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the theoretical treatment of time observables is one of the important loose ends of quantum theory . among them
the time of arrival ( toa ) has received much attention lately @xcite , for earlier reviews see @xcite .
a major challenge is to find the connection between _ ideal _ toa distributions , defined for the particle in isolation , formally independent of the measurement method , and _ operational _ ones , explicitly dependent on specific measurement models and procedures .
it is important to know , for example , what exactly a given operational procedure is measuring , or if and how a given ideal quantity may or may not be obtained with a particular experiment .
modern research on the quantum toa owes much to the seminal work by allcock @xcite . looking for an ideal quantum arrival - time concept
, he considered that arrival - time measuring devices should rapidly transfer any probability that appears at @xmath0 ( @xmath1 being the `` arrival position '' ) from the incident channel into various orthogonal and time - labeled measurement channels .
as a simple model to realize this basic feature he proposed a pulsed , periodic removal , at time intervals @xmath2 , of the wave function for @xmath0 , while the @xmath3 region would not be affected , see fig .
a similar particle removal would provide the distribution of first arrivals for an ensemble of classical , freely - moving particles as @xmath4 .
the difficulties to solve the corresponding mathematical problem lead allcock to study instead a different , continuous model with an absorbing imaginary potential in the right half - line , @xmath5 , @xmath6 , to simulate the detection .
he argued that the two models should lead to similar conclusions with a time resolution of the order @xmath2 in the chopping model or @xmath7 in the complex potential model .
he then solved the schrdinger equation with the complex potential and noticed that for @xmath8 , where @xmath9 is a maximum relevant energy in the wave packet ,     at times @xmath10 , @xmath11 separated by @xmath2 .
figures ( a ) and ( b ) represent two instants immediately before and after the elimination of norm at @xmath0 .
[ f0],width=226 ]    the apparatus response vanishes , @xmath12 , with @xmath13 , because of quantum mechanical reflection .
this is one of the first discussions of the quantum zeno effect , although it was not known by this name at that time .
eight years later misra and sudarshan @xcite generalized this result studying the passage of a system from one predetermined subspace to its orthogonal subspace : the periodic projection method in the limit @xmath4 was presented as a natural way to model a continuous measurement , however it did not lead to a time distribution of the passage but to its suppression @xcite .
this lack of a `` trustworthy algorithm '' to compute toa and related distributions prompted them to put in doubt the completeness of quantum theory and has been much debated since then , see @xcite for review .
for the specific case of a projection onto a region of space , as in the toa procedure proposed by allcock , several works have later emphasized that , in the limit of infinitely frequent measurements , the resulting `` zeno dynamics '' in the original subspace corresponds to hard wall ( dirichlet ) boundary conditions @xcite .
allcock discarded the short @xmath2 ( or high @xmath14 ) limit as useless , and examined the other limit where the measurement may be expected to be @xmath14-independent , @xmath15 , later on .
we shall not treat this limit hereafter but , for completeness , let us briefly recall that he got the current density @xmath16 by deconvolution of the absorbed norm with the apparatus response function , @xmath17 where @xmath18 and @xmath19 are the wavenumber and position operators , @xmath20 is the mass , and the average is computed with the freely moving wave function @xmath21 .
this result makes sense classically , but in quantum mechanics @xmath16 is not positive semidefinite even for states composed only by positive momenta @xcite .
he also got , within some approximation , a positive distribution , @xmath22 , which has been later known as `` kijowski s distribution '' @xcite , @xmath23 in this paper , instead of disregarding the zeno limit because of reflection , as it is customary , we shall look at the small amount of norm @xmath24 detected , i.e. , eliminated in the projection process , and normalize , [ pizeno ] _
zeno=_t0 . in this manner
, a rather simple ideal quantity will emerge : there is , in other words , interesting physical information hidden behind the zeno effect , which can be disclosed by normalization . to fulfill this program
we shall put the parallelism hinted by allcock between the pulsed measurement and the continuous measurement on a firmer , more quantitative basis .
we shall now define formally the pulsed and continuous measurement models mentioned above and also an intermediate auxiliary model @xcite that will be a useful bridge between the two .
@xcite is followed initially although the analysis and conclusions will be quite different .
the `` chopping process '' amounts to a periodic projection of the wave function onto the @xmath3 region at instants separated by a time interval @xmath2 .
there is nothing here beyond standard measurement theory @xcite .
each chopping step eliminates interference terms in the density operator between right and left components , and the right component is separated from the ensamble ( detected ) so that it can not come back to the left .
the wave functions immediately after and before the projection at the instant @xmath10 , are related by @xmath25 the wave at @xmath0 may also be canceled with a `` kicked '' imaginary potential @xmath26 where the subscript `` k '' stands for `` kicked '' and @xmath27 , = -i_i =- iv_0 ( ) , [ v ] provided [ v0 t ] v_0 t .
the general ( and exact ) evolution operator is obtained by repetition of the basic unit [ ue ] _
k(0,t)= e^-i _ 0t / e^-it/ , where @xmath28 . for the continuous model , the evolution under the imaginary potential ( [ v ] ) is given by [ uot ] ( 0,t)&= & e^-i(_0+)t/=e^-i_0t/ e^-it/ + & + & o(t^2[,_0]/^2 ) .
[ condi ] comparing with eq .
( [ ue ] ) we see that the kicked and continuous models agree when t^2||/^2 1 .
[ kc ] at first sight a large @xmath29 , see eq .
( [ v0 t ] ) , seems to be incompatible with this condition so that the three models would not agree @xcite .
in fact the numerical calculations give a different result and show a better and better agreement between the continuous and pulsed models as @xmath30 when @xmath2 and @xmath14 are linked by some predetermined ( large ) constant @xmath31 , @xmath32     ( normalized ) for the projection method and the continuous ( complex potential ) model .
the initial state is a minimum uncertainty product gaussian for @xmath33na atoms centered at @xmath34 m with @xmath35 m and average velocity @xmath36 cm . in all numerical examples
negative momentum components of the initial state are negligible .
[ f1],height=241 ]     ( dashed line ) , @xmath37 ( solid line ) , @xmath38 ( big sparse dots ) , and @xmath39 ( for pulses separated by @xmath40 ns , dotted line ) .
the initial wave packet is a combination @xmath41 of two gaussian states for the center - of - mass motion of a single caesium atom that become separately minimal uncertainty packets ( with @xmath42 m , and average velocities @xmath43 cm / s , @xmath44 cm / s ) at @xmath1 and @xmath45s . [ f2],height=234 ]     with increasing @xmath14 versus time for the same initial wave function as in fig . 2 ( @xmath33na atoms . )
@xmath46 is the initial kinetic energy .
[ f3],height=234 ]    figures [ f1 ] and [ f2 ] illustrate this agreement : in fig .
[ f1 ] ( inset ) the average absorption time , @xmath47 $ ] , is shown versus @xmath2 and @xmath48 for a gaussian wave packet sent from the left towards the origin .
the main figure shows the time averages versus @xmath2 ( chopping ) and @xmath7 ( continuous ) .
the lines bend at high coupling because of reflection . the normalized absorption distribution as @xmath30 was derived in @xcite by making explicit use of the known stationary scattering wave - functions , @xmath49 where @xmath50 is the initial average momentum .
[ f2 ] shows for a more challenging state , a combination of two gaussians , that this ideal distribution becomes indistinguable from the normalized chopping distribution when @xmath2 is small enough .
even the minor details are reproduced , and differ from @xmath51 and @xmath52 , also shown .    to understand the compatibility `` miracle '' of the inequalities ( [ v0 t ] ) and ( [ kc ] )
, we apply the robertson - schrdinger ( generalized uncertainty principle ) relation , ||2 |v_i| h_0 , where @xmath53 denotes the standard deviation . since @xmath54 is rigorously bounded at all times by @xmath55 , is the norm in @xmath0 , @xmath56 which is maximal at @xmath57 . ]
imposing @xmath58 with @xmath59 , a sufficient condition for dynamical agreement among the models is v_0h_0 .
[ condi3 ] for large @xmath14 the packet is basically reflected by the wall so that @xmath60 tends to retain its initial value and eq .
( [ condi3 ] ) will be satisfied during the whole propagation . fig .
[ f3 ] shows the ratio @xmath61 for three values of @xmath14 as a function of time .
this implies in summary that @xmath62 , eq .
( [ pz ] ) , a very remarkable result , which illustrates that an active intervention on the system dynamics may after all provide an ideal quantity defined for the system in isolation .
so far we have discussed the limits @xmath63 in order to find the corresponding time distribution .
in fact a very simple argument relates the pulsed and continuous measurements _ approximately _ for finite , non - zero values of @xmath2 and @xmath64 , when they are sufficiently large to make reflection negligible : the average detection time is delayed with respect to the ideal limit corresponding to @xmath38 as tt_zeno+t/2t_zeno+ , see fig .
[ f1 ] , since , once a particle is in @xmath0 , @xmath65 and @xmath66 are precisely the average life times in the continuous and discrete measuring models , respectively . for optimized straight lines .
reflection at small @xmath2 ( or high @xmath14 ) favors the detection of faster particles and bends the @xmath67 lines towards shorter times , as in fig .
] this suggests an approximate agreement between projection and continuous dynamics provided that the relation @xmath68 is satisfied . for large @xmath14 , this is asymptotically not in contradiction with the requirement of a large @xmath31 since @xmath69 as @xmath70 ; in any case quantum reflection breaks down the linear dependence , see fig .
[ f1 ] .
a similar relation between pulsed and continuous measurements was described by schulman @xcite and has been tested experimentally @xcite .
the simplest model in @xcite may be reinterpreted as a two - level atom in a resonant laser field , with the excited state decaying away from the 2-level subspace at a rate @xmath71 @xcite , @xmath72 .
the relation between pulsed and continuous measurements follows by comparing the exponential decay for the effective 2-level hamiltonian with rabi frequency @xmath73 and excited state lifetime @xmath74 , with the decay dynamics when @xmath75 and the system is projected every @xmath2 into the ground state .
it takes the form @xcite @xmath76 for @xmath77 ( weak driving ) . in our toa model
we have a different set of parameters but a comparison is possible by taking into account that the imaginary potential ( [ v ] ) may be physically interpreted as the effective interaction for the ground state in the weak driving regime , for a localized resonant laser excitation with subsequent decay , @xmath78 .
this gives @xcite @xmath79 , so that @xmath68 becomes t 2 , different from schulman s relation , as it may be expected since the pulsed evolution depends on @xmath73 in schulman s model but not in our case , where it is only driven by the kinetic energy hamiltonian @xmath80 .
the first discussions of the zeno effect , understood as the hindered passage of the system between orthogonal subspaces because of frequent instantaneous measurements , emphasized its problematic status and regarded it as a failure to simulate or define quantum passage - time distributions .
we have shown here that in fact there is a `` bright side '' of the effect : by normalizing the little bits of norm removed at each projection step , a physical time distribution defined for the freely moving system emerges .
( there are other `` positive '' uses of the zeno effect , such as reduction of decoherence in quantum computing , see e.g. @xcite . ) in the case of the projection measurements to determine the toa , this distribution is given by eq .
( [ pz ] ) .
this result is fundamentally different from the current density or from kijowski s toa distribution , ( operational approaches to measure them by fluorescence are described in @xcite ) and corresponds to a classical time - distribution of local kinetic energy density @xcite , rather than a classical toa distribution .
could in principle be obtained in the zeno limit for states with positive momentum support by transforming the initial state @xmath81 , with @xmath82 a constant , as in the operator normalization technique @xcite . ]
experimental realizations of repeated measurements will rely on projections with a finite frequency and pulse duration that provide approximations to the ideal result .
feasible schemes may be based on pulsed localized resonant - laser excitation @xcite , or sweeping with a detuned laser @xcite .
a challenging open question for further research is the possibility to devise alternative ( non periodic ) pulse sequences to enhance the robustness of the zeno effect in a toa measurement , similar in spirit to the ones proposed in the context of quantum information processing @xcite .
nevertheless , the ordinary ( periodic ) sequence has successfully been applied in quantum optical realizations of the quantum zeno effect @xcite .
the proposed normalization method may be applied to other measurements as well , i.e. not only for a toa of freely moving particles , but in general to first passages between orthogonal subspaces , and it will be interesting to find out in each case the ideal time distribution brought out by normalization .
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repeated measurements of a quantum particle to check its presence in a region of space was proposed long ago ( g. r. allcock , ann . phys . * 53 * , 286 ( 1969 ) ) as a natural way to determine the distribution of times of arrival at the orthogonal subspace , but the method was discarded because of the quantum zeno effect : in the limit of very frequent measurements the wave function is reflected and remains in the original subspace . 
 we show that by normalizing the small bits of arriving ( removed ) norm , an ideal time distribution emerges in correspondence with a classical local - kinetic - energy distribution .