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arXiv:0906.1909 · Physical Review A

Cooling a micromechanical resonator to its ground state by measurement and feedback

Christian Bergenfeldt, Klaus Mølmer

Published Oct 27, 2009• 7 citations
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Abstract

We present an analysis of the cooling of a micromechanical resonator by means of measurement and feedback. The measurements are performed via the coupling to a Cooper-pair box, and although the coupling does not lead to net cooling, the extraction of information and hence entropy from the system leads to a pure quantum state. Under suitable circumstances, the states become very close to coherent states, conditioned on the measurement record, and can hence be displaced to the oscillator ground state.

DOI
10.1103/physreva.80.043838

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We present an analysis of the cooling of a micromechanical resonator by means of measurement and feedback. The measurements are performed via the coupling to a Cooper-pair box, and although the coupling does not lead to net cooling, the extraction of information and hence entropy from the system leads to a pure quantum state. Under suitable circumstances, the states become very close to coherent states, conditioned on the measurement record, and can hence be displaced to the oscillator ground state.
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.
We propose an application of a single Cooper pair box (Josephson qubit) for active cooling of nanomechanical resonators. Latest experiments with Josephson qubits demonstrated that long coherence time of the order of microsecond can be achieved in special symmetry points. Here we show that this level of coherence is sufficient to perform an analog of the well known in quantum optics ``laser'' cooling of a nanomechanical resonator capacitively coupled to the qubit. By applying an ac driving to the qubit or the resonator, resonators with frequency of order 100 MHz and quality factors higher than 103{10}^{3}103 can be efficiently cooled down to their ground state, while lower-frequency resonators can be cooled down to micro-Kelvin temperatures. We also consider an alternative setup where dc-voltage-induced Josephson oscillations play the role of the ac driving and show that cooling is possible in this case as well.
Feedback-based control of nano- and micromechanical resonators can enable the study of macroscopic quantum phenomena and also sensitive force measurements. Here, we demonstrate the feedback cooling of a low-loss and high-stress macroscopic SiN membrane resonator close to its quantum ground state. We use the microwave optomechanical platform, where the resonator is coupled to a microwave cavity. The experiment utilizes a Josephson travelling wave parametric amplifier, which is nearly quantum-limited in added noise, and is important to mitigate resonator heating due to system noise in the feedback loop. We reach a thermal phonon number as low as 1.6, which is limited primarily by microwave-induced heating. We also discuss the sideband asymmetry observed when a weak microwave tone for independent readout is applied in addition to other tones used for the cooling. The asymmetry can be qualitatively attributed to the quantum-mechanical imbalance between emission and absorption. However, we find that the observed asymmetry is only partially due to this quantum effect. In specific situations, the asymmetry is fully dominated by a cavity Kerr effect under multitone irradiation.
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.
Feedback-based control of nano- and micromechanical resonators can enable the study of macroscopic quantum phenomena and also sensitive force measurements. Here, we demonstrate the feedback cooling of a low-loss and high-stress macroscopic SiN membrane resonator close to its quantum ground state. We use the microwave optomechanical platform, where the resonator is coupled to a microwave cavity. The experiment utilizes a Josephson traveling-wave parametric amplifier, which is nearly quantum-limited in added noise, and is important for mitigating resonator heating due to system noise in the feedback loop. We reach a thermal phonon number as low as 1.6, which is limited primarily by microwave-induced heating. We also discuss the sideband asymmetry observed when a weak microwave tone for independent readout is applied in addition to other tones used for the cooling. In a typical situation, the asymmetry can be attributed to the quantum-mechanical imbalance between emission and absorption. In specific situations, however, we find that the asymmetry is an artifact due to coupling of different sideband processes by cavity nonlinearity under multitone irradiation.