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Frontiers in Optics

Evading the Schawlow-Townes limit in feedback oscillators

Hudson A. Loughlin, Vivishek Sudhir

Published Jan 1, 2023• 0 citations
Publisher

Abstract

We develop a general quantum theory of feedback oscillators and use it to study the origin of quantum noise in such systems, the limitations posed by quantum noise, and systematic techniques to evade these limitations. For a laser, the quantum noise limit is the Schawlow-Townes limit, which can be evaded using squeezing or entanglement.
DOI
10.1364/fio.2023.fm1b.5

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Feedback oscillators, consisting of an amplifier whose output is partially fed back to its input, provide stable references for standardization and synchronization. Notably, the laser is such an oscillator whose performance can be limited by quantum fluctuations. The resulting frequency instability, quantified by the Schawlow-Townes formula, sets a limit to laser linewidth. Here, we show that the Schawlow-Townes formula applies universally to feedback oscillators beyond lasers. This is because it arises from quantum noise added by the amplifier and out-coupler in the feedback loop. Tracing the precise origin of quantum noise in an oscillator informs techniques to systematically evade it: we show how squeezing and entanglement can enable sub-Schawlow-Townes linewidth feedback oscillators. Our analysis clarifies the quantum limits to the stability of feedback oscillators in general, derives a standard quantum limit (SQL) for all such devices, and quantifies the efficacy of quantum strategies in realizing sub-SQL oscillators.
Feedback oscillators, such as lasers and masers, serve as time references in modern computing, communication, and measurement. Quantum fluctuations ultimately limit their phase stability and ability to keep time precisely; in the absence of quantum engineering, their phase stability is bounded by a standard quantum limit (SQL). Techniques to improve the frequency stability of feedback oscillators beyond the SQL have been theorized, but have not yet been demonstrated. We demonstrate an opto-electronic oscillator (OEO), a type of feedback oscillator, with phase stability approaching the SQL. We then engineer the OEO's quantum state to improve its phase stability, thereby demonstrating the essential principle quantum-enhancement of feedback oscillators. Similar techniques may be employed to evade the SQL in other feedback oscillators such as masers and lasers.
A fully quantum-mechanical model of feedback to the pump source of a four-level laser is developed with a view to predicting the achievable intensity noise reduction. For solid-state lasers, the model shows that quantum noise sources due to laser dynamics have a significant impact on the optimization of the feedback loop. Experimental results obtained with a diode pumped neodymium:yttrium aluminum garnet (Nd:YAG) laser are found to be in good agreement with the theoretical model. The ultimate limit to the noise suppression comes from the quantum noise due to the measurement processes in the feedback loop. A scheme to overcome this limit using a squeezed vacuum is theoretically demonstrated to be a highly efficient method of generating bright intensity squeezed light, particularly in the kHz regime where bright squeezing is otherwise difficult to obtain.
The main advantage of quantum metrology relies on the effective use of entanglement, which indeed allows us to achieve strictly better estimation performance over the standard quantum limit. In this paper, we propose an analogous method utilizing entanglement for the purpose of feedback control. The system considered is a general linear dynamical quantum system, where the control goal can be systematically formulated as a linear quadratic Gaussian control problem based on the quantum Kalman filtering method; in this setting, an entangled input probe field is effectively used to reduce the estimation error and accordingly the control cost function. In particular, we show that, in the problem of cooling an opto-mechanical oscillator, the entanglement-assisted feedback control can lower the stationary occupation number of the oscillator below the limit attainable by the controller with a coherent probe field and furthermore beats the controller with an optimized squeezed probe field.
We study a class of feedback oscillators realized by a phase-insensitive amplifier in positive feedback, where either the amplifier or the feedback element may determine the oscillator's linewidth. The spectral purity of the output of such a device originates from basic demands of quantum mechanics and causality. The resulting expression generalizes the Schawlow-Townes limit, which is itself one component of a standard quantum limit for feedback oscillators. Recently realized bad-cavity oscillators, such as superradiant lasers and solid-state masers, can saturate this generalized Schawlow-Townes limit. This limit can be surpassed through appropriate quantum engineering, for example, by atomic spin squeezing in a superradiant laser.