Precision Measurement Applications

Precision measurements turn small energy differences into accumulated phases and then into count probabilities. Symmetry helps choose transitions, predict which shifts cancel under reversal, and identify amplitudes that vanish in an ideal limit. It also tells us what a null test can establish: a reversal-sensitive signal is evidence for a particular transformation behavior, but backgrounds with the same behavior must still be distinguished.

Helpful background. Larmor precession fixes magnetic-moment signs; selection rules constrains transition amplitudes; time reversal explains antiunitarity. Precision spectroscopy develops the broader uncertainty and frequency-standard framework.

A Ramsey fringe compares an atom with an oscillator

For two levels, the unperturbed transition frequency is $\nu_0=(E_b-E_a)/h$. Ideal short pulses prepare and recombine a coherent superposition, with free evolution for time $T$. Choose pulse phases so that the measured excited-state probability is

$$p_b=\frac12[1+C\cos\phi],\qquad \phi=2\pi(\nu-\nu_0)T+\phi_{\rm ctrl}.$$

Here $\nu$ is the interrogation oscillator frequency, $0<C\leq1$ is the fringe contrast, and $\phi_{\rm ctrl}$ is the relative pulse phase. The convention can be changed by a pulse-phase shift; the observable is a relative phase, not an isolated state’s overall phase. Real pulse duration, dephasing and shifts require a more complete model.

At a fringe’s side, $\phi=\pi/2$, the frequency slope has magnitude $\pi CT$. For $N$ independent binary outcomes, the projection-noise standard deviation of the estimated probability is $1/(2\sqrt N)$ there. Linear error propagation gives

$$\delta\nu=\frac1{2\pi CT\sqrt N}.$$

This is a statistical sensitivity for the stated model, not a complete clock accuracy or stability budget. Increasing $T$ helps only while coherence, oscillator noise and other limitations remain controlled. Atomic beams introduces Ramsey interrogation, and Ludlow and colleagues review the clock setting.

Symmetry-related transitions can cancel a shift

In a weak-field regime with good hyperfine labels, a level has a leading Zeeman shift proportional to $g_Fm_FB$. A transition’s linear shift is the difference between its two level shifts. An $m_F=0$ to $m_F'=0$ transition, when such levels and a suitable probe exist, has no first-order Zeeman shift in this regime. This does not remove quadratic shifts or shifts from electric fields and radiation.

Alternatively, two transitions related by reversal of their magnetic labels may obey

$$\nu_\pm(B)=\nu_0\pm aB+bB^2+\cdots.$$

Their average removes the odd term, while their difference measures it. The common quadratic term remains. Equal weighting and matching experimental conditions are part of the cancellation. Symmetry predicts the dependence on the reversal; it does not supply the coefficients or guarantee that the field was reversed exactly.

Selection rules also concern the drive. A transition forbidden at electric-dipole order may still be driven through another multipole, a two-photon process, or controlled state mixing. A narrow line and a symmetry-forbidden leading amplitude are related experimental opportunities, not identical statements. See atomic spectra.

Refocusing removes signals as well as noise

For $H=-\gamma B_z(t)S_z$, the phase of the up amplitude relative to the down amplitude accumulates as $\gamma\int B_z(t)\,dt$. The positive frequency magnitude in a static field is $|\gamma B_z|$, but the signed phase also records the sign of $\gamma$ and the field direction. A precession measurement needs transverse coherence; an energy eigenstate has no rotating mean spin.

Ideal instantaneous transverse $\pi$ pulses reverse the sign of the longitudinal coupling in the toggling frame. The accumulated phase becomes

$$\phi=\gamma\int_0^T y(t)B_z(t)\,dt, \qquad y(t)=\pm1.$$

One echo pulse at $T/2$ gives $y=+1$ before the pulse and $-1$ afterward. It cancels a constant field contribution, whether that contribution is unwanted noise or the desired DC signal. A suitably timed alternating field can instead accumulate phase with the pulses. Magnetometry therefore chooses a control sequence for the signal’s time dependence; more refocusing does not automatically improve every measurement. See dynamical decoupling for the frequency-selective description.

Parity-odd mixing can be visible before an energy shift

Let $H_0$ preserve parity and take an isolated, nondegenerate parity eigenstate $|n\rangle$. If a weak perturbation is $\lambda V_-$ with $PV_-P^{-1}=-V_-$, then $\langle n|V_-|n\rangle=0$. Its first-order energy shift vanishes, but its state changes by

$$|\delta n\rangle=\lambda\sum_{m\ne n} \frac{|m\rangle\langle m|V_-|n\rangle}{E_n-E_m}.$$

Only opposite-parity states contribute. A parity-odd observable or a previously forbidden transition can consequently have a contribution linear in $\lambda$. Near degeneracy this nondegenerate expansion fails and the nearly degenerate subspace must be diagonalized.

Interference makes a small amplitude accessible. If a controlled reversal produces amplitudes $A_\pm=A_0\pm\epsilon A_1$, with real small $\epsilon$, then

$$|A_+|^2-|A_-|^2 =4\epsilon\operatorname{Re}(A_0^*A_1).$$

The signal depends on relative phase as well as amplitude size. A weak interaction can violate parity, but stray electric fields can also mix atomic parity states by explicitly breaking the symmetry of the apparatus. Reversal signatures and additional controls are needed to distinguish these mechanisms. Tests of fundamental symmetries develops atomic parity-violation experiments.

A spin-aligned intrinsic electric dipole tests parity and time reversal

Within an isolated spin-$J$ multiplet with no additional orientational degree of freedom, rotational covariance makes a vector operator restricted to that multiplet proportional to $\mathbf J$. For $J>0$, define a permanent intrinsic electric dipole coefficient $d$ by

$$\mathbf d=d\frac{\mathbf J}{\hbar J},\qquad H_d=-d\frac{\mathbf J\cdot\mathbf E}{\hbar J}.$$

Here $d$ has units of charge times length and is the dipole expectation in a maximally polarized state. The angular-momentum operator is axial and time-reversal odd; the electric field is polar and time-reversal even. Thus this coupling is odd under both transformations. Its nonzero coefficient would violate parity and time reversal in the intrinsic interaction. “Isolated multiplet” does not mean one nondegenerate spin state: its different magnetic projections are essential to the argument.

A polar molecule’s ordinary body-fixed electric dipole is different. It follows a molecular orientation, an additional polar degree of freedom. That dipole does not by itself violate parity or time reversal. Field-free states can be chosen with definite parity and zero laboratory dipole expectation; an applied electric field mixes opposite-parity states and orients the molecule. Safronova and colleagues explain how such molecules can nevertheless enhance the sensitivity to a distinct intrinsic symmetry-violating interaction.

For spin $1/2$ and parallel laboratory fields, a simple effective model is

$$H=-\left(\frac{\gamma\hbar B}{2}+dE\right)\sigma_z.$$

Define the signed splitting by $\Delta E=E_{-z}-E_{+z}$, rather than always relabeling the lower energy. Then

$$\Delta E(E)=\gamma\hbar B+2dE,\qquad \Delta E(E)-\Delta E(-E)=4dE.$$

An electric-field-correlated magnetic change would imitate part of this difference. Reversing a laboratory field is a useful experimental operation, but is not the complete antiunitary time-reversal transformation. Interpreting the signal requires the effective Hamiltonian and its ordinary backgrounds.

Exercises

1. Why does the Ramsey frequency estimate lose first-order sensitivity at the top of a fringe? What does doubling $N$ do to its projection-noise uncertainty at the fringe’s side?

Solution

At $\phi=0$ the derivative of the cosine vanishes, so a small frequency change produces no linear probability change. At $\phi=\pi/2$ the slope is nonzero. With $C,T$ held fixed, doubling $N$ reduces $\delta\nu$ by $\sqrt2$, not by two.

2. Evaluate the echo phase for a constant field $B_0$, then for a field that is $B_0$ during the first half and $-B_0$ during the second half.

Solution

For the constant field the two signed integrals cancel: $\phi=\gamma B_0(T/2-T/2)=0$. For the reversing field, the control sign and field sign reverse together, giving $\phi=\gamma B_0T$. The sequence rejects a DC field but retains this synchronized signal.

3. In the spin-$1/2$ EDM model, suppose reversing $E$ also changes $B$ from $B_0+\delta B$ to $B_0-\delta B$. Find the signed splitting difference and the false inferred EDM if the magnetic effect is ignored.

Solution

The difference is $4dE+2\gamma\hbar\delta B$. Dividing by $4E$ would give $d_{\rm inferred}=d+\gamma\hbar\delta B/(2E)$. The correlated magnetic shift has the same electric-field reversal signature in this measurement, so another measurement or controlled comparison must constrain it.

References

• D. Budker and M. Romalis, “Optical magnetometry,” Nature Physics 3, 227–234, 2007. DOI.
• A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Reviews of Modern Physics 87, 637–701, 2015. DOI.
• N. F. Ramsey, Molecular Beams, Oxford University Press, 1956.
• M. S. Safronova, D. Budker, D. DeMille, D. F. Jackson Kimball, A. Derevianko, and C. W. Clark, “Search for new physics with atoms and molecules,” Reviews of Modern Physics 90, 025008, 2018, sections IV–V. DOI; author’s copy.