The content of phase information of an arbitrary phase--sensitive measurement is evaluated using the maximum likelihood estimation. The phase distribution is characterized by the relative entropy--a nonlinear functional of input quantum state. As an explicit example the multiple measurement of quadrature operator is interpreted as quantum phase detection achieving the ultimate resolution predicted by the Fisher information.
We propose a non-Gaussianity measure of a multimode quantum state based on the negentropy of quadrature distributions. Our measure satisfies desirable properties as a non-Gaussianity measure, i.e., faithfulness, invariance under Gaussian unitary operations, and monotonicity under Gaussian channels. Furthermore, we find a quantitative relation between our measure and the previously proposed non-Gaussianity measures defined via quantum relative entropy and the quantum Hilbert-Schmidt distance. This allows us to estimate the non-Gaussianity measures readily by homodyne detection, which would othe…
We identify what ideal correlated photon number states are to required to maximize the discrepancy between local realism and quantum mechanics when a quadrature homodyne phase measurement is used. Various Bell inequality tests are considered.
We determine the extreme points of the convex set of covariant phase observables. Such extremals describe the best phase parameter measurements of laser light - the best in the sense that they are free from classical randomness due to fluctuations in the measuring procedure. We also characterize extreme fuzzy rotated quadratures.
We present a setup to perform sub shot noise measurements of the phase quadrature for intense pulsed light without the use of a separate local oscillator. A Mach--Zehnder interferometer with an unbalanced arm length is used to detect the fluctuations of the phase quadrature at a single side band frequency. Using this setup, the non--separability of a pair of quadrature entangled beams is demonstrated experimentally.
This work deals with the problem of photon detection generated from the mirror-induced Dynamical Casimir Effect. Particularly we are interested in measurement of those statistical characteristics of a generated intracavity field which may confirm its nonthermal properties. Here an indirect protocol for quadrature dispersion measurement is presented.
Real-time controls based on quantum measurements are powerful tools for various quantum protocols. However, their experimental realization have been limited by mode-mismatch between temporal mode of quadrature measurement and that heralded by photon detection. Here, we demonstrate real-time quadrature measurement of a single-photon wavepacket induced by a photon detection, by utilizing continuous temporal-mode-matching between homodyne detection and an exponentially rising temporal mode. Single photons in exponentially rising modes are also expected to be useful resources for interactions with…
We exhibit measurements for optimal state estimation which have a finite number of outcomes. This is achieved by a connection between finite optimal measurements and Gauss quadratures. The example we consider to illustrate this connection is that of state estimation on $N$ qubits, all in a same pure state. Extensions to state estimation of mixed states are also discussed.
We demonstrate a contradiction of quantum mechanics with local hidden variable theories for continuous variable quadrature phase amplitude (``position'' and ``momentum'') measurements, by way of a violation of a Bell inequality. For any quantum state, this contradiction is lost for situations where the quadrature phase amplitude results are always macroscopically distinct. We show that for optical realisations of this experiment, where one uses homodyne detection techniques to perform the quadrature phase amplitude measurement, one has an amplification prior to detection, so that macroscopic f…
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, …
We reveal that quadrature squeezing can result in significantly better quantum-estimation performance with quantum heterodyne detection (of H. P. Yuen and J. H. Shapiro) as compared to quantum homodyne detection for Gaussian states, which touches an important aspect in the foundational understanding of these two schemes. Taking single-mode Gaussian states as examples, we show analytically that the competition between the errors incurred during tomogram processing in homodyne detection and the Arthurs-Kelly uncertainties arising from simultaneous incompatible quadrature measurements in heterody…
Finite resolution quantum nondemolition (QND) measurements allow a determination of light field properties while preserving some of the original quantum coherence of the input state. It is thus possible to measure correlations between the photon number and a quadrature component of the same light field mode. Nonclassical features emerge as photon number quantization is resolved. In particular, a strong anti-correlation of quantization and coherence is observable in QND measurements of photon number, and a correlation between measurement induced quantum jumps and quadrature component measuremen…
We suggest an interferometric scheme assisted by squeezing and linear feedback to realize the whole class of field-quadrature quantum nondemolition measurements, from Von Neumann projective measurement to fully non-destructive non-informative one. In our setup, the signal under investigation is mixed with a squeezed probe in an interferometer and, at the output, one of the two modes is revealed through homodyne detection. The second beam is then amplitude-modulated according to the outcome of the measurement, and finally squeezed according to the transmittivity of the interferometer. Using str…
We study the quadrature uncertainty of the quantum elliptical vortex state using the associated Wigner function. Deviations from the minimum uncertainty states were observed due to the absence of the Gaussian nature. In our study of the entropy, we noticed that with increasing vorticity, entropy increases for both the modes. We further observed that, there exists an optimum value of ellipticity which gives rise to maximum entanglement of the two modes of the quantum elliptical vortex states. A further increase in ellipticity reduces the entropy thereby resulting in a loss of information carryi…
We consider how to measure collective spin states of an atomic ensemble based on the recent multi-pass approaches for quantum interface between light and atoms. We find that a scheme with two passages of a light pulse through the atomic ensemble is efficient to implement the homodyne tomography of the spin state. Thereby, we propose to utilize optical pulses as a phase-shifter that rotates the quadrature of the spins. This method substantially simplifies the geometry of experimental schemes.
We present a method of measuring expectation values of quadrature moments of a multimode field through two-level probe ``homodyning''. Our approach is based on an integral transform formalism of measurable probe observables, where analytically derived kernels unravel efficiently the required field information at zero interaction time, minimizing decoherence effects. The proposed scheme is suitable for fields that, while inaccessible to a direct measurement, enjoy one and two-photon Jaynes-Cummings interactions with a two-level probe, like spin, phonon, or cavity fields. Available data from pre…
One of the most direct preparations of a Gottesman-Kitaev-Preskill qubit in an oscillator uses a tunable photon-pressure (also called optomechanical) coupling of the form $g \hat{q} a^{\dagger} a$, enabling to imprint the modular value of the position $\hat{q}$ of one oscillator onto the state of an ancilla oscillator. We analyze the practical feasibility of executing such modular quadrature measurements in a parametric circuit-QED realization of this coupling. We provide estimates for the expected GKP squeezing induced by the protocol and discuss the effect of photon loss and other errors on …
We describe a simple and efficient setup to generate and characterize femtosecond quadrature-entangled pulses. Quantum correlations equivalent to about 2.5 dB squeezing are efficiently and easily reached using the non-degenerate parametric amplification of femtosecond pulses through a single-pass in a thin (0.1 mm) potassium niobate crystal. The entangled pulses are then individually sampled to characterize the non-separability and the entropy of formation of the states. The complete experiment is analysed in the time-domain, from the pulsed source of quadrature entanglement to the time-resolv…
We observe -9.01$\pm$0.14 dB of squeezing and +15.12$\pm$0.14 dB of antisqueezing with a local oscillator phase locked in homodyne measurement. In reference [1], two main factors are pointed out which degrade the observed squeezing level: phase fluctuation in homodyne measurement and intracavity losses of an optical parametric oscillator for squeezing. We have improved the phase stability of homodyne measurement and have reduced the intracavity losses. We measure pump power dependences of the squeezing and antisqueezing levels, which show good agreement with theoretical calculations taking acc…
Variable measurement operators enable the optimization of strategies for testing quantum properties and the preparation of a range of quantum states. Here, we experimentally implement a weak-field homodyne detector that can continuously tune between measuring photon numbers and field quadratures. We combine a quantum signal with a coherent state on a balanced beam splitter and detect light at both output ports using photon-number-resolving transition edge sensors. We observe that the discrete difference statistics converge to the quadrature distribution of the signal as we increase the coheren…
We consider the measuring procedure that in principle allows to avoid the homodyne detection for the simultaneous selection of both quadrature components in the far-field. The scheme is based on the use of the coherent sources of the non-classical light. The possibilities of the procedure are illustrated on the basis of the use of pixellised sources, where the phase-locked sub-Poissonian lasers or the degenerate optical parametric oscillator generating above threshold are chosen as the pixels. The theory of the pixellised source of the spatio-temporal squeezed light is elaborated as a part of …
As an application of the simultaneous and continuous measurement of noncommutative observables formulated in our previous paper [C. Jiang and G. Watanabe, Phys. Rev. A 102, 062216 (2020)], we propose a scheme to generate the pure ideal quadrature-squeezed state in a one-dimensional harmonic oscillator system by the feedback control based on such type of measurement of noncommutative quadrature observables. We find that, by appropriately setting the strengths of the measurement and the feedback control, the pure ideal quadrature-squeezed state with arbitrary squeezedness can be produced. This i…
Until now, Schrödinger's cat states are generated by subtracting single photons from the whole bandwidth of squeezed vacua. However, it was pointed out recently that the achievable purities are limited in such method (J. Yoshikawa, W. Asavanant, and A. Furusawa, arXiv:1707.08146 [quant-ph] (2017)). In this paper, we used our new photon subtraction method with a narrowband filtering cavity and generated a highly pure Schrödinger's cat state with the value of $-0.184$ at the origin of the Wigner function. To our knowledge, this is the highest value ever reported without any loss corrections. The…
A novel scheme to realize the whole class of quantum nondemolition (QND) measurements of a field quadrature is suggested. The setup requires linear optical components and squeezers, and allows optimal QND measurements of quadratures, which minimize the information gain versus state disturbance trade-off.
We study theoretically and experimentally the quantum properties of a type II frequency degenerate optical parametric oscillator below threshold with a quarter-wave plate inserted inside the cavity which induces a linear coupling between the orthogonally polarized signal and idler fields. This original device provides a good insight into general properties of two-mode gaussian states, illustrated in terms of covariance matrix. We report on the experimental generation of two-mode squeezed vacuum on non-orthogonal quadratures depending on the plate angle. After a simple operation, the entangleme…
We present a protocol for measuring the quadrature of a harmonic oscillator (HO). The HO is coupled to a qubit, with an interaction modulated by the qubit control and effectively proportional to the HO quadrature $I$. Repeated measurement of the qubit leads to gradually increasing information on the quadrature $I$, leading to squeezing. We derive an analytical formula for the quadrature variance, $(ΔI)^2 = 1/(1+4ϕ^2 s)$, with $ϕ$ the product of interaction strength and interaction time and $s$ the number of repetitions of the measurement. We discuss the robustness of this scheme against decohe…
We study the squeezing of output quadratures of an electro-magnetic field escaping from a resonator coupled to a general quantum system with arbitrary interaction strengths. The generalized theoretical analysis of output squeezing proposed here is valid for all the interaction regimes of cavity-quantum electrodynamics: from the weak to the strong, ultrastrong, and deep coupling regimes. For coupling rates comparable or larger then the cavity resonance frequency, the standard input-output theory for optical cavities fails to calculate the correct output field-quadratures and predicts a non-negl…
Quantum state tomography aims to determine the quantum state of a system from measured data and is an essential tool for quantum information science. When dealing with continuous variable quantum states of light, tomography is often done by measuring the field amplitudes at different optical phases using homodyne detection. The quadrature-phase homodyne measurement outputs a continuous variable, so to reduce the computational cost of tomography, researchers often discretize the measurements. We show that this can be done without significantly degrading the fidelity between the estimated state …
We examine the moment-reconstruction performance of both the homodyne and heterodyne (double-homodyne) measurement schemes for arbitrary quantum states and introduce moment estimators that optimize the respective schemes for any given data. In the large-data limit, these estimators are as efficient as the maximum-likelihood estimators. We then illustrate the superiority of the heterodyne measurement for the reconstruction of the first and second moments by analyzing Gaussian states and many other significant non-classical states.
The estimation of high order correlation function values is an important problem in the field of quantum computation. We show that the problem can be reduced to preparation and measurement of optical quantum states resulting after annihilation of a set number of quanta from the original beam. We apply this approach to explore various photon bunching regimes in optical states with gamma-compounded Poisson photon number statistics. We prepare and perform measurement of the thermal quantum state as well as states produced by subtracting one to ten photons from it. Maximum likelihood estimation is…
We propose general methodology of deterministic single-mode quantum interaction nonlinearly modifying single quadrature variable of a continuous variable system. The methodology is based on linear coupling of the system to ancillary systems subsequently measured by quadrature detectors. The nonlinear interaction is obtained by using the data from the quadrature detection for dynamical manipulation of the coupling parameters. This measurement-induced methodology enables direct realization of arbitrary nonlinear quadrature interactions without the need to construct them from the lowest-order gat…
The Holevo Cramér Rao bound is a lower bound on the sum of the mean square error of estimates for parameters of a state. We provide a method for calculating the Holevo Cramér-Rao bound for estimation of quadrature mean parameters of a Gaussian state by formulating the problem as a semidefinite program. In this case, the bound is tight; it is attained by purely Guassian measurements. We consider the example of a symmetric two-mode squeezed thermal state undergoing an unknown displacement on one mode. We calculate the Holevo Cramér-Rao bound for joint estimation of the conjugate parameters for t…
An electromagnetic field quadrature measurement, performed on one of the modes of the nonlocal single-photon state $a|1,0>-b|0,1>$, collapses it into a superposition of the single-photon and vacuum states in the other mode. We use this effect to implement remote preparation of arbitrary single-mode photonic qubits conditioned on observation of a preselected quadrature value. The quantum efficiency of the prepared qubit can be higher than that of the initial single photon.
In this paper we investigate the coupling properties of pairs of quadrature observables, showing that, apart from the Weyl relation, they share the same coupling properties as the position-momentum pair. In particular, they are complementary. We determine the marginal observables of a covariant phase space observable with respect to an arbitrary rotated reference frame, and observe that these marginal observables are unsharp quadrature observables. The related distributions constitute the Radon tranform of a phase space distribution of the covariant phase space observable. Since the quadrature…
The eight-port homodyne detector is an optical circuit designed to perform the monitoring of two quadratures of an optical field, the signal. By using quantum Bose fields and quantum stochastic calculus, we give a complete quantum description of this apparatus, when used as quadrature detector in continuous time. We can treat either the travelling waves in the optical circuit, either the observables involved in the detection part: two couples of photodiodes, postprocessing of the output currents. The analysis includes imperfections, such as not perfectly balanced beam splitters, detector effic…
A new simple entanglement measurement method is proposed for the bright EPR beams generated from a non-degenerate optical parametric amplifier operating at deamplification. Due to the output signal and idler modes are frequency degenerate and in phase, the needed phase shift of interference for the measurement of the correlated phase quadratures and anti-correlated amplitude quadratures can be accomplished by a quarter-wave plate and a half wave plate without separating the signal and idler beam. Therefore, phase locking and local oscillators are avoided.
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
We demonstrate that for two canonically conjugate operators $\hat{q},\hat {p} $,the global correlation $\langle \hat{q} \hat {p} + \hat{p} \hat {q} \rangle -2 \langle \hat{q}\rangle \langle \hat {p}\rangle$, and the local correlations $\langle \hat{q} \rangle (p) - \langle \hat{q}\rangle$ and $\langle \hat{p} \rangle (q)-\langle \hat {p}\rangle$ can be measured exactly by Von Neumann-Arthurs-Kelly joint quadrature measurements . These correlations provide a sensitive experimental test of quantum phase space probabilities quite distinct from the probability densities of $ q,p $. E.g. for EPR st…
Quantum phase transitions are often embodied by the critical behavior of purely quantum quantities such as entanglement or quantum fluctuations. In critical regions, we underline a general scaling relation between the entanglement entropy and one of the most fundamental and simplest measure of the quantum fluctuations, the Heisenberg uncertainty principle. Then, we show that the latter represents a sensitive probe of superradiant quantum phase transitions in standard models of photons such as the Dicke Hamiltonian, which embodies an ensemble of two-level systems interacting with one quadrature…
We review a scheme for performing a back-action evading measurement of one mechanical quadrature in an optomechanical setup. The experimental application of this scheme has been limited by parametric instabilities caused in general by a slight dependence of the mechanical frequency on the electromagnetic energy in the cavity. We find that a simple modification to the optical drive can effectively eliminate the parametric instability even at high intracavity power, allowing realistic devices to achieve sub-zero-point uncertainties in the measured quadrature.
The time evolution of a squeezed coherent state conditioned by the results of a single and double heterodyne measurement is discussed. The mean values of quadratures as well as the dynamics of quadrature uncertainties have been obtained within the framework of the theory of continuous measurements based on filtration equations. It has been found that while the mean values depend on the measured noise, the uncertainties in the optical quadratures are deterministic. Explicit solutions for the latter have been provided. Finally, a time development of the squeeze parameter for the posterior squeez…
A method for gaining information about the phonon-number moments and the generalized nonlinear and linear quadratures in the motion of trapped ions (in particular, position and momentum) is proposed, valid inside and outside the Lamb-Dicke regime. It is based on the measurement of first time derivatives of electronic populations, evaluated at the motion-probe interaction time t=0. In contrast to other state-reconstruction proposals, based on measuring Rabi oscillations or dispersive interactions, the present scheme can be performed resonantly at infinitesimal short motion-probe interaction tim…
Two-photon absorption (TPA) is of fundamental importance in super-resolution imaging and spectroscopy. Its nonlinear character allows for the prospect of using quantum resources, such as entanglement, to improve measurement precision or to gain new information on, e.g., ultrafast molecular dynamics. Here, we establish the metrological properties of nonclassical squeezed light sources for precision measurements of TPA cross sections. We find that there is no fundamental limit for the precision achievable with squeezed states in the limit of very small cross sections. Considering the most releva…
We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisen…
Homodyne measurement is a corner-stone of quantum optics. It measures the fundamental variables of quantum electrodynamics - the quadratures of light, which represent the cosine-wave and sine-wave components of an optical field and constitute the quantum optical analog of position and momentum. Yet, standard homodyne, which is used to measure the quadrature information, suffers from a severe bandwidth limitation: While the bandwidth of optical states can easily span many THz, standard homodyne detection is inherently limited to the electrically accessible, MHz to GHz range, leaving a dramatic …
Through simultaneous but unequal electromechanical amplification and cooling processes, we create a method for nearly noiseless pulsed measurement of mechanical motion. We use transient electromechanical amplification (TEA) to monitor a single motional quadrature with a total added noise $-8.5\pm2.0$ dB relative to the zero-point motion of the oscillator, or equivalently the quantum limit for simultaneous measurement of both mechanical quadratures. We demonstrate that TEA can be used to resolve fine structure in the phase-space of a mechanical oscillator by tomographically reconstructing the d…
A scheme of an ultra-sensitive magnetometer in the cavity quantum electromagnonics where the intracavity microwave mode coupled to a magnonic mode via magnetic dipole interaction is proposed. It is shown that by driving both magnonic and microwave modes with external classical fields and controlling the system parameters, one can reduce the added noise of magnetic field measurement below the standard quantum limit (SQL). Surprisingly, we show that beyond the rotating wave approximation (RWA), not only the added noise can be suppressed, but also the output cavity response to the input signal ca…
We analyze the continuous monitoring of a qudit coupled to a cavity using both phase-preserving and phase-sensitive amplification. The quantum trajectories of the system are described by a stochastic master equation, for which we derive the appropriate Lindblad operators. The measurement back-action causes spiraling in the state coordinates during collapse, which increases as the system levels become less distinguishable. We discuss two examples: a two-level system and an $N$-dimensional system and meter with rotational symmetry in the quadrature space. We also provide a comparison of the effe…
A process which strongly amplifies both quadrature amplitudes of an oscillatory signal necessarily adds noise. Alternatively, if the information in one quadrature is lost in phase-sensitive amplification, it is possible to completely reconstruct the other quadrature. Here we demonstrate such a nearly perfect phase-sensitive measurement using a cavity optomechanical scheme, characterized by an extremely small noise less than 0.2 quanta. We also observe microwave radiation strongly squeezed by 8 dB below vacuum. A source of bright squeezed microwaves opens up applications in manipulations of qua…
We develop an interferometric technique for making time-resolved measurements of field-quadrature operators for nonequilibrium ultracold bosons in optical lattices. The technique exploits the internal state structure of magnetic atoms to create two subsystems of atoms in different spin states and lattice sites. A Feshbach resonance turns off atom-atom interactions in one spin subsystem, making it a well-characterized reference state, while atoms in the other subsystem undergo nonequilibrium dynamics for a variable hold time. Interfering the subsystems via a second beam-splitting operation, tim…
