Assisted tunneling of a wave packet between square barriers

0.15 MB·English

by  G. Kalbermann

Assisted tunneling of a wave packet between square barriers
G. K¨albermann∗
Soil and Water department, The Robert H. Smith Faculty of Agriculture,
Food and Environment Hebrew University, Rehovot 76100, Israel
The assisted tunneling of a wave packet between square one dimensional barriers is treated ana-
lytically. Thesurvivalprobabilityiscalculatedexactlyforapotentialmimickingaconstantelectric
field with arbitrary time dependence. The pole spectrum of the properly normalized unperturbed
wavefunctions determines the decay time. The tunneling probability is enhanced by the perturba-
tion. The results are exemplified for a simplified model of nuclear α decay.
PACSnumbers: 03.65.Xp,73.43.Jn23.60.+e
9
0
0 I. INTRODUCTION II. ANALYTICALMETHOD FOR ASSISTED
2 TUNNELING

n
a TheSchr¨odingerequationforaonedimensionalsystem
J Thequantumtunnelingparadigmexplainsdiversepro-
between square barriers is
cesses,ascurrentsinJosephsonjunctionsandnuclearal-
0
3 pha decay. Energeticallyforbiddenregionsfor aclassical
object, become accessible for a quantum system. ∂Ψ 1 ∂2Ψ
] i = − +γ (Θ(d x)Θ(x x ))Ψ (1)
h ∂t 2 m ∂x2 −| | | |− 0
p Gurney and Condon and Gamow[2, 3] , used the tun-
- neling model to reproduce the experimentally measured In the following all our units are either fm for length
nt huge range of α decay lifetimes of similar unstable nu- andtime,andfm−1 forenergies,momentaandpotential
a clei. The scheme has withstood continuous scrutiny in strength γ.
u the intervening years [4, 7]. WeconsideraninitialGaussianwavepacketlocatedt=
q
0 in the region between the barriers.
[

In a previous work we investigated the tunneling of a
2 one dimensionalmetastablestate betweendelta function Ψ(x,t=0)=e−∆x22 (2)
v
4 barriers,excitedbyatimedependentpotential.[1]Byin-
where ∆ is the width parameter of the packet. This
5 specting the modification of the spectrum of the poles
1 ruling the decay process, we estimated a non-negligible packetisareasonablemodelforthemetastablestateand
1 enhancementofthetunnelingprobabilitywhenanexter- facilitates the analytical evaluation of the decay ampli-
. tudes. However,themethodpresentedhereisnotlimited
9 nal time dependent perturbation acts on the system.
to gaussian packets.
0
8 Inside the barrier region the packet disperses if
In the present work we find exact solutions to the as-
0 ∆ < x . ,but,soonenoughswellingcomestoahaltdue
0
: sisted tunneling process, with square barrier potentials to the presence of the barriers. It can then spread only
v
instead of delta functions. Square potentials provide a
i through the tunneling process governed by the barriers.
X more realistic description of the barriers involved in de- ϕ (k)
e
r cay processes. The even χe(x) = √πn (k) and odd χo(x) =
a e
ϕ (k)
o
stationary states of eq.(1) for energies below
Decay times are enormously different form the nat- √πn (k)
o
ural time scales of nuclear and even atomic phenom- the barrier height γ are
ena. It is not possible to follow the decay process nu-
merically. Analytical expressions are then of the utmost
importance[1, 5, 6]. The method andformulaspresented cos(kx); x <x
0
| |
here aim at contributing in this direction. In the next ϕ (k)= A eκ|x|+B e−κ|x| ;d> x >x (3)
e  1 1 | | 0
section we present the model and the analytical solution C cos(kx)+sign(x)D sin(kx); x >d
 1 1
| |
method. In section 3 we solve the equations for a time
harmonic, linear potential and evaluate the effect of the where κ is defined in eq.(7), and sign(x) = 1, 1 for
−
perturbationonthe tunneling process. Some mathemat- x>0,x<0.
ical details are presented in appendix A.
sin(kx); x <x
0
| |
ϕ (k)= sign(x)(A eκ|x|+B e−κ|x|); d> x >x (4)
∗Electronicaddress: [email protected] o  sign(x)C22cos(kx)+2D2sin(kx); x| |>d 0
| |

2
wherewehaveextractedafactorof√π fromthenormal- withµacouplingconstant. Foraspatiallyconstanttime-
izations for convenience. harmonicelectricfieldofintensityE interactingwithan
0
The set of even-odd functions is orthonormal and α particle of charge 2 e, we have
| |
complete.[8, 9] The zeros of the normalization factors
govern the exponential decay of the metastable wave
functions in the inner region[1]. µ G(t)=2 e E sin(ωt) (9)
0
| |
The normalization factors are given by
In order to eliminate the explicit space dependence of
the perturbation term in the Schr¨odinger equation, we
((ne(k))2 = (C1(k)2+D1(k)2) apply the unitary transformation
1
D = c κ e k s
1 2 2 1
−2
(cid:18) Ψ(x,t) = e−iσΦ
c κ2 e c +c κ2 e c
2 2 1 2 1 1
− ζ2 µ2
+ c κ e k s +e k2 s s e κ c k s σ = µ x ζ+ dt
2 1 1 2 1 2 2 1 2
− 2 m
Z
e κ c k s e k2 s s /(κ k e )
1 1 2 1 1 2 3
− − ζ = G(t) dt (10)
(cid:19)
1 Z
C = e k s κ s e
1 2 1 2 2
2 − The Schr¨odinger equation(1) including the perturbation
(cid:18)
κ 2 c s +e κ2 c s +e k s κ s of eq.(8) becomes
1 2 1 1 2 1 1 2
c e k2 s +c e κ c k+c e κ c k
2 2 1 2 2 1 2 1 1
−
∂Φ 1 ∂2Φ
+ c2 e1 k2 s1 /(κ k e3) (5) i ∂t = 2−m ∂x2 +γ (Θ(d−|x|)Θ(|x|−x0))Φ
(cid:19)
µ ∂Φ
+ iζ(t) (11)
m ∂x
(n (k))2 = (C (k)2+D (k)2)
o 2 2
The wave function Φ is expanded in the complete set
1
D = c e κ c k of even and odd states of the unperturbed Schr¨odinger
2 2 2 1
−2 −
(cid:18) equation(1)includingstateswithenergiesabovethebar-
c κ2 e s +c κ2 e s rier
2 2 1 2 1 1
−
c e κ c k e k2 c s e k s κ s
2 1 1 2 1 2 2 1 2
− − −
e k s κ s +e k2 c s /(κ k e ) ∞ −i k2 t
− 1 1 2 1 1 2 3 Φ(x,t)= χi(k,x) ci(k,t) e 2 m dk (12)
(cid:19)
1 iX=e,oZ0
C = e κ c k s
2 2 1 2
−2
(cid:18) Theamplitudesce,o consequentlyobeythetimeevolu-
+ e κ2 s s e κ2 s s tion
2 1 2 1 1 2
−
+ e κ c k s c e k2 c c κ e k s
1 1 2 2 2 1 2 2 1
− −
c κ e k s +c e k2 c / µ k ζ
2 1 1 2 1 1 c˙ (k) = c (k)
− o e
(cid:19) −m ne(k) no(k)
( κ k e ) (6)
3 µ k ζ
c˙ (k) = c (k) (13)
e o
m n (k) n (k)
where e o
κ = 2 m γ k2 e =eκ (x0+d) wherewehaveusedthe superpositionintegralcalculated
3
− in appendix A [12]
e2 = ep2 κ d e1 =e2 κ x0
c = cos(k d) s =sin(k d)
2 2
c1 = cos(k x0) s1 =sin(k x0) I = e−i(k′22−mk2)tχe(k,x)∂χo(k′,x)dx
∂x
(7) Z
k
= δ(k k′) (14)
We assume a time dependent spatially linear pertur- ne(k)no(k) −
bation potential
and a dot denotes a time derivative.
The exactsolutionto eqs.(13)for aninitially symmet-
V(x,t)=µ x G(t) (8) ric wave packet such as the one of eq.(2) is
3
-10
10 /n (k)
e
µ k Y 1.2
c (k,t) = c (k,0) cos( )
e e
m ne no 1.0
µ k Y
c (k,t) = c (k,0) sin( ) 0.8
o e
− m n n
e o
0.6
t
Y(t) = ζ(t′) dt′
0.4
Z0
ce(k,0) = π14n∆e(k) e−4k∆22 (15) 00..02
0.0 0.5 1.0 -1 1.5
k (fm )
wherec (k,0)correspondsawelllocalized∆<<x wave -10
e 0 10 /n (k)
packetofthe formofeq.(2). Otherinitialwavefunctions 0.4 o
merely change the functional dependence of c (k,0).
e
0.3
III. ASSISTED TUNNELINGSOLUTION
0.2
Weareinterestedinthedecayofthewavepacketfrom
0.1
the internal zone leaking out to infinity. As discussed
in [1], the most important contribution to the integral
0.0
in eq.(12) originates from the poles in the amplitudes of
0.0 0.5 1.0 -1 1.5
eq.(15). Thisisindeedcorrectfortheunperturbeddecay k (fm )
process. However, for the perturbed process, the ampli-
tpurdoxesimairtey otosctilhlaetorerya.l mTohmeepnotulems laiexisin. eWxthreenmtehlye cinlotsee- FinIGun.i1t:s1o0f−f1m0n−e1(kfo)−r1a,taynpdic1a0l−α10dneoc(aky)−b1ar,riaesra function of k
grationoverthisvariableiseffected, theargumentofthe
harmonic functions in eq.(15) can oscillate wildly, sup-
pressing the contribution of the poles.(see figures 2 and
the complex momentum plane, whereas its inverse, cor-
3below)However,ifthe argumentofthe harmonicfunc-
responding to the incoming condition, has poles in the
tions is of order one, the poles will still dominate. This
upper half. Consequently, both sets of poles in the up-
canbeachievedbychoosingasufficientlyhighfrequency
per and lower complex momentum plane appear when
for the external perturbation of eq.(9).
incoming and outgoing pieces are present.[10].
Thenormalizationfactorsaroundtheirminimacanbe
As the poles are extremely sharp in position, it was
cast in the form
necessarytouseaquadrupleprecisionalgorithmonanα
cluster mainframe to find them. The imaginary parts of
k2 n2 λ (k2 k 2)2+β (16) thepolesaretypicallymanyordersofmagnitudesmaller
e,o ≈ j,(e,o) − j,(e,o) j,(e,o) than the real parts.
where j enumerates the zeros of ne,no. Figure 1 shows 10−10ne(k)−1, and 10−10no(k)−1 , for
The minima are located close to the parameters m = 3727MeV = 18.88fm−1,x0 =
12fm,d= 22fm,γ = 22MeV = 0.11fm−1, correspond-
ing to a typical α decay barrier. The spikes are due to
(2n+1) π p theextremeclosenessoftheminimaofthenormalization
k
e
≈ 2 (1+p x ) factors to their complex zeros.
0
n π p When eq.(12) is evaluated by contour integration in
k
o ≈ (1+p x ) the complex k2 plane, the contour has to be closed from
0
below. Inthe lowerhalf-planethe convergenceis insured
p = 2 m γ (17)
bytheexponentiale−i k22mt.Thepolesareseparatedfrom
p
Eqs.(17) providestartingvalues for the numericalsearch each other and the integrand drops essentially to zero
of the location of the poles of the wave function. The outside the pole, hence, the wave function consists of
poles in the complex momentum plane, appear in pairs an incoherent sum of the residues of the different poles.
located symmetrically above and below the real momen- The negative imaginary part of each pole induces a time
tumaxis. Thestationarywavefunctionsofeqs.(3,4)con- decaying exponential. Each exponent determines a dif-
sist of incoming and outgoing parts. The standard S ferent decay constant and decay time.[1] If the original
matrix,reflectingthe behaviorofthe outgoingboundary wave function is even in space, as in eq.(2), only even
conditionatspatialinfinity,haspolesinthelowerhalfof poles show up in the unperturbed wave function.
4
The argumentof the harmonic functions in eq.(15) in- first odd minimum second odd minimum
-1
cludes a potentially large denominator near the minima (ne no) kmin= 0.2516280711 kmin=0.5031067133
1.2
of the normalization factors. The product of these even-
1.1
odd factors near a minimum is depicted in figures 2 and
3. This product is not only finite, but of order one. We 0.8
1
were not able to produce an analytical formula for this -n)o
product, although the numerical values are very sugges- 0.7 n e
( 0.4
tive.
first even minimum second even minimum 0.3 0.0
(ne no)-1 kmin= 0.1258231401 (ne no)-1 kmin= 0.3773959279 -2 -1 0 -210 -1 2 -80 -40 0 40-20 -810
k-kmin ( 10 fm ) k-kmin ( 10 fm )
01..82 01..82 (n1e.2 no)-1 t hkmirdin =o0d.d7 5m4i2n6im56u6m6 (n1.e2 no)-1 f okmurinth= 1o.d0d0 4m8i8n3im88u5m
1.0
1.0
0.4 0.4 0.8
0.8
0.6
0.6
0.0 0.0 0.4
-2 -1 0 1 2 -80 -40 0 40 80 0.4
k-kmin ( 10-20fm-1 ) k-kmin ( 10-20fm-1 ) 0.2 0.2
(ne no)-1 t hkmirdin =e v0e.6n2 m87in3im83u4m37 (ne no)-1 f okmurinth= e 0v.e8n7 9m6i5n9im41u4m3 0.0-600 -200 200-20 -1600 0.0-3000 -1000 1000 3000
k-k ( 10 fm ) -20 -1
min k-k ( 10 fm )
1.2 1.2 min
0.8 0.8 FIG.3: no(kj,e) nj,e−1,aroundtheminimaofno(k). Param-
eters as in figure1
0.4 0.4
0.0 0.0 µ q Y(t) 2
-600 -200 200 600 -3000 -1000 1000 3000 u = j,e
k-kmin ( 10-20fm-1 ) k-k ( 10-20fm-1 ) j,e −(cid:18)2 m no(qj,e) βj,e(cid:19)
min
w = qj,e(∆2/4+i tp/(2 m))
−
β
q = (k )2 i j,e
FIG.2: no(kj,e) nj,e−1,aroundtheminimaofne(k). Param- j,e j,e − λj,e
eters as in figure 1 sj,e = √qj,e (19)
Figure 2 depicts (n (k) n (k))−1, around the minima
e o
of n (k). Note the degree of accuracy needed for the
e
location of the minima. The parameters are those of
figure 1. Figure 3 depicts (n (k) n (k))−1 around the
e o
minima of n (k).
o
Expandingtheharmonicfunctionsineq.(15),theexact ∞ (u )n
expressionforthewavefunctionintheregionbetweenthe Φ (x,t) = H (x,t) j,o
oo j (n!)2 (2n+1)
barriers becomes
j n=0
XX
π3/4∆ sin(s x)µ q Y(t) ew
j,o j,o
H (x,t) =
j
Φ(x,t) = Φee(x,t)+Φoo(x,t)+Φeo(x,t)+Φoe(x,t) − 2 m ne(qj,o)2 βj,oλj,o
(18) µ qj,o Y(t) p 2
u =
j,o
− 2 m n (q ) β
where (cid:18) e j,o j,o(cid:19)
β
∞ (u )n qj,o = (kj,o)2 i j,o p
j,e − λ
Φ (x,t) = G (x,t) j,o
ee j (n!)2
j n=0 sj,o = √qj,o (20)
X X
π3/4∆ cos(s x) s ew
j,e j,e
G (x,t) =
j
2 β λ
j,e j,e
p
5
∞ (u )n Fromeq.(24) itis clearthat for longtimes the even-even
j,o
Φ (x,t) = L (x,t)
eo j ((n 1)!)2 n (2 n 1) componentisessentiallythedominantone. Wecannow
j n=1 − −
XX
2π3/4∆ cos(s x) β ew
j,o j,o 1.0
L (x,t) =
j 2 m n (q )2s λ S(t)
e j,o j,po j,o
(21)
0.8
∞ (u )n 0.6
j,e
Φ (x,t) = M (x,t)
oe j (2n+1)(n!)2
j n=0
XX
π3/4µ Y(t) s ∆ cos(s x) ew
M (x,t) = j,e j,e 0.4
j
2 m n (q )2 β λ
o j,e j,e j,e
(22)
p
The seriesin eqs.(19,20,21,22) canbe expressedin terms 0.2
of standard functions
Φee(x,t) = Gj(x,t) J0(uj,e) 0.0
j 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X
π t (sec)
Φ (x,t) = H (x,t) J (u )
oo j 0 j,o
− 2
j (cid:20)
X
FIG.4: S(t)of eq.(25),without, fullline, and with harmonic
J0(uj,o)H1(uj,o)−J1(uj,o)H0(uj,o perturbation, dashed line, see text for parameters
(cid:18) (cid:19)(cid:21)
π
Φeo(x,t) = Lj(x,t) J0(uj,o) evaluate the influence of the perturbation on the decay
− 2
j (cid:20) time. Figure 4 depicts the nonescape probability of the
X
system in the region between the barriers
J0(uj,o)H1(uj,o) J1(uj,o)H0(uj,o)
−
(cid:18) (cid:19)(cid:21)
Φoe(x,t) = Mj(x,t) 2 J0(uj,e)− 2 Ju1(uj,e) −π S(t)= x0 Ψ(x,t)2 dx (25)
Xj (cid:20) j,e (cid:18) Z−x0 | |
J0(uj,e)H1(uj,e) J1(uj,e)H0(uj,e) (23) for the square barrier case, we have used the parameters
−
(cid:19)(cid:21) relevantforαdecaymentionedbeloweq.(1)andanintial
whereJ0,1 denotesthe Besselfunctionofthe order(0,1), wave packet of width ∆ = 4fm. The lower curve corre-
and H0,1 is the Struve H function. [11] spondsto(seeeq.(9))anelectricfieldofE =0.1Volt/m,
For long times we can use the asymptotic expansions andafrequencyofω =10π1017Hz. Theaimofthiswork
of the Bessel and Struve functions to obtain[11] is to present the analyticalsolutions and show the accel-
eration of the decay process. A more realistic choice of
parameters demands a thoroughnumerical evaluation of
2
Φ (x,t) G (x,t) sin(u theintegralsoutsidethepolesandispostponedforfuture
ee j j,e
→ j (cid:18)suj,e π work. Foratimeharmonicperturbation(9),therearetwo
X
π −3 major corrections to the unperturbed decay rate. The
+ )+O(u 2) amplitude of the wave function is to lowest order multi-
4 j,e
(cid:19) µ2 k2
Φ (x,t) H (x,t) 1 +O(u−32) plied by a factor of the form(1−ct2/4), with c≈ m2ω2
oo → j u j,o from the series in u . This factor depletes the wave at
j (cid:18) j,o (cid:19) j,e
X times of the order of the period of the harmonic pertur-
1 −3 bation for moderate values of µ, much faster than the
Φ (x,t) L (x,t) +O(u 2)
eo → j j (cid:18)uj,o j,o (cid:19) actual decay time determined by the poles. A second
X correction already noted in[1], arises when considering
Φ (x,t) M (x,t) 2 +O(u−23) (24) the higher order poles j > 1 in the sum of eq.(19). The
oe → j u j,e
j (cid:18) j,e (cid:19) latter generates a different decay time for each pole[1].
X
6
Figure 5 shows the results for delta function barriers
νδ(x x )+νδ(x+x ),withparameterscorrespondingto
0 0
−
the square barrier case ν =γ(d x ). The unperturbed
1.0 0
S(t) decay time of around 10−16sec−is orders of magnitude
smallerthanthesquarebarrierone. Theassistedtunnel-
0.8 ing is barely noticeable.
In summary, we presented a simple analytical method
to calculate assisted tunneling. The technique can be
0.6 generalized to three dimensions, and to other potentials
and initial metastable states. Such endeavor is worth
pursuing due to the dramatic impact assisted tunneling
0.4 mayhaveonnucleartechnologiesaswellasonsolidstate
devices. The question of electron screening for the nu-
clear decay acceleration must also be addressed to pro-
0.2 duce experimentally meaningful predictions.
0.0 Acknowledgments
0 50 100 150 200 250 300
t ( 10- 1 7 sec)
It is a pleasure to acknowledge the anonymous referee’s
remarks and corrections.
FIG.5: S(t)of eq.(25),without, full line, andwith harmonic
perturbation, dashed line, for δ barriers
[1] G. K¨albermann, Phys.Rev. C77 041601(R)(2008). willbeafinitecontributionthatisnegligibleascompared
[2] R. W. Gurney and E.U. Condon, Phys. Rev. 33,127 totheδ functioncontributionoftheouterregion. Conse-
(1929). quently we can use the outer region expression through-
[3] G. Gamow, Zeit. Phys. 51, 204 (1928). out.
[4] E. L. Medeiros et al. Jour. of Phys. G32, B23 (2006). The functions χ are then replaced for all x by
e,o
[5] Van Dijk W. and Nogami Y., Phys. Rev. C65, 024608
(2002).
[6] Garcia-Calderon G., Maldonado I. and Villavicencio J., √π n (k)φ (k) = (E ei kx+E∗ e−i kx), x>0
e e
Phys.Rev.A76, 012103 (2007). √π n (k)φ (k) = (E e−i kx+E∗ ei kx), x<0
[7] B. Holstein, Am.J. Phys. 64, 1062 (1996). e e
[8] K.R. Brownstein, Am.J. Phys. 43, 173 (1975). √π no(k)φo(k) = (F ei kx+F∗ e−i kx),x>0
[9] S.H. Patil, Am.J. Phys. 68, 712 (2002). √π n (k)φ (k) = (F e−i kx+F∗ ei kx),x<0
o o
[10] A. I. Baz, Ya. B. Zeldovich and A. M. Perelomov, Scat- −
C i D
tering,ReactionsandDecayinNonrelativisticQuantum E = 1− 1
Mechanics, Israel Program for Scientific Translations, 2
Jerusalem (1969). C2 i D2
F = − (26)
[11] M. Abramowitz and I. Stegun, Handbook of mathemat- 2
ical functions, DoverPublications, NY (1972). Taking advantadge of the symmetry properties of the
[12] In ref.[1] there was a sign misprint in the second of
wave functions, the matrix element of eq.(14) becomes
eqs.(11)
Mk′,k = ∞ e−i(k′22−mk2) tχe(k,x)∂χo(k′,x)dx
∂x
Z−∞
IV. APPENDIX A:PERTURBATIONMATRIX
4
ELEMENT 2
= (I (x ) I (x=0))
π n (k) n (k′) i →∞ − i
e o
i=1
X
Thematrixelementofeq.(14)doesnotvanishfortran- ei(k+k′)x
sitions between even and odd states solely. The matrix I (x) = i k′EF˜
1 i (k+k′)
element evidently diverges and we seek to evaluate the
divergent part. The most important contribution comes ei(k−k′)x
I (x) = i k′EF˜∗
fromtheouterregionof x >d. Intheinnerregionthere 2 − i (k k′)
| | −
7
ei(−k+k′)x A lengthy but straightforwardcalculationthen gives the
I (x) = i k′E∗F˜
3 i ( k+k′) result of eq.(14)
−
e−i(k+k′)x
I (x) = i k′E∗F˜∗ (27)
4 − i (k+k′)
−
where the tilde denotes a factor that depends on k’ in- k
stead of k. The all important contribution comes from Mk′,k = δ(k k′) (29)
n (k) n (k) −
k′ k e o
→
δ(k k′)
Mk′,k ≈ 2 i k′(E∗ F −E F∗)n (k)−n (k′) (28)
e o