Observing momentum disturbance in double-slit “which-way” measurements

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 p^, post-selecting on finding it at position x, and averaging the result over many repetitions. This yieldsp(x)=Re[pw(x)](1)where pw(x)=〈x∣p^∣ψ〉/〈x∣ψ〉 is the weak value of the momentum operator (21–25). Note that for a scalar nonrelativistic wave function defined as ψ(x)=ρ(x)eiS(x)/ℏ, with ρ and S real, the above definition conforms to the original definition of the Bohmian momentum as p(x) = ∇ S(x) (13). Applying the above theory to photons might seem problematic (26) since electromagnetic waves correspond to particles with zero rest mass. However, we use a monochromatic beam, with x being the transverse position of the photon. This is well described by the nonrelativistic Schrödinger equation for a particle of mass m = h/(λc), where c is the speed of light and λ is the wavelength. Then, we can reconstruct the motion of this effective nonrelativistic particle using its position and velocity v(x) = p(x)/m (16–18).

In our experiment, we create an effective transverse wave function for the photon that is a superposition of two paths: ψϕ(x)=[fu(x)+eiϕfd(x)]/2. Here, ϕ is the relative phase between the paths, while fu(d)(x)=(w2π)−1/2e−(x∓D/2)2/w2 is the Gaussian wave packet, with a waist of w, of the upper (lower) path, and D ≫ w. An ideal WWM creates a two-component entangled state, correlating the path of the particle with orthogonal basis states of a probe. Measuring this probe in a different complementary basis realizes a quantum eraser (27). After the strong measurement, conditioned on the outcome, the wave function of the system (and its Bohmian particles) evolves independent of the measurement apparatus (14). In Bohmian mechanics, the details of the momentum change depend on the basis in which the WWM is measured because of the theory’s nonlocality, as illustrated in (17, 18). The minimum momentum disturbance occurs when the two outcomes of the quantum eraser measurement project the system state into ψ⁰ and ψ^π, respectively (15). That is, the particle is still in a superposition of its two paths, only with different relative phases, which, when summed, wash out the interference pattern.

We create the above superpositions and consider an ensemble of Bohmian trajectories starting at N transverse positions x_i(z₁), where z₁ represents the initial plane. By reconstructing each trajectory forward, as described above, to plane z_j, we can obtain an ensemble of new transverse photon positions xizj and transverse photon momenta pϕ(xizj). The change in the Bohmian momentum for this single trajectory, as a consequence of inducing a phase ϕ ≠ 0, is then calculated as pi=pϕ(xizj)−p0(xizj). By summing over all initial positions i, with the appropriate weights, the momentum disturbance distribution can be obtained (15)Pzjϕ(p)=1M∑i=2N−1℘(xi(z1))δ(p−pi)(2)where M is a factor that ensures ∫∞∞Pzjϕ(p)dp=1 and ℘(xi(z1))=∣ψϕ(xi(z1))∣2(xi+1z1−xi−1z1)/2. To obtain a smooth distribution with a finite ensemble size M, we approximate δ(p − p_i) by a Gaussian distribution of SD σ = 0.1ℏ/D. The Bohmian momentum disturbance distribution PzjB(p) up to plane z_j due to the WWM can then be calculated asPzjB(p)=η0Pzj0(p)+ηπPzjπ(p)(3)where η⁰ (η^π) represents the weight of the case ϕ = 0 (ϕ = π), with η⁰ + η^π = 1. A WWM with perfect path distinguishability corresponds to η⁰ = η^π = 1/2. The total momentum disturbance distribution, PTB(p), is when z_j is in the far field.

We can quantify the momentum disturbance by〈∣p∣〉zjB=∫PzjB(p)∣p∣dp(4)

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〈∣p∣〉TB≥2ℏπD(1−V)(5)

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 [fu(x)+eiϕfd(x)]/2. In our experiment, we simulate the quantum eraser WWM by inducing the relative phases ϕ = 0 or ϕ = π by rotating half-wave plate 1 (HWP1) to 22.5° or 67.5°, respectively.

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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 xkzj at the z_j plane is recorded by the pixels of the intensified charge-coupled device (ICCD) camera (Andor iStar 334), which is triggered by the electronic signal from the detection of the idler photon. The weak value of the transverse momentum is obtained from many runs by〈p(xkzj)〉w=hλ ζarcsin(NRkj−NLkjNRkj+NLkj)(6)where 1/ζ = 1/336 is the measured dimensionless coupling strength (18) and NRk(Lk)j is the photon count corresponding to the right-hand (left-hand) circular polarization. From this, we obtain the transverse Bohmian velocity as v(xkzj)=〈p(xkzj)〉w/m. A system of three lenses (L1, L2, and L3), with the middle lens translatable in the z direction, allows us to vary z_j from the near field (z₁ = 1.445 m) to the far field (z₁₁₇ = 8.612 m).

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 xkzj+1=xkzj+(zj+1−zj)v(xkzj)/c2−v2(xkzj), but this is practically identical to the formula using the transverse momentum, xkzj+(zj+1−zj)p(xkzj)/mc. Note that although the weak measurement crystal is at a fixed position, the evolution of the effective wave function, described by Schrödinger’s equation with the system of lenses acting as quadratic potentials, ensures that our method yields the weak value of transverse momentum at any desired z plane. Thus, as in (16), we rely upon knowledge of the Schrödinger evolution in this refocusing part of the apparatus, but note that we do not make use of any knowledge of the wave function itself in obtaining the Bohmian momenta and paths.

In the experiment, we consider 198 initial positions x_i(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.

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Following all the trajectories, we can obtain the Bohmian momentum disturbance distribution Pzjϕ(p) and the mean absolute momentum disturbance 〈∣p∣〉zjB at different planes z_j using Eqs. 2 and 4, respectively. The latter as a function of z is shown in Fig. 3A, with η⁰ = η^π = 1/2. This shows that the momentum disturbance is not a momentum kick that occurs at the point of the WWM (15). Rather, it gradually accumulates during the propagation of the photons. This delayed Bohmian momentum disturbance is characteristic of a nonclassical momentum disturbance as defined in (11).

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To further demonstrate the difference, Fig. 3B compares the total Bohmian momentum disturbance distribution Pzjϕ(p) at the last plane (z₁₁₇ = 8.612 m) with that at the first plane (z₁ = 1.445 m) (inset). The two peaks in Pz117π(p) at p ≈ ±3ℏ/(2D) are the dominant contribution to the mean absolute Bohmian momentum disturbance. They are almost absent from Pz1π(p), since in the near field, there has not been sufficient time for the wave function to develop the different phase gradients for the different ϕ that guide the Bohmian photons. The lack of any immediate disturbance to the Bohmian momentum from our WWM reflects the fact that the moments of the far-field momentum distribution are unchanged by this type of WWM (15, 28, 29).

We also look at the trade-off between the mean absolute momentum disturbance 〈∣p∣〉TB and the visibility V by changing the relative weight of the cases of ϕ = 0 and ϕ = π. This corresponds to WWMs with partial information, allowing a nonzero visibility V to remain. The results are shown in Fig. 4. The increase in the momentum disturbance as the visibility is reduced is observed and always exceeds the theoretical bound of inequality (5). The methods to estimate the fringe visibility V can be found in Materials and Methods.

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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, 〈∣p∣〉z117B, but using the Gaussian approximation to the δ-function to get Pz117B(p) and using the same experimental conditions (117 image planes from z₁ = 1.445 m to z₁₁₇ = 8.612 m with 198 initial positions). The result is the black dashed line and now agrees well with the experimental results. In this case, there is a predicted nonzero total mean absolute momentum 〈∣p∣〉zB even for the case of no WWM (V = 1). This simply reflects the SD (σ = 0.1ℏ/D) of the Gaussian distribution, which we use to approximate the δ-function when calculating the Bohmian momentum disturbance distribution.