Tunnelling times, Larmor clock,and the elephant in the room
TTunnelling times, Larmor clock, and the elephant in the room
D. Sokolovski , ∗ and E. Akhmatskaya , Departmento de Qu´ımica-F´ısica, Universidad del Pa´ıs Vasco, UPV/EHU, Leioa, Spain Basque Center for Applied Mathematics (BCAM),Alameda de Mazarredo 14, 48009 Bilbao, Bizkaia, Spain and IKERBASQUE, Basque Foundation for Science,Plaza Euskadi 5, 48009 Bilbao, Spain (Dated: February 8, 2021)
ABSTRACT:
A controversy surrounding the “tunnelling time problem” stems from the seeminginability of quantum mechanics to provide, in the usual way, a definition of theduration a particle is supposed to spend in a given region of space. For this reasonthe problem is often approached from an “operational” angle. Typically, one triesto mimic, in a quantum case, an experiment which yields the desired result for aclassical particle. One such approach is based on the use of a Larmor clock. We showthat the difficulty with applying a non-perturbing Larmor clock in order to “time” aclassically forbidden transition arises from the quantum Uncertainty Principle. Wealso demonstrate that for this reason a Larmor time (in fact, none of the Larmortimes) cannot be interpreted as a physical time interval. We also provide a theoreticaldescription of the quantities measured by the clock. ∗ [email protected] a r X i v : . [ qu a n t - ph ] F e b I. INTRODUCTION
The “tunnelling time” problem which has been with us for nearly a century [1], still hasits share of controversy (for a recent review see [2]), and for a good reason. A prerequisite forany constructive discussion is a possibility to define its subject in a meaningful way. For aclassical particle, a duration spent in a given region of space is indeed a well established anduseful concept. In quantum mechanics, the Uncertainty Principle (UP) forbids answeringthe ”which way?” question if two or more pathways leading to the same final outcomeinterfere [3]. By the same token a duration, readily determined for each path, must remainindeterminate for a process where interference plays a crucial role. This is particularly truein the case of tunnelling.The early attempts to define the duration a quantum particle spends in the barrier byfollowing the evolution of the transmitted wave packet [4]-[5] yielded the so-called Wigner-Smith (WS) time delay, essentially the energy derivative of the phase of the transmissionamplitude. One immediate problem with the method is that if the WS result is used toestimate the time spent by the particle in the barrier, this time turns out to be shorter thanthe 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 accordancewith the Uncertainty Principle, the WS time cannot be interpreted as a physical time intervalspent by a tunnelling particle in the barrier [6]. However, as was noted in [2], the argumentof [6] applies to the “phase time” of [4]-[5]. Would it still be true if the tunnelling time weredefined in a different manner?An alternative approach was proposed by A.I. Baz’ [7], who employed Larmor precession of amagnetic moment (spin) in a magnetic field, small enough not to affect tunnelling seriously[8]. The interest in the Larmor (Baz’) clock was recently renewed after its experimentalrealisation was reported in [9], and in what follows we will analyse it in some detail. Byconstruction, such a clock probes the response of a scattering amplitude to a small variationof the potential, rather than to a variation of the particle’s energy. Thus, the Larmor timewas 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 [10]). DespiteBaz’s assertion in [10], the Larmor clock approach soon encountered its own difficulties. Inparticular, if applied to tunnelling transmission the method yielded not one but two timeparameters, which B¨uttiker [11] proposed to combine into a single “interaction time.” In [12]Sokolovski and Baskin have shown the two Larmor times to be the real and imaginary partsof a “complex time” obtained as an average, in which the usual probabilities were replacedwith quantum probability amplitudes. The lack of clarity about these matters points to amore fundamental problem, which requires further attention.The purpose of this paper is to demonstrate that the difficulty in deducing the durationspent in the barrier, evident in the analysis of the Wigner-Smith time delay [6], persistsalso in the conceptually different Larmor clock approach [7]-[12]. To do so we will againappeal to the Uncertainty Principle, a rule of primary importance for any discussion of thetunnelling time problem, yet rarely mentioned in such discussions. It will also be upon usto 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 shownin Fig.1 in a wave packet state | G (cid:105) , and is detected after exiting the second beam splitter,as shown in Fig.1. Travelling via different arms of the interferometer, the electron spendsdifferent durations, τ and τ , in a region containing constant magnetic field directed alongthe z -axis, B (in an experiment using photons and Faraday’s rotation the field would bedirected along the arms). An additional element (e.g., an extra potential) in the second armensures that an extra phase, φ is acquired there by both spin components. So how muchtime did the electron spend in the magnetic field?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, phasesexp( ± ω L τ , ), where ω L is the Larmor frequency. Thus, beyond the second beam splitterthe wave function is given by (the ˆ σ s are Pauli matrices) (cid:104) x | Φ (cid:105) = [ G ( x, t ) exp( − iω L ˆ σ z τ ) + (1) G ( x, t ) exp( − iω L ˆ σ z τ )] | ↑ x (cid:105) , Y X G x spin M a g n e t i c f i e l d ϕ xy τ τ L x G
0 Barrier φ Z B
FIG. 1. A particle reaches the final position x after passing through an interferometer, and a weeklycoupled Larmor clock is used to determine the duration it spends in the magnetic field. The caseof tunnelling across a potential barrier, shown in the inset, is more complicated, yet conceptuallysimilar. where G , ( x, t ) are the parts of the original wave packets arriving at x via the first and thesecond arm, respectively. One notes that the sum of the rotations in the square bracketsdoes not add up to a single rotation around the z -axis, so no duration can be deduced fromEq.(1) directly. Perhaps, making the field small could help? Indeed, sending ω L → (cid:104) x | Φ (cid:105) ≈ [ G ( x, t ) + G ( x, t )](1 − iω L σ z τ ) | ↑ x (cid:105) , whichnow looks like an overall rotation through a small angle ω L τ . Does this mean that τ ≡ τ G ( x, t ) + τ G ( x, t ) G ( x, t ) + G ( x, t ) ≡ τ α ( τ ) + τ α ( τ ) (2)is a suitable candidate for the duration spent in the field? Not quite so. The quantities G i are the transition amplitudes [3] for an electron, initially in | G (cid:105) , to reach | x (cid:105) via the i -tharm of the interferometer, and τ is complex valued. This new problem can be dealt with byevaluating the mean angle of precession in the xy -plane, ϕ xy , guaranteed at least to be real.The result, ϕ xy ≈ (cid:104) Φ | x (cid:105) ˆ σ y (cid:104) x | Φ (cid:105) = ω L Re[ τ ] , (3)appears to give preference to the real part of τ , and may look satisfactory. (Note thatmeasuring the angle of rotation in the xz -plane would yield also the value of Im[ τ ], but it isnot important to us here.)However, our real problems are only beginning. A non-negative probability distribution, ρ ( z ) ≥
0, has many useful properties. For example, an expectation value (cid:104) z (cid:105) marks roughlythe centre of the region where ρ ( z ) (cid:54) = 0, and the variance gives an estimate of the size of thisregion. 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 ( x, t ) ≈ − G ( x, t ), and make the denominator in Eq.(2) small. A similar cancellationwill not occur in the numerator, and τ can be made as large as one wants. On the otherhand, with both arms of about the same length, (cid:32)L ≈ L ≈ L , the electron spends inmotion approximately ∼ L/v . Now the “duration” in Eq.(3) can easily exceed the totaltime electron was in motion, Re[ τ ] >> L/v ≡ T total . (4)Similarly, τ , and G , could be chosen so that τ = 0, making it look like the electron, knownto move at a speed v in each arm, crosses the field infinitely fast if both arms are consideredtogether. These are serious issues, which should not be ignored. One has to decide whetherto 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 iswrong with the very question asked. It is, indeed, frustrating to have two durations, τ and τ , and to be unable to combine them into anything meaningful if a particle passes throughboth arms of the interferometer in Fig.1.The frustration is of a familiar kind. In a Young’s double-slit experiment, an electronpasses trough one of the two slits, but it is not possible to know which particular slit waschosen. The impossibility of answering the “which way?” question, without destroyinginterference, is the essence of the Uncertainty Principle, without which quantum mechanics “would collapse” [3]. The experiment in Fig.1 is a kind of a double-slit case, with the onlydifference 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. Sincethe only a priory restriction on the in general complex valued relative amplitudes α , inEq.(2) is that they should add up to unity, α + α = 1, one can find suitable α s for any choice of a complex τ , α = τ − τ τ − τ , α = − τ − τ τ − τ . (5)Unable to forbid one to ask the question operationally, quantum theory gives all possibleanswers, suitable and unsuitable, according to the circumstances. Depending on the pa-rameters of the interferometer, the measured real part of τ can be positive, negative, zero,coincide with τ or τ , or lie between them. The answer to a question that should not havean answer can be “anything at all” .One can envisage a following dialogue between an experimentalist Alice ( A ) and a theoreti-cian Bob ( B ). A : I have just measured the mean angle ϕ xy , and divided it by the Larmor frequency. Itfollows that τ Alice = Re[ τ ] = ( τ + τ ) /
2, a perfectly reasonable result. And I was told thistime does not exist. B : It does not. Change your settings, and the same procedure will give you Re[ τ ] <
0. Thetime 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 changingthe setup slowly enough for the electron “to see” its conditions “frozen” during its journeyto the detector. For your τ Alice to serve as a classical time scale you would also need toshow that τ n α ( τ ) + τ n α ( τ ) = τ n , n = 1 , , ... . However, this happens only if one of the α svanishes, in which case either τ or τ 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 thewave packet do not overlap at x . Then, at a given t , you will know which way the electronhas travelled, and also the duration, τ or τ it has spent in the magnetic field. But then, ofcourse, it would be a different experiment. A : And what if I take instead the imaginary part, or the modulus of τ , as was suggested,for example by B¨uttiker [11]? B : Or any real valued combination of Re[ τ ] and Im[ τ ]. You will still encounter “times”which are too long for common sense or too short for Einstein’s relativity, although with τ Alice = | τ | 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 [13]. 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 [14]. B : It is just what Eq.(3) says, τ is a sum of relative probability amplitudes for reachingthe detector via different arms, multiplied by the corresponding durations spent in the field,thus, also an amplitude. And so is every other “weak value” [15]. Your time is just the realpart 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’stranslational 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 valuedcombinations of the system’s probability amplitudes. A : And what is then an amplitude? B : According to Feynman [3], 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 intophysics 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” [3].The case of Ref.[9] 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 (cid:105) , can reachthe final position x , and the corresponding amplitude has two components, A ( x ← G ) = G ( x, t ) + G ( x, t ) ≡ (6) A ( x ← G | τ ) + A ( x ← G | τ ) . For a quantum particle crossing a potential barrier, there are many possible τ s, and manycomponents to the transition amplitude [16], A ( x ← G ) = (cid:90) T total A ( x ← G | τ ) dτ. (7)The mean angle of spin’s rotation in a small magnetic field, confined to the barrier, is givenby an analogue of (3) ϕ xy ≈ ω L Re (cid:34) (cid:82) T total τ A ( x ← G | τ ) dτ (cid:82) T total A ( x ← G | τ ) dτ (cid:35) (8)= ω L Re (cid:20)(cid:90) T total τ α ( τ ) dτ (cid:21) ≡ ω L Re [ τ ] . In the classical limit, highly oscillatory A ( x ← G | τ ) develops a stationary region aroundthe classical duration τ class , where it varies more slowly. This is the only region contributingto the integral in (8), and one recovers the classical result, τ = τ class . But this well definedduration disappears already if A ( x ← G | τ ) 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 ← G | τ ) in Eq.(6)has no stationary regions, and rapidly oscillates throughout the allowed range 0 ≤ τ ≤ T total .The tunnelling amplitude (6) is extremely small for a tall or a wide barrier (see the inset inFig.1). This happens not because A ( x ← G | τ ) is itself small, but because its oscillationscancel each other almost exactly. The delicate balance is easily perturbed, and an attempt todestroy interference between different durations would also destroy the tunnelling transitionone wanted to study. III. DISCUSSION
Finally, if Alice were to repeat also the experiment of Ref.[9], this is what Bob would sayabout her result. “A fundamental problem, arising each time a Larmor clock is applied totunnelling, but often overlooked - the proverbial elephant in the room - has to do with thequantum Uncertainty Principle. According to the Principle, one can have tunnelling, andnot 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 decidebetween knowing the slit chosen by the particle, or having the interference pattern on thescreen, but not both at the same time. You have tried to keep tunnelling intact (your clockperturbs it only slightly), and learn something about the duration spent in the barrier. Youmight expect the UP to make your result always look flawed in one way or another, but thisis not how the UP works. If you consider all possible experiments of this type, some of themwill 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 thecombinations of probability amplitudes which, unlike probabilities, have few restrictions ontheir signs and magnitudes. Though your result of 0 . ms does look plausible you cannotrecommend using it the way you would use a classical time scale just because of this. Afterall, 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 applieseverywhere in equal measure. You cannot say that you resolved the controversy regardinghow long a tunnelling particle spends in the barrier region, or proved that this duration isnon-zero. The controversy, if you wish to call it that, goes to the very heart of the quantumtheory, and must be accepted, rather than resolved.” IV. METHODSA. Probability amplitude to spend a given duration τ in the barrier Consider a particle with a mean momentum p , prepared in a wave packet state G ( (cid:126) = 1), G ( x ) = (cid:90) a ( p − p ) exp( ipx ) dp = exp( ip x ) W ( x ) , (9)where a ( p − p ) discribes the distribution of the particle’s momenta, and W ( x ) is the wavepacket’s envelope. At t = 0 the wave packet lies to the left of a potential barrier V ( x ) of awidth d , as shown in the inset in Fig.1. All momenta p in (9) are such that in order to crossthe 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, A ( x ← G ) = (cid:90) dx (cid:48) (cid:88) paths exp( iS [ x ( t )] G ( x (cid:48) ) , (10)where a path x ( t ) starts in x (cid:48) at t = 0, and ends in x at t = T total . The action functionalis given by the usual S [ x ( t )] = (cid:82) T total [ µ ˙ x / − V ( x )] dt , with µ denoting the particle’s mass.Each path spends a certain amount of time in the barrier region 0 ≤ x ≤ d . Thus durationcan be computed with the help of a “stop-watch” (SW) expression, τ SW [ x ( t )] = (cid:90) T total θ [0 ,d ] ( x ( t )) dt, (11)0where θ [0 ,d ] = 1 for 0 ≤ x ≤ d and 0 otherwise, so that only the time intervals spent in thebarrier are added to the total. It is readily seen that τ SW [ x ( t )] cannot be negative, nor canexceed the time the particle was in motion, hence0 ≤ τ SW [ x ( t )] ≤ T total . (12)A simple cosmetic operation turns the path sum (10) into the sum over durations spent inthe barrier. Restricting the summation to the paths which spend there precisely τ seconds,yields A ( x ← G | τ ) ≡ (cid:90) dx (cid:48) (cid:88) paths δ ( τ SW [ x ( t )] − τ ) exp( iS [ x ( t )] G ( x (cid:48) ) , (13)where δ ( z ) is the Dirac delta, and we have A ( x ← G ) ≡ (cid:90) T total A ( x ← G | τ ) dτ. (14)This is bad news for one’s effort to determine the time actually spent in the potential - allsuch durations interfere. We are back to the Young’s interference experiment, except thatinstead of two paths, each going through one of the slits, we have a continuum of routes, eachlabelled by the value of the τ SW [ x ( t )]. According to the Uncertainty Principle [3] the “whichway?” ( “which τ ?”) question has no answer. The only exception is the classical limit.Typically, A ( x ← G | τ ) is highly oscillatory, but in a classically allowed case, e.g., withthe barrier removed, the oscillations are slowed down near the classical value τ cl = µd/p .If A ( x ← G | τ ) has a unique stationary phase point of this kind, τ cl will appear as theonly time parameter, whenever one evaluates integrals involving A ( x ← G | τ ), and classicalmechanics will apply as a result.The problem with tunnelling is that no such preferred time emerges for a classically forbiddentransition, and all τ s must be treated equally (a similar situation is shown in Fig.3 of [6],although for a different quantity). To make things worse, in tunnelling the amplitude A ( x ← G ) is very small ( ∼ exp[ − (2 µV − p ) / d ] for a rectangular barrier), while A ( x ← G | τ )is not. Thus, the exponentially small tunnelling amplitude results from a highly accuratecancellation between (not small) oscillations of A ( x ← G | τ ). For this reason, any attemptto modify or neglect any part of the integrand in Eq.(13) would considerably change theresult, and destroy the tunnelling.1 B. An uncertainty relation for the duration τ Although the Uncertainty Principle hampers one’s attempts to ascribe a unique barrierduration to a tunnelling transition, there is still one more thing we can do. Writing the δ -function in (13) as δ ( τ SW [ x ( t )] − τ ) = (2 π ) − (cid:90) dλ exp { iλ ( τ − τ SW [ x ( t )]) } , (15)and inserting it into (13), we note that the new action S λ [ x ( t )] ≡ S [ x ( t )] − λ (cid:90) T total dtθ [0 ,d ] ( x ( t )) (16)corresponds to adding to the barrier V ( x ) a rectangular potential λθ [0 ,d ] ( x ( t )), a well or abarrier, depending on the sign of λ . Equation (13) can now be written in an equivalent form, A ( x ← G | τ ) = (2 π ) − (cid:90) ∞−∞ dλ exp( iλτ ) ˜ A ( x ← G | λ ) , (17)where ˜ A ( x ← G | λ ) is the amplitude to reach, at t = T total , the final location x from theinitial state G , while moving in a combined potential V ( x ) + λθ [0 ,d ] ( x ). In other words, toevaluate the amplitude A ( x ← G | τ ) one needs to know the amplitudes of transmission forall composite potentials. And vice versa , to know the amplitude for a given potential oneneeds to know the amplitudes for all durations spent therein.Note next that even the calculation of the full amplitude distribution of the durations spentin a region [0 , d ] for a free particle, V ( x ) = 0, is already a non-trivial task. It involves eval-uation of the transmission amplitudes for all rectangular wells and barriers, and integrationin Eq.(17). However, once A ( x ← G | τ ) is obtained, the distribution for a rectangularpotential V ( x ) = V θ [0 ,d ] ( x ) comes for free, A ( x ← G | τ ) = exp( − iV τ ) A ( x ← G | τ ) . (18)As we mentioned above, in the semiclassical limit, the free amplitude distribution A ( x ← G | τ ) develops a stationary region around τ = µd/p . When the barrier is raised, the factorexp( − iV τ ) destroys the stationary region, A ( x ← G | τ ) rapidly oscillates everywhere, and A ( x ← G ) becomes small for a tunnelling particle.Equation (17) is a kind of uncertainty relation between the duration τ and the potential inthe region of interest. It implies that a device employed to measure the τ must introducesome uncertainty into the potential, the greater the uncertainty, the more accurate themeasurement. Which brings us to the Larmor clock.2 C. 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 ˆ H int = ω L ˆ j z , where ω L is theLarmor frequency. By the time t , an initial state | γ (cid:105) = j (cid:88) m = − j γ m | m (cid:105) , ˆ j z | m (cid:105) = m | m (cid:105) (19)becomes rotated by an angle ω L t around the z -axis, | γ ( t ) (cid:105) = exp( − iω L t ˆ j z ) | γ (0) (cid:105) = j (cid:88) m = − j γ m exp( − imω L t ) | m (cid:105) . (20)Suppose the spin travels with a classical particle moving along a trajectory x ( t ), and thefield exists only in the region 0 ≤ x ≤ d . Then the spin, precessing only when the particleis in the field, 0 ≤ x ( t ) ≤ d , ends up rotated by ω L τ SW [ x ( t )] by t = T total . Quantally, fora particle in the inset of Fig.1, the final (unnormalised) spin’s state can be found simplyby adding up its states, rotated by ω L τ , each multiplied by the probability amplitude ofspending in the field a net duration τ . The result is | γ ( T total ) (cid:105) = (cid:90) T total dτ A ( x ← G | τ ) exp( − iω L τ ˆ j z ) | γ (0) (cid:105) . (21)In general, the r.h.s. of (21) cannot be rewritten as a single rotation around the z -axis byan angle ω L τ (cid:48) , | γ ( T total ) (cid:105) (cid:54) = exp( − iω L τ (cid:48) ˆ j z ) | γ (0) (cid:105) , and no unique time τ (cid:48) can be associatedwith a quantum transition in this way.With the help of Eq.(17), one obtains an equivalent form of Eq.(21), | γ ( T total ) (cid:105) = j (cid:88) m = − j ˜ A ( x ← G | mω L ) γ m | m (cid:105) . (22)This shows that each spin component traverses the barrier as if the potential there were V ( x ) + mω L , so the potential, experienced by the particle as a whole, remains uncertainwithin the range from − jω L to jω L . As was already noted, a viable clock has to introducethis uncertainty, and we may ask what can be learnt about the duration spent in the barrierby applying the Larmor clock.An experiment could consist in detecting, at t = T total , the particle in x and its spin in astate | β (cid:105) = (cid:80) jm = − j β m | m (cid:105) . From (21) the corresponding probability is P ( β, x ← γ, G ) = | (cid:90) T total dτ Γ( τ | ω L , β, γ ) A ( x ← G | τ ) | , (23)3where Γ( τ | ω L , β, γ ) ≡ (cid:104) β | exp( − iω L τ ˆ j z ) | γ (cid:105) = j (cid:88) m = − j β ∗ m γ m exp( − imω L τ ) . (24)Thus, by measuring the probability (23), one can determine the absolute value of the in-tegral in Eq.(23), which involves the amplitude distribution of the durations spent by theparticle inside the barrier in the absence of the clock. Note that little is left of the originaltunnelling transition, where the transmission amplitude A ( x ← G ) is typically small. Asalready mentioned at the end of subsection A, the presence of an additional factor such asΓ( τ | ω L , β, γ ) is likely to alter destructive interference which defines tunnelling. As a result, (cid:82) T total dτ Γ( τ | ω L , β, γ ) A ( x ← G | τ ) could differ from the original tunnelling amplitude inEq.(14) by orders of magnitude. D. A non-perturbing (weak) Larmor clock
One can try to return to tunnelling by sending ω L →
0, and learn something aboutthe tunnelling time from the particle’s response to the clock. (This already bodes ill forone’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 τ ). Nevertheless, we obtainΓ( τ | ω L , β, γ ) ≈ (cid:104) β | γ (cid:105) − iω L τ (cid:104) β | ˆ j z | γ (cid:105) , (25)so that the relative change in the probability (23) with and without the magnetic field is P ( β, x ← γ, G ) ω L − P ( β, x ← γ, G ) ω L =0 P ( β, x ← γ, G ) ω L =0 ≈ Z ( β, γ )]Im[ τ ] + Im[ Z ( β, γ )]Re[ τ ] , (26)where Z ( β, γ ) ≡ (cid:104) β | ˆ j z | γ (cid:105) / (cid:104) β | γ (cid:105) and τ ≡ (cid:82) T total τ A ( x ← G | τ ) dτ (cid:82) T total A ( x ← G | τ ) dτ = (cid:82) T total τ A ( x ← G | τ ) dτA ( x ← G ) ≡ (cid:90) T total τ α ( τ ) dτ (27)is the complex time of Sokolovski and Baskin [12]. The quantity in the l.h.s. of Eq.(27) canbe measured, and by choosing a different | β (cid:105) one can, in principle, determine the values ofRe[ τ ], Im[ τ ], or indeed of their various combinations. Moreover, for (cid:104) β | γ (cid:105) = 0, one has P ( β, x ← γ, G ) ω L ∼ ω L | τ | , (28)so the modulus of τ can also be determined directly.Now there are many real valued time parameters related to the complex time (27), yet none4of them is a suitable candidate for a physical time interval representing the net durationspent in the barrier. The easiest way to demonstrate it is to note that for an improbabletransition, A ( x ← G | τ ) →
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 ← G | τ ) by τ candestroy the cancellation, characteristic of tunnelling. Thus, | τ | may, in principle, exceed thetotal duration of motion, | τ | >> T total . This makes little sense, especially if one recalls thateach and every Feynman path in Eq.(10) spends in the barrier no more than T total . E. The Baz’ clock
Finally we briefly discuss a particular type of a weak Larmor clock, employing a spin-1 / et al in [9]. Now ˆ j z = σ z / σ z is the Pauli matrix), andthe spin’s initial direction is along the x -axis, whose azimuthal and polar angles are φ = 0and θ = π/ | γ ( T total ) (cid:105) = 2 − / (cid:82) T total dτ A ( x ← G | τ ) exp( − iω L τ / (cid:82) T total dτ A ( x ← G | τ ) exp(+ iω L τ / (29) ≈ − / A ( x ← G | τ ) − iω L Re[ τ ] / ω L Im[ τ ] /
21 + iω L Re[ τ ] / − ω L Im[ τ ] / . As it was discussed in subsection C, this cannot in general correspond to a rotation aroundthe z -axis. On the other hand, in any state, a spin-1 / xy -, butalso in the xz -plane. The state of a spin, polarised along a direction making angles δφ and π/ − δθ with the x - and the z -axis, respectively, can be written as exp( − iδφ/
2) cos (cid:0) π − δθ (cid:1) exp(+ iδφ/
2) sin (cid:0) π − δθ (cid:1) ≈ − / − iδφ/ δθ/
21 + iδφ/ − δθ/ . (30)Comparing (29) with (30) we find that the spin has rotated by the (small) angles δφ = ω L Re[ τ ] , (in the xy -plane) and δθ = ω L Im[ τ ] (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 ω L τ class = τ SW [ x class ( t )]. Thus, the first of5Eqs.(31) looks like the classical result, with τ class replaced by Re[ τ ]. The second of Eqs.(31)has no classical analogue, and should serve as a warning that a straightforward extension ofthe classical duration to the quantum case may not be possible. (One already knows thisfrom the Uncertainty Principle.)The appearance of not one, but two rotation angles was first noted by B¨uttiker in [11],albeit in a slightly different context. [Ref. [11] considered transmission of a particle witha known momentum p which, in our language, corresponds to replacing A ( x ← G | τ )with A ( p ← G | τ ) ≡ (cid:82) exp( − ip x ) A ( x ← G | τ ) dx in all formulae, and making G nearlymonochromatic.] In [11] B¨uttiker defined two “times”, τ y ≡ δφ/ω L and τ z ≡ δθ/ω L , whichcorrespond to our R [ τ ] and Im[ τ ], respectively. Ramos et al measured both the real andthe imaginary parts of τ , which can be seen in Fig.3 of [9]. The authors of [9] found bothparameters positive and concluded that their results were “inconsistent with claims thattunnelling takes ’zero time’”. To abide by this conclusion one needs to take for granted thatthe ”time tunnelling takes” exists as a meaningful concept, but this is not the case.The confusion can be traced back to B¨uttiker [11]. When faced with two times parametersinstead of one, he opted for a non-negative combination of the two, τ x ≡ (cid:112) τ y + τ z . Thisequals the modulus of the “complex time” in Eq.(27), τ x = | τ | . At least one point made in[11] requires a comment, if not a correction. In τ x B¨uttiker believed to have found (we readin the Abstract of [11]) “the time interval during which a particle interacts with the barrierif it is finally transmitted.” However, neither Re[ τ ] nor Im[ τ ], nor any combination of thetwo can be interpreted as a physical time interval. A weighted sum of quantum mechanicalamplitudes, τ , may not give a meaningful answer to the question “how much time does atunnelling particle spend within the barrier region?” for the same reason the UncertaintyPrinciple [3] forbids identifying the particle’s path in Young’s double-slit experiment. [1] L.A. MacColl, Note on the transmission and reflection of wave packets by potential barriers ,Phys. Rev. , 621 (1932).[2] U. Satya Sainadh, R.T. Sang, I.V. Litvinyuk, Attoclock and the quest for tunnelling time instrong-field physics , J. Phys. Photonics. , 042002 (2020). [3] R.P. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics III (Dover Publi-cations, Inc., New York, 1989), Ch.1: Quantum Behavior.[4] Wigner, E.P. Lower limit for the energy derivative of the scattering phase shift.
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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 ac-knowledged. EA acknowledges the financial support of the Ministerio de Econom´ıa y7Competitividad (MINECO) of the Spanish Government through BCAM Severo Ochoaaccreditation SEV-2017-0718 and PID2019-104927GB-C22 grant. This work was also sup-ported by the BERC 2018e2021 Program and ELKARTEK Programme (KK-2020/00049and 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.