Abstract

With substantial improvements on the optical arrangement, we show that significant enhancement of the magneto-optical rotation effect in rubidium vapor can be achieved without using complex and expensive heterodyne polarimetric methods. The huge signal-to-noise ratio performance using low cost photo-detectors opens the door to chip-sized device manufacturing and in-situ applications in biomagnetism.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical heterodyne polarimetry, due to its high common-mode noise rejection performance, has been the central element in all atomic magnetometers based on the nonlinear magneto-optical rotation (NMOR) effect [16]. In a typical single-beam $\Lambda -$type (Fig. 1(a)) atomic magnetometer scheme in room temperature the polarization plane rotation is linearly dependent on the third-order polarization at the probe frequency [7,8]

$$P_{{\pm}}^{(3)}\quad \propto \quad \frac{\kappa{|\Omega_{{\mp}}|^2}\Omega_{{\pm}}\Gamma}{(\delta_P+i\Gamma)(\delta_P^2+\Gamma^2)(\mp2\delta_B+i\gamma_{0})}.$$
Here $\delta _B=\mu _Bg_FB_z$ is the magnitude of the magnetic field ($B_z$) induced Zeeman shift and $\gamma _0$ is the corresponding Zeeman resonance line width (i.e., the relaxation rate between states $|1\rangle$ and $|2\rangle$ when collision is neglected). $\kappa$ is proportional to the product of the medium density, length and relevant optical transition strengths. The Rabi frequencies of the circularly-polarized components of the linearly-polarized probe field $\vec {E}_P=\vec {E}_P^{(+)}+\vec {E}_P^{(-)}$ are expressed as $\Omega _{\pm }$. $\delta _P$ is the one-photon laser detuning from the excited electronic state having the resonance line width of $\Gamma$. As the result of initially equally populated states $|1\rangle$ and $|2\rangle$, there are two completely equal two-photon transitions between Zeeman magnetic sublevels, i.e., $|1\rangle \rightleftharpoons |2\rangle$ via the upper electronic state $|3\rangle$. Note that Eq. (1) predicts a rotation of the probe field polarization plane that is linearly dependent on the atom density and medium length [2]. In practice, this rotation angle is very small and difficult to detect, and therefore sophisticated optical and electronics detection schemes such as heterodyne polarimetry must be used.

 figure: Fig. 1.

Fig. 1. (a): Single-beam $\Lambda -$scheme where states $|1\rangle$ and $|2\rangle$ have magnetic quantum number $m_F=\pm 1$ and equal initial populations. (b): Dual-beam inelastic wave-mixing scheme where the probe and WM fields share the same populated Zeeman states and coherence.

Download Full Size | PPT Slide | PDF

Recently, a different type of NMOR scheme has been realized using a nonlinear inelastic wave mixing (WM) method (dual-beam NMOR scheme) [8,9]. This technique has exhibited a significant enhancement to the signal optical power spectra density signal-to-noise ratio (SNR) even with substantially lower probe laser intensity and medium density. However, the complex and expensive optical heterodyne polarimetry is still used for signal detection. From applications viewpoint, especially for chip-size applications, it is very desirable to simplify the detection setup.

Here, we report a significant improvement of this new technique and show that a giant NMOR signal can be directly measured using low cost polarizer, photo-diodes, and simple 774 op-amp chips, a complete elimination of optical heterodyne polarimetry. We emphasize that to date all NMOR-based atomic magnetometers require expensive balanced detector and complex heterodyne detection arrangement simply because at room temperature the resulting optical NMOR signals for most conventional NMOR schemes at nano-Tesla level are generally too weak to be detected using photo-diodes. Therefore, our improvement represents an important step toward engineering applications. In Section 2 we briefly review the principle of nonlinear inelastic optical wave mixing process and the key physics that leads to the highly enhanced NMOR power spectra density SNR. In Section 3 we describe our dual-arm and dual-measurement-branch setup and experimental details. In Section 4 we show side-by-side comparison of magnetic dispersion signals simultaneously acquired using low cost photo-diodes with op-amps and using an elaborate heterodyne scheme with balanced detectors. In Section 5 we discuss signal noise characteristics and finally in Section 6 we present the conclusion on our much simplified optical measurement system and its possible applications in micro-chip based devices.

2. Dual-beam NMOR technique and SNR enhancement

In this section we briefly review the physics of inelastic WM and the key concept that leads to the significant enhancement of the NMOR SNR. Detailed theoretical frameworks and mathematics can be found in Ref. [8]. Figs. 1(a) and 1(b) depict the energy level diagrams with relevant laser couplings used in the conventional and inelastic WM measurement schemes. The conventional scheme, Fig. 1(a), consists of a three-state atomic system interacting with two different circular components of a single linearly polarized probe field. The inelastic WM scheme, Fig. 1(b), on the other hand requires two linearly polarized light fields and it can work with either three or four-state atomic systems. To avoid lengthy mathematics notations and discussions on level beating etc, we use a four-state system as shown in Fig. 1(b) to explain the physics of inelastic WM process.

From the atom-light interaction viewpoint the single-beam $\Lambda -$scheme shown in Fig. 1(a) has two symmetric but mutually-restrictive two-photon excitation channels, i.e., $|1\rangle \rightarrow |3\rangle \rightarrow |2\rangle$ and the reverse transition $|2\rangle \rightarrow |3\rangle \rightarrow |1\rangle$. We emphasize that the two transitions are equally important precisely because that initially population is shared equally in the two lower states $|1\rangle$ and $|2\rangle$. This initially condition is common in nearly all atomic magnetometer schemes. The competition between these two two-photon processes results in a self-limiting effect: i.e., the growth (by emission) of one circular field component requires the decrease (by absorption) of the other field component. As a result, no appreciable change of either field component is permitted by energy conservation since both field components are from the same probe laser field. Consequently, the growth of NMOR effect is limited and the rotation of polarization plane is very small. This is the major draw-back of all conventional single-beam $\Lambda -$scheme based atomic magnetometers.

The second draw back of the conventional single-beam $\Lambda -$scheme can be seen from the numerator of Eq. (1) which shows that the intensity of the probe field can increase the NMOR effect. In fact, this is the reason why this effect is referred as a nonlinear-MOR effect. However, the intensity of the probe field must be limited to avoid ac Stark shift and power broadening of magnetic transitions. Experimentally, this is one of considerations why the probe field intensity is generally restricted to be less than the saturation intensity of the one-photon transition. Typically, the one-photon detuning is chosen to be on the order of one Doppler-broadened line width in order to gain a noticeable enhancement to the NMOR signal and yet to avoid spontaneous emission and Zeeman state power broadening. Clearly, both complications are detrimental to magnetic field detection sensitivity.

The dual-beam inelastic WM scheme introduces a second linearly polarized laser, represented by two red arrows in Fig. 1(b), to remedy these issues. This WM field, $\vec {E}_{WM}$, creates a second two-photon excitation channel by a wave-mixing process via populated intermediate Zeeman states, providing efficient energy transfer from the WM field to the selected branch of the probe field through the shared Zeeman intermediate states [10,11]. The key point of this energy transfer is that it breaks the probe field self-limiting effect in the single laser scheme, resulting in a significant growth of the ground state Zeeman coherence and producing a large NMOR signal even with substantially reduced probe intensity and medium density. Intuitively, this inelastic WM process can be viewed as a two-wave mixing (involving the two components of the probe) process with coherently and dynamically prepared states and Zeeman coherence [10,11] by the WM field. We emphasize that such a large NMOR optical signal amplitude enhancement cannot be achieved using the single-beam $\Lambda -$schemes described in Fig. 1(a) [26], nor can it be produced without the dynamic wave-mixing process [12]. We also stress that this WM technique is neither the pump-probe scheme widely-employed in laser spectroscopy nor the Zeeman-shift based laser locking technique, both yielding only linearized probe response while requiring saturation of one-photon transition. In fact, the energy-providing WM field in our experiment is far-off detuned with very weak power (often even less than that of the probe field), resulting in an excitation rate that is typically a factor of 1000 below the saturation rate used in pump-probe schemes in laser spectroscopy.

3. Experiment

Figure 2 shows the experimental setup which has two symmetric measurement arms, one for each light field. In each arm there are two detection branches, provided by a 50-50 non-polarizing beam splitter, for comparison. One branch consists of a Glan prism and two independent low-cost 774 op-amp based simple low gain photo-detectors attached to two orthogonal ports (solid green line), separately and simultaneously detecting magneto-optical resonance and dispersion. The second arm consists a heterodyne polarimetry detection scheme using a combination of waveplate, beam splitter and balanced detector with matched optical sensors (dashed black line). For the data reported here the probe field generally couples the $D_1$ line transitions with a ref one-photon detuning in the range of -200 MHz to -5 GHz whereas the counter-propagating WM field generally couples the $D_2$ line transitions with similar but slightly smaller red detunings. We have studied all $D_1$ and $D_2$ combinations using this setup.

 figure: Fig. 2.

Fig. 2. Dual-arm dual-detection setup with cross-reference Glan prism (GP) and heterodyne detection arms. The angle between the probe polarization and the wave-mixing field polarization is 45$^\textrm {o}$. WP: half wave plate. BS: 50-50 beam splitter. PBS: polarization beam splitter. BD: balanced detector. PD: low cost photodiode. Glan prisms are adjusted so that without magnetic field both probe and WM light transmit through and no light appears in the reflect ports.

Download Full Size | PPT Slide | PDF

The linearly P-polarized probe field $\vec {E}_P$ propagates along the long axis of an uncoated vapor cell of 6 cm in length and 2 cm in diameter. The cell contains isotopically pure $^{87}$Rb vapor with about 667 Pa Ne buffer gas and is enclosed in a home-made temperature-controlled magnetically-shielded container (30 cm in length and 15 cm in diameter). The magnetic field is generated by a solenoid using a precision current source. The linearly-polarized WM field $\vec {E}_{WM}$, rotated by 45$^\textrm {o}$ with respect to the P-polarized probe, counter-propagates with respect to the probe field. This arrangement avoids spurious optical noise from registering on the relevant detectors. Both probe laser (1.2 mm in diameter) and WM laser (5 mm in diameter) are independently frequency locked to relevant transitions and are transmitted via optical fibers [13]. No optical pumping of non-accessed hyperfine states were used.

4. Comparison of heterodyne polarimetry and non-heterodyne schemes

In Fig. 3 we show NMOR signals obtained using the two detection branches, i.e., the balanced heterodyne detection branch (Figs. 3(a) and 3(b)) and the non-heterodyne detection branch using low cost prism, low gain amplifiers and photo-detectors (Figs. 3(c) and 3(d)). In a typical measurement, we measure the NMOR signal simultaneously using both branches. We first set the probe field power to produce a magnetic resonance signal with a peak-peak value of 4 mV but with the WM field turned off. We scan the magnetic field in the range of $\simeq \pm 2\mu$T and a typical scan takes 25 ms. Since the WM laser is turned off this conventional single-beam $\Lambda -$scheme yields an NMOR signal shown by the blue trace in Fig. 3(a). We then switch on a weak WM field without changing any other experimental parameters. The WM laser typically has an intensity similar to that of the probe laser but has a slight smaller one-photon detuning (usually about 2/3 of the probe detuning). With this weak WM light present we observe a large enhancement of NMOR signal amplitude (red trace in Fig. 3(a) exhibits an enhancement factor of about 200). The fast oscillating noise exhibited in Fig. 3(a) is due to the electronic noise of the oscilloscope and can be substantially reduced by averaging more scans or by using a spectrum analyzer. Figure 3(b) shows a scan in a narrower region of $\simeq \pm 2$nT. In Fig. 3(a) the blue trace takes 3.2 s (consists of 128 scans of 25 ms each) whereas the red trace takes only 16 averaging. In Fig. 3(b) both traces are averaged 128. We note that the slope of the re-scaled magnetic dispersion signals of the two schemes remains the same near the zero magnetic field region (Fig. 3(a) near the center). This indicates that the WM field does not affect the magnetic field sensitivity [14]. This part of measurements, while similar to those given in Ref. [9], is necessary as it provides a traceable cross reference to the key improvement with low cost optics shown below.

To show the remarkable performance of the dual-beam scheme we also record the NMOR signals simultaneously using non-heterodyne branch, as shown in Figs. 3(c) and 3(d). We first note the nearly all atomic magnetometers relying on single-beam three-state systems, whether it is spin exchange relaxation free scheme or optically-pumped scheme, require the use of expensive heterodyne polarimetry and high-speed lock-in amplifier systems to ensure reliable NMOR signal detection. This is precisely because of their extremely small magneto-optical rotation angle and low SNR. In fact, no studies have been reported so far on using a simple non-heterodyne detection technique in single-beam atomic magnetometry. The giant NMOR signal enhancement demonstrated by inelastic WM technique now makes non-heterodyne signal detection possible.

 figure: Fig. 3.

Fig. 3. (a) and (b): Probe NMOR heterodyne signal as detected by the standard polarimetric technique shown in the dashed black frame in Fig. 2. (a): $E_P$: 1.9 mW/cm$^{2}$ ($\delta _P/2\pi = -$400 MHz, $F=2\rightarrow F'=1$). $E_{WM}$: 1.2 mW/cm$^{2}$ ($\delta _{WM}/2\pi = -$120 MHz $F=2\rightarrow F{''}=3$). Blue: the widely-used single-beam $\Lambda -$scheme (WM field is turned off). Red: the inelastic wave mixing scheme. The red trace is rescaled to show the magnetic resonance line shape preservation. (b): Magnetic field scan between $\pm$2 nT without data rescaling. (c) and (d): Probe NMOR signal using the simple prism+low cost diode technique (from the transmission and reflection ports of a Glan prism, solid green frame in Fig. 2). Green: detector background. The dispersion signal in (d) when the WM field is turned off (blue trace) is indistinguishable from the detector background (green trace). The medium temperature is 350 K.

Download Full Size | PPT Slide | PDF

Figure 3(c) shows the magneto-optical resonance [1518] signal detected from the transmission port of the Glan prism whereas Fig. 3(d) exhibits the magnetic dispersion signal simultaneously detect from the reflection port of the Glan prism. The blue/red traces are obtained with the WM turned off/on. The green traces represent the photo-diode background (first measured without magnetic field and then scaned the magnetic field to make sure no magnetic anomaly). It is amply clear that the conventional scheme, which yields blue traces, is incapable of detecting the magnetic dispersion. Without the WM laser the blue trace is non-distinguishable from the green trace by the photo-diode background as shown in Fig. 3(d) under the atom density used in our experiment. In fact, it is precisely because of its poor NMOR optical SNR the single-beam $\Lambda -$scheme based atomic magnetometry never employs non-heterodyne method for weak magnetic field detection.

When a weak WM field is turned on the second excitation pathway is established and the two laser fields form an inelastic WM scheme. A tremendously large two-photon magneto-optical resonance is now easily detected through the transmission port of the Glan prism (Fig. 3(c), red trace). Correspondingly, the reflection port of the Glan prism registers an enormous increase in the magnetic dispersion signal (Fig. 3(d), red trace). Figures 3(c) and 3(d) demonstrate the superior NMOR signal strength and SNR of the nonlinear elastic WM technique, which are easily detected with simple non-heterodyne technique using low cost polarizer and photo-diodes. The enhanced two-photon magneto-optical resonance is a key evidence that the inelastic WM process arises from a large Zeeman coherence introduced by the WM field. In our experiment the magnetic dispersion signal amplitude is routinely enhanced by a factor of 300. Using the data given in Fig. 3(d), we estimate the maximum magneto-optical rotation angle achieved is about 0.12 radian. Clearly, with the WM field this angle is not measurable.

5. Superior SNR performance achievable with the low cost detection system

Detection noise is a critically important performance specification for any precision measurement technology. To demonstrate the superior optical noise immune capability of the WM technique for NMOR measurement we compare in Fig. 4 (top panel) the NMOR optical signal noise spectral densities of the single-beam $\Lambda -$scheme (blue trace) and the WM technique (red trace) at a fixed 10 nT magnetic field. We first adjust the probe field, with the WM field turned off, to obtain a signal with a selected signal amplitude in time domain (i.e. using an oscilloscope). The photo-current from photo-diodes are measured by a low-noise spectrum analyzer and the optical signal power density is plotted as a function of frequency in the range of the spectrum analyzer (20 kHz). This is shown by the blue trace in the top panel in Fig. 4. Without changing any operation parameters of the probe field we now turn on the WM light and and achieve a factor of 200 increase in the NMOR optical signal amplitude. When the photo-diode current is measured again we found that the noise spectrum of the probe light is not affected by the presence of the WM light even though the WM laser has enhanced the NMOR signal amplitude by a factor of more than 200. This is shown by the red trace in the top panel in Fig. 4. We also found that in order for the conventional singe-beam three-state scheme to generate such a large NMOR optical signal amplitude (verified in time domain) the probe laser intensity must be substantially increased. When the photo-current for such a high-power probe excitation is analyzed we found that the noise floor of the NMOR signal is increased by nearly 20 dB (see the black trace in Fig. 4, top panel), indicating substantial performance degradation in measurement sensitivity. Further study by scanning the magnetic field has shown that such a strong excitation condition, which is aimed to match the performance of a weak WM laser, also results in a significantly spread of the magnetic resonance linewidth. This indicates a significant power broadening effect in the high power single-beam $\Lambda -$technique [8,19]. The white dashed lines in contour projection plots in Fig. 4 lower panel trace out the positions of dispersion peaks in both schemes. The large resonance peak broadening in the conventional scheme and the minimal power broadening in the WM scheme show the superior performance of the WM technique.

 figure: Fig. 4.

Fig. 4. Top panel: Comparison of noise spectra densities. Blue: conventional single-beam $\Lambda -$scheme. Red: the WM technique. Clearly, the WM technique does not degrade the probe noise. Black: single-beam $\Lambda -$scheme with higher power probe in order to produce an NMOR signal amplitude that matches that of the WM technique in time domain. Green: detector noise. Lower panel: Numerical calculation of power broadening effects in the single-beam $\Lambda -$technique (left) and the WM technique (right). White dashed lines in contour projection plots trace positions of dispersion peaks.

Download Full Size | PPT Slide | PDF

6. Conclusion

We have demonstrated a robust low cost non-heterodyne NMOR detection scheme based on the dual-beam inelastic WM technique. The SNR performance of the dual-beam inelastic WM technique is the key element that enables such a low-cost, much-simplified detection scheme for weak magnetic field sensing at room temperature, an environment that has prohibited non-heterodyne polarimetric detection method for NMOR applications so far. It opens realm possibilities for micro-chip based devices for bio-magnetism applications. For instance, it is possible to integrate a 1-mm size micro-Glan prism with two photo-diode chips and low-gain op-amps directly packaged on two orthogonal ports, a mini-detection module configuration that is very difficult to achieve with a multi-component heterodyne-based polarimetry technique that requires a well-matched photonsensor pair.

Funding

National Natural Science Foundation of China (11104075, 11774093, 11774262); Science and Technology Commission of Shanghai Municipality (16ZR1409800).

Acknowledgment

We thank Dr. L. Deng of Center for Optics Research and Engineering (CORE) of Shandong University (Qingdao) for many suggestions. The experiment reported here was carried out by Dr. F. Zhou and Dr. E.Y. Zhu while working at NIST under the supervision of Dr. L. Deng.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. A. K. Zvezdin and V. A. Kotov, Modern magnetooptics and magnetooptical materials (CRC, 1997).

2. M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms (Oxford University, 2010).

3. D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. 81(26), 5788–5791 (1998). [CrossRef]  

4. D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999). [CrossRef]  

5. E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

6. D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000). [CrossRef]  

7. Y.-R. Shen, The principles of nonlinear optics (Wiley-Interscience, 1994).

8. C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018). [CrossRef]  

9. F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017). [CrossRef]  

10. K. Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic two-wave mixing,” Phys. Rev. Lett. 98(8), 083604 (2007). [CrossRef]  

11. L. Deng, M. G. Payne, and W. Garrett, “Inelastic wave mixing and multi-photon destructive interference based induced transparency in coherently prepared media,” Opt. Commun. 242(4-6), 641–647 (2004). [CrossRef]  

12. With the probe frequency fixed scanning WM laser frequency reveals many resonance structures with various amplitude enhancement but with the detuning combination reported here showing the large enhancement. Such rich detuning dependent resonance structures are characteristic of nonlinear optical process.

13. While phase-locking the WM laser with respect to the probe laser can improve the performance we have found that the improvement is no significant.

14. V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000). [CrossRef]  

15. A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998). [CrossRef]  

16. D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002). [CrossRef]  

17. M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001). [CrossRef]  

18. C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003). [CrossRef]  

19. J. Ward and A. Smith, “Saturation of two-photon-resonant optical processes in cesium vapor,” Phys. Rev. Lett. 35(10), 653–656 (1975). [CrossRef]  

References

  • View by:

  1. A. K. Zvezdin and V. A. Kotov, Modern magnetooptics and magnetooptical materials (CRC, 1997).
  2. M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms (Oxford University, 2010).
  3. D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. 81(26), 5788–5791 (1998).
    [Crossref]
  4. D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999).
    [Crossref]
  5. E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).
  6. D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
    [Crossref]
  7. Y.-R. Shen, The principles of nonlinear optics (Wiley-Interscience, 1994).
  8. C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
    [Crossref]
  9. F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017).
    [Crossref]
  10. K. Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic two-wave mixing,” Phys. Rev. Lett. 98(8), 083604 (2007).
    [Crossref]
  11. L. Deng, M. G. Payne, and W. Garrett, “Inelastic wave mixing and multi-photon destructive interference based induced transparency in coherently prepared media,” Opt. Commun. 242(4-6), 641–647 (2004).
    [Crossref]
  12. With the probe frequency fixed scanning WM laser frequency reveals many resonance structures with various amplitude enhancement but with the detuning combination reported here showing the large enhancement. Such rich detuning dependent resonance structures are characteristic of nonlinear optical process.
  13. While phase-locking the WM laser with respect to the probe laser can improve the performance we have found that the improvement is no significant.
  14. V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
    [Crossref]
  15. A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
    [Crossref]
  16. D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
    [Crossref]
  17. M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
    [Crossref]
  18. C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
    [Crossref]
  19. J. Ward and A. Smith, “Saturation of two-photon-resonant optical processes in cesium vapor,” Phys. Rev. Lett. 35(10), 653–656 (1975).
    [Crossref]

2018 (1)

C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
[Crossref]

2017 (1)

F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017).
[Crossref]

2007 (1)

K. Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic two-wave mixing,” Phys. Rev. Lett. 98(8), 083604 (2007).
[Crossref]

2004 (1)

L. Deng, M. G. Payne, and W. Garrett, “Inelastic wave mixing and multi-photon destructive interference based induced transparency in coherently prepared media,” Opt. Commun. 242(4-6), 641–647 (2004).
[Crossref]

2003 (1)

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

2002 (1)

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

2001 (1)

M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
[Crossref]

2000 (2)

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

1999 (1)

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999).
[Crossref]

1998 (2)

D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. 81(26), 5788–5791 (1998).
[Crossref]

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

1995 (1)

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

1975 (1)

J. Ward and A. Smith, “Saturation of two-photon-resonant optical processes in cesium vapor,” Phys. Rev. Lett. 35(10), 653–656 (1975).
[Crossref]

Affolderbach, C.

M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
[Crossref]

Aleksandrov, E.

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Andreeva, C.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Auzinsh, M.

M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms (Oxford University, 2010).

Balabas, M.

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Bednar, C.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Bevilacqua, G.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Biancalana, V.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

Budker, D.

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999).
[Crossref]

D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. 81(26), 5788–5791 (1998).
[Crossref]

M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms (Oxford University, 2010).

Cartaleva, S.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Dancheva, Y.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Deng, L.

C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
[Crossref]

F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017).
[Crossref]

K. Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic two-wave mixing,” Phys. Rev. Lett. 98(8), 083604 (2007).
[Crossref]

L. Deng, M. G. Payne, and W. Garrett, “Inelastic wave mixing and multi-photon destructive interference based induced transparency in coherently prepared media,” Opt. Commun. 242(4-6), 641–647 (2004).
[Crossref]

Fleischhauer, M.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Garrett, W.

L. Deng, M. G. Payne, and W. Garrett, “Inelastic wave mixing and multi-photon destructive interference based induced transparency in coherently prepared media,” Opt. Commun. 242(4-6), 641–647 (2004).
[Crossref]

Gawlik, W.

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Graf, L.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

Hagley, E.

C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
[Crossref]

Hagley, E. W.

F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017).
[Crossref]

Ivanov, A.

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Jiang, K.

K. Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic two-wave mixing,” Phys. Rev. Lett. 98(8), 083604 (2007).
[Crossref]

Karaulanov, T.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Kemp, W.

M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
[Crossref]

Kimball, D.

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999).
[Crossref]

Knappe, S.

M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
[Crossref]

Kotov, V. A.

A. K. Zvezdin and V. A. Kotov, Modern magnetooptics and magnetooptical materials (CRC, 1997).

Lukin, M.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Marinelli, C.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Mariotti, E.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

Meschede, D.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

Mikhailov, E.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Moi, L.

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Nagel, A.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

Naumov, A.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

Novikova, I.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Payne, M. G.

K. Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic two-wave mixing,” Phys. Rev. Lett. 98(8), 083604 (2007).
[Crossref]

L. Deng, M. G. Payne, and W. Garrett, “Inelastic wave mixing and multi-photon destructive interference based induced transparency in coherently prepared media,” Opt. Commun. 242(4-6), 641–647 (2004).
[Crossref]

Rochester, S.

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999).
[Crossref]

M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms (Oxford University, 2010).

Sautenkov, V.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Scully, M. O.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Senkov, N.

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Shen, Y.-R.

Y.-R. Shen, The principles of nonlinear optics (Wiley-Interscience, 1994).

Smith, A.

J. Ward and A. Smith, “Saturation of two-photon-resonant optical processes in cesium vapor,” Phys. Rev. Lett. 35(10), 653–656 (1975).
[Crossref]

Stähler, M.

M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
[Crossref]

Velichanskii, V.

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Velichansky, V.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Vershovskii, A.

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Ward, J.

J. Ward and A. Smith, “Saturation of two-photon-resonant optical processes in cesium vapor,” Phys. Rev. Lett. 35(10), 653–656 (1975).
[Crossref]

Weis, A.

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Welch, G. R.

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Wynands, R.

M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
[Crossref]

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

Yakobson, N.

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Yashchuk, V.

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999).
[Crossref]

D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. 81(26), 5788–5791 (1998).
[Crossref]

Zhou, F.

C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
[Crossref]

F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017).
[Crossref]

Zhu, C.

C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
[Crossref]

Zhu, C. J.

F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017).
[Crossref]

Zhu, E. Y.

C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
[Crossref]

Zolotorev, M.

D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. 81(26), 5788–5791 (1998).
[Crossref]

Zvezdin, A. K.

A. K. Zvezdin and V. A. Kotov, Modern magnetooptics and magnetooptical materials (CRC, 1997).

Appl. Phys. B (1)

C. Andreeva, G. Bevilacqua, V. Biancalana, S. Cartaleva, Y. Dancheva, T. Karaulanov, C. Marinelli, E. Mariotti, and L. Moi, “Two-color coherent population trapping in a single cs hyperfine transition, with application in magnetometry,” Appl. Phys. B 76(6), 667–675 (2003).
[Crossref]

Europhys. Lett. (2)

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44(1), 31–36 (1998).
[Crossref]

M. Stähler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, “Picotesla magnetometry with coherent dark states,” Europhys. Lett. 54(3), 323–328 (2001).
[Crossref]

Opt. Commun. (1)

L. Deng, M. G. Payne, and W. Garrett, “Inelastic wave mixing and multi-photon destructive interference based induced transparency in coherently prepared media,” Opt. Commun. 242(4-6), 641–647 (2004).
[Crossref]

Opt. Spectrosc. (1)

E. Aleksandrov, M. Balabas, A. Vershovskii, A. Ivanov, N. Yakobson, V. Velichanskii, and N. Senkov, “Laser pumping in the scheme of an mx-magnetometer,” Opt. Spectrosc. 78, 292–298 (1995).

Phys. Rev. A (2)

D. Budker, D. Kimball, S. Rochester, V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000).
[Crossref]

V. Sautenkov, M. Lukin, C. Bednar, I. Novikova, E. Mikhailov, M. Fleischhauer, V. Velichansky, G. R. Welch, and M. O. Scully, “Enhancement of magneto-optic effects via large atomic coherence in optically dense media,” Phys. Rev. A 62(2), 023810 (2000).
[Crossref]

Phys. Rev. Appl. (1)

C. Zhu, F. Zhou, E. Y. Zhu, E. Hagley, and L. Deng, “Breaking the energy-symmetry-based propagation growth blockade in magneto-optical rotation,” Phys. Rev. Appl. 10(6), 064013 (2018).
[Crossref]

Phys. Rev. Lett. (4)

D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. 81(26), 5788–5791 (1998).
[Crossref]

D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83(9), 1767–1770 (1999).
[Crossref]

K. Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic two-wave mixing,” Phys. Rev. Lett. 98(8), 083604 (2007).
[Crossref]

J. Ward and A. Smith, “Saturation of two-photon-resonant optical processes in cesium vapor,” Phys. Rev. Lett. 35(10), 653–656 (1975).
[Crossref]

Rev. Mod. Phys. (1)

D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Sci. Adv. (1)

F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017).
[Crossref]

Other (5)

Y.-R. Shen, The principles of nonlinear optics (Wiley-Interscience, 1994).

A. K. Zvezdin and V. A. Kotov, Modern magnetooptics and magnetooptical materials (CRC, 1997).

M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms (Oxford University, 2010).

With the probe frequency fixed scanning WM laser frequency reveals many resonance structures with various amplitude enhancement but with the detuning combination reported here showing the large enhancement. Such rich detuning dependent resonance structures are characteristic of nonlinear optical process.

While phase-locking the WM laser with respect to the probe laser can improve the performance we have found that the improvement is no significant.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. (a): Single-beam $\Lambda -$scheme where states $|1\rangle$ and $|2\rangle$ have magnetic quantum number $m_F=\pm 1$ and equal initial populations. (b): Dual-beam inelastic wave-mixing scheme where the probe and WM fields share the same populated Zeeman states and coherence.
Fig. 2.
Fig. 2. Dual-arm dual-detection setup with cross-reference Glan prism (GP) and heterodyne detection arms. The angle between the probe polarization and the wave-mixing field polarization is 45$^\textrm {o}$. WP: half wave plate. BS: 50-50 beam splitter. PBS: polarization beam splitter. BD: balanced detector. PD: low cost photodiode. Glan prisms are adjusted so that without magnetic field both probe and WM light transmit through and no light appears in the reflect ports.
Fig. 3.
Fig. 3. (a) and (b): Probe NMOR heterodyne signal as detected by the standard polarimetric technique shown in the dashed black frame in Fig. 2. (a): $E_P$: 1.9 mW/cm$^{2}$ ($\delta _P/2\pi = -$400 MHz, $F=2\rightarrow F'=1$). $E_{WM}$: 1.2 mW/cm$^{2}$ ($\delta _{WM}/2\pi = -$120 MHz $F=2\rightarrow F{''}=3$). Blue: the widely-used single-beam $\Lambda -$scheme (WM field is turned off). Red: the inelastic wave mixing scheme. The red trace is rescaled to show the magnetic resonance line shape preservation. (b): Magnetic field scan between $\pm$2 nT without data rescaling. (c) and (d): Probe NMOR signal using the simple prism+low cost diode technique (from the transmission and reflection ports of a Glan prism, solid green frame in Fig. 2). Green: detector background. The dispersion signal in (d) when the WM field is turned off (blue trace) is indistinguishable from the detector background (green trace). The medium temperature is 350 K.
Fig. 4.
Fig. 4. Top panel: Comparison of noise spectra densities. Blue: conventional single-beam $\Lambda -$scheme. Red: the WM technique. Clearly, the WM technique does not degrade the probe noise. Black: single-beam $\Lambda -$scheme with higher power probe in order to produce an NMOR signal amplitude that matches that of the WM technique in time domain. Green: detector noise. Lower panel: Numerical calculation of power broadening effects in the single-beam $\Lambda -$technique (left) and the WM technique (right). White dashed lines in contour projection plots trace positions of dispersion peaks.

Equations (1)

Equations on this page are rendered with MathJax. Learn more.

P ± ( 3 ) κ | Ω | 2 Ω ± Γ ( δ P + i Γ ) ( δ P 2 + Γ 2 ) ( 2 δ B + i γ 0 ) .