Excess optical quantum noise in atomic sensors

Enhanced nonlinear optical response of a coherent atomic medium is the basis for many atomic sensors, and their performance is ultimately limited by the quantum fluctuations of the optical readout. Here we demonstrate that the off-resonant interactions, with the aid of the near-resonant process, can significantly modify the quantum noise of a coherent light field, even when its effect on the mean signal is negligible. The altered quantum optical noise distribution results in excess noise in the measurement quantity. We illustrate this concept by using an atomic magnetometer based on the nonlinear Faraday effect. These results show the existence of previously unnoticed factors in fundamental limitations in atomic magnetometry and could have impact in a wide range of atom-light–based precision measurements.

Figure 1

(a) Polarization rotation of a linearly polarized coherent optical field under a magnetic field, depicted in a Poincaré sphere. The initial linear polarization is along $J_x$, and the balanced polarimeter detection measures the value of $J_y$ [7]. Quantum uncertainty in the Stokes vector direction is represented by the noise ellipse (or circle, for a coherent state). (b) Output polarization of a linearly polarized light taking into account off-resonant atom-light interaction without a magnetic field. The total Hamiltonian has a shearing effect on the uncertainty circle, turning it into an ellipse, resulting in a squeezed state of light. By adjusting the phase between $x$ and $y$ polarization component $\left({\varphi }_{\mathrm{PRP}}\right)$, one can rotate the noise ellipse, making possible a sub-shot-noise detection of $J_y$. (c) When a small magnetic field is applied, the noise in the magnetic-field measurement (proportional to $J_y$ fluctuations) is above the shot-noise level, since the long axis of the noise ellipse is almost along the equator of the Poincaré sphere. Rotation of the Stokes vector around the $J_x$ axis by ${\varphi }_{\mathrm{PRP}}$, as in (b), can decrease noise but at the price of a reduced $J_y$ signal value.

Figure 2

Schematics of the experimental setup (see text for abbreviations). Inset shows the detuning of the laser with respect to Rb atomic levels.

Figure 3

(a) Minimum and maximum measured quadrature noise spectra as a function of detection frequency. The optimal conditions to detect squeezing depend on the laser power, similar to the previous results [11]. (b) Atomic density dependence of the squeezed and antisqueezed quadratures. The incident laser power is $P=4$ mW. The noise power is normalized to the shot-noise level (0 dB level); the spectrum analyzer frequency is set 200 kHz with the resolution and video bandwidths of 30 kHz and 30 Hz correspondingly. The error bars are one standard deviation.

Figure 4

Nonlinear Faraday rotation and noise power measurements for various quadratures. Magnetic field is modulated at 170 kHz (vertical dashed line). Individual curves (i)–(iv) are recorded at fixed positions of the phase-retarding plate ${\varphi }_{\mathrm{PRP}}$. The incident laser power is $P=4$ mW; the spectrum analyzer settings are $\text{RBW}=30$ kHz and $\text{VBW}=30$ Hz. The noise power is normalized to the shot-noise level, set to be 0 dB.

Figure 5

(a) Nonlinear Faraday rotation signal for a sinusoidally modulated magnetic field (${\mathrm{f}}_{\mathrm{mod}}=1717$ Hz), measured in the intensity quadrature $\left({\varphi }_{\mathrm{PRP}}=0\right)$; in the squeezing quadrature (${\varphi }_{\mathrm{PRP}}$ is set to minimize the quantum noise), the measured shot-noise level is also shown. The relevant signal is in the box; two neighboring peaks are due to technical noise. For these measurements the spectrum analyzer settings are $\text{RBW}=\text{VBW}=1$ Hz. (b) Measured quantum noise limited (solid diamonds) and estimated shot-noise limited (hollow diamonds) magnetic-field sensitivity as functions of atomic density. Inset: measured intensity quadrature noise and shot noise as a function of atomic density. The lines are to guide the eyes.