Quantum states of light having a Wigner function with negative values represent a key resource in quantum communication and quantum information processing. Here, we present the generation of such a state at the telecommunication wavelength of 1550nm. The state is generated by means of photon subtraction from a weakly squeezed vacuum state and is heralded by the `click' of a single photon counter. Balanced homodyne detection is applied to reconstruct the Wigner function, also yielding the state's photon number distribution. The heralding photons are frequency up-converted to 532nm to allow for the use of a room-temperature (silicon) avalanche photo diode. The Wigner function reads W(0,0)=-0.063 +/- 0.004 at the origin of phase space, which certifies negativity with more than 15 standard deviations.
We derive sampling functions for estimation of quantum state fidelity with Schrödinger cat-like states, which are defined as superpositions of two coherent states with opposite amplitudes. We also provide sampling functions for fidelity with squeezed Fock states that can approximate the cat-like states and can be generated from Gaussian squeezed states by conditional photon subtraction. The fidelities can be determined by averaging the sampling functions over quadrature statistics measured by homodyne detection. The sampling functions are designed such that they can compensate for losses and i…
An analysis of the homodyne tomography process that is often used to determine the Wigner functions of quantum optical states is performed to consider the effects of the spatiotemporal degrees of freedom. The homodyne tomography process removes those parts of the input state that are not associated with the mode of the local oscillator by tracing out those degrees of freedom. Using a functional approach to incorporate all the spatiotemporal degrees of freedom, we find that this reduction in the degrees of freedom introduces distortions in the observed Wigner function. The analysis also shows h…
I present a novel algorithm for reconstructing the Wigner function from homodyne statistics. The proposed method, based on maximum-likelihood estimation, is capable of compensating for detection losses in a numerically stable way.
We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner function and the marginal distribution and discretize Fourier space. We show that this technique solves most technical difficulties encountered with kernel deconvolution based methods and reconstructs overall better and smoother Wigner functions. We also give estimators of the reconstruction errors for both methods and show improvement in noise handling proper…
We present an experimental demonstration of the power of real-time feedback in quantum metrology, confirming a theoretical prediction by Wiseman regarding the superior performance of an adaptive homodyne technique for single-shot measurement of optical phase. For phase measurements performed on weak coherent states with no prior knowledge of the signal phase, we show that the variance of adaptive homodyne estimation approaches closer to the fundamental quantum uncertainty limit than any previously demonstrated technique. Our results underscore the importance of real-time feedback for reaching …
Low-noise, efficient, phase-sensitive time-domain optical detection is essential for foundational tests of quantum physics based on optical quantum states and the realization of numerous applications ranging from quantum key distribution to coherent classical telecommunications. Stability, bandwidth, efficiency, and signal-to-noise ratio are crucial performance parameters for effective detector operation. Here we present a high-bandwidth, low-noise, ultra-stable time-domain coherent measurement scheme based on balanced homodyne detection ideally suited to characterization of quantum and classi…
The accuracy in determining the quantum state of a system depends on the type of measurement performed. Homodyne and heterodyne detection are the two main schemes in continuous-variable quantum information. The former leads to a direct reconstruction of the Wigner function of the state, whereas the latter samples its Husimi $Q$~function. We experimentally demonstrate that heterodyne detection outperforms homodyne detection for almost all Gaussian states, the details of which depend on the squeezing strength and thermal noise.
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on $\mathbb R^2$ which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < η\leq 1$, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the $\mathbb L_\infty$-risk over a class of infinitely differentiable. We compute also the lower …
We investigate exponential phase moments of the s-parametrized quasidistributions (smoothed Wigner functions). We show that the knowledge of these moments as functions of s provides, together with photon-number statistics, a complete description of the quantum state. We demonstrate that the exponential phase moments can be directly sampled from the data recorded in balanced homodyne detection and we present simple expressions for the sampling kernels. The phase moments are Fourier coefficients of phase distributions obtained from the quasidistributions via integration over the radial variable …
We discuss excess noise contributions of a practical balanced homodyne detector in Gaussian-modulated coherent-state (GMCS) quantum key distribution (QKD). We point out the key generated from the original realistic model of GMCS QKD may not be secure. In our refined realistic model, we take into account excess noise due to the finite bandwidth of the homodyne detector and the fluctuation of the local oscillator. A high speed balanced homodyne detector suitable for GMCS QKD in the telecommunication wavelength region is built and experimentally tested. The 3dB bandwidth of the balanced homodyne …
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…
We directly sample the exponential moments of the canonical phase for various quantum states from the homodyne output. The method enables us to study the phase properties experimentally, without making the detour via reconstructing the density matrix or the Wigner function and calculating the phase statistics from them. In particular, combing the measurement with a measurement of the photon-number variance, we verify fundamental number-phase uncertainty.
In this work, we consider the estimation of single mode Gaussian states using four different measurement schemes namely: i) homodyne measurement, ii) sequential measurement, iii) Arthurs-Kelly scheme, and iv) heterodyne measurement, with a view to compare their relative performance. To that end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out involved calculations. We show that the optimal performance of the Arthurs-Kelly scheme and the sequential measurement is equal to the heterodyne measur…
The eight-port homodyne detection apparatus is analyzed in the framework of the operational theory of quantum measurement. For an arbitrary quantum noise leaking through the unused port of the beam splitter, the positive operator valued measure and the corresponding operational homodyne observables are derived. It is shown that such an eight-port homodyne device can be used to construct the operational quantum trigonometry of an optical field. The quantum trigonometry and the corresponding phase space Wigner functions are derived for a signal field probed by a classical local oscillator and a …
We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems. The state is represented through the Wigner function, a generalized probability density on $\mathbb{R}^2$ which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The effect of the losses due to detection inefficiencies, which are always present in a real experiment, is the addition to the tomographic data of independent Gaussian noise. We construct a kernel estimator for the Wigner function, prove tha…
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.
Optical homodyne tomography is discussed in the context of classical image processing. Analogies between these two fields are traced and used to formulate an iterative numerical algorithm for reconstructing the Wigner function from homodyne statistics.
We experimentally demonstrate the reconstruction of a photon number conditioned state without using a photon number discriminating detector. By using only phase randomized homodyne measurements, we reconstruct up to the three photon subtracted squeezed vacuum state. The reconstructed Wigner functions of these states show regions of pronounced negativity, signifying the non-classical nature of the reconstructed states. The techniques presented allow for complete characterization of the role of a conditional measurement on an ensemble of states, and might prove useful in systems where photon cou…
We derive a theoretical framework for the experimental certification of non-Gaussian features of quantum states using double homodyne detection. We rank experimental non-Gaussian states according to the recently defined stellar hierarchy and we propose practical Wigner negativity witnesses. We simulate various use-cases ranging from fidelity estimation to witnessing Wigner negativity. Moreover, we extend results on the robustness of the stellar hierarchy of non-Gaussian states. Our results illustrate the usefulness of double homodyne detection as a practical measurement scheme for retrieving i…
A pulsed balanced homodyne detector has been developed for precise measurements of electric field quadratures of pulsed optical quantum states. A high level of common mode suppression (> 85 dB) and low electronic noise (730 electrons per pulse) provide a signal to noise ratio of 14 dB for the measurement of the quantum noise of individual pulses. Measurements at repetition rates up to 1 MHz are possible. As a test, quantum tomography of the coherent state is performed and the Wigner function and the density matrix are reconstructed with a 99.5% fidelity. The detection system can also be used f…
We study a class of phase-space distribution functions that is generated from a Gaussian convolution of the Wigner distribution function. This class of functions represents the joint count probability in simultaneous measurements of position and momentum. We show that, using these functions, one can determine the expectation value of a certain class of operators accurately, even if measurement data performed only with imperfect detectors are available. As an illustration, we consider the eight-port homodyne detection experiment that performs simultaneous measurements of two quadrature amplitud…
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…
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 funct…
We propose a method for characterizing a photodetector by directly reconstructing the Wigner functions of the detector's Positive-Operator-Value-Measure (POVM) elements. This method extends the works of S. Wallentowitz and Vogel [Phys. Rev. A 53, 4528 (1996)] and Banaszek and Wódkiewicz [Phys. Rev. Lett. 76, 4344 (1996)] for quantum state tomography via weak-field homodyne technique to characterize quantum detectors. The scheme uses displaced thermal mixtures as probes to the detector and reconstructs the Wigner function of the photodetector POVM elements from its outcome statistics. In order …
Telecommunication wavelength with well-developed optical communication technologies and low losses in the waveguide are advantageous for quantum applications. However, an experimental generation of non-classical states called non-Gaussian states at the telecommunication wavelength is still underdeveloped. Here, we generate highly-pure-single-photon states, one of the most primitive non-Gaussian states, by using a heralding scheme with an optical parametric oscillator and a superconducting nano-strip photon detector. The Wigner negativity, the indicator of non-classicality, of the generated sin…
We experimentally demonstrate that a non-classical state prepared in an atomic memory can be efficiently transferred to a single mode of free-propagating light. By retrieving on demand a single excitation from a cold atomic gas, we realize an efficient source of single photons prepared in a pure, fully controlled quantum state. We characterize this source using two detection methods, one based on photon-counting analysis, and the second using homodyne tomography to reconstruct the density matrix and Wigner function of the state. The latter technique allows us to completely determine the mode o…
Many applications in quantum communication, sensing and computation need provably quantum non-Gaussian light. Recently such light, witnessed by a negative Wigner function, has been estimated using homodyne tomography from a single atom dispersively coupled to a high-finesse cavity. This opens an investigation of quantum non-Gaussian light for many experiments with atoms and solid-state emitters. However, at their early stage, an atom or emitter in a cavity system with different channels to the environment and additional noise are insufficient to produce negative Wigner functions. Moreover, hom…
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…
The characterization of quantum features in large Hilbert spaces is a crucial requirement for testing quantum protocols. In the continuous variables encoding, quantum homodyne tomography requires an amount of measurements that increases exponentially with the number of involved modes, which practically makes the protocol intractable even with few modes. Here we introduce a new technique, based on a machine learning protocol with artificial Neural Networks, that allows to directly detect negativity of the Wigner function for multimode quantum states. We test the procedure on a whole class of nu…
One of the central foundational questions of physics is to identify what makes a system quantum as opposed to classical. One seminal notion of classicality of a quantum system is the existence of a non-contextual hidden variable model as introduced in the early work by Bell, Kochen and Specker. In quantum optics, the non-negativity of the Wigner function is a ubiquitous notion of classicality. In this work we establish an equivalence between these two concepts. In particular, we show that any non-contextual hidden variable model for Gaussian quantum optics has an alternative non-negative Wigne…
Radial Coarse Graining Restores Vacuum Majorization in Wigner Phase Space
School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；ICTP-AP, University of Chinese Academy of Sciences（中国科学院大学 ICTP-AP）
We present a multimode theory of non-Gaussian operation induced by an imperfect on/off-type photon detector on a splitted beam from a wideband squeezed light. The events are defined for finite time duration $T$ in the time domain. The non-Gaussian output state is measured by the homodyne detector with finite bandwidh $B$. Under this time- and band-limitation to the quantm states, we develop a formalism to evaluate the frequency mode matching between the on/off trigger channel and the conditional signal beam in the homodyne channel. Our formalism is applied to the CW and pulsed schemes. We expl…
We investigate the possibilities of preserving and manipulating the coherence of atomic two-level systems by ideal projective homodyne detection and feedback. For this purpose, the photon emission process is described on time scales much shorter than the lifetime of the excited state using a model based on Wigner-Weisskopf theory. The backaction of this emission process is analytically described as a quantum diffusion of the Bloch vector. It is shown that the evolution of the atomic wavefunction can be controlled completely using the results of homodyne detection. This allows the stabilization…
We demonstrate the reconstruction of the Wigner function from marginal distributions of the motion of a single trapped particle using homodyne detection. We show that it is possible to generate quantum states of levitated optomechanical systems even under the effect of continuous measurement by the trapping laser light. We describe the opto-mechanical coupling for the case of the particle trapped by a free-space focused laser beam, explicitly for the case without an optical cavity. We use the scheme to reconstruct the Wigner function of experimental data in perfect agreement with the expected …
We present a conditional experiment involving a parametric amplifier and an avalanche photodetector to generate highly nonclassical states of the radiation field. The nonclassicality is robust against amplifier gain, detector efficiency and dark counts. At the output all the generalized Wigner functions have negative values, and this is exploited in order to reveal the nonclassicality through quantum homodyne tomography.
Recently, a non-Gaussian field, which may be a useful basis for entanglement distillation and efficient quantum teleportation, has been experimentally produced by subtracting a photon from a squeezed Gaussian field. We investigate the nonclassicality of such the field using negativity in the Wigner function and the existence of positive well-defined $P$ function. We obtain the condition to see negativity in the Wigner function for the case including the mixed Gaussian incoming field, the threshold photodetection and the inefficient homodyne measurement. We show how similar the photon-subtracte…
We experimentally verify the quantum non-Gaussian character of a conditionally generated noisy squeezed single-photon state with positive Wigner function. Employing an optimized witness based on probabilities of squeezed vacuum and squeezed single-photon states we prove that the state cannot be expressed as a mixture of Gaussian states. In our experiment, the non-Gaussian state is generated by conditional subtraction of a single photon from squeezed vacuum state. The state is probed with a homodyne detector and the witness is determined by averaging a suitable pattern function over the measure…
We demonstrate a state reconstruction technique which provides either the Wigner function or the density matrix of a field mode and requires only avalanche photodetectors, without any phase or amplitude discrimination power. It represents an alternative, of simpler implementation, to quantum homodyne tomography.
The Wigner function for one and two-mode quantum systems is explicitely expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, also the density operator of those systems is written in terms of the marginal distribution of these quadratures. Some examples to apply this formalism, and a reduction to the usual optical homodyne tomography are considered.
It is a long-standing belief, as pointed out by Bell in 1986, that it is impossible to use a two-mode Gaussian state possessing a positive-definite Wigner function to demonstrate nonlocality as the Wigner function itself provides a local hidden-variable model. In particular, when one performs continuous-variable (CV) quadrature measurements upon a routinely generated CV entanglement, namely, the two-mode squeezed vacuum (TMSV) state, the resulting Wigner function is positive-definite and as such, the TMSV state cannot violate any Bell inequality using CV quadrature measurements. We show here, …
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.
A quantitative analysis is conducted on the impacts of experimental imperfections in the input state, the detector properties, and their interactions on photon-subtracted squeezed vacuum states in terms of a quantum non-Gaussian character witness and Wigner function. Limitations of the non-classicality and quantum non-Gaussian characteristic of Schrödinger kitten states are identified and addressed. The detrimental effects of a photon-number detector on the generation of odd Schrödinger kitten state at near-infrared wavelengths ($\sim$ 860 nm) and telecommunication wavelengths ($\sim$ 1550 nm)…
The non-linearity of a conditional photon-counting measurement can be used to `de-Gaussify' a Gaussian state of light. Here we present and experimentally demonstrate a technique for photon number resolution using only homodyne detection. We then apply this technique to inform a conditional measurement; unambiguously reconstructing the statistics of the non-Gaussian one and two photon subtracted squeezed vacuum states. Although our photon number measurement relies on ensemble averages and cannot be used to prepare non-Gaussian states of light, its high efficiency, photon number resolving capabi…
Why do we need quantization to describe vision? What are the quadrature operators of the electromagnetic field? Is it possible to measure them? What are the characteristic functions useful for? In this brief tutorial we provide the theoretical tools needed to describe the generation, manipulation and characterization of optical quantum states and of the main passive (beam splitters) and active (squeezers) devices involved in experiments, such as the Hong-Ou-Mandel interferometer and the continuous-variable quantum teleportation. We also introduce the concept of operator ordering and the descri…
We implement the direct sampling of negative phase-space functions via unbalanced homodyne measurement using click-counting detectors. The negativities significantly certify nonclassical light in the high-loss regime using a small number of detectors which cannot resolve individual photons. We apply our method to heralded single-photon states and experimentally demonstrate the most significant certification of nonclassicality for only two detection bins. By contrast, the frequently applied Wigner function fails to indicate such quantum characteristics for the quantum efficiencies present in ou…
We have studied theoretical un-symmetric multi-photon subtracted twin beam state and demonstrated a method for generating states that resembles to high photon number states with the increase in the number of subtracted photons through Wigner distribution function, which can be reconstructed experimentally by Homodyne measurement. A crucial point is high non-classicality is obtained by photon subtraction when mean photons per mode of twin beam state is low. We have calculated photon statistics from the phase space distribution function and found sub-poissonian behaviour in the same low mean pho…
A fundamental element of quantum information processing with photonic qubits is the nonclassical quantum interference between two photons when they bunch together via the Hong-Ou-Mandel (HOM) effect. Ultimately, many such pure photons must be processed in complex interferometric networks, and for this it is essential to synchronize the arrival times of the flying photons preserving their purity. Here we demonstrate for the first time the HOM interference of two heralded, pure optical photons synchronized through two independent quantum memories. Controlled storage times up to 1.8 microseconds …
