Table Of ContentQuantum Zeno effect in a model multilevel molecule
D. Bruno,† P. Facchi,‡,¶ S. Longo, ,§,† P. Minelli,† S. Pascazio, ,¶ and A.
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IstitutodiMetodologieInorganicheedeiPlasmi,ConsiglioNazionaledelleRicerche, Bari,Italy,
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DipartimentodiMatematica,Universitàdi Bari,Bari,Italy,IstitutoNazionaledi Fisica
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Nucleare, Sezionedi Bari,Bari,Italy,DipartimentodiChimica, Universitàdi Bari,Bari,Italy,
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2 Dipartimentodi Fisica,UniversitàdiBari,Bari,Italy,AbdusSalamInternationalCentrefor
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TheoreticalPhysics,Trieste,Italy, andIstitutoNazionaledi FisicaNucleare, SezionediTrieste,
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E-mail: [email protected]
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§UniversitàdiBari-Chimica
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1
Abstract
Westudy the dynamics of the populations of amodel molecule endowed with two sets of
rotational levels of different parity, whose ground levels are energy degenerate and coupled
by a constant interaction. The relaxation rate from one set of levels to the other one has an
interesting dependence ontheaverage collision frequency ofthemolecules inthegas. Thisis
interpretedasaquantumZenoeffectduetothedecoherenceeffectsprovokedbythemolecular
collisions.
Introduction
The quantum Zeno effect is usually formulated as the hindrance of the evolution of a quantum
system due to frequent measurements performed by a classical apparatus1,2 and is formalized
accordingtovonNeumannprojectionrule.3 Theliteratureofthelastfewyearsonthistopicisvast
and contemplates a variety of physical phenomena, ranging from oscillating (few level) systems4
and alternative proposals5 to bona fide unstable systems,6 where the so-called "inverse" Zeno
effect can takeplace.
The ideas and concepts at the basis of the quantum Zeno effect (QZE) were also successfully
extendedtocontinuousmeasurementprocesses bydifferentauthorsandindifferentcontexts7 and
led to a remarkable explanation of the stability of chiral molecules.8 This was a fertile idea, in
that it explained the behavior of a variety of physical systems in terms of a similar underlying
mechanism.
The QZE is, however, a much more general phenomenon, that takes place when a quantum
system is strongly coupled to another system9 or when it undergoes a rapid dephasing process.
Such a rapid loss of phase coherence ("decoherence") of the quantum mechanical wave function
(for instance as a result of frequent interactions with the environment) is basically equivalent to
a continuous measurement process (the main difference being that the state of the system is not
necessarily explicitlyrecorded byapointer).
2
The quantum Zeno effect is always ultimately ascribable to the short-time features of the dy-
namical evolution law:10 it is only the study of this dynamical problem that determines the range
in which a frequent disturbance or interaction will yield a QZE. The very definition of "frequent"
is a delicate problem, that depends on the features of the interaction Hamiltonian. Moreover, one
shouldalsonoticethatthequantumsystemisnotnecessarilyfrozen initsinitialstate,11 butrather
undergoes a "quantumZeno dynamics",possiblyevolvingaway from its initialstate.13 Thestudy
of such an evolution in the "quantum Zeno subspace"12 is in itself an interesting problem, whose
mathematical and physical aspects, as well as the possible applications to chemistry and physical
chemistry,are notcompletelyclearand requirefurtherstudyand elucidation.
After the seminal experiment by Itano and collaborators,4 the QZE has been experimentally
verifiedinavarietyofdifferentsituations,onexperimentsinvolvingphotonpolarization,14nuclear
spin isomers,15 individual ions,16–19 optical pumping,20 NMR,21 Bose-Einstein condensates,22
thephoton numberof theelectromagneticfield in a cavity,23 and new experimentsare in prepara-
tionwithneutronspin.24,25
WefocushereontheinterestingexampleofQZEproposedin:15thenuclearspindepolarization
mechanisms in 13CH F, due to magnetic dipole interactions and collisions among the molecules
3
in the gas, was experimentally investigatedand interpreted as a QZE. In a few words, the 13CH F
3
molecule has two kinds of angular momentum states, according to the value of the total spin of
the three protons (H nuclei): I =3/2 (ortho) and I =1/2 (para). Transitions between states with
differentparityare(electricdipole)forbidden,sothatspinflipoccursviaaweakcouplingbetween
two levels of different spin parity (this is most effective when there is an accidental degeneracy
between the levels, achievable, for example, via a Stark effect26). One observes a significant
dependence of the spin relaxation on the gas pressure and interprets this as a QZE provoked by
the dephasing due to molecular collisions. Nuclear spin conversion in polyatomic molecules is
reviewedin.27
The aim of this article is to study the occurrence of the QZE in the general framework of
collision-inhibited Rabi-like oscillations between two sets of rotational levels. We shall study
3
the evolution of the level populations in a model multilevel molecule endowed with two sets of
rotational levels of different parity. In particular, we shall concentrate on the interesting effects
that arise as a consequence of the interactions (collisions) with the other molecules in the gas.
The model we shall adopt will be studied both numerically and analytically, and the results will
be compared. One of the main objectives of our investigation will be the analysis of apparently
differentphenomenaintermsofaZeno dynamics.
WeshallintroducethesysteminSection andtheZenoprobleminthepresentcontextinSection
. InSections-westudytheproblemfromananalyticpointofview,byderivingandapproximately
solving a master equation. In Section the analytical result are compared to an accurate numerical
simulation. Weconcludein Section witha fewremarks.
The system
Our model molecule has two subsets of rotational levels (to be called left (L) and right (R) levels
in the following) of different parity, whose ground levels are energy degenerate and coupled by
a constant interaction. (The choice of the ground levels is motivated by simplicity: one could
chooseanyothercoupleofenergy-degeneratelevelsintheL-Rsubspaces.) Themoleculesundergo
collisions with other identical molecules in the gas and we assume that these collisions couple
the rotational energy levels but do not violate spin parity conservation. We shall focus on the
dependence of the relaxation rate on the average collision time or, equivalently, on gas pressure:
a QZE takes place if the transition between the left and right subspaces is inhibited when the
collisionsbecomemorefrequent (i.e., thegas pressureincreases).
A sketch of the system is shown in Figure1. The total Hilbert spaces of each molecule is
madeupoftwosubspacesH (left)andH (right),withN andN levelsrespectively. Collisions
L R L R
cannot provoke L R transitions, so that no transitions are possible between the two subspaces,
↔
except through their ground states. However, collisions with other particles in the gas provoke
transitionswithineach subspace.
4
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Figure1: Poissoniancollisionsin agasofmultilevelmolecules.
5
TheHamiltonianis
H =H +H (t)=H +H +H (t), (1)
f coll 0 1 coll
whereH =H +H is thefree Hamiltonianand
f 0 1
N N
(cid:229) L (cid:229) R
H = E n n + E n n , (2)
0 nL| Lih L| nR| Rih R|
nL=1 nR=1
H = h¯W ( 1 1 + 1 1 ), (3)
1 L R R L
| ih | | ih |
H = h¯(cid:229) d (t t )V, (4)
coll j
−
j
V = a V +a V , (5)
L L R R
N(cid:229)s−1 N(cid:229)s−1
V = V = n n +1 + n +1 n , (6)
s ns | sih s | | s ih s|
ns=1 ns=1
with s=L,R. The energy levels n have energies E (s=L,R) and H provokes L R transi-
| si ns 1 ↔
tions between the two ground states, with (Rabi) frequency W . W is small (in a sense to be made
preciselater),forsuchatransitioniselectric-dipoleforbidden. H accounts fortheeffect ofcol-
coll
lisionswiththegas(environment): thecollisionsaredistributedaccordingtothePoissonstatistics,
appropriateforgasphasewithshort-rangebinary collisions,so thattheyoccurat times
t =t +dt , (7)
j+1 j j
wheredt ’s areindependentrandom variableswithdistribution
j
1
p(dt )= exp( dt /t ) (8)
j t j
−
and (common) average t . The coupling constants a are in general different from each other
L,R
and measure the "effectiveness" of a collision. For the sake of simplicity we assume that colli-
sionsprovoketransitionsonlybetweenadjacentlevels[V in(6)involvesonly"nearestneighbors"
s
6
couplings]. We willassume,forconcreteness, thattheenergy levelsare rotational,so that
E =h¯w n (n +1) (s=L,R) (9)
ns s s s
and 1 and 1 are theonlyresonant pairofstates:
L R
| i | i
E =E , E =E for m ,n >1. (10)
1L 1R mL 6 nR L R
See Figure1. The Hilbert spaces H and H are finite dimensional, with dimensions N and
L R L
N , respectively. This is because, in general, the number of accessible rotational levels is limited
R
to a few tens, since for sufficiently high energies molecules tend to dissociate. This could be
accounted for by introducing two "absorbing" levels N , N .28 However, in our analysis,
L+1 R+1
| i | i
we will explore a time region in which the introduction of absorbing levels is not necessary (in
otherwords,thetimesinvolvedwillnot belongenoughtodisplay"bordereffects").
Zeno effect
Before we start our theoretical and numerical analysis it is convenient to focus on the physics of
themodelintroducedintheprecedingsectionandtoclarifyinwhichsenseweexpectaZenoeffect
totakeplace. Westartfromasimplenumericalexperimentandcalculatethetimeevolutionofthe
populationsbytheMonteCarlo methoddescribed in.29
Consider a uniform gas of identical molecules, having the internal structure described in the
precedingsection. Asinglemoleculefreelywandersinatotalvolumeandundergoesrandomcolli-
sions. Byneglectingthespatialcomponentofthewavefunction,eachmoleculecanberepresented
by an (N +N )-dimensional state vector y (t) that describes its internal state.30 This physical
L R
| i
situation is well schematized by the model described in Sec. . During the free flight the evolution
7
isgovernedby thefree hamiltonian
i
y (t) =exp H t y (0) . (11)
f
| i −h¯ | i
(cid:18) (cid:19)
Since the molecules are immersed in a bath, the collisions are distributed in time according to
the Poisson statistics(7)-(8), with average collisionfrequency (per particle) t 1. Once a collision
−
occurs, acollisiontimeissampledaccording to:
dt = t log(y), (12)
−
y being a random number uniformly distributed in [0,1[. The collisions are modelled as instanta-
neous events and act on the left/right subspaces independently. As a result of a collision,the state
becomes
y (t+0+) =exp i (cid:229) a V y (t) . (13)
s s
| i − | i
s=L,R !
Thematrixexp( i(cid:229) a V )isevaluatednumerically. Itisassumedtobeindependentoftheinternal
s s s
−
stateofthecollidingpartners and oftheirkineticenergy.
Wealso stressthatsinceouraimistoinvestigatetheoccurrence ofaQZEwithintheproposed
level structure, we are not interested in the dissociation of highly excited molecules. To this end,
we must restrict our attention to times such that the molecules do not "see" the upper limit of
the rotational levels, so that "border" effects do not play any significant role. In this way the
dissociationofhighlyexcitedmoleculescan besafelyneglected.
Theafore-mentionedqualitativefeaturesofouranalysiswillbecarefullyscrutinizedandmade
preciseinthefollowingsections. Wenowtakethemforgrantedandgiveafewpreliminaryresults
inorder toget afeelingforthephysicsat thebasisoftheZeno effect.
We set N = N = 40 energy levels, with energies given by (9), where n = 1,...,40, w =
L R s L
1.3 1010 s 1 and w =9.7 109s 1. We always compute the average over an ensemble of 5 103
− R −
· · ·
particles. All particles are initially in the 1 state and we study the temporal behavior of the
L
| i
8
relativepopulationintheleft subspace
(cid:229)
P p , (14)
L ≡ nL
nL
p being theoccupationprobabilityofstate n .
nL | Li
TheresultsofournumericalintegrationareshowninFigure2-Figure3. InFigure2,a =0.2
L
a a
=0.2 , =0
L R
1,00
0,95
0,90
0,85
P
L
0,80 t -1=500*W /2p
t -1=600*W /2p
0,75 t -1=800*W /2p
t -1=1000*W /2p
0,70 t -1=1200*W /2p
t -1=1500*W /2p
0,65
0 0,2 0,4 0,6 0,8 1
Time (Rabi periods)
Figure 2: Temporal evolution of P . The collision frequency t 1 is varied between 500T 1 and
L − R−
1500T 1 (T =2p /W ). We set a =0.2,a =0, so that, in practice, N =40 left energy levels
R− R L R L
are coupled to only N =1 right level. The survival probability in the left subspace increases as
R
the collision frequency is increased: frequent collisions hinder transition to the right subspace, a
manifestationofa("classicallyintuitive")Zeno effect.
and a =0, so that collisions do not provoke transitions among the right states (or, equivalently,
R
the right subspace consists only of state 1 ). It is apparent that when the collision frequency
R
| i
t 1 is increased between 500T 1 and 1500T 1 (T =2p /W being the Rabi period) the survival
− R− R− R
probability in the left subspace increases. If the collisions are viewed as a dephasing process
9
a =0 , a =0.2
L R
1,00
t -1 =500*W/ 2p
t -1 =600*W/ 2p
0,80
t -1 =800*W/ 2p
t -1 =1000*W/ 2p
0,60
P t -1 =1200*W/ 2p
L
t -1 =1500*W/ 2p
0,40
0,20
0,00
0 0,2 0,4 0,6 0,8 1
Time (Rabi periods)
Figure 3: Temporal evolution of P . The collision frequency t 1 is varied between 500T 1 and
L − R−
1500T 1. Unlike in the previous figure, we set a = 0,a = 0.2, so that in practice, N = 1
R− L R L
left level is coupled to N = 40 right levels. Again, the survival probability in the left subspace
R
increases as the collision frequency is increased: frequent collisions hinder transition to the right
subspace,a manifestationofa("classicallycounterintuitive")Zeno effect.
10
