Identifying wave packet fractional revivals by means of information entropy

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by  E. Romera F. de los Santos

Identifying wave packet fractional revivals by means of information entropy
E. Romera1 and F. de los Santos1
1
Instituto Carlos I de F´ısica Te´orica y Computacional,
Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain
(Dated: February 2, 2008)
Wave packet fractional revivals is a relevant feature in the long time scale evolution of a wide
range of physical systems, including atoms, molecules and nonlinear systems. We show that the
sum of information entropies in both position and momentum conjugate spaces is an indicator of
fractional revivals by analyzing three different model systems: (i) the infinite square well, (ii) a
particlebouncingverticallyagainstawallinagravitationalfield,and(iii)thevibrationaldynamics
of hydrogen iodide molecules. This description in terms of information entropies complements the
8 usual onein terms of the autocorrelation function.
0
0 PACSnumbers: 42.50.Md,03.67.-a,3.65.Ge
2

n
Thephenomenonofquantumwavepacketrevivalshas wave function, the smaller is the uncertainty, and the
a
J received wide attention over the last years. It has been higher is the accuracy in predicting the localization of

5 investigatedtheoretically in atomic and molecular quan- theparticle. Momentum-spaceentropymeasurestheun-
1 tum systems [1], and observedexperimentally in, among certainty in predicting the momentum of the particle.
others, Rydberg wave packets in atoms and molecules, Thus,informationentropygivesanaccountofthespread-
]
h molecular vibrational states, and Bose-Einstein conden- ing (high entropy values) and the regenerating (low en-
p sates [2]. Revivals occur when a wave packet solution tropyvalues) ofinitially welllocalizedwavepackets dur-
- of the Schr¨odinger equation evolves in time to a state ing the time evolution. Moreover, if ρ(x) = ψ(x)2
nt that closely reproduces its initial waveform. Fractional and γ(p) = φ(p)2 are respectively the probabilit|y den|-
a | |
revivals appear as the temporal formation of structures sities in position and momentum spaces (where ψ and
u
thataregivenbyasuperpositionofshiftedandrephased φ are the position and momentum wave packets), the
q
[ initial wave packets [3, 4, 5]. It has been shown [3, 6] uncertainty relation for the information entropy implies

1 that the relevant time scales of wave function evolution Sρ+Sγ ≥ 1+lnπ, where Sρ = − ρ(x)lnρ(x)dx and,
v arecontainedinthecoefficientsoftheTaylorseriesofthe analogously,Sγ = γ(p)lnγ(p)dpR. Thisinequalityisa
−
7 energyspectrum,En,aroundtheenergyEn0 correspond- generalizationoftheRstandardvariance-basedHeisenberg
6 ingtothepeakoftheinitialwavepacket. Moreprecisely, uncertaintyrelation[10]. Itissatisfiedasastrictequality
2 thesecond-,third-,andfourth-ordertermsinthisexpan- only for Gaussian wave packets and bounds from below
2
sionareassociatedwith,respectively,theclassicalperiod thesumoftheentropiesto1+lnπ. Duringtheevolution
.
1 of motion T , the quantum revival time-scale T , and ofa Gaussianwavepacket,the entropysum decreasesat
cl rev
0 the so-called superrevival time. Fractional revival times the revival times to reach the above value, which plays
8
can be given in terms of the quantum revival time-scale a similar role to unity in the autocorrelation function.
0
by t = pT /q, with p and q mutually prime [3], and Furthermore,the formationofanumberof‘minipackets’
: rev
v
are usually analyzed using the autocorrelation function of the originalpacket, i.e. fractionalrevivals of the wave
i
X A(t), which is the overlap between the initial and the function, will correspond to the relative minima of the
r time-evolving wave packet [7]. At certain fractional re- total entropy. Notice that it is the sum of the entropies
a vivals, however, the autocorrelation function may be of thatisemployedasanindicatorofthefractionalrevivals,
limitedhelpsincethewavepacketreformsitself,possibly and not either of them separately, because only the sum
intoascaledcopyofitsoriginalshape,inalocationthat embraces both the configurational and the motion as-
does not generally coincide with its initial position. An pects of the wave packet dynamics. In this context, we
expectation value analysis of wave packet evolution has point out that the sum of entropies for the phase and
beenrecentlyproposedbysomeauthors,butitmissesto photon number has been used to study the formation of
fully detect the fractional revivals [5, 8, 9]. macroscopic quantum superposition states from an ini-
tially coherent state via interaction with a Kerr medium
In this Letter we study the wave packet dynamics by
[11]. Additionally, we note that a finite difference eigen-
means of the sum of the information entropies of the
value method has been recently derived from which the
probability density of the wave packet, in both position
variousordersofrevivalscanbedirectlycalculatedrather
and momentum spaces. We shall show that it provides
than searching for them [12].
a natural framework for fractional revival phenomena.
Theposition-spaceinformationentropymeasurestheun- In what follows we investigate the time evolution of
certainty in the localization of the particle in space, so wave packets in three one-dimensional, model systems
the lower is this entropy the more concentrated is the that exhibit fractional revival behavior, namely, the fa-
2
miliar infinite square-well, the quantum ’bouncer’, that A)
20 0 4 2
is, a quantum particle bouncing on a hard surface under 119876 59 4 71 3185179 2
1/1/1/1/1/1/ 1/2/ 1/ 2/3/ 1/5/3/2/5/3/4/ 1/
theinfluence ofgravity,andasuperpositionofmolecular
0.8
wave packets in a Morse potential describing the vibra-
2
tions of hydrogen iodide molecules. The infinite square- )|
t
wellisespeciallywellsuitedfortheunderstandingoffrac- A(0.4
|
tional revival phenomena because of its amenability to
analytical treatment, and has been analyzed by several 0
authors(see,forinstance,[4,5,13]). Onthe otherhand,
the quantum bouncer is the quantum version of a fa- t) 4
(
miliar classical system and has also been studied in the Sγ
+
contextofwavepacketpropagationandrevivalphenom- ) 3
t
(
ena[5,14,15]. Recently,gravitationalquantumbouncers ρ
S
havebeenrealizedusingneutrons[16]andatomicclouds
2
[18], hence endowingthis systemwith greatphysicalsig- 1/121/101/91/81/71/6 1/52/9 1/4 2/73/10 1/35/143/82/55/123/74/9 1/2
nificance. Finally,molecularwavepacketsprovideareal- t/T
B) rev
isticscenariofortheassessmentoftheentropyapproach.
)
Rpaneerdvimivcaaenlnstabalenlydpirfnorabwceatdivoebnypalarcarkenevdtiosvmainls-vpohhlaavvsineegbflvueeiobnrreaostcbieosnencraveleildnevteeexrls-- 2|A(t)|00..048 (t)+S(tργ2.253
ferometry [2, 19] 0 5 10 S 0 2.5 5
Consideraninfinitepotential-welldefinedasV(x)=0 t/T t/T
cl cl
for 0 < x < L and V(x) = + otherwise. The time-
∞
dependent wave function for a localized quantum wave FIG. 1: Time dependence of |A(t)|2 and Sρ(t)+Sγ(t) for
packet is expanded as a one-dimensional superposition an initial Gaussian wave packet with x0 = L/2, p0 = 400π,
of energy eigenstates as andσ=1/10inaninfinitesquare-well. A)Long-timedepen-
dence. The main fractional revivals are indicated by vertical
ψ(x,t)= anun(x)e−iEnt/~, (1) dotted-lines. B) First classical periods of motion. Quantum
Xn wavepacketsarerepresentedbysolid linesandtheirclassical
component by dashed-lines.
where u (x) represent the normalized eigenstates
n
and E the corresponding eigenvalues, u (x) =
n n
2/Lsin(nπx/L),E = n2~2π2/2mL2. Following the
n
cpustomary procedure, the classical period and the re- times,theGaussianwavepacketevolvesquasi-classically,
vival time can be computed as T = 2mL2/~πn and but in a few periods the quantum and classical wave
cl
T =4mL2/~π, respectively. As an example, it is easy packet trajectories start moving apart (Fig. 1B), the
rev
toseebydirectsubstitutionin(1)thatψ(L x,T /2)= classical component of the wave function being defined
rev
ψ(x,0),soattime t=T /2acopyofth−e initialstate asψ (x,t)= a u (x)e−i2πnt/Tcl [5]. Forlongertime
− rev cl n n n
reforms itself, reflected around the center of the well [4]. scales,a largePamplitude modulation is superimposedon
WeshallconsideraninitialGaussianwavepacketwith thequasiperiodicoscillations(Fig. 1A).Inthislong-time
a width σ, centered at a position x0 and with a mo- regime, the wave packet initially spreads and delocalizes
mentum p0, ψ(x,0) = exp[ (x x0)2/2α2~2 +ip0(x while undergoing a sequence of fractional revivals with
− − −
x0)/~]/ α~√π. Assuming that the integration region the creation of sub-packets, each of them similar to the
can bepextended to the whole real axis, the expan- initial one so that the sum of the entropies reaches a
sioncoefficientscanbeapproximatedwithhighaccuracy relative minimum and the modulus of the autocorrela-
by an analytic expression [5]. To calculate the corre- tion function a relative maximum. The most important
sponding time dependent, momentum wave function we fractional revivals are denoted by vertical dashed-lines.
use the Fourier transform of the equation (1), and the It can be observed that the identification of some frac-
momentum-space normalized eigenstates tional revivals from the autocorrelation function is not
soclear-cutascomparedwiththeentropyapproach(see,
~ p
φ (p)= n ( 1)neipL/~ 1 . (2) for example, the cases t=pTrev/q with q =7 or 9).
n rπLp2 p2(cid:20) − − (cid:21)
− n We have also investigated the interesting initial con-
Withoutlossofgenerality,weshallhenceforthtake2m= dition x0 = 0.8L and p0 = 0, for which there is no
~=L=1, and σ =1/10 for the initial wave packet. classical periodic motion. Snapshots of the numerical
In Fig. 1 the sum of entropies, S (t) + S (t), and simulation of the position-space probability density are
ρ γ
theautocorrelationfunction, A(t),areshownforanini- given in Fig. 2 at several times. It is apparent from
| |
tial wave packet with x0 = 0.5 and p0 = 400π. At early the bottom panel of Fig. 3 that the sum of entropies
3
8 t=0 t=Trev/10 t=Trev/8 t=Trev/7 1 1/71/61/51/4 1/3 2/5 1/2 3/5 2/3 3/44/55/6
x)4
ρ(
00 0.4 0.8 0 0.4 0.8 0 0.4 0.8 0 0.4 0.8 2A(t)|0.5
|
t=T /6 t=T /5 t=T /4 t=3T /10
rev rev rev rev
8
0
x)4
ρ(
00 0.4 0.8 0 0.4 0.8 0 0.4 0.8 0 0.4 0.8 S(t)γ 6
t=T /3 t=3T /8 t=2T /5 t=T /2 +
8 rev rev rev rev (t)ρ 4
S
x)4
ρ( 2
00 0.4 0.8 0 0.4 0.8 0 0.4 0.8 0 0.4 0.8 1/71/61/51/4 1/3 2/5 1/2 3/5 2/3 3/44/55/6
x x x x t/T
rev
FIG.2: Snapshotsoftheprobabilitydensityinposition-space FIG.4: Timedependenceof(top panel)|A(t)|2 and(bottom
for an initial Gaussian wave packet with x0 = 0.8L, p0 = panel) Sρ(t)+Sγ(t)for an initial Gaussian wavepacketwith
0, and σ = 1/10 in an infinite square-well and at different z0=100,p0=0,andσ=1inaquantumbouncer. Themain
fractional revivaltimes. fractional revivals are indicated by vertical dashed-lines.
1 1/101/81/71/6 1/5 1/4 3/10 1/3 3/83/82/5 1/2 1 1/81/61/51/4 1/33/8 1/2 5/82/32/3 3/4 5/67/8 1
2|A(t)|0.5 2|A(t)|0.5
0
0
4
(t)+S(t)ργ 2.53 S(t)+S(t)ργ 3
S
2
2 8654 38 2 83 4 668 1
1/101/81/71/6 1/5 t/1/4T 3/10 1/3 3/83/82/5 1/2 1/1/1/1/ 1/3/ t/T1/rev 5/2/ 3/ 5/5/7/
rev
FIG.5: Timedependenceof(top panel)|A(t)|2 and(bottom
FIG.3: Time dependenceof (toppanel)|A(t)|2 and(bottom panel) Sρ(t)+Sγ(t) for an initial superposition of molecular
panel)Sρ(t)+Sγ(t)foraninitialGaussianwavepacketinan wavepacketwithGaussianweightsinaMorsepotential. The
infinitesquare well. Parameters as in Fig. 2. mainfractionalrevivalsareindicatedbyverticaldashed-lines.
has a minimum at the main fractional revivals, denoted is, a particle in a potential V(z) = mgz, if z > 0 and
by vertical dashed-lines, except for the case t = T /7 V(z)=+ otherwise. Upon introducing the character-
where the position-space probabilitydensity has a srheavpe istic gravi∞tational length l = ~/2gm2 1/3 and defining
g
compatible with a collapsedwavefunction (Fig. 2). The z′ = z/lg and E′ = E/mglg,(cid:0)as in [20(cid:1)], the eigenfunc-
autocorrelation function, as plotted in the top panel of tions and eigenvalues are given by E′ = z ,u (z′) =
n n n
Fig. 3, fails to show the fractional revivals occurring Ai(z′ z ) with n = 1,2,3,..., where Ai(z) is the
n n
at, for example, t/Trev = 1/6, 1/8, and 3/10. This dis- NAiryfunc−tion, zndenotesitszeros,and nistheun(z′)
crepancycanbe tracedbacktothe factthatinformation normalization−factor. Very accurate analyNticapproxima-
entropiestakeintoaccountindividualminigaussianpack- tions to z and can be found in [20]. Consider now
n n
ets independently oftheir relativepositions, whereasthe an initial GaussiNan wave packet localized at a height z0
autocorrelationfunctiondependsontherelativeposition above the floor, with a width σ and an initial momen-
between the initial wave packet and the evolved one. tum p0 = 0. The corresponding coefficients in Eq. (1)
Next, we study the behavior of these same quantities canbeobtainedanalytically[20],andtheclassicalperiod
in other physical situations. Consider a particle bounc- and the revival time are Tcl = 2√z0 and Trev = 4z02/π,
ing ona hardsurface under the influence ofgravity,that respectively.
4
The temporal evolution of the total entropy and of proach to systems with two or more quantum numbers.
the autocorrelation function was computed numerically This work was supported by the Spanish projects
for the initial conditions z0 = 100, σ = 1, and p0 = 0 FIS2005-00791/00973and FQM-165/0207.
(see Fig. 4). For that, the corresponding wave packet in
momentum-space was obtained numerically by the fast
Fourier transform method. One can see in the bottom
panel of Fig. 4 that the reformation of subpackets at
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| |
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−
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− − − ∞ ontheautocorrelationfunctionofwavepacketshavebeen
and n=0,1,...,[λ 1/2],[x] being the integer partof x.
− realized in recent experiments. See M. Mehring et al.,
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[11] Note,however,thatfractional revivalsandthetimeevo-
metric about Trev/2 [22]. As an initial wave packet, we lution of coherent states produced by a Kerr media are
shall take a superposition of Morse eigenstates assum- related but quite different phenomena. For instance, at
ing a gaussian population of vibrational levels c 2 = half therevival-time,fractional revivalshaveaGaussian
n
exp (n n0)2/σ /(πσ)1/2, with n0 = 7, σ = 3.| A|s in distribution of state amplitudes, whereas the state evo-
− lution of coherent states in a Kerr medium leads to a
the (cid:0)quantum bou(cid:1)ncer case, the fast Fourier method is
superposition of two distinct, equally weighted coherent
used to compute the entropy in the momentum space.
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1084 (2001).
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