PHYSICAL REVIEW A APRIL 1998VOLUME 57, NUMBER 4 Determination of atom-field observables via resonant interaction A. Luis and L. L. Sa´nchez-Soto Departamento de O´ ptica, Facultad de Ciencias Fı´sicas, Universidad Complutense, 28040 Madrid, Spain ~Received 22 October 1997! We find the observables that can be determined by two practical schemes based on atom-field resonant interaction. In the first example, the interaction is followed by photon-number and atomic-population measure- ments. In the second scheme the directly measured quantity is the atomic deflection. In particular, we show that they provide the measurement of the atom-field relative phase. When the initial field state is known, the atomic density matrix can be reconstructed.@S1050-2947~98!10704-7# PACS number~s!: 42.50.Dv, 03.65.Bz, 42.50.Vk o he - e s ro i s te w , f e tr na t i be e he e te e he e so ee a g tu a io of a se b he be ed ber tly the de, o- ble s t- as of se sta- d, I. INTRODUCTION Quantum mechanics postulates that every self-adjoint erator can be measured~leaving aside superselection rules!. This also applies to positive-operator measures through t Naimark extension@1#. Although the practical implementa tion of such measurements is known in very few cases, th are methods extending the number of observables whose tistics can be determined in practice. For instance, cont lable couplings can relate directly measurable quantities w more involved observables through a suitable data analy The dependence of the statistics on controllable parame is the basis of generalized measurements described positive-operator measures. This also underlies well-kno schemes determining the quantum state, like tomography instance@2#. In this work we reexamine this issue for the measurem of observables of a two-level atom and a one-mode elec magnetic field. Two simple arrangements based on reso interaction are considered. In the first example, resonan teraction is followed by the measurement of photon num and atomic population. In the second arrangement, the m sured observable is the atomic deflection@3,4#. Both schemes provide information about the atom-field state prior to t coupling. The plan of this work is as follows: In Secs. II and III, w analyze the joint atom-field observables that can be de mined from the statistics of the corresponding measurem In Sec. IV, we consider the possibility of measuring t atom-field relative phase@5,6#, including a concrete exampl to illustrate the experimental feasibility of the method. Re nant interaction is highly sensitive to phase relations betw the field and the atomic dipole. The output measurable qu tities depend on the initial relative phase, so the arran ments studied here should allow us to determine the quan statistics of this observable. Given the relevance of ph variables and the difficulties that their quantum descript encounter, we think it is worth examining the possibility their practical measurement@7#. If the initial field state is known, these arrangements c be regarded as generalized measurements of atomic ob ables. In Sec. V we study the atomic variables that can determined in this way. The possibility of reconstructing t density matrix of the atom is examined as well. 571050-2947/98/57~4!/3105~7!/$15.00 p- ir re ta- l- th is. rs by n or nt o- nt n- r a- r- nt. - n n- e- m se n n rv- e II. DETERMINATION OF ATOM-FIELD OBSERVABLES BY PHOTON-NUMBER AND ATOMIC-POPULATION MEASUREMENTS In this section we examine the observables that can determined when atom-field resonant interaction is follow by the simultaneous measurement of field-photon num and atomic-level population. The interaction is convenien described by the Jaynes-Cummings Hamiltonian in rotating-wave approximation~in units \51! @8# H5v~Sz1a†a!1l~a†S21aS1!, ~2.1! where Sz5(ue&^eu2ug&^gu)/2, S15ue&^gu, and S25ug&^eu. The vectorsue& and ug& represent the excited and ground energy levels of the isolated atom,a anda† are the annihilation and creation operators for the field mo and l is the coupling constant. For simplicity, exact res nance has been assumed. We shall consider that the field experiences a controlla phase shiftd before its interaction with the atom, which i equivalent to shift by2d the atomic phase. Ifr in is the initial density matrix of the atom-field system, the final ou put density matrixrout after these two steps can be written rout5Ur inU †, ~2.2! where the unitary operatorU is U5e2 i tHeida†a, ~2.3! andt is the interaction time. The joint probability for the simultaneous measurement photon number and atomic population is Pout~ j ,n,d!5tr~routu j ,n&^ j ,nu!, ~2.4! wherej 5e,g. Since only one fixed interaction timet will be needed,t is not included in the parametrization of the probabilities. Our purpose is to extract from this measurement the tistics of atom-field variables in the initial state. To this en we express the probabilities~2.4! in terms ofr in , Pout~ j ,n,d!5tr~r inU †u j ,n&^ j ,nuU !5tr@r inD~ j ,n,d!#, ~2.5! 3105 © 1998 The American Physical Society th d - tim le ra- s ting ure- ngle eld ted ed o- de- ed the ing be g ess re- e- ob- a s nt. ob- this ou- 3106 57A. LUIS AND L. L. SÁNCHEZ-SOTO where D~g,n,d!5cn 2ug,n&^g,nu1sn 2ue,n21&^e,n21u 1 isncn~eidue,n21&^g,nu 2e2 idug,n&^e,n21u!, D~e,n,d!5sn11 2 ug,n11&^g,n11u1cn11 2 ue,n&^e,nu 2 isn11cn11~eidue,n&^g,n11u 2e2 idug,n11&^e,nu!, ~2.6! and sn5sin~ltAn!, cn5cos~ltAn! ~2.7! are known parameters. The determination of atom-field observables requires inversion of the previous equations, expressingue,n&^e,nu, ug,n&^g,nu, ue,n21&^g,nu, and ug,n&^e,n21u as functions of D(e,n,d) and D(g,n,d). This inversion can be carrie out by means of a discrete Fourier analysis ind. In this case three values ford are enough, for instanced r52pr /3 with r 50,61. This gives the following equations: ue,n&^e,nu5 1 2 @D~e,n,d!1D~g,n11,d!# 1 1 6c4n14 ( r @D~e,n,d r !2D~g,n11,d r !#, ug,n&^g,nu5 1 2 @D~e,n21,d!1D~g,n,d!# 2 1 6c4n ( r @D~e,n21,d r !2D~g,n,d r !#, ug,0&^g,0u5D~g,0!, ue,n21&^g,nu5 2i 3s4n ( r e2 idrD~e,n21,d r ! 5 22i 3s4n ( r e2 idrD~g,n,d r !, ug,n&^e,n21u5 22i 3s4n ( r eidrD~e,n21,d r ! 5 2i 3s4n ( r eidrD~g,n,d r !. ~2.8! It should be noted that@D(e,n21,d)1D(g,n,d)# and D(g,0) are independent ofd, so anyd r can be used when ever they appear. We have assumed that the interaction t is chosen such that s4n5sin(2ltAn)Þ0 and c4n5cos(2ltAn)Þ0. e e Thus Eqs.~2.8! provide the statistics of any observab expressible as an arbitrary linear combination of the ope tors ue,n&^e,nu, ug,n&^g,nu, ue,n21&^g,nu, ug,n&^e,n21u, ~2.9! i.e., of every observable commuting withSz1a†a. To this end, probabilities~2.5! should be known for the three value d r of the phase shift. This can be accomplished by repea the measurement after each phase shiftd r . In addition, this can be achieved in the form of a single generalized meas ment. The three phase shifts can be embodied in a si realization by using a nonresonant interaction of the fi with an auxiliary three-level atom, for example@9#. One of the levels can be too strongly detuned to affect or be affec by the field, while the other two can be sufficiently detun so that the transition probability is negligible. This nonres nant interaction produces a phase shift in the field that pends on the atomic level. If the auxiliary atom is prepar in a superposition of its three states, the detection of energy level after the interaction implies the correspond phase shift in the field, which can be adjusted to d r , r 5061, by properly selecting the detuning, couplin constant, and interaction time. Before considering particular examples, we can expr the result obtained in a slightly different form. The measu ment provides information about the atom-field stater in . Although this information is not complete, we can neverth less express this partial knowledge in terms of a quasipr ability distribution in phase space, for instance theQ func- tion. The atom-fieldQ function is defined as Q~a,q,w!5 1 2p2 tr~r inuq,w&ua&^au^q,wu!, ~2.10! whereua& are field coherent states, and uq,w&5sin~q/2!ug&1eiw cos~q/2!ue&, ~2.11! are SU~2! coherent states for the atom@10#. Instead of the completeQ(a,q,w) function, the measurement provides reducedQ function depending just onuau, q, and the phase difference f5w2u between the atomic phasew and the field phaseu : q~ uau,q,f!5E du Q~a5uaueiu,q,w5f1u!. ~2.12! This is because E duuq,w5f1u&ua5uaueiu&^a5uaueiuu^q,w5f1uu ~2.13! commutes withSz1a†a, so Eqs.~2.8! can be used to expres q(uau,q,f) in terms of the statistics of the measureme Similar results would be obtained by using other quasipr ability distributions. This phase-space picture expresses the main result of section in a more classical way. The atom-field resonant c n- el ho ti ly as od io d o e e el s- m - m b b h nd al b- tial or of tly or r is the how t of eld. ut- f hase fer- ela- cify this se, ssi- e 57 3107DETERMINATION OF ATOM-FIELD OBSERVABLES VIA . . . pling backtransformsq anduau ~which are phase-space cou terparts of the measured operatorsSz anda†a! into functions of q, uau, andf. This, and the determination ofq(uau,q,f) just shown, suggest that the quantum translation of the r tive phase should be included among the observables w statistics can be derived from this measuring scheme. Sec IV is devoted to an examination of this possibility in pure quantum terms. III. DETERMINATION OF ATOM-FIELD OBSERVABLES VIA ATOMIC DEFLECTION Here we analyze a different measurement scheme b also on the resonant coupling of the atom with a single-m field. In this case, we shall consider the atomic deflect produced when the atom passes near the node of a stan electromagnetic wave@4#. The information about atom-field variables will be contained in the change of transverse m mentum` experienced by the atom. As in Sec. II, we assume that, before the atom-field int action, a controllable phase shiftd can be produced on th field. Afterwards, the atom crosses a standing light fi aligned along theX direction, passing through a small tran verse region centered around a node atx50. If this region is small enough, the interaction can be described by the Ha tonian H int52kx~a†S21aS1!, ~3.1! wherek is a coupling constant andx denotes the correspond ing position for the center of mass of the atom. We assu the Raman-Nath regime@11#, where the motion of the atom alongX during its passage through the standing wave may ignored, and the kinetic energy in the Hamiltonian can neglected. In this limit there is a transverse momentum s associated with each eigenvalue ofk(a†S21aS1). The out- put transverse-momentum distributionPout(`,d) for a given field-phase shiftd is Pout~`,d!5Pin~0!Pin~` !1 ( n51,6 ` Pin ~6 !~n,d!Pin~`7ktAn!, ~3.2! wherePin(`) is the initial distribution,t is the interaction time, Pin~0!5tr@r inD~0!#, Pin ~6 !~n,d!5tr@r inD ~6 !~n,d!#, ~3.3! with D~0!5ug,0&^g,0u, D~6 !~n,d!5 1 2 ~ ug,n&^g,nu1ue,n21&^e,n21u 6eidue,n21&^g,nu6e2 idug,n&^e,n21u!, ~3.4! andr in is the initial density matrix representing the field a the internal state of the atom. We have assumed that initi a- se on ed e n ing - r- d il- e e e ift ly ` is uncorrelated with the rest of variables. The output pro abilitiesPout(`,d) can be measured by observing the spa distribution of the atom far away from the standing field. F simplicity, we consider a Gaussian forPin(`), Pin~` !5 1 A2ps` exp2S `2 2s` 2 D . ~3.5! If kt/s` is large enough, the output distribution consists separate peaks centered at the values `56ktAn. ~3.6! According to Eq. ~3.2!, we have Pout(`56ktAn,d) }Pin (6)(n,d), so the peak heights at these points give direc the probabilities~3.3!. As in Sec. II the knowledge of these probabilities f three phase shifts,d r52pr /3, r 50,61, allows us to infer the statistics of any linear combination of ug,n&^g,nu1ue,n21&^e,n21u, ug,0&^g,0u, ue,n21&^g,nu, ug,n&^e,n21u ~3.7! by inverting Eq. ~3.4! with the help of a discrete Fourie analysis. The set of observables that can be determined now smaller than before, since the separate contribution of projectorsug,n&^g,nu and ue,n21&^e,n21u is not available without further measurements. Nevertheless, we shall s in Sec. IV that this scheme also provides the measuremen the phase difference between the atomic dipole and the fi IV. MEASUREMENT OF THE ATOM-FIELD RELATIVE PHASE We have shown that the statistics of observables comm ing with Sz1a†a can be derived from the probabilities o these measurements. This must include the atom-field p difference. The operatorSz1a†a generates identical shifts in the field and atomic-dipole phases, leaving the phase dif ence unchanged. Then the quantum description of the r tive phase must commute withSz1a†a. Besides this general remark, it is still necessary to spe what may be understood as the quantum translation of variable. Because of the problematic description of pha several different approaches are available. Two main po bilities have been examined recently@5,6#. It has been shown that a unitary operatorEf representing the exponential of th phase difference and commuting withSz1a†a is defined by the polar decomposition@5# S2a†5AS2S1a†aEf5EfAS1S2aa†. ~4.1! The simultaneous eigenvectors ofEf andSz1a†a are uf0 ~0!&5ug,0&, uf6 ~n!&5 1 & ~ ug,n&6 i ue,n21&), ~4.2! with eigenvalues fo th e fo ct n en se s ar - s ach sure ced II. la- ent. u- acti- ere effi- ce. he al- and ise qs. n rue in- g e ue 3108 57A. LUIS AND L. L. SÁNCHEZ-SOTO Efuf0 ~0!&5uf0 ~0!&, Efuf6 ~n!&56 i uf6 ~n!&, ~Sz1a†a!uf~n!&5S n2 1 2D uf~n!&. ~4.3! These eigenvectors define a joint probability distribution the phase difference andSz1a†a as Pin~0,f0 ~0!!5tr~r inuf0 ~0!&^f0 ~0!u!, Pin~n,f6 ~n!!5tr~r inuf6 ~n!&^f6 ~n!u!. ~4.4! Focusing on the scheme analyzed in Sec. II, we have the orthogonal projectors~4.4! are linear combinations of th operators~2.9!. Equations~2.8! give uf0 ~0!&^f0 ~0!u5D~g,0!, uf6 ~n!&^f6 ~n!u5 1 2 @D~e,n21,d!1D~g,n,d!# 7 1 3s4n ( r @e2 idrD~e,n21,d r ! 2eidrD~g,n,d r !#. ~4.5! Then, the statistics of the phase-difference operator can extracted from the measured probabilitiesPout( j ,n,d). Similarly, the scheme examined in Sec. III can be used this purpose, since the projection measure in Eq.~4.4! is a linear combination of the operators~3.7!. This shows that this operator is indirectly measured in both schemes. Nevertheless, the arrangement based on atomic defle also allows the direct measurement of this phase-differe operator. To this end, the parameterd in Sec. III can be chosen to bed5p/2. In such a case, the transverse mom tum distribution at the peaks is proportional to the pha difference probabilities~4.4! since, from Eq.~3.4!, we have D~6 !~n,d5p/2!5uf6 ~n!&^f6 ~n!u, ~4.6! and then Pout~`56ktAn,d5p/2!}Pin~n,f6 ~n!!. ~4.7! Besides the operator description, there is also the po bility of describing the phase difference by taking the m ginal distribution for this variable from a joint positive operator measure for the dipole and field absolute pha @6,7,12#. The result is a positive-operator measureL~f! com- muting with Sz1a†a, L~f!5 ( n50 ` L~n,f!, ~4.8! whereL(n,f) has the general form r at be r ion ce - - si- - es L~0,f!5 1 2p ug,0&^g,0u, L~n,f!5 1 2p ~ ue,n21&^e,n21u1ug,n&^g,nu 1mneifue,n21&^g,nu1mn* e2 ifug,n&^e,n21u!, ~4.9! andmn are parameters depending on the particular appro used for the absolute phases. This positive-operator mea is again a linear combination of the operators~2.9! or ~3.7!, so the corresponding probability distribution can be dedu from the statistics of the measurements in Secs. II and I The preceding calculations show how the atom-field re tive phase can be theoretically inferred from measurem However, this inference will be affected by the limited acc racy attainable in real measurements. In the case of a pr cal realization of the arrangement examined is Sec. II, th are several sources of uncertainty such as detection in ciencies or the spread of the interaction time, for instan Therefore, it would be desirable to examine in which way t reconstructed values would deviate from the true input v ues under realistic practical conditions. The consequences of nonunit detection efficiencies the way they can be dealt with have been well studied@13#, so we will consider in some detail the effect of an imprec determination of the interaction time. We can see in E ~4.5! or ~4.9! that the interaction time appears only whe obtaining the nondiagonal matrix elements^e,n21ur inug,n& from Eqs.~2.8!. For instance, we have 2i 3 ( r eidrPout~g,n,d r !5s4n^e,n21ur inug,n&. ~4.10! To calculate the deviation of the inferred values from the t ones, we will assume a Gaussian distribution of possible teraction times, W~t8!5 1 A2pst exp2 ~t82t!2 2st 2 , ~4.11! with average interaction timet and widthst . The measured values ^e,n21ur inug,n&meas are obtained after averagin Pout(g,n,d r) over the distributionW(t8), leading to ^e,n21ur inug,n&meas5 1 s4n 2i 3 ( r eidr P̄out~g,n,d r ! 5 s̄4n s4n ^e,n21ur inug,n&, ~4.12! where P̄out(g,n,d r) and s̄4n denote the corresponding tim averages. Since s̄4n5s4ne22l2st 2n, ~4.13! we have the following relation between inferred and tr values fo t o te un ob l c g er x al of re a m on gi er on no e an o om - r e ft o e s ta e mic ment les . II, to in the d as no er e eld. - 57 3109DETERMINATION OF ATOM-FIELD OBSERVABLES VIA . . . ^e,n21ur inug,n&meas5e22l2st 2n^e,n21ur inug,n&. ~4.14! This relation leads to a measured probability distribution the relative phase which is broader and smoother than true one. This is the deteriorating effect of the spread interaction times. In principle, this effect can be numerically compensa for, provided that the distributionW(t8) were known. How- ever, such compensation will increase the effect of an avoidably imprecise experimental determination of the pr abilities Pout(g,n,d r). Next we evaluate relation~4.14! under realistic practica values for the parameters involved. We will see that, in fa this can be a rather small effect. In principle, any avera interaction timet is valid provided thats4nÞ0, so that Eq. ~4.10! can be inverted. However, from a more practical p spective, the optimum value will be the one leading to ma mum s4n , in order to minimize the effect of experiment noise inPout(g,n,d r) and the lack of a precise knowledge t. If the field state has a mean photon numbern̄, we can imposes4n̄;1, which meansltAn̄;1. Then Eq.~4.14! can be written as ^e,n21ur inug,n&meas5e22~n/n̄!~st /t!2 ^e,n21ur inug,n&. ~4.15! Accuracies in the interaction time of the order of 1% a within the current experimental values, so th (st /t)2;1024, and the exponential in Eq.~4.15! is of the order of unity. To be more specific, we can consider a cromaser configuration: for Rydberg atoms the coupling c stantl can be of the order of 104– 106 s21, and mean atomic velocities are in the range 101– 103 m/s, with a velocity spread of 1%. This leads to average interaction times ran from 1025 to 1023 s with a time spreadst /t;0.01@8,9,14#. With this typical values, the optimumltAn̄;1 can be sat- isfied even in the case of very small photon numbers. Th fore, we conclude that under actual experimental conditi the deviation of the inferred values from the true ones is noticeable. This discussion could be extended also to the arrangem analyzed in Sec. III. In such a case the interaction time its spread will affect the position and width, respectively, the peaks of the spatial distribution of atoms far away fr the standing field. Each peak at̀56ktAn will have a width s` 21(kst) 2n. A numerical evaluation using param eters within the preceding ranges and previously conside values fors` @4# shows that for small photon numbers th corresponding peaks are clearly distinguishable, even a including the interaction-time spread. V. QUANTUM ATOMIC-STATE RECONSTRUCTION So far we have been concerned with a determination atom-field variables assuming that before the measurem the density matrix is completely unknown. It can be intere ing to examine what happens when the initial system s factorizes r in5r in a ^ r in f , and the initial field stater in f is known. In such a case, the information supplied by the m r he f d - - t, e - i- t i- - ng e- s t nt d f ed er f nt t- te a- surement can be regarded as information about the ato state. In other words, this leads to a generalized measure of the atomic system. We shall study the atomic observab that can be determined by using the arrangement of Sec for instance. The knowledge of the initial field state can be taken in account by performing the trace over the field variables Eq. ~2.5!. This leads to a positive-operator measure in Hilbert space of the atom Da~ j ,n,d!5trf@r in f D~ j ,n,d!#, ~5.1! and the statistics of the measurement can be regarde depending only on the atomic state Pout~ j ,n,d!5tra@r in a Da~ j ,n,d!#. ~5.2! This atomic positive-operator measure is given by Da~g,n,d!5cn 2rn,nug&^gu1sn 2rn21,n21ue&^eu 1 isncn~eidrn,n21ue&^gu2e2 idrn21,nug&^eu!, Da~e,n,d!5sn11 2 rn11,n11ug&^gu1cn11 2 rn,nue&^eu 2 isn11cn11~eidrn11,nue&^gu 2e2 idrn,n11ug&^eu!, ~5.3! where the field matrix elementsrn,n85^nur in f un8& are known quantities. As a matter of fact, the photon-number variable carries relevant information, so we can remove it by summing ov n, Da~g,d!5 ( n50 ` Da~g,n,d!5cug&^gu1sue&^eu 1 i 2 ~eiddue&^gu2e2 idd* ug&^eu!, Da~e,d!5 ( n50 ` Da~e,n,d!5I 2Da~g,d!, ~5.4! where I is the identity in the atomic Hilbert space, and th parametersc, s, andd are c5 ( n50 ` cos2~ltAn!^nur in f un&, s5 ( n50 ` sin2~ltAn11!^nur in f un&, d5 ( n50 ` sin~2ltAn11!^n11ur in f un&. ~5.5! This corresponds to perform no measurement on the fi After the interaction only the atomic level is detected. Equations~5.4! can be inverted following the same pro cedure of preceding sections and, provided thatdÞ0 andc Þs, we have th s n th i o ex e e l o in si he can used x of a- to he o- tate . field the be nant trol- e the his the m s a ed ro- on- 3110 57A. LUIS AND L. L. SÁNCHEZ-SOTO ue&^eu5 1 3~s2c! F( r Da~g,d r !23cIG , ug&^gu5 1 3~c2s! F( r Da~g,d r !23sIG , ue&^gu5 22i 3d ( r e2 idrDa~g,d r !, ug&^eu5 2i 3d* ( r eidrDa~g,d r !. ~5.6! Since every operator acting in the Hilbert space of atom ~in particular the atomic density matrixr in a ! is a linear combination of these four operators, this scheme allows u determine the statistics of any atomic observable as a fu tion of the measured probabilities for three values of field-phase shift. This is equivalent to say that the atom state can be reconstructed from the measurement. A simple and common choice for the field state is a c herent oneua&. The coefficientsc, s, andd can be calcu- lated very accurately by means of available analytical pressions@15#. To simplify the result as far as possible, th interaction timet can be chosen such thatlt!1, while the amplitude of the coherent state is high enough as to giv finite value forltuau @16#. In this case we are at the initia stages of the resonant evolution, long before the first c lapse, and we have c.cos2~ltuau!, s.sin2~ltuau!, d.eiu sin~2ltuau!, ~5.7! whereu is the phase of the coherent complex amplitudea. Such a limit corresponds to the semiclassical atom-field teraction. The previous conditions are tantamount to con ering that the atom-field interaction is governed by t Hamiltonian H5vSz1l~a* eivtS21ae2 ivtS1!, ~5.8! ry ev - e to c- e c - - a l- - d- wherea is a classical complex amplitude. The atomic state determination this scheme provides be regarded as equivalent to the tomographic procedure for the reconstruction of field states@2#. State reconstruction for finite-dimensional systems has been studied before@17#. It has been shown that the elements of the density matri an arbitrary spinj are completely determined by the me surement of the spin projection along 4j 11 different direc- tions. The procedure followed in this section corresponds j 5 1 2. In the semiclassical limit, the measurement of t atomic populationSz after the field phase shifts and the res nant interaction is in fact the measurement on the initial s of the three spin projectionsSr , Sr5cos~2ltuau!Sz1sin~2ltuau!sin~d r1u!Sx 1sin~2ltuau!cos~d r1u!Sy , ~5.9! where Sx5(ue&^gu1ug&^eu)/2 and Sy5 i (ug&^eu2ue&^gu)/2 are the spin operators associated with the two-level atom Finally, whenr in a instead ofr in f is known in advance, the schemes studied in this work become measurements of observables. This possibility has been already studied for arrangements considered in Secs. II@18# and III @4#. VI. CONCLUSIONS We have found the atom-field observables that can determined from some available measurements after reso interaction. The dependence of the arrangement on con lable parameters, like phase shifts, can be used to deduc statistics of different observables. 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