CERN-PH-TH/2007-065
On the over-barrier reflection in quantum mechanics
with multiple degrees of freedom
D.G. Levkova1, A.G. Panina,b2, S.M. Sibiryakovc,a3
aInstitute for Nuclear Research of the Russian Academy of Sciences,
60th October Anniversary prospect 7a, Moscow 117312, Russia.
bMoscow Institute of Physics and Technology,
Institutskii per. 9, Dolgoprudny 141700, Moscow Region, Russia.
cTheory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland.
Abstract
We present an analytic example of two dimensional quantum mechanical system,
where the exponential suppression of the probability of over–barrier reflection changes
non-monotonically with energy. The suppression is minimal at certain “optimal” ener-
gies where reflection occurs with exponentially larger probability than at other energies.
1 Introduction
Tunneling and over–barrier reflection are the characteristic non–perturbative phenomena in
quantum mechanics. They typically occur with exponentially small probabilities,
P ∝ e−F/~ , (1)
where F is the suppression exponent; still, the above phenomena are indispensable in under-
standing a wide variety of physical situations, from the generation of baryon number asym-
metry in the early Universe [1] to chemical reactions [2] and atom ionization processes [3].
During the last decades extensive investigations of tunneling processes in systems with
many degrees of freedom have been performed [2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. These studies
1levkov@ms2.inr.ac.ru
2panin@ms2.inr.ac.ru
3Sergey.Sibiryakov@cern.ch, sibir@ms2.inr.ac.ru
http://arxiv.org/abs/0704.0409v2
revealed a rich variety of features of multidimensional tunneling which are in striking contrast
to the properties of one–dimensional tunneling and over–barrier reflection. In particular, the
following phenomenon has been observed: the probability of tunneling may depend non-
monotonically on the total energy of the system and exhibit resonance-like peaks. One can
envisage three physically different mechanisms of this phenomenon. The first mechanism,
present already in one-dimensional case, is tunneling via creation of a metastable state.
In this case the tunneling probability at the maximum of the resonance is exponentially
higher than at other energies. On the other hand, the resonance width ∆E is exponentially
suppressed; so, after averaging with an energy distribution of a finite width the effect of the
resonance is washed out in the semiclassical limit ~ → 0. The second possible mechanism of
non-monotonic behavior of P(E) is quantum interference [7, 13] (see also [14]). In this case
the peak value of the tunneling probability is only by a factor of order one higher than the
average value, while the width of the resonance scales as ∆E ∝ ~. Again, the resonances
become indiscernible in the semiclassical limit. In both these cases the resonances can
be attributed to the subleading semiclassical corrections, i.e. non-monotonic behavior of
the pre-exponential factor omitted in Eq. (1). The third possibility is that the suppression
exponent F (E) is non-monotonic. In this case the existence of the “resonances” is the leading
semiclassical effect: the optimal tunneling probability at the maximum of the resonance is
exponentially higher than the probability at other energies. At the same time the resonance
width scales as4 ∆E ∝
~. This last possibility of “optimal tunneling” is definitely of
interest; yet, it did not receive much attention in literature. We are aware of only a few works
mentioning non–monotonic dependence of the suppression exponent on energy [15, 16, 14].
It is worthwhile studying this phenomenon in detail; this can provide a new insight into the
dynamics of multidimensional tunneling.
In this paper we consider the process of over–barrier reflection in a simple model with two
degrees of freedom. Our setup is interesting in two respects. First, the model under study
is essentially non–linear and the variables cannot be separated; still, over–barrier reflections
in this model can be described analytically within the semiclassical framework. Thus, this
model can serve as an analytic laboratory for the study of multidimensional tunneling. Sec-
ond, the suppression exponent F of the reflection process behaves non–monotonically as the
4This follows from the representation
P(E) ∝ exp
F (Eo)
F ′′(Eo)(E − Eo)2
of the tunneling probability in the vicinity of the maximum.
total energy E changes. We demonstrate that the function F (E) possesses a number of
local minima E = Eo, where reflection is optimal. We stress that the process we study is
exponentially preferable at “optimal” energies as compared to other energies.
Our model describes the motion of a quantum particle in the two dimensional harmonic
waveguide (see Refs. [8, 10, 14] for similar models). The Hamiltonian is
w2(x, y) ,
where x, y are the Cartesian coordinates and m is the mass of the particle. The function
U = mω2w2/2 represents the waveguide potential in two dimensions: a particle with small
energy is bound to move along the line w(x, y) ≈ 0. We do not introduce a potential barrier
across the waveguide and consider the case when the line w = 0 stretches all the way from
x → −∞ to x → +∞. We also assume that the function w(x, y) is linear in the initial
asymptotic region,
w(x, y) → y as x→ −∞ .
In the present paper we consider two particular cases of the function w(x, y) describing
waveguides with one and two sharp turns5, see Fig. 1.
The motion of the particle at x → −∞ is a superposition of free translatory motion in
x direction and oscillations of frequency ω along y coordinate; the state of such a particle is
fully characterized by two quantum numbers, the total energy E and y–oscillator excitation
number N . The particle sent into the waveguide from the asymptotic region x→ −∞ with
given E, N may either continue to move towards x → +∞, or reflect back into the region
x→ −∞. We are interested in the probability P(E,N) of reflection.
Let us discuss reflections at the classical level. [Note that the classical counterpart of N
is the energy of transverse oscillations.] Consider first the waveguide with one sharp turn
(Fig. 1a). One observes that the outcome of the classical evolution, i.e. whether or not
the particle reflects from the turn, depends not only on the total energy E, but on other
dynamical quantities as well. In particular, the direction of the momentum of the particle
in the vicinity of the turn (point C on the graph) is important. This means that the entire
dynamics in the waveguide should be taken into account in order to determine the possibility
of classical reflection. This is in sharp contrast with the situation in one–dimensional case,
where reflection from the potential barrier (or transition through it) is ensured by the value
of the conserved energy of the particle.
5The explicit expressions for the waveguide functions w(x, y) will be presented in the subsequent sections.
Figure 1: The equipotential contour U = E for the waveguides with (a) one and (b) two
sharp turns. An example of classical trajectory is shown in the case (b).
Now, consider the waveguide with two turns. The model is characterized by the angles
of the turns and the distance L between them (see Fig. 1b). Suppose the particle starts
moving classically from x → −∞ with N = 0 along the valley w = 0. Then, the transverse
oscillations get excited only after the particle crosses the first turn, point C ′ on the plot, so
that at the time of arrival to the second turn (point C) approximately ωτ/2π oscillations
are made, where τ ∼ L
m/2E is the time of motion between the two turns. The state of
the particle (coordinates and momenta) at which it comes across the second turn depends
periodically on the phase of transverse oscillations ωτ . Hence, one expects that the regime
of motion of the classical particle can change from transmission to reflection and back as the
energy grows (τ decreases); the energies where it happens can be roughly estimated as
mω2L2
2(2πn)2
. (2)
We will see that this is indeed the case for the waveguides with certain angles of the turns.
At some values of E, N the reflection process cannot proceed classically. Then, at the
quantum mechanical level its probability is exponentially suppressed, F (E,N) > 0. It is
natural to call such a process “over–barrier reflection”6. The central quantity to be studied
below is the suppression exponent F (E,N) of this process. The above discussion suggests
that F (E,N), being determined by the entire dynamics in the waveguide, may be a highly
non–trivial function. For the particular case of the waveguide with alternating regimes of
6By this term we want to emphasize that the process is classically forbidden. Recall, however, that there
is no actual potential barrier across the waveguide in our setup.
classical reflections and transmissions F should oscillate: F = 0 at the energies where the
classical reflections are allowed, and F > 0 at the energies where the reflections are classically
forbidden. One can expect that the similar oscillatory behavior of the suppression exponent
persists for other two–turn models as well. Now, instead of reaching zero, F may possess
a number of local positive minima implying that the reflection at the “optimal” energies is
still a tunneling process.
Let us emphasize the difference of the “optimal tunneling” from quantum interference
and resonance phenomena in our two–turn model. The interference of the de Broglie waves
reflected from the two turns can, in principle, lead to oscillations in the reflection proba-
bility P(E). One can estimate the positions of the interference peaks by equating the De
Broglie wavelength of the particle to an integer fraction of the distance between the turns,
2mE ∼ L/n. This yields the energies of the interference peaks,
Eintn ∼
(2πn)2~2
This formula is completely different from Eq. (2) for the peaks due to “optimal tunneling”.
In particular, the distance between the adjacent inteference peaks,
∆Eint ∼
scales proportional to ~. Thus, these peaks should be averaged over in the semiclassical
limit. Besides, the amplitude of the interference peaks is at most of order one and does
not affect the suppression exponent. Indeed, the exponential increase of the scattering
amplitude can arise due to quantum interference only in the presence of a resonant state
with exponentially long life–time. This state should be supported somewhere in between the
turns and should be classically stable. In Sec. 4.2 we show that such states are absent in
our system. One concludes that the peak–like structure of the probability P(E) of “optimal
tunneling” is caused by completely different physical reasons as compared to the case of
resonance scattering in quantum theory.
It is worth noting that the phenomenon of “optimal tunneling” has an important imple-
mentation in field theory. Recently it was argued [17] (see also Ref. [16]) that the probability
of tunneling induced by particle collisions [18, 19] reaches its maximum at a certain “op-
timal” energy and stays constant7 at higher energies. This result, if generic, provides the
7As opposed to the quantum mechanical case, the tunneling probability does not decrease at energies
higher than the “optimal” one. This is due to the possibility, specific to the field theoretical setup, to emit
the excess of energy into a few hard particles, so that tunneling effectively occurs at the “optimal” energy.
answer to the long–standing question [20] about the high–energy behavior of the probability
of collision–induced nonperturbative transitions in field theory. The quantum mechanical
model presented here supports the generic nature of the phenomenon of “optimal tunnel-
ing”; the simplicity of our model enables one to get an intuitive insight into the nature of
this phenomenon.
The paper is organized as follows. In Sec. 2 we review the semiclassical method of complex
trajectories, which is exploited in the rest of the paper. Reflections in the waveguides with
one and two turns are considered in Secs. 3 and 4 respectively. We discuss our results in
Sec. 5. In appendix we analyze the validity of some assumptions made in the main body of
the paper.
2 The semiclassical method
We start by describing the semiclassical method8 of complex trajectories which will be used
in the study of over–barrier reflections. We concentrate on the derivation of the formula
for the suppression exponent F (E,N) (see Refs. [2, 8, 9] for the details of the method and
Ref. [19] for the field theory formulation). In what follows we use the system of units
~ = m = ω = 1 ,
where the Hamiltonian takes the form,
p2x + p
y + w
2(x, y)
. (3)
One starts with the amplitude of reflection into the state with definite coordinates
xf < 0 , yf ,
A = 〈xf , yf |e−iĤ(tf−ti)|E, N〉 . (4)
Here |E, N〉 is the initial state of the particle moving in the asymptotic region xi → −∞
with fixed translatory momentum p0 =
2(E −N) and the oscillator excitation number N .
Semiclassically,
〈xi, yi|E, N〉 = eip0xi cos
′)dy′ + π/4
, (5)
8Note that the method has been confirmed by the explicit comparison with the exact quantum mechanical
results in Refs. [8, 9, 14]; specifically, the recent check [14] deals with the case when the dependence of the
suppression exponent on energy is not monotonic.
where xi, yi denote initial coordinates,
2N − y′2 , (6)
and we omitted the pre-exponential factor which is irrelevant for our purposes. Using Eq. (5),
one rewrites the amplitude (4) as a path integral,
dxidyi
[dx][dy]
xf , yf
xi, yi
eiS+ip0xi cos
′)dy′ + π/4
, (7)
where S is the classical action of the model (3).
In the semiclassical case the integral (7) is dominated by the (generically complex) saddle
point. Note that, as we continue the integrand in Eq. (7) into the plane of complex coor-
dinates, one of the exponents constituting the initial oscillator wave function grows, while
the other becomes negligibly small. Within the validity of our approximation, we omit the
decaying exponent by writing
′)dy′ + π/4
→ exp
′)dy′
, (8)
with the standard choice9 of the branch of the square root in Eq. (6).
One proceeds by finding the saddle point for the integral (7) with the substitution (8).
Extremization with respect to x(t), y(t) leads to the classical equations of motion,
ẍ = −wwx , ÿ = −wwy . (9)
Differentiating with respect to xi ≡ x(ti), yi ≡ y(ti), one obtains,
ẋi = p0 =
2(E −N) , ẏi = py(yi) =
2N − y2i .
The latter equations are equivalent to fixing the total energy E and initial oscillator energy
N of the complex trajectory,
ẋ2i +N , (10a)
ẏ2i + y
. (10b)
9 The correct branch is fixed by drawing a cut between the oscillator turning points y = ±
2N , and
choosing Im py > 0 at y ∈ R, y >
2N , see, e.g., Refs. [21].
Substituting the saddle–point configuration10 into Eq. (7), one obtains the amplitude of the
process with exponential accuracy,
A ∝ eiS+iB(xi, yi) ,
where the term
B(xi, yi) = p0xi +
′)dy′ (11)
is the initial–state contribution. For the inclusive reflection probability one writes,
dxfdyf |A|2 ∝
dxfdyf e
iS−iS∗+iB−iB∗ .
The integral over the final states can also be evaluated by the saddle point technique; ex-
tremization with respect to xf ≡ x(tf ), yf ≡ y(tf) fixes the boundary conditions in the
asymptotic future,
Im ẋf = Im xf = 0 , Im ẏf = Im yf = 0 . (12)
In this way one obtains the expression (1) for the reflection probability, where the suppression
exponent F is given by the value of the functional
F (E, N) = 2 ImS + 2 ImB(xi, yi)
evaluated on the saddle–point configuration — a complex trajectory satisfying the boundary
value problem (9), (10), (12).
The contribution B(xi, yi) of the initial state is simplified after one uses the asymptotic
form of the solution at t→ −∞ (xi → −∞),
x = p0t + x0 , y = ae
−it + āeit . (13)
Equations (10) guarantee that the quantities p0 =
2(E −N) and 2aā = N are real, since
E, N ∈ R. Therefore, one may introduce two real parameters T , θ as follows,
2 Im x0 = −p0T , ā = a∗eT+θ . (14)
One finds for the initial term (11),
2 Im B(xi, yi) = Im
2p0xi − 2Narccos(yi/
2N) + yi
2N − y2i
= −p20T −N(T + θ) + Im(yiẏi) ,
10For simplicity we assume that the saddle–point configuration is unique. Otherwise, one should take the
saddle point corresponding to the weakest exponential suppression.
and thus
F = 2 Im S̃ −ET −Nθ , (15)
where S̃ is the classical action of the system (3) integrated by parts,
S̃ = −1
xẍ+ yÿ + w2(x, y)
. (16)
Let us comment on the physical meaning of the parameters T , θ. Consider two trajectories
which are solutions to the boundary value problem (9), (10), (12) at neighbouring values of
E, N . The differential of the quantity 2 Im S̃ as one deforms one trajectory into the other is
d (2 Im S̃) = d Im(2S + xiẋi + yiẏi) = Im(xidẋi − ẋidxi + yidẏi − ẏidyi) = EdT +Ndθ ,
where in the last equality we used the asymptotic form (13), (14) of the solution. Then,
from Eq. (15) one finds,
dF (E,N) = −TdE − θdN . (17)
Thus, the parameters T and θ are (up to sign) the derivatives of the suppression exponent
with respect to energy E and initial oscillator excitation number N respectively.
Our final remark is that the boundary value problem (9), (10), (12) is invariant with
respect to the trivial time translation symmetry,
t→ t+ δt , δt ∈ R , (18)
which can be fixed in any convenient way.
3 The model with one turn
To warm up, we consider the simplest model, where the waveguide has one sharp turn,
w = y θ(−x+ y tg β) + cos β (x sin β + y cos β) θ(x− y tg β) . (19)
Here θ(x) is the step function. It is convenient to use the rotated coordinate system,
cos β − sin β
sin β cos β
The waveguide function takes the form,
w = η cos β − ξ sin β θ(−ξ) . (20)
Figure 2: The equipotential contour w2(x, y) = 2N for the waveguide (20) and the trajectory
of the critical solution with energy N/ cos2 β.
The equipotential contour w2(ξ, η) = const is shown in Fig. 2. One observes that the motion
of the particle in two regions, ξ < 0 and ξ > 0, decomposes into the translatory motion and
oscillations in the coordinates x, y and ξ, η respectively (see. Eqs. (19) and (20)); the
frequency of η–oscillations in the latter case is cos β.
Due to the presence of the step function, the first derivatives of the potential (20) are
discontinuous11 at ξ = 0. Strictly speaking, the semiclassical method is not applicable in
this situation [21]. Thus, the formula (20) should be regarded as an approximation to some
waveguide function with smooth turn. Generically the width of the smoothened turn is
characterized by a parameter b; the sharp–turn approximation (20) corresponds to b → 0.
An example of smoothening is provided by the following substitution in Eq. (20),
θ(ξ) → θb(ξ) =
1 + e−ξ/b
. (21)
The semiclassical description can be used as long as the de Broglie wavelength of the particle
is small compared to the linear size of the potential12, 1/
E ≪ b. We conclude that the
sharp–turn and semiclassical approximations are valid simultaneously for smooth waveguides
1 ≫ b≫ 1/
E . (22)
11Note that the potential itself is continuous.
12Another semiclassical condition is that the energy is sufficient to excite a lot of oscillator levels, E ≫ 1.
It is satisfied provided Eq. (22) holds.
An important property of the model (20) is invariance of the classical equations of motion
(9) under the rescaling of the coordinates,
x→ Λx , y → Λy . (23)
Using the transformation (23), one may express a solution x(t), y(t) with energy E in terms
of the “normalized” one,
x = x̃
E , y = ỹ
where the solution x̃(t), ỹ(t) has unit energy; its initial oscillator excitation number is
ν = N/E .
The suppression exponent (15) takes the form,
F (E, N) = Efβ(ν) , (24)
where fβ(ν) is the exponent for the “normalized” solution. Substituting the expression (24)
into Eq. (17), one obtains,
fβ(ν) = −T − θν . (25)
We will exploit Eq. (25) in the end of this section. Now, we proceed to finding the “normal-
ized” trajectories.
At certain initial data ν > νcr the particle can reflect from the turn classically, so that
fβ(ν > νcr) = 0 .
Let us find the value of νcr. In the region ξ < 0 the classical solution takes the form,
x(t) = p0t+ x0 , (26a)
y(t) = A0 sin(t + ϕ) . (26b)
Having crossed the line ξ = 0 (line AB in Fig. 2), the classical particle can never return
back into the region ξ < 0. Indeed, in this case it moves at ξ > 0 with constant momentum
pξ > 0. Thus, the particle can reflect classically only if its trajectory touches the line
ξ = 0. The potential of our model has ill–defined derivatives at ξ = 0, and the fate of the
particle moving along the line AB depends on the particular choice of the smoothening of
the potential. In appendix we consider the motion of the classical particle in the case when
nonzero smoothening of width b is switched on. For a class of smoothenings we show that
in the small vicinity (δξ ∼ b) of any trajectory touching the line ξ = 0 there exists some
“smoothened” trajectory, which reflects classically from the turn. Consequently, below we
associate the trajectories touching the line ξ = 0 with the classical reflected solutions.
One notices that the inclination of the trajectory (26) is bounded from above
therefore, the classical trajectory of the particle can touch the line ξ = 0, that is, y/x = ctg β
only at
A0/p0 ≥ ctg β . (27)
From Eqs. (27), (26), (10) one extracts the condition for the particle to reflect classically
from the turn,
ν ≥ νcr = cos2 β . (28)
The critical classical solution at ν = νcr touches the line ξ = 0 at η = 0 (point C in Fig. 2),
where its trajectory
xcr(t) =
2t sin β , (29)
ycr(t) =
2 sin t cos β .
has the largest inclination.
We now turn to the classically forbidden reflections at ν < νcr, which are described by
the boundary value problem (9), (10), (12). One makes the following important observation.
The waveguide function (20) has the form of two analytic functions glued together at ξ = 0.
Hence, the equations of motion (9) can be continued analytically to the complex values of
coordinates in two different ways, starting from the regions ξ < 0 and ξ > 0 respectively. In
this way one obtains two complex solutions, ξ−(t), η−(t) and ξ+(t), η+(t). These solutions
and their first derivatives should be matched at some moment of time t1, ξ(t1) = 0. [Note
that the matching time t1 does not need to be real.] Below we conventionally refer to these
solutions as the ones belonging to the regions ξ < 0 and ξ > 0.
By the same reasoning as above we find that once the particle arrives into the region
ξ > 0, it never reflects back to ξ < 0, unless pξ = 0. So, in the region ξ > 0 one writes,
ξ+(t) = 0 , (30a)
η+(t) =
cos β
sin(t cos β + ϕη) , (30b)
where the “normalization” condition E = 1 has been used explicitly. Due to the conditions
in the asymptotic future, Eqs. (12), the parameter ϕη is real. We use the translational
invariance (18) to set ϕη = 0. Note that we again associate the trajectory going along the
line ξ = 0 with the reflected one.
The physical picture of over–barrier reflection that comes to mind matches with the new
mechanism of multidimensional tunneling proposed recently in Refs. [9, 11]. The process
proceeds in two steps. The first step, which is exponentially suppressed, is formation of the
periodic classical orbit (30) oscillating along the line ξ = 0. This orbit is unstable. At the
second step of the process the unstable orbit decays classically forming a trajectory going
back to x → −∞ at t → +∞. Clearly, the second step does not affect the suppression
exponent of the whole process, and we do not consider it explicitly. In what follows we
concentrate on the determination of the tunneling trajectory describing the first step of the
process.
One should find the solution at ξ < 0 and impose the boundary conditions (10). Note,
however, that the energy of our solution is fixed already. As for the initial oscillator excitation
number ν, it does not change during the evolution in the region ξ < 0. Thus, one may fix it
at the matching time t = t1. One writes,
(ẏ2 + y2)
= cos2 β + sin2 β sin2(t1 cos β) .
This complex equation allows one to express t1 as
sin(t1 cos β) = −i
νcr − ν
sin β
, (31)
where the choice of the sign is dictated by the condition in footnote 9. It is convenient to
introduce notation t1 = iT1, T1 ∈ R.
In order to find the suppression exponent fβ(ν), one needs to evaluate the parameters
T (ν), θ(ν). At ξ < 0 the solution has the form,
x−(t) = p0(t− iT/2) + x′0 , (32a)
y−(t) = ae
−it + a∗eT+θ+it , (32b)
where the definitions (13), (14) have been taken into account explicitly, so that p0, x
0 ∈ R.
One evaluates p0, x
0, a, T , θ by matching the coordinates x±, y± and their first derivatives
νcr0.20.150.10.050
Figure 3: The suppression exponent fβ(ν) for the waveguide (20); β = π/3.
ẋ±, ẏ± at t = iT1; this yields
x′0 = 0 , p0 =
2(1− ν) , a = i
1− ν/ cos2 β
T + θ
cos2 β − ν
sin β
The last two equations, together with Eq.(25), define the function fβ(ν),
fβ(ν) =
cos β
arcsh
νcr − ν
sin β
− ν cos β arcsh
νcr − ν
sin β
(νcr − ν)(1− ν)
this finction is plotted in Fig. 3. One observes that at ν → νcr the quantities T1, T, θ, fβ
tend to zero, and the complex trajectory tends to the classically allowed critical solution, cf.
Eqs. (29),
2 sin β , a→ i√
cos β .
At ν = 0 one has,
fβ(0) = −2 +
cos β
arcth (cos β) . (33)
To summarize, we obtained the suppression exponent for the reflection of a particle in the
simplest waveguide with one sharp turn.
Figure 4: The equipotential contour w2(x, y) = 2N ′ for the waveguide (35) and the trajectory
of the critical solution with energy N ′/ cos2 β > EB. The matching points C, C
′ are shown
by the thick black dots.
4 The model with two turns
4.1 Introducing the system
In the model of the previous section the suppression exponent was proportional to energy
because of the coordinate rescaling symmetry (23). Now, we are going to demonstrate that
small violation of this symmetry results in highly non–trivial graph for F (E).
One introduces a second turn into the waveguide, see Fig. 4. We want to consider this
turn as a small perturbation, so, we assume its angle α to be smaller than β. It is convenient
to introduce two additional coordinate systems, x′, y′ and ξ, η, bound to the central and
rightmost parts of the waveguide respectively. They are related to the original coordinate
system x, y as follows,
cosα sinα
− sinα cosα
cos β − sin β
sin β cos β
x′ − L
Note that the origin of the coordinate system ξ, η is shifted by the distance L. The waveguide
function is
w = θ(−x′)θ(−ξ)y + θ(−ξ)θ(x′)y′ cosα + θ(ξ)η cosα cos β ; (35)
it consists of three pieces glued together continuously at x′ = 0 and ξ = 0 (lines A′B′ and
AB in Fig. 4 respectively). At t→ −∞ the particle comes flying from the asymptotic region
x′ < 0, where w = y. In the intermediate region x′ > 0, ξ < 0 the particle moves in the x′
direction oscillating along the y′ coordinate with the frequency cosα. Finally, in the region
ξ > 0 its motion is free in the coordinates ξ, η; the frequency of η–oscillations is cosα cos β.
The model (35) no longer possesses the symmetry (23): rescaling of coordinates changes
the length L of the central part of the waveguide. In what follows it is convenient to work
in terms of the rescaled dynamical variables,
x̃ = x/L , ỹ = y/L .
In new terms the parameter L disappears from the classical equations of motion, entering
the theory through the overall coefficient L2 in front of the action. The initial–state quantum
numbers are also proportional to L2,
E = L2Ẽ , N = L2Ñ . (36)
Thus, the conditions (22) for the validity of the semiclassical approximation are satisfied in
the limit
L→ ∞ , Ẽ, Ñ = fixed .
The suppression exponent takes the form
F (E,N) = L2F̃ (Ẽ, Ñ) . (37)
To simplify notations, we omit tildes over the rescaled quantities in the rest of this sec-
tion. Rescaling back to the physical units can be easily performed in the final formulae by
implementing Eqs. (36), (37).
4.2 Classical evolution
Let us begin this subsection by demonstrating that there are no stable classical solutions
localized in the region between the turns. This is important for the determination of the
tunneling probability, since such stable solutions could lead to exponential resonances in the
tunneling amplitude. The argument proceeds as follows. Any trajectory which is localized in
the intermediate region should reflect from the line AB infinitely many times. Each reflection
involves touching the unstable orbit living at the line AB. This implies that the trajectory
itself is unstable.
We proceed by determining the region of initial data E, N , which correspond to the
classical reflections. [For brevity we will refer to this region as the “classically allowed
region”, as opposed to the “classically forbidden region” where reflections occur only at the
quantum mechanical level. We stress that these are the regions in the plane of quantum
numbers E, N .] Let us search for the critical classical solutions which correspond to the
smallest initial oscillator number N = Ncr(E) at given energy E. As in the previous section,
one finds that the particle must get stuck at the line13 AB for some time in order to reflect
back. Let us first make an assumption inspired by the study of the one-turn model that the
critical solutions touch the line AB at their maximum inclination point (point C in Fig. 4).
We will see shortly that this is true only at energies above a certain value EB, see Eq. (50).
Still, the analysis based on the above assumption enables one to catch the qualitative features
of the critical line N = Ncr(E). Besides, the analysis is considerably simplified in this case;
we postpone the accurate study until the end of this subsection. Keeping in mind the above
remarks, one writes for the solution in the intermediate region,
x′cr(t) = t
2E sin β + 1 , (38a)
y′cr(t) =
cos β
sin(t cosα) . (38b)
Before entering the intermediate region, the particle crosses the line A′B′ (point C ′ in Fig. 4).
The initial oscillator number N is most conveniently calculated at the moment
t = t0 ≡ −
2E sin β
of crossing. Using the relations (34) one obtains,
ẋcr(t0) =
sin β cosα− cos β sinα cos
cosα√
2E sin β
, (39)
and thus
Ncr(E) = E −
ẋ2cr(t0) = E − E
sin β cosα− cos β sinα cos
cosα√
2E sin β
, E > EB .
 0.05
 0.15
0.90.80.70.60.50.40.30.10
Figure 5: The boundary N = Ncr(E) of the classically allowed region at E > EB for the
waveguide model (35); β = π/3, α = π/30. The region of the classically allowed initial data
lies above this boundary. The empty circles correspond to the energies E = En, where the
curve N = Ncr(E) touches its lower envelope N = E cos
2(β + α).
As an example, we show in Fig. 5 the region of the classically allowed initial data for β = π/3,
α = π/30. One observes that the function Ncr(E) oscillates between two linear envelopes,
E cos2(β + α) and E cos2(β − α); the period of oscillations decreases as E → 0. Moreover,
the curve Ncr(E) has a number of minima at the points E = E
n . This means that the
energies E = Ecrn are optimal for reflection: in the vicinity of any point E = E
n , N =
Ncr(E
n ) reflections become exponentially suppressed independently of whether the energy
gets increased or decreased. This feature is particularly pronounced in the case α+β = π/2,
when the lower envelope coincides with the line N = 0. Then, the classical reflections (i.e.
reflections with the probability of order 1) at N = 0 are possible only in the vicinities of the
points
8π2(n− 1/2)2
This is the case we used in Introduction to illustrate the effect.
The minima E = Ecrn exist at other values of the parameters as well. For instance, let
13We do not consider reflections from the line A′B′. They disappear at larger values of N than reflections
from the line AB if α is small enough.
us find the positions of these minima in the case α ≪ 1. One differentiates Eq. (40) with
respect to energy and obtains,
Ecrn = En
π(n− 1/2)
arcsin
ctg β
2πα(n− 1/2)
+O(α2)
, (41)
where
8π2(n− 1/2)2 sin2 β
are the points where the curve N = Ncr(E) touches its lower envelope. The argument of
arcsine in Eq. (41) should be smaller than one, so, the minima Ecrn exist only at large enough
n ≥ n0 ≡
ctg β
+ 1 , (43)
where [·] stands for the integer part.
Let us make several comments. First, note that n0 ∼ O(1/α), consequently, all the
optimal points Ecrn lie in the region of small energies E ∼ 1/n20 ∼ O(α2). Second, as we
pointed out before, the formula (40) for the function Ncr(E) holds at E > EB. Comparing
the expressions (42), (43) and (50), one observes that En0 > EB if tg β > 1. So, there
does exist a range of energies where the non-monotonic behavior of the function Ncr(E) can
be inferred from the formula (40). In fact, the conclusion about the existence of the local
minima of Ncr(E), as well as the expressions (41), (42), (43) determining their positions,
remain valid also at E < EB. This follows from the rigorous analysis of the boundary of the
classically allowed region to which we turn now. The reader who is more interested in the
tunneling processes may skip this part and proceed directly to subsection 4.3.
Now, we do not appeal to the Ansatz (38). Instead, we start with the general solution
in the intermediate region,
x′ = p′0(t− t0) , (44a)
y′ = A′0 sin [(t− t0) cosα+ ϕ′] . (44b)
It is convenient to parametrize it by the total energy E = p′20 /2 + cos
2 αA′20 /2 and the
“inclination” γ defined by the relation
p′0/A
0 = tg γ cosα .
Expressions (44) take the following form,
2E (t− t0) sin γ , (45a)
cos γ
sin [(t− t0) cosα + ϕ′] . (45b)
The constants t0 and ϕ
′ are fixed by demanding the trajectory (45) to reflect classically from
the second turn, i.e. touch the line ξ = 0 at t = 0,
(x′ − 1) cos β − y′ sin β
= 0 ,
= ctg β .
These conditions imply,
t0 = −
2E sin γ
tg2 β
tg2 γ
− 1 , (46a)
ϕ′ = −
cosα√
2E sin γ
tg2 β
tg2 γ
− 1− arccos
. (46b)
One sees that the classical reflections are possible only at γ ∈ [0; β]; the boundary value
γ = β reproduces the solution (38).
In order to find Ncr(E), one should minimize the value of the incoming oscillator exci-
tation number with respect to γ at fixed E. At t = t0, when the particle crosses the first
turn,
p0 ≡ ẋ(t0) =
2E(cosα sin γ − sinα cos γ cosϕ′) . (47)
Since N = E − p20/2, one can maximize the value of the translatory momentum p0 instead
of minimizing N(γ). Formula (39) represents the value γ = β lying at the boundary of the
accessible γ–domain; this value should be compared to p0(γ) taken at local maxima.
Let us consider the case α ≪ 1. At large enough energies, E ∼ 1, Eq. (47) is dominated
by the first term, which grows with γ, so that the maximum of p0(γ) is indeed achieved at
γ = β. At small energies, however, the second term in Eq. (47) becomes essential because of
the quickly oscillating cosϕ′ multiplier: the frequency of cosϕ′ oscillations grows as E → 0,
and at E ∼ α2, in spite of the small magnitude proportional to sinα, the second term
produces the sequence of local maxima of the function p0(γ).
One expects the parameters of the trajectory at small α not to be very different from the
ones at α = 0 (the latter case was considered in Sec. 3). So, we write,
γ = β − δγ ,
where 0 < δγ ≪ 1. Expanding the expressions (46), (47) and taking into account that
E ∼ α2 one obtains,
ϕ′ = −
2E sin β
(1 + δγ ctg β) , (48a)
2E(sin β − δγ cos β − α cos β cosϕ′) . (48b)
Now, the local maxima of the initial translatory momentum can be obtained explicitly by
differentiating Eqs. (48) with respect to δγ. One finds the sequence of them,
δγn = −tg β +
sin2 β
cos β
2πn− π − arcsin
2E sin2 β
α cos β
. (49)
Only the maxima with δγn > 0 should be taken into account. The local maxima exist when
E ≤ EB ≡
α2 cos2 β
2 sin4 β
. (50)
Substituting Eq. (49) into the expressions (48), one evaluates the values of p0 at the local
maxima,
p0,n(E) =2
2E sin β − 2E sin2 β
2πn− π − arcsin
2E sin2 β
α cos β
2E cos β
1− 2E sin
α2 cos2 β
The graphs Nn(E) = E − p20,n(E)/2 are shown in Fig. 6 for the case β = π/3, α = π/30.
Each graph is plotted for the energy range E > EAn restricted by the condition δγn > 0.
They are presented together with the curve given by the formula (40). By definition, the
critical solution corresponds to the lowest of these graphs. Clearly, for each “local” curve
representing the n-th local minimum of N(γ) there is a range of energies EAn < E < EBn
where it lies lower than the “global” curve (40). This means that the parameter γ of the
critical solution changes discontinuously across the points E = EBn . Correspondingly, the
curve Ncr(E) has a break at these points. On the other hand, the function Ncr(E) is smooth
at the points An as the “local” graphs end up exactly at δγ = 0, where the parameters of
the n-th “local” solution coincide with the ones of the “global” solution.
To summarize, we have observed that the boundary of the classically allowed region is
given by a collection of many branches of classical solutions, each branch being relevant in
its own energy interval. We will see that a similar branch structure is present in the complex
trajectories describing over–barrier reflections in the classically forbidden region of E, N .
4.3 Classically forbidden reflections
In this subsection we demonstrate that the suppression exponent F (E, N) viewed as a
function of energy at fixed N exhibits oscillations deep inside the classically forbidden region
 0.02
 0.04
 0.06
 0  0.05  0.1  0.15  0.2
Figure 6: The graphs Nn(E) corresponding to the local minima of the function N(γ) (dashed
lines) plotted together with the “global” curve, Eq. (40) (solid line); β = π/3, α = π/30.
The critical curve N = Ncr(E) is obtained by taking the minimum among all the graphs.
of initial data. This result comes without surprise if one takes into account the non-monotonic
behavior of the boundary Ncr(E) of the classically allowed region. Indeed, the curve N =
Ncr(E) coincides with the line F (E,N) = 0. One has,
= −∂EF
N=Ncr(E)
so that
(Ecrn , N
n ) = 0 .
We conclude that the points E = Ecrn are the local minima of the function F (E) at fixed
N = N crn . It is natural to expect that such local minima of F (E) exist at other values of
N as well. To illustrate this fact explicitly, we study the complex trajectories, solutions to
Eqs. (9), (10), (12).
Following the tactics of the previous section, we find solutions in three separate regions:
initial region x′ < 0, final region ξ > 0, and the intermediate region x′ > 0, ξ < 0. These
solutions, together with their first derivatives, should be glued at t = t0, when the complex
trajectory crosses the line x′ = 0, and at t = t1, when ξ = 0. Besides, we are looking for the
tunneling solution which ends up oscillating along the line AB, see Fig. 4. As discussed in
Sec. 3 this assumes existence of the second step of the process: classical decay of the unstable
orbit living at ξ = 0; the latter decay is described by a real trajectory14 going to x → −∞
at t→ +∞.
The solution in the final region ξ > 0 is (cf. Eqs. (30)),
ξ+(t) = 0 , (51a)
η+(t) =
cosα cos β
sin(t cosα cos β) , (51b)
where we used the time translation invariance (18) to fix the final oscillator phase ϕη = 0.
In the intermediate region x′ > 0, ξ < 0 one writes,
x′(t) = p′0t + x
0 , (52a)
y′(t) = a′e−it cosα + ā′eit cosα . (52b)
Note that the final solution (51) does not contain free parameters; thus, the matching of x′,
ẋ′, y′, ẏ′ at t = t1 enables one to express all the parameters in Eqs. (52) in terms of one
complex variable t1,
p′0 =
2E sin β cos φ1 , (53a)
x′0 = 1 +
[sin φ1 − φ1 cos φ1] , (53b)
eiφ1/ cos β [sinφ1 + i cos β cosφ1] , (53c)
ā′ =
e−iφ1/ cos β [sinφ1 − i cos β cos φ1] , (53d)
where we introduced φ1 = t1 cosα cos β.
As the energy of the solution has been fixed already, the only remaining initial condition
involves initial oscillator excitation number at x′ < 0, see Eqs. (10). It is convenient
to impose this condition at the matching point t = t0. One recalls the definition of the
matching time t0,
p′0t0 + x
0 = 0 ,
14One wonders why this trajectory does not reflect from the turn A′B′ on its way back. This concern is
removed by the observation that the trajectory produced in the decay of the unstable orbit is not unique:
in appendix we show that the decay can occur at any point of the segment AC giving rise to a whole bunch
of potential decay trajectories. Most of these trajectories pass through the turn A′B′ without reflection.
which, after taking into account the expressions (53a), (53b), leads to the following equation,
cosα√
2E sin β
sinφ1
cos β
− cosφ1∆φ = 0 , (54)
where ∆φ = cosα(t1 − t0). At t = t0 one has,
ẋ(t0) = p
0 cosα− ẏ′(t0) sinα =
2(E −N) ,
and thus √
= ctgα sin β cosφ1 − sin φ1 sin∆φ− cos β cosφ1 cos∆φ . (55)
As before, ν = N/E.
Two complex equations (54), (55) determine the matching times t0, t1, and, consequently,
the complex trajectory. Although these equations cannot be solved explicitly, they can
be simplified in the case α ≪ 1, which we consider from now on. For concreteness, we
study reflections at N = 0. It is important to keep in mind that in the region of interest
E ∼ Ecrn ∼ O(α2); thus, one should regard all the momenta p and oscillator amplitudes a,
ā, as the quantities of order O(α). At the same time, for the distances along the waveguide
one has x ∼ O(1), so that the real parts of time intervals may be parametrically large,
Re t ∼ x/p ∼ O(1/α).
Further on, it will be convenient to work in terms of real variables, so, we represent φ1
and ∆φ as
φ1 = cosα cos β(τ1 + iT1) , ∆φ = cosα(τ + i∆T ) .
Note that τ and ∆T are the real and imaginary parts of the time interval t1 − t0 which the
particle spends in the intermediate region. Now, equation (54) enables one to express
2E sin βch(T1 cos β)
+O(α) , (56)
τ1 = −
τ cos β
cos β
−∆T cth(T1 cos β)
+O(α3) . (57)
Note that τ1 ∼ O(α), τ ∼ O(1/α). Then, the real part of Eq. (55) implies that
ch(T1 cos β) =
sin β
1 + α ctgβ cos τe∆T
+O(α2) . (58)
While deriving this formula we imposed T1 < 0 which follows from the requirement that
in the limit α → 0 equation (31) should be recovered; besides, we assumed e∆T ∼ O(1).
Substituting Eq. (58) into Eq. (56) and the imaginary part of Eq. (55), we obtain the final
set of equations,
2E = α ctgβ cos τe∆T +O(α2) , (59a)
(1 + ∆T )e−∆T = α ctgβτ sin τ +O(α) . (59b)
These two nonlinear equations, still, cannot be solved explicitly. Nevertheless, one can get
a pretty accurate idea about the structure of their solutions.
Before proceeding to the analysis of the above equations, let us derive a convenient
expression for the suppression exponent F0(E) ≡ F (E,N = 0). Note that on general
grounds one expects to obtain an expression of the form,
F0(E) = E(fβ(0) +O(α)) ,
where fβ(0) is given by Eq. (33). We are interested in the O(α) correction in this expression,
so, one must be careful to keep track of the subleading terms during the derivation.
Making use of the equations of motion, one obtains for the incomplete action (16) of the
system,
2 Im S̃ = Im p′0 =
2E sin β Im(cosφ1) .
Substitution of Eqs. (56), (57), (58) into this formula yields
2 Im S̃ = 2E
−1−∆T − α ctg β cos τe∆T
cos2 β
+ 2∆T
+O(α2)
For the parameter T one has (see Eqs. (14)),
T = −
2 Im x0
2 Im(x(t0)− p0t0)
= 2(T1 −∆T ) +
sinα Im y′(t0) , (60)
where in the last equality we used Eqs. (34) and x′(t0) = 0. The quantity Im y
′(t0) is
evaluated by using Eqs. (52b), (53) and (58); one finds,
Im y′(t0) = −
ctg β cos τe∆T +O(α)
Substituting everything into the formula (15), we obtain,
F0(E) = E
fβ(0)− 4α ctg β cos τ ∆T e∆T +O(α2)
. (61)
This expression implies that determination of the O(α) correction to the suppression expo-
nent involves finding τ , ∆T with O(1)–accuracy. This is precisely the level of accuracy of
Eqs. (59). Below we will also need the following formulae, which can be easily obtained by
using T = −F.
and Eq. (60),
= fβ(0) + 2(∆T + 1) +O(α) , (62)
2(∆T + 1 +O(α))
. (63)
Note that, though the suppression exponent differs from that in the one–turn case only by
O(α) correction, its derivative gets modified in the zeroth order in α.
Now, we are ready to analyze Eqs. (59). One begins by solving Eq. (59b) graphically,
see Fig. 7. The important property of this equation is as follows. One notices that the l.h.s.
of Eq. (59b) is always smaller than 1, the maximum being achieved at ∆T = 0. Therefore,
the solutions to this equation are confined to the bands
τ sin τ <
This corresponds to
τ ∈ [0; 2π(n1 − 1) + δτn1 ] or τ ∈ [2πn− π − δτn; 2πn + δτn] , n ≥ n1 (64)
where
δτn = arcsin
2πα(n− 1/2)
+O(α) ,
+ 1 , (65)
with [·] in the last formula standing for the integer part. The forbidden bands, where
τ sin τ > tgβ/α, are marked in Fig. 7 by yellow shading. The property (64) introduces a
topological classification of the solutions τ , ∆T to Eqs. (59). Namely, these solutions fall
into a set of continuous branches: the “local” branches τn(E), ∆Tn(E) living inside the
strips τ ∈ [2πn − π − δτn; 2πn + δτn], n ≥ n1, and the “global” branch τg(E), ∆Tg(E)
inhabiting the very first band τ ∈ [0; 2π(n1 − 1) + δτn1 ]. As follows from the definition of
τ , the topological number n counts the number of y′–oscillations during the evolution in the
intermediate region.
Let us consider the “global” branch. From Eqs. (59) one has,
τg → 2π(n1 − 1) +O(α lnα) , ∆Tg → ln(tg β/α) , E → 0 ,
τg → 0 , ∆Tg → −1 , E → +∞ .
10π 9π 8π 7π 6π  5π4π 3π 2π π 0
g 4 5
10π 9π 8π 7π 6π  5π4π 3π 2π π 0
g 4 5
Figure 7: Curves representing solutions to Eq. (59b); β = π/3, α = π/30.
By inspection of Fig. 7 one can work out the qualitative behavior of the functions τg(E),
∆Tg(E). Alternatively, these functions can be found numerically. They are plotted in Fig. 8
for the case β = π/3, α = π/30 (the curves marked with “g”). One observes that at
high enough energies the function ∆Tg(E) exhibits oscillations around the line ∆T = −1.
According to the formula (63) this means that the function F0(E)/E is non-monotonic, it
attains local minima at the points
E ′n =
8π2(n− 1/2)2
1 + 2αe−1ctgβ +O(α2)
. (66)
Moreover, if
n ≥ n′0 ≡
fβ(0) exp
fβ(0)
+ 1 (67)
there exist Eon = E
n(1 + O(α)), such that ∆T (E
n) = −1 − fβ(0)/2. Then, according to
Eq. (62) the points Eon are the “optimal” energies corresponding to the local minima of the
suppression exponent F0(E).
At low energies the function ∆Tg(E) ceases to oscillate and becomes large and positive.
According to Eq. (62) this means that the suppression exponent F0,g(E) of the “global”
solution becomes negative at low energies15, see Fig. 9. This is a clear signal that the
15It is worth mentioning that Eqs. (59) and the expression (61) for the suppression exponent become
 0  0.1  0.2  0.3
 0  0.1  0.2  0.3
 0  0.1  0.2  0.3
E’4E’5
Figure 8: Several first branches of solutions to Eqs. (59): “global” branch (“g”) and two
“local” branches (“4”, “5”); β = π/3, α = π/30.
 0.05
 0.15
 0.25
 0  0.2  0.4  0.6
 0.02
 0.04
0.120.110.100.09
Figure 9: The suppression exponent F0(E) for the “global” and first “local” (n = 4) branches;
β = π/3, α = π/30. The vicinity of intersection of the graphs is enlarged in the upper right
corner.
“global” solution becomes unphysical at these energies and its contribution to the reflection
probability should be discarded: negative suppression exponent contradicts the unitarity
requirement16, P < 1. One is forced to conclude that at low energies reflection is described
by the “local” solutions. Let us study them in detail.
For the n-th branch one obtains,
τn → 2πn+O(α lnα) , ∆Tn → ln(tg β/α) , E → 0 ,
τn → 2πn− π , ∆Tg → +∞ , E → +∞ .
From Fig. 7 one learns that the n-th solution passes through the points
∆Tn = −1 , τ = 2πn or τ = 2πn− π . (68)
inapplicable at large ∆T : the assumption e∆T ∼ O(1) which was used in the derivation of these equations
gets violated. Nevertheless, by analyzing the full equations (54), (55) one can show that dF0,g/dE = −Tg
is large and positive at E → 0. This is sufficient for concluding that F0,g(E) is negative in the low–energy
domain.
16Another indication that the “global” solution is unphysical at small E is that the function τg(E) is
bounded from above. Indeed, τ is the time interval the particle spends in the intermediate part of the
waveguide, one expects it to tend to infinity as E → 0 for a physically relevant solution.
Thus, each curve ∆Tn(E) has one sharp dip, its minimum is smaller than −1, see Fig. 8. As
in the case with the “global” branch, the points (68) represent the extrema of the functions
F0,n(E)/E; the positions of the local minima are again given by Eq. (66).
Making use of Eq. (61), we find that the suppressions F0,n(E) of the “local” branches are
large and positive at high energies. Hence, these solutions give subdominant contributions
to the reflection probability at such E as compared to the “global” solution. As energy
decreases, F0,n(E) also decreases, then makes one oscillation and drops to negative values at
small E. The latter property means that each “local” branch becomes unphysical at small
enough energies. The suppression exponent of the first “local” branch (corresponding to
n = 4 in the case β = π/3, α = π/30) is presented in Fig. 9.
An alert reader may have already guessed that we have met here the typical Stokes
phenomenon [21]. In fact, the Stokes phenomenon is specific to the situations where some
integral (e.g., the path integral (7) in our case) is evaluated by the saddle–point method.
Essentially, it means the following: as one gradually changes the parameters of the integral
in question, a given saddle point may become non–contributing after the values of these
parameters cross a certain curve drawn in the parameter space, the Stokes line. Since the
result of the computation should be continuous, this phenomenon occurs only for subdomi-
nant saddle points (saddle–point trajectories in our case). Unfortunately, apart from several
heuristic conjectures [21, 12], sometimes rather suggestive [13], there is presently no general
method of dealing with the Stokes phenomenon in the semiclassical calculations. However,
in the situation encountered above it suffices to use the simplest logic lying at the heart of
all other approaches17.
When gathering the final result for the suppression exponent, we follow two guidelines.
First, it is clear that, as energy decreases, each branch becomes unphysical before F0,n(E)
crosses zero. On the other hand, at high energies one should pick up the branch corresponding
to the smallest value of the suppression exponent. Looking at Fig. 9, one notes that the
curves F0,g(E), F0,4(E) have two intersections, A and B. At E > EB one chooses the
“global” branch. In the region EA < E < EB we switch to the first “local” branch, because
in this region F0,4(E) < F0,g(E). Naively, at E = EA one should jump back to the “global”
branch; however, in order to preserve unitarity at small energies, we suppose that somewhere
in between the points B and A the “global” branch becomes non–contributing, so that one
should stay at the “local” branch at E < EA. Similarly, the adjacent “local” branches have
17The simplification in the present case is related to the fact that we concentrate on the dominant semi-
classical contribution, leaving aside the subdominant ones.
two intersections; as the energy decreases, we switch from n-th branch to n + 1-th at the
first intersection, and stay there until the intersection with the n+2-th branch. Overall, one
obtains the graph for the suppression exponent plotted in Fig. 10. The suppression exponent
 0.02
 0.04
 0.06
 0.08
 0  0.05  0.1  0.15  0.2  0.25
Figure 10: The final result for the suppression exponent F0(E) in the region of small energies;
β = π/3, α = π/30. The points where different branches merge are shown with thick black
dots.
oscillates between two linear envelopes, F = E(fβ(0) ± 4e−1α ctg β); oscillations pile up in
the region of low energies. The reflection process is optimal in the vicinities of the minima
of the function F0(E).
5 Discussion
By considering a class of two–dimensional waveguide models, we have demonstrated explicitly
that the probability of over–barrier reflection can be non–monotonic function of energy. The
origin of the effect lies in the classical dynamics: the parameters of the complex trajectory
describing over–barrier reflection change quasi-periodically as the energy gets decreased.
This results in the oscillatory behavior of the suppression exponent. Reflection occurs with
exponentially larger probability in the vicinities of “optimal” energies (local minima of the
suppression exponent) while being highly suppressed in between.
Our results are obtained for a fairly specific class of waveguides, namely, the ones with
very sharp turns. However, the qualitative features observed in this paper should be valid
for quite general waveguide models: a classical particle with high energy feels any large–scale
turn of the waveguide as a sharp one18; if two turns are separated by a long interval of free
motion, one arrives to the model (35). We remark that the phenomenon of optimal tunneling
has been observed also in numerical investigation of a smooth waveguide, see Ref. [14].
The branch structure of solutions observed in the region of small energies is interesting
from the mathematical point of view. We have shown that there exists an infinite sequence of
complex trajectories marked by the topological number n. Each branch produces physically
consistent result for the suppression exponent in some energy interval; outside of this interval
the n-th branch would correspond either to highly suppressed transitions (high energies) or
to violation of unitarity (low energies). We collected the final graph for the suppression
exponent basing on the empirical considerations, which hardly may be acknowledged as
satisfactory. Our study clearly shows that the method of complex trajectories should be
equipped with a convenient rule to pick up the physical trajectory among the discrete set of
solutions to the boundary value problem (9), (10), (12) (in other words, the method to deal
with the Stokes phenomenon). Presently, such a rule is absent.
We note that the described physical phenomenon of optimal tunneling is present inde-
pendently of the way the branches of solutions are glued together. The result at relatively
high energies is given by the “global” branch, which displays a large number of local minima
if n′0 > n1, see Eqs. (67), (65). This is the case for the illustrative example considered
throughout this paper, see Fig. 9.
As a final remark, we point out some open issues. We have calculated the suppression
exponent of reflection using the sharp–turn approximation. It would be instructive to extend
our analysis by finding corrections due to the finite turn widths. The motivation is twofold.
First, the analysis performed in appendix implies existence of a rich variety of distinctive
semiclassical solutions contributing almost equally into the reflection probability. This fea-
ture might be a manifestation of chaos [7] which is present in our system but hidden by the
sharp–turn approximation. [Note that chaos is inherent in a very similar waveguide model
18More precisely, one should compare the width b of the turn to the quantity 2π
, where p0 is the
translatory momentum of the particle and ω stands for the frequency of transverse oscillations; if b≪ 2πp0
one is in the class of models with sharp turns.
with smooth potential, see Ref. [14].] Clearly, the structure of solutions in the vicinities of
the turns is worth further investigation.
Second, it was proposed recently in Refs. [9, 11] that the process of dynamical tunneling
in quantum systems with multiple degrees of freedom (including field theoretical models,
see Refs. [19]) can proceed differently from the ordinary case of one–dimensional tunnel-
ing. Namely, classically unstable state can be created during the process; this state decays
subsequently into the final asymptotic region. The analysis performed in the present paper
naturally conforms with this tunneling mechanism: all our complex trajectories are matched
with the unstable orbit living at the turn. Still, the sharp–turn approximation does not allow
to distinguish between the truly unstable trajectories staying at the turn forever and those
which reflect from the turn in a finite time. To decide whether the tunneling mechanism of
Refs. [9, 11] is indeed realized in our model one needs to go beyond the sharp–turn approx-
imation. Then, the candidate for the “mediator” unstable state is the “excited sphaleron”,
the solution considered in the appendix. Presumably, in our model one can answer analyti-
cally to the question of whether or not the “excited sphaleron” acts as an intermediate state
of the tunneling process. This study is quite beyond the scope of the present paper and we
leave it for future investigations.
Acknowledgments. We are indebted to F.L. Bezrukov and V.A. Rubakov for the en-
couraging interest and helpful suggestions. This work is supported in part by the Russian
Foundation for Basic Research, grant 05-02-17363-a; Grants of the President of Russian
Federation NS-7293.2006.2 (government contract 02.445.11.7370), MK-2563.2006.2 (D.L.),
MK-2205.2005.2 (S.S.); Grants of the Russian Science Support Foundation (D.L. and S.S.);
the personal fellowship of the “Dynasty” foundation (awarded by the Scientific board of
ICFPM) (A.P.) and INTAS grant YS 03-55-2362 (D.L.). D.L. is grateful to Universite Libre
de Bruxelles and EPFL (Lausanne) for hospitality during his visits.
A Classical motion near the turn
In this appendix we analyze the motion of the particle near the sharp turn of the waveguide
(20) at nonzero smoothening of the turn, see, e.g., Eq. (21). We suppose that in the small
vicinity of the turn the function w(ξ, η) can be represented in the form
w(ξ, η) = cos β (η − bv(ξ/b)) , (69)
where v(ψ) does not depend explicitly on b. Moreover, we consider the case when v(ψ) has
a maximum19,
v′(ψ0) = 0 . (70)
Due to the property (70) one immediately obtains the exact periodic solution to the
equations of motion (9), which we call “excited sphaleron” [9],
ξsp = bψ0 , ηsp = Aη sin(t cos β + ϕη) + bv(ψ0) . (71)
We are going to show that this solution is unstable: a small perturbation above it grows
with time and the particle flies away to either end of the waveguide. In particular, there are
solutions that describe the decay of the sphaleron to ξ → −∞ both at t → ±∞. Clearly,
such solutions correspond to reflections from the turn.
In the vicinity of the sphaleron the trajectory of the particle can be represented in the
form,
ξ = bψ(t) , η = ηsp(t) + bρ(t) , (72)
where ψ, ρ ∼ O(1). Writing down the classical equations of motion (9) in the leading order
in b, one obtains,
Aη sin(2s)v
′(ψ) , (73)
+ 4ρ = 4[v(ψ)− v(ψ0)] , (74)
where s = (t cos β +ϕη)/2. It is worth noting that the right hand side of Eqs. (73), (74) are
of different order in b. We will see that due to this difference ρ = 0 in the leading order in b.
Let us first consider the linear perturbations above the excited sphaleron,
ψ = ψ0 + δψ , δψ ≪ 1 .
Equation (73) can be linearized with respect to δψ leading to the Mathieu equation
δψ + 2q sin(2s)δψ = 0 ,
with canonical parameter q = −2v′′0Aη/b > 0. As q ∼ O(1/b) ≫ 1, one can apply the WKB
formula,
A cosW
dW/ds
, (75)
19For the smoothening (21), the properties (69), (70) hold with v(ψ) = ψtgβ
, ψ0 ≈ 1.28.
where |A| ≪ 1, and
sin(2s′) .
Note that we have chosen the solution symmetric with respect to time reflections,
δψ(π/2− s) = δψ(s) . (76)
At s ∈ [0; π/2] the exponent W is real and the particle gets stuck at ψ ≈ ψ0, oscillating
around this point with high frequency dW/ds ∼ O(b−1/2). At s < 0 the solution (75) grows
exponentially, meaning that the particle flies away from the excited sphaleron,
δψ(s < 0) =
A cos(W (0)− π/4)
|dW/ds|
e|W (s)−W (0)| .
In what follows, we choose A cos(W (0)− π/4) < 0, so that δψ < 0 at s < 0. Let us denote
by s1 < 0 the point where δψ becomes formally equal to −1,
A cos(W (0)− π/4)
|dW/ds|
e|W (s1)−W (0)| = −1 .
In what follows we suppose that s1 ∼ O(1), hence, A is exponentially small. Then, in the
vicinity of this point, |s− s1| ≪ 1, one has,
δψ = − exp
−2q sin(2s1)(s1 − s)
= − exp
4v′′0Aη sin(2s1)
(s1 − s)√
. (77)
We notice that δψ evolves from exponentially small values to δψ ∼ O(1) during the charac-
teristic time |s− s1| ∼ O(
When δψ ∼ O(1) the linear approximation breaks down and one has to solve the nonlinear
equation (73). Using s = s1 +O(
b) one writes
Aη sin(2s1)v
′(ψ) . (78)
This equation permits to draw a useful analogy with one–dimensional particle moving in
the effective potential Veff (ψ) = −4b−1Aη sin(2s1)v(ψ) (see Fig. 11). This auxiliary particle
starts in the region near the maximum of the potential at (s − s1)/
b → +∞ with energy
E ≈ Vmax and rolls down toward ψ → −∞ at (s−s1)/
b→ −∞. In this limit v(ψ) → ψ tg β
and the solution takes the form
ψ = C1 + C2(s− s1) + 2b−1Aη sin(2s1) tg β (s− s1)2 .
Vmax 
Figure 11: The effective potential for Eq. (78).
Note that the coefficients C1, C2 here are not independent: they are determined by the
parameter s1 through matching of the solution with Eq. (77) at (s− s1)/
b→ +∞. We do
not need their explicit form, however.
Let us argue that the function ρ remains small during the whole evolution of the particle
in the vicinity of the sphaleron. Indeed, in the linear regime one has δψ ≪ 1 and the r.h.s.
of Eq. (74) is small. So, ρ does not get excited. On the other hand, the nonlinear evolution
of ψ proceeds in a short time interval ∆s = O(
b); so, again, ρ is suppressed by some power
of b.
The trajectory (72) found in the vicinity of the sphaleron should be matched at
1 ≫ |s− s1| ≫
with the free solution in the asymptotic region ξ < 0, see Eqs. (26). It is straightforward to
check that matching can be performed up to the second order in (t− t1), which is consistent
with our approximations. In this way one determines the free asymptotic solution which, up
to corrections of order O(b), coincides with the sinusoid coming from ξ → −∞ at t → −∞
and touching the line ξ = 0 at t = t1.
Now we recall that, by construction, the obtained solution is symmetric with respect to
time reflections,
ξ(s) = ξ(π/2− s) , η(s) = η(π/2− s) .
This means that it satisfies ξ → −∞ at t → ±∞. This solution describes reflection of the
particle from the turn.
The reasoning presented in this appendix puts considerations of the main body of this
paper on the firm ground: we have found the “smoothened” solutions which reflect classically
from the turn, and in the limit b→ 0 coincide with the free solutions of Sec. 3 touching the
line ξ = 0.
It is worth mentioning that, apart from the reflected solution we have found, in the
vicinity of any trajectory touching the line ξ = 0 there exists a rich variety of qualitatively
different motions. First of all, one may successfully search for solutions which are odd with
respect to time reflections (Eq. (76) with minus sign). Such solutions, though close to the
reflected ones at t < 0, describe transmissions of the particle through the sharp turn into the
asymptotic region ξ → +∞. Relaxing the time reflection symmetry, one can find solutions
leaving the vicinity of the turn at any point η < 0, which is different, in general, from the
starting point η = η(s1). Yet another types of solutions are obtained in the case when the
amplitude A of δψ–oscillations at s ∈ [0; π/2] is so small that δψ does not reach the values
of order one during the time period s ∈ [−π/2; 0]. If the particle is still in the vicinity of the
point ψ0 at s = −π/2, it remains for sure in this vicinity at s ∈ [−π; −π/2], because the r.h.s.
of Eq. (73) is positive again. In this way one obtains solutions, which spend two, three, etc.
sphaleron periods at ψ ≈ ψ0 before escaping into the asymptotic regions ψ → ±∞. In the
leading order in b all these solutions correspond to the identical initial state, and (in the case
of classically forbidden transitions) to the same value of the suppression exponent. However,
an accurate study of the dynamics in the vicinity of the the sphaleron is generically required
to obtain the correct value of the suppression exponent in the case b ∼ 1, cf. Ref. [14].
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	Introduction
	The semiclassical method
	The model with one turn
	The model with two turns
	Introducing the system
	Classical evolution
	Classically forbidden reflections
	Discussion
	Classical motion near the turn

  We present an analytic example of two dimensional quantum mechanical system,
where the exponential suppression of the probability of over-barrier reflection
changes non-monotonically with energy. The suppression is minimal at certain
"optimal" energies where reflection occurs with exponentially larger
probability than at other energies.

Introduction
Tunneling and over–barrier reflection are the characteristic non–perturbative phenomena in
quantum mechanics. They typically occur with exponentially small probabilities,
P ∝ e−F/~ , (1)
where F is the suppression exponent; still, the above phenomena are indispensable in under-
standing a wide variety of physical situations, from the generation of baryon number asym-
metry in the early Universe [1] to chemical reactions [2] and atom ionization processes [3].
During the last decades extensive investigations of tunneling processes in systems with
many degrees of freedom have been performed [2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. These studies
1levkov@ms2.inr.ac.ru
2panin@ms2.inr.ac.ru
3Sergey.Sibiryakov@cern.ch, sibir@ms2.inr.ac.ru
http://arxiv.org/abs/0704.0409v2
revealed a rich variety of features of multidimensional tunneling which are in striking contrast
to the properties of one–dimensional tunneling and over–barrier reflection. In particular, the
following phenomenon has been observed: the probability of tunneling may depend non-
monotonically on the total energy of the system and exhibit resonance-like peaks. One can
envisage three physically different mechanisms of this phenomenon. The first mechanism,
present already in one-dimensional case, is tunneling via creation of a metastable state.
In this case the tunneling probability at the maximum of the resonance is exponentially
higher than at other energies. On the other hand, the resonance width ∆E is exponentially
suppressed; so, after averaging with an energy distribution of a finite width the effect of the
resonance is washed out in the semiclassical limit ~ → 0. The second possible mechanism of
non-monotonic behavior of P(E) is quantum interference [7, 13] (see also [14]). In this case
the peak value of the tunneling probability is only by a factor of order one higher than the
average value, while the width of the resonance scales as ∆E ∝ ~. Again, the resonances
become indiscernible in the semiclassical limit. In both these cases the resonances can
be attributed to the subleading semiclassical corrections, i.e. non-monotonic behavior of
the pre-exponential factor omitted in Eq. (1). The third possibility is that the suppression
exponent F (E) is non-monotonic. In this case the existence of the “resonances” is the leading
semiclassical effect: the optimal tunneling probability at the maximum of the resonance is
exponentially higher than the probability at other energies. At the same time the resonance
width scales as4 ∆E ∝
~. This last possibility of “optimal tunneling” is definitely of
interest; yet, it did not receive much attention in literature. We are aware of only a few works
mentioning non–monotonic dependence of the suppression exponent on energy [15, 16, 14].
It is worthwhile studying this phenomenon in detail; this can provide a new insight into the
dynamics of multidimensional tunneling.
In this paper we consider the process of over–barrier reflection in a simple model with two
degrees of freedom. Our setup is interesting in two respects. First, the model under study
is essentially non–linear and the variables cannot be separated; still, over–barrier reflections
in this model can be described analytically within the semiclassical framework. Thus, this
model can serve as an analytic laboratory for the study of multidimensional tunneling. Sec-
ond, the suppression exponent F of the reflection process behaves non–monotonically as the
4This follows from the representation
P(E) ∝ exp
F (Eo)
F ′′(Eo)(E − Eo)2
of the tunneling probability in the vicinity of the maximum.
total energy E changes. We demonstrate that the function F (E) possesses a number of
local minima E = Eo, where reflection is optimal. We stress that the process we study is
exponentially preferable at “optimal” energies as compared to other energies.
Our model describes the motion of a quantum particle in the two dimensional harmonic
waveguide (see Refs. [8, 10, 14] for similar models). The Hamiltonian is
w2(x, y) ,
where x, y are the Cartesian coordinates and m is the mass of the particle. The function
U = mω2w2/2 represents the waveguide potential in two dimensions: a particle with small
energy is bound to move along the line w(x, y) ≈ 0. We do not introduce a potential barrier
across the waveguide and consider the case when the line w = 0 stretches all the way from
x → −∞ to x → +∞. We also assume that the function w(x, y) is linear in the initial
asymptotic region,
w(x, y) → y as x→ −∞ .
In the present paper we consider two particular cases of the function w(x, y) describing
waveguides with one and two sharp turns5, see Fig. 1.
The motion of the particle at x → −∞ is a superposition of free translatory motion in
x direction and oscillations of frequency ω along y coordinate; the state of such a particle is
fully characterized by two quantum numbers, the total energy E and y–oscillator excitation
number N . The particle sent into the waveguide from the asymptotic region x→ −∞ with
given E, N may either continue to move towards x → +∞, or reflect back into the region
x→ −∞. We are interested in the probability P(E,N) of reflection.
Let us discuss reflections at the classical level. [Note that the classical counterpart of N
is the energy of transverse oscillations.] Consider first the waveguide with one sharp turn
(Fig. 1a). One observes that the outcome of the classical evolution, i.e. whether or not
the particle reflects from the turn, depends not only on the total energy E, but on other
dynamical quantities as well. In particular, the direction of the momentum of the particle
in the vicinity of the turn (point C on the graph) is important. This means that the entire
dynamics in the waveguide should be taken into account in order to determine the possibility
of classical reflection. This is in sharp contrast with the situation in one–dimensional case,
where reflection from the potential barrier (or transition through it) is ensured by the value
of the conserved energy of the particle.
5The explicit expressions for the waveguide functions w(x, y) will be presented in the subsequent sections.
Figure 1: The equipotential contour U = E for the waveguides with (a) one and (b) two
sharp turns. An example of classical trajectory is shown in the case (b).
Now, consider the waveguide with two turns. The model is characterized by the angles
of the turns and the distance L between them (see Fig. 1b). Suppose the particle starts
moving classically from x → −∞ with N = 0 along the valley w = 0. Then, the transverse
oscillations get excited only after the particle crosses the first turn, point C ′ on the plot, so
that at the time of arrival to the second turn (point C) approximately ωτ/2π oscillations
are made, where τ ∼ L
m/2E is the time of motion between the two turns. The state of
the particle (coordinates and momenta) at which it comes across the second turn depends
periodically on the phase of transverse oscillations ωτ . Hence, one expects that the regime
of motion of the classical particle can change from transmission to reflection and back as the
energy grows (τ decreases); the energies where it happens can be roughly estimated as
mω2L2
2(2πn)2
. (2)
We will see that this is indeed the case for the waveguides with certain angles of the turns.
At some values of E, N the reflection process cannot proceed classically. Then, at the
quantum mechanical level its probability is exponentially suppressed, F (E,N) > 0. It is
natural to call such a process “over–barrier reflection”6. The central quantity to be studied
below is the suppression exponent F (E,N) of this process. The above discussion suggests
that F (E,N), being determined by the entire dynamics in the waveguide, may be a highly
non–trivial function. For the particular case of the waveguide with alternating regimes of
6By this term we want to emphasize that the process is classically forbidden. Recall, however, that there
is no actual potential barrier across the waveguide in our setup.
classical reflections and transmissions F should oscillate: F = 0 at the energies where the
classical reflections are allowed, and F > 0 at the energies where the reflections are classically
forbidden. One can expect that the similar oscillatory behavior of the suppression exponent
persists for other two–turn models as well. Now, instead of reaching zero, F may possess
a number of local positive minima implying that the reflection at the “optimal” energies is
still a tunneling process.
Let us emphasize the difference of the “optimal tunneling” from quantum interference
and resonance phenomena in our two–turn model. The interference of the de Broglie waves
reflected from the two turns can, in principle, lead to oscillations in the reflection proba-
bility P(E). One can estimate the positions of the interference peaks by equating the De
Broglie wavelength of the particle to an integer fraction of the distance between the turns,
2mE ∼ L/n. This yields the energies of the interference peaks,
Eintn ∼
(2πn)2~2
This formula is completely different from Eq. (2) for the peaks due to “optimal tunneling”.
In particular, the distance between the adjacent inteference peaks,
∆Eint ∼
scales proportional to ~. Thus, these peaks should be averaged over in the semiclassical
limit. Besides, the amplitude of the interference peaks is at most of order one and does
not affect the suppression exponent. Indeed, the exponential increase of the scattering
amplitude can arise due to quantum interference only in the presence of a resonant state
with exponentially long life–time. This state should be supported somewhere in between the
turns and should be classically stable. In Sec. 4.2 we show that such states are absent in
our system. One concludes that the peak–like structure of the probability P(E) of “optimal
tunneling” is caused by completely different physical reasons as compared to the case of
resonance scattering in quantum theory.
It is worth noting that the phenomenon of “optimal tunneling” has an important imple-
mentation in field theory. Recently it was argued [17] (see also Ref. [16]) that the probability
of tunneling induced by particle collisions [18, 19] reaches its maximum at a certain “op-
timal” energy and stays constant7 at higher energies. This result, if generic, provides the
7As opposed to the quantum mechanical case, the tunneling probability does not decrease at energies
higher than the “optimal” one. This is due to the possibility, specific to the field theoretical setup, to emit
the excess of energy into a few hard particles, so that tunneling effectively occurs at the “optimal” energy.
answer to the long–standing question [20] about the high–energy behavior of the probability
of collision–induced nonperturbative transitions in field theory. The quantum mechanical
model presented here supports the generic nature of the phenomenon of “optimal tunnel-
ing”; the simplicity of our model enables one to get an intuitive insight into the nature of
this phenomenon.
The paper is organized as follows. In Sec. 2 we review the semiclassical method of complex
trajectories, which is exploited in the rest of the paper. Reflections in the waveguides with
one and two turns are considered in Secs. 3 and 4 respectively. We discuss our results in
Sec. 5. In appendix we analyze the validity of some assumptions made in the main body of
the paper.
2 The semiclassical method
We start by describing the semiclassical method8 of complex trajectories which will be used
in the study of over–barrier reflections. We concentrate on the derivation of the formula
for the suppression exponent F (E,N) (see Refs. [2, 8, 9] for the details of the method and
Ref. [19] for the field theory formulation). In what follows we use the system of units
~ = m = ω = 1 ,
where the Hamiltonian takes the form,
p2x + p
y + w
2(x, y)
. (3)
One starts with the amplitude of reflection into the state with definite coordinates
xf < 0 , yf ,
A = 〈xf , yf |e−iĤ(tf−ti)|E, N〉 . (4)
Here |E, N〉 is the initial state of the particle moving in the asymptotic region xi → −∞
with fixed translatory momentum p0 =
2(E −N) and the oscillator excitation number N .
Semiclassically,
〈xi, yi|E, N〉 = eip0xi cos
′)dy′ + π/4
, (5)
8Note that the method has been confirmed by the explicit comparison with the exact quantum mechanical
results in Refs. [8, 9, 14]; specifically, the recent check [14] deals with the case when the dependence of the
suppression exponent on energy is not monotonic.
where xi, yi denote initial coordinates,
2N − y′2 , (6)
and we omitted the pre-exponential factor which is irrelevant for our purposes. Using Eq. (5),
one rewrites the amplitude (4) as a path integral,
dxidyi
[dx][dy]
xf , yf
xi, yi
eiS+ip0xi cos
′)dy′ + π/4
, (7)
where S is the classical action of the model (3).
In the semiclassical case the integral (7) is dominated by the (generically complex) saddle
point. Note that, as we continue the integrand in Eq. (7) into the plane of complex coor-
dinates, one of the exponents constituting the initial oscillator wave function grows, while
the other becomes negligibly small. Within the validity of our approximation, we omit the
decaying exponent by writing
′)dy′ + π/4
→ exp
′)dy′
, (8)
with the standard choice9 of the branch of the square root in Eq. (6).
One proceeds by finding the saddle point for the integral (7) with the substitution (8).
Extremization with respect to x(t), y(t) leads to the classical equations of motion,
ẍ = −wwx , ÿ = −wwy . (9)
Differentiating with respect to xi ≡ x(ti), yi ≡ y(ti), one obtains,
ẋi = p0 =
2(E −N) , ẏi = py(yi) =
2N − y2i .
The latter equations are equivalent to fixing the total energy E and initial oscillator energy
N of the complex trajectory,
ẋ2i +N , (10a)
ẏ2i + y
. (10b)
9 The correct branch is fixed by drawing a cut between the oscillator turning points y = ±
2N , and
choosing Im py > 0 at y ∈ R, y >
2N , see, e.g., Refs. [21].
Substituting the saddle–point configuration10 into Eq. (7), one obtains the amplitude of the
process with exponential accuracy,
A ∝ eiS+iB(xi, yi) ,
where the term
B(xi, yi) = p0xi +
′)dy′ (11)
is the initial–state contribution. For the inclusive reflection probability one writes,
dxfdyf |A|2 ∝
dxfdyf e
iS−iS∗+iB−iB∗ .
The integral over the final states can also be evaluated by the saddle point technique; ex-
tremization with respect to xf ≡ x(tf ), yf ≡ y(tf) fixes the boundary conditions in the
asymptotic future,
Im ẋf = Im xf = 0 , Im ẏf = Im yf = 0 . (12)
In this way one obtains the expression (1) for the reflection probability, where the suppression
exponent F is given by the value of the functional
F (E, N) = 2 ImS + 2 ImB(xi, yi)
evaluated on the saddle–point configuration — a complex trajectory satisfying the boundary
value problem (9), (10), (12).
The contribution B(xi, yi) of the initial state is simplified after one uses the asymptotic
form of the solution at t→ −∞ (xi → −∞),
x = p0t + x0 , y = ae
−it + āeit . (13)
Equations (10) guarantee that the quantities p0 =
2(E −N) and 2aā = N are real, since
E, N ∈ R. Therefore, one may introduce two real parameters T , θ as follows,
2 Im x0 = −p0T , ā = a∗eT+θ . (14)
One finds for the initial term (11),
2 Im B(xi, yi) = Im
2p0xi − 2Narccos(yi/
2N) + yi
2N − y2i
= −p20T −N(T + θ) + Im(yiẏi) ,
10For simplicity we assume that the saddle–point configuration is unique. Otherwise, one should take the
saddle point corresponding to the weakest exponential suppression.
and thus
F = 2 Im S̃ −ET −Nθ , (15)
where S̃ is the classical action of the system (3) integrated by parts,
S̃ = −1
xẍ+ yÿ + w2(x, y)
. (16)
Let us comment on the physical meaning of the parameters T , θ. Consider two trajectories
which are solutions to the boundary value problem (9), (10), (12) at neighbouring values of
E, N . The differential of the quantity 2 Im S̃ as one deforms one trajectory into the other is
d (2 Im S̃) = d Im(2S + xiẋi + yiẏi) = Im(xidẋi − ẋidxi + yidẏi − ẏidyi) = EdT +Ndθ ,
where in the last equality we used the asymptotic form (13), (14) of the solution. Then,
from Eq. (15) one finds,
dF (E,N) = −TdE − θdN . (17)
Thus, the parameters T and θ are (up to sign) the derivatives of the suppression exponent
with respect to energy E and initial oscillator excitation number N respectively.
Our final remark is that the boundary value problem (9), (10), (12) is invariant with
respect to the trivial time translation symmetry,
t→ t+ δt , δt ∈ R , (18)
which can be fixed in any convenient way.
3 The model with one turn
To warm up, we consider the simplest model, where the waveguide has one sharp turn,
w = y θ(−x+ y tg β) + cos β (x sin β + y cos β) θ(x− y tg β) . (19)
Here θ(x) is the step function. It is convenient to use the rotated coordinate system,
cos β − sin β
sin β cos β
The waveguide function takes the form,
w = η cos β − ξ sin β θ(−ξ) . (20)
Figure 2: The equipotential contour w2(x, y) = 2N for the waveguide (20) and the trajectory
of the critical solution with energy N/ cos2 β.
The equipotential contour w2(ξ, η) = const is shown in Fig. 2. One observes that the motion
of the particle in two regions, ξ < 0 and ξ > 0, decomposes into the translatory motion and
oscillations in the coordinates x, y and ξ, η respectively (see. Eqs. (19) and (20)); the
frequency of η–oscillations in the latter case is cos β.
Due to the presence of the step function, the first derivatives of the potential (20) are
discontinuous11 at ξ = 0. Strictly speaking, the semiclassical method is not applicable in
this situation [21]. Thus, the formula (20) should be regarded as an approximation to some
waveguide function with smooth turn. Generically the width of the smoothened turn is
characterized by a parameter b; the sharp–turn approximation (20) corresponds to b → 0.
An example of smoothening is provided by the following substitution in Eq. (20),
θ(ξ) → θb(ξ) =
1 + e−ξ/b
. (21)
The semiclassical description can be used as long as the de Broglie wavelength of the particle
is small compared to the linear size of the potential12, 1/
E ≪ b. We conclude that the
sharp–turn and semiclassical approximations are valid simultaneously for smooth waveguides
1 ≫ b≫ 1/
E . (22)
11Note that the potential itself is continuous.
12Another semiclassical condition is that the energy is sufficient to excite a lot of oscillator levels, E ≫ 1.
It is satisfied provided Eq. (22) holds.
An important property of the model (20) is invariance of the classical equations of motion
(9) under the rescaling of the coordinates,
x→ Λx , y → Λy . (23)
Using the transformation (23), one may express a solution x(t), y(t) with energy E in terms
of the “normalized” one,
x = x̃
E , y = ỹ
where the solution x̃(t), ỹ(t) has unit energy; its initial oscillator excitation number is
ν = N/E .
The suppression exponent (15) takes the form,
F (E, N) = Efβ(ν) , (24)
where fβ(ν) is the exponent for the “normalized” solution. Substituting the expression (24)
into Eq. (17), one obtains,
fβ(ν) = −T − θν . (25)
We will exploit Eq. (25) in the end of this section. Now, we proceed to finding the “normal-
ized” trajectories.
At certain initial data ν > νcr the particle can reflect from the turn classically, so that
fβ(ν > νcr) = 0 .
Let us find the value of νcr. In the region ξ < 0 the classical solution takes the form,
x(t) = p0t+ x0 , (26a)
y(t) = A0 sin(t + ϕ) . (26b)
Having crossed the line ξ = 0 (line AB in Fig. 2), the classical particle can never return
back into the region ξ < 0. Indeed, in this case it moves at ξ > 0 with constant momentum
pξ > 0. Thus, the particle can reflect classically only if its trajectory touches the line
ξ = 0. The potential of our model has ill–defined derivatives at ξ = 0, and the fate of the
particle moving along the line AB depends on the particular choice of the smoothening of
the potential. In appendix we consider the motion of the classical particle in the case when
nonzero smoothening of width b is switched on. For a class of smoothenings we show that
in the small vicinity (δξ ∼ b) of any trajectory touching the line ξ = 0 there exists some
“smoothened” trajectory, which reflects classically from the turn. Consequently, below we
associate the trajectories touching the line ξ = 0 with the classical reflected solutions.
One notices that the inclination of the trajectory (26) is bounded from above
therefore, the classical trajectory of the particle can touch the line ξ = 0, that is, y/x = ctg β
only at
A0/p0 ≥ ctg β . (27)
From Eqs. (27), (26), (10) one extracts the condition for the particle to reflect classically
from the turn,
ν ≥ νcr = cos2 β . (28)
The critical classical solution at ν = νcr touches the line ξ = 0 at η = 0 (point C in Fig. 2),
where its trajectory
xcr(t) =
2t sin β , (29)
ycr(t) =
2 sin t cos β .
has the largest inclination.
We now turn to the classically forbidden reflections at ν < νcr, which are described by
the boundary value problem (9), (10), (12). One makes the following important observation.
The waveguide function (20) has the form of two analytic functions glued together at ξ = 0.
Hence, the equations of motion (9) can be continued analytically to the complex values of
coordinates in two different ways, starting from the regions ξ < 0 and ξ > 0 respectively. In
this way one obtains two complex solutions, ξ−(t), η−(t) and ξ+(t), η+(t). These solutions
and their first derivatives should be matched at some moment of time t1, ξ(t1) = 0. [Note
that the matching time t1 does not need to be real.] Below we conventionally refer to these
solutions as the ones belonging to the regions ξ < 0 and ξ > 0.
By the same reasoning as above we find that once the particle arrives into the region
ξ > 0, it never reflects back to ξ < 0, unless pξ = 0. So, in the region ξ > 0 one writes,
ξ+(t) = 0 , (30a)
η+(t) =
cos β
sin(t cos β + ϕη) , (30b)
where the “normalization” condition E = 1 has been used explicitly. Due to the conditions
in the asymptotic future, Eqs. (12), the parameter ϕη is real. We use the translational
invariance (18) to set ϕη = 0. Note that we again associate the trajectory going along the
line ξ = 0 with the reflected one.
The physical picture of over–barrier reflection that comes to mind matches with the new
mechanism of multidimensional tunneling proposed recently in Refs. [9, 11]. The process
proceeds in two steps. The first step, which is exponentially suppressed, is formation of the
periodic classical orbit (30) oscillating along the line ξ = 0. This orbit is unstable. At the
second step of the process the unstable orbit decays classically forming a trajectory going
back to x → −∞ at t → +∞. Clearly, the second step does not affect the suppression
exponent of the whole process, and we do not consider it explicitly. In what follows we
concentrate on the determination of the tunneling trajectory describing the first step of the
process.
One should find the solution at ξ < 0 and impose the boundary conditions (10). Note,
however, that the energy of our solution is fixed already. As for the initial oscillator excitation
number ν, it does not change during the evolution in the region ξ < 0. Thus, one may fix it
at the matching time t = t1. One writes,
(ẏ2 + y2)
= cos2 β + sin2 β sin2(t1 cos β) .
This complex equation allows one to express t1 as
sin(t1 cos β) = −i
νcr − ν
sin β
, (31)
where the choice of the sign is dictated by the condition in footnote 9. It is convenient to
introduce notation t1 = iT1, T1 ∈ R.
In order to find the suppression exponent fβ(ν), one needs to evaluate the parameters
T (ν), θ(ν). At ξ < 0 the solution has the form,
x−(t) = p0(t− iT/2) + x′0 , (32a)
y−(t) = ae
−it + a∗eT+θ+it , (32b)
where the definitions (13), (14) have been taken into account explicitly, so that p0, x
0 ∈ R.
One evaluates p0, x
0, a, T , θ by matching the coordinates x±, y± and their first derivatives
νcr0.20.150.10.050
Figure 3: The suppression exponent fβ(ν) for the waveguide (20); β = π/3.
ẋ±, ẏ± at t = iT1; this yields
x′0 = 0 , p0 =
2(1− ν) , a = i
1− ν/ cos2 β
T + θ
cos2 β − ν
sin β
The last two equations, together with Eq.(25), define the function fβ(ν),
fβ(ν) =
cos β
arcsh
νcr − ν
sin β
− ν cos β arcsh
νcr − ν
sin β
(νcr − ν)(1− ν)
this finction is plotted in Fig. 3. One observes that at ν → νcr the quantities T1, T, θ, fβ
tend to zero, and the complex trajectory tends to the classically allowed critical solution, cf.
Eqs. (29),
2 sin β , a→ i√
cos β .
At ν = 0 one has,
fβ(0) = −2 +
cos β
arcth (cos β) . (33)
To summarize, we obtained the suppression exponent for the reflection of a particle in the
simplest waveguide with one sharp turn.
Figure 4: The equipotential contour w2(x, y) = 2N ′ for the waveguide (35) and the trajectory
of the critical solution with energy N ′/ cos2 β > EB. The matching points C, C
′ are shown
by the thick black dots.
4 The model with two turns
4.1 Introducing the system
In the model of the previous section the suppression exponent was proportional to energy
because of the coordinate rescaling symmetry (23). Now, we are going to demonstrate that
small violation of this symmetry results in highly non–trivial graph for F (E).
One introduces a second turn into the waveguide, see Fig. 4. We want to consider this
turn as a small perturbation, so, we assume its angle α to be smaller than β. It is convenient
to introduce two additional coordinate systems, x′, y′ and ξ, η, bound to the central and
rightmost parts of the waveguide respectively. They are related to the original coordinate
system x, y as follows,
cosα sinα
− sinα cosα
cos β − sin β
sin β cos β
x′ − L
Note that the origin of the coordinate system ξ, η is shifted by the distance L. The waveguide
function is
w = θ(−x′)θ(−ξ)y + θ(−ξ)θ(x′)y′ cosα + θ(ξ)η cosα cos β ; (35)
it consists of three pieces glued together continuously at x′ = 0 and ξ = 0 (lines A′B′ and
AB in Fig. 4 respectively). At t→ −∞ the particle comes flying from the asymptotic region
x′ < 0, where w = y. In the intermediate region x′ > 0, ξ < 0 the particle moves in the x′
direction oscillating along the y′ coordinate with the frequency cosα. Finally, in the region
ξ > 0 its motion is free in the coordinates ξ, η; the frequency of η–oscillations is cosα cos β.
The model (35) no longer possesses the symmetry (23): rescaling of coordinates changes
the length L of the central part of the waveguide. In what follows it is convenient to work
in terms of the rescaled dynamical variables,
x̃ = x/L , ỹ = y/L .
In new terms the parameter L disappears from the classical equations of motion, entering
the theory through the overall coefficient L2 in front of the action. The initial–state quantum
numbers are also proportional to L2,
E = L2Ẽ , N = L2Ñ . (36)
Thus, the conditions (22) for the validity of the semiclassical approximation are satisfied in
the limit
L→ ∞ , Ẽ, Ñ = fixed .
The suppression exponent takes the form
F (E,N) = L2F̃ (Ẽ, Ñ) . (37)
To simplify notations, we omit tildes over the rescaled quantities in the rest of this sec-
tion. Rescaling back to the physical units can be easily performed in the final formulae by
implementing Eqs. (36), (37).
4.2 Classical evolution
Let us begin this subsection by demonstrating that there are no stable classical solutions
localized in the region between the turns. This is important for the determination of the
tunneling probability, since such stable solutions could lead to exponential resonances in the
tunneling amplitude. The argument proceeds as follows. Any trajectory which is localized in
the intermediate region should reflect from the line AB infinitely many times. Each reflection
involves touching the unstable orbit living at the line AB. This implies that the trajectory
itself is unstable.
We proceed by determining the region of initial data E, N , which correspond to the
classical reflections. [For brevity we will refer to this region as the “classically allowed
region”, as opposed to the “classically forbidden region” where reflections occur only at the
quantum mechanical level. We stress that these are the regions in the plane of quantum
numbers E, N .] Let us search for the critical classical solutions which correspond to the
smallest initial oscillator number N = Ncr(E) at given energy E. As in the previous section,
one finds that the particle must get stuck at the line13 AB for some time in order to reflect
back. Let us first make an assumption inspired by the study of the one-turn model that the
critical solutions touch the line AB at their maximum inclination point (point C in Fig. 4).
We will see shortly that this is true only at energies above a certain value EB, see Eq. (50).
Still, the analysis based on the above assumption enables one to catch the qualitative features
of the critical line N = Ncr(E). Besides, the analysis is considerably simplified in this case;
we postpone the accurate study until the end of this subsection. Keeping in mind the above
remarks, one writes for the solution in the intermediate region,
x′cr(t) = t
2E sin β + 1 , (38a)
y′cr(t) =
cos β
sin(t cosα) . (38b)
Before entering the intermediate region, the particle crosses the line A′B′ (point C ′ in Fig. 4).
The initial oscillator number N is most conveniently calculated at the moment
t = t0 ≡ −
2E sin β
of crossing. Using the relations (34) one obtains,
ẋcr(t0) =
sin β cosα− cos β sinα cos
cosα√
2E sin β
, (39)
and thus
Ncr(E) = E −
ẋ2cr(t0) = E − E
sin β cosα− cos β sinα cos
cosα√
2E sin β
, E > EB .
 0.05
 0.15
0.90.80.70.60.50.40.30.10
Figure 5: The boundary N = Ncr(E) of the classically allowed region at E > EB for the
waveguide model (35); β = π/3, α = π/30. The region of the classically allowed initial data
lies above this boundary. The empty circles correspond to the energies E = En, where the
curve N = Ncr(E) touches its lower envelope N = E cos
2(β + α).
As an example, we show in Fig. 5 the region of the classically allowed initial data for β = π/3,
α = π/30. One observes that the function Ncr(E) oscillates between two linear envelopes,
E cos2(β + α) and E cos2(β − α); the period of oscillations decreases as E → 0. Moreover,
the curve Ncr(E) has a number of minima at the points E = E
n . This means that the
energies E = Ecrn are optimal for reflection: in the vicinity of any point E = E
n , N =
Ncr(E
n ) reflections become exponentially suppressed independently of whether the energy
gets increased or decreased. This feature is particularly pronounced in the case α+β = π/2,
when the lower envelope coincides with the line N = 0. Then, the classical reflections (i.e.
reflections with the probability of order 1) at N = 0 are possible only in the vicinities of the
points
8π2(n− 1/2)2
This is the case we used in Introduction to illustrate the effect.
The minima E = Ecrn exist at other values of the parameters as well. For instance, let
13We do not consider reflections from the line A′B′. They disappear at larger values of N than reflections
from the line AB if α is small enough.
us find the positions of these minima in the case α ≪ 1. One differentiates Eq. (40) with
respect to energy and obtains,
Ecrn = En
π(n− 1/2)
arcsin
ctg β
2πα(n− 1/2)
+O(α2)
, (41)
where
8π2(n− 1/2)2 sin2 β
are the points where the curve N = Ncr(E) touches its lower envelope. The argument of
arcsine in Eq. (41) should be smaller than one, so, the minima Ecrn exist only at large enough
n ≥ n0 ≡
ctg β
+ 1 , (43)
where [·] stands for the integer part.
Let us make several comments. First, note that n0 ∼ O(1/α), consequently, all the
optimal points Ecrn lie in the region of small energies E ∼ 1/n20 ∼ O(α2). Second, as we
pointed out before, the formula (40) for the function Ncr(E) holds at E > EB. Comparing
the expressions (42), (43) and (50), one observes that En0 > EB if tg β > 1. So, there
does exist a range of energies where the non-monotonic behavior of the function Ncr(E) can
be inferred from the formula (40). In fact, the conclusion about the existence of the local
minima of Ncr(E), as well as the expressions (41), (42), (43) determining their positions,
remain valid also at E < EB. This follows from the rigorous analysis of the boundary of the
classically allowed region to which we turn now. The reader who is more interested in the
tunneling processes may skip this part and proceed directly to subsection 4.3.
Now, we do not appeal to the Ansatz (38). Instead, we start with the general solution
in the intermediate region,
x′ = p′0(t− t0) , (44a)
y′ = A′0 sin [(t− t0) cosα+ ϕ′] . (44b)
It is convenient to parametrize it by the total energy E = p′20 /2 + cos
2 αA′20 /2 and the
“inclination” γ defined by the relation
p′0/A
0 = tg γ cosα .
Expressions (44) take the following form,
2E (t− t0) sin γ , (45a)
cos γ
sin [(t− t0) cosα + ϕ′] . (45b)
The constants t0 and ϕ
′ are fixed by demanding the trajectory (45) to reflect classically from
the second turn, i.e. touch the line ξ = 0 at t = 0,
(x′ − 1) cos β − y′ sin β
= 0 ,
= ctg β .
These conditions imply,
t0 = −
2E sin γ
tg2 β
tg2 γ
− 1 , (46a)
ϕ′ = −
cosα√
2E sin γ
tg2 β
tg2 γ
− 1− arccos
. (46b)
One sees that the classical reflections are possible only at γ ∈ [0; β]; the boundary value
γ = β reproduces the solution (38).
In order to find Ncr(E), one should minimize the value of the incoming oscillator exci-
tation number with respect to γ at fixed E. At t = t0, when the particle crosses the first
turn,
p0 ≡ ẋ(t0) =
2E(cosα sin γ − sinα cos γ cosϕ′) . (47)
Since N = E − p20/2, one can maximize the value of the translatory momentum p0 instead
of minimizing N(γ). Formula (39) represents the value γ = β lying at the boundary of the
accessible γ–domain; this value should be compared to p0(γ) taken at local maxima.
Let us consider the case α ≪ 1. At large enough energies, E ∼ 1, Eq. (47) is dominated
by the first term, which grows with γ, so that the maximum of p0(γ) is indeed achieved at
γ = β. At small energies, however, the second term in Eq. (47) becomes essential because of
the quickly oscillating cosϕ′ multiplier: the frequency of cosϕ′ oscillations grows as E → 0,
and at E ∼ α2, in spite of the small magnitude proportional to sinα, the second term
produces the sequence of local maxima of the function p0(γ).
One expects the parameters of the trajectory at small α not to be very different from the
ones at α = 0 (the latter case was considered in Sec. 3). So, we write,
γ = β − δγ ,
where 0 < δγ ≪ 1. Expanding the expressions (46), (47) and taking into account that
E ∼ α2 one obtains,
ϕ′ = −
2E sin β
(1 + δγ ctg β) , (48a)
2E(sin β − δγ cos β − α cos β cosϕ′) . (48b)
Now, the local maxima of the initial translatory momentum can be obtained explicitly by
differentiating Eqs. (48) with respect to δγ. One finds the sequence of them,
δγn = −tg β +
sin2 β
cos β
2πn− π − arcsin
2E sin2 β
α cos β
. (49)
Only the maxima with δγn > 0 should be taken into account. The local maxima exist when
E ≤ EB ≡
α2 cos2 β
2 sin4 β
. (50)
Substituting Eq. (49) into the expressions (48), one evaluates the values of p0 at the local
maxima,
p0,n(E) =2
2E sin β − 2E sin2 β
2πn− π − arcsin
2E sin2 β
α cos β
2E cos β
1− 2E sin
α2 cos2 β
The graphs Nn(E) = E − p20,n(E)/2 are shown in Fig. 6 for the case β = π/3, α = π/30.
Each graph is plotted for the energy range E > EAn restricted by the condition δγn > 0.
They are presented together with the curve given by the formula (40). By definition, the
critical solution corresponds to the lowest of these graphs. Clearly, for each “local” curve
representing the n-th local minimum of N(γ) there is a range of energies EAn < E < EBn
where it lies lower than the “global” curve (40). This means that the parameter γ of the
critical solution changes discontinuously across the points E = EBn . Correspondingly, the
curve Ncr(E) has a break at these points. On the other hand, the function Ncr(E) is smooth
at the points An as the “local” graphs end up exactly at δγ = 0, where the parameters of
the n-th “local” solution coincide with the ones of the “global” solution.
To summarize, we have observed that the boundary of the classically allowed region is
given by a collection of many branches of classical solutions, each branch being relevant in
its own energy interval. We will see that a similar branch structure is present in the complex
trajectories describing over–barrier reflections in the classically forbidden region of E, N .
4.3 Classically forbidden reflections
In this subsection we demonstrate that the suppression exponent F (E, N) viewed as a
function of energy at fixed N exhibits oscillations deep inside the classically forbidden region
 0.02
 0.04
 0.06
 0  0.05  0.1  0.15  0.2
Figure 6: The graphs Nn(E) corresponding to the local minima of the function N(γ) (dashed
lines) plotted together with the “global” curve, Eq. (40) (solid line); β = π/3, α = π/30.
The critical curve N = Ncr(E) is obtained by taking the minimum among all the graphs.
of initial data. This result comes without surprise if one takes into account the non-monotonic
behavior of the boundary Ncr(E) of the classically allowed region. Indeed, the curve N =
Ncr(E) coincides with the line F (E,N) = 0. One has,
= −∂EF
N=Ncr(E)
so that
(Ecrn , N
n ) = 0 .
We conclude that the points E = Ecrn are the local minima of the function F (E) at fixed
N = N crn . It is natural to expect that such local minima of F (E) exist at other values of
N as well. To illustrate this fact explicitly, we study the complex trajectories, solutions to
Eqs. (9), (10), (12).
Following the tactics of the previous section, we find solutions in three separate regions:
initial region x′ < 0, final region ξ > 0, and the intermediate region x′ > 0, ξ < 0. These
solutions, together with their first derivatives, should be glued at t = t0, when the complex
trajectory crosses the line x′ = 0, and at t = t1, when ξ = 0. Besides, we are looking for the
tunneling solution which ends up oscillating along the line AB, see Fig. 4. As discussed in
Sec. 3 this assumes existence of the second step of the process: classical decay of the unstable
orbit living at ξ = 0; the latter decay is described by a real trajectory14 going to x → −∞
at t→ +∞.
The solution in the final region ξ > 0 is (cf. Eqs. (30)),
ξ+(t) = 0 , (51a)
η+(t) =
cosα cos β
sin(t cosα cos β) , (51b)
where we used the time translation invariance (18) to fix the final oscillator phase ϕη = 0.
In the intermediate region x′ > 0, ξ < 0 one writes,
x′(t) = p′0t + x
0 , (52a)
y′(t) = a′e−it cosα + ā′eit cosα . (52b)
Note that the final solution (51) does not contain free parameters; thus, the matching of x′,
ẋ′, y′, ẏ′ at t = t1 enables one to express all the parameters in Eqs. (52) in terms of one
complex variable t1,
p′0 =
2E sin β cos φ1 , (53a)
x′0 = 1 +
[sin φ1 − φ1 cos φ1] , (53b)
eiφ1/ cos β [sinφ1 + i cos β cosφ1] , (53c)
ā′ =
e−iφ1/ cos β [sinφ1 − i cos β cos φ1] , (53d)
where we introduced φ1 = t1 cosα cos β.
As the energy of the solution has been fixed already, the only remaining initial condition
involves initial oscillator excitation number at x′ < 0, see Eqs. (10). It is convenient
to impose this condition at the matching point t = t0. One recalls the definition of the
matching time t0,
p′0t0 + x
0 = 0 ,
14One wonders why this trajectory does not reflect from the turn A′B′ on its way back. This concern is
removed by the observation that the trajectory produced in the decay of the unstable orbit is not unique:
in appendix we show that the decay can occur at any point of the segment AC giving rise to a whole bunch
of potential decay trajectories. Most of these trajectories pass through the turn A′B′ without reflection.
which, after taking into account the expressions (53a), (53b), leads to the following equation,
cosα√
2E sin β
sinφ1
cos β
− cosφ1∆φ = 0 , (54)
where ∆φ = cosα(t1 − t0). At t = t0 one has,
ẋ(t0) = p
0 cosα− ẏ′(t0) sinα =
2(E −N) ,
and thus √
= ctgα sin β cosφ1 − sin φ1 sin∆φ− cos β cosφ1 cos∆φ . (55)
As before, ν = N/E.
Two complex equations (54), (55) determine the matching times t0, t1, and, consequently,
the complex trajectory. Although these equations cannot be solved explicitly, they can
be simplified in the case α ≪ 1, which we consider from now on. For concreteness, we
study reflections at N = 0. It is important to keep in mind that in the region of interest
E ∼ Ecrn ∼ O(α2); thus, one should regard all the momenta p and oscillator amplitudes a,
ā, as the quantities of order O(α). At the same time, for the distances along the waveguide
one has x ∼ O(1), so that the real parts of time intervals may be parametrically large,
Re t ∼ x/p ∼ O(1/α).
Further on, it will be convenient to work in terms of real variables, so, we represent φ1
and ∆φ as
φ1 = cosα cos β(τ1 + iT1) , ∆φ = cosα(τ + i∆T ) .
Note that τ and ∆T are the real and imaginary parts of the time interval t1 − t0 which the
particle spends in the intermediate region. Now, equation (54) enables one to express
2E sin βch(T1 cos β)
+O(α) , (56)
τ1 = −
τ cos β
cos β
−∆T cth(T1 cos β)
+O(α3) . (57)
Note that τ1 ∼ O(α), τ ∼ O(1/α). Then, the real part of Eq. (55) implies that
ch(T1 cos β) =
sin β
1 + α ctgβ cos τe∆T
+O(α2) . (58)
While deriving this formula we imposed T1 < 0 which follows from the requirement that
in the limit α → 0 equation (31) should be recovered; besides, we assumed e∆T ∼ O(1).
Substituting Eq. (58) into Eq. (56) and the imaginary part of Eq. (55), we obtain the final
set of equations,
2E = α ctgβ cos τe∆T +O(α2) , (59a)
(1 + ∆T )e−∆T = α ctgβτ sin τ +O(α) . (59b)
These two nonlinear equations, still, cannot be solved explicitly. Nevertheless, one can get
a pretty accurate idea about the structure of their solutions.
Before proceeding to the analysis of the above equations, let us derive a convenient
expression for the suppression exponent F0(E) ≡ F (E,N = 0). Note that on general
grounds one expects to obtain an expression of the form,
F0(E) = E(fβ(0) +O(α)) ,
where fβ(0) is given by Eq. (33). We are interested in the O(α) correction in this expression,
so, one must be careful to keep track of the subleading terms during the derivation.
Making use of the equations of motion, one obtains for the incomplete action (16) of the
system,
2 Im S̃ = Im p′0 =
2E sin β Im(cosφ1) .
Substitution of Eqs. (56), (57), (58) into this formula yields
2 Im S̃ = 2E
−1−∆T − α ctg β cos τe∆T
cos2 β
+ 2∆T
+O(α2)
For the parameter T one has (see Eqs. (14)),
T = −
2 Im x0
2 Im(x(t0)− p0t0)
= 2(T1 −∆T ) +
sinα Im y′(t0) , (60)
where in the last equality we used Eqs. (34) and x′(t0) = 0. The quantity Im y
′(t0) is
evaluated by using Eqs. (52b), (53) and (58); one finds,
Im y′(t0) = −
ctg β cos τe∆T +O(α)
Substituting everything into the formula (15), we obtain,
F0(E) = E
fβ(0)− 4α ctg β cos τ ∆T e∆T +O(α2)
. (61)
This expression implies that determination of the O(α) correction to the suppression expo-
nent involves finding τ , ∆T with O(1)–accuracy. This is precisely the level of accuracy of
Eqs. (59). Below we will also need the following formulae, which can be easily obtained by
using T = −F.
and Eq. (60),
= fβ(0) + 2(∆T + 1) +O(α) , (62)
2(∆T + 1 +O(α))
. (63)
Note that, though the suppression exponent differs from that in the one–turn case only by
O(α) correction, its derivative gets modified in the zeroth order in α.
Now, we are ready to analyze Eqs. (59). One begins by solving Eq. (59b) graphically,
see Fig. 7. The important property of this equation is as follows. One notices that the l.h.s.
of Eq. (59b) is always smaller than 1, the maximum being achieved at ∆T = 0. Therefore,
the solutions to this equation are confined to the bands
τ sin τ <
This corresponds to
τ ∈ [0; 2π(n1 − 1) + δτn1 ] or τ ∈ [2πn− π − δτn; 2πn + δτn] , n ≥ n1 (64)
where
δτn = arcsin
2πα(n− 1/2)
+O(α) ,
+ 1 , (65)
with [·] in the last formula standing for the integer part. The forbidden bands, where
τ sin τ > tgβ/α, are marked in Fig. 7 by yellow shading. The property (64) introduces a
topological classification of the solutions τ , ∆T to Eqs. (59). Namely, these solutions fall
into a set of continuous branches: the “local” branches τn(E), ∆Tn(E) living inside the
strips τ ∈ [2πn − π − δτn; 2πn + δτn], n ≥ n1, and the “global” branch τg(E), ∆Tg(E)
inhabiting the very first band τ ∈ [0; 2π(n1 − 1) + δτn1 ]. As follows from the definition of
τ , the topological number n counts the number of y′–oscillations during the evolution in the
intermediate region.
Let us consider the “global” branch. From Eqs. (59) one has,
τg → 2π(n1 − 1) +O(α lnα) , ∆Tg → ln(tg β/α) , E → 0 ,
τg → 0 , ∆Tg → −1 , E → +∞ .
10π 9π 8π 7π 6π  5π4π 3π 2π π 0
g 4 5
10π 9π 8π 7π 6π  5π4π 3π 2π π 0
g 4 5
Figure 7: Curves representing solutions to Eq. (59b); β = π/3, α = π/30.
By inspection of Fig. 7 one can work out the qualitative behavior of the functions τg(E),
∆Tg(E). Alternatively, these functions can be found numerically. They are plotted in Fig. 8
for the case β = π/3, α = π/30 (the curves marked with “g”). One observes that at
high enough energies the function ∆Tg(E) exhibits oscillations around the line ∆T = −1.
According to the formula (63) this means that the function F0(E)/E is non-monotonic, it
attains local minima at the points
E ′n =
8π2(n− 1/2)2
1 + 2αe−1ctgβ +O(α2)
. (66)
Moreover, if
n ≥ n′0 ≡
fβ(0) exp
fβ(0)
+ 1 (67)
there exist Eon = E
n(1 + O(α)), such that ∆T (E
n) = −1 − fβ(0)/2. Then, according to
Eq. (62) the points Eon are the “optimal” energies corresponding to the local minima of the
suppression exponent F0(E).
At low energies the function ∆Tg(E) ceases to oscillate and becomes large and positive.
According to Eq. (62) this means that the suppression exponent F0,g(E) of the “global”
solution becomes negative at low energies15, see Fig. 9. This is a clear signal that the
15It is worth mentioning that Eqs. (59) and the expression (61) for the suppression exponent become
 0  0.1  0.2  0.3
 0  0.1  0.2  0.3
 0  0.1  0.2  0.3
E’4E’5
Figure 8: Several first branches of solutions to Eqs. (59): “global” branch (“g”) and two
“local” branches (“4”, “5”); β = π/3, α = π/30.
 0.05
 0.15
 0.25
 0  0.2  0.4  0.6
 0.02
 0.04
0.120.110.100.09
Figure 9: The suppression exponent F0(E) for the “global” and first “local” (n = 4) branches;
β = π/3, α = π/30. The vicinity of intersection of the graphs is enlarged in the upper right
corner.
“global” solution becomes unphysical at these energies and its contribution to the reflection
probability should be discarded: negative suppression exponent contradicts the unitarity
requirement16, P < 1. One is forced to conclude that at low energies reflection is described
by the “local” solutions. Let us study them in detail.
For the n-th branch one obtains,
τn → 2πn+O(α lnα) , ∆Tn → ln(tg β/α) , E → 0 ,
τn → 2πn− π , ∆Tg → +∞ , E → +∞ .
From Fig. 7 one learns that the n-th solution passes through the points
∆Tn = −1 , τ = 2πn or τ = 2πn− π . (68)
inapplicable at large ∆T : the assumption e∆T ∼ O(1) which was used in the derivation of these equations
gets violated. Nevertheless, by analyzing the full equations (54), (55) one can show that dF0,g/dE = −Tg
is large and positive at E → 0. This is sufficient for concluding that F0,g(E) is negative in the low–energy
domain.
16Another indication that the “global” solution is unphysical at small E is that the function τg(E) is
bounded from above. Indeed, τ is the time interval the particle spends in the intermediate part of the
waveguide, one expects it to tend to infinity as E → 0 for a physically relevant solution.
Thus, each curve ∆Tn(E) has one sharp dip, its minimum is smaller than −1, see Fig. 8. As
in the case with the “global” branch, the points (68) represent the extrema of the functions
F0,n(E)/E; the positions of the local minima are again given by Eq. (66).
Making use of Eq. (61), we find that the suppressions F0,n(E) of the “local” branches are
large and positive at high energies. Hence, these solutions give subdominant contributions
to the reflection probability at such E as compared to the “global” solution. As energy
decreases, F0,n(E) also decreases, then makes one oscillation and drops to negative values at
small E. The latter property means that each “local” branch becomes unphysical at small
enough energies. The suppression exponent of the first “local” branch (corresponding to
n = 4 in the case β = π/3, α = π/30) is presented in Fig. 9.
An alert reader may have already guessed that we have met here the typical Stokes
phenomenon [21]. In fact, the Stokes phenomenon is specific to the situations where some
integral (e.g., the path integral (7) in our case) is evaluated by the saddle–point method.
Essentially, it means the following: as one gradually changes the parameters of the integral
in question, a given saddle point may become non–contributing after the values of these
parameters cross a certain curve drawn in the parameter space, the Stokes line. Since the
result of the computation should be continuous, this phenomenon occurs only for subdomi-
nant saddle points (saddle–point trajectories in our case). Unfortunately, apart from several
heuristic conjectures [21, 12], sometimes rather suggestive [13], there is presently no general
method of dealing with the Stokes phenomenon in the semiclassical calculations. However,
in the situation encountered above it suffices to use the simplest logic lying at the heart of
all other approaches17.
When gathering the final result for the suppression exponent, we follow two guidelines.
First, it is clear that, as energy decreases, each branch becomes unphysical before F0,n(E)
crosses zero. On the other hand, at high energies one should pick up the branch corresponding
to the smallest value of the suppression exponent. Looking at Fig. 9, one notes that the
curves F0,g(E), F0,4(E) have two intersections, A and B. At E > EB one chooses the
“global” branch. In the region EA < E < EB we switch to the first “local” branch, because
in this region F0,4(E) < F0,g(E). Naively, at E = EA one should jump back to the “global”
branch; however, in order to preserve unitarity at small energies, we suppose that somewhere
in between the points B and A the “global” branch becomes non–contributing, so that one
should stay at the “local” branch at E < EA. Similarly, the adjacent “local” branches have
17The simplification in the present case is related to the fact that we concentrate on the dominant semi-
classical contribution, leaving aside the subdominant ones.
two intersections; as the energy decreases, we switch from n-th branch to n + 1-th at the
first intersection, and stay there until the intersection with the n+2-th branch. Overall, one
obtains the graph for the suppression exponent plotted in Fig. 10. The suppression exponent
 0.02
 0.04
 0.06
 0.08
 0  0.05  0.1  0.15  0.2  0.25
Figure 10: The final result for the suppression exponent F0(E) in the region of small energies;
β = π/3, α = π/30. The points where different branches merge are shown with thick black
dots.
oscillates between two linear envelopes, F = E(fβ(0) ± 4e−1α ctg β); oscillations pile up in
the region of low energies. The reflection process is optimal in the vicinities of the minima
of the function F0(E).
5 Discussion
By considering a class of two–dimensional waveguide models, we have demonstrated explicitly
that the probability of over–barrier reflection can be non–monotonic function of energy. The
origin of the effect lies in the classical dynamics: the parameters of the complex trajectory
describing over–barrier reflection change quasi-periodically as the energy gets decreased.
This results in the oscillatory behavior of the suppression exponent. Reflection occurs with
exponentially larger probability in the vicinities of “optimal” energies (local minima of the
suppression exponent) while being highly suppressed in between.
Our results are obtained for a fairly specific class of waveguides, namely, the ones with
very sharp turns. However, the qualitative features observed in this paper should be valid
for quite general waveguide models: a classical particle with high energy feels any large–scale
turn of the waveguide as a sharp one18; if two turns are separated by a long interval of free
motion, one arrives to the model (35). We remark that the phenomenon of optimal tunneling
has been observed also in numerical investigation of a smooth waveguide, see Ref. [14].
The branch structure of solutions observed in the region of small energies is interesting
from the mathematical point of view. We have shown that there exists an infinite sequence of
complex trajectories marked by the topological number n. Each branch produces physically
consistent result for the suppression exponent in some energy interval; outside of this interval
the n-th branch would correspond either to highly suppressed transitions (high energies) or
to violation of unitarity (low energies). We collected the final graph for the suppression
exponent basing on the empirical considerations, which hardly may be acknowledged as
satisfactory. Our study clearly shows that the method of complex trajectories should be
equipped with a convenient rule to pick up the physical trajectory among the discrete set of
solutions to the boundary value problem (9), (10), (12) (in other words, the method to deal
with the Stokes phenomenon). Presently, such a rule is absent.
We note that the described physical phenomenon of optimal tunneling is present inde-
pendently of the way the branches of solutions are glued together. The result at relatively
high energies is given by the “global” branch, which displays a large number of local minima
if n′0 > n1, see Eqs. (67), (65). This is the case for the illustrative example considered
throughout this paper, see Fig. 9.
As a final remark, we point out some open issues. We have calculated the suppression
exponent of reflection using the sharp–turn approximation. It would be instructive to extend
our analysis by finding corrections due to the finite turn widths. The motivation is twofold.
First, the analysis performed in appendix implies existence of a rich variety of distinctive
semiclassical solutions contributing almost equally into the reflection probability. This fea-
ture might be a manifestation of chaos [7] which is present in our system but hidden by the
sharp–turn approximation. [Note that chaos is inherent in a very similar waveguide model
18More precisely, one should compare the width b of the turn to the quantity 2π
, where p0 is the
translatory momentum of the particle and ω stands for the frequency of transverse oscillations; if b≪ 2πp0
one is in the class of models with sharp turns.
with smooth potential, see Ref. [14].] Clearly, the structure of solutions in the vicinities of
the turns is worth further investigation.
Second, it was proposed recently in Refs. [9, 11] that the process of dynamical tunneling
in quantum systems with multiple degrees of freedom (including field theoretical models,
see Refs. [19]) can proceed differently from the ordinary case of one–dimensional tunnel-
ing. Namely, classically unstable state can be created during the process; this state decays
subsequently into the final asymptotic region. The analysis performed in the present paper
naturally conforms with this tunneling mechanism: all our complex trajectories are matched
with the unstable orbit living at the turn. Still, the sharp–turn approximation does not allow
to distinguish between the truly unstable trajectories staying at the turn forever and those
which reflect from the turn in a finite time. To decide whether the tunneling mechanism of
Refs. [9, 11] is indeed realized in our model one needs to go beyond the sharp–turn approx-
imation. Then, the candidate for the “mediator” unstable state is the “excited sphaleron”,
the solution considered in the appendix. Presumably, in our model one can answer analyti-
cally to the question of whether or not the “excited sphaleron” acts as an intermediate state
of the tunneling process. This study is quite beyond the scope of the present paper and we
leave it for future investigations.
Acknowledgments. We are indebted to F.L. Bezrukov and V.A. Rubakov for the en-
couraging interest and helpful suggestions. This work is supported in part by the Russian
Foundation for Basic Research, grant 05-02-17363-a; Grants of the President of Russian
Federation NS-7293.2006.2 (government contract 02.445.11.7370), MK-2563.2006.2 (D.L.),
MK-2205.2005.2 (S.S.); Grants of the Russian Science Support Foundation (D.L. and S.S.);
the personal fellowship of the “Dynasty” foundation (awarded by the Scientific board of
ICFPM) (A.P.) and INTAS grant YS 03-55-2362 (D.L.). D.L. is grateful to Universite Libre
de Bruxelles and EPFL (Lausanne) for hospitality during his visits.
A Classical motion near the turn
In this appendix we analyze the motion of the particle near the sharp turn of the waveguide
(20) at nonzero smoothening of the turn, see, e.g., Eq. (21). We suppose that in the small
vicinity of the turn the function w(ξ, η) can be represented in the form
w(ξ, η) = cos β (η − bv(ξ/b)) , (69)
where v(ψ) does not depend explicitly on b. Moreover, we consider the case when v(ψ) has
a maximum19,
v′(ψ0) = 0 . (70)
Due to the property (70) one immediately obtains the exact periodic solution to the
equations of motion (9), which we call “excited sphaleron” [9],
ξsp = bψ0 , ηsp = Aη sin(t cos β + ϕη) + bv(ψ0) . (71)
We are going to show that this solution is unstable: a small perturbation above it grows
with time and the particle flies away to either end of the waveguide. In particular, there are
solutions that describe the decay of the sphaleron to ξ → −∞ both at t → ±∞. Clearly,
such solutions correspond to reflections from the turn.
In the vicinity of the sphaleron the trajectory of the particle can be represented in the
form,
ξ = bψ(t) , η = ηsp(t) + bρ(t) , (72)
where ψ, ρ ∼ O(1). Writing down the classical equations of motion (9) in the leading order
in b, one obtains,
Aη sin(2s)v
′(ψ) , (73)
+ 4ρ = 4[v(ψ)− v(ψ0)] , (74)
where s = (t cos β +ϕη)/2. It is worth noting that the right hand side of Eqs. (73), (74) are
of different order in b. We will see that due to this difference ρ = 0 in the leading order in b.
Let us first consider the linear perturbations above the excited sphaleron,
ψ = ψ0 + δψ , δψ ≪ 1 .
Equation (73) can be linearized with respect to δψ leading to the Mathieu equation
δψ + 2q sin(2s)δψ = 0 ,
with canonical parameter q = −2v′′0Aη/b > 0. As q ∼ O(1/b) ≫ 1, one can apply the WKB
formula,
A cosW
dW/ds
, (75)
19For the smoothening (21), the properties (69), (70) hold with v(ψ) = ψtgβ
, ψ0 ≈ 1.28.
where |A| ≪ 1, and
sin(2s′) .
Note that we have chosen the solution symmetric with respect to time reflections,
δψ(π/2− s) = δψ(s) . (76)
At s ∈ [0; π/2] the exponent W is real and the particle gets stuck at ψ ≈ ψ0, oscillating
around this point with high frequency dW/ds ∼ O(b−1/2). At s < 0 the solution (75) grows
exponentially, meaning that the particle flies away from the excited sphaleron,
δψ(s < 0) =
A cos(W (0)− π/4)
|dW/ds|
e|W (s)−W (0)| .
In what follows, we choose A cos(W (0)− π/4) < 0, so that δψ < 0 at s < 0. Let us denote
by s1 < 0 the point where δψ becomes formally equal to −1,
A cos(W (0)− π/4)
|dW/ds|
e|W (s1)−W (0)| = −1 .
In what follows we suppose that s1 ∼ O(1), hence, A is exponentially small. Then, in the
vicinity of this point, |s− s1| ≪ 1, one has,
δψ = − exp
−2q sin(2s1)(s1 − s)
= − exp
4v′′0Aη sin(2s1)
(s1 − s)√
. (77)
We notice that δψ evolves from exponentially small values to δψ ∼ O(1) during the charac-
teristic time |s− s1| ∼ O(
When δψ ∼ O(1) the linear approximation breaks down and one has to solve the nonlinear
equation (73). Using s = s1 +O(
b) one writes
Aη sin(2s1)v
′(ψ) . (78)
This equation permits to draw a useful analogy with one–dimensional particle moving in
the effective potential Veff (ψ) = −4b−1Aη sin(2s1)v(ψ) (see Fig. 11). This auxiliary particle
starts in the region near the maximum of the potential at (s − s1)/
b → +∞ with energy
E ≈ Vmax and rolls down toward ψ → −∞ at (s−s1)/
b→ −∞. In this limit v(ψ) → ψ tg β
and the solution takes the form
ψ = C1 + C2(s− s1) + 2b−1Aη sin(2s1) tg β (s− s1)2 .
Vmax 
Figure 11: The effective potential for Eq. (78).
Note that the coefficients C1, C2 here are not independent: they are determined by the
parameter s1 through matching of the solution with Eq. (77) at (s− s1)/
b→ +∞. We do
not need their explicit form, however.
Let us argue that the function ρ remains small during the whole evolution of the particle
in the vicinity of the sphaleron. Indeed, in the linear regime one has δψ ≪ 1 and the r.h.s.
of Eq. (74) is small. So, ρ does not get excited. On the other hand, the nonlinear evolution
of ψ proceeds in a short time interval ∆s = O(
b); so, again, ρ is suppressed by some power
of b.
The trajectory (72) found in the vicinity of the sphaleron should be matched at
1 ≫ |s− s1| ≫
with the free solution in the asymptotic region ξ < 0, see Eqs. (26). It is straightforward to
check that matching can be performed up to the second order in (t− t1), which is consistent
with our approximations. In this way one determines the free asymptotic solution which, up
to corrections of order O(b), coincides with the sinusoid coming from ξ → −∞ at t → −∞
and touching the line ξ = 0 at t = t1.
Now we recall that, by construction, the obtained solution is symmetric with respect to
time reflections,
ξ(s) = ξ(π/2− s) , η(s) = η(π/2− s) .
This means that it satisfies ξ → −∞ at t → ±∞. This solution describes reflection of the
particle from the turn.
The reasoning presented in this appendix puts considerations of the main body of this
paper on the firm ground: we have found the “smoothened” solutions which reflect classically
from the turn, and in the limit b→ 0 coincide with the free solutions of Sec. 3 touching the
line ξ = 0.
It is worth mentioning that, apart from the reflected solution we have found, in the
vicinity of any trajectory touching the line ξ = 0 there exists a rich variety of qualitatively
different motions. First of all, one may successfully search for solutions which are odd with
respect to time reflections (Eq. (76) with minus sign). Such solutions, though close to the
reflected ones at t < 0, describe transmissions of the particle through the sharp turn into the
asymptotic region ξ → +∞. Relaxing the time reflection symmetry, one can find solutions
leaving the vicinity of the turn at any point η < 0, which is different, in general, from the
starting point η = η(s1). Yet another types of solutions are obtained in the case when the
amplitude A of δψ–oscillations at s ∈ [0; π/2] is so small that δψ does not reach the values
of order one during the time period s ∈ [−π/2; 0]. If the particle is still in the vicinity of the
point ψ0 at s = −π/2, it remains for sure in this vicinity at s ∈ [−π; −π/2], because the r.h.s.
of Eq. (73) is positive again. In this way one obtains solutions, which spend two, three, etc.
sphaleron periods at ψ ≈ ψ0 before escaping into the asymptotic regions ψ → ±∞. In the
leading order in b all these solutions correspond to the identical initial state, and (in the case
of classically forbidden transitions) to the same value of the suppression exponent. However,
an accurate study of the dynamics in the vicinity of the the sphaleron is generically required
to obtain the correct value of the suppression exponent in the case b ∼ 1, cf. Ref. [14].
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	Introduction
	The semiclassical method
	The model with one turn
	The model with two turns
	Introducing the system
	Classical evolution
	Classically forbidden reflections
	Discussion
	Classical motion near the turn
