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Characterization of quantum states of light by means of homodyne detection and reconstruction of Wigner functions

Andrés Martínez Silva

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

In this work, a continuous variable quantum tomography is presented. This is done using two approaches, one using simulated data and another one using data from measurements performed in the laboratory. The Radon inverse transform, under the approximation of filtered back projection, alongside the details on the implementa- tion of balanced homodyne detection needed to perform the tomography are clearly explained. Density matrices in the quadrature and number representation are also obtained from the Wigner function. Additionally, complete documentation of the computational implementation and a user-friendly interface for the tomography is reported

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In this work, a continuous variable quantum tomography is presented. This is done using two approaches, one using simulated data and another one using data from measurements performed in the laboratory. The Radon inverse transform, under the approximation of filtered back projection, alongside the details on the implementa- tion of balanced homodyne detection needed to perform the tomography are clearly explained. Density matrices in the quadrature and number representation are also obtained from the Wigner function. Additionally, complete documentation of the computational implementation and a user-friendly interface for the tomography is reported
This chapter begins with a review of classical tomography, which reconstructs hidden structures inside an object by successive transmissions of an X-ray beam at different angles and lateral displacements. By interpreting the Wigner distribution as an analogue of the density distribution in a physical object, the mathematical methods of classical tomography — the Radon transform, the Fourier slice theorem, and the inverse Radon transform — are adapted to perform a high resolution determination of a quantum state of light. In optical homodyne tomography, the classical transmission angle is replaced by the adjustable phase of a local oscillator, and the successive lateral displacements are replaced by a series of measurements of the quadrature operator defined by the local oscillator phase. Experimental data is presented showing various properties of the Wigner distribution for a coherent state.
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.
This thesis reports the realization and applications of a versatile wideband balanced homodyne detector operating in the time domain and intended for Quantum Optical Tomography. The detector can work with high repetition rate lasers used for the production and measurement of the electric field quadratures of quantum states of light. Full characterization of the detector by shot-noise measurements and by verifying its capability to reconstruct the density matrix and Wigner function of various states of interest is given. By overcoming photodiodes response mismatch, high frequency electronics issues and by choosing an appropriate optical set-up, a high level of common mode suppression (52.4dB) and a low electronic noise gave a shot-to-electronic noise ratio of I3dB for a Local Oscillator with 6.2 x 108 photons-per-pulse on a ~ 100MHz bandwidth. Results for the characterization of a single-photon Fock state are presented. An average efficiency of 56% was obtained for the generation/detection of the state.
The efficient transfer of a quantum state from photons to matter qubits in order to momentarily store information has become a central problem in quantum information processing. A quantum memory turns out to be an essential tool to achieve advanced technologies such as quantum networks, quantum repeaters, deterministic single photon sources or linear optics quantum computers. The realization of a quantum interface has been investigated in various forms, among which one can find solid-state atomic ensembles, color centers in crystal lattices, cold atomic gases, optical phonons in diamond and many others. Here we focus on a broadband quantum interface for high repetition rate (76 MHz) ultrafast optical pulses (250 fs) at telecommunication wavelength (1530 nm) based on the photon echo process occurring in semiconductor quantum dots. We evaluated the quantum state of photonic qubits in order to characterize the impact of the storage on the transmitted signal. Homodyne traces corresponding to projections of the Wigner function of the signal on rotated quadrature components were obtained using broadband balanced homodyne detection, i.e. mixing the ultrafast optical pulses to analyze with a high repetition rate pulsed local oscillator. The reconstruction of the Wigner function from the homodyne traces was performed using three algorithms: the inverse Radon transform, the minimax adaptive reconstruction and the maximum likelihood estimation. The three methods lead to similar results, concluding that for an input pulse in a coherent state, the reemitted photon echo is also in a coherent state.
We present a continuous-variable experimental analysis of a two-photon Fock state of free-propagating light. This state is obtained from a pulsed nondegenerate parametric amplifier, which produces two intensity-correlated twin beams. Counting two photons in one beam projects the other beam in the desired two-photon Fock state, which is analyzed by using a pulsed homodyne detection. The Wigner function of the measured state is clearly negative. We developed a detailed analytic model which allows a fast and efficient analysis of the experimental results.
We present an analysis of the time domain measurement of temporally resolvable quantum states using balanced homodyne detection. Our approach outlines a formalism of detecting quantum states in arbitrary temporal modes via projection of the temporal mode onto a natural detector basis. We then present an algorithm for simulating the resultant photocurrent of continuous homodyne detection in the presence of a temporally resolved mode, and use this algorithm to explore the effects of realistic measurement errors on marginal reconstruction and quantum state tomography. A complete implementation of the method is provided through open source code on a GitHub repository.