A single-photon Fock state has been generated by means of conditional preparation from a two-photon state emitted in the process of spontaneous parametric down-conversion. A recently developed high-frequency homodyne tomography technique has been used to completely characterize the Fock state by means of a pulse-to-pulse analysis of the detectors' difference photocurrent. The density matrix elements of the generated state have been retrieved with a final detection efficiency of about 57%. A comparison has been performed between the phase-averaged tomographic reconstructions of the Wigner function as obtained from the measured density-matrix elements and from a direct Abel transform of the homodyne data. The ability of our system to work at the full repetition rate of the pulsed laser (82 MHz) substantially simplifies the detection scheme, allowing for more ``exotic'' quantum states to be generated and analyzed.
We present a homodyne detection scheme to verify Bell's inequality on correlated optical beams at the output of a nondegenerate parametric amplifier. Our approach is based on tomographic measurement of the joint detection probabilities, which allows high quantum efficiency at detectors. A self-homodyne scheme is suggested to simplify the experimental set-up.
Homodyne tomography provides a way for measuring generic field-operators. Here we analyze the determination of the most relevant quantities: intensity, field, amplitude and phase. We show that tomographic measurements are affected by additional noise in comparison with the direct detection of each observable by itself. The case of of coherent states has been analyzed in details and earlier estimations of tomographic precision are critically discussed.
We propose to experimentally test the nonclassicality of quantum states through homodyne tomography. For single-mode states we check violations of inequalities involving the photon-number probability. For two-mode states we test the nonclassicality by reconstructing some suitable number-operator functions. The test can be performed with available quantum efficiency of homodyne detection, by measuring the pertaining quantities on the corresponding noisy states.
We present a continuous-variable experimental analysis of a two-photon Fock state of free-propagating light. This state is obtained from a pulsed non-degenerate 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.
Using a recent result of Albini et al. to represent quantum homodyne tomography in terms of a single observable (as a normalized positive operator measure) we construct a generalized Markov kernel which transforms (the measurement outcome statistics of) this observable into (the measurement outcome statistics of) a covariant phase space observable. We also consider the inverse question. Finally, we add some remarks on the quantum theoretical justification of the experimental implementations of these observables in terms of balanced homodyne and 8-port detection techniques, respectively.
We show that data from homodyne-like detection based on photon-number-resolving (PNR) detectors may be effectively exploited to reconstruct quantum states of light using the tomographic reconstruction techniques originally developed for homodyne detection based on photodiodes. Our results open new perspectives to quantum state reconstruction, and pave the way to the use of PNR-based homodyne-like detectors in quantum information science.
Displaced Fock states of the electromagnetic field have been synthesized by overlapping the pulsed optical single-photon Fock state |1> with coherent states on a high-reflection beamsplitter and completely characterized by means of quantum homodyne tomography. The reconstruction reveals highly non-classical properties of displaced Fock states, such as negativity of the Wigner function and photon number oscillations. This is the first time complete tomographic reconstruction has been performed on a highly non-classical optical state.
Quantum properties of optical modes are typically assessed by observing their photon statistics or the distribution of their quadratures. Both particle- and wave-like behaviours deliver important information, and each may be used as a resource in quantum-enhanced technologies. Weak-field homodyne detection provides a scheme which combines the wave- and particle-like descriptions. Here we show that it is possible to observe a wave-like property such as the optical coherence across Fock basis states in the detection statistics derived from discrete photon counting. We experimentally demonstrate …
We propose a general method for measuring an arbitrary observable of a multimode electromagnetic field using homodyne detection with a single local oscillator. In this method the local oscillator scans over all possible linear combinations of the modes. The case of two modes is analyzed in detail and the feasibility of the measurement is studied on the basis of Monte-Carlo simulations. We also provide an application of this method in tomographic testing of the GHZ state.
We show that the full statistics of the two detectors outputs in a balanced homodyne detection setup involving a local oscillator in an ideal coherent state is 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 [Kühn and Vogel, Phys. Rev. A, \textbf{98}, 013832 (2018)] two-detector correlation probability for Fock states is presented for states $\vert 1 \rangl…
Pulsed homodyne quantum tomography usually requires a high detection efficiency limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency ($<50\%$) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian type. This result is obtained by applying the so-called "minimax" adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our…
Random-phase homodyne tomography of the field intensity is a concrete example of the Quantum Roulette of Helstrom. In this paper we give the explicit POM of such measurement and compare it with direct photodetection and heterodyne detection. Effects of nonunit quantum efficiency are also considered. Naimark extensions for the roulette POM are analyzed and its experimental realization is discussed.
The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The coherent states tomograms are shown to be described by normal distributions with varying dispersions and means. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments.
We propose a method to couple metastable flux-based qubits to superconductive resonators based on a quantum-optical Raman excitation scheme that allows for the deterministic generation of stationary and propagating microwave Fock states and other weak quantum fields. Moreover, we introduce a suitable microwave quantum homodyne technique, with no optical counterpart, that enables the measurement of relevant field observables, even in the presence of noisy amplification devices.
Time-domain balanced homodyne detection is performed on two well-separated temporal modes sharing a single photon. The reconstructed density matrix of the two-mode system is used to prove and quantify its entangled nature, while the Wigner function is employed for an innovative tomographic test of Bell's inequality based on the theoretical proposal by Banaszek and Wodkiewicz [Phys. Rev. Lett. 82, 2009 (1999)]. Provided some auxiliary assumptions are made, a clear violation of Banaszek-Bell's inequality is found.
We derive various sampling functions for multimode homodyne tomography with a single local oscillator. These functions allow us to sample multimode s-parametrized quasidistributions, density matrix elements in Fock basis, and s-ordered moments of arbitrary order directly from the measured quadrature statistics. The inevitable experimental losses can be compensated by proper modification of the sampling functions. Results of Monte Carlo simulations for squeezed three-mode state are reported and the feasibility of reconstruction of the three-mode Q-function and s-ordered moments from 10^7 sample…
We derive quantum trajectories (also known as stochastic master equations) that describe an arbitrary quantum system probed by a propagating wave packet of light prepared in a continuous-mode Fock state. We consider three detection schemes of the output light: photon counting, homodyne detection, and heterodyne detection. We generalize to input field states that are superpositions and or mixtures of Fock states and illustrate the formalism with several examples.
We experimentally demonstrate the generation of multi-photon Fock states with up to three photons in well-defined spatial-temporal modes synchronized with a classical clock. The states are characterized using quantum optical homodyne tomography to ensure mode selectivity. The three-photon Fock states are probabilistically generated by pulsed spontaneous parametric down conversion at a rate of one per second, enabling complete characterization in 12 hours.
Standard quantum state reconstruction techniques indicate that a detection efficiency of $0.5$ is an absolute threshold below which quantum interferences cannot be measured. However, alternative statistical techniques suggest that this threshold can be overcome at the price of increasing the statistics used for the reconstruction. In the following we present numerical experiments proving that quantum interferences can be measured even with a detection efficiency smaller than $0.5$. At the same time we provide a guideline for handling the tomographic reconstruction of quantum states based on ho…
Heralded single photons are prepared at a rate of ~100 kHz via conditional measurements on polarization-nondegenerate biphotons produced in a periodically poled KTP crystal. The single-photon Fock state is characterized using high frequency pulsed optical homodyne tomography with a fidelity of (57.6 +- 0.1)%. The state preparation and detection rates allowed us to perform on-the-fly alignment of the apparatus based on real-time analysis of the quadrature measurement statistics.
We have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The single-photon Fock state |1> was prepared using conditional measurements on photon pairs born in the process of parametric down-conversion. A probability distribution of the phase-averaged electric field amplitudes with a strongly non-Gaussian shape is obtained with the total detection efficiency of (55+-1)%. The angle-averaged Wigner function reconstructed from this distribution shows a strong dip reaching classically impossible negat…
We report the implementation of a high-rate source of single and two-photon states. By combining the advantages of short pulses and cavities, heralding rates up to 250kHz with 88% fidelity have been obtained for the single photons as well as 800Hz with 82% fidelity for the two-photon states. Furthermore, we developed a setup where the homodyne measurement is conditioned by the heralding of the quantum states, enabling the detection of most of the heralded events. This allows a faster characterization of the photon source leading to an increase in the fidelities up to 91% and 88% respectively f…
For the visualization of quantum states, the approach based on Wigner functions can be very effective. Homodyne detection has been extensively used to obtain the density matrix, Wigner functions and tomographic reconstructions of optical fields for many thermal, coherent or squeezed states. Here, we use time-domain optical homodyne tomography for the quantum state recognition and reconstruction of the femtosecond optical field from a nonequilibrium superradiant coherent electron-hole state formed in a semiconductor laser structure. We observe severe deviations from the Poissonian statistics of…
Ideal photon-number-resolving detectors form a class of important optical components in quantum optics and quantum information theory. In this article, we theoretically investigate the potential of multiport devices having reconstruction performances approaching that of the Fock-state measurement. By recognizing that all multiport devices are minimally complete, we first provide a general analytical framework to describe the tomographic accuracy (or quality) of these devices. Next, we show that a perfect multiport device with an infinite number of output ports functions as either the Fock-stat…
Detecting entanglement of non-Gaussian continuous-variable states from single-copy homodyne measurements
National University of Singapore（新加坡国立大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit（MajuLab，法国国家科学研究中心-尼斯大学-新加坡国立大学-南洋理工大学国际联合研究单位）；National Institute of Education, Nanyang Technological University（南洋理工大学国立教育学院）
We discuss the spectral structure and decomposition of multi-photon states. Ordinarily `multi-photon states' and `Fock states' are regarded as synonymous. However, when the spectral degrees of freedom are included this is not the case, and the class of `multi-photon' states is much broader than the class of `Fock' states. We discuss the criteria for a state to be considered a Fock state. We then address the decomposition of general multi-photon states into bases of orthogonal eigenmodes, building on existing multi-mode theory, and introduce an occupation number representation that provides an …
We investigate the capabilities of loss-tolerant quantum state characterization using a photon-number resolving, time-multiplexed detector (TMD). We employ the idea of probing the Wigner function point-by-point in phase space via photon parity measurements and displacement operations, replacing the conventional homodyne tomography. Our emphasis lies on reconstructing the Wigner function of non-Gaussian Fock states with highly negative values in a scheme that is based on a realistic experimental setup. In order to establish the concept of loss-tolerance for state characterization we show how lo…
Using a continuous-wave type-II optical parametric oscillator below threshold, we have demonstrated a novel source of heralded single-photons with high-fidelity. The generated state is characterized by homodyne detection and exhibits a 79% fidelity with a single-photon Fock state (91% after correction of detection loss). The low admixture of vacuum and the well-defined spatiotemporal mode are critical requirements for their subsequent use in quantum information processing.
Using cavity-enhanced non-degenerate parametric downconversion, we have built a frequency tunable source of heralded single photons with a narrow bandwidth of 8 MHz, making it compatible with atomic quantum memories. The photon state is 70% pure single photon as characterized by a tomographic measurement and reconstruction of the quantum state, revealing a clearly negative Wigner function. Furthermore, it has a spectral brightness of ~1,500 photons/s per MHz bandwidth, making it one of the brightest single photon sources available. We also investigate the correlation function of the down-conve…
We demonstrate, theoretically and experimentally, that statistical mixtures of the vacuum state |0> and the single-photon Fock state |1> are nonclassical according to the Vogel criterion (W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)), regardless of the vacuum fraction. The ensembles are synthesized via conditional measurements on biphotons generated by means of parametric downconversion, and their quadrature statistics are measured using balanced homodyne detection. A comparative review of various quantum state nonclassicality criteria is presented.
A wide range of experiments studying microwave photons localized in superconducting cavities have made important contributions to our understanding of the quantum properties of radiation. Propagating microwave photons, however, have so far been studied much less intensely. Here we present measurements in which we reconstruct the Wigner function of itinerant single photon Fock states and their superposition with the vacuum using linear amplifiers and quadrature amplitude detectors. We have developed efficient methods to separate the detected single photon signal from the noise added by the ampl…
The production of conditional quantum states and quantum operations based on the result of measurement is now seen as a key tool in quantum information and metrology. We propose a new type of photon number detector. It functions non-deterministically, but when successful, it has high fidelity. The detector, which makes use of an n-photon auxiliary Fock state and high efficiency Homodyne detection, allows a tunable tradeoff between fidelity and probability. By sacrificing probability of operation, an excellent approximation to a photon number detector is achieved.
We report on the preparation of entangled two mode squeezed states of yet unseen quality. Based on a measurement of the covariance matrix we found a violation of the Reid and Drummond EPR-criterion at a value of only 0.36\pm0.03 compared to the threshold of 1. Furthermore, quantum state tomography was used to extract a single photon Fock state solely based on homodyne detection, demonstrating the strong quantum features of this pair of laser-beams. The probability for a single photon in this ensemble measurement exceeded 2/3.
We employ the quantum state of a single photon entangled with the vacuum (|1,0>-|0,1>), generated by a photon incident upon a symmetric beam splitter, to teleport single-mode quantum states of light by means of the Bennett protocol. Teleportation of coherent states results in truncation of their Fock expansion to the first two terms. We analyze the teleported ensembles by means of homodyne tomography and obtain fidelities of up to 99 per cent for low source state amplitudes. This work is an experimental realization of the quantum scissors device proposed by Pegg, Phillips and Barnett (Phys. Re…
Conventional Glauber coherent states (CS) can be defined in several equivalent ways, e.g., by displacing the vacuum or, explicitly, by their infinite Poissonian expansion in Fock states. It is well known that these definitions become inequivalent if applied to finite $d$-level systems (qudits). We present a comparative Wigner-function description of the qudit CS defined (i) by the action of the truncated displacement operator on the vacuum and (ii) by the Poissonian expansion in Fock states of the Glauber CS truncated at $(d-1)$-photon Fock state. These states can be generated from a classical…
We address conditional de-Gaussification of continuous variable states by inconclusive photon subtraction (IPS) and review in details its application to bipartite twin-beam state of radiation. The IPS map in the Fock basis has been derived, as well as its counterpart in the phase-space. Teleportation assisted by IPS states is analyzed and the corresponding fidelity evaluated as a function of the involved parameters. Nonlocality of IPS states is investigated by means of different tests including displaced parity, homodyne detection, pseudospin, and displaced on/off photodetection. Dissipation a…
A detailed theoretical analysis of the spatiotemporal mode of a single photon prepared via conditional measurements on a photon pair generated in the process of parametric down-conversion is presented. The maximum efficiency of coupling the photon into a transform-limited classical optical mode is calculated and ways for its optimization are determined. An experimentally feasible technique of generating the optimally matching classical mode is proposed. The theory is applied to a recent experiment on pulsed homodyne tomography of the single-photon Fock state (A. I. Lvovsky et al., Phys Rev. Le…
Quantum tomography is the standard method of reconstructing the Wigner function of quantum states of light by means of balanced homodyne detection. The reconstruction quality strongly depends on the photodetectors quantum efficiency and other losses in the measurement setup. In this article we analyse in detail a protocol of enhanced quantum tomography, proposed by Leonhardt and Paul in 1994, which allows one to reduce the degrading effect of detection losses. It is based on phase sensitive parametric amplification, with the phase of the amplified quadrature being scanned synchronously with th…
The development of useful photon-photon interactions can trigger numerous breakthroughs in quantum information science, however this has remained a considerable challenge spanning several decades. Here we demonstrate the first room-temperature implementation of large phase shifts ($\approxπ$) on a single-photon level probe pulse (1.5us) triggered by a simultaneously-propagating few-photon-level signal field. This process is mediated by $Rb^{87}$ vapor in a double-$Λ$ atomic configuration. We use homodyne tomography to obtain the quadrature statistics of the phase-shifted quantum fields and per…
We demonstrate the generation of multi-photon quantum states of light by cavity-enhanced parametric down-conversion in the high-repetition-rate pulsed regime. An external enhancement cavity resonant with the spectral comb of modes of a mode-locked pump laser provides a coherent build-up of the pump intensity and greatly enhances the parametric gain without sacrificing its high repetition rate and comb structure. We probe the parametric gain enhancement by the conditional generation and tomographic analysis of two-photon Fock states. Besides its potential impact to efficiently generate highly-n…
