We present a general model to account for the multimode nature of the quantum electromagnetic field in projective photon-counting measurements. We focus on photon-subtraction experiments, where non-gaussian states are produced conditionally. These are useful states for continuous-variable quantum information processing. We present a general method called mode reduction that reduces the multimode model to an effective two-mode problem. We apply this method to a multimode model describing broadband parametric downconversion, thereby improving the analysis of existing experimental results. The main improvement is that spatial and frequency filters before the photon detector are taken into account explicitly. We find excellent agreement with previously published experimental results, using fewer free parameters than before, and discuss the implications of our analysis for the optimized production of states with negative Wigner functions.
An oscillatory correlation function has been observed by the coincidence counting of multimode two-photon pairs produced with a degenerate optical parametric oscillator far below threshold. The coherent superposition of the multimode two-photon pairs provides the oscillation in the intensity correlation function. The experimental data are well fitted to a theoretical curve.
We address the reconstruction of the full photon distribution of multimode fields generated by seeded parametric down-conversion (PDC). Our scheme is based on on/off avalanche photodetection assisted by maximum-likelihood (MaxLik) estimation and does not involve photon counting. We present a novel constrained MaxLik method that incorporates the request of finite energy to improve the rate of convergence and, in turn, the overall accuracy of the reconstruction.
We present a method for computing the action of conditional linear optical transformations, conditioned on photon counting, for arbitrary signal states. The method is based on the Q-function, a quasi probability distribution for anti normally ordered moments. We treat an arbitrary number of signal and ancilla modes. The ancilla modes are prepared in an arbitrary product number state. We construct the conditional, non unitary, signal transformations for an arbitrary photon number count on each of the ancilla modes.
We investigate the implementation of binary projective measurements with linear optics. This problem can be viewed as a single-shot discrimination of two orthogonal pure quantum states. We show that any two orthogonal states can be perfectly discriminated using only linear optics, photon counting, coherent ancillary states, and feedforward. The statement holds in the asymptotic limit of large number of these physical resources.
Experimental reconstructions of photon number distributions of both continuous-wave and pulsed light beams are reported. Our scheme is based on on/off avalanche photodetection assisted by maximum-likelihood estimation and does not involve photon counting. Reconstructions of the distribution for both semiclassical and quantum states of light are reported for single-mode as well as for multimode beams.
Lossless filtering of a single coherent (Schmidt) mode from spatially multimode radiation is a problem crucial for optics in general and for quantum optics in particular. It becomes especially important in the case of nonclassical light that is fragile to optical losses. An example is bright squeezed vacuum generated via high-gain parametric down conversion or four-wave mixing. Its highly multiphoton and multimode structure offers a huge increase in the information capacity provided that each mode can be addressed separately. However, the nonclassical signature of bright squeezed vacuum, photo…
We develop a multimode theory of direct homodyne measurements of quantum optical quasidistribution functions. We demonstrate that unbalanced homodyning with appropriately shaped auxiliary coherent fields allows one to sample point-by-point different phase space representations of the electromagnetic field. Our analysis includes practical factors that are likely to affect the outcome of a realistic experiment, such as non-unit detection efficiency, imperfect mode matching, and dark counts. We apply the developed theory to discuss feasibility of observing a loophole-free violation of Bell's ineq…
We investigate the possibility of implementing a given projection measurement using linear optics and arbitrarily fast feedforward based on the continuous detection of photons. In particular, we systematically derive the so-called Dolinar scheme that achieves the minimum error discrimination of binary coherent states. Moreover, we show that the Dolinar-type approach can also be applied to projection measurements in the regime of photonic-qubit signals. Our results demonstrate that for implementing a projection measurement with linear optics, in principle, unit success probability may be approa…
Quantum state discrimination is a central problem in quantum measurement theory, with applications spanning from quantum communication to computation. Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more pow…
The 'Ghost' interference experiment is analyzed when the source of entangled photons is a multimode Optical Parametric Amplifier(OPA) whose weak limit is the two-photon Spontaneous Parametric Downconversion(SPDC) beam. The visibility of the double-slit pattern is calculated, taking the finite coincidence time window of the photon counting detectors into account. It is found that the coincidence window and the bandwidth of light reaching the detectors play a crucial role in the loss of visibility on coincidence detection, not only in the 'Ghost' interference experiment but in all experiments in…
In this thesis we present a direct scheme for measuring quasidistribution functions of light. This scheme, based on photon counting, is derived from a simple relation linking the Wigner function with photon statistics. We develop a full multimode theory of the scheme, and show that the principle of the measurement can be straightforwardly generalized to the detection of the multimode radiation. We also discuss practical aspects of the scheme, including statistical error, non-unit detection efficiency, and imperfect interference visibility. Next, we report experimental realization of the scheme…
A complete characterization of quantum fluctuations in many-body systems is accessible through the full counting statistics. We present an exact computation of statistical properties of light in a basic model of light-matter interaction: a multimode photonic field coupled to a single two-level emitter. We mostly consider an initial coherent state in a given mode and demonstrate how the original Poissonian statistics gets modified because of quantum many-body scattering effects leading to non-Poissonian distributions. We argue that measuring this statistics in a simple quantum optical setup pro…
In contrast to the fully projective limit of strong quantum measurement, where the evolution is locked to a small subspace (quantum Zeno dynamics), or even frozen completely (quantum Zeno effect), the weak non-projective measurement can effectively compete with standard unitary dynamics leading to nontrivial effects. Here we consider global weak measurement addressing collective variables, thus preserving quantum superpositions due to the lack of which path information. While for certainty we focus on ultracold atoms, the idea can be generalized to other multimode quantum systems, including va…
We derive a set of criteria to decide whether a given projection measurement can be, in principle, exactly implemented solely by means of linear optics. The derivation can be adapted to various detection methods, including photon counting and homodyne detection. These criteria enable one to obtain easily No-Go theorems for the exact distinguishability of orthogonal quantum states with linear optics including the use of auxiliary photons and conditional dynamics.
We theoretically study electro-optic light modulation based on the quantum model where the linear electro-optic effect and the externally applied microwave field result in the interaction between optical cavity modes. The model assumes that the number of interacting modes is finite and effects of the mode overlapping coefficient on the strength of the intermode interaction can be taken into account through dependence of the coupling coefficient on the mode characteristics. We show that, under certain conditions, the model is exactly solvable and, in the semiclassical approximation where the mi…
We present a method for spatially resolved multiphoton counting based on an intensified camera with the retrieval of multimode photon statistics fully accounting for non-linearities in the detection process. The scheme relies on one-time quantum tomographic calibration of the detector. Faithful, high-fidelity reconstruction of single- and two-mode statistics of multiphoton states is demonstrated for coherent states and their statistical mixtures. The results consistently exhibit classical values of Mandel and Fano parameters in contrast to raw statistics of camera photo-events. Detector operat…
We propose a simple and efficient method for generating metrologically useful quantum entanglement in an ensemble of spin-1 atoms that interacts with a high-finesse optical cavity mode. It requires straightforward preparation of $N$ atoms in the $m_F=0$ sublevel, tailoring of the atom-field interaction to give an effective Tavis-Cummings model for the collective spin-1 ensemble, and a photon counting measurement on the cavity output field. The photon number provides a projective measurement of the collective spin length $S$, which, for the chosen initial state, is heavily weighted around value…
We investigate the action of local unitary operations on multimode (pure or mixed) Gaussian states and single out the minimal number of locally invariant parametres which completely characterise the covariance matrix of such states. For pure Gaussian states, central resources for continuous-variable quantum information, we investigate separately the parametre reduction due to the additional constraint of global purity, and the one following by the local-unitary freedom. Counting arguments and insights from the phase-space Schmidt decomposition and in general from the framework of symplectic an…
Determining an unknown quantum state from an ensemble of identical systems is a fundamental, yet experimentally demanding, task in quantum science. Here we study the number of measurement bases needed to fully characterize an arbitrary multi-mode state containing a definite number of photons, or an arbitrary mixture of such states. We show this task can be achieved using only linear optics and photon counting, which yield a practical though non-universal set of projective measurements. We derive the minimum number of measurement settings required and numerically show that this lower bound is s…
We propose an enhanced optical interferometer based on tailored non-classical light generated by nonlinear dynamics and projective measurements in a three-level atom cavity QED system. A coherent state in the cavity becomes dynamically entangled with two ground states of the atom and is transformed to a macroscopic superposition state via a projective measurement on the atom. We show that the resulting highly non-classical state can improve interferometric precision measurements well beyond the shot-noise limit once combined with a classical laser pulse at the input of a Mach-Zehnder interfero…
One of the peculiar features in quantum mechanics is that a superposition of macroscopically distinct states can exits. In optical system, this is highlighted by a superposition of coherent states (SCS), i.e. a superposition of classical states. Recently this highly nontrivial quantum state and its variant have been demonstrated experimentally. Here we demonstrate the superposition of coherent states in quantum measurement which is also a key concept in quantum mechanics. More precisely, we propose and implement a projection measurement onto the arbitrary superposition of the SCS bases in opti…
State measurement of a quantum harmonic oscillator is essential in quantum optics and quantum information processing. In a system of trapped ions, we experimentally demonstrate the projective measurement of the state of the ions' motional mode via an effective cross-Kerr coupling to another motional mode. This coupling is induced by the intrinsic nonlinearity of the Coulomb interaction between the ions. We spectroscopically resolve the frequency shift of the motional sideband of the first mode due to presence of single phonons in the second mode and use it to reconstruct the phonon number dist…
In this article, we introduce a framework for quantum state tomography of qutrits by projective measurements. The framework is based on photon-counting with measurement results distorted due to the Poisson noise and dark counts. Two different measurement schemes are investigated numerically and compared in terms of their efficiency for distinct numbers of photons per measurement. The accuracy of state reconstruction is quantified by figures of merit which are presented on graphs versus the amount of noise.
Phase-insensitive optical amplifiers uniformly amplify each quadrature of an input field and are of both fundamental and technological importance. We find the quantum limit on the precision of estimating the gain of a quantum-limited phase-insensitive optical amplifier using a multimode probe that may also be entangled with an ancilla system. In stark contrast to the sensing of loss parameters, the average photon number $N$ and number of input modes $M$ of the probe are found to be equivalent and interchangeable resources for optimal gain sensing. All pure-state probes whose reduced state on t…
Rydberg arrays merge the collective behavior of ordered atomic arrays with the controllability and optical nonlinearities of Rydberg systems, resulting in a powerful platform for realizing photonic many-body physics. As an application of this platform, we propose a protocol for quantum non-demolition (QND) photon counting. Our protocol involves photon storage in the Rydberg array, an observation phase consisting of a series of Rabi flops to a Rydberg state and measurements, and retrieval of the stored photons. The Rabi frequency experiences a $\sqrt{n}$ collective enhancement, where $n$ is the…
The article undertakes the problem of pure state estimation from projective measurements based on photon counting. Two generic frames for qubit tomography are considered -- one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases (MUBs). Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements. The accuracy of each frame is quantified by the average fidelity and purity. The efficiency of the frames is compared and discussed. The method can be generalized to hi…
