Research ArticlePHYSICS

Symmetry-breaking inelastic wave-mixing atomic magnetometry

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Science Advances  01 Dec 2017:
Vol. 3, no. 12, e1700422
DOI: 10.1126/sciadv.1700422


  • Fig. 1 Energy levels and laser couplings of the inelastic optical WM-enhanced NMOR effect.

    (A) Simplified single-beam Λ scheme in the presence of a magnetic field and a linearly polarized probe field. The lower three states are the magnetic sublevels of a generic F = 1 manifold (from left to right: MF = −1, 0, +1) of an alkali atom. (B) Simplified optical WM scheme with linearly polarized probe and wave-mixing fields. For mathematical simplicity, only excited states |2〉 and |4〉 are considered.

  • Fig. 2 Numerical calculations of inelastic optical WM-enhanced NMOR.

    NMOR effect of the optical WM technique (A) and the single-beam Λ technique (B) as functions of δB31 at z = 1 cm. Parameters are chosen to show a representative 120-fold NMOR optical SNR enhancement. When the blue trace in (B) is rescaled vertically by a factor of 120, the magnetic resonance line shape is indistinguishable from that of the red trace, attesting to the fact that the WM scheme does not alter the magnetic resonance line shape. (C) NMOR as a function of the normalized propagation distance z/L for the single-beam Λ scheme (blue arrowed circle indicates the left scale) and the WM scheme (red arrowed circle indicates the right scale) at δB/2π = 5 Hz. (D and E) Probe NMOR angle as a function of z/L and δB31 with and without the WM field. Parameters: Embedded Image = 200 kHz, Embedded Image = 100 kHz, δp/2π = −5 GHz, δ4/2π = −2 GHz, Γ/2π = 300 MHz, γ0/2π = 10 Hz, κ = 109/(cm s), Embedded Image = 0.5. Inset in (E): Magnetic resonance line shape of the single-beam Λ scheme is identical to that shown in (D) after rescaling the vertical axis by a factor of 120.

  • Fig. 3 Theoretical optical noise–limited sensitivity, Zeeman populations, and ground-state Zeeman coherence.

    (A) Optical noise–limited sensitivity enhancement factor M as a function of Embedded Image for B ≈ 0 and z = 1 cm. Inset: sensitivity enhancement factor (dashed line) M → 1 as Embedded Image. The dotted line is the theoretical sensitivity limit of the single-beam Λ scheme (3). Parameters: κ = 109/(cm s), Embedded Image = 300 kHz, δp/2π = −5 GHz, and δ4/2π = −3 GHz. The arrow on the horizontal axis labels the value of Embedded Image that maximizes the enhancement. (B). Zeeman state populations (black solid and dashed curve; arrowed circles indicate the left scale) and ground-state Zeeman coherence (green dashed curve; arrowed circle indicates the right scale) as functions of the Zeeman frequency shift δB for the WM method. Notice that the ground-state Zeeman coherence is close to 30% of the maximum Zeeman coherence, which is significant for such a weak continuous-wave WM excitation process [Embedded Image = 200 kHz]. This contrasts sharply with the usual single-beam Λ method where the ground-state Zeeman coherence is two orders of magnitude smaller (not shown).

  • Fig. 4 Optical WM-enhanced NMOR signal in 87Rb.

    (A) Ip = 580 μW/cm2 with δp/2π = −5 GHz (F = 2 → F′ = 2) and IWM = 80 μW/cm2 with δ4/2π = −2 GHz (F = 2 → F′ = 2). The blue trace is scaled up by a factor of 270 to match the NMOR signal of the red trace. (B) Ip = 30 μW/cm2 with δp/2π = −5 GHz and IWM = 20 μW/cm2 with δ4/2π = −2 GHz. At this low probe intensity, no NMOR signal can be detected using the single-beam Λ method (blue trace). The NMOR signal is clearly seen using the optical WM method (red trace). All traces are averaged over 16 scans, each with a 5-ms duration. Temperature is 311 K. The dominant fast noise is oscilloscope electronic noise. The asymmetry in resonance line shape is mainly due to the large one-photon detunings. The vertical axis (voltage) is a measure of the photocurrent derived from a standard polarimetry detector using a precision resistor. The angle between the linear probe and WM field polarizations is 45°.

  • Fig. 5 NMOR optical noise power spectral densities.

    Noise power spectral density of data shown in Fig. 4A. Blue trace (dotted): single-beam Λ method. Red trace (solid): optical WM method. Note that the noise power spectral density of the optical WM method is comparable to the single-beam Λ method. Pink trace (top): noise power spectral density of the single-beam Λ method with 50 times higher probe intensity to match the NMOR signal of the optical WM method. Green trace: detector electronic noise. Black trace [spectrum analyzer (SA)]: the intrinsic noise of the SA.

  • Fig. 6 Signature and threshold of deep-inelastic WM and scattering process.

    (A) Normalized WM field loss (at the cell exit) as a function of WM field input intensity. Ip = 22 μW/cm2 with δp/2π = −3 GHz and δ4/2π = −2 GHz. Temperature is 311 K. Black dots and arrow indicate the right scale, and the green triangles and arrow indicate the left scale. (B) The threshold behaviors of a coherent WM process at different probe intensities. The lines are guides to the eye to show the presence of a “WM-triggered lasing threshold,” which is expected from the onset of coherent directional energy flow in the WM process. Detunings and temperature are the same as in (A). (C and D) Numerical calculation of energy loss of the circularly polarized WM field components. The significant and simultaneous energy loss of both components is characteristic of a deep-inelastic WM and scattering process. Parameters used in calculation are the same as in Fig. 2.

  • Fig. 7 Experimental setup for the inelastic optical WM-enhanced NMOR.

    Symmetric experimental setup allows simultaneous measurements of both the probe and WM light polarization components. M, mirror; BS, 50-50 beam splitter; PO, polarization optics; GL, Glen prism; BL det., balanced detector.