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.
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.
The technique of dc-balanced, pulsed homodyne detection for the purpose of determining optical-field statistics on short time scales is analyzed theoretically. Such measurements provide photon-number and phase distributions associated with a repetitive signal light field in a short time window. Time- and space-varying signal and local-oscillator pulses are treated, thus generalizing earlier treatments of photoelectron difference statistics in homodyne detection. Experimental issues, such as the effects of imperfect detector balancing on (time-integrated) dc detection and the consequences of background noise caused by non-mode-matched parts of the multimode signal field, are analyzed. The Wigner, or joint, distribution for the two field-quadrature amplitudes during the sampling time window can be directly determined by tomographic inversion of the measured photoelectron distributions. It is pointed out that homodyne detection provides a new method for the simultaneous measurement of temporal and spectral information. Although the theory is generally formulated, with both signal and local-oscillator fields being quantized, emphasis is placed on the limit of a strong, coherent local-oscillator field, making semiclassical interpretation possible.
Driven by single photon detection requirements, the theory of arrays of off-on detectors has been developed and applied. However, for a comprehensive characterization of nonclassicality, one also needs phase sensitive properties. This missing link is introduced with the derived theory of phase sensitive click counting measurements. It unifies the balanced homodyne detection for high intensities with the click detection in the few-photon regime. We formulate a hierarchy of nonlinear squeezing conditions to probe quantum effects beyond standard squeezing. Imperfections stemming from fluctuations, detector efficiency, and dark count rates are considered. Experimentally accessible sampling formulas are given. Our theory paves the way towards novel applications of light in quantum metrology.
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.
The eight-port homodyne detector is an optical circuit designed to perform the monitoring of two quadratures of an optical field: the signal. By using quantum Bose fields and quantum stochastic calculus, we give a complete quantum description of this apparatus, when used as quadrature detector in continuous time. We can treat either the traveling waves in the optical circuit or the observables involved in the detection part: two couples of photodiodes, postprocessing of the output currents\dots{}. The analysis includes imperfections, such as not perfectly balanced beam splitters, detector efficiency, electronic noise, phase and intensity noise in the laser acting as local oscillator; this last noise is modeled by using mixtures of field coherent states as statistical operator of the laser component. Due to the monitoring in continuous time, the output is a stochastic process and its full probability distribution is obtained. When the output process is sampled at discrete times, the quantum description can be reduced to discrete mode operators, but at the price of having random operators, which contain also the noise of the local oscillator. Consequently, the local oscillator noise has a very different effect on the detection results with respect to an additive noise, such as the noise in the electronic components. As an application, the problem of secure random-number generation is considered, based on the local oscillator shot noise. The rate of random bits that can be generated is quantified by the min-entropy; the possibility of classical and quantum side information is taken into account by suitable conditional min-entropies. The final rate depends on which parts of the apparatus are considered to be secure and on which ones are considered to be exposed to the intervention of an intruder. In some experimentally realistic situations, the entropy losses are computed, depending on the values of the parameters quantifying the imperfections.
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:=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 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 and 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 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.
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.