The phantom slit effect James Q. Quach∗ and Maciej Lewenstein ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain arXiv:1709.09851v1 [quant-ph] 28 Sep 2017 We show that under the weak measurement scheme, the double-slit experiment can produce an interference pattern even when one of the slits is completely blocked. The initial and final states are corpuscular, whilst the intermediate states are wave-like, in that it exhibits an interference pattern. Remarkably, the interference pattern is measured to be vertically polarised, whilst simultaneously the individual photons are measured to be horizontally polarised. We call this the phantom slit effect. The phantom slit is the dual of the quantum Cheshire cat. The conventional view of measurement in quantum mechanics is that it is a destructive process that irrevocably projects the system into an eigenstate of the observed variable. Weak measurements provides a formal non-destructive measurement scheme by weakly coupling the system to an ancilla, and performing a measurement (projection) on the ancilla in some appropriate basis [1]. Operationally, the ancilla is a measurement device with a pointer; the interaction of the system and the ancilla shifts the pointer state proportional to the magnitude of the observed variable. As the ancilla interacts only very weakly with the system, the state can evolve without appreciable disturbance. Weak values seek to represent observables of intermediate states, as the system evolves from an initial to a final or post-selection state. Judiciously choosing the postselection state onto which to project the ancilla (and system), have been shown to produce anomalous phenomena that have sparked debate over the interpretation of weak values [2–9]. Amongst the anomalous phenomena is the quantum Cheshire cat (QCC) effect [10], where the position of a photon exists in one arm of an interferometer, whilst its polarisation exists in the other arm; the whimsical name alludes to Lewis Carroll’s Cheshire cat, whose grin (polarisation) could exist without its body (photon). Backdropping the spirited debate are many experiments that have used weak values to measure several observables in different quantum systems [11–17]; this includes the famous Kocsis et. al experiment [18], which observed the individual trajectories of photons as they formed the interference pattern in the double-slit experiment. In this experiment, the photon momentum was weakly measured, and the post-selection was the photon position. Here we show an interesting new effect, where the second slit in a double-slit experiment can still affect an interference pattern even after its has been blocked or removed - we dub it the phantom slit. Weak values. Prior to measurement the initial system state |ψi i and pointer state |mi i are uncoupled. In the weak measurement scheme, the interaction Hamiltonian between the system and pointer is Ĥint = g(t)Ô P̂ , (1) FIG. 1. The phantom slit effect. Slit B is blocked, so that light emanates only from slit A. A weak value detector moves up and down in a line parallel to the slit plane. It measures a position dependent interference pattern given by hσ̂V iw = cos φ , revealing the wave-like nature of the intermediate states, between the initial and final states. Although the intermediate states are wave-like, the initial ψi (x) = hx|ψi i and final states ψf (x) = hx|ψf i are corpuscular. The interference pattern is vertically polarised, whilst the photons are horizontally polarised. d is inter-slit spacing, and θ is the angle from the normal of the slit plane. φ = kd sin θ, where k is the wavenumber of the light source. which couples the system’s observable Ô to the pointer momentum P̂ . The interaction with the pointer occurs for a short time, outside of which coupling constant g is zero, R so that the evolutionary operator is Û = exp(− ~i Ĥint dt) = exp(− ~i g ÔP̂ ). After the interaction with the pointer, the system undergoes a projective measurement where only a subset of the measured states are chosen. Labelling this post-selected state |ψf i, the final pointer state is (~ = 1) |mf i = hψf | exp(−ig ÔP̂ )|ψi i|mi i  hf |Ô|ii  P̂ |mi i ≈ hψf |ψi i 1 − ig hψf |ψi i (2) ≈ hψf |ψi i exp(−ighÔiw P̂ )|mi i where hÔiw ≡ hψf |Ô|ψi i hψf |ψi i is known as the weak value of Ô. (3) 2 The pointer momentum P̂ is conjugate to the pointer position Q̂. Let us now write R the initial pointer state in the position basis, |mi i = dq|qiϕ(q), where ϕ(q) ≡ hq|mi i. The final pointer state in the position basis then is Z |mf i ≈ hψf |ψi i exp(−ighÔiw P̂ ) dq|qiϕ(q) Z (4) = hψf |ψi i dq|qiϕ(q − ghÔiw ) where we have used the fact that P̂ acts as a translation operator that shifts the pointer state in the conjugate q-basis by ghÔiw . What this means is that if the pointer states were the positions of a needle on a measuring device, the interaction of the measurement device with the system will shift the position of the needle by a distance proportional to hÔiw , thereby giving us a measurement of observable Ô. Double-slit experiment. The interference pattern in the double-slit experiment is due to the phase differences φ of the paths emanating from the slits. Let |Ai and |Bi form an orthonormal basis representing the state of particles that went through slit A and B. For a point on the screen identified by √ the angle θ in Fig. 1, the state is |ψi = (|Ai + eiφ |Bi)/ 2 (we work in the regime were higher-order paths which pass through both slits are negligible [19]). φ = kd sin θ, where k is the wavenumber of the light source and d is the slit spacing. We will as usual assume that the double-slit experiment is in a steady-state. The polarisation of the photon is described by a linear combination of the horizontal |Hi and vertical |V i basis states. We also√define right-circular polarisation |Ri ≡ (i|Hi + |V i)/ 2, √and left-circular polarisation states |Li ≡ (i|Hi − |V i)/ 2. For coherent interference, the particles from slit A and B must have the same polarisation. Without loss of generality we assume right-circular polarisation, √ |ψo i = (|Ai + eiφ |Bi)|Ri/ 2 . (5) Projecting onto a position basis (for clarity, we distinguished the pointer position basis |qi from the system position basis |xi), the probability of detection of the particle at the detection screen is P (x) = hx|ψo ihψo |xi 1 = (|ψA (x)|2 + |ψB (x)|2 + 2|ψA (x)||ψB (x)| cos φ) . 2 (6) This produces the well known interference pattern of Young’s double-slit experiment. The cos φ term encapsulates the interference component of the pattern: destructive interference occurs when the √ paths are out of phase [φ = (2n+1)π], |−i ≡ (|Ai−|Bi)/ 2; constructive interference occurs wihen √ the paths are in phase (φ = 2nπ), |+i ≡ (|Ai + |Bi)/ 2, where n is an integer. One can non-destructively reveal these phase differences by performing a weak measurement between the slits and the detection screen. We measure the interference with the following detector for vertical polarisation: σ̂V = (|+ih+| − |−ih−|)|V ihV | (7) The detector for horizontal polarisation σ̂H , is similarly defined. The non-trivial eigenvalues of the detector are -1 for destructive interference and 1 for constructive interference. The effect of σ̂V on the state is to switch the |Ai and |Bi states of the vertical-polarisation component (similarly for σ̂H ), σ̂V (cA |Ai + cB |Bi)|V i = (cA |Bi + cB |Ai)|V i, (8) where ci are complex co-efficients. The expectation value of σ̂V reveals the interference component of the double slit experiment: hψo |σ̂V |ψo i = cos φ . (9) Operationally, if σ̂V was a localised measuring device, as it moved up and down a plane parallel to the slit plane, it would measure interference as shifts in its pointer between and including 1 and -1, depending on its position as determined by angle φ. Note that σ̂H will measure the same values, hψo |σ̂H |ψo i = cos φ. Phantom slit. Now we close slit B so that the initial state is (|Ai|Ri ≡ |A, Ri) |ψi i = |A, Ri . (10) Let us choose the post-selection state √ |ψf i = (|A, Hi − ieiφ |B, V i)/ 2 . (11) From Eq. (3), the weak value of σ̂V and σ̂H for this system is hσ̂V iw = cos φ , hσ̂H iw = 0 . (12) We have taken only the real component of the weak values, as this is the value that is represented by the shift in the pointer; the imaginary component represents the momentum of the pointer [1]. Another (more operational) interpretation of the imaginary component is given in Ref. [20]. Eq. (12) shows that even though slit B does not physically exists, under the weak measurement scheme, we can detect an interference pattern as if it does exists and is vertically polarised. In other words, if σ̂V were a localised measure device, as it as it moved up and down a plane parallel to the slit plane, it would successively measure constructive and destructive interference, in exactly the same pattern as in the double-slit experiment [Eq. (9)]. The key difference here however, is that only measurements which correspond to the postselection state |ψf i are selected. σ̂H in contrast would register no interference, and therefore the interference 3 polarisation of the photons (V̂ ≡ |V ihV |, Ĥ ≡ |HihH|): hV̂ iw = 0 , (a) (b) BS PBS BS BS PS HWP A PS B P2 BS P1 |ψi〉 (c) σV V H σH φ=0 π/2 π 3π/2 2π FIG. 2. (a) An interferometer setup of the double-slit experiment. A beam splitter (BS) splits the light into two paths, representing the paths from two slits. A phase shifter (PS) introduces a phase difference eiφ between the two arms. A final BS combines the two paths to coherently interfere, as detected by the (CCD) camera. (b) An interferometer setup of the phantom slit effect. Polarisers at P1 implement σ̂V and σ̂H ; polarisers at P2 implement V̂ and Ĥ. A half-way plate (HWP), PS, BS, and polarising beam-splitter (PBS) ensure that states orthogonal to |ψf i are not selected. The measurements are weak, so that the original source beam is undisturbed (dotted line) by the polarisers. (c) depicts the spot size at a given level of intensity as detected by a camera in (b), if all four weak measurements (σ̂V , σ̂H , V̂ , Ĥ) were applied simultaneously for different values of φ. The elements constituting σ̂V (σ̂H ) slightly deflects the beam up (down), and V̂ (Ĥ) slightly deflects the beam left (right). hσ̂V iw is at a maximum at φ = 0, 2π (constructive interference) and disappears at φ = π (destructive interference). Whilst the interference pattern is vertically polarised, the photons are horizontally polarised, hĤiw = 1. Note hσ̂H iw = hV̂ iw = 0 pattern comes from the vertical polarisation component only. In phantom slit effect, although the intermediate states exhibit interference, the initial and final states remarkably contain no interference terms: Pi (x) = |ψA (xi )|2 , Pf (xf ) = 21 (|ψA (xf )|2 + |ψB (xf )|2 ). Fig. 1 schematically shows the corpuscular nature of the initial and final states, whilst the intermediate states are wave-like. Another important feature arises when we measure the hĤiw = 1 . (13) Eq. (13) shows that the photons of the intermediate states are horizontally polarised. Weak measurements are a non-destructive process, and therefore all measurements commute and can be made simultaneously. Therefore, whilst the interference pattern of the intermediate state is vertically polarised, the photons themselves are simultaneously measured to be horizontally polarised. Interferometer. The double-slit experiment may be implemented as an interferometer. A beam splitter (BS) splits the light into two paths, representing the paths from two slits. The phase differential can be controlled with a phase shifter (PS) which introduces a phase difference eiφ between the two arms [Fig. 2(a)]. Viewing the phantom slit effect in an interferometer helps remove the mysteries of the effect, and weak values in general. To implement the phantom slit effect one needs to be able to post-select the state |ψf i. |ψf i is in fact the QCC post-selection state generalised with a phase differential. One may consider the phantom slit effect as the dual of the QCC effect, with the roles of polarisation and position reversed. The corresponding schematic of an interferometric setup for post-selection is given in Fig. 2(b). The post-selection state is selected with a halfway plate (HWP), PS, BS, and polarising beam splitter (PBS). The HWP switches polarisation |Hi ↔ |V i. The PS shifts the phase by ie−iφ . The PBS transmits horizontal polarisation and reflects vertical polarisation. With path A (B) in Fig. 2(b) corresponding to state |Ai (|Bi), one sees that under these transformations, the cameras in Fig. 2(b) detects |ψf i with 100% probability. In an interferometric setup, σ̂V (σ̂H ) can be implemented as a polariser that reflects vertical (horizontal) polarisation, placed at location P1 in Fig. 2(b). The vertical (horizontal) polariser is slightly tilted up (down) to produce a small vertical displacement of the beam. Similarly, a polariser that reflects vertical (horizontal) polarisation would implement V̂ (Ĥ), placed at location P2 in Fig. 2(b). However here the vertical (horizontal) polariser is slightly tilted left (right) to produce a small horizontal displacement of the beam. Placed at these locations, these polarises have the desired effect on the initial state, i.e. √ √ σ̂H |ψi i = |BHi/ 2 , (14) σ̂V |ψi i = |BV i/ 2 , √ √ Ĥ|ψi i = |AHi/ 2 , V̂ |ψi i = |AV i/ 2 , (15) For weak measurements, the polarisers should only interact weakly, letting most of the light through [dotted line in Fig 2(b)]. Furthermore, the deflection brought about by these polarisers should be less than the characteristic cross-section width of the beam, so that it is uncertain whether an individual photon has been deflected 4 of not. The detection screen could be charge-coupled device cameras, as for example used in the Koscsis et. al double-slit interferometer experiment; the relative intensity at the detection screen acts as the pointer. The tilt of the polarisers are chosen so that the deflected beam is incident on a location of the source beam spot that is of approximately the same order of intensity. Therefore after recombination with the source beam, the normalised intensity detected by the cameras above the background intensity is given by Eq. (12) and Eq. (13) for the corresponding detectors. Fig. 2(c) depicts the spot sizes at a given level of intensity above the background, if all four weak measurements were applied simultaneously for different values of φ. The interpretation of weak values remains an intensely debated topic. The phantom slit effect should open up further discussion in the context of the double-slit experiment, which serves as a golden-standard of foundational quantum interpretation. In the Kocsis et al. double-slit experiment, weak measurements were used to map the individual paths of photons, as they moved from the slits to the detection screen. In further theoretical and experimental work, it would be interesting to map the weak value trajectories of individual photons of the phantom slit effect, along the lines of the Kocsis et al. experiment. Acknowledgements. The authors thank M. Bera, A. Riera, and S. Quach for discussions and checking the manuscript. This work was financially supported by the European Commission’s Marie Curie Actions for Cofunding of Regional, National and International Programmes (COFUND), the Spanish Ministry of Economy and Competitiveness (MINECO), Fundacio Privada Cellex, Spanish Ministerio de Economı́a y Competitividad grants (Severo Ochoa SEV-2015-0522 and FOQUS FIS2013-46768-P), and Generalitat de Catalunya (2014 SGR 874). [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] ∗ [1] [2] [3] [4] quach.james@gmail.com Yakir Aharonov, David Z. Albert, and Lev Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988). Atsushi Nishizawa, Kouji Nakamura, and Masa-Katsu Fujimoto, “Weak-value amplification in a shot-noise-limited interferometer,” Phys. Rev. A 85, 062108 (2012). P. Ben Dixon, David J. Starling, Andrew N. Jordan, and John C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009). Tatsuhiko Koike and Saki Tanaka, “Limits on amplifi- [18] [19] [20] cation by aharonov-albert-vaidman weak measurement,” Phys. Rev. A 84, 062106 (2011). Daniel Rohrlich and Yakir Aharonov, “Cherenkov radiation of superluminal particles,” Phys. Rev. A 66, 042102 (2002). A. Matzkin, “Observing trajectories with weak measurements in quantum systems in the semiclassical regime,” Phys. Rev. Lett. 109, 150407 (2012). Holger F Hofmann, “Sequential measurements of noncommuting observables with quantum controlled interactions,” New Journal of Physics 16, 063056 (2014). Justin Dressel and Alexander N. Korotkov, “Avoiding loopholes with hybrid bell-leggett-garg inequalities,” Phys. Rev. A 89, 012125 (2014). L. Vaidman, “Tracing the past of a quantum particle,” Phys. Rev. A 89, 024102 (2014). Yakir Aharonov, Sandu Popescu, Daniel Rohrlich, and Paul Skrzypczyk, “Quantum cheshire cats,” New Journal of Physics 15, 113015 (2013). G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photon polarization,” Phys. Rev. Lett. 94, 220405 (2005). Masataka Iinuma, Yutaro Suzuki, Gen Taguchi, Yutaka Kadoya, and Holger F Hofmann, “Weak measurement of photon polarization by back-action-induced path interference,” New Journal of Physics 13, 033041 (2011). Jeff S Lundeen, Brandon Sutherland, Aabid Patel, Corey Stewart, and Charles Bamber, “Direct measurement of the quantum wavefunction,” arXiv preprint arXiv:1112.3575 (2011). J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot, G. Johansson, and L. DiCarlo, “Partial-measurement backaction and nonclassical weak values in a superconducting circuit,” Phys. Rev. Lett. 111, 090506 (2013). Dawei Lu, Aharon Brodutch, Jun Li, Hang Li, and Raymond Laflamme, “Experimental realization of postselected weak measurements on an nmr quantum processor,” New Journal of Physics 16, 053015 (2014). Luis José Salazar-Serrano, Davide Janner, Nicolas Brunner, Valerio Pruneri, and Juan P. Torres, “Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification,” Phys. Rev. A 89, 012126 (2014). Stephan Sponar, Tobias Denkmayr, Hermann Geppert, Hartmut Lemmel, Alexandre Matzkin, Jeff Tollaksen, and Yuji Hasegawa, “Weak values obtained in matterwave interferometry,” Phys. Rev. A 92, 062121 (2015). Sacha Kocsis, Boris Braverman, Sylvain Ravets, Martin J. Stevens, Richard P. Mirin, L. Krister Shalm, and Aephraim M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011). James Q. Quach, “Which-way doubleslit experiments and born-rule violation,” Phys. Rev. A 95, 042129 (2017). J. Dressel and A. N. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).