⃗ p$_{E}$ = ∂H ∂ ⃗ E$_{M}$ = − 4 πα$_{E}$ ⃗ E$_{M}$ − µ 2 M ⃗ K, (A1) 〈 ⃗ p$_{E}$ 〉 ≡ 〈 N ( ⃗ v ) | ⃗ p$_{E}$ | N ( ⃗ v ) 〉 = − 4 πα$_{E}$ ⃗ E$_{M}$ + µ 〈 ⃗ σ 〉 2 M × ⃗ v + . . . , (A2) 〈 E 〉 = ⃗ E$_{M}$ · ∫ E$_{M}$ 0 〈 ⃗ p$_{E}$ ′ 〉 dE ′ M = − 1 2 4 πα$_{E}$ ⃗ E 2 M + µ 〈 ⃗ σ 〉 2 M · ( ⃗ v × ⃗ E$_{M}$ ) . (A3) Δ E = 〈 N ( ⃗ 0) ∣ µ 2 M ⃗ K · ⃗ E$_{M}$ ∣ N ( ⃗ v ) 〉 1 M − k$_{0}$ 〈 N ( ⃗ v ) ∣ µ 2 M ⃗ K · ⃗ E$_{M}$ ∣ N ( ⃗ 0) 〉 . (A4) Appendix B: Analysis of Unpolarized Neutron Correlation Functions 18 g ( x$_{4}$, n )$_{i}$ = √ g ( x$_{4}$, n )$_{i}$ g ( x$_{4}$, − n )$_{i}$ , (B1) Here we present the analysis of unpolarized neutron correlation functions. On each configuration, for each value of the electric field strength, we form the unpolarized, source- averaged lattice correlation function, g ( x$_{4}$, n )$_{i}$. Parity invariance is enforced by taking the geometric mean of correlators obtained for a given field value and its negative. Specifically we form The off-diagonal matrix elements in the non-relativistic limit evaluate to half the value of the diagonal matrix elements in that limit. We can thus write the energy shift as Δ E = 1 4 ⃗ m · ( ⃗ v × ⃗ E$_{M}$ ) 1 0 − 1 $_{2}$M⃗ v 2 ⃗ m · ( ⃗ v × ⃗ E$_{M}$ ) . (A5) In the limit of zero velocity, a non-vanishing contribution from the motion-induced electric dipole moment emerges. This contribution is the same as that derived in Eq. (6). Without neutron motion, contributions from the motion-induced electric dipole mo- ment na¨ıvely vanish. At second order, however, the motion-induced dipole can interact with itself via neutron propagation. We must employ a limiting procedure to handle the nucleon pole. In a quantum mechanical notation, the shift due to an intermediate-state neutron with energy k$_{0}$ = M + 1 $_{2}$M⃗ v 2 has the form The second term is readily identified as the interaction energy of the magnetic moment with magnetic field seen in the neutron’s rest frame: 〈 ⃗ m 〉 · ⃗ B , with ⃗ m = µ⃗σ 2 $_{M}$, and ⃗ B = ⃗ v × ⃗ E$_{M}$ . This explains why one sees a motion-induced electric dipole moment in the frame in which neutron moves with velocity ⃗ v . we see that the additional term corresponds to a motion-induced dipole moment [43]. In the external field, the electric dipole moment contributes to the total energy in the form that does not vanish when the electric field is turned off. This extra contribution is a motion-induced effect as can be seen from neutron matrix elements. For a neutron moving non-relativistically, effective action for the neutron that appears in Eq. (4). The Minkowski space electric field we denote by ⃗ E$_{M}$ . Without subtracting the magnetic moment contribution, the electric dipole operator picks up an additional contribution