Observation of asymmetric line shapes in precision microwave spectroscopy of the positronium $2{}^{3}S_1\to 2{}^{3}P_J$ ($J=1,2$) fine-structure intervals

We report new measurements of the positronium (Ps) $2{}^{3}S_1\to 2{}^{3}P_J$ fine-structure intervals, ${\nu }_J$ ($J=0,1,2$). In the experiments, Ps atoms, optically excited to the radiatively metastable $2{}^{3}S_1$ level, flew through microwave radiation fields tuned to drive transitions to the short-lived $2{}^{3}P_J$ levels, which were detected via the time spectrum of subsequent ground-state Ps annihilation radiation. Both the ${\nu }_1$ and ${\nu }_2$ line shapes were found to be asymmetric, which, in the absence of a complete line-shape model, prevents accurate determination of these fine-structure intervals. Conversely, the ${\nu }_0$ line shape did not exhibit any significant asymmetry; the observed interval, however, was found to disagree with QED theory by 4.2 standard deviations.

Figure 1

Energy level diagram of the $n=1$ and 2 levels in positronium. The energies of the $O\left(m{\alpha }^2\right)$ (Bohr) levels are indicted by the dashed horizontal lines. The calculated transition frequencies for the $n=1$ hyperfine interval (${\nu }_{\mathrm{HFS}}$) and the $2{}^{3}S_1\to 2{}^{3}P_J$ ($J=0,1,2$) intervals (${\nu }_J$) are also shown [15]. Radiative or annihilation lifetimes for all levels are also indicated, according to the primary decay mode.

Figure 2

Schematic of the Ps target, UV laser, and a WR-75 microwave guide with microwave radiation off resonance (a) and on resonance (b). A potential of $-3.5$ kV was applied to the target electrode ($V_T$) to control the beam implantation energy. The angular distribution selection in the $x$ direction due to the UV laser bandwidth is also shown. (c) The placement of the four LYSO detectors around the target chamber with respect to the position of the waveguide.

Figure 3

Zeeman structure of Ps in the $2{}^{3}S_1(M_J$) and $2{}^{3}P_{J=0,1,2}(M_J$) levels relative to the $n=2$ Bohr [i.e., $O\left(m{\alpha }^2\right)]$ level. The arrows represent the allowed $\mathrm{\Delta }{M}_J=0$ transitions to the $2{}^{3}P_J$ states. Note the different energy scales of the various sections in this figure.

Figure 4

(a) Lifetime spectra measured using D4 for three cases as outlined in the text. (b) Difference spectra demonstrating $2{}^{3}S_1$ production (Sig. 1–Sig. B) with and without the electric field switched off after the positron beam has been implanted into the silica target, as indicated. (c) Difference spectrum demonstrating the microwave radiation induced ${\nu }_2$ transition (Sig. 2–Sig. 1).

Figure 5

Example line shapes of the ${\nu }_0$ (a), ${\nu }_1$ (b), and ${\nu }_2$ (c) transitions, measured in a magnetic field of 32 G using detector D4. Both Lorentzian and Fano fits were applied to all line shapes. The dotted (green) vertical lines at $\mathrm{\Delta }\nu =0$ indicate the Zeeman-shifted theoretical transition frequencies, shown in the legends. The insets show the allowed $\mathrm{\Delta }{M}_J=0$ transitions (solid arrows) to the $2{}^{3}P_J$ states, followed by $\mathrm{\Delta }{M}_J=0,\pm 1$ radiative decay (dashed arrows) to the ground state.

Figure 6

(a) The microwave-induced transition signal, $S_{\gamma }$, measured on resonance for different applied microwave powers. (b) Measured linewidths ${\mathrm{\Gamma }}_{\mathrm{m}}$ obtained from Lorentz fits and (c) Fano asymmetry parameter $q$, obtained from Fano fits to data measured as a function of applied microwave power. The curves in each panel are fits to the functions shown in the corresponding legends, as explained in the text. These measurements were made for the ${\nu }_1$ transition in a magnetic field of 82 G. The vertical dashed line indicates the power (4.3 mW) at which the ${\nu }_1$ data presented elsewhere in this article were recorded.

Figure 7

The ${\nu }_0$ (a), ${\nu }_1$ (b), and ${\nu }_2$ (c) transition frequencies, relative to the calculated zero-field frequencies, measured in different magnetic fields. The ${\nu }_{\mathrm{R}}$ values from Lorentz and Fano fits to the line shapes are indicated by the blue squares and red circles, respectively, and the quadratic fits are indicated by the solid curves. The calculated transition frequencies as a function of the magnetic field are represented by the dashed lines. In the case of the ${\nu }_2$ transition, the dashed line represents the average of the two individual $\mathrm{\Delta }{M}_J=0$ transitions, weighted by the relative transition strengths.

Figure 8

The measured Lorentz linewidths ${\mathrm{\Gamma }}_{\mathrm{m}}$ for the ${\nu }_0$ (a), ${\nu }_1$ (b), and ${\nu }_2$ (c) transitions as a function of the magnetic field $B$. The horizontal shaded bars represent the average value of ${\mathrm{\Gamma }}_{\mathrm{m}}$, and the solid lines represent a linear fit to the data.

Figure 9

The measured Fano asymmetry parameter, $q$, for the ${\nu }_0$ (a), ${\nu }_1$ (b), and ${\nu }_2$ (c) transitions as a function of the magnetic field $B$. The horizontal shaded bars represents the average value of $q$, and the line represent a linear fit to the data.

Figure 10

Schematic diagram of the scenarios considered when calculating QI effects on (a) the $2{}^{3}S_1\to 2{}^{3}P_0$, i.e., ${\nu }_0$, (b) the $2{}^{3}S_1\to 2{}^{3}P_1$, i.e., ${\nu }_1$, and (c-i) and (c-ii) the $2{}^{3}S_1\to 2{}^{3}P_2$, i.e., ${\nu }_2$ transitions. The value of $|M_J|$ of the sublevels in each case is indicated in brackets.

Figure 11

Calculated line shapes showing the shift in the resonance frequency arising from the presence of far-detuned off-resonance transitions. The line shapes are fitted with Lorentz and Fano functions. Both functions yield identical results because of the lack of asymmetry. The ${\nu }_0$ and ${\nu }_1$ resonances are pulled by the ${\nu }_2$ resonance yielding a negative frequency shift as shown in (a) and (b). The ${\nu }_2$ resonance (c) is shifted positively. The width of the line shapes is $\approx 60$ MHz.

Figure 12

(a) An example of a Doppler-broadened line shape of the $1{}^{3}S_1\to 2{}^{3}P_{\mathrm{J}}$ transition centered around 243 nm, where the natural linewidth is $\approx 50$ MHz, but the observed linewidth is $\approx 900$ GHz. (b) Relative shift in the $2{}^{3}S_1\to 2{}^{3}P_1$ transition as a function of UV detuning. The dashed vertical lines in (a) indicate the UV laser wavelengths used for the measurements shown in (b), and the shaded bar indicates the 100 GHz bandwidth of the UV laser.

Figure 13

(a) Estimated magnitude of stray electric fields in the waveguides as a function of distance from the grids. (b) Stark shift of the ${\nu }_J$ transitions as a function of electric field.

Figure 14

Constrained parameter space for a scalar mediator [73]. The solid blue curve denotes the lower bound for a shift of 2.77 MHz for the $2S\to 2P$ transition. The solid black curve is the same observable obtained for a measurement of the Ps $1S\to 2S$ interval [74]. The dashed curves represent the constraints that would be obtained if the corresponding measurements were performed with uncertainties equal to the current QED uncertainties [73]. The red curve is the bound obtained from measurements of the electron gyromagnetic factor, $a_e$ [75]. The shaded gray region represents the parameter space constrained by astrophysical observations [76].

Figure 15

Constrained parameter space for a pseudoscalar mediator [73]. The solid blue curve denotes the lower bound for a shift of 2.77 MHz for the $2{}^{3}S_1\to 2{}^{3}P_0$ transition. The solid black curve is the same observable obtained for a measurement of the Ps $1{}^{1}S_0\to 1{}^{3}S_1$ transition [77, 78, 79]. The dashed curves represent the constraints that would be obtained if the corresponding measurements were performed with uncertainties equal to the current QED uncertainties [73]. The red curve is the bound obtained from measurements of the electron gyromagnetic factor, $a_e$ [75]. The shaded gray region represents the parameter space constrained by astrophysical observations [76].

Figure 16

Comparison of the previous and present measurements with theory for the (a) ${\nu }_0$, (b) ${\nu }_1$, and (c) ${\nu }_2$ transitions. Results from Lorentz ($L$) and Fano ($F$) line profile fits are shown for the current measurements of ${\nu }_1$ and ${\nu }_2$ intervals. Also shown for these transitions are the maximum (peak) values obtained from the Fano profiles, as discussed in the text. The theoretical values with uncertainties are indicated by the vertical solid lines. Statistical and systematic uncertainties for the experimental data have been added in quadrature.