数理解析研究所講究録
1493 巻 2006 年 118-134

118

Complex dynamics and quantum tunnelingin the presence of chaos
Akira Shudo

首藤啓 (首都大学東京物理)Department of Physics, Tokyo Metropolitan University,

1-1 Minami-Ohsawa, Hachioji, Tokyo 192-OS97, Japan

shudo@phys.metro-u.ac.jp

1

Introduction‘TUnneling phenomenon is peculiar to quantum mechanics and no counterparts exist
in classical mechanics. Features of tunneling are nevertheless strongly influenced byunderlying classical dynamics [1, 2, 3, 4, 5]. One of the most efficient methods toanalyze quantum tunneling processes is the WKB or semiclassical method in which
the classical orbits are employed as inputs of approximation. However, since quantum
tunneling is an essentially classically forbidden process, real classical orbits have noability to describe it, instead the complex classical orbits play central roles.
It should be recalled that using complex orbits itself is not new in the WKB theory,
rather it goes back to the beginning of the WKB theory, especially a text book example of quantum tunneling in one dimension. Also, a well-known technique using thecomplex space is the so-called instanton method in which tunneling penetration is evaluated by a classical path moving along the imaginary time [6]. A natural extension tohigher-dimensional systems would be possible to a certain extent as long as the system
is completely integrable $[7, 8]$ . Complexified tori can bridge classically disconnected
regions and a semiclassical argument can be developed on them. However, entirely
different situations emerge even when one performs analogous extension to the systemslightly perturbed from intergrable systems. This is because complexified tori are destroyed no matter how small the strength of perturbation is and the natural boundary
may appear in complex plane.

2Time-domain Semiclassical Analysis

First, we briefly sketch how one can describe quantum tunneling using the semiclassicaltechnique.

Dynamical Tunnelng
The system we are concerned with is the area-preserving map:$F$

: $\mapsto(q+H’(p)-V’(q)H’(p)-V’(q))$ .

(1)

Qualitative feature of phase space $(q,p)$ is controlled by the choice of kinetic term $H(p)$and also the potential term $V(q)$ .

119
In generic cases, phase space is composed of quatiperiodic regions (2) and chaotic
regions $(C)$ , and these are invariant objects in themselves and disconnected by any classical orbits. However, in quantum mechanics, the tunneling effect allows the transition
between two any disconnected components of 2 and , and quantum tunneling oversuch dynamical barrierss is particularly called dynamical tunneling in the literature.
More precise specification or definition of quantum tunneling in the present setting isprovided later,$C$

Quantum Propagator

A standard recipe to construct quantum mechanics of the area-preserving map is firstto express unitary operator in the discretized Feynman path integral form. In the
momentum (p-) representation, for example, the -step quantum propagator is written$n$as$K(p_{0};p_{n})=<p_{n}|U^{n}|p_{0}>= \int\cdots\int\prod_{j}dq_{j}\prod_{j}dp_{j}\exp[\frac{i}{\hslash}S(\{q_{j}\}, \{p_{j}\})]$

,

(2)

where
denotes the discretized action along each path The classical map(1) is recovered by imposing the variational condition.
$S(\{q_{j}\}, \{p_{j}\})$$\delta S(\{q_{j}\}, \{p_{j}\})=0$

Semiclassical Approximation
In order to include the complex orbits that are inevitable to describe classically forbidden processes, instead of applying the stationary phase method, we evaluate thequantum propagatorthe saddle point method. The final expression
for the semiclassical propagator in p–representation takes the form as$<p_{n}|\hat{U}^{n}|p_{0}>\mathrm{b}\mathrm{y}$$K^{\epsilon c}(p_{0};p_{n})= \sum_{\gamma}A_{\gamma}(p_{0},p_{n})\exp\{\frac{i}{\hslash}S_{\gamma}(p_{0},p_{n})\}$

.

,

(3)

$\mathrm{a}11\mathrm{c}1\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}1\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{s}\gamma \mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}y\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{f}\mathrm{i}_{\overline{2}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}}^{\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}1\mathrm{a}\mathrm{n}\mathrm{d}}$$\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}1\mathrm{m}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a},p_{0}=\alpha \mathrm{a}\mathrm{n}\mathrm{d}p_{n}=\beta A_{\gamma}(\mathrm{p}0,p_{n})=[2\pi\hslash(\partial p_{n}/\partial q\mathrm{o})_{p0}]^{-}\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}$amplitude factor as sociated with the stability of each orbit 7, andcorresponding classical action.

$S_{\gamma}(p_{0},p_{n})$is theInitial Value Representation

The semiclassical sum (3) contains not only real classical orbits but also complex classical ones, the latters would be responsible for the tunneling process. One physicalrequirement is that since we take the p–representation,and should be real-valued
since they both are observables. This implies that a canonical conjugate variable
does not have any constraint and may take not only real values but also complex ones,
may be complex. To include complex orbits, wetoso itinerary from, we have a. For given
extend the initial angle as $q0=\xi+i\eta$representation for semiclassically contributing complex paths as$p_{0}$$p_{n}$$q_{0}$$(p_{0}, q_{0})$$(p_{n}, q_{n})$$(\xi, \eta\in \mathbb{R})$Mg,$\beta_{\equiv\{(p_{0},q_{0}=\xi+i\eta)\in \mathbb{C}^{2}}|p_{0}=\alpha,$$\alpha,$$p_{n}=\beta$}.

$\beta\in \mathbb{R}$(4)

If there exist no classical orbits on the real phase space connecting two states
and $p_{n}=\beta$ , we should say that this process is classically forbidden, and bridged only
by complex classical orbits$p_{0}=\alpha$

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3

Tunneling Orbits in the Linear map

Before going to chaotic maps, we examine the linear map. The linear map is helpful
to understand what classical objects represent quantum tunneling and to know what
are new ingredients in chaotic maps.
Solutions
The linear map is given as$F:\mapsto$ ,where$(p_{0}, q_{0})$$\omega$(5)

is a fixed rotation number. It is easy to show that$(p_{n}, q_{n})$as$p_{n}=p_{0}+K_{n}\sin(q_{0}+n\omega/2)$$q_{n}=q_{0}+n\omega$are expressed by,,where $K_{n}=K \sin(n\omega/2)/\sin\frac{(d}{2}$ . For a given initial condition
have
the initial coordinate $q_{0}=\xi+i\eta$ should satisfy$p_{0}=\alpha\in \mathbb{R}$, in order to$q_{n}\in \mathbb{R}$$\xi=(k+\frac{1}{2})\pi-\frac{n\omega}{2}$( : integer) or$k$$\eta=0$.

(6)

Tunneling Branches

Figure 1 illustrates a set of initial conditions that contribute to the semiclassical sum(3)$\mathcal{M}_{n}^{\alpha,*}=\{q_{0}=\xi+i\eta|p_{0}=\alpha, -\infty<p_{n}<\infty\}$.

(7)

Here we set $\alpha=0$ . The orbits satisfying the former condition in (6) are real orbits,
and the latter are complex ones.
Note that tunieling branches appear in a pair-wise manner as shown in Fig. 1; theone giving exponentially decaying and the other exponentially blowing up contribution.
The former is a correct tunneling branch and the latter is unphysical and should be
removed as a result of Stokes phenomenon. The Stokes phenomenon is quite importantin the complex WKB analysis, however, we do not discuss this issue here. (See [9] forexample.)Figure (2) is a schematic sketch demonstrating that pairs of tunneling branchesemanating from the real Lagrangian manifold. Note that tunneling tails in the wave-

function are reproduced by tunneling branches.
An important remark is that, in order to go into deep tunneling regions, one should. That is, the amount of imaginary part necessary to allow thetake a large
tunneling transition is gained in the imaginary part of the initial condition. We cansay, otherwise, that no dynamics is involved in the tunneling process of linear map andplay anonly the initial condition controls it. It is worthwhile to note that
analogous role of imaginary time that is used in the so-called instanton method [6].
The instanton orbit runs in the imaginary direction in complex time domain, and thehigher potential barriers one wants to go beyond, the deeper imaginary time domain
one has to use.
${\rm Im} q_{0}=\eta$${\rm Im} q0=\eta$121

Figure 1: The initial value plane introduced as (7). The lines running in the vertical direction
give the tunneling contributions.

4

Tunneling Orbits in Ideally Chaotic Maps

In contrast to the previous one, tunneling orbits in chaotic maps behave quite differently. We first see how tunneling orbits appear in the most generic situations, mixed
systems.
hnneling Orbits in The Standard Map

There are a variety of chaotic maps in the form (1) that realize mixed-type phase space.
We here consider the standard map: $H(p)=p^{2}/2,$ $V(q)=K\sin q$ . Figure 3 shows atypical mixed phase space with some moderate parameter value and the corresponding
initial value set. In the semiclassical propagator, the initial state is taken as anellipse approximating a KAM curve, not $p_{0}=const$ as in the linear map. In the initial, two curves running in the vertical direction that exactly corresponds to thesettunneling branches shown in Fig. 1. The mechanism of tunneling induced by suchbranches are essentially the same as the one taking place in the linear map. However,in case of the standard map, there are a huge number of branches not touching on thereal axis, $\eta=0$ . All these are due to that the system is not completely integrable andthe map generates chaos not only in real but also in complex plane.
We skip all the details concerning the mechanism creating tunneling tails in quantum wavefunctions [4]. Of significantly importance is that specific sequences of branches,
which, as shown in Fig. 3, usually form successive chain-like structures, control thetunneling processes, meaning that we have only to focus on these specific structures.
We remark that the existence of these chained objects is not limited to the standard
map.
$\mathcal{M}_{n}^{\alpha,*}$$\mathcal{M}_{n}^{\alpha,*}$The H\’enon Map
Next, we consider the H\’enon map, in which both kinetic and potential terms are polyno-

122

Figure 2: (a) The real Lagrangian manifold and tunneling branches. An exact form of thereal Lagrangian manifold is expressed as $p_{n}=p_{0}+K_{n}\sin(q_{n}-n\omega/2)$ .
The wavefunction$(\mathrm{b})$in the p- representation (schematic). (c) The real Lagrangian manifold and pairs of tunnelingbranches (schematic).

mial functions. There are several reasons to employ the H\’enon map: the most relevant
one is that complex dynamics has been and is being most extensively investigated, apart of which is introduced below. This is important because the understanding ofwhat is going on in complex phase space is crucial.
A canonical form of the H\’enon map is given as : $(x, y)\mapsto(y, y^{2}-x+a)$ , but
an alternative form obtained by an affine change of variables as $(p, q)=(y-x, x-1)$
together with a parameter $c=1-a$ fits the present purpose:$f$$F:-\rangle$ ,$V(q)=- \frac{q^{3}}{3}-cq$.

(8)

The H\’enon map has a nonlinear paremeter (or ) controlling the dynamics qualitatively. When $a\gg 1$ , the so-called horseshoe condition is satisfied and the mapping is
conjugate to the symbolic dynamics with the binary full shift [10]. All the stable andunstable manifolds intersect transversally when the horsehoe is realized and the system
keeps hyperbolicity up to the first tangency point [11]. Non-wondering set forms fractal
repeller on the real plane.
$a$

$c$

123

Figure 3: (a) Phase space for standard map with $K=1.5$ . An ellipse in the lower torus
region is the initial condition. (b) The initial value set$\mathcal{M}_{n}^{\alpha,*}$at $n=6$ and its magnification.

The Horseshoe Limit: An Ideal Setting

The horseshoe limit provides an ideal situation where the chained structure observedin generic case appears in a genuine manner.
Figure 4 demonstrates the set
when the parameter is within the horseshoe
locus. We notice that a similar chained structure as found in Fig. 3 is formed. A
relation between
and invariant objects in dynamical systems is read by putting
the intersection of the (complex) stable manifoldof periodic orbits with theinitial value plane. The intersection points are aligned along
the chain and the labels inserted in the figure denote the binary coding for periodic
orbits from where the stable manifolds $W^{s}(p)$ emanate. That is, for example, the dotlabeled as (000) is a point at which the stable manifold of the periodic orbit whosebinary coding is (000) intersects with the initial value plane . Recall that, in thehorseshoe parameter locus, all the periodic orbits are contained in . The chained
structure is associated with invariant structures in phase space in this way. As a result,the itineracy of the orbits launched on the chained structure is governed by the stablemanifold attached to it. As shown in Fig. 4(b), the orbits whose initial conditions
are put on the set
approach exponentially to the real plane, reflecting that the$c$$\mathcal{M}_{n}^{\alpha,*}$$\mathcal{M}_{n}^{\alpha,*}$$W^{\mathit{8}}(p)$$p$$\mathcal{I}=\{(q,p)\in \mathbb{C}^{2}|p_{0}=\alpha\}$$\mathcal{I}$$\mathbb{R}^{2}$$\mathcal{M}_{n}^{\alpha,*}$124

Figure 4: (a) The initial value set
at $n=3$ for the H\’enon map in the horseshoe
parameter locus. The dots represent the primary intersection points (see the definition in thetext) of (complex) stable manifolds and the initial value plane T. (b) The distance from thereal plane as a function of the time step . The orbits are launched from the points onand close to the intersection points.
$\mathcal{M}_{n}^{\alpha,*}$$n$)$\mathcal{M}_{n}^{\alpha*}$orbits closely follow the orbits on the stable manifolds. Otherwise stated, the tunnelingorbits forming chained structures are guided by the stable manifolds of the periodic
orbits [12]. In a similar manner, on the chained structure found in generic mixed case(Fig. 3), the orbits starting from it also approaches the real plane exponentially. Other
orbits, not forming the chain, do not shows such behavior and are not attracted fromthe real plane.
Tree Structure in Hyperbolic Case
It is instructive to see the situation where the system is hyperbolic but not in thehorseshoe locus. Such a situation may not be so common. For instance, in the standard map, except for the so-called anti-integrable limit, phase space is composed ofquasiperiodic and chaotic components. In the H\’enon map, there indeed exist parameter loci in which the dynamics on the real plane is hyperbolic but cannot be reduced
to the binary full shift as the horseshoe limit does so. The existence of such parameterloci was suggested in [14], and recently shown rigorously via computer-assisted proof[15].
The reason why hyperbolic but not horseshoe cases are so important in our issue isthat it can be a model to examine how tunneling orbits behave when the non-wondering
set is no more confined on the real plane, and thereby real and complex saddles coexist.
In the horseshoe locus, as depicted in Fig. 4, each chain runs only in the vertial
direction. However, as seen in Fig. 5, the tree structure begins to develop in. Furthermore, though not shown here, as the time step proceeds, we can see thatthe number of generation increases. Here we use the term generation as a rank in thehierarchical tree structures. The chained branch running in the perpendicular direction$\mathcal{M}_{n}^{\alpha,*}$125
is the first generation and horizontal one the second generation and so on.

Figure 5: (a) The initial value set
at $n=20$ for the H\’enon map in the parameter
locus at which the map is hyperbolic but not has the horseshoe structure. The dots shown
are some of primary intersection points of (complex) stable manifolds and the initial value
plane Z. The right hand figure is a magnification of the left one.
$\mathcal{M}_{n}^{\alpha,*}$Primary Intersection Points
To explain the implication of the tree structure, we introduce the ordering for intersection points of stable manifolds and the initial value plane Z.
Here, the primary intersection is defined as an intersection point at which the stablemanifold $W^{s}(p)$ emanating from a saddle (period ) first intersects with Z. Moreprecisely, we specify the ordering of the intersection as follows: it is known that there
is a conjugation map from to $W^{s}(p)$ such that
$n$

$p$

$\Phi$$\mathbb{C}$$\Phi^{-1}F^{n}\Phi=F_{\epsilon n}$

where

(9)

$F_{sn}(\zeta)=\lambda^{-1}\zeta$

for
C. Here A denotes the maximal eigenvalue of the tangent map ofat .
The coordinate is normalized in the sense that the map
is expressed by a linear
transformation. Putting an appropriate domain, say on the -plane, containing as
the origin of -plane and $D’=D-F_{sn}(D)$ , the plane is decomposed into a familyof disjoint domains $D_{m}=F_{sn}^{-m}(D’)(m\in \mathrm{Z})$ . Here, the primary intersection is defined
as an intersection point between andinwith the minimalin thecoordinate.
The dots shown in Fig. 5 are some of primary intersection points so defined. As in
the horseshoe case, each intersection point is attached to a curve of. In contrast“
to the horseshoe case, however, the orbits launched at points on the tree-shaped
do not approach the real plane monotonically. To see the difference, we observe thebehavior of the orbits whose initial conditions are close to the primary intersection
points. We plot in Fig. 6(a) the distance from the real plane as a function of time
step and the corresponding schematic representation of the set
is given as Fig.
$\zeta\in$$F^{n}$$p$$F^{n}$$\zeta$$D$$\zeta$$p$$\zeta$$\zeta$$\mathcal{I}$$W^{s}(p)$$D_{m}$$m$$\zeta$$\mathcal{M}_{n}^{\alpha,*}$$\mathcal{M}_{n’}^{\alpha}$

$n$

6(b).

$M_{n}^{\alpha,*}$126

Figure 6: (a) The distance from the real plane as a function of time step . The orbits are
and close to the primary intersection points in the tree
launched from the points on
structure. (b) Tree structure and the primary intersection points (schematic).
$n$$\mathcal{M}_{n}^{\alpha,*}$Itineracy in The Complex Phase Space

For the interpretation of the origin of the behavior observed in Fig. 6(a), the following
theorem is crucial:[$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}$$\mathrm{B}\mathrm{e}\mathrm{d}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{d}$-Smillie]

For any saddle$p,$$\overline{W^{\epsilon}(p)}=J^{+}$

and also$\overline{W^{u}(p)}=J^{-}$

and $K^{\pm}=$ { $(p,$ $q)|$
is bounded $(narrow\pm\infty)$ }. Since
in the hyperbolic case, the above theorem claims that any stable manifold.
is dense in the forward bounded orbits
Now, letand be the saddles in the real and complex plane, respctively. We
then introduce the primary intersections between
andof these two kinds of(saddles, say,they are denoted byandfi Z) and
respectively (see Fig. 7).
Consider a side branch in the second genetation, which is attached to the intersec. Since is located close to , the trajectories
tion point denoted byandtrace similar itinerary in the very initial time stage. But thestarting at
finally approaches
by definition and cannot reach the realorbit launched at the
implies that in a very close neighborhoodphase space. However, the fact
there exists an intersectionof a real saddleofcan be, then the trajectory fromand be. Let the primary intersection of
converges directly to , whereas the trajectory fromfirst approaches
becauseand next converge to , which means thatis very close toiswith . In other words, the trajectory fromthe secondary intersection of
makes a side trip in the complex domain before accessing to the real plane.
which is located just at theIn the same way, there should exist

Here$J^{\pm}=\partial K^{\pm}$$||F^{n}(p,$ $q)||$$K^{\pm}=J^{\pm}$$K^{+}$$\mathrm{r}_{i}$$\mathrm{c}_{j}$$W^{\epsilon}$$\mathcal{I}$$\mathrm{r}_{1}$$\mathrm{X}_{1}$$\mathrm{c}_{1;}$$\mathrm{Y}_{1}=W^{s}(\mathrm{c}_{1})\cap \mathcal{I}$$\mathrm{X}_{1}$$\mathrm{Y}_{1}(\in W^{\delta}(\mathrm{c}_{1})\cap \mathcal{I})$$\in W^{\epsilon}(\mathrm{r}_{1})$$\mathrm{Y}_{1}$$\mathrm{X}_{1}$$\mathrm{Y}_{1}$$\mathrm{Y}_{1}$$\mathrm{c}_{1}$$\overline{W^{s}(p)}=J^{+}$$\mathrm{X}_{2}’\in W^{\epsilon}(\mathrm{r}_{2})\cap \mathcal{I}$$\mathrm{Y}_{1}$$\mathcal{I}$$W^{s}(\mathrm{r}_{2})$$\mathrm{r}_{1})$$\mathrm{X}_{2}$$\mathrm{X}_{2}’$$\mathrm{r}_{2}$$\mathrm{X}_{2}’$$\mathrm{r}_{2}(\mathrm{r}_{2}$$\mathrm{X}_{2}$$\mathrm{Y}_{1}=W^{s}(\mathrm{c}_{1})\cap \mathcal{I}$$W^{\epsilon}(\mathrm{r}_{2})$$\mathrm{c}_{1}$$\mathrm{r}_{2}$$\mathrm{X}_{2}’$$\mathcal{I}$$\mathrm{X}_{2}’$$\mathrm{Y}_{2}’=W^{\epsilon}(\mathrm{c}_{2})\cap \mathcal{I}$127side of X’2’ where denotes another complex saddle. The side chain of the third orderis realized very close to the tirtiary intersectionof the intersction, if
is taken large enough. Inductively, the m-th order generation, germinates from the$(m-1)$ -th order generation.
$\mathrm{c}_{2}$$\mathrm{X}_{3}’’$$\mathcal{I}\cap W^{s}(\mathrm{r}_{3})$$n$Figure 7: Initial value plane$\mathcal{I}$and stable manifolds for real saddles (schematic).

As predicted by the theorem
and numerical observations imply that
there exist a rich variety of tunneling paths wandering over complex phase space before
reaching close to the real plane. Ishii gave a rigorous statement supporting such an
aspect [12]. It assures the existence of orbits that allow trajectories exhibiting chaotic
itinerancy over the complex saddles:$\overline{W^{\partial}(p)}=J^{+}$

Theorem[Ishii] Let
and let $p_{i}(1\leq i\leq N)$ be saddle periodic
points in. Take any positive integers and any neighborhood of $p:(1\leq$$i\leq N)$ . Then, there exists a pointsuch that its orbit stays inat least-times iterates and.
$0<h_{\mathrm{t}\mathrm{o}\mathrm{p}}(F|_{\mathrm{R}^{2}})<\log 2$$\mathbb{C}^{2}\backslash \mathbb{R}^{2}$$k_{i}$$z\in \mathbb{C}^{2}\backslash \mathbb{R}^{2}$$k_{i}$$U_{i}$$U_{i}$$\lim_{narrow\infty}d(F^{n}(z), \mathbb{R}^{2})=0$

The convergent theorem of currents is a central result of Bedford and Smillie and many
results follow from it [17, 16, 18]. The above theorem essentially uses the results of [18].
An important fact is, as mentioned below, that the convergent theorem applies not only
to hyperbolic but also to any cases including mixed phase space. Therefore, it is natural
to expect that the behavior just sketched above is not limited to the hyperbolic case.
In fact, the orbits launched at a chained branch in the higher generation, for example
as given in Fig. 3, exhibit essentially in the same behavior.

128
Why Are Chained Structures So $Important^{Q}$
Up to now, we only saw the behavior of complex orbits that contribute to the semiclassical propagator (3). The readers may have questions as to why the chained structures
found commonly in the initial value plane is so important in our tunneling problem.
We have entirely skipped this topic.
The main reason is that, as shown in Fig. 4, the orbits forming the chain structure
approach the real plane exponentially. Correspondingly, the imaginary part of action,, converges rapidly and we may expect thattakes the smallest values ascompared with those for the orbits taking side trip, unless some nontrivial cancellation
mechanism works in itinerating orbits. Since the weight of each term in the semiclassical, the orbits that gain the minimalsum (3) is almost controlled byare most
dominant contributors in the sum. This is a rough reasoning for the importance of thechained objects. It is not yet clear that the orbits wondering in complex space, as foundin Fig. 6(a), play some roles in the tunneling problem. If they make non-negligible
contributions, we may say that genuine complex chaos appears in quantum tunneling.
More detailed aruguemts of observability of complex chaos in quantum tunneling will
be discussed in [13].
${\rm Im} S_{n}$$|{\rm Im} S_{n}|$${\rm Im} S_{n}$5${\rm Im} S_{n}$Tunneling Orbits in Mixed Phase Space

Initial and Final Manifolds in Semiclassical Dynamics

As often emhpasized, the most natural and important situation is that KAM curves
and chaotic regions coexist in phase space. That is, the transition from KAM regions
to chaotic seas has particularly to be investigated. To focus our problem more sharply,
we again make clear the setting of the semiclassical argument.
So far, we have taken -representation like eq. (2). However, we can freely replace
it to other representations. For example, we may take the coherent representation$K(q_{n},p_{n};q_{0},p_{0})=<q_{n},p_{n}|\hat{U}^{n}|q_{0},p_{0}>$. Herethe coherent statewith $a=(q+ip)/\sqrt{2}$ . We have a similar semiclassical expression as$\Psi$$|q,p>=|a>\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{s}$$K^{sc}(q_{n},p_{n};q_{0},p_{0})= \sum_{\gamma}A_{\gamma}(p_{0}, q_{0})\exp\{\frac{i}{\hslash}S_{\gamma}(p_{0}, q_{0})\}$

,

(10)

where the sum is taken over all classical paths satisfying given initial and final coherent
states, $i.e.,$ $q0+ip_{0}=q_{\alpha}+ip_{\alpha},$ $q_{n}-ip_{n}=q_{\beta}+ip_{\beta}$ . Note that the variables
can take complex ones $[19, 20]$ . Introducing thetake real values whereas, semiclassically contributing
variables $Q=q+ip$ and $P=q– ip$ wherecomplex paths are given as$q_{\alpha},p_{\alpha},$$q_{\beta},p_{\beta}$$q_{0},p0,$ $q_{0},p_{0}$$q,p\in \mathbb{C}$$M_{n}^{\alpha,\beta}\equiv\{(Q_{0}, P_{0})\in \mathbb{C}^{2}|Q_{0}=\alpha, P_{n}=\beta, \alpha=q_{\alpha}+ip_{\alpha}, \beta=q_{\beta}-ip_{\beta}, \}$

.

(11)

Note that in any representation the manifold of initial or final states is one-dimensional
complex manifold, thereby the space of the search parameter forms one-dimensional
complex plane. This is interpreted as a manifestation of uncertainty principle of quantum mechanics.
Convergent Theorem and Mixing Property in Complex Phase Space129
The semiclassical dynamics is therefore just to follow the time evolution of a onedimensional complex manifold with a boundary condition imposed on the final state.
The final state is also expressed as one-dimensional complex manifold. We again recall
that the asymptotic behavior of wide classes of manifolds is well controlled using potentially theoretic arguments of complex dynamical systems [17, 16, 18], More precisely,
the following convergent theorem of currents tells us how one-dimensional manifold
behaves asymptotically:

Theorem[Bedford-Smillie]
For a complex one-dimensional locally closed sub-manifold $M$ in either
braic variety, there is a constant $\gamma>0$ so that

$J^{\pm}$or an alge(12)$\lim_{narrow+\infty}\frac{1}{2^{n}}[F^{\mp n}M]=\gamma\cdot dd^{c}G^{\pm}(x, y)$in the sense of current, where $[M]$ is the current of integration of $M$ , i.e. $[M](\phi)\equiv$. In this statement,
represents the Green function given by$G^{\pm}(x, y)$$\int_{M}\phi|_{M}$$G^{\pm}(x, y) \equiv\lim_{narrow\pm\infty}\frac{1}{2^{n}}\log^{+}||F^{n}(x, y)||$where$dd^{c}$,(13)is the complex Laplacian,(14)$dd^{\mathrm{c}}u \equiv 2i\sum_{j,k}\frac{\partial u}{\partial z_{j}\partial\overline{z}_{k}}dz_{j}\wedge d\overline{z}_{k}$The statement claims that an arbitrary algebraic curve in , for example our initial
manifold given as, converges to theof $dd^{c}G^{\pm}(x, y)$ . Moreover,$\mathbb{C}^{2}$$p_{0}=\alpha\in \mathbb{R}$[$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}$Here$\mathrm{B}\mathrm{e}\mathrm{d}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{d}$$\mu\equiv\mu^{+}\wedge\mu^{-}$-Smillie]and$\mu^{\pm}$$\mu$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}o\mathrm{r}\mathrm{t}$is mixing and the hyperbolic measure.

is induced by the Green function as$\mu^{\pm}\equiv\frac{1}{2\pi}dd^{c}G^{\pm}(x, y)$

.

(15)

The complex equilibrium measure thus defined becomes a unique maximal entropy
probability measure [17, 16, 21, 18].
$\mu$

Implications in The Tunneling Problem

Since the asymptotic behavior is so described by the convergent theorem, we obtaintheorems on the relation between tunneling orbits and the Julia set in the forward
direction by taking into account the final states also [12]. The result indeed supports
and is consistent with numerical observations illustrated in Figs. 3, 4, and 6. In thissense, we can say that, in contrast to tunneling orbits in the linear map, the complex
orbits behind tunneling processes in chaotic systems have truly different characters.
As stressed in section 3, tunneling penetration does not occur as a result of dynamics in the linear map, but gaining the imaginary part of the initial condition makes it
possible. On the hand hand, in the H\’enon map, due to the mixing property statedabove, arbitrary neighborhoods of the points in the Julia set are connected by the
dynamics in complex plane. To see more explicitly how the mixing property plays

130
the role, the coherent state representation is useful since the coherent state is a minimal wavepacket, and is an object closes to an orbit in classical dynamics. For any,
neighborhoodof initial and final points,
andin the coherent
representation, there exist a time step $N$ such that
. Since the neighborhood
should be taken as an open set in , the orbit connecting between
and. However; we can find anothermay not be contained in the initial manifold
initial statethat can be taken arbitrarily close to the original point
which contains a desired orbit. In other words, although one cannot say that a set
always contains a connecting orbit, there is a wavepacket arbitrarily close to thecontains such a connecting orbit. The tunnelingoriginal one whose initial planetransition, reflecting the mixing property of the complex measure , takes place in this$U_{\alpha}$$(q_{\alpha},p_{\alpha})$$U_{\beta}$$(q_{\beta},p_{\beta})$$f^{N}(U_{\alpha})\cap U_{\beta}\neq\emptyset$$\mathbb{C}^{2}$$U_{\alpha}$$U_{\alpha}$$\mathcal{M}_{n}^{\alpha,\beta}$$U_{\beta}$$(q_{\alpha},p_{\alpha})$$(q_{\alpha}’, p_{\alpha}’)$$M_{n}^{\alpha,\beta}$$\mathcal{M}_{n}^{\alpha’,\beta’}$

$\mu$

way.
Complexified $KAM$ Curves and Natural Boundaries

As mentioned, theorems of Bedford and Smillie are suggestive and certainly give afundamental principle for the semiclassical description of quantum tunneling processes.
However, only to know the existence of invariant measure having such nice properties
is not enough to get further predictive view in tunneling phenomena. since the mostimportant process, that is, the transition from the region dominated by KAM curves
to chaotic regions, necessarily involves many delicate issues in the problem of nearly
integrable Hamiltonian dynamics, so more precise information on complex structuresin such regions is strongly required.
In particular, we need to discuss the role of complexified $KAM$ curves. To thisend, the works done by several authors who studied the domain of analyticity of complexified KAM curves become significant clues $[22, 23]$ . The aim in those works was
at which the last KAM
to specify a critical value of the perturbation parametercircle disappears, and also to study the universality of the critical function: thebreakdown threshold of the KAM curve with rotation number . Here we will see that
this subject is indeed of fundamental importance in the tunneling problem.
A standard recipe to consider the analyticity domain of KAM curves is first to
express KAM curves parametrically as$K_{c}$$K(\omega)$$\omega$$C_{\omega}$where$u(\varphi, \omega)$:$=(2\pi\omega+u(\varphi, \omega)-u(\varphi-2\pi\omega, \omega)\varphi+u(\varphi,\omega))$

,

(16)

is determined by the following functional equation:.

$u(\varphi+2\pi\omega, \omega)-2u(\varphi, \omega)+u(\varphi-2\pi\omega, \omega)=V’(\varphi+u(\varphi,\omega))$(17)

is given in the -variable as a constant rotation
The dynamics on the curve. For a given rotation number , the existence of an analytic KAM curve isequivalent to the existence of a positive radius of convergency of the Lindstedt series,$C_{\mathrm{t}d}$$\varphi_{n+1}=$$\varphi$$\varphi_{n}+2\pi\omega$$\omega$$u( \varphi, \omega)\equiv\sum_{k=1}^{\infty}K^{k}\sum_{\nu\leq k}\mathrm{e}^{i\nu\varphi}u_{\nu}^{(k)}(\omega)$

.

(18)

As first suggested in Ref. [22], making analytical continuation of the Lindstedt series
to the complex plane, one can do it at most to a certain domain of -plane and there
possibly exist a natural boundary. The existence of the natural boundary implies that$\varphi$

131

natural boundary for KAM circle
with a rotation number$\omega$$\nearrow$

Figure 8: Natural boundaries of KAM circles (schematic). The complex orbits which describe
tunneling processes are also inserted in the figure. Those orbits launched almost aroundnatural boundaries and go down to the real plane. The distance from the real plane decreases
exponentially, just as shown in Fig. 4(b) and the bold line of Fig. 6(a).

KAM circles cannot be globally invariant in complex plane. Numerical observations
suggest the shape of the boundary is fractal in $(q,p)$ -plane. We schematically draw an
aspect in Fig. 8.
Mixing in the Complexified $KAM$ Regions ?
$l$

The orbits on KAM curves are bounded both in the forward and backward directions,so they are contained in the filled-in Julia set $K=K^{+}\cap K^{-}$ Note that numericaltests suggest thathave no interior points [9]. There is no rigorous proof validating
those numerical results, but ifhave no interior points, $J=K$ follows.
Next we ask the relation between and. In case the system is hyperbolic, $J=J^{*}$ holds, but in general it was at most proved that
. In non-hyperbolicor mixed systems, both possibilities remain: (i) $J=J^{*}$ or (ii) $J\neq J^{*}$ . If the former
holds, it immediately follows that {KAM curves}. Since the mixing property
holds on , a more specified situation just mentioned above is realized, that is the
mixing property of complex dynamics allows the connection between two separated
regions via the real dynamics. On the other hand, if the case (ii) is true, it can happen
that such orbits do not necessarily exist.
It is not an easy task to check which situation is really the case even numerically.
What we have presented in Fig. 9 is a typical behavior of the orbits sandwiched between
complexified KAM curves. Here we put an initial pointsuch thatis on a certain KAM circle and. The orbit so located very close to
the real plane initially rotate along a KAM curve that is closest to the initial point but
it gradually leaves the real plane. Then it moves in complex phase space in a spiralway. In almost all cases, however, the orbits moving up in such a way diverge to infinity
finally. More precisely, until a certain finite boundary, orbits rotate along theof$K^{\pm}$$K^{\pm}$$J$$J^{*}=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mu$$J^{*}\subset J$$\subset J^{*}$$J^{*}$$(q_{0},p_{0})$$|{\rm Im} q_{0}|,$$({\rm Re} q_{0}, {\rm Re} p_{0})$$|{\rm Im} p_{0}|\ll 1$$\mathrm{w}\mathrm{a}\mathrm{U}$132
complexified KAM curves as shown in Fig. 8, but once they reach the boundary, theyquickly fly away to infinity. The fact that typical orbits behave in this way is entirely
has no interior points because arbitrary chosen initialconsistent with the fact that.
points in complex plane should belong toOn the other hand, if the initial point is chosen very carefully, we can find the orbits
such that they initially leave the real plane and tend to boundaries in a similar way
as above, but they again go back to and approaches the real plane. Figure 9 exactly
demonstrates such an example. An important fact is that, as shown in Fig. 9(b), oncethe orbit goes back close to the real plane, it rotates along the KAM curve that isdifferent from the initially located one. Several different circles observed in Fig. 9(b) is
projections of an orbit belonging to different time intervals. This numerical experimentstrongly implies that KAM curves on the real domain are indeed bridged via complex
orbits in the Julia set.
We should note, however, that this result does not necessarily suggest $J=J^{*}$ . If
$J=J^{*}$ , due to the result of Bedford and Smillie again, there must exist saddle periodic
points in any close neighborhood of complexified KAM circles, which is just the wall
surface of cylinders shown in Fig. 8. Such an aspect cannot not be so easily verified
even numerically.
$K^{\pm}$$\mathbb{C}^{2}\backslash K$

Figure 9: (a) Temporal behavior of an orbit that is put close to the real plane.
is almost zero,is on a KAM curve on the real plane. In the temporal behavior, whentakes also almost zero. Thus, the orbit is very close to the real plane each time focusing
plane.
ofoccurs. (b) Projection of an orbit on to$({\rm Re} q_{0}, {\rm Re} p_{0})$${\rm Im} q_{n}$${\rm Im} p_{n}$${\rm Im} q_{n}$$(q_{n},p_{n})$Natural Bounda ries of $KAM$ Curves and The Julia Set
As mentioned, a semiclassical state is identified as a one-dimensional complex manifold.
For the moment, we forget restrictions originating from quantum mechanics and concentrate only on the semiclassical dynamics, that is, the dynamics of one-dimensional
complex manifolds.
Suppose, as a Gendanken Experiment, an initial semiclassical manifold put exactly
on a certain KAM curve with a rotation number . The complexification is made by$\omega$

133
extending the angle variable in the Lindstedt series (18) to the complex plane as. The initial value plane thus complexified is nothing but an analyticalextension of the KAM curve with a given rotation number . Since KAM curves withdifferent rotation numbers give different invariant sets, the transition between themis still forbidden so long as the Fourier expansion (18) provides a complex analytic
function. However, natural boundaries possibly appear, thereby we cannot extend theinitial value plane beyond them. The best possible way to make the orbits be confinedwithin a KAM curve would be the procedure described here.
On the other hand, the convergent theorem applies at least to complex algebraic
curves. If we take a complex algebraic curve as an initial semiclassical set, forward
iteration of it, roughly speaking, tends toand the backward to. Therefore,such classes of complex one-dimensional curves are not confined in KAM curves andnecessarily spread over chaotic seas.
If not only algebraic curves but also any semiclassical states cannot stay within
KAM curves, we may say that the transition between (real) classically forbidden regions
takes place in the purely classical dynamics. To make the situation more transparent,
we have to clarify the relation between natural boundaries of KAM curves and theJulia set. This would become a core question.
Estimating the imaginary actionis the most relevant task in the semiclassicalanalysis of quantum tunneling. Numerical experiments, together with an analysis forthe linear model discussed in section 3, suggest that this is roughly proportional to thelength of integrable branches in the initial value plane. More precisely, as shownin Fig. 3 two integrable branches emanating from the real branch $\eta=0$ extends inthe imaginary direction and they disappear in the aggregated region in which chained
structure are hidden. The length of integrable branches is then almost equal to theimaginary part of the lower boundary of those aggregated branches. It was also foundnumerically that natural boundaries of complexified KAM curves are almost locatedalong the lower boundary of aggregated branches [13]. Therefore,
must be closely
connected with the width of the analyticity domain of KAM curves, that is, the radiusof convergence of Lindstadt series must control the tunneling action. If this is indeed
the case, it can happen that the tunneling probability varies irregularly as a functionof the rotation number since the radius changes in a fractal manner. So, it would
become extremely important for the quantum tunneling problem to study how thenature of natural boundaries affect the tunneling action.
$\varphi$$\varphi=\varphi’+i\varphi’’$$\omega$$J^{-}$$J^{+}$${\rm Im} S_{n}$$M_{n}^{\alpha,*}$${\rm Im} S_{n}$$\omega$${\rm Im} S_{n}$The present note is written on the basis of the collaboration with Y. Ishii and K.S.
Ikeda. The author thanks E. Bedford for his stimulating suggestions.

References[1] S. C. Creagh, in Tunneling in complex systems ed. by S. Tomsovic (World Scientific, Singapore, 1998) pp. 35- 100.
[2] 0. Bohigas, S. Tomsovic and D. Ullmo Phys. Rep. 223, 43-133 (1993); S. Tomsovic
and D. Ullmo Phys. Rev E50, 145-162 (1994).

[3] S.D. Frischat and E. Doron, Phys. Rev. E57, 1421-1443 (1999).

134[4] A. Shudo and K. S. Ikeda, Phys. Rev. Lett. 74, 682-685 (1995); Physica D115,
234-292 (1998); T. Onishi, A. Shudo, K. S. Ikeda and K. Takahashi, Phys. Rev.
68, 056211-1-22 (2003).

$\mathrm{E}$[5] K. Takahashi and K.S. Ikeda, Ann. Phys. (NY)284, $94-$ 140 (2000); J. Phys. A
36 (2003) 7953; EuroPhys. Lett. 71, 193- 199 (2005).
[6] J.S. Langer, Ann. Phys. 54, 258-275 (1969); S. Coleman, in The Whys of Nuclear
Physics, A.Zichin(ed.) (Academic, N.Y., 1977); W.H. Miller, J. Chem. Phys. 53,
1949-1959 (1970); Adv. Chem. Phys. 25, 69-177 (1974),[7] M. Wilkinson, Physica D21. 341-354 (1986); M. Wilkinson and J. H. Hannay,
Physica D27, 201-212 (1987).
[8] S. C. Creagh, J. Phys. A 27, 4969-4993 (1994).
[9] A. Shudo, RIMS K\^oky\^uroku, 1424184-199 (2005).
[10] R. Devaney and Z.Nitechi, Commun. Math. Phys., 67, 137-146 (1979).
[11] E. Bedford and J. Smillie, Ann. Math., 160, 1-26 (2004).
[12] A. Shudo, Y. Ishii and K.S. Ikeda, J. Phys. A 35, L225-L231 (2002). A. Shudo,
Y. Ishii and K.S. Ikeda, to be submitted.
[13] A. Shudo, Y. Ishii and K.S. Ikeda, to be submitted.
[14] M.J. Davis, R. S. MacKay,and A. Sannami, Physica D52171-178 (1991).

[15] Z. Arai, On hyperbolic plateaus of the Hdon maps, preprint (2005).

[16] E. Bedford, and J. Smillie, J. Amer. Math. Soc. 4, 657-679 (1991).
[17] E. Bedford, and J. Smillie, Invent. Math. 103, 69-99 (1991).
[18] E. Bedford, M. Lyubich and J. Smillie, Invent. Math. 112, 77-125 (1993).
[19] J.R. Klauder, In Path Integrals, edited by G.J. Papadopoulos and J.T. Devreese,NATO Advanced Summer Institute (Plenum, New York, 1978), p. 5; J.R. Klauder,in Random Media, edited by G. Papanicolauou (Springer-Verlag, New York, 1987)
p. 163.
[20] S. Adachi, Ann. Phys. 195, 45- (1989).

[21] J. Smillie, Ergod. Th. and Dynam. Sys. 10, 823-827 (1990).

[22] J.M. Greene and I.C. Percival, Physica D3, 530-548 (1981); I.C. Percival, Physica
D6, 67-77 (1982).
[23] S. Marmi, J. Phys. A 23, 3447-3474 (1990); A. Berretti, and L. Chierchia, Nonlin$39- 44(\mathrm{l}990.$$\mathrm{e}_{1992);\mathrm{A}.\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i},l_{\mathrm{C}\mathrm{e}11\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i},\mathrm{L}.\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{a}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{C}.\mathrm{F}\mathrm{a}1\mathrm{c}\mathrm{o}1\mathrm{i}\mathrm{n}\mathrm{i},\mathrm{J}.\dot{\mathrm{S}}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{P}\mathrm{h}\mathrm{y}\mathrm{s}.66}^{;\mathrm{A}.\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{S}.\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{i},\mathrm{P}\mathrm{h}\mathrm{y}\mathrm{s}.\mathrm{R}\mathrm{e}\mathrm{v}.\mathrm{L}\mathrm{e}\mathrm{t}\mathrm{t}68,.1443- 1446}}(\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y},3,$1613-1630 (1992).

,