The stability of the two lock points are plotted as a function of squeezing factor in Figure 3.
2.1 Case (a) Noise Locking squeezed vacuum on a Homodyne detector 2.1.1 Derivation of the noise locking error signal
This signal is then demodulated and low pass filtered to give the error signal. The band pass filtered output is sent to an envelope detector which gives an output proportional to the real envelope of the input. All the electronics used to derive the error signal from the bandpass filter onwards are identical. Here only one output port of the beamsplitter is detected and the output of the photodetector is sent to the BPF. In the case of NL two coherent fields, the input fields ^ a ( t ) and ^ b ( t ) are set to have similar coherent amplitude. The output of the balanced homodyne detector is sent to the bandpass filter (BPF). In the case of NL of squeezed vacuum on the homodyne detector, ^ a ( t ) is the squeezed state and ^ b ( t ) the local oscillator (LO). For the two NL cases considered in this paper, case (a) the squeezed state on the homodyne detector and case (b) the phase of two coherent fields, the input fields and photodetection differs slightly. The output fields, ^ d ( t ) and ^ c ( t ), are incident on the photodetectors PD1 and PD2. These operators satisfy the standard commutation relations. Figure 1 shows the input fields, ^ a ( t ) and ^ b ( t ), with relative phase denoted θ that interfere on a balanced beamsplitter. In this section, the error signal and lock stability of NL are derived theoretically. The output of the envelope detector is demodulated and low pass filtered (LPF). To derive the NL error signal, the output of the homodyne is bandpass filtered (BPF) then envelope detected (ED). The output fields, ^ d ( t ) and ^ c ( t ), are incident on the photodetectors, PD1 and PD2. Here the local oscillator beam ^ b ( t ) passes through a phase modulator (PM) with applied sinusoidal modulation at frequency Ω / 2 π. 2 Theory of Noise Locking Figure 1: Setup of a balanced homodyne detector with input fields, ^ a ( t ) and ^ b ( t ), interfering with relative phase, θ, on a balanced beamsplitter. We conclude with a discussion of NL and its applications.
In Section 3.2, experimental results of a squeezed vacuum spectrum taken using NL for homodyne phase control are shown and we present error signal spectra which agree qualitatively with the results derived in Section 2. We also analyze the NL stability by measuring the error signal spectra and then compare it to that of a coherent modulation locking (CML) technique. In Section 3.1, we show results from the coherent NL experiment of the error signals and lock acquisition. In Section 3, we analyze NL experimentally, using both the coherent NL experiment setup and the squeezed vacuum on a homodyne detector. We also show the performance is degraded by losses and detector inefficiency, as uncorrelated vacuum fluctuations couple into the signal. Perhaps fortunately, we find the lock stability to be superior when locked to the squeezed quadrature rather than the anti-squeezed quadrature. The stability is dependent on the level of squeezing/anti-squeezing (fringe visibility in the case of coherent field NL) and also on which quadrature is locked to. We find the NL lock stability improves weakly with increasing detection bandwidth ( Δ ω 1 / 4 dependence) in contrast to standard coherent locking techniques, where increasing detection bandwidth reduces the lock stability.
Then we present theoretical results for the NL of coherent fields. We derive the NL error signal and calculate the theoretically achievable stability. In Section 2, we theoretically analyze the control of a squeezed vacuum state on a balanced homodyne detector, although the formalism is general enough to apply to other systems.
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