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QUANTUM CONTINUOUS MEASUREMENTS: THE SPECTRUM OF THE OUTPUT

Alberto Barchielli, Matteo Gregoratti

Published Oct 1, 2008• 4 citations
Publisher

Abstract

When a quantum system is monitored in continuous time, the result of the measurement is a stochastic process. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. A typical continuous measurement for quantum optical systems is the so called homodyne detection. In this paper we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible homodyne spectra and we discuss how this is related to the typical quantum phenomenon of squeezing.

DOI
10.1142/9789812835277_0005

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When a quantum system is monitored in continuous time, the result of the measurement is a stochastic process. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. A typical continuous measurement for quantum optical systems is the so called homodyne detection. In this paper we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible homodyne spectra and we discuss how this is related to the typical quantum phenomenon of squeezing.
When a quantum system is monitored in continuous time, the result of the measurement is a stochastic process. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. A typical continual measurement for quantum optical systems is the so called homodyne detection. In this paper we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible homodyne spectra and we discuss how this is related to the typical quantum phenomenon of squeezing.
Abstract The stochastic Schrödinger equation, of classical or quantum type, allows to describe open quantum systems under measurement in continuous time. In this paper we review the link between these two descriptions and we study the properties of the output of the measurement. For simplicity we deal only with the diffusive case. Firstly, we discuss the quantum stochastic Schrödinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schrödinger equation by diagonalization of suitable commuting quantum observables. Then, we give the a posteriori state, the conditional system state at time t given the output up to that time, and we link its evolution to the classical stochastic Schrödinger equation. Moreover, the relation with quantum filtering theory is shortly discussed. Finally, we study the output of the continuous measurement, which is a stochastic process with probability distribution given by the rules of quantum mechanics. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. In particular we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible spectra and we discuss how this is related to the typical quantum phenomenon of squeezing. We use a simple quantum system, a two-level atom stimulated by a laser, to discuss the differences between homodyne and heterodyne detection and to explicitly show squeezing and anti-squeezing of the homodyne spectrum and the Mollow triplet in the fluorescence spectrum.
Quantum Open System Theory is nowadays a highly developed field which involves, on the mathematical side, many mathematical concepts such as operator theory, semigroups of operators on certain Banach spaces (algebras), classical and quantum stochastic differential equations. Inside Quantum Open System Theory there is the theory of quantum continual measurements: a quantum system is monitored with continuity in time. This is surely an open system because, apart from possible interactions with the external environment like a thermal bath, the system is interacting with the measuring apparatus. The typical field in which a theory of continual measurements is needed is quantum optics: when light emitted by some system is observed (by counting of photons - direct detection - or through some interference mechanism - heterodyne and homodyne detection) a continual measurement is performed.
Spectral components of continuous squeezed fields are entangled. In this article we review and clarify this phenomenon by analyzing systematically the relations between the correlations of modes filtered from stationary continuous fields and the cross-power spectrum between the operators of the corresponding spectral components. Moreover, we study the specific spectral components that are filtered in homodyne or heterodyne detections and their entanglement properties. In particular, we establish the equivalence between two-mode squeezing variance and logarithmic negativity for the spectral components of continuous stationary fields, thereby demonstrating that the measurement of the homodyne or heterodyne spectrum is, in fact, a direct measurement of the logarithmic negativity between specific spectral modes. As an illustrative example, we apply these concepts to the analysis of entanglement in ponderomotive squeezing.
Recent advances, particularly in the area of quantum optics, have caused heightened interest in the fundamental limitations on the achievable accuracy that may be obtained in measuring quantum mechanical systems. We present a quantum correlation result and a noise commutation relation for correlated quantum systems. The measurement of quantum system observables requires the correlation of these micro-observables and the measuring apparatus (usually a macroscopic system). The noise commutation relation is applied to quantum measurements leading to a generalized Heisenberg uncertainty relation. The generalized Heisenberg uncertainty relation yields a lower bound on the inherent unavoidable extra noise in quantum measurements, which is due to the measuring process itself. To indicate the utility of the noise commutation relation, two different applications in optics are discussed. The first, an application of the generalized Heisenberg uncertainty relation, indicates how a model-independent noise limit for balanced homodyne detection can be obtained. Another application is discussed in which the noise commutation relation is applied to develop a model-independent lower bound for the inherent noise of a quantum optical linear amplifier.
Measuring the power spectral density of a stochastic process, such as a stochastic force or magnetic field, is a fundamental task in many sensing applications. Quantum noise is becoming a major limiting factor to such a task in future technology, especially in optomechanics for temperature, stochastic gravitational wave, and decoherence measurements. Motivated by this concern, here we prove a measurement-independent quantum limit to the accuracy of estimating the spectrum parameters of a classical stochastic process coupled to a quantum dynamical system. We demonstrate our results by analyzing the data from a continuous-optical-phase-estimation experiment and showing that the experimental performance with homodyne detection is close to the quantum limit. We further propose a spectral photon-counting method that can attain quantum-optimal performance for weak modulation and a coherent-state input, with an error scaling superior to that of homodyne detection at low signal-to-noise ratios.