Since the coordinates of the wavefunctions now conform to classical mechanics, this form has a wholly classical, statistical interpretation. An ensemble of particles with probability density |-( ˜ *p*, 0)| 2, defining the probability of occurrence of a certain initial momentum *p*, move on classical trajectories and hence the ensemble members evolve to the position probability density |-(*r*, *t*)| 2.

The factor *dp*/*dr* is the *classical* trajectory density of finding the system in the volume element *dr* given that it started with a momentum *p* in the volume element *dp* (see the books by Gutzwiller [29] or Heller [30]). Quantum mechanics provides the initial ensemble momentum distribution located at a microscopic distance *r*- ≈ 0. Each element of the initial momentum wave function is then imaged onto the spatial wave function at large distance *r*, where the coordinates are related by classical mechanics.

That is, from Eq. (6) one has the asymptotic equality of probabilities in initial momentum space and final position space, i.e.

$$|\Psi(\mathbf{r}(t))|^2 \, d\mathbf{r} = |\tilde{\Psi}(\mathbf{p}, 0)|^2 \, d\mathbf{p}.\tag{7}$$

This shows that the loci of points of equal probability of particle detection are classical trajectories. Nevertheless, according to Eq. (5), the wavefunction remains intact.

Clearly, the IT can only be interpreted in the ensemble picture. The wavefunction spreading corresponds to the natural divergence of an ensemble of classical trajectories of differing initial momentum emanating from a microscopic volume and being detected after traversing a macroscopic distance. Nevertheless, estimates of the *r* and *t* values at which the semi-classical approximation becomes valid (Ref. [6]) show that this occurs for values which are still microscopic, typically only tens of atomic units, the precise value dependent upon particle masses and energies.

It is to be emphasised that the IT describes classical evolution of the wavefunction variables and the transition to this property arises from unitary quantum propagation i.e. the transition to classical behaviour is autonomous; external interactions are unnecessary. This justifies a routine assumption of experimentalists that one can use classical mechanics to trace a trajectory back from a point on the detector to the quantum reaction zone and is valid even for light particles such as electrons.

The consideration of the quantum to classical transition from a more mathematical viewpoint, so-called semi-classical quantum mechanics, began with the early WKB approximations and Van Vleck's work on time propagators [31]. It was formulated initially for scattering theory, for example by Mott and Massey [17], by Ford and Wheeler [32] and by Brink [33]. A completely general theory emerged later in the work of Berry and Mount [34] and of Miller [35] and Heller [30], for example. Major contributions made by Gutzwiller are to be found in Ref. [29].

In semi-classical scattering theory one examines the transition to a classical crosssection which occurs when the collision energy is much greater than the interaction energies of the collision complex, see for example, [36, 37]. This is to be contrasted with the IT in which quantum systems of atomic dimension are described fully by quantum mechanics but the transition to macroscopic distances by the semiclassical approximation. Then the semi-classical description is valid for all energies, after distances are traversed such that the classical action far exceeds -. This is the autonomous aspect of the quantum to classical transition.

#### **4 The Quantum to Classical Transition**

#### *4.1 Historical Context*

The question of the transition from quantum to classical mechanics in the motion of particles is as old as wave mechanics itself. In the SP picture it is required that in the classical limit the wavefunction of a single particle describes a classical trajectory i.e. a narrow wavepacket. In the ensemble picture the limit is, as embodied in the IT, that the wavefunction describes an ensemble of particles following classical trajectories.

#### *4.2 Schrödinger, Heisenberg and Kennard.*

Schrödinger, immediately following his invention of wave mechanics in a sequence of papers in 1926, investigated the classical limit of wave mechanics. In a paper [38] entitled "On the continuous transition from micro- to macro-mechanics" he gave an example of how a packet of waves describing the harmonic oscillator can move in such a way that the displacement of the wavepacket as a whole follows the well-known classical dynamics of the one-dimensional harmonic oscillator. In this calculation Schrödinger repeatedly draws the analogy of superpositions of oscillator eigenfunctions to wavepackets formed from classical normal modes on an oscillating string.

The important point to note here is that Schrödinger was seeking, through the wave equation, to represent a *single particle* as a packet of quantum waves which is so localised in space that it can be perceived as a classical particle. Nevertheless he recognized the limitations of his model, pointing out, for example, that a nondispersive packet can only be built from *bound* eigenfunctions and any admixture of continuum states will result in an expanding wavepacket as in the optical case.

This latter point was taken up by Heisenberg [39] in a lengthy paper on the interpretation of the new quantum mechanics and its relation to classical mechanics. In a section also called "the transition from micro- to macro-mechanics", Heisenberg criticises the relevance of bound states in connection with classical mechanics. To illustrate the difficulty with continuum motion Heisenberg showed that an initial Gaussian wavepacket moving freely will spread in space as a function of time and so cannot represent a single material particle.

A more precise demonstration of the classical aspects of quantum motion can be traced back to 1927 in a paper by Kennard [40]. Kennard showed that the *centroid* of quantum "probability packets" moves according to classical mechanics. In retrospect, Kennard probably deserves recognition for the "Ehrenfest" theorem, but perhaps this is denied him since he couched his proof in the language of matrix mechanics, whereas Ehrenfest [41] used Schrödinger wave mechanics.

Kennard's paper is a very important landmark in the development of the meaning of the wavefunction. Interestingly, this is one of the last papers to utilise predominantly the Born, Heisenberg, Jordan [42] theory of matrix mechanics. Kennard defines a "probability amplitude" *M* (*q*) for a variable *q* in matrix mechanics, which is later shown to be equivalent to the Schrödinger wavefunction ψ(*q*).

He considers the motion of "probability packets" and shows that, for the cases of free motion or motion in constant electric or magnetic fields, the centroid obeys classical mechanics.

As perhaps the first to emphasise the ensemble picture, Kennard shows that Heisenberg's "proof" of the uncertainty principle is properly formulated as the statistical spread of momentum and position measured on an ensemble of identical systems. The spread, for the particular case of a free wavepacket, is calculated using the probability *MM* <sup>∗</sup> *dq* which is identical to Born's probability interpretation of the Schrödinger wavefunction.

As mentioned above, the case of free motion had been solved already by Heisenberg [39] who showed that a Schrödinger free wavepacket spreads in time. Kennard, although he shows that his probability amplitude *M* is the same as a Schrödinger wavefunction ψ, uses this spreading as an argument against the superiority of Schrödinger wave mechanics with respect to matrix mechanics.

Kennard raises objections to the Schrödinger wave equation by pointing out that a spreading wavefunction of an electron must correspond to a spreading of charge density. Note that here, in contrast to his view of the *M* of matrix mechanics, in interpreting Schrödinger's ψ, Kennard is assuming that the SP picture applies to this wavefunction. Then he points out that a detection of the electron must localise its full charge at a point. Hence, because of the measurement, the original diffuse wavepacket "loses any further physical meaning" and must be replaced by "a new, smaller wavepacket". Kennard is using the necessity, in the particle picture, to invoke a "collapse of the wavefunction" as an argument against the use of a Schrödinger wavefunction.

Following this objection to the collapse scenario, Kennard then advances the ensemble interpretation of the probability amplitude of matrix mechanics. He writes "the wavepacket spreads, for example, like a charge of shot, in which each pellet describes a trajectory dependent upon its initial position and motion and the whole charge spreads in time as a consequence of differences in these initial conditions", precisely as described by the IT Eq. (6). In the ensemble picture, as distinct from the SP picture, there is no problem with the spreading of the wavepacket. Classical particles with different initial momenta will spread out as they move from micro- to macroscopic distances.

#### *4.3 Ehrenfest and Einstein*

Ehrenfest's paper was published a few months after Kennard's. Apparently, the clarification of the connection of quantum to classical mechanics received an enormous boost with this publication. Ehrenfest used the Schrödinger equation to prove the theorem that quantum position and momentum *expectation values* obey a law similar to Newton's law of classical mechanics. In one dimension, using Ehrenfest's notation, it is expressed as

$$m\frac{d^2\langle\mathbf{x}\rangle}{dt^2} = \int d\mathbf{x}\,\Psi\Psi^\* \left(-\frac{\partial V}{\partial \mathbf{x}}\right) = -\left(\frac{\partial V}{\partial \mathbf{x}}\right) \tag{8}$$

As often remarked, however, this is not Newton's Law which would require−∂*V* /∂*x* to appear on the r.h.s.. However, it turns out that for the cases *V* = *a*, *V* = *ax* and *V* = *ax*2, where *a* is a real constant, the theorem is the same as Newton's law. The spreading of wavepackets remains a problem since, if the wavepacket occupies a macroscopic volume of space, little meaning can be attributed to an average position. Also, for all other potentials with terms higher than quadratic one does not have motion according to Newton's law. Hence, for these two reasons and despite its appealing form, in general Ehrenfest's theorem cannot be considered as describing the transition to classical mechanics, as emphasised by Ballentine [43, 44].

Mindful of Heisenberg's proof of free wavepacket spreading, Ehrenfest is careful to stress that, within the particle picture, the motion of the mean value according to Newtonian mechanics is meaningful only "for a small wavepacket which remains small (mass of the order of 1gm.)". Clearly he was thinking of a single particle described by a small wave packet. The ideas that narrow wavepackets and Ehrenfest's theorem embody the nature of the quantum to classical transition for a single particle, pervade most elementary text books on quantum mechanics even today.

The SP and ensemble pictures were hotly discussed at the Fifth Solvay conference in October 1927. Einstein gave an example of electrons emerging from a small hole to impinge on a distant screen. He pointed out that in the ensemble picture the wave function simply gives the probability of electron detection at a given point. However, in the SP picture of the wave function, the wave which has spread to occupy a macroscopic space, must "collapse" to a point on the screen. Einstein objected to this and commented "one can only remove the objection in this manner, that one does not describe the process by the Schrödinger wave only but at the same time one localises the particle during propagation". Remarkably, after more than ninety years, we recognise that the IT wave function fulfills *exactly the property that Einstein was seeking*, a Schrödinger wave whose variables follow classical trajectories.

#### **4.3.1 After 1927**

It is interesting, although understandable in the first years of quantum and wave mechanics, that the SP and ensemble pictures are confused continually. This applies not only to Kennard, as outlined above, but also to Heisenberg and Schrödinger themselves. In discussing the uncertainty principle, Heisenberg describes exclusively measurements on a single particle, as is discussed in great detail in the accompanying paper by Schmidt-Böcking et.al. [12]. This is despite the Kennard paper quoted above and, most importantly, Robertson's proof [13] of the uncertainty principle. Both papers make clear that the spread of measured values of a variable refers to an ensemble statistical spread and not the uncertainty in measuring that property on a single particle. Similarly, Schrödinger, although a confirmed advocate of the SP picture, still admits the validity of Born's statistical interpretation and the necessity to consider a sequence of measurements, see the discussion of Mott's problem given below.

Although the Ehrenfest Theorem and narrow wavepackets are used as the classical limit in many elementary text books, reminders have been given continually since 1927 of the problems involved with this picture and the essential interpretation of a wavefunction as representing an ensemble and not a single particle.

Kemble in 1935 [49], comments that the interpretation of quantum mechanics "asserts that the wavefunctions of Schrödinger theory have meaning primarily as descriptions of the behaviour of (infinite) assemblages of identical systems similarly prepared".

Writing in 1970, Ballentine [43] advances several arguments "in favour of considering the quantum state description to apply only to an ensemble of similarly prepared systems, rather than supposing as is often done, that it exhaustively represents a single physical system". In addition, in 1972 [44] Ballentine, considers "Einstein's interpretation of quantum mechanics" and advances convincing evidence that Einstein was a firm proponent of the ensemble picture. Indeed, Ballentine must be considered as a prophet of the ensemble picture and many of his ideas are corroborated by the arguments advanced in the present paper.

In a scholarly essay in 1980, on the "Probability interpretation of quantum mechanics", Newton [45] emphasises that "the very meaning of probability implies the ensemble interpretation".

In 1994, Ballentine et al. [46] examined the Ehrenfest theorem from the point of view of the quantum/classical transition and concluded that "the conditions for the applicability of Ehrenfest's theorem are neither necessary nor sufficient to define the classical regime." Furthermore, in connection with the ensemble or SP pictures they pointed out that "the classical limit of a quantum state is an ensemble of classical orbits, not a single classical orbit." A comprehensive account of ensemble interpretations of quantum mechanics is given by Home and Whitaker [47].

#### **5 Consequences of the IT and the Ensemble Picture**

In this section three classic problems of quantum theory are analysed briefly within the IT and related ensemble picture. The problems are the subject of countless papers and the ensemble aspects have been discussed before. However, the consequences of the IT illuminate further the simplicity of the ensemble explanation. Then the reconciliation of the classical trajectory aspect of IT with the quantum interference effect is presented.

#### *5.1 The Schrödinger Cat*

The mere posing of this question by Schrödinger [50] attests to his adherence to the SP interpretation of the wavefunction. As has been observed earlier, in the ensemble picture the interpretation is trivial, as explained by Ballentine [48]. Since the wavefunction applies to many observations, one finds that half the cats are alive and half are dead. No meaning can be attached to the observation of a single cat, unless successive measurements are made over time and feline re-incarnation is allowed.

In the same paper, Schrödinger comments on the apparent problem that radioactive decay described by a spherically-symmetric wave does not lead to uniform illumination of a spherical screen but rather to individual points which slowly are seen to be uniformly distributed. However, although he states that "it is impossible to carry out the experiment with a single radioactive atom" he does not concede that this requires an ensemble interpretation of the wavefunction. This is precisely the problem of Mott which is considered next.

#### *5.2 The "Mott Problem" of Track Structure*

One of the oldest "problems" of the interpretation of a wavefunction for material particles is that posed by Schrödinger [50] and addressed in 1929 by Mott [51]. Certainly Mott's paper was at the instigation of his mentor Darwin, a confirmed adherent of the SP picture [52]. This is one of the most striking examples of erroneously assigning a wavefunction to a single particle. Mott remarked,

"In the theory of radioactive disintegration, as presented by Gamow, the α-particle is represented by a spherical wave which slowly leaks out of the nucleus. On the other hand, the α-particle, once emerged, has particle-like properties, the most striking being the ray tracks that it forms in a Wilson cloud chamber. It is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space."

Mott presents a detailed argument based on scattering theory to argue that only atoms lying on the same straight line will be ionised successively by an α-particle emitted in a spherical wave. Although Mott repeatedly refers to the *probability* of ionisation he interprets the wavefunction as applying to a single α-particle.

However, according to the IT and the ensemble interpretation the proof of Mott is completely superfluous. There is absolutely no mystery attached to "how it is that an outgoing spherical wave can produce a straight track". This apparent dichotomy of wave mechanics is explained by the dual nature of the semi-classical wavefunction of Eq. (5); quantum wavefunction with classically-connected coordinates. Each coordinate of the initial momentum wavefunction corresponds to a specific momentum and therefore to a specific position *r*(*t*) along the classical trajectory. The spherical *S* wavefunction applies to the ensemble as a whole and specifies equal probability of emission in all directions, i.e. uniform distribution of *p* on the unit sphere. Each α-particle is launched with a certain momentum *p* distributed according to the initial momentum wavefunction and, according to the IT, the position on a macroscopic cloud chamber scale follows a *straight line* classical trajectory. Hence it is obvious that only atoms lying along this trajectory can be ionised and the usual straight track in the cloud chamber is observed.

This is a prime example of the principle that what one perceives, in this case directed motion (a set of classical trajectories) or a spherically uniform distribution (a quantum S-wave probability) depends upon the nature and duration of the experiment.

#### *5.3 Entanglement and Wavefunction Collapse*

That wavefunction superposition applies to an ensemble is made clear also by the process of radioactive decay discussed above. Although usually thought of in the time domain, the stationary picture is simpler. The wave function of an ensemble of nuclei is described by a superposition of the state of a bound nucleus and the state of two separated product nuclei at the same total energy. The intrusion of a measuring device simply detects which state a given member of the ensemble occupies. The absence of a signal in a measuring device denotes undecayed state and a signal denotes a decay. The half-life is interpreted from a sequence of measurements on the ensemble. It is not a property of a single nucleus, although colloquially the half-life is often so ascribed. This aspect is emphasised particularly in the very clear exposition of Rau [53].

The paper of Einstein, Podolsky and Rosen [54], whose result often is referred to as the "EPR paradox", has been the subject of an enormous number of works on the subject of reality, action at a distance etc. Throughout the EPR paper appears the SP viewpoint of a partial wavefunction describing an independent particle.

Already in the first replies to the EPR paper, by Schrödinger [55] and Bohr [56], it was pointed out that it is essential to consider the *two-particle* commuting operators, ignored by EPR. Nevertheless the reply papers did not apply these considerations directly to the EPR entangled wavefunctions.

Here we infer the ensemble picture and show that the recognition of good twoparticle quantum numbers is essential. Then, in the pure states considered in EPR, a good quantum number ensures that every pair of the ensemble will give the same value of the corresponding two-particle property upon measurement.

EPR consider a two-particle eigenstate written in the entangled form

$$
\Psi(\mathbf{x}\_1, \mathbf{x}\_2) = \int \psi\_p(\mathbf{x}\_2) \mu\_p(\mathbf{x}\_1) \, dp \tag{9}
$$

where

*up*(*x*1) <sup>=</sup> *<sup>e</sup> <sup>i</sup> px*<sup>1</sup> and <sup>ψ</sup>*p*(*x*2) <sup>=</sup> *<sup>e</sup>*<sup>−</sup> *<sup>i</sup> <sup>p</sup>*(*x*2−*x*0) (10)

are eigenfunctions of one-particle operators *p*1, *p*<sup>2</sup> with eigenvalues *p* and −*p* respectively. The constant *x*<sup>0</sup> is arbitrary. Note that the single-particle momentum *p* can take any value.

The *p* integral in this equation can be carried out to give

$$\begin{split} \Psi(\mathbf{x}\_1, \mathbf{x}\_2) &= 2\pi \delta(\mathbf{x}\_1 - \mathbf{x}\_2 + \mathbf{x}\_0) \\ &= 2\pi \int \delta(\mathbf{x}\_1 - \mathbf{x}) \delta(\mathbf{x} - \mathbf{x}\_2 + \mathbf{x}\_0) \, d\mathbf{x} \end{split} \tag{11}$$

which is an entangled state in position space. However, again, all *x* values are possible.

Thus it has been shown that one and the same two-particle function can be expanded in terms of eigenfunctions of observables of particle 2, in this case *p* and *x*, which do not commute.

As shown by Schrödinger [55] and Bohr [56], the conserved quantities emerge from a transformation to relative and centre-of-mass (CM) coordinates for equal mass *m* particles. We define relative *xr* and CM position *X* as

$$\mathbf{x}\_r = \mathbf{x}\_1 - \mathbf{x}\_2 \quad \text{and} \quad X = (\mathbf{x}\_1 + \mathbf{x}\_2)/2 \tag{12}$$

and correspondingly relative and CM momenta

$$p\_r = (p\_1 - p\_2)/2 \quad \text{and} \quad P\_{\text{CM}} = p\_1 + p\_2 \tag{13}$$

Immediately one sees from Eq. (11), that -(*x*1, *x*2) *is* an eigenfunction of the relative position coordinate *xr* = *x*<sup>1</sup> − *x*<sup>2</sup> with eigenvalue *xr* = −*x*0. Similarly, from Eq. (9) with Eq. (10) one sees it is simultaneously an eigenfunction of CM momentum *PCM* with eigenvalue zero. This is in order since these two operators commute. However it is readily checked, as must be, that -(*x*1, *x*2) *is not* an eigenfunction of *X* or *pr* since these do not commute with *PCM* and *xr* respectively.

In summary, the two-particle wavefunction of EPR fixes the CM momentum at zero and the relative position of the particle pair is equal to −*x*0. This is the only information in the two-particle wavefunction. One has, however, the clear requirement that the two-particle wavefunction should propagate intact to the detectors. In any measurement the two corresponding two-particle observables have the same precise value for all members of the ensemble of pairs.

Now one has two possible scenarios characterising entanglement.

The measured two-particle eigenvalues are sharp, *PCM* = *p*<sup>1</sup> + *p*<sup>2</sup> = 0 and *xr* = *x*<sup>1</sup> − *x*<sup>2</sup> = *x*<sup>0</sup> for all members of the ensemble with no statistical spread, in accordance with their commutation. Note that the specification of *two-particle* conserved observables allows one to assign precise values to both non-commuting *one-particle* observables. Hence there is absolutely no barrier to measuring both position and momentum of one or both of the particles with arbitrary accuracy. This is emphasised by the analysis of the Uncertainty Relation in the accompanying paper of Schmidt-Böcking et.al. [12]. Of course the *single-particle p* values have a distribution of probability predicted by projection of the one-particle probability amplitude out of the two-particle wavefunction.

Both scenarios require non-local information. The measurement in (b) requires communication between the two separated detectors to ensure coincidence. In case (a) only one detector is required but the non-local information is in the knowledge of the two-particle quantum numbers which are conserved for all particle separations.

The simultaneous fixing of position and momentum becomes apparent within the IT if, as is normal, detection is made at large distances from the volume from which the correlated pair is created. According to the IT there is a classical connection between position and initial momentum for detection of particles 1 and 2 at times *t*<sup>1</sup> and *t*<sup>2</sup> respectively. Then the space wavefunction can be written

$$
\Psi(x\_1, x\_2) \propto \Psi(p\_1, p\_2 = -p\_1). \tag{14}
$$

In particular the IT gives the classical relation

$$\mathbf{x}\_1 = p\_1 t\_1 / m \text{ and } \mathbf{x}\_2 = -p\_2 t\_2 / m \tag{15}$$

so that from the second conservation law *x*<sup>2</sup> = *x*<sup>1</sup> + *x*<sup>0</sup> one has the restriction

$$\mathbf{x}\_0 = -(p\_1 t\_1 + p\_2 t\_2)/m. \tag{16}$$

Single-particle *x* and *p* can be measured simultaneously with sub- accuracy, see Ref. [12].

A striking manifestation of such entanglement, which has been well-studied in experiments, is the full fragmentation of the helium atom by a single photon. This example is given since it comprises simultaneously both the momentum and position (continuous variable) entanglement of EPR *and the discrete variable spin entanglement*, as envisaged by Bohm [2], in a pure two-electron state. Furthermore, from the IT, the electrons can be assigned classical trajectories *within the two-electron quantum wavefunction*. This is not a "Gedankenexperiment" but a real measured system [57].

The two electrons emerging can be detected in coincidence and occupy a <sup>1</sup>*P<sup>o</sup>* two-electron continuum state (this means their state is a spin singlet, has total orbital angular momentum one unit and odd parity). A selection rule [58] says that electrons of the same energy cannot be ejected back-to-back i.e at 180◦ such that *p*<sup>1</sup> = −*p*2. That is, the *two-electron* state has a node for the EPR configuration as the coincidence experiments confirm.

If one of the electrons is left undetected a counter will register electrons of a given energy at a particular angle. However, if a detector diametrically opposed is switched on to detect electrons of the same energy in coincidence, the counts in both detectors will be zero. This coherent state can be made incoherent by switching off one of the detectors when electrons will be measured again. The essence is that this pure effect of wavefunction entanglement is evident, even though according to the IT, the electrons are moving on classical trajectories after they exit the reaction zone with well-defined momenta.

In interpreting the wavefunction, as in EPR, it is crucial that the ensemble is viewed as an ensemble of *two-electron* systems. This two-electron wavefunction is the single quantum entity and it must be transmitted to the macroscopic detection zone unchanged. Then there is no wavefunction interpretation problem within the ensemble picture. The wavefunction node says that there is zero probability that a given member of the ensemble (a coincident pair of electrons) will be emitted in the forbidden configuration.

The coincident detection of position and momentum extracts the information from the wavefunction of the ensemble of two-electron states. The non-coincident detection of electrons extracts information only on the ensemble of single electrons. The effect of entanglement is non-local simply because the two-electron wavefunction is non-local.

A comprehensive discussion of the implications of entanglement for the uncertainties in measured properties relevant to quantum information in the case of the continuous variable description of light rather than material particles, is given by Braunstein and van Loock [22].

#### *5.4 Quantum Interference*

The Davisson-Germer experiment of 1927 on electron-beam diffraction established the validity of the description of particle ensembles by a wave function. The diffraction of heavy neutral particle beams was confirmed in the pioneering experiments of Stern and co-workers as early as 1929 [59]. The demonstration of interference even of large molecules has been achieved recently in the remarkable experiments of Arndt and his group [60], reported in Chap. 24 of this volume.

The explanation of interference patterns in terms of semi-classical wavefunctions and the underlying classical trajectories has been given in great detail by Kleber and co-workers [61] and will not be repeated here. Based upon the IT (see eq.(1) of Kleber [62]), their theory is used to interpret experiments such as those of Blondel et.al. [63]. Here the "photoionisation microscope" exhibits interference rings of electrons ionised from a negative ion in the presence of an extracting electric field. In the semi-classical explanation electrons can occupy two classical trajectories. Either they proceed directly to the detector or, initially they are ejected moving away from the detector but are turned around in the electric field. The imaging of the spatial wavefunction squared is obtained by detection on a fixed flat screen i.e. only the position of electrons is detected. Then an interference pattern from the two trajectories is observed.

However, were the vector position *and* vector momentum of the electrons to be observed, that would correspond to a "which way" determination and the perception would be of two distinct non-interfering classical trajectories. Interestingly, as distinct from entanglement, in this case it is a lack of information which gives rise to wave perception. Blondel et al. [63] remark also that for ionisation from neutral atoms the interference rings are there but are too small to be detected, again showing that perception depends upon resolution.

#### **6 The Imaging Theorem and Decoherence Theory: IT and DT**

As stated in the Introduction, the suppression of state superposition, entanglement and interference through environmental interaction can be seen as a requirement on the way to a classical limit of quantum mechanics and has come to be known as "decoherence theory" (DT). It is viewed as a universal phenomenon, extending even to the classical limit of quantum gravity [64, 65] (for an interesting discussion see Ref. [66]). In the following the transition from quantum to classical perception is discussed.

#### *6.1 Decoherence*

There is an enormous literature on DT and associated theories describing "spontaneous localisation" due to stochastic interaction. Space does not permit a discussion of the many and varied aspects of these theories, so here consideration is given to those features relevant to the quantum to classical transition embodied in the IT and to the SP or ensemble interpretation of the wavefunction.

The essence of DT is given in the famous paper of Zurek [24] and in more detail in the reviews of Schlosshauer [25, 26]. A more exhaustive treatment with discussion of the dependence of the environmental interaction terms is to be found in the stochastic Schrödinger equation approach [67]. Here the simpler original density matrix version of Zurek [24] is sufficient as illustration.

The basic mechanism of DT by which certain quantum aspects are eliminated is quite straightforward, accounting for the universality of this phenomenon. In the simplest case presented in Ref. [25], a one-dimensional two-state quantum system *S*, with states| ψ*<sup>n</sup>* , is assumed to become entangled with an "environment" with corresponding states | *En* . Limiting to two-state quantum systems, the ensuing entangled state vector is

$$\left|\left|\Psi\right> = \alpha \left|\left|\psi\_1\right>\right| E\_1\left> + \beta \left|\left|\psi\_2\right>\right| E\_2\left>\right>\right.\tag{17}$$

and gives a total density matrix ρ = | - - |. According to Ref. [25], "the statistics of all possible local measurements on *S* are exhaustively encoded in the reduced density matrix ρ*<sup>S</sup>* ", given by

$$\begin{split} \rho\_{\mathcal{S}} &= \mathcal{T} \boldsymbol{r}\_{E} \rho = |\boldsymbol{\alpha}|^{2} |\psi\_{1}\rangle \langle \psi\_{1} | + |\beta|^{2} |\psi\_{2}\rangle \langle \psi\_{2} | \\ &+ \alpha \beta^{\*} |\psi\_{1}\rangle \langle \psi\_{2} | \langle E\_{2} | E\_{1} \rangle + \alpha^{\*} \beta |\psi\_{2}\rangle \langle \psi\_{1} | \langle E\_{1} | E\_{2} \rangle. \end{split} \tag{18}$$

Then a measurement of the particle's position is given by the diagonal element,

$$\begin{split} \rho\_S(\mathbf{x}, \mathbf{x}) &= \left| \alpha \right|^2 \left| \psi\_1(\mathbf{x}) \right|^2 + \left| \beta \right|^2 \left| \psi\_2(\mathbf{x}) \right|^2 \\ &+ 2 \text{Re} [\alpha \beta^\* \psi\_1(\mathbf{x}) \psi\_2^\*(\mathbf{x}) \langle E\_2 \mid E\_1 \rangle ] \end{split} \tag{19}$$

where "the last term represents the interference contribution". The assumption of DT is that in general the states of the environment are orthogonal and so the interference term disappears. More importantly, from Eq. (18) the off-diagonal terms disappear and one has a diagonal density matrix only. From Eq. (19) this has two "classical" terms interpreted as classical probabilities.

A slightly different model is adopted in Ref. [24] in that the two states comprising the system *S* are taken as two spatially-separated Gaussian wavefunctions. The corresponding system density matrix exhibits four peaks. This density matrix is propagated in time subject to a temperature-dependent environment interaction. The result is to give a density matrix of diagonal form with only two peaks along the diagonal.

In this case the decoherence reduces the off-diagonal elements to zero and the diagonal term does not contain the "interference" contribution since the Gaussians do not overlap. This removal of coherence between different spatial parts of the wavefunction is considered to correspond to the emergence of classicality. In connection with the classical transition Schlosshauer writes [26] "the interaction between a macroscopic system and its environment will typically lead to a rapid approximate diagonalisation of the reduced density matrix in position space and thus to spatially localised wavepackets that follow (approximately) Hamiltonian trajectories". This following of classical trajectories however, is not proven in detail.

Implicit is the SP picture in which the diagonal elements represent narrow wavepackets giving classical behaviour via Ehrenfest's theorem. The ultimate spreading of these wavepackets is not considered, although suitable environmental interaction can lead to the wavepackets remaining narrow. In short, the transition to classicality is viewed as an elimination of quantum coherence effects and the vital feature of the emergence of classical dynamics according to Newton is not shown.

#### *6.2 Unitary Evolution*

In appendix A, following the example of Ref. [24], the free *unitary* propagation of two, initially narrow, Gaussian wavepackets within the IT is calculated. It is shown that, under low detector resolution, the density matrix also assumes the diagonal form

$$\rho(\mathbf{x}, \mathbf{x}, t) = \frac{1}{\sqrt{\pi} \eta(t)} \left( e^{-(\mathbf{x} - \mathbf{X}\_1)^2 / \eta^2} + e^{-(\mathbf{x} - \mathbf{X}\_2)^2 / \eta^2} \right) \tag{20}$$

where *X*1, *X*<sup>2</sup> are the centres of the wavepackets and the time-dependent width is η = ˜σ*t*/μ, for initial width σ˜ and particle mass μ. Hence, the intrinsic spreading of the wavepacket with time emerges as expected in the ensemble picture. In this picture there is no problem of interpretation of the two probabilities; 50% of the ensemble members will be detected near to *X*<sup>1</sup> and 50% near to *X*2. Wavefunction collapse is unnecessary. Most important however, in the IT, the propagation of the co-ordinates of the diagonal density matrix is according to *classical* mechanics. Nevertheless, if the resolution is on the microscopic scale then interference and manifestations of quantum propagation resulting from finite off-diagonal elements can be detected. Just as in optics, the perception of particle trajectory (ray) or wave is decided by the sharpness of vision.

The study of collision complexes in nuclear, atomic and molecular physics has long been concerned with the questions of measurement of interference and entanglement effects [18–20]. Coincidence detection of several collision fragments in entangled states are performed with increasing sophistication (see, for example, Ref. [11]). In line with the IT, classical motion of the collision fragments outside the reaction zone is shown to be appropriate. Nevertheless quantum coherence is preserved showing that environmental decoherence does not occur in such experiments.

The degree of decoherence assigned to a many-body entangled state depends upon which particles are *not* observed or even which dynamical properties are observed and which are not. Coherence can be fully or partially removed according to the experiment. In the language of the experimentalist, either one registers the "coincidence" spectrum or the "singles" spectrum. Again this illustrates that perception of quantum effects depends upon the measurement. Non-detection of collision variables corresponds to a partial trace of the full density matrix, as in DT.

#### **7 Conclusions**

The imaging theorem corresponds only to the ensemble interpretation. According to the IT, an initial momentum wavefunction decides the spatial wavefunction at macroscopic distance. The modulus squared of the initial momentum wavefunction corresponds to an ensemble of classical particles with the same initial momentum distribution. Each particle appears to move along a classical trajectory to be registered at well-defined position at a distant screen. Nevertheless the quantum wavefunction is preserved so that the loci of points of equal probability are the classical trajectories but that probability is given by the quantum position wavefunction.

Indeed, all collision experiments support the ensemble picture. One counts many particles at different locations and times on a detector and so builds an image of the initial momentum distribution. Particularly striking in this respect is the observation of the gradual assembly of an interference pattern. Using electron diffraction through a pair of slits, it has been shown [4] that the wave interference pattern is built up slowly by registering many hundreds of hits of individual electrons on a detector screen. Even more strikingly, the experiment has been performed with very large organic molecules [69]. This shows convincingly that it is the ensemble of hits at the detector that builds up the wave interference pattern.

In contrast, the SP picture is that the wavefunction, extending over macroscopic distance, is carried by each molecule and the wavefunction collapses at different points on the screen. That is, the detector is required to instigate decoherence leading to instantaneous wavefunction collapse (from macroscopic to microscopic extent) and the electron being registered at a single localised point on the detector. Again, one is faced with the dilemma of Kennard and Einstein in accepting the plausibility of such a transition.

To summarise, it has been shown that;

(3) Standard measurement techniques, either on single or multiple particles, can lead to perception or otherwise of the quantum properties of interference and entanglement according to the information registered. The inference of classical or quantum behaviour depends ultimately upon the resolution and detail of the measurement performed.

Without environment influence, within the IT, unitary evolution of quantum systems results in effective decohering effects. This "decoherence" is of a different nature than in DT. It occurs due to cancellation of oscillating terms of different phase, which leads to non-resolution of oscillatory terms in the propagation of the density matrix to macroscopic times, as in Eq. (27) and Eq. (28). Hence, lack of sufficient resolution results in effective decoherence although paradoxically it arises from the very terms, oscillatory phase factors, which are the hallmark of quantum coherence in the wavefunction.

Already in 1951, long before the formulation of DT, in his discussion of the motion of atom beams in the Stern-Gerlach experiment [1], Bohm [2] points out the decoherence arising from interaction with a macroscopic detector. Interestingly, he attributes the lack of interference also to the impossibility of resolving oscillatory energy phase factors in the unitary time propagation (exactly as in the Appendix) but does not really emphasise the distinction between this and the DT interactions.

The preservation of the wavefunction can lead to interference. However, the perception of interference patterns, or not, again depends upon the nature of the measurement performed. The observation of interference patterns implies that, although resolution is high, incomplete information as to the different trajectories encoded in the wavefunction variables is extracted by the measurement. That is, a "which way" detection is not performed. Then, whether one perceives quantum or classical dynamics depends simply upon the precision of the measurement performed and the amount of information extracted from the wavefunction. This is all in close analogy, both physically and mathematically, to the optical case of perception of particle, wave or ray properties.

In the case of the detection of the effects of particle entanglement it is necessary to treat the ensemble entity as corresponding to the *many-particle* wavefunction and its quantum numbers. Incomplete extraction of the information encoded in the many-particle ensemble wavefunction, for example detection of only some of the particles or incomplete specification of vector variables, corresponds to an effective decoherence.

**Acknowledgements** The author is extremely grateful to Prof. James M. Feagin for several years of close cooperation on the derivation and meaning of the imaging theorem. The insight of Prof. Horst Schmidt-Böcking on experimental methods and their significance for the uncertainty relation, described in the accompanying paper (Chap. 12 of this volume), is acknowledged also.

#### **8 Appendix**

As in the discussion of decoherence by Zurek [24] the time development of a onedimensional single-particle ensemble wavepacket is considered. The wavepacket is composed of two Gaussians centred at *x* = *X*<sup>1</sup> and *x* = *X*<sup>2</sup> with width such that there is essentially no overlap at *t* = 0. The initial state is then

$$\Psi(\mathbf{x}, t=0) = (\pi \sigma^2)^{-1/4} \sum\_{i=1,2} e^{-\mathbf{x}\_i^2/(2\sigma^2)} \tag{21}$$

where *xi* ≡ *x* − *Xi*. For *t* > *t*<sup>0</sup> this initial wavefunction propagates freely in time and has the exact form

$$\begin{split} \Psi(\mathbf{x}, t) &= (\sigma^2/\pi)^{1/4} \left(\sigma^2 + \frac{i\hbar t}{\mu}\right)^{-1/2} \\ &\quad \times \sum\_{i=1,2} \exp\left[-\frac{x\_i^2}{2\left(\sigma^2 + \frac{i\hbar t}{\mu}\right)}\right], \end{split} \tag{22}$$

where μ is the particle mass. The IT condition emerges in the limit of large times and distances. Large times corresponds to *t*/μ σ2. Then the spatial wavefunction assumes the IT form,

$$\begin{split} \Psi(\mathbf{x},t) &\approx \left(\frac{\sigma^2}{\pi}\right)^{1/4} \left(\frac{\mu}{i\hbar t}\right)^{1/2} \\ &\times \sum\_{i=1,2} e^{-\left(\mu\mathbf{x}\_i\sigma/(\sqrt{2}\hbar t)\right)^2} e^{i\mu\mathbf{x}\_i^2/(2\hbar t)} \end{split} \tag{23}$$

The IT limit giving the classical trajectory is such that *xi* and *t* both become large but the ratio is a constant classical velocity. To emphasise this we introduce the momenta *pi* = μ*xi*/*t*. We also define, as the width of the Gaussian in momentum space, <sup>σ</sup>˜ <sup>≡</sup> -/σ. Then we can simplify the asymptotic spatial wavefunction using

$$(\mu \mathbf{x}\_i \sigma / (\sqrt{2} \hbar t))^2 \equiv p\_i^2 / (2\tilde{\sigma}^2) \tag{24}$$

and the energy phases

$$
\mu \mathbf{x}\_i^2 / (2\hbar t) = p\_i^2 t / (2\mu \hbar). \tag{25}
$$

The asymptotic spatial wavefunction is then,

$$\Psi(\mathbf{x},t) \approx \left(\frac{\mu}{i\sqrt{\pi}\tilde{\sigma}t}\right)^{1/2} \sum\_{i=1,2} e^{-p\_i^2/(2\tilde{\sigma}^2) + ip\_i^2 t/(2\mu\hbar)}\tag{26}$$

which looks exactly like a pair of free momentum Gaussians propagating in time and corresponds to the 1D form of the IT of Eq. (5), with *dpi*/*dxi* = μ/*t* for free motion.

The diagonal element of the density matrix is defined as ρ(*x*, *x*, *t*) = -<sup>∗</sup>(*x*, *t*)-(*x*, *t*) and is

$$\begin{split} \rho(\mathbf{x}, \mathbf{x}, t) &= \frac{\mu}{\sqrt{\pi}\tilde{\sigma}t} \sum\_{i=1,2} e^{-p\_i^2/\tilde{\sigma}^2} \\ &+ 2\cos\left[ (p\_1^2 - p\_2^2)t/(2\mu\hbar) \right] e^{-(p\_1^2 + p\_2^2)/(2\tilde{\sigma}^2)} \end{split} \tag{27}$$

The off-diagonal density matrix is defined as ρ(*x*, *x*- , *t*) = -<sup>∗</sup>(*x*, *t*)-(*x*- , *t*) and consists of four terms,

$$\rho(\mathbf{x}, \mathbf{x}', t) = \frac{\mu}{\sqrt{\pi}\tilde{\sigma}t} \sum\_{i, j = 1, 2} e^{-(p\_i^2 + p\_j'^2)/(2\tilde{\sigma}^2)} e^{-i(p\_i^2 - p\_j'^2)t/(2\mu\hbar)} \tag{28}$$

At *t* = 0 this gives rise to four Gaussian peaks, as in Ref.[24]. It reduces to the diagonal element when *pi* = *p*- *i* , i.e. *x* = *x*as it should.

One sees that the diagonal matrix element shows two peaks at *p*<sup>1</sup> = 0, *p*<sup>2</sup> = 0 or equivalently *x* = *X*1, *x* = *X*2. There is also an interference term. In the off-diagonal element there are four peaks, with the two additional peaks at *x*- = *X*<sup>1</sup> and *x*- = *X*2. These also contain oscillatory phase factors giving interference.

Clearly, to observe interference effects the temporal resolution must typically be less than one oscillation, i.e. *t* < 4μπ/(*p*<sup>2</sup> <sup>1</sup> − *p*<sup>2</sup> <sup>2</sup>). Consider that the particles are electrons with mass unity in atomic units (a.u.). If we take the two peaks to be separated by 1 a.u. of distance, then we have *t* < 4π ≈ 10−<sup>16</sup> s. However, typical resolutions are nanoseconds, that is seven orders of magnitude larger than this. If the resolution is δ*t* ≡ τ then the measurement must be integrated over this time period. Typically the oscillatory terms will then give, omitting constants

$$\int\_{-\pi/2}^{\pi/2} e^{i(p\_1^2 - p\_2^2)t} \, dt \approx \delta(p\_1^2 - p\_2^2). \tag{29}$$

and similarly for the off-diagonal element when *pi* is replaced by *p*- *i* . In other words, the oscillations will average to zero under low resolution of measurement on an atomic time scale. From Eq. 27 this implies that the density matrix will exhibit only two diagonal gaussian peaks for such measurements,

$$\rho(\mathbf{x}, \mathbf{x}, t) = \frac{\mu}{\sqrt{\pi}\tilde{\sigma}t} \left( e^{-p\_1^2/\tilde{\sigma}^2} + e^{-p\_1^2/\tilde{\sigma}^2} \right) \tag{30}$$

with *pi* = μ(*x* − *Xi*)/*t*. For the off-diagonal elements, from Eq. (28), all the terms will average to zero under normal time resolution to give zero off-diagonal elements. This is exactly the limit, elimination of off-diagonal density matrix elements, given by Zurek [24] as the classical limit and resulting from time propagation in the presence of an interacting environment. However, we emphasise again that the wavepackets on the diagonal are spreading and only in the limit that particles are macroscopically massive can this be ignored to give localised single particles as envisaged in [25].

By contrast the IT proves that classicality emerges from unitary Hamiltonian propagation under low temporal resolution, in that the density matrix then has only two diagonal peaks . Quantum coherence is lost except where the temporal and spatial resolution are extremely high. The peaks represent an ensemble of classical particles spreading on classical trajectories and distributed according to the initial Gaussian momentum wavefunction.

#### **References**

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#### **Chapter 12 The Precision Limits in a Single-Event Quantum Measurement of Electron Momentum and Position**

#### **H. Schmidt-Böcking, S. Eckart, H. J. Lüdde, G. Gruber, and T. Jahnke**

**Abstract** A modern state-of-the-art "*quantum measurement*" [The term "quantum measurement" as used here implies that parameters of atomic particles are measured that emerge from a single scattering process of quantum particles.] of momentum and position of a **single** electron *at a given time* ["at a given time" means directly after the scattering process. (It should be noticed that the duration of the reaction process is typically extremely short => attoseconds).] and the precision limits for their experimental determination are discussed from an **experimentalists point of view**. We show—by giving examples of actually performed experiments—that in a single reaction between quantum particles *at a given time* only the momenta of the emitted particles but not their positions can be measured with sub-atomic resolution. This fundamental disparity between the conjugate variables of momentum and position is due to the fact that during a single-event measurement only the total momentum but not position is conserved as function of time. We highlight, that (other than prevalently perceived) Heisenberg's "Uncertainty Relation" UR [1] does not limit the achievable resolution of momentum in a **single-event measurement**. Thus, Heisenberg's statement that in a single-event measurement only either the position or the momentum (velocity) of a quantum particle can be measured with high precision contradicts a real experiment. The UR states only a correlation between the mean statistical fluctuations of a large number of repeated single-event measurements of two conjugate variables. A detailed discussion of the real measurement process and its precision with respect to momentum and position is presented.

H. Schmidt-Böcking (B) · S. Eckart · G. Gruber · T. Jahnke Institut für Kernphysik, Universität Frankfurt, 60438 Frankfurt, Germany e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
#### **1 Introduction**

Otto Stern was the pioneer in high-resolution momentum spectroscopy of atoms and molecules moving in vacuum. Gerlach and Stern performed between 1920 to 1922 in Frankfurt their famous Stern-Gerlach experiment (SGE). They obtained for Ag atoms a sub-atomic momentum resolution in the transverse direction of 0.1 a.u. [2]. Today, modern state-of-the-art spectrometer devices such as the Scienta electron spectrometers [3] or the COLTRIMS Reaction Microscope C-REMI [4] can provide even a much better resolution. The imaging system C-REMI can even measure several particles in coincidence by detecting the momenta of all charged fragments emitted in a quantum process. Thus the complete entangled dynamics of such a single quantum process can be visualized. However, in such high resolution experiments the experimenter cannot obtain any direct information on the relative positions of particles. For a single event the absolute and also relative positions inside the quantum reaction are not measureable. The purpose of this paper is to illustrate for a single-event scattering measurement, as discussed by Heisenberg [1], the precision limits of electron momentum and position by presenting experimental examples.

The goal of a quantum measurement, e.g. scattering of a quantum projectile on a target atom in vacuum, is to obtain information on the quantum mechanical collision process. How can such a measurement be performed? The experimenter must prepare projectiles and target objects in a well-defined momentum and as far as possible also in a well-defined position state. This is typically achieved by classical methods. As shown below the momentum state of projectile and target object can be prepared with sub-atomic precision, but positions at a given time can never be controlled with atomic size accuracy. The reason is, no particle in vacuum can be brought completely at rest in the system of measurement and thus positions are not conserved with time. In other words the experimenter cannot predict with sub-atomic precision the impact parameter of the collision and the impact parameters are statistically distributed. Thus numerous single event measurements one after the other have to be summed up to obtain a statistical distribution.

The statistical distribution contains two sources of errors: First, the systematical error of each single-event measurement. This is de facto the horizontal error bar which is given by the quality of preparation and of the classical detection device only. This error bar depends very little on Heisenberg's uncertainty relation. Second, there is a statistical error in the ordinate values, which depends on the number of detected events and which does not depend on the precision of the single-event measurement. The sum of all single-event measurements, i.e. the statistical distribution, is relevant for comparison with theory.

What are the precision limits for parameters in the quantum experiment? The detection apparatus delivers only auxiliary values, from which then information on the quantum process can be deduced. Such auxiliary values are: The time, when the collision occurs. It can be determined by classical methods (see below) with about 50 pico-second precision. During the collision electrons and ionic fragments can be emitted each with a so-called final momentum. Immediately after emission they move in a spectrometer device in which the charged quantum particles exchange momentum due to the electro-magnetic force with the macroscopic detection device. Finally the particles impact on a detector, which can be placed at any distance from the collision zone. The auxiliary values, that are measured by the detector, are: detector impact position and time. Typically they are measured with a precision of 50 μm and 50 pico-seconds. It is to be noticed that these auxiliary quantities allow the experimenter to deduce the particle trajectories in the detection device and to determine the final momentum in the laboratory system from the trajectory of the particle (see below).

To obtain sub-atomic momentum precision (laboratory system) in a single event, the velocity vector (i.e. the total momentum) of the center-of-mass of the single-event collision system must be known from the method of preparation. Conservation laws are therefore of fundamental importance for the implementation of a single-event quantum measurement. An observable can only be measured with sub-atomic accuracy if time-dependent conservation properties are strictly fulfilled during the generally very short duration of the measurement. Total linear momentum, total angular momentum and total energy are conserved but not location. The measured momenta of all fragments can then be corrected for the center-of-mass motion because the total momentum is conserved. For position no conservation law exists, thus a large uncertainty in the location measurement cannot be avoided. Therefore Heisenberg's suggestions that a high resolution position measurement is possible and this position measurement would be even the basis of any quantum measurement completely contradicts real experiments.

Since 1927 numerous papers have been published discussing the consequences of the UR on a quantum measurement within the wave-picture. To the best of our knowledge there is no publication available, where the constraints and the purely classical experimental limits of a single-event quantum measurement are analyzed from the view of an experimenter. Although in the introduction of his paper [1], Heisenberg considered the kinematics and mechanics of a single particle and the measurement of the position and the velocity (momentum) of a single electron "at a given moment", Heisenberg's UR (x · p ≥ è) applies, however, only for the mean statistical fluctuations of a large number of repeated single-event measurements of two conjugate variables and can be viewed to be a prediction of the future particle properties.

We deploy therefor the following two statements:

#### **Statement 1. The UR applies for the statistical distribution of a large ensemble or for repeated measurements but not for the resolution of a single-event measurement.**

This statement is in line with previous work, that revisited this discussion, as well. For example, Ballentine [5], Park and Margenau [6] as well as Briggs [7, 8] contradicted the single-event interpretation of Heisenberg and concluded that it applies only to a large ensemble of similarly prepared systems.

**Statement 2. A single-event measurement can only provide information on the particle's properties back in the past but never allow a prediction of future properties, since the impact of a particle on the detector changes the particles momentum and position state.**

Which parameters of a quantum reaction are measurable and what is the achievable precision? We discuss this question by illustrating the concept of a "real" quantum experiment. As paradigm example for a typical quantum measurement we have chosen the scattering of an electron or ion on a gaseous target atom followed by the coincident detection of all reaction fragments with modern "state-of-the-art" detection devices.

We will discuss the following three findings:

The UR imposes, in contradiction to Heisenberg's claim, no limit on the achievable momentum (velocity) resolution of a single quantum measurement. The UR affects the resolution of such a measurement only indirectly, as it has an impact on the quality of preparation of the pre-collision states of projectiles and target atoms. This has already been highlighted by Kennard in 1927 [9] who theoretically considered the passage of scattered electrons in a classical detection device and concluded: *"In den hier behandelten Fällen haben wir keinerlei quantentheoretische Abweichung gefunden von den klassischen Ergebnissen. Die einzige quantenhafte Eigentümlichkeit in solchen einfachen Fällen liegt in der durch das Heisenberg'sche Unbestimmtheitsgesetz festgesetzten prinzipiellen Unbestimmtheit der Anfangswerte" ("In the cases discussed here, we did not find any quantum theoretical deviation from the classically calculated values. The only quantum influence in such cases originates from the effect that the preparation state values are indeterminate in accordance to Heisenberg's Uncertainty Relation.").* Today, the debate over the statistical versus single-event interpretation is still not converged (see [5–16] for proponents of the single-event interpretation and for papers opposing this interpretation).

In the following chapters we discuss the **purely experimental aspects** and the limits of experimental precision in a single-event measurement of momentum (velocity) and position and present examples:

In Sect. 2: The **scheme and time evolution of a single-event measurement** is discussed beginning with the preparation of the measurement followed by the quantum reaction process and concluding with the detection of the charged fragment in a classical measurement apparatus.

In Sect. 3: The **electron momentum (velocity) measurement by Time-of-Flight (TOF) trajectory imaging** is presented.We consider realistic experimental scenarios for electrons based on experimental results.

In Sect. 4: The determination of the **angular momentum state** of a single electron by a multi-fragment coincidence technique.

In Sect. 5: The **experimental limits for an electron position measurement** are discussed. We also show that Heisenberg's "Gedankenexperiment" on the γ-microscope is not feasible.

In Sect. 6: We consider the **product of precisions in momentum and position measurement of a freely moving single electron**. New experimental techniques for measuring momentum and position of a freely moving electron simultaneously in a one-step approach are provided for the moment of impact on a detector. Within this approach the product of the experimental error bars in electron momentum and detector impact position can be below è by several orders of magnitude.

#### **2 Scheme of a Quantum Measurement**

We consider an experiment where a projectile beam intersects in ultra-high vacuum with a gaseous target to ensure controlled single-event conditions i.e. that only one reaction process occurs during each measurement period. Because of the statistical nature of quantum measurements (to yield statistical distributions) one must prepare numerous projectiles in the "nearly identical" pre-collision state and numerous target objects in controlled "nearly identical" momentum and position states. In the preparation of the pre-collision state "nearly identical" means this preparation is still limited by Heisenberg's UR with respect to the large ensemble projectile and target momentum and position fluctuation widths. E.g. in an ion-atom collision the experimenter cannot precisely adjust the impact parameter to obtain the same deflection angle. The selection of impact parameters is of pure statistical nature. Thus, the experiment has to be repeatedly performed with numerous of such single projectiles and target objects. Finally summing over a huge number of single-events the experimenter obtains a statistical distribution that allows for the retrieval of the final-state fluctuation width (with the help of theory also quantum mechanical properties or properties of the wave function).

#### *2.1 Time Evolution of a Quantum Measurement*

In Fig. 1 the scheme of a single-event quantum experiment and the time evolution of such a complete quantum measurement process are shown. The measurement may be separated in three sequential steps: the time of preparation (pre-collision step, zone A), the time of reaction (zone B), and the time after the reaction (post-collision step, zone C) before the reaction products impact on the detector. In the view of the experimenter the momenta and trajectories of the particles in the macroscopic preparation stage A (pre-collision) as well as in the macroscopic spectrometer system C (post-collision) can be treated by the laws of classical physics. The very tiny reaction region B (typically of atomic to micrometer size) is a purely quantum mechanical region and must be treated accordingly. The dynamics in region B cannot be directly observed by the experimenter. The classical behavior in A and C is justified theoretically by the Imaging Theorem of the accompanying papers [7, 8]. This result shows

**Fig. 1** Time evolution of a quantum measurement. A indicates the time interval before the interaction of projectile (1) and target (2), B is the very short time interval of the quantum scattering process (3) (occurring at the time t0) and C the time interval in which the emitted reaction particles (4) are travelling inside the classical detection setup. The particle is finally detected on a detector (6). The detector yields an electronic signal (7) (typically a nanosecond long) providing time information on the quantum scattering event, which is stored electronically in a computer (8)

that, after propagation to or from macroscopic distances, the position and momentum variables of the quantum wave function obey classical relations.

The reaction products emitted in the quantum reaction are interacting with the macroscopic measurement apparatus in zone C. In the macroscopic apparatus they can be treated as classical particles with classically defined momenta (moving on classical trajectories) since they exchange in zone C de facto only momentum with the measurement device due to applied electric or magnetic spectrometer fields. Any interaction of the fragments with the rest gas in the spectrometer can be excluded because of the very low vacuum pressure (typically below 10−<sup>8</sup> millibar). At the end of the macroscopic detection device position-sensitive detectors measure the impact position in the laboratory system of each fragment and also the time of impact for each fragment separately (if required all fragments can be measured in coincidence).

As Popper pointed out [17], after completion of a measurement the experimenter determines always the kinematical parameters of the "past" for each single event, whereas the UR makes predictions into the future for the outcome of statistical distributions of many repeated single-event measurements.

#### **3 Electron Momentum (Velocity) Measurement by Time-of-Flight (TOF) Trajectory Imaging**

#### *3.1 The Experimental Scheme for Momentum (Velocity) Measurement*

In the following we describe a quantum measurement of charged particles from an ionization process using a momentum-imaging approach. After leaving the reaction zone B (see Fig. 1 at time *t*01n) the charged fragments begin to move in zone C on "quasi-classical" trajectories (see Refs. [7, 8]) with classically defined momenta, since in zone C they nearly exclusively exchange momentum with the spectrometer via classical forces. The distance d from the reaction point, from where one can neglect quantum mechanical post-collision interaction, can be crudely estimated by comparing the strengths of interacting forces, i.e. the magnitude of momentum exchange. In zone B the force between electron and ion dominates and in zone C the force imposed on the charged particles by the spectrometer fields is dominating. This is because the force between electron and ion depends on their distance d. Assuming the ion is singly charged then the electron-ion force is Fion = e2/d2 (in a.u.). For d = 1000 a.u. one obtains Fion = 10−<sup>6</sup> a.u., for d = 1 μm one obtains Fion = 2.8 × 10−<sup>9</sup> a.u. The strength of the classical force in the fields of the measurement device can be estimated from the electric field strength in the spectrometer. The field is typically larger than 10 V/cm, thus for an electron the acting spectrometer force is FeS > e · 10 V/cm = 1.92 × 10−<sup>9</sup> a.u. Therefore, for distances d larger than a few tens of micrometers the electron-ion force strength can be neglected and the

**Fig. 2** Scheme of trajectory imaging technique for charged quantum particles in a classical spectrometer [4]. The electron momentum vector (blue arrow) is the so-called "final momentum", with which the electron is emitted from the collision process with respect to the center-of-mass system of the reaction process

emitted fragments are only interacting with the spectrometer field yielding a welldefined classical trajectory due to momentum conservation (charged fragment plus spectrometer are entangled). The momentum change and thus the classical trajectory of the fragment in the spectrometer depend on the electric-magnetic field design and on the final fragment momentum *pfn*.

A static electric field accelerates electrons and positively charged ionic fragments into opposite directions. The fragments are finally detected by two position- and time-sensitive detectors placed in opposite directions (only one direction is shown in Fig. 2). Since the spectrometer provides for positively and negatively charged particles nearly a 4π-detection efficiency it can capture a complete image of the reaction process in momentum space.

The measurement of the final momentum of an emitted fragment can thus be achieved through a precise determination of the particle trajectory in part C in the classical detection device. To determine the complete classical trajectory of each particle one has to measure only the classical location parameters *r0***= (x0, y0, z0)** and *rfn***=** (**xfn**, **yfn**, **zfn**) as well as times **t0** and **tfn** (see Fig. 2). Both time parameters can be determined with a precision of about 50 pico-seconds, **t0** can be measured by using a timed-bunched projectile beam and **tfn** by using a "state-of-the-art" classical detection device [4]. Target location and position of impact on the detector can be measured with a precision of better than 50 μm (even 10 μm are achievable). Knowing (or calibrating) the electro-magnetic field configuration and measuring the above listed parameters, the final momentum vector of the fragment can easily be deduced by using simple classical equations [4]. **Although all auxiliary parameters**

**Fig. 3** Scheme of the C-REMI [4] which can image with 4πsolid angle all emitted charged particles (ions: red trajectory, electrons: blue trajectory) in coincidence. A projectile beam intersects in the center of the C-REMI with a super-sonic gas jet (from below) inducing the quantum reaction process. The applied electric field super-imposed by a magnetic field (see the brown coils) projects all charged fragments/electrons on position- and time-sensitive detectors

#### **are measured with macroscopic accuracy only, sub-atomic resolution for the electron and ion momenta (velocities) can be obtained**.

The C-REMI [4] is such a "state-of-the-art" momentum-imaging device. In Fig. 3 the scheme of such a detection approach is presented. The reaction takes place within the tiny intersection region of projectile and target beams (e.g. internally very cold super-sonic gas jet). The blue and red curves in Fig. 3 indicate the classical trajectories of ionic fragments (red line) and electrons (blue line) in the spectrometer. With the help of electric and magnetic fields nearly all fragments are projected on positionsensitive detectors yielding a very high multi-coincidence detection efficiency.

Before we discuss a real experimental scenario, we first define "good" and "bad" resolution in a single-event quantum measurement with respect to the standard dimensions in an atomic system. The standard sizes of atomic parameters are defined by the classical features of an electron in a hydrogen atom. The classical K-shell radius is *rK* = 5.29 × 10−<sup>9</sup> cm, which is used to define the atomic unit of length (a.u.). The classical electron velocity of the electron in the hydrogen K-shell is *vK* = 2.18 × 108 cm/s, which defines 1 a.u. of velocity. The classical momentum of the electron in the hydrogen K-shell is *p* = *mevK* = 1 a.u. An atomic unit of time is defined by the ratio of the hydrogen K-shell radius divided by the corresponding electron velocity, or 5.29 × 10−<sup>9</sup> cm divided by 2.18 × 108 cm/s yielding 24 attoseconds. Furthermore, the electron charge *e* and mass are also set to 1 a.u. and hence è results to be 1 a.u., too.

Thus, it appears very reasonable when we define resolution of single-event quantum measurements with respect to these atomic units. "Good" sub-atomic resolution is on the order of a few percent of one a.u. and "very good" resolution is on the order of a per mill or even better. Bad resolution is larger than one a.u.

#### *3.2 Momentum (Velocity) Measurement and Its Achievable Resolution for an Electron*

The achievable experimental precisions for momentum (velocity) are discussed here for two quantum processes. First, the transfer ionization process which is

$$10\,\text{keV}\,\text{He}^{2+} + \text{He} = \sim\,\text{He}^{1+} + \text{He}^{2+} + \text{e}\_z$$

investigated by Schmidt et al. [18]. This experiment was performed to search for vortices in the electron current which should be visible in the velocity/momentum distribution of the emitted electrons. To visualize such effects in the electron momentum distribution, a high experimental momentum resolution (δp=0.01 a.u.) in a single event is required. Additionally, a coincidence measurement with the ejected ions is necessary in order to determine the orientation of the quasi-molecule during the collision. This was achieved with the C-REMI approach. During such slow collisions quasi-molecular orbitals are formed and electrons are promoted to the continuum via a few selected angular momentum states.

In Fig. 4 the measured electron-momentum distributions are shown together with the achieved single-event resolution δp (black square) and with one example of a momentum fluctuation width <p> (varies with electron energy). It is to be noticed that in this experiment of Schmidt et al. the electron-detector distance from the intersection region (gas jet-projectile beam) was only 3 cm due to other experimental requirements. This short distance limits the momentum resolution, because of the very short TOF. Nevertheless a resolution of 0.01 a.u. was obtained. The resolution can be improved by increasing this distance.

To demonstrate the high resolving power for electron momenta of the C-REMI a numerical example, a kind of "*Gedankenexperiment*", i.e. the process of electron impact ionization of He

$$\text{e} + \text{He} = \text{\textdegree He}^{1+} + 2\text{e}$$

is discussed here. Today such an experiment would be feasible.

In Appendix A the preparation of the required electron beam quality is described which enables the high required momentum accuracy of the projectiles. To yield the required excellent "Time-of-Flight" TOF resolution the detectors should be located as far as possible from the zone B, i.e. the spectrometer should be as large as possible. For a trajectory length inside region C (from zone B to the electron detector surface)

**Fig. 4 a** Measured electron velocity distribution in the nuclear collision plane in units of the projectile velocity vp = 0.63 a.u. for small nuclear scattering angles <1.25 mrad. **b** Perpendicular to the nuclear collision plane. **c**, **d** corresponding theoretical predictions. An electron moving with the projectile velocity vp = 0.63 a.u. has a momentum of 0.63 a.u. [18]. The experimental resolution in a single event of δp = 0.01 a.u. corresponds to an energy resolution of approximately 1 meV

of 2 m the angular resolution of the trajectory measurement is of the order of the sum of the intersection width of projectile beam and target beam and detector position resolution divided by the trajectory length. This ratio is about 2 · 50 μm/200 cm ≈ 0.5 × 10−4. This geometrical ratio limits the transverse momentum resolution in x- and y-direction. The longitudinal momentum resolution (in z-direction) depends on the TOF resolution. An electron moving with 2 a.u. momentum has a velocity of 4.38 × 10+8 cm/sec and its total TOF inside C is 200 cm/4.38 × 10+8 (cm/sec) = 450 ns. Thus the relative TOF resolution -TOF/TOF is about 10-4 yielding an overall momentum precision for an electron of 2 a.u. momentum of 2 × 10−<sup>4</sup> a.u..

We would like to notice, that in C-REMI the velocities and masses of moving particles are measured, which yield directly the momenta. The velocities are macroscopically large and therefore directly measurable with macroscopic classical TOF devices. Heisenberg considered the measurement of the velocity (momentum) of an electron bound in an atom too. His approach will be discussed in Appendix B together with the possibility of momentum measurements of bound electrons via the process of Compton scattering.

#### **4 Measurement of Angular Momentum of a Single Electron**

Any bound electron usually has an orbital angular momentum in addition to its own spin. Due to the spin-orbit coupling, all electrons in an atom form one unit providing a quantized total angular momentum. If an experimenter can only measure the momentum of only one emitted electron (so-called single parameter measurement), then an experimenter can hardly make any statement about the quantum state in which the electron was originally bound. In case of single parameter measurement only from the electron momentum distribution of a large amount of identical ionization processes one can make a statement about the type of multipole distribution and thus on the angular momentum transfer involved. Thus the angular momentum of a single freely-moving electron emitted in a quantum reaction appears undetectable.

However, if the electron kinetic energy can be assigned to a single transition between well-defined quantum states, whose quantum numbers can be deduced. Furthermore in an ion-atom collision process and in the case of a complete multiparticle coincidence measurement, when the nuclear collision plane is determined too, this additional information can be employed in some cases to deduce the angular momentum states of a single ejected electron. In a slow ion-atom collision process, quasi-molecular electronic orbitals are formed during the collision, which are sharply angularly quantized with respect to the nucleus-nucleus scattering plane. Thus different angular momentum states are emitted due to space quantization only into distinct regions like in the Stern-Gerlach experiment (see e.g. data in Fig. 4 of this paper). If e.g. in a transfer-ionization process an electron passes over from these quasi-molecular states into the continuum [18] then the electrons in the x-y plane perpendicular to the nucleus-nucleus scattering plane are emitted with discrete transverse momenta (Fig. 4) and the different quasi-molecular orbitals e.g. 1 and 2 in Fig. 4 can clearly distinguished. Just as in the Stern-Gerlach experiment, these discrete transverse momenta correspond to certain angular momentum states which can be discerned in a coincidence measurement.

This clearly proves (Fig. 4: comparison of experiment and theory) that in a coincidence experiment the directional quantization of the quasi-molecular states becomes measureable and thus in selected collision systems the angular momentum states of single emitted electrons can be determined too.

#### **5 Electron-Position Measurement and Achievable Resolution**

Heisenberg described the position measurement of single electrons at a given moment as the foundation of any parameter measurement. He proposed to measure the velocity by detecting the electron positions at two succeeding moments. He explained his view on position measurements by thought experiments: "*If one wants to understand, what the definition of 'position of a particle', e.g. of the electron (relative to the reference system of measurement) means, one must describe well-defined experimental approaches, how the 'position of an electron' can be measured; otherwise the definition of position is meaningless. He continued:* "*There is no shortage of such experimental approaches, which can measure the 'position of an electron' with unlimited precision.*" (page 174) [1]. Therefore he viewed a trajectory as a discontinuous path because of discontinuous observations. On page 185 he continued: "*I believe that the appearance of a classical trajectory is manifested by its observation*".

Heisenberg proposed to use a so-called γ-microscope to measure the position of a quantum object, e.g. an electron at a given moment. He ascertained [1]: "*The resolution of the light microscope is only limited by the wave length of the light. Using short wave length x-rays the resolution should have no limitation.*" The scheme of such a photon microscope measurement can only be explained in the wave-picture (thus many photons must be detected). But one has to make sure that the object is not changing its position during the exposure time of the measurement. With the help of such a microscope (combination of lenses) one can magnify tiny objects and project their image on a detector, e.g. photo plate. There is an one-to-one correspondence between position on the object and the position on the detector (only valid in the transverse plane). Thus with the help of lenses relative positions on very small quantum objects can be enlarged and thus become observable. It should be noted, that a "microscope" device for magnifying the geometrical size of an atom (about 10−<sup>8</sup> cm diameter) and also magnifying the relative positions of atoms in a molecule to the macroscopic size of 1 mm must have a magnification factor of more than 106.

Heisenberg was convinced that the position of an electron at a given time could be measured even with "ultimate" precision using the technique of such a light microscope if the wavelength of the light would be small enough to resolve subatomic structure.<sup>1</sup> At a "given time" means always an exposure time period in which

<sup>1</sup>Several reasons prevent that Heisenberg's so-called γ-microscope can measure the position of one selected electron inside an atom at a given time with a required resolution of 10−<sup>10</sup> cm or even better: First: Since the focus of the γ-pulse is of macroscopic size (larger than 1 μm2) the scattered photons of the γ-pulse interact with different electrons in the atom or molecule and the measurement can on principal not identify which photons were scattered on the one special electron. Second: The Compton cross sections for scattering photons with a wave length of 10−<sup>10</sup> cm (or h<sup>ν</sup> <sup>=</sup> 1.2 MeV) on an electron are smaller than 10−<sup>25</sup> cm2. Therefore it requires per attosecond pulse more than 10+19 photons in a focus of 1 μm<sup>2</sup> to scatter about 100 photons on this electron. Such a photon pulse carries an energy power equivalent with 1% of the total energy emission of the sun. Third a technical reason: The γ-microscope needs a high precision lens system for 1 MeV photons to magnify the 1 μm2 focus size to make the different electrons on a macroscopic detector distinguishable.

the location of a moving electron must be considered as "frozen". Such a time period for an electron detection must be shorter than one attosecond.

Therefore, in order to obtain an image of the position of an electron with subatomic resolution using a γ-microscope, one would have to scatter on the **same electron** numerous photons in a one attosecond "exposure" time period (since the electron is moving with a typical velocity of 1% of the speed of light). Because these γ-scattering cross sections (Compton scattering) are of nuclear size the photon pulse intensity in one attosecond must exceed 1019 photons per pulse in a focus of 1 μm diameter. A further problem in such a measurement is that the experimenter has no control on which electron in the target atom or molecule the photons are scattered. Both effects make such a γ-microscope measurement physically not feasible.

Furthermore, each Compton scattering process, as mentioned above, is destructive for the electronic state, thus the electronic state changes immediately. This disturbing effect of momentum transfer to the electron and thus changing the electron's position subsequently was already realized by Bohr [19]. These arguments show that Heisenberg's γ-microscope is not suited to measure the position of an electron at a given moment.

In one attosecond exposure time because of the tiny cross sections at most one photon might be scattered on the same electron. Thus the only information the experimenter obtains with Compton scattering is the detection of only one single photon providing one momentum vector. Even if this photon momentum vector is measured with sub-atomic precision the location of the reaction can never be deduced from this one vector with a precision better than the preparation of the target position before the scattering.

In contrast, position-measurements of heavy nuclei or atoms can be performed with a γ-microscope, since the velocities of atoms or nuclei are typically a factor of 10.000 smaller. Thus, the heavy particle position can be considered as "frozen" even for an exposure time of a few femtoseconds. Such relative position measurements of heavy atoms in molecules are now routinely performed with FEL X-ray pulses [20], where a lateral position resolution of about 5 Å is achieved. A slightly better resolution of about 3 Å is achieved with CRYO-electron microscopy [21].

One may expect that when performing a multi-coincidence measurement, i.e. measuring the momentum vectors of several fragments of the same reaction with excellent resolution, one could deduce the position, where the reaction took place,

Heisenberg proposed also to use energetic α-particles as scattering projectiles (because of their even shorter de Broglie wave-length) for a super high-resolution microscope and estimated even a position resolution of 10−<sup>12</sup> cm as possible. He wrote on page 175: "*When two very fast particles succeeding each other scatter in a very short time distance* Δ*t on the same electron, then the distance between the positions of both collisions is* Δ*l. From the scattering laws, which has been observed for* α*-particles, we can conclude, that* Δ*l can be made as small as* 10−<sup>12</sup> *cm, if* Δ*t can be made sufficiently small and the* α*-particles fast enough.*"

For a several MeV α-particle beam this requires a relative distance of the α-particles in the beam of about 10−<sup>11</sup> cm. This relative distance of the α-particles is about one thousand times smaller than the normal inter-nuclear distance in a hydrogen molecule. α-particles with a relative distance of 10−<sup>11</sup> cm would repel each other with a huge Coulomb repulsion force creating huge non-controllable transverse momenta of the α-beam.

by reconstructing the intersection point of all momentum vectors. Even if the impact positions of all fragments on the detector could be measured with atomic position resolution, the momentum vectors have still a finite angular uncertainty limited again by the target preparation. Because of the macroscopic dimensions of the detection device even a tiny angular uncertainty of these vectors would spoil any precise position measurement of the reaction region within the laboratory frame.

#### **6 Product of Precisions in Momentum and Precision in Position in a Real Measurement of a Freely Moving Single Electron**

The paradigmatic demonstration experiment for the UR given in textbooks for measuring simultaneously position and momentum of an electron (wave picture) is the scattering of this "wave" on a narrow slit (first step). The scattered wave yields an interference pattern of the electron wave on a screen (second step). According to most of the textbooks position and momentum of these electrons can only be measured in such a **two step-approach**, where in the first step the position is measured by the slit width and in the second-step by the interference pattern on the screen the momentum. The electron is theoretically described by wave functions which are different before and after the slit: before passing the slit the electron is described as plane wave with well-defined momentum eigenvalue but not localized in x-position; just after the slit the electron is described as a wave packet with some distribution in position and momentum. Thus on its way to the screen the electron is in a state which is not an eigenstate of the momentum operator. In both of the two time steps the UR is fulfilled. This is a result of the fact that the two operators do not commute. Thus the wave function is disturbed and a conceptually unavoidable uncertainty in the second-step measurement (momentum measurement) is generated. From the interference structure in the transverse momentum distribution of many single-event measurements the de Broglie wave length λ and thus electron momentum *p* can be determined.

We will now estimate how small the product of the two precision widths *x* · *px* ≥ è can be made in a **single-event process** by using a modern state-of-theart detection device. We are in particular interested in whether the product of the experimental position resolution times the experimental momentum resolution can be made smaller than è. With today's detection technique the two-step detection scheme can be replaced for single electron detection by a quasi **one-step detection approach**, where the narrow slit is "upgraded" to a very small pixel detector, which measures position and time of impact too. Thus we consider the momentum and position at the time when a single moving electron impacts on the position-sensitive detector. One can construct detectors which can measure the impact position of the electron on the detector with a few a.u. precision δxy = 2 a.u. (see Appendix C).

In a single event this detector provides, at the instant of electron impact, also a very fast electronic signal (time resolution <50 pico-seconds) which yields precise information on the electron velocity. If furthermore the location of the interaction region and the interaction time, from where this electron is emitted, are known with macroscopic precision, one can determine the electron Time-of-Flight TOF. Knowing precisely the distance d between emission point to detector (e.g. d = 2 m and d = 0.1 mm) one can precisely calculate the electron velocity: Assuming the measured TOF is e.g. 456.6 ± 0.1 ns and d is 200.00 ± 0.01 cm the electron velocity is then ve = 2 · 200.00 cm/456.6 nanosecond = 4.3800 × 10+8 cm/s with an error bar of ±0.025%. Transforming the velocity in a.u. we obtain for ve = pz = 2.0021 ± 0.001 a.u. (pz is the electron momentum in flight direction). In perpendicular direction the errors in momentum are δpx, y = (0.1 mm/2000 mm) · 2.0021 a.u. ≈ 10−<sup>4</sup> a.u. Thus, in case of a single event measurement the product of the experimental error bars in the momentum and position measurement can be made δpxy · δxy ≈ 10−<sup>4</sup> a.u. · 2 a.u. = 2 × 10−<sup>4</sup> a.u. which is much smaller than è.

One could argue, however, that the detection plus preparation is still a two-step measurement. But nevertheless in a single event the product of precisions can be made much smaller than è. Thus, once the particle has been detected, the trajectory, that the particle has travelled on in the past, can be defined such that the product of precisions of momentum and position measurement of this freely moving single electron is not limited by è.

#### **7 Conclusion**

We have shown that in **a single-event quantum measurement** the momenta of emitted electrons or ions can be measured with high sub-atomic precision and the limits of precision for the momentum measurement are not restricted by Heisenberg's UR if one assigns trajectories to particles that have been detected. The precision in measuring positions in a single event can never approach or being better than 1 a.u. in a single-event measurement because the two conjugate parameters position and momentum do not have the apparent physical symmetry suggested by the UR, i.e., there exists a disparity in momentum compared to position measurement. The fundamental reason is: in a single event momenta are conserved with time (i.e. they are dynamically entangled), but positions are not conserved. This fundamental difference between momentum and position measurement as function of time in a quantum reaction is also apparent from the wave description (see Appendix D). The position wave functions broaden with increasing time even during a very short single-event measurement.

For a single freely moving particle in the moment of impact on the detector momentum and also position on the detector can be simultaneously detected in a single-step approach using position-sensitive detectors combined with a time-offlight measurement. The product of the experimental momentum resolution δp times position resolution δx on the detector can be made much smaller than è.

**Acknowledgements** We are indebted to John Briggs, Tilman Sauer, Bretislav Friedrich, Reinhard Dörner, Robert Griffiths and Lothar Schmidt for many very valuable discussions and corrections to the manuscript.

#### **Appendix A**

Electrons with a very well-defined momentum pz = 2.000 a.u. can be created by photoionization. The primary photon energy h · ν is chosen to be 78.988 eV ± 0.007 eV (linearly polarized photons). These photons can singly ionize the He atom. In our example a single electron with a kinetic energy of 54.392 ± 0.007 eV and He-recoil ion with a kinetic energy of 7.4 meV are emitted back-to-back. In the He center-of-mass system both freely moving particles, electron and recoil ion, have the identical momentum with opposite direction (pe = −prec = 2.000 a.u. with a precision of about 10−<sup>3</sup> a.u.). The angular distribution of electrons emitted by photon ionization and recoil ions is a perfect dipole distribution.

Since the ionized He atom is not at rest in the laboratory system at the moment of ionization, one must correct the electron momentum vector for the motion of the He-atom in the Lab frame. The final electron velocity is about 5000 km/sec and the internal velocity spread of the cold super-sonic He jet is below 50 m/sec. Because the He jet is moving along the negative y direction, the correction for the electron in the z-direction can be performed with sufficient precision. Furthermore, the very small momentum kick of the incoming photon (pphoton ≈ 0.021 a.u.) to the centerof-mass of the He atom changes the He velocity by only about 10 m/sec. Thus the absolute value of the electron momentum is known with about 10−<sup>3</sup> a.u. precision. The direction of the final electron beam can now be defined by a collimation system (collimation in x and y direction). Using a double slit system at 2 m distance in the zdirection with slit widths of 10 microns each (in x- and y- direction) the so-collimated electron beam has an angular divergence of δ = 20/(2 × 10+6) rad = 10−<sup>5</sup> rad. The momentum exchange with electrons inside the slits due to image-charge formation is also insignificant. Thus the momenta of the electron beam have a width **px, y, z < 10 <sup>−</sup><sup>3</sup> a.u.**, i.e. each electron in this electron beam has a momentum of 2.000 a.u. in the z-direction (with a precision of about 10−<sup>3</sup> a.u. in all three dimensions x, y, and z). By changing the primary photon energy, the electron beam's momentum can be varied. Similar momentum resolution results can be also obtained for photon beams and slow ion beams, too.

#### **Appendix B**

Heisenberg described also a way to measure the velocity (momentum) of an electron bound in an atom. Heisenberg's approach to measure the velocity of a bound electron was copied from classical physics where the velocity of a particle is determined from the quotient of measured distance divided by measured time difference. Heisenberg's concept was to detect by γ-photon scattering at two instants in time the electron locations (for a bound electron separated by less than 10−<sup>11</sup> cm). This would require a very precise simultaneous measurement of location and time within sub-attosecond time separation. As we discussed in Sect. 4 such a measurement of position and time with the required resolution is physically impossible to perform. Furthermore, Heisenberg worried that the electron velocity (momentum) just before and just after the photon (Compton) scattering process is not the same one due to the momentum kick by the photon and thus the velocity measurement before the kick seemed to be not accessible.

In 1927, Heisenberg was not aware that one could measure nevertheless both—the electron momenta just before and after the scattering by performing **a coincidence measurement of the scattered Compton-photon and the ejected electron**.

Both electron momenta just before and just after scattering can be determined with very high resolution (δp = 10−<sup>2</sup> to 10−<sup>3</sup> a.u.) due to momentum conservation during the scattering process. This is possible at high photon energies and large photon momentum transfers where the "impulse approximation" is well justified [22]. In the "impulse approximation" for Compton ionization the ejected electron is treated as a quasi-unbound electron and thus the momentum change of the whole atom by Compton ionization is rather small and the remaining ion acts only as a spectator. The coincidence measurement allows therefore for a precise determination of *P***e ionized** which is the momentum vector of the ejected electron after the Compton scattering and *P***e bound**, which is the unknown momentum vector of the bound electron just before scattering (see vector equation in Fig. 5). When Heisenberg wrote his paper in 1927, the experimental techniques to study quantum processes were in an "archaic" state compared with today. Precise timing measurements in the nanosecond regime required for coincidence measurements were beyond imagination. The first generation of coincidence technique was just invented in 1924 by Bothe and Geiger [23], but Bothe's and Geiger's time resolution was limited to a fraction of a millisecond. Instead of using high-energetic photon impact one can perform the same kind of momentum spectroscopy of bound electrons by using very fast ion [24] or electron impact [25].

Heisenberg also considered a velocity measurement of a bound electron by using the Doppler-effect (wave length-shift) of scattered red light. Heisenberg wrote on page 177 [1]: "*The velocity of a particle can easily be defined by a measurement, if the particle velocity is constant (no acting forces). One can scatter red light on the particle und measures by the Doppler shift its velocity. The measurement becomes more precise, as longer the wave length of the light is, since then the velocity change of the particle per photon becomes smaller. The determination of position becomes accordingly more unprecise predicted by equation (1) (UR). To measure the velocity at a given moment, the Coulomb forces of the nucleus and the of the other electrons must suddenly disappear, to ensure from this moment a constant velocity, which is necessary to perform the measurement.*" The Doppler-effect approach would, however, only allow the determination of the particle velocity component in the

**Fig. 5** Momentum vector diagram for Compton scattering. The dotted vectors *PSum initial* and *PSum final* represent the sum vectors in the initial state of impacting photon *P<sup>γ</sup>* (prepared and precisely known) and of bound electron *Pe bound* (unknown) as well as in the final state of ejected electron *Pe ionized* and of scattered photon *P<sup>γ</sup>* **-**(both precisely measured)

direction of the incoming red light. Furthermore, the red light photon would be scattered on the whole atom and not on a single electron and would thus probe the atom velocity only.

#### **Appendix C**

Today, in principle one could build a macroscopic position-sensitive electron or ion detector with better than 10 a.u. position and 50 pico-seconds timing resolution. This detector can be a very small pixel detector or a large area position sensitive detector. Such a detector can be reassembled from two components: a commercially available position-sensitive channel-plate detector with a standard position resolution of 50 μm in x and y direction, respectively and a very thin (nanometer thickness) mask of regularly positioned holes of <10 a.u. diameter each, where electrons and ions can only be detected when they impact into such a hole (distance mask to detector surface a few nano-meter only). Then they induce in the detector an electronic signal. From this single electronic signal simultaneously position and timing information is obtained. The distance from hole to hole is 100 μm, thus the detection device is able to determine each particle impact on an absolute scale in the laboratory system for the position measurement with 10 a.u. precision. Therefore, one can detect the location of this particle impact in x and y direction with δx ≈ δy ≈ 10 a.u. resolution.

#### **Appendix D**

Time-dependent conservation laws are of fundamental importance for the implementation of a high-resolution single-event quantum measurement. This conservation is valid for the total linear momentum, for the total angular momentum and for the total energy, but not for the location. This is obvious for a particle picture, but also in the wave picture. In Fig. 6 one can see that the spatial wave function widens linearly over time, but the momentum wave function maintains its narrow width, which is determined by the preparation of the measurement. If several fragments are emitted in the reaction process, momentum conservation applies to each of the fragments on their flight to the detector. The momentum exchange with the classical electro-magnetic fields of the measuring apparatus can be determined from the classically measured trajectory of each fragment and thus corrected with high resolution.

The asymmetry as function of time for position and momentum space of the freely moving particle is also apparent in the wave approach and is due to the time propagation with *U* = exp(-*T* · *E*) with *E*(*p*) = *p*<sup>2</sup>/2 that breaks the symmetry of position vs. momentum space.

**Fig. 6** Fourier limited wave function in position (**a**) and momentum (**b**) space. The phase is flat in position and momentum space and the amplitude is a Gaussian distribution. Wave function **c** from **a** after propagation for the time -*T* = 10 a.u. The resulting wave function **d** in momentum space is <sup>2</sup> = <sup>1</sup> · exp(-*<sup>T</sup>* · *<sup>E</sup>*) with *<sup>E</sup>*(*p*) <sup>=</sup> *<sup>p</sup>*2/2. <sup>2</sup> has the same amplitude in momentum space as in **b** but a quadratic phase. In position space the amplitude distribution broadens compared to **a**

#### **References**

and double ionization induced by the virtual photon field of fast heavy ions x-ray and innershell processes. AIP Conf. Proc. **389**, 153 (1996); J. Ullrich, et al., Recoil ion momentum spectroscopy. J. Phys. B **30**, 2917 (1997) Topical Review

25. E. Weigold, I. McCarthy, Ion Electron Momentum Spectroscopy (Springer, 1999). ISBN 978- 1-4615-477

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#### **Chapter 13 Precision Physics in Penning Traps Using the Continuous Stern-Gerlach Effect**

**Klaus Blaum and Günter Werth**

#### **1 Introduction**

"A single atomic particle forever floating at rest in free space" (H. Dehmelt) would be the ideal object for precision measurements of atomic properties and for tests of fundamental theories. Such an ideal, of course, can ultimately never be achieved. A very close approximation to this ideal is made possible by ion traps, where electromagnetic forces are used to confine charged particles under well-controlled conditions for practically unlimited time. Concurrently, sensitive detection methods have been developed to allow observation of single stored ions. Various cooling methods can be employed to bring the trapped ion nearly to rest. Among different realisations of ion traps we consider in this chapter the so-called Penning traps which use static electric and magnetic fields for ion confinement. After a brief discussion of Penning-trap properties, we consider various experiments including the application of the "continuous Stern-Gerlach effect", which have led recently to precise determinations of the masses and magnetic moments of particles and antiparticles. These serve as input for testing fundamental theories and symmetries.

#### **2 Penning-Trap Properties**

The Penning traps used in the experiments described herein consist of a symmetric stack of cylindrical electrodes as shown in Fig. 1.

K. Blaum

Max-Planck-Institut für Kernphysik, Heidelberg, Germany

G. Werth (B) Johannes-Gutenberg-Universität, Mainz, Germany e-mail<EMAIL_ADDRESS>
**Fig. 1** Schematic of our cylindrical Penning trap holding a single highly charged ion stored in this electrode configuration

A static voltage is applied between the connected outer electrodes and the central ring. At a positive polarity at the endcaps a positively charged particle is confined in the axial direction. Escape in the radial direction is precluded by a homogeneous magnetic field directed along the trap's axis. Additional voltages applied to the socalled correction electrodes, placed between the ring and endcaps, serve to provide an electric potential which depends—at least for small amplitudes of the trapped particle—only on the square of the distance from the trap's centre. This is a prerequisite for achieving high resolution in the determination of the motional frequencies of the trapped ions. It is further required to avoid perturbation of the confined ion by collisions with background gas molecules. In our case, the trap electrodes and their container box are in thermal contact with a liquid helium bath. The cryopumping results in a residual pressure of less than 10−<sup>16</sup> mbar. As a result, not even a single collision of the trapped ion with a neutral molecule was observed during typical trapping times of several months.

The ion's motion in the trap arrangement described above can be calculated analytically. As a result one obtains a superposition of three harmonic oscillations: An axial one at frequency νz, which depends on the ions mass *M*, the size of the trap, and the voltage, and two radial oscillations with frequencies ν<sup>+</sup> and ν−. The frequency ν<sup>+</sup> is near the cyclotron frequency ν*<sup>c</sup>* = *qB/*(*2*π*M*) of the free ion with charge *q* in the magnetic field *B*, slightly perturbed by the presence of the electric trapping field. The centre of this motion orbits around the trap centre at a low "magnetron frequency" ν−. Figure 2 shows a sketch of the ion's motion. An important relation connects the motional frequencies to the free ion's cyclotron frequency [1]:

**Fig. 2** Sketch of the ion's motion in a Penning trap. For details see text

$$
\nu\_+^2 + \nu\_-^2 + \nu\_z^2 = \nu\_\circledast \tag{l}
$$

Further details on Penning traps can be found in Refs. [2, 3].

#### **3 Single Ion Detection by Induced Image Currents**

Achieving high precision in trap experiments requires the use of single trapped particles to avoid perturbations by Coulomb interaction with other ions. The standard way to detect single trapped particles is by observation of laser induced fluorescence. This requires, however, optical transitions in the ions which are in reach of laser wavelengths. This is not the case for highly charged ions or elementary particles such as electrons or protons and their antiparticles. In these cases, detection can be performed by the image current that the oscillating ion induces in the trap's electrodes [4]. This current is on the order of a few fA and requires very sensitive detection methods. This can be realized by a superconducting high-Q tank circuit, kept at the temperature of the surrounding He-bath and tuned to the resonance frequency of the ion oscillation. Figure 3 shows a scheme of the detection of a radial frequency. The noise power of the circuit can be amplified and Fourier analysed. In case of

**Fig. 3** Scheme for detection of the radial ion oscillation by a superconducting tank circuit attached to segments of the trap's split ring electrode

an ion present in the trap, the Fourier spectrum shows a maximum at the resonance frequency of the ion oscillation, as shown in Fig. 4.

The ion signal as shown in Fig. 4 indicates that the ion's kinetic energy is well above the thermal energy of the circuit. In order to reduce the ion's energy as required for high-precision measurements, the ion is kept in resonance with the circuit. The extra energy which the hot ion transfers into the circuit is then dissipated into the helium bath (resistive cooling) [5]. As consequence the ion adopts the temperature of the environment. The signal is then converted into a minimum in the noise spectrum as shown in Fig. 5. This can be understood based on the fact that the equivalent electronic circuit of the oscillating ion is a series resonance circuit which shortcuts the noise power of the detection circuit at its resonance frequency.

#### **4 The Masses of the Proton and Antiproton**

The ion signals such as those shown in Fig. 5 can be used for high-precision mass measurements. As recent examples, we consider the atomic masses of the proton and the antiproton. Their comparison serves as a test of the CPT invariance theorem. Quite general determination of atomic masses requires the comparison with the standard atomic mass, the carbon atom. The comparison is performed through measurements of the cyclotron frequencies of the particle under investigation ν*c*(*P*) = *eB*/(*2*π*MP*) (e.g. the proton *P* with mass *MP*) and that of a carbon ion of charge state *q:* ν*c*(*C<sup>q</sup>*+) = *qB*/(*2*π*Mq*<sup>+</sup> *<sup>C</sup>* ) in the same magnetic field *B* provided by a superconducting solenoid. In the case of the proton we used *C*6+ and correct the measured frequency by the masses of the missing electrons and their respective binding energies:

$$M\_P = \frac{e}{q} \frac{\nu\_C(C)}{\nu\_C(P)} M\_C \tag{2}$$

By resolving the central part of the ion detection signal as shown in Fig. 5, we can determine the centre frequency with an uncertainty of a few mHz. In our experiments we measure exclusively the axial frequency. In order to determine the radial frequency as required for the determination of the cyclotron frequencies according to Eq. (1), we couple the radial frequencies to the axial one by an additional r.f. field applied to the trap electrodes at their difference frequency to the axial. This leads to a split of the axial detection signal which allows to determine the radial frequencies with the same uncertainty.

In order to perform the measurements of the respective cyclotron frequencies at the same position of the magnetic field, we extend our Penning trap by a number of additional electrodes. In different potential minima we can store simultaneously a single proton and a single carbon ion. By changing the potentials at the electrodes we can transport one ion into the central part of the trap structure where the homogeneity of the magnetic field is highest, while the other ion is stored in one of the remaining potential minima. Frequent exchange of protons and carbon ions eliminates to a large extent the influence of possible time variations of the magnetic field which is provided by a superconducting magnet. Figure 6 shows the complete setup.

Our result for the proton's atomic mass including the statistical and systematic uncertainties is [6]

$$M\_P = 1.007\ 276\ 466\ 583(15)(29)\ \text{u.}\tag{3}$$

i.e. a relative mass uncertainty of 3 × 10−<sup>11</sup> has been achieved. It improves earlier results by a factor of 3 and determines the proton mass value in the most recent CODATA compilation of fundamental constants [7] (Fig. 7).

A similar experiment as described above using a nearly identical setup has been performed at CERN/Geneva, where single antiprotons have been confined and their cyclotron frequency measured [8]. The main difference was that, for comparison with a reference mass, not a carbon ion could be used since it would require a change

**Fig. 6** Setup of the trap configuration for the determination of proton's atomic mass. The left part serves for the creation of protons and carbon ions by electron bombardment of a target. After ion creation and removal of unwanted species, single ions are transferred to one of the potential minima. Measurements are performed in the so-called measurement trap located at the most homogeneous part of the magnetic field. Shown are the resonance circuits attached to the measurement trap's electrode tuned to the different axial frequencies for the proton and C6+. The trap configuration is placed in a hermetically sealed container in thermal contact with a liquid-He bath. The low temperature of the container walls provides a vacuum below 10−<sup>16</sup> mbar by cryofreezing. Collisions of the stored particles with background molecules are absent for the period of several months which allows a nearly infinitely long perturbation-free storage time

of the voltage sign at the trap because of the different charge signs of the particles. This, in turn, could lead to uncontrollable errors. Instead, the CERN experiment used the negative hydrogen ion *H*−. Taking into account the masses of the 2 additional electrons, their binding energies *Ea* and *Eb* as well as the polarizability α*pol*,*H*<sup>−</sup> of *H*−, the antiproton's *map* mass can be compared to the protons mass through

13 Precision Physics in Penning Traps Using the Continuous … 253

$$m\_{(H^{-})} = m\_{ap} \left( 1 + 2 \frac{m\_e}{m\_p} + \frac{\alpha\_{pol, H^{-}B\_0^2}}{m\_p} - \frac{E\_b}{m\_p} - \frac{E\_a}{m\_p} \right) \tag{4}$$

The result of the proton/antiproton mass ratio is

$$(q/m)\_{\rm ap}/(q/m)\_{\rm p} = -1.000\ 000\ 000\ 001\ (69)\tag{5}$$

at a relative uncertainty level of 7 × 10−11. This result represents the most stringent test of the CPT invariance in the baryon sector.

#### **5 The** *g***-Factor of the Bound Electron**

The *g*-factor of the electron is a dimensionless constant which relates the electron's magnetic moment μ*<sup>s</sup>* to the spin *S* and the Bohr magnetron μ*B*:

$$
\mu\_x = \text{g}\,\mu\_B\text{S} \tag{6}
$$

Dirac's relativistic treatment of the free electron predicts *g* = 2. Experimentally a deviation from *2* is found, which among others gave rise to the theory of Quantum Electrodynamics (QED), which describes the interaction of charged particles with electromagnetic fields by the exchange of virtual photons. Evaluation of Feynman diagrams to high orders allows calculating the *g*-factor of the free electron to extremely high precision [9]:

(g − 2)/2 = 0.001 159 652 181 78 (77). It agrees well with the experimental value [10]: (g − 2)/2 = 0.001 159 652 180 73 (28). The agreement represents the best test of QED in weak external fields.

In contrast to a free electron, an electron bound to an atomic nucleus experiences an extremely strong electric field. The strength of the field for a single electron bound in the 1S-state of a hydrogen-like ion of nuclear charge *Z* ranges from 109 V/cm in the helium ion (Z = 2) to > 1015 V/cm in H-like uranium (Z = 92) (Fig. 8). This gives rise to a variety of new effects. The largest change of the bound electron's *g*-factor was analytically derived by Breit (1928) from the Dirac equation: *gBreit* <sup>=</sup> <sup>2</sup> 3 1 + 2 1 − (*Z*α) 2 ≈ 2 <sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>3</sup> (*Z*α) 2 [11], with α ≈ 1/137 the fine structure constant. The extremely high electric fields within atoms also require different methods to be used for calculating the QED contributions to the electron's magnetic moment. Feynman diagrams have to be calculated using the solution of the Dirac equation as an electron propagator. Contributions of high orders in (*Z*α) have been calculated by several authors [12]. In addition, nuclear structure and recoil effects must be considered [13]. Figure 9 summarises the different contributions to the electron's *g*-factor in the ground state of H-like ions as function of the nuclear charge *Z*.

**Fig. 9** Contributions from bound-state QED, nuclear size, structure, and recoil to the *g*-factor of the electron bound in hydrogen-like ions as function of the nuclear charge *Z* (*courtesy* Z. Harmann)

#### **6 The Continuous Stern-Gerlach Effect**

The determination of the electron's *g*-factor requires the measurement of the spin precession (Larmor) frequency ω*L*:

$$
\omega\_L = \frac{g}{2} \left. \frac{e}{m\_e} B \right. \tag{7}
$$

The magnetic field strength *B* can be derived from the measurement of the three motional frequencies as described in Sect. 4. The spin precession, however, does not influence the ion's motion in a homogeneous magnetic field and consequently cannot be detected by observation of the axial ion resonance as only observable in our experimental set-up (see Sect. 3). The required coupling of the spin motion to the ion's oscillation can be provided by an inhomogeneity of the magnetic field. The force of *F* = μ*SgradB* of a *B*-field inhomogeneity *gradB* on the magnetic moment μ*<sup>S</sup>* associated with the spin increases or reduces the electric trapping force of the Penning trap depending of the spin's direction. Consequently, a change in the spin direction leads to a change in the ion's axial frequency. This method has been first employed by Dehmelt in his experiment on the *g*-factor of the free electron and termed "continuous Stern-Gerlach effect" [14]. It was later adapted to experiments on highly charged ions [15].

The size of the change in the axial frequency upon a change in the spin direction is given by

$$
\Delta\omega\_{\rm t} = \frac{\text{g}\,\mu\_{\rm B}\,B\_{2}}{M\omega\_{\rm t}}\tag{8}
$$

where *M* is the ions mass and *B*<sup>2</sup> is the quadratic part of the series expansion of the magnetic field *BZ* = *B0* + *B2 z<sup>2</sup>* + …. The magnetic field inhomogeneity in our experiments is produced by a ferromagnetic central ring electrode (nickel) in the Penning trap assembly. It produces a bottle shaped *B*-field. Odd terms in the series expansion vanish. The measured value of *B2* in our trap is 10 mT/mm2. The calculated change in the axial frequency, e.g., for H-like 28Si13+ is 240 mHz in a total oscillation frequency of 412 kHz. The detection of such small frequency changes requires high stability of the electric trapping field. Figure 10 shows that changes in the spin direction, induced by a microwave field in the trap, can be unambiguously detected with nearly 100% probability.

#### **7 Measurement of** *g***-Factors**

Our first application of the continuous Stern-Gerlach effect was the determination of the *g*-factor of the electron bound in hydrogen-like ions.We monitor induced spin flips as shown in Fig. 10 while varying the microwave field's frequency. The maximum spin flip probability occurs when the microwave frequency ω coincides with the Larmor precession frequency ω*L*. The *g*-factor can be derived from Eq. (5) when the magnetic field *B* is known. *B* is obtained using the single ion detection signals at the ion's oscillation frequencies as described in Sects. 3 and 4. The *g*-factor then follows from the measurement of ω*<sup>L</sup>* and ω*c*:

$$\text{kg} = 2 \frac{\alpha\_L}{\alpha\_c} \frac{q\_{ion}}{m\_{ion}} \frac{m\_e}{e} = 2 \Gamma \frac{q\_{ion}}{m\_{ion}} \frac{m\_e}{e} \tag{9}$$

 is the ratio of the applied microwave frequency ω to the measured cyclotron frequency. However, in order to obtain high precision in the *g*-factor determination we are faced with conflicting requirements: The continuous Stern-Gerlach effect requires a strong inhomogeneity of the *B*-field at the ions position in order to detect induced spin flips with high probability. High accuracy for *B*-field determination on the other side requires a very homogeneous field at the ion's position. In order to resolve this conflict we used a Penning-trap configuration similar to the one shown in Fig. 6 but modified by introducing a nickel ring electrode in one of the storage traps that provides the required *B*-field inhomogeneity. Step 1 of the measurement procedure is the determination of the ion's spin direction in the inhomogeneous trap by introduction of a spin flip and observation of the change in the axial oscillation frequency as illustrated in Fig. 10. With a known spin direction, the ion is then transported into the measurement trap where the oscillation frequency ω*<sup>c</sup>* is determined. Simultaneously the ion is irradiated by a microwave field attempting to change the spin direction. Then the ion is transported back into the inhomogeneous trap and, as in step 1, its spin direction is determined again. A successfully induced spin flip in the measurement trap is monitored in the inhomogeneous trap by the corresponding change in the axial frequency. Frequent repetition of this procedure with varying microwave frequencies and monitoring the spin flip probability for different frequencies results in a resonance curve with a maximum at the Larmor precession frequency ω*L*. The fact that ω*<sup>L</sup>* and ω<sup>c</sup> are measured at the same position as well at the same time eliminates to a large extent the uncertainties due to time fluctuations of the *B*-field. Figure 11 shows an example of measurements on hydrogen-like 28Si13+, where the spin flip probability is plotted against the ratio Γ of ω and ω*c.*. The maximum is at Γ = ω*L*/ω*C.*.

The result of the experiment on 28Si13+ for the *g*-factor of the single bound electron is gexp = 1.995 348 958 7 (5)(3)(8) [16]. The numbers in parenthesis correspond respectively to the systematic and statistical uncertainties and the error of the electron mass taken from the CODATA 2012 tables of fundamental constants [17]. The result agrees well with the theoretical value *g*theo = 1.995 348 958 0 (17) [18] and represents the most stringent test of QED calculations in strong external fields.

**Fig. 11** Measured spin flip probability of 28Si13+ as a function of the frequency ratio <sup>=</sup> <sup>ω</sup>/ω*C.*. The maximum occurs at Γ = ω*L*/ω*C*. Γ *theo* is the theoretical predicted value. Shifts caused by systematic effects are not yet corrected for

Similar results with comparable precision have been obtained on H-like 12C5+ [19] and 16O7+ [20]. Also experiments using lithium-like ions 40Ca17+ and 48Ca17+ [21], and 28Si11+ [22] have been performed. Here the interaction of the additional 2 bound electrons modify slightly the value of the *g*-factor. The comparisons between theory and experiment test calculations of the inter-electronic interaction. Most recently the *g*-factor of the boron-like ion 40Ar13+ has been determined with high precision [23]. The experimental result distinguishes between the conflicting predictions of the contribution of the electron-electron interaction.

Analogously to the electronic *g*-factors, the magnetic moments of the proton and antiproton were obtained. The particular challenge in these experiments had to do with the fact that the magnetic moment of these particles is about 500 times smaller than that of the electron. According to Eq. (6), the corresponding change in the axial frequency is significantly smaller than in the case of hydrogen-like ions. Its observation requires extremely high stability in the trap parameters. A measurable frequency change upon a spin flip was obtained by making the trap diameter about 3 times smaller than in the previous experiments and replacing the Ni-ring by a CoFe-ring and thus obtaining a larger magnetic field inhomogeneity. The result for the proton's and antiproton's magnetic moments are μ*<sup>p</sup>* = 2.792 847 344 62(82) μ<sup>N</sup> [24] and μ*antip* = −2.792 847 344 1(42) μ<sup>N</sup> [25], with μ<sup>N</sup> the nuclear magnetron. The agreement of the two values within the error bars represents a test of the CPT invariance theorem.

#### **8 The Electron Mass**

As evident from the result of the *g*-factor determination in 28Si13+, the largest contribution in the error budget arises from the uncertainty in the electron mass taken from the CODATA 2012 compilation of fundamental constants. The QED contributions to the electronic *g*-factor of H-like ions scale approximately with the square of the nuclear charge Z. Since in the case of 28Si13+ with *Z* = 14 we find an agreement between theory and experiment on the level of 10−<sup>10</sup> it can be reasonably assumed that in the case of H-like 12C5+ with *Z* = 6 the QED contributions to the *g*-factor have been calculated correctly. We can therefore rewrite Eq. (9) between the *g*-factor and the electron mass as

$$m\_{\epsilon} = \frac{g}{2} \frac{e}{q} \frac{v\_{\text{cyc}}}{v\_{\text{L}}} m\_{\text{ion}} \equiv \frac{g}{2} \frac{e}{q} \frac{1}{\Gamma} m\_{\text{ion}} \tag{10}$$

We take now the *g*-factor from theory and determine the electron mass. 12C5+ is the natural choice as ion since there is virtually no uncertainty in its mass. In our experiment [26] we obtained as new value for the electron mass *me* = 0.000 548 579 909 067 (14)(9)(2) a.u. The first two errors are the statistical and systematic uncertainties of the measurement, respectively, and the third one represents the uncertainties of the theoretical prediction of the *g*-factor and the electron binding energies in the carbon ion. The new value surpasses that of the CODATA 2012 literature value by a factor of 13 and represents the basis for the most recent adjustment of fundamental constants [7].

#### **9 What Comes Next?**

At the Max-Planck-Institut für Kernphysik in Heidelberg an electron beam ion trap (EBIT) [27] will allow production of hydrogen-like ions of high nuclear charge *Z* up to 208Pb81+. They can be extracted from the EBIT and injected into an improved Penning-trap arrangement (ALPHATRAP) placed at the center of a superconducting magnet [28]. Here Larmor-to-cyclotron frequency ratio measurements can be performed at the *dg*/*g* ≈ 10−<sup>12</sup> level of accuracy. This will provide more stringent tests of bound state QED contributions to the electron's *g*-factor in an extremely strong electric field as evident from Figs. 8 and 9. In addition a new determination of the fine structure constant α by comparison of theoretical and experimental results seems possible. Figure 12 shows a sketch of the setup.