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Quantum Feedback Cooling of a Mechanical Oscillator Using Variational Measurements:Tweaking Heisenberg's Microscope

Habibi, H., Zeuthen, E., Ghanaatshoar, M., Hammerer, K.

Published Jan 1, 2016• 15 citations
OpenAlex

Abstract

We revisit the problem of preparing a mechanical oscillator in the vicinity of its quantum-mechanical ground state by means of feedback cooling based on continuous optical detection of the oscillator position. In the parameter regime relevant to ground state cooling, the optical back-action and imprecision noise set the bottleneck of achievable cooling and must be carefully balanced. This can be achieved by adapting the phase of the local oscillator in the homodyne detection realizing a so-called variational measurement. The trade-off between accurate position measurement and minimal disturbance can be understood in terms of Heisenberg's microscope and becomes particularly relevant when the measurement and feedback processes happen to be fast within the quantum coherence time of the system to be cooled. This corresponds to the regime of large quantum cooperativity $C_{\\text{q}}\\gtrsim1$, which was achieved in recent experiments on feedback cooling. Our method provides a simple path to further pushing the limits of current state-of-the-art experiments in quantum optomechanics.

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•Jul 18, 2014•1 citation
We revisit the problem of preparing a mechanical oscillator in the vicinity of its quantum-mechanical ground state by means of feedback cooling based on continuous optical detection of the oscillator position. In the parameter regime relevant to ground state cooling, the optical back-action and imprecision noise set the bottleneck of achievable cooling and must be carefully balanced. This can be achieved by adapting the phase of the local oscillator in the homodyne detection realizing a so-called variational measurement. The trade-off between accurate position measurement and minimal disturbance can be understood in terms of Heisenberg's microscope and becomes particularly relevant when the measurement and feedback processes happen to be fast within the quantum coherence time of the system to be cooled. This corresponds to the regime of large quantum cooperativity Ctextqgtrsim1C_{\\text{q}}\\gtrsim1Ctextq​gtrsim1, which was achieved in recent experiments on feedback cooling. Our method provides a simple path to further pushing the limits of current state-of-the-art experiments in quantum optomechanics.
Abstract We revisit the problem of preparing a mechanical oscillator in the vicinity of its quantum-mechanical ground state by means of feedback cooling based on continuous optical detection of the oscillator position. In the parameter regime relevant to ground-state cooling, the optical back-action and imprecision noise set the bottleneck of achievable cooling and must be carefully balanced. This can be achieved by adapting the phase of the local oscillator in the homodyne detection realizing a so-called variational measurement. The trade-off between accurate position measurement and minimal disturbance can be understood in terms of Heisenberg’s microscope and becomes particularly relevant when the measurement and feedback processes happen to be fast within the quantum coherence time of the system to be cooled. This corresponds to the regime of large quantum cooperativity C q ≳ 1 , which was achieved in recent experiments on feedback cooling. Our method provides a simple path to further pushing the limits of current state-of-the-art experiments in quantum optomechanics.
Why are classical theories often sufficient to describe the physics of our world even though everything around us is entirely composed of microscopic quantum systems? The boundary between these two fundamentally dissimilar theories remains an unsolved problem in modern physics. Position measurements of small objects allow us to probe the area where the classical approximation breaks down. In quantum mechanics, Heisenbergâ s uncertainty principle dictates that any measurement of the position must be accompanied by measurement induced back-action---in this case manifested as an uncertainty in the momentum. In recent years, cavity optomechanics has become a powerful tool to perform precise position measurements and investigate their fundamental limitations. The utilization of optical micro-cavities greatly enhances the interaction between light and state-of-the-art nanomechanical oscillators. Therefore, quantum mechanical phenomena have been successfully observed in systems far beyond the microscopic world. In such a cavity optomechanical system, the fluctuations in the position of the oscillator are transduced onto the phase of the light, while fluctuations in the amplitude of the light disturb the momentum of the oscillator during the measurement. As a consequence, correlations are established between the amplitude and phase quadrature of the probe light. However, so far, observation of quantum effects has been limited exclusively to cryogenic experiments, and access to the quantum regime at room temperature has remained an elusive goal because the overwhelming amount of thermal motion masks the weak quantum effects. This thesis describes the engineering of a high-performance cavity optomechanical device and presents experimental results showing, for the first time, the broadband effects of quantum back-action at room temperature. The device strongly couples mechanical and optical modes of exceptionally high quality factors to provide a measurement sensitivity ∼ ⁣104\sim\!10^4∼104 times below the requirement to resolve the zero-point fluctuations of the mechanical oscillator. The quantum back-action is then observed through the correlations created between the probe light and the motion of the nanomechanical oscillator. A so-called â variational measurementâ , which detects the transmitted light in a homodyne detector tuned close to the amplitude quadrature, resolves the quantum noise due to these correlations at the level of 10\% of the thermal noise over more than an octave of Fourier frequencies around mechanical resonance. Moreover, building on this result, an additional experiment demonstrates the ability to achieve quantum enhanced metrology. In this case, the generated quantum correlations are used to cancel quantum noise in the measurement record, which then leads to an improved relative signal-to-noise ratio in measurements of an external force. In conclusion, the successful observation of broadband quantum behavior on a macroscopic object at room temperature is an important milestone in the field of cavity optomechanics. Specifically, this result heralds the rise of optomechanical systems as a platform for quantum physics at room temperature and shows promise for generation of ponderomotive squeezing in room-temperature interferometers.
We present a detailed study to show the possibility of approaching the quantum ground state of a hybrid optomechanical quantum device formed by a Bose–Einstein condensate (BEC) confined inside a high-finesse optical cavity with an oscillatory end mirror. Cooling is achieved using two experimentally realizable schemes: back-action cooling and cold damping quantum feedback cooling. In both the schemes, we found that increasing the two-body atom–atom interaction brings the mechanical oscillator to its quantum ground state. It has been observed that back-action cooling is more effective in the good cavity limit, while the cold damping cooling scheme is more relevant in the bad cavity limit. It is also shown that in the cold damping scheme, the device is more efficient in the presence of a BEC than in the absence of a BEC.
We revisit the stochastic master equation approach to feedback cooling of a quantum mechanical oscillator undergoing position measurement. By introducing a rotating wave approximation for the measurement and bath coupling, we can provide a more intuitive analysis of the achievable cooling in various regimes of measurement sensitivity and temperature. We also discuss explicitly the effect of backaction noise on the characteristics of the optimal feedback. The resulting rotating wave master equation has found application in our recent work on squeezing the oscillator motion using parametric driving and may have wider interest.
cavityoptomechanical coupling, and realizes a measurement of the oscillator’s displacement with an imprecision 40 dB below that at the standard quantum limit, while maintaining an imprecision-backaction product within a factor of 5 of the Heisenberg uncertainty limit. As a demonstration of its utility, we use the measurement to feedback cool the oscillator to an phonon occupation of 5.4 0.7 (i.e, a ground state probablity of 16%). Our results establish a new benchmark for the performance of a linear position sensor, and signal the emergence of engineered mechanical oscillators as practical subjects for measurement-based quantum control.
We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling --- an example of coherent feedback control --- and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing --- that is, for the same interaction strength --- and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.