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Modified Two-Slit Experiments and Complementarity Journal of Quantum Information Science, 2012, 2, 35-40 http://dx.doi.org/10.4236/jqis.2012.22007 Published Online June 2012 (http://www.SciRP.org/journal/jqis) Modified Two-Slit Experiments and Complementarity Tabish Qureshi Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi, India Email: [email protected] Received March 17, 2012; revised April 25, 2012; accepted May 5, 2012 ABSTRACT Some modified two-slit interference experiments claim to demonstrate a violation of Bohr’s complementarity principle. A typical such experiment is theoretically analyzed using wave-packet dynamics. The flaw in the analysis of such ex- periments is pointed out and it is demonstrated that they do not violate complementarity. In addition, it is quite gener- ally proved that if the state of a particle is such that the modulus square of the wave-function yields an interference pat- tern, then it necessarily loses which-path information. Keywords: Complementarity; Wave-Particle Duality; Two-Slit Experiment 1. Introduction pattern, one cannot get the which-way information. The authors have a clever scheme for establishing the exis- It is generally accepted that the wave and particle aspects tence of the interference pattern without actually observ- of quantum systems are mutually exclusive. Niels Bohr ing it. First, the exact locations of the dark fringes are elevated this concept, which is probably born out of the determined by observing the interference pattern. Then, uncertainty principle, to the status of a separate principle, thin wires are placed in the exact locations of the dark the principle of complementarity [1]. fringes. The argument is that if the interference pattern Since then, the complementarity principle has been exists, sliding in wires through the dark fringes will not demonstrated in various experiments, the most common affect the intensity of light on the two detectors. If the of which is the two-slit interference experiment. It has interference pattern is not there, some photons are bound been shown that if the which-way information in a dou- to hit the wires, and get scattered, thus reducing the pho- ble-slit experiment is stored somewhere, the interference ton count at the two detectors. This way, the existence of pattern is destroyed, and if one chooses to “erase” the interference pattern can be established without actually which-way information after detecting the particle, the disturbing the photons in any way. This is similar in interference pattern comes back. This phenomenon goes spirit to the so called “interaction-free measurements” by the name of quantum erasure [2,3]. where the non-observation of a particle along one path Recently, some interesting experiments were proposed establishes that it followed the other possible path, with- and carried out which claim to violate the complement- out actually measuring it [7]. The authors carried out the tarity principle [4-6]. A schematic diagram of a typical such experiment is shown in Figure 1. Basically, it con- sists of a standard two-slit experiment, with a converging B’ DB lens L behind the conventional screen for obtaining the A interference pattern. Although the experiments use pin- holes instead of slits, we will continue to refer to them as B slits. If the screen is removed, the light passes through DA the lens and produces two images of the slits, which are A’ captured on two detectors DA and DB respectively. Open- S G L ing only slit A results in only detector DA clicking, and opening only slit B leads to only DB clicking. The authors Figure 1. A schematic representation of the modified two- slit experiment. Light emerges from two pin-holes (not slits) argue that the detectors DA and DB yield information about which slit, A or B, the particle initially passed A and B and interferes. Thin wires are placed carefully in the exact locations of the dark fringes of the interference through. If one places a screen before the lens, the inter- pattern. The lens L collects the light and obtains the images ference pattern is visible. of the two slits A’ and B’, respectively. The detectors DA Conventionally, if one tries to observe the interference and DB collect the photons for these two images. Copyright © 2012 SciRes. JQIS 36 T. QURESHI experiment and found that sliding in wires in the ex- two experiments if putting in the wires is not changing pected locations of the dark fringes, doesn’t lead to any the results. One would say, the interference is out there in significant reduction of intensity at the detectors. Hence the middle, and one can check out that the photons are they claim that they demonstrated a violation of com- not passing through certain regions, the dark fringes. plementarity. Let us understand what is happening in the experiment The complementarity principle is at the heart of quan- slightly better by simplifying the experiment. One might tum mechanics, and its violation will be deeply disturb- argue that a two-slit experiment is the simplest experi- ing to its established understanding. As expected, there ment one can imagine. But this experiment still has a has been scepticism towards the modified two-slit ex- large number of degrees of freedom, and the Hilbert periment, and a heated debate followed [8-16]. Interest- space is large. We will show in the following that the ingly, different criticisms do not agree with each other on simplest interference (thought) experiment can be just why complementarity is not violated in this experiment. carried out using a spin-1/2 particle. None of the criticisms has been satisfactorily able to point out any flaw in the interpretation of the experiments. Most 2. A Simplified “Two-Slit” Experiment agree that if the introduction of wires has no effect on the Let there be a spin-1/2 particle traveling along x-axis. intensity, it shows that interference exists. Almost every- For our purpose, its physical motion is unimportant—we body seems to agree that detecting a photon at (say) DA will only be interested in the dynamics of its spin. In the means that it came from slit A. However, just because two-slit experiment, the particle emerges from the slits in blocking slit B leads to only detector DA clicking and a superposition of two physically separated, localized blocking slit A leads to only detector DB clicking, quan- wave-packets, which are orthogonal to each other. Their tum mechanics doesn’t say that when both slits are open, physical separation guarantees their orthogonality. Any detector DA clicking implies that the photon came from subsequent unitary evolution will retain their orthogo- slit A. This has also been pointed out by Kastner [9]. nality. We assume the initial state of the spin to be Some authors [13,15] have tried to argue along the 1 following line. Blocking dark fringes also blocks out . (1) 0 parts of the bright fringes. They believe, when both slits 2 are open, the photons contain full which-slit information, Here, the states and are the eigenstates of which is partially destroyed by partially blocking the S , and play the role of the two wave-packets that emerge bright fringes. They argue that since without the wires Z from the double-slit. The time evolution, in the conven- completely blocking the dark fringe, one cannot infer the tional two-slit experiment, spreads the wave-packets so existence of the interference pattern, when one tries to that they overlap. In our thought experiment, we employ increase the information about the existence of the inter- a homogeneous magnetic field B, acting along the y-axis, ference, the which-slit information is proportionately to evolve the two states and . Thus the Hamil- decreased. However, in these works, the calculation shows tonian of the system is H BS . the existence of full interference, without any blocking y wires. Without the blocking wires, the photons are claimed The time evolution operator can be written as to have full which-slit information (distinguishability = 1, i Ut expBSyt in their language). So, if one were to take their calcula- tion of distinguishablity as correct, then the existence of (2) full interference in the calculation (in the sense of Bt Bt 2 cosi y sin x yielding an interference pattern) seems to imply 22 that quantum formalism allows existence of interference where y is the usual Pauli matrix. The time evolution pat- tern for photons for which full which-slit informa- operator, for a time π 2B has the form tion exists. This clearly goes against the spirit of com- plement- tarity. In our view, the calculation of distin- 1 Ui 1 y . (3) guishability in these works is fundamentally flawed. 2 Many authors have fallen back to a more formal inter- It is straight forward to see that the time evolution pretation of Bohr’s principle, namely that the wave and transforms the states and as particle nature cannot be seen in the same experiment, 1 for the same photons. They argue that authors have to do U not one, but two experiments to prove their point—one 2 (4) without the wires, one with the wires. Some argue that 1 the existence of fringes has already been assumed, and U () 2 that the argument of the experiment is circular. However, a reader who respects empirical facts, doesn’t see it as A further evolution through a time will transform Copyright © 2012 SciRes. JQIS T. QURESHI 37 the states as 3. Interference and Which-Way Information U 2 3.1. Which-Way Information Destroys (5) Interference U 2 There have been some recent developments in under- After an evolution through a total time 2 , if one puts standing the origins of complementarity.
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