Tunneling of Bound Systems at Finite Energies: Complex Paths Through Potential Barriers

4 Pages·0.12 MB·English

by  G. F. Bonini

Tunneling of Bound Systems at Finite Energies: Complex Paths Through Potential
Barriers.
G.F. Boninia, A.G. Cohenb, C. Rebbib and V.A. Rubakovc
aInstitut fu¨r theoretische Physik, University of Heidelberg, D-69120 Heidelberg, Germany
bDepartment of Physics, Boston University, Boston, MA 02215, USA
cInstitute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia
(January 19, 1999)
Weadaptthesemiclassicaltechnique,asusedinthecontextofinstantontransitionsinquantumfield
theory,tothedescription oftunnelingtransmissions at finiteenergiesthroughpotentialbarriers by
complexquantummechanicalsystems. Evenforsystemsinitiallyintheirgroundstate,notgenerally
describable in semiclassical terms, the transmission probability has a semiclassical (exponential)
form. The calculation of the tunneling exponent uses analytic continuation of degrees of freedom
9
intoacomplexphasespaceaswellasanalyticcontinuationoftheclassical equationsofmotioninto
9
9 thecomplex timeplane. Wetest thissemiclassical techniquebycomparingits resultswith thoseof
1 a computational investigation of thefull quantummechanical system, findingexcellent agreement.

n
PACS: 03.65.Sq, 02.70.-c
a
J

1
2
1. Tunneling phenomena are inherent in numerous about an instanton [6,7].

quantum systems, from atoms to condensed matter to The purpose of this paper is two fold. First, we adapt
1
v quantum field theory. Even in systems with a small the technique of Refs. [3,4] to tunneling of quantum me-
2 parameter—coupling constant—a quantitative descrip- chanicalboundsystemsthroughhighandwide potential
6 tion of tunneling is possible only in a limited number barriers. As an example, we consider a system of two
0
of cases. Perhaps the best known example is the WKB degrees of freedom with linear binding force. We find
1
approximation familiar from one-dimensional wave me- that if the bound system is initially in a highly excited
0
9 chanics; similar techniques, such as the “most probable state, the tunneling exponent is indeed calculable in a
9 escape path” and instanton methods [1,2], are used to semiclassicalway. This resultishardlysurprising,asthe
/ study tunneling from the bottom of potential wells. In initialstateitselfcanbedescribedinsemiclassicalterms.
h
p the latter cases the calculation of the tunneling proba- We formulate the complexified classical boundary value
- bility may be reduced to the solution of classical equa- problem relevant to the calculation of the exponent in
t
n tions of motion for real generalized coordinates in imag- this case.
a inary (“Euclidean”) time, supplemented by the analysis Second, the real strength of this formalism is that it
u
of small fluctuations about this classical Euclidean tra- also enables one to treat barrier penetration when the
q
: jectory. However, these methods often fail in describing bound system is initially in a low lying state, e.g. the
v
tunneling of systems with more than one degreesof free- ground state. This is far from obvious, as this initial
i
X dom at finite energies. state cannot be described semiclassically. Nevertheless,
r It has been suggested recently [3,4], in the context of we argue that in this case the tunneling exponent can
a instanton transitions in quantum field theory,that semi- be obtained by an appropriate limiting procedure. The
classicaltechniquesmaybeusedforcalculatingtheexpo- resulting technique is less-well justified, so we have cho-
nential suppression factors in a class of processes where sen to test it by direct computation of the transmission
multi-dimensional systems tunnel at finite energies. The probability in the full quantum theory. We briefly de-
proposal involves a double analytic continuation: the scribe the numerical methods involved, and present the
degrees of freedom are continued into a complex phase resultsofboththefullquantummechanicalandsemiclas-
space, and the equations of motion are solved along a sicalanalyses. Wefindgoodagreementbetweenthetwo,
contour in complex time. The tunneling exponent is de- confirming the validity of the semiclassical approach.
termined by an appropriate solution of the calssical, al-
2. Tobespecific,letusconsideraquantummechanical
beit complexified, equations of motion. Computation by
systemoftwoparticlesofequalmassm=1/2movingin
numericalmethodsis thenfeasibleevenforsystemswith
one dimension. Let these particles be bound by the har-
alargenumberofdegreesoffreedom,ashasalreadybeen
monicpotential(ω2/8)(x x )2,andoneoftheseparti-
demonstrated in a field theoretic model [5]. A problem 1− 2
clesberepelledfromtheoriginbyapositivesemidefinite
with the formalism of Refs. [3,4] is that its derivation
potential V(x ) that vanishes as x (we could
from first principles is still lacking, although its plausi- 1 1 → ±∞
of course allow V to depend on x as well, provided it
bility has been supported by perturbative calculations 2
couples to the internal degree of freedom). We take this
1
potential to have the form V(x1)= g−2U(gx1), where g T0(E)=C0(ǫ)e−g12F0(ǫ) (3)
is a small constant. We set ¯h = 1, so the classical limit
correspondsto g 0. Inwhatfollowswe presentthe re- and that the exponent is obtained by taking the limit
→
sults of numerical calculations for ω =1/2 and gaussian
potential,U(x)=exp( x2/2),althoughthetreatmentof F0(ǫ)=limν 0F(ǫ,ν) (4)
− →
otherpotentialswouldbesimilar. Intermsofthecenter-
One of the main purposes of this paper is to check this
of-mass and relative coordinates, X = (x +x )/2 and
1 2
limiting procedure by comparison with a fully quantum
y =(x x )/2, the Lagrangianreads
1 2
− mechanical calculation.
1 1 1 1
L= X˙2+ y˙2 ω2y2 U[g(X +y)] (1) 3. To see that Eq.(2) is indeed valid, and to obtain
2 2 − 2 − g2
theprocedureforcalculatingtheexponentF(ǫ,ν),letus
Far from the origin (X ), the center-of-mass and consider the transmission amplitude A(Xf,yf;P,n) =
internaldegrees of fre|edo|m→d∞ecouple and the system can Xf,yf exp[ iH(tf ti)]P,n , where Xf (> 0) and yf
h | − − | i
be characterizedby its center-of-massmomentum P and arethecoordinatesattimetf,andweeventuallytakethe
oscillatorexcitationnumbern,or,equivalently,bynand limit (tf ti) . This amplitude may be written as
− → ∞
thetotalenergyE =P2/2+ω(n+1/2). Wewishtocalcu- aconvolutionofthe evolutionoperatorinthe coordinate
late the probablityT (E) for transmissionof the system basis and the wave function of the initial state. The for-
n
through the barrier V. Of particular interest is T (E), mer is given by the path integral Xf,yf exp[ iH(tf
0 h | − −
thetransmissionprobabilityofthis systeminitiallyinits ti)]Xi,yi = [dX][dy] expiS wheretheintegrationruns
| i
oscillator ground state. over paths saRtisfying (X,y)(ti) = (Xi,yi), (X,y)(tf) =
It is convenientto introduce rescaled total energy and (Xf,yf). For an initial state with P g−1, n g−2,
∝ ∝
occupation number ǫ = g2E and ν = g2n. With our the initial wave function is semiclassical and has the ex-
choiceofU,thetopofthebarriercorrespondstoapoten- ponentialform. Inthecaseofaharmonicbindingpoten-
tial energy ǫ=1. For ǫ<1 transmissionis possible only tial, this follows from the integral representation in the
via tunneling. For ǫ just above 1 classical over-barrier coherent state formalism:
dtreaends,ittihonerseareexipstosssaibnleufnosrtavberley,sspteactiiaclcinlaitsisaiclasltastoelsu.tiIonn- Xi,yi P,n = eiPXi dzdz¯e−z¯z z¯n e−21z2−12ωyi2+√2ωzyi
h | i √2π Z 2πi √n!
with both particles stationary at the top of the barrier,
x =x =0, so that ǫ=1. If one perturbs this solution
1 2 (One may replace z¯n/√n! by exp(nlogz¯/√n + n/2)
by giving an arbitrarily small, common positive velocity
at large n.) By introducing the rescaled integration
tobothparticles,theywillmovetowardX = . There-
∞ variables X gX, y gy, etc., we observe that
versed evolution takes the system to X = , with the → →
−∞ A(Xf,yf;P,n) is given by an integral of an exponential
classical oscillator characterized by a certain excitation
of the form exp( g 2Γ) where Γ depends only on the
energy ǫosc ων (and a certain phase of the classical − −
0 ≡ 0 rescaled integration variables, ν and ǫ, and does depend
oscillator). The combined evolution is the classicaltran-
explicitly on g2. This allows for a semiclassical analysis:
sition over the barrier from this particular asymptotic
we find stationary points of Γ and evaluate the integrals
state. By solving the (real time) classical equations of
using a stationary phase approximation. We outline the
motion numerically, we found that ν 0.9 for ω =1/2.
0 ≈ mainstepsinthederivationofthestationarypointequa-
The classical evolution of the system initially in the
tions.
classical oscillator ground state (x = x ) leads to the
1 2 Variation of Γ with respect to X(t) and y(t) for t <
i
excitation of the oscillator as it approaches the barrier.
t < t leads to the conventional classical equations of
f
Classical transition over the barrier occurs in this case
motion. When classical transitions are forbidden, there
only if the total energy exceeds some critical value. In
will be no real solutions satisfying the boundary condi-
our example we found numerically ǫ =1.8.
crit tions. Nevertheless there will be solutions with complex
If ǫ and ν are such that classical transitions over the
valuesoftheintegrationvariables. Whenperformingthe
barrierarenotpossible,thesystemhastotunnel. Wewill
analytic continuation we will, in general, encounter sin-
shortlyseethatatǫandν fixed,andg 0(i.e.,atlarge
→ gularities. To deal with this problem, we note that the
total energy and initial occupation number, E,n g 2)
∝ − time contour, originally the real axis, can be distorted
the transmission probability has the semiclassical form
into the complex plane, keeping the end points t , t
i f
Tn(E)=C(ǫ,ν)e−g12F(ǫ,ν) (2) fiavxoeidd.tThhesisedsienfogrumlaartitioiens.oTf thhues,tiomuer sctornattoeugryaisllotwosseuasrctho
forcomplexsolutionsoftheclassicalequationsofmotion
Thecaseoftheinitialoscillatorgroundstateismoresub-
along a contour ABCDE in the complex time plane, as
tle. In analogy to Refs. [3,4] we suggest that the trans-
shown in Fig. 1.
mission probability at n=0 has the form
2
There are further stationary point equations coming ∂(2ImS0) ∂(2ImS0)
=ǫ, =ν
from variation of Γ with respect to the integration vari- ∂T ∂θ
ables at the end point t . It is convenient to formu-
i i.e. the pairs (ǫ,ν) and (T,θ) are Legendre-conjugate.
late these equations along part B of the contour, where
We have solved the equations of motion numerically
t = iT/2+t, t = real (this is possible because
′ ′ → −∞ alongthe contourBCDE subject to the boundary condi-
theequationsofmotiondecoupleintheasymptoticpast).
tions (i)–(iii). In particular, we have evaluated the limit
Instead of ǫ and ν we introduce new real parameters T
(4). The result of this semiclassical calculation is shown
and θ; T enters the problem through the shape of the
in Fig. 3.
contour. The general complex solution at large negative
t′ is X(t′) = X0 + pt′, y(t′) = ue−iωt′ + veiωt′ where 4. To check this semiclassical procedure, we have per-
X0, p, u and v are complex parameters. The stationary formed a numerical analysis of the full quantum system
point equations at the initial time lead to the following defined by (1). This is conveniently done in a basis of
boundary conditions: (i) X(t′) is real (i.e. p is real and center-of-mass coordinate X eigenstates and oscillator
T maybechosensothatX0 isalsoreal),(ii)thepositive excitationnumbern. Inthisbasisthestateisrepresented
and negative frequency parts of y(t′) are related to each by a multi-component wave function ψn(X) X,nΨ ,
other by v =u∗eθ. and the time-independent Schr¨odinger equati≡onhread|s i
More boundary conditions appear when one evalu-
ates the total transmission probability, i.e. integrates ∂2ψn(X) 1
+ n+ ωψ (X)
A(X ,y ;P,n)2 over X and y , again in a gaussian − ∂X2 (cid:18) 2(cid:19) n
f f f f
| |
approximation. These conditions involve the final time
+ Vnn′(X)ψn′(X)=Eψn(X) (5)
and simply require that (iii) X(t) and y(t) are real on
Xn′
the DE part of the contour.
where Vnn′(X)= nV(X+y)n′ . Ourchoice ofa gaus-
h | | i
sian potential V enables us to calculate Vnn′(X) by a
numerical iteration procedure. Equation (5) is supple-
Im t
mented with the standard boundary conditions: (a) the
incoming wave (X ) is in a state of given center-
→ −∞
of-massmomentumP andexcitationnumbern; (b)only
outgoing waves exist at X + .
→ ∞
B C Tosolvethesystem(5)numerically,weintroducealat-
tice with equal spacing, X = ka, and discretize eq. (5)
k
⊗
using the Numerov–Cowling algorithm (which reduces
A D E Re t the discretization error to O(a6)). We also truncate the
systemtoa finite numberofoscillatormodes n N . In
0
≤
order to insure good accuracy of the solution, we have
chosenthenumberoflatticesites2N andthecutoffN
X 0
aslargeas2N =2 4096,N =400. Thiscorrespondsto
X 0
·
Fig 1. Complex time contour used to find the stationary over 3 million coupled complex equations. To deal with
point solutions. them, we take advantage of the special form of Eq. (5).
Indeed,byinvertingasetof(N +1) (N +1)matrices,
0 0
×
At given T and θ these three boundary conditions are whichiscomputationallyfeasible,Eq.(5)canberecastin
sufficient to specify the complex solution of the classical the form ψn(Xk) = n′[Lkψn′(Xk 1)+Rkψn′(Xk+1)].
equations of motion on the contour BCDE (up to time The elimination of ψPn at definite X−k leads to a system
translationsalongtherealaxis). Giventhis solution,the of similar form for the remaining variables (with suit-
exponentforthetransmissionprobability(2)isthevalue ably redefined L and R), againafter (N +1) (N +1)
0 0
×
of 2ReΓ at the stationary point. Explicitly, we find matrixalgebraandmatrixinversion. Inthiswaywepro-
gressively eliminate variables at intermediate values of
F(ǫ,ν)=2ImS0−ǫT −νθ Xk and ultimately obtain a system that linearly relates
ψ atthe endpoints X = N a andX =+N a. With
n X X
where −
a discretized version of the boundary conditions (a) and
1 1 ω2 (b), this final system is straightforward to solve. The
S0 =−Z dt (cid:20)2X∂t2X + 2y∂t2y+ 2 y2+U(X+y)(cid:21) transmission probability is then determined by ψn 2 at
BCDE | |
the end point X =N a.
X
is the (rescaled) classicalaction for the complex solution We performed a series of checks of this numerical pro-
of the above boundary value problem. The total energy ceduretoinsurethatourcalculationsaresufficientlypre-
and excitation number are related to T and θ by ciseandthattheresultsareclosetothecontinuumlimit.
3
0 hand, limitations of the semiclassical computations are
far less severe. The generalization of the semiclassical
−5
approachtoquantum-mechanicalsystemswithharmonic
binding of more than two particles in more than one
−10
space dimension is straightforward, and we also expect
g2 E = 1.1
log T0−15 gg22 EE == 11..23 tilhaartwoatyheprrobviniddeidngthpeoitresnetmiailcslamssaiycablewtarveeatfeudncitnioanssiamre-
g2 E = 1.4
−20 g2 E = 1.5 known. Indeed, in all such cases the transmission ampli-
g2 E = 1.6 tudes with highly excited initial states will be given by
−25 g2 E = 1.7 (path)integralsofexponentialfunctions,andthetunnel-
g2 E = 1.8
ingexponentswillbedeterminedbyappropriatestation-
−30
20 40 60 80 100 ary points. The latter will be complex solutions to clas-
1/g2 sical field equations on contours in complex time, with
Fig2. Logarithmofthetransmissionprobabilityvs. 1/g2. boundaryconditionsdependingonthebindingpotential.
AlimitanalogoustoEq.(4)willthendeterminethetun-
WepresentinFigs.2and3theresultsofthefullquan- nelingexponentforincomingsystemsinlowlyingbound
tum mechanical computation of the transmission proba- states.
bilityforthesysteminitiallyinitsoscillatorgroundstate. The semiclassical calculability of pre-exponential fac-
ThepotentialV isgaussian,andω =1/2. Figure2shows tors is less clear. While it is plausible that these fac-
that the transmission probability T0(E) indeed has the torsaregivenbyfunctionaldeterminantsaboutcomplex
functional form (3): at fixed ǫ g2E, the logarithm of classical solutions for highly excited incoming states (fi-
≡
T0 isverywellfitbyalinearfunctionofg−2. Weusethis nite ν in our model), we do not expect that a limiting
fit to obtain the exponent F0(ǫ). Both the full quantum property similar to Eq. (4) will continue to hold for the
mechanicalresults for F0(ǫ) and the semiclassicalresults pre-exponents. The calculation of such pre-exponential
(the latter obtained by implementing the limiting pro- factors for low lying states remains an interesting open
cedure (4)) are shown in Fig. 3. Clearly, there is good problem.
agreement between the two. (The slight discontinuities
in the quantum mechanical results are an artifact of the Ackowledgements. We are indebted to P. Tinyakov
energy dependence of the g2 range from which we can for helpful discussions. This research was supported in
extract F . They provide an indication of the errorsdue part under DOE grant DE-FG02-91ER40676, Russian
0
to higher order effects.) We conclude that the validity Foundation for Basic Research grant 96-02-17449a and
of the semiclassical approach is confirmed by the direct by the U.S. Civilian Research and Development Foun-
quantum mechanical computation. dation for Independent States of FSU (CRDF) award
RP1-187. Two of the authors (C.R. and V.R.) would
like to thank Professor Miguel Virasoro for hopsitality
atthe Abdus SalamInternationalCenterforTheoretical
Physics, where part of this work was carried out.
quantum
1 semiclassical
0.8
F0 0.6
0.4 [1] T.Banks,M.BenderandT.T.Wu,Phys.Rev.D8(1973)
3346; T. Banks and M. Bender, Phys. Rev. D8 (1973)
0.2 3366;.
[2] S.Coleman, The uses of instantons, In: The whys of sub-
0 nuclear physics. Proc. 1977 Int. School on Subnuclear
1 1.2 1.4 1.6 1.8 2
Physics, ed. A. Zichichi, Plenum, N.Y.,1979.
g2 E [3] V.A. Rubakov and P.G. Tinyakov, Phys. Lett. B279
Fig 3. Quantum mechanical and semiclassical results.
(1992) 165.
[4] V.A.Rubakov,D.T. Son,andP.G. Tinyakov,Phys.Lett.
B287 (1992) 342.
5. Fullquantummechanicalcomputations(analyticor
[5] A.N.KuznetsovandP.G.Tinyakov,Mod.Phys.Lett.A11
numerical) of barrierpenetration probabilities are rarely (1996) 479; Phys. Rev.D56 (1997) 1156.
possible. Even for our simplified system, values of g [6] P.G. Tinyakov, Phys.Lett. B284 (1992) 410.
smaller than 0.1 are difficult to study, as one has to deal [7] A.H.Mueller, Nucl. Phys. B401 (1993) 93.
with very small transmission coefficients. On the other
4