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Physical Review A

Unbalanced homodyning for quantum state measurements

Sascha Wallentowitz, W. Vogel

Published Jun 1, 1996• 301 citations
Publisher

Abstract

We propose the reconstruction of the quantum state of light from the homodyne photocounting statistics of a single, realistic photodetector. Contrary to the development of homodyning over the last decade, our approach is based on unbalanced detection with a weak local oscillator. Representing the quantum state in terms of s-parametrized quasiprobability distributions, the method even allows one to determine the Wigner function provided the quantum efficiency of the detector is sufficiently large. We show that perturbing effects due to classical noise of the local oscillator are small. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.4528

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We propose the reconstruction of the quantum state of light from the homodyne photocounting statistics of a single, realistic photodetector. Contrary to the development of homodyning over the last decade, our approach is based on unbalanced detection with a weak local oscillator. Representing the quantum state in terms of s-parametrized quasiprobability distributions, the method even allows one to determine the Wigner function provided the quantum efficiency of the detector is sufficiently large. We show that perturbing effects due to classical noise of the local oscillator are small. \textcopyright{} 1996 The American Physical Society.
A method is introduced that allows one to measure normal-ordered moments of the displaced photon-number operator up to higher orders, without the need of photon-number resolving detectors. It is based on unbalanced homodyne correlation measurements, with the local oscillator being replaced by a displaced dephased laser. The measured moments yield a simple approximation of quasiprobabilities, representing the full quantum state. Quantum properties of light are efficiently certified through normal-ordered observables directly accessible by our method, which is illustrated for a weakly squeezed vacuum and a single-photon-added thermal state.
Multichannel unbalanced homodyning is proposed for measuring the density matrix of a single-mode optical field in the photon-number representation. Combining the signal beam and the local oscillator by a hightransmittance beam splitter, the interfering field is detected using multichannel photocounting. The density matrix is determined by direct statistical sampling of the coincident events recorded for various values of the local-oscillator phase. The usefulness of the method is demonstrated by computer simulations of measurements including error estimations.
The experimental schemes for measuring quantum-mechanical phase properties of light suggested and partly also realized thus far, namely (i) amplification, with the help of a quantum amplifier, of the microscopic field before phase measurement, (ii) heterodyning the field with a strong local oscillator, and (iii) performing two separate homodyne measurements on the field after beam splitting, are compared from a theoretical point of view.They share the common feature that undesired noise enters the experimental setup, which makes the measurement fuzzy.It will be pointed out that all three schemes amount to measuring the Q function of the original field, and hence are fully equivalent.Since the Q function can be interpreted as a smoothed Wigner function, one may associate with the introduced noise a smoothing process in which intrinsically quantum-mechanical features displayed by finer details of the Wigner function -especially by the occurrence of negative values -are lost.As a consequence, the measured phase distribution will be broader than the "true" one based on the concept of a quantum-mechanical phase operator.In realistic experiments, the nonunit detection efficiency further deteriorates the measuring results.It will be shown that also this echect can be properly described by an (additional) smoothing process leading to a certain s-parametrized quasiprobability distribution, with a parameter s that is connected with the detection efficiency in a simple way, as the distribution that is actually measurable.
The determination of the quantum properties of a single-mode radiation field by heterodyne or double homodyne detection is studied. The realistic case of not fully efficient photodetectors is considered. It is shown that a large amount of quite precise information is available, whereas the completeness of such information is also discussed. Some examples are given and the special case of states expressed as a finite superposition of number states is considered in some detail. \textcopyright{} 1996 The American Physical Society.
The Wigner function is a quasi-probability distribution of quantum states of light that describes the probability amplitude of any state through their marginal distribution. The Wigner function can be reconstructed through quantum tomography which uses the information obtained from light quadratures to reconstruct a 3d image of the Wigner function. Light quadratures can be obtained by means of balanced homodyne detection (BHD). BHD uses a controlled interference between the quantum field to be characterized and a local oscillator, which is a coherent source with well-defined frequency and phase. The main feature of a BHD is its ability to measure extremely small noise signals, even below the quantum vacuum limit for squezeed light. To obtain precise measurements in BHD it is necessary to characterize the noise sources in the electronics of the BHD such as Johnson noise and fundamental noises such as shot noise. To characterize the noise in the BHD, time based and frequency-based measurements were performed to find the quantum to classical noise ratio (QCNR). QCNR measures the capacity of the BHD to distinguish quantum noise. A high QCNR allows us to make reliable measurements of the light quadratures. For this project we developed a circuit based on the one presented by Lang et al. in low noise balanced homodyne detector for quantum noise measurements.
Systems of on-off detectors are well established for measuring radiation fields in the regime of small photon numbers. We propose to combine these detector systems with unbalanced homodyning with a weak local oscillator. This approach yields phase-space functions, which represent the click counterpart of the s parametrized quasiprobabilities of standard photoelectric detection theory. This introduced class of distributions can be directly sampled from the measured click-counting statistics. Therefore, our technique visualizes nonclassical effects without further data processing. Surprisingly, a small number of on-off diodes can yield more insight than perfect photon number resolution. Quantum signatures in the particle and wave domain of the quantized radiation field, as shown by photon number and squeezed states, respectively, will be uncovered in terms of negativities of the sampled phase-space functions. Application in the vast fields of quantum optics and quantum technology will benefit from our efficient nonclassicality characterization approach.