The Salecker-Wigner-Peres clock, Feynman paths, and a tunnelling time that should not exist
TThe Salecker-Wigner-Peres clock, Feynman paths, and a tunnelling time that shouldnot exist
D. Sokolovski a,b a Departmento de Qu´ımica-F´ısica, Universidad del Pa´ıs Vasco, UPV/EHU, Leioa, Spain and b IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Spain
The Salecker-Wigner-Peres (SWP) clock is often used to determine the duration a quantum particleis supposed to spend is a specified region of space Ω. By construction, the result is a real positivenumber, and the method seems to avoid the difficulty of introducing complex time parameters, whicharises in the Feynman paths approach. However, it tells little about about the particle’s motion. Weinvestigate this matter further, and show that the SWP clock, like any other Larmor clock, correlatesthe rotation of its angular momentum with the durations, τ , which the Feynman paths spend inΩ, thereby destroying interference between different durations. An inaccurate weakly coupled clockleaves the interference almost intact, and the need to resolve the resulting ”which way?” problemis one of the main difficulties at the centre of the ”tunnelling time” controversy. In the absenceof a probability distribution for the values of τ , the SWP results are expressed in terms of moduliof the ”complex times”, given by the weighted sums of the corresponding probability amplitudes.It is shown that over-interpretation of these results, by treating the SWP times as physical timeintervals, leads to paradoxes and should be avoided. We also analyse various settings of the SWPclock, different calibration procedures, and the relation between the SWP results and the quantumdwell time. The cases of stationary tunnelling and tunnel ionisation are considered in some detail.Although our detailed analysis addresses only one particular definition of the duration of a tunnellingprocess, it also points towards the impossibility of uniting various time parameters, which may occurin quantum theory, within the concept of a single ”tunnelling time”. PACS numbers: 03.65.Xp,73.40.Gk
I. INTRODUCTION
Recent progress in attosecond science [1] has returnedto prominence the nearly hundred years old [2] question”how long does it take for a particle to tunnel?”. Thereare serious disagreements, e.g., between the authors of[3], who claimed that ”optical tunnelling is instanta-neous”, and the conclusions of [4] suggesting that ”theelectron spends a non-vanishing time under the potentialbarrier”. An overview of the ”tunnelling time problem”,in its relation to attosecond physics, can be found, forexample, in [5].The tunnelling time problem was extensively investigatedin the last decade of the previous century, mostly in thecontext of tunnelling across stationary potential barriers,closely related to the then-fashionable subject of carriertransport in heterostructures (for review see [6]- [10]).The problem also has a more fundamental aspect. An of-ten cited difficulty in defining a tunnelling time is the ab-sence of the corresponding hermitian operator, an the im-possibility of performing a standard von Neumann mea-surement [11] in order to determine it. However, this isnot a major obstacle, since the von Neumann procedurecan be extended to measuring quantities represented bycertain types of functionals on the Feynman paths of themeasured system [12]- [15], by making the meter monitorthe system over an extended period of time.One time parameter, represented by such a functional, isthe net time a quantum particle spends in the specified region of space. It is intimately related to Larmor preces-sion, and following Buettiker [16] we will refer to it as the traversal time . With the functional specified, the prob-lem becomes one in quantum measurement theory. It wasstudied in some depth in [17]-[25]. The main conclusionof these studies, which we maintain to date, is as follows.Traversal time can be measured by an extended von Neu-mann procedure, and the relevant meter is a variant ofa Larmor clock, a spin, whose angle of rotation corre-lates with the duration spent in the magnetic field [20].However, a quantum measurement is significantly morecomplicated than its classical counterpart, largely due tothe trade-off between its accuracy, and the perturbationthe measurement produces. Larmor clocks with spins ofdifferent sizes, observed in different states and subjectedto different magnetic fields, will all produce different re-sults. These results, although perfectly tractable, lackthe universality of the classical traversal time. To put itdifferently, analysis of the quantum traversal time prob-lem is worthy from the general point of view, but itsresult is bound to disappoint a practitioner wishing toknow only ”how many seconds does it take to tunnel, af-ter all?”.Among many possible versions of the Larmor clock [16],[26]-[32], one stands out, and has been the subject ofmany recent and not so recent studies [33]-[44]. TheSalecker-Wigner-Peres (SWP) clock was first consideredas a quantum tool for measuring space-time distancesin the general relativity [33], and was later adopted byPeres [34] for timing events in non-relativistic quantum a r X i v : . [ qu a n t - ph ] A ug mechanics. Specifications of the SWP clock include thechoice of its initial and final states, the size of the spin,the strength of the field, and the particular way in whichthe result of the measurement is calculated. The re-sulting time can represented as the average value of the”clock time” operator and is, by construction, a real pos-itive number. The SWP result is often taken to be thedefinition of the time a particle spends in the magneticfield contained in the region of interest. One reason whythe analysis must not stop there is because such a resulttells little about the particle’s motion. Timing a classicalparticle by means of a classical stopwatch, and getting aresult of one second, implies that the particle has actu-ally spent one second in the region Ω, plus all practicalconsequences one can draw from this information. Theimplications of measuring one second with a quantumclock remain unclear, until one considers its precise rela-tion to the particle’s Feynman paths.Like every Larmor clock, the SWP clock modifies the con-tributions the Feynman paths make to a transition am-plitude, depending on the final state in which the clockis found. As one would expect, a nearly classical clock,equipped with a very large spin or angular momentum,destroys the interference between the paths spending dif-ferent durations in Ω almost completely. In this case,having found the initial state of the clock rotated by anangle φ , one can be certain that the particle did spendin Ω φ/ω L seconds, where ω L stands for the Larmor fre-quency [20]. Choosing a weaker field, or a smaller angularmomentum, would leave certain amount of the interfer-ence intact, and reduce the accuracy of the measurement.Even so, by varying the accuracy, one can probe certainaspects of the particle’s motion. For example, in the caseof resonance tunnelling across a double barrier, a mea-surement of a medium accuracy allows one to identifythe long delays associated with the exponential decayof the barrier’s metastable state [see Fig.8 of [21]]. Im-proving the accuracy, one finds the evidence of the parti-cle ”bouncing” between the potential walls [see Fig.9 of[21]]. Both the decay and the ”bounces” are often associ-ated with the particle being trapped in a metastable well.Both can be observed, but not at the same time [21]. Oneproblem with quantum time measurements is that the re-strictions on the Feynman paths, imposed by the clock,tend to perturb the transition the particle is supposed tomake. Thus, if an accurate clock is employed, the parti-cle may either not reach its final state at all, or be seen tospend no time in Ω [21]. Similarly, resonance tunnelling,even in the presence of a relatively inaccurate Larmorclock, will not be the same, as without it. Yet whenone asks ”how long it takes to tunnel?”, he/she usuallymeans ”unperturbed”. This is a well known difficulty inquantum mechanics, where ”to know” often implies ”todisturb”.A natural way to avoid the unwelcome perturbation is toreduce the coupling between the clock and the system,and try to interpret whatever information can be gainedin this manner. The purpose of this paper is to anal- yse the results obtained by an SWP clock in the limit ω L →
0, and relate them to the time parameters describ-ing the motion of a quantum particle, involved in a tran-sition between known initial and final states. This bringsthe discussion into the realm of the inaccurate, ”indirect”[20], or ”weak” [45] measurements of the traversal time.In the ”weak” regime, we can expect a weak SWP clockto make a rather poor job of destroying interference be-tween different values of the traversal time. We will alsoneed to heed D. Bohm’s warning [46] that ”if the interfer-ence were not destroyed”, ”the quantum theory could beshown to lead to absurd results”, and see what it meansfor the quest to find ”the tunnelling time”.The rest of the paper is organised as follows.In Section II we discuss various time parameters describ-ing the motion of a classical particle.Section III lists some of the quantum time parameterswhich are not discussed in this paper.In Sect. IV we define the quantum traversal time, andits amplitude distribution, for a particle pre- and post-selected in the known initial and final states.In Sect. V we introduce the ”complex times”, whichare likely to arise in any weakly perturbing measurementscheme.In Sect. VI we cast the complex times into a more famil-iar operator form.In Sect. VII we describe the family of Larmor clocks,and their relation to the amplitude distribution of thequantum traversal time.In Sect. VIII we introduce the SWP clock as a particularmember of the family.In Sect. IX we reduce the coupling, and show that thetime measured by a weakly coupled SWP clock is natu-rally expressed in terms the moduli of the complex timeof Sect. V.In Sect. X we study the calibration procedure proposedin [37], and demonstrate that it can lead to ”absurd re-sults” predicted by Bohm.In Sect. XI we try to make sense of these ”absurd re-sults”, and establish a connection between the complextimes and the weak values of quantum measurement the-ory.In Sect. XII we revisit the dwell time and show it to bea particular case of the ”complex times” of Sect. V.In Sect. XIII we ask whether the Peres’ clock would mea-sure the dwell time, and find that it would not.In Sect. XIV we apply our general analysis to tunnellingacross a stationary potential barrier.In Sect. XV we apply the analysis to a simple model oftunnel ionisation.Section XVI contains our conclusions.
II. WHICH CLASSICAL TIME?
We start by reiterating the three questions which, inour opinion, one might want to answer before performinga quantum measurement. These are:(i) What is being measured?(ii) By what means is it being measured?(iii) To what accuracy is it being measured?The first question arises already in classical mechanics,when we discuss the time parameters describing the pres-ence of a particle, moving along a trajectory x cl ( t ), in aspecified region of space, Ω. One obvious choice is thenet duration the particle spends in Ω. It is given by theintegral [17] τ Ω [ x cl ( t )] = (cid:90) t t Θ Ω ( x cl ( t )) dt, (1)where Θ Ω ( z ) has the value of one for a z inside Ω, andzero otherwise. Another choice would be, for example,the time interval between the moments t in , when theparticle enters Ω for the first time, and t out , when itleaves it for the last time, τ in/out [ x cl ( t )] = t out [ x cl ( t )] − t in [ x cl ( t )] . (2)The choice depends on the question we want to ask.If the particle has a tendency to change colour fromwhite to black proportionally to the net duration spentin Ω, to predict the shade of grey acquired we need τ Ω [ x cl ( t )]. If, on the other hand the temperature in Ωchanges with a frequency ω , and we need the particleto experience no change, the required condition wouldbe ωτ in/out [ x cl ( t )] <<
1, and not ωτ Ω [ x cl ( t )] <<
1. Ingeneral, these two parameters are different. τ Ω [ x cl ( t )] (cid:54) = τ in/out [ x cl ( t )] . (3)Even classically, different time parameters require differ-ent measurement procedures. To measure τ Ω [ x cl ( t )], wecan equip the particle with a magnetic moment whichprecesses in the magnetic field introduced in Ω. Dividingthe final angle of precession by the Larmor frequency, ω L , we obtain the value of τ Ω [ x cl ( t )]. It appears thatno similar procedure exists for τ in/out [ x cl ( t )], or even for t in [ x ( t )] in Eq.(2) [14]. The difficulty is in stopping theclock after the first entry in Ω, and preventing it fromrunning again should the path leave the region and thenre-enter. In classical mechanics we can simply plot theparticle’s trajectory, and determine t in [ x cl ( t )] from thegraph. In the quantum case, there is no trajectory todraw, and the absence of a meter is a serious problem[14].The accuracy of a measurement is of no great importancein the classical case, where a meter (a clock) can monitora particle with any precision, without altering its tra-jectory x cl ( t ). It plays a much more important role inthe quantum case, where there is a tradeoff between theaccuracy of the measurement, and the perturbation themeter exerts on the particle’s motion.Throughout the rest of the paper we will try to an-swer the following question: What is the total amountof time a quantum particle starting in a known state | ψ I (cid:105) at t = t , and then observed on a state | ψ F (cid:105) at t = t , had spent in a specified region Ω between t and t ? . Tomeasure it we will employ a highly inaccurate Salecker-Wigner-Peres clock, specifically designed to perturb thestudied quantum transition as little as possible. The ex-periment we have in mind is like this. A particle is pre-pared in | ψ I (cid:105) , coupled to an SWP clock, and then de-tected in | ψ F (cid:105) . If the detection is successful, we ”read”the clock in some manner, record the result, and drawconclusions about the duration spent in Ω. Although weconsider one-dimensional scattering, most of our resultscan be extended to two or three dimensions. III. OTHER QUANTUM TIMES BEYOND THESCOPE OF THIS PAPER
In quantum mechanics there are many different waysto introduce quantities measured in units of time. Beforeproceeding with our main task, we briefly discuss someof the time parameters, which describe a scattering (tun-nelling) process and are not a subject of of this paper.The simplest way to probe the tunnelling delay is to pre-pare a particle in a wave packet state on one side of thebarrier, choose a location x on its other side, and eval-uate the probability P ( x, t ) = | ψ ( x, t ) | . Using P ( x, t )as a probability distribution, one can construct the realnon-negative mean time [47], also known as the ”time ofpresence” [48] (cid:104) t ( x ) (cid:105) = (cid:90) tP ( t, x ) dt/ (cid:90) P ( x, t ) dt. (4)This mean time can be measured by performing N >> x and x + dx at a time t . If in N cases the particle isfound there, the ratio N /N dx would yield an approx-imate value of P ( x, t ). Repeating the checks at vari-ous times, allows one to reconstruct P ( x, t ) and, withit, (cid:104) t ( x ) (cid:105) . This is, however, different from what we in-tend to do here, as explained in Sect. 18 of [49].A slightly different method was recently proposed by Pol-lack in Refs. [50], [51]. There the particle is prepared ina thermal mixed stateˆ ρ I = exp( − β ˆ H/ | x (cid:105)(cid:104) x | exp( − β ˆ H/ , (5)where x is some initial location, ˆ H is the Hamiltonian,and β is the inverse temperature. The state is evolveduntil some t , ˆ ρ ( t ) = exp( i ˆ Ht )ˆ ρ I exp( − i ˆ Ht ), and the prob-ability to find the particle at a location x on the otherside of the barrier, P ( x, t ) = tr {| x (cid:105)(cid:104) x | ˆ ρ ( t ) } , is insertedinto Eq.(4) for the mean transit time. The mean timecan then be measured as discussed above. This is alsonot what we wish to discuss below, if only because herewe are not interested in systems in thermal equilibrium.Finally, the authors of [4] proposed using the probabilitycurrent evaluated at two locations on the opposite sidesof the barrier, x and x , and define the mean transittime as the difference between the moments the out- andin-going probability currents at x and x reach theirmaxima. Measuring, albeit indirectly, this time wouldrequire a different experiment, e.g., the one in which thepresence of the particle is checked at all times to theright of x , and then at x , the evaluated probabilitiesare differentiated with respect to time to yield the cur-rents, and the maxima of the two curves are identified.This procedure is not our subject either.The list of possible quantum time parameters can beextended, and new times will, undoubtably, be proposedin future studies. It is not our intention to compare rela-tive merits or defects of the approaches discussed in thisSection. (Except, perhaps, citing some of well knownproblems with defining quantum arrival times [48], or re-lying on the probability current in order to determinetimes, or time intervals [52]). Rather, we note that mea-surements of different quantum times require differentexperimental procedures, and should not be expected togive the same result. To some extent this is true alreadyin classical mechanics, as was pointed out in the pre-vious Section. Thus, A may propose, and perform, anexperiment in which a time parameter associated with atunnelling transition vanishes, and claim tunnelling to bean ”infinitely fast” process. B can do something differ-ent, obtain a non-zero answer, and state ”that tunnellingdoes take time after all”. The argument between A andB will never have a meaningful resolution, since bothclaims rely on the assumption that there is a single timetunnelling ”takes”, and there is overwhelming evidencethat this assumption is false. In this paper, to add tothis evidence, we consider a particular classical time (1),and see what will happen if it is generalised to the fullquantum case. IV. TRAVERSAL TIME FOR QUANTUMMOTION
A classical particle of a mass µ in a potential V ( x, t )goes from some initial position x I at t = t to a final po-sition x F at t = t along a smooth continuous trajectory x cl ( t ). There is a single value of the duration spent in Ω,and it is given by the functional τ Ω [ x ( t )] in Eq.(1).The quantum case is more complex. A quantum parti-cle can make a transition from an initial state | ψ I (cid:105) at t = t to a final state | ψ F (cid:105) at t = t . To proceed, weneed to choose a representation. Since we are interestedin a spacial region Ω, the coordinate representation is theappropriate one. Now a point particle can be thought ofas being at some location x ( t ) at any time t ≤ t ≤ t ,and a possible scenario for reaching ψ F from ψ I is byfollowing a Feynman path x ( t ), which is continuous, butnot smooth [54]. The path is virtual, and is equippedonly with a probability amplitude (we use ¯ h = 1) A ([ x ( t )] , ψ I , ψ F ) = (cid:104) ψ F | x F (cid:105) exp { iS [ x ( t )] }(cid:104) x I | ψ I (cid:105) , (6)where S [ x ( t )] = (cid:82) t t [ ˙ x / µ − V ( x, t )] dt is the classicalaction. The full transition amplitude to reach | ψ F (cid:105) from | ψ I (cid:105) is given by the Feynman path integral [54], whichwe symbolically write as A ( ψ F , ψ I , t , t ) = (cid:88) paths A ([ x ( t )] , ψ I , ψ F ) . (7)Note that the set of Feynman paths in Eq.(7) is alwaysthe same. What changes, with the change of the poten-tial in which a particle moves, are the path amplitudes A ([ x ( t )] , ψ I , ψ F ). The classical dynamics emerges fromEq.(7) when the contribution to the path integral comesfrom the vicinity of the path x cl ( t ) on which S [ x ( t )] isstationary [54].What can be said about the duration a particle spendsin Ω is dictated by the basic rules of quantum mechan-ics. The functional τ Ω [ x ( t )] can be evaluated for eachof the Feynman paths. The paths can be combined andrecombined into new pathways, just as the superpositionprinciple allows us to recombine vectors in Hilbert spaceinto a new vector [15]. Combining together all the pathswhich share the same value τ of τ Ω [ x ( t )], we create a newvirtual pathway, for reaching | ψ F (cid:105) from | ψ I (cid:105) , and spend-ing τ seconds in Ω along the way , and sacrifice to inter-ference all other information contained in the individualFeynman paths. The amplitude for the new pathway is A ( ψ F , ψ I , t , t | τ ) = (8) (cid:88) paths A ([ x ( t )] , ψ I , ψ F ) δ ( τ Ω [ x ( t )] − τ ) , where δ ( z ) is the Dirac delta. Integrating Eq.(8) overall possible τ ’s restores the full transition amplitude A ( ψ F , ψ I , t , t ) in Eq.(7).In addition, we have A ( ψ F , ψ I , t , t | τ ) ≡ , for τ < τ > t − t , since non-relativistic Feynman paths may not spend in Ωa duration which is either negative, or exceeds the totalduration of motion.The situation is a standard one in quantum mechanics.For given initial and final states of the particle, we havenot one, but infinitely many values of the traversal time τ . To each value we can ascribe a probability ampli-tude, but not the probability itself. This is not differentfrom what happens in Young’s two-slit experiment [20].The expectation that there must, after all, be a singletraversal time associated with a quantum transition, isas good, or as bad, as the assumption that each electronmust have actually gone through one slit or another. Ac-cording to Feynman [55], the latter assumption should beabandoned, and the rule for adding amplitudes must beaccepted as the basic axiom of quantum theory instead.Throughout the rest of the paper, we will maintain thispoint of view, despite possible objections from the pro-ponents of the Bohmian version of quantum theory [39],[53], [56].Thus, our virtual pathways, labelled by the value of τ ,interfere just like individual Feynman paths they com-prise, and should together be considered a single indivis-ible pathway connecting | ψ I (cid:105) and | ψ F (cid:105) [15]. Interferencebetween them can be destroyed by an accurate meter[20], registering the actual value of τ each time the tran-sition is observed, but the probability P acc ( ψ F , ψ I , t , t )to reach the final state, while being observed, will change, P acc ( ψ F , ψ I , t , t ) ≡ (cid:90) t − t dτ | A ( ψ F , ψ I , t , t | τ ) | (cid:54) = (10) | (cid:90) t − t dτ A ( ψ F , ψ I , t , t | τ ) | = | (cid:88) paths A ([ x ( t )] , ψ I , ψ F ) | . This simple discussion should help to establish the sta-tus of the time parameter represented by the functional(1) within the standard quantum theory. We note thesimilarity between the quantum traversal time problemand the Young’s double-slit experiment. If we were ableto construct a unique traversal time, or even a probabil-ity distribution for such times, we could, in principle, alsodetermine the slit through which the electron has passed,with the interference pattern on the screen intact. Ac-cording to Feynman [55], the latter is an impossible task.
V. THE COMPLEX TIMES
Even before going into the details of a particular mea-surement, we can guess what would happen if the meter’sinteraction with the particle has been deliberately madesmall (weak), in order to preserve the interference, andminimise the perturbation produced on the particle’s mo-tion. There is a fashionable view that the result would bea ”weak value”, a new type of quantum variable, capableof providing a new insight into physical reality. (For arecent review see [57], the term ”weak measurement ele-ments of reality” was coined in [58]).Recently we argued against over-interpretation of the”weak values”, and offered a more prosaic explanation[15], [59]. In the absence of probabilities, any weaklyperturbing scheme is bound to give a result, expressed interms of the probability amplitudes. A scheme set up toweakly measure a quantity would typically yield a real result expressed, in one way or another, in terms of thecomplex valued sums of the corresponding amplitudes,weighed by the values of the measured quantity and, oc-casionally, the amplitudes themselves [15], [59]. Far fromrepresenting a new type of ”reality”, these results onlygive us this limited information about the particular setof virtual pathways connecting the initial and final states.They would, for example, shed no new light on the mech-anism of the two-slit experiment, mentioned in the previ-ous Section, beyond what is known from the textbooks.In the case of the traversal time (1) one such weighted sum is the ”complex time” introduced in [17], τ Ω ( ψ I , ψ F ) = (11) (cid:88) paths τ Ω [ x ( t )] A ([ x ( t )] , ψ I , ψ F ) / (cid:88) paths A ([ x ( t )] , ψ I , ψ F )= (cid:90) t − t dτ τ A ( ψ F , ψ I , t , t | τ ) /A ( ψ F , ψ I , t , t ) . The quantities of these type, first introduced by Feyn-man [54] as ”transition elements of functionals”, reduceto the weak values of [57], if the functional in questionis the instantaneous value of a variable A ( t ) at a time t < t < t .The quantity in Eq.(11) was often dismissed as a can-didate for the duration quantum particle performing atransition (e.g., tunnelling transmission across a poten-tial barrier) spends in a specified region of space, on ac-count of it being complex valued. For example, in Ref.[6]we read: ”...common sense dictates that to the questionof the duration of a tunneling process, the answer, if itexists at all, must be a real time”. The key words hereare ”if it exists”, and in the previous Section we explainedin what sense the answer should not exist. As a conse-quence, only complex valued combinations of transitionamplitudes similar to (11) will be found in the analy-sis of a non-perturbing weakly coupled meter. It is truethat the result of a physical measurement must be real,but there is no contradiction. Different setups, employedto weakly measure the quantum traversal time (1), mayyield Re τ Ω , Im τ Ω or | τ Ω | , as demonstrated in the tablein [18].Finally, we can define expressions similar to (11) forhigher powers of the functional τ [ x ( t )], τ n Ω ( ψ I , ψ F , t , t ) ≡ (cid:82) t − t dτ τ n A ( ψ F , ψ I , t , t | τ ) A ( ψ F , ψ I , t , t ) , (12)where n = 2 , .... . We will require some of these quan-tities in what follows. In the following Sections we illus-trate what has been said so far, using the example of aweakly coupled SWP clock. VI. COMPLEX TIMES IN OPERATORNOTATIONS
It may be convenient to formulate the problem in termsof partial evolution operators acting in the particle’sHilbert space. Let ˆ U part [ x ( t )] = | x F (cid:105) exp { iS [ x ( t )] }(cid:104) x I | be an operator which evolves the (part)icle along a sin-gle Feynman path x ( t ), which starts in x I at t ,and endsin x F at t . Summing over all paths which spend exactly τ seconds in Ω, (summation over x I and x F included),we obtain an evolution operator conditioned by the re-quirement that the particle spend exactly τ seconds inthe region of interest,ˆ U part ( t , t | τ ) = (cid:88) paths ˆ U part [ x ( t )] δ ( τ [ x ( t )] − τ ) . (13)Summing Eq.(13) over all τ ’s restores the full evolutionoperator, ˆ U part ( t , t ), (cid:90) t − t ˆ U part ( t , t | τ ) dτ = (cid:88) paths ˆ U part [ x ( t )] (14)= ˆ U part ( t , t ) . Next we introduce an operator ˆ U part ( t , t | λ ) as a Fouriertransform of ˆ U part ( t , t | τ ),ˆ U part ( t , t | τ ) = (2 π ) − / (cid:90) exp( iλτ ) ˆ U part ( t , t | λ ) dλ, (15)and, with it, an operator family,ˆ U ( n ) part ( t , t ) = (cid:90) t − t τ n ˆ U part ( t , t | τ ) dτ (16)= ( i ) n ∂ nλ ˆ U part ( t , t | λ ) | λ =0 , n = 0 , , ..., where ˆ U (0) part ( t , t ) = ˆ U part ( t , t ).Now the complex ”averages” in Eqs. (11) and (12), canalso be written as τ n Ω ( ψ F , ψ I , t , t ) = (cid:104) ψ F | ψ ( n ) (cid:105)(cid:104) ψ F | ψ (0) (cid:105) , (17)where | ψ ( n ) ( t , t , ψ I ) (cid:105) ≡ ˆ U ( n ) part ( t , t ) | ψ I (cid:105) , (18)and τ Ω ( ψ I , ψ F ) = τ (1)Ω ( ψ I , ψ F ).The usefulness of this approach becomes more evidentas we realise that ˆ U part ( t , t | λ ) coincides with the evo-lution operator for a particle moving in the original po-tential V ( x, t ) plus an additional potential which equals λ inside Ω, and vanishes outside it, ˆ U part ( t , t | λ ) =exp[ − i (cid:82) t t ˆ H part ( t, λ ) dt ], where the particle’s Hamilto-nian, ˆ H part ( t, λ ), is given byˆ H part ( t, λ ) ≡ − ∂ x / µ + V ( x, t ) + λ Θ Ω ( x ) . (19)This result follows by noting that if one writes δ ( τ [ x ( t )] − τ ) as (2 π ) − (cid:82) exp { iλ ( τ − τ [ x ( t )]) } dλ and inserts it in(8), the action S [ x ( t )] in Eq.(6) is modified by the term − λ (cid:82) t t Θ Ω ( x ( t )) dt , which corresponds to adding an extrapotential λ Θ Ω ( x ). The operators ˆ U ( n ) part ( t , t ) can now beevaluated by expanding ˆ U part ( t , t | λ ) in powers of λ withthe help of the perturbation theory [54]. For example, forˆ U (1) part ( t , t ) we have ˆ U (1) part ( t , t ) = (20) (cid:90) t t dt (cid:48) (cid:90) Ω dx (cid:48) ˆ U part ( t , t (cid:48) ) | x (cid:48) (cid:105)(cid:104) x (cid:48) | ˆ U part ( t (cid:48) , t ) , so that τ Ω ( ψ I , ψ F ) = (cid:82) t t dt (cid:48) (cid:82) Ω dx (cid:48) ψ ∗ F ( t (cid:48) , x (cid:48) ) ψ I ( t (cid:48) , x (cid:48) ) (cid:104) ψ F | ˆ U part ( t , t ) | ψ I (cid:105) , (21) where ψ I ( t (cid:48) , x (cid:48) ) ≡ (cid:104) x (cid:48) | ˆ U part ( t (cid:48) , t ) | ψ I (cid:105) and ψ F ( t (cid:48) , x (cid:48) ) ≡(cid:104) x (cid:48) | ˆ U † part ( t , t (cid:48) ) | ψ F (cid:105) . With the help of Eq.(17) is easy toprove an identity (cid:104) ψ ( m ) | ψ ( n ) (cid:105) = τ n Ω ( ψ ( m ) , ψ I ) τ m Ω ∗ ( ψ (0) , ψ I ) , (22)which we will use in what follows.Finally, the Fourier transform (15), relatingˆ U part ( t , t | τ ) to ˆ U part ( t , t | λ ), suggests that τ and λ are, in some sense, ”conjugate variables”. They mustsatisfy an uncertainty relation [21], so that a narrowamplitude distribution of τ would imply a broad rangeof λ ’s, and vice versa. Thus, in order to know theduration τ spent in Ω, one must make the potential in Ωuncertain. Conversely, if the potential is sharply defined, τ cannot, in general, be known exactly. For example,the choice ˆ U part ( t , t | τ ) = ˆ U part ( t , t ) δ ( τ − τ ) makesin Eq.(15) the effective range of integration over λ infinite. Therefore an evolution for which τ is knownexactly, must be represented as a sum of evolutions forall possible potentials λ Θ Ω ( x ) added to V ( x, t ). It alsomeans that in order to measure τ , a meter would needto introduce at least some uncertainty in the potentialin the region of interest. This can be done by equippingthe particle with a magnetic moment, proportional to itsspin, or angular momentum, so that each component ofthe spin would experience a different potential inside Ω,where a constant magnetic field is introduced. There arevarious ways for preparing the spin degree of freedom,and we will discuss them next. VII. THE FAMILY OF LARMOR CLOCKS
Any spin-rotating (Larmor) quantum clock relies onthe fact that a magnetic moment, proportional to a spinof a size j , undergoes in a magnetic field Larmor pre-cession with an angular frequency ω L . Let the field bedirected along the z -axis, and | m (cid:105) denote the state inwhich the projection of the spin on the axis is m , so thatthe spin’s Hamiltonian is given by ˆ H spin = ω L ˆ j z , withˆ j z | m (cid:105) = m | m (cid:105) . Then, after a time t , an arbitrary (2 j +1)-component initial spin state | γ I (cid:105) = (cid:80) jm = − j γ Im | m (cid:105) , willend up rotated around the z -axis by an angle ω L t , | γ I (cid:105) →| γ ( t ) (cid:105) = (cid:80) jm = − j γ Im exp( − imω L t ) | m (cid:105) .If the magnetic field is introduced only in the region Ω,and the spin is travelling with a classical particle whichfollows a trajectory x cl ( t ) for t ≤ t ≤ t , the state ro-tates only while the particle remains inside Ω. The finalangle of rotation is ω L τ Ω [ x cl ( t )], and we have | γ ( t ) (cid:105) = exp {− iω L τ Ω [ x cl ( t )]ˆ j z }| γ I (cid:105) . (23)Generalisation to the case where the particle is quan-tum, rather than classical, is now straightforward. Atransition between | ψ I (cid:105) and | ψ F (cid:105) involves a range of du-rations, each occurring with the probability amplitude(8). Hence, the final state of the spin is a superpositionof all possible rotations weighted by the correspondingamplitudes, | γ ( t ) (cid:105) = (cid:90) t − t A ( ψ F , ψ I , t , t | τ ) (24) × exp {− iω L τ ˆ j z }| γ I (cid:105) dτ. The amplitude to find the spin in some state | β (cid:105) = (cid:80) jm = − j β m | m (cid:105) takes a particularly simple form [20], (cid:104) β | γ ( t ) (cid:105) = (cid:90) t − t G ( ω L τ | j, β, γ I ) A ( ψ F , ψ I , t , t | τ ) dτ, (25)where G ( ω L τ | j, β, γ I ) ≡ (cid:104) β | exp( − iω L τ ˆ j z | γ I (cid:105) (26)= j (cid:88) m = − j β ∗ m γ Im exp( − imω L τ ) . Choosing orthonormal bases | β k (cid:105) , k = 0 , , .. j , and | N (cid:105) to describe the spin and the particle [60], respectively,we easily reconstruct the state into which the system,initially described by the product | ψ I (cid:105)| γ I (cid:105) , evolves by t = t , | Ψ( t ) (cid:105) = (cid:88) N (cid:88) k (cid:90) t − t G ( ω L τ | j, β k , γ I ) (27) × A ( N, ψ I , t , t | τ ) dτ | β k (cid:105)| N (cid:105) Expanding G in a Taylor series around ω L = 0, and usingthe operators of the previous Section, we can rewrite thisas (cid:104) β k | Ψ( t ) (cid:105) = ∞ (cid:88) n =0 ( − iω L ) n n ! (cid:104) β k | ˆ j nz | γ I (cid:105) ˆ U ( n ) part ( t , t ) | ψ I (cid:105) . (28)We note that in the classical limit, where there is asingle classical trajectory x cl ( t ), and a single classicalduration τ cl = τ Ω [ x cl ], a Larmor clock ceases to af-fect the particle’s motion, and is driven by it. Indeed,with A ( N, ψ I , t , t | τ ) = A ( N, ψ I , t , t ) δ ( τ − τ cl ), fromEq.(27) we have | Ψ( t ) (cid:105) = exp( − iω L τ cl ˆ j z ) | γ I (cid:105) ˆ U part ( t , t ) | ψ I (cid:105) (29)Transition to the classical limit is discussed, for ex-ample, in [21]. In this limit A ( N, ψ I , t , t | τ ) becomeshighly oscillatory everywhere except in a small vicinityof τ = τ Ω [ x cl ( t )] ≡ τ cl , making A ( N, ψ I , t, t | τ ) tend to A ( N, ψ I , t , t ) δ ( τ − τ cl ). This is the only case in whicha uniquely defined traversal time can be ascribed to aquantum transition.Thus,we have a family of Larmor clocks, each defined bya particular choice of ω L , j , | β k (cid:105) , and | γ I (cid:105) . VIII. THE SALECKER-WIGNER-PERESCLOCK
A particular choice due to Salecker and Wigner [33],and also to Peres [34], defines the Salecker-Wigner-Peresclock. After the measurement, the spin is to be observedin one of the orthogonal states, obtained from its initialstate, | γ I (cid:105) = | β (cid:105) = (2 j + 1) − / j (cid:88) m = − j | m (cid:105) , (30)by rotation through one of the angles φ k = 2 πk/ (2 j + 1), k = 0 , , ..., j , | β k (cid:105) = exp( − i ˆ j z φ k ) | β (cid:105) = j (cid:88) m = − j exp( − imφ k )(2 j + 1) − / | m (cid:105) . (31)The function G in Eq.(26) is, therefore, given by [20], [21] G SW P ( ω L τ | j, β k , β ) = (2 j + 1) − × (32)sin[(2 j + 1)( φ k − ω L τ ) / φ k − ω L τ ) / . A particle may be observed (post-selected) in a particularsub-space N of its Hilbert space, specified by a projectorˆΠ( N ) = (cid:88) N ∈ N | N (cid:105)(cid:104) N | . (33)The options range from detecting the particle in a singlefinal state | N (cid:105) of the chosen basis, ˆΠ( N ) = | N (cid:105)(cid:104) N | , tobeing completely ignorant of its final state, thus choosingˆΠ( N ) = 1. In all cases, the measured time is defined asthe average of the times corresponding to rotations bythe angles φ k , τ k = φ k /ω L , weighed by the probabilities, P ( k, N ), for finding the spin in the rotated state,T Ω ( N , ψ I ) ≡ j (cid:88) k =0 τ k P ( k, N ) = j (cid:88) k =1 πk (2 j + 1) ω L P ( k, N ) , (34)where P ( k, N ) = (cid:104) Ψ( t ) | β k (cid:105)(cid:104) β k | ˆΠ( N ) | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | ˆΠ( N ) | Ψ( t ) (cid:105) , (35)with | Ψ( t ) (cid:105) given by Eq.(27).The rational behind Eqs.(34) and (35) is simple. Aftera time t , the hand of a classical clock rotates by a welldefined angle, and points at the hour. The final positionof a quantum state, which replaces the classical hand,appears to be distributed, pointing at different ”hours”with different probabilities. Equation (34) represents the”mean time” measured in this way, and associated withthe passage of the particle from the state | ψ I (cid:105) to any-where in the part of its Hilbert space denoted as N .There are at least three remarks to be made. Firstly,finding the clock in a state | β k (cid:105) , rotated by φ k , by nomeans guarantees that the particle has indeed spent aduration φ k /ω L in Ω. Unless G in Eq.(32) is propor-tional to δ ( τ − φ l /ω L ), various durations τ continue tointerfere, and the precise time the particle spends in Ωremains, in general, indeterminate.Secondly, if we are not interested in the final state ofthe particle, ˆΠ( N ) = 1, T Ω ( all, ψ I ) can be written as anexpectation value of an operator ˆ T Ω = (cid:80) jk =0 | β k (cid:105) τ k (cid:104) β k | ,T Ω (all , ψ I ) = (cid:104) Ψ( t ) | ˆ T Ω | Ψ( t ) (cid:105) . (36)It is tempting to conclude that ˆ T Ω represents the ”traver-sal time operator”, and that with it the ”time problem”has been brought into the framework of standard quan-tum mechanics. This is not quite so, since in quantummeasurements the measured operator acts on the vari-ables of the studied system, whereas ˆ T Ω acts on the vari-able of the clock.Finally, being coupled to the clock perturbs the parti-cle’s motion, and whatever information is obtained, nolonger refers to the particle ”on its own”. The obviousway out of this last difficulty is to try to reduce the cou-pling as much as it is possible. In the next Section we willshow that this would inevitably lead to ”complex times”,whose appearance we have already anticipated in Sect.V. IX. A NON-PERTURBING (WEAK) SWPCLOCK
We still need to specify the values of ω L and j , whichdetermine the accuracy of the measurement. In [20] itwas shown that if j → ∞ while ω L is kept finite, the func-tion G ( ω L τ | j, β k , β ) in Eq.(26) becomes proportional to δ ( τ − τ k ), so that the spin can be found in | β k (cid:105) if, andonly if, the particle has actually spent in Ω a duration τ . This is a very accurate measurement, and P ( k, N ) inEq.(34) becomes also the probability with which a dura-tion τ would occur. But, as Eq.(10) demonstrates, bygaining in accuracy we destroyed the original transition,and will have learnt very little about the duration spentin Ω with the interference intact.To pin down this elusive duration we may try to keep j finite, while making the magnetic field, and with it ω L ,very small. This will reduce the perturbation which af-fects the particle’s motion. Indeed, and as ω L →
0, fromEq.(28) we have (cid:104) β k | Ψ( t ) (cid:105) ≈ (cid:104) β k | β (cid:105) ˆ U part ( t , t ) | ψ I (cid:105) (37)= δ k ˆ U part ( t , t ) | ψ I (cid:105) . The particle moves unimpeded, the spin does not rotate,and the clock provides no information at all.We need to find the first correction to this result. Trun-cating the expansion (28) after the terms linear in ω L ,and inserting the result into Eq.(34) yieldsT Ω ( N , ψ I ) = ω L Q ( j ) T SW P ( N , Ω , ψ I ) + O ( ω L ) , (38) with Q ( j ) ≡ j (cid:88) k =0 φ k |(cid:104) β k | ˆ j z | β (cid:105)| . (39)In Eq.(38), the new time parameter T SW P ( N , Ω , ψ I ),called the SWP time until a better name is found, isgiven by [61] T SW P ( N , Ω , ψ I ) = W ( N , ψ I ) − × (40) (cid:88) N ∈ N W ( N, ψ I ) | τ Ω ( N, ψ I ) | , where W ( N, ψ I ) ≡ |(cid:104) N | ˆ U part ( t , t ) | ψ I (cid:105)| (41)is the probability for the particle to be found in | N (cid:105) inthe absence of the clock, and W ( N , ψ I ) ≡ (cid:88) N ∈ N W ( N, ψ I ) (42)is this probability for the whole of the chosen subspace N .Equation (40) is the central result of our discussion so far.In its l.h.s., we have an average, obtained for an ensem-ble weakly coupled SWP clocks, which is the observedresult of the measurement. In the r.h.s., the only quan-tity describing the particle is T SW P ( N , ψ I ). It is givenby the weighted sum of the squared moduli of the com-plex times defined in (11), evaluated for the transitionsinto all orthogonal states spanning the chosen subspace N of the particle’s Hilbert space. This is an illustrationof the general principle discussed in Sect. V: in a weaklyperturbing inaccurate measurement the system is alwaysrepresented by combinations of the relevant probabilityamplitudes, given in this case by τ Ω ( N, ψ F ) of Eq.(11).We note also that the squares of the SWP times, ratherthan the SWP times themselves, are additive. For twodisjoint subspaces N and N (cid:48) , ˆΠ( N ) ˆΠ( N (cid:48) ) = 0, equation(34) gives T SW P ( N ∪ N (cid:48) , Ω , ψ I ) = [ W ( N , ψ I ) + W ( N (cid:48) , ψ I )] − × (43)[ W ( N , ψ I ) T SW P ( N , Ω , ψ I ) + W ( N (cid:48) , ψ I ) T SW P ( N (cid:48) , Ω , ψ I ) . Next we look at the SWP times from a slightly differentprospective.
X. CALIBRATION AND THE UNCERTAINTYPRINCIPLE
A perhaps less direct way to arrive at the SWP time T SW P ( N , Ω , ψ I ) is to calibrate SWP clock by using a pro-cedure similar to the one proposed by Leavens in [37].Consider first introducing a magnetic field everywhere inspace, rather than just inside Ω. Now all Feynman pathsspend in the field the same duration τ = t − t , we have A ( ψ F , ψ I , t , t | τ ) = A ( ψ F , ψ I , t , t ) δ ( τ − t + t ) for all | ψ I (cid:105) and | ψ F (cid:105) , and the clock decouples from the particle’smotion. Thus, in the limit ω L → free ( t − t ) = ω L Q ( j )( t − t ) . (44)Next, let us measure T Ω ( N , ψ I ) in the case where themagnetic field exists only inside Ω. Let us also assumethat there is some hypothetical duration τ ” in Ω” which theparticle spends in Ω, and during which the spin rotates. Ifso, the resulting value of T Ω ( N , ψ I ) should be the sameas T free ( τ ” in Ω” ), obtained for a spin that has been infree rotation for τ ” in Ω” seconds. Equating T Ω ( N , ψ I ) inEq.(38) to T free ( τ ” in Ω” ) shows that we must identify thesought τ ” in Ω” with the SWP time, τ ” in Ω” = T SW P ( N , Ω , ψ I ) . (45)This may seem reasonable, since T SW P ( N , Ω , ψ I ) is a realvalued positive quantity, yet there is a serious concern.Apparently, we impose a single duration while, accordingto Sect. V, there shouldn’t be one. It ought to be pru-dent to make further checks.First we consider a classical case, where the magneticfield is confined to Ω, and the single path which con-nects | ψ F (cid:105) with | ψ I (cid:105) , spends τ cl seconds in Ω. Clearly, A ( ψ F , ψ I , t , t | τ ) = A ( ψ F , ψ I , t , t ) δ ( τ − τ cl ), and weobtain the correct result τ ” in Ω” = τ cl . (46)The problems begin once quantum interference starts toplay a role. To see it, suppose that there are exactly twovirtual paths, via which | ψ F (cid:105) can be reached from | ψ I (cid:105) (perhaps in a situation similar to the one shown in Fig.1,or in a setup where the particle’s wave packet is split intotwo parts, which pass via different optical fibres, and arelater recombined). The paths spend in Ω τ and τ sec-onds, and have the probability amplitudes A and A ,respectively. How much time does the particle spend inΩ? Before proceeding, we recall the Uncertainty Prin-ciple, already outlined in Sect. IV. Interference mergesthe two virtual paths into a single route connecting theparticle’s initial and final states. The duration spent inΩ is, therefore, truly indeterminate [15]. It is not τ or τ , nor any other similar duration. It should not exist.Noting that now A ( ψ F , ψ I , t , t | τ ) = A δ ( τ − τ ) + A δ ( τ − τ ), [in the situation shown in Fig.1, the ampli-tude A ( x F , x I , t , t | τ ) has two stationary regions around τ and τ , and is highly oscillatory elsewhere], and using(45) we find τ ” in Ω” = (cid:12)(cid:12)(cid:12)(cid:12) A τ + A τ A + A (cid:12)(cid:12)(cid:12)(cid:12) (47)There are no a priori restrictions on the magnitude orthe sigh of the ratio A /A . Suppose the transition takesthree seconds, t − t = 3, and the paths spend in Ω 1 and 2 seconds, respectively. Choosing A = 0 . A = − A + 0 .
001 we find τ ” in Ω” = 4998s >> t − t = 3s , (48)which is strange. Further, choosing A = 0 . A = − .
25, yields τ ” in Ω” = 0 , (49)which is also strange, especially if | ψ I (cid:105) and | ψ F (cid:105) are, asin Fig.1, localised on the opposite sides of the region Ω,which the particle, therefore, has to cross. XI. COMPLEX TIMES AND THE ”WEAKMEASUREMENTS”
Both experiments described at the end of the previousSection, can be performed, at least in principle. We must,therefore, decide on an interpretation of the results (48)and (49). There are two possibilities. Either, (A), the τ ” in Ω” represents a physical duration, and has furtherimplications for our understanding of quantum motion.Or, (B), it is something else, in which case we need toexplain what it is precisely. Next we look at both options.A. The first option may appear either absurd [46] orintriguing, depending on the reader’s viewpoint. Indeed,should the 4998s in Eq.(48) be ”physical”, we must con-clude that quantum mechanics allows a particle ”to spendmore than an hour in some place during a journey thatlast only three seconds”. By taking the 0s result inEq.(48) literally, we defy Einstein’s relativity by lettingthe particle ”cross the region infinitely fast”.Both conclusions are reminiscent of other ”surprising” re-sults obtained within the so-called ”weak measurements”approach [57]. Among these one encounters the notionsof ”negative kinetic energy” [62], ”negative number ofparticles” [63], ”having one particle in several places si-multaneously” [64], a ”photon disembodied from its po-larisation” [65], an ”electron with disembodied chargeand mass” [65], ”an atom with the internal energy disem-bodied from the mass” [65], and ”photons found in placesthey neither enter or leave” [66], [67]. Perhaps closestto the subject of this paper is the concept of ”chargedparticles moving faster than light through the vacuum”,introduced in [68]. In all these examples, the analysis re-lies on obtaining the ”weak value” (cid:104) ˆ B (cid:105) w , of an operatorˆ B = (cid:80) i | b i (cid:105) B i (cid:104) b i | , defined for a system prepared (pre-selected) in a state | ψ I (cid:105) and then found (post-selected)in a state | ψ F (cid:105) , (cid:104) ˆ B (cid:105) w = (cid:104) ψ F | ˆ B | ψ I (cid:105)(cid:104) ψ F | ψ I (cid:105) . (50)The conclusions of Refs. [62]-[68], mentioned above, arethen drawn from the properties of the complex valuedquantity (cid:104) ˆ B (cid:105) w .0B. Another explanation available to us is more downto earth (see also [15], [59], [69], [70]). The results (48)and (49) may simply illustrate the Uncertainty Princi-ple (above) by showing that it is impossible to ascribea meaningful duration to a situation where two or moredurations interfere to produce the result. Response of aquantum system to an attempt to measure the traversaltime without perturbing the system’s motion results inevaluation of the sum of the probability amplitudes (8),which can, in principle, take any value at all. The weakvalue (50) is another illustration of this general principle.Indeed, the final state | ψ F (cid:105) can be reached via passingthrough the eigenstates | b i (cid:105) , and the amplitude for the i -th virtual path is A ( i, ψ f , ψ i ) = (cid:104) ψ F | b i (cid:105)(cid:104) b i | ψ I (cid:105) . Thus,Eq.(50) can be cast in the form similar to (11) (cid:104) ˆ B (cid:105) w = (cid:88) paths B i A ( i, ψ I , ψ F ) / (cid:88) paths A ( i, ψ I , ψ F ) . (51)where the eigenvalues of ˆ B on the routes | ψ F (cid:105) ← | b i (cid:105) ←| ψ I (cid:105) , B i , replace the values of τ Ω [ x ( t )] on the virtualFeynman paths. We could call the complex time (11)”the weak value of the traversal time functional”, bear-ing in mind that it is just a particular combination ofthe probability amplitudes A ( ψ F , ψ I , t , t | τ ). No otherinterpretation of the complex times should be possible,as the Uncertainty Principle does not allow to view prob-ability amplitudes, or their combinations, as the actualvalues of a physical quantity [15].We strongly advocate the second explanation. Indeed,there is nothing strange about the result (47) itself, andthe only thing at fault is our desire to impose, throughthe calibration procedure, a single physical duration | τ Ω | where there shouldn’t be one. Not surprisingly, the re-sult is unsatisfactory. Accordingly, with Eq.(48), theclock has not ”aged” by more than an hour in a spanof three seconds. Its final state | γ ( t ) (cid:105) is a superpositionof the states | γ ( t | τ ) (cid:105) and | γ ( t | τ ) (cid:105) , rotated by the angles ω L τ and ω L τ , respectively. It is certainly not equal to | γ ( t | τ ” in Ω” ) (cid:105) , and cannot, in general, be obtained by arotation through any angle ω L τ X , A | γ ( t | τ ) (cid:105) + A | γ ( t | τ ) (cid:105) (cid:54) = const | γ ( t | τ X ) (cid:105) , (52)since higher orders in ω L would involve quantities τ n Ω inEq.(12), and τ n Ω (cid:54) = ( τ Ω ) n .We must conclude that the calibration of a weak SWPclock fails to define a meaningful traversal time for aquantum particle. This is in agreement with the Uncer-tainty Principle. XII. THE DWELL TIME
Despite the difficulties outlined in the previous Section,much of the discussion about tunnelling times contin-ues to be built around a tacit assumption that a singleclassical-like duration, which characterises a classicallyforbidden transition, exists and simply has not yet been x tx I ,t x F ,t (cid:2) (cid:2) (cid:1) time c oo r d i na t e A A FIG. 1: (Color online) A semiclassical particle can reach thefinal state | x F (cid:105) from | x I (cid:105) directly, and by having been re-flected off a wall at x = 0. The particle is heavy, so there aretwo ”classical” trajectories trajectories, which minimise theaction S in Eq.(6). The trajectories interfere, and are trav-elled with the amplitudes A and A , spending in the regionΩ τ τ and τ seconds, respectively. All in all, how muchtime does the particle spend in Ω? This is the ”which way?”question at the centre of the traversal time controversy. found [5]. One candidate for the role of this parame-ter is the dwell time , a special case of the complex time(11), evaluated for the final state | ψ F (cid:105) obtained by un-perturbed evolution of the initial state | ψ I (cid:105) , τ dwell Ω ( ψ I ) ≡ τ Ω ( ˆ U part ( t , t ) ψ I , ψ I ) . (53)It can be written is several different ways. Expanding | ˆ U part ( t , t ) ψ I (cid:105) is some basis | N (cid:105) , we can express τ dwell Ω in a form similar to Eq.(40), τ dwell Ω ( ψ I ) = (cid:88) N W ( N, ψ I ) τ Ω ( N, ψ I ) . (54)In the operator notations of Sect. VI, the dwell timebecomes τ dwell Ω ( ψ I ) = (cid:104) ψ I | ˆ U † part ( t , t ) ˆ U (1) part ( t , t ) | ψ I (cid:105) , (55)and using Eq.(20) yields a derived result, often mistakenfor the definition of τ dwell Ω ( ψ I ), [16, 17, 37, 38], τ dwell Ω = (cid:90) t t dt (cid:48) (cid:90) Ω dx | ψ I ( x, t (cid:48) ) | . (56)The dwell time possesses several attractive properties.Firstly, like its classical counterpart, it is non-negative,and never exceeds the total duration of motion, τ dwell Ω ≤ t − t , thus avoiding the problems encountered in the pre-vious Section. Secondly, written as in Eq.(56), it appearsto have a simple probabilistic structure, with the contri-bution from the interval dt (cid:48) proportional to the probabil-ity to find the particle in Ω. Thirdly, the same expressionarises in approaches as different as the Bohm trajectories1method [39], and the Feynman path approach consideredhere.Before accepting τ dwell Ω as the long sought classical-like duration, we note that certain questions about itremain unanswered. The interference between differentdurations, which contribute to the transition, remains in-tact, and the conflict with Uncertainty Principle contin-ues unresolved. Also, the said properties do not extendto the individual terms τ Ω ( N, ψ I ) in Eq.(54), which re-main complex-valued, and should not be confused withmeaningful durations, as was shown in the previous Sec-tion. Finally, it may be that, for some unknown reason,a classical-like duration can only be defined for a quan-tum system, which follows uninterrupted evolution alongits ”orbit” in the Hilbert space, | ψ ( t ) (cid:105) = ˆ U part ( t, t ) | ψ i (cid:105) .But then, to be a true analogue of the classical traversaltime, τ dwell Ω ( ψ I ) should arise whenever a time measure-ment perturbs the particle only slightly, and no post-selection is performed on the particle in the end. Shouldit not be so, the appealing form of Eq.(56) would be for-tuitous, and have no further physical consequences. Wewill test this last assumption next. XIII. WOULD THE SWP CLOCK MEASURETHE DWELL TIME?
An example at hand is the SWP clock in the ω L → t and t without controlling the particle’s final state,evaluate the average T SW P ( all, ψ I ) by choosing N to co-incide with all of the particle’s Hilbert space, and use thecalibration procedure of Sect. X. Will the result coincidewith the dwell time (56) as was assumed in [44]? Thequestion was studied also in [37].The answer is yes, provided the system evolves along sin-gle classical path. We have already shown [see Eq.(46)]that the SWP time coincides with the τ cl , evaluated forthis path. With an (almost) classical particle representedby a very narrow wave packet crossing the region Ω, theequality τ dwell Ω = τ cl follows directly from the ”stopwatchexpression” (56). Thus, τ cl is the unique duration whicharises from both approaches in the classical limit.However, in the full quantum case, the answer is no.From Eq.(40) we have T SW P ( all, Ω , ψ I ) = (cid:104) ψ I | ˆ U (1) † part ( t , t ) | ˆ U (1) part ( t , t ) | ψ I (cid:105) / , (57)and the application of (22) yields T SW P ( all, Ω , ψ I ) = (cid:113) τ dwell Ω ( ψ I ) × τ Ω ( ψ (1) ( t ) , ψ I ) (58)which, in general, is not the same as τ dwell Ω ( ψ I ).It is easy to see the reason for this discrepancy. Accord-ing to Eq.(55) the dwell time must involve the productˆ U † part ( t , t ) ˆ U (1) part ( t , t ), and could only appear in the lin-ear in ω L corrections to P ( k, all) in Eq.(35). But with thechoice of the states | β k (cid:105) in Eq.(31), all such corrections vanish, for k (cid:54) = 0 because (cid:104) β k | β (cid:105) = 0, and for k = 0since (cid:104) β | ˆ j z | β (cid:105) = 0. Neither would τ dwell Ω ( ψ I ) appearin the higher order corrections to P ( k, all), since none ofthese corrections contain the required term ˆ U † part ( t , t ).In general, the dwell time plays no role in the analysisof the SWP clock, as defined in Sect. VIII. However,in some special cases, T SW P ( all, Ω , ψ I ) may accidentallyreduce to τ dwell Ω ( ψ I ), as we will show in the next Section. XIV. STATIONARY TUNNELLING AND THELEAVENS’ ANALYSIS
All that was said above applies to tunnelling of a par-ticle prepared in a wave packet state, shown in the dia-gram in Fig.2. The particle’s initial state at t = t is asuperposition of the plane waves with positive momenta p > ψ I ( x ) = (cid:104) x | ψ I (cid:105) = (cid:90) ∞ A ( p ) exp( ipx ) dp, (59)located to the left of a barrier of a finite width, V ( x ), oc-cupying the region [0 , d ]. The wave packet moves towardsthe barrier, and the energies E ( p ) = p / µ are chosen allto lie below the barrier height, so that in order to betransmitted the particle has to tunnel. At a sufficientlylarge time t , the scattering is complete, and the wavepacket is divided into the transmitted (T), ψ T ( x, t ), andreflected (R), ψ R ( x, t ), parts, (cid:104) x | ψ ( t ) (cid:105) ≡ (cid:104) x | ˆ U part ( t − t ) | ψ I (cid:105) = ψ T ( x, t ) + ψ R ( x, t ) =(2 π ) − / (cid:90) ∞ dpT ( p ) A ( p ) exp[ px − iE ( t − t )] +(2 π ) − / (cid:90) ∞ dpR ( p ) A ( p ) exp[ − ipx − iE ( t − t )] , (60)where T ( p ) and R ( p ) are the transmission and reflectionamplitudes, respectively.We are interested in the duration spent in the barrierregion, and choose Ω ≡ [0 , d ]. In the limit t , → ∓∞ ,matrix elements of the operators in Sect. VI betweenthe plane waves | p (cid:105) , (cid:104) x | p (cid:105) = exp ( ipx ), are given by ( n =0 , , , ... ) (cid:104) p (cid:48) | ˆ U ( n ) part ( t − t ) | p (cid:105) = (61) i n exp[ − iE ( p )( t − t )] ∂ nλ T ( p, λ = 0) δ ( p − p (cid:48) ) , for p > p (cid:48) > , and i n exp[ − iE ( p )( t − t )] ∂ nλ R ( p, λ = 0) δ ( | p | − | p (cid:48) | ) , for p > p (cid:48) < , where T ( R )( p, λ ) denote the transmission (reflection)amplitudes for a composite barrier V ( x ) + λ Θ [0 ,d ] ( x ).From Eq.(17), for the complex tunnelling and reflection2times of a particle with an initial momentum p we have τ [0 ,d ] ( p, p ) = i∂ λ ln T ( p, λ = 0) , (62)and τ [0 ,d ] ( − p, p ) = i∂ λ ln R ( p, λ = 0) . (63)In the following we will be interested only in whetherthe particle is transmitted, or reflected, and employ theprojectors ˆΠ(tunn) = (cid:82) ∞ | p (cid:105)(cid:104) p | and ˆΠ(refl) = (cid:82) −∞ | p (cid:105)(cid:104) p | ,on all positive and all negative momenta, to distinguishbetween the two outcomes.Suppose next that a weak SWP clock is used to measurethe time the particle spends in Ω. To obtain the SWPtime for transmission, we replace in Eq.(40) summationover N by integration over p , and ˆΠ( N ) with ˆΠ(tunn).We find T SW P (tunn , [0 , d ] , Ψ I ) = (64) W (tunn) − / (cid:20)(cid:90) ∞ dp | T ( p ) | | A ( p ) | | τ [0 ,d ] ( p, p ) | (cid:21) / , and similarly for reflection, T SW P (refl , [0 , d ] , Ψ I ) = (65) W (refl) − / (cid:20)(cid:90) ∞ dp | R ( p ) | | A ( p ) | | τ [0 ,d ] ( − p, p ) | (cid:21) / , where W (tunn) ≡ (cid:82) ∞ | T ( p ) | | A ( p ) | dp and W (refl) ≡ (cid:82) ∞ | R ( p ) | | A ( p ) | dp , are the tunnelling and reflectionprobabilities, respectively. Finally, if we do not carewhether the particle is transmitted or reflected, fromEqs.(40) and (61) we find the calibrated SWP result,without post-selection, to be T SW P (all , [0 , d ] , Ψ I ) = (66) (cid:26)(cid:90) ∞ dp | A ( p ) | [ | ∂ λ T ( p, λ = 0) | + | ∂ λ R ( p, λ = 0) | ] (cid:27) / Returning to the question of the previous Section, wemay want to compare this with the dwell time which,according to Eqs.(54), (62) and (63), is given by τ dwell [0 ,d ] ( ψ I ) = (67) i (cid:90) ∞ | A ( p ) | [ T ∗ ( p ) ∂ λ T ( p, λ = 0) + R ∗ ( p ) ∂ λ R ( p, λ = 0)] dp As expected, the SWP result in Eq.(66) is different fromthe dwell time in (67). Leavens [37] studied, mostly nu-merically, the case where the incident particle has a defi-nite momentum p . To arrive at his results from Eqs.(66)and (67) it is sufficient to choose a nearly monochromaticwave packet, so narrow in the momentum space, that thetransmission and reflection amplitudes and their deriva-tives can be approximated by their values at p . Since (cid:82) ∞ | A ( p ) | dp = 1, this yields T SW P (all , [0 , d ] , p ) = (68) (cid:2) | ∂ λ T ( p , λ = 0) | + | ∂ λ R ( p , λ = 0) | (cid:3) / and τ dwell [0 ,d ] ( p ) = (69) i [ T ∗ ( p ) ∂ λ T ( p , λ = 0) + R ∗ ( p ) ∂ λ R ( p , λ = 0)]In [37] it was shown that a good agreement between theSWP result T SW P (all , [0 , d ] , p ) and the dwell time (69)is achieved for free motion, if the width of the region, d , issufficiently large. Good agreement between the two wasalso found for a barrier turned into a potential step, e.g.,if d is sent to infinity, thus making transmission impossi-ble. The latter result follows immediately by putting inEqs.(68) and (69) T ( p , λ ) ≡ | R ( p ) | λ ≡
1, whichgives T SW P (all , [0 , ∞ ] , p ) = (70) − ∂ λ Arg [ R ( p , λ = 0)] = τ dwell [0 , ∞ ] ( p ) . The case of a free particle crossing the region of a width d , requires a little more attention. We can neglect thereflection term in Eq.(69) but not in (68), and must eval-uate both derivatives instead. Using Eq.(20) we easilyfind T SW P (all , [0 , d ] , p ) τ dwell [0 ,d ] ( p ) = (cid:115) (cid:12)(cid:12)(cid:12)(cid:12) sin( p d ) p d (cid:12)(cid:12)(cid:12)(cid:12) p d →∞ → , which explains the good agreement found by Leavens forbroad regions. Finally, another minor point regardingthe analysis of [37] is consigned to the Appendix.To summarise, we can agree with Leavens on the generaldiscrepancy between the dwell time, and what is mea-sured by a calibrated SWP clock. We also have dex-plained why this discrepancy must arise. However, wedisagree with the final conclusion of [37] that ”it is onlythe dwell time, which does not distinguish between trans-mitted and reflected particles, that is a meaningful con-cept in conventional interpretations of quantum mechan-ics”. The dwell time is, we argue, just a special case ofthe ”complex time” and is no more, and no less, meaning-ful than the tunnelling and reflection times in Eqs.(62)and (63). XV. TUNNEL IONISATION
Our general analysis applies also to the case of tun-nel ionisation, where the tunnelling time problem hasattracted recent theoretical interest [5]. In an ionisa-tion experiment, an initially bound electron has a chanceto escape by tunnelling across a potential barrier brieflycreated by a time dependent external field. One maybe interested in the duration the escaped electron hasspent in the classically forbidden region, and attempt tomeasure it by means of a weak SWP clock, perturbingtunnelling as little as possible. A realistic calculation ofsuch a measurement can be found, for example, in [44],and here we will limit ourselves to the formulation of the3 barr
BCD i n c i den t ba rr i e rr e f l e c t ed t unne ll ed (cid:1) (arb.units) P o t en t i a l FIG. 2: (Color online) An incident wave packet impacts ona potential barrier, and is divided into the transmitted (tun-nelled) and reflected parts. What is the duration the particlehas spent in the region Ω which contains the barrier? problem, and the identification of the time parameterssuch a measurement would produce.A one-dimensional sketch of the setup is shown in Fig.3.Bound at t = t in the single bound state of a potentialwell, | ψ I (cid:105) = | ψ (cid:105) , the particle can escape to the con-tinuum while an external field is converting the bindingpotential into a potential barrier. Long after the fieldis switched off, at some t = t , its wave function isdivided into the ”bound” part, describing the particleswhich failed to leave the well, and the ”free” part, de-scribing the escaped particles moving away from it. We,therefore, haveˆ U part ( t , t ) | ψ I (cid:105) = | ψ bound (cid:105) + | ψ free (cid:105) ≡ (71) C ( t , t )] | ψ (cid:105) + (cid:90) ∞ B ( p, t , t ) | p (cid:105) dp. and find the ionisation probability to be given by, W (ion) = (cid:104) ψ free | ψ free (cid:105) = (cid:90) ∞ | B ( p, t , t ) | dp. (72)Let the particle be monitored by a weak SWP clock, withthe magnetic field is localised in the classically forbiddenregion Ω, as shown in Fig.3. We will also have at our dis-posal a perfect remote detector, capable of determiningwhether the particle has escaped, and if it has, and ableto evaluate its momentum p . With this, we can choose topost-select the particle in the free state, and record theclock’s reading only if the particle was seen to escape.We can also post-select it in the bound state, and keepthe readings only in the case the remote detector has notfired. Alternatively, we can choose not to post-select atall, and retain all of the clock’s readings.There is a set of complex times which, as discussed inSect. V, are related to the response of the system tothe introduction of a constant potential λ Θ Ω ( x ) in theregion of interest. If such a potential is introduced, thewave function at t retains the form (71), but its co-efficients should depend on λ , C ( t , t ) → C ( λ, t , t ), B ( p, t , t ) → B ( p, λ, , t , t ). Thus, for an escaped parti-cle with a momentum p we can define the complex time(11) and other complex ”averages” (12) as [we omit thetime dependence of the coefficients C and B , and recallthat τ Ω ≡ τ ] τ n Ω ( p, ψ , t , t ) = ( i ) n B ( p, λ = 0) − ∂ nλ B ( p, λ = 0) . (73)Similarly, for a particle which remained in the well, wehave τ n Ω ( ψ , ψ , t , t ) = ( i ) n C ( λ = 0) − ∂ nλ C ( λ = 0) . (74)There is also a real valued dwell time, which does notdistinguish between the particles which have escaped andthose which remained bound, τ dwell Ω ( ψ ) = | C ( λ = 0) | τ Ω ( ψ , ψ , t , t ) (75)+ (cid:90) ∞ | B ( p, λ = 0) | τ Ω ( p, ψ , t , t ) dp. What is measured in an experiment depends on how theclock is prepared and read. If the weak SWP clock ofSect. IX is used, and the calibration procedure of Sect. Xis applied, the time found for the particles which remainbound in the potential well is T SW P (bound , Ω , ψ ) = | τ Ω ( ψ , ψ , t , t ) | (76)= | ∂ λ ln C ( λ = 0) | . For the particles which leave the well with unspecifiedmomentum, the measurement will yield T SW P (free , Ω , ψ ) = (77) W (ion) − / (cid:20)(cid:90) ∞ | τ Ω ( p, ψ , t , t ) | | B ( p ) | dp (cid:21) / = W (ion) − / (cid:20)(cid:90) ∞ | ∂ λ B ( p, λ = 0) | dp (cid:21) / Finally, if the final state of the particles is not controlled,from (43) we have T SW P (all , Ω , ψ ) = { [1 − W (ion)] × (78) T SW P (bound , Ω , ψ ) + W (ion) T SW P (free , Ω , ψ ) } / = (cid:20) | ∂ λ C ( λ = 0) | + (cid:90) ∞ | ∂ λ B ( p, λ = 0) | dp (cid:21) / , which is not the same as the dwell time in Eq.(75).If, on the other hand, we follow Leavens [37] in choosing | γ I (cid:105) = | β j (cid:105) , (see Appendix), the sum in the r.h.s. ofEq.(39) will vanish, and to evaluate the new SWP time T (cid:48) SW P , we would need to go to the next order in ω L inEq.(28). For example, instead of (77) from Eq.(83), wewill have T (cid:48) SW P (free , Ω , ψ ) = (79)4 W (ion) − / (cid:26)(cid:90) ∞ Re[ τ Ω ( p, ψ ) τ ∗ ( p, ψ )] | B ( p ) | dp (cid:27) / = W (ion) − / (cid:26)(cid:90) ∞ Im[ ∂ λ B ( p, λ = 0) ∂ λ B ∗ ( p, λ = 0) dp (cid:27) / . and, as before, will not recover the dwell time (75) in thecase no post-selection is made.The dwell time would, however, occur naturally if insteadof evaluating the averages (34) or (83), we would employa more general Larmor clock, described in Sect. VII, andconsider a small difference in the probability P ( k, all) ≡(cid:104) Ψ( t ) | β k (cid:105)(cid:104) β k | Ψ( t ) (cid:105) for the clock to be found in a state | β k (cid:105) before and after it interacts with the particle. Asimple calculation, using Eq.(28), shows that this changeis proportional to τ dwell Ω ( ψ ) δP ( k, all) ≡ P ( k, all) − |(cid:104) β k | γ I (cid:105)| = (80)2 ω L Im[ (cid:104) γ I | β k (cid:105)(cid:104) β k | ˆ j z | γ I (cid:105) ] τ dwell Ω ( ψ ) + O ( ω L )Defining the measured mean value as δ T Ω (all , ψ I ) ≡ (cid:80) jk =0 τ k δP ( k, all), we obtain δ T Ω (all , ψ I ) = Q (cid:48) ( j ) τ dwell Ω ( ψ ) + O ( ω L ) , (81)with Q (cid:48) ( j ) = 2Im (cid:110)(cid:80) jk =0 φ k (cid:104) γ I | β k (cid:105)(cid:104) β k | ˆ j z | γ I (cid:105) (cid:111) . If themagnetic field is introduced everywhere in space, we find δ T free Ω ( t − t ) = Q (cid:48) ( j )( t − t ). Using this relation tocalibrate the result (81), as was done in Sect. X, showsthat for this particular clock, the duration imposed inthe quantum case is τ dwell Ω ( ψ ). This is the case of linearcalibration, studied by Leavens and McKinnon in [38].Thus, also in the case of tunnel ionisation, applicationof a weak SWP clock does not yield a single real dura-tion the particle is supposed to spend in the classicallyforbidden region, but rather a variety of complex valuedtime parameters, through which the real valued result ofthe measurement is expressed. These parameters differfor different settings of the clock, and reduce to a uniqueclassical value only in the primitive semiclassical limit,where a single classical trajectory connects the initial andfinal states. In the next Section we give our conclusions. XVI. CONCLUSION AND DISCUSSION
Mathematical exercises presented above do not them-selves form a basis for a discussion about ”the amount oftime a tunnelling particle spends in the barrier”. Theyonly illustrate the far more general principle at stake.Most of the quantum transitions, and certainly tun-nelling, are interference phenomena, which require contri-butions from many virtual Feynman paths. Each Feyn-man path spends certain amount of time, τ Ω [ path ], insidethe region of interest Ω. We can group together the pathssharing the same value of τ Ω [ path ], and see a transitionas a result of interference between all traversal times in-volved. The difficulty in determining the duration, spent ion1 BBBB P o t en t i a l f r eebound (cid:1) x (arb.units) FIG. 3: (Color online) at first the particle is trapped in theground state of a potential well. A time dependent externalfield turns the potential step into a barrier, and then restoresit to its original shape. The particle’s wave function is di-vided into the part still trapped, and the escaped part, freelypropagating away from the well. What is the duration theparticle has spent in the classically forbidden region Ω? by a quantum particle in Ω, is then the well known diffi-culty in answering the ”which way?” (”which τ ?) ques-tion in the presence of interference, the only mystery inquantum mechanics, according to Feynman [71]. In thispaper we have examined in detail one particular way oftrying to answer the question, while leaving the inter-ference intact. Arguably, the general conclusions, whichcan be drawn from our analysis, are more important thanany of its technical details. We will formulate these con-clusions in a perhaps unusual form of attempting to askthe most relevant questions, and then trying to answerthem the best we can. a) What is measured by the SWP clock? Like everyclock of the Larmor family, the SWP clock measures thenet time τ the particle’s Feynman paths spend in theregion of interest. b) How is this time measured? By modifying the con-tributions of different τ ’s to the particle’s transition am-plitude, depending on final state in which the clock isobserved. c) Does the SWP analysis come up with an ”opera-tor for the tunnelling time”? Strictly speaking, no. Theoperator (36), often quoted in that capacity, acts on thevariables of the clock, and not on the variables of the par-ticle. It defines, therefore, a von Neumann measurementwhich needs to me made on the spin. d) To what accuracy is it measured?
If the function G SW P in Eq.(26) limits the values of τ , which contributeto the transition | ψ I (cid:105)| β (cid:105) → | ψ F (cid:105)| β k (cid:105) , to a region of awidth ∆ τ around some value τ k , we can say that by ob-serving the clock in | β k (cid:105) , we have measured a value τ k toan accuracy ∆ τ . A weak ( ω L → ∞ ) SWP clock, whosemain purpose is to perturb the transition as little as pos-sible, does not discriminate between different times inthis way. Rather, it studies the response of a particle to5the small variations of the probability amplitudes definedin Eq.(8), and its accuracy is very poor. e) Is there a probability distribution for the traversaltime in the case of tunnelling? Not unless it is cre-ated by an accurate clock, which destroys the inter-ference between different values of τ . If a weak clockis employed, only the probability amplitude distribution A ( ψ F , ψ I , t , t | τ ) in Eq.(8) is available. f ) Are complex traversal times inevitable? Interfering(virtual) pathways should together be considered a sin-gle route connecting the initial and final states of thesystem. By the Uncertainty Principle [71], virtual path-ways cannot be distinguished without destroying interfer-ence between them. Accordingly, the response of a sys-tem to a weakly perturbing measurement of the traver-sal time functional (1) is always formulated in terms ofthe complex valued sum of the corresponding amplitudes, A ( ψ F , ψ I , t , t | τ ) in Eq.(8), weighted by the values of thefunctional, τ [15]. This is a general result behind the so-called weak measurement theory [57], [59]. The complextime τ Ω in (11) is the ”weak value” of the functional (1). g) Can complex times be measured? Certainly, for ex-ample by a weak SWP clock discussed above, and thefact that they are complex valued is no major obstacle.However, since the result of a measurement must be real,it is impossible to say apriori whether a particular ex-periment would yield Re τ Ω , Im τ Ω , | τ Ω | , or, indeed, anyother real valued combination of Re τ Ω and Im τ Ω (seealso [18]). Our detailed analysis of the SWP clock usedby Peres [34], shows that what it measures is, in fact, | τ Ω | . h) Are complex times related to physical time inter-vals? In general, they are not. Any attempt at over-interpretation, by treating parts of τ Ω as if they wereactual durations, would lead to insurmountable difficul-ties. For example, in the case described in Sect. X, onewould face not only the chance of faster-than-light travel,but also the possibility of spending a month on the beachduring a one-week leave from office. None of the two areoffered by elementary quantum mechanics. i) What are the complex times then? Just what theirdefinition tells us. A complex time is what one wouldobtain by multiplying the amplitude to reach | ψ F (cid:105) from | ψ I (cid:105) and spend a duration τ in Ω by τ , and sum over allthe τ ’s which contribute to the transition. j) Is the dwell time more meaningful than other com-plex times? No, it is a particular case of a complex time,whose attractive properties can be traced to the fact thatthe operator ˆ U part ( t , t | λ ) in Eq.(15) is hermitian for anyreal λ [24]. In the quantum case, it does not always takethe place of the classical duration, as was shown in Sect.XII. The Uncertainty Principle does not forbid the weakvalues to look appealing in particular cases. Rather, itguarantees the existence of ”unappealing” results, shoulddifferent initial and final states be chosen instead [69]. Itis these other results which should warn one against giv-ing too much credit to the nice exceptions. k) Does the SWP clock measure the dwell time? Asdefined in [34] and in Sect. VIII, it does not. The choiceof the states in which the clock is observed is such that theterms which add up to the dwell time do not contributeto the result, even if the final state of the particle isnot controlled. With a different Larmor clock it would,however, be possible to evaluate τ dwell Ω ( ψ I ) [38]. l) Does tunnelling particle spend a finite amount oftime in the barrier? We could equally ask whether theelectron in the Young’s double slit experiment reachesthe screen by passing through the holes in the screen?All Feynman paths which contribute to tunnelling spendsome time in the barrier. Moreover, replacing theSchroedinger equation with a relativistic Klein-Gordonone [72], leaves only the paths spending in Ω a time longerthan the width of the regionspeed of light [73]. In every virtual sce-nario (i.e., the one to which we can ascribe an amplitude,but not the probability [15] ) the electron goes throughone of the holes, and the particle spends a reasonableduration inside the barrier. m) How much time does a tunnelling particle spend inthe barrier?
We could equally ask ”which hole did theelectron go through?”. In standard (Feynman) quantummechanics it goes through both, and through neither onein particular [71]. In the same sense, the particle spendsin the barrier all durations at the same time. The ques-tion is meaningless in a very strong sense, and an attemptto force it brings an unsatisfactory answer, τ Ω ( ψ I , ψ F ).Consider two researchers using two weak Larmor clocks,but one determining Re τ Ω , and the other | τ Ω | , for a tran-sition where Re τ Ω is zero, but | τ Ω | is not. To the firstresearcher the transition takes no time in Ω, to the sec-ond researcher this time is finite. Their subsequent argu-ment would have no resolution, as both would be rightabout their results, but both will be wrong in their finalconclusions. n) Can one expect the complex time (11) to occur inother applications? Only where the quantity of in-terest can be obtained by integrating the amplitudes A ( ψ F , ψ I , t , t | τ ) over τ . Some examples were given in[18] and [21]. o) Can there be other definitions of the tunnellingtime? In quantum mechanics, the failure to define oneunique tunnelling time does not mean that such timescannot be defined at all. On the contrary, it means thatthere are more possible time parameters, than in the clas-sical case [74]. Firstly, there are Re τ Ω , Im τ Ω , | τ Ω | alreadymentioned. Then there are weak values of other func-tionals, e.g., of τ in/out [ x cl ( t )] in Eq.(2). There are alsotimes not related to Feynman paths. One famous ex-ample is the phase time [9], which can be interpreted asthe weak value of the spacial shift with which the parti-cle leaves the scatterer, divided by the particle’s velocity[49]. Moreover, one can define other times, e.g., as themoments the front, the maximum, the rear, or the centreof mass of a wave packet passes through a chosen surfacein space [6]. The Pollack and Miller time [75], and thetimes mentioned in Sect. III, provide further examples.6 p) Can there be a unique tunnelling time scale? Thatis, could one leave aside all the details of the previousdiscussion, and simply be assured that tunnelling takesapproximately τ approx microseconds, so that all devicesusing it should not go faster that τ approx ? The answerin standard (Feynman) quantum mechanics appears tobe ”no”. If there were such a time scale, it could befound by examining the corresponding amplitude distri-bution A ( ψ F , ψ I , t , t | τ ). For example, for a particle ofa given energy, tunnelling across a rectangular barrier,the amplitude distribution is oscillatory, and exhibits afractal behaviour [21]. Hence, its Fourier spectrum con-tains all frequencies, and we cannot associate with itany specific time scale a priori . In a particular applica-tion, A ( ψ F , ψ I , t , t | τ ) may be integrated with a smoothfunction G ( τ ), whose width ∆ τ determines which of thehigher frequencies would be neglected. However, the pro-cess of making ∆ τ smaller will never converge to a re-sult which no longer depends on ∆ τ . Thus, we argue,any new tunnelling time measured in an experiment, orfound theoretically, should be used strictly in the partic-ular context it was obtained. For instance, a statement”the peak of the tunnelled wave packet has arrived at thedetector 1 fs. earlier than that of a free propagating one”is correct. Its extension ”... and, therefore, the particlehas spent 1 fs. less in the barrier” is unwarranted. Anyclaim to find the universal tunnelling time, or time scale,is likely to be misleading. q) And the classical time scale? One exception to p ) is the (semi) classical limit, where rapidly oscillat-ing A ( ψ F , ψ I , t , t | τ ) develops a very narrow stationaryregion around single classical value τ cl [21]. If so, thecontribution to any (within reason) integral over τ , in-volving A ( ψ F , ψ I , t , t | τ ), comes from the vicinity of τ cl .Appearance of a single stationary region signals, there-fore, return to the classical description. r) Could an extension, or alternative formulation ofquantum mechanics help define the traversal time in adifferent way? Such a theory will have also solved the”which way?” problem for the double-slit experiment. s) Did Bohm’s trajectories approach achieve that?.
One approach which claims to achieve that is the Bohm’causal interpretation [56], [76]. In Bohm’s theory, aparticle moves along a streamline of a probability cur-rent calculated with a time dependent wave function ψ ( x, t ), and its initial position is distributed accordingto | ψ ( x, t = 0) | . The streamlines cannot cross, and aBohm’s trajectory leading to a given point on the screenin the Young’s experiment always passes through one ofthe slits. Similarly, a particle crossing a region of spacealways spends there a unique amount of time. A de-tailed comparison between the Bohm’s trajectory and theFeynman path approaches to the tunnelling time problemwas made in [39], where the author concluded that thetwo approaches are incompatible. It is not our purposeto continue this discussion, and we will limit ourselvesto just two comments. Firstly, the unperturbed Bohm’strajectories do not help us with the analysis of the SWP clock, while the Feynman paths do. Bohm’s trajecto-ries are formulated in the absence of a measuring device,and must change once such a device is introduced, in or-der to describe its effects. Secondly, by using Feynmanamplitudes, one can define the time any quantum sys-tem spends in an arbitrary subspace of its Hilbert space.For example we can define and measure the time a qubitspends in one of its states [77], [78]. It is unclear howBohm’s approach can be extended to cover these cases.In summary, we have analysed the work of a weaklyperturbing Salecker-Wigner-Peres clock in terms of vir-tual Feynman paths, and related it to the complex traver-sal time first introduced in [17]. We have shown that inthe standard (Feynman) quantum mechanics the appear-ance of complex times in an inevitable consequence of theUncertainty Principle. We also explained why these com-plex times, or their real valued combinations, should notbe interpreted as physical durations, and tried to drawsome of more general conclusions about the state of thetunnelling problem in quantum theory. XVII. APPENDIX: A DIFFERENT CHOICE OFTHE INITIAL STATE FOR AN SWP CLOCK
It is worth clarifying one difference between our re-sults of Sect. X and those of [37]. According to Eq.(26)of [37], for a free running clock, as ω L →
0, we musthave T free Ω ∼ ω L , whereas according to our Eq. (44)is should be proportional to ω L . The reason is thatin [37] Leavens considered also choosing a different ini-tial state for the clock, replacing ( j is an integer) | β (cid:105) with | β j (cid:105) , and effectively postulated a negative duration τ (cid:48) k − j = ( φ k − φ j ) /ω L < | β k (cid:105) with 0 ≤ k < j . In this case, from Eq.(32)we have G SW P ( ω L τ | j, β k , β j ) = G SW P ( ω L τ | j, β k − j , β ),and | β k − j (cid:105) ≡ exp[ − i ˆ j z ( φ k − φ j )] | β (cid:105) , so that Eq.(34)becomes T (cid:48) Ω ( N , ψ I ) = j (cid:88) k =0 τ (cid:48) k − j P ( k − j, N ) , (82)which is also Eq.(20) of [37]. Proceeding as in Sect.IX, we find that, with this choice, the contribution toT (cid:48) Ω ( N , ψ I ), linear in ω L , vanishes, leaving T (cid:48) Ω ( N , ψ I )proportional to ω L as ω L →
0. For a freely runningclock, with the magnetic field introduced everywhere inspace, we have T (cid:48) free Ω ( t − t ) ∼ ( t − t ) . CalculatingT (cid:48) Ω ( N , ψ I ) to the first non-vanishing order in ω L , andcomparing the result with T (cid:48) free Ω ( t − t ), we find that thetime T (cid:48) SW P ( N , Ω , ψ I ), measured by the modified clock, isgiven by T (cid:48) SW P ( N , Ω , ψ I ) = W ( N , ψ I ) − / × (83) (cid:40) (cid:88) N ∈ N W ( N, ψ I )Re[ τ Ω ( N, ψ I ) τ ∗ ( N, ψ I )] (cid:41) / , XVIII. ACKNOWLEDGEMENTS
Support of MINECO and the European Regional De-velopment Fund FEDER, through the grant FIS2015- 67161-P (MINECO/FEDER) is gratefully acknowledged. [1] F. Krausz and M. Ivanov, Rev. Mod. Phys., . , 163(2009).[2] L. A. MacColl, Phys. Rev. , 621 (1932).[3] L.Torlina, F. Morales, J. Kaushal, I. Ivanov, A. Kheifets,A. Zielinski, A. Scrinzi, H. G. Mulle, S. Sukiasyan, M.Ivanov, and O. Smirnova, Nat. Phys., , 503 (2015).[4] N. Teeny, C. H. Keitel, and H. Bauke, Phys. Rev. A, ,022104 (2016).[5] A. Landsman and U. Keller, Phys. Rep. , 1, (2015).[6] E. H. Hauge and J. A. Stoevneng, Rev. Mod. Phys. ,917 (1989);[7] R. Landauer and Th. Martin, Rev. Mod. Phys. , 217(1994);[8] C. A. de Carvalho, H. M. Nussenzweig, Phys. Rep. ,83 (2002);[9] V. S. Olkhovsky, E. Recami and J. Jakiel, Phys. Rep. , 133 (2004);[10] H. G. Winful, Phys. Rep. , 1 (2006).[11] J. v. Neumann, Mathematical Foundations of QuantumMechanics (Princeton University Press, Princeton, 1955),p. 183.[12] D. Sokolovski and R. Sala Mayato, Phys. Rev. A ,042101 (2005); , 052115 (2006); , 039903(E) (2006).[13] D. Sokolovski, Phys. Rev. A , 062117 (2011).[14] D. Sokolovski, Phys. Rev. D , 076001 (2013).[15] D. Sokolovski, Mathematics, , 56 (2016), (Special Issue Mathematics of Quantum Uncertainty , open access.][16] M. Buettiker, Phys. Rev. B, , 6178, (1983).[17] D. Sokolovski and L. M. Baskin, Phys. Rev. A , 4604(1987).[18] D. Sokolovski and J. N. L. Connor, Phys. Rev. A ,6512 (1990).[19] D. Sokolovski and J. N. L. Connor, Phys. Rev. A ,1500( 1991).[20] D. Sokolovski and J. N. L. Connor, Phys. Rev. A , 4677(1993).[21] D. Sokolovski, S. Brouard. and J.N.L. Connor, Phys.Rev. A, , 1240, (1994).[22] D. Sokolovski, Phys. Rev. A, , R5(R), (1995).[23] D. Sokolovski, Phys. Rev. Lett., , 4946, (1997).[24] D. Sokolovski, Phys. Rev. A, , 042125, (2007).[25] D. Sokolovsky, in Time in Quantum Mechanics , edited byJ.G. Muga, R. Sala Mayato and I.L. Egusquiza (Seconded., Springer, Berlin, Heidelberg, New York, 2008), p.195.[26] A.I. Baz’, Yad. Fiz. , 252 (1966) [Sov. J. Nucl. Phys. ,182 (1967)].[27] A.I. Baz’, Yad. Fiz. , 229 (1967) [Sov. J. Nucl. Phys. ,161 (1967)].[28] V. F. Rybachenko, Yad. Fiz. , 895 (1967) [Sov. J. Nucl. Phys. , 635 (1967)].[29] J.P. Falck and E.H. Hauge, Phys. Rev. B, , 3287(1988).[30] C. R. Leavens and G. C. Aers, Phys. Rev. B, , 5387(1989).[31] C.S.Park, Phys. Lett. A , 741 (2013).[32] J. Kausha, F. Morales1, L. Torlina, M. Ivanov, and O.Smirnova, J. Phys. B: Atomic, Molecular and OpticalPhysics, , 234002 (2015).[33] H. Salecker and E.P. Wigner, Phys. Rev., , 571,(1958).[34] A. Peres, Am. J. Phys, , 553 (1980).[35] P.C.W. Davies, J. Phys. A , 2115 (1986).[36] C. Foden and K.W.H. Stevens, IBM J. Res. Dev., , 99(1988).[37] C.R. Leavens, Solid State Comm, , 781 (1993).[38] C.R. Leavens, W.R. McKinnon, Phys. Lett. A , 12(1994).[39] C.R. Leavens, Found. Phys., , 229 (1995).[40] D. Alonso, R. Sala Mayato and J.G. Muga, Phys. Rev.A, , 032105 (2003).[41] R. Sala Mayato, D. Alonso and I.L. Egusquiza, in Timein Quantum Mechanics , (Ref. 25), p. 235.[42] M. Calcada, J.T. Lunardi and L.A. Manzoni, Phys. Rev.A, , 012110 (2009).[43] J.T. Lunardi, L.A. Manzoni and A.T. Nystrom, Phys.Lett. A, , 415 (2011).[44] N. Teeny, C.H. Keitel, and H. Bauke, arXiv:1608.02854[quant-ph] (2016).[45] T. Zimmermann, S. Mishra, B.R. Doran, D.F. Gor-don, and A.S. Landsman, Phys. Rev.Lett., , 233603(2016).[46] D. Bohm, Quantum Theory , (Addison?Wesley, Reading,MA, 1965).[47] Y. Japha, and G. Kuritzki, Phys. Rev. A, , 586 (1996).[48] J.G. Muga and C.R. Leavens, Phys. Rep,, , 353,(2000).[49] D. Sokolovski and E. Akmatskaya, Ann. Phys. , 307(2013).[50] E. Pollak, Phys. Rev. Lett., , 070401, (2017).[51] E. Pollak, J.Phys.Chem.Lett., , 352, (2017).[52] V. Delgado, S. Brouard, and J.G. Muga, Solid StateComm., , 979, (1995).[53] C.R. Leavens, Superlattices and Microstructures., ,795 (1998).[54] R. P. Feynman and A. R. Hibbs, Quantum Mechanicsand Path Integrals , (McGraw-Hill, New York, 1965).[55] R. P. Feynman,
The Character of Physical Law , (MITpress, 1985)[56] P.R. Holland,
The Quantum Theory of Motion , (Cam- bridge University Press, Cambridge 1993).[57] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, andR. W. Boyd, Rev. Mod. Phys, , 307 (2014).[58] L. Vaidman, Found. Phys. , 895 (1996).[59] D. Sokolovski, Phys. Lett. A, , 1593 (2016).[60] To simplify the notations we assume that the particle isin a large box, so its states | N (cid:105) are discrete. As the sizeof the box tends to infinity, the sums over N should bereplaced by the corresponding integrals.[61] Note that W ( N, ψ I ) = 0 does not guarantee that thecorresponding term in the sum in Eq.(40) vanishes, since | τ Ω ( N, ψ I ) | = (cid:104) ψ (1) | ψ (1) (cid:105) /W ( N, ψ I ).[62] Y. Aharonov, S. Popescu, D. Rorhlich and L. Vaidman,Phys. Rev. A , 4084 (1993).[63] Y. Aharonov, A. Botero,S. Popescu, B. Reznik and J.Tollaksen, Phys. Lett. A , 130 (2002).[64] Y. Aharonov and L. Vaidman, in Time in Quantum Me-chanics , (Ref. 25), Vol. 1.[65] Y. Aharonov, S. Popescu, D. Rohrlich, and P.Skrzypczyk, New J. Phys., , 113015 (2013).[66] L. Vaidman, Phys. Rev. A , 052104 (2013). [67] A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman,Phys. Rev. Lett. , 240402 (2013).[68] D. Rohrlich, and Y. Aharonov, , Phys. Rev. A , 042102(2002).[69] D. Sokolovski, Phys. Lett. A, , 1097 (2015).[70] D. Sokolovski, Phys. Lett. A, , 227 (2017).[71] R. P. Feynman, R. Leighton, and M. Sands, The Feyn-man Lectures on Physics III (Dover Publications, Inc.,New York, 1989).[72] F.E. Low, Annln Phys. Leipzig , 660 (1998).[73] D. Sokolovski, Proc. Roy. Soc. A, , 4999, (2004).[74] E. Steinberg, in Time in Quantum Mechanics , (Ref. 25),p. 350.[75] E. Pollak, and W.H. Miller, Phys. Rev. Lett. , 115(1984).[76] D. Bohm, B. J. Hiley and P. N. Kaloyerou, Phys. Rep. , 323 (1987).[77] D. Sokolovski, Proc. R. Soc. Lond. A, , 1505, (2004).[78] D. Sokolovski, Phys. Rev. Lett.,102