Cambridge University Press eBooks

Nonlinear systems for continuous measurements

Vladimir B. Braginsky, Farid Ya Khalili, Kip S. Thorne

Published Sep 24, 1992• 0 citations

Abstract

Fluctuational and dynamical back action of the measuring device During a continuous monitoring of a quantum object, the measuring device can act back on the object in general in two different ways: First, it can influence the evolution of the expectation value of the measured observables. This is called the “dynamical back action,” and it is a regular, predictable effect. (An example is the dynamical back action produced by the dynamical rigidity of a capacity sensor, which will be discussed in chapter X.) Second, the measuring device can perturb the observables in a random way, increasing their uncertainties, i.e. producing random deviations from their expectation values. A key feature of linear, continuous measurements, as discussed in the last chapter, is that for them these two types of back action are completely independent. The fluctuational back action is fundamental and irremovable, while the dynamical back action can be avoided entirely by a suitable construction of the measuring device. For nonlinear systems the situation is completely different, as one can readily see from a typical example of a nonlinear, continuous measurement: a monitoring of an oscillator's energy. In accord with the uncertainty relation, the energy measurement produces perturbations of the oscillator's phase, and as the measurement proceeds continuously, these perturbations cause the phase to diffuse in a Brownian-motion type way.

DOI: 10.1017/cbo9780511622748.010

Related papers

Emergent classicality in continuous quantum measurements

Juha Javanainen, Janne Ruostekoski

We develop a classical theoretical description for nonlinear many-body dynamics that incorporates the back-action of a continuous measurement process. The classical approach is compared with the exact quantum solution in an example with an atomic Bose–Einstein condensate in a double-well potential where the atom numbers in both potential wells are monitored by light scattering. In the classical description, the back-action of the measurements appears as diffusion of the relative phase of the condensates on each side of the trap. When the measurements are frequent enough to resolve the system dynamics, the system behaves classically. This happens even deep in the quantum regime, and demonstrates how classical physics emerges from quantum mechanics as a result of measurement back-action.

arXiv:1104.0820New Journal of PhysicsJan 3, 201328 citations

On Lossless Approximations, the Fluctuation- Dissipation Theorem, and Limitations of Measurements

Henrik Sandberg, Jean‐Charles Delvenne, John C. Doyle

In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics, and use the results to derive limitations of classical measurements. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear system is dissipative if, and only if, it can be approximated by a linear lossless system over arbitrarily long time intervals. Hence lossless systems are in this sense dense in dissipative systems. A linear active system can be approximated by a nonlinear lossless system that is charged with initial energy. As a by-product, we obtain mechanisms explaining the Onsager relations from time-reversible lossless approximations, and the fluctuation-dissipation theorem from uncertainty in the initial state of the lossless system. The results are applied to measurement devices and are used to quantify limits on the so-called observer effect, also called back action, which is the impact the measurement device has on the observed system. In particular, it is shown that deterministic back action can be compensated by using active elements, whereas stochastic back action is unavoidable and depends on the temperature of the measurement device.

arXiv:1009.2830IEEE Transactions on Automatic ControlJul 19, 201026 citations

Diffusive Dynamics and Continuous Monitoring

Alessio Serafini

In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.

CRC Press eBooksJun 1, 20230 citations

Quantum Back-Action of an Individual Variable-Strength Measurement

Michael Hatridge, Shyam Shankar, Mazyar Mirrahimi +9 more

Measuring a quantum system can randomly perturb its state. The strength and nature of this back-action depend on the quantity that is measured. In a partial measurement performed by an ideal apparatus, quantum physics predicts that the system remains in a pure state whose evolution can be tracked perfectly from the measurement record. We demonstrated this property using a superconducting qubit dispersively coupled to a cavity traversed by a microwave signal. The back-action on the qubit state of a single measurement of both signal quadratures was observed and shown to produce a stochastic operation whose action is determined by the measurement result. This accurate monitoring of a qubit state is an essential prerequisite for measurement-based feedback control of quantum systems.

arXiv:1903.11732ScienceJan 10, 2013278 citations

Avoiding back-action in quantum measurements

Ian S. Osborne

Quantum Mechanics The very process of measuring a quantum system has an influence on the system through the process of back-action. Suh et al. used a back-action evasion scheme to monitor the motion of a miniature oscillator without influencing its motion (see the Perspective by Bouwmeester). The scheme should help in the understanding of the fundamental limits associated with measurement and will have practical implications in providing a low-temperature thermometer and a probe of extremely weak forces. Science , this issue p. [1262][1] [1]: /lookup/doi/10.1126/science.1253258

ScienceJun 12, 20140 citations

Linear measurements

Vladimir B. Braginsky, Farid Ya Khalili, Kip S. Thorne

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.

Cambridge University Press eBooksSep 24, 19920 citations