Novel approach to neutron electric dipole moment search using weak measurement

Since CP violation arises from only the phase of the Cabibbo–Kobayashi–Maskawa matrix in the standard model (SM) and it is tiny [1], CP-violating observables have provided good measurement sensitive to physics beyond the SM. In particular, measurement of the electric dipole moment (EDM) of the neutron, d_n, can give a clear signal of new physics (NP), and has been a big subject for the last seventy years [2].

The neutron EDM arises from three-loop short-distance [3–5], two-loop long-distance [6, 7], one-loop contributions from the QCD theta term [8], and tree-level charm-quark contributions [9] within the SM, while it can arise from one-loop diagrams in general NP models, such as multi-Higgs bosons [10–13], supersymmetric particles [14–17], leptoquark [18–20], and models with dynamical electroweak symmetry breaking [21, 22]. In addition, observed matter-antimatter asymmetry in the Universe requires new CP-violating sources [23, 24], which could be verified by the measurement of the EDM, e.g. reference [25].

So far, although much effort has been devoted to search for the EDMs, they have not been observed yet. One of the most severe limits comes from the neutron EDM search [26]

$$\begin{equation}{\left({d}_{\text{n}}\right)}_{\text{exp}}\enspace {< }3.0{\times}1{0}^{-26}\enspace \mathrm{e}\enspace \mathrm{c}\mathrm{m}\enspace \left(90\%\enspace \mathrm{C}\mathrm{L}\right),\end{equation} \tag{ 1 }$$

by measuring a neutron resonant frequency of ultracold neutrons (UCNs) based on the separated oscillatory field method (the so-called Ramsey method) [27, 28] ⁶ . This limit is five orders of magnitude larger than the SM prediction ${\left({d}_{\text{n}}\right)}_{\text{SM}}\sim 1{0}^{-\left(31-32\right)}\enspace \mathrm{e}\enspace \mathrm{c}\mathrm{m}$ [6, 7, 9]. Nevertheless, it severely constrains the NP scenarios that include additional CP violation.

In the early stage of the neutron EDM experiments, not the UCNs but a polarized neutron beam had been utilized [31–34]. However, it was known that there was a large systematic uncertainty in neutron beam experiment which comes from relativistic effects. The relativistic effects arise from the motion of neutrons (velocity v) through the electric field E, as (see, e.g. reference [35] for a derivation)

$$\begin{equation}\mathbf{B}=\frac{\mathbf{E}{\times}\mathbf{v}}{{c}^{2}}.\end{equation} \tag{ 3 }$$

Even if the neutron beam is shielded from the external magnetic field which we will assume in this paper, the external electric field does generate the magnetic filed depending on the velocity (it can be interpreted as the relativistic transformation of F_μν ), and the sensitivity of the experiment becomes dull because of the large spin magnetic moment interaction.

In order to avoid large uncertainties, current experiments and new proposed projects are using the UCNs [36–44]. Moreover, most of the experiments have employed the Ramsey method [27, 28]. The main reasons why the UCNs are preferred are the following two [45]: first, the systematic uncertainty from the relativistic effects can be neglected because of its small velocity of the UCNs. Second, the UCNs can have longer interaction times with the external electric field because they can be trapped easily, and the statistical uncertainty is suppressed.

On the other hand, in the polarized neutron beam experiments, although severe systematic effects come from the relativistic E × v corrections [33, 45], one can prepare a much larger amount of neutrons, which can reduce statistical fluctuations. Also, one can use stronger external electric fields, because the neutron beams are not covered by an insulating wall unlike the UCNs. Besides, there are several ideas that the relativistic E × v corrections can be suppressed in the neutron beam experiments with the Ramsey method [45] or with a spin rotation in non-centrosymmetric crystals [46–49]. The latter idea has been realized in an experiment [50].

In this paper, we propose a novel experimental approach in the search for the neutron EDM by applying not the Ramsey method but a method of weak measurement [51–53], and discuss conditions how our setup can overtake the current upper limit in equation (1). This work is the first application of the weak measurement to the neutron EDM measurement. For instance, the spin magnetic moment interaction had been measured by the weak measurement [52, 54], where a tremendous amplification of signal (a component of a spin) emerged. Also, the weak measurement using an optical polarizer with a laser beam has been realised [55, 56]. We also show that the relativistic E × v corrections are suppressed in this approach.

In the weak measurement, two quantum systems are prepared, and then the initial and final states are properly selected in one of the quantum systems, which are called pre- (|ψ_i⟩) and post-selections (⟨ψ_f|), respectively. A weak value, corresponding to an observable $\hat{A}$, is defined as

$$\begin{equation}{\langle \hat{A}\rangle }^{W}\equiv \frac{\langle {\psi }_{\mathrm{f}}\vert \hat{A}\vert {\psi }_{\mathrm{i}}\rangle }{\langle {\psi }_{\mathrm{f}}\vert {\psi }_{\mathrm{i}}\rangle }\in \mathbb{C},\end{equation} \tag{ 4 }$$

and can be amplified by choosing the proper selections of the states: ⟨ψ_f|ψ_i⟩ ∼ 0, which is called weak value amplification (for reviews see, e.g. references [57, 58]). Since the weak value is obtained as an observable quantity corresponding to $\hat{A}$ in an intermediate measurement between |ψ_i⟩ and ⟨ψ_f| without disturbing the quantum systems, measurement of the weak value plays an important role in the quantum mechanics itself. In addition, the weak value provides new methods for precise measurements [59, 60]. In fact, the weak value amplification was applied to precise measurements such as the spin Hall effect of the light (four orders of magnitude amplified) [61] and the beam deflection in a Sagnac interferometer (two orders of magnitude amplified) [62].

In our setup (see figure 1), as explained in details at the next section, unlike the Ramsey method with using the UCN, we consider a polarized neutron beam with the velocity of ∼10³ m/ sec . We investigate the motion of a neutron bunch in the external electric field with spatial gradient, and apply methods of the weak measurement which leads to amplification of the signal.

Zoom In Zoom Out Reset image size

Very interestingly, it will be shown in our setup that the systematic uncertainty from the relativistic E × v effect can be irrelevant compared to a neutron EDM signal. This fact is expected as a new virtue of the weak measurement because our finding implies that the weak measurement itself is useful in quantum systems to suppress the systematic uncertainty such as the relativistic effect.

This paper is organized as follows. In section 2, we propose an experimental setup for the neutron EDM search using the weak measurement. In section 3, we analytically calculate an expected observable in this setup. Especially, the weak value is introduced. Numerical results are evaluated in section 4. We will show the weak value amplification and a potential sensitivity to the neutron EDM signal in this setup. Finally, section 5 is devoted for the conclusions. In appendix A, a general setup of an external electric field is considered. In appendix B, a formalism of a full-order calculation of the expected observable is provided.

We consider application of the weak measurement method [52] to the neutron EDM measurement. Because reference [52] utilizes an external magnetic field with a spatial gradient for measuring the spin magnetic moment, one should use the polarized neutron beam and an external electric field with a spatial gradient.

In order to obtain a signal amplification in a weak measurement (weak value amplification), two important selections are necessary: the pre-selection and the post-selection. The pre-selection is equivalent to preparation of the initial state in the conventional quantum mechanics. On the other hand, the post-selection is extraction of a specific quantum state at the late time [51], and it makes to understand the weak value amplification difficult because of lack of counterparts in the conventional quantum mechanics. In a nutshell, the role of the post-selection is filtering where only events that the observable (such as the position of the neutron) takes a large value are collected. Note that the weak value amplification originates from the quantum interference [63–66], so that classical filtering is not suitable. In our setup, we impose selections of the spin polarization of the neutrons as the pre- and post-selections.

Figure 1 shows our proposed experimental setup. The detailed explanation is given in the figure caption and the following paragraph, especially the electric field with the spatial gradient α_x is represented by the orange arrows. We set xyz axis as follows: The neutrons fly along the z axis. The external electric field has the gradient along x axis. A spin direction of the pre-selection at the first spin polarizer is y axis.

The entire process of the setup is divided into four stages:

• (a)  
  Pre-selection. At t = 0, the neutron bunch emitted from the neutron source is polarized at the first polarizer (the yellow box in figure 1), and the quantum state of the total system is represented by the following density matrices

  $$\begin{equation}{\hat{\rho }}_{\text{ini}}^{\text{total}}={\hat{\rho }}_{\text{ini}}^{P}\otimes {\hat{\rho }}_{\text{ini}}^{S},\end{equation} \tag{ 5 }$$

  where ${\hat{\rho }}_{\text{ini}}^{P}$ is the initial state of the neutron position and ${\hat{\rho }}_{\text{ini}}^{S}$ is the pre-selected spin polarization.
• (b)  
  Time evolution in external electric field. For 0 < t < T, the neutron bunch goes through the external electric field with spatial gradient in figure 1, where the total system evolves by the Hamiltonian equation (9) as

  $$\begin{equation}{\hat{\rho }}_{\text{ini}}^{\text{total}}\left(t\right)={\text{e}}^{-\text{i}\hat{H}t}{\hat{\rho }}_{\text{ini}}^{\text{total}}{\text{e}}^{\text{i}\hat{H}t}.\end{equation} \tag{ 6 }$$

  Note that in this paper, although we do not use the natural units, ℏ is discarded for simplicity.
• (c)  
  Post-selection. After the time evolution of the total system in the external electric field, at t = T, the neutrons are selected in the second polarizer (the pink box in figure 1) where a specific spin polarization state (${\hat{\rho }}_{\text{fin}}^{S}$) of the neutron can pass. Then, the neutron position is represented by a density matrix

  $$\begin{equation}{\hat{\rho }}_{\text{fin}}^{P}={\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{\rho }}_{\text{ini}}^{\text{total}}\left(T\right)\right].\end{equation} \tag{ 7 }$$
• (d)  
  Measurement of the center of mass. After the post-selection, position shifts of the neutron along x-axis are measured ⁷ . Here, by taking the average of those shifts, one can measure a position shift of the center-of-mass of the neutrons passing the post-selection. The expectation value of the position shift is expressed as

  $$\begin{equation}{{\Delta}}_{x}^{W}=\frac{{\mathrm{Tr}}_{P}\left[\hat{x}{\hat{\rho }}_{\text{fin}}^{P}\right]}{{\mathrm{Tr}}_{P}\left[{\hat{\rho }}_{\text{fin}}^{P}\right]}.\end{equation} \tag{ 8 }$$

The detail evaluations of these processes will be discussed in the next section.

The external electric field is set between the pre-selection and the post-selection. As discussed in appendix A, one can define the x axis as the direction of spatial gradient of the external electric field. Therefore, x dependence of y and z components of the external electric field is negligible without loss of generality. According as the size of the neutron EDM and the spin polarization, displacement of the neutron along x axis occurs. We would like to maximize this displacement by the weak value amplification.

In this setup, the single neutron can be described by the non-relativistic Hamiltonian [67]:

$$\begin{equation}\hat{H}=\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}-{d}_{\text{n}}\hat{\mathbf{E}}\cdot \hat{\boldsymbol{\sigma }}-\frac{{\mu }_{\text{n}}}{{m}_{\text{n}}{c}^{2}}\left(\hat{\mathbf{E}}{\times}\hat{\mathbf{p}}\right)\cdot \hat{\boldsymbol{\sigma }},\end{equation} \tag{ 9 }$$

where $\hat{\mathbf{p}}$ is the momentum operator of a neutron, m_n is the neutron mass, d_n is the neutron EDM, μ_n is the neutron magnetic moment, $\hat{\mathbf{E}}$ is an operator of the external electric field vector, and $\hat{\boldsymbol{\sigma }}$ is the spin operator corresponding to the polarized neutron: the operation on spin-states is defined as ${\hat{\sigma }}_{i}\left\vert {{\pm}}_{i}\right\rangle ={\pm}\left\vert {{\pm}}_{i}\right\rangle \enspace \mathrm{f}\mathrm{o}\mathrm{r}\enspace i=x,y,z$. This Hamiltonian is defined in the Hilbert space $\mathcal{H}={\mathcal{H}}_{P}\otimes {\mathcal{H}}_{S}$, where ${\mathcal{H}}_{P}$ is the Hilbert space of the position (P) of the neutron, while ${\mathcal{H}}_{S}$ is that of the spin (S) of the neutron ⁸ . Note that although the external magnetic field is zero in the setup, the magnetic filed is generated by the relativistic effect in equation (3).

In this section, we derive an analytic formula of an expectation value of deviation of the neutron position from x = 0, which is shown as ${{\Delta}}_{x}^{W}$ in figure 1.

To analytically study our strategy for the neutron EDM search based on the weak value amplification, we adopt two assumptions as follows:

Assumption 1. We consider the following external electric field operator:

$$\begin{equation}\hat{\mathbf{E}}=\left({E}_{x}^{0}+\alpha \hat{x},\enspace {E}_{y}^{0},\enspace 0\right),\end{equation} \tag{ 10 }$$

where ${E}_{x}^{0},\enspace {E}_{y}^{0}$, and α are constants (namely α = dE_x /dx) ⁹ . $\hat{x}$ is an operator corresponding to the x coordinate of the neutron. A necessary condition of this form is discussed in appendix A. Even if the E_z component is nonzero, the effect is irrelevant in this setup. Although E_z generates B_x,y via the relativistic effects in the third terms of equation (9), they are significantly suppressed by small neutron momenta, v_x and v_y .

Assumption 2. We consider the following neutron initial state in the Hilbert space ${\mathcal{H}}_{P}$:

$$\begin{equation}{\hat{\rho }}_{\text{ini}}^{P}\equiv \left\vert {G}_{{p}_{x0}}\otimes {p}_{y0}\otimes {p}_{z0}\right\rangle \left\langle {G}_{{p}_{x0}}\otimes {p}_{y0}\otimes {p}_{z0}\right\vert ,\end{equation} \tag{ 11 }$$

where we defined as

$$\begin{align}\hfill \langle x\vert {G}_{{p}_{x0}}\rangle & =\frac{1}{{\left(2\pi {d}^{2}\right)}^{1/4}}{\text{e}}^{\text{i}{p}_{x0}\cdot x}{e}^{-\frac{{x}^{2}}{4\enspace {d}^{2}}},\hfill \end{align} \tag{ 12 }$$

$$\begin{align}\hfill \langle j\vert {p}_{j0}\rangle & =\frac{1}{\sqrt{2\pi }}{\text{e}}^{\text{i}{p}_{j0}\cdot j}\enspace ,\quad \mathrm{f}\mathrm{o}\mathrm{r}\enspace j=y,\enspace z.\hfill \end{align} \tag{ 13 }$$

Here, p_x0, p_y0 and p_z0 are the initial neutron momenta, and $\vert {G}_{{p}_{x0}}\rangle ,\enspace \vert {p}_{y0}\rangle$ and |p_z0⟩ are the quantum states of x, y and z directions, respectively. We are interested in the spatial displacement of the neutron along x axis. As explained below, in the weak measurement, the expectation value of the neutron position depends on only variance of the distribution. Therefore, we assume the Gaussian wave packet |G⟩ as the a quantum state of x direction, for simplicity [68, 69]. In the distribution, we regard d as a standard deviation of the neutron beam and assume that the neutron beam diameter is 2d for the x direction ¹⁰ . On the other hand, for y and z directions, we assume the plane wave.

In addition to above assumptions, we use several numerical approximations in this section. These approximation are reasonable when one takes input values which will be used in section 4. Note that these numerical approximations are not used in the final plot of section 4.

Based on the assumption 1 in equation (10), the Hamiltonian in equation (9) can be expressed as

$$\begin{align}\hfill \hat{H}& =\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}-{g}_{\mu }\left[\chi \left({E}_{x}^{0}+\alpha \hat{x}\right)+{E}_{y}^{0}{\hat{n}}_{{p}_{z}}\right]\otimes {\hat{\sigma }}_{x}\hfill \\ \hfill & \quad -{g}_{\mu }\left[\chi {E}_{y}^{0}-\left({E}_{x}^{0}+\alpha \hat{x}\right){\hat{n}}_{{p}_{z}}\right]\otimes {\hat{\sigma }}_{y}\hfill \\ \hfill & \quad -{g}_{\mu }\left[\left({E}_{x}^{0}+\alpha \hat{x}\right){\hat{n}}_{{p}_{y}}-{E}_{y}^{0}{\hat{n}}_{{p}_{x}}\right]\otimes {\hat{\sigma }}_{z},\hfill \end{align} \tag{ 14 }$$

where we defined ${p}_{0}\equiv \sqrt{{p}_{x0}^{2}+{p}_{y0}^{2}+{p}_{z0}^{2}}$, g_μ ≡ μ_n p₀/(m_n c²), χ ≡ d_n/g_μ , and ${\hat{n}}_{{p}_{i}}\equiv {\hat{p}}_{i}/{p}_{0}$ (i = x, y, z) for convenience in the following analysis. All interactions are normalized by g_μ , and the EDM interaction is represented as χg_μ . Note that χ is dimensionless real quantity.

The time evolution operator is ${\mathrm{e}}^{-\mathrm{i}\hat{H}t}$. Using the Baker–Campbell–Hausdorff formula, and $\left[\hat{x},{\hat{p}}_{x}\right]=i$ and $\left[\hat{x},{\hat{p}}_{x}^{2}\right]=2\mathrm{i}{\hat{p}}_{x}$, we obtain

$$\begin{align}\hfill \text{exp}\left(-\mathrm{i}\hat{H}t\right)& =\text{exp}\left(-\mathrm{i}\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}t\right)\enspace \text{exp}\left[-\mathrm{i}\left({\hat{H}}_{0}+{\hat{H}}_{\chi }\right)t\right.\hfill \\ \hfill & \left.\quad +\enspace \mathcal{O}\left({t}^{3}{g}_{\mu }^{2}\frac{{\alpha }^{2}}{{m}_{\text{n}}}\enspace ,{t}^{3}{g}_{\mu }^{2}\alpha {E}_{y}^{0}\frac{{\hat{p}}_{x}}{{m}_{\text{n}}}\right)\right],\hfill \end{align} \tag{ 15 }$$

where the interaction Hamiltonian with the background (${\hat{H}}_{0}$) and with the EDM (${\hat{H}}_{\chi }$) are

$$\begin{align}\hfill & {\hat{H}}_{0}\equiv -{g}_{\mu }{E}_{x}^{0}\left[\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{\hat{n}}_{{p}_{z}}\otimes {\hat{\sigma }}_{x}-{\hat{n}}_{{p}_{z}}\otimes {\hat{\sigma }}_{y}\right.\hfill \\ \hfill & \left.+\enspace \left({\hat{n}}_{{p}_{y}}-\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{\hat{n}}_{{p}_{x}}\right)\otimes {\hat{\sigma }}_{z}\right]+{g}_{\mu }\alpha \left(\hat{x}+\frac{t}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\hfill \\ \hfill & {\times}\enspace \left({\hat{n}}_{{p}_{z}}\otimes {\hat{\sigma }}_{y}-{\hat{n}}_{{p}_{y}}\otimes {\hat{\sigma }}_{z}\right),\hfill \end{align} \tag{ 16 }$$

$$\begin{align}\hfill {\hat{H}}_{\chi }\equiv & -\chi {g}_{\mu }\left[{E}_{x}^{0}+\alpha \left(\hat{x}+\frac{t}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\right]\otimes {\hat{\sigma }}_{x}\hfill \\ \hfill & -\enspace \chi {g}_{\mu }{E}_{y}^{0}\otimes {\hat{\sigma }}_{y}.\hfill \end{align} \tag{ 17 }$$

We have checked that the $\mathcal{O}\left({t}^{3}\right)$ terms in equation (15) are numerically negligible in the following analysis. Moreover, the last term in equation (17) is totally screened by ${g}_{\mu }{E}_{x}^{0}{\hat{n}}_{{p}_{z}}\otimes {\hat{\sigma }}_{y}$ in ${\hat{H}}_{0}$.

Then, we expand the interaction Hamiltonian by $\alpha \hat{x}$, and obtain the following analytic form:

$$\begin{align}\hfill \text{exp}\left[-\mathrm{i}\left({\hat{H}}_{0}+{\hat{H}}_{\chi }\right)t\right]& ={\hat{U}}_{0}\left(t\right)+\alpha \left(\hat{x}+\frac{t}{2{m}_{\text{n}}}{\hat{p}}_{x}\right) {\hat{U}}_{1}\left(t\right)\hfill \\ \hfill & \quad +\enspace \mathcal{O}\left({\alpha }^{2}{\left(\hat{x}+\frac{t}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)}^{2}\right),\hfill \end{align} \tag{ 18 }$$

with

$$\begin{align}\hfill {\hat{U}}_{0}\left(t\right)& ={I}_{2}\enspace \mathrm{cos}\left({g}_{\mu }{E}_{x}^{0}t\right)\hfill \\ \hfill & \quad +\enspace \left(\begin{matrix}\hfill -\mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{x0}}+\mathrm{i}{n}_{{p}_{y0}}\hfill & \hfill -{n}_{{p}_{z0}}+\mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{z0}}+i\chi \hfill \\ \hfill {n}_{{p}_{z0}}+\mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{z0}}+\mathrm{i}\chi \hfill & \hfill +\mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{x0}}-\mathrm{i}{n}_{{p}_{y0}}\hfill \end{matrix}\right)\hfill \\ \hfill & \quad {\times}\enspace \mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}t\right)+\mathcal{O}\left({\left(\frac{{E}_{y}^{0}}{{E}_{x}^{0}}\right)}^{2}\right),\hfill \end{align} \tag{ 19 }$$

$$\begin{align}\hfill {\hat{U}}_{1}\left(t\right)=& \left(\begin{matrix}\hfill -\mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{x0}}+i{n}_{{p}_{y0}}\hfill & \hfill \mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{z0}}-{n}_{{p}_{z0}}+\mathrm{i}\chi \hfill \\ \hfill \mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{z0}}+{n}_{{p}_{z0}}+\mathrm{i}\chi \hfill & \hfill \mathrm{i}\frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{x0}}-i{n}_{{p}_{y0}}\hfill \end{matrix}\right){g}_{\mu }t\hfill \\ \hfill & {\times}\enspace \mathrm{cos}\left({g}_{\mu }{E}_{x}^{0}t\right)\hfill \\ \hfill & +\left(\begin{matrix}\hfill i\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}{n}_{{p}_{x0}}-{g}_{\mu }t\hfill & \hfill -\mathrm{i}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}{n}_{{p}_{z0}}\hfill \\ \hfill -\mathrm{i}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}{n}_{{p}_{z0}}\hfill & \hfill -\mathrm{i}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}{n}_{{p}_{x0}}-{g}_{\mu }t\hfill \end{matrix}\right)\hfill \\ \hfill & {\times}\enspace \mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}t\right)+\mathcal{O}\left({\left(\frac{{E}_{y}^{0}}{{E}_{x}^{0}}\right)}^{2}\right),\hfill \end{align} \tag{ 20 }$$

where I₂ is the 2 × 2 unit matrix. Hereafter, we assume ${\hat{n}}_{{p}_{i}}\to {n}_{{p}_{i0}}={p}_{i0}/{p}_{0}\enspace \left(i=x,\enspace y,\enspace z\right)$ and ${n}_{{p}_{x0}}^{2}+{n}_{{p}_{y0}}^{2}+{n}_{{p}_{z0}}^{2}=1$ for simplicity of calculations. The higher-order terms $\mathcal{O}\left({\left({E}_{y}^{0}/{E}_{x}^{0}\right)}^{2}\right)$ are numerically irrelevant when ${E}_{y}^{0}\ll {E}_{x}^{0}$. The ${\hat{U}}_{0}\left(t\right)$ and ${\hat{U}}_{1}\left(t\right)$ satisfy the unitarity condition:

$$\begin{equation}\begin{gathered}{c}{\hat{U}}_{0}\left(t\right){\hat{U}}_{0}{\left(t\right)}^{{\dagger}}={I}_{2}\enspace ,\quad {\hat{U}}_{1}\left(t\right){\hat{U}}_{1}{\left(t\right)}^{{\dagger}}={\left({g}_{\mu }t\right)}^{2}{I}_{2}\enspace ,\\ {\hat{U}}_{0}\left(t\right){\hat{U}}_{1}{\left(t\right)}^{{\dagger}}+{\hat{U}}_{1}\left(t\right){\hat{U}}_{0}{\left(t\right)}^{{\dagger}}=0,\end{gathered}\end{equation} \tag{ 21 }$$

up to the following higher-order corrections:

$$\begin{equation}\mathcal{O}\left(\chi \frac{{E}_{y}^{0}}{{E}_{x}^{0}},\enspace {\chi }^{2},\enspace {\left(\frac{{E}_{y}^{0}}{{E}_{x}^{0}}\right)}^{2},\enspace {n}_{{p}_{x0}}^{2},\enspace \frac{{E}_{y}^{0}}{{E}_{x}^{0}}{n}_{{p}_{x0}}{n}_{{p}_{y0}}\right).\end{equation} \tag{ 22 }$$

Now, we obtain the compact form of the time evolution operator,

$$\begin{align}\hfill \text{exp}\left(-\mathrm{i}\hat{H}t\right)& \simeq \text{exp}\left(-\mathrm{i}\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}t\right)\hfill \\ \hfill & \quad {\times}\enspace \left[{\hat{U}}_{0}\left(t\right)+\alpha \left(\hat{x}+\frac{t}{2{m}_{\text{n}}}{\hat{p}}_{x}\right) {\hat{U}}_{1}\left(t\right)\right].\hfill \end{align} \tag{ 23 }$$

As mentioned previous section, the weak measurement requires pre- and post-selections, and we select the neutron spin polarization. Since the neutron polarization rate is not perfect in practical spin polarizers, we include an impurity effect in the pre- and post-selections as mixed spin states of the neutron. As will be shown later, the final result significantly depends on the impurity effect. This is because neutron passing probability at the post-selection is sensitive to the impurity effect in this setup, and large passing probability dulls the neutron position shift. We consider the following pre-selected state in the Hilbert space ${\mathcal{H}}_{S}$:

$$\begin{align}\hfill {\hat{\rho }}_{\text{ini}}^{S}& =\left(1-{\epsilon}\right)\vert \psi \rangle \langle \psi \vert +{\epsilon}\vert \phi \rangle \langle \phi \vert \hfill \\ \hfill & =\frac{1}{2}\left(\begin{matrix}\hfill 1\hfill & \hfill -\mathrm{i}\left(1-2{\epsilon}\right)\hfill \\ \hfill \mathrm{i}\left(1-2{\epsilon}\right)\hfill & \hfill 1\hfill \end{matrix}\right),\hfill \end{align} \tag{ 24 }$$

where two polarization states are (see figure 1)

$$\begin{align}\hfill \vert \psi \rangle & =\vert {+}_{y}\rangle =\frac{1}{\sqrt{2}}\left(\begin{matrix}\hfill i\hfill \\ \hfill -1\hfill \end{matrix}\right),\hfill \end{align} \tag{ 25 }$$

$$\begin{align}\hfill \vert \phi \rangle & =\vert {-}_{y}\rangle =\frac{1}{\sqrt{2}}\left(\begin{matrix}\hfill i\hfill \\ \hfill 1\hfill \end{matrix}\right).\hfill \end{align} \tag{ 26 }$$

Here, |±_y ⟩ are eigenstates of the spin operator ${\hat{\sigma }}_{y}$, and (0 < ≪ 1) stands for the selection impurity.

After the pre-selection, the quantum state of the total system at the initial time t = 0 can be expressed as direct-product ${\hat{\rho }}_{\text{ini}}^{\text{total}}={\hat{\rho }}_{\text{ini}}^{P}\otimes {\hat{\rho }}_{\text{ini}}^{S}$, where ${\hat{\rho }}_{\text{ini}}^{P}$ is defined in equation (11), which is the assumption 2.

The late-time quantum state of the total system at t = T (see figure 1) just before the post-selection is given as

$$\begin{align}\hfill {\hat{\rho }}_{\text{ini}}^{\text{total}}\left(T\right)& ={\text{e}}^{-\text{i}\hat{H}T}{\hat{\rho }}_{\text{ini}}^{\text{total}}{\text{e}}^{\text{i}\hat{H}T}\hfill \\ \hfill & ={\text{e}}^{-\text{i}\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}T}\left[{\hat{U}}_{0}\left(T\right)+\alpha \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right) {\hat{U}}_{1}\left(T\right)\right]{\hat{\rho }}_{\text{ini}}^{\text{total}}\hfill \\ \hfill & \quad {\times}\enspace \left[{\hat{U}}_{0}{\left(T\right)}^{{\dagger}}+\alpha \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right) {\hat{U}}_{1}{\left(T\right)}^{{\dagger}}\right]{\text{e}}^{\text{i}\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}T}.\hfill \end{align} \tag{ 27 }$$

Next, we consider the following post-selected state:

$$\begin{align}\hfill {\hat{\rho }}_{\text{fin}}^{S}& =\left(1-{\epsilon}\right)\vert {\phi }_{\delta }\rangle \langle {\phi }_{\delta }\vert +{\epsilon}\vert {\psi }_{\delta }\rangle \langle {\psi }_{\delta }\vert \hfill \\ \hfill & =\frac{1}{2}\left(\begin{matrix}\hfill 1+\left(1-2{\epsilon}\right)\mathrm{sin}\delta \hfill & \hfill \mathrm{i}\left(1-2{\epsilon}\right)\mathrm{cos}\delta \hfill \\ \hfill -\mathrm{i}\left(1-2{\epsilon}\right)\mathrm{cos}\delta \hfill & \hfill 1-\left(1-2{\epsilon}\right)\mathrm{sin}\delta \hfill \end{matrix}\right),\hfill \end{align} \tag{ 28 }$$

with

$$\begin{align}\hfill \vert {\psi }_{\delta }\rangle & \equiv {\text{e}}^{\text{i}\frac{\delta }{2}{\hat{\sigma }}_{x}}\vert {+}_{y}\rangle =\frac{1}{\sqrt{2}}\left(\begin{matrix}\hfill \mathrm{i}\left(\mathrm{cos}\enspace \frac{\delta }{2}-\mathrm{sin}\enspace \frac{\delta }{2}\right)\hfill \\ \hfill -\left(\mathrm{cos}\enspace \frac{\delta }{2}+\mathrm{sin}\enspace \frac{\delta }{2}\right)\hfill \end{matrix}\right),\hfill \\ \hfill \vert {\phi }_{\delta }\rangle & \equiv {\text{e}}^{\text{i}\frac{\delta }{2}{\hat{\sigma }}_{x}}\vert {-}_{y}\rangle =\frac{1}{\sqrt{2}}\left(\begin{matrix}\hfill \mathrm{i}\left(\mathrm{cos}\enspace \frac{\delta }{2}+\mathrm{sin}\enspace \frac{\delta }{2}\right)\hfill \\ \hfill \mathrm{cos}\enspace \frac{\delta }{2}-\mathrm{sin}\enspace \frac{\delta }{2}\hfill \end{matrix}\right).\hfill \end{align} \tag{ 29 }$$

Here, δ is a polarization angle around x-axis for the post-selection (see figure 1). It is known that small δ angle is preferred for the weak value amplification [52]. The selection impurity is also included in the post-selection, and we assume its quality is the same as the pre-selection, for simplicity.

After the post-selection, the final state of the total system is written as ${\hat{\rho }}_{\text{fin}}^{\text{total}}={\hat{\rho }}_{\text{fin}}^{P}\otimes {\hat{\rho }}_{\text{fin}}^{S}$. Using the late-time state in equation (27), we obtain the neutron final state in the Hilbert space ${\mathcal{H}}_{P}$,

$$\begin{align}\hfill {\hat{\rho }}_{\text{fin}}^{P}& \equiv {\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{\text{total}}\right]={\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{\rho }}_{\text{ini}}^{\text{total}}\left(T\right)\right]\hfill \\ \hfill & ={\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}{\left(T\right)}^{{\dagger}}\right]{\text{e}}^{-\text{i}\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}T}\hfill \\ \hfill & \quad {\times}\enspace \left\{{\hat{\rho }}_{\text{ini}}^{P}+\chi {g}_{\mu }\alpha T\left[W{\hat{\rho }}_{\text{ini}}^{P}\left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\right.\right.\hfill \\ \hfill & \left.\left.\quad +\enspace {W}^{{\ast}}\left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right){\hat{\rho }}_{\text{ini}}^{P}\right]\right\}{\text{e}}^{\text{i}\frac{{\hat{\mathbf{p}}}^{2}}{2{m}_{\text{n}}}T}\hfill \\ \hfill & \quad +\mathcal{O}\left({\left({g}_{\mu }\alpha T\right)}^{2}\right),\hfill \end{align} \tag{ 30 }$$

where we defined the following dimensionless complex quantity W,

$$\begin{equation}W\equiv \frac{1}{\chi {g}_{\mu }T}\frac{{\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{1}{\left(T\right)}^{{\dagger}}\right]}{{\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}{\left(T\right)}^{{\dagger}}\right]}.\end{equation} \tag{ 31 }$$

The W corresponds to the weak value. For the third term of equation (30), using equations (24) and (28), we used

$$\begin{align}\hfill & {\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{1}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}{\left(T\right)}^{{\dagger}}\right]\hfill \\ \hfill & \quad ={\mathrm{Tr}}_{S}\left[{\hat{U}}_{0}{\left(T\right)}^{{\ast}}{\left({\hat{\rho }}_{\text{ini}}^{S}\right)}^{T}{\hat{U}}_{1}{\left(T\right)}^{T}{\left({\hat{\rho }}_{\text{fin}}^{S}\right)}^{T}\right]\hfill \\ \hfill & \quad ={\mathrm{Tr}}_{S}{\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{1}{\left(T\right)}^{{\dagger}}\right]}^{{\ast}}.\hfill \end{align} \tag{ 32 }$$

Similarly, one can easily find ${\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}{\left(T\right)}^{{\dagger}}\right]={\mathrm{Tr}}_{S}{\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}{\left(T\right)}^{{\dagger}}\right]}^{{\ast}}=$ real.

In an ideal experimental setup limit, ${E}_{y}^{0}/{E}_{x}^{0}\to 0$, ${n}_{{p}_{x0,y0}}\to 0$, and → 0, the weak value W is expressed as

$$\begin{align}\hfill W& =-\mathrm{i}\frac{\langle \psi \left(T\right)\vert {\hat{\sigma }}_{x}\vert {\phi }_{\delta }\rangle }{\langle \psi \left(T\right)\vert {\phi }_{\delta }\rangle }+\frac{\mathrm{i}}{\chi }\frac{\langle \psi \left(T\right)\vert {\hat{\sigma }}_{y}\vert {\phi }_{\delta }\rangle }{\langle \psi \left(T\right)\vert {\phi }_{\delta }\rangle }\hfill \end{align} \tag{ 33 }$$

$$\begin{align}\hfill & =\frac{\mathrm{i}}{\chi }-{\mathrm{e}}^{-2\mathrm{i}{g}_{\mu }{E}_{x}^{0}T}\enspace \mathrm{cot}\left(\frac{\delta }{2}\right)+\mathcal{O}\left(\chi \right),\hfill \end{align} \tag{ 34 }$$

where we define

$$\begin{align}\hfill \vert \psi \left(T\right)\rangle & ={\hat{U}}_{0}\left(T\right)\vert \psi \rangle \hfill \\ \hfill & =\left\{\mathrm{cos}\left({g}_{\mu }{E}_{x}^{0}T\right){I}_{2}+\mathrm{i}\enspace \mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}T\right)\left[\chi {\hat{\sigma }}_{x}-{\hat{\sigma }}_{y}\right]\right\}\vert \psi \rangle \hfill \end{align} \tag{ 35 }$$

$$\begin{align}\hfill & ={\mathrm{e}}^{-\mathrm{i}{g}_{\mu }{E}_{x}^{0}T}\vert \psi \rangle +\mathrm{i}\enspace \mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}T\right)\chi {\hat{\sigma }}_{x}\vert \psi \rangle .\hfill \end{align} \tag{ 36 }$$

According to the definition of the weak value in equation (4), we find $W=-\mathrm{i}{\langle {\hat{\sigma }}_{x}\rangle }^{W}+\frac{\mathrm{i}}{\chi }{\langle {\hat{\sigma }}_{y}\rangle }^{W}$ in the ideal experimental limit, and show that W is amplified by cot(δ/2) for small δ region [52]. One should note that since W is always multiplied by χ in equation (30), the first term in equation (34) is not singular in χ → 0 limit. In other words, there is a contribution in equation (30) that is independent of χ (signal) and sensitive to the weak value W, especially ${\langle {\hat{\sigma }}_{y}\rangle }^{W}$. We will show that such a contribution corresponds to a background effect (from the relativistic E × v effect). It would be interesting possibility to measure the weak value from the background effect, even if one cannot measure the neutron EDM signal. It is noteworthy that proposed setup is valuable for not only the neutron EDM search but also the quantum mechanics itself.

In practical experimental setup, since ${E}_{y}^{0}\ne 0$, ${n}_{{p}_{x0,y0}}\ne 0$, and ≠ 0, $\mathcal{O}\left(1/\chi \right)$ term survives in W that induces χ-independent contributions in equation (30). This means that the neutron magnetic moment, which should be χ independent, behaves as a background effect against the neutron EDM signal in the weak measurement.

Using ${\hat{\rho }}_{\text{fin}}^{P}$ in equation (30), one can consider ${\mathrm{Tr}}_{P}\left[{\hat{\rho }}_{\text{fin}}^{P}\right]$ and ${\mathrm{Tr}}_{P}\left[\hat{x}{\hat{\rho }}_{\text{fin}}^{P}\right]$ as follows:

$$\begin{align}\hfill {\mathrm{Tr}}_{P}\left[{\hat{\rho }}_{\text{fin}}^{P}\right]=& \mathrm{Tr}\left[{\hat{\rho }}_{\text{fin}}^{\text{total}}\right]\hfill \\ \hfill =& {\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}^{{\dagger}}\left(T\right)\right]\left(1+\chi {g}_{\mu }\alpha \frac{{p}_{x0}}{{m}_{\text{n}}}{T}^{2}\text{Re}W\right)\hfill \\ \hfill & \quad +\mathcal{O}\left({\left({g}_{\mu }\alpha T\right)}^{2}\right),\hfill \end{align} \tag{ 37 }$$

$$\begin{align}\hfill {\mathrm{Tr}}_{P}\left[\hat{x}{\hat{\rho }}_{\text{fin}}^{P}\right]& =\mathrm{Tr}\left[\hat{x}{\hat{\rho }}_{\text{fin}}^{\text{total}}\right]\hfill \\ \hfill & ={\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}^{{\dagger}}\left(T\right)\right]\left({\mathrm{Tr}}_{P}\left[\hat{x}{\text{e}}^{-\text{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T}{\hat{\rho }}_{\text{ini}}^{P}{\text{e}}^{\text{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T}\right]\right.\hfill \\ \hfill & \quad +\chi {g}_{\mu }\alpha T\left\{W{\mathrm{Tr}}_{P} \left[\hat{x}\enspace {\text{e}}^{-\text{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T}{\hat{\rho }}_{\text{ini}}^{P}\left( \hat{x} + \frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x} \right){\text{e}}^{\text{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T}\right]\right.\hfill \\ \hfill & \quad \left.\left.+{W}^{{\ast}}{\mathrm{Tr}}_{P}\left[\hat{x}\enspace {\text{e}}^{-\text{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T}\left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right){\hat{\rho }}_{\text{ini}}^{P}{\text{e}}^{\text{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T}\right]\right\}\right)\hfill \\ \hfill & \quad +\mathcal{O}\left({\left({g}_{\mu }\alpha T\right)}^{2}\right)\hfill \\ \hfill & ={\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}{\hat{U}}_{0}\left(T\right){\hat{\rho }}_{\text{ini}}^{S}{\hat{U}}_{0}^{{\dagger}}\left(T\right)\right]\hfill \\ \hfill & \quad {\times}\left(\frac{{p}_{x0}}{{m}_{\text{n}}}T+2\chi {g}_{\mu }\alpha T\left({d}^{2}+\frac{1+4{d}^{2}{p}_{x0}^{2}}{4{d}^{2}}\frac{{T}^{2}}{2{m}_{\text{n}}^{2}}\right)\text{Re}W\right.\hfill \\ \hfill & \quad \left.-\chi {g}_{\mu }\alpha \frac{{T}^{2}}{2{m}_{\text{n}}}\text{Im}W\right)+\mathcal{O}\left({\left({g}_{\mu }\alpha T\right)}^{2}\right).\hfill \end{align} \tag{ 38 }$$

Here, we used the following relations (the assumption 2 in equation (11)):

$$\begin{equation}\begin{gathered}{c}{\mathrm{Tr}}_{P}\left[{\hat{\rho }}_{\text{ini}}^{P}\right]=1\enspace ,\quad {\mathrm{Tr}}_{P}\left[\hat{x}{\hat{\rho }}_{\text{ini}}^{P}\right]=0\enspace ,\quad {\mathrm{Tr}}_{P}\left[{\hat{x}}^{2}{\hat{\rho }}_{\text{ini}}^{P}\right]={d}^{2},\\ {\mathrm{Tr}}_{P}\left[{\hat{p}}_{x}{\hat{\rho }}_{\text{ini}}^{P}\right]={p}_{x0}\enspace ,\quad {\mathrm{Tr}}_{P}\left[{\hat{p}}_{x}^{2}{\hat{\rho }}_{\text{ini}}^{P}\right]=\frac{1+4\enspace {d}^{2}{p}_{x0}^{2}}{4{d}^{2}}\enspace ,\\ {\mathrm{Tr}}_{P}\left[\hat{x}{\hat{p}}_{x}{\hat{\rho }}_{\text{ini}}^{P}\right]=\frac{i}{2},\end{gathered}\end{equation} \tag{ 39 }$$

and

$$\begin{equation}\left[\hat{x},\enspace {\mathrm{e}}^{-\mathrm{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T}\right]=\frac{T}{{m}_{\text{n}}}{\hat{p}}_{x}\enspace {\mathrm{e}}^{-\mathrm{i}\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}T},\end{equation} \tag{ 40 }$$

from $\left[\hat{x},{\hat{p}}_{x}^{2}\right]=2\mathrm{i}{\hat{p}}_{x}$. Note that the neutron passing probability at the post-selection is expressed by $\enspace \mathrm{Tr}\left[{\hat{\rho }}_{\text{fin}}^{\text{total}}\right]$. Using equation (37), we obtain

$$\begin{equation}\enspace \mathrm{Tr}\left[{\hat{\rho }}_{\text{fin}}^{\text{total}}\right]\approx 2{\epsilon}+\frac{1}{4}{\delta }^{2}\ll 1.\end{equation} \tag{ 41 }$$

This corresponds to reduction of the neutron beam intensity after the post-selection.

Eventually, we obtain an expectation value of the position shift of the neutron after the post-selection as (see figure 1),

$$\begin{align}\hfill {{\Delta}}_{x}^{W}& \equiv \frac{{\mathrm{Tr}}_{P}\left[\hat{x}{\hat{\rho }}_{\text{fin}}^{P}\right]}{{\mathrm{Tr}}_{P}\left[{\hat{\rho }}_{\text{fin}}^{P}\right]}\hfill \\ \hfill & =\frac{{p}_{x0}}{{m}_{\text{n}}}T+2\chi {g}_{\mu }\alpha T\left[{d}^{2}+\frac{1}{2{d}^{2}}{\left(\frac{T}{2{m}_{\text{n}}}\right)}^{2}\right]\text{Re}W\hfill \\ \hfill & \quad -\enspace \chi {g}_{\mu }\alpha \frac{{T}^{2}}{2{m}_{\text{n}}}\text{Im}W+\mathcal{O}\left({\left({g}_{\mu }\alpha T\right)}^{2}\right).\hfill \end{align} \tag{ 42 }$$

Note that the ${\left(T/2{m}_{\text{n}}\right)}^{2}/2{d}^{2}$ term in the third term is numerically negligible. In the limit of ${n}_{{p}_{x0,y0}}\to 0$, we obtain the following analytical formula of the expected position shift:

$$\begin{align}\hfill {{\Delta}}_{x}^{W}& ={{\Delta}}_{x}^{W}\left(\mathrm{E}\mathrm{D}\mathrm{M}\right)+{{\Delta}}_{x}^{W}\left(\mathrm{B}\mathrm{G}\right)\hfill \\ \hfill & \quad +\enspace \mathcal{O}\left({{\epsilon}}^{2},\enspace {\delta }^{2},\enspace {\left(\frac{{E}_{y}^{0}}{{E}_{x}^{0}}\right)}^{2},\enspace {\left({g}_{\mu }\alpha T\right)}^{2}\right),\hfill \end{align} \tag{ 43 }$$

with

$$\begin{align}\hfill {{\Delta}}_{x}^{W}\left(\mathrm{E}\mathrm{D}\mathrm{M}\right)=& -\chi {g}_{\mu }\alpha T{d}^{2}\delta \frac{1-3{\epsilon}}{2{\epsilon}}\enspace \mathrm{cos}\left(2{g}_{\mu }{E}_{x}^{0}T\right)\hfill \\ \hfill & +\enspace \chi \alpha {d}^{2}\frac{1-{\epsilon}}{{\epsilon}}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}\enspace \mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}T\right)\hfill \\ \hfill & {\times}\enspace \left[2{g}_{\mu }{E}_{x}^{0}T\enspace \mathrm{cos}\left({g}_{\mu }{E}_{x}^{0}T\right)-\mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}T\right)\right],\hfill \end{align} \tag{ 44 }$$

$$\begin{align}\hfill {{\Delta}}_{x}^{W}\left(\mathrm{B}\mathrm{G}\right)& =\alpha \frac{T}{{m}_{\text{n}}}\delta \frac{1-{\epsilon}}{8{\epsilon}}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}\enspace {\mathrm{sin}}^{2}\left({g}_{\mu }{E}_{x}^{0}T\right)\hfill \\ \hfill & \quad -\alpha {d}^{2}\delta \frac{1-3{\epsilon}}{4{\epsilon}}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}\left[2{g}_{\mu }{E}_{x}^{0}T\enspace \mathrm{cos}\left(2{g}_{\mu }{E}_{x}^{0}T\right)\right.\hfill \\ \hfill & \quad \left.-\mathrm{sin}\left(2{g}_{\mu }{E}_{x}^{0}T\right)\right].\hfill \end{align} \tag{ 45 }$$

Here, the ${{\Delta}}_{x}^{W}\left(\mathrm{E}\mathrm{D}\mathrm{M}\right)$ corresponds to the EDM signal, while the ${{\Delta}}_{x}^{W}\left(\mathrm{B}\mathrm{G}\right)$ is the shift by the background effect which stems from the neutron magnetic moment (the relativistic E × v effect). The relativistic effects ${{\Delta}}_{x}^{W}\left(\mathrm{B}\mathrm{G}\right)$ mimic the neutron EDM signal ${{\Delta}}_{x}^{W}\left(\mathrm{E}\mathrm{D}\mathrm{M}\right)$.

Surprisingly, we find that such the relativistic effect is dropped in small T region when ${E}_{y}^{0}\ll {E}_{x}^{0}$ and/or δ ≈ 0. For instance, when one takes δ = 0 in the post-selection, the expected position shift is

$$\begin{align}\hfill {{\Delta}}_{x}^{W}\left(\mathrm{E}DM\right){\vert }_{\delta \to 0}& =\chi \alpha {d}^{2}\frac{1-{\epsilon}}{{\epsilon}}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}\enspace \mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}T\right)\hfill \\ \hfill & \quad {\times}\enspace \left[2{E}_{x}^{0}{g}_{\mu }T\enspace \mathrm{cos}\left({g}_{\mu }{E}_{x}^{0}T\right)\right.\hfill \\ \hfill & \left.\quad -\mathrm{sin}\left({g}_{\mu }{E}_{x}^{0}T\right)\right],\hfill \\ \hfill {{\Delta}}_{x}^{W}\left(\mathrm{B}G\right){\vert }_{\delta \to 0}& =0.\hfill \end{align} \tag{ 46 }$$

In this limit which one can realize by setting the first and second polarizers to be turned the opposite directions, the background shift from the relativistic E × v effect is dropped. Note that for large T region, the expansion of ${\hat{\rho }}_{\text{fin}}^{P}$ with respect to g_μ αT does not work, and suppression of ${{\Delta}}_{x}^{W}\left(\mathrm{B}G\right)$ no longer occurs.

We observe that a dimensionless combination ${g}_{\mu }{E}_{x}^{0}T$ is given by

$$\begin{align}\hfill {g}_{\mu }{E}_{x}^{0}T\simeq & -1.0{\times}1{0}^{-1}\left(\frac{{v}_{\text{n}}}{1{0}^{3}\enspace \mathrm{m}\cdot \enspace {\mathrm{sec}}^{-1}}\right)\left(\frac{{E}_{x}^{0}}{1{0}^{7}\enspace \mathrm{V}\cdot {\mathrm{m}}^{-1}}\right)\hfill \\ \hfill & {\times}\enspace \left(\frac{T}{1{0}^{-2}\enspace \mathrm{sec}}\right),\hfill \end{align} \tag{ 47 }$$

where we define the neutron velocity v_n as p₀ = m_n v_n. For $\vert {g}_{\mu }{E}_{x}^{0}T\vert \ll 1$ region, the expected position shift in equations (44) and (45) is given by

$$\begin{align}\hfill {{\Delta}}_{x}^{W}\left(\mathrm{E}\mathrm{D}\mathrm{M}\right)\simeq & -\chi {g}_{\mu }\alpha T{d}^{2}\delta \frac{1-3{\epsilon}}{2{\epsilon}}\left[1-2{\left({g}_{\mu }{E}_{x}^{0}T\right)}^{2}\right]\hfill \\ \hfill & +\enspace \chi \alpha {d}^{2}\frac{1-{\epsilon}}{{\epsilon}}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}{\left({g}_{\mu }{E}_{x}^{0}T\right)}^{2},\hfill \end{align} \tag{ 48 }$$

$$\begin{align}\hfill {{\Delta}}_{x}^{W}\left(\mathrm{B}\mathrm{G}\right)& \simeq \alpha \frac{T}{{m}_{\text{n}}}\delta \frac{1-{\epsilon}}{8{\epsilon}}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}{\left({g}_{\mu }{E}_{x}^{0}T\right)}^{2}\hfill \\ \hfill & \quad +\enspace 2\alpha {d}^{2}\delta \frac{1-3{\epsilon}}{3{\epsilon}}\frac{{E}_{y}^{0}}{{\left({E}_{x}^{0}\right)}^{2}}{\left({g}_{\mu }{E}_{x}^{0}T\right)}^{3}.\hfill \end{align} \tag{ 49 }$$

Using $\vert {E}_{y}^{0}/{E}_{x}^{0}\vert \ll 1$ and ≪ 1, eventually we obtain an approximation formula,

$$\begin{equation}{{\Delta}}_{x}^{W}\approx {{\Delta}}_{x}^{W}\left(\mathrm{E}\mathrm{D}\mathrm{M}\right)\approx -\frac{\chi {g}_{\mu }\alpha T{d}^{2}\delta }{2{\epsilon}}=-{d}_{\text{n}}\frac{\alpha T{d}^{2}\delta }{2{\epsilon}}.\end{equation} \tag{ 50 }$$

As we will show in the next section, choosing suitable input parameters such like , δ, T, and ${E}_{y}^{0}/{E}_{x}^{0}$, the weak value ReW can be significantly amplified, and it is just the weak value amplification.

Although we analytically obtain the expectation value of the deviation of the neutron position from x = 0 as ${{\Delta}}_{x}^{W}$ up to corrections of $\mathcal{O}\left({\left({g}_{\mu }\alpha T\right)}^{2}\right)$, we will also give the full-order result in appendix B. In the leading-order analysis in equation (42), the EDM signal induced by χ can be enhanced by large value of αTd². We find, however, that the EDM signal is not amplified by the large value of αTd² in the full-order analysis, because an additional damping factor appears as we will discuss in the next section.

It is important to distinguish the neutron EDM signal from the background shift. Since the background effect depends on many parameters and is complicated in our setup, we evaluate it numerically in the next section.

Before closing this section, let us comment on a special setup in which $\hat{\mathbf{E}}=0$ with pre- and post-selections. In such a case, ${{\Delta}}_{x}^{W}=\left({p}_{x0}/{m}_{\text{n}}\right)T$ is predicted. Therefore, the first term equation (42) can be subtracted by using data of a setup where the external electric field is turned off.

In this section, we show numerical results with varying many input parameters. Here and hereafter, we assume the following neutron beam velocity:

$$\begin{equation}{v}_{\text{n}}=1{0}^{3}\enspace \mathrm{m}\enspace {\mathrm{sec}}^{-1}\enspace ,\quad {n}_{{p}_{x0}}={n}_{{p}_{y0}}=0\enspace ,\quad {n}_{{p}_{z0}}=1,\end{equation} \tag{ 51 }$$

and the neutron beam size

$$\begin{equation}d=0.1\enspace \mathrm{m},\end{equation} \tag{ 52 }$$

where v_n and d are reasonable values in the J-PARC neutron beam experiment [70, 71], while ideal values of ${n}_{{p}_{x0,y0}}$ are taken.

First, we show the weak value ReW in equation (31) in figure 2. In both panel, the black dotted lines represent ReW = ±1, which corresponds to the eigenvalues of ${\hat{\sigma }}_{x}$. Hence, the weak value gives amplification of the signals for the regions with |ReW| > 1. In the left panel, we investigate δ and dependence, where T = 10⁻³ sec, ${E}_{x}^{0}=1{0}^{7}$ V m⁻¹, ${E}_{y}^{0}=10\enspace$ V m⁻¹, and d_n = 10⁻²⁶ e cm are taken.

Zoom In Zoom Out Reset image size

Here, we classify the weak measurement in the setup. Taking δ = π (α = 0 in reference [52]) corresponds to an ordinary indirect measurement, where the final beam shift is given by the eigenvalue of the spin operator of the EDM direction $\langle \psi \vert {\hat{\sigma }}_{x}\vert \psi \rangle$, in addition to the classical motion (the first term in equation (42)). In this setup, |ψ⟩ is set as |+_y ⟩ omitting the impurity , so that a zero eigenvalue is obtained as $\langle {+}_{y}\vert {\hat{\sigma }}_{x}\vert {+}_{y}\rangle =0$, which implies that the beam shift occurs only by the classical motion. As you can see in the left panel of figure 2, the weak value W is significantly suppressed around δ = π. Also, one can consider a different setup: δ = π with |ψ⟩ = |+_x ⟩. In this case, the beam shift is given by $\langle {+}_{x}\vert {\hat{\sigma }}_{x}\vert {+}_{x}\rangle =1$ term in addition to the classical motion. The black dotted line in figure 2 shows this latter setup.

It is shown that the weak value amplification, |ReW| > 1, occurs when ≲ 10⁻², and the amplification is maximized for small δ regions. As you can see, two orders of magnitude amplification is possible by small and δ. In the right panel, ${E}_{y}^{0}$ dependence of the weak value is investigated, where δ = 0.01π is fixed. It is found that ${E}_{y}^{0}$ dependence is negligible. Note that we also observed that the weak value is insensitive to T, ${E}_{x}^{0}$, and d_n for $\vert {g}_{\mu }{E}_{x}^{0}T\vert \ll 1$ region (see equation (47)). Above results show that the weak value amplification can be controlled by only the polarization angle δ and the selection impurity in the pre- and post-selections.

Here, we comment about a weak-measurement approximation, which is evaluation up to the first order of χ. For d_n = 10⁻²⁶ e cm, the parameter χ is 1.49 × 10⁻⁷, and the real part of the weak value Re W is smaller than 10³ according to the left panel of figure 2. Consequently, our evaluation based on the weak-measurement approximation is valid.

Next, we compare the leading-order approximation with respect to g_μ αT, which is shown in the previous section, with the full-order analysis. Since equations for the full-order analysis are lengthy, we put them on appendix B. From the second term of equation (42), the expected position shift of the center-of-mass of the neutron bunch is nearly proportional to αTd² within the leading-order approximation. Thus, it is expected that one can amplify the EDM signal by adopting a large value of αTd². This fact is, however, incorrect in the full-order analysis. As shown in appendix B, the full-order results include a damping factor $\text{exp}\left\{-{\left({g}_{\mu }\alpha T\right)}^{2}\left[{d}^{2}+{\left(T/2{m}_{\text{n}}\right)}^{2}/4{d}^{2}\right]/2\right\}$ with respect to αTd² appeared in equations (B17) and (B18). Since this damping factor becomes significant for a region of

$$\begin{equation}{g}_{\mu }\alpha Td{ >}1,\end{equation} \tag{ 53 }$$

the EDM signal cannot be amplified by the large value of αTd². This factor comes from a Gaussian integral,

$$\begin{equation}\int \mathrm{d}x\frac{1}{\sqrt{2\pi {d}^{2}}}{\text{e}}^{\text{i}\left({g}_{\mu }\alpha T\right)x}{\mathrm{e}}^{-\frac{{x}^{2}}{2\enspace {d}^{2}}}={\mathrm{e}}^{-\frac{1}{2}{\left({g}_{\mu }\alpha T\right)}^{2}{d}^{2}},\end{equation} \tag{ 54 }$$

and makes the transition probability in equation (37) finite even when δ → 0 and the ideal experimental setup limit are taken: ${E}_{y}^{0}/{E}_{x}^{0}\to 0$, ${n}_{{p}_{x0,y0}}\to 0$, and → 0.

In figure 3, we show the expected position shift of the center-of-mass of the neutron bunch ${{\Delta}}_{x}^{W}$ as a function of T. In both panels, the solid lines stand for the EDM signal parts ${{\Delta}}_{x}^{W}\left(\mathrm{E}\mathrm{D}\mathrm{M}\right)$, which are proportional to χ, with d_n = 10⁻²⁵ e cm. On the other hand, the dashed lines are for the background effects ${{\Delta}}_{x}^{W}\left(\mathrm{B}\mathrm{G}\right)$, which are χ independent. The blue and red lines correspond to the leading-order calculations and the full-order ones, respectively. We take = 0, δ = 10⁻³, ${E}_{y}^{0}=10$ V m⁻¹, α = 10⁸ V m⁻², and ${E}_{x}^{0}=1{0}^{6}$ (10⁷) V m⁻¹ for the left (right) panel.

Zoom In Zoom Out Reset image size

It is shown that, the leading- and the full-order calculations are well consistent with each other in small T regions. On the other hand, ${{\Delta}}_{x}^{W}$ is significantly suppressed for large T regions. This figures also show that the background effects in small T region are smaller than the EDM signal contributions by several orders of magnitude, which has been shown analytically in the previous section. In order to suppress the background effects, we adopt T = 0.01 sec in following estimations. We also find that the EDM signal contribution is insensitive to ${E}_{x}^{0}$, while the background effect is sensitive. Note that the background effect is also scaled by ${E}_{y}^{0}$ (see equation (49)).

Finally, we show a potential sensitivity of the weak measurement that can probe the neutron EDM signal. In this setup, the neutron EDM can be probed by precise measurement of ${{\Delta}}_{x}^{W}$. In current technology, it is possible to measure the neutron position by several methods with spatial resolutions of 100 nm [72], 1 μm [73], 2 μm [74], 5 μm [75, 76], 22 μm, [77], and 50 μm [78]. These spatial resolutions determine the potential sensitivity to the neutron EDM signal. The detector schemes of references [75–78] are examined for the neutron beam, and a thermal neutron is examined in reference [74]. On the other hand, the ones of references [72, 73] are examined for the UCN, but they can also be utilized for cold neutron (neutron beam) with small detection efficiency [79]. Recently, the detector scheme of reference [72] has been improved [79], and the detection efficiency becomes $\mathcal{O}\left(1\%\right)$ with spatial resolution of 1–2 μm for the cold neutron. Although the emitted neutron beam size is significantly larger than the spatial resolution of the detector, it would not raise a matter. Rather, the spatial resolution should be compared with the statistical uncertainty of the beam size, which will be discussed in the end of this section.

By requiring a condition $\vert {{\Delta}}_{x}^{W}\vert { >}1\enspace \mu \mathrm{m}$ as a reference value, we show the sensitivity to the neutron EDM in figure 4, where ${{\Delta}}_{x}^{W}$ includes both the EDM signal and the background shift, and the full-order formalism is used. Here, notice that the information necessary for this analysis is only a time-averaged shift of the center-of-mass of the neutrons passing the post-selection. The sensitivity is shown as a contour on the –δ plane, here we take T = 0.01 sec, ${E}_{x}^{0}=1{0}^{7}$ V m⁻¹, ${E}_{y}^{0}=10$ V m⁻¹, and α = 10⁸ V m⁻². Based on parameters in references [45, 80] and discussions with an experimentalist at the J-PARC [79], these parameters are chosen. In the figure 4, the red (dashed) line corresponds to the current (improved) neutron EDM bound in equation (1) (equation (2)), and the larger d_n region is excluded. Moreover, we find the background effect is negligible on this plane.

Zoom In Zoom Out Reset image size

We show that the impurity effect changes the sensitivity drastically, and find that the neutron EDM signal can be probed for a very small impurity region, < 10⁻⁵. It is two orders of magnitude smaller than the current technology, e.g. = (6 ± 1_stat. ± 3_sys.) × 10⁻⁴ [81], which is shown as the vertical blue dotted line in figure 4: the setup with = 6 × 10⁻⁴ and δ = 0.1 = 5.7° could probe a region d_n > 3 × 10⁻²⁵ e cm.

We comment on contributions from nonzero ${n}_{{p}_{x0}}$ and ${n}_{{p}_{y0}}$ values. We find that if ${n}_{{p}_{x0,y0}}$ are smaller than 10⁻⁵, these effects do not appear in above numerical evaluations. We also find that the effect of ${n}_{{p}_{y0}}$ is more significant than ${n}_{{p}_{x0}}$: $\mathcal{O}\left(1\right)$ contributions are produced for ${n}_{{p}_{y0}}\sim 1{0}^{-4}$ region.

We also comment on a statistical condition for measuring non-zero ${{\Delta}}_{x}^{W}$, where we compare the spatial resolution of the detector with the statistical uncertainty of the neutron beam. In such an experiment, one has to reject a null hypothesis of the neutron beam following the Gaussian distribution with the average position 0 and the variance d². If one measures the time-averaged shift of the center-of-mass of the neutrons by $\mathcal{N}$ neutrons, the statistical condition for measuring non-zero ${{\Delta}}_{x}^{W}$ at nσ level is expressed as ${{\Delta}}_{x}^{W}{ >}n\cdot d/\sqrt{\mathcal{N}}$. If one considers a case that a resolution of 1 μm for ${{\Delta}}_{x}^{W}$ and d = 0.1 m, the condition is $\mathcal{N}{ >}1{0}^{10}{n}^{2}$. In this setup, the number of neutrons $\mathcal{N}$ is

$$\begin{equation}\mathcal{N}\simeq {\mathcal{N}}_{0}\cdot 2{\epsilon}\cdot {\varepsilon }_{\text{eff}},\end{equation} \tag{ 55 }$$

where ${\mathcal{N}}_{0}$ represents the number of emitted neutrons, 2 is the neutron passing probability at the post-selection in equation (41), and ɛ_eff is the detection efficiency at the detector. Then, the statistical condition is

$$\begin{equation}{\mathcal{N}}_{0}{ >}\frac{1{0}^{10}{n}^{2}}{2{\epsilon}{\varepsilon }_{\text{eff}}}.\end{equation} \tag{ 56 }$$

Since the $\mathcal{O}\left(1{0}^{9}\right)$ neutrons can be generated per second in the experiment [71], a required time is t > 5n²/(ɛ_eff) sec. Even if the detection efficiency for the neutron beam is $\mathcal{O}\left(1\%\right)$ [73, 79], the statistical condition for n = 2 is satisfied by $\mathcal{O}\left(2000/{\epsilon}\right)$ seconds neutron beam.

In this paper, we proposed a novel approach in a search for the neutron EDM by applying the weak measurement, which is independent from the Ramsey method. Although the relativistic E × v effect provides a severe systematic uncertainty in the neutron EDM experiment in which the neutron beam is used, we find such a contribution is numerically irrelevant in the weak measurement. This is quite unexpected result, and we believe that this fact would provide a new virtue of the weak measurement.

To investigate a potential sensitivity to the neutron EDM search, we included the effect from the selection impurities in the pre- and post-selections. Our study showed that the size of the impurity crucially determines the sensitivity.

We found that the weak measurement can reach up to d_n > 3 × 10⁻²⁵ e cm within the current technology. This is one order of magnitude less sensitive that the current neutron EDM bound, where the UCNs based on the Ramsey method are used.

In addition, our approach could provide a new possibility to measure the weak value of the neutron spin polarization from the background effect. This fact makes our study fascinating in the point of view of the quantum mechanics.

The detailed study about the Fisher information based on reference [82] would be one of future directions of this study, where one can explore whether the weak-value amplification in the neutron EDM measurement outperforms the conventional Ramsey method one. Also, the perturbative effects from the Gaussian beam profile [83] could give additional systematic error in the weak measurement [84].

Although the small impurity, < 10⁻⁵, for probing the neutron EDM is difficult at the present time, we hope several improvements on the experimental technology, e.g. the sensitivity can be amplified by α and the resolution of ${{\Delta}}_{x}^{W}$ measurement, and anticipate that this kind of experiment will be performed in future.

We are grateful to Marvin Gerlach for collaboration in the early stage of this project. We would like to thank Joseph Avron, Yuji Hasegawa, Masataka Iinuma, Oded Kenneth, and Izumi Tsutsui for worthwhile discussions of the weak measurements. We also thank Kenji Mishima for helpful discussions about current neutron beam experiments. We greatly appreciate many valuable conversations with our colleagues, Gauthier Durieux, Masahiro Ibe, Go Mishima, Yael Shadmi, Yotam Soreq, and Yaniv Weiss. D U would like to thank all the members of Institute for Theoretical Particle Physics (TTP) at Karlsruhe Institute of Technology, especially Matthias Steinhauser, for kind hospitality during the stay. D U also appreciates being invited to conferences of the weak value and weak measurement at KEK. The work of T.K. is supported by the Israel Science Foundation (Grant No. 751/19) and by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 19K14706.

In this appendix, we consider a general setup of the external electric field with spatial gradients. Let us consider an external electric field with gradients $\partial \hat{\mathbf{E}}/\partial x=\left({\alpha }_{x},\enspace {\alpha }_{y},\enspace 0\right)$:

$$\begin{equation}\hat{\mathbf{E}}\left(x\right)=\left({E}_{x}^{0}+{\alpha }_{x}x,\enspace {E}_{y}^{0}+{\alpha }_{y}x,\enspace 0\right),\end{equation} \tag{ A1 }$$

where ${E}_{x,y}^{0}$ and α_x,y are constants. Note that the setup is insensitive to E_z component and its relativistic effect.

When one considers a rotation of the coordinate that all spatial dependence go to single electric filed component, the following $\hat{\mathbf{E}}$ with a coordinate x' are obtained:

$$\begin{align}\hfill {\hat{\mathbf{E}}}^{\prime }{\left({x}^{\prime },{y}^{\prime }\right)}^{T}& =R{\hat{\mathbf{E}}}^{T}=\frac{1}{\sqrt{{\alpha }_{x}^{2}+{\alpha }_{y}^{2}}}\hfill \\ \hfill & \quad {\times}\enspace \left(\begin{matrix}\hfill {\alpha }_{x}{E}_{x}^{0}+{\alpha }_{y}{E}_{y}^{0}+\left({\alpha }_{x}^{2}+{\alpha }_{y}^{2}\right)x\hfill \\ \hfill -{\alpha }_{y}{E}_{x}^{0}+{\alpha }_{x}{E}_{y}^{0}\hfill \\ \hfill 0\hfill \end{matrix}\right)\hfill \end{align} \tag{ A2 }$$

$$\begin{align}\hfill & =\left(\begin{matrix}\hfill \frac{1}{\sqrt{{\alpha }_{x}^{2}+{\alpha }_{y}^{2}}}\left({\alpha }_{x}{E}_{x}^{0}+{\alpha }_{y}{E}_{y}^{0}\right)+{\alpha }_{x}{x}^{\prime }-{\alpha }_{y}{y}^{\prime }\hfill \\ \hfill \frac{1}{\sqrt{{\alpha }_{x}^{2}+{\alpha }_{y}^{2}}}\left(-{\alpha }_{y}{E}_{x}^{0}+{\alpha }_{x}{E}_{y}^{0}\right)\hfill \\ \hfill 0\hfill \end{matrix}\right),\hfill \end{align} \tag{ A3 }$$

and

$$\begin{equation}{\mathbf{x}}^{\prime }\equiv \left(\begin{matrix}\hfill {x}^{\prime }\hfill \\ \hfill {y}^{\prime }\hfill \\ \hfill {z}^{\prime }\hfill \end{matrix}\right)=R\mathbf{x}=\frac{1}{\sqrt{{\alpha }_{x}^{2}+{\alpha }_{y}^{2}}}\left(\begin{matrix}\hfill {\alpha }_{x}x+{\alpha }_{y}y\hfill \\ \hfill -{\alpha }_{y}x+{\alpha }_{x}y\hfill \\ \hfill z\hfill \end{matrix}\right),\end{equation} \tag{ A4 }$$

where the rotation matrix R is

$$\begin{equation}R=\left(\begin{matrix}\hfill \mathrm{cos}\enspace \theta \hfill & \hfill \mathrm{sin}\theta \hfill & \hfill 0\hfill \\ \hfill -\mathrm{sin}\enspace \theta \hfill & \hfill \mathrm{cos}\enspace \theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right)\enspace ,\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\quad \mathrm{cos}\enspace \theta =\frac{{\alpha }_{x}}{\sqrt{{\alpha }_{x}^{2}+{\alpha }_{y}^{2}}}.\end{equation} \tag{ A5 }$$

The spatial dependence of y' in ${\hat{\mathbf{E}}}^{\prime }\left({x}^{\prime },{y}^{\prime }\right)$ would provide us with a spatial displacement of the neutron along y' axis. In order to obtain the experimental setup in equation (10), α_y ≪ α_x are thus required.

In this appendix, we give building blocks of the full-order calculation for ${{\Delta}}_{x}^{W}$ with respect to g_μ αT. The expected position shift of the center-of-mass of the neutron bunch is defined as (see equations (37), (38) and (42) for the leading-order analysis):

$$\begin{equation}{{\Delta}}_{x}^{W}=\frac{\mathrm{Tr}\left[\left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right){\hat{\rho }}_{\text{fin}}^{\text{total}}\right]}{\mathrm{Tr}\left[{\hat{\rho }}_{\text{fin}}^{\text{total}}\right]},\end{equation} \tag{ B1 }$$

with

$$\begin{align}\hfill \mathrm{Tr}\left[{\hat{\rho }}_{\text{fin}}^{\text{total}}\right]& =\langle {G}_{{p}_{x0}}\vert {\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}\enspace {\mathrm{e}}^{-\mathrm{i}T\hat{H}}{\hat{\rho }}_{\text{ini}}^{S}\enspace {\text{e}}^{\text{i}T\hat{H}}\right]\vert {G}_{{p}_{x0}}\rangle ,\hfill \end{align} \tag{ B2 }$$

$$\begin{align}\hfill \mathrm{Tr}\left[\left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right){\hat{\rho }}_{\text{fin}}^{\text{total}}\right]& =\langle {G}_{{p}_{x0}}\vert \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right){\mathrm{Tr}}_{S}\hfill \\ \hfill & \qquad {\times}\left[{\hat{\rho }}_{\text{fin}}^{S}\enspace {\mathrm{e}}^{-\mathrm{i}T\hat{H}}{\hat{\rho }}_{\text{ini}}^{S}\enspace {\text{e}}^{\text{i}T\hat{H}}\right]\vert {G}_{{p}_{x0}}\rangle .\hfill \end{align} \tag{ B3 }$$

In the full-order analysis, we discard only $\mathcal{O}\left({\chi }^{2}\right)$ contributions. Using equations (15)–(17) for the time evolution operator, ${\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}\enspace {\mathrm{e}}^{-\mathrm{i}T\hat{H}}{\hat{\rho }}_{\text{ini}}^{S}\enspace {\text{e}}^{\text{i}T\hat{H}}\right]$ is expanded as

$$\begin{align}\hfill {\mathrm{Tr}}_{S}\left[{\hat{\rho }}_{\text{fin}}^{S}\enspace {\mathrm{e}}^{-\mathrm{i}T\hat{H}}{\hat{\rho }}_{\text{ini}}^{S}\enspace {\mathrm{e}}^{\mathrm{i}T\hat{H}}\right]=& {\;\;\mathrm{e}}^{-\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\left\{-\frac{1}{2{\hat{E}}_{x}^{2}}\left[{\hat{E}}_{x}^{2}\left(-1+{\left(1-2{\epsilon}\right)}^{2}\enspace \mathrm{cos}\left(\delta \right)\right)\enspace {\mathrm{cos}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)-{\left({E}_{y}^{0}{n}_{{p}_{x0}}-{\hat{E}}_{x}{n}_{{p}_{y0}}\right)}^{2}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}gT\right)\right.\right.\hfill \\ \hfill & \left.\left.-\left({\hat{E}}_{x}^{2}+{\left({E}_{y}^{0}\right)}^{2}\right){n}_{{p}_{z0}}^{2}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)-{\left(1-2{\epsilon}\right)}^{2}{\left({E}_{y}^{0}{n}_{{p}_{x0}}-{\hat{E}}_{x}{n}_{{p}_{y0}}\right)}^{2}\enspace \mathrm{cos}\left(\delta \right)\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\right.\right.\hfill \\ \hfill & \left.\left.-{\left(1-2{\epsilon}\right)}^{2}\left(-{\hat{E}}_{x}+{E}_{y}^{0}\right)\left({\hat{E}}_{x}+{E}_{y}^{0}\right){n}_{{p}_{z0}}^{2}\enspace \mathrm{cos}\left(\delta \right)\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\right.\right.\hfill \\ \hfill & \left.\left.-2{\left(1-2{\epsilon}\right)}^{2}{\hat{E}}_{x}\left({E}_{y}^{0}{n}_{{p}_{x0}}-{\hat{E}}_{x}{n}_{{p}_{y0}}\right){n}_{{p}_{z0}}\enspace \mathrm{sin}\left(\delta \right)\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)+{\left(1-2{\epsilon}\right)}^{2}{\hat{E}}_{x}{E}_{y}^{0}{n}_{{p}_{z0}}\enspace \mathrm{sin}\left(\delta \right)\enspace \mathrm{sin}\left(2{\hat{E}}_{x}{g}_{\mu }T\right)\right]\right.\hfill \\ \hfill & \left.-\chi \frac{1}{2{\hat{E}}_{x}^{2}}\left[-2{\hat{E}}_{x}{E}_{y}^{0}{n}_{{p}_{z0}}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)-2{\left(1-2{\epsilon}\right)}^{2}{\hat{E}}_{x}{E}_{y}^{0}{n}_{{p}_{z0}}\enspace \mathrm{cos}\left(\delta \right)\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\right.\right.\hfill \\ \hfill & \left.\left.+{\left(1-2{\epsilon}\right)}^{2}{\hat{E}}_{x}^{2}\enspace \mathrm{sin}\left(\delta \right)\enspace \mathrm{sin}\left(2{\hat{E}}_{x}{g}_{\mu }T\right)\right]\right\}\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}+\mathcal{O}\left({\chi }^{2}\right),\hfill \end{align} \tag{ B4 }$$

where ${\hat{E}}_{x}$ is defined as ${\hat{E}}_{x}={E}_{x}^{0}+\alpha \left(\hat{x}+T{\hat{p}}_{x}/2{m}_{\text{n}}\right)$. To calculate equations (B2) and (B3), the following building blocks are needed:

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\enspace {\mathrm{cos}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =\frac{1}{4}\left[{f}_{1}\left(2{g}_{\mu }\right)+{f}_{1}\left(-2{g}_{\mu }\right)+2\right],\hfill \end{align} \tag{ B5 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =-\frac{1}{4}\left[{f}_{1}\left(2{g}_{\mu }\right)+{f}_{1}\left(-2{g}_{\mu }\right)-2\right],\hfill \end{align} \tag{ B6 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\frac{1}{{\hat{E}}_{x}^{2}}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle ={T}^{2}{\int }_{0}^{{g}_{\mu }}d{g}_{1}{\int }_{0}^{{g}_{1}}d{g}_{2}\left[{f}_{1}\left(2{g}_{2}\right)+{f}_{1}\left(-2{g}_{2}\right)\right],\hfill \end{align} \tag{ B7 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\frac{1}{{\hat{E}}_{x}}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =\frac{T}{2i}{\int }_{0}^{{g}_{\mu }}d{g}_{1}\left[{f}_{1}\left(2{g}_{1}\right)-{f}_{1}\left(-2{g}_{1}\right)\right],\hfill \end{align} \tag{ B8 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\enspace \mathrm{sin}\left(2{\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =\frac{1}{2i}\left[{f}_{1}\left(2{g}_{\mu }\right)-{f}_{1}\left(-2{g}_{\mu }\right)\right],\hfill \end{align} \tag{ B9 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\frac{1}{{\hat{E}}_{x}}\enspace \mathrm{sin}\left(2{\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =T{\int }_{0}^{{g}_{\mu }}d{g}_{1}\left[{f}_{1}\left(2{g}_{1}\right)+{f}_{1}\left(-2{g}_{1}\right)\right],\hfill \end{align} \tag{ B10 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\enspace {\mathrm{cos}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =\frac{1}{4}\left[{f}_{2}\left(2{g}_{\mu }\right)+{f}_{2}\left(-2{g}_{\mu }\right)+\frac{T}{2{m}_{\text{n}}}{p}_{x0}\right],\hfill \end{align} \tag{ B11 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =-\frac{1}{4}\left[{f}_{2}\left(2{g}_{\mu }\right)+{f}_{2}\left(-2{g}_{\mu }\right)-\frac{T}{2{m}_{\text{n}}}{p}_{x0}\right],\hfill \end{align} \tag{ B12 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\frac{1}{{\hat{E}}_{x}^{2}}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle ={T}^{2}{\int }_{0}^{{g}_{\mu }}d{g}_{1}{\int }_{0}^{{g}_{1}}d{g}_{2}\left[{f}_{2}\left(2{g}_{2}\right)+{f}_{2}\left(-2{g}_{2}\right)\right],\hfill \end{align} \tag{ B13 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\frac{1}{{\hat{E}}_{x}}\enspace {\mathrm{sin}}^{2}\left({\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =\frac{T}{2i}{\int }_{0}^{{g}_{\mu }}d{g}_{1}\left[{f}_{2}\left(2{g}_{1}\right)-{f}_{2}\left(-2{g}_{1}\right)\right],\hfill \end{align} \tag{ B14 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\enspace \mathrm{sin}\left(2{\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =\frac{1}{2i}\left[{f}_{2}\left(2{g}_{\mu }\right)-{f}_{2}\left(-2{g}_{\mu }\right)\right],\hfill \end{align} \tag{ B15 }$$

$$\begin{align}\hfill & \langle {G}_{{p}_{x0}}\vert \left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right)\enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\frac{1}{{\hat{E}}_{x}}\enspace \mathrm{sin}\left(2{\hat{E}}_{x}{g}_{\mu }T\right)\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle =T{\int }_{0}^{{g}_{\mu }}d{g}_{1}\left[{f}_{2}\left(2g1\right)+{f}_{2}\left(-2{g}_{1}\right)\right],\hfill \end{align} \tag{ B16 }$$

with

$$\begin{align}\hfill & {f}_{1}\left({g}_{\mu }\right)\equiv \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\enspace {\text{e}}^{\text{i}{\hat{E}}_{x}{g}_{\mu }T}\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle ={\text{e}}^{\text{i}\left({g}_{\mu }{E}_{x}^{0}T-{g}_{\mu }\alpha T\frac{T}{2{m}_{\text{n}}}{p}_{x0}\right)}\enspace {\mathrm{e}}^{-\frac{{\left({g}_{\mu }\alpha T\right)}^{2}}{2}\left[{d}^{2}+\frac{1}{4{d}^{2}}{\left(\frac{T}{2{m}_{\text{n}}}\right)}^{2}\right]},\hfill \end{align} \tag{ B17 }$$

$$\begin{align}\hfill & {f}_{2}\left({g}_{\mu }\right)\equiv \langle {G}_{{p}_{x0}}\vert \enspace {\text{e}}^{-\text{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\left(\hat{x}+\frac{T}{2{m}_{\text{n}}}{\hat{p}}_{x}\right){\text{e}}^{\text{i}{\hat{E}}_{x}{g}_{\mu }T}\enspace {\mathrm{e}}^{+\mathrm{i}T\frac{{\hat{p}}_{x}^{2}}{2{m}_{\text{n}}}}\vert {G}_{{p}_{x0}}\rangle \hfill \\ \hfill & \quad =\left\{-\frac{T}{2{m}_{\text{n}}}\left({p}_{x0}+{g}_{\mu }\alpha T\right)+\mathrm{i}{g}_{\mu }\alpha T\left[{d}^{2}+\frac{1}{4{d}^{2}}{\left(\frac{T}{2{m}_{\text{n}}}\right)}^{2}\right]\right\}{\text{e}}^{\text{i}\left({g}_{\mu }{E}_{x}^{0}T-{g}_{\mu }\alpha T\frac{T}{2{m}_{\text{n}}}{p}_{x0}\right)}\enspace {\mathrm{e}}^{-\frac{{\left({g}_{\mu }\alpha T\right)}^{2}}{2}\left[{d}^{2}+\frac{1}{4{d}^{2}}{\left(\frac{T}{2{m}_{\text{n}}}\right)}^{2}\right]}.\hfill \end{align} \tag{ B18 }$$

Combining equations (B2)–(B18), one can numerically calculate the expected position shift of the center-of-mass of the neutron bunch at the full order. See e.g. references [53, 85] for detailed discussions of the damping factors in equations (B17) and (B18).