fulltext

The tunnelling time for a wave packet as measured

with a physical clock

A. BEGLIUOMINI(1) and L. BRACCI(1) (2)

(1_{) Dipartimento di Fisica, Università di Pisa - Piazza Torricelli 2, I-56126 Pisa, Italy} (2_{) INFN, Sezione di Pisa - Pisa, Italy}

(ricevuto il 29 Maggio 1996; approvato il 30 Agosto 1996)

Summary. — We study the time required for a wave packet to tunnel beyond a

square barrier, or to be reflected, by envisaging a physical clock which ticks only when the particle is within the barrier region. The clock consists in a magnetic moment initially aligned with the x-axis which in the barrier region precesses around a constant magnetic field aligned with the z-axis, the motion being in the y direction. The values of the x and y components of the magnetic moment beyond or in front of the barrier allow to assign a tunnelling or reflection time to every fraction of the packet which emerges from the barrier and to calculate tunnelling times tT , x and tT , yand reflection times tR , xand tR , y. The times tT , xand tT , y(tR , xand tR , y) are remarkably equal, and independent of the initial position (in front of the barrier) of the packet.

PACS 03.65.Bz – Foundations, theory of measurement, miscellaneous theories (including Aharonov-Bohm effect, Bell inequalities, Berry’s phase).

PACS 73.40.Gk – Tunneling.

1. – Introduction

An old problem in Quantum Mechanics which has attracted the attention of several authors (see [1-3] for extensive reviews) is that of a satisfactory definition of the time required for such a peculiarly quantum-mechanical phenomenon as tunnelling. In the previous paper in this issue [4] we investigated the behaviour of a wave packet impinging onto a square barrier and proposed definitions of the interaction time (or dwell time) tD, as well as of the transmission time tT and the reflection time tR which

were based on the time dependence of the probabilities Pi (i 41, 2, 3) of finding the

particles in the region x E2d, NxNEd, xDd, respectively, the potential barrier lying between x 42d and x4d.

According to these definitions, tD4





(

)



(

)

each of the times tD, tTand tRare weighted sums of the time intervals dt. For tD the

weight is the probability of finding the particle within the barrier, for tT and tR the

weight is the fraction of the total transmitted (reflected) packet which at time t has not yet been transmitted (reflected), T 4P3(Q) and R4P1(Q) being the probability that

the particle is definitely transmitted or reflected. tD is independent of the choice of the

lower extremum t2in eq. (1), whereas tTand tRdepend on the values of t1and t3, which

should be chosen as the time when the packet begins to interact with the barrier. In ref. [4] we fixed t1and t3as the time tesuch that the contribution to the dwell time of the

interval [ 0 , te] is equal to that fraction e of the total dwell time which can be regarded

as the resolution of our procedure. We estimated this resolution as 0 .01 tD.

It is desirable, however, to remove also the slight arbitrariness implicit in the above choice, and to have a more objective definition of the times involved in the tunnelling process. To this purpose, we resort to the well-known device [5-7] of attaching to the particle a clock which runs only when the particle is in the barrier region. This clock can be provided by a magnetic moment which precesses around a constant magnetic field B confined within the barrier region. In order for this interaction not to affect substantially the barrier height, it is necessary to consider the limit of infinitesimal magnetic field, which will cause an infinitesimal precession. The measure of this precession, i.e. of the average of the spin of the particle in the region beyond the barrier (or the average over the reflected component) should yield the time the particle has spent in the barrier before being transmitted (or reflected). So far, the use of a physical clock for determining the tunnelling time has been limited to the case of stationary solutions.

In order to work out this program, we have studied the evolution of a wave packet representing a particle with spin-(1O2), polarized in the x direction, initially located in front of the barrier, which travels in the y direction towards the barrier. The average kinetic energy is lower than the barrier height. In the barrier region the particle interacts with a magnetic field B directed in the z direction. As a consequence, the transmitted and reflected packets will be no longer polarized in the x direction.

A magnetic moment m initially aligned in the x direction which emerges at time t from the barrier region after interacting a time t(t) with a constant magnetic field has a

y component my4 2 m sin (vt), where (in units with ˇ 4 1 ) v 4 mB is the Larmor

frequency. In turn, the x component is mx(t) 4m cos(vt). We extract t(t) from the

average, at time t, of the operators mx and my over the transmitted packet which are

denote as aSx(t)b3 and aSy(t)b3. We solve for t(t) either of the equations aSx(t)b34 1 2



cos [vt(x) ] dP3Odx dxOP3(t) ,

(4) aSy(t)b34 2 1 2



sin [vt(x) ] dP3Odx dxOP3(t) .

(5)

The above equations are to be viewed as the definition of t(t). The tunnelling time is calculated as follows: tT4



In other words, we weight every increment dP3of the probability of having the particle

beyond the barrier with the time t(t) that same fraction dP3of the transmitted packet

has spent within the barrier, the time being measured through the spin precession. Throughout the calculation we take advantage of the infinitesimal value of v, so that we are content with the first non-vanishing term.

We notice that in principle eqs. (4) and (5) yield different definitions of t(t), i.e. we have two different observables, Sx and Sy, whose measurement allows an indirect

measurement of t(t). Thus, eq. (6) could yield two different results for the tunnelling time tT. As a matter of fact, the results obtained starting from eq. (4) or eq. (5) are

essentially equal, well within the accuracy of the calculation, although, as will be discussed in sect. 4, the values of t(t) are not exactly equal (but they are practically equal starting from sufficiently early times). Moreover, the results are independent of the location of the packet at time t 40.

With analogous considerations we derive the reflection time tR, starting from the

average of Sx and Sy in the region in front of the barrier:

aSx(t)b14 1 2



cos [vt(x) ] dP1Odx dxOP1(t) ,

(48) aSy(t)b14 2 1 2



sin [vt(x) ] dP1Odx dxOP1(t) ,

(58)

where the label 1 denotes the average over the region in front of the barrier. Also in this case the values obtained by choosing either of the values of t(t) extracted from eqs. (48) and (58) tR4



are very near and do not depend on the initial position of the wave packet.

In conclusion, the prescriptions embodied in eqs. (4)-(6), (48)-(68) appear to give a rather objective measurement of the tunnelling time tT(of the reflection time tR). The

ambiguity in the choice of the lower extremum in eqs. (2) and (3), i.e. the time when the particle begins to interact with the potential.

In sect. 2 we present the problem, and in sect. 3 we derive the time t(t) and the consequent values for tR and tT. We comment our results in sect. 4, while sect. 5 is

devoted to the conclusions.

2. – Definition of the problem

We consider a wave packet describing a particle of mass m, moving in the y direction, where a potential barrier lies between y 42d and y4d:

V(y) 4V04 k02O2 m NyNE d .

(7)

A homogeneous magnetic field B directed in the z direction is confined in the barrier region. The particle has a magnetic moment m 4gS which in the barrier region is coupled to the magnetic field by the interaction mB 4gBSz. Consequently, the

Hamiltonian of the particle is (we use units with ˇ 41):

H 4./ ´ ( p2 O2 m 1 V0) 2 (vO2) sz, p2 O2 m , NyNE d , NyND d . (8)

The initial state of the particle is decribed by the wave function

c(y , 0 ) 4f(y) x ,

(9)

where x is the spinor

(

)

located in front of the barrier and peaked at y 4y0. We choose for f( y) the same wave

function whose evolution was studied in ref. [4],

f(y) 4



(10)

where ck( y) are the eigenfunctions of the Hamiltonian (8) with v 40 and

a(k) 4 (2d2O4 p3)1 O4_{exp [2(k2k}av)2d2] exp [iky0] .

(11)

The parameters involved in the problem are chosen as follows:

m 41 , d 4k2 , kav4 9 .9 , d 42, k04 10 , y04 215 .

(12)

In order to study the time evolution of the wave function (9) under the action of the Hamiltonian (8) we must expand (9) in the basis of the eigenfunctions c_{k 1}( y) and

c_{k 2}( y) with Sz4 1 O2 and Sz4 21 O2 , respectively. These functions are the

eigen-functions of the square barrier Hamiltonian, where the barrier height is V02 vO2 and

V01 vO2 , respectively. We note, however, that the initial wave function is located in the

region y E2d, where the dependence of ck 1( y) and ck 2( y) on Szis only through the

reflection coefficient A(k). The contribution of this part of the wave function to the coefficients a_{1(k) 4}



k 1( y)* f( y) dy and a2(k) 4



due to the Gaussian shape of (11). Consequently, we can safely assume the coefficients

The time evolution of the wave function (9) is given by

c(y , t) 4

.

/

´

1 Ok2



tO2] dk ,

1 Ok2



tO2] dk .

(13)

3. – Evaluation of the tunnelling time and the reflection time

We derive the tunnelling and reflection times according to the program outlined in the introduction. We focus our attention on the calculation of the tunnelling time, the case of the reflection time being quite similar.

Let P3(t) be the probability of finding the particle beyond the barrier ( y Dd) at

time t (u is the Heavyside function):

P3(t) 4

(

)

(14)

We want to associate to the increment dP34 ( dP3Odt) dt of this probability the time t(t)

which this fraction of the transmitted packet has spent in the barrier region. The tunnelling time for the packet will be calculated according to the equation

tT4

y



Q

t(t) dP3Odt dt

z

N

(15)

The time t(t) will be extracted from the average of the spin component Sxor Syat time t

in the region beyond the barrier:

aSx(t)b34

(

)

(16a)

aSy(t)b34

(

)

(16b)

The key for deriving t(t) is the spin precession which occurs when the particles is in the barrier region where a constant magnetic field aligned with the z-axis is located. For a magnetic moment m aligned with the x-axis at time t 40 and interacting with the field, the time dependence of its x- and y-component is given by

mx4 m cos (vt) , my4 2m sin (vt) ,

(17)

where v 4mB. The same time dependence holds for the x and y components of the spin angular momentum:

Sx4 1 O2 cos (vt) , Sy4 21 O2 sin (vt) .

(18)

If the fraction dP3of the packet emerging beyond the barrier in the time interval dt

at time t has spent a time t(t) in the barrier region, we assign to this fraction a contribution to aSx(t)b3 and aSy(t)b3 equal to

dSx4 dP3Odt 1 O2 cos

(

)

(19a)

dSy4 2dP3Odt 1 O2 sin

(

)

Consequently, the values of aSx(t)b3 and aSy(t)b3 will be given as

aSx(t)b34 1 O2



t

dP3Odp cos

(

)

(20a)

aSy(t)b34 21 O2



t

dP3Odp sin

(

)

(20b)

We have to extract t(t) from either of eqs. (20). This can be done by multiplying both sides of the equations by P3(t) and taking the derivative with respect to t. It is useful to

observe, however, that we need to consider the limit when v tends to zero, since we are interested in the tunnelling time beyond a barrier of height V0, and a non-infinitesimal

magnetic field would alter the barrier height. If we write

aSx(t)b34 Nx(t) OP3(t) , aSy(t)b34 Ny(t) OP3(t) ,

(21) we get

dNx(t) Odt41O2 dP3Odt cos

(

)

(22a)

dNy(t) Odt421O2 dP3Odt sin

(

)

(22b)

In eqs. (22a) we approximate cos (vt) B12 (1O2) v2_t2_{, and in eq. (22b) we put}

sin (vt) Bvt. We note also that Nx and P3 are even in v, whereas Ny is odd in v.

Moreover, at order 0 in v we have Nx4 ( 1 O2 ) P3, Ny4 0 . By putting Nx4 Nx( 0 )1

v2Nx( 2 ), P34 Nx( 0 )1 v2P3( 2 ), Ny4 vNy( 1 ), from eqs. (22) we get

dNx( 2 )Odt 4 1 O2 [ dP3( 2 )Odt 2 1 O2 t(t)2dNx( 0 )Odt] ,

(23a)

dNy( 1 )Odt 4 21 O2 dNx( 0 )Odt t(t) .

(23b)

According to the above equations, we have two different possible prescriptions to derive the time t(t) which must be inserted into eq. (15). We denote by tT , xthe result

obtained from eq. (15) using eq. (23a), which yields for t(t)

t(t) 4 [2( dP3( 2 )Odt 2 2 dNx( 2 )Odt)

O

(24a)

and by tT , y the result for t obtained from eq. (23b):

t(t) 422[ dNy( 1 )Odt]

O

(24b)

The calculation of tT , x requires the calculation of the integral

tT , x4



values at t 4Q of Ny( 1 ) and P3, and is given immediately by tT , y4

y



z

N

Performing the calculation, we find

tT , x4 3 .53 , tT , y4 3 .52 .

(26)

In order to find tR we repeat the above calculations starting from the values

of aSx(t)b14 Nx(t) OP1(t) and aSy( 1 )b14 Ny(t) OP1(t), where now Nx(t) 4

(

u(2y2d) Sxc( y , t)

)

(

)

extrac-tion of t(t) according to formulae analogous to eqs. (24), where of course P3is

substitut-ed with P1. The reflection times tR , x and tR , y are given by the equations

tR , x4



Q

] 2[ dP1( 2 )Odt dNx( 0 )Odt 2 2 dNx( 2 )Odt dNx( 0 )Odt](1 O2dtOP1(Q) ,

(27a) tR , y4

y



z

N

The results are as follows:

tR , x4 0 .52 , tR , y4 0 .56 .

(28)

Finally, we observe that, while eqs. (25b) and (27b) directly connect the average value at t 4Q of Syin the region in front of or beyond the barrier with tR , yor tT , y, this

connection does not hold for the x component, i.e. we do not have the relation, which holds in the stationary problem [7],

aSxb 41O2(12v2tx2O2 ) .

(29)

As a consequence, there is no use in introducing tzas tz4 2aSzb Ov, since the condition

tz24 t2x1 t2y, which is derived from the identity aSxb21 aSyb21 aSzb24 1 O4 , no longer

holds, due to the failure of eq. (29).

4. – Comments

The results obtained in the previous section for both the tunnelling and the reflection time using tx(t) or ty(t) are close to each other, and are remarkably similar

to the values found in ref. [4], where we found tT4 3 .39 , tR4 0 .55 . This supports the

reliability of the definition of the tunnelling and reflection times proposed therein and the conclusions about the impossibility of extending the notion of tunnelling (reflection) time proposed for stationary problems by several authors to the case of wave packets. For a comparative discussion of some results of this kind we defer the reader to ref. [4].

Some comments are in order with regard to the dependence of the tunnelling and reflection time on the initial position of the packet and on the different way to evaluate this time (i.e. the choice between tT , x and tT , y, or between tR , x and tR , y).

Concerning the first point, it is easy to see that the integral in eq. (25b) can be written as

tT , y4



N



(30)

where D(k) is the transmission coefficient and the derivative with respect to v is taken at v 40. No dependence on the value y0 where the packet is peaked at t 40 is

left.

The comparison between tT , x and tT , y (tR , x and tR , y) is more delicate. We will

focus our discussion on the tunnelling time, the considerations for the reflection time being quite similar. Equations (22) show that the condition for the identity of the values of t(t) derived from eqs. (22a) and (22b), respectively is

[ dNx(t) Odt]21 [ dNy(t) Odt]24 1 O4 [ dP3Odt]2.

(31)

This condition is not verified for any value of t. However, the ratio

R 4 ]4[ dNx(t) Odt]21 4[ dNy(t) Odt]22 [ dP3Odt]2(

O

(32)

is less than 0.1 starting from t B2 (less than 0.01 starting from tB2.7), where the main contribution to the integral (15) comes from. This can be easily understood. With the parameters of the problem, the packet travels with a velocity approximately equal to

kavOm B 10 . Given that the initial peak is located at y04 215 , up to t B 1 .5 the fraction

of the packet in the barrier region is practically zero.

As to the reasons why the l.h.s. of eq. (31) is very nearly equal to the r.h.s., while not being exactly equal, they can be traced to the following observations. Each term in the numerator of eq. (32) can be written in terms of the corresponding flux calculated at

y 4d. If c1(c2) and c 81(c 82) are the values in y 4d of the component with Sz4 1 O2

(Sz4 21 O2 ) of the wave function and its derivative with respect to y, the numerator N

in eq. (32) reads N 4Nc8* c1 22 c* c1 8 N2 2 2 Nj11 j2N 2_, (33)

j₁ and j₂ being the currents due to c₁and c_{2, calculated in y 4d. If the function}

c1(c2) is expressed in polar form, c14 r1exp [iu1], (c24 r2exp [iu2] ), we have

j_{14 r}2

1u 81 ( j24 r 2

2u 82) and N can be written as follows:

N 4 (r2r18 2 r1r8 )2 2 2 (r1 2 _u 1 822 r2 2 _u 2 82)(r₁2 2 r2 2 _{) .} (34)

At order 0 in v , N vanishes, whereas the second-order contribution in v is only due to the first-order variation, i.e. it can be obtained by writing r_{14 r 1 v dr ,}

r_{24 r 2 w dr , and the like for the other terms. As a consequence, eq. (32) can be}

written as follows:

R 4v2] 4(r 8 dr 2 rdr 8 )22 16 r2u 8 dr(u8 dr1rdu8)(O(r4u 82) . (35)

On the other hand, using eqs. (22) and denoting by txthe value of t in eq. (22a) and by

ty the value of t in eq. (22b), at second order in v , R can be written as

R 4v2(t2y2 t2x) 42v2tavDt ,

Dt being ty2 tx and tav the average of tx and ty. As a consequence,

DtOtav4 ] 2(r 8 dr 2 rdr 8 )22 8 r2u 8 dr(u8 dr1rdu8)(O(r4u 82tav2 ) .

(37)

Since drOr and the analogous terms appearing in eq. (37) are much smaller than 1, we have DtOtb1, and, consequently, tT , x4 tT , y.

5. – Conclusions

We have studied the process of a wave packet tunnelling beyond a potential barrier which has a clock running only in the barrier region. This is achieved by the well-known device, so far envisaged only for stationary problems, of considering the particle endowed with a magnetic moment coupled to a magnetic field confined in the barrier region. The time is measured by means of the precession of the magnetic moment around the field direction.

The particle is initially polarised in a direction orthogonal to the direction of motion and that of the field. So, for any fraction of the packet emerging beyond the barrier (or reflected by it), the value of each of the two components of the magnetic moment beyond the barrier (or in front of it) orthogonal to the field can yield a measurement of the time spent in the barrier region. We find that the results obtained using any of these components are quite similar, within the accuracy of the calculation. The results for the tunnelling and reflection times obtained by means of this method are in good agreement with the results obtained for the same packet, based on the consideration of the time behaviour of the probabilities of having the particle in front of or within or beyond the barrier [4].

The tunnelling and reflection times obtained in this way are independent of the position at time t 40 of the packet, provided it is located sufficiently far from the barrier edge. Thus, the definition of the tunnelling or reflection time by means of a physical clock looks a promising approach when one wants to go beyond the monochromatic approximation.

R E F E R E N C E S

[1] HAUGE E. H. and STOVNENG J. A., Rev. Mod. Phys., 61 (1989) 917. [2] OLKHOVSKY V. S. and RECAMIE., Phys. Rep., 214 (1992) 339. [3] LANDAUER R. and MARTIN TH., Rev. Mod. Phys., 66 (1994) 217. [4] BEGLIUOMINI A. and BRACCI L., Nuovo Cimento B, 112 (1997) 591. [5] BAZ8 A. I., Sov. J. Nucl. Phys., 4 (1967) 182.

[6] RYBACHENKOV. F., Sov. J. Nucl. Phys., 5 (1967) 635. [7] BUTTIKER M., Phys. Rev. B, 27 (1983) 6178.