Application Cases

Experiment Name: Experimental Study of a Linear Coherent Feedback Control System

Test Equipment: High-voltage amplifier, oscilloscope, low-pass filter, controller, photodetector, piezoelectric ceramic, etc.

Experimental Process:

Figure 1: Phase locking scheme. PZT1–PZT2: Piezoelectric ceramics; HVA1–HVA2: High-voltage amplifiers; S1: 10 Hz scanning signal; S2: 2 kHz modulation signal; Mixer: Mixer; Splitter: Splitter; VF: Voltage follower; LPF: Low-pass filter; PID: Proportional-integral-derivative control circuit; OSC: Oscilloscope. D1–D2: Photodetectors; OPA: Four-wave mixing process; C: Controller; Pr: Probe light; Conj: Conjugate light.

Figure 2: Waveforms of the optical field after applying the modulation signal. 1-Errorsignal: Error signal; 2-Pr and 3-Conj: Waveforms of the probe light and conjugate light after applying the 2 kHz modulation signal, respectively.

The experimental procedure is illustrated in Figure 1. First, a 2 kHz modulation signal S2 is generated, amplified by the high-voltage amplifier HVA2, and used to drive the piezoelectric ceramic PZT2. Additionally, a 10 Hz scanning signal S1 is generated, amplified by the high-voltage amplifier HVA1, and used to drive the piezoelectric ceramic PZT1. By observing the output waveforms of the modulated optical field on the oscilloscope OSC, as shown in Figure 2, the electrical signals of the probe light beam (2-Pr) and the conjugate light field (3-Conj) resemble Lorentzian waveforms. The distance between each spike is determined by the frequency of the scanning signal, and the sawtooth-like waveform superimposed on the signal is the modulation signal. By adjusting the voltage of the modulation signal, the magnitude of the sawtooth waveform can be controlled. It is important to note that during the phase locking process, appropriate parameters for the scanning and modulation signals must be selected to ensure proper modulation of the optical field.

The DC signal detected by D2 is fed into port 1 of the splitter, where it is duplicated into two identical signals, output from ports 2 and 3 of the splitter. The signal from port 3 is directly input to the oscilloscope for observation after passing through a voltage follower VF. The voltage follower here serves to prevent signal interference from subsequent electronic components, thereby ensuring signal fidelity. The DC signal from port 2 is mixed with the scanning signal after passing through the voltage follower VF. After filtering out the high-frequency components of the mixed signal using a low-pass filter, the error signal (1-Errorsignal) is obtained. It can be observed that the maximum slope of the error signal corresponds to the position of the output signal spike. The error signal is then input to the proportional-integral-derivative controller PID, where the phase point can be directly locked by adjusting the parameters. Due to the relatively long optical path of the system, to ensure effective locking, a windbreak plate is added to mitigate air flow disturbances.

Experimental Results:

Figure 3: Intensity difference squeezing of the system output optical field under different feedback coefficients k, while keeping the amplifier pump power constant. The vertical axis represents noise power, and the horizontal axis represents scanning time. The pump power is 85 mW, and the injected probe light power is 30 µW. Figures (a), (b), and (c) show the intensity difference noise spectra (curve A) and the corresponding shot noise limits (curve B) of the system output optical field when the feedback coefficients k are 0, 0.2, and 0.4, respectively. The red lines represent the average noise levels of each curve.

The experimental results are shown in Figure 3. The black curve B represents the normalized shot noise limit, the blue curve A represents the normalized intensity difference noise of the output optical field, and the red lines indicate the average normalized noise levels of each curve. As shown in Figure 3(a), when k = 0, the system operates without feedback, which can be regarded as a single non-phase-sensitive four-wave mixing process. At this point, the system produces an intensity difference squeezing of -2.30 dB. When k = 0.2, the system produces an intensity difference squeezing of -4.03 dB, as shown in Figure 3(b). When the feedback is further increased to k = 0.4, the system produces an intensity difference squeezing of -1.19 dB, as shown in Figure 3(c). It can be observed that as the feedback strength increases, the intensity difference noise of the system also changes. Additionally, under certain feedback conditions, feedback control can enhance the intensity difference squeezing of the system. However, further increasing the feedback strength reduces the degree of intensity difference squeezing, even falling below the level without feedback, indicating over-feedback.

Figure 4: Intensity difference squeezing of the system output optical field under different feedback coefficients k, with amplifier pump powers of 65 mW, 75 mW, and 85 mW. The vertical axis represents intensity difference squeezing (IDS), and the horizontal axis represents the feedback coefficient k. The solid lines represent the experimentally obtained intensity difference squeezing, while the dashed lines represent the theoretical intensity difference squeezing considering system losses. Assuming the transmission coefficients of the probe light and conjugate light in the atomic cell are Ta = 0.9 and Tb = 0.95, respectively, the optical path transmission loss is l = 0.18, and the detection efficiency is η = 0.9. Red dots: 65 mW; Blue squares: 75 mW; Green diamonds: 85 mW.

As shown in Figure 4, under different pump powers, there exists an optimal feedback parameter k∗ that maximizes the quantum characteristics of the system. When P = 65 mW, k∗ lies in the range [0.25, 0.35], and the maximum intensity difference squeezing of the ensemble is |2.65 dB. When P = 75 mW, k∗ lies in the range [0.20, 0.30], and the maximum intensity difference squeezing of the ensemble is |3.54 dB. When P = 85 mW, k∗ lies in the range [0.15, 0.25], and the maximum intensity difference squeezing of the ensemble is |4.03 dB. Measurements under other pump powers reveal that higher pump powers correspond to smaller optimal feedback coefficients k∗, which can also be derived from theoretical calculations. The actual measurement results align with the theoretical simulation trends, but there is an approximate 1 dB deviation in absolute values, possibly due to other losses.

High-Voltage Amplifier Recommendation: ATA-7020

Figure: ATA-7020 High-Voltage Amplifier Specifications

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