Semiclassical propagation of coherent states and wave packets: hidden saddles
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Semiclassical propagation of coherent states and wave packets: hidden saddles
Huichao Wang and Steven Tomsovic
Department of Physics and Astronomy, Washington State University, Pullman, WA. USA 99164-2814
(Dated: August 11, 2021)
Semiclassical methods are extremely important in the subjects of wave packet and coherent state
dynamics. Unfortunately, these essentially saddle point approximations are considered nearly im-
possible to carry out in detail for systems with multiple degrees of freedom due to the difficulties
of solving the resulting two-point boundary value problems. However, recent developments have
extended the applicability to a broader range of systems and circumstances. The most important
advances are first to generate a set of real reference trajectories using appropriately reduced dimen-
sional spaces of initial conditions, and second to feed that set into a Newton-Raphson search scheme
to locate the exposed complex saddle trajectories. The arguments for this approach were based
mostly on intuition and numerical verification. In this paper, the methods are put on a firmer theo-
arXiv:2107.08799v2 [quant-ph] 9 Aug 2021
retical foundation and then extended to incorporate saddles hidden from Newton-Raphson searches
initiated with real trajectories. This hidden class of saddles is relevant to tunneling-type processes,
but a hidden saddle can sometimes contribute just as much as or more than an exposed one.
I. INTRODUCTION dles contributing to the same position. As time increases
beyond the Ehrenfest time, the number of physically rel-
The evolutions of Glauber coherent states for bosonic evant saddles increases, but it is always finite in number
many-body systems and the mathematically nearly- at some fixed propagation time. One might say that a
identically related Gaussian wave packets arise in a multi- set of measure zero saddles from the infinite set must be
tude of physical contexts. Examples abound in quantum selected as the physically relevant saddles. Furthermore,
optics [1, 2], far out-of-equilibrium dynamics in bosonic these trajectories necessarily involve complexified posi-
many-body systems [3, 4], molecular spectroscopy [5, 6], tion and momenta. This analytic continuation of real
femto-chemistry [7], and attosecond physics [8]. Theoret- classical dynamics has many nontrivial features, such as
ical work encompasses, for example, coherent state repre- branch cuts associated with singular runaway trajecto-
sentations of path integrals [9] and in the context of many ries [10].
bosons or the short wavelength limit, the semiclassical Considerable progress has been made in developing a
approximation [9–11]. This approximation is fundamen- practical method of identifying physically relevant sad-
tal to studies of quantum-classical correspondence [12], dles directly without encountering any of the irrelevant
and pre- and post-Ehrenfest-time-scale dynamics [13–15] saddles [17–20]. The basic idea builds on earlier tech-
as well. niques of identifying a single real reference trajectory
A complete semiclassical approximation for coherent for each classical transport pathway that exists in the
state dynamics may be obtained by the saddle point ap- nonlinear dynamics of the system [21–24]. Then using
proximation applied to coherent state path integrals [9, a Newton-Raphson algorithm, a unique saddle point is
11]. In the context of wave packets the essentially identi- identified for each transport pathway and it accounts for
cal approximation is known as generalized Gaussian wave the pathway’s contributions to the dynamics. The tech-
packet dynamics (GGWPD) and it has been proven to niques have been developed using Gaussian wave pack-
be equivalent to a complexified version of time-dependent ets, but it applies equally well to bosonic coherent states
WKB theory [10]. In either case, its implementation in expressed in quadratures; see [15]. In many cases the
practice leads to considerable technical difficulties for any method can be extended to systems with many degrees of
system possessing nonlinear dynamics. freedom by identifying and neglecting directions in phase
To begin with, the classical trajectories that define the space that do not lead to diverging initial conditions [20].
saddles’ properties are solutions of a two-point boundary Using these techniques, a complex saddle is identified
value problem, which is highly nontrivial for nonlinear with each classically allowed transport pathway. Let us
dynamical systems. If there are many degrees of free- dub these “exposed saddles”. Amongst the infinity of
dom, this problem may be effectively impossible to solve. exposed saddles, almost all contribute too little to be
It appears that for any propagation time t 6= 0, there concerned with, but it is straightforward to restrict the
exists an infinity of saddles; see Figure 1 ahead. At the search for real transport pathways that have sufficient
shortest evolution time scale, typically only one saddle is amplitude to be relevant. Nevertheless, there is still a
physically relevant; this is the Ehrenfest time regime [16]. great deal more to be done. These works were justified
Parenthetically, physically relevant means: i) the saddle intuitively and left open the question of how to iden-
is on the correct side of the Stokes lines; and ii) its con- tify physically relevant saddles associated with classically
tribution to the wave function at the relevant position non-allowed transport pathways. In contrast to exposed
is significant enough to be larger than any of the errors saddles, consider these “hidden saddles”, in part because
due to the saddle point approximations from other sad- they are not directly discoverable using real trajectory2
input into a Newton-Raphson search scheme. The focus The appropriate projection of the coherent state re-
of this work is to examine these techniques in greater sults in a Gaussian wave packet form [1] with parame-
detail, add additional justifications where possible, and ters which can be straightforwardly mapped onto those
develop a method to locate the physically relevant hid- of such a wave packet. One exposition of the parame-
den saddles. This extends the techniques to incorporate ter mapping is given in Appendix A of [20]. Suffice it
tunneling-like phenomena. to mention here that the complex parameter z can be
The structure of the paper is as follows, Sect. II de- mapped onto momentum and position centroids, and the
fines the critical quantitites, sets notations, introduces ground state links the shape parameters. There are suf-
the purely quartic oscillator for simple illustrations, and ficient wave packet parameters to squeeze the coherent
discusses the background of various semiclassical approx- states and rotate them in quadratures. The focus now
imation methods and their interrelationships. This is turns to Gaussian wave packets.
followed in Sect. III by a discussion of the justification
of the existing methods, and develops a technique for
identifying hidden saddles . The paper concludes with a B. Gaussian wave packets
summary of the work and possible future lines of related
research. A normalized Gaussian wave packet may be parame-
terized as follows:
bα 2 ipα i
II. BACKGROUND φα (x) = exp − ( x − qα ) + (x − qα ) + pα qα
2~ ~ 2~
1/4
bα + b∗α
In order to elucidate the method for implementing a × (2)
search for hidden saddles and justify certain procedures, 2π~
it is helpful to start with some background on the com- where the subscript α is a label for the parameters that
plete semiclassical approximation along with certain par- define the particular wave packet, x is the position vari-
tial or imperfect versions of the semiclassical approxima- able for the quantum system, and (q, p) are the canon-
tion. As the mathematical results and manipulations can ically conjugate position-momentum phase space vari-
be made to appear essentially identical through the appli- ables for the analogous classical system. The real cen-
cation of quadratures, it is unnecessary to treat evolution troid is given by (qα , pα ) and the width by bα (if bα is
of Glauber coherent states and wave packets separately. complex, the wave packet has a chirped phase depen-
The discussion is presented in the language of wave pack- dence). This form has the advantage that ~ does not ex-
ets, but it is understood that all the results carry over plicitly appear in the equations for the two-point bound-
to coherent states. As a final note, for simplicity the ary value problem given ahead. Note that this form has
discussion and equations given here are reduced to their the exact same phase convention as the coherent state
single degree of freedom forms. Although, the interest is of Eq. (1). It also leaves the overall shape of its Wigner
in multi-degree-of-freedom systems, it is much easier to transform independent of ~, other than the volume (over-
illuminate the basic ideas with a simple example and the all scale). Similarly to a coherent state, for nonlinear
equations reduced to their one degree of freedom forms. dynamical systems the evolved wave packet φα (x; t) is
All the necessary multi-degree-of-freedom equations can generally not Gaussian. Note that a matrix element of
be found elsewhere, for example in [19, 20], and the ad- the coherent state path integral could be expressed in
ditional ideas presented here extend to the many degrees quadratures as
of freedom case. Z ∞
A βα (t) = dx φ∗β (x)φα (x; t) (3)
−∞
A. Coherent states and which could be thought of as a transport coefficient
for wave packets. If β = α, it would be a diagonal element
A Glauber coherent state describing a bosonic many- or a return amplitude.
body system takes the normalized form
1. Lagrangian manifolds
!
2
|z|
|zi = exp − + z↠|0i . (1)
2
Lagrangian manifolds play a central role in semiclassi-
cal approximations, i.e. WKB theory [25]. They provide
It’s evolved form |z(t)i does not remain a coherent state, a very geometric picture of the application of the sad-
but the overlap with the bra vector version of another dle point approximation. Using the parameterization of
coherent state hz 0 | can be viewed as a matrix element of a Eq. (2), the appropriate manifold for a Gaussian wave
coherent state path integral, hz 0 |z(t)i. In the language of packet was identified in [10] as
wave packets, this is often termed a correlation function
or transport coefficient. bα (q − qα ) + i (p − pα ) = 0 (4)3
where (p, q) are chosen from the sets of all complex po- Each solution of these equations is a potential saddle
sitions and conjugate momenta satisfying this equation. point in the theory for transport coefficients. If instead,
This generates a complex line in a two dimensional com- one is interested in the evolution of the wave packet itself,
plex phase space (complex position and momentum), the equations would be given by
somewhat akin to a plane embedded in a four dimen-
sional space. Likewise for the complex conjugate wave bα (q0 − qα ) + i (p0 − p~α ) = 0
packet (dual version), the equation is qt − x = 0 (7)
b∗α (q − qα ) − i (p − pα ) = 0 (5) where x is the position argument of the evolved wave
packet (a momentum representation is also possible, but
For a position ket or bra vector, the Lagrangian mani- not used here). For either boundary value problem, non-
fold would be the value of the position and the set of all linear dynamical systems lead to an infinite number of
complex momenta. solutions to these equations. Almost all of them must be
One consequence of the Lagrangian manifold being thrown away due to boundary conditions or due to their
necessarily complex is that it blurs the distinction be- contributions being vastly smaller than the errors inher-
tween classically allowed and non-allowed processes. In ent in the semiclassical approximation. Excluding them
ordinary WKB, the tori used for constructing wave func- from the search algorithm by design is highly desirable
tions are real Lagrangian manifolds, the intersections of and the basis of the works [19, 20].
two manifolds generate stationary phase points (not sad-
dles), and the exponentially decaying tails of wave func-
tions require propagation with imaginary time or some C. The purely quartic oscillator
analytic continuation, for example, the introduction of
imaginary momenta [26]. Here it is the hidden saddles The one degree of freedom purely quartic oscillator has
which correspond to classically non-allowed processes. a number of simple features that makes it ideal for illus-
Since both hidden and exposed saddles are linked to com- trating the ideas discussed in this paper. Its analytically
plex trajectories, “real” versus “complex” is not the distin- continued form [complex (p, q)] is given by
guishing factor. There is however the intuitive notion of
complex phase points close or near to being real; see [17] p2
and references therein. Since distance is not defined in H(p, q) = + λq4 (8)
2
phase spaces, a priori, the concept of “near” is not well de-
fined. Even worse, a saddle can have an arbitrarily large where ~ = m = 1 and λ = 0.05 are the values taken for all
imaginary component in its initial condition and still be the illustrations shown in this paper. The corresponding
exposed. Part of our work here is to add some preci- Schrödinger equation is given by
sion to this concept, which ends up helping to categorize
∂2
the saddles into exposed or hidden groups. Just as the ∂ 4
i φ(x; t) = − + λx φ(x; t) (9)
distinction of real and complex trajectories is physically ∂t 2∂x2
significant in the case of ordinary WKB theory, so is the
exposed/hidden saddle distinction important physically. Being a homogeneous Hamiltonian, the real classical dy-
It also alters significantly how one must search for the namics of this system leads to simple scaling relations
saddles, which is treated ahead. amongst the quantities(q, p; t; E). For example, propa-
gating an initial condition (qE0 , pE0 ) for a time t that
belongs on the E0 energy surface leads to a scaled tra-
2. Two-point boundary value problem jectory replica on any desired second energy surface E
according to:
The two-point boundary value problem that arises in
qE (t/γ) = γqE0 (t)
the saddle point approximation can be succinctly ex-
pressed using these manifolds. In essence, the manifold pE (t/γ) = γ 2 pE0 (t) (10)
corresponding to the ket vector is propagated to time t
according to the dynamical equations of motion, and its where γ = (E/E0 )1/4 . Classical actions, defined as S =
Ldt, scale as γ 3 , and the period of a closed orbit, τE0 ,
R
intersections with the manifold associated with bra vec-
tor give the trajectories that define the saddle points’ scales as τE = τE0 /γ. Therefore, if γ = n±1 , where n is
properties. Letting (~p0 , ~q0 ) represent the complex initial any positive integer, the two periodic orbits on different
condition for a trajectory and (~ pt , ~qt ) the complex phase energy surfaces would have periods that are an integer
point that is generated by evolving this initial condition multiple of each other. In the time that the longer period
to time t, the two-point boundary value problem can be orbit closes once, the shorter period orbit retraces itself
written as n times.
The potential generates nonlinear dynamical equations
bα (q0 − qα ) + i (p0 − p~α ) = 0 that lead to sufficiently complicated behaviors for our
b∗β (qt − qβ ) − i (pt − pβ ) = 0 (6) purposes. As a first example, see Fig. 1, which illustrates4
t=τ D. Wigner transform
20
The Wigner transform of a wave packet (ket-like) gen-
10 erates a multivariate Gaussian function, which can be
Im q
0 thought of as a density of real classical initial conditions
0 that underlie a quantum wave packet and account for
the uncertainty principle. It can be used to help under-
−10 stand the partial semiclassical approximation known as
linearized wave packet dynamics [13], and is essential for
−20 discussing the basis of an enhanced approximation known
as an off-center, real trajectory method [22–24, 27, 28].
−0.2 −0.1 0.0 0.1 0.2 The Wigner transform of a wave packet parametrized
Re q0 as in Eq. (2) is given by
20 t = 2τ 1 Aα p − pα
W(p, q) = exp − (p − pα , q − qα ) · ·
π~ ~ q − qα
10 (11)
Im q where Aα is
0
0
1/c d/c
Aα = Det [Aα ] = 1 (12)
−10 d/c c + d2 /c
−20 with the association
bα = c + id (13)
−0.2 −0.1 0.0 0.1 0.2
Re q0 The 2 × 2 dimensional matrix Aα is real and symmetric.
If bα is real, there are no correlations between p and q (d
FIG. 1. Initial conditions of exposed saddles satisfying Eq. (7) vanishes); i.e. the wave packet is not chirped. The off-
for a wave packet, Eq. (2), centered at (qα , pα ) = (0, 20) of diagonal elements (blocks in more degrees of freedom) of
width parameter bα = 32 and propagated with the Hamilto- Aα disappear.
nian, Eq. (8). The complex momentum of a saddle’s initial
conditions follows from the complex position. In the upper
panel, the propagation time is the period of the orbit with ini- E. Partial semiclassical theories
tial condition (q0 , p0 ) = (0, 20) and in the lower panel twice
this period. The density of exposed saddles increases with
increasing time, but even in the limit of small times, the total There are two important partial semiclassical approx-
number of saddles is infinite. The only exception is t = 0 imations known as linearized wave packet dynamics and
off-center, real trajectory methods, respectively, in order
of increasing sophistication. The former linearizes the
dynamics completely, and the latter contains the nonlin-
ear dynamical information, which is much, much closer
the placement of exposed saddle initial conditions in a to GGWPD.
plane for the evolution of an initial wave packet evalu-
ated at x = 0 and some fixed time t. There is no limiting
domain within the Lagrangian manifold for the solutions, 1. Linearized wave packet dynamics (LWPD)
and an infinity of solutions is implied over the manifold’s
infinite domain. In particular, note that these exposed The main idea involves solving the equations of motion
saddles can have imaginary parts of their initial condi- for the parameters that define an evolved Gaussian, in-
tions that extend out to ± infinity. What alters with cluding the center, width, and phase/normalization fac-
propagation time is the density of solutions. As time tor in a global fashion. Its validity is constrained to cases
increases, double the propagation time is shown in the where the quadratically expanded potential around the
lower panel, the density increases. At t = 0, there is ef- central trajectory is a good approximation to the dy-
fectively only one saddle left with all the other saddles namics within the phase space volume defined by the
pushed out to infinity. Keep in mind that these exposed Wigner transform of the evolving wave packet. As the
saddles are only a small subset of all the saddles. There wave packet spreads out with increasing time, it soon ex-
are solutions everywhere throughout the complex q plane, tends well beyond the domain of validity for the quadratic
not just approximately along a line, which is where the expansion, which leads to the breakdown of the approx-
exposed saddles find themselves. imation.5
The validity is limited to pre-Ehrenfest time scales. 40
One method for determining this time scale in practice is t=τ
to consider two evolved densities. The first is the initial
wave packet’s Wigner transform evolved classically (nu- 20
merically), and the second is the Wigner transform of the p
LWPD evolved wave packet. Let the overlap integral of
these two densities at t = 0 be normalized to unity. Then 0
LWPD remains an accurate approximation until a time
such that the overlap drops precipitously from one. This
is because LWPD has only linearized dynamical infor- −20
mation about the central trajectory whereas the density
of initial conditions incorporates nonlinear dynamical ef-
−40
fects relative to the central trajectory. See Fig. 2 for an
−10 0 q 10
illustration. The one and three standard deviation (σ)
ellipses of LWPD appear straight and are superposed on
40
top of the 1, 3, 5 σ curves of the initial state’s propagated
Wigner transform. After one period of the central trajec-
tory of the density’s motion some information about the t=τ
20
wave packet central region may still be given correctly by
LWPD, but beyond the 1σ curve, the dynamical informa- p
tion is no longer accurate. This illustrates the breakdown 0
of LWPD at short time scales for nonlinear dynamical
systems.
−20
2. Off-center, real trajectory method
−40
−10 0 10
The important idea underlying this method is that for q
any nonlinear dynamical system over time a localized
density of initial conditions disperses and especially in FIG. 2. The evolving Wigner density associated with the wave
a bounded system, the dispersed evolved density must packet defined in the Fig. 1 caption and the LWPD approxi-
fold over or one might say create foliations. In some lo- mation. Initially an ellipse, the 1, 3, 5 standard deviation (σ)
cal region of phase space, these foliations slice through contours of the propagated Wigner density are given by the
(with increasing density as time increases), and each one curvy figure after just one period of the central trajectory’s
contains an infinite collection of nearly identically behav- motion. Superposed in the upper panel is the 1σ contour
of the LWPD approximation, and in the lower panel, the 3σ
ing trajectories; see the study of the production of whorls
contour. Only phase points within 1σ are still approximately
and tendrils in [12]. In other words, the stability matrix linearly related to the central trajectory by this propagation
of one member of the local set predicts rather well the be- time.
havior of its neighbors within the foliation. On the other
hand, each foliation has a completely different dynami-
cal character and represents a unique transport pathway.
By identifying a member from each foliation, which nec- can be used to generate a contribution to the quantity
essarily involves off-center initial conditions [those ini- of interest. Furthermore, each of these off-center, real
tial conditions within the Wigner density, but not equal reference trajectories can be used in a Newton-Raphson
to (qα , pα )], and following the essential prescription of search to locate a unique exposed saddle, which generates
LWPD, a wave is constructed whose information is lo- an important subset of the saddles of GGWPD [19].
cally correct. Summing over the contributions of all the It also lends itself to a prescription for identifying the
foliations incorporates the nonlinear dynamical features. phase space directions of fastest dispersion thereby indi-
This technique locally reconstructs the features of the cating the directions that are unnecessary for developing
evolving wave packet including quantum interference be- the foliations for the quantity of interest [20]. In this way,
tween transport pathways, and its validity goes well be- problems with large numbers of degrees of freedom can
yond the Ehrenfest time scale [21, 27, 29]. be reduced to vastly fewer numbers. This effective di-
The Wigner transformed initial density plays a more mensional reduction makes it possible to treat fully more
critical role for this method since it is essentially being complicated dynamical systems. Even if many degrees of
used to identify the foliations themselves as a function of freedom are active in this sense, they usually separate out
time. This is illustrated in Fig. 3 where each foliation can in time scales. Thus, one can successively add directions
be used to identify a single real reference trajectory about as time increases, but start with just one or two. For
which one can linearize the dynamics locally and that many problems, being able to go to intermediate time6
40 portant dynamical quantities such as position, momen-
t=τ tum and classical action are not multi-valued functions
of time. This is consistent with it possessing the Painlevé
20 property in which solutions of a differential equation have
p no movable branch points. In this sense, for the pure
quartic oscillator the singularities are isolated poles in a
0 complex propagation time space.
On the other hand, these singularities must come in
continuous families. Consider a trajectory which be-
−20 comes singular exactly at a propagation time t0 . Dif-
ferentially nearby will be other trajectories that become
−40 singular at real propagation times of t0 ± . By following
−10 −5 0 5 10 the changing initial conditions for a singular trajectory
q continuously as the propagation time varies from t = 0
40 to t = ∞, they can be associated to a continuously shift-
9 t = 3τ ing set of initial conditions. This is illustrated for the
quartic oscillator in Fig. 4. For the real and imaginary
20 7 parts of positions on the initial manifold, the real part
of the final position is contoured and shown as a den-
p 5 sity plot. The blank lines correspond to sets of initial
0 conditions for trajectories that become singular at any
4 time up to the final propagation time of three periods
of the motion for the initial conditions (q, p) = (0, 20).
−20 6 All the blank lines appear to be emanating from one of
two limiting points and converging towards that partic-
8 ular point as longer propagation times are considered.
−40 However, the precise structure of the dynamics for q in
−10 −5 0 5 10 the neighborhood of (0, 0.625) is quite complicated and
q
difficult to discern precisely. Modulo the difficulties in
the immediate neighborhood of this point for long times,
FIG. 3. The relationship between foliations developing in the
dynamics and exposed saddle points for the x = 0 point of this Lagrangian manifold for the initial coherent state
the propagated wave packet defined in the Fig. 1 caption. Its defined in the Fig. 1 caption can be partitioned into an
Wigner density is shown propagated for one and three periods infinity of subregions by delineating the boundary of each
of the central trajectory’s motion. The real parts of the final region with the lines of singular trajectory initial condi-
position and momentum variables of the exposed saddle tra- tions. Within any given subregion, the initial conditions
jectories have a one-to-one correspondence to the foliations can be varied continuously without encountering a singu-
of the Wigner density generated by the dynamics. The folia- lar trajectory. In the next section, these subregions play
tions are enumerated (encircled integers) by switching labels an important role.
at the turning point caustics and form a complete set of those
foliations associated with initial conditions within 5σ of the
initial wave packet centroid. Due to lack of space the first
three foliations are not labeled in the central region. G. Features and accuracy of the semiclassical
approximation
ranges beyond the Ehrenfest scale is sufficient [15]. Even though semiclassical methods seem to be applied
most often to make qualitative physical arguments, it
is important to point out that their full implementation
F. Complications due to complexifying the classical in detail leads to very quantitatively accurate approxi-
dynamics mations for systems in the appropriate limits, e.g. short
wavelength or large particle number limits. For exam-
There are a number of complications arising if classi- ple, coherent state propagations in Bose-Hubbard lattices
cal dynamics are analytically continued to complex phase with many sites are accurately reproduced well past the
space variables. One of the most significant consider- Ehrenfest time scale [15].
ations is the existence of singular orbits, i.e. orbits that The accuracy of the semiclassical approximation is il-
acquire infinite momenta in finite propagation times [10]. lustrated in Fig. 5 for the much simpler quartic oscillator
If time is also allowed to be complex so that a path can where it is possible to compare the full propagated wave
be defined to avoid the singular real time point, there function with its GGWPD approximation. For this ex-
may exist multi-sheeted functions describing the dynam- ample it is easier to ensure that all the physically relevant
ics [30, 31]. Numerically for the quartic oscillator, im- saddles have been identified, i.e. those associated with the7
1.0 0.50
0.25
Re φ(x,t)
0.00
0.8 −0.25
QM
−0.50 GGWPD
−10 −5 0 5 10
x
1
10
0.6 1
2 8 7 5 3
3 5
Im q0 Im S
4 0
2
0.4 9 6 4
−5
5 −10 −5 0 x 5 10
1000
100 9 8 5
0.2 Re S 4
6 10 7 6
3
1
8
2
8 1
4 −8 0
−4
0.0
0 −10 −5 0 5 10
0 7 x
−4
4
−8 8
FIG. 5. Comparison of the quantum propagation of a wave
−8 packet and the GGWPD approximation. The particular wave
8 packet illustrated is the one defined in the Fig. 1 caption prop-
−4 8
4 agated for three periods of the motion of its central trajectory.
-0.2 The foliation labels associated with the saddles are shown
-0.3 -0.2 -0.1 0.0 0.1 0.2 in Fig. 3. The lower two panels show the imaginary and
Re q 0
real parts of the classical actions, respectively. The dashed
and dotted lines indicate where saddles become hidden. The
FIG. 4. Density plot of the final position’s real part for initial dashed line indicates a saddle has crossed a Stokes line, which
conditions on the Lagrangian manifold of the wave packet de- is indicated here by a crossing of the real parts of two classical
fined in the Fig. 1 caption. The thin blank lines correspond actions, and must be excluded. The dotted lines indicate hid-
to initial conditions for trajectories which become singular den saddles, which can be included, and may be physically
before three periods of the central trajectory’s motion. The relevant if the imaginary parts of their actions are not too
initial conditions of the saddles as a function of the propa- large.
gated wave function’s position x appear as continuous lines
labelled by their foliation number. The line is solid where the
saddle is exposed, and dashed or dotted where hidden. The
dashes indicate the saddle crossed a Stokes line and no longer 9 foliations seen in Fig. 3, and that saddles on the wrong
is included. The transition from exposed to hidden occurs at side of Stokes lines are excluded. Their initial conditions
the avoided crossing of two foliations, which occurs near the are shown along with their foliation label in Fig. 4. In
classical turning point. The avoided crossing also marks the Fig. 5, the real and imaginary parts of the contributing
transition in how quickly the real and imaginary parts of the saddles’ classical actions are shown as a function of po-
classical action are changing as a function of x, respectively sition and labeled by their respective associated foliation
below the comparison; note that the foliation label re-
mains valid beyond the turning point caustic positions8
where exposed saddles become hidden. QM
0.4
It turns out that all of the physically relevant exposed GGWPD
and hidden saddles reside within the central vertical sub- 0.2
region of Fig. 4; in fact, the set of saddles associated with Re φ(x,t)
the nine foliations are all within the part of the central 0.0
subregion below the structure near Imag(q) ∼ 0.6. Let’s
call it the classical zone. A curious feature of the classical −0.2
zone boundary is that each point on it can be thought
t = 3τ
of as a limiting or accumulation point of an infinite se- −0.4
quence of points from an infinity of lines passing by get-
ting ever closer to the boundary. At least for the quartic −0.6
oscillators, the saddles tend to be well away from these −10 −8 x −6 −4
complicated boundary zones. Not a single physically rel-
10.0
evant saddle comes from any other subregion; we suspect
that this is true more generally than just for the quartic 7.5
oscillator. 8 7 5
Just for emphasis, all exposed saddles reside within 5.0
the classical zone and that means an infinity, even those Im S
contributing vastly too little to be physically relevant. 2.5
Since they are exposed though, it is straightforward to
implement a cutoff on their physical relevance based on 0.0
the weighting of the classical reference trajectories used
to locate them. This has the practical consequence of −2.5
9 6 4
eliminating almost all of the Lagrangian manifold from
−5.0
the search technique to be used to solve the two-point −10 −8 −6 −4
boundary value problem, and therefore is a great and x
essential simplification. 9
In the central region of the propagated wave function 1000
of Fig. 5, the exposed saddles dominate the contributions 8
of the hidden saddles. Even some of the exposed saddles 800
included are contributing with very small magnitudes. Re S
600
The exposed saddles are the ones whose imaginary parts
of the action are changing relatively slowly (real parts rel- 400 7
atively quickly) as a function of position. Also, the con-
tribution of the hidden saddles are imperceptible there, 200 6
i.e. the saddles on the correct side of Stokes lines and pos- 5
sessing the opposite behavior in their real and imaginary
0 4
parts as the exposed saddles’ actions. −10 −8 −6 −4
x
Away from the center, the number of exposed saddles
diminishes as various turning points in the real classi-
cal dynamics are surpassed. Some of the hidden saddles FIG. 6. Expanded view of the left side of Fig. 5. The propa-
gated wave function decreases quite a bit to the left of where
begin to get comparable in magnitude to the remaining
foliations ○6 and ○ 7 switch from exposed to hidden. Just
exposed saddles and even further out dominate the con- to the left of this switch, the hidden saddle ○’s7 contribu-
tributions to the wave function. The left side of the wave tion is just as significant as those of the exposed saddles ○
8
function is magnified to illustrate these points in Fig. 6. and ○,9 whereas ○ 6 has crossed a Stokes line and ceases to
The dots representing the GGWPD approximation only contribute. The semiclassical inaccuracies seen where ○ 6 and
deviate from the quantum mechanical results at just a ○9 cross Stokes lines can be healed (uniformized) following
few places where a formerly physically relevant saddle Berry’s prescription [32].
has crossed a Stokes line and must be abruptly thrown
away causing a discontinuity in the approximation. This
occurs here where two saddles nearly coalesce and the real
parts of their actions cross [10]. Unlike in ordinary WKB
where the coalescence is perfect at classical turning point improved upon following a uniformization prescription
caustics, for coherent states and wave packets generally due to Berry [32]. The upshot is that GGWPD carried
the real and imaginary parts of the classical actions do out in full is highly accurate for the entire propagated
not cross at the same point and in that sense these are wave function including the exponentially decaying tails,
just near coalescences that appear as avoided crossings. and even the most troublesome locations near caustics
Even these small inaccuracies and discontinuities can be can be improved through uniformization techniques.9
III. JUSTIFICATIONS FOR the Gaussian saddle integration gives the same re-
IMPLEMENTATION STRATEGIES AND sult as the Gaussian integration using the related
HIDDEN SADDLES neighboring real reference trajectory as an expan-
sion point and incorporating the non-vanishing lin-
The focus of this section is justifications of off-center, ear term. Adopting this definition, the off-center
real trajectory methods [22, 24], implementations of GG- real reference trajectory method is partly a poor
WPD [15, 17–19], and incorporating hidden saddles. man’s GGWPD in which one does not bother with
These techniques have used reasonable, physically mo- performing the Newton-Raphson search to find the
tivated, but heuristic arguments and numerical evidence true saddle point, but each contribution from the
to support their developments. However, certain con- Gaussian integrals expanded about a real trajec-
cepts such as the already mentioned idea of a complex tory gives essentially the same result as the Gaus-
phase space point being near [17] a real one are some- sian integral performed at the real trajectory’s as-
what vague or misleading. It is possible to add some sociated saddle point. It is important to note that
precision to such ideas. this does not imply that the complex saddle is near
a real point. It can have very large imaginary com-
ponents of position and momentum. If a real clas-
A. Nearness of complex to real trajectories sical transport pathway gives an extremely small
contribution, then the complex saddle necessarily
We have thought of three somewhat related possible has a significant imaginary part of the complex
avenues of adding precision to the concept of nearness classical action attenuating its contribution equiva-
between a complex and real trajectory. lently. It is also true that this necessarily excludes
hidden saddles from the method. Therefore, one
1. The first route is to insist that ”nearby” means a expects accurate approximations using the tech-
Newton-Raphson search beginning from a real ref- niques of [21, 22, 24] where the quantum dynamics
erence trajectory taken from some particular fo- is not dominated by hidden saddle processes even
liation generated by the dynamics converges to a though the true saddles are not being used.
unique complex saddle trajectory that can conse-
quently be associated with that foliation. Afterall, 3. A final third route for defining nearness begins
with some small technical modifications, this is how with the observation that even though the saddles
the exposed saddles are identified [15, 17, 19]. for wave packets generally do not coalesce exactly
where a caustic is encountered, there would typ-
2. A second route, which we find a bit more com- ically be a near coalescence. As parameters are
pelling, is related to how the saddle point approxi- varied, a saddle remains exposed until a caustic of
mation is carried out. It involves a locally quadratic some kind is encountered and it switches to hid-
expansion of some very complicated action function den status beyond the caustic. This would be ex-
about a point for which the linear term vanishes. pected to make the first route fail as well since
However, inside a domain in which this function is a Newton-Raphson search beginning from a real
behaving quadratically enough, an expansion about trajectory would no longer have a unique saddle
any point inside that domain is possible to describe towards which it would converge. As one might
the behavior of that function. For the action func- have expected, the behavior of the real and imag-
tions of dynamical systems these domains tend to inary parts of the saddle’s action function changes
be highly asymmetric. One direction in the do- character beyond a caustic. Whereas, the real part
main may be very "compressed", whereas another of the action is varying rapidly along the exposed
is quite "elongated", which, of course, complicates part of a foliation and the imaginary part relatively
the indeterminate idea of “near”. There are exposed slowly, the opposite occurs for the hidden saddle,
saddles with imaginary parts of their initial con- the imaginary part varies relatively rapidly, but not
ditions approaching infinity as well. Nevertheless, the real part. The saddle’s contribution switches
the quadratic expansion is used as an approxima- from a rapidly varying phase to a rapidly dimin-
tion locally for the purposes of integration (usually ishing magnitude as a function of the parameter.
with the limits extended to ±∞). The only distinc- In Figs. 5,6 it is clear where the transition between
tion between the two approaches is that, except for exposed and hidden occurs as the position is varied.
the expansion made exactly at the saddle point,
there exists a linear term. Incorporating the linear
term into the Gaussian integral being performed
would lead to the exact same value for a perfectly B. Incorporating hidden saddles
quadratic function, and gives nearly the same re-
sult if the function is sufficiently quadratic locally. The semiclassical methods of [15, 17, 19, 20] are de-
Therefore, a second definition of the complex sad- signed to locate the physically relevant subset of exposed
dle trajectory being near a real trajectory is that saddles. The question which remains is how to identify10
physically relevant hidden saddles in cases where they in this paper are problems in which part of the analysis
must be found. They are particularly relevant where ex- concerns the propagation of Gaussian wave packets or
ponentially decaying behavior is dominant, such as in coherent states in bosonic many-body systems. For non-
tunneling problems, but even in the simple quartic os- linear dynamical systems, the saddle point approxima-
cillator example, there are locations where hidden sad- tion leads to a two-point boundary value problem with
dles give contributions as large as those coming from the an infinity of solutions, almost all of which are physi-
most relevant exposed saddles, as mentioned previously. cally irrelevant. The practical problems with implement-
In such cases, they also need to be incorporated into the ing the theory only mount further as a system’s number
methods. of degrees of freedom increases or the dynamics become
The technique proposed here relies on the general fea- chaotic. Two techniques, i.e. off-center real trajectory
ture that the saddles do not coalesce exactly in the neigh- methods [22, 24] and using those resultant trajectories
borhood of caustics. The idea is to begin with a com- to identify physically relevant saddles [15, 17, 19, 20], are
plete set of exposed saddles, and then follow the com- placed on a firmer foundation by showing how they are
plex saddles continuously as a function of the natural related to dynamical structure that partitions the La-
parameters. Since the saddles do not collide precisely, grangian manifolds underlying wave packet and coherent
they can be followed through and beyond the caustic re- state propagation.
gions without ambiguity. As the parameters are changed For a simple one degree of freedom dynamical system,
slightly, the previous saddle is to be used as the input the pure quartic oscillator, the wave packet / coherent
for a Newton-Raphson search instead of some real tra- state related Lagrangian manifolds can be partitioned
jectory. If as a parameter shifts, a particular saddle into an infinite number of subregions by the lines of ini-
crosses the boundary between being exposed and hidden, tial conditions of singular trajectories. One region in par-
the Newton-Raphson search using real trajectory input ticular, here dubbed the classical zone, contains all the
would fail whereas a search using the neighboring com- exposed and hidden saddles discussed here. The exposed
plex saddle as input has no relevance to the concept of saddles can be placed in a one-to-one correspondence
nearness to real dynamics. Thus, there is no convergence with all the real classical transport pathways, and are
problem as one steps through the parameter variation. straightforwardly located with a Newton-Raphson search
This is illustrated in Fig. 4 where the initial conditions technique fed by a single reference trajectory from each
for saddles associated with the foliations over a range of transport pathway. A simple criterion based on the real
final position are shown. For each near approach of two reference trajectory’s initial conditions and final phase
exposed saddles, which is near a turning point caustic in point can be used to determine which saddles contribute
this case, the exposed saddles become hidden and one of sufficiently for them to be considered physically relevant.
them crosses a Stokes line. The curvature of the avoided It only involves the differences between the initial phase
crossing introduces no problems to a Newton-Raphson space centroid and the initial conditions, and the final
search using the neighboring saddle as a starting point. phase point and final centroid.
For the quartic oscillator this method identifies the The hidden saddles, where relevant, cannot be identi-
most physically relevant hidden saddles, they all remain fied this way. However, one can begin with the max-
within the classical zone, and they even maintain a la- imal set of exposed saddles (for the quartic oscillator
belling with respect to the real classical foliations. The this means setting the final position to zero) and vary
homogeneity of the quartic oscillator Hamiltonian makes the relevant parameters smoothly, here position. Follow-
it a particularly simple case. The idea of following ex- ing each exposed saddle continuously with final position
posed saddles through caustic regions is more general leads eventually to a regime where that saddle cannot be
than just the turning point caustic example encountered found with a Newton-Raphson search using a real trajec-
here and extends to systems with more degrees of free- tory as initial input. Nevertheless, one can follow each
dom, whatever their dynamics, integrable, chaotic or saddle as it moves from an exposed region into a hidden
mixed [15, 19, 20]. Nevertheless, the hidden saddles are one. Some of the hidden saddles found this way do cross
certainly a more complicated story in general as even Stokes lines and need to be thrown away, but that can be
the double well creates a number of foreseeable complica- evaluated by calculating their complex action functions.
tions. For foliations whose trajectories pass near a caus- For this simple dynamical example, all the physically rel-
tic near the barrier multiple times in its history, it will evant hidden saddles necessary to calculate the propagat-
be necessary to consider the crossing through the caustic ing wave packet / coherent state out in the exponentially
(or barrier) at each instance, leading to a much larger decreasing tail regions are from the classical zone and
multiplicity of hidden saddles. could be found through real parameter variation of final
position. This method would work in an identical way
if the quantity of interest were the overlap of the prop-
IV. CONCLUSIONS agated wave function with a final coherent state/ wave
packet. The parameter(s) varied would be the centroid
The semiclassical approximation can be extremely im- of the final wave packet.
portant in many physical contexts. Of particular interest The dynamics of the pure quartic oscillator are ex-11
tremely simple, not only because it is a one-degree-of- locate the saddles in the bald spots [33]. Thus, depending
freedom system, but also because the potential is homo- on the system, complex time may be unavoidable.
geneous. This gives rise to simple scaling relations be- Further complications may arise with the introduc-
tween trajectories on any pair of energy surfaces. More tion of multiple degrees of freedom, where the dynamics
general dynamical systems might be expected to give rise may include KAM tori, chaotic regions, Arnol’d diffu-
to further challenges not encountered in this simple ex- sion, etc... The off-center trajectory and exposed sad-
ample. Indeed, one issue that has been identified is the dle methods have been shown to work in higher dimen-
so-called bald spot problem [10, 33]. This arises when a sional systems. Nevertheless, it is not known whether the
system possesses movable branch points as a function of partitioning of the Lagrangian manifold found here has
initial conditions, which in turn may block the Newton- a straightforward multi-degree of freedom generalization
Raphson search scheme for exposed or hidden saddles be- nor is it known whether the appearance of chaotic dy-
yond some point as dynamical quantities are varied (such namics fundamentally alters the picture of a partitioned
as the position variable in the propagated wave function). Lagrangian manifold. One natural extension of the cur-
In these circumstances, complex time contours can be in- rent work would be to investigate the possibility of such
troduced in order to circumvent the branch points and higher dimensional partitionings.
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