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. It has been ar- a spin detector, one will either get a “spin-down” or a gued, and demonstrated, that one need not actually carry “spin-up”. In the beginning, if we started out with a state out a which-way measurement in order that the interfe- , the detector at the end will register a state. On reence disappears. Mere existence of which-way infor- the other hand, if we started with a state, the detec- mation in the state is sufficient to destroy any potential tor at the end will register a state. Thus, the detec- interference [3,17]. In other words, one can say that mere tor at the end obtains a which-initial-state information existence of the possibility of getting which-way infor- about the spin, exactly as the detectors in the modified mation is enough to destroy the interference pattern. This two-slit experiment obtain a which-slit information. can be simply demonstrated as follows. Suppose that the So, now we carry out our thought experiment, with the state of a particle having passed through a double-slit, initial state , as given by (1). After a time we 0 just before hitting the screen, is given by get into a state which is equivalent to the interference x x x (9) region in the modified two-slit experiment: AB 11 where x and x represent the amplitude of U . (6) A B 0 22 the particle passing through slit 1 and 2, respectively. Probability of finding the particle at a point x on the Equation (6) represents an interference pattern, be- screen, is given by cause the “down-spins” cancel out to give destructive interference, while the “up-spins” add up to give con- 222 xxx AB structive interference. So, in our two-state interference (10) ** experiment, there is one dark fringe and one bright fringe. ABx xx BA x After parts of the states have been destroyed by the “de- The last two terms represent interference. Now, let us structive interference”, what is left is just suppose that in a slightly modified version of this ex- 11 U . (7) periment, the state is given by 0 22 xx AB 1 x2 (11) Now, these two contributions from the two initially orthogonal states are identical. States which are not or- where 1 and 2 are certain orthonormal eigenstates thogonal, are naturally not distinguishable. Thus the of a suitable observable, which get entangled with the par- which-way or which-initial-state information is lost at ticle, and hence carry which-way information. A mea- this stage. It might serve some useful purpose by keeping surement of that observable yielding 1 will lead to a the two contributions separate and evolving them for a definitive conclusion that the particle passed through slit further time to see what they yield at the spin detector. 1, and likewise for 2 . In this case the probability of finding the particle at a point x, is given by 11 U 2 (8) 222 0 22 22 x ABxx . (12)
So, we see that each part of the initial state leads to a The two terms which would have given interference, superposition of and states. So, although the are killed by the orthogonality of 1 and 2 . One spin-detector at the end still gives either a “spin-up” or a would notice that a which-way measurement is not even “spin-down”, it gives no information about whether the needed here—mere existence of which-way information, initial state was a or . Yet, if one started out or mere possibility of a which-way measurement, is with either a or initial state, which is equiva- enough to destroy interference. lent of closing one slit, (5) tell us that each will go to only its corresponding detector, thus giving the which- 3.2. Interference Destroys Which-Way initial-state information. Information Taken in isolation, this over-simplified thought experi- ment may not prove anything substantial, but it does pro- In the following we will prove that the converse of which- vide a clue to where the problem lies in the modified way information necessarily destroys interference is also two-slit experiments. It appears that the very existence of true. This would mean that a quantum state which has a interference destroys which-way information. form required to yield interference, cannot contain any
Copyright © 2012 SciRes. JQIS 38 T. QURESHI which-way information. AA BB In any variant of the conventional two-slit experiment, | | | | the initial state has to be in a superposition of two or- thogonal states. This follows simply from the fact that the | | | | (16) two slits are distinguishable. In the case of a Mach- | | | | Zehnder interferometer, a half-silvered mirror splits the beam into superposition of two spatially separated beams. | | 2 Thus the initial (unnormalized) state can be writ- 0 Using orthogonality of A and B , we get ten as ||1|. (17) . (13) 00AB0 Substituting this in (16) yields These states then evolve and reach the region where ||||| 4. (18) they spatially overlap. and evolve to A 0 B 0 A and B respectively, so that in the region of In the region of overlap the state has the actual form overlap, the state looks like . Using this, the norm of can be written as AB . (14) |||||. (19) The time evolution being unitary, these two parts re- tain their orthogonality, so that AB 0 . Let us Combining (18) and (19), we get assume that in the region where they interfere, they have || 2. (20) the form: From the above equation it is obvious that and AB, , (15) can never be orthogonal. Hence in overlap where is chosen to be normalized, keeping A region contains no which-way information. and B unnormalized, which can always be done. This general analysis shows that if the state has a form Here, , and need not be simple states— which could yield interference, it cannot contain any each may involve a multitude of degrees of freedom. which-way information. In other words, existence of in- However, A and B have to have this form, so terference necessarily destroys which-way information. that there are parts from the two which cancel out exactly, So, complementarity is robust, and cannot be violated to give the so-called dark-fringes. In the case of the con- in any such interference experiment where one tries to would constitute the ventional two-slit experiment, look for which-way information after interference. pattern involving all the dark fringes (a specific example of this appears in the next section). For the case of a 4. Two-Slit Experiment with Wave-Packets Mach-Zehnder interferometer, represents the part of the state from one beam, reaching one of the two detec- Coming back to the modified two-slit experiment, let us tors. Interference happens when parts reaching one see what implication the preceding analysis has on it. detector, from both the beams, cancel out. The result is, Clearly, in the modified two-slit experiment, after the that particular detector does not detect any particles (dark particle passes through the interference region, the fringe). Thus, if a state described by (14), (15) has a which-way information is lost. The detector DA clicking non-zero , one can be sure that a measurement will doesn’t mean that the particle came from slit A. It may lead to an interference pattern. One can thus associate a appear hard to visualize that a wave-packet which travels non-zero with the existence of interference, even in a straight line from slit A can contribute to a click on without doing an actual measurement. detector DB , which is not in its direct path. However, the If complementary could indeed be violated, the parts argument of momentum conservation will hold only if of the two initial states which are left (after the destruc- we knew that the particle started out from slit A—in that tive interference), namely and , should be or- situation it would never reach detector DB . In order to thogonal, so that they can contain a which-way informa- demonstrate the validity of these arguments for the tion. States which are not orthogonal, cannot lead to dis- modified two-slit experiment, let us look at the two-slit tinguishable outcomes in a measurement. In order that experiment in more detail. describes the pattern of dark fringes, it should be We carry out the analysis for massive particles, to orthogonal to and , so that it is distinguishable show that the argument hold not just for photons, but for from the other parts. However, the following analysis any quantum particle. Consider the particle to be moving goes through even without that restriction. along the x-axis, and the slit plane parallel to the y-axis. The norm of is given by We will only be interested in the dynamics of the state in
Copyright © 2012 SciRes. JQIS T. QURESHI 39
22 the y-direction, whereas the x-axis motion just serves the yy00 yy 1 purpose of transporting the particle from the slits to the yt, aCt eΩΩtt e detectors. Let us assume that when the particle emerges 2 from the double-slit, its state is given by a superposition (25) 22 yy00 yy of two distinct, spatially localized wave-packets. The 1 ΩΩtt state at this time, which we choose to call t 0 , can be bC t e e 2 written as y,0 In this state, the parts of the state coming from the two slits are equal. So, there is no which-way information in 22 yyyy 00 (21) the state any more. If a lens is used after this stage, which 1 22 1/4 ae be , 2 π 2 takes the part ey yt0 /Ω() to one detector and the other part to the other detector, one can easily see from (25) where is the width of the wave-packets, 2y0 is the slit that a part coming from one slit, becomes a superposition separation, and a and b are the amplitudes of the two of two parts going to the two detectors. So, each detector wave-packets. receives equal contribution from the two slits, and regis- The wave-packets evolve in time, during which they tering a particle gives no information on which slit the spread, and reach the region where they overlap, and thus, particle came from. To a mind used to classical way of interfere. This state can just be obtained by evolving (21), thinking, it might appear that the particle received at a using the Hamiltonian H pm2 2 . Hence, the state at particular detector came from a particular slit, but the time t can be written as preceding analysis shows that this presumption is erro- neous. 2 Note however, that if one of the slits is closed, say, if a y y0 2 yt,exp aCt y y0 /Ω()t 2it = 1 and b = 0 the state at time t will be Cte , 2 m which goes to only one detector when a lens is used. (22) A critic might argue that instead of cancelling out some y y 2 parts of the two wave-packets, one might just evolve bC t exp0 , them separately, and because they were initially orthogo- 2it 2 nal, they will give distinct result at the end. The answer m to that is, if one really looks for interference by putting where thin wires, one is blocking those very parts of the wave- 1 packets which are cancelling out. So, those parts will not Ct 1/4 . (23) reach the detectors, even if one insists on not cancelling π 22 itm them out. On the other hand, the argument of bringing in Now, this state can also be rewritten as wires is not really needed. The state (25) as such, has no which path information. yt,
yy22 5. Conclusion 0 Ωt 22yy yy aC t e cosh00 sinh (24) Although the modified two-slit experiments do have ge- ΩΩtt nuine interference, as shown by introducing thin wires,
yy22 the detectors detecting the photons behind the converging 0 Ωt 22yy00 yy lens, do not yield any which-way information. It was not bC t e cosh sinh ΩΩtt easy to see this without mathematically rigorous argu- ments. The wave-packet analysis makes this fact trivially where Ωtit2 2m. The term involving obvious. Many earlier analysis of complementarity have concentrated on showing how existence of which-way 2yy 0 information destroys interference. We have taken the sinh in this expression is an example of Ωt reverse approach, as demanded by the modified two-slit introduced in the preceding section. experiment. We have shown that if a state is such that the In the usual case of a two-slit experiment, the ampli- modulus square of its wave-function yields an interfe- tudes coming from the two slits are approximately equal, rence pattern, it cannot contain any which-way informa- i.e., ab12. In this case, the sinh terms cancel out tion. The complementarity principle is thus robust and to give the dark fringes, and what is left is cannot be violated in any experiment which is a variant
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