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arXiv:1805.02550 · Physical Review A

Full statistics of homodyne correlation measurements

B. Kühn, W. Vogel

Published Jul 18, 2018• 5 citations
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Abstract

We derive the full statistics of the product events in homodyne correlation measurements, involving a single mode signal, a local oscillator, a linear optical network, and two linear photodetectors. This is performed for the regime of high intensities impinging on the detectors. Our description incorporates earlier proposed homodyne correlation measurement schemes, such as the homodyne cross-correlation and homodyne intensity-correlation measurements. This analysis extends the amount of information retrieved from such types of measurements, since previously attention was paid only to the expectation value of the correlation statistics. As an example, we consider the correlation statistics of coherent, Gaussian, and Fock states. Moreover, nonclassical light is certified on the basis of the variance of the measurement outcome.

DOI
10.1103/physreva.98.013832

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We derive the full statistics of the product events in homodyne correlation measurements, involving a single mode signal, a local oscillator, a linear optical network, and two linear photodetectors. This is performed for the regime of high intensities impinging on the detectors. Our description incorporates earlier proposed homodyne correlation measurement schemes, such as the homodyne cross-correlation and homodyne intensity-correlation measurements. This analysis extends the amount of information retrieved from such types of measurements, since previously attention was paid only to the expectation value of the correlation statistics. As an example, we consider the correlation statistics of coherent, Gaussian, and Fock states. Moreover, nonclassical light is certified on the basis of the variance of the measurement outcome.
We show that the full statistics of the outputs of two detectors in a balanced homodyne detection setup involving a local oscillator in an ideal coherent state are experimentally accessible despite the excess noise existing in actual laser sources. This possibility is illustrated using phase randomized coherent states’ signals from which the statistics of Fock states can accurately be obtained. The experimental verification of the recently predicted [Phys. Rev. A98, 013832 (2018)PLRAAN1050-294710.1103/PhysRevA.98.013832] two-detector correlation probability for Fock states |1⟩ and |2⟩ is presented.
The statistics of the difference events in a balanced homodyne-detection scheme is studied without using the standard assumption of a strong, classical local oscillator. Starting from the quantum theory of photon counting, we derive the probability distribution for the difference events in the two detection channels. In the limit of the local oscillator being strong compared with the signal, the difference-number statistics tends to the statistics of the electric-field strength of the signal field. For weak signals, this limit may be obtained already far from a classical behavior of the local oscillator. While changing the local oscillator intensity, a transition of the observable concerning the signal field occurs from the photon-number difference towards the electric-field strength. Such a measurement scheme renders it possible to observe the quantum features of a coherent state or a single-photon state from the point of view of the field-strength statistics, a picture which is closely related to classical optics. The possibility to get some insight in the statistical properties of the phase difference of two microscopic fields is discussed.
Based on proposals to measure squeezing by homodyne cross correlation [Ou, Hong, and Mandel, Phys. Rev. A 36, 192 (1987)] and by homodyne intensity correlation measurements with a weak local oscillator [Vogel, Phys. Rev. Lett. 67, 2450 (1991)], the method of homodyne correlation measurements is considered in detail for the case of weak local oscillators. Aside from the feasibility of measuring squeezing in low efficiency detection, it appears possible to record the correlation of the noise of two noncommuting observables, namely, the intensity and the electric field strength of the signal field. The separation of the different contributions to the measured correlations is considered and spectral correlation measurements are analyzed. Local oscillator noise, which is not balanced out in such a scheme, is found to be attenuated effectively by decreasing the local oscillator field. In particular, in the homodyne cross correlation scheme, it is of interest to use a beam splitter that significantly differs from a 50%:50% partition.
We discuss homodyne and heterodyne statistics when squeezed light from a parametric oscillator is mixed with coherent light from a local oscillator. We also consider the case where the fields or the intensities of the two nondegenerate beams from a parametric oscillator are combined. By using a dynamic model of these beams we discuss photon sequences underlying these beams in terms of photoelectric pulse sequences recorded by a detector. Analytic expressions for quantities directly measurable in photon counting experiments are derived, using the method developed in our earlier investigations.1
The balanced homodyne detector is the paradigm of phase-sensitive detection schemes in quantum optics. We study the effects of a quantum local oscillator in a coherent state for this detector when it is coupled to a single signal mode. We calculate explicit general expressions for the output statistics of the difference current of this detector (applicable even to non-classical states), and compare them to the statistics of the so-called quadrature-phase amplitude. We show that the naive condition that the number of quanta in the local oscillator be much larger than the number in the signal is not sufficient to ensure that the homodyne detector approximates an ideal detectorof the quadrature-phase amplitude, further, we derive explicit conditions for this approximation to be achieved when the signal is a coherent state or a superposition of coherent states.
Homodyne detection is the most effective detection scheme employed in quantum optics to characterize quantum states. It is based on mixing at a beam splitter the signal to be measured with a coherent state, called the “local oscillator,” and on evaluating the difference of the photocurrents of two photodiodes measuring the outputs of the beam splitter. If the local oscillator is much more intense than the field to be measured, the homodyne signal is proportional to the signal-field quadratures. If the local oscillator is less intense, the photodiodes can be replaced with photon-number-resolving detectors, which have a smaller dynamics but can measure the light statistics. The resulting new homodyne-like detector acquires a hybrid nature, being it capable of yielding information on both the particle-like (statistics) and wave-like (phase) properties of light signals. The scheme has been tested in the measurement of the quadratures of coherent states, bracket states and phase-averaged coherent states at different intensities of the local oscillator.