Timescales of tunneling decay of a localized state

1.2 MB·English

by  Yue Ban

Timescales of tunneling decay of a localized state
Yue Ban1, E. Ya. Sherman1,2, J. G. Muga1, and M. Bu¨ttiker3
1Department of Physical Chemistry, Universidad del Pa´ıs Vasco UPV-EHU, 48080 Bilbao, Spain
2 IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain
3D´epartement de Physique Th´eorique, Universit´e de Gen´eve, CH-1211 Gen´eve 4, Switzerland
Motivated byrecent timedomain experimentson ultrafast atom ionization, weanalyze thetran-
sients and timescales that characterize, besides the relatively long lifetime, the decay by tunneling
ofalocalized state. Whilethetunnelingstartsimmediately, some timeisrequiredfor theoutgoing
flux to develop. This short-term behavior depends strongly on the initial state. For the initial
state tightly localized so that the initial transients are dominated by over-the-barrier motion, the
timescale for the flux propagation through the barrier is close to the Bu¨ttiker-Landauer traversal
time. Then a quasistationary, slowly decay process follows, which sets ideal conditions for observ-
1
ing diffraction in time at longer times and distances. To define operationally a tunnelling time at
1
the barrier edge, we extrapolate backwards the propagation of the wave packet escaped from the
0
potential. This extrapolated time is considerably longer than the timescale of the flux and density
2
buildupat thebarrier edge.
n
a
PACSnumbers: 03.65.Nk,03.65.Xp,03.75.-b
J

5
I. INTRODUCTION pointing out that the multiple time scales correspond to

] differentquantizationsofthe classicaltransmissiontime,
h
p Tunneling, being one of the most fundamental con- duetothenoncommutingobservablesandpossibleorder-
- cepts in quantum mechanics, remains a source of strong ings involved[16]. This clarifiedthe meaning ofdifferent
t
n theoretical and experimental controversies on the rele- partitions of the dwell time into transmitted, reflected
a and interference components. Another research line has
vant timescales of the process [1, 2]. A typical tunneling
u beentheinvestigationofcharacteristictimesforthetran-
setting is the scattering process, where a wavepacket is
q
sientdynamicsofthewavefunction(e.g. theforerunners)
[ reflected from and transmitted through a barrier higher
than the incident particle energy. In this case, the group in a plethora of potential configurations and initial con-
2 ditions using asymptotic methods [17–19]. For specially
delay[3],definedintermsoftheenergyderivativesofthe
v prepared states, in particular for confined or semicon-
reflection or transmission phase shift, describes the mo-
3 fined initial waves with a flat density, these transients
tion of the wave packet peak. It was also found that the
5
showdiffractionintime[19,20],i.e. temporaloscillations
8 group delay for particles tunneling through an opaque
reminiscentofspatialFresnel-diffractionbya sharpedge
3 barrier is independent of the barrier width (the “Hart-
. man effect” [4]). [20]. Also, the analysis of the partial density of states
8
[21] provides another approach to the understanding of
0 The measurement of the time spent by a tunneling
the tunneling process.
0 particle in the classically forbidden region can be based
1 on the approach of Baz and Rybachenko [5, 6], which Animportanttunneling-dependentphenomenonisthe
v: uses the Larmorprecessionof a particle with a magnetic decay of a metastable system [22–25], related, e.g., to
i moment in a weak magnetic field or in effective fields in the state ionization in optics and to the discharging of
X
solids due to the spin-orbit coupling [7, 8]. In classically a capacitor in mesoscopic physics [26–32]. Compared
r forbidden regions there is not only precession but also a to the full scattering problem, the decay configuration,
a
rotation[9] ofthe moment into the direction ofthe field. or“half-scattering”problem,hasbeenfrequentlyconsid-
Bu¨ttiker and Landauer [10] analyzed tunneling through ered unproblematic because of the absence of the trans-
an oscillating rectangular barrier and found an interac- mitted/reflected wavepacket splitting and the existence
tion time which is closely related to the rotation of the of a well known characterization in terms of resonance
magneticmomentinamagneticfield[9,10]. Atunneling lifetimes. In fact, one may still pose classically minded
time has been measured for electromagnetic waves pass- questions on the tunnelling time similar to the ones in
ing through inhomogeneous optical structures [11, 12] the scattering configuration, however, without obvious
andwaveguides[13,14]. Interestingly,ingraphene,asin- answers. A particle may wander in the trapping well for
gle layerof carbonatoms,which providesanother exam- a while and then escape through the barrier. For such a
pleofmasslessparticles,thetransportoccursviaevanes- history the lifetime would be a waiting time in the well
cent waves, and the Wigner-Smith delay is linear in the plus a tunneling time. Can this quantity be defined and
tunneling distance [15]. measured in a sensible way? The understanding can be
Brouard,Sala,andMuga,using scatteringtheorypro- basedontheanalysisofquantuminterferenceamongthe
jectors for to-be-transmitted/reflected wavepacket parts Feynman paths [33] or on the considerationof the quan-
andforlocalizingthe particleatthe barrier,setaformal tities defined by operational procedures, as presented in
framework that unified many of the existing proposals this paper.
2
Inrecentyears,the techniquesofatomionizationby a wave function Ψ(x,t) can be expressed as
strong laser field and attosecond probing of electron dy-
namics opened a new venue for experimental studies of ∞ ik2t
Ψ(x,0<t<t )= G(k)φ (x)exp dk,(2)
the tunneling times. The measurement of ionization of 0 Z k (cid:18)− 2 (cid:19)
0
He atoms [34] holds the promise to observe the tunnel-
ing delay of electrons in real time. In the experiment with
access to the dynamics at the tunneling time scale is
∞
gained through extrapolation of long time measurement G(k)= Ψ(x,0)φ (x)dx, (3)
k
Z
to shorter times by assuming that a particle that has 0
escaped through an energetically forbidden into a classi-
where φ (x) are delocalized real states in the potential
cal allowed region, follows classical dynamical laws. Us- k
U (x)withtheenergyE =k2/2. Weusehere¯h m 1
2
ing these laws and scattering data the moment of es- ≡ ≡
units, where m is the electron mass.
cape from the tunneling region into the classical region
is determined. A related experiment [35] measures the
perpendicular distribution of the nascent quantum me-
chanical momentum uncertainty of the initial state as it
is revealed by tunneling.
The aim of this paper is to study the dynamics of the
tunneling-induceddecayofalocalizedstateandtoinves-
tigate to what extent we can extract information on the
shorttimebehaviorfromthelongtimedynamics. Wein-
troduceasimplequantummechanicalmodel,whichsim-
ulates the experimental measurement of tunneling time
and allows for opening and closing the barrier [34]. We
calculate the probability density and flux to study the
short-term dynamics near the barrier and the long-term
dynamics far away from it. For an opaque barrier we
observea relativelyshorttransient time scale, where the
tunneling isdevelopedintoaquasi-steadydecayprocess.
The details of the initial state have an important im-
pact on the transients. At large distances, the flux and
the density start to grow at long times, reach a peak
and then decreaseby diffraction-in-time oscillations. We
found that the tunneling time obtained by extrapolat-
ing the particle motion from the position of the remote
detector to the right edge of the barrier is not directly
related to the timescale of the outgoing flux buildup.
II. GENERAL DESCRIPTION OF TUNNELING
AND MODEL POTENTIAL
To study the tunneling dynamics, we consider the po- FIG. 1. The time-dependent potential U1(x) at t = 0 and
t > t0 and U2(x) within the interval 0 < t < t0. (a) shows
tential U(x,t), infinite at x < 0. At t < 0 the poten-
thatU1(x)isastepinthepositive-halfplane,whichis0from
tial holds bound states with wavefunction ϕ (x) and en-
j x = 0 to x = a1 and U0 from x = a1 to infinity, while (b)
ergy Ej; it changes at t = 0 to allow the tunneling, and presentsabarrierwhichextendsfromx=a1 tox=a2andis
changesbackatt=t0 toitsinitialform. Theentiretime ofheightofU0. Thepotentialisalwaysinfiniteinthenegative
dependence is halfplane.
U (x) (t<0 and t>t ),
U(x,t)= 1 0 (1)
(cid:26)U2(x) (0<t<t0). After closing the barrier at t = t , the potential gets
0
itsoriginalform,andthewavefunctioncanbeexpressed
The initial state Ψ(x,t = 0) prepared at t = 0 begins
as
to evolve at t > 0 and the probability to find the elec-
troninside the potentialdecreases. Thisis similarto the
Ψ(x,t>t )= B ϕ (x)exp[ iE (t t )] (4)
ionization of an atom by a strong laser field, which lets 0 j j − j − 0
a valence electron to tunnel through the barrier. At the Xj
closing time t0, the potential becomes U1(x) again and + ∞ B(k)ϕ (x)exp ik2(t−t0) dk,
the decay terminates. In the time interval0<t<t0 the Z√2U0 k (cid:20)− 2 (cid:21)
3
where summation is extended over the bound states of U (x) is
2
the initial potential, and
C(k)sin(kx), (0<x<a )
1
B = ∞Ψ(x,t )ϕ (x)dx, (5) φk(x)=D(k)e−κkx+F(k)eκkx, (a1 <x<a2)(8)
j Z0 0 j  2/πsin[kx+θ(k)], (x>a2)
p
B(k)=Z ∞Ψ(x,t0)ϕk(x)dx, (6) with the normalization condition hφk′|φki = δ(k −k′).
0 The coefficients C(k), D(k), F(k) and the phase θ(k)
satisfy the boundary conditions of the potential U (x),
with ϕ (x) being the continuum states in the potential 2
k andκ =√2U k2. Inthetunnelingregimek <√2U ,
U1(x). With Eqs. (2) and (4) we have fully specified the whilekin the p0ro−pagating regime k > √2U , and iκ 0is
time evolution of the escape problem. 0 k
substituted by q defined below Eq. (7).
1 III. TUNNELING DYNAMICS AT SHORT AND
LONG TIMES.
)
k A. Decay of the state
( 0
G
The decay rate is described by the probability to find
the particle in the well from x=0 to x=a defined as
1
-1
a1
w (t)= Ψ (x,t)Ψ(x,t)dx. (9)
1 ∗
Z
0 1 2 3 4 5 6 0
k The lifetime t , whichalsocanbe seenas the dwelltime,
l
an important time scale introduced to characterize the
decayrate,isthe time thatittakesforthe relativeprob-
FIG. 2. (Color online) The matrix element G(k) of the
initial state with the delocalized states of the potential U2. ability in the well w1(t)/w1(0) to decay to 1/e. For an
k is the wave vector of the delocalized states. The solid line opaque barrier the decay time can be written approxi-
corresponds to the initial state Ψ(x,0) = ϕ0(x), which is a mately as tl ≈ ATk0exp(2κd), with κ = 2(U0−E0)
bfrooumnda1st=ate1otfoUa12.=F1o.r4.thTishceadseasthheedblianreriecrorirnesUp2onedxstetnodas arinodddof=mao2t−iona1f,owrhaeprearkt0ic=le√in2tEh0e,wTke0ll,=ex2pp/k(02isκtdh)epe1-
sinusoidal initial state Ψ(x,0) = √2sin(πx) and to a barrier − ≪
is the tunneling rate, and A is a barrier-dependent coef-
in U2 extending from a1 = 1 to a2 = 1.42. In both cases, ficient of order 1. Therefore, κd is a significant physical
κd=2 and U0 =16.
parameterwith regardto the decayprocessandthe tun-
neling time. Analysis of different examples where this
simple expressionforthe lifetime doesholdcanbe found
To specify the model, we assume that at t < 0 the
in Ref. [25].
potential U (x) is a step in the positive half-axis, being
1 To study the dynamics, we first set the initial state
zero at 0 < x < a and U at x > a , as shown in
1 0 1 as the ground state of the Hamiltonian correspondingto
Fig.1(a). At t=0 the potential changes to U (x), which
2 U (x) for a typical potential. For example, for U = 16
1 0
is a rectangular barrier of the height U extended from
0 there are two bound eigenstates: the ground ϕ (x) (en-
0
a to a , as shown in Fig. 1(b). The rectangular bar-
1 2 ergy E ) and the excited state ϕ (x) (energy E ). The
0 1 1
rier being the simplest example of an everywhere finite
energy E for Ψ(x,0) = ϕ (x) is 3.52. By choosing dif-
0 0
potential, where the wavefunction and its derivative are
ferenta ,onecanmodifythetransparencyofthebarrier.
2
continuous,allowsfortheexactanalysisofthedynamics.
For example, if κd = 2, corresponding to a = 1.4, this
2
We put here a 1 and measure the time, energy, and
1 ≡ barrier is a moderately opaque one. The correspond-
momentum in the corresponding units.
ing G(k) is shown in Fig. 2. Under the conditions of
The delocalized eigenstates of the Hamiltonian corre-
Ψ(x,0)=ϕ (x)the lifetime inthispotentialist =17.5,
0 l
sponding to U (x) have the form:
1 as illustrated in Fig. 3(a). Calculated real and imag-
inary parts of Ψ(x,t) after some short transients show
C (k)sin(kx), (0<x<a )
ϕk(x)=(cid:26) 12/πsin[qx+θ1(k)], (x>a1) 1 (7) fTahstisobsceihllaavtiioornsimwpitlhiesthtehaetnvinelotperemfusnocftitohneepxople(−s tin/2tthl)e.
p complex energy plane, the states we consider correspond
where q = √k2 2U0, k > √2U0 with C1(k) and θ1(k) to the simple Breit-Wigner resonances. Detailed analy-
−
determined by the boundary conditions at x = a1, and sis of various types of resonances and their relation to
the norm is determined by ϕk′ ϕk =δ(q q′). different timescales can be found in Refs. [36, 37]. To
h | i −
The eigenstate of the Hamiltonian corresponding to clarify the influence of the initial state on tunneling, we
4
1.0 0.08
0 (a)
]
(t) -0.1 0.06
g
0.8 n[ -0.2
) l -0.3 t) 0.04
(t 0 1 2 3 4 ,1
1 t a
w ( 0.02
J
0.6
0.00
(a)
0.4 -0.02
0 4 8 12 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
t t
1.0
0 (b)
)] -0.1 0.06
g(t -0.2
[
0.8 n -0.3
l )
(t) -0.4 0 1 2 3 4 ,t20.04 0.5
w10.6 t J(a 0.02 ρ(x,t) 000...243 xx==aa12 (x15)
0.1
0.0 0.5 1.0 1.5 2.0
(b) t
0.4 0.00
0 4 8 12 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
t t
FIG. 3. (Color online) The probability to find the particle FIG. 4. (Color online) (a) Flux at the left edge a1. The
in the well from 0 to a1. (a) The solid and the dashed lines solid, dashed anddot-dashedlines correspond tothebarriers
are for two initial states Ψ(x,0) = ϕ0(x), one for U0 = 16 U0 =24, U0 =16, and U0 =10, respectively. (b) Flux at the
andtheotherforU0 =10. (b)Thesolidandthedashedlines rightedgea2. Theinsetpresentsthedensityata1(solid)and
represent the barriers U0 = 16, and U0 = 10 with the same multipliedbyfifteenata2(dashed),withthebarrierU0 =16.
initial state Ψ(x,0)=√2sin(πx). The insets exhibit ln[g(t)] All lines correspond to the same parameters as those in (a).
at small time scale, where g(t) w1(t)/w1(0). For all curves For both plots, κd=2 and Ψ(x,0)=ϕ0(x).
≡
κd=2.
B. Short-term dynamics
alteritintothegroundstateofaninfinite-wallpotential,
Ψ(x,0) = √2sin(πx). Therefore, E is π2/2 and a be-
0 2 Inthissubsectionweaddressthebeginningofthetun-
comes 1.42 to keep κd unchanged. As shown in Fig. 2,
neling by studying the short-time dynamics of the flux
the coefficient G(k) in Eq. (3) has two contributions,re-
1 ∂Ψ(x,t) ∂Ψ (x,t)
latedtotheresonancescorrespondingtotheboundstates J(x,t)= Ψ∗(x,t) ∗ Ψ(x,t)(1,0)
of the initial potential. For Ψ(x,0) = ϕ (x), the second 2i(cid:20) ∂x − ∂x (cid:21)
0
one, corresponding to the first excited state with a fast and the density,
decay,isextremelyweak. Thepresenceofmorethanone
ρ(x,t)=Ψ (x,t)Ψ(x,t). (11)
bound state in the initial potential combined with the ∗
condition of low transparency of the barrier leads to im- These two quantities satisfy the continuity equation
portant consequences for the short-time scale tunneling
∂ρ(x,t) ∂J(x,t)
dynamics. + =0. (12)
∂t ∂x
Although all the probabilities decrease exponentially
at long times, showing the general feature of the pro- Based on Eq. (12), we obtain the flux at the edges,
cess, the ones with Ψ(x,0) = √2sin(πx) in Fig. 3(b)
dw (t) dw (t)
1 2
oscillatefast, while thosewith Ψ(x,0)=ϕ0(x) in (a)de- J(a1,t)=− dt , J(a2,t)= dt , (13)
cay smoothly. This is because Ψ(x,0) = √2sin(πx) has
where
a significant contribution of the second anomaly shown
in Fig. 2 leading to interference with the “ground state ∞
w (t)= Ψ (x,t)Ψ(x,t)dx (14)
resonance”. 2 Z ∗
a2
5
0.15 T ofaparticleintheinitialwellwithpotentialU . The
(a) k0 1
periodT determinestheprefactoroftheescapetimeas
k0
a =1.3 κd=1.5 discussed below Eq. (9). Therefore, classically speaking,
2
the decay by tunneling as a steady process begins when
0.10
)
t the electron hits the barrier.
,
2
a a =1.4 κd=2 TheinsetinFig. 4(b)showsthebehaviorofthedensity
(
J 2 attheedge. ContrarytoJ(a1,0)=J(a2,0)=0,ρ(a1,0)
0.05
and ρ(a ,0) are not zero, as the ground state for the
2
a =1.5 κd=2.5 potential U (x) is not fully localized in the well, and the
2 1
plateau in ρ(a ,0) is clearly seen.
2
0.00 Eventhoughthenumericallyestimatedtimesaremuch
0 0.2 0.4 0.6 0.8 1 1.2 1.4
shorter, a similar trend in variation with the height of
t
the barrier is shown by the Bu¨ttiker-Landauer time (BL
time), providedthat κd remains unchanged. The traver-
saltime of Bu¨ttiker and Landauer[10], t =d/κ, given
(b) BL
0.15 by the barrierwidth d divided by the ”semiclassical”ve-
locity κ = [2(U E )]1/2, is an important time scale,
0 0
−
) U =10.5 κd=1.5 especially in opaque conditions. With the conservation
,t 0.10 0 of κd, tBL is proportional to 1/κ2.
2
a The dependence of the flux at the right edge on time
J( U =16 κd=2 withdifferentwidthbutkeepingtheheightofthebarrier
0
0.05 is shown in Fig. 5(a). It is interesting to find that the
U =23 κd=2.5 time that the flux begins to form a plateau is almost
0
equal for the different widths, although the flux value
0.00 at the plateau changes strongly, roughly as exp( 2κd).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 −
In Fig. 5(b), this characteristic time t also remains
t pl
almost unchanged for different U by keeping the same
0
widtha . Weconcludethattheobservedscaleofπ/(2E )
2 0
FIG.5. (Coloronline) (a)Fluxattherightedgeontimewith is universal and does not depend on the details of the
different width: solid line, κd=1.5, dashedline, κd=2, and potential.
dot-dashed line, κd = 2.5, U0 = 16. (b) Flux at the right For comparison, we illustrate in Fig. 6 the flux at a
edge on time with different height, by remaining a2 = 1.4:
short time scale for a transparent barrier κd = 0.25. As
solid line, κd = 1.5, dashed line κd = 2, and dot-dashed
a matter of fact, the flux and the density have similar
line, κd = 2.5, respectively. The ground state energy E0
dependsonlyweaklyon thepotentialheight U0.Forall plots profiles. They grow from initial to the maximum value
Ψ(x,0)=ϕ0(x). and decrease rapidly, contrary to the opaque behavior
without an obvious peak. The peak is positioned at t
≈
0.4, similar to the time of plateau development in Fig.5.
For the initial state Ψ(x,0) = √2sin(πx) the flux
is the probability to find the particle outside the poten- is enhanced by an order of magnitude and oscillates
tial. stronger compared to the initial Ψ(x,0) = ϕ (x), be-
0
InFig. 4weillustratethetimedependenceoftheedge causesin(πx)containslargercontributionsfromdifferent
flux for different barriers during a short time scale for eigenstates. Different from that with Ψ(x,0) = ϕ (x),
0
Ψ(x,0) = ϕ (x), where the lines shift to the right by the density is zero at t = 0 outside the barrier. The
0
lowering the barrier. It can be seen from both panels time-dependence of the density at the edges with the
of Fig. 4 that J(a ,t) and J(a ,t), basically, increase initial state Ψ(x,0) = √2sin(πx) is presented in Fig.
1 2
during a short time interval and then reach at a charac- 7 for a typical barrier. For various system parameters,
teristictimet aroughplateau,whichshowsasmoother the delay time between the maximum of the density at
pl
behaviorfor J(a ,t), while J(a ,t)dives foraveryshort the right and the left edges is in a good agreement with
2 1
time to a negative value and oscillates more strongly. the Bu¨ttiker-Landauertime t for these potentials. We
BL
As a result, no feature can be clearly distinguished as concludethatt manifests itselfasa delaytime ifover-
BL
a precise instant when tunneling begins. By using the the-barrier motion becomes essential due to the choice
continuity equations (13), the decay time t can be reli- of the initial state. The momentum components that
l
ably estimated as 1/J (a ), where J (a ) is the typical matter at first are the ones larger than √2U , as the
pl 2 pl 2 0
value of the flux at the plateau. The time scale when momentum distribution of sin(πx) is broad. Moreover,
J(a ,t)developsaplateaualsobecomeslarger,although its average local velocity at the right edge of the barrier
2
it cannot be defined precisely. A crude estimate for the v (a ,t) = J(a ,t)/ρ(a ,t) decreases from a large value
B 2 2 2
scale at which the plateau forms is t π/(2E ), ap- with some oscillations to a relatively stable one close to
pl 0
∼
proximately a factor of 2 less than the oscillation period √2E during a short interval, after which the real tun-
0
6
1.2
1.0
1.0
0.8
) 0.8 )
t t
, ,
a20.6 (x 0.6
( ρ
J
0.4 0.4
0.2 0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
t t
FIG.6. (Coloronline)Timeevolutionofthefluxattheright FIG. 7. Density evolution with the barriers U0 = 10, a2 =
edge, where the solid line represents the flux, when U0 = 16 1.63 (solid line at a2 and dashed line at a1). As the density
and a2 = 1.05. The dashed line corresponds to the flux for at a2 is relatively small, it is multiplied by the factor of ten.
U0 = 10 and a2 = 1.06. The two barriers are transparent as The barrier is opaque with κd = 2 and the initial state is
κd=0.25, and theinitial state is ϕ0(x). Ψ(x,0)=√2sin(πx).
neling starts. This means that the forerunners just go a sharp maximum, and then decreases at timescales of
above the barrier instead of tunneling. By contrast, the t with the sequential oscillations due to the diffraction
l
tunnelingprocessforΨ(x,0)=ϕ (x)occursindeedfrom in time phenomenon [20]. The profiles of the flux and
0
the instant that the decay initiates, because vB(a2,t) is density are very similar, with J(X,t)≈√2E0ρ(X,t).
always smaller than √2U . In the attosecond experiment analysis [34] the tunnel-
0
As discussed in [38, 39] with analytical models, t ingtimewasdefinedasthedelaybetweenthetime when
BL
is a characteristic time describing different phenomena, the barrierbegins to be loweredand the time when elec-
among them over-the-barrier transients. This is rather tronexperiencestheaccelerationbyexternalfield,ascan
paradoxical,consideringitsassociationwith“tunnelling” be extracted from the long-time behavior. Similarly, we
inthedefiningformula,andhassurelynotbeenfullyap- can use the operational phenomenological approach to
preciated. A more intuitive understanding of this unex- define a tunneling time as
pected role is still missing.
X a
2
ttun,1 =tX − , (15)
− v
p
C. Long-term dynamics where tX is the time when the edetector measures the
strongestfirstpeakoftheflux,v = 2E isthevelocity
p 0
q
We have now established that the short time dynam- with which the particle moves out of the potential, and
ics of the probability flux is governed by robust time the energy e e
scales. We next investigatethe long term dynamics with
the goal to find out whether a suitable extrapolation of E =E ∞U ϕ2(x)dx (16)
the long term scattering data can be used to gain infor- 0 0−Z 0 0
a2
mation on the short time dynamics. We find that af- e
isslightlylessthanE ,asthepotentialchangessuddenly
ter formation of the steady tunneling, the particle prob- 0
from U (x) to U (x). As t is a measurable quantity
ability density shows two distinct features. The first 1 2 X
the tunneling time can be calculated by Eq. (15). For
one is an almost uniform change in ρ(x,t) for x < a
1
instance, for Ψ(x,t = 0) = ϕ (x), U = 16, a = 1.4,
with ρ(x,t) ρ(x,0)exp( t/t ). The second one can 0 0 2
l
≈ − when the detector is at X = 120, then t = 50.3, and
beviewedasabroadbump(wavepacket)withrelatively X
smalldensitypropagatingwiththevelocitycloseto√2E0 resulting ttun,1 =5.62. Another approach,similar to Eq.
(15), is to calculate the tunneling time as
andspreadingintime. Inthissubsection,weconsiderthe
dynamics of the bump at long time scale and trace it to
shorttimestoobtaintheoperationallydefinedtunneling X a
2
times. ttun,2 =tX − , (17)
− v
X
WeassumethatthefluxJ(X,t)andthedensityρ(X,t)
are measured by a detector located at X a . It is where v is the velocity of the flux peak. For example,
2 X
≫
shown in Fig. 8 for Ψ(x,0)=ϕ (x), that the density at v = 2.517 as defined by the motion from X = 120 to
0 X
X a is nearly zero up to some time, then grows to X = 122. According to Eq. (17), the corresponding
2
≫
7
0.025 16
JHx,tL
X=60 X=90 X=120 14 0.07
0.020
12
)
t 0.015
,
x 10
(
ρ x
0.010 8
6
0.005
-0.02
4
0.000
10 20 30 40 50 60 70 2
t HaL 0 5 10 15
t
FIG. 8. Time evolution of density ρ(X,t) at given positions
X = 60,90,120, provided that Ψ(x,0) = ϕ0(x), U0 = 16, JHx,tL
a2 =1.4. Astrong initial peak isfollowed byoscillations due 0.07
to thediffraction in time process. 125
tunneling time is ttun,2 = 3.18. As it can be assumed x120
that the velocity of the electron is constant outside the
barrier in a classical manner, we can conclude that the
tunneling time extracted from the measurements by a
-0.02
remote detector is X-independent. Both Eq. (15) and 115
Eq. (17) extrapolate electron motion from distance to
the exit of the tunneling process. However, these two
timesarenotequaleithertothetime offormationofthe HbL 45 50 55
steady tunneling in Fig. 4 or to the decay time t . t
l
Fig. 9 depicts how the flux evolves with time and dis-
tance at two timescales for Ψ(x,0) = ϕ (x). The cur-
0 FIG. 9. (Color online) The flux versus time and position
vature of the maximum flux region in Fig. 9(a) shows after a short time near the barrier (a) and after a long time
thatittakessometime forthefluxtodevelopaconstant farawayfromthebarrier(b). Alltheparametersarethesame
speed in free space. In addition, Fig. 9(a) demonstrates as in Fig. 8. The dashed straight lines serve as the guide for
that a peak of the flux at a appears at t 1, smaller the eye only. The short-dash line in (a) demonstrates the
2
≈
than t in Eq. (17). nonlinearity of the peak position as a function of time. The
tun,2
It is expected in some models that the wave function long-dashlinein(b)showsthatthisdependenceisveryclose
to linear at these times and distances.
at long times and distances can be obtained with the
stationary phase approximation. Eq. (2) in this limit
can be recast as
Ψ(x>a ,t) (18)
2
≈
1 ∞dkG(k)exp i θ(k)+kx k2t , in Fig. 10, and tl is the exponential decay time intro-
√2πiZ0 (cid:26) (cid:20) − 2 (cid:21)(cid:27) duced above. The peak of the flux with shorter closing
time t appears earlier than that with longer t , as a
0 0
and the phase θ(k) + kx k2t/2 can be expanded result of the increased energy spreading [40]. After the
−
near the stationary point K satisfying the equation closingtime t , the probabilityto findthe particleinthe
0
(dθ(k)/dk)k=K+x Kt=0. However,G(k)inourcalcu- well remains almost unchanged. This shows that com-
−
lations is not a sufficiently smooth function near the K- ponents with largermomenta tunnel throughthe barrier
points due to the resonancesshowninFig. 2. Therefore, first, and therefore, closing the barrier can decrease the
the stationary phase approximation does not provide a operationally defined times t and t . The mea-
tun,1 tun,2
satisfactory description of the peak propagation. surement of the total number of particles escaped from
Another factor that affects the tunneling time is the the potentialas the function ofthe closing time canhelp
closingtimet ,whenthepotentialturnsbackfromU (x) to find the timescale of the flux formation t : with the
0 2 pl
toU (x) andthe tunneling is interrupted. The time evo- increase of t through this region the number of escaped
1 0
lution of the flux observed at the same remote position particles which is quadratic in t for short times changes
0
of detector with different closing times is demonstrated at t >t to a linear dependence.
0 pl
8
0.06 ing starts instantly, however, some time of the order of
π/(2E ), where E is the ground state energy in the ini-
0 0
0.05
X=120 tial potential, is required to develop the outgoing flux
eventually proportionalto the characteristic exponential
0.04
) decay rate 1/t . As expected, there is a time delay be-
t l
,
x 0.03 tweenthefluxdevelopmentattheleftandtherightedges
(
J ofthebarrier. Iftheinitialstateisthegroundstateofthe
0.02 potential at t<0, the timescale of the flux development
ismuchlongerthantheBu¨ttiker-Landauertraversaltime
0.01 expected for the given barrier parameters. However, if
the initial state is more tightly localized, the Bu¨ttiker-
0.00
Landauertimemanifestsitselfasatime delayoftheflux
30 40 50 60 70
t and density maxima between the left and right edges of
the barrier.
At long times we have considered the propagation of
FIG.10. (Coloronline) Fluxasafunctionoftimedetectedat
the escaped wave packet at distances much larger than
X =120 with different closing time t0 = (solid), t0 =tl/3
∞ thescaleofthepotential. Fromtheoperationaldefinition
(dashed), t0 = tl/6 (dot-dashed), t0 = tl/9 (dotted), tl is
of the escape time, related to the position of the maxi-
the life time for the particle in the case of t0 = . Other
parameters are thesame as those in Fig. 8. ∞ mum of the wave packet density, we have estimated the
time the particles escaped from the potential. This time
is also much longer than the traversaltime for the given
barrier. To determine the effect of the time dependence
ofthebarrier,weimplementedescapetimewindowscon-
IV. CONCLUSIONS
siderably shorter than the tunneling decay time t . The
l
increasing importance of faster components for shorter
Motivatedbyrecentexperimentsontheultrafastatom time windows leads to the extrapolated time estimated
ionization by optical fields, we have studied numerically for the closing potentials smaller than that for the po-
exactly, for a rectangular barrier, the time-resolved tun- tentials permanently open.
neling for short time scales and the further propagation
at long times of an initially localized particle. The bar-
rier we consider is opaque, however, not extremely so, V. ACKNOWLEDGEMENT
to ensure a reasonable tunneling probability. The prob-
ability density evolution on a short time scale much less E.Y.S. and J.G.M. are grateful to the support of Uni-
thanthedecaytime,dependsstronglyontheinitialstate. versityofBasqueCountryUPV-EHU(GrantGIU07/40),
Dependentonhowthisstateisprepared,this short-term Basque Country Government (IT-472-10), and Ministry
motioncanincludebothunder-the-barriertunnelingand of Science and Innovation of Spain (FIS2009-12773-C02-
over-the-barrier propagation, as seen in the evolution of 01). M. B. is supported by the Swiss NSF, MaNEP, and
the density and flux at the barrier edges. The tunnel- the European ITN, NanoCTM.
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