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arXiv:2610.25727

Measuring the Quantum Phase of Multiphoton Systems

Ian Baum, Riley B. Dawkins, Jannatul Ferdous, Omar S. Magana-Loaiza

quant-phphysics.opticsSubmitted Oct 19, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

The quantum phase of a multiparticle system has remained a long-standing mystery in quantum physics. The absence of a phase operator that faithfully represents this physical quantity has prevented the formulation of a meaningful uncertainty relation with particle number. Here, we resolve this problem by measuring, for the first time, the quantum phase of multiparticle systems containing up to fifteen particles. We achieve this through a new quantum state tomography scheme that combines homodyne interferometry with photon-number-resolving detection. This approach enables us to reconstruct complementary photon-number and phase representations and establish the corresponding number-phase uncertainty relation. Our measurements further provide access to the mutually unbiased basis of a multiparticle system, a fundamental resource for multiparticle quantum information processing. Furthermore, we report the spatially resolved characterization of the quantum phase. These results provide an operational framework for measuring quantum phase in multiparticle systems, with implications for fundamental quantum physics and scalable multiparticle quantum technologies.

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The quantum phase of a multiparticle system has remained a long-standing mystery in quantum physics. The absence of a phase operator that faithfully represents this physical quantity has prevented the formulation of a meaningful uncertainty relation with particle number. Here, we resolve this problem by measuring, for the first time, the quantum phase of multiparticle systems containing up to fifteen particles. We achieve this through a new quantum state tomography scheme that combines homodyne interferometry with photon-number-resolving detection. This approach enables us to reconstruct complementary photon-number and phase representations and establish the corresponding number-phase uncertainty relation. Our measurements further provide access to the mutually unbiased basis of a multiparticle system, a fundamental resource for multiparticle quantum information processing. Furthermore, we report the spatially resolved characterization of the quantum phase. These results provide an operational framework for measuring quantum phase in multiparticle systems, with implications for fundamental quantum physics and scalable multiparticle quantum technologies.
Experiments have been performed to determine the Wigner distribution and the density matrix (and for pure states the wave function) of a light mode, by using tomographic inversion of a set of measured probability distributions for quadrature amplitudes.From these measurements the quantum distributions of optical phase and photon number have been obtained.The measurements of quadrature-amplitude distributions for a temporal mode of the electromagnetic field are carried out using balanced homodyne detection.We refer to this new method as optical homodyne tomography.Given the measured density matrix, one can experimentally infer any of the various quantum distributions of optical phase, in particular the Pegg-Barnett (or, equivalently, Shapiro-Shepard) phase distribution, the marginal Wigner distribution, and the Vogel-Schleich operational phase distribntion.We have used this approach to make measurements of the number-plase uncertainty relation for coherent-state fields.The coherent states do not attain the minimum value for the number-phase uncertainty product, as set by the expectation value of the commutator of the number and phase operators; this is true theoretically and experimentally.
We have experimentally determined all of the quantities involved in the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a coherent state of small average photon number. This is accomplished by determining the quantum state of the field using optical homodyne tomography, which uses measured distributions of electric-field quadrature amplitude to determine the Wigner function and hence the density matrix. The measured state is then used to calculate the uncertainty product for the number and phase, as well as the expectation value of the commutator of the number and phase operators. The experimental results agree with the quantum-mechanical predictions. We also present measured phase- and photon-number distributions for these weak coherent states, as well as their measured complex wave functions.
Abstract Entanglement is a key ingredient for quantum technologies and a fundamental signature of quantumness in a broad range of phenomena encompassing many-body physics, thermodynamics, cosmology and life sciences. For arbitrary multiparticle systems, entanglement quantification typically involves nontrivial optimisation problems, and it may require demanding tomographical techniques. Here, we develop an experimentally feasible approach to the evaluation of geometric measures of multiparticle entanglement. Our framework provides analytical results for particular classes of mixed states of N qubits, and computable lower bounds to global, partial, or genuine multiparticle entanglement of any general state. For global and partial entanglement, useful bounds are obtained with minimum effort, requiring local measurements in just three settings for any N. For genuine entanglement, a number of measurements scaling linearly with N are required. We demonstrate the power of our approach to estimate and quantify different types of multiparticle entanglement in a variety of N-qubit states useful for quantum information processing and recently engineered in laboratories with quantum optics and trapped ion setups.
The quantum wave function of multiple particles provides additional information which is inaccessible to detectors working alone. Here, we introduce the coincidence wave-front sensing (CWS) method to reconstruct the phase of the multiphoton transverse spatial wave function. The spatially resolved coincidence photon counting is involved. Numerical simulations of two-photon cases using the weak measurement wave-front sensor are performed to test its correctness, and the phase information hidden in the correlation is revealed. Our work provides a direct spatial way to characterize multipartite quantum systems and leads to fundamental studies like experimental Bohmian mechanics and applications in quantum optical technologies.
Quantum optics is central to emerging quantum technologies such as quantum metrology and quantum information. The continuous variable (CV) approach to encoding information in a physical system, for example the electromagnetic field quadratures has significant advantages over discrete variables (DV) encoding. CV offers higher density encoding in a single optical state and on-demand state preparation for example. However, techniques for CV preparation, manipulation and measurement are not as well developed as the corresponding DV approach. In this thesis we present methods to characterize quantum detectors that operate in the CV regime, and discuss their application to CV state preparation. A quantum detector is described by its positive operator valued measure (POVM) set, with one operator for each measurement outcome. We present an experimental scheme for detector tomography, that is reconstruction of the POVM set of an unknown detector by means of measurement outcomes for a set of tomographically complete input states. We use this scheme to experimentally determine the POVM set of an avalanche photodiode (APD) and a time-multiplexed-detector (TMD) and compare the results to models of the detectors. We introduce a weak-homodyne photon-number-resolving detector (PNRD) , which uses a weak local oscillator beam to set a phase reference for the detection process. We develop a theoretical description of the sensitivity regimes for this detector using the Wigner phase space picture. We also present experimental detector tomography for a weak-homodyne detector based on two APDs. In many cases it is sufficient to calibrate the efficiencies of previously characterized PNRDs. We experimentally demonstrate a scheme using correlated photons produced by spontaneous parametric downconversion (SPDC) to determine the efficiencies of PNRDs, based on the photon number statistics of the source. The same SPDC source is also used to demonstrate preparation of photonic states in the photon number basis and for a first proof-of-principle experiment with a weak-homodyne PNRD based on a TMD.
Photon-number-resolving measurements allow one to address quantum optics from the corpuscular angle of quantum physics. A number of classically non-intuitive features are typically expected from working in the photon-number Fock eigenbasis, such as nonpositive Wigner functions. In this talk, we report on the progress of two applications of photon-number-resolving measurements in quantum optics: quantum interferometry with photon-subtracted twin beams and quantum state tomography of Fock states created by heralded parametric downconversion.