Collapse of the quantum wavefunction and Welcher-Weg (WW) experiments

Transcription

1 Coapse of the quantum wavefunction Wecher-Weg (WW) experiments Y.Ben-Aryeh Physics Department, Technion-Israe Institute of Technoogy, Haifa, 3000 Israe e-mai: Absstract The 'coapse' of the wave function in a genera measuring process is anayzed by a pure quantum mechanica (QM) approach. The probem of the deayed choice Wecher-Weg (WW) experiments is anayzed for Mach-Zehnder (MZ) interferometer. The WW effect is reated to compementarity principe orthogonaity of wavefunctions athough it produces sma momentum changes between the eectromagnetic (EM ) fied the beam-spitters ('s). For QM states for which we have a superposition of states interference effects there is no reaity before measurement, but by excuding such effects the system gets a rea meaning aready before measurement. Keywords: Wavefunction coapse; Deayed-Choice; Wecher-Weg (WW); reaity; measurement ; entangement.

2 . Introduction Fundamenta issues in physics are the reations between Quantum Mechanica (QM ) Cassica effects. Whie QM processes are produced by reversibe unitary processes in cassica theories the effects are quite often irreversibe. The interaction between a microscopic QM system a measuring macroscopic apparatus eads to the 'coapse' of the wavefunctions [,] which is basicy an irreversibe process, its interpretation by pure QM might raise some probems. A fundamenta probem in QM arises by the caim that reaity is obtained ony after measurement [3]. Especiay interesting in this connection are the deayed choice experiments since as Wheeer has commented on this probem: "the most starting is that seen in the deayed choice experiment. It reaches back into the past in apparent opposition to the norma order of time" [4]. I woud ike to treat in the present paper these probems using ony pure QM approach, compare my anaysis with those of other authors. Let me expain further these probems the methods which I choose for their sions. The basic probem in the interpretation of quantum measurement can be expained as foows: If the state is not an eigenstate of the observabe, no determinate vaue is attributed to the observabe. However, the measurement is described by a proection postuate [,] which characterizes the 'coapse' of the quantum state of the system into an eigenstate of the measured observabe. According to "orthodox quantum mechanics (QM)" the observer gets a determinate vaue as the come of measurement [3]. Athough this assumption, referred to as a coapse, seems to be in agreement with the experimenta measurements, conceptuay, it raises the probem of nonunitary evion, which is not incuded in pure QM. Wheeer expained Bohr attitude using a singe simpe sentence: "No eementary phenomenon is a phenomenon unti it is registered (observed) phenomenon" [4]. Conceptuay it raises the question if the quantum word has an obective meaning before measurement. I woud ike to anayze in the present paper the interaction between the microscopic quantum system the macroscopic apparatus of measurement. In such interaction entanged states are produced in which the system wavefunctions are correated with a certain degree of freedom of the apparatus, depending on the chosen operator of measurement. This interaction is a competey unitary process. The 'rea' measurement is made on the macroscopic degree of freedom of the apparatus giving information on the quantum state due to correation. After measurement, the macroscopic degree of freedom is excuded by a partia trace operation over the

3 entanged states. There are two options for such trace operations by which the system wavefunction coapses which are anayzed both mathematicay physicay in Section. Reated to these measurement effects we can examine Wheeer [4] proposa of "deayed-choice" Gedanken Experiment in interferometer in which the choice of which property wi be observed is made after the photon has aready passed the first beam spitter. "Thus one decides whether the photon sha have come by one re or by both res after it has aready done its trave" [4]. There is a ot of iterature verifying experimentay the deayed choice phenomena. (see e.g. [5]). The distinction between cassica quantum states can resove this probem. The idea that the photon has aready done its trave is not correct. As ong as we have superposition of quantum states, entangement interference effects there is not any reaity before measurement. This is in agreement with Bohr attitude [4]. However, when we add the Wecher-Weg (WW) detector or omitting in an interferometer the second beam spitter, this eads to orthogonaity reation between the two res of the optica system (i.e. eiminating the interference terms) then the system gets a cassica meaning as the 'which-way' the photon traveed is fixed aready before measurement. The WW phenomena have raised conceptua probems. Scuy his coegues [6,7] have caimed that the WW phenomena foows from the compementarity principe. They expained the compementary principe as foows : "We say that two observabes are 'compementary' if precise knowedge of one of them impies that a possibe comes of measuring the other one are equay probabe". They have given [6,7] exampes iustrating this property. In the WW experiments compementarity means that simutaneous observation of wave partice behaviour is prohibited. Storey et a. [8] caimed that "interference is ost by the transfer of momentum to the partice whose path is determined ". Engert, Scuy Wather have responded [9] in the negative caiming that compementarity must be accepted as independent principe in quantum mechanics, rather then a consequence of the position-momentum uncertainty reation. Storey et a. by making certain cacuations remained in their opinion [0]: "As momentum is conserved in every interaction in quantum mechanics, any process that changes one pattern into another must invove momentum transfer". Experiments [,] have shown that the

4 WW apparatus that "determines" which way might not incude any momentum transfer, this is in contradiction with the caims made by storey et a [8,0]. Engert [3] has found a way to quantify which way using a certain inequaity but as pointed by him :"The derivation of the inquaity does not make use of Heisenberg 's uncertainty reation in any form". Scuman [4] has anayzed the "two-sit experiment" showing the determination of the sit transversed destroys interference. As expained by him " the destruction of interference comes ab because the detected portion of the wavefunction is orthogona to that which comes from the other sit. The wavefunction passing through becomes entanged with degrees of freedom of the detector". Whie such caims are in agreement with our previous anaysis [5,6] there is a certain difference. As pointed by him you "must take into account the arger Hibert space, the detector as we". But in the tota Hibert space one needs to take into account the recoi of the 's in the interferometers or the wa at -sit experiment. As shown in our previous papers [5,6 ] the measurement in one ocation (the WW detector) infuences the wavefunction in other ocations fixes momentum at that second pace, exacty ike that of EPR. It is true that for bodies which are macroscopic the change in their wavefunctions is quite sma they cannot be considerd as the cause for the WW effect [6-7,9]. But, as many photons are invoved in the interference effects Storey et a. are right in their caim that momentum transfers shoud be invoved in the WW experiments. I give in Section 3 detaied cacuations for such momentum transfers in Mach-Zehnder (MZ) interferometer using the methods deveoped in our previous pubications [5-6], showing that momentum transfers indeed occur but that they are the resut of the orthogonaity reations [4], or equivaenty of the compementarity properties [6,7,9,]. In Section 4 the present resuts concusions are summarized.. Coapse of the wavefunction in the measurement process There are two stages in the measurement process: ) In the first stage, the interaction of the measuring apparatus with the quantum system eads to correations between the quantum states of the system a certain degree of freedom of the measuring apparatus. This stage is obtained by a competey unitary process eading to the entanged quantum state

5 a () where are the eigenstates of the quantum system 'measuring' operator ˆ O ( ) with eigenvaues n, correspondingy, a are the ampitudes of the corresponding entanged states mutipications, are the 'correated' measuring device wavefunctions. We assume that the apparatus quantum states are orthonorma. One shoud notice that we have chosen here a specia operator ˆ O referred here by the superscript eading to a specia form of the entangement given by Eq. (). Choosing another measuring operator a the entangement parameters wi vary correspondingy. We express the entanged state of Eq. () by the density operator a a k k k k. () As pointed by Peres [7] the interaction between the system the apparatus produces entanged state due to a unitary interaction between the microscopic partices the quantum wavefunctions of apparatus: "- it cannot be anything ese, if quantum theory is correct". However, the mechanism of entangement described by the above anaysis is different from that of [7]. ) In the second stage of measurement the pure density operator of Eq. () 'coapses'. Let us describe first the 'mathematica' mechanism which is behind the process of coapse then expain its physica meaning impications. We have two options to do that : a) By tracing over a macroscopic correated states assuming we get k 0 for k for k a ' (3). (4) We find in this case that after measurement the quantum state which has been described originay as a superposition of quantum states is converted to a "statistica state". The physica meaning behind this trace operation is that we can measure the 'probabiities' for getting the apparatus quantum states which are equa to

6 probabiities of having the correated quantum system as given by Eq.(4). The physica concusion is that we can predict the come obtained by the apparatus of measurement ony with a certain probabiity. In order to reaize the compete statistics coming from the measurements we have to repeat the preparation of the system under the same conditions average the resuts over the comes so that the probabiity for getting eigenvaue corresponding to the eigenstates which are the eigenstates of a certain operator of measurement ˆ O is given by a. b) If, however, for one sampe we have the resut of a measurement over the macroscopic degree of freedom corresponding to the eigenvaue (or by eimination a resuts which do not give ) then we have ( ) n a. (5) In Eq. (5) the origina quantum state has been proected into a specia quantum state which after normaization is given by. Such state wi start its new time evion as a pure state. Whie such resuts are in agreement with the ordinary description of "the coapse of the wavefunction" [,8], we find that the essentia source for such 'coapse' comes from the entangement of the quantum system wavefunctions with those of the apparatus, obtaining the information from the measurements of the apparatus system reducing the density operator by the trace operation over macroscopic states, excuding them. Let us iustrate the present approach to the coapse of the wavefunctions by anayzing Stern-Gerach experiments for / spin system using it as a prototype simpe case of measurement. The entanged state is produced by a magnetic fied gradient where the eectron beam with / spin is spitted into two beams with different directions. With going into a the detais of measurement of this system [9] we note that by the above first stage of measurement we get the entanged state a a, (6) where the superscript (n) refers to the axis of quantization, correspond to the spin states with spin eigenvaues of / - /,

7 are the apparatus states representing the directions of the beams which are entanged with the system spin states, a a are the ampitudes for the entanged states, respectivey. The entanged state of Eq. (6) corresponds to Eq. () as a specia simpe case. Let us expain the two physica options of measurements: a) The detectors described by the macroccopic wave functions measure the number of partices m n going in each direction,respectivey, so that m / ( m n) a, n / ( m n) a. Then the reduced density operator is given by Eq. (4). b) In another possibiity: After separating the spin states so that they are going in different directions et us assume that we put the detectors ony in one direction, excuding those spin partices going in this direction by detecting them. Then the spin states which are going in the other direction have not been detected can be considered as pure states after normaizing with we defined spin eigenvaue ( with eigenvaue / or with eigenvaue -/). It shoud be pointed that the above anaysis has treated ony the "von Neumann measurements" (often caed as "measurements of the first kind"). There is a more genera type of measurement known as "a positive operator vaue measure (POVM)" [8]. The fundamenta issue of irreversibiity is the same for a these measurements [7]. 3. Wecher-Weg (WW) experiments, compementarity, orthogonaity, momentum transfers in Mach-Zehnder (MZ) interferometer The WW effects in MZ interferometer can be reated to the coapse of the wavefunction foowing our previous anaysis [5,6]. It wi be shown here that the WW detector eads to a transfer of momentum between the eectromagnetic fied the 's athough there might not be any change of momentum of the WW detector itsef. Assuming creation annihiation operators inserted into the ut ports of the MZ interferometer, then the transformation of the first beamspitter ( ) is given by [6]:

8 t r exp p ˆ b. (7) r exp p t ˆ b Here t r are the transmission refection coefficient of p is the momentum change of a photon refected from the transferred into momentum changes of, respectivey. which has been with opposite direction, in agreement with conservation of momentum. The unitary matrix transformation of Eq. (7) exp p of momentum transation operating on the incudes the operator macroscopic wavefunction of. We get, however [5]: exp p. (8) Here is the macroscopic wavefunction of. The resut (8) foows from the fact the photon momentum is very sma in comparison to the uncertainty in the momentum of macroscopic obect. This is aso the reason for negecting the momentum transation operators of Eq. (7) in conventiona treatment of MZ interferometer. We wi, however, keep the momentum transation operators in our anaysis as it can ater resove the controversy between Scuy et a [6-7,9,] Storey et a. [8,0] ab momentum transfers in WW experiments. Omitting first the effect of WW detector we have two more transformations. The difference in phase for the two modes entering optica path, can be represented by [6]: Here bˆ exp( i) 0b ˆ 0 b b b due to a difference in. (9) b are the creation mode operators entering. A simiar transformation to that of Eq. (7) is obtained for [6]: b t r exp p ˆ a 3 b r exp p t 4. (0)

9 Combining the transformations (7) (9) (0) we obtain our MZ transformation expressing the ut creation operators operators 3 4 as function of the put creation C C 3 exp i /, () C C 4 where C t t exp( i / ) r r exp p exp p exp( i / ),. (). C t r exp( i / ) exp p rt exp p exp( i / ) If we eiminate t, r 0. then the above equations wi be vaid by assuming Any two mode radiation state entering the two ut ports of the interferometer can be described as a function of the operators operating on the vacuum state. By using the transformations (-) one can transform this functions into a function of 3 4 operating on the vacuum state of the put modes. Using this straightforward method one can find the effect of the MZ interferometer on the transmitted radiation. In [6] this method has been appied for coherent states. Here for simpicity we assume that one photon is entering into one ut port of the MZ interferometer giving the ut state as a 0 exp i / C a 0 C a 0, (3) where the subscript refers to ut port subscripts 3 4 refer to put ports 3 4, respectivey. The wavefunction in put port 3 is orthogona to that of 4 due to orthogona space dependence. The probabiity for measurement of the photon in put port 3 is given by C t t r r t t r r exp( i ) exp p exp p C. C., (4) that in put port 4 is given by C t r r t t r r t exp( i ) exp p e xp p C. C.. (5)

10 The ast term in Eq. (4) which is with opposite sign to the ast term of Eq. (5) represents the interference term. We shoud take into account that the momentum transation operators with subscripts operate on the macroscopic bodies, respectivey, due to the refection of the photon from these 's. The change in the 's momentum wavefunctions due to the photon refection is very sma reative to their momentum uncertainty, as expressed for by using a simiar equation for by Eq. (8). Therefore, in the ordinary treatment of MZ interferometer a the transationa momentum operators can be negected Eqs. (4-5) are reduced to the conventiona MZ anaysis. Let us see now what wi be the effect of inserting the WW detector. Assuming for simpicity that the WW detector denoted as operator O ˆ WW with the radiation refected from. Then Eq. (7) is exchanged into is interacting ˆ t r exp p ˆ O WW b. (6) r exp p Oˆ ˆ WW t b Repeating again a the transformations with O ˆ WW we find that in the expressions for C C we shoud exchange exp p by exp p exp p Oˆ WW. (7) Then in Eq. () terms which incude O ˆ WW are orthogona to terms which do not incude O ˆ WW. We find that Eqs. (4) (5) are exchanged into C t t r r ; C t r r t. (8) Due to the WW detector a interference terms are eiminated. By comparing Eqs. (4-5) with Eq. (8) we find that the WW detector has eiminated aso the effects of the momentum transation operators. We find, therefore, that the WW detector has ed to exchange of momentum [8,0] between the EM fied the entanged macroscopic 's. However, such exchange of momentum is the resut of the WW detector which is responsibe to the which way effects [6,7,9,]. Athough many photons are invoved in the interference effect this exchange of momentum has a negigibe effect on the macroscopic 's which have a arge momentum uncertainty. Mometum transfer effects might be important ony if micro-'s are reaizabe. The intention of

11 the present anaysis is, however, to show that athough sma momentum exchanges occur in the WW experiments the cause of the WW effects is the introduction of orthogonaity by WW detector [4] or equivaenty by the compementarity of partice wave properties [6,7,9,]. 4. Summary concusion In the present work it has been shown in Section that QM measurements are reated to entangement processes produced between the eigenstates of certain QM measurement operators the macroscopic states of the apparatus of measurement as described in Eq. (). In this expression ony the degree of freedom of the macroscopic measuring device which is entanged with the QM system is taken into account whie other degrees of freedom of the macroscopic system are disregarded. After producing such entangement, the 'measurement' is made on the macroscopic entanged degree of freedom. Then using this information the tota density operator is reduced to that of the QM system by tracing over the entanged states, excuding the macroscopic states. Two options for using such process as described by Eqs. (4) (5). The method has been demonstrated for a simpe Stern-Gerach experiment. In Section 3 the MZ transformations has been generaized so that they incude momentum transation operators operating on the macroscopic 's wavefunctions. This has ed to fina MZ transformation given by Eqs. (-). For simpicity of discussion we have assumed that one photon is inserted into one ut port as described by Eq. (3). Then the probabiity for measuring the photon in put ports 3 4 are given by Eqs. (4) (5), respectivey. By inserting the WW detector in the refected beam Eq. (7) is changed into Eq. (6) eading to orthogonaity between the two res of the interferometer excuding the interference terms. Equivaent resuts are transmitted from obtained if we wi put the WW detector in the beam. By comparing Eqs, (4-5) with (8) we find that the eimination of the interference terms by the WW eads aso to exchange of momentum between the EM fied the 's. However, we find that the effect of the momentum transation operators on the macroscopic wave function simiary on ( ), as expressed by Eq. (8), are negigibe, so that they cannot be regarded as the cause for the 'which way' effect, but ony as a resut of the coapse of

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