Shifts due to neighboring resonances for microwave measurements of the 2 ${}^{3}P$ fine structure of helium

Quantum-mechanical interference with far-off-resonance neighboring states is found to cause systematic shifts for the measurements of the 2 ${}^{3}P$ fine-structure intervals. The shifts depend on the type of experiment used to measure the intervals. Here the shifts are calculated for measurements using a single microwave pulse and for measurements using the Ramsey method of separated oscillatory fields. The shifts are small, but are large enough to affect the continuing program of determining the fine-structure constant from a comparison between accurate experimental measurements and theoretical calculations of the interval energies. The separated-oscillatory-field shifts are found to be much smaller than the single-pulse shifts.

Figure 1

The $n=2$ triplet energy levels of helium. For this work the population is assumed to start in the $|1\rangle$ state and an applied microwave field either drives the $|1\rangle \to |2\rangle$ transition (solid arrow) or the $|1\rangle \to |0\rangle$ transition (dashed arrow). The allowed radiative decay paths from states $|1\rangle$, $|2\rangle$, and $|0\rangle$ are indicated. When driving one of the resonances (the solid arrow or the dashed arrow), a far-off-resonant transition amplitude from the other resonance causes a shift in the measured line center.

Figure 2

Timing for the microwave pulses. (a) A single microwave pulse is considered. (b, c) The in-phase and ${180}^{\circ }$-out-of-phase cases for two pulses are represented. The difference of signal obtained in (b) and (c) allows for subnatural line widths using the Ramsey SOF technique. A switching time of $t_s=1$ ns is included in the calculations.

Figure 3

Line shapes of the $|1\rangle \to |2\rangle$ resonance obtained from numerical integrations for (a) a single microwave pulse with a duration $D=200$ ns, and for (b) SOF with two microwave pulses of $D=100$ ns, separated by $T=500$ ns. These line shapes are for a microwave magnetic field amplitude of $B_0=0.2$ gauss and show that the initially empty $|b\rangle$ state of Fig. 1 is populated when the microwave transition is driven. The fits (solid lines) are obtained using Eqs. (4) and (5). The fits are used to determine small shifts in the resonance line centers, as shown in Fig. 4 and Tables and .

Figure 4

Shifts of the $|1\rangle \to |2\rangle$ resonance versus microwave field strength. The shifts for (a) a single microwave pulse with a duration $D=200$ ns and for (b) SOF with two microwave pulses of $D=100$ ns, separated by $T=500$ ns, are obtained by fits similar to those shown in Fig. 3. On each plot, the squares represent the full shift and the circles the shifts that would result in the absence of the ${\gamma }_{20\to b}$ interference term in Eq. (1). The quadratic fits shown [Eq. (6)] are used to extrapolate to zero microwave power.