- Email: [email protected]
Quantum Zeno-effect with polarized neutrons
Physica B 297 (2001) 299}302
Quantum Zeno-e!ect with polarized neutrons H. Rauch* Atominstitut der OG sterreichischen Universita( ten, Stadionallee 2, A-1020 Wien, Austria
Abstract Neutron spin rotation can be achieved by proper spin rotation stages. A division into several rotation stages does not change the outcome of the rotation as long as the same interaction power acts on the quantum system. This situation changes drastically when a state separation is made behind each stage. In this case the transition between quantum states becomes hindered and a kind of quantum state freezing appears. Various unavoidable quantum loss factors caused by the parasitic e!ects with any interaction potential and the wave packet structure give rise to various constraints for repetitive measurements and prevents a complete freezing of a quantum state in Zeno-like experiments. Additionally, a spectral change occurs, due to the dispersive action of the interaction, which has to be taken into account when many repetitive measurements are considered. A dedicated proposal for a repetitive neutron spin rotation experiment within a perfect crystal resonator will be analyzed in detail. Unavoidable losses due to quantum phenomena can be separated from losses caused by experimental imperfections. 2001 Elsevier Science B.V. All rights reserved. Keywords: Zeno e!ect; Interaction-free measurement; Neutron quantum optics
1. Introduction The well-known Zeno-phenomenon (paradoxon) can be taken as a characteristic example of a temporal or spatial evolution of a quantum system which is kept under frequent observation and which becomes frozen in the initial state [1}2]. Thus, an unstable quantum state which is continuously observed is never found to decay or a `watched kettle never boilsa. It is also an example of the fact that simpli"ed considerations show a completely di!erent e!ect than a realistic physical
* Corresponding author. Tel.: #43-1-588-01-141-00; fax: #43-1-588-01-141-99. E-mail address: [email protected] (H. Rauch).
one [5}6]. Proposals for related experiments and some realizations of them exist for various decay processes [8] for light polarization rotation experiments [9,10] and for neutron polarization rotation experiments [5,11}12], which will be discussed in more detail in this paper. The topic is closely related to the non-exponential decay of quantum states for very short times where a quadratic time dependence is expected and to so-called `interaction-freea measurements [10,14,15]. Here we discuss several neutron optical experiments which show unavoidable losses due to parasitic re#ection e!ects or due to the wave packet structure of realistic wave functions. It will also be shown that a general coupling in phase space exists which prohibits the view onto one parameter space of the system only. In the literature a so-called `anti-Zeno-e!ecta is mentioned as well [16]. It means a faster
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 8 5 2 - 8
300
H. Rauch / Physica B 297 (2001) 299}302
transition between quantum states and needs another not yet known interaction.
2. Zeno-e4ect experiments with neutrons The quantum Zeno-e!ect describes the hindrance of the transition between quantum states when a large number of successive measurements are performed to check whether the system is still in the initial state or not [1}2,17]. A quadratic behaviour of the survival probability for short times exists for the upper (or lower) Zeeman level when a magnetic "eld perpendicular to the polarization direction of a neutron beam is applied (e.g. Ref. [18]),
l P "cos * , > 2v
(1)
where l/v denotes the time the neutron with velocity v spends within such a "eld of length l and "2B/ is the Larmor rotation frequency. * When the thickness of the rotation coil ful"lls the condition l"l "(2m#1)v/ , complete spin * reversal occurs (P "0 for m"1,2, 2), which is > used in many polarized neutron experiments [19]. The same relation holds when the coil is separated into n shorter coils (l "nl) and no magnetic "elds act between the coils, as shown in Fig. 1. Similarly, more rotation stages (or more transitions through one coil) with correspondingly lower magnetic "elds can be used.
Fig. 2. Survival probability in the upper spin state in spin-#ip (n"1) and Zeno-like situations.
A completely di!erent situation exists when, behind each coil, a "ltering process takes place where only spin-up neutrons can pass through whereas spin-down neutrons are detected. A polarized He "lter could ful"ll these requirements because it acts simultaneously as "lter and detector for #ipped neutrons (e.g. Ref. [20]). When one uses the same "eld con"guration as discussed for the case without "lters, one gets
l P " cos * > 2v
+ cos
2n
P1 , (2) L i.e. no transition will occur and the initial state seems to be frozen (Fig. 2). This relation also holds when wave packets are used to describe the spatial part of the wave function. There may be deviations when the coherence length exceeds the dimen sion l of the "eld regions because then passage time and evolution time become di!erent [5}6,21]. In any case the magnetic interaction <"B is needed for any spin rotation. Each potential causes besides the transmitted a back re#ected wave, whose mean re#ectivity can be written as (e.g. Ref. [22]) 1 < ! , RM
! 2 2E
Fig. 1. Arrangement for successive spin rotation by several DC-coils (above) and arrangement for the observation of the neutron Zeno-phenomenon.
L
(3)
where < denotes the interaction potential of the ! # and ! spin state and E the kinetic energy of the neutrons. In the case of thermal neutrons this energy is orders of magnitudes larger than < and > < . The use of the average re#ectivity is justi"ed \ when the barrier produces spatial phase shifts larger than the coherence length of the beam.
H. Rauch / Physica B 297 (2001) 299}302
The coherence length follows from its momentum distribution k as k" (e.g. Ref. [23]). Al though RM is usually very small it has to be taken ! into account in the case of many repetitative measurements. As in the case of multiple barrier transmission, one should notice that each rotation stage also causes at least small parasitic beams which can add up in the case of large n, or when resonance conditions cause an essential change of the situation discussed above. These losses are velocity (momentum) dependent and change (narrow) the initial spectrum, which causes longer coherence lengths and an enforced collective action of the rotation stages. Thus, a situation may exist where the spin state does not change because the transition probability goes to zero (Eq. (2)) but the intensity changes due to parasitic quantum losses, which means that a perfect Zeno-e!ect is, in principle, unphysical. When n independent successive barriers are taken into account, the intensity of the # polarized beam becomes reduced at least as (Eq. (3)) I "P ¹M L"P (1!RM )L > > > 1 < L > "P 1! > 2 2E
n < > #2 && 0
1! 2 2E L
(4)
and an even enhanced reduction is expected when a collective action of the barriers occurs. Thus, a state variation is expected in any case. Nevertheless, there is no question about the existence of the Zeno-like phenomena for reasonably low n values, but the nPR limit is still under discussion. In this respect, an intrinsic contradiction concerning the existence of the classical Zeno-e!ect exists, which justi"es new experimental work. Such an experiment is in progress at the pulsed ISIS spallation source where we are using the features of our newly developed perfect crystal neutron resonator (Fig. 3) [24,25]. This instrument allows a multiple (up to 4000 times) transmission of neutrons through a spin rotation coil and a continuous measurement of the negative spin state due to the spin-dependent transmission of the exit perfect crystal. This spin-dependent transmission is
301
Fig. 3. Perfect crystal resonator system adapted for a neutron Zeno-e!ect experiment.
caused by the spin-dependent Zeeman shift of the neutrons inside the magnetic "eld applied to that crystal. The spin rotation angle in the #ipper coil can be controlled by the strength of a static rotation "eld, or by the amplitude of the oscillating #ipper "eld.
3. Discussion It has been shown that any interaction of a neutron beam with a potential produces unavoidable parasitic beams which are characteristic for this kind of interaction. Multiple potential barriers do not only produce an addiative e!ect of losses, but also show enhanced losses due to various resonance e!ects. Therefore, the Zeno-phenomenon, which is based on a repetitive interaction and observation of a quantum system, has to be discussed in a new light including unavoidable quantum losses. In this respect, the imperfect Zeno-phenomenon and the limit of a non-completely frozen state for an in"nite number of stages can be seen as a "ngerprint of quantum mechanics. In the limiting case, when the coherence length exceeds the
302
H. Rauch / Physica B 297 (2001) 299}302
dimension of the spin rotation stages ( *d), a de viation from Eq. (2) can be expected when the evolution time (l/v) has to be replaced by the passage time of the packet ( /v). In this case, the survival probability would also tend to zero. This limit is still being debated in the literature [5}6,15,21]. Related experiments with a perfect crystal neutron resonator are in progress.
Acknowledgements Very fruitful discussions with E. Jericha (Vienna), S. Pascazio (Bari) and Z. Hradil (Olomouc) are gratefully acknowledged. This work has been supported by the Austrian Science Foundation (project P13332-PHY).
References [1] B. Misra, E.C.G. Sudershan, J. Math. Phys. 18 (1977) 756. [2] L. Fonda, G.C. Ghirardi, A. Rimini, Rep. Progr. Phys. 41 (1978) 587. [3] D. Home, M.A.B. Whitaker, Phys. Lett. A 173 (1993) 327. [4] E. Joos, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin, 1996, (Chapter 3). [5] H. Nakazato, M. Namiki, S. Pascazio, H. Rauch, Phys. Lett. A 199 (1995) 27.
[6] A. Venugopalan, R. Ghosh, Phys. Lett. A 204 (1995) 11. [7] A. Pati, Phys. Lett. A 215 (1996) 7. [8] W.H. Itano, D.J. Heinzen, J.J. Bollinger, D.J. Wineland, Phys. Rev. A 41 (1990) 2295. [9] A. Peres, Am. J. Phys. 48 (1980) 931. [10] P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. Kasevich, Phys. Rev. Lett. 74 (1995) 4763. [11] S. Pascazio, M. Namiki, G. Badurek, H. Rauch, Phys. Lett. A 179 (1993) 155. [12] K. Machida, H. Nakazato, S. Pascazio, H. Rauch, S. Yu, Phys. Rev. A 60 (1999) 3448. [13] Z. Hradil, H. Nakazato, M. Namiki, S. Pascazio, H. Rauch, Phys. Lett. A 239 (1998) 333. [14] A.C. Elitzur, L. Vaidman, Found. Phys. 23 (1993) 987. [15] S.H. Simon, P.M. Platzman, Phys. Rev. A 61 (2000). [16] A.P. Balachandran, S.M. Roy, Phys. Rev. Lett. 84 (2000) 4019. [17] M.A.B. Whitaker, Phys. Lett. A 244 (1998) 502. [18] F. Mezei, Z. Physik 255 (1972) 146. [19] W.G. Williams, Polarized Neutrons, Clarendon Press, Oxford, 1988. [20] R. Surkau, J. Becker, M. Ebert, T. Grossmann, W. Heil, D. Hofmann, H. Humboldt, M. Leduc, E.W. Otten, D. Rohe, K. Siemensmeyer, M. Steiner, F. Tasset, N. Trautmann, Nucl. Instr. and Meth. A 384 (1997) 444. [21] J.-W. Pan, Y. Zhang, Q. Dong, Phys. Lett. A 238 (1998) 405. [22] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Vol. I, Wiley, New York 1977. [23] H. Rauch, H. Wolwitsch, H. Kaiser, R. Clothier, S.A. Werner, Phys. Rev. A 53 (1996) 902. [24] M. Schuster, H. Rauch, E. Seidl, E. Jericha, C.J. Carlile, Phys. Lett. A 144 (1990) 297. [25] E. Jericha, D.E. Schwab, C.J. Carlile, M.R. Jaekel, R. Loidl, H. Rauch, Nucl. Instr. and Meth. A 440 (2000) 597.
Quantum Zeno-e!ect with polarized neutrons H. Rauch* Atominstitut der OG sterreichischen Universita( ten, Stadionallee 2, A-1020 Wien, Austria
Abstract Neutron spin rotation can be achieved by proper spin rotation stages. A division into several rotation stages does not change the outcome of the rotation as long as the same interaction power acts on the quantum system. This situation changes drastically when a state separation is made behind each stage. In this case the transition between quantum states becomes hindered and a kind of quantum state freezing appears. Various unavoidable quantum loss factors caused by the parasitic e!ects with any interaction potential and the wave packet structure give rise to various constraints for repetitive measurements and prevents a complete freezing of a quantum state in Zeno-like experiments. Additionally, a spectral change occurs, due to the dispersive action of the interaction, which has to be taken into account when many repetitive measurements are considered. A dedicated proposal for a repetitive neutron spin rotation experiment within a perfect crystal resonator will be analyzed in detail. Unavoidable losses due to quantum phenomena can be separated from losses caused by experimental imperfections. 2001 Elsevier Science B.V. All rights reserved. Keywords: Zeno e!ect; Interaction-free measurement; Neutron quantum optics
1. Introduction The well-known Zeno-phenomenon (paradoxon) can be taken as a characteristic example of a temporal or spatial evolution of a quantum system which is kept under frequent observation and which becomes frozen in the initial state [1}2]. Thus, an unstable quantum state which is continuously observed is never found to decay or a `watched kettle never boilsa. It is also an example of the fact that simpli"ed considerations show a completely di!erent e!ect than a realistic physical
* Corresponding author. Tel.: #43-1-588-01-141-00; fax: #43-1-588-01-141-99. E-mail address: [email protected] (H. Rauch).
one [5}6]. Proposals for related experiments and some realizations of them exist for various decay processes [8] for light polarization rotation experiments [9,10] and for neutron polarization rotation experiments [5,11}12], which will be discussed in more detail in this paper. The topic is closely related to the non-exponential decay of quantum states for very short times where a quadratic time dependence is expected and to so-called `interaction-freea measurements [10,14,15]. Here we discuss several neutron optical experiments which show unavoidable losses due to parasitic re#ection e!ects or due to the wave packet structure of realistic wave functions. It will also be shown that a general coupling in phase space exists which prohibits the view onto one parameter space of the system only. In the literature a so-called `anti-Zeno-e!ecta is mentioned as well [16]. It means a faster
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 8 5 2 - 8
300
H. Rauch / Physica B 297 (2001) 299}302
transition between quantum states and needs another not yet known interaction.
2. Zeno-e4ect experiments with neutrons The quantum Zeno-e!ect describes the hindrance of the transition between quantum states when a large number of successive measurements are performed to check whether the system is still in the initial state or not [1}2,17]. A quadratic behaviour of the survival probability for short times exists for the upper (or lower) Zeeman level when a magnetic "eld perpendicular to the polarization direction of a neutron beam is applied (e.g. Ref. [18]),
l P "cos * , > 2v
(1)
where l/v denotes the time the neutron with velocity v spends within such a "eld of length l and "2B/ is the Larmor rotation frequency. * When the thickness of the rotation coil ful"lls the condition l"l "(2m#1)v/ , complete spin * reversal occurs (P "0 for m"1,2, 2), which is > used in many polarized neutron experiments [19]. The same relation holds when the coil is separated into n shorter coils (l "nl) and no magnetic "elds act between the coils, as shown in Fig. 1. Similarly, more rotation stages (or more transitions through one coil) with correspondingly lower magnetic "elds can be used.
Fig. 2. Survival probability in the upper spin state in spin-#ip (n"1) and Zeno-like situations.
A completely di!erent situation exists when, behind each coil, a "ltering process takes place where only spin-up neutrons can pass through whereas spin-down neutrons are detected. A polarized He "lter could ful"ll these requirements because it acts simultaneously as "lter and detector for #ipped neutrons (e.g. Ref. [20]). When one uses the same "eld con"guration as discussed for the case without "lters, one gets
l P " cos * > 2v
+ cos
2n
P1 , (2) L i.e. no transition will occur and the initial state seems to be frozen (Fig. 2). This relation also holds when wave packets are used to describe the spatial part of the wave function. There may be deviations when the coherence length exceeds the dimen sion l of the "eld regions because then passage time and evolution time become di!erent [5}6,21]. In any case the magnetic interaction <"B is needed for any spin rotation. Each potential causes besides the transmitted a back re#ected wave, whose mean re#ectivity can be written as (e.g. Ref. [22]) 1 < ! , RM
! 2 2E
Fig. 1. Arrangement for successive spin rotation by several DC-coils (above) and arrangement for the observation of the neutron Zeno-phenomenon.
L
(3)
where < denotes the interaction potential of the ! # and ! spin state and E the kinetic energy of the neutrons. In the case of thermal neutrons this energy is orders of magnitudes larger than < and > < . The use of the average re#ectivity is justi"ed \ when the barrier produces spatial phase shifts larger than the coherence length of the beam.
H. Rauch / Physica B 297 (2001) 299}302
The coherence length follows from its momentum distribution k as k" (e.g. Ref. [23]). Al though RM is usually very small it has to be taken ! into account in the case of many repetitative measurements. As in the case of multiple barrier transmission, one should notice that each rotation stage also causes at least small parasitic beams which can add up in the case of large n, or when resonance conditions cause an essential change of the situation discussed above. These losses are velocity (momentum) dependent and change (narrow) the initial spectrum, which causes longer coherence lengths and an enforced collective action of the rotation stages. Thus, a situation may exist where the spin state does not change because the transition probability goes to zero (Eq. (2)) but the intensity changes due to parasitic quantum losses, which means that a perfect Zeno-e!ect is, in principle, unphysical. When n independent successive barriers are taken into account, the intensity of the # polarized beam becomes reduced at least as (Eq. (3)) I "P ¹M L"P (1!RM )L > > > 1 < L > "P 1! > 2 2E
n < > #2 && 0
1! 2 2E L
(4)
and an even enhanced reduction is expected when a collective action of the barriers occurs. Thus, a state variation is expected in any case. Nevertheless, there is no question about the existence of the Zeno-like phenomena for reasonably low n values, but the nPR limit is still under discussion. In this respect, an intrinsic contradiction concerning the existence of the classical Zeno-e!ect exists, which justi"es new experimental work. Such an experiment is in progress at the pulsed ISIS spallation source where we are using the features of our newly developed perfect crystal neutron resonator (Fig. 3) [24,25]. This instrument allows a multiple (up to 4000 times) transmission of neutrons through a spin rotation coil and a continuous measurement of the negative spin state due to the spin-dependent transmission of the exit perfect crystal. This spin-dependent transmission is
301
Fig. 3. Perfect crystal resonator system adapted for a neutron Zeno-e!ect experiment.
caused by the spin-dependent Zeeman shift of the neutrons inside the magnetic "eld applied to that crystal. The spin rotation angle in the #ipper coil can be controlled by the strength of a static rotation "eld, or by the amplitude of the oscillating #ipper "eld.
3. Discussion It has been shown that any interaction of a neutron beam with a potential produces unavoidable parasitic beams which are characteristic for this kind of interaction. Multiple potential barriers do not only produce an addiative e!ect of losses, but also show enhanced losses due to various resonance e!ects. Therefore, the Zeno-phenomenon, which is based on a repetitive interaction and observation of a quantum system, has to be discussed in a new light including unavoidable quantum losses. In this respect, the imperfect Zeno-phenomenon and the limit of a non-completely frozen state for an in"nite number of stages can be seen as a "ngerprint of quantum mechanics. In the limiting case, when the coherence length exceeds the
302
H. Rauch / Physica B 297 (2001) 299}302
dimension of the spin rotation stages ( *d), a de viation from Eq. (2) can be expected when the evolution time (l/v) has to be replaced by the passage time of the packet ( /v). In this case, the survival probability would also tend to zero. This limit is still being debated in the literature [5}6,15,21]. Related experiments with a perfect crystal neutron resonator are in progress.
Acknowledgements Very fruitful discussions with E. Jericha (Vienna), S. Pascazio (Bari) and Z. Hradil (Olomouc) are gratefully acknowledged. This work has been supported by the Austrian Science Foundation (project P13332-PHY).
References [1] B. Misra, E.C.G. Sudershan, J. Math. Phys. 18 (1977) 756. [2] L. Fonda, G.C. Ghirardi, A. Rimini, Rep. Progr. Phys. 41 (1978) 587. [3] D. Home, M.A.B. Whitaker, Phys. Lett. A 173 (1993) 327. [4] E. Joos, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin, 1996, (Chapter 3). [5] H. Nakazato, M. Namiki, S. Pascazio, H. Rauch, Phys. Lett. A 199 (1995) 27.
[6] A. Venugopalan, R. Ghosh, Phys. Lett. A 204 (1995) 11. [7] A. Pati, Phys. Lett. A 215 (1996) 7. [8] W.H. Itano, D.J. Heinzen, J.J. Bollinger, D.J. Wineland, Phys. Rev. A 41 (1990) 2295. [9] A. Peres, Am. J. Phys. 48 (1980) 931. [10] P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. Kasevich, Phys. Rev. Lett. 74 (1995) 4763. [11] S. Pascazio, M. Namiki, G. Badurek, H. Rauch, Phys. Lett. A 179 (1993) 155. [12] K. Machida, H. Nakazato, S. Pascazio, H. Rauch, S. Yu, Phys. Rev. A 60 (1999) 3448. [13] Z. Hradil, H. Nakazato, M. Namiki, S. Pascazio, H. Rauch, Phys. Lett. A 239 (1998) 333. [14] A.C. Elitzur, L. Vaidman, Found. Phys. 23 (1993) 987. [15] S.H. Simon, P.M. Platzman, Phys. Rev. A 61 (2000). [16] A.P. Balachandran, S.M. Roy, Phys. Rev. Lett. 84 (2000) 4019. [17] M.A.B. Whitaker, Phys. Lett. A 244 (1998) 502. [18] F. Mezei, Z. Physik 255 (1972) 146. [19] W.G. Williams, Polarized Neutrons, Clarendon Press, Oxford, 1988. [20] R. Surkau, J. Becker, M. Ebert, T. Grossmann, W. Heil, D. Hofmann, H. Humboldt, M. Leduc, E.W. Otten, D. Rohe, K. Siemensmeyer, M. Steiner, F. Tasset, N. Trautmann, Nucl. Instr. and Meth. A 384 (1997) 444. [21] J.-W. Pan, Y. Zhang, Q. Dong, Phys. Lett. A 238 (1998) 405. [22] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Vol. I, Wiley, New York 1977. [23] H. Rauch, H. Wolwitsch, H. Kaiser, R. Clothier, S.A. Werner, Phys. Rev. A 53 (1996) 902. [24] M. Schuster, H. Rauch, E. Seidl, E. Jericha, C.J. Carlile, Phys. Lett. A 144 (1990) 297. [25] E. Jericha, D.E. Schwab, C.J. Carlile, M.R. Jaekel, R. Loidl, H. Rauch, Nucl. Instr. and Meth. A 440 (2000) 597.