# Tunnelling times, Larmor clock, and the elephant in the room

D. Sokolovski1,3<EMAIL_ADDRESS>E. Akhmatskaya2,3 1 Departmento de
Química-Física, Universidad del País Vasco, UPV/EHU, Leioa, Spain  2 Basque
Center for Applied Mathematics (BCAM),
Alameda de Mazarredo 14, 48009 Bilbao, Bizkaia, Spain  3 IKERBASQUE, Basque
Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain

###### Abstract

ABSTRACT:

A controversy surrounding the “tunnelling time problem” stems from the seeming
inability of quantum mechanics to provide, in the usual way, a definition of
the duration a particle is supposed to spend in a given region of space. For
this reason the problem is often approached from an “operational” angle.
Typically, one tries to mimic, in a quantum case, an experiment which yields
the desired result for a classical particle. One such approach is based on the
use of a Larmor clock. We show that the difficulty with applying a non-
perturbing Larmor clock in order to “time” a classically forbidden transition
arises from the quantum Uncertainty Principle. We also demonstrate that for
this reason a Larmor time (in fact, none of the Larmor times) cannot be
interpreted as a physical time interval. We also provide a theoretical
description of the quantities measured by the clock.

###### pacs:

03.65.Ta, 03.65.AA, 03.65.UD

##  I Introduction

The “tunnelling time” problem which has been with us for nearly a century
McColl , still has its share of controversy (for a recent review see Rev ),
and for a good reason. A prerequisite for any constructive discussion is a
possibility to define its subject in a meaningful way. For a classical
particle, a duration spent in a given region of space is indeed a well
established and useful concept. In quantum mechanics, the Uncertainty
Principle (UP) forbids answering the ”which way?” question if two or more
pathways leading to the same final outcome interfere FeynL . By the same token
a duration, readily determined for each path, must remain indeterminate for a
process where interference plays a crucial role. This is particularly true in
the case of tunnelling.
The early attempts to define the duration a quantum particle spends in the
barrier by following the evolution of the transmitted wave packet WE -Smith
yielded the so-called Wigner-Smith (WS) time delay, essentially the energy
derivative of the phase of the transmission amplitude. One immediate problem
with the method is that if the WS result is used to estimate the time spent by
the particle in the barrier, this time turns out to be shorter than the
barrier width divided by the speed of light. This apparently “superluminal
behaviour” does not lead to a conflict with Einstein’s relativity for the
simple reason that, in accordance with the Uncertainty Principle, the WS time
cannot be interpreted as a physical time interval spent by a tunnelling
particle in the barrier DSnat . However, as was noted in Rev , the argument of
DSnat  applies to the “phase time” of WE -Smith . Would it still be true if
the tunnelling time were defined in a different manner?
An alternative approach was proposed by A.I. Baz’ Baz1 , who employed Larmor
precession of a magnetic moment (spin) in a magnetic field, small enough not
to affect tunnelling seriously Baz2 . The interest in the Larmor (Baz’) clock
was recently renewed after its experimental realisation was reported in Stein
, and in what follows we will analyse it in some detail. By construction, such
a clock probes the response of a scattering amplitude to a small variation of
the potential, rather than to a variation of the particle’s energy. Thus, the
Larmor time was found to disagree with the Wigner and Smith result, and
proposed to be the “correct” estimate of the duration of a scattering process
(see the footnote on p.169 of Baz3 ). Despite Baz’s assertion in Baz3 , the
Larmor clock approach soon encountered its own difficulties. In particular, if
applied to tunnelling transmission the method yielded not one but two time
parameters, which Büttiker Butt  proposed to combine into a single
“interaction time.” In SB  Sokolovski and Baskin have shown the two Larmor
times to be the real and imaginary parts of a “complex time” obtained as an
average, in which the usual probabilities were replaced with quantum
probability amplitudes. The lack of clarity about these matters points to a
more fundamental problem, which requires further attention.
The purpose of this paper is to demonstrate that the difficulty in deducing
the duration spent in the barrier, evident in the analysis of the Wigner-Smith
time delay DSnat , persists also in the conceptually different Larmor clock
approach Baz1 -SB . To do so we will again appeal to the Uncertainty
Principle, a rule of primary importance for any discussion of the tunnelling
time problem, yet rarely mentioned in such discussions. It will also be upon
us to answer the question “does a Larmor clock measure a physical time
interval and, if not, then what does it measure?”

##  II Results

To lay bare the conceptual difficulty, we start by considering a simple
thought experiment, where an electron, with its spin polarised along the
$x$-axis, enters an interferometer shown in Fig.1 in a wave packet state
$|G_{0}{\rangle}$, and is detected after exiting the second beam splitter, as
shown in Fig.1. Travelling via different arms of the interferometer, the
electron spends different durations, $\tau_{1}$ and $\tau_{2}$, in a region
containing constant magnetic field directed along the $z$-axis, $B$ (in an
experiment using photons and Faraday’s rotation the field would be directed
along the arms). An additional element (e.g., an extra potential) in the
second arm ensures that an extra phase, $\phi$ is acquired there by both spin
components. So how much time did the electron spend in the magnetic field?

Figure 1: A particle reaches the final position $x$ after passing through an
interferometer, and a weekly coupled Larmor clock is used to determine the
duration it spends in the magnetic field. The case of tunnelling across a
potential barrier, shown in the inset, is more complicated, yet conceptually
similar.

The question is more difficult that it may seem. If the wave packet travelling
at a velocity $v$ is fast, and the field is not too strong, the two spin
components acquire, in each arm, phases $\exp(\pm\omega_{L}\tau_{1,2})$, where
$\omega_{L}$ is the Larmor frequency. Thus, beyond the second beam splitter
the wave function is given by (the $\hat{\sigma}$s are Pauli matrices)

$\displaystyle{\langle}x|\Phi{\rangle}=[G_{1}(x,t)\exp(-i\omega_{L}\hat{\sigma}_{z}\tau_{1})+$
(1) $\displaystyle
G_{2}(x,t)\exp(-i\omega_{L}\hat{\sigma}_{z}\tau_{2})]|\uparrow_{x}{\rangle},$

where $G_{1,2}(x,t)$ are the parts of the original wave packets arriving at
$x$ via the first and the second arm, respectively. One notes that the sum of
the rotations in the square brackets does not add up to a single rotation
around the $z$-axis, so no duration can be deduced from Eq.(1) directly.
Perhaps, making the field small could help? Indeed, sending $\omega_{L}\to 0$
and keeping only the linear terms, one finds
${\langle}x|\Phi{\rangle}\approx[G_{1}(x,t)+G_{2}(x,t)](1-i\omega_{L}\sigma_{z}\overline{\tau})|\uparrow_{x}{\rangle}$,
which now looks like an overall rotation through a small angle
$\omega_{L}\overline{\tau}$. Does this mean that

$\displaystyle\overline{\tau}\equiv\frac{\tau_{1}G_{1}(x,t)+\tau_{2}G_{2}(x,t)}{G_{1}(x,t)+G_{2}(x,t)}\equiv\tau_{1}\alpha(\tau_{1})+\tau_{2}\alpha(\tau_{2})$
(2)

is a suitable candidate for the duration spent in the field? Not quite so. The
quantities $G_{i}$ are the transition amplitudes FeynL  for an electron,
initially in $|G_{0}{\rangle}$, to reach $|x{\rangle}$ via the $i$-th arm of
the interferometer, and $\overline{\tau}$ is complex valued. This new problem
can be dealt with by evaluating the mean angle of precession in the
$xy$-plane, $\varphi_{xy}$, guaranteed at least to be real. The result,

$\displaystyle\varphi_{xy}\approx{\langle}\Phi|x{\rangle}\hat{\sigma}_{y}{\langle}x|\Phi{\rangle}=\omega_{L}\text{Re}[\overline{\tau}],$
(3)

appears to give preference to the real part of $\overline{\tau}$, and may look
satisfactory. (Note that measuring the angle of rotation in the $xz$-plane
would yield also the value of $\text{Im}[\overline{\tau}]$, but it is not
important to us here.)
However, our real problems are only beginning. A non-negative probability
distribution, $\rho(z)\geq 0$, has many useful properties. For example, an
expectation value ${\langle}z{\rangle}$ marks roughly the centre of the region
where $\rho(z)\neq 0$, and the variance gives an estimate of the size of this
region. This is no longer true for the distributions which change sign, and
the “average” in Eq.(2) is of this latter type. Adjusting the phases and
lengths, one can ensure that $G_{2}(x,t)\approx-G_{1}(x,t)$, and make the
denominator in Eq.(2) small. A similar cancellation will not occur in the
numerator, and $\overline{\tau}$ can be made as large as one wants. On the
other hand, with both arms of about the same length, $\L_{1}\approx
L_{2}\approx L$, the electron spends in motion approximately $\sim L/v$. Now
the “duration” in Eq.(3) can easily exceed the total time electron was in
motion,

$\displaystyle\text{Re}[\overline{\tau}]>>L/v\equiv T_{total}.$ (4)

Similarly, $\tau_{1,2}$ and $G_{1,2}$ could be chosen so that
$\overline{\tau}=0$, making it look like the electron, known to move at a
speed $v$ in each arm, crosses the field infinitely fast if both arms are
considered together. These are serious issues, which should not be ignored.
One has to decide whether to allow a quantum particle to spend more time that
it has at its disposal, and hail Eq.(4) as a new triumph of quantum theory.
The other possibility is to conclude that something is wrong with the very
question asked. It is, indeed, frustrating to have two durations, $\tau_{1}$
and $\tau_{2}$, and to be unable to combine them into anything meaningful if a
particle passes through both arms of the interferometer in Fig.1.
The frustration is of a familiar kind. In a Young’s double-slit experiment, an
electron passes trough one of the two slits, but it is not possible to know
which particular slit was chosen. The impossibility of answering the “which
way?” question, without destroying interference, is the essence of the
Uncertainty Principle, without which quantum mechanics “would collapse” FeynL
. The experiment in Fig.1 is a kind of a double-slit case, with the only
difference that the “which way?” question has been disguised as a “how much
time?” query.
It is instructive to see how quantum mechanics implements the Principle in
practice. Since the only a priory restriction on the in general complex valued
relative amplitudes $\alpha_{1,2}$ in Eq.(2) is that they should add up to
unity, $\alpha_{1}+\alpha_{2}=1$, one can find suitable $\alpha$s for any
choice of a complex $\overline{\tau}$,

$\displaystyle\alpha_{1}=\frac{\overline{\tau}-\tau_{2}}{\tau_{1}-\tau_{2}},\quad\alpha_{2}=-\frac{\overline{\tau}-\tau_{1}}{\tau_{1}-\tau_{2}}.$
(5)

Unable to forbid one to ask the question operationally, quantum theory gives
all possible answers, suitable and unsuitable, according to the circumstances.
Depending on the parameters of the interferometer, the measured real part of
$\overline{\tau}$ can be positive, negative, zero, coincide with $\tau_{1}$ or
$\tau_{2}$, or lie between them. The answer to a question that should not have
an answer can be “anything at all”.
One can envisage a following dialogue between an experimentalist Alice (A) and
a theoretician Bob (B).

A: I have just measured the mean angle $\varphi_{xy}$, and divided it by the
Larmor frequency. It follows that
$\tau_{Alice}=\text{Re}[\overline{\tau}]=(\tau_{1}+\tau_{2})/2$, a perfectly
reasonable result. And I was told this time does not exist.
B: It does not. Change your settings, and the same procedure will give you
$\text{Re}[\overline{\tau}]<0$. The time parameter you measure is not a
meaningful duration.

A: Let us just forget about the cases where something goes wrong. Surely, in
my case it is the time an electron spends in the magnetic field.
B: Just don’t tell that Carol-the-engineer. What she wants, is a time scale
for changing the setup slowly enough for the electron “to see” its conditions
“frozen” during its journey to the detector. For your $\tau_{Alice}$ to serve
as a classical time scale you would also need to show that
$\tau_{1}^{n}\alpha(\tau_{1})+\tau_{2}^{n}\alpha(\tau_{2})=\overline{\tau}^{n}$,
$n=1,2,...$. However, this happens only if one of the $\alpha$s vanishes, in
which case either $\tau_{1}$ or $\tau_{2}$ is the time scale Carol would be
happy with.

A: But this time scale is a very well known and useful concept. How can it not
exist?
B: It is also an essentially classical concept, useful when there is no
interference involved. Make one arm of the interferometer much longer than the
other, so that the two parts of the wave packet do not overlap at $x$. Then,
at a given $t$, you will know which way the electron has travelled, and also
the duration, $\tau_{1}$ or $\tau_{2}$ it has spent in the magnetic field. But
then, of course, it would be a different experiment.

A: And what if I take instead the imaginary part, or the modulus of
$\overline{\tau}$, as was suggested, for example by Büttiker Butt ?
B: Or any real valued combination of $\text{Re}[\overline{\tau}]$ and
$\text{Im}[\overline{\tau}]$. You will still encounter “times” which are too
long for common sense or too short for Einstein’s relativity, although with
$\tau_{Alice}=|\overline{\tau}|$ you would not need to worry about negative
durations.

A: So what is my “time” good for?
B: It does describe the response of the electron to a small perturbation of a
particular type, a small rectangular potential, introduced by the constant
magnetic field. A different “time” would arise if the response to a small
oscillating potential were to be studied instead BL .

A: So, if my time is not a “meaningful duration”, what is it? It looks like
one of the “weak values” we heard so much about recently WVrev .
B: It is just what Eq.(3) says, $\overline{\tau}$ is a sum of relative
probability amplitudes for reaching the detector via different arms,
multiplied by the corresponding durations spent in the field, thus, also an
amplitude. And so is every other “weak value” DSann . Your time is just the
real part of a particular probability amplitude.

A: But I have just measured it.
B: Not quite, you just measured the spin, and then tried to learn something
about electron’s translational degree of freedom. In doing so, you relied on
the first-order perturbation theory. Response of a system to a small
perturbation is commonly described in terms of real valued combinations of the
system’s probability amplitudes.

A: And what is then an amplitude?
B: According to Feynman FeynL , it is a basic concept in our description of
quantum behaviour.

A: This does not tell me very much. Can you be more specific?
B: I am afraid not. Nor, I suspect, can anybody else, unless a radically new
insight into physics of the double-slit experiment is gained in future. In
Feynman’s words, at the moment “no one will give you any deeper description of
the situation” FeynL .

The case of Ref.Stein  is similar to the one just discussed, if not more
involved (see Methods). In Fig.1, there are only two routes by which an
electron, starting in a state $|G_{0}{\rangle}$, can reach the final position
$x$, and the corresponding amplitude has two components,

$\displaystyle A(x\leftarrow G_{0})=G_{1}(x,t)+G_{2}(x,t)\equiv$ (6)
$\displaystyle A(x\leftarrow G_{0}|\tau_{1})+A(x\leftarrow G_{0}|\tau_{2}).$

For a quantum particle crossing a potential barrier, there are many possible
$\tau$s, and many components to the transition amplitude PRA ,

$\displaystyle A(x\leftarrow G_{0})=\int_{0}^{T_{total}}A(x\leftarrow
G_{0}|\tau)d\tau.$ (7)

The mean angle of spin’s rotation in a small magnetic field, confined to the
barrier, is given by an analogue of (3)

$\displaystyle\varphi_{xy}\approx\omega_{L}\text{Re}\left[\frac{\int_{0}^{T_{total}}\tau
A(x\leftarrow G_{0}|\tau)d\tau}{\int_{0}^{T_{total}}A(x\leftarrow
G_{0}|\tau)d\tau}\right]\quad$ (8)
$\displaystyle=\omega_{L}\text{Re}\left[{\int_{0}^{T_{total}}\tau\alpha(\tau)d\tau}\right]\equiv\omega_{L}\text{Re}\left[\overline{\tau}\right].$

In the classical limit, highly oscillatory $A(x\leftarrow G_{0}|\tau)$
develops a stationary region around the classical duration $\tau_{class}$,
where it varies more slowly. This is the only region contributing to the
integral in (8), and one recovers the classical result,
$\overline{\tau}=\tau_{class}$. But this well defined duration disappears
already if $A(x\leftarrow G_{0}|\tau)$ has two, rather than just one,
stationary regions, and we are back to the situation similar to the one shown
in Fig.1.
Quantum tunnelling is a destructive interference phenomenon, where
$A(x\leftarrow G_{0}|\tau)$ in Eq.(6) has no stationary regions, and rapidly
oscillates throughout the allowed range $0\leq\tau\leq T_{total}$. The
tunnelling amplitude (6) is extremely small for a tall or a wide barrier (see
the inset in Fig.1). This happens not because $A(x\leftarrow G_{0}|\tau)$ is
itself small, but because its oscillations cancel each other almost exactly.
The delicate balance is easily perturbed, and an attempt to destroy
interference between different durations would also destroy the tunnelling
transition one wanted to study.

##  III Discussion

Finally, if Alice were to repeat also the experiment of Ref.Stein , this is
what Bob would say about her result. “A fundamental problem, arising each time
a Larmor clock is applied to tunnelling, but often overlooked - the proverbial
elephant in the room - has to do with the quantum Uncertainty Principle.
According to the Principle, one can have tunnelling, and not know the time
spent in the barrier, or know this duration, but have tunnelling destroyed.
One faces precisely the same choice in the double slit experiment, where
he/she must decide between knowing the slit chosen by the particle, or having
the interference pattern on the screen, but not both at the same time. You
have tried to keep tunnelling intact (your clock perturbs it only slightly),
and learn something about the duration spent in the barrier. You might expect
the UP to make your result always look flawed in one way or another, but this
is not how the UP works. If you consider all possible experiments of this
type, some of them will give seemingly reasonable outcomes, whereas other
‘times’ would be negative, too short, too long, etc. This is necessary, and is
possible because such ‘times’ can be expressed as the combinations of
probability amplitudes which, unlike probabilities, have few restrictions on
their signs and magnitudes. Though your result of $0.61$ $ms$ does look
plausible you cannot recommend using it the way you would use a classical time
scale just because of this. After all, in a double-slit experiment one cannot
cherry pick the points on the screen, where the ‘which way?’ question can be
answered meaningfully, since the Uncertainty Principle applies everywhere in
equal measure. You cannot say that you resolved the controversy regarding how
long a tunnelling particle spends in the barrier region, or proved that this
duration is non-zero. The controversy, if you wish to call it that, goes to
the very heart of the quantum theory, and must be accepted, rather than
resolved.”

##  IV Methods

###  IV.1 Probability amplitude to spend a given duration $\tau$ in the
barrier

Consider a particle with a mean momentum $p_{0}$, prepared in a wave packet
state $G_{0}$ ($\hbar=1$),

$\displaystyle
G_{0}(x)=\int{\color[rgb]{0,0,0}a}(p-p_{0})\exp(ipx)dp=\exp(ip_{0}x){\color[rgb]{0,0,0}W(x)},$
(9)

where ${a}(p-p_{0})$ discribes the distribution of the particle’s momenta, and
$W(x)$ is the wave packet’s envelope. At $t=0$ the wave packet lies to the
left of a potential barrier $V(x)$ of a width $d$, as shown in the inset in
Fig.1. All momenta $p$ in (9) are such that in order to cross the barrier the
particle has to tunnel. The probability amplitude to detect the particle at
$x$ close to the maximum of the transmitted wave packet, after it has been in
motion for $T_{total}$ seconds, can be represented as a sum over Feynman
paths,

$\displaystyle A(x\leftarrow G_{0})=\int
dx^{\prime}\sum_{paths}\exp(iS[x(t)]G_{0}(x^{\prime}),$ (10)

where a path $x(t)$ starts in $x^{\prime}$ at $t=0$, and ends in $x$ at
$t=T_{total}$. The action functional is given by the usual
$S[x(t)]=\int_{0}^{T_{total}}[{\mu\dot{x}^{2}}/{2}-V(x)]dt$, with $\mu$
denoting the particle’s mass. Each path spends a certain amount of time in the
barrier region $0\leq x\leq d$. Thus duration can be computed with the help of
a “stop-watch” (SW) expression,

$\displaystyle{\tau_{SW}}[x(t)]=\int_{0}^{T_{total}}\theta_{[0,d]}(x(t))dt,$
(11)

where $\theta_{[0,d]}=1$ for $0\leq x\leq d$ and $0$ otherwise, so that only
the time intervals spent in the barrier are added to the total. It is readily
seen that ${\tau_{SW}}[x(t)]$ cannot be negative, nor can exceed the time the
particle was in motion, hence

$\displaystyle 0\leq{\tau_{SW}}[x(t)]\leq T_{total}.$ (12)

A simple cosmetic operation turns the path sum (10) into the sum over
durations spent in the barrier. Restricting the summation to the paths which
spend there precisely $\tau$ seconds, yields

$\displaystyle A(x\leftarrow G_{0}|\tau)\equiv\int
dx^{\prime}\sum_{paths}\delta({\tau_{SW}}[x(t)]-\tau)\exp(iS[x(t)]G_{0}(x^{\prime}),$
(13)

where $\delta(z)$ is the Dirac delta, and we have

$\displaystyle A(x\leftarrow G_{0})\equiv\int_{0}^{T_{total}}A(x\leftarrow
G_{0}|\tau)d\tau.$ (14)

This is bad news for one’s effort to determine the time actually spent in the
potential - all such durations interfere. We are back to the Young’s
interference experiment, except that instead of two paths, each going through
one of the slits, we have a continuum of routes, each labelled by the value of
the ${\tau_{SW}}[x(t)]$. According to the Uncertainty Principle FeynL  the
“which way?” ( “which $\tau$?”) question has no answer. The only exception is
the classical limit. Typically, $A(x\leftarrow G_{0}|\tau)$ is highly
oscillatory, but in a classically allowed case, e.g., with the barrier
removed, the oscillations are slowed down near the classical value
$\tau_{cl}=\mu d/p$. If $A(x\leftarrow G_{0}|\tau)$ has a unique stationary
phase point of this kind, $\tau_{cl}$ will appear as the only time parameter,
whenever one evaluates integrals involving $A(x\leftarrow G_{0}|\tau)$, and
classical mechanics will apply as a result.
The problem with tunnelling is that no such preferred time emerges for a
classically forbidden transition, and all $\tau$s must be treated equally (a
similar situation is shown in Fig.3 of DSnat , although for a different
quantity). To make things worse, in tunnelling the amplitude $A(x\leftarrow
G_{0})$ is very small ($\sim\exp[-(2\mu V-p_{0}^{2})^{1/2}d]$ for a
rectangular barrier), while $A(x\leftarrow G_{0}|\tau)$ is not. Thus, the
exponentially small tunnelling amplitude results from a highly accurate
cancellation between (not small) oscillations of $A(x\leftarrow G_{0}|\tau)$.
For this reason, any attempt to modify or neglect any part of the integrand in
Eq.(13) would considerably change the result, and destroy the tunnelling.

###  IV.2 An uncertainty relation for the duration $\tau$

Although the Uncertainty Principle hampers one’s attempts to ascribe a unique
barrier duration to a tunnelling transition, there is still one more thing we
can do. Writing the $\delta$-function in (13) as

$\displaystyle\delta({\tau_{SW}}[x(t)]-\tau)=(2\pi)^{-1}\int
d\lambda\exp\\{i\lambda(\tau-{\tau_{SW}}[x(t)])\\},$ (15)

and inserting it into (13), we note that the new action

$\displaystyle S_{\lambda}[x(t)]\equiv
S[x(t)]-\lambda\int_{0}^{T_{total}}dt{\color[rgb]{0,0,0}\theta_{[0,d]}(x(t))}$
(16)

corresponds to adding to the barrier $V(x)$ a rectangular potential
$\lambda\theta_{[0,d]}(x(t))$, a well or a barrier, depending on the sign of
$\lambda$. Equation (13) can now be written in an equivalent form,

$\displaystyle A(x\leftarrow
G_{0}|\tau)=(2\pi)^{-1}\int_{-\infty}^{\infty}d\lambda\exp(i\lambda\tau){\tilde{A}}(x\leftarrow
G_{0}|\lambda),$ (17)

where ${\tilde{A}}(x\leftarrow G_{0}|\lambda)$ is the amplitude to reach, at
$t=T_{total}$, the final location $x$ from the initial state $G_{0}$, while
moving in a combined potential $V(x)+\lambda\theta_{[0,d]}(x)$. In other
words, to evaluate the amplitude $A(x\leftarrow G_{0}|\tau)$ one needs to know
the amplitudes of transmission for all composite potentials. And vice versa,
to know the amplitude for a given potential one needs to know the amplitudes
for all durations spent therein.
Note next that even the calculation of the full amplitude distribution of the
durations spent in a region $[0,d]$ for a free particle, $V(x)=0$, is already
a non-trivial task. It involves evaluation of the transmission amplitudes for
all rectangular wells and barriers, and integration in Eq.(17). However, once
$A_{0}(x\leftarrow G_{0}|\tau)$ is obtained, the distribution for a
rectangular potential $V(x)=V\theta_{[0,d]}(x)$ comes for free,

$\displaystyle A(x\leftarrow G_{0}|\tau)=\exp(-iV\tau)A_{0}(x\leftarrow
G_{0}|\tau).$ (18)

As we mentioned above, in the semiclassical limit, the free amplitude
distribution $A_{0}(x\leftarrow G_{0}|\tau)$ develops a stationary region
around $\tau=\mu d/p_{0}$. When the barrier is raised, the factor
$\exp(-iV\tau)$ destroys the stationary region, $A(x\leftarrow G_{0}|\tau)$
rapidly oscillates everywhere, and $A(x\leftarrow G_{0})$ becomes small for a
tunnelling particle.
Equation (17) is a kind of uncertainty relation between the duration $\tau$
and the potential in the region of interest. It implies that a device employed
to measure the $\tau$ must introduce some uncertainty into the potential, the
greater the uncertainty, the more accurate the measurement. Which brings us to
the Larmor clock.

###  IV.3 The Larmor clock

The clock consists of a magnetic moment, proportional to an angular momentum
(spin) of a size $j$, coupled to a magnetic field along the $z$-axis via
$\hat{H}_{int}=\omega_{L}\hat{j}_{z}$, where $\omega_{L}$ is the Larmor
frequency. By the time $t$, an initial state

$\displaystyle|\gamma{\rangle}=\sum_{m=-j}^{j}\gamma_{m}|m{\rangle},\quad\hat{j}_{z}|m{\rangle}=m|m{\rangle}$
(19)

becomes rotated by an angle $\omega_{L}t$ around the $z$-axis,

$\displaystyle|\gamma(t){\rangle}=\exp(-i\omega_{L}t\hat{j}_{z})|\gamma(0){\rangle}=\sum_{m=-j}^{j}\gamma_{m}\exp(-im\omega_{L}t)|m{\rangle}.$
(20)

Suppose the spin travels with a classical particle moving along a trajectory
$x(t)$, and the field exists only in the region $0\leq x\leq d$. Then the
spin, precessing only when the particle is in the field, $0\leq x(t)\leq d$,
ends up rotated by $\omega_{L}{\tau_{SW}}[x(t)]$ by $t=T_{total}$. Quantally,
for a particle in the inset of Fig.1, the final (unnormalised) spin’s state
can be found simply by adding up its states, rotated by $\omega_{L}\tau$, each
multiplied by the probability amplitude of spending in the field a net
duration $\tau$. The result is

$\displaystyle|\gamma(T_{total}){\rangle}=\int_{0}^{T_{total}}d\tau
A(x\leftarrow
G_{0}|\tau)\exp(-i\omega_{L}\tau\hat{j}_{z})|\gamma(0){\rangle}.$ (21)

In general, the r.h.s. of (21) cannot be rewritten as a single rotation around
the $z$-axis by an angle $\omega_{L}\tau^{\prime}$,
$|\gamma(T_{total}){\rangle}\neq\exp(-i\omega_{L}\tau^{\prime}\hat{j}_{z})|\gamma(0){\rangle}$,
and no unique time $\tau^{\prime}$ can be associated with a quantum transition
in this way.
With the help of Eq.(17), one obtains an equivalent form of Eq.(21),

$\displaystyle|\gamma(T_{total}){\rangle}=\sum_{m=-j}^{j}{\tilde{A}}(x\leftarrow
G_{0}|m\omega_{L})\gamma_{m}|m{\rangle}.$ (22)

This shows that each spin component traverses the barrier as if the potential
there were $V(x)+m\omega_{L}$, so the potential, experienced by the particle
as a whole, remains uncertain within the range from $-j\omega_{L}$ to
$j\omega_{L}$. As was already noted, a viable clock has to introduce this
uncertainty, and we may ask what can be learnt about the duration spent in the
barrier by applying the Larmor clock.
An experiment could consist in detecting, at $t=T_{total}$, the particle in
$x$ and its spin in a state
$|\beta{\rangle}=\sum_{m=-j}^{j}\beta_{m}|m{\rangle}$. From (21) the
corresponding probability is

$\displaystyle
P(\beta,x\leftarrow\gamma,G_{0})=|\int_{0}^{T_{total}}d\tau\Gamma(\tau|\omega_{L},\beta,\gamma)A(x\leftarrow
G_{0}|\tau)|^{2},$ (23)

where

$\displaystyle\Gamma(\tau|\omega_{L},\beta,\gamma)\equiv{\langle}\beta|\exp(-i\omega_{L}\tau\hat{j}_{z})|\gamma{\rangle}=\sum_{m=-j}^{j}\beta^{*}_{m}\gamma_{m}\exp(-im\omega_{L}\tau).$
(24)

Thus, by measuring the probability (23), one can determine the absolute value
of the integral in Eq.(23), which involves the amplitude distribution of the
durations spent by the particle inside the barrier in the absence of the
clock. Note that little is left of the original tunnelling transition, where
the transmission amplitude $A(x\leftarrow G_{0})$ is typically small. As
already mentioned at the end of subsection A, the presence of an additional
factor such as $\Gamma(\tau|\omega_{L},\beta,\gamma)$ is likely to alter
destructive interference which defines tunnelling. As a result,
$\int_{0}^{T_{total}}d\tau\Gamma(\tau|\omega_{L},\beta,\gamma)A(x\leftarrow
G_{0}|\tau)$ could differ from the original tunnelling amplitude in Eq.(14) by
orders of magnitude.

###  IV.4 A non-perturbing (weak) Larmor clock

One can try to return to tunnelling by sending $\omega_{L}\to 0$, and learn
something about the tunnelling time from the particle’s response to the clock.
(This already bodes ill for one’s task, since the uncertainty introduced in
the potential will also tend to zero, which, according to subsection B, should
lead to a large uncertainty in $\tau$). Nevertheless, we obtain

$\displaystyle\Gamma(\tau|\omega_{L},\beta,\gamma)\approx{\langle}\beta|\gamma{\rangle}-i\omega_{L}\tau{\langle}\beta|\hat{j}_{z}|\gamma{\rangle},$
(25)

so that the relative change in the probability (23) with and without the
magnetic field is

$\displaystyle\frac{P(\beta,x\leftarrow\gamma,G_{0})_{\omega_{L}}-P(\beta,x\leftarrow\gamma,G_{0})_{{\omega_{L}}=0}}{P(\beta,x\leftarrow\gamma,G_{0})_{{\omega_{L}}=0}}\approx
2\text{Re}[Z(\beta,\gamma)]\text{Im}[\overline{\tau}]+\text{Im}[Z(\beta,\gamma)]\text{Re}[\overline{\tau}],$
(26)

where
$Z(\beta,\gamma)\equiv{{\langle}\beta|\hat{j}_{z}|\gamma{\rangle}/{\langle}\beta|\gamma{\rangle}}$
and

$\displaystyle\overline{\tau}\equiv\frac{\int_{0}^{T_{total}}\tau
A(x\leftarrow G_{0}|\tau)d\tau}{\int_{0}^{T_{total}}A(x\leftarrow
G_{0}|\tau)d\tau}=\frac{\int_{0}^{T_{total}}\tau A(x\leftarrow
G_{0}|\tau)d\tau}{A(x\leftarrow
G_{0})}\equiv\int_{0}^{T_{total}}\tau\alpha(\tau)d\tau$ (27)

is the complex time of Sokolovski and Baskin SB . The quantity in the l.h.s.
of Eq.(27) can be measured, and by choosing a different $|\beta{\rangle}$ one
can, in principle, determine the values of $\text{Re}[\overline{\tau}]$,
$\text{Im}[\overline{\tau}]$, or indeed of their various combinations.
Moreover, for ${\langle}\beta|\gamma{\rangle}=0$, one has

$\displaystyle
P(\beta,x\leftarrow\gamma,G_{0})_{\omega_{L}}\sim\omega_{L}^{2}|\overline{\tau}|^{2},$
(28)

so the modulus of $\overline{\tau}$ can also be determined directly.
Now there are many real valued time parameters related to the complex time
(27), yet none of them is a suitable candidate for a physical time interval
representing the net duration spent in the barrier. The easiest way to
demonstrate it is to note that for an improbable transition, $A(x\leftarrow
G_{0}|\tau)\to 0$, the denominator of (27) can be very small. At the same
time, the numerator does not have to be small, since multiplication of
$A(x\leftarrow G_{0}|\tau)$ by $\tau$ can destroy the cancellation,
characteristic of tunnelling. Thus, $|\overline{\tau}|$ may, in principle,
exceed the total duration of motion, $|\overline{\tau}|>>T_{total}$. This
makes little sense, especially if one recalls that each and every Feynman path
in Eq.(10) spends in the barrier no more than $T_{total}$.

###  IV.5 The Baz’ clock

Finally we briefly discuss a particular type of a weak Larmor clock, employing
a spin-$1/2$ in a weak magnetic filed. It was introduced by A.I. Baz’ more
than fifty years ago Baz , and recently implemented by Ramos et al in Stein .
Now $\hat{j}_{z}=\sigma_{z}/2$ ($\sigma_{z}$ is the Pauli matrix), and the
spin’s initial direction is along the $x$-axis, whose azimuthal and polar
angles are $\phi=0$ and $\theta=\pi/2$ respectively. According to (21) the
final (unnormalised) state of the spin is given by

$\displaystyle|\gamma(T_{total}){\rangle}=2^{-1/2}\begin{pmatrix}\int_{0}^{T_{total}}d\tau
A(x\leftarrow G_{0}|\tau)\exp(-i\omega_{L}\tau/2)\\\ \int_{0}^{T_{total}}d\tau
A(x\leftarrow G_{0}|\tau)\exp(+i\omega_{L}\tau/2)\end{pmatrix}$ (29)
$\displaystyle\approx 2^{-1/2}A(x\leftarrow
G_{0}|\tau)\begin{pmatrix}1-i\omega_{L}\text{Re}[\overline{\tau}]/2+\omega_{L}\text{Im}[\overline{\tau}]/2\\\
1+i\omega_{L}\text{Re}[\overline{\tau}]/2-\omega_{L}\text{Im}[\overline{\tau}]/2)\end{pmatrix}.$

As it was discussed in subsection C, this cannot in general correspond to a
rotation around the $z$-axis. On the other hand, in any state, a spin-$1/2$
must point along some direction on the Bloch sphere. Thus, we expect the state
(29) to be rotated not only in the $xy$-, but also in the $xz$-plane. The
state of a spin, polarised along a direction making angles $\delta\phi$ and
$\pi/2-\delta\theta$ with the $x$\- and the $z$-axis, respectively, can be
written as

$\displaystyle\begin{pmatrix}\exp(-i\delta\phi/2)\cos\left(\frac{\pi}{4}-\frac{\delta\theta}{2}\right)\\\
\exp(+i\delta\phi/2)\sin\left(\frac{\pi}{4}-\frac{\delta\theta}{2}\right)\end{pmatrix}\approx
2^{-1/2}\begin{pmatrix}1-i\delta\phi/2+\delta\theta/2\\\
1+i\delta\phi/2-\delta\theta/2)\end{pmatrix}.$ (30)

Comparing (29) with (30) we find that the spin has rotated by the (small)
angles

$\displaystyle\delta\phi=\omega_{L}\text{Re}[\overline{\tau}],\quad\text{(in
the
$xy$-plane)}\quad\text{and}\quad\delta\theta=\omega_{L}\text{Im}[\overline{\tau}]\quad\text{(in
the $xz$-plane)}.$ (31)

We recall further that a spin travelling with a classical particle along a
trajectory $x_{class}(t)$ would rotate only in the $xy$-plane by an angle
$\omega_{L}\tau_{class}={\tau_{SW}}[x_{class}(t)]$. Thus, the first of
Eqs.(31) looks like the classical result, with $\tau_{class}$ replaced by
$\text{Re}[\overline{\tau}]$. The second of Eqs.(31) has no classical
analogue, and should serve as a warning that a straightforward extension of
the classical duration to the quantum case may not be possible. (One already
knows this from the Uncertainty Principle.)
The appearance of not one, but two rotation angles was first noted by Büttiker
in Butt , albeit in a slightly different context. [Ref. Butt  considered
transmission of a particle with a known momentum $p_{0}$ which, in our
language, corresponds to replacing $A(x\leftarrow G_{0}|\tau)$ with
$A(p_{0}\leftarrow G_{0}|\tau)\equiv\int\exp(-ip_{0}x)A(x\leftarrow
G_{0}|\tau)dx$ in all formulae, and making $G_{0}$ nearly monochromatic.] In
Butt  Büttiker defined two “times”, $\tau_{y}\equiv\delta\phi/\omega_{L}$ and
$\tau_{z}\equiv\delta\theta/\omega_{L}$, which correspond to our
$R[\overline{\tau}]$ and $\text{Im}[\overline{\tau}]$, respectively. Ramos et
al measured both the real and the imaginary parts of $\overline{\tau}$, which
can be seen in Fig.3 of Stein . The authors of Stein  found both parameters
positive and concluded that their results were “inconsistent with claims that
tunnelling takes ’zero time’”. To abide by this conclusion one needs to take
for granted that the ”time tunnelling takes” exists as a meaningful concept,
but this is not the case.
The confusion can be traced back to Büttiker Butt . When faced with two times
parameters instead of one, he opted for a non-negative combination of the two,
$\tau_{x}\equiv\sqrt{\tau_{y}^{2}+\tau_{z}^{2}}$. This equals the modulus of
the “complex time” in Eq.(27), $\tau_{x}=|\overline{\tau}|$. At least one
point made in Butt  requires a comment, if not a correction. In $\tau_{x}$
Büttiker believed to have found (we read in the Abstract of Butt ) “the time
interval during which a particle interacts with the barrier if it is finally
transmitted.” However, neither $\text{Re}[\overline{\tau}]$ nor
$\text{Im}[\overline{\tau}]$, nor any combination of the two can be
interpreted as a physical time interval. A weighted sum of quantum mechanical
amplitudes, $\overline{\tau}$, may not give a meaningful answer to the
question “how much time does a tunnelling particle spend within the barrier
region?” for the same reason the Uncertainty Principle FeynL  forbids
identifying the particle’s path in Young’s double-slit experiment.

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Acknowledgements

Financial support of MCIU, through the grant
PGC2018-101355-B-100(MCIU/AEI/FEDER,UE) (DS), and of the Basque Government
through Grant No IT986-16 (DS) is gratefully acknowledged. EA acknowledges the
financial support of the Ministerio de Economía y Competitividad (MINECO) of
the Spanish Government through BCAM Severo Ochoa accreditation SEV-2017-0718
and PID2019-104927GB-C22 grant. This work was also supported by the BERC
2018e2021 Program and ELKARTEK Programme (KK-2020/00049 and KK-2020/00008)
funded by the Basque Government.

Author contributions D.S. and E.A. both wrote the paper, and reviewed it.

Competing financial interests The authors declare no competing financial
interests.

Corresponding authors Correspondence to Dmitri Sokolovski,
<EMAIL_ADDRESS>