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International Journal of Quantum Information

Homodyne-like detection scheme based on photon-number-resolving detectors

Alessia Allevi, Matteo Bina, Stefano Olivares, Maria Bondani

Published Dec 1, 2017• 13 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749917400160

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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.
The measurement of quantum states of light is fundamental for the investigations of Quantum Optics and for the applications of Quantum Information. Here we present the implementation of a novel detection apparatus, that is a homodyne-like detection scheme based on photon-number-resolving detectors. Thanks to its hybrid nature, the apparatus is useful to investigate both the particle-like and wave-like properties of light. We tested the scheme in the measurement of the quadratures of different optical states, namely coherent states and mixtures of coherent states. In particular, we investigate the performance of our scheme compared to that of a standard homodyne detector.
In the standard homodyne configuration, an unknown optical state is combined with a local oscillator (LO) on a beam splitter (BS). Good quadrature measurements require a high-amplitude LO and two high-efficiency photodiodes whose signals are subtracted and normalized. By changing the LO phase, it is then possible to infer the optical state in the mode matching the LO. For quantum information processing, the states of interest are in well-separated modes, corresponding to a pulsed configuration with one relevant LO mode per measurement. We theoretically investigate what can be learned about the unknown optical state by counting photons in one or both outgoing paths after the BS, keeping the LO mode fixed but choosing its phase and magnitude. We consider measurement configurations where the BS acts differently on different sets of matching modes. When the BS acts identically on all matching modes it is possible to determine the content of the unknown optical state in the mode matching the LO conditional on each number of photons in the orthogonal modes on the same path. In particular, if both the phase and the intensity of the LO can be varied, then the statistics of just one of the counters is enough to infer these parameters, while in the case of an LO with fixed intensity both detectors are needed to accomplish this. Our results are derived by demonstrating a bijection, or lack thereof, between the probability distributions over the space of outcomes of the counter(s) and certain parameters of the unknown state for different measurement configuration. We report an experiment that was conducted to demonstrate the theory in the case where the BS acts differently depending on the polarization.
We investigate the quantization of Kramers-Kronig detection technique initially developed for classical optical communications. It consists in mixing the unknown field with a strong monochromatic local oscillator on an unbalanced beam splitter. A single output of the beam splitter undergoes a direct detection of the optical intensity by means of a single photodiode. When the measured output verifies signal processing constraints, namely, the minimal phase and the single-sideband constraints, Kramers-Kronig detection reconstructs the phase of the signal from the intensity measurements via a digitally computed Hilbert transform. The local oscillator being known, Kramers-Kronig detection allows for reconstructing the quadratures of the unknown field. We show that this result holds in the quantum regime up to first order in the local oscillator amplitude and thus that Kramers-Kronig detection acts as a coherent detection able to measure both quadratures, making it a Gaussian measurement similar to double homodyne detection. We also study in detail the phase information measured by Kramers-Kronig detection for bosonic coherent states, monomode pure states, and mixed states. Finally, we propose and investigate a spectral tomography protocol for single-photon states that is inspired by Kramers-Kronig detection and relies on a spectral engineering of the single photon.
The homodyne detection is one of the most basic tools for identifying the quantum state of light. It has been used to detect useful non-local properties, such as entanglement for the quantum teleportation and distillability of a secret key in quantum key distribution. In so doing, the detection scheme employs a bright optical pulse, called the local oscillator (LO) pulse, and the LO pulse is usually transmitted along with the signal pulses. The LO pulse is presumed to be a coherent state with an infinite intensity. However, it is difficult in practice to hold this presumption owing to noise in the optical transmission channels or an intervention by a malicious third party. As a result, the implementation may no longer be the homodyne detection, and those outcomes may merely disguise successful detection of entanglement or a secret key. Here, we present an alternative scheme that works as the homodyne detection to detect the non-local properties of light in a verifiable manner, without any presumption for the LO pulses. This scheme is essentially based on the same setup as the conventional implementation for the homodyne detection. This result contributes to close any possible loophole in the homodyne detection caused by the deviation from the ideal LO pulses.
We propose a homodyne-like detection scheme involving photon-number-resolving detectors to discriminate between two coherent states affected by either uniform or gaussian phase noise. A proof-of-principle experiment is performed employing two hybrid photodetectors, whose outputs are used in post processing to calculate the shot-by-shot photon-number differences. The performance of the strategy is quantified in terms of the error probability in discriminating the noisy coherent signals as a function of the characteristic noise parameters.