Experimental Comparison of Bohm-like Theories with Different Ontologies
Arthur O. T. Pang, Hugo Ferretti, Noah Lupu-Gladstein, Weng-Kian Tham, Aharon Brodutch, Kent Bonsma-Fisher, J. E. Sipe, Aephraim M. Steinberg
EExperimental Comparison of Bohm-like Theories with Different Ontologies
Arthur O. T. Pang, ∗ Hugo Ferretti, Noah Lupu-Gladstein, Weng-Kian Tham, AharonBrodutch, † Kent Bonsma-Fisher,
1, 2, ‡ J. E. Sipe, § and Aephraim M. Steinberg
1, 3, ¶ Department of Physics and Centre for Quantum Information Quantum ControlUniversity of Toronto ,
60 St George St , Toronto , Ontario , M5S 1A7 , Canada National Research Council of Canada ,
100 Sussex Dr , Ottawa , Ontario , K1A 0R6 , Canada Canadian Institute for Advanced Research , Toronto , Ontario , M5G 1M1 , Canada (Dated: October 30, 2019)The de Broglie-Bohm theory is a hidden variable interpretation of quantum mechanics whichinvolves particles moving through space with definite trajectories. This theory singles out positionas the primary ontological variable. Mathematically, it is possible to construct a similar theorywhere particles are moving through momentum space, and momentum is singled out as the primaryontological variable. In this paper we experimentally show how the two theories lead to differentontological descriptions. We construct the putative particle trajectories for a two-slit experiment inboth the position and momentum space theories by simulating particle dynamics with coherent light.Using a method for constructing trajectories through the primary and derived (i.e. non-primary)spaces, we compare the ontological pictures offered by the two theories and show that they do notagree. This contradictory behaviour brings into question which ontology for Bohmian mechanics isto be preferred.
I. INTRODUCTION
Bohm’s hidden variable interpretation of quantum me-chanics [1, 2], also known as Bohmian mechanics or deBroglie-Bohm theory [3, 4], is an alternative formulationof quantum mechanics with a clear deterministic ontol-ogy, and experimental predictions that match those ofquantum theory. The theory continues to attract atten-tion [5–10], perhaps due to the fact that it allows physi-cists to visualize the dynamics of quantum systems. Asis the case in classical physics, Bohmian particles havewell defined properties at all times. In Bohmian the-ory all properties can be determined from the particle’sactual position and the guiding wave, giving positionspecial ontological significance. Wiseman [11] showedthat it is possible to experimentally extract the veloc-ities attributed to Bohmian particles by taking condi-tional averages of weak measurements on an ensembleof post-selected systems. Extending his ideas, some ofthe present authors and others were recently able to con-struct the putative Bohmian trajectories in various two-slit experiments [6, 7, 9, 10]. These results have beenconsidered by some as evidence for the validity of Bohm’sinterpretation and the preferred status of position [12].The choice of position as the preferred ontological vari-able introduces an asymmetry which is foreign to bothclassical and quantum mechanics. In classical Hamil-tonian mechanics, position and momentum act as thecanonical phase space variables and are both equally im-portant in formulating the theory, and in orthodox quan- ∗ [email protected] † [email protected] ‡ kent.bonsma-fi[email protected] § [email protected] ¶ [email protected] tum mechanics position and momentum are treated onequal footing. So the importance placed on position inthe Bohmian approach was one of the main criticismsof Bohm’s work by the pioneers of quantum theory [13].Shortly after Bohm’s paper appeared, Epstein [14, 15]pointed out that there is nothing inherent in the formu-lation that requires position to be the preferred variable,and that other possible choices can lead to other results,i.e. different ontological descriptions, while still yieldingexperimental predictions identical to those of quantumtheory.In this paper, we demonstrate how different choicesof the primary variable can lead to qualitatively differ-ent trajectories. Using light to simulate the mechanicsof massive particles [16], in a double-slit setup similar tothose used earlier [6, 7] we construct the trajectories inboth Bohm’s theory (which we refer to as x -Bohm) andan alternative theory in which momentum is the preferredvariable ( p -Bohm). The differences between the trajecto-ries in the two theories illustrate why the results of pre-vious experiments [6, 7] should be understood as specificinstances of the many possible ontological descriptions ofthe same system. This multitude of possible theories, andcorresponding ontological pictures, makes it difficult todecide which theory, if any, should be ascribed to reality,emphasizing one of the weaknesses in Bohm’s approach.We begin in Sec. II by describing some of the basicfeatures of the x -Bohm and p -Bohm theories, and themethod for constructing trajectories through a sequenceof weak and strong measurements. In Sec. III we lay outthe details of our experiment, including the specifics ofthe lens system and the weak measurement procedure.The results of the experiments, including plots of thetrajectories and phase space snapshots at the near andfar field are presented in Sec. IV for both the x -Bohmand p -Bohm theories. The implications of our results arediscussed in Sec. V. a r X i v : . [ qu a n t - ph ] O c t II. BOHMIAN THEORY
Contrary to classical mechanics, which allows for thedeterministic prediction of the motion of particles, quan-tum mechanics only offers statistical predictions of theresults of measurements. Yet in 1952 David Bohm in-troduced [1, 2] a deterministic dynamical theory thatits advocates argue provides an underlying descriptionmore fundamental than quantum mechanics [17, 18]. Inhis generalization and extension of earlier ideas by deBroglie [18], the positions of particles play the role ofhidden variables; their motion is characterized by well-defined trajectories, as the particles are “guided” by theSchr¨odinger wave. In this approach position variables,together with the Schr¨odinger wave, have a special sig-nificance as the primary ontological variables; the mo-menta of particles simply follow from their velocities,determined by the gradient of the Schr¨odinger wave atthe positions of the particles. The symmetry of posi-tion and momentum that characterizes orthodox quan-tum mechanics is broken, with position variables morefundamental than momentum variables.Shortly after Bohm’s work appeared, Epstein [14, 15]noted that different choices of the primary ontologicalvariable can lead to different theories. In particular,one could work with the momentum representation ofthe wave function and build a theory where particlesare characterized fundamentally by their momenta . Incontrast to Bohm’s original theory, which we refer to as“ x -Bohm,” in Epstein’s proposal, which we refer to as a“ p -Bohm” theory, it is momentum that has primary onto-logical status. In his reply to Epstein [19], Bohm pointedout technical difficulties in implementing a “ p -Bohm” ap-proach when the Hamiltonian involved the Coulomb po-tential. But he also argued that an “ x -Bohm” approach,where particle position and the coordinate representationof the wave function are the primary ontological vari-ables, seemed more favored because “in all fields otherthan the quantum theory, space and time have thus farstood out as the natural frame for the description of theprogress of physical phenomena.” Nevertheless, alternateapproaches were developed further a few decades later byBohm and his collaborators [20], and a general frameworkfor such theories was discussed by Holland [4, 21, 22] andothers [23, 24].In the rest of this section we sketch both x -Bohm and p -Bohm theory, discuss the trajectories that follow fromeach, and show how – under the assumption that one ofthe theories is correct – its associated trajectories can berevealed by weak measurements. We begin with trajec-tories of the primary ontological variable of the particles– position for x -Bohm and momentum for p -Bohm – andthen turn to the trajectories that can be associated with The possibility of a velocity-based theory had already been raisedby Pauli at the 1927 Solvay conference in response to de Broglie’spilot wave theory [1, 18]. non-primary variables. This allows us to compare the twotheories by contrasting their predictions for trajectoriesin the same space. We focus on the one-dimensional mo-tion of a single particle, where the classical Hamiltonianas a function of position and momentum is H ( x, p ), anddenote the coordinate wave function by ψ ( x, t ) and themomentum wave function by ˜ ψ ( p, t ). The Schr¨odingerequations for these two wave functions are i ¯ h ∂∂t ψ ( x, t ) = H (cid:18) x, − i ¯ h ∂∂x (cid:19) ψ ( x, t ) (1) i ¯ h ∂∂t ˜ ψ ( p, t ) = H (cid:18) i ¯ h ∂∂p , p (cid:19) ˜ ψ ( p, t ) . (2) A. Position Ontological Bohmian Theory ( x -Bohm) In Bohm’s original theory [1, 2], the particle’s posi-tion and the wave function ψ ( x, t ) constitute the objec-tively real elements from which all other properties canbe derived . In describing an ensemble of experimen-tal runs, at some initial time ( t = 0) each particle isassumed to have a definite position according to a prob-ability distribution function | ψ ( x, | , and each particleis guided through space by the wave function. Writing ψ ( x, t ) = R x ( x, t ) exp [ iS x ( x, t ) / ¯ h ], where R x ( x, t ) and S x ( x, t ) are real functions of position and time, for aHamiltonian of the form H ( x, p ) = p / m + V ( x ) theguidance equation is v x ( x, t ) = 12 m ∂S x ( x, t ) ∂x , (3)and the trajectory for each particle is given by dx ( t ) dt = v x ( x ( t ) , t ) . (4)Since the expression (3) for the velocity v x ( x, t ) can alsobe written as [4] v x ( x, t ) = j x ( x, t ) | ψ ( x, t ) | , (5)where j x ( x, t ) is the usual probability current density oforthodox quantum mechanics, j x ( x, t ) = ¯ h mi (cid:18) ψ ∗ ( x, t ) ∂ψ ( x, t ) ∂x − ψ ( x, t ) ∂ψ ∗ ( x, t ) ∂x (cid:19) , (6) For an N -particle system the wave function is a function overthe 3 N -dimensional configuration space of the system, and sincethe wave function is granted ontological significance that config-uration space must be taken as the underlying arena of reality;the wave function and a point in this configuration space, iden-tifying the positions of all N particles, are best taken to identifythe ontology of the theory. it follows that as the particles in the ensemble move,and as ψ ( x, t ) evolves according to Schr¨odinger’s equation(1), the evolution of the distribution function character-izing the positions of the particles follows the evolutionof | ψ ( x, t ) | .Although the Bohmian trajectories had been studiedtheoretically and discussed in the literature since 1952(see, e.g., Philippidis et al. [5]), it seems it was not untilWiseman’s work in 2007 [11] that a strategy for identi-fying them experimentally was investigated. Wisemanpointed out that the expression (3) for the velocity of aparticle at x , which can be written as [4] v x ( x, t ) = 1 m Re (cid:20) (cid:104) x | ˆ p | ψ ( t ) (cid:105)(cid:104) x | ψ ( t ) (cid:105) (cid:21) , (7)where ˆ p is the momentum operator ( (cid:104) x | ˆ p | x (cid:48) (cid:105) = − i ¯ h∂δ ( x − x (cid:48) ) /∂x ), can be connected with the theory of weak mea-surements introduced by Aharonov, Albert, and Vaid-man (AAV) [25]. Weak measurements are those withsmall back action and consequently high uncertainty, andWiseman noted that the expression (7) corresponds tothe operational prescription of a weak momentum mea-surement followed immediately by a strong (projective)position measurement. The apparent simultaneous mea-surement of two conjugate variables respects the un-certainty relations since the momentum measurement isweak, and consequently the measurement scenario mustbe repeated many times with the averaging done sepa-rately for every final value of position. This fits neatlyinto the Bohmian perspective in general: Since all vari-ables in the theory are uniquely determined by the pri-mary ontological variable, it could be argued that ensem-ble averaging can be justified as long as post-selectiononto the primary ontological value for each experimentalrun is sufficiently accurate and the measurement backaction for the weak measurement is sufficiently small.Of course, the identification of the right-hand-side of(7) with a weak momentum measurement followed by astrong position measurement can be made operationally,independent of any proposed explanation of quantum me-chanics in terms of a deeper theory. Nonetheless, thetrajectories that are predicted by x -Bohm theory can beformally constructed from the results of weak measure-ments; this has been done by Kocsis et al. [6] for a singleparticle in a double-slit interferometer, and by Mahler etal. [7] for entangled light. Advocates of x -Bohm theorythen identify these constructed trajectories with trajec-tories that are held to really exist. B. Momentum Ontological Bohmian Theory( p -Bohm) In p -Bohm theory one adopts momentum as the pri-mary ontological variable, and the fundamental dynamicstake place in momentum space. Here, one relies on themomentum representation of the wave function ˜ ψ ( p, t ),and for Hamiltonians of the form H ( x, p ) = p / m + V ( x ) there is no general expression for the time derivative v p ( p, t ) of the momentum of a Bohmian particle, dp ( t ) dt = v p ( p ( t ) , t ) , (8)which would be analogous to the corresponding expres-sion (3) for the time derivative v x ( x, t ) of the positionof a Bohmian particle in x -Bohm theory. This can betraced to the fact that all such Hamiltonians exhibit thesame dependence on p but, depending on the potential,can have very different dependences on x ; thus, whileSchr¨odinger’s equation in coordinate space (1) involvesonly second derivatives with respect to x , Schr¨odinger’sequation in momentum space (2) can involve any num-ber of derivatives with respect to p . Nonetheless, onecan look for an expression for v p ( p, t ) analogous to theexpression (5) for v x ( x, t ), writing v p ( p, t ) = j p ( p, t ) (cid:12)(cid:12)(cid:12) ˜ ψ ( p, t ) (cid:12)(cid:12)(cid:12) , (9)where j p ( p, t ) is a current density in momentum space.In a one-dimensional problem it must satisfy ∂j p ( p, t ) ∂p = − ∂∂t (cid:18)(cid:12)(cid:12)(cid:12) ˜ ψ ( p, t ) (cid:12)(cid:12)(cid:12) (cid:19) , (10)and since the right-hand-side is determined by theSchr¨odinger equation in momentum space (2), a unique j p ( p, t ) can be identified, j p ( p, t ) = 2¯ h (cid:90) p −∞ Im (cid:18) ˜ ψ ∗ ( p (cid:48) , t ) (cid:18) V ( i ¯ h ∂∂p (cid:48) ) ˜ ψ ( p (cid:48) , t ) (cid:19)(cid:19) dp (cid:48) , (11)under the physically reasonable assumption that j p ( p, t ) → | p | → ∞ [23]. The situation is morecomplicated in higher dimensions; in three dimensions,for example, the continuity equation for a momentumcurrent density j p ( p , t ) only restricts the divergence of j p ( p , t ) and not its curl, and it is not immediately clearhow it should be assigned. The range of possible choicesfor current densities in general de Broglie-Bohm theo-ries, and the criteria one might want to apply in makinga choice, have been investigated by Struyve and Valen-tini [23]. Our focus in this paper will be on free particles( V ( x ) = 0), where the distribution (cid:12)(cid:12)(cid:12) ˜ ψ ( p, t ) (cid:12)(cid:12)(cid:12) of Bohmianparticles in momentum space is time independent, andfrom (9,11), and in agreement with physical intuition,we have v p ( p, t ) = 0. C. Trajectories in a non-primary space
Consider then a p -Bohm theory for free particles. In anensemble of experimental runs there would be a distribu-tion of particles characterized by (cid:12)(cid:12)(cid:12) ˜ ψ ( p, t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˜ ψ ( p, (cid:12)(cid:12)(cid:12) in momentum space, and each Bohmian particle wouldmaintain its momentum. Is the question of what eachparticle is doing in real space even meaningful? After all,momentum is here the primary ontological variable, andso the arena of reality is momentum space. Each particlehas a momentum and it is constant; there seems nothingelse that can be said. The theory does not concern itselfwith trajectories in spacetime, despite the fact that, atleast as argued by Bohm, that is “the natural frame forthe description of the progress of physical phenomena.”For a Bohm-like theory with a given primary onto-logical variable, Holland [4, 21] suggested a strategy foridentifying the values of variables other than the primaryontological variable. For one-dimensional systems and inour notation, if we consider a “ ξ -Bohm theory,” wherehere ξ is an eigenvalue of a Hermitian operator ˆ ξ thatwe take to have continuous eigenvalues, the value ω of acontinuous variable associated with a Hermitian operatorˆ ω is taken to be ω ξ ( ξ, t ) = Re (cid:20) (cid:104) ξ | ˆ ω | ψ ( t ) (cid:105)(cid:104) ξ | ψ ( t ) (cid:105) (cid:21) (12)at time t , if the ket is | ψ ( t ) (cid:105) and the primary ontologicalvariable has value ξ . Holland did not take this suggestionto be at the level of a new postulate, and even consideredother approaches for some physical systems. Nonetheless,the proposal has the physically comforting feature thatthe average of the values granted to a variable ω over anensemble of Bohmian particles described by a ξ -Bohmtheory does agree with the expectation value of the op-erator associated with that variable in the ket describingthe ensemble, (cid:104) ψ ( t ) | ˆ ω | ψ ( t ) (cid:105) = (cid:90) ω ξ ( ξ, t ) |(cid:104) ξ | ψ ( t ) (cid:105)| dξ. (13)As an example, consider momentum in an x -Bohm the-ory. For a particle at position x at time t , from (12) wesee that the value of momentum that would be assignedis p x ( x, t ) = Re (cid:20) (cid:104) x | ˆ p | ψ ( t ) (cid:105)(cid:104) x | ψ ( t ) (cid:105) (cid:21) . (14)Comparing with the x -Bohm expression (7) for v x ( x, t ),we find p x ( x, t ) = mv x ( x, t ) , (15)as would be physically expected. Yet we can now also as-sign evolving position variables to particles in a p -Bohmtheory, for the prescription (12) gives x p ( p, t ) = Re (cid:20) (cid:104) p | ˆ x | ψ ( t ) (cid:105)(cid:104) p | ψ ( t ) (cid:105) (cid:21) , (16)and following x p ( p, t ) as t advances allows us to assign atrajectory in real space to a Bohmian particle in p -Bohmtheory with momentum p . Furthermore – and somewhat remarkably! – Holland’sprescription (12) is precisely that which operationallycharacterizes a weak ˆ ω measurement followed by a strongˆ ξ measurement. Thus, just as velocities of particles in an x -Bohm theory can be constructed by weak momentummeasurements followed by strong position measurements,so the positions of particles in a p -Bohm theory can beconstructed by weak position measurements followed bystrong momentum measurements. And so we have aroute to identifying trajectories of Bohmian particles in“non-primary” spaces, by which we mean spaces associ-ated with variables other than the primary ontologicalvariable. This is done by first constructing the trajecto-ries of the ontological variable, leading to an equation for ξ ( t ) and then using Eq. (12) to construct the trajectorygiven by ω ( ξ ( t ) , t ). We can experimentally construct tra-jectories in momentum space for particles in an x -Bohmtheory, as indeed has already implicitly been done [6, 7]relying on (15), and we can also experimentally constructtrajectories in position space for particles in a p -Bohmtheory. We turn to this in the following sections. III. EXPERIMENTAL SCHEME
In our experiment, we simulate the evolution of a mas-sive particle under x -Bohm and p -Bohm theories withlight from a laser diode, using the fact that light propa-gating in the paraxial regime can be modelled with theSchr¨odinger equation. The propagation of monochro-matic light can be modelled with the Helmholtz equation[26] ∇ A + | k | A = 0 , (17)where A is the vector potential and k = ( k x , k y , k z ) isthe wave vector. In the paraxial regime where | k |≈ k z ,this equation can be reduced to i ∂∂z u = − ∂ ∂x u | k | , (18)where A = u · exp ( ik z z ), u is the envelope function of thepropagating light, z is the longitudinal position, and the y coordinate is factored out through a separation of vari-ables. Equation (18) has the form of the one-dimensionalSchr¨odinger equation (1) for a free particle. Defining aneffective mass through | k | = mc/ ¯ h , the correspondence ofvariables between optical and massive particle regimes issummarized in Table I. Note that we use the transverseangle θ = k x / | k | , equivalent to a normalized momentum,when plotting results.Drawing the analogy between Eq. (18) and theSchr¨odinger equation we simulate the trajectories of aparticle double-slit experiment by sending 915 nm laserlight through a double-slit apparatus, employing the ex-perimental setup outlined in Figures 1 and 2. In the restof this section we describe the details of this setup, be-ginning with the gadget used for the weak momentum Paraxial Light Particle in 1D[plotted units] normalized unitsTransverse position: x [mm] Position: x Longitudinal position: z [m] Time: tc Transverse angle: θ = k x | k | [rad] Momentum: pmc TABLE I. Variable correspondence between paraxial light anda particle in 1D. Note that in the derivations we use k x and | k | while the normalized momentum θ is used in the plots. X P
Slit setup System of lenses Imaging setup
X P
Beam couplerTranslation stage QWP/HWP/PolarizerLensesBeam splitterMirror Calcite positionx-Bohm/p-Bohm Beam displacerCamera
FIG. 1. Illustration of experimental setup in three parts. Theslit setup is used to initialize the double-slit experiment. Adiagonally-polarized beam is split into two co-propagatingbeams using a displaced Sagnac interferometer, where thebeam separation (set to 2 mm throughout the experiment)can be tuned by translating one of the mirrors. The systemof lenses is used to simulate propagation of the light along z over a large range (see Fig. 7 and Fig. 8 for further details).The two plots above the lens system indicate the intensityprofiles of the beam before and after the lens transformation.A thin piece of calcite is used to weakly couple momentumto polarization (the weak measurement). The calcite is po-sitioned before the lenses for the weak momentum measure-ment in the x -Bohm experiment and between the lenses forweak position measurement in the p -Bohm experiment. Theimaging setup, consisting of a polarizing beam displacer anda CCD camera, is used to obtain two interference patterns,one for each polarization. measurement and the procedure employed in the exper-imental construction of position trajectories in x -Bohm,which closely follow those outlined earlier [6, 7]. We thendescribe how the same procedure is used to constructmomentum trajectories in x -Bohm theory, and how thesetup is modified for constructing position trajectories in p -Bohm theory. A. Weak momentum measurements
A weak measurement is performed by coupling the de-sired observable to a pointer variable, often a different de-gree of freedom of the same physical system, followed by a strong pointer variable measurement [27, 28]. Here weuse polarization as the pointer. Our observable of interestis momentum, which maps to k x for the light beam. Weuse ˆ k x to denote the operator form of k x . Specifically, inthe position representation (cid:104) x | ˆ k x | x (cid:48) (cid:105) = − i∂δ ( x − x (cid:48) ) /∂x .As described below, the shift in polarization will be pro-portional to the weak value which can then be extractedthrough a standard polarization measurement. We usethe notation | H (cid:105) , | V (cid:105) for horizontal and vertical polar-ization respectively, | D (cid:105) = ( | H (cid:105) + | V (cid:105) ) / √ | A (cid:105) = ( | H (cid:105) −| V (cid:105) ) / √ | R (cid:105) = ( | H (cid:105) + i | V (cid:105) ) / √ | L (cid:105) = ( | H (cid:105) − i | V (cid:105) ) / √ | D (cid:105) . Polar-ization is coupled to the transverse momentum of thelight using a thin calcite crystal. The interaction can bedescribed by the Hamiltonian H I = ¯ hg ˆ k x | k | ˆ σ z (19)whereˆ σ z = 12 ( | H (cid:105) (cid:104) H | − | V (cid:105) (cid:104) V | ) = 12 ( | D (cid:105) (cid:104) A | + | A (cid:105) (cid:104) D | ) , (20)and g is the coupling strength. If the joint state of thetransverse position and polarization before the calcite is | Ψ (cid:105) = | ψ (cid:105)⊗| D (cid:105) , then with a sufficiently weak interaction,i.e., sufficiently small ζk x = gtk x (cid:28) k x , the joint state after the calcite is | Ψ (cid:48) (cid:105) ≈ | ψ (cid:105) ⊗ | D (cid:105) − i ζ k x | k | | ψ (cid:105) ⊗ | A (cid:105) . (21)The interaction is followed by a projective x measure-ment, post-selected on the result, x f . The state of thepointer following this post-selection is (cid:104) x f | Ψ (cid:48) (cid:105) ≈ ψ ( x f ) (cid:16) e − i ζ | k | (cid:104) ˆ k x (cid:105) w | H (cid:105) + e i ζ | k | (cid:104) ˆ k x (cid:105) w | V (cid:105) (cid:17) , (22)where (cid:68) ˆ k x (cid:69) w = (cid:104) x f | ˆ k x | ψ (cid:105)(cid:104) x f | ψ (cid:105) (23)is the weak value, in general a complex number. The realpart of the weak value shows up as a phase shift between H and V polarizations which can be extracted by theprojective measurement ˆ σ y = ( | R (cid:105) (cid:104) R | − | L (cid:105) (cid:104) L | ). Thiscorresponds to making measurements of the right- andleft-circular intensities, resulting in (cid:104) ˆ σ y (cid:105) = sin (cid:18) ζ | k | Re (cid:16)(cid:68) ˆ k x (cid:69) w (cid:17)(cid:19) . (24) Calcite Lens 2Focal plane TranslatableElement QWP Lens 3 ImagingSetupLens 2Lens 1 f f +f +d ∞
0m 1m x -Bohm Lens 3Focal plane +f f f d ∞
0m 1m p -Bohm FIG. 2. Illustration of the system of lenses configured tomake measurements for x -Bohm (top) and p -Bohm (bottom)theories. The focal lengths of the lenses are f = 15 cm and f = f = 10 cm from left to right, and the total length, fromLens 1 to the imaging setup, is 55 cm. Lens 1 focuses thebeam and remaps the position variable of planes from 0 m toinfinity onto the position variable of planes from 0 cm to 15 cmafter the lens, with a scaling factor. The grey axis indicatesthe correspondence between the location of the focus of Lens2 and the effective propagation distance being imaged by thelens setup, and a detailed plot of the propagation distance vsthe displacement d of Lens 2 is plotted in Figure 7. Top: Lens2 and Lens 3 map the position variable at the dotted line tothe imaging setup (solid line); Bottom: Lens 2 and Lens 3 areset to be 20 cm away from each other, forming a one-to-onetelescope. B. Position trajectories in x -Bohm theory A double-slit pattern is generated by separating aGaussian beam (1 /e diameter of 0.55 mm) into two, us-ing a horizontally-displaced Sagnac interferometer (slitsetup in Figure 1) that gives an effective slit separationof 2 mm. The light is then diagonally polarized and sentthrough a thin calcite crystal (0.2 mm, cut at 45 degrees)to weakly couple the transverse momentum of the light topolarization via a birefringent phase shift (see Sec. III Aabove). Importantly, the interaction Hamiltonian (19)commutes with the Hamiltonian for free propagation.This implies that the calcite crystal can remain fixed ata single z position before the lens system independent ofthe plane of interest.Next, the co-propagating beams traverse a system ofthree lenses (Fig. 1, middle pane), labelled Lens 1, 2,3 with respective focal lengths 15 cm, 10 cm, 10 cm (seeFigure 2). By translating Lens 2 along the z -axis, wesimulate different propagation distances for the light, re-sulting in effective distances ranging from 0 .
66 m to 3 . . The calibration of the lens sys-tem is discussed in Appendix B.Finally, the co-propagating beams enter the imagingsetup (Fig. 1, right pane), where the resulting intensitypatterns at the end of the lens system were measured ona CCD camera. In addition to the intensity of the inter-ference pattern, the polarization is measured by a quar-ter wave plate and a polarization beam displacer thateffectively separates the left- and right-circularly polar-ized light in the vertical direction. Since the interferenceoccurs along the horizontal transverse axis , the interfer-ence pattern of the left- and right-circular polarizationscan be measured independently. The intensity patternsof the two polarizations ( | u | in Eq. (18)), given by I R and I L , differ by an amount directly related to the realpart of the weak value of transverse momentum, whichin the limit of an infinitely weak measurement can beextracted asRe (cid:16)(cid:68) ˆ k x (cid:69) w (cid:17) | k | = 1 ζ (cid:20) sin − (cid:18) I R − I L I R + I L (cid:19) − φ (cid:21) , (25)where the sin − term comes from Eq. (24) and φ is amomentum-independent phase shift acquired in the cal-cite crystal, set by tilting the calcite; ζ = 134 . ± .
13 isthe coupling strength which depends on the length of thecalcite. The calibration of ζ is discussed in Appendix B.Examples of measured intensity patterns for near- andfar-field propagation distances are shown in the top rowof Figure 3, while measured values of the momentum, inthe same two planes, are shown in the third row of Fig-ure 3. By performing this measurement for each z -plane,we extracted ensemble-average values of the transversemomentum as a function of position, from which we con-struct particle trajectories. Experimentally constructed x -Bohm position trajectories are shown in the top row ofFigure 4, with theoretically calculated trajectories shownin the bottom row. We will discuss all experimental re-sults in greater detail in Section IV. C. Momentum trajectories in x -Bohm theory To construct the momentum trajectories in x -Bohm wefollow the procedure outlined in Sec. II C, where again weuse Eq. (14). In our case the momentum is proportionalto the velocity (see Eq. (15)), which implies that the mea-surement for the x trajectories in Section III B suffices for With our experimental parameters, λ = 915 nm, slit separation s = 2 mm, and slit width w = 0 .
55 mm, we expect the near-to-farfield transition to occur at s / ( λ/ ( πw/ .
77 m. This is the axis that is horizontal and perpendicular to the axisof propagation. The quantity ζ corresponds to rotation imparted to the polar-ization of light per transverse angle of the light, and is hencedimensionless. constructing the momentum trajectories. The resultingtrajectories are shown in the first row of Figure 5.Note that proportionality between velocity and mo-mentum is only valid when the potential term in theHamiltonian is independent of p . In cases where the po-tential has p and/or p terms, Eq. (7) is no longer valid,while Eq. (14) remains valid generically. D. Momentum trajectories in p -Bohm theory As described in Sec. II B, the conservation of momen-tum for a free particle implies that the p -Bohm momen-tum trajectories follow lines of constant p . There is noneed to construct these trajectories experimentally; how-ever, the relative probabilities (or density of trajectories)can be measured by making a strong p measurement. Inpractice this is accomplished by strong x measurementsin the far field, using the fact that momentum maps toposition at infinity (see Figure 8 in Appendix B). Thesame setup is used to calibrate the coupling strength ofthe weak measurement (again, see Appendix B). The re-sulting trajectories are shown in the second row of Fig-ure 5.We emphasize that p -Bohm theory in three dimensionsis not unique and that different theories lead to differentvelocities, v p ( p , t ) [23]. However, in one dimension thecontinuity constraint (10) essentially identifies (11) as thecurrent density in momentum space, which in the limit ofa free particle leads via Eq. (9) to the conservation of the p -Bohm momentum. The results in Fig. 5 are thereforefree of any ambiguity that would affect p -Bohm theoriesin higher dimensions. E. Position trajectories in p -Bohm theory Position is a derived variable in p -Bohm theory, andso position trajectories can be constructed following theprocedure in Sec. II C using Eq. (16). Operationally,this amounts to making a weak position measurementfollowed by a post-selection on momentum. This isachieved by using lens transformations to map betweenposition and momentum since a lens performs a position-momentum Fourier transform at its focus. Practically,Lens 2 and Lens 3 are kept at a fixed distance from oneanother, making a one-to-one telescope, and are trans-lated together (Figure 2). In this way, the transversemomentum between Lens 2 and Lens 3 corresponds totransverse position in a fixed propagation plane. As such,the calcite crystal is placed in between Lens 2 and Lens 3to perform a weak position measurement. Additionally,the one-to-one telescope relates the light one focal lengthbefore Lens 2 to the light one focal length after Lens 3by an identity transformation. This effectively places thefar field of the interfering beams onto the imaging setup,causing it to perform a strong momentum measurement.To read out the weak measurement, the quarter wave plate and polarization beam displacer, once again, areused to separate the left- and right-circularly polarizedcomponent of the beam in the vertical direction and, withprocedures similar to those in Section III B, we can ex-tract the weak position value post-selected on momen-tum. The fourth row of Figure 3 shows results of thecorresponding weak measurements in near- and far-fieldplanes. Position trajectories, with momentum as the on-tological variable, are constructed in the same manner asbefore. Constructed trajectories are shown in the secondrow of Figure 4. IV. COMPARISON OF x -BOHM AND p -BOHMTRAJECTORIES We now consider the x -Bohm and p -Bohm particle tra-jectories in detail. The trajectories are constructed byinterpolating data points taken at discrete z -planes rang-ing from an effective distance of 0 .
66 m to 3 . A. Single-time position-momentum snapshots
For a given z -plane, which corresponds to an instantin time, data were taken by fixing the lenses and post-selecting on the preferred ontological variable producinga complete description of the functions p ( x, t ) and x ( p, t )for the x -Bohm and p -Bohm theories respectively. Re-sults for two of these instants of time, one in the near-field and one at far-field, are presented in Figure 3 andcompared with theoretical predictions. To illustrate thedifference between the two theories we begin with a nu-merically simulated plot (Figure 3, second row), where weoverlay two ontological momentum-position snapshots,based on Eq. (12). Experimental results are shown inthe third and fourth rows of Figure 3.In x -Bohm, peaks in the momentum p ( x, t ) appearwhen a particle approaches a minima in the double-slitinterference pattern (i.e. the minima in the top row ofFigure 3). These peaks get progressively narrower, withwidth approaching zero, as the measurement is taken fur-ther into the far field. The asymptotic large x behaviorof the function p ( x ) corresponds to p/m = x − sgn( x ) w/ t , (26)with t being propagation time and w = 2 mm being theslit separation. This can be roughly interpreted as theconsequence of the guiding wave ψ ( x, t ) at the near fieldhaving two distinguishable parts with a small overlap so -3 -2 -1 0 1 2 3 Position (mm) -2-1012 M o m en t u m ( m r ad ) Weak values theoretical, near field -5 0 5
Position (mm) -2-1012 M o m en t u m ( m r ad ) Weak values theoretical, far field -3 -2 -1 0 1 2 3
Position (mm) I n t en s i t y ( a r b . un i t s ) Intensity distribution experimental, near field -5 0 5
Position (mm) I n t en s i t y ( a r b . un i t s ) Intensity distribution experimental, far field -3 -2 -1 0 1 2 3
Position (mm) -2-1012 M o m en t u m ( m r ad ) x -Bohm experimental result, near field -3 -2 -1 0 1 2 3 Position (mm) -2-1012 M o m en t u m ( m r ad ) p -Bohm experimental result, near field -5 0 5 Position (mm) -2-1012 M o m en t u m ( m r ad ) x -Bohm experimental result, far field -5 0 5 Position (mm) -2-1012 M o m en t u m ( m r ad ) p -Bohm experimental result, far field FIG. 3. Near field and far field snapshots. Near field (left column) data were taken at an effective 0 .
70 m from the slits,and far field (right column) taken at 3 . x -Bohm (red) and p -Bohm (blue) theories in the second row showqualitative differences between the predictions of the two theories, in particular in the near field. The results of the two theoriesare expected to converge at infinity. Third and fourth rows show the measured position-momentum profiles for x -Bohm and p -Bohm theories respectively. A slight difference in the slope of the p -Bohm experimental near field weak value comparing tothat of the theoretical weak value can be found. This is due to systematic uncertainties not reflected in the error bars whilecalibrating the Gaussian beam width and effective propagation distance. A qualitative difference between the predictions of thetwo theories can be seen, notably the presence of strong peaks in the near-field data for x -Bohm and not for p -Bohm. Error barscorrespond to the standard deviation between weak values given by different calcite tilt angles. Error bars are larger in areaswhere the overall intensity of the interference is small (see Section IV A). Note that in the p -Bohm experiment post-selectionis always at infinity so that the minima do not correspond to those in the intensity profiles in the first row. that particles away from the overlap are effectively guidedby one or the other, leading to behaviour similar to whatone would observe if only one slit were open.In p -Bohm theory, we expect a linear relation x ( p ) = p · t/m . It is important to note that, experimentally, datawith momentum post-selection always projects the farfield onto the imaging setup. Similarly, the guiding wave˜ ψ ( p, t ) has the form of the far-field interference pattern.Due to the nature of our measurement, the weak valueof the variable of interest is very sensitive to backgroundnoise when the post-selection probability is small. Back-ground light and other systematic errors dominate themeasured signal, and as a consequence the probability ofregistering a measurement in the left- and right-circularpolarization basis becomes roughly equal. As a conse-quence, and by referring to Eq. (25), one can see thatthe weak value tends to the incorrect result of − ζ − φ near the minima of the interference patterns. As men-tioned in Section III B, the value of φ in our experimentwas controlled by the horizontal tilt angle of the calcitecrystal. In an idealized noiseless measurement, the valueof φ exists purely as a calibration parameter of the weakmeasurement (see Appendix B) and does not affect themeasured weak value. However, with some amount ofnoise present in the measurements this is not the case.To account for this imperfection, we measure weak valuesusing various calcite tilts. The final weak value at eachtime is tabulated by averaging measurement results withdifferent calcite tilts. Error bars in the third and fourthrow of Figure 3 correspond to the standard deviation ofthe measurement given a set of values for φ . The effectsof the calcite tilt are particularly pronounced in p -Bohmexperiments, where the measurements at the minima govery close to zero and the standard deviation betweenmeasurement results increases significantly. B. Constructing Trajectories
We construct a set of trajectories for both position andmomentum in both x -Bohm and p -Bohm theory, chosenso that the density of the selected trajectories in the pri-mary ontological space corresponds to particle distribu-tion probabilities. This is possible due to the fact thatthe velocities are defined through the probability current(see Sec. II). However, there is no a priori reason to ex-pect this feature to be preserved in the derived space,and indeed we will see that it is not. Similarly, the tra-jectories in the primary space cannot cross, since givena wave function, the velocity is uniquely defined by thevalue of the primary ontological variable. As we will see,the p ( x ) trajectories in x -Bohm do cross. The trajecto-ries for x -Bohm and p -Bohm experiments are shown inFigures 4 and 5 respectively along with theoretical tra-jectories derived from a numerical simulation.
1. Position Trajectories
The x -Bohm position trajectories shown in the top rowof Figure 4 are very similar in nature to those obtainedearlier [6]. These trajectories originate from one of thetwo slits and, while providing the signatures of interfer-ence for the probability density, they generally divergeaway from x = 0 while displaying a rapid ‘acceleration’through each region of destructive interference, where thedensity becomes low and the ratio of flux to density cor-respondingly large. As required by the Bohmian formal-ism, the trajectories of the primary ontological variabledo not cross.For p -Bohm, the position trajectories (Figure 4 mid-dle) originate from a single point in between the two slitsand spread out in a manner that preserves momentum,resulting in straight lines given by x = pm/t . A cross-ing (or in this case convergence to a point at t = 0) ispossible since these are not the trajectories of the pri-mary ontological variable. Due to imperfect translationof Lens 2 and Lens 3, causing some transverse displace-ment, a systematic error is introduced to the weak valuemeasurement and the trajectories are displaced by a dif-ferent amount at each plane. This causes the trajectoriesto shift in the y-axis of Figure 4, resulting in the ex-perimental weak values of position deviating from simplestraight lines.Apart from the obvious discrepancy between the posi-tion trajectories in x -Bohm and p -Bohm, the p -Bohm po-sition trajectories exhibit the potentially surprising phe-nomenon of originating at x = 0 rather than in eitheror both of the slits. That is, the initial position forall the particles according to p-Bohm theory is a posi-tion which has vanishingly low probability according tox-Bohm; moreover, a detector placed at that positionwould never be expected to register a photon. Placinga detector at this point and not registering a detectionis, however, consistent with the p -Bohm description ofquantum mechanics.
2. Momentum Trajectories
Next, we construct particle momentum trajectories,tracking the change of momentum over time (see Fig-ure 5). As the conservation of momentum of light infree space is an assumption used in the alignment, the p -Bohm momentum trajectories are constructed fromtheory as flat lines with a distribution derived fromthe strong momentum measurement (position at the farfield).The x -Bohm momentum trajectory functions are pro-portional to the time derivative of the position trajec-tory functions. The peaks observed in x -Bohm momen-tum trajectories correspond to time intervals when theposition trajectories are crossing the minima of the in-terference pattern (as emphasised in Fig. 6). The timeinstances at which these peaks appear are highly sensi-0 Propagation distance (m) -4-2024 T r an sv e r s e d i s p l a c e m en t ( mm ) Experimental p -Bohm position trajectories Propagation distance (m) -4-2024 T r an sv e r s e d i s p l a c e m en t ( mm ) Experimental x -Bohm position trajectories Propagation distance (m) -4-2024 T r an sv e r s e d i s p l a c e m en t ( mm ) Theoretical x -Bohm and p -Bohm position trajectories FIG. 4. Constructed position trajectories based on x -Bohm (red) and p -Bohm (blue) theories. Top (middle) plot corresponds to x -Bohm ( p -Bohm) position trajectories constructed experimentally. Bottom plot corresponds to numerical simulation of both x -Bohm (red solid line) and p -Bohm (blue dotted line) position trajectories overlaid. x -Bohm trajectories originate from thelocation of the two slits, while p -Bohm trajectories originate from mid-point between the two slits. In the far field, trajectoriesfrom x -Bohm and p -Bohm theory converge to the same values as expected. The x -Bohm trajectories are also plotted in Figure6 (top) with one position trajectory highlighted. Propagation distance (m) -2-1012 T r an sv e r s e m o m en t u m ( m r ad ) Experimental p -Bohm momentum trajectories Propagation distance (m) -2-1012 T r an sv e r s e m o m en t u m ( m r ad ) Experimental x -Bohm momentum trajectories Propagation distance (m) -2-1012 T r an sv e r s e m o m en t u m ( m r ad ) Theoretical x -Bohm and p -Bohm momentum trajectories FIG. 5. Constructed momentum trajectories based on x -Bohm (red) and p -Bohm (blue) theories. Top(middle) plot correspondsto x -Bohm( p -Bohm) momentum trajectories constructed experimentally. Bottom plot corresponds to numerical simulation ofboth x -Bohm (red solid line) and p -Bohm (blue solid line) momentum trajectories overlaid. At far fields both sets of trajectoriesbunch in the manner of a far field interference pattern. The momentum trajectories for x -Bohm are equal to the first derivativeof the position trajectories under x -Bohm, whereas the momentum trajectories for p -Bohm are flat lines due to the conservationof momentum. Peaks in the x -Bohm trajectories corresponds to the crossing of the particle over an interference minimum, andtheir time of occurrence is highly sensitive to the initial conditions of the ontological variable, causing inconsistencies betweenthe numerically simulated and experimentally constructed trajectories. The alignment procedure for our system of lenses andthe calibration of the weak measurement strength ζ rely on the conservation of momentum. Similarly, the flat lines given by p -Bohm are derived from the conservation of momentum and a single measurement at the far field which gives the momentumdistribution, as reflected in the distribution of lines. The x -Bohm trajectories are also plotted in Figure 6 (bottom) with onemomentum trajectory highlighted. Propagation distance (m) -1-0.500.51 T r an sv e r s e d i s p l a c e m en t ( mm ) Experimental x -Bohm position trajectories, highlighted Propagation distance (m) -2-1012 T r an sv e r s e m o m en t u m ( m r ad ) Experimental x -Bohm momentum trajectories, highlighted FIG. 6. x -Bohm experimental trajectories for position (top)and momentum (bottom) with a single initial setting high-lighted for clarity. Note that the highlighted, non-primaryvariable, trajectory of momentum (bottom) corresponds tothe highlighted ontological trajectory of position (top). Peaksin the momentum trajectories correspond to the ontologicalposition trajectories crossing a minimum in the two-slit inter-ference pattern. tive to the initial conditions of the x -Bohm position tra-jectories, and as such, they do not align with the peakpositions in the numerical simulation.To further explain the trajectory behaviour at thesepeaks, we highlight a single trajectory line in Figure 6,where the top and bottom plots correspond to a posi-tion and momentum trajectory in x -Bohm for the sameinitial conditions. A peak in the momentum trajectorydirectly corresponds to the portion of the position trajec-tory where the particle crosses a minimum in the double-slit interference pattern. V. DISCUSSION
When everyone is somebody, then noone’s anybody.
W.S. Gilbert, The Gondoliers
The lack of an ontological interpretation has been crit-icised as a serious drawback of quantum theory sinceits early days [18], for without such an interpretationthe visualization of quantum dynamics is not possible.Apart from any philosophical considerations, such visu-alizations are arguably essential for developing the intu-ition necessary for scientific development. At the sametime, incorrect visualizations (such as those involving theaether in electrodynamics) can lead us astray. Bohm’sinterpretation, with its deterministic particle trajecto-ries, presents an attractive visual picture of quantum dynamics at the cost of some non-trivial assumptions.Among these is an assumption of the role of positionand the coordinate representation of the wave functionas the fundamental variables that determine the dynam-ics. Indeed, in contrast to classical physics, where boththe initial position and momentum are necessary for pre-dicting the dynamics that follows, in an x -Bohm theorythe initial conditions are just the initial position of theparticle, together with the initial wave function. Whilethe theory ensures that in current experiments the realposition remains hidden and that the dynamics appearsnon-deterministic, the significance of this hidden variablefor the real dynamics hints at a fundamental asymmetryin nature. It is therefore tempting to view the identifica-tion of this asymmetry as a profound discovery, suggest-ing that position is indeed more important than othervariables. One could even hope that the realization thatposition plays a special role would lead to new experimen-tal predictions. However, as emphasized in this work, thespecific choice of position is not unique, leaving us withan infinite number of possible ontological variables – eachallegedly more fundamental than all the others – or, asGilbert’s line above implies, with none.Our main results show that a variation of Bohm’s the-ory ( p -Bohm) can be used to construct a very differentontological picture, one in which the dynamics are basedon the momentum variable. In this theory the funda-mental trajectories are paths through momentum space,and the equations of motion take a form which is closer tothat of Newton’s laws, with a first time derivative for mo-mentum. If the underlying ontological pictures of boththeories were the same, one might say that the symme-try between position and momentum had been restored,removing a non-trivial assumption from Bohm’s theory.The pictures are, however, very different (see Figures 3,4 and 5) and so, the asymmetry in a Bohm-like theory isconfirmed but ambiguous. One is left to wonder whichof the two theories, or indeed of the infinite intermedi-ate theories with other ontology, is correct, and possiblymore importantly what is the preferred variable.A striking example of this conundrum arises when weconsider a harmonic oscillator, where the Hamiltonianoperator ˆ H can be written in terms of the usual raisingand lowering operators ˆ a † and ˆ a ,ˆ H = ¯ hω (cid:18) ˆ a † ˆ a + 12 (cid:19) . (27)We can also write ˆ H = 12 ˆ p θ + 12 ω ˆ x θ , (28)for any real θ , where the operators ˆ x θ and ˆ p θ are definedas ˆ x θ = (cid:114) ¯ h ω (cid:0) ˆ a † e iθ + ˆ ae − iθ (cid:1) , (29)ˆ p θ = i (cid:114) ¯ hω (cid:0) ˆ a † e iθ − ˆ ae − iθ (cid:1) . (30)3Since [ˆ x θ , ˆ p θ ] = i ¯ h , we can construct what might be calleda θ − Bohm theory by taking ˆ x θ to be the “position oper-ator” for the particular θ chosen; in this representationthe Schr¨odinger equation is i ¯ h ∂∂t ψ ( x θ , t ) = − ¯ h ∂ ψ ( x θ , t ) ∂x θ + 12 ω x θ ψ ( x θ , t ) , (31)and following the usual Bohmian procedure and taking ψ ( x θ , t ) = R θ ( x θ , t ) exp( iS θ ( x θ , t ) / ¯ h ), with R θ ( x θ , t ) and S θ ( x θ , t ) both real, the guidance equation is v θ ( x θ , t ) = 12 ∂S θ ( x θ , t ) ∂x θ . (32)At least following Holland’s suggestion [4], this would betaken as the value of the non-primary variable associatedwith ˆ p θ (compare Eq. (12)). Here for each θ a differentphysical picture emerges, and weak p θ measurements fol-lowed by strong x θ measurements would allow the con-struction of trajectories for each θ -Bohm theory, yieldingentirely different visualizations.The significance of this is apparent if one considersthe Hamiltonian (27) to describe a mode of the radiationfield associated with a standing wave. Then in a stan-dard treatment [29] the operators ˆ x and ˆ p are (withinfactors) associated with the electric and magnetic fieldsrespectively. Thus in the 0-Bohm theory the ground state(or indeed any energy eigenstate) would be associatedwith an ensemble of different values of the electric fieldbut, following Holland’s suggestion, the magnetic fieldwould vanish in each member of the ensemble. On theother hand, in the π/ p π/ that would correspond to the electric field and ˆ x π/ withthe magnetic field, and so a description of the ground state (or indeed any energy eigenstate) in the π/ θ . Considering more general quantum states ofthe radiation field, weak measurements associated withone field quadrature followed by strong measurements as-sociated with the complementary quadrature would allowfor the formal construction of very different sets of tra-jectories.What would be the physical motivation for granting re-ality to one set of trajectories or the other? For massiveparticles, Wiseman has suggested that the question canbe settled in favour of the usual position variable if onetries to construct the trajectories of particles in a poten-tial with an x dependence more than quadratic [30]. Un-fortunately the measurement of trajectories in such the-ories remains experimentally challenging even with thesimplification of a photonic simulation. We expect thatcontinued work in this direction, ideally experiments in-volving massive particles, would lead to results that shedfurther light on the question. For states of the radiationfield, weak and strong measurements of field quadratureswould extend the discussion of the kind of issues raisedhere to Bohmian descriptions of field theories. ACKNOWLEDGMENTS
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To calculate the effective imaging plane, we employ the ray transfer matrix analysis on our lens system. Thepropagation and thin lens matrix is given by M prop ( d ) = (cid:18) d (cid:19) , (A1a) M lens ( f ) = (cid:18) − f (cid:19) . (A1b)To see the effective imaging plane at some distance y from the slits after the Lens 1, we analyze the ray matrix oflight being back propagated for a distance of y and then forward propagated through a lens and some distance f − d .This results in a ray matrix of (cid:32) − f − df − y (cid:16) − f − df (cid:17) − d + f − f yf + 1 (cid:33) . (A2)For the position distribution of the plane after the lens to be equivalent to the effective imaging plane, we require d = f y/f . (A3)This results in the transformation matrix (cid:32) f f + y − f f + yf (cid:33) , (A4)which corresponds to a transformation that relates the position of the beam as a function of the position of the beambefore the transformation only and does not depend on the momentum of the beam before the transformation. Theresulting transformation also results in a scaling of f / ( f + y ) in the position variable from the effective image planeto the plane at distance f − d after Lens 1. Placing the focus of Lens 2 at a distance of f − d after Lens 1 results ina transformation matrix of (cid:32) f ( f − d ) f ( f + y ) − f f f ( f + y ) f − f f f + f y (cid:33) (A5)where d is the distance after the second lens. As one can conclude from inspecting this matrix, the momentum ofthe light after the second lens is independent of the momentum at the effective imaging plane and proportional to itsposition distribution. Placing a calcite to perform a weak momentum measurement after Lens 2 thus results in theweak position measurement at the effective image plane.Lens 3 controls whether position or momentum of the effective imaging plane is projected onto the imaging setup.When placed one focal length away from the imaging system it transforms momentum after Lens 3, which reflects theposition of the effective imaging plane, onto the position distribution at the imaging setup. This is the configurationfor x -Bohm measurement where post-selection was performed on position. Alternatively, Lens 3 could be placed at f + f away from Lens 2. The resulting transformation with all three lenses and back propagation combined is (cid:18) − f f − f + yf (cid:19) . (A6)Note the absence of f in the equation. This is due to the fact that Lens 2 and Lens 3 are identical and they wereconfigured such that a one-to-one telescope is formed. The transformation effectively places the focus of Lens 1 ontothe imaging setup, performing a momentum post-selection for measurements in p -Bohm.5 Lens 2 distance from equilibrium (mm) E ff e c t i v e i m age p l ane p r opaga t i on d i s t an c e ( m ) FIG. 7. Calibration of the position of the second lens from the left and the corresponding propagation distance of the effectiveimage plane. The origin position of the second lens is the place where far field is projected onto the camera. The blue verticalbars indicate the calibrated distance with uncertainty, where the uncertainty mostly originates from the beam profile of theindividual slit not being perfectly Gaussian. The orange dotted line is the fitted curve of the calibration data.
Appendix B: Calibration1. Lens system
The system of lenses ( Figure 2) must be calibrated in order to infer the effective propagation distance. This isdone by noting the magnification of the setup given the position of lenses. By measuring the waist of the beam froman individual slit, as well as the distance between the centroid of individual beams from the two slits, the effectivepropagation distance and magnification can be calculated. The results of the lens calibration is shown in Figure 7.
2. Weak measurement calibration
Calibration of the strength of the weak measurement ζ was determined by performing weak measurement andpost-selection on the same variable, where ω and ξ in Eq. (12) were both set to be either position or momentumwhen calibrating for x -Bohm or p -Bohm, respectively. The corresponding setup and calibration results can be foundin Figure 8.6 Weak X CalibrationWeak P Calibration
Effective propagation distance (m) W ea k m ea s u r e m en t s t r eng t h (r ad m - ) FIG. 8. Illustration of calcite location for calibration of weak measurement strength ζ . Top (middle) figure corresponds tothe setup where we calibrate the strength of weak position (momentum) measurement, where position (momentum) of theeffective plane was measured both weakly and strongly. While the strength of weak momentum measurement is constant( ζ = 134 . ± ..