Variational approach to the tunneling-time problem

Variational approach to the tunneling-time problem

Christian Bracher, Manfred Kleber, and Mustafa Riza

Physik-Department T30, Technische Universita¨t Mu¨nchen, James-Franck-Straße, D-85747 Garching, Germany 共Received 23 February 1999兲

Tunneling problems are characterized by different quantum time scales of motion. In this paper, we identify a tunneling time scale, which is based on a simple variational principle. The method utilizes the stationary eigenfunctions for a given one-dimensional potential structure, and it provides a truly local definition of the tunneling time, independent of the asymptotic shape of the potential. We express the minimum tunneling time in terms of the more common time scales obtained from the Larmor clock setup. Asymptotic formulas for both the extreme quantum and the semiclassical limit are presented. As an experimental verification of the varia-tional approach we demonstrate that the minimum tunneling time governs the time a particle requires to traverse the barrier in a symmetric double-well structure.关S1050-2947共99兲04209-2兴

PACS number共s兲: 03.65.⫺w, 73.40.Gk

I. INTRODUCTION

In recent years, one of the most controversial debates re-garding the foundations of physics dealt with the time spent by a quantum particle traversing a given sector in space. Even though the problem is not limited to potential barrier penetration, the issue became notorious as the ‘‘tunneling time problem.’’ Despite decades of discussion, no unani-mously accepted solution emerged; rather, differing propos-als for the quantum-mechanical sojourn time abound关1–3兴. Lately, experiments indicating superluminal transmission of photons through ‘‘tunneling barriers’’ built of mismatched wave guides 关3–5兴 and multilayer mirrors 关6兴 stirred re-newed interest in the problem.

Thus, we are faced with the curious situation that to a seemingly definite and simple question, various answers pre-vail that are not necessarily compatible with each other. It seems that much of the trouble in defining the tunneling time is rooted in our conception of a ‘‘clock.’’ The notion of an external stopwatch timing some process without influencing the event is an intrinsically classical idea, and one should not be surprised that this classical picture breaks down in the quantum limit. In fact, there is no unifying ‘‘clock principle’’ in the quantum realm, but every attempt to identify the evo-lution of some physical observable with the elapsed time leads to its own proprietary set of quantum time scales of motion. Evidently, the readings of these quantum clocks are supposed to match in the classical limit. However, there is no way to reverse this process and select a unique ‘‘proper’’ quantum clock. Rather, the tunneling time depends on how one sets out to measure it.

Our subjective assessment of the situation delivers a mixed message. On the one hand, it obviously implies that the quest for a definite tunneling time is doomed to failure. Yet, it also opens the way for further alternative definitions of the tunneling time. In order to be useful, rather than being mere theoretical constructs these newly proposed quantities should be based on a physical property of the tunneling pro-cess under consideration.

The possibility of tunneling is almost inevitably con-nected with the process of reflection. This wave-mechanical feature entangles the tunneling-time problem with the prob-ability and the duration of particle reflection from a potential

structure. For example, in the case of symmetric potential barriers, the Wigner phase time for tunneling 关7兴 and for reflection are identical quantities. The same property is valid for the corresponding phase times obtained from the Larmor clock approach 关8–10兴. Such a result is not easily inter-preted, in particular for extended barriers, where reflection dominates and particles apparently are accelerated in tunnel-ing 共Hartman effect兲. However, there is one situation where tunneling without reflection occurs 关11兴: In resonant tunnel-ing through the central barrier of a symmetric double-well potential, the quantum particle may oscillate between the two wells. This problem is unique in the sense that this transmis-sion process periodically takes place with unit probability, while reflection is absent.

In our contribution, we elucidate how the switching time, which is observed in an experiment as a splitting of degen-erate energy levels in the symmetric double well, is con-nected to a general expression␶min(E) for the tunneling time that we will denote the minimum tunneling time as it yields a time scale for stationary tunneling processes founded upon a simple variational principle. Having introduced the time scale, which is related in concept to the dwell time ␶S(E) originally devised by Smith 关12兴, we proceed to derive ex-plicit expressions for ␶min(E), including a representation in terms of the set of Larmor clock times, the workhorse for comparisons between different quantum clocks. Subse-quently, we inquire into the properties of the minimum tun-neling time, in particular, its asymptotic behavior. Finally, the results of the theory are illustrated by means of some simple examples.

II. COMMON APPROACHES

It is instructive to first present a brief overview of some common definitions for the tunneling time, which we will use to motivate our proposal of ␶min(E). We also state the results of the Larmor clock model in a compact fashion 关8–10兴 for reference purposes. 共As indicated above, nothing is implied by not mentioning some of the major approaches to the tunneling-time problem in this section.兲

A. The Bohmian dwell time

Here, we are concerned with a nonrelativistic description of one-dimensional stationary quantum motion. Hence, our PRA 60

object of interest is the continuous, doubly degenerate spec-trum of eigenstates ⌿E(x) of the stationary Schro¨dinger equation 共for the sake of simplicity, let us omit any vector potential兲:

冋

⫺ ប 2 2m ⳵2 ⳵x2⫹U共x兲

册

⌿E共x兲⫽E⌿E共x兲. 共1兲

To each solution⌿E(x), we assign as a functional the 共spa-tially constant兲 probability current j关⌿E兴 defined as usual by

j关⌿E兴⫽ ប

mIm

冋

⌿E*共x兲

⳵

⳵x⌿E共x兲

册

. 共2兲

Now, a conceptually simple definition for the time spent by a quantum particle in the interval a⬍x⬍b may be given by the following scheme:

␶D关⌿E兴⫽ 1 兩 j关⌿E兴兩

冕

a

b

dx兩⌿E共x兲兩2. 共3兲

Formally, ␶D关⌿E兴 denotes the time required by the current

j关⌿E兴 关Eq. 共2兲兴 to replace the particles present in the interval 共the barrier兲 a⬍x⬍b. This scaling invariant expression is therefore inspired by a hydrodynamical model of quantum mechanics, so it should not come as a surprise that in the framework of the Bohm interpretation of quantum mechan-ics, the tunneling time takes on form共3兲 关13兴.

B. Smith’s dwell time

The original approach by Smith 关12兴 subtly differs from the above development. Smith was interested in isolated po-tential barriers, which are limited to the range a⬍x⬍b; oth-erwise, U(x) should vanish 共see Fig. 1兲. In this situation, those special solutions ⌿_E˜(x), ⌿_E—(x) of Eq. 共1兲, which behave as outgoing waves for x˜⫾⬁, play a prominent role. Clearly, outside the barrier these eigenstates are entirely fixed by the reflection and transmission amplitudes of the potential U(x) (k2⫽2mE/ប2):

⌿E˜共x兲⬀

再

eikx_{⫹兩R共E兲兩e}i␳(E)_e⫺ikx _共x⬍a兲

兩T共E兲兩ei␦(E)_eikx _共x⬎b兲, 共4兲 ⌿E—共x兲⬀

再

兩T共E兲兩ei␦(E)_e⫺ikx _共x⬍a兲

e⫺ikx⫹兩R共E兲兩ei␴(E)eikx 共x⬎b兲. 共5兲

In passing we note that the complex reflection and transmis-sion amplitudes R(E) and T(E) are not completely indepen-dent, but subject to restrictions imposed by unitarity require-ments. These not only enforce the equality of the left-hand

and right-hand transmission amplitudes T(E) in Eqs.共4兲 and 共5兲, but also interrelate reflection and transmission quantities 关10兴:

兩R共E兲兩2_{⫹兩T共E兲兩}2_{⫽1, 共6兲} ␳共E兲⫹␴共E兲⫽␲⫹2␦共E兲. 共7兲 Later on, we will connect these relations to the readings of the Larmor clock.

In the asymptotic sectors x⬍a and x⬎b we may interpret the total current j关⌿_E兴 for eigenstates 共4兲 and 共5兲 as the difference of an incoming current jincand a reflected current

jrefl. In Smith’s original definition of the dwell time, in Eq. 共3兲 the total current j关⌿E兴 is replaced by the incoming cur-rent jinc. Consequently, Smith’s dwell times ␶S˜(E) and ␶S—_{(E) 共which generally differ for nonsymmetric barriers兲}

are linked to our definition of ␶D关⌿E兴 via the transmission probability兩T(E)兩2 of the barrier:

␶S˜_共E兲⫽ 1

jinc

冕

a b

dx兩⌿_E˜共x兲兩2⫽兩T共E兲兩2␶D关⌿_E˜兴. 共8兲

关An analogous relation holds for␶S—(E).兴

Obviously, unlike prescription 共3兲, Smith’s original defi-nition of the dwell time ␶S(E) works only for finite-range potential barriers and outgoing waves, i.e., it implicitly de-pends on the asymptotic behavior of⌿E(x). Hence, it is not a local time scale in the sense that knowledge of U(x) in the range a⬍x⬍b suffices to determine the tunneling time, a criterion that is met by the definition of ␶D关⌿_E兴 in Eq. 共3兲. Thus, we will continue to work with the latter expression, which we nevertheless refer to as dwell time in the follow-ing.

C. The Larmor clock

One of the most fruitful approaches to the tunneling-time problem is the Larmor clock model first put forward by Baz’ 关14兴. Here, we employ the notation used in refined treatments of this gedanken experiment 关8–10兴. Optical analogues of these time scales have been accessed in experiments using frustrated internal reflection of light 关15,16兴.

Formally, we obtain the Larmor time scales through the following setup. Consider the finite-range potential U(x) de-picted in Fig. 1. We now perturb the barrier potential by superimposing an infinitesimal variational step potential

V(x) covering the barrier:

V共x兲⫽V⌰共x⫺a兲⌰共b⫺x兲. 共9兲

Then, the Larmor times are defined as the logarithmic de-rivatives of the reflection and transmission amplitudes

R(E,V) and T(E,V) of the barrier with respect to the

per-turbation strength V at V⫽0. It is somewhat surprising to see that this linear-response theory leads to a set of two complex, or, respectively, four real tunneling time scales:

FIG. 1. A finite-range potential barrier U(x) extending from

␶R y ,˜ 共E兲⫹i␶R z 共E兲⫽ iប R˜共E兲 ⳵R˜共E,V兲 ⳵V

冏

V⫽0

⫽⫺ប⳵␳_⳵共E兲_V ⫹iប⳵ln兩R共E兲兩_⳵V , 共10兲

␶T y_{共E兲⫹i␶} T z_共E兲⫽ iប T共E兲 ⳵T共E,V兲 ⳵V

冏

_V⫽0

⫽⫺ប⳵␦_⳵共E兲_V ⫹iប⳵ln兩T共E兲兩_⳵V . 共11兲 Equivalent definitions hold for waves ⌿_E—(x) 关Eq. 共5兲兴 im-pinging on the right-hand side of the barrier. Unitarity re-strictions共6兲 and 共7兲 imposed on their parent amplitudes im-ply corresponding sum rules for the Larmor time scales:

兩R共E兲兩2_␶ R z_{共E兲⫹兩T共E兲兩}2_␶ T z_{共E兲⫽0, 共12兲} ␶R y ,˜_{共E兲⫹␶} R y ,—_{共E兲⫽2␶} T y_{共E兲. 共13兲} Therefore, only three independent Larmor time scales exist. We also note that for symmetric potential barriers U(x) ⫽U(⫺x), Eq. 共13兲 implies that the Larmor clock readings for reflected and transmitted waves coincide: ␶_Ry ,˜(E) ⫽␶R

y ,—

(E)⫽␶_Ty(E).

Although it appears difficult to assign an unambiguous physical interpretation to individual Larmor times, their ver-satility renders them a powerful tool in the analysis of the tunneling-time problem. Most proposals for the tunneling time may be restated as various combinations of the Larmor time scales ␶_Ry(E),␶_Ty(E),␶_Rz(E), and ␶_Tz(E), which quali-fies them for comparative studies of quantum clocks. For example, Smith’s dwell time␶_S˜(E)共8兲 adopts the form of a weighed ‘‘y ’’ Larmor time average:

␶S˜共E兲⫽兩R共E兲兩2␶R

y ,˜_{共E兲⫹兩T共E兲兩}2_␶ T

y_{共E兲. 共14兲} 共In the case of symmetric potential barriers, ␶S(E) and ␶T

y_{(E) are also identical.兲 In a similar vein, the minimum} tunneling time␶min(E), which we are about to define, is ame-nable to a description in terms of the Larmor times.

III. A MINIMUM TUNNELING TIME

Keeping these preliminary remarks in mind, we now pro-ceed to define a variationally determined tunneling time scale␶min(E). We start out our discussion from the Bohmian result for the dwell time␶D关⌿E兴 关Eq. 共3兲兴 that we motivated in Sec. II A. There, we noted that this particular time scale is a functional of the eigenstate⌿E(x) of Schro¨dinger equation 共1兲. However, it appears preferable to deal with a tunneling time that, apart from the particle energy E, depends solely on the potential relief U(x) in the interval of interest a⬍x⬍b 关17兴. 共After all, this property holds true for classical trans-mission over the barrier.兲 In order to transform the functional ␶D关⌿E兴 into a time scale independent of our choice of wave function ⌿E(x) in Eq.共3兲, we employ a simple variational principle: Clearly, ␶D关⌿E兴 is a positively definite quantity, which must be bounded from below; it continuously varies in

the two-dimensional state space of eigenfunctions ⌿E(x) in the interval a⬍x⬍b. Hence, for some special solutions ⌿min

E

(x) of Eq.共1兲 the dwell time␶D关⌿E兴 assumes its mini-mum value, which we shall denote as the minimini-mum tunnel-ing time␶min(E) in the potential structure U(x) for the inter-val a⬍x⬍b:

␶min共E兲⫽ min H_⌿⫽E⌿

冋

1 兩 j关⌿兴兩

冕

a

b

dx兩⌿共x兲兩2

册

. 共15兲 Correspondingly, we call any eigenstate ⌿_minE (x) of Eq. 共1兲 that minimizes Eq.共15兲 a minimal wave function of the po-tential barrier U(x) in the interval a⬍x⬍b. We should point out that no maximum value of the dwell time functional ␶D关⌿E兴 exists as it diverges for solutions ⌿E(x) that do not carry any current, j关⌿_E兴⫽0. This happens, e.g., for real solutions of Eq. 共1兲. For the sake of clarity, we remark that the term ‘‘minimal tunneling time’’ refers to the origin of this time scale in a variational procedure. We do not claim that ␶min(E) presents a universal lower bound for tunneling-time proposals that are based upon a different principle.

As an immediate consequence of its definition 共15兲, we note that the minimum tunneling time ␶min(E) does not present an additive quantity. Assume that the interval a⬍x ⬍b is split into two subintervals a⬍x⬍c and c⬍x⬍b. Then, from the variation in Eq. 共15兲 the inequality readily follows:

␶min共a,b;E兲⭓␶min共a,c;E兲⫹␶min共c,b;E兲. 共16兲 Obviously, equality in Eq. 共16兲 should occur in the limit of classical motion 关EⰇU(x)兴. For particle tunneling, i.e., E ⰆU(x), the additivity property does not nearly hold. 共See also, Sec. V B.兲

Unlike most other candidates for the tunneling time, our contender ␶min(E) shows the advantage of being a locally determined quantity. Let us elaborate this notion: The com-mon definitions of Smith’s dwell time ␶_S˜(E) 关Eq. 共8兲兴 and the Larmor times ␶R

y ,˜ (E),␶T y (E),␶R z (E), and␶T z (E) 关Eqs. 共10兲 and 共11兲兴 are all founded upon outgoing-wave solutions and thus implicitly depend on imposed boundary conditions. Any change of the potential U(x) outside the range a⬍x ⬍b will cause a different selection of outgoing wave states; as a consequence, Smith’s dwell time and the Larmor times for quantum motion in the interval a⬍x⬍b will be affected. Hence, these time scales depend on global properties of

U(x). In contrast, by definition共15兲 the minimum tunneling

time ␶min(E) is wholly determined by the set of eigenfunc-tions⌿E(x) in the examined interval a⬍x⬍b. But this set is completely fixed by the choice of energy E and the topogra-phy of the potential U(x) in this interval. Thus, both␶min(E) and the corresponding set of minimal wave functions ⌿min

E

(x) are truly local quantities. This property makes them applicable in situations where no outgoing wave solutions exist; we present an example in Sec. VI.

conjuga-tion of the wave funcconjuga-tion corresponds to a time reversal op-eration, which merely changes the sign of the current j关⌿_E兴 in Eq.共2兲.兴

IV. EXPLICIT EXPRESSIONS

From the variational principle stated in Eq.共15兲, we now derive an explicit representation for the minimum tunneling time ␶min(E) and the corresponding wave functions⌿_minE (x) in terms of an arbitrary basis of eigenfunctions in the barrier region a⬍x⬍b. For a finite-range barrier, we also may ex-press ␶min(E) in terms of the common Larmor time scales 共Sec. II C兲.

A. Representation by eigenfunctions

First, we show how to obtain␶min(E) and⌿_minE (x) from a pair of linearly independent real eigenstates c(x) and s(x) of Hamiltonian共1兲 with energy E. Our treatment, which is very general, only requires that the Wronskian determinant of these solutions be normalized:

W 关s,c兴⫽s共x兲

⬘

c共x兲⫺s共x兲c共x兲

⬘

⫽1. 共17兲

This property may always be achieved by a simple scaling operation on s(x).共Note that this normalization procedure, if

performed symmetrically on c(x) and s(x), leads to dimen-sional units 关length兴1/2 for these wave functions. Although unusual, this normalization scheme has the advantage that

c(x) and s(x) do not depend on the interval boundaries a

and b.兲

To find the minimum tunneling time and minimum wave functions, we form a general linear combination of the basis functions ⌿E(x)⫽␣c(x)⫹␤s(x), introduce it into the de-fining equation of dwell time 关Eq. 共3兲兴, and perform the variation of the complex parameters ␣,␤ 关Eq. 共15兲兴. This straightforward procedure leads to simple and elegant ex-pressions for the requested quantities. For ␶min(E), we find

␶min共E兲⫽2m_ប

再

冕

a b dx c共x兲2

冕

a b dx s共x兲2 ⫺

冋

冕

a b dx s共x兲c共x兲

册冎

1/2 . 共18兲

Note that the Cauchy-Schwarz inequality for integrals 关18兴 guarantees a positive definite radicand in this expression, which is furthermore independent of the actually employed set of basis functions c(x) and s(x), provided that condition 共17兲 is satisfied. A possible choice for a pair of conjugate complex minimal wave functions ⌿min

E (x) is given by the linear combination, ⌿min E _{共x兲⬀c共x兲⫺}

冉

冕

a b d␰c共␰兲2

冕

a b d␰s共␰兲2

冊

1/2

exp

冦

⫾i arccos

冕

a b d␰s共␰兲c共␰兲

冑

冕

a b d␰c共␰兲2

冕

a b d␰s共␰兲2

冧

s共x兲. 共19兲

关We remark that this formula contains as phase the abstract angle between the eigenstates c(x) and s(x) in the normed space L2(a,b) of square-integrable functions.兴 For the spe-cial case of symmetric barriers 关U(x)⫽U(⫺x), a⫽⫺b兴, expressions 共18兲 and 共19兲 may be considerably simplified. Exploiting the symmetry properties of the potential, we may select even and odd-parity eigenstates as basis functions

c(x)⫽c(⫺x) and s(x)⫽⫺s(⫺x) in these formulas. This

procedure yields ␶min共E兲⫽4m_ប

冑

冕

0 b dx c共x兲2

冕

0 b dx s共x兲2_{, 共20兲} ⌿min E 共x兲⬀

冑

冕

0 b d␰s共␰兲2c共x兲⫾i

冑

冕

0 b d␰c共␰兲2s共x兲. 共21兲

B. Connection to the Larmor clock

If we restrict ourselves to potential barriers U(x) that are confined to the interval a⬍x⬍b 共Fig. 1兲, we may use the set of outgoing waves ⌿_E˜(x),⌿_E—(x) 关Eqs. 共4兲 and 共5兲兴 as a basis in the state space of eigenfunctions ⌿E(x). Note that the integral appearing in the definition of the dwell time

␶D关⌿E兴 关Eq. 共3兲兴 may be expressed in terms of Larmor times 共10兲 and 共11兲. Hence, by variation of the wave function, we are able to represent the minimum tunneling time␶min(E) in terms of the transmission probability of the barrier and its set of Larmor clock time scales. A tedious calculation finally leads to a fairly compact expression:

␶min共E兲⫽_{兩T共E兲兩}1 兵兩R共E兲兩2_␶ R y ,˜ 共E兲␶R y ,— 共E兲 ⫹兩T共E兲兩2_␶ T y_共E兲2_⫹␶ R z_共E兲␶ T z_共E兲 其1/2_{. 共22兲} Note that the whole set of left-hand and right-hand Larmor time scales enters this functional. In the same manner, the minimal wave function ⌿min

E

(x) might be represented in terms of the Larmor time scales and the outgoing waves ⌿E˜(x) and ⌿E—(x). The relevant expression is, however, extraordinarily complicated, and we refrain from displaying it here.

V. ASYMPTOTIC BEHAVIOR

establish the behavior of this tunneling time scale in the lim-iting cases of extreme quantum dynamics and semiclassical motion. To define these regimes, we may employ the modu-lus of the classical action functional Scl(a,b;E) as a charac-teristic quantity of the barrier potential U(x):

Scl共a,b;E兲⫽

冕

a b

dx

冑

2m关E⫺U共x兲兴. 共23兲 If 兩S_cl(a,b;E)兩Ⰶប, the wave function ⌿_E(x) is hardly af-fected by the barrier. In the opposite regime 兩Scl(a,b;E)兩 Ⰷប, the problem is usually amenable to a treatment in the framework of the WKB approximation.

A. The narrow barrier limit

Let us first study ultrathin barriers with兩Scl(a,b;E)兩Ⰶប. In this case, the radius of curvature of the wave function ⌿E(x) in the barrier is large compared to the extension d ⫽b⫺a of the interval of interest, so we may neglect qua-dratic and higher terms in d in the Taylor expansion of ⌿E(x) with respect to x⫽a. Under these circumstances, for the calculation of␶min(E) we may always employ the pair of approximate eigenfunctions c(x)⫽1 and s(x)⫽x⫺a as a normalized function basis in Eq. 共18兲. This linear approxi-mation immediately yields

␶min共E兲⫽

_冑

1 3

md2

ប ⫹O共d4兲. 共24兲 In the narrow barrier limit, the maximum transmission veloc-ity vmin⫽d/␶min(E) is, therefore, given by the universal ex-pressionvmin⫽

冑

3ប/md. Note that vmindepends, apart from natural constants, only on the barrier width d, but is indepen-dent of the energy E. This property may be viewed as a consequence of the Heisenberg uncertainty relation; indeed, we may restate Eq. 共24兲 in the alternative form (mv_min)d

⬇

冑

3ប. 关As d˜0, vmin may assume arbitrarily large values. This unphysical behavior could be corrected by starting with the proper relativistic wave equation instead of Schro¨dinger equation共1兲.兴

B. The semiclassical limit

For兩Scl(a,b;E)兩Ⰷប, application of a semiclassical theory is in order. We may distinguish between two entirely differ-ent regimes, viz., quasiclassical transmission over the barrier for EⰇU(x), and the case of barrier penetration where E ⰆU(x).

For convenience, we introduce the action functional

S(x;E)⫽兩Scl(a,x;E)兩 and the abbreviation⌺(x;E):

S共x;E兲⫽

冏

冕

a x d␰

冑

2m关E⫺U共␰兲兴

冏

, 共25兲 ⌺共x;E兲⫽

冋

ប 2 2m兩E⫺U共x兲兩

册

1/4 . 共26兲

Let us now first inquire into the quasiclassical limit E ⰇU(x). Then, a pair of WKB solutions cWKB(x) and

sWKB(x) is given by

cWKB共x兲⫽⌺共x;E兲cos关S共x;E兲/ប兴, 共27兲

s_WKB共x兲⫽⌺共x;E兲sin关S共x;E兲/ប兴. 共28兲

Note that even though these functions are only approximate solutions to the Schro¨dinger equation, they show a spatially constant Wronskian determinant which is already normalized in the sense of Eq. 共17兲. Thus we may insert the pair

cWKB(x) and sWKB(x) into Eq.共18兲 to obtain the WKB ap-proximation to the minimum tunneling time in the limit of quasiclassical transmission. After a few transformations, we find ␶min WKB_共E兲⫽

冑

_␶cl共E兲2_⫺

再

冕

a b _dx vcl共x;E兲 cos

冋

2 បS共x;E兲

册

冎

2 ⫺

再

冕

a b _dx vcl共x;E兲 sin

冋

2 បS共x;E兲

册

冎

2 . 共29兲

Here,␶cl(E) denotes the classical time of flight from x⫽a to

x⫽b, whereas v_cl(x;E) represents the classical particle ve-locity in the potential U(x). From Eq. 共29兲 we infer that ␶min(E) indeed presents a valid quantum clock since it coin-cides with the classical transmission time␶cl(E) in the limit ប˜0. As expected,␶min

WKB

(E)⭐␶cl(E) holds.

Let us finally examine the tunneling case EⰆU(x). In this case, we have to replace oscillating functions 共27兲 and 共28兲 by evanescent waves:

c_WKB共x兲⫽1

2⌺共x;E兲exp关⫺S共x;E兲/ប兴, 共30兲

s_WKB共x兲⫽⌺共x;E兲exp关S共x;E兲/ប兴. 共31兲

Using this normalized pair of basis functions, from Eq. 共18兲 we arrive after some reorderings at the following WKB ap-proximation for the minimum tunneling time in the thick barrier penetration limit:

tunneling time is roughly inversely proportional to the modu-lus of the barrier transmission amplitude兩T(E)兩.

VI. APPLICATION TO DOUBLE-WELL STRUCTURES

So far, our considerations regarding the minimum tunnel-ing time ␶min(E) have been of an entirely mathematical na-ture. Yet the preponderance of theoretical reflections over practical applications has always been a troublesome aspect of the discussion about the tunneling-time problem. There-fore, it appears important to identify physical processes that permit us to extract the quantum time scales of motion in an actual experiment rather than to content oneself with a ‘‘gedankenversuch.’’ The time scale ␶min(E) quite naturally emerges in the quantum mechanics of double-well potentials as is elaborated below.

A. Level splitting

Let us examine the following setup共Fig. 2兲: The symmet-ric potential U(x) is composed of two wells centered around

x⫽⫾b, which are separated by a central potential barrier. As x˜⫾⬁, outer barriers isolate the double-well structure from

the environment. Each well gives rise to a discrete spectrum of eigenstates with energies En, but since the structure is assumed symmetric, the states in the left and right wells are in resonance, and the double degeneracy of the eigenstates will be lifted by the possibility of electron tunneling through the central barrier. This causes a small shift of the eigenen-ergies En, which will split into close doublets E˜n⫾⌬En/2 separated by the level splitting⌬En. Because these doublets are composed of states of opposite parity, transitions be-tween them may be enforced by electromagnetic dipole ra-diation, which allows for a precise determination of the level splitting. As a typical example, we refer to the fine structure of the states of the ammonia molecule caused by oscillation, which may be observed as maser radiation. A simple presen-tation is given in Feynman’s textbook关19兴.

Let us become more specific. Assume that Hwell denotes the Hamiltonian of a single well, say, the right one. Then, the well eigenstates are given by

Hwell⌿n共x兲⫽En⌿n共x兲. 共33兲 Due to the presence of the other well, these eigenstates split into states of definite parity⌿n

(e) (x) and⌿n (o) (x) of the com-plete system: ⌿n (e) 共x兲⬇␣关⌿n共x兲⫹⌿n共⫺x兲兴, 共34兲 ⌿n (o)_{共x兲⬇␤关⌿} n共x兲⫺⌿n共⫺x兲兴. 共35兲

Here, we fix the prefactors␣ and␤ by the requirement that at the center of symmetry x⫽0, ⌿_n(e)(0)⫽1 and ⌿_n(o)(0)

⬘

⫽1 holds. With this choice, the Wronskian determinant of both functions will be normalized at x⫽0, which obviously eases the connection to the developments of Sec. IV. States 共34兲 and 共35兲 are eigenstates of the complete Hamiltonian H:

H⌿_n(e)共x兲⫽

冉

E˜_n⫺⌬En 2

冊

⌿n (e)_{共x兲, 共36兲} H⌿_n(o)共x兲⫽

冉

E˜n⫹ ⌬En 2

冊

⌿n (o) 共x兲. 共37兲

The deformation of the well potential leads to a deviation from the original eigenenergy E˜_n⫺E_n that is, however, ex-ponentially small in the WKB sense. The same statement holds for the splitting ⌬En caused by tunneling.

We now have to look for the connection with the tunnel-ing time through the central barrier. To this end, we examine the evolution of a state that starts out in the right potential well. From Eqs. 共34兲 and 共35兲 we infer that such a state for

t⫽0 is initially given by ⌿_E(r)(x,0)⫽␤⌿_n(e)(x)⫹␣⌿_n(o)(x); its evolution is governed by Eqs. 共36兲 and 共37兲:

⌿E

(r)_{共x,t兲⫽2␣␤e}⫺iE˜_nt/ប_兵⌿

n共x兲cos ⍀nt

⫹i⌿n共⫺x兲sin ⍀n其t, 共38兲 where ⍀n⫽⌬En/ប. It is seen that the electron population oscillates between both wells with a period Tn⫽2␲/⍀n, which means that the electron is forced through the central potential barrier in a time Tn/2. Clearly, Tn/2 is a time scale that characterizes quantum tunneling. Let us now show that

T_n is indeed intimately related to the minimum tunneling time␶min(E) for the double-well potential structure.

B. Connection to␶min„E…

Let us first represent the oscillation period Tn in terms of eigenfunctions共34兲 and 共35兲 of the potential U(x). A simple calculation starting from Eqs.共36兲 and 共37兲 shows that 关20兴

冕

0 ⬁ dx⌿_n(e)共x兲⌿_n(o)共x兲⫽⫺ប 2 2m W 关⌿n (o)_共x兲,⌿ n (e)_共x兲兴 0 ⬁

E_n(o)⫺E_n(e) .

共39兲 The wave functions ⌿_n(e,o)(x) are bound states, so their Wronskian determinant vanishes for x˜⬁; at the lower limit

x⫽0, we chose the normalization of these functions in a way

as to guarantee a unit Wronskian. Since En (o)_⫺E

n (e) ⫽2␲ប/Tn, we obtain from Eq.共39兲:

Tn⫽ 4␲m ប

冕

0 ⬁ dx⌿n (e)_共x兲⌿ n (o)_{共x兲. 共40兲} This expression is still exact. However, the functions in the integrand implicitly depend共via their eigenenergies兲 on Tn. To get rid of this dependence, we note that their true eigenenergies deviate from their unperturbed value En only by an exponentially small amount, so it appears appropriate to replace the bound states in Eq. 共40兲 by a pair of

functions c(x) and s(x) to U(x) with energy En that are normalized in the same way, i.e., c(0)⫽s(0)

⬘

⫽1 and

c(0)

⬘

⫽s(0)⫽0. Then, they show a unit Wronskian

deter-minant,W 关s,c兴⫽1 关Eq. 共17兲兴, and we obtain approximately

Tn⬇ 4␲m

ប

冕

0 d

dx c共x兲s共x兲. 共41兲

Note that this operation enforces the introduction of a cutoff

d — unlike their counterparts ⌿_n(e,o)(x), the functions c(x) and s(x) are not genuinely bound states of the potential and thus diverge exponentially for x˜⬁. 关In the example of Sec. VII C, we found it practical to choose d in a way as to mini-mize the wave function c(x)2 at x⫽d.兴

In form共41兲, the period Tnmay be expressed by the mini-mum tunneling time ␶min(E_n). We first remark that␶min(E_n) in the symmetric interval⫺d⬍x⬍d is given by Eq. 共20兲:

␶min共⫺d,d;En兲⫽ 4m ប

冑

冕

0 d dx c共x兲2

冕

0 d dx s共x兲2. 共42兲 关Here, we exploited the parity properties of c(x) and s(x).兴 In connection with Eq. 共18兲, this allows us to express Eq. 共41兲 in terms of the time scales ␶min(0,d;E_n) and ␶min (⫺d,d;En):

Tn⬇␲

冑

␶min共⫺d,d;En兲2⫺4␶min共0,d;En兲2. 共43兲 Finally, we note that␶min(0,d;En) for opaque tunneling bar-riers is exponentially small compared to the minimum tun-neling time for the entire structure ␶min(⫺d,d;En)关Eq. 共32兲兴 and thus may be omitted. This yields the estimate

Tn⬇␲␶min共⫺d,d;En兲. 共44兲 For tunneling barriers, the beat period Tn in the double well is strictly coupled to␶min(En). Alternatively, we find for the level splitting the expression ⌬En⬇2ប/␶min(⫺d,d;En). 关Again, we emphasize that there are proposals for the tunnel-ing time, which may lead to results much smaller in value than␶min(⫺d,d;E_n).兴

VII. EXAMPLES

Let us now turn our attention to a few model potentials

U(x) in order to illustrate the theory of the minimum

tunnel-ing time␶min(E). We present some—occasionally surprising — results regarding quantum motion through simple square barriers and double-spike structures. Furthermore, we assess the quality of the minimum tunneling-time estimate for the level splitting in double well structures共Sec. VI兲 by numeri-cal numeri-calculations for a parabolic double-well potential.

A. Square barrier

We begin with a study of the minimum tunneling time ␶min(E) for the most popular, yet to some extent pathological barrier potential, the symmetric rectangular barrier:

U共x兲⫽U⌰共x⫺b/2兲⌰共b/2⫺x兲. 共45兲

Let us introduce the wave number ␬ in the barrier,

␬⫽_ប1

冑

2m兩E⫺U兩, 共46兲 and use a pair of fundamental solutions c(x) and s(x) of definite parity in the barrier region 兩x兩⬍b/2, which is nor-malized in the sense of Eq. 共17兲:

c共x兲⫽cos共␬x兲, s共x兲⫽1

␬sin共␬x兲. 共47兲 关Here, we assume E⬎U; for E⬍U, the trigonometric func-tions in Eq. 共47兲 should be replaced by the corresponding hyperbolic ones.兴 Then, we obtain from Eq. 共20兲 for␶min(E):

␶min共E兲⫽

冦

m ប␬2

冑

sinh 2_{␬b⫺共␬b兲}2 _共E⬍U兲 1

冑

3 mb2 ប 共E⫽U兲 m ប␬2

冑

共␬b兲 2_⫺sin2_{␬b 共E⬎U兲.} 共48兲

In contrast, the corresponding dwell time ␶_S˜(E) 关Eq. 共8兲兴 关which for this symmetric barrier coincides with the Larmor phase time␶_Ty(E)兴 is only defined for E⭓0 and reads, using the abbreviations k2⫽2mE/ប2 and k₀2⫽2mU/ប2 关8兴,

␶S˜共E兲⫽

冦

mk ប␬ 2␬b共␬2⫺k2兲⫹k₀2sinh 2␬b 4␬2k2⫹k0 4 sinh2␬b 共0⬍E⬍U兲 mb បk0 1⫹k₀2b2/3 1⫹k₀2b2_/4 共E⫽U兲 mk ប␬ 2␬b共␬2⫹k2兲⫺k0 2 sin 2␬b 4␬2k2⫹k₀4sin2␬b 共E⬎U兲. 共49兲 Figure 3 shows that the minimum tunneling time␶min(E) for the square barrier smoothly rises with decreasing energy E. As expected, in the classical limit EⰇU,␶min(E) merges into the classical time of flight ␶cl(E)⫽b/vcl(E), whereas in the limit of particle tunneling (EⰆU), the minimum tunneling time grows exponentially with both␬ and b, thus confirming the asymptotic formulas put forward in Sec. V. For E ⫽U,␶min(E) passes smoothly through its universal value for narrow barriers 共24兲. We note that the usual dwell time ␶S˜(E)关Eq. 共49兲兴 in the tunneling regime is characterized by a radically different behavior.

B. Double delta spike

We now switch to a further example where the minimum tunneling time␶min(E) shows a rather peculiar behavior. The model potential barrier we have in mind consists merely of two delta ‘‘spikes’’ forming a symmetric quantum well:

U共x兲⫽U关␦共x⫺b/2兲⫹␦共x⫹b/2兲兴. 共50兲

we should thus expect that these resonances reveal them-selves also in the tunneling-time spectrum. Indeed, this prop-erty holds for Smith’s dwell time ␶_S˜(E) 关Eq. 共8兲兴, which here equals the Larmor time scale␶_Ty(E)关Eq. 共14兲兴, and for this particular barrier is given by

␶S˜共E兲⫽ 2m

ប

kb共␬2_⫹2k2_{兲⫹␬sin kb共2k sin kb⫺␬cos kb兲} 4k4⫹␬2共2k cos kb⫹␬sin kb兲2 .

共51兲 Here, k2⫽2mE/ប2, and ␬⫽2mU/ប2 is a measure for the strength of the spike potentials.

From Sec. IV we infer that the minimum tunneling time ␶min(E) for the well solely depends on the wave functions ⌿E(x) in the interior of the barrier兩x兩⬍b/2, which are com-pletely unaffected by the presence of the delta spikes! There-fore, ␶min(E) is independent of the spike strength U and hence given by its free-particle value共48兲:

␶min共E兲⫽ m បk2

冑

共kb兲

2_⫺sin2_{kb. 共52兲} In Fig. 4, we display both tunneling time scales as a function of E. As anticipated, ␶min(E) depends on the internal struc-ture of the potential U(x) rather than its transmission prop-erties T(E), which affect Smith’s dwell time ␶_S˜(E) in a considerable manner共Sec. II B兲. 关The Larmor time scales are entirely based on the transmission and reflection amplitudes

T(E) and R(E) of U(x), see Eqs.共10兲 and 共11兲.兴 C. Parabolic double well

Finally, we would like to verify the connection between the minimum tunneling time ␶min(En) and the level splitting ⌬En in a symmetric double-well potential U(x)⫽U(⫺x) that we established in Sec. VI. For this purpose, we construct

such a double well in a comparatively simple manner by matching parts of parabolic potentials:

U共x兲⫽

冦

m␻2 2 共x⫹b兲 2 _{共x⬍⫺c兲} U0⫺ m␻2 2 x 2 _{共兩x兩⬍c兲} m␻2 2 共x⫺b兲 2 _共x⬎c兲. 共53兲

For a continuously differentiable potential function U(x), we have to set c⫽b/2 and U0⫽m␻2b2/4 in Eq.共53兲. The shape of this parabolic double well is depicted in Fig. 2.

Here, we are interested in a calculation of the exact level splitting ⌬E0 and corresponding minimum tunneling-time estimate 共44兲 for the lowest-lying doublet, which emerges from the unperturbed oscillator well ground state at E₀ ⫽ប␻/2. Clearly, the height U0and the extension of the cen-tral tunneling barrier may be tuned by changing the well separation 2b. As a useful first approximation to ⌬E0, we employ a semiclassical formula taken from Landau’s text-book关21兴: ⌬E0⫽ ប␻ ␲ exp

再

⫺ 1 ប 兩Scl共⫺a,a;E0兲兩

冎

. 共54兲 Here,⫾a denote the turning points of classical motion, and

Scl(⫺a,a;E0) represents the classical action for barrier pen-etration共23兲. 关Note that our less sophisticated estimate 共32兲 is not directly applicable here as it has not been corrected for the effects of turning points. However, both expressions agree in their exponential dependence.兴

Despite its simple structure, the eigenstates ⌿_n(e,o)(x) 关Eqs. 共34兲 and 共35兲兴 of the double-well potential U(x) 关Eq. 共53兲兴 are available only through numerical computation. They may be represented in terms of parabolic cylinder func-tions关18兴 properly matched at x⫽⫾c. In a recursive proce-dure, we determined the exact eigenenergies E₀(e)and E₀(o)of the lowest levels of U(x) and compiled a list of the corre-sponding level splittings ⌬E0 for several values of the di-mensionless separation parameter␤⫽

冑

2m␻/បb. Numerical values for the minimum tunneling time ␶min(⫺d,d;E₀) 关Eq.

FIG. 3. Minimum tunneling time ␶min(E) 关Eq. 共48兲兴 for a

rect-angular barrier of width b⫽6 Å and height U⫽2 eV as a function of the particle energy E共solid line兲. For comparison, the classical time of flight ␶cl(E) (E⬎U), and the WKB approximation

␶min WKB

(E) 关Eq. 共32兲兴 for barrier penetration (E⬍U) are additionally plotted 共short lines兲. The entirely different behavior of the corre-sponding dwell time ␶S˜(E)⫽␶T

y

(E) 关Eq. 共49兲兴 in the tunneling regime is evident共long dashed line兲.

FIG. 4. Tunneling times for motion through a symmetric poten-tial well demarcated by two identical delta spikes U␦(x⫾b/2) as a function of energy E. Solid line: ␶min(E) 关Eq. 共52兲兴. Dashed line:

␶S˜(E)⫽␶T y

(E)关Eq. 共51兲兴. Note that␶min(E) may exceed the dwell

共42兲兴 were established in a second series of calculations. To this end, carrying out the program of Sec. VI B we first de-termined properly normalized pairs of eigenfunctions c(x) and s(x) to U(x) with energy E₀⫽ប␻/2. Next, we selected cutoffs d for the integration in Eq.共41兲: Our simple criterion identified d with the minimum of the absolute value of the approximate solution c(x) in the exterior sector x⬎b, i.e., we chose the largest value of d so that c(d)c(d)

⬘

⫽0. 关These cutoffs are usually located far in the outer barriers of the double well, thus indicating good performance of approxi-mation共41兲, see Fig. 2.兴 The minimum tunneling time then was computed by integration 共42兲 and subsequently con-verted into an estimate for⌬E0 by means of Eq.共44兲.

Table I gathers results of these calculations for parabolic double wells of different separation and barrier penetrability. For comparison, the exact level splittings ⌬E0 are comple-mented with corresponding minimum tunneling-time esti-mates共44兲 and the less accurate semiclassical formula 共54兲. We infer that except for very shallow barriers 共small values of␤), the tunneling-time estimate indeed presents a splendid approximation to the true level splitting 共extreme right col-umn in Table I兲, thus confirming the significance of␶min(E) as a measure for the duration of quantum motion in these structures.

VIII. CONCLUSION

Pursuing a variational approach to the long-standing tunneling-time problem, in this paper we presented the con-cept of the minimum tunneling time ␶min(E), a time scale of quantum motion that holds for stationary one-dimensional problems. It is defined as the minimum amount of time re-quired by the probability current in order to replace the par-ticle number present in a given interval of space a⬍x⬍b. Therefore, ␶min(E) bears a close relationship to the idea of the dwell time originally proposed by Smith, which essen-tially represents an analogous quantity for outgoing waves. Both approaches are rooted in the hydrodynamical interpre-tation of quantum mechanics advocated in particular by Bohm.

As a major feature of the minimum tunneling-time ansatz, we emphasized that the quantity ␶min(E) represents a genu-inely local time scale of motion that, apart from the particle energy E, solely depends on the topography of the potential

U(x) in the sector of interest a⬍x⬍b, whereas most

con-tenders for the rank of the tunneling time scale, including the dwell time and the ubiquitous set of Larmor time scales, implicitly rely on boundary conditions imposed onto the un-derlying wave functions. 共Nevertheless, the minimum tun-neling time may be expressed as a functional of the set of Larmor clock readings.兲 The variational principle leads to simple and elegant formulas for␶min(E). Asymptotic expres-sions for the minimum tunneling time are available in the semiclassical limit: In the case of classically allowed motion, ␶min(E) gradually merges into the classical time of flight, and thus matches a basic demand for a valid tunneling time scale. For the opposite case of barrier penetration, the minimum tunneling time grows exponentially with the classical action for barrier traversal 共32兲. Qualitatively speaking, ␶min(E) is much shorter than the lifetime of a corresponding metastable state: Whereas␶min(E) is inversely proportional to the modu-lus 兩T(E)兩 of the semiclassical transmission amplitude, the lifetime of the metastable state grows with兩T(E)兩⫺2. At the same time, ␶min(E) generally lasts much longer than the semiclassical instanton or bounce tunneling time, which is defined as the classical traversal time for a particle of energy ⫺E moving in the inverted potential barrier ⫺U(x). The results of Sec. VI suggest that the variationally defined tun-neling time ␶min(E) corresponds to a periodical process where a particle tunnels through a barrier in a coherent fash-ion, whereas the decay of a metastable state proceeds in an incoherent, irreversible manner.

Apart from the formal elegance of the variational method, the minimum tunneling-time approach has the advantage of being related to experimentally accessible quantities. As a physical process that allows for the determination of␶min(E), we identified the energy level splitting in a symmetric double-well potential caused by tunneling of particles through the central potential barrier separating both quantum wells. These level splittings, and thus the minimum tunnel-ing time for the double-well structure, are available from spectroscopic measurements. We finally remark that a quite similar situation is realized in nature by electron exchange during the scattering of protons on neutral hydrogen atoms 关22兴.

ACKNOWLEDGMENT

One of us 共C.B.兲 has benefited from the support of the ‘‘Studienstiftung des Deutschen Volkes.’’

TABLE I. Numerical results for the level splitting in a symmetric parabolic double well关Eq. 共53兲兴. The first column specifies a dimensionless measure ␤⫽

冑

2m␻/បb for the separation of both wells; next, the corresponding crest U0of the central potential barrier is given in terms ofប␻. The following three columns

display the level splitting⌬E0 for the oscillator ground state (n⫽0) in units of ប␻. The exact value is

presented in the left column, whereas the approximation to⌬E0 gained via the tunneling-time scheme is

stated in the central column. Semiclassical estimate共54兲 for ⌬E0 is shown in the right column. To the

extreme right, the relative error of the minimum tunneling time approach is displayed.

␤ U0 ⌬E0共exact兲 ⌬E0关via␶min(E)兴 ⌬E0共semicl.兲 Relative error

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ex-pected.

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关15兴 M. Deutsch and J.E. Golub, Phys. Rev. A 53, 434 共1996兲. 关16兴 Ph. Balcou and L. Dutriaux, Phys. Rev. Lett. 78, 851 共1997兲. 关17兴 One may raise the objection that all available information con-cerning a physical system is contained in its quantum wave function. Although this statement is obviously correct, it is not an easy task to tailor the wave function to the needs of various tunneling-time proposals, including the dwell and Larmor times 共Sec. II兲, which implicitly employ outgoing-wave solu-tions. Furthermore, there exist quantum processes whose evo-lution is governed by characteristic time scales, which are in-dependent of the actual choice of the wave function. Resonant tunneling in symmetric double-well barrier structures共Sec. VI兲 presents a notable example.

关18兴 Handbook of Mathematical Functions, edited by M. Abramowitz and I.A. Stegun共Dover, New York, 1972兲. 关19兴 R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman

Lectures on Physics共Addison-Wesley, Reading, 1965兲, Vol. 3, Chap. 9.

关20兴 A. Messiah, Quantum Mechanics 共North-Holland, Amsterdam, 1964兲, Vol. I, p. 99.

关21兴 L.D. Landau and E.M. Lifshitz, Quantum Mechanics 共Perga-mon, London, 1958兲, p. 177.