Quantum breaking time near classical equilibrium points

4 Pages·0.18 MB·English

by  Fabrizio Cametti

Quantum breaking time near classical equilibrium points
Fabrizio Cametti1 and Carlo Presilla1,2,3
1Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, Piazzale A. Moro 2, Roma 00185, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1
3Istituto Nazionale per la Fisica della Materia, Unit`a di Roma 1 and Center for Statistical Mechanics and Complexity
(Dated: January 11, 2002)
By using numerical and semiclassical methods, we evaluate the quantum breaking, or Ehrenfest
timeforawavepacketlocalized aroundclassical equilibriumpointsofautonomousone-dimensional
systems with polynomial potentials. We find that the Ehrenfest time diverges logarithmically with
the inverse of the Planck constant whenever the equilibrium point is exponentially unstable. For
stableequilibrium points,wehaveapowerlaw divergencewith exponentdeterminedbythedegree
of thepotential nearthe equilibrium point.
2
0 PACSnumbers: 03.65.Sq,05.45.Mt,47.52.+j
0
2
n The question of estimating how long classical and towards the classical equilibrium point (p0,q0) the pe-
a quantum evolutions stay close is one of the main prob- riodofmotiondiverges,sothatthe evolutionofaphase-
J lems of semiclassical analysis. The evolution of a quan- space distribution function localized around this point

1 tumobservablecanfollowthatofthecorrespondingclas- must show a continuous frequency distribution around
3
sicaloneuptoafinitetime,thesocalledquantumbreak- ν = 0. On the other hand, in the quantum case, due to

ing, or Ehrenfest, time. As initially conjectured in [1, 2] the discrete nature of the spectrum, the frequency dis-
1
v and rigorously proved in [3, 4, 5], whenever the classical tribution is characterized by a gap between zero and a
7 flow is chaotic, the Ehrenfest time diverges logarithmi- minimal frequency. We call this minimal frequency the
4 callyin~. Thisresultiseasilyunderstood. Startingfrom Ehrenfest frequency, ν . In fact, its inverse, ν−1, is an
E E
1 aninitialvalue∆(~) ~/I,whereI isacharacteristicac- upper bound to the time at which the quantum-classical
1 ∼
tionofthesystem,thedifferencebetweenaclassicalflow, correspondenceofthe evolutionofanyobservablebreaks
0
2 with Lyapunov exponent λ > 0, and the corresponding down. We estimate the Ehrenfest time as νE−1.
0 quasi-periodic quantum flow increases as ∆(~)exp(λt). By using numerical and semiclassical methods, we
/ The two flows depart at t λ−1log(I/~). The situation study the behavior of ν (~) aroundclassicalequilibrium
h E
∼
p isdifferentforaregularclassicalflow. Inthis case,start- points,bothstableandunstable,forseveralautonomous
- ing from the work [6], it was suggested in [7] that the one-dimensional systems. We find that ν−1(~) diverges
nt Ehrenfest time grows algebraicallyas ~−δ. The determi- logarithmicallyfor~ 0whenevertheequEilibriumpoint
→
a nation of the value of δ and its possible universal nature is exponentially unstable. In all the other cases, the
u
is still an open problem. See [8] and references therein Ehrenfest time follows a power law with exponent re-
q
for recent results. lated to the degree of the potential near the equilibrium
:
v The ~-scaling of the Ehrenfest time is usually investi- point.
i
X gated for classical flows which are completely chaotic or In the following, we consider systems described by the
r regular. However,itisinterestingtostudythequantum- Hamiltonians
a
classical correspondence in systems having isolated un-
p2 q2α q2β
stable orbits embedded in a completely regular phase- H(p,q)= +A +B , (1)
2m 2α 2β
space. The simplest example is given by the ubiquitous
double-wellsystemdefinedbytheHamiltonianH(p,q)=
withA 0,B >0andβ >α 1. Byproperlyrescaling
p2 q2 + q4. For this system, there is only one unstable position≤, momentum and time≥, we can always reduce to
2 − 2 4
periodicorbit,namelythatassociatedtotheequilibrium the case B =1, m=1 and either A=0 or A= 1 [16].
−
point (p0,q0) = (0,0), with positive Lyapunov exponent For A = 0, we have single-well systems with a classical
λ=1. IsitpossibletohavealogarithmicEhrenfesttime stable equilibriumpoint (p ,q )=(0,0)atenergyε=0.
0 0
in proximity of an isolated exponentially unstable point A more interesting situation occurs for A = 1. In this
−
like (p0,q0)? case the systems are double-welloscillatorsand the clas-
The usualway of studying the Ehrenfesttime consists sicalequilibriumpoint(p ,q )=(0,0)atenergyε=0is
0 0
in comparing the evolution of classical observables with unstable. In the particular case α = 1, the equilibrium
the quantum expectation value of the corresponding op- point is exponentially unstable. In both cases, A = 0 or
erators,eitherinthe coherentstaterepresentation[3],or A = 1, the periodic orbits near the equilibrium point
−
intheframeworkofWeylquantization[4]. Inthepresent at ε=0 have a period which diverges for ε 0.
→
case, we follow a simpler approach based on the analy- On the quantum mechanical side, in order to repre-
sis of the quantum spectrum. We know that on going senta state localizednearthe classicalequilibriumpoint
2
100 ~=10−4 2 ε42 =0.0008148
)
q
10−5 (2 1
4
φ
10−10
) ~=10−8 0
(ν 100
P
10−5 −1
10−10 2
ε40 =−0.0054148
100 ~=10−12 (q)
0 1
4
10−5 φ
0
10−10
10−4 10−3 10−2 10−1 100
ν −1
−1.5 −1 −0.5 0 0.5 1 1.5
q
FIG.1: FouriertransformofthesurvivalprobabilityP(ν)for
different values of ~ in the case A = 0, β = 2. The dashed
line is the ~→0 limit distribution given by Eq. (14). FIG.2: Eigenfunctionsφ40andφ42correspondingtothemin-
imal frequency ν in the double-well case α = 1, β = 2 for
E
~ = 10−2. The dashed curve is the initial wavefunction (2)
with (p0,q0)=(0,0).
(p ,q ) we consider the following initial wavefunction
0 0
1 (q q )2 p q
q ψ(0) = exp − 0 exp i 0 . (2) By using semiclassical and numerical techniques, we
h | i (π~)41 (cid:20)− 2~ (cid:21) (cid:16) ~ (cid:17) nowshowthattheFouriertransformofthesurvivalprob-
ability,
The associated Wigner function,
∞ ∞
Wψ(p,q)= π1~exp(cid:20)−(p−~p0)2(cid:21)exp(cid:20)−(q−~q0)2(cid:21), (3) P(ν) = nX=0mX=0|cn|2|cm|2δ(ν−νnm), (8)
forsufficientlysmallvaluesof~ischaracterizedbyagap,
can be interpreted as a phase-space probability distri-
large with respect to the typical level spacing, between
bution centered around the point (p ,q ) and has the
0 0
ν = 0 and a frequency which we call the Ehrenfest fre-
property
quency, defined as
limW (p,q)=δ(p p )δ(q q ). (4)
ψ 0 0
~→0 − − νE = min νnm. (9)
n6=m
Intheseexpressions~istheadimensionalrescaledPlanck |cn|2|cm|26=0
constant, whichvanishes when, for instance, the mass m
In the simple case A = 0, by using standard WKB
of the system is taken larger and larger.
approximations,we have
Instead of considering the evolution of a specific ob-
servable, we study the simpler survival probability 2β
1 β+1
ε = n+ ~δ(β) , (10)
n
(t)= ψ(0)ψ(t) 2, (5) (cid:20)(cid:18) 2(cid:19) (cid:21)
P |h | i|
with
whichcontainsthesamegrossdynamicalinformation. In
the basis of the eigenstates of the Hamiltonian, Γ 1 3+ 1
π 2 β
δ(β)= h (cid:16) (cid:17)i , (11)
H|φni=εn|φni, n=0,1,2,... , (6) r2 Γ 1+ 21β (2β)21β
(cid:16) (cid:17)
the survival probability (t) can be written as
P and
∞ ∞
P(t) = nX=0mX=0|cn|2|cm|2exp(iνnmt), (7) |cε|2 = 2√πΓ((212+βββ))−21β+~12sεin−σ2(1βε,e~−;β2)~ε , (12)
Γ(1)Γ(1+ 1 ) σ(ε,~;β)
2 2β
where c = ψ(0)φ and ν =(ε ε )/(2π~). Note
k k nm n m
thatck =0fhorko|ddi,duetothesymm−etryofthesystem with σ(ε,~;β) = 2√2(2β)21β~−1εβ2+β1. The behavior of
and of the initial wavefunction. (ν)obtainedbyusingtheseexpressionsforε and c 2
P n | εn|
3
100 ~=10−2
100
10−5
10−10
10−5 ) ~=10−3
2|n P(ν 100
|c 10−5
10−10
10−10 ~=10−4
100
10−5
10−10
10−15 10−2 10−1 100
−20 −10 0 10 20 ν
ε /~
n
FIG.4: FouriertransformofthesurvivalprobabilityP(ν)for
FIG. 3: Superposition coefficients |c |2 as a function of ε /~ different values of ~ in thedouble-well case α=1, β=2.
n n
for~=10−3inthedouble-wellcasesα=1,β=2(×),α=2,
β =4 (2), and α=3, β=6 (3).
Fortheassociatedeigenfunctionsonlymicrolocalexpres-
sions are available [10] which do not allow for a direct
isshowninFig.1inthecaseβ =2. Weseethatfor~ 0
→ determination of the superposition coefficients cε 2. For
the frequencydistribution (ν) approachesacontinuous | |
P this reason, in all cases α 1 we determine numerically
limit given by ≥
the eigenvalues and eigenfunctions of the system. With
standardnumericaltechniques, this representsanunsur-
ε ε
1 2
P0(ν)=~li→m0Z dε1dε2 p(ε1)p(ε2)δ(cid:18)ν− 2−π~ (cid:19),(13) mountable task since the interesting eigenstates, namely
those close to energy ε = 0, have a quantum number n
where p(ε) = c 2dn and n(ε) is obtained by inverting which diverges quickly for ~ 0. We bypass the prob-
| ε| dε lem by using the algorithm [→12] which allows to evalu-
ε=ε . By using (10) and (12), we find
n
ate selected eigenstates having a very large number of
(ν)=4K (4π ν ), (14) nodes. In Fig. 2 we show, as an example, the couple of
0 0
P | |
even eigenfunctions with energy closest to ε = 0 in the
where K0 is the Bessel function of zero-th order. Fig- double-well case α = 1, β = 2 evaluated for ~ = 10−2.
ure 1 alsoshows the presence of the gapatν =0 andits Note that, already for this still relatively large value of
shrinking as ~ 0. Since the level spacingεn+1 εn in- ~, the corresponding quantum number is n 40. In our
→ − ∼
creasesbyincreasingn,theEhrenfestfrequency(9)turns numerical calculations we go beyond n 104.
out to be νE = (ε2 ε0)/(2π~). According to (10), its In Fig. 3 we show the superposition∼coefficients eval-
−
inverse diverges as uated for different double-well systems for ~ = 10−3.
We seethat c 2 decreasesexponentiallydepartingfrom
νE−1 ∼~11−+ββ. (15) ε = 0. For |smna|ller values of ~, the superposition coef-
ficients c 2 follow approximately the same exponential
n
We now consider double-well systems, i. e., the case behavior| as| a function of ε /~ and become denser and
n
A = 1. For these systems, the standard WKB ap- | |
− denser.
proximation fails near the unstable equilibrium point at
TheFouriertransformofthesurvivalprobability(8)is
energy ε = 0. Only in the particular case α = 1, a reg-
determined by using the eigenvalues and the superposi-
ularizedsemiclassicalapproximationhas been developed
tion coefficients obtained numerically. In Fig. 4 we show
[9, 10, 11] and the quantization condition for the energy
(ν) in the case α = 1, β = 2 for different values of
levels reads ~P. As in the single-well case, at ν = 0 we have a gap
1 whose width shrinks as ~ 0. The width of this gap,
=cos(φ(ε,~)), (16) →
namelytheEhrenfestfrequency,isyieldedbyacoupleof
1+exp2πε
~ evenconsecutiveeigenvalues,closetotheenergyε=0of
q
the classical equilibrium point. This can be understood
where
roughly in the following way. Consider the number of
4 ε ~ 1 ε states, ,intheenergyrange[ε ~,ε+~]. Thefrequen-
φ(ε,~)= log argΓ +i π. (17) Nε −
3~ − ~ 16 − (cid:18)2 ~(cid:19)− ciesassociatedtotheeigenvaluesinthisenergyrangeare
4
spectruminturncoincidewiththe energiesofthe bound
states of the corresponding confining inverted potential.
ν−1 ∼~−1/2
100 E In conclusion, we have shownthat the presence of iso-
lated exponentially unstable orbits is sufficient to break
thequantum-classicalcorrespondenceatatimescalelog-
ν−1 ∼~−1/3 arithmicin~−1. Thisfeaturemayberelevantinallmeso-
E
1
−E scopic systems which are modeled by one-dimensional
ν
multi-well Hamiltonians [13, 14]. In these systems the
50
Ehrenfest time behavior is related to experimentally de-
ν−1 ∼log~−1 tectable properties as the classical to quantum crossover
E
of the shot noise [15].
WewouldliketothankThierryPaulforverystimulat-
0 ingdiscussions. Thisresearchwaspartiallysupportedby
10−9 10−7 10−5 10−3 10−1
Cofinanziamento MURST protocollo MM02263577 001.
~
−1
FIG.5: InverseoftheEhrenfestfrequency,ν ,asafunction
E
of ~ in the double-well cases α=1, β =2 (×), α=2, β =4
(2), and α= 3, β = 6 (3). The solid line is the regularized
WKBpredictionbasedon(16-17),whilethedashedanddot-
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This is the same scaling law which we would obtain, as
[16] For A = 0, the physical energy and the Planck con-
described by Eq. (15), in the case of a single-well poten-
stant are given in terms of the corresponding rescaled
tialV(q)=q2α/(2α). Thisfactcanbeunderstoodinthe − β 1 2β
following way. For ~ 0, the discrete eigenvalues of the quantitiesviathesubstitution ε→εm β−1Bβ−1τβ−1,
→ ~ → ~ m−β−β1Bβ−11τββ−+11, where τ is an arbi-
double-wellaboveε=0correspondtotheenergiesofthe
trary time scale unit. In the double-well case, A <
continuous spectrum of the barrier q2α/(2α) at which
the transmission coefficient is maxim−um. According to 0, we have ε → ε (−A)−β−βαBβ−αα and ~ →
WKBapproximation,theseresonancesofthecontinuous ~ m−12(−A)−2(ββ+−1α)B2(αβ+−1α).