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arXiv:quant-ph/0112128

Squeezing and feedback

Howard M. Wiseman

Published Dec 21, 2001• 0 citations
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Abstract

Electro-optical feedback has many features in common with optical nonlinearities and hence is relevant to the generation of squeezing. First, I discuss theoretical and experimental results for traveling-wave feedback, emphasizing how the ``in-loop'' squeezing (also known as ``squashing'') differs from free squeezing. Although such feedback, based on ordinary (demolition) photodetection cannot create free squeezing, it can be used to manipulate it. Then I treat feedback based on nonlinear quantum optical measurements (of which non-demolition measurements are one example). These {\em are} able to produce free squeezing, as shown in a number of experiments. Following that I discuss theories showing that intracavity squeezing can be increased using ordinary feedback, and produced using QND-based feedback. Finally, I return to ``squashed'' fields and present recent results showing that the reduced in-loop fluctuations can suppress atomic decay in a manner analogous to the effect for squeezed fields.

DOI
10.48550/arxiv.quant-ph/0112128

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Electro-optical feedback has many features in common with optical nonlinearities and hence is relevant to the generation of squeezing. First, I discuss theoretical and experimental results for traveling-wave feedback, emphasizing how the in-loop'' squeezing (also known as squashing'') differs from free squeezing. Although such feedback, based on ordinary (demolition) photodetection cannot create free squeezing, it can be used to manipulate it. Then I treat feedback based on nonlinear quantum optical measurements (of which non-demolition measurements are one example). These {\em are} able to produce free squeezing, as shown in a number of experiments. Following that I discuss theories showing that intracavity squeezing can be increased using ordinary feedback, and produced using QND-based feedback. Finally, I return to ``squashed'' fields and present recent results showing that the reduced in-loop fluctuations can suppress atomic decay in a manner analogous to the effect for squeezed fields.
Using a feedback loop it is possible to reduce the fluctuations in one quadrature of an in-loop field without increasing the fluctuations in the other. This effect has been known for a long time, and has recently been called `squashing' (Buchler et al 1999 Opt. Lett. 24 259), as opposed to the `squeezing' of a free field in which the conjugate fluctuations are increased. In this paper I present a general theory of squashing, including simultaneous squashing of both quadratures and simultaneous squeezing and squashing. I show that a two-level atom coupled to the in-loop light feels the effect of the fluctuations as calculated by the theory. In the ideal limit of light squeezed in one quadrature and squashed in the other, the atomic decay can be completely suppressed.
This thesis considers two types of applications of quantum feedback control; feedback creation of nonclassical states of light, and controlling nonclassical properties of an ensemble of atoms. An electro-optical feedback loop will create an in-loop field with nonclassical photon statistics similar to squeezed light, resulting in fluorescence line-narrowing of a two-level atom coupled to such light. We extend this theory to study a three-level atom coupled to broadband squashed light, and confirm the two-level atom line-narrowing using a more realistic non-Markovian description of the feedback loop. The second type of application utilizes continuous QND measurement of atomic ensembles. If we measure the collective spin, then the system experiences conditional spin squeezing dependent on the measurement results. We show that feedback based on these results can continuously drive the system into the same conditioned state, resulting in deterministically reproducible spin squeezing. If we measure the atom number fluctuations of a BEC, then, due to the nonlinearity of atomic self interactions, this is also information about phase fluctuations. We show that feedback based on this information can greatly reduce the collisional broadening of the linewidth of an atom laser out-coupled from the condensate.
We examine the quantum-limited behavior of an electro-optical intensity feedback loop and present a simple theory and experimental data showing excellent agreement. We show that, although the light incident upon the in-loop detector may be sub-Poissonian, this light has unique properties different from those of a freely propagating beam of intensity squeezed light. We support this by presenting the results of homodyne measurements of the phase noise of light extracted from the loop. The utility of the in-loop field is discussed, and it is shown that in all cases in which linear optical components are used, no advantage in signal-to-noise ratio is gained by taking measurements by using this light rather than a coherent source. We also discuss effects seen in the extracted or out-of-loop light. We demonstrate the existence of, and derive an expression for, an optimum gain for suppressing low-level classical noise. Conversely, in the high-gain limit, we demonstrate that the extra noise seen in the out-of-loop photocurrent that is due to the feedback process is expressible purely in terms of the mean photocurrents involved. Lastly, we introduce the novel concept of using an intensity feedback loop in conjunction with a squeezed source and show that the feedback loop has the capability of electronically transferring squeezing from one light beam to another.
Electro-optical feedback can produce an in-loop photocurrent with arbitrarily low noise. This is not regarded as evidence of ``real'' squeezing because squeezed light cannot be extracted from the loop using a linear beam splitter. Here I show that illuminating an atom (which is a nonlinear optical element) with ``in-loop'' squeezed light causes line narrowing of one quadrature of the atom's fluorescence. This has long been regarded as an effect that can only be produced by squeezing. Experiments on atoms using in-loop squeezing should be much easier than those with conventional sources of squeezed light.
Coherent feedback is a non-measurement based, hence a back-action free, method of control for quantum systems. A typical application of this control scheme is squeezing enhancement, a purely non-classical effect in quantum optics. In this technical note we report its first experimental demonstration that well agrees with the theory taking into account time delays and losses in the coherent feedback loop. The results clarify both the benefit and the limitation of coherent feedback control in a practical situation.
Quantum nonlinear feedback control is developed for traveling optical fields. We first describe the discretization of the traveling optical fields. The discrete-time formulation is used to describe the stochastic master equation subject to homodyne measurement. Nonlinear feedback is formulated by directly feeding the measurement outcomes back to the traveling field through a multiplicative action. Since the measurement outcomes have a correlation with the system, the multiplicative feedback control can create nonlinear effects in the traveling field. In this formulation, a time delay is naturally introduced in the feedback loop. This is essentially different from instantaneous feedback in a continuous-time setting. As an example of the feedback scheme, a quantum nondemolition sum gate is considered. Numerical results show that quantum superposition state can be created by applying the feedback to a squeezed state.