We investigate the Cramer-Rao bound on optical tweezers based particle position measurement with coherent light, which defines the ultimate classical resolution limit achievable with such devices. Methods to surpass this limit using quantum resources are examined, and comparison made to current state-of-the-art devices used in the biological and physical sciences.
We investigate the Cramer-Rao bound on optical tweezers based particle position measurement with coherent light, which defines the ultimate classical resolution limit achievable with such devices. Methods to surpass this limit using quantum resources are examined, and comparison made to current state-of-the-art devices used in the biological and physical sciences.
Summary form only given. Optical techniques are widely used in many areas of science and technology to make accurate measurements. The precision of these measurements suffers from limits due to the unavoidable quantum fluctuations of light. When the light is in the quasi-classical coherent state, this limit is called ”standard quantum limit” and scales as 1/√N with N the number of photons. While better scaling can be achieved with exotic quantum states, coherent light is experimentally interesting as we are able to produce it with large N. We consider light as a probe for the measurement of a parameter p that affects the medium or the length through which the light propagates. We derived the ultimate limit of sensitivity, the so-called quantum Cramér-Rao bound, for the measurement of p [1]: for u(ω, p) the normalized mean field mode of the light, the minimal fluctuation of p that can be measured is given in Fig. 1. We also showed that this ultimate limit of sensitivity can be reached experimentally in a homodyne detection scheme, where the intense local oscillator is taken in the mode wp o ∂u ∂p, called detection mode associated with p. This measurement scheme can be applied to space-time positioning experiments such as ranging or clock synchronization [2]. The basic method for space-time positioning relies on the exchange of regular light pulses between two emitters. For coherent broadband light pulses, such as the ones delivered by a mode-locked femtosecond laser, of central frequency ω0 and of spectral width Δω, the quantum limits of the measurements of the delay t are given by δtphase = 2√Nω0 1 for interferometric measurements and δttof = 2√NΔω 1 for time-of-flight measurements. However, these two methods do not reach the quantum Cramer-Rao bound; this ultimate limit is achieved by a homodyne detection with local oscillator w(ω) o iωu(ω), and is given by 1 δtmin = (1) 2√N . Δω2 + ω2 0 For usual characteristics of a Titane:Sapphire femtosecond laser and 1 ms integration time, δtmin = 10-23s. We demonstrate this limit experimentally by using pulse shaping techniques to shape the local oscillator in the detection mode. We also study the influence of the intensity and phase noise of a mode-locked femtosecond laser on this limit. This technique can also be applied in the case of ranging in a dispersive medium such as air. In this case, dispersive effects due to environmental parameters such as pressure, temperature or humidity affect the precision of measurements. Usually, this problem is tacked by measuring separately the environmental parameters or using several wavelengths, which can be difficult and require post-processing. We show that using the same technique as previously shown, the parasitic effects due to the medium in a distance measurement can be canceled by shaping properly the local oscillator [3]: we choose for the local oscillator a mode that is orthogonal to the environmental effects, so the homodyne detection is insensitive to them, at the expense of a lower sensitivity. This scheme is an all-optical and real-time measurement of distance in a dispersive medium.
Quantum enhanced sensitivity in optical tweezers based particle tracking was recently demonstrated. This has provided the necessary tool for quantum metrology to play an important role in biological measurements. Here we introduce the basic theory relevant to such optical tweezers experiments, and overview the significance of sub-shot noise limited sensitivity to practical experiments. In particular, biophysical experiments are subject to optical power constraints, which therefore limits the absolute sensitivity which is classically achievable. Quantum enhanced particle tracking can overcome this limit, and is therefore likely to play an important role in such biophysical experiments in the near future.
For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.
We establish a simple method to assess the quantum Fisher information required for resolving two incoherent point sources with an imaging system. The resulting Cramér-Rao bound shows that the standard Rayleigh limit can be surpassed by suitable coherent measurements. We explicitly find these optimal strategies and present a realization for Gaussian and slit apertures. This involves a projection onto the optimal bases that is accomplished with digital holographic techniques and is compared with a CCD position measurement. Our experimental results unequivocally confirm unprecedented sub-Rayleigh precision.
With the rapid advancement of science and technology, phase measurement plays a crucial role in various fields such as optics, communication, and radar. Traditional phase measurement methods, relying on classical light sources, produce results following the central limit theorem, limiting measurement precision to the standard quantum limit. However, recent in-depth studies of quantum theory and quantum enhanced experimental techniques are breaking this barrier. The use of specially designed non-classical light sources for phase measurement not only provides the possibility of breaking through the limits of measurement precision but also meets the growing demand for high-precision measurements.This paper started from a statistical perspective, introduced key indicators for evaluating measurement precision, including the Cramér-Rao Boundary (CRB) and Fisher Information, and extends to experimental approaches beyond the standard quantum limit. Subsequently, by introducing quantum light sources such as squeezed states, entangled states, and methods like harmonic amplification to achieve amplifier the phase and optical detection, the paper provides a comprehensive review of recent research achievements in super-resolution phase measurement techniques. Finally, we select several technological categories with a promising engineering outlook, analyze the technical challenges limiting their maturity, and provide a prospective outlook.
We experimentally demonstrate optomechanical motion and force measurements near the quantum precision limits set by the quantum Cramér-Rao bounds. Optical beams in coherent and phase-squeezed states are used to measure the motion of a mirror under an external stochastic force. Utilizing optical phase tracking and quantum smoothing techniques, we achieve position, momentum, and force estimation accuracies close to the quantum Cramér-Rao bounds with the coherent state, while the estimation using squeezed states shows clear quantum enhancements beyond the coherent-state bounds.