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arXiv:2101.11838

Vacuum fluctuations and balanced homodyne detection through ideal multi-mode photon number or power counting detectors

Kouji Nakamura

Published Jan 28, 2021• 0 citations
arXivHTMLPDF

Abstract

The balanced homodyne detection as a readout scheme of gravitational-wave detectors is carefully examined, which specifies the directly measured quantum operator in the detection. This specification is necessary to apply the quantum measurement theory to gravitational-wave detections. We clarify the contribution of vacuum fluctuations to the noise spectral density without using the two-photon formulation. We found that the noise spectral density in the two-photon formulation includes vacuum fluctuations from the main interferometer but does not includes those from the local oscillator which depends on the directly measured operators.

DOI
10.48550/arxiv.2101.11838

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The balanced homodyne detection as a readout scheme of gravitational-wave detectors is carefully examined, which specifies the directly measured quantum operator in the detection. This specification is necessary to apply the quantum measurement theory to gravitational-wave detections. We clarify the contribution of vacuum fluctuations to the noise spectral density without using the two-photon formulation. We found that the noise spectral density in the two-photon formulation includes vacuum fluctuations from the main interferometer but does not includes those from the local oscillator which depends on the directly measured operators.
Abstract The balanced homodyne detection as a readout scheme of gravitational-wave detectors is carefully examined from the quantum field theoretical point of view. The readout scheme in gravitational-wave detectors specifies the directly measured quantum operator in the detection. This specification is necessary when we apply the recently developed quantum measurement theory to gravitational-wave detections. We examine the two models of measurement. One is the model in which the directly measured quantum operator at the photodetector is Glauber’s photon number operator, and the other is the model in which the power operator of the optical field is directly measured. These two are regarded as ideal models of photodetectors. We first show these two models yield the same expectation value of the measurement. Since there is consensus in the gravitational-wave community that vacuum fluctuations contribute to the noises in the detectors, we also clarify the contributions of vacuum fluctuations to the quantum noise spectral density without using the two-photon formulation which is used in the gravitational-wave community. We found that the conventional noise spectral density in the two-photon formulation includes vacuum fluctuations from the main interferometer but does not include those from the local oscillator. Although the contribution of vacuum fluctuations from the local oscillator theoretically yields the difference between the above two models in the noise spectral densities, this difference is negligible in realistic situations.
In the context of the readout scheme for gravitational-wave detectors, the "double balanced homodyne detection" proposed in [K.~Nakamura and M.-K.~Fujimoto, arXiv:1709.01697.] is discussed in detail. This double balanced homodyne detection enables us to measure the expectation values of the photon creation and annihilation operators. Although it has been said that the operator b^θ:=cos⁡θb^1+sin⁡θb^2\hat{b}_θ:=\cosθ\hat{b}_{1}+\sinθ\hat{b}_{2}b^θ​:=cosθb^1​+sinθb^2​ can be measured through the homodyne detection in literature, we first show that the expectation value of the operator b^θ\hat{b}_θb^θ​ cannot be measured as the linear combination of the upper- and lower-sidebands from the output of the balanced homodyne detection. Here, the operators b^1\hat{b}_{1}b^1​ and b^2\hat{b}_{2}b^2​ are the amplitude and phase quadrature in the two-photon formulation, respectively. On the other hand, it is shown that the above double balanced homodyne detection enables us to measure the expectation value of the operator b^θ\hat{b}_θb^θ​ if we can appropriately prepare the complex amplitude of the coherent state from the local oscillator. It is also shown that the interferometer set up of the eight-port homodyne detection realizes our idea of the double balanced homodyne detection. We also evaluate the noise-spectral density of the gravitational-wave detectors when our double balanced homodyne detection is applied as their readout scheme. Some requirements for the coherent state from the local oscillator to realize the double balanced homodyne detection are also discussed.
Motivated by the readout scheme in interferometric gravitational-wave detectors, we consider the device which measures the expectation value of the photon annihilation and creation operators for output optical field from the main interferometer. As the result, the eight-port homodyne detection is rediscovered as such a device. We evaluate the noise spectral density in this measurement. We also briefly discuss on the application of our results to the readout scheme of gravitational-wave detectors. We call this measurement scheme to measure these expectation values as "double balanced homodyne detection."
The possibility of using squeezed states and balanced homodyne detection of gravitational waves is discussed. It is shown that the quantum noise due to high laser intensities in Michelson interferometer for gravitational waves detection can be reduced by sending squeezed vacuum states to the 'dark' port of the interferometer. The present analysis describes photon statistics measurements effects related to quadrature balanced homodyne detection showing the advantage of using this scheme for detecting gravitational waves.
Vacuum fluctuation at audio frequencies is very important and interesting in many research fields, such as the gravitational wave detection, ultra-weak magnetic field measurement, and the research of quantum metrology, etc. Since the generation of squeezed light in 1985, most of the squeezed light have been generated and measured at radio frequencies (～MHz) as there has not been much technical noise at higher frequencies. In the Michelson-interferometer-based gravitational wave detection, the detection band has frequencies from a few to tens of thousands Hz. Measuring vacuum noise at such low frequencies is a challenge since we have to stabilize and control all the audio noises and the interferences from a variety of mechanical and electronic noises, therefore a very high classical noise suppression is needed when the measurement time increases. In order to measure the squeezed light of low frequencies, the standard vacuum noise at audio frequencies must be measured. In this paper, a balanced homodyne detection system for measuring the low-frequency quantum vacuum noises is reported. It is not trivial to extend the detected frequency to very low analysis frequencies. Through a self-made self-subtraction balanced homodyne configuration, which can eliminate the DC component of each photocurrent from the photodiode and the classical common-mode technical noise, the standard vacuum noise has been detected. The linearity of the vacuum noise power has been validated by varying the local oscillator power, showing that the saturation power of light incidence is about 3.2 mW. When the incident-light power is 400 W, the standard vacuum noise is 11 dB higher than the electronic noise at 80 Hz. In the regime of about 80 Hz to 400 kHz, the linearity of the standard noise power as a function of incident laser power is verified. However, when the measurement is carried out at even lower frequencies, for example, 50 Hz, we may encounter some excess and non-stationary noises and find that the measured noise power is not proportional to the incident light power any more. These non-stationary noises are the main technical obstacle at low frequencies. The average common mode rejection ratio in the test frequency range from 10 Hz to 400 kHz is 55 dB and its maximum 63 dB at 100 Hz is obtained, implying a high suppression of the technical noise. This self-made homodyne vacuum noise detector can be widely used for precision measurement in quantum metrology and quantum optics.
We discuss a balanced homodyne detection scheme with imperfect detectors in the framework of the operational approach to quantum measurement. We show that a realistic homodyne measurement is described by a family of operational observables that depends on the experimental setup, rather than a single field quadrature operator. We find an explicit form of this family, which fully characterizes the experimental device and is independent of a specific state of the measured system. We also derive operational homodyne observables for the setup with a random phase, which has been recently applied in an ultrafast measurement of the photon statistics of a pulsed diode laser. The operational formulation directly gives the relation between the detected noise and the intrinsic quantum fluctuations of the measured field. We demonstrate this with two examples: the operational uncertainty relation for the field quadratures, and the homodyne detection of suppressed fluctuations in photon statistics.