Atomic mixer based on phase control without ac Zeeman shift

Atom-based mixers have been shown to be very useful in performing the absolute measurement of the magnitude of a radio-frequency (rf) field by using Autler-Townes (AT) splitting. However, there has been less success in measuring the phase of a rf field with the AT splitting in a magnetic-resonance system. The phase of the rf magnetic field plays a very important role in imaging applications. Here, we design an atomic mixer for measuring the phase of the rf field by detecting the transmission spectra of a laser-detected magnetic-resonance system based on the interference between Raman and cascade two-photon processes for $F_g=4$ of the $D_1$ line of cesium atoms. A scheme of measuring the rf phase can be realized with the elimination of the ac Zeeman shift of the system by adjusting the amplitude ratio of a transverse fundamental-wave field and its third-harmonic field. Theoretical and experimental results show that the scheme with cancellation of the ac Zeeman shift is superior in linear measurement of the longitudinal rf component. Our results provide schemes for a magnetic sensor based on quantum interference of nonlinear processes for the absolute measurement of the relative phase of rf fields.

Figure 1

Theoretical model geometry of the system in a laboratory frame $xyz$ where the quantization axis lies along the direction of light polarization $\varepsilon$ and in a rotating frame $x^{\prime }{y}^{\prime }{z}^{\prime }$ where the quantization axis lies along the direction of the offset field $B_0$.

Figure 2

Schematic of two-photon transition processes. The Zeeman splitting caused by the offset field is ${\mathrm{\Omega }}_0$, and the ac Zeeman shift caused by the transverse rf fields is $\mathrm{\Delta }{\mathrm{\Omega }}_0$. The left section indicates that when the condition ${\omega }_{\sigma \left(\pi \right)}-{\omega }_{\pi \left(\sigma \right)}={\mathrm{\Omega }}_0+\mathrm{\Delta }{\mathrm{\Omega }}_0$ is met, the atoms in the ground-state sublevels will absorb a $\sigma$ (or $\pi$) rf photon and then release a $\pi$ (or $\sigma$) rf photon to form a Raman two-photon process. The right section indicates that when the condition ${\omega }_{\sigma }+{\omega }_{\pi }={\mathrm{\Omega }}_0+\mathrm{\Delta }{\mathrm{\Omega }}_0$ is met, the atoms in the ground-state sublevels will absorb a $\pi$ rf photon and a $\sigma$ rf photon to form a cascade two-photon process.

Figure 3

Experimental setup. DL pro, grating-stabilized tunable single-mode diode laser; HWP, half-wave plate; PBS, polarization beam splitter; FM, flat mirror; Digilock 110, feedback controller Digilock 110; GTP, Glan-Taylor prism; cell 1, vacuum chamber filled with cesium atoms; cell 2, cesium atomic cell coated with paraffin; PD, photodetector; DDS, direct digital frequency synthesizer; PCS, precision current source; OSC, digital storage oscilloscope. All details can be found in the text.

Figure 4

AT spectra vs the frequency of the probing field ${\mathit{B}}_{\mathit{p}}$. (a) The pink circles represent the measured data of $\frac{\mathrm{\Delta }{\mathrm{\Omega }}_0}{2\pi }\approx -50$ Hz; the resonant frequencies of the left and right sidebands are about 99 427 and 100 796 Hz, respectively. (b) The blue circles represent the measured data of $\frac{\mathrm{\Delta }{\mathrm{\Omega }}_0}{2\pi }\approx 0$ Hz; the resonant frequencies of the left and right sidebands are about 99 450 and 100 872 Hz, respectively. (c) The purple circles represent the measured data of $\frac{\mathrm{\Delta }{\mathrm{\Omega }}_0}{2\pi }\approx 50$ Hz; the resonant frequencies of the left and right sidebands are about 99 477 and 100 946 Hz, respectively. All data are measured under the condition of $\mathrm{\Delta }\varphi =\pi$. The fitting curves are the line shapes with ${\gamma }_1\approx 2\pi \times 45\mathrm{Hz}$ from Eq. (9).

Figure 5

Sideband interval vs relative phase. The blue solid dots represent the measured data with a step of $\pi /9$. All data are measured under the resonance condition with ${\mathrm{\Omega }}_0=2\pi \times 100.161\mathrm{kHz}$, ${\mathrm{\Omega }}_L=2\pi \times 6.236\left(5\right)\mathrm{kHz}$, ${\mathrm{\Omega }}_{T1}=2\pi \times 9.992\left(5\right)\mathrm{kHz}$, and ${\mathrm{\Omega }}_{T2}=2\pi \times 12.910\left(5\right)\mathrm{kHz}$. The red fitting curve is the theoretical fit of Eq. (6).

Figure 6

Sideband interval vs longitudinal Rabi frequency. The blue diamonds represent the measured data with $\mathrm{\Delta }\varphi =\pi$; the purple circles represent the measured data with $\mathrm{\Delta }\varphi =\pi /2$. All data are measured under the same resonance condition with ${\mathrm{\Omega }}_0=2\pi \times 100.161\mathrm{kHz}$, ${\mathrm{\Omega }}_L=2\pi \times 6.236\left(5\right)\mathrm{kHz}$, ${\mathrm{\Omega }}_{T1}=2\pi \times 9.992\left(5\right)\mathrm{kHz}$, and ${\mathrm{\Omega }}_{T2}=2\pi \times 12.910\left(5\right)\mathrm{kHz}$. The fitting curves are the theoretical fits of Eq. (6).

Figure 7

Theoretical model geometry of scheme II with only one transverse rf field.

Figure 8

Sideband interval in scheme II vs relative phase. The blue solid dots represent the measured data with a step of $\pi /9$. All data are measured under the resonance condition with ${\mathrm{\Omega }}_0=2\pi \times 100.160\mathrm{kHz}$, ${\mathrm{\Omega }}_{L1}=2\pi \times 2.630\left(5\right)\mathrm{kHz}$, ${\mathrm{\Omega }}_{L2}=2\pi \times 8.040\left(5\right)\mathrm{kHz}$, and ${\mathrm{\Omega }}_T=2\pi \times 3.875\left(5\right)\mathrm{kHz}$. The red fitting curve is the theoretical fit of Eq. (C6).

Figure 9

Sideband interval vs longitudinal Rabi frequency. The blue diamonds represent the measured data with $\mathrm{\Delta }\varphi =0$; the purple circles represent the measured data with $\mathrm{\Delta }\varphi =\pi /2$. (a) is the measurement under the fitting value of ${\mathrm{\Omega }}_{L2}=2\pi \times 7.480\left(5\right)$ kHz; (b) is the measurement under the fitting value of ${\mathrm{\Omega }}_{L1}=2\pi \times 3.256\left(5\right)\mathrm{kHz}$. All data are measured under resonance conditions with ${\mathrm{\Omega }}_0=2\pi \times 100.110\mathrm{kHz}$, ${\mathrm{\Omega }}_T=2\pi \times 7.750\left(5\right)\mathrm{kHz}$, and $\frac{\mathrm{\Delta }{\mathrm{\Omega }}_0^{\left(\mathrm{II}\right)}}{2\pi }\approx 200\mathrm{Hz}$. The fitting curves are the theoretical fits of Eq. (C6).