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Interrupted fluorescence experiments, and hidden variables
PhysicsLettersA 181 (1993) 114—118 North-Holland
PHYSICS LETTERS A
Interrupted fluorescence experiments, and hidden variables D. Home Department of Physics, Bose Institute, Calcutta 700009, India
and M.A.B. Whitaker Department ofPure and Applied Physics, The Queen’s University ofBelfast, Belfast BT7 INN, Northern Ireland, UK Received 26 April 1993; acceptedfor publication 13 August 1993 Communicated by J.P. Vigier
A recent theory of interrupted fluorescence experiments does not make use of collapse, but recognises the orthogonality of different branches ofthe wavefunction resulting from differences in photon number. However this argument suggests that a single system may, for a substantial period of time, lie in one branch of its many-component wavefunction. It is suggested that hidden variables may help to clarify the situation.
The phenomenon of interrupted fluorescence has been of interest for a number of years. The experiments are performed using trapped laser-cooled ions in the so-called V-configuration of atomic physics [1]. Transitions from a highly fluorescent state 3> to the ground state I 1> may readily be detected. The presence of another level 12>, which decays to the ground state much more slowly than I 3>, is mdicated by the occurrence of substantial periods oftime, “dark periods”, during which no photons are observed. Typically in these experiments the Ii> —13> transition is strongly driven, and the I 1>— I 2> transition weakly driven, but the phenomenon is rather insensitive to the precise experimental conditions; in particular, it was initially expected that the phenomenon would only occur under incoherent illuminalion, but it was later shown that is should occur even when the illumination was coherent [2]. It is stressed that the experiments are carried out using single ions. A variety of theoretical approaches have been made. Cook and Kimble [3] assumed from the outset that sudden transitions between light and dark periods did occur. An intuitive view, according to which the ion is always in only one of the three 1evels, would predict periodic “shelving” ofthe atom at 114
level 12>, and hence the occurrence of long dark penods. Schenzle and Brewer [41, however, pointed out that a more apparently sophisticated approach would suggest that the state of the system would be, at all times, a coherent superposition of all three states of the ion. This might lead one to assume that fluorescence would be continuous in time, but reduced in intensity by the metastable state 12>. These authors tackled the problem using a different theoretical technique, that of photon-counting statistics, and obtained the answer corresponding to the “intuitive” rather than the “sophisticated” approach, and this agreed also with experiment. This does not provide any direct understanding about the state of the system. In an attempt to provide this, several authors [5—7] postulated a collapse of state vector. If a photon corresponding to the I 3>—Il> transition is emitted, it is said that cotlapse takes place so that the state vector immediately afterwards is a linear combination of states I 1> and I 3>; matrix elements P22, P12, P2I~P23 and P32 will be zero at that time. If, on the other hand, no photon is emitted in a short period of time, &, the state vector collapses to state 12>; the only nonzero matrix element will be P22• This constitutes a negative-result Elsevier Science Publishers B.V.
Volume 181, number 2
PHYSICS LETTERS A
effect. One will expect a period of fluorescence to ensue in the first case, a dark period in the second. Such a position explains the observed results in an adequate fashion. However it appears open to considerable objection in principle. Let us first assume that the collapse occurs only if a photon is detected, or if a negative result is observed. This is the “onthodox” von Neumann interpretation of quantum theory. It has been subject to severe criticism, in particular by Ballentine [8], and in many papers of Bell, collected in ref. [9]. It suggests that quantum evolution follows two distinct paths usually that given by the Schrodinger equation, but, at a “measurement”, by the entirely different procedure of cotlapse. Yet the behaviour of the particles involved in the measurement, including those constituting the macroscopic “measuring apparatus”, could presum—
ably be described by the Schrödinger equation, and this does not allow the collapse process. As stressed by Bell, “measurement” is not a primary term; whether it occurs, and if it does so, the time of occurrence, is not well defined, and therefore occurrence or otherwise should not be described as affecting the evolution of the system. The argument above assumes that the state vector is to be interpreted objectively as providing information about the system itself. It may be suggested that if the state vector is interpreted subjectively as providing knowledge about the system, the problem would be removed, since clearly knowledge available about a system does change instantaneously at a measurement. In this way the state vector is interpreted epistemologically rather than ontologically; Heisenberg, for example, sometimes seems close to such a position [10]. Such an argument cannot be satisfactory, however; at best it merely avoids the real problems. It might appear attractive if it were felt that all observables had precise values, but only some of these could be known at any time. Bell’s theorem, though, tells us that there are definite limits on which cornbinations of observables may have distinct values simultaneously. This point must be one of the central problems of quantum theory, in particular one may ask how observables obtain or lose distinct values at “measurements”, but a subjective interpretation cannot address the problems, since it cannot discuss
4 October 1993
whether observables have distinct values, only our knowledge of such. In addition to these general arguments, there are further difficulties for the idea of collapse on measurement in the present case. It suggests that the atomic process itself is driven by the act of observation. Yet the observation itself, the registration of the photon, may be delayed at will by placing the detector a large way from the ion. (It could, at least in principle, take place in a far-away galaxy at a time when the apparatus containing the ions has been dismantled.) Such a suggestion cannot be ruled out; it lies at the heart of the quantum Zeno prediction, for example. However it implies that the behaviour of the system will be markedly different depending on whether it is observed or not. Even leaving aside the question of whether this seems inherently reasonable, we may return to the results of Schenzle and Brewer [41 mentioned above. Their analysis is independent of observation of the photons, and it therefore appears totally unsatisfactory that the method of collapse on measurement only gives the same results if observation does occur, giving totally different results if there is no observation. It could be suggested that we must not even discuss what would happen in the absence of observation, since clearly such predictions may not be checked empirically. Such a position perhaps smacks rather of logical positivism than of Copenhagen. more The restrictions placed on consideration of unperformed experiments by the principle of cornplementarity forbid the simultaneous analysis of experiments which may be carried out only by the use of mutually exclusive experimental arrangements [11,121. They do not forbid discussion per se of the results of (single) experiments that could be performed. For example, Murdoch [11, p. 105] writes that “observation in quantum physics reveals properties of the object that are not created by the process of observation”; Folse [12, p. 2431 writes that “the phenomenonalist interpretation of complernentanity misses the point that what is most revolutionary about the framework is the suggestion that in the combination of complementary descriptions of phenomenal objects, we convey information about an independent physical reality [our emphasis]”. ...
115
Volume 181, number 2
PHYSICS LETTERS A
The other possibility is that collapse occurs when the photon is emitted (or when no photon is emitted in a given time, in the negative result case) irrespective of observation. Such a position lies outside any of the “orthodox” ideas of Bohr or von Neumann, where collapse must always result from interaction with a macroscopic device. It perhaps appears rather more implausible than collapse on measurement, because it seems even more unlikely that there should be a departure from the Schrödinger equation when only microscopic entities are involved. There are many other situations involving atoms and/or photons where, to obtain agreement with experiment, it is essential to maintain the superposition of wavefunctions, and there seems no obvious reason why the opposite should apply in this case, It was because the collapse suggestion appeared unsatisfactory that we put forward an alternative approach [131, which explained the observations without requiring a collapse. This approach merely foltowed the Schrodinger equation, taking explicit account of the state of the electromagnetic field by including in the state vector the number of photons of different types. It is well known that states with different numbers of photons are orthogonal to each other. Then we write a general state of the system as
I~
(t)
=
~
c~~2~3(t) Ip, n2, n3> ,
(1)
P.fl2,fl3
where p takes the values 1, 2 and 3 corresponding to the three ionic states, and n2 and n3 give the number of photons with frequencies corresponding to the 12>—I 1> and I 3>—Il> transitions, respectively, As the system evolves, state (1) remains pure, and the full density matrix, involving joint states of the ion and the photon numbers, remains idempotent. However, the reduced density matrix for ionic states only will be nonidempotent, having very small matrix elements between 12>, and each of I 1> and I 3>. This is because there will be no values of n2 and n3 for which both c2~~3 and either c1~2~3, or c3~2~3 are substantially different from zero, This reduced density matrix will compare very closely with the 3 x 3 density matrix produced by the collapse theory above. In the latter, states I 1> and I 3> are “cut off” from 12> by the refusal to allow a linear superposition to be maintained; in ref. [13], 116
4 October 1993
the same effect is achieved by realising that 2> is coupled to almost totally different states of the electromagnetic field from Ii> and I 3>. We believe that the description given in ref. [131 and described above must be regarded as the most satisfactory one available because it merely follows the standard rules of quantum development, rather than introducing a second type of process the cotlapse which lies outside the Schrodinger equation, and takes place at a time which cannot be precisely defined. Nevertheless it is the purpose of the present Letter to point out that this description has consequences which appear to lie outside the usual approaches to quantum interpretation. The difficulties are largely obscured in a density-matrix description, because a density matrix may naturally be considered to refer to an ensemble, but they become central when one focuses attention on a single system. The implication is that, on the one hand, the system is represented by the state vector of (1), but, on the other, it is either emitting a stream of photons, or remaining dark. The immediate feeling is that the state vector cannot contam all information concerning the individual systern; in other words, it provides an incomplete description of the system. Yet that is precisely the form of words usually used to introduce the idea ofhidden variables. Hidden variables, it may be suggested, give —
—
the information about which branch of (1) the individual system is actually “in”, and thus serve to relate the equation for the state vector in (1) to the actual behaviour of the system. Objections to this suggestion may be broadly along the lines ofthe earlier discussion in this paper. If one discusses the events purely in terms of observation, it is clear that successive observations will yield information pertaining to the same branch of the state vector. This may be achieved, of course, by a cotlapse of state vector at a measurement (and we would regard this as, at least, a more acceptable concept than collapse at emission of a photon, or failure to emit a photon, as in the papers discussed above). Alternatively, in the absence of a collapse at measurement, the correlation of states ofobserved system to (mutually orthogonal) states of observing system at a measurement, will also produce consistency between the first and the subsequent observation. However we do not regard this as a successful an-
Volume 181, number 2
PHYSICS LETTERS A
gument against our suggestion, and for reasons practically the same as those given earlier in the paper. We do not believe it makes physical sense for observations (possibly again made after the experiment has been shut down) to control such experiments as the ones being considered. We would, in any case, prefer to discuss the behaviour of the systern in the absence ofexperiment, and do not feel such discussion is illegitimate. (A comparison may be made with the quantum Zeno predictions. It is true that here observation, including negative-result observation, does play an important role, though this role is usually regarded with suspicion, and certainly contributes to the common use of the term quantum Zeno paradox. Nevertheless, in analysis of these postulated effects, the central feature is comparison ofthe behaviour when the system is observed, with that when no observation takes place, and there is no suggestion that consideration of the tatter is not allowed.) One may also ask whether the situation discussed in this paper is entirely novel. It could be argued that any atom in, for example, a discharge tube, may go through a series of transitions, absorbing energy, and emitting photons. The same question may be asked in this case: When a photon is emitted, may we safety say that the state vector ofthe atom concerned should correspond to the tower of the levels connected by the transition? If so, should this fact be registered in the state vector of the system? And, if again so, how? It is true that our case may not differ from this one in essence. However, we feet that there are factors which make the situation discussed in this paper not perhaps qualitatively different, but more demanding of attention. First, the study is of a single ion, and secondly, the behaviour is steady (fluorescing or not fluorescing) over a substantial period of time. There may be no question of avoiding the issue by averaging over many systems, or over a period of time, and thus regarding the density matrix, or the superposition of (1) as describing the situation adequately. We may also perhaps anticipate a suggestion that the situation described wilt appear straightforward inside an ensemble interpretation of quantum theory, as championed by Ballentine [8]. The reply would be that such a remark is correct only for some classes of ensemble interpretation, which are themselves equivalent to hidden variable theories of —
4 October 1993
quantum mechanics. Thus we would regard this point as an argument in favour, rather than against, our position. This is argued fully by Home and Whitaker [14]. We suggest, therefore, that the hidden variable approach to the analysis ofthe experiments appears to be the least open to criticism. In this paper we make no detailed attempt to discuss what type of hidden variables will be most useful. The well-known quantum potential approach of Bohm [15—17],though, has certain advantages. In particular Malik [18] has performed some explicit calculation of the behaviour of the Bohm type of hidden variables in such situations as EPR. He has shown that, white in generat circumstances these variables will not be directly related to quantities of macroscopic interest, in extreme cases such an identification may be possibte. Similarly, calculation of scattering and interference phenomena show that the hidden variables may have direct physical meaning after, though not during, these events [19,20]. This is of direct relevance to our case where we may expect hidden vanabtes to be present at all times, but only to manifest themselves directly in certain circumstances, following, for example, emission of a photon. To conclude, then, we point out that our arguments may be rather unusual in the hidden variable literature. Nearly always perhaps, hidden variables are proposed as more satisfying solutions of problems already dealt with by conventional approaches. In this paper they are postulated in response to a conceptual problem related directly to experimental results. An approach which recognises that the interrupted fluorescence experiments are on single systems, and which avoids a subjective approach to the state vector, while also avoiding the problems of collapse, appears to lead to an incomplete state vector, and hence to a prediction of hidden variables.
References [1] H. Dehmelt, Bull. Am. Phys. Soc. 20 (1975) 60. [21D.T. Pegg, R. Loudon and P.L. Knight, Phys. Rev. A 33 (1986) 4085. [3]R.J. Cook and H.J. Kimble, Phys. Rev. Lett. 54 (1985) 1023 [4] A. Schenzle and R.G. Brewer, Phys. Rev. A 34 (1986) 3127. [5] M. Porrati and S. Putterman, Phys. Rev. A 36 (1987) 929.
117
Volume 181, number 2 [6] [7] [8] [9]
PHYSICS LETTERS A
D.T. Pegg and P.L. Knight, Phys. Rev. A 37 (1988) 4303. D.T. Pegg and P.L. Knight, J. Phys. D 21(1988) 5128. L.E. Ballentine, Rev. Mod. Phys. 42 (1970) 358. J.S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge Univ. Press, Cambridge, 1987). [10] W. Heisenberg, Physics and philosophy (Allen and Unwin, London, 1959). [I 1] D. Murdoch, Niels Bohr’s philosophy ofphysics (Cambridge Univ. Press, Cambridge, 1987). [12] H.J. Folse, The philosophy of Niels Bohr (North-Holland, Amsterdam, 1985).
118
[13] [14] [15] [16] [17] [18]
4 October 1993
D. Home and M.A.B. Whitaker, J. Phys. A 19 (1992)1847. D. Home and M.A.B. Whitaker, Phys. Rep. 210 (1992) 223. D. Bohm, Phys. Rev. 85 (1952) 166. D. Bohm, Phys. Rev. 85 (1952)180. D. Bohm and B.J. Hiley, Phys. Rep. 144 (1987) 323. Z. Malik, UK Workshop on Conceptual problems in physics, Portsmouth (1992). [191 C. Dewdney and B.J. Hiley, Found. Phys. 12 (1982) 27. [20] C. Dewdney, Phys. Lett. A 109 (1985) 377.
PHYSICS LETTERS A
Interrupted fluorescence experiments, and hidden variables D. Home Department of Physics, Bose Institute, Calcutta 700009, India
and M.A.B. Whitaker Department ofPure and Applied Physics, The Queen’s University ofBelfast, Belfast BT7 INN, Northern Ireland, UK Received 26 April 1993; acceptedfor publication 13 August 1993 Communicated by J.P. Vigier
A recent theory of interrupted fluorescence experiments does not make use of collapse, but recognises the orthogonality of different branches ofthe wavefunction resulting from differences in photon number. However this argument suggests that a single system may, for a substantial period of time, lie in one branch of its many-component wavefunction. It is suggested that hidden variables may help to clarify the situation.
The phenomenon of interrupted fluorescence has been of interest for a number of years. The experiments are performed using trapped laser-cooled ions in the so-called V-configuration of atomic physics [1]. Transitions from a highly fluorescent state 3> to the ground state I 1> may readily be detected. The presence of another level 12>, which decays to the ground state much more slowly than I 3>, is mdicated by the occurrence of substantial periods oftime, “dark periods”, during which no photons are observed. Typically in these experiments the Ii> —13> transition is strongly driven, and the I 1>— I 2> transition weakly driven, but the phenomenon is rather insensitive to the precise experimental conditions; in particular, it was initially expected that the phenomenon would only occur under incoherent illuminalion, but it was later shown that is should occur even when the illumination was coherent [2]. It is stressed that the experiments are carried out using single ions. A variety of theoretical approaches have been made. Cook and Kimble [3] assumed from the outset that sudden transitions between light and dark periods did occur. An intuitive view, according to which the ion is always in only one of the three 1evels, would predict periodic “shelving” ofthe atom at 114
level 12>, and hence the occurrence of long dark penods. Schenzle and Brewer [41, however, pointed out that a more apparently sophisticated approach would suggest that the state of the system would be, at all times, a coherent superposition of all three states of the ion. This might lead one to assume that fluorescence would be continuous in time, but reduced in intensity by the metastable state 12>. These authors tackled the problem using a different theoretical technique, that of photon-counting statistics, and obtained the answer corresponding to the “intuitive” rather than the “sophisticated” approach, and this agreed also with experiment. This does not provide any direct understanding about the state of the system. In an attempt to provide this, several authors [5—7] postulated a collapse of state vector. If a photon corresponding to the I 3>—Il> transition is emitted, it is said that cotlapse takes place so that the state vector immediately afterwards is a linear combination of states I 1> and I 3>; matrix elements P22, P12, P2I~P23 and P32 will be zero at that time. If, on the other hand, no photon is emitted in a short period of time, &, the state vector collapses to state 12>; the only nonzero matrix element will be P22• This constitutes a negative-result Elsevier Science Publishers B.V.
Volume 181, number 2
PHYSICS LETTERS A
effect. One will expect a period of fluorescence to ensue in the first case, a dark period in the second. Such a position explains the observed results in an adequate fashion. However it appears open to considerable objection in principle. Let us first assume that the collapse occurs only if a photon is detected, or if a negative result is observed. This is the “onthodox” von Neumann interpretation of quantum theory. It has been subject to severe criticism, in particular by Ballentine [8], and in many papers of Bell, collected in ref. [9]. It suggests that quantum evolution follows two distinct paths usually that given by the Schrodinger equation, but, at a “measurement”, by the entirely different procedure of cotlapse. Yet the behaviour of the particles involved in the measurement, including those constituting the macroscopic “measuring apparatus”, could presum—
ably be described by the Schrödinger equation, and this does not allow the collapse process. As stressed by Bell, “measurement” is not a primary term; whether it occurs, and if it does so, the time of occurrence, is not well defined, and therefore occurrence or otherwise should not be described as affecting the evolution of the system. The argument above assumes that the state vector is to be interpreted objectively as providing information about the system itself. It may be suggested that if the state vector is interpreted subjectively as providing knowledge about the system, the problem would be removed, since clearly knowledge available about a system does change instantaneously at a measurement. In this way the state vector is interpreted epistemologically rather than ontologically; Heisenberg, for example, sometimes seems close to such a position [10]. Such an argument cannot be satisfactory, however; at best it merely avoids the real problems. It might appear attractive if it were felt that all observables had precise values, but only some of these could be known at any time. Bell’s theorem, though, tells us that there are definite limits on which cornbinations of observables may have distinct values simultaneously. This point must be one of the central problems of quantum theory, in particular one may ask how observables obtain or lose distinct values at “measurements”, but a subjective interpretation cannot address the problems, since it cannot discuss
4 October 1993
whether observables have distinct values, only our knowledge of such. In addition to these general arguments, there are further difficulties for the idea of collapse on measurement in the present case. It suggests that the atomic process itself is driven by the act of observation. Yet the observation itself, the registration of the photon, may be delayed at will by placing the detector a large way from the ion. (It could, at least in principle, take place in a far-away galaxy at a time when the apparatus containing the ions has been dismantled.) Such a suggestion cannot be ruled out; it lies at the heart of the quantum Zeno prediction, for example. However it implies that the behaviour of the system will be markedly different depending on whether it is observed or not. Even leaving aside the question of whether this seems inherently reasonable, we may return to the results of Schenzle and Brewer [41 mentioned above. Their analysis is independent of observation of the photons, and it therefore appears totally unsatisfactory that the method of collapse on measurement only gives the same results if observation does occur, giving totally different results if there is no observation. It could be suggested that we must not even discuss what would happen in the absence of observation, since clearly such predictions may not be checked empirically. Such a position perhaps smacks rather of logical positivism than of Copenhagen. more The restrictions placed on consideration of unperformed experiments by the principle of cornplementarity forbid the simultaneous analysis of experiments which may be carried out only by the use of mutually exclusive experimental arrangements [11,121. They do not forbid discussion per se of the results of (single) experiments that could be performed. For example, Murdoch [11, p. 105] writes that “observation in quantum physics reveals properties of the object that are not created by the process of observation”; Folse [12, p. 2431 writes that “the phenomenonalist interpretation of complernentanity misses the point that what is most revolutionary about the framework is the suggestion that in the combination of complementary descriptions of phenomenal objects, we convey information about an independent physical reality [our emphasis]”. ...
115
Volume 181, number 2
PHYSICS LETTERS A
The other possibility is that collapse occurs when the photon is emitted (or when no photon is emitted in a given time, in the negative result case) irrespective of observation. Such a position lies outside any of the “orthodox” ideas of Bohr or von Neumann, where collapse must always result from interaction with a macroscopic device. It perhaps appears rather more implausible than collapse on measurement, because it seems even more unlikely that there should be a departure from the Schrödinger equation when only microscopic entities are involved. There are many other situations involving atoms and/or photons where, to obtain agreement with experiment, it is essential to maintain the superposition of wavefunctions, and there seems no obvious reason why the opposite should apply in this case, It was because the collapse suggestion appeared unsatisfactory that we put forward an alternative approach [131, which explained the observations without requiring a collapse. This approach merely foltowed the Schrodinger equation, taking explicit account of the state of the electromagnetic field by including in the state vector the number of photons of different types. It is well known that states with different numbers of photons are orthogonal to each other. Then we write a general state of the system as
I~
(t)
=
~
c~~2~3(t) Ip, n2, n3> ,
(1)
P.fl2,fl3
where p takes the values 1, 2 and 3 corresponding to the three ionic states, and n2 and n3 give the number of photons with frequencies corresponding to the 12>—I 1> and I 3>—Il> transitions, respectively, As the system evolves, state (1) remains pure, and the full density matrix, involving joint states of the ion and the photon numbers, remains idempotent. However, the reduced density matrix for ionic states only will be nonidempotent, having very small matrix elements between 12>, and each of I 1> and I 3>. This is because there will be no values of n2 and n3 for which both c2~~3 and either c1~2~3, or c3~2~3 are substantially different from zero, This reduced density matrix will compare very closely with the 3 x 3 density matrix produced by the collapse theory above. In the latter, states I 1> and I 3> are “cut off” from 12> by the refusal to allow a linear superposition to be maintained; in ref. [13], 116
4 October 1993
the same effect is achieved by realising that 2> is coupled to almost totally different states of the electromagnetic field from Ii> and I 3>. We believe that the description given in ref. [131 and described above must be regarded as the most satisfactory one available because it merely follows the standard rules of quantum development, rather than introducing a second type of process the cotlapse which lies outside the Schrodinger equation, and takes place at a time which cannot be precisely defined. Nevertheless it is the purpose of the present Letter to point out that this description has consequences which appear to lie outside the usual approaches to quantum interpretation. The difficulties are largely obscured in a density-matrix description, because a density matrix may naturally be considered to refer to an ensemble, but they become central when one focuses attention on a single system. The implication is that, on the one hand, the system is represented by the state vector of (1), but, on the other, it is either emitting a stream of photons, or remaining dark. The immediate feeling is that the state vector cannot contam all information concerning the individual systern; in other words, it provides an incomplete description of the system. Yet that is precisely the form of words usually used to introduce the idea ofhidden variables. Hidden variables, it may be suggested, give —
—
the information about which branch of (1) the individual system is actually “in”, and thus serve to relate the equation for the state vector in (1) to the actual behaviour of the system. Objections to this suggestion may be broadly along the lines ofthe earlier discussion in this paper. If one discusses the events purely in terms of observation, it is clear that successive observations will yield information pertaining to the same branch of the state vector. This may be achieved, of course, by a cotlapse of state vector at a measurement (and we would regard this as, at least, a more acceptable concept than collapse at emission of a photon, or failure to emit a photon, as in the papers discussed above). Alternatively, in the absence of a collapse at measurement, the correlation of states ofobserved system to (mutually orthogonal) states of observing system at a measurement, will also produce consistency between the first and the subsequent observation. However we do not regard this as a successful an-
Volume 181, number 2
PHYSICS LETTERS A
gument against our suggestion, and for reasons practically the same as those given earlier in the paper. We do not believe it makes physical sense for observations (possibly again made after the experiment has been shut down) to control such experiments as the ones being considered. We would, in any case, prefer to discuss the behaviour of the systern in the absence ofexperiment, and do not feel such discussion is illegitimate. (A comparison may be made with the quantum Zeno predictions. It is true that here observation, including negative-result observation, does play an important role, though this role is usually regarded with suspicion, and certainly contributes to the common use of the term quantum Zeno paradox. Nevertheless, in analysis of these postulated effects, the central feature is comparison ofthe behaviour when the system is observed, with that when no observation takes place, and there is no suggestion that consideration of the tatter is not allowed.) One may also ask whether the situation discussed in this paper is entirely novel. It could be argued that any atom in, for example, a discharge tube, may go through a series of transitions, absorbing energy, and emitting photons. The same question may be asked in this case: When a photon is emitted, may we safety say that the state vector ofthe atom concerned should correspond to the tower of the levels connected by the transition? If so, should this fact be registered in the state vector of the system? And, if again so, how? It is true that our case may not differ from this one in essence. However, we feet that there are factors which make the situation discussed in this paper not perhaps qualitatively different, but more demanding of attention. First, the study is of a single ion, and secondly, the behaviour is steady (fluorescing or not fluorescing) over a substantial period of time. There may be no question of avoiding the issue by averaging over many systems, or over a period of time, and thus regarding the density matrix, or the superposition of (1) as describing the situation adequately. We may also perhaps anticipate a suggestion that the situation described wilt appear straightforward inside an ensemble interpretation of quantum theory, as championed by Ballentine [8]. The reply would be that such a remark is correct only for some classes of ensemble interpretation, which are themselves equivalent to hidden variable theories of —
4 October 1993
quantum mechanics. Thus we would regard this point as an argument in favour, rather than against, our position. This is argued fully by Home and Whitaker [14]. We suggest, therefore, that the hidden variable approach to the analysis ofthe experiments appears to be the least open to criticism. In this paper we make no detailed attempt to discuss what type of hidden variables will be most useful. The well-known quantum potential approach of Bohm [15—17],though, has certain advantages. In particular Malik [18] has performed some explicit calculation of the behaviour of the Bohm type of hidden variables in such situations as EPR. He has shown that, white in generat circumstances these variables will not be directly related to quantities of macroscopic interest, in extreme cases such an identification may be possibte. Similarly, calculation of scattering and interference phenomena show that the hidden variables may have direct physical meaning after, though not during, these events [19,20]. This is of direct relevance to our case where we may expect hidden vanabtes to be present at all times, but only to manifest themselves directly in certain circumstances, following, for example, emission of a photon. To conclude, then, we point out that our arguments may be rather unusual in the hidden variable literature. Nearly always perhaps, hidden variables are proposed as more satisfying solutions of problems already dealt with by conventional approaches. In this paper they are postulated in response to a conceptual problem related directly to experimental results. An approach which recognises that the interrupted fluorescence experiments are on single systems, and which avoids a subjective approach to the state vector, while also avoiding the problems of collapse, appears to lead to an incomplete state vector, and hence to a prediction of hidden variables.
References [1] H. Dehmelt, Bull. Am. Phys. Soc. 20 (1975) 60. [21D.T. Pegg, R. Loudon and P.L. Knight, Phys. Rev. A 33 (1986) 4085. [3]R.J. Cook and H.J. Kimble, Phys. Rev. Lett. 54 (1985) 1023 [4] A. Schenzle and R.G. Brewer, Phys. Rev. A 34 (1986) 3127. [5] M. Porrati and S. Putterman, Phys. Rev. A 36 (1987) 929.
117
Volume 181, number 2 [6] [7] [8] [9]
PHYSICS LETTERS A
D.T. Pegg and P.L. Knight, Phys. Rev. A 37 (1988) 4303. D.T. Pegg and P.L. Knight, J. Phys. D 21(1988) 5128. L.E. Ballentine, Rev. Mod. Phys. 42 (1970) 358. J.S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge Univ. Press, Cambridge, 1987). [10] W. Heisenberg, Physics and philosophy (Allen and Unwin, London, 1959). [I 1] D. Murdoch, Niels Bohr’s philosophy ofphysics (Cambridge Univ. Press, Cambridge, 1987). [12] H.J. Folse, The philosophy of Niels Bohr (North-Holland, Amsterdam, 1985).
118
[13] [14] [15] [16] [17] [18]
4 October 1993
D. Home and M.A.B. Whitaker, J. Phys. A 19 (1992)1847. D. Home and M.A.B. Whitaker, Phys. Rep. 210 (1992) 223. D. Bohm, Phys. Rev. 85 (1952) 166. D. Bohm, Phys. Rev. 85 (1952)180. D. Bohm and B.J. Hiley, Phys. Rep. 144 (1987) 323. Z. Malik, UK Workshop on Conceptual problems in physics, Portsmouth (1992). [191 C. Dewdney and B.J. Hiley, Found. Phys. 12 (1982) 27. [20] C. Dewdney, Phys. Lett. A 109 (1985) 377.