Average clock times for scattering through asymmetric barriers
AAverage clock times for scattering through asymmetric barriers
Bryce A. Frentz , Jos´e T. Lunardi , and Luiz A. Manzoni Department of Physics, Concordia College, 901 8th St. S., Moorhead, MN 56562, USA Departamento de Matem´atica e Estat´ıstica, Universidade Estadual de Ponta Grossa. Avenida General Carlos Cavalcanti,4748. Cep 84030-000, Ponta Grossa, PR, BrazilReceived: / Revised version:
Abstract.
The reflection and transmission Salecker-Wigner-Peres clock times averaged over the post-selected reflected and transmitted sub-ensembles, respectively, are investigated for the one dimensionalscattering of a localized wave packet through an asymmetric barrier. The dwell time averaged over thesame post-selected sub-ensembles is also considered. The emergence of negative average reflection timesis examined and we show that while the average over the reflected sub-ensemble eliminates the negativepeaks at resonance for the clock time, it still allows negative values for transparent barriers. The saturationof the average times with the barrier width (Hartman effect) is also addressed.
PACS.
The search for a sensible definition of quantum tunneling times is one of the most enduring problems in quantummechanics, despite numerous efforts in the last few decades (see, e.g., [1,2,3,4,5,6,7,8,9,10] and references therein).The difficulty in obtaining a time scale for the tunneling problem lies in the well-known impossibility of defining a self-adjoint time operator canonically conjugated to the (bounded from below) Hamiltonian. Attempts to use a “tempus”operator that is canonically conjugated to the Hamiltonian but not given by the time evolution of the system, havelead to complex stationary times [11,12] (also see [13] for a review of methods attempting to define time operators).A more common approach to this problem is to obtain operational definitions of a parameter with the dimensionsof time and investigate its properties. These attempts have rendered several definitions of time that, while useful inspecific situations, are generally not considered the definitive answer to the question of how long it takes for a particleto tunnel through a potential barrier. Among the most common (and useful) stationary time scales considered in theliterature are the much studied phase time [14,15], dwell time [16,17], Larmor time [17,18,19] (also see [20,21]) andthe Salecker-Wigner-Peres (SWP) clock times [22,23,24,25].Recently, the SWP clock has been reconsidered and it was shown that, contrary to previous claims in the literature,it can be directly applied to interacting particles ( i.e., without the need for “calibration” [26,27]) provided that onefollows Peres’ [23] approach to associate the clock’s hand with the peak of the clock’s wave function [24]. In particular,this approach allowed the derivation, directly from Schr¨odinger’s equation, of a relationship between the dwell timeand the SWP clock reflection and transmission times without an interference term [24], thus ending the controversyabout the compatibility of such relationship with standard quantum mechanics (see, e.g., [1,2,3,4] and referencestherein).The SWP clock has also been used to treat the scattering of a wave packet by a potential and to obtain, by makinguse of a post-selection of the final state, a definition for an average traversal (reflection) time [28]. It was shown that,for wave packets of finite width, such average transmission time does not saturate in the limit of opaque barriers forsymmetric potentials, which is an important result associated with the Hartman effect [15]. One must notice, however,that the absence of saturation is not enough to prevent apparent superluminal speeds in the relativistic case [29] –but, as pointed by Davies [30], the SWP clock times can be interpreted as weak values [31,32], and such results arenot unexpected for weak values when they are associated with small probabilities. a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] a r X i v : . [ qu a n t - ph ] D ec Bryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers
The results in [28,29] were obtained considering symmetric potentials, in which case the average transmission(reflection) SWP times coincide with the average of the dwell time over the transmitted (reflected) sub-ensemble [28](as a consequence of the identity between the stationary SWP and dwell times for such potentials). Therefore, in orderto better understand the properties of the average
SWP clock times introduced in [28], it is necessary to analyze asituation in which they differ from the dwell time averaged over the corresponding sub-ensemble , such as for asymmetric potentials.Tunneling through asymmetric potentials has important applications in semiconductor heterostructures (see [33]and references therein) and it has been investigated due to the possibility of negative reflection times. In this context,the authors of [34,35] show that the asymmetry of the potential can give rise to negative reflection phase times inthe stationary regime (see also [36] for an investigation involving asymmetric photonic band gaps). A partial analysisof the stationary SWP clock times for an asymmetric barrier appeared in [37], in which the authors considered acomparison between phase, dwell and the B¨uttiker (or Larmor) times – the stationary SWP clock time coincides withone of Buttiker’s characteristic times (namely, τ y in [37]).In this work, we extend the approach introduced in [28] and investigate the properties of the average reflectionand transmission SWP times for a non-relativistic particle tunneling through an asymmetric potential barrier. Theparticle is represented by a gaussian wave packet (in a fully time-dependent treatment), and for comparison we alsoconsider the average of the dwell time over the post-selected transmitted and reflected sub-ensembles.This paper is organized as follows. In Section 2 we briefly review the essentials of the SWP clock to measurereflection and transmission times for a wave packet scattered by a potential barrier, introducing the modificationsnecessary to deal with the asymmetric potential that interest us here. Section 3 presents our results for the averageclock times and the comparison with the dwell time averaged over a final sub-ensemble. Finally, Section 4 is reservedto our concluding remarks. The Salecker-Wigner-Peres clock [22,23] is essentially an external quantum rotor coupled to a particle, which measuresthe time interval spent by the particle in a certain region of interest. This is accomplished by introducing an interactionbetween the clock and the particle given by H int = P ( z ) (cid:0) − i (cid:126) ω ∂∂θ (cid:1) , where 0 ≤ θ < π and ω ≡ π/ ( N τ ). The parameter τ corresponds to the clock’s resolution and N = 2 j + 1 is the Hilbert’s space dimension ( j is a non-negative integer orhalf-integer); P ( z ) = 1 if z ∈ ( z , z ) and zero otherwise. It can be shown [23] that the effect of this coupling to thestationary state of a particle with energy E interacting (in one dimension) with a potential V ( z ) is the addition of apotential barrier V m = m (cid:126) ω in the region z ∈ ( z , z ), with m = − j, . . . , + j .Restricting ourselves to potentials V ( z ) with constant (but not necessarily equal) values in the regions ( −∞ , z ),( z , z ), and ( z , ∞ ), which will be named respectively as regions I, II and III, and assuming the particle to be incidentfrom the left of z , the solution of the time-independent Schr¨odinger equation with potential V ( m ) ( z ) ≡ V ( z )+ V m P ( z )can written as ψ ( m ) I ( z ) = e ik z + B ( m ) ( k ) e − ik z ; ψ III ( z ) = C ( m ) ( k ) e ik z , (1)where ψ I and ψ III indicate the wave function in regions I and III, respectively. In (1) we defined k ≡ ( k , k , k ), with k , k and k being the respective wave numbers in regions I, II, and III.The transmission time measured by the SWP clock in the stationary case is then [23,24] t Tc ( k ) = − (cid:126) (cid:18) ∂∂ V m ϕ ( m ) T ( k ) (cid:19) V m =0 , (2)where ϕ ( m ) T is the phase of the amplitude C ( m ) ( k ) in the presence of the perturbation. A similar expression is obtainedfor the stationary reflection time by using the phase of the reflection amplitude B ( m ) ( k ).In [24] it was shown that, as a consequence of Schr¨odinger equation, the stationary SWP clock times and the dwelltime ( τ D ) satisfy the following relationship τ D ( k ) = T ( k ) t Tc ( k ) + R ( k ) t Rc ( k ) , (3)which will prove useful in what follows. In this expression T ( k ) is the weak perturbation limit ( V m →
0) of T ( m ) ( k ) = k k (cid:12)(cid:12) C ( m ) ( k ) (cid:12)(cid:12) , which is the transmission coefficient in the presence of the clock’s perturbation. In the same way, R = lim V m → R ( m ) , with R ( m ) being the reflection coefficient in the presence of the clock . Notice that in [24], T and R refer to transmission and reflection amplitudes instead of coefficients as here. This slight changein notation is convenient in order that (3) applies immediately also to asymmetric potentials.ryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers 3 In this work we will be mainly concerned with the average tunneling times for localized incident particles representedby a wave packet of the form Φ ( z, t ) = (cid:90) dk √ π A ( k ) e i ( k z − E ( k ) t/ (cid:126) ) . The initial wave function representing the particle-clock system is given by Ψ in ( z, t, θ ) = Φ ( z, t ) v ( θ ), where the clockis chosen to be in a state v ( θ ) strongly peaked at θ = 0 [23]). The wave function transmitted to the right of thepotential will have the clock and the particle in an entangled state which, assuming a weak coupling between the clockand the potential, is given by [28] Ψ tr ( z, t, θ ) = (cid:90) dk √ π A ( k ) C ( k ) e i ( k z − E ( k t (cid:126) ) v ( θ − ωt Tc ( k )) . The average transmission time can be obtained by post-selecting Ψ tr as the final state, in addition to the above pre-selection of the initial state, tracing out the the particle’s degrees of freedom long after the interaction has occurredand, finally, taking the expectation value of the operator giving the peak of the clock’s eigenfunctions (for details see[28]). The resulting average transmission time is (cid:104) t Tc (cid:105) = (cid:90) dk √ π ρ T ( k ) t Tc ( k ) , (4)where ρ T ( k ) is the probability density that a particle with wave number k be found in the ensemble of transmittedparticles, which is given by ρ T ( k ) = | A ( k ) | T ( k ) (cid:82) dk √ π | A ( k ) | T ( k ) . (5)The corresponding expression for the average reflection time can be obtained through a similar procedure, in whichone post-select the reflected asymptotic sub-ensemble, and the probability density that the particle of wave number k be found in this ensemble is obtained from (5) by making the change T → R .It should be noticed that an analysis of the Larmor clock for wave packets leads to a result similar to (4) [38]as one of the possible time scales. Similarly, (4) is the real part of the complex average time obtained in [39]. Otherformalisms exist leading to complex stationary times whose real part coincides with (2), such as the one presentedby Fertig in [40] (in fact, the complex time scale obtained in [40] is the same introduced in [41]) – these approachespresumably would also lead to (4) as the real part of a complex average time when applied to wave packets. Theadvantage of the SWP clock formalism when compared with those is that it provides a simple way to obtain a unique ( real ) average time scale.It is important to mention that other approaches exist in the literature to treat the tunneling of wave packetsthrough a potential barrier that lead to average times not directly related to the above mentioned averages. In [42]the authors formally develop the center-of-mass clock, which to leading order leads to the phase time for a wavepacket narrow in the k -space. A method developed in [43,44] by using Green functions and which renders a complexstationary time similar to B¨uttiker’s (but with sensitivity to the energy E rather than to the potential) was extendedin [45] to treat wave packets, with the result of obtaining an average of the phase time. The latter result is similar tothe results obtained in [38], with the main difference that in [45] the authors adopt a cutoff in energy at V – sucha cutoff poses difficulties with respect to the localization of the tunneling particle [28,38,42]. For an analysis of thephase time averaged over the transmitted wave packet for symmetric potentials, see [46] (we know of no such analysisfor asymmetric potentials).As it is well-known, for symmetric potentials the stationary SWP transmission and reflection times coincide [17,38], and from (3) both these times coincide with the dwell time. As a consequence, for symmetric potentials the aboveaverage transmitted (reflected) time is just the dwell time averaged over the transmitted (reflected) sub-ensemble [28].However, the fact that for asymmetric potentials the stationary SWP clock times are different from the dwell time[38] has important consequences for the sub-ensemble averages above defined. Thus, in order to fully understand theproperties of the average
SWP clock times, it is important to investigate how they deviate from an average of thedwell time over the same sub-ensembles, namely (cid:104) τ D (cid:105) T ( R ) = (cid:90) dk √ π ρ T ( R ) ( k ) τ D ( k ) . (6)For symmetric potentials these average dwell times coincide with the average times defined in (4)-(5) (and the corre-sponding for the reflected sub-ensemble), but in the case considered here (asymmetric potentials) they have differentproperties (see next section). In particular, the average times defined in (6) are also good candidates to be inter-preted as transmission (reflection) times, since they clearly distinguish between the transmitted and reflected particlesthrough the post-selection of the final state. Bryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers a zV V Fig. 1.
An asymmetric potential of the type considered in this work, with | V | < V ; V can be positive or negative. Ensemble averages similar to (4)–(6) can, in principle, be obtained for any well defined stationary time scale (see,e.g., [38,39,46,47]). Finally, it is worth noticing that in the limit of spatially very wide wave packets the averages(4)-(6) tend to the corresponding stationary times.
Let us consider the scattering of a non-relativistic particle of mass µ and energy E ( k ) ≡ (cid:126) k µ by an asymmetricpotential of the form V ( z ) = V Θ ( z ) Θ ( a − z ) + V Θ ( z − a ), where V is assumed to be a positive constant and V > | V | , with V also constant (see Fig. 1). We consider that the particle is incident from the left and that theSWP clock runs only while the particle is in the region (0 , a ) – in this case, the derivative with respect to V m in (2)coincides with the derivative with respect to V . Also, as it is well known, for this potential the transmission andreflection coefficients are T ( k ) = k k | C ( k ) | and R ( k ) = | B ( k ) | , respectively; the wave numbers in the potential andtransmitted regions are related to k by k = (cid:126) (cid:112) µ [ E ( k ) − V ] and k = (cid:126) (cid:112) µ [ E ( k ) − V ].The transmission and reflection amplitudes for the asymmetric potential above are easily calculated and canbe found, for example, in [35,37]. The phase ϕ T ( k ) of the transmission amplitude can be obtained from C ( k ) = | C ( k ) | e i [ ϕ T ( k ) − k a ] and, for propagating modes , it is given by ϕ T ( k ) = tan − (cid:20) ( k + k k ) k ( k + k ) tan( k a ) (cid:21) . (7)Following a notation similar to that in [35], the reflection amplitude can be written as B = CGe ik a and, if ϕ ( k )indicates the phase of G ( k ), it follows that B ( k ) = | C ( k ) G ( k ) | e iϕ R ( k ) , where the reflection phase is given by ϕ R ( k ) = ϕ T ( k ) + ϕ ( k ). The explicit expression for ϕ is obtained from (7) by substituting k → − k .From (2) and (7) it follows that the stationary SWP transmission time for propagating energies is given by t Tc ( k ) = µ ( k + k ) (cid:126) k (cid:2) k a ( k + k k ) sec ( k a ) + ( k − k k ) tan( k a ) (cid:3)(cid:2) k ( k + k ) + ( k + k k ) tan ( k a ) (cid:3) . (8)The corresponding expressions for evanescent modes can be obtained from the above equations by making the sub-stitution k → iq , with q = (cid:126) (cid:112) µ [ V − E ( k )], as usual, provided that E ( k ) > V . If V > E ( k ) < V thetransmission time is not defined, since there is no transmitted wave at z > a .The SWP clock reflection time can be obtained by defining t ( k ) ≡ − (cid:126) ∂ϕ ∂V and noticing that t Rc ( k ) = t Tc ( k ) + t ( k ) . (9)The auxiliary time t ( k ), which can be obtained from (8) by making k → − k , vanishes in the symmetric case andallows the separation of the contributions to t Rc ( k ) due to the asymmetry of the potential (analogous to what happensfor the phase time [35]). In addition, from (3) and conservation of probability, one can write the dwell time as τ D ( k ) = t Tc ( k ) + R ( k ) t ( k )= t Rc ( k ) − T ( k ) t ( k ) , (10)which makes clear that the dwell and SWP clock times differ only in the asymmetric case. ryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers 5 a - times,R H a L R, V > < t D , V > t D , V < t cR , V > t cR , V < Fig. 2.
Typical behavior of the stationary reflection clock time and the dwell time (thick curves) with respect to the barrierwidth a . The reflection coefficient (thin curve) is also shown in plots (a) (scaled by a factor 200) and in plot (c) (scaled by afactor 2). All quantities are expressed in atomic units (a.u.), µ = 1 and (cid:126) = 1, with V = 0 .
30 and | V | = 0 .
15. (a) propagatingcase, with E ( k ) = 0 .
35. (b) detail of plot (a) for small values of the barrier thickness. (c) evanescent case, with E ( k ) = 0 . The SWP clock provides a “residence” time, i.e., it reads the time spent by particle under the potential, similarly tothe dwell time. However, while the dwell time averages over the reflected and transmitted channels, eqs. (9)-(10) showthat the SWP clock has the advantage of distinguishing between the final channels in the stationary case – evidently,such distinction can only occur for asymmetric potentials, since by symmetry the two channels must give the sameresults for symmetric potentials.It is interesting to notice that a relationship can be established between the dwell time and ν ( E ), the averageelectronic density of energy per unit length [43,44,48,49]. Temporarily indicating the explicit dependence of amplitudesand times in terms of the energy and making use of (3), such relationship can be generalized as2 π (cid:126) aν ( E ) = 2 T ( E ) t Tc ( E ) + R − ( E ) t Rc − ( E ) + R + ( E ) t Rc + ( E ) , where the − (+) indicates incidence from the left (right) and we used the fact that the transmission amplitude isnot affected by the incidence direction. For symmetric potentials the above expression clearly reduces to t T ( R ) c ( E ) = τ D ( E ) = π (cid:126) aν ( E ).A partial analysis of the stationary SWP times for asymmetric potentials appears in [37], which only considers theequivalent to t Tc ( k ), as one of the time scales associated with the Larmor time. Hence, before considering the averagetimes introduced in the previous section it is important to complement those investigations by analyzing the propertiesof t Rc ( k ).Figure 2 displays typical plots of the stationary SWP reflection and dwell times; in addition, the coefficient ofreflection R ( k ) is also shown. Figure 2(a) considers a propagating energy and Fig. 2(b) shows a detail of 2(a) fortransparent barriers. We observe that at the vicinity of the resonances, which occur at k a = nπ with n a positiveinteger [35,37], the reflection time shows large peaks that can be positive or negative depending on the sign of V .In fact, at the resonance t Rc ( k ) = − (2 µk a/ (cid:126) k )( V − V ) /V and we see that the peak has the opposite sign of V .Figure 2(c) shows a plot for evanescent energies and, as expected, t Rc ( k ) and τ D ( k ) saturate to the same value foropaque barriers regardless of the sign of V [see (10)]. In the limit of transparent barriers, k a (cid:28)
1, the SWP reflectiontime behaves as t Rc ( k ) ∼ (cid:126) k a/V , and it is negative for V <
0, for both propagating and evanescent modes, as it isclearly shown in Figs. 2(b) and 2(c) (it is worth noticing that this is exactly the opposite of what happens with thephase time, which is negative for transparent barriers only if V > Bryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers Fig. 3.
Behavior of the average transmission clock time, (cid:104) t Tc (cid:105) , and the average dwell time, (cid:104) τ D (cid:105) T , with respect to the barrierwidth a . The stationary times t Tc ( k ) and τ D ( k ), corresponding to the central wave number k , and the total probability oftransmission of the wave packet ( P T ; scaled by a factor 30) are also shown. All quantities are expressed in atomic units: σ = 10, z = − σ , V = 0 .
30 and | V | = 0 .
15. (a) E ( k ) = 0 . V >
0. (b) E ( k ) = 0 . V <
0. (c) E ( k ) = 0 . V >
0. (d) E ( k ) = 0 . V < is small [equivalently, large T ( k )], i.e., for transparent barriers and close to resonances – this is, of course, a directconsequence of (10).Let us turn to the main subject of this work and focus on the average times and consider a right-moving Gaussian wave packet centered around a tunneling wave number k > σ , and whichat t = 0 is spatially centered around z <
0. The k -space representation of such wave packet is given by A ( k ) =(2 σ /π ) exp (cid:8) − iz ( k − k ) − σ ( k − k ) (cid:9) .In Fig. 3 we compare the behavior of the average clock transmission time, (cid:104) t Tc (cid:105) , and the dwell time averagedover the transmitted sub-ensemble, (cid:104) τ D (cid:105) T . Figures 3(a) and 3(b) refer to a wave packet with central energy E inthe tunneling region. The corresponding stationary times evaluated at the central wave number, t Tc ( k ) and τ D ( k ),are shown to provide a reference as to the saturation width and we also show the total transmission probability P T = (cid:82) dk √ π | A ( k ) | T ( k ) associated with the wave packet. It can be seen that (cid:104) t Tc (cid:105) and (cid:104) τ D (cid:105) T behave qualitatively invery similar ways, with small quantitative differences, especially for thin barriers (in which case, their behavior is alsovery close to the corresponding stationary times). It is observed that for a finite dispersion σ neither (cid:104) t Tc (cid:105) nor (cid:104) τ D (cid:105) T saturate with the barrier width “ a ” but, in fact, both average times tend to behave linearly with the barrier widthin the extreme opaque regime – however, both times show a residual consequence of the Hartman effect in the formof a very slow growth behavior for intermediate regions of the barrier width. It is worth noticing that the larger σ the larger will be this intermediate region (for σ → ∞ one recovers the saturated result of the stationary case). Theabove results are consistent with those obtained in [28], which analyzed (cid:104) t Tc (cid:105) for symmetric potentials (also see [46],which considered an analogous average for the phase time). It should be noticed that the slow growth region, whichstarts around the saturation width for the stationary time, is characterized by relatively small probabilities for theparticle transmission. Figures 3(c) and 3(d) correspond to an initial wave packet centered on a propagating energy.Again, we observe that (cid:104) t Tc (cid:105) and (cid:104) τ D (cid:105) T behave similarly over the whole range of barrier widths and tend to coincidefor thin barriers.The quantitative difference between (cid:104) τ D (cid:105) T and (cid:104) t Tc (cid:105) is given by the average of R ( k ) t ( k ) over the transmittedensemble, since from (6) and (10) it follows that (cid:104) τ D (cid:105) T = (cid:104) t Tc (cid:105) + (cid:104) Rt (cid:105) T . As a consequence of the product R ( k ) T ( k )in the numerator of the average (cid:104) Rt (cid:105) T , the differences between (cid:104) τ D (cid:105) T and (cid:104) t Tc (cid:105) tend to be more pronounced in the ryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers 7 Fig. 4.
Behavior of (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R with respect to the barrier width a for an initial wave packet centered on a tunnelingenergy E ( k ). Except for the dispersion σ , all the parameters are the same as in Figs. 3(a) and 3(b). (a) σ = 10, V >
0. (b) σ = 20, V >
0. (c) σ = 10, V <
0. (d) σ = 20, V <
0. (e) detail of the behavior of (cid:104) t Rc (cid:105) for thin barriers and V <
0, for severalvalues of the dispersion σ . In (a)-(d) we also plotted the total probability P R for the particle reflection (scaled by a factor 4)and the stationary times t Rc ( k ) and τ D ( k ). extreme opaque region, when it is safe to assume R ( k ) (cid:39)
1. Then, in the extreme opaque limit (cid:104) t Tc (cid:105) > (cid:104) τ D (cid:105) T for V > (cid:104) Rt (cid:105) T becomes negative in this limit – see Figs. 3(a) and 3(c). For V < (cid:104) Rt (cid:105) T reverses and (cid:104) t Tc (cid:105) < (cid:104) τ D (cid:105) T in the extreme opaque limit, as can be observed in Figs. 3(b) and 3(d). For transparentbarriers (cid:104) τ D (cid:105) T is slightly larger (smaller) than (cid:104) t Tc (cid:105) for V > V < | V | (cid:28) V and | V | (cid:28) E the sign of V has little influence and the averages (cid:104) t Tc (cid:105) and (cid:104) τ D (cid:105) T are virtuallyidentical, as one would expect from the fact that in the symmetric case the SWP transmission time and the dwell timecoincide [see eq. (10) and notice that t ( k ) → (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R with respect to the barrier width a , for an initialwave packet peaked on a tunneling energy [the central energy E ( k ) is such that V > E ( k ) > V ]. Figures 4(a) and4(b) correspond to V >
0, and in this case we observe that both (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R are non-negative, as expected sincethey consist of an average over non-negative stationary times; for thin barriers both these average times are close tothe respective stationary times evaluated at the central wave number, because in this case the relative wave numbercomposition of the reflected wave packet does not change significantly. For opaque barriers it follows from (10) that thestationary times coincide ( T ∼ (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R also coincide. Figures 4(c) and 4(d) assume V <
0, and their most striking feature is that (cid:104) t Rc (cid:105) in general assumes negative values for transparent barriers. Thisfollows from the fact that, as mentioned above, for thin barriers the average time (cid:104) t Rc (cid:105) closely resembles the stationarytime, t Rc ( k ), which is negative for very thin barriers whenever V <
0. Notice that the region in which the averageclock reflection time can be negative is characterized by relatively small probabilities of reflection.
Bryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers Fig. 5.
Typical behavior of the average times (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R versus the barrier width a , for an initial wave packet centeredon a propagating energy. The corresponding stationary times and the total probability of reflection ( P R ; scaled by a factor 100)are also plotted for reference. The parameters in (a) and (b) are the same as in Figs. 3(c) and 3(d), respectively. Figure 4 also illustrates the dependence of (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R with the spatial dispersion σ . It is seen that for initialwave packets sharply peaked at a tunneling mode k [large σ ; see Figs. 4(b) and 4(d)] both average times tend tosaturate to the same value as the corresponding stationary times. On the other hand, for smaller σ ’s the contributionsof the over-the-barrier modes become important, leading to the slow growth of both these average times [see Figs. 4(a)and 4(c)]. In addition, Fig. 4(e) shows that for transparent barriers, with V < (cid:104) t Rc (cid:105) is little affected by the wavepacket spatial dispersion and it is very close to the corresponding stationary time for σ between 3 a.u. and 10 a.u.Significant deviations from t Rc ( k ) occur only for very small values of σ but in this case negative wave numbers begin toplay an important role in the wave packet composition invalidating the derivation of (cid:104) t Rc (cid:105) , which assumes only positivewave numbers [28]. Finally, from Figs. 4(a)-(d) we see that when the total probability of reflection is high ( P R (cid:39) (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R tend to coincide.Figure 5 shows the typical behavior of the average times (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R with respect to the barrier width a , foran initial wave packet centered on a propagating energy, E ( k ). A finite dispersion σ has the effect of smoothing outthe peaks (negative or positive) of (cid:104) t Rc (cid:105) at resonances (with respect to k ), in such a way that for thick barriers thequalitative behavior of (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R is similar [the values of these two averages tend to be closer as the probabilityof reflection increases, as expected from (10)]. For wave packets sufficiently localized in the configuration space (cid:104) t Rc (cid:105) is always positive for thick barriers. In the region of very thin barriers both (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R behave similarly to thecorresponding stationary times, and this implies that (cid:104) t Rc (cid:105) assumes negative values for a sufficiently thin barrier when V < (cid:46) We considered the one dimensional scattering of a wave packet through a static asymmetric potential and investigatedthe behavior of the average transmission and reflection SWP clock times, which were compared to the behavior of thedwell time averaged over the transmitted and reflected sub-ensembles, respectively.We evaluated the average transmission time for an initial wave packet centered on a tunneling energy and verifiedthat it does not saturate in the opaque regime; in fact, (cid:104) t Tc (cid:105) increases linearly with the barrier width in the extremeopaque limit, a result similar to the one obtained for symmetric potentials. We observed that (cid:104) t Tc (cid:105) behaves qualitativelyin a very similar way to (cid:104) τ D (cid:105) T , the dwell time averaged over the transmitted sub-ensemble, for wave packets centered onboth evanescent and propagating energies – both these average times provide good scales for the particle transmissionacross an asymmetric barrier.The clock’s reflection time was also investigated. In the stationary case it allows for negative values in two situations:for V < V > (cid:104) t Rc (cid:105) for a localized wave packet and concluded that the negative values persist in the average time forthin barriers, because in this case the average behaves very similarly to the stationary time (such a result is associatedwith a relatively small probability of reflection). On the other hand, (cid:104) t Rc (cid:105) behaves well at the vicinities of resonancesof its central component; in fact, the average has the effect of smoothing out the (positive or negative) peaks foundin the stationary case. This can be understood by observing that resonant modes have a very small probability ofreflection, and then do not contribute significantly to such an average.In general, for non-transparent barriers, the average clock time of reflection behaves in a (qualitatively) similarway to the dwell time averaged over the reflected sub-ensemble (although, of course, there are quantitative differenceswhich could only be settled one way or the other on an experimental basis). Thus, based on the properties described ryce A. Frentz et al.: Average clock times for scattering through asymmetric barriers 9 in the previous section, both (cid:104) t Rc (cid:105) and (cid:104) τ D (cid:105) R are potentially good time scales to describe the reflection of well localizedparticles by asymmetric barriers. The fact that (cid:104) t Rc (cid:105) may lead to negative values for transparent barriers, dependingon the parameters, does not discard it as a good scale since these values are in general associated with relativelylow probabilities and may be explained using the weak measurement theory [31,32,50] – it is well known that the stationary SWP times are weak values [30,51]; however, an in-depth study of the SWP clock for wave packets inthe context of the weak measurement theory is still necessary to clarify this point. Finally, (cid:104) t R ( T ) c (cid:105) has the advantage(with respect to (cid:104) τ D (cid:105) R ( T ) ) that in the stationary limit the SWP clock times do distinguish between the reflected andtransmitted channels, while the dwell time is an average over these channels. The authors would like to thank two anonymous referees for several references and suggestions to improve the manuscript.This work was partially supported by NASA Minnesota Space Grant Consortium (B.A.F. and L.A.M.) and NSF STEP grant
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