The search for the neutron electric dipole moment

Figure 1: © Studio HübnerBraun

The neutron Electric Dipole Moment (nEDM), $d_n$, is an intrinsic property of the neutron, defined by the potential energy $W = − \vec d_n \cdot \vec E$ in an external electric field $\vec E$. This is the electric analogue of the magnetic moment. The measurement of the neutron EDM provides an important probe of fundamental discrete symmetries, as a nonzero nEDM would violate both parity (P) and time-reversal (T) symmetries. Since the first direct searches in the 1950s, experimental sensitivity has improved by six orders of magnitude. The current most precise measurement, using ultracold neutrons, is still compatible with zero: $d_n = (0.0 \pm 1.1) \times 10^{-26} \, e \, \text{cm}$.

Basic definitions, physical meaning, associated phenomenon

Classical concept of the electric dipole moment

The electric dipole moment (EDM) of a classical system made of a discrete or continuous assembly of electric charges quantifies the separation between positive and negative charges. This notion can be made precise by considering how such a system, described by a charge distribution $\rho(r)$, interacts with an external electrostatic potential $V(r)$. Assuming that the external potential is weak enough not to perturb the charge distribution (which remains rigid), the total electrostatic energy is given by: $$W = \int \rho(r) V(r) dr.$$ Expanding the potential in a Taylor series around the center of the system (taken as the origin), \begin{array}{lcl} V(r) & = & V(0) + r_i \partial_i V(0) + \frac{1}{2} r_i r_j \partial_i \partial_j V(0) + \cdots \\ & = & V(0) + \vec r \cdot \vec E + \frac{1}{2} \vec r \cdot \vec \partial \vec E \cdot \vec r + \cdots, \end{array} and substituting into Eq. (1) gives: $$W = q V(0) - \vec d \cdot \vec E + \cdots, $$ where $q = \int \rho(r) dr$ is the electric charge, and $$\vec d = \int \vec r \rho(r) dr $$ is the electric dipole moment. This classical expression, with units of charge times distance, defines how a charge distribution couples linearly to an external electric field.

Definition of the magnetic and electric dipole moments for a spin 1/2 particle

The structure of the interaction term $-\vec d \cdot \vec E$ in the electrostatic energy is not limited to classical systems. In fact, this form provides a more fundamental definition of the electric dipole moment, even when no internal charge distribution can be meaningfully defined—such as for elementary particles like the neutron or the electron. With this generalization, the EDM characterizes the linear response of a system to an external electric field, just as the magnetic dipole moment $\vec \mu$ characterizes the response to a magnetic field. In nonrelativistic quantum mechanics, this linear coupling is encoded in the Hamiltonian operator $\hat H = -\vec \mu \cdot \vec B - \vec d \cdot \vec E$. This expression constitutes the operational definition of $\vec \mu$ and $\vec d$, as vector operators acting on the internal degrees of freedom of the system.

For a spin 1/2 particle, the internal state is a two component spinor $| \psi \rangle = a | + \rangle + b | - \rangle$, where $| + \rangle$ and $| - \rangle$ are the eigenstates of spin projection along a chosen quantization axis. Equivalently, the internal state can be represented by the polarization vector $$ \vec p = \langle \psi | \vec \sigma | \psi \rangle. $$ where $\vec \sigma = (\sigma_x, \sigma_y, \sigma_z)$ are the $2\times2$ Pauli matrices. The polarization vector lies on the Bloch sphere $|\vec p | = 1$ and physically represents the spin orientation. It fully determines the quantum state up to a global phase.

By rotational symmetry and the Wigner–Eckart theorem, all vector operators acting on spin-1/2 states must be proportional to $\vec \sigma$. This implies that both the magnetic and electric dipole moments operators of the neutron can be written as $ \vec \mu_n = \mu_n \vec \sigma$ and $\vec d_n = d_n \vec \sigma$, so that the Hamiltonian for a neutron in external electric and magnetic fields takes the form: $$ \hat H = - \mu_n \vec \sigma \cdot \vec B - d_n \vec \sigma \cdot \vec E $$ This expression serves as the operational definition of the magnetic and electric dipole moments of the neutron. Both $\mu_n$ and $d_n$ are intrinsic properties of the neutron that characterize how its spin couples to external fields.

Spin dynamics: Bloch equations and Larmor precession

The spin dynamics is governed by the Schrödinger equation: $$ i \hbar \frac{d}{dt} | \psi \rangle = \hat H | \psi \rangle,$$ which describes the time evolution of the probability amplitudes $a(t)$ and $b(t)$. Equivalently, the evolution can be described in terms of the polarization vector by the Bloch equations: $$ \frac{d \vec p}{dt} = \vec p \times \left( \frac{2\mu_n}{\hbar} \vec B + \frac{2 d_n}{\hbar} \vec E \right). $$ For spin-1/2 systems, the Bloch equations provide a complete description of the quantum dynamics, fully equivalent to the Schrödinger equation.

Note: for higher-spin systems, one can still define a polarization vector as the normalized expectation value of the spin, it will also satisfy the Bloch equations, however, the polarization vector does not encode the full quantum state and therefore the Bloch equations are not equivalent to the Schrödinger equation.

Let us now consider the important case of a pure static magnetic field, with direction defined a the $z$ axis: $$\vec B = B \, \vec e_z , \text{and} \ \vec E = 0. $$ We define the $x$ axis as the direction of the initial transverse polarization: $\vec p (0) = p_x(0) \vec e_x + p_z(0) \vec e_z$. Under these conditions, the Bloch equations admit the analytic solution: $$ \vec p (t) = p_x(0) ( \cos \omega t \ \vec e_x + \sin \omega t \ \vec e_y) + p_z(0) \vec e_z, $$ with $$ \omega = \frac{2 \mu_n}{\hbar} B = \gamma_n B. $$ The solution describes a rotation of the polarization vector around the magnetic field axis which is called Larmor precession:

The longitudinal component of the polarization, $p_z$, is conserved. In particular, if the spin is initially aligned along the $z$-axis, the polarization remains stationnary, $\vec p(t) = \vec p(0)$. This reflects the fact that the stationary eigenstates $|+\rangle$ and $|-\rangle$ of the Hamiltonian $\hat{H}$ are indeed not evolving.
The transverse component of the polarization vector undergoes uniform circular motion in the $x$–$y$ plane at the frequency $f_n = \frac{\omega}{2 \pi}$ called the Larmor frequency. The Larmor frequency is directly proportional to the magnetic field. It is conventional to define the gyromagnetic ratio $\gamma_n = \frac{2 \mu_n}{\hbar}$, so that the Larmor frequency is given by $f_n = \frac{\gamma_n}{2 \pi} B$.

Note: the gyromagnetic ratio for a general spin is defined by the linear relation between the magnetic moment observable $\vec \mu$ and the spin observable $\vec S$ by $\vec \mu = \gamma \vec S$. In the case of a spin 1/2 we have $\vec S = \hbar/2 \vec \sigma$.

Energy splitting and Larmor frequency in B and E fields

In the presence of parallel static magnetic and electric fields, the solution of the Bloch equations yields: $$\hbar \omega = 2 \mu_n B + 2 d_n E. $$ It should be noted that the right hand side of this equation corresponds to the energy splitting between the two stationary states: $$\hat{H} | + \rangle = (\mu_n B + d_n E) | + \rangle, \quad \hat{H} | - \rangle = -(\mu_n B + d_n E) | - \rangle. $$ Thus, the angular precession frequency $\omega$ coincides with the transition frequency between the quantum states $| + \rangle$ and $| - \rangle$: $$\hbar \omega = \langle + | \hat{H} | + \rangle - \langle - | \hat{H} | - \rangle. $$

The Larmor frequency $\frac{\omega}{2\pi} = f_B + f_E$ receives both magnetic and electric contributions, whose typical magnitudes are summarized below.

The magnetic contribution is $f_B = \frac{\mu_n B}{\pi \hbar}$. The natural unit for nuclear magnetic moments is the nuclear magneton $\mu_N = \frac{e \hbar}{2 m_p}$. The magnetic moment of the neutron is of order unity in natural units, and it is known with high precision: $$\mu_n = -1.913 \, 042 \, 76(45) \, \mu_N.$$

The corresponding gyromagnetic ratio is $\gamma_n = (2 \pi) \times 29.164 \, 6935(69) \ \text{Hz}/\text{µT}.$ For example, in the Earth’s magnetic field ($B \approx 50 \, \text{µT}$), the neutron spin precesses at $f_B = \gamma_n B / (2\pi) \approx 1 500$ Hz.

The electric contribution is $f_E = \frac{d_n E}{\pi \hbar}$. The most recent measurement gives

$d_n = (0.0 \pm 1.1)\times 10^{-26} \, e \, \text{cm}.$ Although traditionally expressed in $e \, \text{cm}$, this is not a natural unit for the neutron. A natural scale is $\mu_N / c \approx 0.1 \, \text{fm}$, yielding: $$d_n = (0 \pm 1) \times 10^{-12} \, \mu_N / c.$$ This value is compatible with zero and extremely small even in natural units, suggesting the existence of a symmetry that strongly suppresses the EDM. This point will be discussed in the next section. In typical experiments, the maximum applied electric field is $E \approx 10 \, \text{kV/cm}$. The resulting electric contribution to the precession frequency is $$f_E = \frac{d_n E}{\pi \hbar} \approx (0 \pm 5) \times 10^{-8} \, \text{Hz},$$ corresponding to a full precession cycle in at least 100 days (at 2σ confidence level).

By comparison, an EDM of $2 \times 10^{-26} \, e \, \text{cm}$ in an E-field of $10 \, \text{kV/cm}$ would produce a change in precession frequency that corresponds to a magnetic field of only 3 fT. This underscores the need for extreme control of the magnetic field in the experiments searching for the nEDM. See section on magnetic control.

Quantum sensitivity limit of the measurement

A general estimate of the statistical sensitivity to the neutron EDM can be derived from basic quantum principles, following the approach of Khriplovich and Lamoreaux. Consider an idealized experiment in which polarized neutrons, initially aligned along the $x$ axis, are exposed to an electric field $E \, \vec e_z$, in the absence of any magnetic field, for a duration of $T$. The EDM induces a spin precession at the angular frequency $\omega= 2 d_n E / \hbar$. In the limit $\omega T \ll 1$, the final polarization vector becomes: $$\vec p(T) = \cos \omega T \, \vec e_x + \sin \omega T \, \vec e_y \approx \vec e_x + \omega T \, \vec e_y.$$ To extract the EDM-induced phase, one measures the spin component along $y$, which is the most sensitive direction to the small transverse rotation. This corresponds to a quantum measurement of the observable $\sigma_y$, whose eigenvalues are $\pm 1$. For a single neutron in the sate $| \psi \rangle$, the outcome of the measurement is either $-1$ or $+1$, with statistical mean and variance given by the following expectation values: $$\langle \sigma_y \rangle = \langle \psi | \sigma_y | \psi \rangle = p_y = \omega T ,$$ $$(\Delta \sigma_y)^2 = \langle \psi | \sigma_y^2 | \psi \rangle - (\langle \psi | \sigma_y | \psi \rangle)^2 = 1 - p_y^2 \approx 1. $$ The resulting uncertainty on the angular frequency from a single measurement is therefore $\delta \omega = 1/T$, in agreement with Heisenberg's time–energy uncertainty principle.

It is important to note, however, that a single measurement is not sufficient to verify the small-angle approximation $\omega T \ll 1$ which underlies the linear relation between $p_y = \langle \sigma_y \rangle$ and $\omega$. This discussion of uncertainty is only meaningful when $p_y$ is extracted statistically by repeating the measurement on a large ensemble $N$ of neutrons, providing $N$ independent measurements. In that case the average result will fluctuate around the expectation value $p_y$ and the statistical uncertainty is then given by: $$\delta \omega = \frac{1}{\sqrt{N} T}, \ \delta d_n = \frac{\hbar}{2 E T \sqrt{N}}.$$ This expression defines the quantum projection limit for the statistical sensitivity of a neutron EDM experiment.

Comments on the sensitivity limit:

In real experiments, the precession frequency is measured using a spin resonance technique. A magnetic field is applied, and the initial polarization is longitudinal, not transverse. We will see that the quantum projection limit can in fact be reached. See section on spin resonance
The precision of the measurement is controlled by the exposure time $T$. In the first experiment performed by Smith under the direction of Ramsey and Purcell, a beam of thermalized neutrons (speed about 3000 m/s) traversed an apparatus of lenght $L = 135 \, \text{cm}$, resulting in a passage time of $T \approx 0.5 \, \text{ms}$. More recent experiment use ultracold neutrons (UCNs) that can be stored for duration approaching the beta decay lifetime (880 s). In the most recent experiment neutrons here exposed to the electric field during $T = 180 \, \text{s}$. See section on UCNs

Theoretical implications

Discrete symmetries

As first noted by Ramsey and Purcell, the measurement of the neutron EDM provides a test of fundamental discrete symmetries.

Here we discuss why the EDM violates P and T from basic considerations with the nonrelativistic coupling.

Link to QFT

From the point of view of Quantum Field Theory, the EDM of a fermionic field $f$ corresponds to the imaginary part of teh so-called dipole operator coupling the fermion to the electromagnetic field: $$\mathcal{L} = \frac{1}{2} (\delta \mu + i d) \bar f_L \sigma_{\mu \nu} f_R F^{\mu \nu} + h.c. $$ Indeed, in the non-relativistic limit, one recovers the Hamiltonian $\hat H = -\mu \vec \sigma \cdot B - d \vec \sigma \cdot \vec E $.

Then we sketch the EFT ladder, with the messagge that the EDM receives virtual contributions from CP violating interactions involving all fields in the theory.

contributions in the SM, $\theta$ and weak phase.
strong CP problem
sensitivity to new physics

Spin resonance method

In section Spin Dynamics a homogeneous, static magnetic field along the $z$ axis $\vec B_0 = B_0 \, \vec e_z$ was introduced.

It became clear that an initial, transverse spin polarization $\vec p(0) - p_z(0) \vec e_z$ will precess in the $(x,y)$ plane at the angular frequency $\omega_0 = \frac{2 \mu_n}{\hbar} B_0 = \gamma_n B$.

(Remark: The term 'Larmor frequency' is here reserved for $f_0 = \omega_0 / 2\pi$, see above, while in other articles and books the same name is also used for $\omega_0$.)

In the following an angular frequency $\omega = \omega_0 + \Delta \omega$, with a small $\Delta \omega$ will also be used.

Rotating frame

The full quantum mechanical treatment of the spin dynamics in the static $B_0$ field and an additional time varying $B_1$ (usually oscillatory or rotating) field is straight forward and e.g. found in Ramsey's textbook 'Molecular Beams' (Ramsey NF, 1956). The same textbook also gives more intuitive and classical approaches. For the case of a weaker $B_1$ rotating perpendicularly to $B_0$, a transition to a rotating reference frame considerably simplifies the treatment and gives more intuitive insights. The use of the rotating coordinate systems for such problems was introduced by Rabi, Ramsey and Schwinger (Rabi II, Ramsey NF, Schwinger J, 1954) and will be used here in an even simpler variant.

With only $B_0 \vec e_z$ applied, a transformation from the initial coordinate system $R$ into a reference frame $R'$, itself rotating at angular frequency $\omega$ about the $z = z'$ axis, results in a stationary polarization $\vec p'$ for $\Delta \omega = 0$. In fact, the transformation to $R'$ reduces the magnetic field to $B_0' = B_0 - \omega / \gamma_n$ which disappears for the perfect matching condition $\Delta \omega = 0$.

Here, $R'$ is chosen such that at $t = 0$, $R$ and $R'$ and therefore all coordinate axes coincide.

An initial polarization $\vec p (0) = p\, \vec e_z = p\, \vec e_z'$ is considered and an additional magnetic field $\vec B_1(t) = B_1 (\sin \omega t \ \vec e_x + \cos \omega t \ \vec e_y)$ which rotates in $R$ and can be switched on and off. As described, $\omega = \omega_0 + \Delta \omega$ is used and particularly $\Delta \omega \ll \gamma_n B_1$.

Obviously, for $\Delta \omega = 0$, $\vec B_1' = B_1 \vec e_y'$ is a static magnetic field in the rotating frame $R'$. In that case, $\vec p'$ will precess in $R'$ around $\vec e_y'$ with angular frequency $\gamma_n B_1$.

Consider showing a plot for the rotating frame transformation

Rabi and Ramsey spectroscopy

When switching $\vec B_1$ on for a time $[ 0, 2\tau ]$ such that $2\gamma_n B_1 \tau = \pi$ one obtains $\vec p'(2\tau) = -p\, \vec e_z'$ and for just half that time $[ 0, \tau]$, $\vec p'(\tau) = p\, \vec e_x'$.

The first is referred to as applying a $\pi$ flip, the latter as applying a $\pi/2$ flip of the polarization vector. For finite small $ \Delta \omega \ll \gamma_n B_1$, the $\pi$ and $\pi/2$ flips will still work (however, less efficient as $\Delta \omega$ grows bigger). To make sure the wording is understood, the application of the $B_1$ field for a certain time to induce a $\pi$ or $\pi/2$ flip is also refereed to a $\pi$ or $\pi/2$ 'pulse'. It is the combination of field strength, frequency and duration that matters and not the effect on the polarization, i.e. one can of course apply the pulse without getting the respective flip, depending on the polarization state.

For the sake of simplicity, it is here first assumed that the duration $\tau$ is very (infinitely) short. As only the finite product $\tau B_1$ matters for the effect on the polarization, this corresponds to a very (infinitely) large $B_1$. Both is neither desirable nor practical in a real experiment, however, we will comment later on the implication of a finite $\tau$.

Historically, determining the resonance frequency needed for a $\pi$ flip in a well-known magnetic field was introduced by Rabi to determine nuclear magnetic moments, in particular also of systems more complex than the here considered spin 1/2 neutrons. Today, the method is generally referred to as 'Rabi spectroscopy' to measure the transition frequency from one state to another (here: spin/polarization 'up' (along $z$) to spin/polarization down). Of course, the initial and final states should be treated fully quantum mechanically and the transition in the $\pi$ flip must be associated with a suitable frequency-dependent, complex transition amplitude. The transition probability will be computed as the modulus squared of that amplitude and in its simplest form display an intensity pattern (as function of frequency) equivalent to that of single slit diffraction (as a function of spatial coordinates).

If one for a moment continues the analogy with interferometry, the resolution of a single-slit interferometer can be much improved by going to at least two slits. In two-slits interference, amplitudes from both slits contribute to form interference fringes on a screen. The resolution of the double slit setup can be improved by increasing the spatial separation of the two slits. Coming back to the polarization vector, the idea is to separate the $\pi$ flip into two $\pi/2$ flips to do the same transition. The analogy is complete, and a separation in time of the two $\pi/2$ pulses will significantly improve the resolution in frequency space. In essence, the separation of the transition into a two-step process (the two $\pi/2$ pulses instead of the one $\pi$ pulse) is referred to as 'Ramsey spectroscopy'. Ramsey also pointed out the analogy of his method and interferometry (Ramsey NF, 1993). He also stressed the difference which is that the quantum treatment of the spin 1/2 system has a complete classical analogon which cannot be made for the interferometry. As seen before in section Spin dynamics: Bloch equations and Larmor precession the treatment of the spin 1/2 system with classical Bloch equations is completely equivalent to the quantum mechanical treatment, the intuitively simpler approach to investigate the dynamics of the polarization vector is further pursued.

Here we would show pi and pi/2 pulses on Bloch spheres

Polarization after Ramsey sequence and finite pulse length

In the case of finite small $ \Delta \omega \ll \gamma_n B_1$, a magnetic field $\vec B' = \Delta \omega / \gamma_n \vec e_z \neq 0$ remains in the rotating frame. Applying the $\pi/2$ flip as described before prepares a transverse polarization: $\vec p'(\tau) = p\, \vec e_x'$. For the case of $\Delta \omega = 0$ the transverse polarization was static and a second $\pi/2$ flip at any time would result in the same polarization as for one continuous $\pi$ flip.

However, now the remaining $\vec B'$ causes a (slow) precession at rate $\Delta \omega$ of the transverse component of $\vec p'$. As $\tau \approx 0$ was chosen, after a time $T \gg \tau$ the polarization vector will evolve to

$\vec p'(\tau + T) \approx \vec p'(T) = p\, (\vec e_x' cos (\Delta \omega \, T) + \vec e_y' sin (\Delta \omega \, T)).$

Consider showing Bloch spheres for two different Ts

For the second $\pi/2$ pulse, it is important that the rotation with angular frequency $\omega$, gated in as $B_1$ pulses, was running without phase shift during the time $T$. The second $\pi/2$ flip will again turn a component of the polarization by $\pi/2$, because of the phase condition of the rotating field, still around the $y'$ axis. In $R'$, $p'$ accumulated a phase angle of $\Delta \omega T \bmod 2\pi$ with respect to the $x'$ axis. The $x'$ component of the polarization $p_x= cos (\Delta \omega T)$ will thus be turned into the $- z' = -z$ direction: $p_z = - cos (\Delta \omega \, T)$.

Up to here, $\tau$ was assumed to be negligibly small. In fact, with a $\tau$ of finite length, the spin dynamics will be slightly different. The polarization will not only evolve during T when it is precessing around $z'$ but also during the two $\pi/2$ pulses. To first order, one could guess that the last equation should be complemented by the time of the two $\pi/2$ pulses $p_z = - cos (\Delta \omega (T+2\tau))$.

However, the polarization during a $\pi/2$ pulse is not precessing around the $z'$ axis but around the vector sum $\Delta \omega / \gamma_n \vec e_z' + B_1 \vec e_y'$. Therefore, the additional phase angle in the $(x',y')$ plane is a projection of a sequence of phase angles around different 3D axes that does not simply add up. For an optimized Ramsey sequence one can show with a more tedious treatment that $4\tau/\pi$ is a correction which is numerically a very good approximation (May DJR, 1998):

$$p_z = - cos (\Delta \omega (T+4\tau/\pi)) = - cos (\Delta \omega \, \tilde{T}).$$

Here $\tilde{T} = T + 4 \tau /\pi$ is an effective precession time, introduced for more convenient notation.

Comment on the precision of the correction:

In a real experiment with UCN, one might use on order $T \approx 200$s and $\tau \approx 2$s. This means neglecting or wrongly correcting the precession time leads to 1-2% errors on the cosine argument, while the numerical precision of the appropriate analytic approximation is better than 1E-8, but may still be of concern.

Spin detection, polarization asymmetry and central fringe line-shape

In the section about UCN it is discussed how the neutrons and their spins are detected. The essential point is that one does a counting experiment and sorts individual neutrons according to whether their spins are up $(\uparrow)$ or down $(\downarrow)$ with respect to the $z$ axis.

With the evaluation of the $p_z$ component of the polarization, we in principle have for an individual neutron the probability of being counted as spin up or spin down.

After a Ramsey sequence the number of neutrons detected is $N = N_\uparrow + N_\downarrow$.

From detecting $N_\uparrow$ and $N_\downarrow$ independently, one can calculate the spin asymmetry $$A = \frac{N_\uparrow - N_\downarrow}{N_\uparrow + N_\downarrow}.$$

One must take into account that starting from a fully polarized sample may not be possible, that depolarization effects may occur during the measurement and that the spin detection process is not fully efficient. After a Ramsey sequence $p_z \le 1$ and the detectable spin asymmetry will always be diluted $\vert A \vert < 1$. For the maximal obtainable value of $p_z$ after a Ramsey sequence one can detect $N_\uparrow = N_+ $ and $N_\downarrow = N_-$ (or vice versa). The maximum spin asymmetry that the apparatus can produce is then described by the visibility $\alpha$:

$$\alpha = \frac{\vert {N_+ - N_-} \vert}{N_+ + N_-},$$

and the detected asymmetry is

$$A = - \alpha \, cos (\Delta \omega \, \tilde{T}).$$

This is a good description of the Ramsey fringe pattern close to $\Delta \omega = 0$ (central fringe). The frequency distance to the next fringe from the $\Delta \omega =0$ position is at

$$\Delta \omega = \frac{2\pi}{\tilde{T}}.$$

Here we would show a Ramsey fringe pattern

Measurement sensitivity

In a neutron EDM experiment, one aims at measuring the position of the central Ramsey fringe, i.e. of the spin precession frequency, correlated with the applied electric field at highest possible sensitivity. To measure the frequency at the highest precision, means to measure the change in asymmetry to the highest precision. For the cosine function of the asymmetry, this means measuring at $\Delta \omega \, \tilde{T}$ close to $\pm \pi/2$ where the slope is the largest. This can be done by either shifting the phase of the second $\pi/2$ flip in the Ramsey sequence, and by that obtaining a sine function around $\Delta \omega = 0$ or by adjusting the frequency of the rotating $B_1$ field. If one chooses to adjust the frequency, one can measure at four so-called 'working points', two close to $\Delta \omega = \pi/2$ and two to $-\pi/2$. Measuring the asymmetry in these four points and knowing $\alpha$ and $\tilde{T}$ allows fitting of the frequency of the central fringe.

The sensitivity of the method described can be derived as follows:

Starting from $A = - \alpha \, cos (\Delta \omega \, \tilde{T})$, $\Delta \omega \, \tilde{T} \approx \pi/2$, and $sin(\Delta \omega) \approx 1$ one gets

$$\frac{dA}{d(\Delta \omega)} = \alpha \tilde{T} \, sin (\Delta \omega \, \tilde{T}) \approx \alpha \tilde{T}.$$

As the interest is in the uncertainty of the frequency, one can immediately transform to obtain:

$$ \sigma(\Delta \omega) = \frac{\sigma(A)}{\alpha \tilde{T}}$$

Now one is left with the determination of $\sigma(A)$. As the choice was to measure the asymmetry $A = (N_\uparrow - N_\downarrow)(N_\uparrow + N_\downarrow)$ close to its largest slope, at around $A\approx0$, one has the total number of counted neutrons $N = N_\uparrow + N_\downarrow$ and both spin counts about the same, $N_\uparrow \approx N_\downarrow \approx N/2$. One can then easily show $\sigma(A)=1 / \sqrt(N)$

resulting in

$$ \sigma(\Delta \omega) = \frac{1}{\alpha \tilde{T} \sqrt{N}}.$$

Comment:

One should note that the quantum projection limit derived in section: Quantum sensitivity limit of the measurementis reached, up to the factor of $\alpha$ which ideally would be 1 and experiments try to maximize.

UCNs

Ultracold neutrons (UCNs) are free neutrons of extremely low total energy (<300 neV), and, as we will see, can be confined by magnetic or material traps for durations on the order of the neutron lifetime. This enables fundamental precision physics experiments to take advantage of long observation times. UCNs also experience all fundamental forces. In this section we will review the strength of these forces, how to transport and store UCNs, how to detect UCNs, and mention different production mechanisms.

- add reference books that I follow and why - put grav./mag. after strong interaction for sense of scales - give a tldr before diving into the derivation

Gravitational interaction

Owning to their low energy, gravity plays a large role in transporting and storing UCNs. The gravitational potential energy at the surface can be approximated as:

$$V_{grav} = m_N g h \approx 102.5$$ neV/m,

which means that UCNs will lose energy during elevation rise (or conversely gain energy if falling). Therefore, experiments can tune and change the energy of UCNs for measurements by raising or lowering UCNs in Earth's gravitational field.

Electromagnetic interaction

The magnetic moment of the neutron enables UCNs to be manipulated with external magnetic fields,

$$V_{mag} = -\vec{\mu} \cdot \vec{B}(\vec{r})$$,

which can be as large as $\approx 60.3$ neV/T (depending on the direction of spin and magnetic field).

PROBABLY TALK ABOUT ADIABATIC VS SUDDEN APPROXIMATION

Weak interaction

Free neutrons will undergo beta decay into a proton, electron, and electron anti-neutrino

$$n \rightarrow p + e + \bar{\nu}_e$$,

with an average lifetime of roughly $880$s. There is still a discrepancy between material/gravitational traps and beam experiments on the order of $10$s, known as the `Neutron Lifetime Puzzle'. Bound neutrons are stable due to the strong nuclear interaction.

Strong interaction

Neutrons are bound, and stable, in nuclei thanks to the strong nuclear (or $NN$) interaction. The $NN$ interaction is an effective interaction that emerges from non-perturbative quantum chromodynamics, and dominates bound neutron interactions inside the nucleus (on the scale of fm). But to a free, unbound, and ultracold neutron, the $NN$ interaction appears as a very narrow potential energy well of depth $U\sim$MeV. Consider a free UCN of $<300$ neV and how it interactions with the nucleus: the de Broglie wavelength of a UCN is $\lambda = h / \sqrt{2 m E} > 50$ nm, whereas typical nuclear sizes where the $NN$ interaction dominates (mediated by meson exchange) are $R=R_0 A^{1/3} \sim $ 10s of fm. As $\lambda \gg R$, and $U \gg E$ a UCN sees the nucleus as a very narrow and deep well.

At these energies, we would like to know how UCNs interact with the atomic nuclei of a surface material. Let's first consider a single-nucleus scatterer. Using the time-independent Schrodinger equation in integral form, the scattered UCN wavefunction is given exactly by:

$$\psi(\vec{r}) = \psi_0(\vec{r}) - \frac{m}{2 \pi \hbar^2} \int \frac{e^{i k |\vec{r} - \vec{r}_0 | }}{|\vec{r} - \vec{r}_0 |} V(\vec{r}_0) \psi (\vec{r}_0) d^3 \vec{r}_0$$

where $\psi_0(\vec{r})$ is the free UCN wavefunction before the scattering. This is form is interpreted as a superposition of spherical waves scattering at various positions $\vec{r}_0$ by the potential $V(\vec{r}_0)$

Now we can consider the case of our deep and narrow nuclear potential $V(\vec{r}_0)$. As the potential is well-localized at the nucleus ($\vec{r}_0=0$) and drops rapidly to zero on the distance scales of the UCN, we look far away from the scattering center and approximate the scattered wavefunction for $| \vec{r} | \gg | \vec{r}_0 |$. In this limit ($r_0/r \ll 1$)

$$\frac{e^{i k |\vec{r} - \vec{r}_0 | }}{|\vec{r} - \vec{r}_0 |} \approx \frac{ e^{i k r} }{ r } e^{i \vec{k} \cdot \vec{r}_0} $$.

Taking the free UCN wavefunction before scattering as a plane wave $\psi_0(\vec{r}) = e^{ikz}$, the Schrodinger equation becomes:

$$\psi(\vec{r}) = e^{ikz} - \frac{m}{2 \pi \hbar^2} \frac{ e^{i k r} }{ r } \int e^{i \vec{k} \cdot \vec{r}_0} V(\vec{r}_0) \psi (\vec{r}_0) d^3 \vec{r}_0$$.

At this stage, we can compare the above equation to the general solution to the Schrodinger equation for spherically symmetrical potentials (like $V(\vec{r}_0)$ here)

$$\psi(r, \theta , \phi ) = e^{i k z} + \sum_{l,m} C_{l,m} h_l^{(1)} (kr) Y_l^m(\theta,\phi) \xrightarrow{r\ large} e^{i k z} + f( \theta, \phi) \frac{ e^{i k r} }{r} $$

where $h_l^{(1)}$ are the spherical Hankel functions of the first kind, $Y_l^m$ are the spherical harmonics, and $C_{l,m}$ are the partial wave amplitudes, and $f( \theta, \phi) = \frac{1}{k} \sum_{l,m} (-i)^{l+1} C_{l,m} Y_l^m(\theta ,\phi )$. $f( \theta, \phi)$ is referred to as the scattering amplitude, as it can be related to the differential cross-section, $\frac{d \sigma}{d \Omega} = |f(\theta, \phi)|^2$.

Putting things together, we can see that the scattering amplitude in our case is

$$f(\theta, \phi) = - \frac{m}{2 \pi \hbar^2} \int e^{i \vec{k} \cdot \vec{r}_0} V(\vec{r}_0) \psi (\vec{r}_0) d^3 \vec{r}_0 $$

In order to continue, we have to deal with the wavefunction inside the integral. Due to the energy difference between the UCN and potential ($E \ll U$), we assume that our potential does not alter the energy-state of the UCN, that is, $\psi(\vec{r}_0) \approx \psi_0 (\vec{r}_0) = e^{i (\vec{k}' \cdot \vec{r}_0) }$, where $\vec{k}'=k \hat{z}$. This is known as the first-order Born approximation, valid for weak scattering potentials, and interpreted as the wavefunction undergoing a single scattering. With this approximation, the scattering amplitude becomes

$$f(\theta, \phi) = - \frac{m}{2 \pi \hbar^2} \int e^{i (\vec{k}- \vec{k}') \cdot \vec{r}_0} V(\vec{r}_0) d^3 \vec{r}_0$$.

We additionally argue that for a UCN, $k\sim 10^{8}$ 1/m whereas $r_0 \sim 10^{-15}$ m, so $kr\sim10^{-7}$ and thus $e^{i (\vec{k}- \vec{k}') \cdot \vec{r}_0} \approx 1$. Thus, we have a very simple form for the scattering amplitude:

$$f(\theta, \phi) = - \frac{m}{2 \pi \hbar^2} \int V(\vec{r}) d^3 \vec{r}$$.

At this point, we still need to know the function of the potential $V(\vec{r})$. We can reasonably assume that the the UCN sees the nuclear potential as constant, and infinitely deep, due to the energy and wavelength scales. We then construct a pseudopotential:

$$V(\vec{r}) = b \ \frac{2 \pi \hbar^2}{m}\ \delta(\vec{r})$$.

We have conveniently chosen $b$ such that the scattering amplitude becomes

$$f(\theta, \phi) = - b $$

and refer to $b$ as the scattering length. Note that our choice of potential means there is no angular dependence in the scattering amplitude, and is equivalent to s-wave scattering only, $l=0$, in the partial wave expansion earlier. Thus the differential cross-section for a UCN scattering off a single nucleus is given by $\frac{d \sigma}{d \Omega} = b^2$.

To expand this to scattering off many nuclei $N_j$ at distances $\vec{R}_j$, we can write the pseudopotential as

$$V(\vec{r}) = \frac{2 \pi \hbar^2}{m} \sum_j b_j \ \delta(\vec{r}-\vec{R}_j)$$

We then calculate the scattering amplitude off many nuclei (where we have looked now before the approximation $e^{i (\vec{k}- \vec{k}') \cdot \vec{r}_0} \approx 1$ as we are now scattering off many nuclei across a larger distance):

$$f(\theta, \phi) = - \frac{m}{2 \pi \hbar^2} \int e^{i (\vec{k}- \vec{k}') \cdot \vec{r}_0} V(\vec{r}_0) d^3 \vec{r}_0 = - \int e^{i (\vec{k}- \vec{k}') \cdot \vec{r}_0} \sum_j b_j \ \delta(\vec{r}_0-\vec{R}_j) d^3 \vec{r}_0 = - \sum_j b_j e^{i (\vec{k}- \vec{k}') \cdot \vec{R}_j} $$

Then the differential cross-section can be expressed as

$$\frac{d \sigma}{d \Omega} = \sum_{i,j} b_i b_j e^{i (\vec{k}- \vec{k}') \cdot (\vec{R}_i - \vec{R}_j)} $$

where we allow for a mixture of isotopes $i,j$ with scattering lengths $b_i, b_j$. Since we cannot distinguish which isotopes are at specific positions, we would like to convert this into the average of the scattering length of the material, $\bar{b}$:

$$\sum_{i,j} b_i b_j e^{i (\vec{k}- \vec{k}') \cdot (\vec{R}_i - \vec{R}_j)} = \sum_{i=j} b_i^2 + \sum_{i\neq j} b_i b_j e^{i (\vec{k}- \vec{k}') \cdot (\vec{R}_i - \vec{R}_j)} = N \Big( (\bar{b^2}) - (\bar{b})^2 \Big) + (\bar{b})^2 \sum_{i,j} e^{i (\vec{k}- \vec{k}') \cdot (\vec{R}_i - \vec{R}_j)}$$

The first term is referred to as incoherent scattering (scattering is uniform in all directions) and the second as coherent scattering (scattering depends on the direction $\vec{k}-\vec{k}'$). Note that if all the nuclei are identical, the incoherent scattering vanishes.

In the coherent scattering limit, we can relate the measured average scattering length $(\bar{b})$ to the psuedopotential earlier. We smear out the potential over an atomic volume and define the bulk material Fermi-potential, $V_F$:

$$V_F \equiv \int V\vec{r} d^3 \vec{r} = \bar{b}\ n \ \frac{2 \pi \hbar^2}{m} $$

where $n$ is the atomic number density of the scattering material for which $\bar{b}$ was measured. This is valid so long as $ k R_\textrm{atomic} \ll 1$. In this limit, the UCN sees a homogeneous potential that it is either refracted from or reflected from.

If the initial UCN energy $E_0$ is greater than the Fermi-potential $V_F$, refraction occurs with the refractive index defined as $n=k'/k$. In the potential, $E'=E_0-V_F$ and so $ \frac{\hbar^2 \ k'^2}{2 m} = \frac{\hbar^2 \ k}{2 m} - \bar{b}\ n \ \frac{2 \pi \hbar^2}{m}$. Thus,

$$n = 1 - \frac{2 \pi n \bar{b} }{k^2}$$

On the other hand, if $E_0 < V_F$, it will be totally reflected if the momentum perpendicular to the surface is less than the Fermi-potential. For a grazing angle to the surface $\theta$, total reflection is given by:

$$E_0 \sin^2 {\theta} < V_F$$

We now arrive at a definition for ultracold neutrons -- neutrons that are totally reflected from a surface for any angle of incidence, $E_0 < V_F$.

Quantum mechanically, the UCN wavefunction will not be perfectly reflected from the surface potential, but penetrate the surface, decaying exponentially. In this penetration, neutron absorption or inelastic up-scattering can occur, and thus UCN losses are possible. These processes can be captured mathematically by taking the surface potential as complex: $V = V_F(1 - i \eta)$, where $\eta$ is defined as a loss coefficient. In this case, the wavefunction isn't perfectly reflected. In the limit $\eta \ll 1$, the reflection probability is given by

$$|R|^2 = 1 - 2 \eta (\frac{E_0 \sin^2 \theta}{V_F - E_0 \sin^2})^{1/2}$$,

where we additionally define the loss probability per bounce, dependent on the energy and angle:

$$\mu(E_0, \theta) = 2 \eta (\frac{E_0 \sin^2 \theta}{V_F - E_0 \sin^2})^{1/2}$$.

We can now define materials that are appealing for UCNs -- those with large $V_F$ and with small loss probabilities per bounce. The table below gives some example values of typical materials used for UCN storage and transport.

Table 1: Common material Fermi-potential and loss parameters.
Material	$V_F$ (neV)	$\eta$
Ni$^{58}$	$XXX$	$X\cdot 10^{-5}$
X	$XXX$	$X\cdot 10^{-5}$

discuss how much more we really need to go into or what's important. I'm not sure all is relevant for the nEDM article.

Production

- where we get neutrons

- where do we do these experiments

Transport and Storage

Detection

Magnetic field control

Magnetic shield

quantum magnetometr

Alternative methods

beam EDM

SFHe

crystal EDM

Citing references

Groups of authors larger than 2 can be cited with "et al.".

As proven in (Albero A, 1999).

References

Ramsey, Norman F. (1956 (reprinted 1963, 1969, ..., 2005)). Molecular Beams. Oxford University Press, London.

Rabi, II; Ramsey, NF and Schwinger, J (1954). Use of rotating coordinates in magnetic resonance problems. Rev. Mod. Phys. 26: 167.

Ramsey, NF (1993). Complementarity with neutron two-path interferences and separated-oscillatory-field resonances. Phys.Rev. A 48: 80.

May, DJR (1998). A High Precision Comparison Of The Gyromagnetic Ratios Of The $^{199}$Hg Atom And The Neutron PhD thesis , University of Sussex.

Albero, Antony (1999). Pizza Margherita. Journal of pizza eaters 19(3): 13. arXiv:0808.000

Mohr, Peter J. (2025). CODATA recommended values of the fundamental physical constants: 2022. Rev. Mod. Phys. 97: 025002. arXiv:2409.03787

Mohr, Peter J.; SURNAME2, FORENAME2; SURNAME3, FORENAME3; SURNAME4, FORENAME4 and SURNAME5, FORENAME5 (2025). CODATA recommended values of the fundamental physical constants: 2022. Rev. Mod. Phys. 97: 025002.