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Abstract
A novel noise-suppressing and lock-free interferometer is proposed and experimentally demonstrated in the study of the quantum non-destructive (QND) interaction of cold atoms. A QND measurement based on far-off resonant dispersive probing is usually carried out by a Mach-Zehnder type interferometer. It is an experimental challenge in its own right to reduce the classical noise, such as acoustic noise, phase noise and amplitude noise of lasers, and to lock the interferometer at the white-light position that corresponds to a nearly zero path-length difference. Here, we report an interferometer with an inserted acousto-optic modulator (AOM). It is noise immune and lock-free in principle. The experiments show that the new interferometer is able to measure cold atoms for more than 30 minutes and reduce the phase noise by about 30 dB.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spin squeezed states (SSS) of atomic ensembles have been used to dramatically reduce the measurement uncertainty limited by quantum projection noise in atomic clocks [1–4]. Such states can be generated using an off-resonant quantum non-destructive (QND) interaction with a coherent light beam [5–7]. The QND measurement of the clock state population difference is usually realized by detecting the state dependent phase shift of the off resonant probe laser beams using a Mach-Zehnder interferometer (MZI) [8–10]. In these researches, a stable and sensitive interferometer is essential to measure the phase shift of probe light. The main noise source influencing the phase shift is classical noise, such as classical amplitude and classical phase noise of laser, and acoustic noise. The classical laser amplitude noise can be canceled by locking the interferometer at a half fringe. The classical phase noise of laser can be suppressed by aligning the Mach-Zehnder interferometer (MZI) closed to zero optical path-length difference, which is called the white-light position [11]. Acoustic noise is caused by any disturbances which perturb the path length difference between the two interferometer arms, including mechanical vibration of mirrors, optical mounts, air fluxes etc. These noises have a great influence on the accuracy of light phase measurement.
In recent years, some techniques and instruments have been developed for the acoustic noise suppression for interferometers [8,12]. A commonly used technique is isolating the surroundings and interferometer platform as much as possible. For example, in order to isolate the free-space interferometer from acoustic noise, one can shield an interferometer in a box of a sound damping material glued on aluminum sheets or make use of locking technique that exploits an off-resonant CW laser propagating through the interferometer collinear with the probe beam. Another solution is to use differential interferometer based on Mach-Zehnder type interferometer [12]. However, in principle, these methods cannot perfectly eliminate noise.
In this paper, we report a new interferometer structure, which is similar to Sagnac interferometer [13–15]. Compared to the MZI structure as shown in Fig. 1(b), it can completely eliminate acoustic noise in principle. We place an AOM working at central frequency of 1 GHz in the interferometer structure. The two output light beams in the interferometer have the same frequency and the same path. Therefore, AOM-assisted interferometer (AAI) structure is immune to vibration and always keep the interferometer operating in white-light position without any need of locking process.
Fig. 1. (a) A sketch of the AOM-assisted interferometer. BS, beam splitters; PD1 and PD2, photodetectors; MOT, Rubidium atoms trapped in MOT; HR, high-reflection mirror; PZT, piezoelectric tube; L1 and L2, f=150 mm lenses; AOM, Free Space Acousto-Optic Modulator, its frequency shift is 1 GHz. (b) Sketch of a conventional interferometer for phase shift detection in previous experiments.
2. Experimental setup and theory
The scheme of our experimental setup is shown in Fig. 1(a). The incident light is divided into two beams by beamsplitter (BS). The incident light is detuned by a certain frequency chosen through theoretical calculation. The transmission light as probe light (red line) is injected into magneto-optical trap (MOT) and then interacts with cold atoms. However, the reflected light as reference light (blue line) passes through the AOM (Brimrose, 410-472-7070) working at central frequency of 1 GHz. After diffraction by AOM, the 0th order diffraction light is blocked and the +1st order diffraction light is produced with about 1 GHz frequency shift. The PZT is placed in the optical path to facilitate testing the noise suppression performance of the interferometer. We can use white noise as the source to drive PZT, so as to obtain the spectrum characteristics of the interferometer on the dynamic signal analyzer. Quarter-wave plate before the PZT can rotate the polarization direction of the incident linearly polarized light by 90 degrees to obtain linearly polarized light orthogonal to the polarization of the incident light. Due to the frequency shift of 1 GHz, the laser incident in this direction basically does not interact with atoms. After passing through the atoms, the probe light and the reference light are adjusted to completely coincide. According to the optical reversibility, it can be known that the +1st order diffraction light after AOM frequency shift of the probe light also coincides with the reference optical path. Finally, the wavelengths of probe and reference beams at the output port of interferometer are completely same, and the optical paths traveled are exactly the same. Therefore, we find that this interferometer is always in the white light position, no matter how the optical path of the interferometer changes. In experiment, we adjust the interferometer to keep the interferometer in white-light position where it is insensitive to classical phase noise and sensitive to the atom induced phase shift. The phase shift of probe light influenced by atoms is detected by the change of voltage in PD1 and PD2.
In experiments, the atomic population can influence the phase of an optical probe field on a transition between the hyperfine ground states and an excited state. The probe light is used as a meter to measure the cold atoms. We analyze the detection principle of the AAI, and obtain the phase shift of a probe beam propagating through the cloud of 87Rb atom. The expressions of the phase shift and absorption of light passing through atoms were derived in Ref. [8]. In the case of alkali metal D line $J \to J^{\prime}$ transition, between states having total electronic angular momentum of J and $J^{\prime}$, the phase shift and absorption of light passing through this refractive index medium are expressed as follows:
As shown in Fig. 2, We calculate the phase shift (red curve) and absorption (blue curve) of the light imposed by the cold 87Rb atoms. Through theoretical calculations, we obtain the interaction strength between atoms and light at different frequencies. In experiment, we choose the probe optical frequency around 384228.235 GHz that is about 120 MHz detuned from the cycling transition ${5^2}{S_{{1 / 2}}}({F = 2} )\to {5^2}{P_{{3 / 2}}}({F^{\prime} = 3} )$, where the absorption is close to zero, and the phase shift is 0.023 rad when a million atoms are trapped in MOT. The reference light is detuned by the frequency shift about 1.12 GHz. The phase shift and absorption of reference light influenced by atoms is negligible. According to experimental conditions, the density ${N_F}$ = 1×109 cm−3, and the length of cold atomic clouds $l$=2 mm. The phase shift is induced by an ensemble of cold multi-level atom. For the monitoring of the atomic phase shift, we consider the experimental situation illustrated in Fig. 1(b), where an AOM-assisted interferometer is placed around the atomic sample.
Fig. 2. Phase shift (red curve) and absorption (blue curve) of cold 87Rb atoms at different frequencies of light. The left part is $F = 2 \to F^{\prime} = 3$ transition, the right part is $F = 1 \to F^{\prime} = 2$ transition. The probe light is detuned by 120 MHz and the reference light is detuned by 1.12 GHz.
3. Experimental results and discussion
In the experiment, the atomic sample is prepared in a standard six-beam Rb MOT. The dispersive measurement of the state population is realized by detecting the state dependent phase shift of the off-resonant probe laser light. As shown in Fig. 3, the ability of this configuration for the acoustic noise suppression is demonstrated by the interference signals with driving the PZT by white noise. We measure the power spectral density (PSD) of the phase noise of MZI and AAI driven by the same white noise source. As shown in Fig. 3(a), When the amplitude of white noise is set to 1 V, we obtained the PSD of the phase noise of MZI and AAI below the frequency of 12.8 KHz. We find the new interferometer structure reduces the noise by about 30 dB in the frequency range of 5 Hz to 100 Hz. We also find that the ability of noise suppression of AAI gradually decreases with the increase of frequency due to the limitation of PZT response frequency. As shown in Fig. 3(a), the PSD of the phase noise of the MZI and AAI signal is shown when the white noise voltage is set to 0.1 V and 1 V. We attribute the noise at the frequency range around 100 Hz and below 10 Hz in AAI to the changes in mirrors tilt angle and rotation of optical platform. In the time domain, as shown in Fig. 3(b), the interferometric signal of the new interferometer structure with the PZT driven by 2 Hz with 8 µm displacement is almost the same as the signal without PZT modulation. Figure 3(c) shows that this configuration enables us to perform stable phase shift measurements in a sufficiently long-time for cold atoms. According to the fluctuation of the voltage in the PD2 shown in Fig. 3(c), the stability of the phase shift of the interferometer can be deduced. Figure 3(d) shows the Allan deviation of the phase shift in 30 minutes. When the averaging time reaches 100 seconds, the stability of phase shift can be 4.5×10−4 rad. When the averaging time increases over 100 seconds, the influence of some low frequency fluctuation in the lab gradually appears.
Fig. 3. (a) The power spectral density (PSD) of the phase noise of MZI and AAI driven by the white noise. When the driving voltage of PZT is set to 1 V, we obtained the PSD of the phase noise of MZI (red line) and AAI (green line) below the frequency of 12.8 KHz. When the driving voltage is set up to 0.1 V, the PSD of the phase noise of MZI (blue line) and AAI (pink line) is shown. The noise of the AAI (black line) and MZI (cyan line) without driving voltage. (b) The interferometric signal in PD2. When PZT is driven by the frequency of 2 Hz with 8 µm displacement (red line) and PZT is not driven (blue line). (c) The output signal of AAI (blue line). (d) The Allan Deviation of the phase shift derived from (c).
In order to further characterize the stability of the system for the phase shift measurement, we applied the interferometer to measure the 87Rb atomic population in ground state F=2. We applied the probe light with the frequency around 384228.235 GHz that is about 120 MHz blue-detuned from the cycling transition ${5^2}{S_{{1 / 2}}}({F = 2} )\to {5^2}{P_{{3 / 2}}}({F^{\prime} = 3} )$. In the experiment, the intensities of the probe light and reference light are 0.5 mw/cm2 and 0.025 mw/cm2 respectively. This set corresponds to a lowest photon scattering rate per atom of 0.14 /s, which is orders of magnitude lower than that of a typical fluorescence measurement with resonant light. And the light shift of atomic transition caused by the reference light is about 0.047 MHz. The atom-induced phase shift due to this light shift is about 9.1×10−7 rad which is negligible in measurement. As shown in Fig. 4(a), we obtained the loading and releasing process of trapped atoms in MOT in a long period time of 200 seconds. In this measuring process, we turned on and off the magnetic field and the MOT light simultaneously to release and recapture atoms. According to the Eq. (1), the phase shift of the probe light induced by atoms is detected by PD1 and PD2 along with the switching of the magnetic field and MOT light. In this long period time, the detected phase shift signal is on the same level. This result shows that AAI has the ability to stably monitor atomic population distribution for a long time. As shown in Fig. 4(b), we also studied the data of the phase shift measurements in the case of switching off the MOT light and magnetic coils at different detunings of the probe light. The results fit the theoretical model shown in Fig. 2. The detuning and intensity of the probe light can be set to appropriate values to meet requirements of scattering rate in different experiments.
Fig. 4. Loading process and release process of cold atoms in MOT measured by the AAI in a long period time about 200 seconds. (a) The phase shift of probe light (red line) influenced by atoms with the switching of the magnetic field and the MOT light simultaneously (blue line). (b) The phase shift measurements in the case of switching off the MOT light and magnetic coils at different detunings of the probe light. The experiment data (diamond) and theoretical simulation (red line).
4. Conclusion
A novel noise-suppressing and lock-free interferometer is report. The probe light and the reference light incident into the MOT with different frequencies as the probe light is diffracted by AOM working at central frequency of 1 GHz, while the output frequency of the two beams is the same value. Because the two light beams go through the same optical path, the interferometer always operated in the white-light position. When the length of interferometer changes, the phase shift of two light beams is completely the same. Therefore, the AOM-assisted interferometer keeps the phase stabilization of output light, enabling measurements of the atoms in a long period time and reduces the noise about 30 dB compared with the Mach-Zehnder type interferometer. This configuration makes the interferometer work in the white-light position without any need of locking methods. For QND measurements of the SSS in atomic ensemble, this interferometer can be used to achieve or even surpass the shot noise limit in the field of light-atom interaction such as atomic clocks, atom interferometers and optical lattices [16–19].
Funding
National Key Research and Development Program of China (2017YFA0304202); National Natural Science Foundation of China (11474254, 11804298); Fundamental Research Funds for the Central Universities (2017QN81005).
Disclosures
The authors declare no conflicts of interest.
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