Preparation and detection of states with simultaneous spin alignment and selectable molecular orientation in PbO

We are pursuing an experiment to measure the electric dipole moment of the electron using the molecule PbO. This measurement requires the ability to prepare quantum states with selectable orientation of the molecular axis and, simultaneously, alignment of the electron spin perpendicular to this axis. It also requires efficient detection of the evolution of the spin-alignment direction within such a state. We describe a series of experiments that have achieved these goals, and the features and limitations of the techniques. Our results include observation of and explanation for a small electric-field dependence of the magnetic $g$ factor in the oriented states. We also discuss possible approaches for improved efficiency in this and similar systems.

Figure 1

(Color online) Energy level structure of the $a\left(1\right)$ $J=1$ manifold. (a) Structure with no electric field $\left(\mathcal{E}=0\right)$. Zeeman shifts ${\delta }_Z$ are shown. (b) Structure with a strong electric field such that ${\mu }_a\mathcal{E}⪢{\Delta }_{{\Omega }_1}$. Here sublevels with $\left|M\right|=1$ have mixed parity. Solid lines indicate the energy levels with only ${\delta }_{St}$ taken into account. Dashed (blue) lines indicate levels with both Stark and Zeeman shifts included. Dotted (red) lines indicate the additional effects of a nonzero $d_e$.

Figure 2

(Color online) Cross sectional top view of the apparatus. The apparatus consists of: (a) up to four layers of nearly cylindrical magnetic shielding; (b) the aluminum vacuum chamber; (c) heat shields; (d) the quartz and tantalum oven; (e) the alumina and sapphire vapor cell; (f) re-entrant gold electrodes; (g) quartz light pipes; (h) fluorescence collection filters and detectors; and (i) microwave electronics and horn. Applied electric and magnetic fields are perpendicular to the plane of the page.

Figure 3

Allowed decay channels for the $a\left(1\right)|J=1⟩$ states. (a) Decays of the $|J=1,M=\pm 1,P=-1⟩$ levels. Only final states with $P^{\prime }=+1={\left(-1\right)}^{J^{\prime }}$ are allowed. (b) Decays of oriented molecule states such as $|J=1,M=\pm 1,N=+1⟩$. These mixed-parity states can decay to final states of both parities. In both cases, only decays to common $\left(M^{\prime }=0\right)$ sublevels give rise to quantum beats.

Figure 4

(Color online) Typical Zeeman quantum beats, measured in the absence of an applied electric field $\left(\mathcal{E}=0\right)$. Both data (blue points, averaged over 70 laser pulses) and the fit function described in the text (red line) are shown. For this data, $c=9.2\left(1\right)\%$, ${\omega }_b=2\pi \times 299.8\left(1\right)\text{kHz}$, and $\varphi =5.02\left(1\right)\text{radians}$. The background/signal ratio $d/\alpha \cong 1.10$ is evident from the vertical scale.

Figure 5

Plot of difference in $g$-factors $\Delta g$ between higher and lower energy $\Omega$-doublet states, as a function of applied electric field $\mathcal{E}$. Points are data; the line is the result of the analytic calculation discussed in the text. The excellent agreement is evident.

Figure 6

Quantum beats with an incoherent superposition of oriented molecular states. The data shown here correspond to the pure beat signal $S_b\left(T\right)$ (see text). The interference between the two beat frequencies, as described in the text, is evident. For this data, $B=540\text{mG}$, corresponding to the average beat frequency ${\overline{\omega }}_b\cong 2\pi \times 1.35\text{MHz}$; the electric-field amplitude $\mathcal{E}=118\text{V}/\text{cm}$; and we determine $\Delta {\omega }_b\cong 2\pi \times 11.4\text{kHz}$.

Figure 7

(Color online) Schematic of the laser-rf double-resonance method for population of oriented molecular states. The laser pulse populates the $|J=1,M=0,P=-1⟩$ state. Then, a short rf pulse is tuned to drive the population to either $|{\psi }_{N_{-}}^{\left(x\right)}⟩$ (blue with frequency ${\omega }_{{\text{rf}}_H}$) or $\left(|{\psi }_{N_{+}}^{\left(x\right)}⟩\right)$ (red with frequency ${\omega }_{{\text{rf}}_L}$).

Figure 8

(Color online) Quantum beat signals with the laser-rf double-resonance scheme. The quantum beat amplitude $A\equiv c\alpha$ is plotted as a function of Stark shift ${\delta }_{St}$. (a) Appearance of quantum beats following rf population of oriented molecular states. The upper black (lower red) trace is at high (low) rf power, as indicated in the legend. Even at the highest powers, the contrast is only about $10–20\%$ of that seen with direct laser state preparation (corresponding to $A\sim 3$ on this plot). The lines broaden less than expected as the power is increased. (b) Disappearance of quantum beats following rf depletion of laser-excited states in the presence of $\mathcal{E}$. Here the lower black (upper red) trace is at high (low) rf power. Beat depletion as large as 70% was observed, and the width of these resonances was consistent with expectations due to rf power broadening.

Figure 9

(Color online) Laser-microwave Raman double-resonance state preparation. In (b) and (d) the rms amplitude spectrum $\left|{\tilde{S}}_b\left({\omega }_b\right)\right|$ of the pure beat signal $S_b\left(T\right)$ (see Fig. 6) is plotted as a function of the beat frequency ${\omega }_b$. Black solid traces show data with $\widehat{x}$ laser polarization preparing an incoherent superposition of oriented states; this provides a baseline for comparison to the case where oriented states are prepared. (a) Energy level diagram for the $zx$ pulse sequence. (b) Results from the $zx$ pulse sequence. The dashed blue trace shows the beat spectrum after both pulses. (No beats were observed after the first pulse.) For this data, the first (second) pulse had a frequency sweep centered around ${\omega }_{\mu 1}=2\pi \times 28.220\text{GHz}$ $\left({\omega }_{\mu 2}=2\pi \times 28.193\text{GHz}\right)$. (c) Energy level diagram for the $xz$ pulse sequence. (d) Results from the $xz$ pulse sequence. The dashed blue (dot-dashed green) trace shows the spectrum after the first (second) pulse. After the first pulse, beats appear at the frequency corresponding to the Zeeman splitting between $|J=2,M=\pm 1⟩$ states. The first (second) pulse was centered around ${\omega }_{\mu 1}=2\pi \times 28.217\text{GHz}$ $\left({\omega }_{\mu 2}=2\pi \times 28.190\text{GHz}\right)$. For both plots, $\mathcal{E}\cong 39\text{V}/\text{cm}$. Each pulse had nominal power of $\sim 1\text{W}$ and duration $2\mu \text{s}$, during which the frequency was swept linearly over a range $d\omega =2\pi \times 11\text{MHz}$. The second pulse arrived $0.5\mu \text{s}$ after the first ended.

Figure 10

Confirmation of the microwave “erasure” method. (a) Plot of quantum beat amplitude $A=c\alpha$ versus microwave frequency, showing two dips at the expected resonant frequencies. The dashed horizontal line corresponds to the full beat amplitude seen with no microwaves applied. For these data, 0 dB microwave power corresponds to $\sim 1\text{W}$ transmitted through the vapor cell, and the pulse duration was $9\mu \text{s}$. (b) Plot of $A$ versus microwave attenuation, at the (resonant) microwave frequency indicated by the arrow in (a). Initially increasing the power increases the efficiency of “erasure” until the beats from one pair of states are completely erased and $A$ has dropped by one half. Further power broadens the lines further, until the second pair of states is also “erased.” Nearly pure oriented states are prepared at the plateau in the range of powers $-15\leftrightarrow -10\text{dB}$.

Figure 11

(Color online) Optical double-resonance probe schemes using the $a\text{−}{C}^{\prime }$ transition. (a) General features of the scheme, illustrated for the case of $\mathcal{E}=0$. Here both excitation and probe lasers have $\widehat{x}$ polarization, and a single sublevel of the $C^{\prime }$ state is populated. (b) Probe excitation of an oriented molecular state. Both $P=+1$ and $P=-1$ sublevels of the $C^{\prime }$ state are coherently populated by the probe laser. (c) Decays from the $|C^{\prime },J=1,M=0,P=\pm 1⟩$ superposition state created by probing oriented molecular states. Population oscillates between the $P=+1$ (dots) and $P=-1$ (dashes) states. By collecting only $\widehat{z}$ polarized fluorescence (orange arrows), only the $P=+1$ population is detected and quantum beats with high contrast $\left(c\cong 1\right)$ should be observed.

Figure 12

Observed quantum beats using optical double-resonance probe detection, with no applied electric field. The $C^{\prime }\to X$ fluorescence signal $S_{C^{\prime }}\left(T\right)$ is plotted as a function of time delay between the $\widehat{x}$ polarized excitation laser [driving the $X\to a\left(1\right)$ transition] and the $\widehat{x}$ polarized probe laser [driving the $a\left(1\right)\to C^{\prime }$ transition]. Points are data and the line is a guide to the eyes. As expected, near 100% contrast is observed. The dc background level in this data is believed due to light leakage through or around the low-quality blue-transmitting color glass filter used here. Subsequent measurements indicated that negligible blackbody radiation is present over the range of wavelengths corresponding to $C^{\prime }\to X$ fluorescence.

Figure 13

Detection of $|a,J=1⟩$ state molecules by microwave absorption on the $|a,J=1⟩\to |a,J=2⟩$ transition (with $\mathcal{E}=0$). The plot shows peak fractional absorption of the microwave power as a function of microwave frequency. Points are data (averaged over 512 laser pulses); the line is a fit to a Gaussian. For this data, the fitted amplitude is $A_{\mu }=3\times {10}^{-4}$; the width is $\Gamma =2\pi \times 52\text{kHz}$ FWHM, in reasonable agreement with the expected Doppler width ${\Gamma }_D=2\pi \times 42.5\text{kHz}$.