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Coherence, Interference and Visibility

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Coherence, Interference and Visibility Coherence, Interference and Visibility Tabish Qureshi Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi, India. E-mail: [email protected] Editors: Louis Marchildon & Danko Georgiev Article history: Submitted on May 10, 2019; Accepted on June 13, 2019; Published on June 17, 2019. he interference observed for a quanton, travers- dark bands on the screen signifying interference [1]. This ing more than one path, is believed to character- was a follow-up of the wave theory of light which he Tize its wave nature. Conventionally, the sharp- had been developing. Interference was understood as ness of interference has been quantified by its visibil- resulting from superposition of two waves, which add ity or contrast, as defined in optics. Based on this vis- constructively or destructively at different locations. It ibility, wave-particle duality relations have been for- was soon realized that in order to produce an observable mulated for two-path interference. However, as one pattern of interference fringes, the two waves must have generalizes the situation to multi-path interference, a constant phase difference between them. This property it is found that conventional interference visibility is of having a constant phase difference between two waves not a good quantifier. A recently introduced measure was called coherence. If the phase difference between two of quantum coherence has been shown to be a good waves is not constant, one has to specify how much does quantifier of the wave nature. The subject of quan- the phase difference vary with time, or the degree of co- tum coherence, in relation to the wave nature of quan- herence. The degree of coherence decides how distinctly tons and to interference visibility, is reviewed here. It visible is the interference pattern. is argued that coherence can be construed as a more As the field of classical optics developed, coherence general form of interference visibility, if the visibil- was precisely defined in terms of a mutual coherence ity is measured in a different manner, and not as con- function, which is essentially a correlation function [2, 3]. trast. If one considers two fields E1 and E2 emanating from two Quanta 2019; 8: 24–35. slits, there is a time difference, say τ, between their arrival at a point on the screen. The intensity on the screen due to the two slits can be represented as 1 Introduction ∗ ∗ arXiv:1905.00917v4 [quant-ph] 19 Jun 2019 I = I1 + I2 + hE1(t)E2(t + τ)i + hE1(t)E2(t + τ)i ∗ The phenomenon of interference was discovered long = I1 + I2 + 2<(hE1(t)E2(t + τ)i): (1) back in 1801 when Thomas Young performed his sem- where the angular brackets denote an averaging over t, inal experiment by making a light beam pass through and I , I represent the respective intensities of the two two spatially separated paths, and observing bright and 1 2 fields at a point on the screen. If t is the time taken for the field to reach the screen from the slits, it can be represented in terms of the field at the slits, at an earlier This is an open access article distributed under the terms of the Creative Commons Attribution License CC-BY-3.0, which time: E1(t) = K1E1(0); E2(t + τ) = K2E2(τ), where permits unrestricted use, distribution, and reproduction in any medium, K1; K2 are time-indendent propagation functions. One provided the original author and source are credited. can define the normalized coherence function, in terms Quanta j DOI: 10.12743/quanta.v8i1.87 June 2019 j Volume 8 j Issue 1 j Page 24 D Source 1 d y 2 x Double slit Screen Figure 1: Schematic diagram of a two-slit interference experiment. There are two possible paths a quanton can take, in arriving at the screen. of the field at the slits, as states and density operators. In such situations, Eq. (4) is ∗ ∗ generalized to hE1(0)E2(τ)i hE1(0)E2(τ)i γ12(τ) = = p ; (2) p 2 2 V = 2jρ12j; (5) hjE1(0)j ihjE2(τ)j i I1I2 The visibility of the interference fringes is convention- where ρ12 represents the off-diagonal part of the density ally defined as [2] matrix of the quanton, in the basis of states representing quanton in one arm of the interferometer or the other. Imax − Imin V = ; (3) The diagonal parts of the density matrix, ρ and ρ , I + I 11 22 max min represent the probability of the quanton passing through where the notations have the usual meaning. It can be one or the other arm of the interferometer. easily seenp that in this case the visibility turns out to be V = 2 I1I2jγ12j , because K ; K cancel out from the I1+I2 1 2 expression. For identical width slits which are equally 2 Wave-particle duality illuminated, the visibility reduces to The fringe visibility given in Eq. (3) formed the basis of V = jγ j: (4) 12 all later work on wave-particle duality. It was taken for Thus we see that this fringe visibility is a straightforward granted that the fringe visibility captures the wave nature, measure of coherence of waves coming from the two slits. and hence the coherence, of the interfering quanton, in Later the field of quantum optics was developed and a all interference experiments. When Bohr proposed his quantum theory of coherence was formulated [4,5]. How- principle of complementarity in 1928 [7], the two-slit in- ever, the quantum theory of coherence closely followed terference experiment (see Figure 1) became a testbed for the earlier classical formulation, except that the classical it. For the two-slit experiment, Bohr’s principle implies fields were replaced by field operators and the averages that if one set up a modified interference experiment in were replaced by quantum mechanical averages [6]. The which one gained complete knowledge about which of coherence function was still the correlation function of the two slits the quanton went through, the interference fields. In analyzing a double-slit interference experiment would be completely destroyed. Acquiring knowledge using quantum optics, the fringe visibility continued to be about which slit the quanton went through, would imply related to the mutual coherence function γ12 via Eq. (4). that the quanton behaved like a particle, by going through For theoretically analyzing interference experiments one particular slit. The wave nature was of course repre- done with quantum particles, e.g. electrons, and experi- sented by the interference pattern. Wootters and Zurek [8] ments like Mach–Zehnder interferometer where paths tra- were the first ones to look for a quantitative statement of versed are discrete, it is more convenient to use quantum Bohr’s principle. They studied the effect of introducing Quanta j DOI: 10.12743/quanta.v8i1.87 June 2019 j Volume 8 j Issue 1 j Page 25 a path-detecting device in a two-slit interference exper- lead to an increase in the fringe contrast or visibility, as iment. They found that acquiring partial information given by (3). The results seemed to fly in the face of the about which slit the quanton went through, only partially basic idea of complementarity that any increase in path destroys the interference pattern. This work was later knowledge, should lead to a degradation of interference. extended by Englert who derived a wave-particle duality However, the authors concluded that the fringe contrast relation [9] or visibility, as given by (3), is not sufficient to quantify D2 + V2 ≤ 1; (6) sharpness of interference. Based on the results of this ex- periment, Luis [20] went to the extent of claiming that for D where is the path-distinguishability, a measure of the multi-path quantum interferometers the visibility of the V particle nature, and the visibility of interference, as interference and ‘which-path’ information are not always defined by Eq. (3). The inequality saturates to an equality complementary observables, and consequently, there are if the state of the quanton and path-detector is pure, unaf- path measurements that do not destroy the interference. fected by any external factors like environment induced decoherence. The issue of wave-particle duality was looked at, using 3.1 Contrast in multi-beam interference a different approach, by Greenberger and Yasin [10]. In an interference experiment which is distinctly asymmetric, 3.1.1 Three-path interference either because of unequal width of the slits, or because of In an interesting work, Bimonte and Musto [21] analyzed the source being unsymmetrically placed with respect to the visibility given by (3) for multi-beam interference the two slits, the quanton would be more likely to pass experiments. They argued that the traditional notion of through one of the slits, than the other. One could make a visibility is incompatible with any intuitive idea of com- prediction about which slit the quanton went through, and plementarity, but for the two-beam case [21]. In the would be right more than 50% of the times. They argued following we rephrase their argument, as it expounds the that the predictability means the quanton is partially be- need for a new visibility. having like a particle. They derived the following duality Consider a quanton passing through n paths, such that relation for such an experiment [10] the state corresponding to the k’th path is j ki. This can P2 + V2 ≤ 1; (7) happen because of a beam-splitter or because of the quan- ton passing through a multi-slit. In general, its density where P is the path-predictability, and V the visibility of operator may be written as interference, given by Eq.
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