Non-Markovian quantum feedback from homodyne measurements : the effect of a non-zero feedback delay time

Content Provider	Semantic Scholar
Author	Giovannetti, Vittorio Tombesi, Paolo Vitali, David
Copyright Year	1999
Abstract	Although feedback schemes have been used for long times to control noise, a general theory of feedback for quantum systems has been developed only some years ago by Wiseman and Milburn [1–3]. Interesting possibilities are opened by the ability to control systems at the quantum level using appropriate feedback loops and some of them have been showed in a series of papers [4–9]. Ref. [4] has shown that an electro-optical feedback loop based on homodyne measurements of a cavity mode provides an affordable way to realize a squeezed bath for the mode. As a consequence, homodyne-mediated feedback can be used to get squeezing [5], and, in the case of optical cavities with an oscillating mirror, it can be used to significantly cool the mirror. This fact can be extremely useful for the interferometric detection of gravitational waves [8]. The application of a feedback loop realizes an effective “reservoir engineering” [10] and therefore it can be useful also for decoherence control, which is a rapidly expanding field since decoherence is the main limiting factor for quantum information processing [11]. Refs. [5–7,9] have already shown that the decoherence induced by photon leakage in electromagnetic cavities can be significantly suppressed with appropriate feedback loops, using the homodyne photocurrent in [5,6] and direct photodetection and atomic injection in [7,9]. However, all the relevant applications considered up to now always assume the zero feedback delay time limit τ → 0, which is much easier to handle because the problem becomes Markovian and the effect of feedback can be expressed in terms of an effective master equation [1–3]. The presence of a non-zero delay has been considered briefly only in [3], where the spectrum of a homodyne measurement has been evaluated for a simple case. The Markovian treatment is justified whenever the feedback delay time is much smaller than the typical cavity timescale. If one considers squeezing or some other stationary state phenomenon, the feedback delay time τ has to be compared with the cavity relaxation time γ−1 and for sufficiently good cavities the Markovian condition γτ 1 is usually satisfied. However, if one considers the feedback scheme for decoherence control, the delay τ has to be negligible with respect to the decoherence time tdec ' (γn̄)−1, which can be much shorter than the damping time when the cavity mean photon number n̄ is large. In these cases, the unavoidable non-zero feedback delay time may have important effects and it would be important to deal with the exact non-Markovian problem with τ 6= 0. There is in fact a renewed interest in non-Markovian effects, which can play an important role when considering quantum optics in high-Q cavities and in photonic bandgap materials. For this reason, non-Markovian trajectory theories have been recently developed in Refs. [12–14]. The quantum theory of feedback has been developed by Wiseman and Milburn in [1,2] using quantum trajectory theory [15], and only later Wiseman showed an equivalent derivation based on the input-output theory [16–18] in Ref. [3]. However Ref. [3] proved the equivalence between the two approaches in the perfect detection η = 1 case only. In this paper we shall see how to extend the quantum Langevin approach of the input-output theory to the non-unit efficiency case and we shall see that this theoretical framework is best suited to deal with the non-Markovian case of non-zero feedback delay time. We shall consider the non-Markovian effects by completely solving the dynamics of a cavity mode in the presence of a homodyne-mediated electro-optical feedback loop, which has been already considered (in the zero delay limit only) in Ref. [6]. The paper is organized as follows. In Section II we shall reconsider the quantum theory of feedback in the case of homodyne measurements, adopting the input-output theory of Gardiner and Collett [16–18] and we shall see how to introduce the non-unit detection efficiency in this framework. In Section III we shall completely solve the nonMarkovian dynamics in the presence of a non-zero feedback delay by considering the time evolution of the probability
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Language	English
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Subject Keyword	Arabic numeral 0 Body cavities Control system Dental caries Extravasation Feedback Fock state Information processing Linear programming relaxation Optics Paper Photons Quantum Theory Quantum decoherence Quantum information science Quantum system Reservoir Device Component Small Spectral leakage Stationary state Turing completeness biopsychosocial AOD use disorder theory
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Resource Type	Article