Using a semiclassical model of photodetection with Poissonian noise and insights from quantum metrology, we prove that linear optics and photon counting can optimally estimate the separation between two incoherent point sources without regard to Rayleigh's criterion. The model is applicable to weak thermal or fluorescent sources as well as lasers.Lord Rayleigh suggested in 1879 that two incoherent optical point sources should be separated by a diffraction-limited spot size for them to be resolved [1]. This criterion has since become the most influential measure of imaging resolution. Under the modern advent of rigorous statistics and image processing, Rayleigh's criterion remains a curse. When the image is noisy, necessarily so owing to the quantum nature of light [2], and Rayleigh's criterion is violated, it becomes much more difficult to estimate the separation accurately by conventional imaging methods [3][4][5]. Modern superresolution techniques in microscopy [6][7][8] can circumvent Rayleigh's criterion by making sources radiate in isolation, but such techniques require careful control of the fluorescent emissions, making them difficult to use for microscopy and irrelevant to astronomy.Here we show that, contrary to conventional wisdom, the separation between two incoherent optical sources can be estimated accurately via linear optics and photon counting (LOPC) even if Rayleigh's criterion is severely violated. Our theoretical model here is based on the semiclassical theory of photodetection with Poissonian noise, which is a widely accepted statistical model for lasers [2] as well as weak thermal [9,10] or fluorescent [5,11] light in astronomy and microscopy. The semiclassical model is consistent with the quantum model proposed in Ref. [12] for weak incoherent sources and the mathematical formalisms are similar, but the semiclassical model has the advantage of being applicable also to lasers, which are important sources for remote-sensing, testing, and proof-of-concept experiments. The semiclassical theory also avoids a quantum description of light and offers a more pedagogical perspective. Compared with the full semiclassical theory in Ref. [13], the Poissonian model is invalid for strong thermal sources but more analytically tractable.Consider J optical modes and a column vector of complex field amplitudes α = (α 1 , . . . , α J ) ⊤ within one coherence time interval. The amplitudes are normalized such that |α j | 2 is equal to the energy in each mode in units of quanta. The central quantity in statistical optics is the mutual coherence matrix [2,9]

Smoothing is an estimation technique that takes into account both past and future observations and can be more accurate than filtering alone. In this Letter, a quantum theory of smoothing is constructed using a time-symmetric formalism, thereby generalizing prior work on classical and quantum filtering, retrodiction, and smoothing. The proposed theory solves the important problem of optimally estimating classical Markov processes coupled to a quantum system under continuous measurements, and is thus expected to find major applications in future quantum sensing systems, such as gravitational wave detectors and atomic magnetometers.

Ever since the inception of gravitational-wave detectors, limits imposed by quantum mechanics to the detection of time-varying signals have been a subject of intense research and debate. Drawing insights from quantum information theory, quantum detection theory, and quantum measurement theory, here we prove lower error bounds for waveform detection via a quantum system, settling the long-standing problem. In the case of optomechanical force detection, we derive analytic expressions for the bounds in some cases of interest and discuss how the limits can be approached using quantum control techniques.Comment: v1: first draft, 5 pages; v2: updated and extended, 5 pages + appendices, 2 figures; v3: 8 pages and 3 figure

Using a flow chart representation of quantum optomechanical dynamics, we design coherent quantum-noise-cancellation schemes that can eliminate the backaction noise induced by radiation pressure at all frequencies and thus overcome the standard quantum limit of force sensing. The proposed schemes can be regarded as novel examples of coherent feedforward quantum control.

Quantum mechanics is potentially advantageous for certain information-processing tasks, but its probabilistic nature and requirement of measurement backaction often limit the precision of conventional classical information-processing devices, such as sensors and atomic clocks. Here we show that, by engineering the dynamics of coupled quantum systems, it is possible to construct a subsystem that evades the measurement backaction of quantum mechanics, at all times of interest, and obeys any classical dynamics, linear or nonlinear, that we choose. We call such a system a quantum-mechanics-free subsystem (QMFS). All of the observables of a QMFS are quantum-nondemolition (QND) observables; moreover, they are dynamical QND observables, thus demolishing the widely held belief that QND observables are constants of motion. QMFSs point to a new strategy for designing classical information-processing devices in regimes where quantum noise is detrimental, unifying previous approaches that employ QND observables, backaction evasion, and quantum noise cancellation. Potential applications include gravitational-wave detection, optomechanical-force sensing, atomic magnetometry, and classical computing. Demonstrations of dynamical QMFSs include the generation of broadband squeezed light for use in interferometric gravitational-wave detection, experiments using entangled atomic-spin ensembles, and implementations of the quantum Toffoli gate. According to quantum mechanics, a measurement of the position of an object must introduce uncertainty to its momentum, called the measurement backaction noise. Since position is coupled to momentum, as the object evolves in time, the backaction noise can perturb the position and contaminate subsequent position measurements. Scientists studying gravitational-wave detection, concerned that this dynamical effect of measurement backaction would place a fundamental limit to the detectors, proposed a general solution: If a quantum observable, represented by a self-adjoint operator OðtÞ in the Heisenberg picture, can be made to commute with itself at times t and t 0 when the observable is measured, viz.,then O can be measured repeatedly with no quantum limits on the predictability of these measurements. In particular, this means that quantum mechanics does not limit the detection of a classical signal that affects O.An observable that obeys Eq. (1) (1) is satisfied, the conjugate observables do not feed back onto the QND observable at the times of interest.The most well-known QND observables are ones that remain static in the absence of classical signals, viz.,

We obtain the ultimate quantum limit for estimating the transverse separation of two thermal point sources using a given imaging system with limited spatial bandwidth. We show via the quantum Cramér-Rao bound that, contrary to the Rayleigh limit in conventional direct imaging, quantum mechanics does not mandate any loss of precision in estimating even deep sub-Rayleigh separations. We propose two coherent measurement techniques, easily implementable using current linear-optics technology, that approach the quantum limit over an arbitrarily large range of separations. Our bound is valid for arbitrary source strengths, all regions of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. The measurement schemes can be applied to microscopy, optical sensing, and astrometry at all wavelengths.PACS numbers: 42.30.-d, 42.50.-p, 06.20.-f The Rayleigh criterion for resolving two incoherent optical point sources [1] is the most widely used benchmark for the resolving power of an imaging system. According to it, the sources can be resolved by direct imaging only if they are separated by at least the diffraction-limited spot size of the point-spread function of the imaging system. While the criterion is heuristic and does not take into account the intensity of the sources or the measurement shot noise, recent work [2-5] has made it rigorous by taking as resolution measure the classical Cramér-Rao lower bound (CRB) of estimation theory [6] on the mean squared error (MSE) of any unbiased estimate of the separation of the sources using spatially-resolved imageplane photon counting. These works showed that if the detected average photon number per mode N s ≪ 1, the MSE of any unbiased estimator based on direct imaging diverges as the source separation decreases to zero over an interval comparable to the Rayleigh limit. This phenomenon, dubbed Rayleigh's curse in [7], stems from the indistinguishability between the photons coming from the two sources and imposes a fundamental limitation of direct imaging in resolving sources much closer than the spot size, even when the measured photon number is taken into account. Recent developments in far-field microscopy [8] sidestep Rayleigh's curse by preventing multiple sources from emitting simultaneously, but control over the emission properties of sources is unavailable in target sensing or astronomical imaging.While the development of novel quantum states of light and measurement techniques has given rise to the vast field of quantum imaging [9], fundamental quantum limits in resolving two incoherent sources have been largely neglected since the early days of quantum estimation theory [10,11]. Recently, the coherent [12] and incoherent [7] two-source resolution problems were revisited using the quantum Cramér-Rao bound (QCRB) [11,13] that accounts for all (unbiased) measurement techniques allowed by quantum mechanics. Under a weak-source assumption similar to that in [2][3][4][5], it was found in [7] that the QCRB showed no depe...

The quantum dynamics of the coupling between a cavity optical field and a resonator microwave field via the electro-optic effect is studied. This coupling has the same form as the opto-mechanical coupling via radiation pressure, so all previously considered opto-mechanical effects can in principle be observed in electro-optic systems as well. In particular, I point out the possibilities of laser cooling of the microwave mode, entanglement between the optical mode and the microwave mode via electro-optic parametric amplification, and back-action-evading optical measurements of a microwave quadrature.

We derive a quantum Cramér-Rao bound (QCRB) on the error of estimating a time-changing signal. The QCRB provides a fundamental limit to the performance of general quantum sensors, such as gravitational-wave detectors, force sensors, and atomic magnetometers. We apply the QCRB to the problem of force estimation via continuous monitoring of the position of a harmonic oscillator, in which case the QCRB takes the form of a spectral uncertainty principle. The bound on the force-estimation error can be achieved by implementing quantum noise cancellation in the experimental setup and applying smoothing to the observations.