VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S week ending1 OCTOBER 2004

Coherent Classical-Path Description of Deep Tunneling

Dong H. Zhang1,2 and Eli Pollak3

1Center for Theoretical and Computational Chemistry and State Key Laboratory of Chemical Reaction Dynamics,Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, People’s Republic of China 116023

2Department of Computational Science, National University of Singapore, Singapore 1192603Chemical Physics Department, Weizmann Institute of Science, 76100, Rehovot, Israel

(Received 13 April 2004; published 27 September 2004)

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A central challenge to the semiclassical description of quantum mechanics is the quantum phenome-non of ‘‘deep’’ tunneling. Here we show that real time classical trajectories suffice to account correctlyeven for deep quantum tunneling, using a recently formulated semiclassical initial value representationseries of the quantum propagator and a prefactor free semiclassical propagator. Deep quantum tunnelingis effected through what we term as coherent classical paths which are composed of one or more classicaltrajectories that lead from reactant to product but are discontinuous along the way. The end and initialphase space points of consecutive classical trajectories contributing to the coherent path are close to eachother in the sense that the distance between them is weighted by a coherent state overlap matrix element.Results are presented for thermal and energy dependent tunneling through a symmetric Eckart barrier.

DOI: 10.1103/PhysRevLett.93.140401 PACS numbers: 03.65.Sq, 03.65.Xp

Tunneling is one of the most fascinating aspects ofquantum mechanics. A (Gaussian) wave packet, localizedinitially on one side of a barrier with a mean energy andan energy variance substantially smaller than the barrierheight is scattered off the barrier, but parts of it aretransmitted through the barrier. This is a classically dis-allowed process; classical trajectories whose energy isbelow the barrier will of course be reflected by it. Yetthe semiclassical description of quantum mechanics hascome up with a variety of beautiful scenarios for ‘‘ex-plaining’’ quantum tunneling with the aid of classicalmechanics.

Perhaps the simplest approach is to represent the initialwave packet in phase space through its Wigner transformand then propagate it classically in time. The wave packetwill always have a tail whose energy is larger than thebarrier height. Classical trajectories initiated at thesephase space points will be transmitted since their energyis above the barrier. The probability for such a trans-mission is exponentially small and so one has a qualita-tive description of tunneling via classical trajectories [1].In fact, this classical path description of tunneling is exactfor a parabolic barrier. However, as noted by Maitra andHeller [2], it will fail for ‘‘deep’’ tunneling. When thebarrier is not parabolic, or more specifically when it goesto a constant value at plus or minus infinity, thenthe classical path contribution becomes too small.Tunneling is then dominated by nonclassical paths con-necting two manifolds of classical trajectories lying closeto the separatrix between transmitted and reflected tra-jectories. Kay [3] reanalyzed the tunneling problem andshowed that ‘‘good’’ estimates of the tunneling probabil-ity may be obtained with single trajectories, provided thatone uses a SemiClassical Initial Value Representation(SCIVR) of the propagator with coherent states that

0031-9007=04=93(14)=140401(4)$22.50

have a complex time dependent width. Grossmann hasrecently shown [4] that an estimate of the tunnelingprobability of similar quality to that of Kay may beobtained by splitting the propagator into a product oftwo equal time propagators and then estimating eachhalf time propagator using an SCIVR propagator. In hiscomputation, tunneling was effected through a combina-tion of two classical trajectory segments. Burant andBatista continued in this vein using time slicing of thepropagator [5].

An alternative description of tunneling is through useof classical trajectories evolving on the upside downpotential energy surface. The tunneling process is thenenvisaged as a classically allowed motion up to the bar-rier, an imaginary time trajectory running through thebarrier, and then again a real time classical trajectoryleading from the turning point to the asymptotic region[6]. This description necessitates the use of complextrajectories.

A compromise between these two approaches has beensuggested by Ankerhold and Saltzer [7,8]. Tunneling iseffected by ‘‘large’’ fluctuations between families of pe-riodic orbits. This description leads to a good estimationof deep tunneling probabilities.

We will present here a perspective of deep tunneling,which may be considered in some sense to include acombination of Kay’s [3] and Grossmann’s [4] insightsto the problem. The path leading from reactants to prod-ucts is composed of segments of classical trajectories. Theend point in phase space of one segment and the initialpoint of the following segment are close to each other, asdetermined by an overlap of coherent state matrix ele-ments. We may thus describe deep tunneling as effectedby ‘‘coherent classical paths’’ leading from reactants toproducts. The number of segments needed is a function of

2004 The American Physical Society 140401-1

VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S week ending1 OCTOBER 2004

the depth of the tunneling: the deeper the tunneling, themore segments one needs. In contrast to the previoussemiclassical approaches, the coherent classical path de-scription presented here leads to the exact quantum tun-neling probability. The methodology is also readilyapplicable to systems with many degrees of freedom,although here we will limit ourselves to one-dimensionaltunneling through a symmetric Eckart barrier V�q� �V0cosh

�2 qq0

with the Hamiltonian H � p2=2m� V�q�.Our starting point is a prefactor free SCIVR of the

quantum propagator whose form is [9]:

K 0�t� �Z 1

�1

dpdq2 �h

ei�hS�p;q;t�jg�p; q; t�ihg�p; q; 0�j; (1)

and we use the ‘‘hat’’ notation to denote quantum opera-tors. The coordinate representation of the coherent state attime t is

hxjg�p; q; t�i ��r�p; q; t�

�14exp

���p; q; t�

2x� q�t��2

�i�hp�t�x� q�t��

�; (2)

where p�t�; q�t� are the classically evolved to time tmomentum and coordinate p and q. Since we are dealingwith a scattering problem in which much of the motion isin the asymptotic region, we will choose the width pa-rameter appropriate for a free particle, that is, �t� �

1�i �ht with as yet an arbitrary positive number and rand i denote the real and imaginary parts of �t� re-spectively. The action is S�p; q; t� �

Rt0 dt

0p�t0�2

2m �eVq�t0�� � �h22 r�t

0�� and the coherent state averaged

potential is defined as eVq�t�� � �r�t� �1=2 R1�1 dxe

�r�t�x�q�t��2V�x�. This specific form of theSCIVR propagator has the feature that on the average itobeys the Heisenberg equation of motion at each point inphase space and time [9]. At the initial time, it reduces to

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the identity operator and it is exact for free particlemotion. If the width parameter is chosen to be a constant(�p; q; t� � const) then the propagator given in Eq. (1) isidentical to Heller’s frozen Gaussian SCIVR propagator[10].

The time evolution equation for the SCIVR propagatoris readily seen to be [11]

i �h@K0

@t� HK0 � C; (3)

where the correction operator is [9]

C�t� �Z 1

�1

dpdq2 �h

�V�q; t�ei�hS�p;q;t�jg�p; q; t�ihg�p; q; 0�j;

(4)

and �V�q; t� � eVq�t�� � V0q�t��q� q�t�� � V�q�. Theformal solution of the time evolution equation is [12]

K 0�t� � K�t� �1

i �h

Z t

0dsK�t� s�C�s�; (5)

where K�t� is the exact quantum propagator which obeysthe equation of motion i �h @K

@t � H K and K�0� � I.Expanding the exact propagator in a power series in thecorrection operator gives the SCIVR series representationof the exact propagator:

K�t� �X1j�0

Kj�t�; Kn�t� �i�h

Z t

0dsKn�1�t� s�C�s�;

n �1: (6)

One then notes that, for example, the first order correctionK1�t� involves a trajectory evolving from time 0 to time sand then a second trajectory evolving from the time s tothe time t. The final phase space point p�s�; q�s� of the firstsegment is related to the initial phase space point of thesecond segment �p0; q0� through the overlap

hg�p0; q0; 0�jg�p; q; s�i ��

4r�s�

� �s��2

�14e�

12��s���s�q

0�q�s��2�p0�p�s��2

�h2�� i

�hp�s���s�p0

��s� q0�q�s��; (7)

demonstrating that the coherent state overlap forces thefinal and initial points to be ‘‘close’’ to each other. In thissense, the sum of the two trajectories gives a ‘‘coherent’’path which evolves up to the time t. Clearly the nth termin the SCIVR series revolves about a coherent path com-posed of n� 1 coherent trajectory segments. In this way,the exact propagator is decomposed into contributions ofcoherent paths, whose number of discontinuities is relatedto the order of the term in the series. From a practicalpoint of view, we shall show below that typically one neednot go beyond the first few terms in the SCIVR series toobtain a very good description of the deep tunnelingprocess.

We first consider the thermal rate for transmissionthrough the symmetric Eckart barrier. The formalexpression for the rate is given in terms of theeigenfunctions j��Fi of the symmetrized thermal fluxoperator F�q�; �� � e��H=2 1

2m p��q� q�� � ��q�

q��p�e��H=2. Here q� is the location of the dividingsurface taken to be 0 for the symmetric Eckart barrierand � � 1=kBT is the inverse temperature. The eigenval-ues of the operator are �F; for explicit expressions interms of matrix elements of the Boltzman operator see,for example, Ref. [13]. For our purposes, we note that theflux eigenfunctions and eigenvalues are computed nu-

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VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S week ending1 OCTOBER 2004

merically accurately. The thermal tunneling factor (unityin the classical high temperature limit but greater thanunity in the deep tunneling region) for the barrier trans-mission probability is then defined as:

��� � 2 �h�FZ 1

0dtCF�t�;

CF�t� ���F

��������Ky�t�F�qz; 0�K�t�����������F

�

�

��F

��������Ky�t�F�qz; 0�K�t�����������F

� :

(8)

where K�t� � e�i= �h�Ht� is the exact quantum real timepropagator and CF�t� is the thermal flux autocorrelationfunction.

Using the same parameters for the Eckart barrier asstudied in Ref. [14], we plot in Fig. 1 the flux autocorre-lation function for three different temperatures. Thewidth parameter was chosen optimally by minimizingthe expectation value of the correction operator, as de-scribed in Ref. [15]. Panel (a) compares the numericallyexact correlation function with the results obtained usingthe zeroth and first order terms in the SCIVR series at the(inverse) temperature �h�! � 4, which is well above thecrossover temperature ( �h�! � 2) between thermal ac-tivation and below barrier tunneling. The tunneling factor

0 500 1000 1500T(time)

0.0

0.4

0.8

1.2

Cff(x

10−1

5 )

QM0

th IVR

1st IVR

2nd

IVR

0 200 400 6000.0

1.0

2.0

3.0

4.0

Cff(x

10−1

3 )

0 100 200 300 400 500

0.0

0.5

1.0

1.5

Cff(x

10−9

)

(c)

(b)

(a)

FIG. 1. The thermal flux autocorrelation function for anEckart barrier at three different temperatures. Panels (a)-(c)correspond to the reduced inverse temperatures �h�! �4; 10; 20, respectively. The notation for the various lines ap-pears in the legend on the figure. For further details, see thetext.

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at this temperature is ��� � 2:07 and so perhaps it is notsurprising that already the first order term converges tothe exact result. Panel (b) of the figure shows the fluxcorrelation function at a lower temperature �h�! � 10 forwhich the tunneling factor is ��� � 162. Panel (c) showsthe same but for very deep tunneling, the (inverse) tem-perature is �h�! � 20 and the tunneling factor is ��� �5:34 108. Even for this deep tunneling case, deep in thesense that the temperature is well below the crossovertemperature of �h�! � 2, one sees that already thesecond term in the SCIVR series converges to the exactanswer. As argued by Maitra and Heller, the zeroth orderterm is insufficient to give the correct tunneling factor.But adding in the coherent paths, which include one andtwo jumps between classical paths suffices for an excel-lent description of the tunneling dynamics. We note thatthe results presented in Fig. 1 are a rather stringent test forthe SCIVR series method. Not only does the transmissionfactor come out correct but so does the time dependentflux correlation function.

It is also of interest to study the tunneling dynamics inthe energy domain. In Fig. 2, we plot the energy depen-dent transmission probability for the same Eckart barrier.The probability is obtained by Fourier transformation ofthe time evolved thermal flux eigenfunction j��Fi whosetemperature was chosen to be �h�! � 4. The fact that thezeroth order term gives probabilities which are greaterthan unity should not be of any special concern. The

0 0.2 0.4 0.6 0.8E(eV)

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

P(E

)

QM0

th IVR

1st IVR

2nd

IVR

0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

1.5

P(E

)

(a)

(b)

FIG. 2. The energy dependent transmission probability for asymmetric Eckart barrier. Panel (a) shows results on a linearscale, panel (b) on a logarithmic scale. The notation for thevarious lines appears in the legend on the figure. Note theconvergence to the numerically exact result even in the lowenergy deep tunneling region.

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-10.0 -8.0 -6.0 -4.0 -2.0 0.00.00

0.02

0.04

0.06

0.08

0.10|ψ|(γ=0.5)

QM0th-order IVR1st-order IVR2nd-order IVR3rd-order IVR

2.0 4.0 6.0 8.0 10.00.00

0.10

0.20

0.30

0.40

0.50

0.60

X

FIG. 3. Reflection and transmission of a Gaussian wavepacket incident on a symmetric Eckart barrier from the right.The right panel of the figure shows the reflected wave packet,the left panel the transmitted. Note the change of scale in theamplitude for each panel. The notation of the lines appears inthe legend on the figure. For other details, see the text.

VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S week ending1 OCTOBER 2004

SCIVR propagator is in general not a unitary operator.What is important to note is that for almost the wholeenergy range, it takes three terms in the SCIVR series toconverge to the exact result. At the lowest energy shown(0.01 eV), the exact probability is 1:29 10�9. Resultsbased on the second, third and fourth order terms are2:51 10�9; 1:40 10�9, and 1:31 10�9, respectively.Since the second order term suffices for all other energies,we do not plot the higher order results in Fig. 2. We alsostress that in all our computations, we have found that theSCIVR series is well behaved and one does not find anydivergence when going to higher orders in the series.

Finally, in Fig. 3, we plot a time evolved Gaussian wavepacket whose initial energy was half the barrier height(0.5 eV) and whose variance in space was 1.5 a.u., whichimplies an energy variance of 0.029 eV. The wave packetwas initiated to the right of the barrier. Its fate afterscattering is shown in the figure. Most of it is reflectedas shown in the right panel. The amplitude of the trans-mitted part, shown in the left panel, is much smaller (notethe change of scale in the figure). However, it takes onlyfour terms in the SCIVR series to obtain the correctreflected and transmitted wave packet.

In summary, we have demonstrated that deep quantumtunneling is well described in terms of coherent classicalpaths which involve only real time classical trajectories.Tunneling occurs through coherent classical paths, con-sisting of a few trajectory segments related to each otherby coherent state overlap matrix elements. Interestingly,

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even deep tunneling does not necessitate more that a fewjumps in the path. We have employed a prefactor freeSCIVR of the propagator. One could repeat the samecomputation using the Herman-Kluk propagator [16].Past experience shows that this would lead to even fasterconvergence of the SCIVR series [9]. In other words, thenumber of trajectory segments needed to describe thetunneling does depend on the SCIVR propagator used.Since it is much more difficult to compute the Herman-Kluk propagator in many dimensions because of theprefactor, we limited ourselves here to the prefactor freepropagator. Although this Letter deals only with one-dimensional tunneling, there is nothing in the SCIVRseries method that limits its dimensionality in principle,and one should expect that the coherent classical pathpicture presented here will remain valid also for multi-dimensional tunneling problems.

We thank Professor J. Shao for stimulating discussions.This work has been supported by grants of the U.S. IsraelBinational Science Foundation and the Israel ScienceFoundation. D. H. Zhang acknowledges an academic re-search grant from the Ministry of Education and Agencyfor Science and Technology Research, Republic ofSingapore.

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[2] N.T. Maitra and E. J. Heller, Phys. Rev. Lett. 78, 3035(1997).

[3] K. G. Kay, J. Chem. Phys. 107, 2313 (1997).[4] F. Grossmann, Phys. Rev. Lett. 85, 903 (2000).[5] J. C. Burant and V. S. Batista, J. Chem. Phys. 116, 2748

(2002).[6] T. F. George and W. H. Miller, J. Chem. Phys. 56, 5722

(1972); ibid. 57, 2458 (1972); W. H. Miller, J. Chem. Phys.62, 1899 (1975).

[7] J. Ankerhold and M. Saltzer, Phys. Lett. A 305, 251(2002).

[8] M. Saltzer and J. Ankerhold, Phys. Rev. A 68, 042108(2003).

[9] S. Zhang and E. Pollak, J. Chem. Phys. 121, 3384 (2004).[10] E. J. Heller, J. Chem. Phys. 75, 2923 (1981).[11] J. Ankerhold, M. Saltzer and E. Pollak, J. Chem. Phys.

116, 5925 (2002).[12] S. Zhang and E. Pollak, Phys. Rev. Lett. 91, 190201

(2003).[13] E. Pollak, J. Chem. Phys. 107, 64 (1997).[14] E. Pollak and B. Eckhardt, Phys. Rev. E 58, 5436 (1998).[15] S. Zhang and E. Pollak, J. Chem. Phys. 119, 11058 (2003).[16] M. F. Herman and E. Kluk, Chem. Phys. 91, 27 (1984).

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