2 ν2 + ( 1(t ) (t ) ν1 ν2 + h.c.). |B | | ih | B B2 | ih | 7

Èa\ Èa\ 4.1. Single atom of type I Èb\ Let us start with the classic example of QBs, namely an atom in a coherent superposition of three states a , b , and c , where the first two states are above and| closelyi | i | i Ω Ω spaced compared to c . This setting is drawn on the left ab ac | i Ωac Ωbc panel of Fig. 3 and is referred to as “type I”. First note that the three levels correspond to different (but fixed) eigenvalues of the energy and are hence orthogonal vec- tors in Hilbert space. This is not at all changed by an Èb\ energy uncertainty which, however, makes it possible to Èc\ Èc\ have a coherent superposition of the three states. Ini- tially, we assume the atom to be in such a superposition of these states, but having emitted no photon yet. Ac- cordingly, the photon state can only be the vacuum 0 γ. Figure 3: Type I (left) and type II (right) of the Quantum | i Beats settings. Then, the initial state of this system can be written as

Ψ(0) = 0 a 0 γ + 0 b 0 γ + 0 c 0 γ , (26) | i A | i| i B | i| i C | i| i If we want to know the probability to detect, e.g., a muon 2 2 2 where 0 + 0 + 0 = 1. If this system undergoes neutrino, ν = U 1 ν1 + U 2 ν2 , the corresponding µ µ µ a time-evolution,|A | |B | the lower|C | state might be populated by projection| operatori is| giveni by | i de-excitation of the upper ones, which is done by photon † µ = νµ νµ , (23) emission. If the state 1x γ = ax 0 γ is assumed to de- P | ih | | i | i scribe a state with one photon of frequency ωx, then the and the probability to detect this state is state at time t can be written as Pµ = Tr( µρ)= ν1 µρ ν1 + ν2 µρ ν2 . (24) P h |P | i h |P | i Ψ(t) = (t) a 0 γ + (t) b 0 γ + (t) c 0 γ + | i A | i| i B | i| i C | i| i Note, however, that the neutrino states ν1,2 will always + 1(t) c 1ac γ + 2(t) c 1bc γ, (27) be orthogonal, since they correspond to eigenstates| i of dif- C | i| i C | i| i ferent masses (like an electron is in that sense orthogonal where (0) = 0, (0) = 0, (0) = 0, 1,2(0) = 0, A 2 A B2 B2 C 2C C 2 to a muon). The result is and (t) + (t) + (t) + 1(t) + 2(t) = 1. |A | |B | |C | |C | |C | 2 ′ 2 2 ′ 2 Under the assumption that all levels are equally pop- Pµ = Uµ1 1(t ) + Uµ2 2(t ) + | | |B | | | |B | ulated, the radiated intensity will be proportional to ∗ ∗ ′ ′ 2 +[Uµ1Uµ2 1 (t ) 2(t )+ c.c.], (25) Ψ(t) E (0,t) Ψ(t) , where B B h | | i whose second line contains oscillatory contributions. E x −ikx † +ikx These oscillation are indeed physical: Eq. (23) is a de- ( ,t)= ǫk,λ ak,λe + ak,λe (28) Xk scription of a detector that is sensitive to νµ’s only. If ,λ it could not distinguish different neutrino flavours, the is the electric field operator and ǫk is the electric field oscillation would vanish again. ,λ per photon of momentum k and polarization λ. Note that the creation and annihilation operators have only one † 4. QUANTUM BEATS non-trivial commutation relation, namely [ak,λ,ak′,λ′ ]= δk,k′ δλ,λ′ . In our case we obtain effectively: The last point to discuss are the so-called Quantum E 0 2 2 † ( ,t) = ǫac(1+2aacaac) + (29) Beats (QBs) [40]. This phenomenon is known from Quan- 2 † † i∆t † −i∆t tum Optics and has often been mentioned as possible +ǫbc(1+2abcabc)+2ǫacǫbc(aacabce + abcaace ), explanation for the GSI anomaly. As we will see, the where ∆ = ωac ωbc. Here, we have already used that corresponding language can be equally used to describe −2 terms like, e.g., aac give no contribution with Ψ from the GSI-experiment and (of course) yields the same re- Eq. (27). Remember now, that the atomic states| arei or- sult as already obtained. Still, it is also useful to consider thonormal. This means that one can, e.g., combine a the experiment from this point of view in order not to be term proportional to b in Ψ(t) only with the corre- misled by claims that erroneously make QBs arising from sponding term b in hΨ(| t) .h The| corresponding combi- a splitting in the final state responsible for the observa- nation of amplitudes| i | (t) 2i does, however, not oscillate, tion at GSI. since any phase will be|B killed| by the absolute value. This Normally, one considers atomic levels for this discussion, is also true for every term involving one of the constant ∗ and we will stick to that here for illustrative purposes and parts of Eq. (29): E.g. the term proportional to (t) 1(t) give the relation to the GSI-experiment at the end of each can involve a factor C C section. This way also easily clarifies the analogies to the † † Quantum Optics formulation. γ 0 a aaca 0 γ =0, (30) h | ac ac| i 8

† † because of aac acting on the left. There are, however, which follows immediately from the action of aac ∗ i∆t remaining oscillatory terms such as 1 (t) 2(t)e , which to the left and of aac to the right, respectively. is proportional to C C 0-photon state coupled with 1-photon state: † † † † γ 0 aaca abca 0 γ = γ 0 (1+a aac)(1+a abc) 0 γ =1. • h | ac bc| i h | ac bc | i This is the only possibility, which is left. If we take for instance the term ∗(t) ′(t), this will oscillate These terms cause the Quantum Beats for a type I atom. anyway, so we will alsoB haveB to check the constant Actually, one could have expected this result intuitively: terms in Eq. (29). The ones proportional to 1 are Both of the coherently excited upper levels can decay into † naturally zero, γ 0 1ab γ = γ 0 a 0 γ = 0. The the same state c via the emission of a photon. Hence, h | i h | ab| i one cannot in any| i way determine the photon energy with- other terms are out measuring it directly. Without such a measurement, † † interference terms will appear. γ 0 aac aac 1ab γ =0, γ 0 aab aab 1ab γ =0, (32) How is the situation for the GSI-experiment? In this h | | i h | | i 0← 0← case one simply has to replace the photon by the neu- |{z}† |{z} † γ 0 a aab 1ab γ =0, and γ 0 a aac 1ab γ =0, trino. As explained in Ref. [17] for instance, a split- h | ac | i h | ab | i ting in the initial state could lead to an oscillatory be- 0← 0← havior. This splitting, however, would have to be tiny, |{z} |{z} 10−15 eV, a value which can hardly be explained. Fur- ∼ where the action of the operators to give zero is thermore, there exists preliminary data on the lifetimes always indicated by the arrow. The argumentation 142 60+ + of Pm with respect to β -decay that shows no os- is analogous for the complex conjugated term. cillatory behavior [12]. An initial splitting in the nucleus would lead to an oscillatory rate in this case, too. Ac- Hence, there can be no Quantum Beats for a single atom cordingly, if such a splitting is present in the initial state, of type II! The intuitive reason is that, by waiting long it could be in the levels of the single bound electron, since enough, one could reach an accuracy in energy that is this would then affect EC-decays while leaving β+-decays good enough to distinguish the possible final states b untouched. and c . This would then be a way to determine the| i energy| i of the emitted photon without disturbing it. To give an analogous argument for the GSI- 4.2. Single atom of type II experiment, one has to turn the comparison given in Sec. 4.1 round and replace the atom by the neutrino and We can study a similar setting, namely an atom of the photon by the ion. The reason is that what is claimed type II, shown on the right panel of Fig. 3. The cor- to interfere in this situation is the neutrino states them- responding initial state would again be described by selves (see, e.g., Ref. [2]). This neutrino is not expected Eq. (26), but its time-evolution would now look like to interact, before losing its coherence (cf. Sec. 2.3). How- ever, once it interacts, it has to decide for a certain mass Ψ(t) = (t) a 0 + (t) b 0 + (t) c 0 + γ γ γ eigenstate. By monitoring this interaction, it would in | ′ i A | i| i ′ B | i| i C | i| i + (t) b 1ab γ + (t) c 1ac γ , (31) B | i| i C | i| i principle be no problem to determine the neutrino’s mass ′ where (0) = 0, (0) = 0, (0) = 0, (0) = 0, (e.g., by exploiting the spatial separation of the mass ′(0) =A 0, andA (tB) 2 + (Bt) 2 C+ (t) 2C + B ′1(t) 2 + eigenstates far away from the source) and from this one C ′(t) 2 = 1. The|A square| of|B the| electric|C | field|B has again| could easily reconstruct the kinematics of the daughter |Cthe form| of Eq. (29), just with bc ab. Due to the ion in the GSI-experiment. Accordingly, no QBs are to orthogonality of the atomic states, there→ are not too many be expected in this situation. combinations which are possible:

0-photon state coupled with itself: 4.3. Two atoms of type II • If we take, e.g., the term (t) 2, it does not oscillate anyway. Hence, only the|A time-dependent| parts in Eq. (29) (with bc ab) could lead to oscillations. On the other hand, there is a situation in which we But they are proportional→ to can expect QBs even for atoms of type II, namely if we have two of them. If these two atoms are separated by † † γ 0 aacaab 0 γ = γ 0 aabaac 0 γ =0. a distance which is smaller than the wavelength of the h | | i h | | i emitted photons, there is no way to resolve their sepa- 1-photon state coupled with itself: ration in space and we have to write down a combined • ′(t) 2 does not oscillate, too, and the time- initial state for both atoms, 1 and 2: dependent|B | terms from the electric ﬁeld yield Ψ(0) = 0 a 1 a 2 0 γ + 0 b 1 b 2 0 γ + 0 c 1 c 2 0 γ † † † | i A | i | i | i B | i | i | i C | i | i | i γ 1ab a aab 1ab γ = γ 0 aaba aaba 0 γ = 0 and h | ac | i h | ac ab| i + 1,0 a 1 b 2 0 γ + 2,0 b 1 a 2 0 γ + 1,0 a 1 c 2 0 γ + † † † D | i | i | i D | i | i | i E | i | i | i γ 1ab a aac 1ab γ = γ 0 aaba aaca 0 γ =0, + 2,0 c 1 a 2 0 γ + 1,0 b 1 c 2 0 γ + 2,0 c 1 b 2 0 γ. h | ab | i h | ab ab| i E | i | i | i F | i | i | i F | i | i | i 9

2π~c −12 The corresponding time-evolution Ψ(t) looks a bit com- λ = Ec 10 m, while the average distance between plicated: | i two ions should∼ be of the order of the storage ring [50], which is roughly 100 m [47]. Hence, this possibility is (t) a 1 a 2 0 γ + (t) b 1 b 2 0 γ + (t) c 1 c 2 0 γ + A | i | i | i B | i | i | i C | i | i | i excluded for the GSI-experiment. + 1(t) a 1 b 2 0 γ + 2(t) b 1 a 2 0 γ + D | i | i | i D | i | i | i + 1(t) a 1 c 2 0 γ + 2(t) c 1 a 2 0 γ + E | i | i | i E | i | i | i + 1(t) b 1 c 2 0 γ + 2(t) c 1 b 2 0 γ + F | i | i | i F | i | i | i 5. CONCLUSIONS + 1(t) b 1 a 2 1ab γ + 2(t) a 1 b 2 1ab γ + G | i | i | i G | i | i | i + 1(t) c 1 a 2 1ac γ + 2(t) a 1 c 2 1ac γ + H | i | i | i H | i | i | i A comparison of the GSI-experiment with several other + 1(t) b 1 b 2 1ab γ + 2(t) c 1 c 2 1ac γ + processes (the Double Slit experiment with photons, I | i | i | i I | i | i | i e+e− µ+µ− scattering, and charged pion decay) has + 1(t) b 1 c 2 1ab γ + 2(t) c 1 b 2 1ab γ + J | i | i | i J | i | i | i been given.→ By using the language of QFT as well as + 1(t) b 1 c 2 1ac γ + 2(t) c 1 b 2 1ac γ . (33) K | i | i | i K | i | i | i the intuitive formulation with probability amplitudes, I ∗ −i∆t have shown that the situation at GSI cannot lead to any One oscillatory term would then be, e.g., 1 1e , which is proportional to J K oscillation of the decay rate, if the correct treatment is chosen and no additional assumptions (as, e.g., a split- † † † γ 1ab a aac 1ac γ = γ 0 aaba aaca 0 γ = (34) ting in the initial state) are taken into account. Also h | ab | i h | ab ac| i † † the frequently mentioned possibility of Quantum Beats = γ 0 (1 + a aab)(1 + a aac ) 0 γ = γ 0 0 γ =1. h | ab ac | i h | i of the ﬁnal state cannot explain the observed oscillations, 0← →0 at least not in the standard picture. Hopefully this ar- |{z} |{z} If the spatial separation is less than the photon wave- ticle will contribute to the clariﬁcation of the physical length, one cannot determine the photon energy, because situation in the experiment that has been performed at one does not know which atom has emitted the radiation. GSI. Accordingly, we expect QBs. Acknowledgements For the GSI-case, this possibility has to be taken into account, because even for runs with one single EC only, I would like to thank A. Blum and M. Lindner for there can have been more ions in the ring that were lost useful discussions, as well as K. Blaum and especially or decayed via β+. In this case (comparing the neutrino J. Kopp for carefully reading the manuscript and giving again with the photon), one has to replace the wave- valuable comments. I am furthermore grateful to the length of the photon by the de Broglie wavelength of the referee of this paper for making valuable suggestions that neutrino. The neutrino energy should be of the same or- have improved the manuscript considerably. This work der as the Q-value of the EC-reaction, which is roughly has been supported by Deutsche Forschungsgemeinschaft 1 MeV [1]. The corresponding wavelength is, however, in the Sonderforschungsbereich Transregio 27.

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