Muga G. Time in Quantum Mechanics

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3 Post-Pauli’s Theorem 61

3.9 Conclusion

Since the clariﬁcation of Pauli’s theorem and the introduction of the self-adjoint and

conjugate quantization of the classical free time of arrival for a spatially conﬁned

particle in [19], we have already made progress toward a better understanding of

the quantum time problem. Our solution to the quantum time of arrival problem via

the self-adjoint and conjugate conﬁned time of arrival operator represents such a

progress. However, our progress has opened a cache of issues confronting standard

quantum mechanics at the foundational level.

Our investigations on the conﬁned quantum time of arrival operators show that

the spectral properties of a time operator can acquire unambiguous interpretation

independent of the postulates of standard quantum measurement theory. In particu-

lar, the spectral properties may be tied with the dynamics of the system, such as the

eigenfunctions of the conﬁned quantum time of arrival operators unitarily arriving

at some point at their respective eigenvalues. This calls for a certain modiﬁcation

of the standard quantum measurement theory, which makes no exceptions on the

interpretation of the spectral properties of observables. How the modiﬁed quantum

theory of measurement will eventually look like is not yet clear. But we foresee that

the modiﬁed theory will require classiﬁcation of quantum observables into, which

we can provisionally call, temporal and non-temporal observables, with the former

falling into the standard quantum measurement theory and the latter falling into the

more general measurement theory that accommodates time as a dynamical observ-

able. But how can this classiﬁcation be made? We suspect that temporal observables

comprise the Hilbert space solutions to the time–energy canonical commutation

relation for a given Hamiltonian.

The most surprising aspect of the conﬁned time of arrival operators is that it

connects the quantum time of arrival problem and the problem of the appearance

of particle in quantum mechanics. In standard quantum mechanics, the appear-

ance of particle arises out of the collapse of the wave function brought about by

position measurement, and the collapse occurs not before but at the moment the

particle appears on our detectors. However, the dynamics of the conﬁned time of

arrival operators suggests that the appearance of particle arises as a combination

of the collapse of the initial wave function into one of the eigenfunctions of the

time of arrival operator, followed by the unitary Schr

odinger evolution of the eigen-

function. This implies that particles, at least within quantum arrival setting, do not

arise out of position measurements but out of time of arrival measurements and that

the collapse of the wave function on the appearance of particle is not fundamental

but decomposable into a series of casually separated processes. This invites us to

reconsider our beliefs on quantum measurement theory.

Finally, the clariﬁcation of “Pauli’s theorem” has brought us the realization

that the time–energy canonical commutation relation has a large class of physi-

cal solutions, contrary to expectations precipitated by Pauli’s theorem. We have

already demonstrated that physically meaningful Hilbert space solutions to the

canonical commutation relation split into categories. But that is as far as we know.

There is a need to pursue the investigation of the whole class of solutions to the

62 E.A. Galapon

canonical commutation relation in general and the relationship of these solutions to

the quantum time problem. We note that only one class of solutions, the dense cate-

gory canonical pairs that form a system of imprimitivities, has been given attention

to in physics literatures, and the rest of the solutions are ignored or are assumed

not to exist. The quantum time problem is multifaceted. With our solution to the

quantum time arrival problem in the form of a solution to the time–energy CCR, it is

not unreasonable to entertain the idea that the different solutions to the time–energy

canonical commutation relation may be identiﬁed with the different facets of the

quantum time problem.

Acknowledgments The works reported here were funded by the National Research Council of the

Philippines through Grant No. I-81-NRCP, the UP Ofﬁce of the Vice Chancellor for Research and

Development through UP-OVCRD Outright Research Grants PNSE-050509 and PNSE-070703,

and by the Ofﬁce of the Vice Chancellor for Academic Affairs through UP System Grants 2004 and

2007. The theory of conﬁned time of arrival operators was developed with the help of R. Caballar,

R. Bahague, R. Vitancol, and A. Villanueva. The idea of taking the limit of the conﬁning length

to inﬁnity arose out of a collaboration with J.G. Muga, I. Egusquiza, and F. Delgado. The author

especially acknowledges J.G. Muga for his encouragements that have become a major motivation

for the author’s personal crusade to understand time in quantum mechanics.

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Chapter 4

Detector Models for the Quantum Time

of Arrival

Andreas Ruschhaupt, J. Gonzalo Muga, and Gerhard C. Hegerfeldt

There is so much in that moment!

John Asherby

4.1 The Time of Arrival in Quantum Mechanics

Quantum particles are characterized, for a given preparation, by a fundamental

stochasticity of their observable features, such as positions, momenta, energies, or

times, e.g., times of arrival at a detector in time-of-ﬂight experiments. In quantum

theory the preparation stage is encoded in a wave function,

whereas averages or

statistical moments of the observables are calculated by a well-known prescription

(the expectation value integral) from self-adjoint operators and their powers: this is

at least the case for position, momentum, or energy. In fact, the entire statistical dis-

tributions are given by the square modulus of the overlap of the wave function with

the corresponding eigenstates. It may be argued that, for a general observable, the

choice of operator is as much of an art as a science, since none of the “rules of quan-

tization” known is of universal validity; there is not necessarily a one-to-one relation

between classical and quantum observables, so that a single classical quantity may

be related to several (observable!) quantum quantities; and, there is frequently no

obvious trace of the apparatus or experimental procedure in the formalism. In spite

of these difﬁculties, the validity of the chosen operator is eventually conﬁrmed or

A. Ruschhaupt (B)

Institut f

ur Mathematische Physik, TU Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig,

Germany, a.ruschhaupt@tu-bs.de

J.G. Muga

Departamento de Qu

ımica F

ısica, UPV-EHU, Apdo 644, 48080 Bilbao, Spain,

jg.muga@ehu.es

G.C. Hegerfeldt

Institut f

ur Theoretische Physik, Universit

at G

ottingen, Friedrich–Hund–Platz 1, D-37077

ottingen, Germany, hegerf@theorie.physik.uni-goettingen.de

More generally in a density operator, but such a generalization is not essential to us here.

Ruschhaupt, A. et al.: Detector Models for the Quantum Time of Arrival. Lect. Notes Phys. 789,

65–96 (2009)

DOI 10.1007/978-3-642-03174-8

 Springer-Verlag Berlin Heidelberg 2009

66 A. Ruschhaupt et al.

denied by comparison with experiment, and by consistency arguments, as for posi-

tion, momentum, or energy. The fate of time as a random variable, realized, for

instance, by the instantaneous clicks of a detector, has been, however, much more

problematic. In fact, amazingly, the theoretical study of time observables has been

in practice abandoned, if not banned, for decades, except for a handful of disperse

contributions (e.g., by Wigner, Aharonov and Bohm, Allcock, Kijowski, Srinivas,

and Werner) [69]. The main reason for this strange state of affairs has been the

dominance of the paradigm imposed by Pauli’s argument against the existence of

self-adjoint time operators, according to which “time is only a parameter” in quan-

tum mechanics. Instead of doing something about it, given the appalling evidence

that time is also a random variable in the laboratories, most physicists have instead

preferred to repeat the above mantra.

There are also complementary reasons that explain, at least partially, the neglect

of time observables. One is that assuming classical translation of the particles has

been an excellent approximation in practice, but in modern atom optics experiments

with ultra-cold atoms this is not necessarily the case. Other reason is that in many

experiments the signal is actually proportional to the (unproblematic) position prob-

ability density ρ(x, t), rather than to a proper time-of-arrival density Π(x, t). There

is unfortunately a widespread confusion about this distinction, see Sect. 4.5.3 below

for more details, and the expression “time of ﬂight” is nowadays applied quite stan-

dardly in the cold-atom community to experiments in which essentially the position

probability density, and not a time-of-arrival (TOA) density, is monitored. We are

thus far from being free from constraining prejudices on time observables and much

remains to be done on experiments and theory, but progress has been made. The fun-

damental theoretical lacuna for time observables has in more recent times struck the

attention of several researchers

and led us, in particular, to embark on an ongoing

research program.

The arrival time has been investigated at a fundamental level without making

the detector explicit, i.e., idealized arrival time distributions – based on positive

operator-valued measures (POVM) instead of a self-adjoint time operator – have

been derived. These ideal theories use symmetry or consistency arguments (e.g., a

correct classical limit), but need the support of experiments and the relation to aux-

iliary operational models which capture the essential aspects of these experiments.

In reverse, it is important to know what a particular experiment is really mea-

suring and, again, the operational models play a useful intermediate role in this

regard. An early attempt to incorporate the apparatus in an explicit, though certainly

simpliﬁed manner is due to Allcock, by means of a complex, localized imaginary

potential that absorbs the quantum wave packet [4–6]. The rationale for its use is

that the incident particle in the measurement process is subjected to an irreversible

transformation from the incident channel into some other ﬁnal channels. It could,

for example, be ionized or emit a photon after having been excited by a laser. The

For reviews on the arrival time see [63, 69] and for more recent work [31, 83, 7, 47, 39, 81, 22,

40, 58, 8, 49, 12, 50, 3, 88, 43].

4 Detector Models for the Quantum Time of Arrival 67

2−level atom

laser

Fig. 4.1 Scheme of the atom-laser model

wave function component in the incident channel would thus lose norm at a rate

proportional to the detection signal, a process mimicked by a complex absorbing

potential.

The subject of this chapter is a more detailed model which has been developed

since 2000 [64, 31, 32, 47, 48, 81, 82, 70, 33]: in essence it consists of a two-level

atom wavepacket in the ground state sent toward a laser-illuminated region where

the atom is excited, see Fig. 4.1; the detection time of the ﬁrst spontaneous photon

provides an operational deﬁnition of the arrival time that can be compared to more

abstract theories and ideal quantities for the particle. We will show that the complex

potential detector model can be justiﬁed in some limit from the atom-laser model.

Moreover, we shall see that by means of different limiting operations of the laser

ﬁeld shape or intensity and manipulations of the time-dependent signal, such as

“deconvolutions” and normalizations at the level of expectation values, or of the

initial state, via “operator normalization,” a wealth of information on a moving atom

wavepacket is available: not only TOA distributions but also probability densities,

kinetic energy densities, current densities, and other local densities.

The structure of the chapter is the following: in the rest of this section Allcock’s

papers [4–6] and the local density concept are reviewed; in Sect. 4.2 the basic atom-

laser model is examined in more detail; the complex potential detector models and

their connection to the atom-laser model will be revisited in Sect. 4.3; Sect. 4.4

deals with the concept of operator normalization and its application to arrival times;

and Sect. 4.5 will show how other local densities, in particular the kinetic energy

density, can be measured by the atom-laser model in certain limits; ﬁnally, pulsed

measurement of the quantum time of arrival will be examined in Sect. 4.6, while

Sect. 4.7 provides a summary.

4.1.1 The Papers of Allcock

Modern research on the quantum TOA owes much to Allcock’s seminal work

[4–6]. Looking for an ideal quantum arrival time concept, he considered, for a par-

ticle moving in one dimension, that arrival time measuring devices should rapidly

transfer any probability that appears at x > 0(x = 0 being the “arrival position”)

from the incident channel into various orthogonal and time-labeled measurement

68 A. Ruschhaupt et al.

channels. As a simple model to realize this basic feature he proposed a pulsed,

periodic removal, at time intervals δt, of the wave function ψ(x)forx > 0, while

the x < 0 region would not be affected, see Fig. 4.2. A similar particle removal

would provide the distribution of ﬁrst arrivals for an ensemble of classical, freely

moving particles as δt → 0. We shall elaborate on this idea in Sect. 4.6.

The difﬁculties to solve the corresponding mathematical problem lead Allcock to

study instead a different, continuous model with an absorbing imaginary potential

in the right half line, −iV

Θ(x), V

> 0, to simulate the detection. He argued that

the two models should lead to similar conclusions with a time resolution of the

order δt in the chopping model or /2V

in the complex potential model. He then

solved the Schr

odinger equation with the complex potential and noticed that for

|E|

max

, where E

max

is a maximal relevant energy in the wave packet, the

apparatus response vanishes, −δ N(t)/δt → 0, with N(t) =ψ(t)|ψ(t), because

of quantum mechanical reﬂection. This is one of the ﬁrst discussions of the quantum

Zeno effect, not yet known by this name at that time.

Fig. 4.2 Schematic representation of the time-of-arrival measurement by periodic projection of the

wave function onto the subspace x < 0 at times t

= jδt, j = ...,−1, 0, 1,.... Figures (a)and

(b) represent two instants immediately before and after the elimination of norm at x > 0

Eight years later Misra and Sudarshan [62] generalized this result studying the passage of a system

from one predetermined subspace to its orthogonal subspace: the periodic projection method in the

limit δt → 0 was presented as a natural way to model a continuous measurement, but it did not

lead to a time distribution of the passage but to its suppression [62], imposing Dirichlet boundary

conditions [35]. This lack of a “trustworthy algorithm” to compute TOA and related distributions

has been much debated since then, see [54, 34, 57] for a review.

4 Detector Models for the Quantum Time of Arrival 69

Allcock thus disregarded the strong absorption limit (equivalently the short δt

limit) because of reﬂection, and only considered the weak absorption limit, in which

the detection takes a long time but all particles are eventually detected. Under the

assumption that the measured arrival time distribution Π was a convolution of an

ideal distribution Π

and an apparatus function R (the distribution for the particle

survival at rest in the imaginary potential, R(t) = 2V

Θ(t)e

−2V

t/

/),

Π = Π

∗ R ,

he suggested as an approximate solution for the unknown ideal distribution the pos-

itive expression

(t) =



ψ

(t)|

1/2

δ(.x)

1/2

|ψ

(t) , (4.1)

where

k and .x are the wave number and position operators, m is the mass, and the

averages are computed with the freely moving wave function ψ

, without the com-

plex potential (unless stated otherwise the initial states have only positive momenta).

He also obtained the (not positive semideﬁnite) current density as the exact solution

for Π

in the V

→ 0 limit,

J(t) =



ψ

(t)|[

kδ(.x) + δ(.x)

k]|ψ

(t) . (4.2)

This last result makes sense classically, but in quantum mechanics J(t) is not posi-

tive semideﬁnite even for states composed only of positive momenta [4–6, 63].

(t) has been later rederived in different ways and is nowadays known as

“Kijowski’s distribution” [55, 63, 47]. Kijowski [55] inferred it from an axiomatic

approach inspired by the arrival time of classical particles. Much more recently, the

distribution Π

has been related to the positive operator-valued measure (POVM)

generated by the eigenstates of the Aharonov–Bohm (maximally symmetric) time-

of-arrival operator [1, 63, 41]. This method emphasizes the fact that self-adjointness

is not necessary to generate quantum probability distributions.

Werner generalized Kijowski’s distribution characterizing all possible covariant

arrival time observables [89] and also showed a connection to absorption processes

described by a semigroup [90]. Kijowski’s distribution has also been compared to

other approaches and generalized for multi-particle systems and for systems subject

to interaction potentials [67, 68, 13, 15, 14, 16, 17].

4.1.2 Local Densities

The quantum mechanical probability ﬂux (4.2) and Kijowski’s distribution (4.1)

are two quantizations of the same classical local density. Other local densities will

Operators will wear a hat when confusion is possible with corresponding functions or c-numbers.

70 A. Ruschhaupt et al.

appear along the chapter so a lightning review of this concept is in order. For a

classical dynamical variable A(q, p) of position and momentum, its local density,

(x), is

(x) =



dpD(x, p)A(x, p) =



dqdpD(q, p)δ(x −q)A(q, p) ,

where D(q, p) is the phase space density. To quantize this expression one can use

δ(x −.x) =|xx|,

and consider, for a point x, the operator

A(x) =

A|xx|, or rather one of its many

symmetrizations, as a quantum density (operator) for the observable

A. For a given

state |ψ, the expectation value ψ|

A(x)|ψ would then be a candidate for the value

of the local density at the point

x of the observable

A. The simplest case is

A = 1,

and it corresponds to the particle density, ρ(x) =|x|ψ|

.If

A does not com-

mute with |xx|, there are inﬁnitely many “combinations” (orderings) to construct

a quantum density [26, 66], for example,

1/2

|xx|

1/2

, (4.3)



A|xx|+|xx|



, (4.4)



1/2

|xx|

1/2





A|xx|+|xx|



. (4.5)

Different symmetrizations may have a perfectly respectful status as physically

observable and measurable quantities, and different orderings may be associated

with latent properties realized via different experimental measurement procedures

as we shall see. They may also be related more indirectly to observables and yet

carry valuable physical information.

In addition to the case of the ﬂux and Π

already mentioned, in which

A = .p/m,

a second important example of this quantum non-uniqueness for a single classical

quantity is the kinetic energy density, [27, 28, 77], with

A = .p

/2m as the kinetic

energy operator. Operational procedures devised to achieve a TOA distribution may

actually provide a kinetic energy density, see Sects. 4.5.1 and 4.5.2. It may also be

possible to ﬁnd simple approximate relations between ρ, Π

, J , and kinetic energy

densities for wave packets with rather well-deﬁned momentum, see Sect. 4.5.3, but,

in general, they are fundamentally different functions.

4.2 The Basic Atom-Laser Model

Apart from Allcock’s work, several “toy models” for arrival time measurements

have been put forward [2, 63, 16, 21, 78, 44, 80]. Instead, the aim of Damborenea

et al. in [31], following [64], was to work out a more realistic model describing

a TOA measurement by quantum optical means. The basic idea is that a region

4 Detector Models for the Quantum Time of Arrival 71

of space may be illuminated by a laser and upon entering the region an atom will

start emitting photons. The ﬁrst-photon emission can be taken as a measure of the

arrival time of the atom in that region, see Fig. 4.1.

The pumping of the atom to

the excited state and the photon emission takes some time and therefore produce

delays with respect to some “ideal” arrival time of the atom. Increasing the laser

intensity, and considering shorter lifetimes, does not solve the problem because the

atom tends to be reﬂected from the laser region without ever emitting a photon. This

is further discussed in Sect. 4.2.1 where the details of the model will be given.

A way out of these difﬁculties is to “subtract” the delays from the ﬁrst-photon

probability density by means of a deconvolution with an atom at rest, see Sect. 4.2.2.

This results in a distribution which, for shorter and shorter lifetime of the atomic

level, converges to J, the quantum mechanical probability ﬂux, and provides a way

to measure it.

4.2.1 Effective One-Dimensional Equations

We assume that a region is illuminated by a laser sheet, treated semiclassically, on

the y−z plane with wave vector in y-direction. There will be no z-dependence in the

full Hamiltonian so this coordinate may be ignored. We will use the quantum jumps

approach [51–53, 76] which is essentially equivalent to quantum trajectories [25]

and to the Monte Carlo wave function approach [30]. Using also the rotating-wave

approximation, and in a laser adapted interaction picture to get rid of an explicit time

dependence, the effective two-dimensional Hamiltonian for the undetected atom

which results from the quantum jump approach after tracing over the photon part is





0 Ω(x)e

−ik

Ω(x)e

−2Δ

−iγ



with |1≡





, |2≡





, and where .p

=−i

∂

∂x

, .p

=−i

∂

∂y

are the

momentum operator components in x −y space, Ω is the Rabi frequency, k

is the

laser wave number, and γ is the Einstein coefﬁcient of level 2, i.e., its decay rate or

inverse lifetime. We will examine the stationary Schr

odinger equation

χ(x, y) = H

χ(x, y) (4.6)

for wave functions with the atom incident in the ground state with well-deﬁned

momentum and energy, E







A way to realize this atom-laser model experimentally is by using a Lambda system in which the

laser drives the transition between levels 1 and 2 whereas 2 decays very fast and preferentially to a

third non-interacting sink state 3 [74].

Muga G. Time in Quantum Mechanics - Vol. 2

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