Proposed experiment of which-way detection by longitudinal momentum transfer in Young's double slit experiment

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1 Proposed experiment of wic-way detection by longitudinal momentum transfer in Young's double slit experiment Masanori Sato Honda Electronics Co., Ltd., 20 Oyamazuka, Oiwa-co, Toyoasi, Aici , Japan Abstract Te momentum of a poton may reveal te answer to te "wic way" problem of Young's double slit experiments. A poton passing troug te boundary between two media, in wic a poton travels at different velocities, undergoes a momentum cange according to te law of conservation of momentum. Te momentum of te poton is transferred locally to te medium, and te boundary between te media receives stress, wic determines te poton trajectory. An experiment is performed using a crystal plate tat can transform te stress to electric carge. We are able to detect te electric carge after te detection of te poton on te screen, and control te sensitivity of poton detection. By means of tis proposed experiment it is determined weter or not an attempt to detect te "wic way" of poton travel destroys te interference patterns. PACS numbers: 03.65Bz, 42..-p Key words: Young's double slit experiments, momentum transfer, locality, Crystal plate, Heisenberg relations 1. Introduction Heisenberg's uncertainty relations and complementarity ave been discussed previously [1-12] as being te reasons wy it as not been possible to determine te direction in wic a poton travels in Young's double slit experiments [1-3]. Te destruction of interference patterns can be explained in terms of uncontrolled momentum kicks to te particle [13]. In recent years, Scully and co-workers [4, 5] ave claimed tat teir sceme destroys te interference patterns witout transferring any transverse momentum to te particle. It is said tat an attempt to detect te direction of poton travel will destroy te interference patterns, because of complementarity and uncertainty. Two groups ave been debating te relative priority of complementarity [4, 5] and uncertainty [6-9]. Te local momentum transfer (wic occurs te particle is localized at a single slit) versus non-local momentum transfer (wic if te particle is delocalized at bot slits) as been discussed [13]. It is also discussed tat te Wignerian analysis and te Bomian analysis clearly distinguis between a local momentum transfer and a nonlocal momentum transfer [14]. Active measurement, for example, te electromagnetic stimulation of electrons [4,5], destroys interference patterns, but we tink tat passive measurement, for example, local momentum transfer, does not destroy te interference patterns. Terefore, tere is a possibility of detecting simultaneously te trajectory and interference patterns, if we detect te poton trajectory by te longitudinal momentum. In tis letter, te momentum discussed is longitudinal rater tan transverse, wic simplifies te argument. Terefore, a newly proposed Young s double slit interference experiment is presented and te metod of 1

2 detection of "wic way" of poton travel in te interference experiment is described. In tis experiment, classical and local momentum kicks to te particle can be controlled linearly. Terefore, we are able to test weter or not te local momentum transfer destroys te interference patterns. 2. Momentum conservation law and stress In tis experiment, te poton momentum k is used. Here, k is te wave number and = ( is Planck s constant). A poton canges 2π its momentum according to te medium toug wic it is propagating. It also canges at te boundary between te media, resulting in a of conservation of momentum is k 1 = k 2 + k m. (1) Here, k 1 is te poton wave number in medium A, k 2 is te wave number in medium B, and k m is te wave number corresponding to te momentum transferred to te medium. Te momentum transfer of equation (1) is a local penomenon. Te directions of k 1, k 2 and k m are normal to te surface, and te momentum transferred to te medium generates pressure (or stress) at te boundary between te two media. Te trajectory of te poton is manifested in te medium as pressure. 3. Proposed Young's double slit experiment In tis proposed experiment, te detection of te Medium A k m Dielectric medium (medium B) Poton Medium A k m -k m Crystal plate Poton -k m Wave number k 2 Transparent electrode Resistor Wave number k 2 Switc Meter Fig. 1 depends on te speed of ligt Te difference in te wave number (te speed of ligt) causes radiation pressure at te surfaces of te two media. pressure increase tere. A poton as energy ω and momentum k; ere ω is te frequency. Te velocity of a poton canges according to te medium troug wic te poton travels, as do te wave number and te momentum. Te transfer of te momentum difference to te medium is te cause of te pressure increase at te boundary. Figure1 sows te local momentum transfer of a poton tat propagates from medium A to medium B, and te stress at te boundary. Te law Fig. 2 Frequency redsifter using a AT-cut crystal plate Te stress on te surface induces te carges. We could detect te carges after te poton arrived at te screen by using te switc. Te sensitivity of poton detection is controlled using te variable resistor. momentum transferred locally to te medium is discussed. Figure 2 sows an experimental setup, using AT cut crystal plates wit transparent electrodes. If a poton passes troug te crystal plate, te stress on te surface induces carges on te electrodes. We can measure te carges by means of a switc and a current meter, after poton detection on te screen. It is already known tat te interference pattern appears if te carges are 2

3 not detected. Of course, it is impossible to determine weter or not a poton passes troug te crystal plate, because te signal is idden in termal noise. However, we can determine weter or not te attempt of detection destroys te interference patterns. In tis experiment, a poton loses some energy due to te detection of te carges, so te redsift ω red occurs. Te law of conservation of energy is ω 1 = ω 2 + ω red + ω red (2) Here, ω 1 is te poton frequency in medium A, ω 2 is te frequency in medium B, ω red is te redsift (i.e., discarged energy), and ω red is te deviation of te redsift. We assume tat te detection of te carges causes an uncontrolled momentum kick to a poton, wic is presented as te deviation of te redsift ω red. Here, te detection of te carges is performed by te discarge tat is disturbed by termal noise, wic causes deviation of te redsift ω red. However, we can modify te redsift by controlling te sensitivity of te carge detection. Te sensitivity is controlled using a variable resistor in te circuit. At a ig resistance te carges are not completely discarged and terefore we cannot obtain te Te setup for te experiment is sown in Fig. 3. If we place te crystal plates as dielectric media in te pat of a poton, one of te dielectric media will obtain momentum according to te passage of te poton. In tis argument, two pats are observed wit distinct separation. According to te law of conservation of momentum, te trajectory of a poton, wic is indicated by te pressure on te boundary of te dielectric media, is sown. Terefore, te proposed experiments test weter or not te detection of poton energy, wic is equivalent to te stress, destroys te interference patterns. Wit tis experiment, we can test te correlation between te pat detection and te interference pattern generation. Te pat detection procedure may result in a redsift, wic is controlled by te sensitivity of te circuit. Terefore, we ave two metods wit wic to control te interference pattern, one by controlling te sensitivity of te circuit, and te oter by controlling te distance between te crystal plate and te screen, wic is indicated by L in Fig. 3, because te redsift ω red and te distance L cause te pase sift of a poton on te screen. Poton trajectory 4. Application of te Heisenberg relations We can testify tis argument, using te Heisenberg relations for te position and te momentum of a poton, wic is represented Poton source Double slit L Screen x p, (3) 4π were x is te position, p is te poton momentum, and means a deviation. We discuss te condition tat a poton as te position and te momentum, using te Heisenberg relations. Here we discuss te momentum p m, wic is te momentum tat is transferred to te medium, and if te ligt speed is 20% decreased in te dielectric medium Fig. 3 Proposed Young's double slit experiment required information, but at lower resistance, we can obtain complete information. p p m 0.2 ( p m ), (4) 5 ere is wavelengt. If we try to obtain te 3

4 momentum p m wit a 10% deviation. Ten assuming p 0. 1 m p m p =, (5) we obtain te condition of te deviation for position x. (6) Equation (5) sows tat we need te distance, at least, equal to te deviation x =, wen we try to measure te momentum p m wit a 10% deviation. So if we take x 1 x, (7) were, x 1 is te distance between te two slits. Tis condition will be fulfilled in tis experiment, wic guarantees tat we can distinguis te momentum p m teoretically. It indicates tat we could distinguis te "wic way" by te momentum p m, if we take te distance x 1 in Eq.(8). For example, using te 630nm laser, te distance between te two slits x µm is enoug to distinguis wic way a poton as passed. In tis proposed experiments, tese discussions indicate tat te mecanism wic interferes to detect te "wic way" of poton travel is not te Heisenberg relations but termal noise disturbance. 5. Discussions In tis letter, we argued te detection of te "wic way" of poton travel, but we did not discuss te mecanism of te interference pattern generations. We tink Bom's quantum potential [15], wic simultaneously determine te poton trajectories and interference patterns, is one of te most suitable teoretical backgrounds of tis experiment. We tink te feasibility of tese experiments depends on te tecnology of termal noise reduction. We do not tink te feasibility of tese experiments are not restricted by te Heisenberg relations. 6. Conclusions In tis letter, we proposed a feasible experiment, wic demonstrates te possibility of simultaneously determining te poton trajectory and te interference pattern. In tis experiment, if te switces are off and te measurement is not performed, te interference patterns can be detected. However, te sensitivity of poton detection can be controlled using a variable resistor. Of course, poton detection by te crystal plate cannot be performed, because of te termal noise disturbance, but not te Heisenberg relations. However, we can determine weter te attempt to detect te poton trajectory destroys te interference patterns, and weter te pase sift of te poton according to te frequency redsift ω red and te deviation ω red can be restricted by adjusting te distance L. Parameters suc as te sensitivity of poton detection and te distance L can be modified linearly. In tis experiment, longitudinal rater tan transverse momentum is discussed. In oter words, te correlation between te detection of te poton trajectory and interference pattern generation is discussed, i.e., we do not discuss te relation between transverse momentum transfer and interference pattern generation. We could obtain information on complementarity and uncertainty wit regard to te "wic way" problem of Young's double slit experiments. References [1] N. Bor, in Albert Einstein: Pilosoper-Scientist, edited by P. A. Sclipp (Library of living Pilosopers, Evaston, 1949), [reprinted in Quantum Teory and Measurement, edited by J. A. Weeler and W. H. Zurek (Prinston University Press, Prinston, 1983)]. [2] R. P. Feynman, R. B. Leigton, and M. Sands, Te Feynman Lectures on Pysics (Addison Wesley, Reading, MA, 1965), Vol. 3. 4

5 [3] C. Coen-Tannoudji, B. Diu, and F. Laloe, Quantum mecanics (Wiley, New York, 1977). [4] M. O. Scully, B. G. Englert and H. Walter, "Quantum optical tests of complementarity" Nature, 351, (1991), [5] B. G. Englert, M. O. Scully, and H. Walter, "Complementarity and uncertainty " Nature, 375, (1995), [6] E. P. Storey, S. M. Tan, M. J. Collett and D. F. Walls, "Pat detection and te uncertainty principle" Nature, 367, (1994), [7] E. P. Storey, S. M. Tan, M. J. Collett and D. F. Walls, "Complementarity and uncertainty" Nature, 375, (1995), [8] E. P. Storey, S. M. Tan, M. J. Collett and D. F. Walls, Pys. Rev. A, 47, (1993), [9] H. M. Weisman and F. E. Harrison, "Uncertainty over complementarity" Nature, 377, (1995), 584. [10] S. M. Tan and D. F. Walls, "Loss of coerence in interferometry" Pys. Rev. A, 47, (1993), [11] P. Kwiat, H. Weinfurter, T. Herzof, A. Zeilinger, and M. A. Kasevic, "Interaction-Free Measurement" Pys. Rev. Lett., 74, (1995), [12] U. Eicmann, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, W. M. Itano, D. J. Wineland, and M. G. Reizen, "Young's Interference Experiment wit Ligt Scattered from Two Atoms" Pys. Rev. Lett., 70, (1993), [13] H. M. Weisman, F. E. Harisson, M. J. Collet, S. M. Tan, D. F. Walls, and R. B. Killip, "Nonlocal momentum transfer in welcer Weg measurements" Pys. Rev. A, 56, (1997), [14] H. M. Wiseman, "Bomian analysis of momentum transfer in welcer Weg measurements" Pys. Rev. A, 58, (1998), [15] D. Bom and B. J. Hiley, Te undivided universe, (Routledge, London, 1993),