Table Of ContentSemiclassical propagator of the Wigner function
Thomas Dittrich,1 Carlos Viviescas,2 and Luis Sandoval1
1Departamento de F´ısica, Universidad Nacional, Bogot´a D.C., Colombia,
2Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Straße 38, 01187 Dresden, Germany
(Dated: February 1, 2008)
Propagation of the Wigner function is studied on two levels of semiclassical propagation, one
based on the van-Vleck propagator, the other on phase-space path integration. Leading quantum
corrections to the classical Liouville propagator take the form of a time-dependent quantum spot.
Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic.
It can be interpreted as the result of interference of a pair of classical trajectories, indicating how
quantum coherences are to be propagated semiclassically in phase space. The phase-space path-
integral approach allows for a finerresolution of thequantumspot in terms of Airy functions.
6
0 PACSnumbers: 03.65.Sq,31.15.Gy, 31.15.Kb
0
2
Quantum propagationin phase space has always been diagonal elements of the density matrix in the relevant
n
in the shadow of propagation in conventional (position, representation,cangiverisetoacompletefailureofsemi-
a
momentum)representations. Yetitissuperiorinvarious classical propagation of the Wigner function. Quantum
J
respects,particularlyinthesemiclassicalrealm: Itavoids coherences are reflected in the Wigner function as “sub-
4
2 all problems owing to projection,such as singularities at Planckian” oscillations [10]. They plague semiclassical
caustics. Canonical invariance of all classical quantities approximations by their small scale and by propagating
2 involved is manifest. Boundary conditions are imposed alongpathsthatcandeviatebyanydegreefromclassical
v consistentlyatasingle(initialorfinal)time,thusremov- trajectories.
7 ing the so-called root-search problem and allowing for In this Letter, we point out how Heller’s objections
5
initial-value representations. Semiclassical approxima- are resolved by considering pairs of trajectories as ba-
0
8 tions to the quantum-mechanical propagator have pre- sis of semiclassical approximations, and present corre-
0 dominantly been seeked in the form of coherent-state sponding expressions for the propagator of the Wigner
5 pathintegrals[1,2,3]. CloselyrelatedareHeller’sGaus- function. The concept of trajectory pairs has been in-
0 sianwavepacketdynamics[4]anditsnumerousmodifica- troduced in the present context by Rios and Ozorio de
h/ tions. Bynow,abroadchoiceofphase-spacepropagation Almeida [11, 12], albeit working in a strongly restricted
p schemesis availablewhichscoreverywellif comparedto space of semiclassical Wigner functions. We here give
- other semiclassical techniques. a general derivation of the propagator, independently of
t
n Almost all of these developments refer to the prop- any initial or final states.
a agation of wavefunctions in some Hilbert space. Less Moreover, we go beyond the level of approximations
u
attention has been paid to the propagation of Wigner based on stationary phase. Employing a phase-space
q
and Husimi functions. They live in projective Hilbert path-integraltechnique, we construct an improved semi-
:
v space, i.e., represent the density operator and are bilin- classical Wigner propagator in terms of Airy functions.
Xi ear in the wavefunction. Besides their popularity, they It resolvesallsingularities andcontains the semiclassical
have a crucial virtue in common: An extension to non- approximations based on trajectory pairs as a limiting
r
a unitary time evolution is immediate. This opens access case. The interference patterns we obtain depend, up
to a host of applications that combine complex quan- to scaling, only on the nature of the underlying classi-
tum dynamics, where a phase-space representationfacil- calphase-spaceflow—ellipticvs.hyperbolic—andinthis
itates the comparison to the corresponding classical mo- sense are universal. While living in projective Hilbert
tion, with decoherence or dissipation: quantum optics space, this result is superior to Gaussian wave-packet
and quantum chemistry, nanosystems in biophysics and propagation in that it allows Gaussians to evolve into
electronics, quantum measurement and computation. non-Gaussians.
By the scales involved, many of them call for semi- In order to fix units and notations, define the
classical approximations. However, only few such stud- Wigner function corresponding to a density operator ρˆ
ies exist, for specific systems predominantly in quantum as W(r) = dfq′exp(−ıp·q′/¯h)hq+q′/2|ρˆ|q−q′/2i
chaos [5], including dissipative systems [6, 7]. By con- where r = R(p,q) is a vector in 2f-dimensional phase
trast, Ref. [8] discusses a new method, Wigner-function space. Its time evolution is generated by a Hamiltonian
propagationanalogoustothesolutionofclassicalFokker- Hˆ(pˆ,qˆ)throughthe equationofmotion(∂/∂t)W(r,t)=
Planck equations. {H(r),W(r,t)} , involvingthe WeylsymbolH(r)of
Moyal
As a major challenge, any attempt to directly prop- the Hamiltonian Hˆ. The Moyal bracket {.,.} [13]
Moyal
agate Wigner functions requires an appropriate treat- converges to the Poisson bracket for h¯ → 0. As this
ment of quantum coherences. As early as 1976, Heller equationof motionis linear, the evolutionofthe Wigner
[9]arguedthat the “dangerouscrossterms”,i.e., the off- function over a finite time can be expressed as an inte-
2
gral kernel, W(r′′,t′′) = d2fr′G(r′′,t′′;r′,t′)W(r′,t′), (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)r’j+’
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
defining the Wigner propaRgator G(r′′,t′′;r′,t′). For au- (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)r’’=r’j’
tonomousHamiltonians,itinducesaone-dimensionaldy- (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)A (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) rcl(rj’,t)
namical group parameterized by t=t′′−t′ (in what fol- (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)j+(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)r’j−’
lows, we restrict ourselves to this case and use t as the (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)A(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
r’ j
only time argument). This implies, in particular, the j+ (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)A(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)j−(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
initial condition G(r′′,r′;0) = δ(r′′−r′) and the com- r’=r’j (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
position (Chapman-Kolmogorov) equation G(r′′,r′,t) = (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
d2frG(r′′,r,t−t′)G(r,r′,t′). (cid:0)(cid:1)r’j−(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
R The Wigner propagator can be expressed in terms of
the Weyl transform of the unitary time-evolution op- FIG.1: Thereducedaction(shaded)oftheWignerpropaga-
erator U(r,t) = dfq′exp(−ıp·q′/¯h) hq+q′/2| Uˆ(t) torinvan-Vleckapproximation,Eq.(5),isthesymplecticarea
|q−q′/2i, calledRWeyl propagator,as a convolution, etwncolotsreadnsbveetrwseeevnetchtoertswro′j+cla−ssric′j−al tarnadjerct′j′+ori−esrr′j′j−±((ts)chaenmdatthice
G(r′′,r′,t)=Z d2fRe−h¯ı(r′′−r′)∧RU∗(˜r−,t)U(˜r+,t), (1) drcrla(wr′i,ntg)),.theTbhreokfeunlllicneentisratlhelinperoipsagthateiocnlapssaitchal¯rjt(rra′j,etc)t.ory
with ˜r ≡(r′+r′′±R)/2. It serves as a suitable start-
±
ing point for a semiclassical approximation, invoking an
with¯r (t)≡(r (t)+r (t))/2,R (t)≡r (t)−r (t),
j j− j+ j j+ j−
expressionfortheWeylpropagatorsemiclassicallyequiv-
and S ≡ A (˜r ,t)−H (˜r ,t)t. The reduced ac-
j± j± j± j± j±
alent to the van-Vleck approximation[14, 15], tion A = tds ¯r˙ (s)∧R (s) is the symplectic area en-
j 0 j j
U(r,t)=2f exp(ıSj(r,t)/¯h−ıµjπ/2). (2) closed betwReen the two trajectory sections and the vec-
Xj |det(Mj(r,t)+I)| tEoqr.s(r4′j)+, a−s ar′jf−unacntidonr′jo′+f r−′′,r′jd′−esc(rFiibgess.a1,d2ibs)t.ribIuntigoennethraalt,
p
The sum includes all classical trajectories j connecting extends from the classical trajectory into the surround-
phase-space points r′, r′′ in time t such that r = ˜r ≡ ingphasespace,forminga“quantumspot”(Fig.3b)with
j j j
(r′ +r′′)/2. M is the corresponding stability matrix, a characteristic oscillatory pattern that results from the
j j j
µ its Maslov index. The action S (r ,t) = A (r ,t)− interference of the contributing classical trajectories. In
j j j j j
H(r ,t)t,withA ,thesymplecticareaenclosedbetween general, it fills only a sector with opening angle < 2π
j j
thetrajectoryandthestraightline(chord)connectingr′ (Fig. 2c), where the sum contains two trajectory pairs
j
and r′′ [14] (hashed areas A in Fig. 1). (fourstationarypoints). Outsidethis“illuminatedarea”,
j j±
Substituting Eq. (2) in Eq. (1) leads to a sum over stationarity cannot be fulfilled, that is, the “shadow re-
pairs j , j , of trajectories whose respective chord cen- gion” is not accessible even for mean paths ¯r (t). The
+ − j
ters ˜r are separated by the integration variable R. border is formed by phase-space caustics along which
j±
Otherwise, the two trajectories are unrelated. A cou- there is exactly one solution (two stationary points). As
pling between them, as expected on classical grounds, r′′ approaches the classical trajectory r (r′,t) starting
cl
comesaboutonlyuponevaluatingtheR-integralbysta- at r′, from the illuminated sector, the two solutions j−,
tionary phase. Stationary points are given by r′′−r′ = j+ coalesce so that M → M , and Eq. (4) becomes
j− j+
(r′′ −r′ +r′′ −r′ )/2. Together with the conditions singular. If the potential is purely harmonic, all mean
j− j− j+ j+
for the two chords,r′+r′′±R=r′ +r′′ , this implies paths coincide with r (r′,t), and the classical Liouville
j± j± cl
propagator,G(r′′,r′,t)=δ(r′′−r′′(r′,t)),isretained. In
r′ =¯r′ ≡(r′ +r′ )/2, r′′ =¯r′′ ≡(r′′ +r′′ )/2. (3) cl
j− j+ j− j+ all other cases, Eq. (4), though based on the van-Vleck
Stationarypoints are thus givenby pairsofclassicaltra- propagator, reflects the structure of stationary points of
jectories such that the initial (final) argument of the the action including third-order terms, with one pair of
propagatoris in the middle between their respective ini- extremaandonepairofsaddlepoints. Itisformulatedin
tial (final) points (Fig. 2b). This does not require these terms of canonically invariant quantities related to clas-
trajectoriestobeidentical! Theydocoincideaslongasr′ sical trajectories and thus generalizes immediately to an
andr′′ areonthe sameclassicaltrajectory,butbifurcate arbitrary number of degrees of freedom. The propaga-
as r′′ moves off the classical trajectory r (r′,t) starting tion of Wigner functions defined semiclassically in terms
cl
at r′, if the dynamics is not harmonic. of Lagrangianmanifolds [11] is contained in Eq. (4) as a
TheresultingsemiclassicalapproximationfortheWig- special case.
ner propagator is (dot indicating time derivative) WearenowabletoresolveHeller’sobjections[9]: Ifthe
twotrajectoriesj−,j+aresufficientlyseparatedandthe
G(r′′,r′,t) = 4f 2cos(Sj(r′′,r′,t)/¯h−fπ/2),(4) potential is sufficiently nonlinear, then (i), the propaga-
hf |det(M −M )| tionpath¯r(r′,t) candiffer arbitrarilyfromr (r′,t), and
Xj j+ j− cl
p (ii), the phase factor in (4) exhibits sub-Planckian oscil-
S (r′′,r′,t) = (˜r −˜r )∧(r′′−r′)+S −S
j j+ j− j+ j− lations. They would couple resonantly to corresponding
t featuresin the initial Wigner function, generatingasim-
= ds¯r˙ (s)∧R (s)−H (r )+H (r ) ,(5)
Z j j j+ j+ j− j− ilarpatternin the finalWignerfunctionaroundthe end-
0 (cid:2) (cid:3)
3
Two paths, r(t) and R(t), have to be integrated over.
The former is subject to boundary conditions r(0) = r′
and r(t)=r′′, the latter is free. The path action is
t
S({r},{R})= ds r˙(s)∧R(s)
Z
0 (cid:2)
+H (r(s)+R(s)/2)−H (r(s)−R(s)/2) . (7)
W W
(cid:3)
Equation (4) is recovered upon evaluating the path-
integral representation in stationary-phase approxima-
tion: Defining r (t) ≡ r(t) ± R(t)/2, with boundary
±
conditions analogous to Eq. (3), and requiring station-
FIG.2: ClassicalbuildingblocksenteringtheWignerpropa-
arityleadstotheHamiltonequationofmotionforr (t):
gator according to Eqs. (4,5), for a stable trajectory starting ±
at r′ = (0.636,0) near the minimum of the cubic potential We againfind pairs of classicaltrajectoriesthat straddle
V(q) = 0.329q3 −0.69q (panel a). (b) Classical trajectory the propagationpath as stationary solutions.
rcl(r′,t) (black line), a pair of auxiliary trajectories rj±(t) We will now include cubic terms in the action, with
(greenlines)andcorrespondingpropagationpath¯rj(r′,t)(red respect to variations of the path variables. To keep
dashed line). The yellow target pattern is the grid of aux- technicalities at a minimum, we restrict ourselves from
iliary initial points r′j± around r′, parameterized by polar now on to a single degree of freedom and to Hamiltoni-
coordinates. Propagated classically over time t, it deforms ans of the standard form H(p,q) = T(p)+V(q), where
into the turquoise pattern around r′′. The red/blue cone is T(p)=p2/2mwhilethepotentialV(q)maycontainnon-
formed by midpoints ¯r′j′ = (r′j′−+r′j′+)/2 that correspond to linearities of arbitrary order. With this form, H (r)
W
extrema/saddles of the action. Its boundaries form caustics
separating the region accessed by two midpoints¯r′′ from the coincides with the Hamiltonian function “quantized” by
unaccessible rest. (c) Enlargement of the area around r′′, merely replacing operators with classical variables. As
the path integral readily allows to treat time-dependent
indicatingthenumberoftrajectorypairsthataccesseachre-
gion. potentials,chaoticclassicalmotionremainswithinreach.
Expanding the action (7) around r(t) = r (r′,t) and
cl
R(t) ≡ (P(t),Q(t)) = 0, there remain only linear terms
inP andlinearandcubic termsinQ. EvaluatingtheR-
point of the non-classical propagationpath. In this way,
sectorofthepathintegralthusresultsinanAiryspread-
quantum coherences are faithfully propagated within a
ing of the propagator, with a rate ∼ V′′′(q (t)), in the
semiclassical approach. cl
p-direction. It is superposed to the classical phase-space
Equations (4,5) translate into a straightforward algo-
flow around the trajectory, i.e., rotation (shear) if it is
rithm for the numerical calculation of the propagator
elliptic (hyperbolic). As aconsequence,aspotofthe full
(Fig.2): (i)Definealocalgrid(e.g.,inpolarcoordinates)
phase-space dimension develops. Scaling ρ = (η,ξ) =
around the initial argument r′ of the propagator,identi-
(µ1/4p,µ−1/4q), with µ = T′′(p )/V′′(q ), we express
fying pairs of auxiliary initial points r′ , r′ with r′ in cl cl
j− j+ the linearized classical motion as a dimensionless map,
their middle. (ii) Propagate trajectory pairs r (t) clas-
j±
sically, keeping track of the symplectic area Aj between cos φ(t) −sin φ(t)
them. (iii) Findthe amplitude andphasecontributedby M(φ(t))=(cid:20) sin φ(t) cos φ(t) (cid:21). (8)
eachtrajectorypair andassociatethemto the finalmid-
points ¯r′′. They constitute a deformed cone, projected These maps form a group parameterized by the angle
j
ontophasespace(Fig.2c). Its“lower”(“upper”)surface φ(t) = tds T′′(pcl(s))V′′(qcl(s)). It is real (imagi-
0
(red (blue) in Fig. 2c) corresponds to pairs of extrema nary) ifRthe linpearized dynamics is elliptic (hyperbolic).
(saddles) of the action, respectively: (iv) Superpose the This allows to evaluate also the r-sector of the path
contributionsofthe twosurfaces,aftersmoothingampli- integral. Transforming the Wigner function to Fourier
tude and phase over midpoints ¯r′j′ within each of them. phasespace,W(γ)≡(FW)(γ)=(2π)−1 d2rexp(−ıγ∧
ThecausticsinEq.(4)resultfromapplyingstationary- r) W(r), and the propagator accordinglyR, G˜ = FGF−1,
f
phaseintegrationinasituationwherepairsofstationary we obtain (γ′ ≡(α′,β′))
points can come arbitrarily close to one another. Since
the underlying van-Vleck propagator admits only up to G(γ′′,γ′,t)=δ(γ′′−M(φ′′)γ′)
quadratic terms in the phase, we seek a superior ap- a a
proach, corresponding to a uniform approximation. It eexp −ı( 30α′3+a21α′2β′+a12α′β′2+ 03β′3) . (9)
3 3
(cid:16) (cid:17)
is available in the form of a path-integral representation
of the Wigner propagator [16], in close analogy to the Thecoefficientsa = tdsσ(s)(sinφ(s))j(cosφ(s))k de-
jk 0
Feynman path integral, pend on where along tRhe classical trajectory how much
quantum spreading ∼ σ(t) = (µ(t))3/4¯h2V′′′(q (t))/8 is
cl
1 pickedupandthusonthespecificsystemandinitialcon-
G(r′′,r′,t)= Dr DRe−ıS({r},{R})/h¯. (6)
hf Z Z ditions. The Fourier transform from Eq. (9) back to the
4
correspondingexactquantum-mechanicalresult(Fig.3a)
for the quantum spot, obtained by expanding the prop-
agator in energy eigenstates [18], the path-integral solu-
tionresolvesthecausticsfarbetterthanEq.(4)(Fig.3c).
The hyperbolic case is obtained replacing trigonometric
by the corresponding hyperbolic functions. As a result,
along unstable trajectories there are no periodic recur-
rences as in the elliptic case; the spot continues expand-
ingintheunstableandcontractinginthestabledirection
(Fig. 3d). Isolated unstable periodic orbits embedded in
a chaotic region of phase space exhibit a degeneracy of
the Weyl propagator [14]. It allows to account for scar-
ring in terms of the Wigner propagator [17].
We have obtained a consistent picture of incipient
quantum effects in the Wigner propagator, both in the
van-Vleck approach and in the path-integral formalism:
(i) for anharmonic potentials, the delta function on the
FIG. 3: Quantum spot replacing the classical delta function
on a stable (elliptic) trajectory near the minimum of a cubic classicaltrajectoryisreplacedbyaquantumspotextend-
potentialasshowninFig.2a,att=1.8(φ≈2π/3). Panel(a) ing into phase space, (ii) its structure shows a marked
shows the exact quantum result for the Wigner propagator, time dependence, qualitatively different for elliptic and
(b)and(c)aresemiclassicalapproximationsbasedonEqs.(4) hyperbolic dynamics, (iii) it exhibits interference fringes
and (9), respectively, all for ¯h = 0.01. Frames coincide with arising as a product of Airy functions, (iv) it can be ex-
thatofFig.2c. (d)Quantumspot,accordingtoEq.(4),foran pressed in terms of canonically invariant quantities as-
unstable classical trajectory near the maximum of the same
sociated to pairs of underlying classical trajectories, (v)
potential, at t = 1.0. Crosses mark the classical trajectory.
withineachlevelofsemiclassicalapproximationused,the
Colour code ranges from red (negative) to blue(positive).
propagator retains its dynamical-group structure. Open
issues include: extension to higher dimensions and to
higher-ordertermsintheaction,performanceinthepres-
original Wigner propagator can be done analytically, af-
ence of tunneling, application to unstable periodic or-
ter transformingthe third-orderpolynomialinthe phase
bits andimplications for scars,trace formulae,andspec-
to a normal form [17].
tral statistics, regularizationof the ballistic nonlinear σ-
The internal structure and the time evolution of the
model, semiclassical propagation of entanglement, and
quantum spot described by Eq. (9) are qualitatively dif-
generalizationto non-unitary time evolution.
ferent for elliptic and hyperbolic classical trajectories
(real and imaginary φ, resp.). In the elliptic case, the
spotisaperiodicfunctionofφ. Inparticular,itcollapses We enjoyed discussions with S. Fishman, F. Groß-
approximately to a point whenever φ = 2lπ, l integer. mann,F.Haake,H.J.Korsch,A.M.OzoriodeAlmeida,
Close to these nodes, it shrinks and grows again along a H. Schanz, K. Scho¨nhammer, B. Segev, T. H. Seligman,
straightlineinthep-direction,reflectingthefactthatfor M. Sieber, and U. Smilansky. Financial support by Col-
shorttime,thequantumAiryspreading∼t1/3 outweighs ciencias, U. Nal. de Colombia, VolkswagenStiftung, and
the classical rotation ∼ t. Only sufficiently far from the Fundaci´onMazdaisgratefullyacknowledged. TDthanks
nodes, while rotating around the trajectory by φ(t)/2, forthehospitalityextendedtohimbyCIC(Cuernavaca),
theone-d.distributionfansoutintoatwo-d.interference Max Planck Institutes in Dresden and G¨ottingen, Inst.
pattern formed as the overlap of the bright (oscillatory) Theor. Phys. at Technion (Haifa), Ben-Gurion U. of the
sides of two Airy functions, with a sharp maximum on Negev (Beer-Sheva), U. of Technology Kaiserslautern,
theclassicaltrajectory(Fig.3b). Incomparisonwiththe and Weizmann Inst. of Sci. (Rehovot).
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