The so-called standard quantum limits (SQL) of measurement errors of coordinate, momentum, amplitude of oscillations, energy, force etc. are due to back action of the meter on the system under test, whenever the meter responds to the coordinate of the system. These SQL are not fundamental and can be surmounted by various methods. In particular, in a coordinate measurement the SQL can be overcome by means of an appropriate correlation of conjugate meter variables. Conditions of quantum nonperturbing (nondemolition) and quasi-nonperturbing measurements of the energy of electromagnetic waves are discussed. Possible methods of these measurements are reviewed. Conditions for overcoming the SQL of wave energy measurement by the optical Kerr effect are analysed. The quantum limit of error of this measurement is discussed. The effects of dissipation, dispersion and generation of combination waves are considered. Results of experiments reported in the literature are discussed. The dependence of the quantum limit of detection of an external action upon a system on the initial state of the system is considered. The relation between the measurement error of an observable A and a perturbation of an observable B, when [A,B] is an operator, is examined.
The so-called standard quantum limits (SQL) of measurement errors of coordinate, momentum, amplitude of oscillations, energy, force etc. are due to back action of the meter on the system under test, whenever the meter responds to the coordinate of the system. These SQL are not fundamental and can be surmounted by various methods. In particular, in a coordinate measurement the SQL can be overcome by means of an appropriate correlation of conjugate meter variables. Conditions of quantum nonperturbing (nondemolition) and quasi-nonperturbing measurements of the energy of electromagnetic waves are discussed. Possible methods of these measurements are reviewed. Conditions for overcoming the SQL of wave energy measurement by the optical Kerr effect are analysed. The quantum limit of error of this measurement is discussed. The effects of dissipation, dispersion and generation of combination waves are considered. Results of experiments reported in the literature are discussed. The dependence of the quantum limit of detection of an external action upon a system on the initial state of the system is considered. The relation between the measurement error of an observable A and a perturbation of an observable B, when [A,B] is an operator, is examined.
In quantum mechanics, the Heisenberg uncertainty principle places a fundamental limit in the measurement precision for certain pairs of physical quantities, such as position and momentum, time and energy or amplitude and phase. Due to the Heisenberg uncertainty principle, any attempt to extract certain information from a quantum object would inevitably perturbitinanunpredictableway. This raises one question,"What is the precision limit in such quantum measurements?" The answer, standard quantum limit (SQL), has been obtained by Braginsky to figure out the fundamental quantum limits of displacement measurement in the context of gravitational wave detection. To circumvent the unavoidable quantum back-action from the priori measurement, quantum non-demolition measurement (QND) methods were introduced by Braginsky and Thorne. To surpass the SQL of the displacement measurement in an interferometer, one can measure only one quadrature of the mechanical motion while give up the information about the other canonically conjugated quadrature. Such measurements can be performed by periodic driving the mechanical oscillator, i.e. the back-action evading (BAE) measurement. Cavity optomechanics provides an ideal table-top platform for the testing of the quantum measurement theory. Mechanical oscillator is coupled to electromagnetic field via radiation pressure, which is enhanced by an optical micro-cavity. Over the last decade, laser cooling has enabled the preparation of mechanical oscillator in the ground state in both optical and microwave systems. BAE measurements of mechanical motion have been allowed in the microwave electromechanical systems, which led to the observations of mechanical squeezing and entanglement. However, despite the theoretical proposal almost 40 years ago, the sub-SQL measurements still remain elusive. This thesis reports our efforts approach the sub-SQL with a highly sideband-resolved silicon optomechanical crystal (OMC) in a 3He buffer gas environment at 2K. The OMC couples an optical mode at telecommunication wavelengths and a colocalized mechanical mode at GHz frequencies. The Helium3 buffer gas environment allows sufficient thermalization of the OMC despite the drastically decreased silicon thermal conductivity. We observe Floquet dynamics in motional sideband asymmetry measurement when employing multiple probe tones. The Floquet dynamics arises due to presence of Kerr-type nonlinearities and gives rise to an artificially modified motional sideband asymmetry, resulting from a synthetic gauge field among the Fourier modes. We demonstrate the first optical continuous two-tone backaction-evading measurement of a localized GHz frequency mechanical mode of silicon OMC close to the ground state by showing the transition from conventional sideband asymmetry to backaction-evading measurement. We discover a fundamental two-tone optomechanical instability and demonstrate its implications on the back-action evading measurement. Such instability imposes a fundamental limitation on other two-tone schemes, such as dissipative quantum mechanical squeezing. We demonstrate state-of-art laser sideband cooling of the mechanical motion to a mean thermal occupancy of 0.09 quantum, which is 7.4dB of the oscillator's zero-point energy and corresponds to 92% ground state probability. This also enables us to observe the dissipative mechanical squeezing below the zero-point motion for the first time with laser light.
State-of-the-art linear sensors of force, motion, and magnetic fields have reached the sensitivity where the quantum noise of the meter is significant or even dominant. In particular, the sensitivity of the best optomechanical devices has reached the standard quantum limit (SQL), which directly follows from the Heisenberg uncertainty relation and corresponds to balancing the measurement imprecision and the perturbation of the probe by the quantum back-action of the meter. The SQL is not truly fundamental and several methods for its overcoming have been proposed and demonstrated. At the same time, two quantum sensitivity constraints which are more fundamental are known. The first limit arises from the finiteness of the probing strength (in the case of optical interferometers---of the circulating optical power) and is known as the energetic quantum limit or, in a more general context, as the quantum Cram\'er-Rao bound (QCRB). The second limit arises from the dissipative dynamics of the probe, which prevents full efficacy of the quantum back-action evasion techniques developed for overcoming the SQL. No particular name has been assigned to this limit; we propose the term dissipative quantum limit (DQL) for it. Here we develop a unified theory of these two fundamental limits by deriving the general sensitivity constraint for stationary, linear systems from which they follow as particular cases. Our analysis reveals a phase transition occurring at the boundary between the QCRB-dominated and the DQL regimes, manifested by the discontinuous derivatives of the optimal spectral densities of the meter quantum noise. This leads to the counterintuitive (but favorable) finding that quantum-limited sensitivity can be achieved with meter noise cross-correlations having a nonzero imaginary component, a feature arising in lossy meter systems. Finally, we show that the DQL originates from the nonautocommutativity of the internal thermal noise of the probe and that it can be overcome in nonstationary measurements.
A fundamental peculiarity of gravitational waves is the fact that they couple very weakly to the matter. For this reason the response of a detector can be so small to be comparable with quantum undeterminations. An important limitation for the sensitivity of future generation interferometric detectors of gravitational waves is the so-called standard quantum limit, which makes impossible to do repeated, arbitrary precision measurements of the relevant observables. It can be explained as the consequence of the measurement process' back action on the detector's dynamics. A relevant theoretical and experimental effort has been done in these years in order to study the possibility of avoiding this limitation, which is no more considered a fundamental one. In this talk after a short introduction to the SQL problem from a general point of view I discuss some proposals for detection strategies designed to evade it.
In recent years advances in technology, especially in the field of quantum optics, have made possible repetitive quantum measurements on a single quantum object, an issue that remained an academic curiosity until 25 years ago. This book, first published in 1992, provides a tutorial introduction to this field. The book is coauthored by an experimentalist and a theorist: the former - Braginsky - was the first experimentalist after Weber to develop techniques for gravitational-wave detection; the latter - Khalili - contributed with Braginsky to develop the formal apparatus to describe the experimental devices for measurement schemes. After the first part, intended as a pedagogical appendix of a standard textbook, and devoted to the modern approach to quantum measurement, the second part is a review of some new methods of quantum measurements for macroscopic objects. The central problem is the detection of a small time-dependent classical force that acts on a quantum probe. This is the typical situation that occurs in gravitational-wave detection, where the probe is a macroscopic moving mass weakly coupled to a classical wave. The problem of gravitational-wave detection and the limitation to accuracy of measurement of position of a moving mass - the so-called standard quantum limit (SQL), which is a consequence of the random back-action of the measuring device on the mass - triggered the development of new quantum nondemolition (QND) measurements, i.e. measurements where the measured quantity is left unperturbed after the measurement. SQL and QND measurements (both Braginsky inventions) permeate the whole book as a transversal chapter. The aim is to give simple criteria and recipes to design a measuring device in order to optimize the accuracy of the measurement, and, at the same time, to minimize the perturbation that the device exerts on the measured object, so that it will not interfere with successive measurements on the same object. The continuous measurements are regarded as the limiting case of a sequence of repeated discrete measurements, when the interval between the measurements is much smaller than the characteristic time-scale on which the measured quantity changes. The analysis of continuous measurements give to the authors the opportunity to illustrate the watchdog effect and the quantum Zeno paradox (real effect), while deriving the master equation for the evolution of the object state. Then, an entire chapter is devoted to the analysis of devices for continuous measurement of small mechanical displacements. The book is easy to understand, and can serve as a complementary reading for graduate level courses in quantum mechanics. The material of the book can be useful for all scientists and engineers involved in researches related to high-sensitive quantum measurements, quantum information theory and quantum computation. The reader, however, should be aware that this book presents a very particular point of view, with no room left to very relevant concepts in quantum measurement theory, as, for example, squeezing, quantum amplification, joint-measurements of non-commuting observables and phase-measurements. More regrettably, the list of references is unsatisfactorily incomplete, with no mention of alternative developments, for example, regarding the SQL for measuring the position of a free mass. Here, schemes to breach the SQL have been proposed in the literature (Yuen H P 1983 Phys. Rev. Lett. 51 719, Ozawa M 1988 Phys. Rev. Lett. 60 385), whereas from the book one could conclude that such SQL would be unsurpassable. Also, a criticism is in order of the way of deriving some SQL's, based solely on the Heisenberg relation, even in consideration that the aim of such derivations is mostly pedagogical.
We found that the measurement sensitivity of an optical integrating gyroscope is fundamentally limited due to ponderomotive action of the light leading to the standard quantum limit of the rotation angle detection. The uncorrelated quantum fluctuations of power of clockwise and counterclockwise electromagnetic waves result in optical power-dependent uncertainty of the angular gyroscope position. We also show that, on the other hand, a quantum back action evading measurement of angular momentum of a gyroscope becomes feasible if proper measurement strategy is selected. The angle is perturbed in this case. This observation hints on fundamental inequivalency of integrating and rate gyroscopes.
The measurement process and the uncertainty relation In the unstarred sections of previous chapters, the back action of the measuring device on the measured object was analyzed by simply invoking the Heisenberg uncertainty relation for the perturbation and the measurement error. Is this simple method of analysis justified? The measurement process does not show up in the standard formulation of the uncertainty relations. Instead, the standard formulation, which follows from the fundamental postulates of quantum mechanics (section 2.1), couples the intrinsic rms uncertainties of observables as defined by the quantum state. Nevertheless, the examples of indirect measurements discussed above (the Heisenberg microscope and Doppler speed meter in chapter I, and the electron probe in chapter III) exhibit a tight coupling between the measurement process and the uncertainty relations. In particular, both the measurement error and the perturbation of the object can be traced to the uncertainty properties of the probe's initial state, and from the fact that this initial state is subject to the uncertainty relations, one can show that the measurement error and perturbation are also subject to it. Is it also possible, starting from the uncertainty relations as properties of the object's quantum state, to obtain the uncertainty relations for the measurement process? To answer this question, let us discuss the following thought experiment: We suppose that a generalized coordinate of some quantum object is measured, and that before the measurement the object was in a state with a very well defined momentum.