Method for the differential measurement of phase shifts induced by atoms in an optical ring cavity

We demonstrate a method of light phase shift measurement using a high-finesse optical ring cavity which exhibits reduced phase noise due to cavity length fluctuations. Two laser beams with a frequency difference of one cavity free spectral range are simultaneously resonant with the cavity, demonstrating noise correlations in the error signals due to the common-mode cavity length fluctuations. The differential error signal shows a 30 dB reduction in cavity noise down to the noise floor in a frequency range up to half the cavity linewidth ($\delta \nu /2\simeq 30$ kHz). Various noise sources are analyzed and their contributions to the noise floor are evaluated. Additionally, we apply this noise-reduced phase shift measurement scheme in a simulated spin-squeezing experiment where we have achieved a factor of 40 improvement in phase sensitivity with a phase resolution of 0.7 mrad, which may remove one important barrier against attaining highly spin-squeezed states. The demonstrated method provides a flexible situation by using an optical ring cavity and two independent beams. This method can find direct application to nondestructive measurements in quantum systems, such as for the generation of spin-squeezed states in atom interferometers and atomic clocks.

Figure 1

(a) Nondestructive phase shift measurement with reduced noise due to cavity length fluctuations. Laser cooled atoms (red circle) interact with the fundamental mode of an optical ring cavity and induce a shift $\mathrm{\Delta }$ of the cavity resonance frequency. Two beams (Ref and Probe; solid red and dotted blue lines, respectively) are coupled to the cavity in counterpropagating directions and the reflections are collected by two photodetectors (PD1 and PD2). (b) Simplified level diagram of the reference and probe beams with respect to the atomic transitions. The probe is close to the atomic resonance while the reference is far detuned. (c) Two beams are resonant with two modes of the cavity therefore the PDH error signals display common cavity-length fluctuations. In the differential scheme the atom-induced phase shift $\mathrm{\Delta }$ can be resolved while the common-mode cavity noise can be suppressed.

Figure 2

Experimental setup. BD1: Two AOMs induce a frequency difference $\mathrm{FSR}\simeq 1.43$ GHz between the two beams, indicated by the solid red (upper) and dotted blue (lower) lines, respectively. AOM2 is driven by RF2, to which the FSK modulation can be applied. Two EOMs are driven at 10.5 MHz from the same source but with tunable relative phase and amplitude. BD2: Cavity, mode-matching optics, and detection system. See text for details. Abbreviations: BD, breadboard; PBS, polarizing beamsplitter; AOM, acousto-optic modulator; EOM, electro-optic modulator; RF, radio frequency; FSK, frequency shift key; OI, optical isolator; PD, photodetector; CCD, charge-coupled device; SG, signal generator; LO, local oscillator; MOD, phase modulation; SP, splitter; MX, mixer; PI, proportional-integral controller; BPF, band-pass filter; LPF, low-pass filter; ADC, analog-to-digital converter; M, mirror; E, error signal.

Figure 3

Cavity noise cancellation results. Traces 1 and 2 (solid red and solid green, respectively) show the frequency PSD of the error signals before and after the noise cancellation, showing about 30 dB noise reduction in a frequency range up to half the cavity linewidth ($\delta \nu /2\simeq 30$ kHz). The noise of the canceled error signal is close to the noise floor (trace 3, solid orange) determined when the laser is off-resonance with the cavity. Also shown are the residual noise due to the difference in the frequency response of the two channels (trace 4, solid gray), the laser relative intensity noise (dotted blue trace), the quantization noise (dashed black trace) due to ADC, and the fiber phase noise (dash-dotted purple trace).

Figure 4

FSK modulation and sensitivity to laser frequency shift. (a) FSK modulation with 2 kHz laser frequency shift. The black trace is the trigger of the FSK; P1 and P2 are $T_m=200\text{−}\mu \mathrm{s}$-long probes with a delay time of 1 ms, representing the measurement sequence of a typical squeezing experiment. (b) A series of frequency shifts ranging from 20 Hz to 2 kHz is applied, each with ten acquisitions. The blue squares and red circles show the value of $\delta P$ for $E_2$ and $E$, respectively; error bars represent the standard deviation of 10 acquisitions. The red line is a linear fit of the red circle data. Inset shows the calculated phase resolutions of $E_2$ and $E$, respectively.

Figure 5

Photodetector noise measured with incident thermal light at different power levels ($60\mu \mathrm{W}, 40\mu \mathrm{W}, 20\mu \mathrm{W}$) and the background noise floor without light (top to bottom). The PSDs of PSN at different power levels are indicated by the dashed lines. Inset shows the simplified schematic of the photodetector; $R_f=100\mathrm{k}\mathrm{\Omega }$.