Total Quantum Zeno Effect and Intelligent States for a Two Level System in a Squeezed Bath

(1)PHYSICAL REVIEW A 74, 052107 共2006兲. Total quantum Zeno effect and intelligent states for a two-level system in a squeezed bath D. Mundarain,1 M. Orszag,2 and J. Stephany1. 1. Departmento de Física, Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080A, Venezuela 2 Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile 共Received 18 August 2006; published 14 November 2006兲. In this work we show that, by frequent measurements of adequately chosen observables, a complete suppression of the decay in an exponentially decaying two-level system interacting with a squeezed bath is obtained. The observables for which the effect is observed depend on the squeezing parameters of the bath. The initial states that display total Zeno effect are intelligent states of two conjugate observables associated to the electromagnetic fluctuations of the bath. DOI: 10.1103/PhysRevA.74.052107. PACS number共s兲: 03.65.Xp, 03.65.Ta. I. INTRODUCTION. II. TOTAL ZENO EFFECT IN UNSTABLE SYSTEMS. Frequent measurements modify the dynamics of a quantum system. This result of quantum measurement theory is known as the quantum Zeno effect 共QZE兲 关1–5兴 and has attracted much attention since first discussed. The QZE is related to the suppression of induced transitions in interacting systems or the reduction of the decay rate in unstable systems. It has been also pointed out that the opposite effect, i.e., the acceleration of the decay process, may also be caused by frequent measurement, and this effect is known as anti-Zeno effect 共AZE兲. The experimental observation of QZE in the early days was restricted to oscillating quantum systems 关6兴, but recently, both QZE and AZE were successfully observed in irreversible decaying processes 关7–9兴. The quantum theory of measurement predicts a reduction in the decay rate of an unstable system if the time between successive measurements is smaller than the Zeno time, which, in general, is smaller than the correlation time of the bath. The effect is universal in the sense that it does not depend on the measured observable whenever the time between measurements is very small. This observation does not preclude the manifestation of the Zeno effect for larger times 共for example, in the exponentially decaying regime兲 for wellselected observables in a particular bath. Recently, it was shown in Ref. 关10兴 that reduction of the decay rate occurs in an exponentially decaying two-level system when interacting with a squeezed bath. In this case, variations on the squeezing phase of the bath may lead to the appearance of either the Zeno or anti-Zeno effect when continuously monitoring the associated fictitious spin. In this work, we show that for the same system it is possible to select a couple of observables whose measurement for adequately prepared systems lead to the total suppression of the transitions, i.e., total Zeno effect. This paper is organized as follows: In Sec. II, we discuss some general facts and review some results obtained in Ref. 关10兴 that are needed for our discussion. In Sec. III, we define the system we deal with and identify the observables and the corresponding initial states that are shown to display the total Zeno effect. In Sec. IV, we show that the initial states that show the total Zeno effect are intelligent spin states, i.e., states that saturate the Heisenberg uncertainty relation between the 共fictitious兲 spin operators. Finally, we discuss the results in Sec. V.. Consider a closed system with Hamiltonian H and an observable A with a discrete spectrum. If the system is initialized in an eigenstate 兩an典 of A with eigenvalue an, then the probability of survival in a sequence of S measurements 共that is, the probability that in all measurements one gets the same result an兲 is. 1050-2947/2006/74共5兲/052107共5兲. 冉. Pn共⌬t,S兲 = 1 −. 冊. ⌬t2 2 S ⌬H , ប2 n. 共1兲. where ⌬2nH = 具an兩H2兩an典 − 具an兩H兩an典2. 共2兲. and ⌬t is the time between consecutive measurements. In the limit of continuous monitoring 共S → ⬁, ⌬t → 0, and S⌬t → t兲, Pn → 1 and the system is locked in the initial state. For an unstable system and considering evolution times larger than the correlation time of the reservoir with which it interacts, the irreversible evolution is well described in terms of the Liouville operator L兵␳其 by the master equation. ⳵␳ = L兵␳其. ⳵t. 共3兲. If measurements are done frequently, then the master equation 共3兲 is modified. The survival probability is time dependent and shown to be given by 关10兴 Pn共t兲 = exp兵具an兩L兵兩an典具an兩其兩an典t其.. 共4兲. This expression is valid only when the time between consecutive measurements is small enough but greater than the correlation time of the bath. For mathematical simplicity in what follows, we consider the zero correlation time limit for the bath and then one is allowed to take the limit of continuous monitoring. From Eq. 共4兲, one observe that the total Zeno effect is possible when 具an兩L兵兩an典具an兩其兩an典 = 0.. 共5兲. Then, for times larger than the correlation time, the possibility of having total Zeno effect depends on the dynamics of the system 共determined by the interaction with the baths兲, on the particular observable to be measured, and on the initial state.. 052107-1. ©2006 The American Physical Society.

(2) PHYSICAL REVIEW A 74, 052107 共2006兲. MUNDARAIN, ORSZAG, AND STEPHANY. The concept of zero correlation time of the bath ␶D is, of course, an idealization. If this time is not zero, then Eq. 共4兲 is only approximate, since ⌬t cannot be strictly zero and at the same time be larger than ␶D. Also, if Eq. 共5兲 is satisfied, then Eq. 共4兲 must be corrected, taking the next nonzero contribution in the expansion of ␳共⌬t兲. In that case, a decay rate proportional to ⌬t appears and the decay time is ⬀ ␥21⌬t , which is, in general, a number much larger than the typical evolution time of the system. Note that as the spectrum of the squeezed bath gets broader, ␶D becomes smaller and one is able to choose a smaller ⌬t, approaching in this way, the ideal situation and the total Zeno effect.. F. 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4. 6 5 4. 0 0.5. 3. φ. 1. 2. 1.5 2. 1. 2.5 0. θ. 3. III. TOTAL ZENO OBSERVABLES. In the interaction picture, the Liouville operator for a twolevel system in a broadband squeezed vacuum has the following structure 关11兴:. FIG. 1. F共␾ , ␪兲 for N = 1 and ␺ = 0.. 冋. L兵␳其 = 21 ␥共N + 1兲共2␴−␳␴+ − ␴+␴−␳ − ␳␴+␴−兲 + 21 ␥N共2␴+␳␴− − ␴−␴+␳ − ␳␴−␴+兲 − ␥ Mei␾␴+␳␴+ − ␥ Me−i␾␴−␳␴− ,. 共7兲. with ␴x, ␴y, and ␴z the Pauli matrices. Let us introduce the Bloch representation of the two-level density matrix. ␳ = 21 共1 + ␳ជ · ␴ជ 兲.. 共8兲. ␳z共t兲 = ␳z共0兲e−␥共2N+1兲t +. +. + cos共␺兲␴y兴. 共9兲. and has the following solutions for Bloch vector components, which give the behavior of the system without measurements. 冋. ␳x共t兲 = ␳x共0兲 sin2. 冋. 冉冊. 冉 冊 冉 冊册. ␺ ␺ ␺ + ␳y共0兲 sin cos 2 2 2. 冉冊. e−␥共N+1/2−M兲t. 共12兲. 共13兲 The eigenstates of ␴␮ are. 冉冊 冉冊. 兩 + 典␮ = cos. ␪ ␪ 兩 + 典 + sin exp共i␾兲兩− 典, 2 2. 共14兲. 兩− 典␮ = − sin. ␪ ␪ 兩 + 典 + cos exp共i␾兲兩− 典. 2 2. 共15兲. If the system is initialized in the state 兩 + 典␮, then the survival probability at time t is P␮+ 共t兲 = exp兵F共␪, ␾兲t其,. 共16兲. F共␪, ␾兲 = ␮具+ 兩L兵兩 + 典␮␮具+ 兩其兩 + 典␮ .. 共17兲. where In this case, the function F共␪ , ␾兲 has the structure F共␪, ␾兲 = − 21 ␥共N + 1兲关␳z共0兲 + ␳z2共0兲 + 21 ␳2x 共0兲 + 21 ␳2y 共0兲兴 + 21 ␥N关共␳z共0兲 − ␳z2共0兲 − 21 ␳2x 共0兲 − 21 ␳2y 共0兲兴. 冉 冊 冉 冊册. ␺ ␺ ␺ + ␳x共0兲cos2 − ␳y共0兲sin cos 2 2 2 ⫻e−␥共N+1/2+M兲t ,. 1 关e−␥共2N+1兲t − 1兴. 2N + 1. ␴␮ = ␴ជ · ␮ˆ = ␴x cos共␾兲 sin共␪兲 + ␴y sin共␾兲 sin共␪兲 + ␴z cos共␪兲.. + 21 ␥N关共1 − ␳z兲␴z − 21 ␳x␴x − 21 ␳y␴y兴 − 21 ␥ M ␳x关cos共␺兲␴x − sin共␺兲␴y兴. 共11兲. Now consider the Hermitian operator ␴␮ associated to the fictitious spin component in the direction of the unitary vecˆ = 关cos共␾兲 sin共␪兲 , sin共␾兲 sin共␪兲 , cos共␪兲兴 defined by the tor ␮ angles ␪ and ␾. ⳵␳ = − 21 ␥共N + 1兲关共1 + ␳z兲␴z + 21 ␳x␴x + 21 ␳y␴y兴 ⳵t. 1 2 ␥ M ␳ y 关sin共␺兲␴x. ␺ ␾ ␺ − ␳x共0兲sin cos 2 2 2. 冉冊 冉冊. In this representation, the master equation takes the form. e−␥共N+1/2−M兲t. ⫻e−␥共N+1/2+M兲t ,. where ␥ is the vacuum decay constant, and N , M = 冑N共N + 1兲 and ␺ are the parameters of the squeezed bath. Here ␴− and ␴+ are the two ladder operators. ␴− = 21 共␴x + i␴y兲,. 冉 冊 冉 冊册 冉 冊 冉 冊册. ␺ ␺ ␺ + ␳x共0兲sin cos 2 2 2. + ␳y共0兲sin2. 共6兲. ␴+ = 21 共␴x + i␴y兲,. 冉冊 冋 冉冊. ␳y共t兲 = ␳y共0兲cos2. − 21 ␥ M ␳x共0兲关cos共␺兲␳x共0兲 − sin共␺兲␳y共0兲兴 共10兲 052107-2. + 21 ␥ M ␳y共0兲关sin共␺兲␳x共0兲 + cos共␺兲␳y共0兲兴,. 共18兲.

(3) PHYSICAL REVIEW A 74, 052107 共2006兲. TOTAL QUANTUM ZENO EFFECT AND INTELLIGENT… 1.02. 1.02 without measurements with measurements. 1.01 1. 1 < σµ > (t). < σµ > (t). without measurements with measurements. 1.01. 0.99. 1. 1. 0.99 0.98. 0.98. 0.97. 0.97. 0.96. 0.96. 0.95. 0.95 0. 0.2. 0.6. 0.4. 0.8. 1. 0. γ t. 0.2. 0.4. 0.6. 0.8. 1. γ t. FIG. 2. 具␴␮1共t兲典 for N = 1 and ␺ = 0. Solid circles: no measurements. Empty circles: with measurements. One measures ␴␮1, and the initial state is 兩 + 典␮1.. FIG. 3. 具␴␮1共t兲典 for N = 1 and ␺ = 0. Solid circles: no measurements. Empty circles: with measurements. One measures ␴␮1, and the initial state is 兩−典␮1.. ˆ is a function of the angles. where now ␳ជ 共0兲 = ␮ In Fig. 1, we show F共␾ , ␪兲 for N = 1 and ␺ = 0 as function of ␾ and ␪. The maxima correspond to F共␾ , ␪兲 = 0. For arbitrary values of N and ␺, there are two maxima corresponding to the following angles:. cussion of frequent measurements, the system is frozen in the state 兩 + 典␮1 共total Zeno effect兲. In Fig. 3, we show the time evolution of 具␴␮1典 when the initial state is 兩−典␮1 without measurements and with measurements of the same observable as in the previous case. One observes that with measurements the system evolves from 兩−典␮1 to 兩 + 典␮1. In general for any initial state, the system under frequent measurements evolves to 兩 + 典␮1, which is the stationary state of Eq. 共21兲 whenever we do measurements in ␴␮1. Analogous effects are observed if one measures ␴␮2. In contrast, for measurements in other directions different from ˆ 1 or ␮ ˆ 2, the system evolves to states that those defined by ␮ are not eigenstates of the measured observables.. ␾1M =. ␲−␺ 2. cos共␪ M 兲 = −. and. 1 共19兲 2共N + M + 1/2兲. and. ␾2M =. ␲−␺ +␲ 2. and. cos共␪ M 兲 = −. 1 . 2共N + M + 1/2兲. 共20兲. ˆ1 These preferential directions given by the vectors ␮ = 关cos共␾1M 兲 sin共␪ M 兲 , sin共␾1M 兲 sin共␪ M 兲 , cos共␪ M 兲兴 and ␮ˆ 2 = 关cos共␾2M 兲 sin共␪ M 兲 , sin共␾2M 兲 sin共␪ M 兲 , cos共␪ M 兲兴 define the operators ␴␮1 and ␴␮2, which show total Zeno effect if the initial state of the system is the eigenstate 兩 + 典␮1 or, respectively, 兩 + 典␮2. To be specific let us consider measurements of the observជ · ␮ˆ 1 共analogous results are obtained for ␴␮2兲. The able ␴␮1 = ␴ modified master equation with measurements of ␴␮1 is given by 关10兴. ⳵␳ = P␮+ L兵␳其P␮+ + 共1 − P␮+ 兲L兵␳其共1 − P␮+ 兲, 1 1 1 1 ⳵t. 共21兲. P␮+ = 兩 + 典␮1 ␮ 具+ 兩. 共22兲. where 1. 1. and L兵␳其 is given by 共6兲. Besides of the total Zeno effect obtained in the cases specified above, it is also very interesting to discuss the effect of measurements for other choices of the initial state. This can be done numerically. In Fig. 2, we show the evolution of 具␴␮1典, which is the mean value of observable ␴␮1, when the system is initialized in the state 兩 + 典␮1 without measurements 关master equation 共6兲兴 and with frequent monitoring of ␴␮1 关master equation 共21兲兴. Consistently with our dis-. IV. INTELLIGENT STATES. Aragone et al. 关12兴 considered well-defined angular momentum states that satisfy the equality 共⌬Jx⌬Jy兲2 = 41 兩 具Jz典兩2in the uncertainty relation. They are called “intelligent states” in the literature. The difference with the coherent or squeezed states, associated to harmonic oscillators, is that these intelligent states are not minimum uncertainty states 共MUS兲, since the uncertainty is a function of the state itself. In this section, we show that the states 兩 + 典␮1 and 兩 + 典␮2 are intelligent states of two observables associated to the bath fluctuations. The master equation 共6兲 can be written in an explicit Lindblad form. ⳵␳ ␥ = 兵2S␳S† − ␳S†S − S†S␳其, ⳵t 2. 共23兲. using only one Lindblad operator S, S = 冑N + 1␴− − 冑N exp兵i␺其␴+ = cosh共r兲␴− − sinh共r兲exp兵i␺其␴+ .. 共24兲. Obviously, the eigenstates of S satisfy the condition 共5兲. Moreover, the states 兩␾1典 = 兩 + 典␮1 and 兩␾2典 = 兩 + 典␮2 are the two eigenstates of S with eigenvalues ␭± = ± i冑M exp兵i␺ / 2其. 052107-3.

(4) PHYSICAL REVIEW A 74, 052107 共2006兲. MUNDARAIN, ORSZAG, AND STEPHANY. S兩␾1,2典 = ␭±兩␾1,2典.. Consider now the standard fictitious angular momentum operators for the two-level system are 兵Jx = ␴x / 2 , Jy = ␴y / 2 , Jz = ␴z / 2其 and also two rotated operators J1 and J2, which are consistent with the electromagnetic bath fluctuations in phase space 共see Fig. 2 in Ref. 关10兴兲 and which satisfy the same Heisenberg uncertainty relation that Jx and Jy. They are. 再 冎 再 冎. J1 = exp. − i␺ i␺ Jxexp 2Jz 2Jz. 冉冊. =cos. 冉冊. ␺ ␺ Jx − sin Jy , 2 2. ⌬ 2J 1⌬ 2J 2 =. with exp兵␤其 =. 冉冊. 1 2. 兲t其 ,. 共28兲. 具J2典共t兲 = 具J2典共0兲exp兵− ␥共N − M +. 1 2. 兲t其 .. 共29兲. Note that the above averages decay with maximum and minimum rates, respectively. In terms of J1 and J2, we have. 再冎 再 冎. i␺ 共J1 − iJ2兲, J− = ␴ = 共Jx − iJy兲 = exp 2 − i␺ 共J1 + iJ2兲. 2. 再冎. i␺ 关cosh共r兲 − sinh共r兲兴共J1 − i␣J2兲 S = exp 2. cosh共r兲 + sinh共r兲 ␣= = exp兵2r其. cosh共r兲 − sinh共r兲. 共31兲. 共32兲. 冑. 共38兲. sinh共r兲 . cosh共r兲. 共39兲. exp兵␤r其 =. 冑. ␲ , 2. 共40兲. 冉 冊. sinh共r兲 N = cosh共r兲 N+1. 1/4. ,. 共33兲. 共34兲. 再冎 再 冎 再 冎 再冎 再 冎 再 冎 再 冎. S = 2i冑M exp. ␲ i␺ exp i Jz exp兵␤rJz其J1 2 2. ␲ ⫻exp − i Jz exp兵− ␤rJz其. 2. 共42兲. Finally, S may be written in the form S = 2i冑M exp. with. i␺ UJzU−1 2. 共43兲. ␲ ␺ ␲ U = exp i Jz exp兵␤rJz其exp i Jz exp − i Jy . 2 2 2 共44兲. Then the eigenstates of S could be obtained from the eigenstates of Jz using U as 共45兲. where C0 is a normalization constant. It is quite clear that 兩␾1,2典 are intelligent states of the observables J1 , J2, which are rotated versions of Jx , Jy. They are also quasi-intelligent states of the original observables Jx , Jy 关13兴. One can verify the above result, by finding directly the eigenstates of S, 兩␾1,2典 =. so that S = exp兵i␺/2其关cosh共r兲 − sinh共r兲兴共1 − ␣2兲1/2J−共␣兲. 共35兲. 共41兲. S takes the form,. 兩␾1,2典 = C0U兩 ± 典,. Following Rashid et al. 关13兴, we define a non-Hermitian operator J−共␣兲. After some algebra, one obtains then that. 1−␣ =i 1+␣. ␤i =. 共30兲. with. 共J1 − i␣J2兲 共1 − ␣2兲1/2. 冑. 共27兲. 具J1典共t兲 = 具J1典共0兲exp兵− ␥共N + M +. J −共 ␣ 兲 =. 共37兲. In terms of the real and imaginary parts of ␤ = ␤r + i␤i,. ␺ ␺ Jx + cos Jy . 2 2. J+ = ␴† = 共Jx + iJy兲 = exp. 兩具Jz典兩2 . 4. J−共␣兲 = exp兵␤Jz其J1exp兵− ␤Jz其. 共26兲. These two operators are associated respectively with the major and minor axes of the ellipse, which represents the fluctuations of the bath. Using Eqs. 共10兲 and 共11兲, one can show that the mean values of these operators have the following exponentially decaying evolution:. S has the form. The eigenstates of S are then eigenstates of J−共␣兲 with eigenvalues ±1 / 2. The eigenstates of J−共␣兲 are also shown to be intelligent states, i.e., they satisfy the equality condition in the Heisenberg uncertainty relation for J1 and J2. − i␺ i␺ Jyexp 2Jz 2Jz. 冉冊. =sin. 共36兲. The operator J−共␣兲 is obtained from J1 by the following transformation:. 再 冎 再 冎. J2 = exp. S = 2␭+J−共␣兲.. 共25兲. 冑. N 兩+典±i N+M. 冑. M −i␺/2 e 兩− 典. N+M. 共46兲. Finally, when the system is initialized in one of these states, the mean value of J1 is zero, 具J1典共t兲 = 0. Then, the term. 052107-4.

(5) PHYSICAL REVIEW A 74, 052107 共2006兲. TOTAL QUANTUM ZENO EFFECT AND INTELLIGENT…. with the biggest decaying rate does not appear in the mean value of the measured observable ␴␮, which becomes 具␴␮典共t兲 = 具J2典共t兲sin共␪ M 兲 + 具Jz典共t兲cos共␪ M 兲.. 共47兲. Using the definition of the angle ␪ , one can prove that M. d具␴␮典 共0兲 = 0, dt. 共48兲. which is a necessary condition in order to obtain total Zeno effect when one is measuring the observable ␴␮ 共see Fig. 2兲. V. DISCUSSION. We have shown that total Zeno effect is obtained for two particular observables ␴␮1 or ␴␮2, for which the azimuthal phases in the fictitious spin representation depend on the phase of the squeezing parameter of the bath and the polar phases depend on the squeeze amplitude r. In this sense, the parameters of the squeezed bath specify some definite atomic directions. When performing frequent measurements on ␴␮1, starting from the initial state 兩 + 典␮1, the system freezes at the initial state as opposed to the usual decay when no measurements are done. On the other hand, if the system is initially prepared in the state 兩−典␮1, the frequent measurements on ␴␮1 will make it evolve from the state 兩−典␮1 to 兩 + 典␮1. More gen-. 关1兴 B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 共1977兲. 关2兴 A. Peres and A. Ron, Phys. Rev. A 42, 5720 共1990兲. 关3兴 R. Passante, Phys. Rev. A 57, 1590 共1998兲. 关4兴 A. D. Panov, Ann. Phys. 249, 5720 共1990兲. 关5兴 A. G. Kofman and G. Kuritzki, Nature 共London兲 405, 546 共2000兲. 关6兴 W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295 共1990兲. 关7兴 S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Miu, B. Sudaram, and M. G. Raizen,. erally, when performing the measurements on ␴␮1, any initial state evolves to the same state 兩 + 典␮1, which is the steady state of the master equation 共21兲 in this situation. The above discussion could appear surprising at first sight. However, taking a more familiar case of a two-level atom in contact with a thermal bath at zero temperature, if one starts from any initial state, the atom will necessarily decay to the ground state. This is because the time evolution of 具␴z典 is the same with or without measurements of ␴z. In both cases, the system goes to the ground state, which is an eigenstate of the measured observable ␴z. In the limit N , M → 0, ␴␮1 → −␴z, and the state 兩 + 典␮1→ 兩−典z, which agrees with the known results. Finally, we also found that the eigenstates of S are also quasi-intelligent states of the observables Jx , Jy, i.e., intelligent states of the rotated version of the observables; that is, of J1 , J2. Starting from an eigenstate of ␴z, these intelligent states are obtained by applying the transformation defined by U. ACKNOWLEDGMENTS. Two of the authors 共D.M. and J.S.兲 were supported by Did-Usb Grant No. Gid-30 and by Fonacit Grant No. G-2001000712. M.O. was supported by Fondecyt Grant No. 1051062 and Nucleo Milenio Grant No. ICM共P02-049兲.. Nature 共London兲 387, 575 共1997兲. 关8兴 M. C. Fischer, B. Gutierrez-Medina, and M. G. Raizen, Phys. Rev. Lett. 87, 040402 共2001兲. 关9兴 P. E. Toschek and C. Wunderlich, Eur. Phys. J. D 14, 387 共2001兲. 关10兴 D. F. Mundarain and J. Stephany, Phys. Rev. A 73, 042113 共2006兲. 关11兴 C. W. Gardiner, Phys. Rev. Lett. 56, 1917 共1986兲. 关12兴 C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 共1976兲. 关13兴 M. A. Rashid, J. Math. Phys. 19, 1391 共1978兲.. 052107-5.


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A primeira é que existe uma relação de continuidade e ruptura entre as associações e sindicatos estudados; a segunda é que, diante da recuperação de momentos significativos na