Abstract
Making a “which-way” measurement (WWM) to identify which slit a particle goes through in a double-slit apparatus will reduce the visibility of interference fringes. There has been a long-standing controversy over whether this can be attributed to an uncontrollable momentum transfer. Here, by reconstructing the Bohmian trajectories of single photons, we experimentally obtain the distribution of momentum change. For our WWM, the change we see is not a momentum kick that occurs at the point of the WWM, but rather one that nonclassically accumulates during the propagation of the photons. We further confirm a quantitative relation between the loss of visibility consequent on a WWM and the total (late-time) momentum disturbance. Our results emphasize the role of the Bohmian momentum in giving an intuitive picture of wave-particle duality and complementarity.
INTRODUCTION
The single-particle Young’s double-slit experiment is the quintessential example of the wave-particle duality of quantum mechanics (1, 2). If one performs a position measurement to determine which slit a quantum particle traverses (particle-like property), then the interference pattern (wave-like property) is damaged. The more “which-way” information one obtains, the lower the visibility of the interference fringes (3–6). However, there has been a vigorous debate on whether the which-way measurement (WWM) destroys interference by disturbing the momentum of the particle (7–12).
Opposite conclusions were obtained by two research groups. In 1991, Scully, Englert, and Walther (SEW) (7) proposed a WWM scheme to prove that one can perform a position measurement with sufficient precision to identify which slit the particle goes, without apparently disturbing its momentum at all. They attributed the loss of visibility to the correlations between particles and the detectors. However, soon after, Storey, Tan, Collett, and Walls (STCW) (8) provided a general formalism, which appeared to show that the detection of path information necessarily involves some momentum transfer to the particles. A careful analysis (11) resolved this contradiction by showing that SEW and STCW were using different concepts of momentum transfer: “classical” and “quantum,” respectively. That is, their analyses were complementary. SEW’s scheme could not be explained by a classical probability distribution for momentum kicks, while the STCW theorem did correctly establish that there must be a nonzero probability amplitude for a momentum change of the expected size.
To study the paradigm of particle-wave duality in more depth, we need a more robust way to quantify the momentum disturbance. Neither SEW nor STCW gave such a measure for general situations. The difficulty is that we cannot unambiguously determine the momentum change to the quantum particle if the particle is not initially in a momentum eigenstate, which is the situation we face in a two-slit experiment. Bohmian mechanics, however, offers a way to solve this difficulty, as it posits that a particle has a definite position and momentum at all times and hence follows a deterministic trajectory (13, 14). The Bohmian probability distribution for momentum transfer was introduced in (15) and showed to be well suited to characterizing the momentum transfers in a WWM, both classical (immediate) and quantum (delayed) (15). It is a true probability distribution and, moreover, can be experimentally observed using established techniques (16–18). This means that it is possible to experimentally explore the relation between the size of the momentum disturbance and the degree of visibility loss in a WWM.
In this work, we sent a triggered single photon through a birefringent double-slit apparatus and reconstructed its Bohmain trajectories using the technique of weak measurement (16–20). Then, we obtained the distribution of Bohmian momentum transfer to the particle in a WWM by comparing all the photon’s trajectories in the free case and disturbed (WWM) case. We showed that the momentum change gradually accumulates during the propagation of the photons, which is negligible at short times. We further demonstrated the mean of the absolute value of the total (late-time) Bohmian momentum transfer
RESULTS
Bohmian momentum disturbance
According to Bohmian mechanics (13, 14), an individual particle has a definite position x and momentum p(x) (see also section SI in the Supplementary Materials). The Bohmian particle’s momentum is determined by its position. It can be obtained experimentally by performing a weak measurement of the quantum mechanical momentum operator
In our experiment, we create an effective transverse wave function for the photon that is a superposition of two paths:
We create the above superpositions and consider an ensemble of Bohmian trajectories starting at N transverse positions xi(z1), where z1 represents the initial plane. By reconstructing each trajectory forward, as described above, to plane zj, we can obtain an ensemble of new transverse photon positions
We can quantify the momentum disturbance by
It was shown theoretically (15) that, for WWMs achieving only partial distinguishability, resulting in a nonzero fringe visibility V, the total mean absolute momentum disturbance is bounded below
This relates the loss of interference in a WWM to the particle’s momentum change. Moreover, the WWM that achieves this bound corresponds to that in Eq. 3, with η0 = (1 + V)/2.
Experimental setup and results
Figure 1 shows our experimental setup. The generation of heralded signal photons is described in section SII of the Supplementary Materials. The signal photon is separated by a beam displacer into its horizontally and vertically polarized components, separated by about 3 mm. By rotating the polarization of one of these beams and compensating the difference in their optical paths, they become distinguishable only by their transverse location, describable by a wave function
Heralded signal photons (SPs) are separated into two paths by a beam displacer (BD 30). A half-wave plate HWP1 is used to change the relative phase between these two paths, while HWP2 is used to make the polarization of both paths the same. A birefringent crystal (PC) is inserted into one of the paths to compensate the difference in the optical length. The photon is prepared in the diagonal polarization state by HWP3 and then goes through a thin calcite crystal to perform weak measurement. The optic axis of the calcite crystal is in the xz plane oriented at 42° to the z axis. A quarter wave plate (QWP) and a beam displacer (BD 40) are used to detect the polarization of the photon. A combination of three lenses, L1 (plano-convex), L2 (aspherical and moveable), and L3 (plano-convex cylindrical), is used to image different planes on the intensified charge-coupled device (ICCD) camera.
The signal photon is then sent to the transverse momentum (or velocity) measurement setup, which consists of a 0.7-mm-thick piece of calcite with its optic axis in the xz plane oriented at 42° to the z axis, followed by a quarter wave plate and a beam displacer. The photon’s position
By measuring the transverse momenta or velocities (weakly) and positions (strongly) at a sequence of different imaging planes, we reconstruct the photon’s Bohmian trajectories. For computational convenience, we follow the method in (18) to reconstruct the trajectories via
In the experiment, we consider 198 initial positions xi(z1), with 99 for each slit. The initial positions are chosen to equally sample the Gaussian distribution of ∣ψ(x)∣2 across each slit (see Materials and Methods for details). The weak value of the transverse momentum at z117 = 8.612 m is shown in Fig. 2A as a function of x and for both phases ϕ. The relative phase difference of π yields the complementary pattern in momenta. Figure 2B shows the reconstructed trajectories beginning at the same place (two places are chosen: x = ±1.02 mm) for the two phases. For ϕ = 0, the trajectories converge to form the zero-order fringe, while for ϕ = π, they diverge toward the two first-order fringes.
(A) The weak value of the transverse velocities or momenta (v/c = λp/h) at z117 = 8.612 m. The red squares and blue dots represent experimental data with the relative phase ϕ being 0 and π, respectively. (B) Trajectories beginning at the same initial condition, x = ±1.02 mm, for ϕ = 0 (red) and ϕ = π (blue). The trajectories are reconstructed from 117 imaging planes. If a trajectory locates on a point that is not at the center of a pixel, then a cubic spline interpolation between neighboring momentum values is used.
Following all the trajectories, we can obtain the Bohmian momentum disturbance distribution
(A) The mean absolute momentum disturbance
To further demonstrate the difference, Fig. 3B compares the total Bohmian momentum disturbance distribution
We also look at the trade-off between the mean absolute momentum disturbance
The blue dots represent the experimental data for various partial WWMs kinds of measurements in the plane z117 = 8.612 m. The red solid line represents the theoretical prediction (5) under ideal conditions. The black dashed line represents the theoretical prediction, calculated with the same experimental conditions. Error bars are estimated from the counting statistics.
For the WWM we implement, this bound should be achievable. However, this bound is calculated considering an infinite number of initial positions at the plane z = 0, whereas in our experiment, we have 198 initial positions at the plane z1 = 1.445 m. Thus, to compare with our experiment, we also calculate V and, in the framework of Bohmian mechanics,
DISCUSSION
In this work, we used the Bohmian probability distribution (15) to experimentally quantify the momentum disturbance arising from a WWM, which destroys (or partially destroys) two-slit interference. In particular, we measured the mean of the absolute value of the Bohmian momentum disturbance
Last, we note that there are other methods to characterize the momentum transfer (28, 29)—the latter having been realized experimentally (31)—which also reflect the difference between classical and nonclassical momentum disturbance. While the momentum disturbance distributions in these methods have the advantage of being independent of the basis used for reading out the WWM device, they are not true probability distributions: They take negative values for nonclassical cases. By contrast, in our experiment, we measured a family of true probability distributions, which quantitatively captured the relationship between momentum disturbance and fringe visibility and which also enabled us to show the nonclassicality of that disturbance quantitatively for the first time. Thus, treating the momentum as an element of reality in Bohmian mechanics arguably provides the most useful method to understand the change of photon’s momentum in a WWM. Moreover, it gives an intuitive picture of part of the “uncontrollable change in the momentum” (1), which enforces complementarity: Although Bohmian dynamics is fully deterministic, the momentum transfer experienced by a particle depends on its initial position within the wave function, and that cannot be controlled by the experimenter.
MATERIALS AND METHODS
The selection of initial transverse positions xi(z1)
The probability distributions of the signal photon in the up and down paths at z1 plane were represented as Pu(x) = ∣fu(x)∣2 and Pd(x) = ∣fd(x)∣2, respectively. One hundred ninety-eight trajectories are reconstructed in the experiment, with 99 for each beam. The initial transverse positions xi(z1) of each beam were chosen to satisfy
Estimation of the interference visibility V
The intensities detected in the ICCD camera are denoted as
For the comparison between experiment and theory (black dashed line in Fig. 4), the visibility was further theoretically estimated with the same experimental conditions in the framework of quantum theory. The intensity distribution was calculated as
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/6/eaav9547/DC1
Section SI. Momentum in Bohmian mechanics and related theories
Section SII. Single photon generation
Fig. S1. Experimental setup for single-photon generation.
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REFERENCES AND NOTES
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