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arXiv:0809.2024

Creation of a quantum oscillator by classical control

S. L. Danilishin, H. Müller‐Ebhardt, H. Rehbein, Kentaro Somiya, Roman Schnabel, K. Danzmann, T. R. Corbitt, Christopher C. Wipf, N. Mavalvala, Yanbei Chen

Published Sep 11, 2008• 11 citations
arXivHTMLPDF

Abstract

As a pure quantum state is being approached via linear feedback, and the occupation number approaches and eventually goes below unity, optimal control becomes crucial. We obtain theoretically the optimal feedback controller that minimizes the uncertainty for a general linear measurement process, and show that even in the absence of classical noise, a pure quantum state is not always achievable via feedback. For Markovian measurements, the deviation from minimum Heisenberg Uncertainty is found to be closely related to the extent to which the device beats the free-mass Standard Quantum Limit for force measurement. We then specialize to optical Markovian measurements, and demonstrate that a slight modification to the usual input-output scheme -- either injecting frequency independent squeezed vacuum or making a homodyne detection at a non-phase quadrature -- allows controlled states of kilogram-scale mirrors in future LIGO interferometers to reach occupation numbers significantly below unity.

DOI
10.48550/arxiv.0809.2024

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Measurements in quantum mechanics cannot perfectly distinguish all states and necessarily disturb the measured system. We present and analyze a proposal to demonstrate fundamental limits on quantum control of a single qubit arising from these properties of quantum measurements. We consider a qubit prepared in one of two nonorthogonal states and subsequently subjected to dephasing noise. The task is to use measurement and feedback control to attempt to correct the state of the qubit. We demonstrate that projective measurements are not optimal for this task, and that there exists a nonprojective measurement with an optimum measurement strength which achieves the best trade-off between gaining information about the system and disturbing it through measurement backaction. We study the performance of a quantum control scheme that makes use of this weak measurement followed by feedback control, and demonstrate that it realizes the optimal recovery from noise for this system. We contrast this approach with various classically inspired control schemes.

arXiv:quant-ph/0608037•Physical Review A•Jan 24, 2007•79 citations
As a pure quantum state is being approached via linear feedback, and the occupation number approaches and eventually goes below unity, optimal control becomes crucial. We obtain theoretically the optimal feedback controller that minimizes the uncertainty for a general linear measurement process, and show that even in the absence of classical noise, a pure quantum state is not always achievable via feedback. For Markovian measurements, the deviation from minimum Heisenberg Uncertainty is found to be closely related to the extent to which the device beats the free-mass Standard Quantum Limit for force measurement. We then specialize to optical Markovian measurements, and demonstrate that a slight modification to the usual input-output scheme -- either injecting frequency independent squeezed vacuum or making a homodyne detection at a non-phase quadrature -- allows controlled states of kilogram-scale mirrors in future LIGO interferometers to reach occupation numbers significantly below unity.
Abstract The measurement of minuscule forces and displacements with ever greater precision is inhibited by the Heisenberg uncertainty principle, which imposes a limit to the precision with which the position of an object can be measured continuously, known as the standard quantum limit1–4. When light is used as the probe, the standard quantum limit arises from the balance between the uncertainties of the photon radiation pressure applied to the object and of the photon number in the photoelectric detection. The only way to surpass the standard quantum limit is by introducing correlations between the position/momentum uncertainty of the object and the photon number/phase uncertainty of the light that it reflects5. Here we confirm experimentally the theoretical prediction5 that this type of quantum correlation is naturally produced in the Laser Interferometer Gravitational-wave Observatory (LIGO). We characterize and compare noise spectra taken without squeezing and with squeezed vacuum states injected at varying quadrature angles. After subtracting classical noise, our measurements show that the quantum mechanical uncertainties in the phases of the 200-kilowatt laser beams and in the positions of the 40-kilogram mirrors of the Advanced LIGO detectors yield a joint quantum uncertainty that is a factor of 1.4 (3 decibels) below the standard quantum limit. We anticipate that the use of quantum correlations will improve not only the observation of gravitational waves, but also more broadly future quantum noise-limited measurements.
In many optical and atomic systems it is now possible to monitor an individual quantum system with high signal to noise and feed back by altering the system dynamics in real time. This motivates further development of quantum mechanical descriptions of feedback control. We discuss recent work on closed loop control of open quantum systems, focusing on general linear systems for which the statistics of the problem are Gaussian. Such systems may be realized with linear optics, parametric amplifiers and homodyne detection. This problem allows a direct comparison with the classical linear, quadratic cost, Gaussian noise (LQG) optimal control problem. This highlights the key distinction between the quantum and classical theories for linear systems: that increased measurement sensitivity may run counter to the control objectives since it increases the backaction noise. While in an idealized classical control problem it is always preferable to obtain a better sensor, quantum mechanical problems generically have an optimal sensitivity since quantum mechanical measurements irreducibly disturb the system. The general theory will be illustrated by reference to specific simple examples.
The LIGO-II gravitational-wave interferometers (ca. 2006--2008) are designed to have sensitivities near the standard quantum limit (SQL) in the vicinity of 100 Hz. This paper describes and analyzes possible designs for subsequent LIGO-III interferometers that can beat the SQL. These designs are identical to a conventional broad band interferometer (without signal recycling), except for new input and/or output optics. Three designs are analyzed: (i) a squeezed-input interferometer (conceived by Unruh based on earlier work of Caves) in which squeezed vacuum with frequency-dependent (FD) squeeze angle is injected into the interferometer's dark port; (ii) a variational-output interferometer (conceived in a different form by Vyatchanin, Matsko and Zubova), in which homodyne detection with FD homodyne phase is performed on the output light; and (iii) a squeezed-variational interferometer with squeezed input and FD-homodyne output. It is shown that the FD squeezed-input light can be produced by sending ordinary squeezed light through two successive Fabry-P\'erot filter cavities before injection into the interferometer, and FD-homodyne detection can be achieved by sending the output light through two filter cavities before ordinary homodyne detection. With anticipated technology (power squeeze factor ${e}^{\ensuremath{-}2R}=0.1$ for input squeezed vacuum and net fractional loss of signal power in arm cavities and output optical train ${\ensuremath{\epsilon}}_{*}=0.01)$ and using an input laser power Io{I}_{o}Io​ in units of that required to reach the SQL (the planned LIGO-II power, ISQL),{I}_{\mathrm{SQL}}),ISQL​), the three types of interferometer could beat the amplitude SQL at 100 Hz by the following amounts $\ensuremath{\mu}\ensuremath{\equiv}\sqrt{{S}_{h}}/\sqrt{{S}_{h}^{\mathrm{SQL}}}$ and with the following corresponding increase $\mathcal{V}=1/{\ensuremath{\mu}}^{3}$ in the volume of the universe that can be searched for a given noncosmological source: $\mathrm{Squeezed}\mathrm{input}---\ensuremath{\mu}\ensuremath{\simeq}\sqrt{{e}^{\ensuremath{-}2R}}\ensuremath{\simeq}0.3$ and $\mathcal{V}\ensuremath{\simeq}{1/0.3}^{3}\ensuremath{\simeq}30$ using Io/ISQL=1.{I}_{o}{/I}_{\mathrm{SQL}}=1.Io​/ISQL​=1. $Variational\ensuremath{-}output---\ensuremath{\mu}\ensuremath{\simeq}{\ensuremath{\epsilon}}_{*}^{1/4}\ensuremath{\simeq}0.3$ and $\mathcal{V}\ensuremath{\simeq}30$ but only if the optics can handle a ten times larger power: ${I}_{o}{/I}_{\mathrm{SQL}}\ensuremath{\simeq}1/\sqrt{{\ensuremath{\epsilon}}_{*}}=10.$ $\mathrm{Squeezed}\mathrm{varational}---\ensuremath{\mu}{=1.3(e}^{\ensuremath{-}2R}{\ensuremath{\epsilon}}_{*}{)}^{1/4}\ensuremath{\simeq}0.24$ and $\mathcal{V}\ensuremath{\simeq}80$ using Io/ISQL=1;{I}_{o}{/I}_{\mathrm{SQL}}=1;Io​/ISQL​=1; and $\ensuremath{\mu}\ensuremath{\simeq}{(e}^{\ensuremath{-}2R}{\ensuremath{\epsilon}}_{*}{)}^{1/4}\ensuremath{\simeq}0.18$ and $\mathcal{V}\ensuremath{\simeq}180$ using ${I}_{o}{/I}_{\mathrm{SQL}}=\sqrt{{e}^{\ensuremath{-}2R}/{\ensuremath{\epsilon}}_{*}}\ensuremath{\simeq}3.2.$
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement. Using a quantum noise model to describe the stochastic evolution of the open quantum system and working in parallel with classical indeterministic control theory, we present the theory of nonlinear optimal quantum feedback control. The resulting quantum Bellman equation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities with the corresponding classical control problem.
In this paper we propose an experimentally viable scheme to enhance the sensitivity of force detection in a hybrid optomechanical setup assisted by squeezed vacuum injection, beyond the standard quantum limit (SQL). The scheme is based on a combination of the coherent quantum noise cancellation (CQNC) strategy with a variational homodyne detection of the cavity output spectrum in which the phase of the local oscillator is optimized. In CQNC, realizing a negative-mass oscillator in the system leads to exact cancellation of the backaction noise from the mechanics due to destructive quantum interference. Squeezed vacuum injection enhances this cancellation and allows sub-SQL sensitivity to be reached in a wide frequency band and at much lower input laser powers. We show here that the adoption of variational homodyne readout enables us to enhance this noise cancellation up to 40dB40\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}40dB compared to the standard case of detection of the optical output phase quadrature, leading to a remarkable force sensitivity of the order of ${10}^{\ensuremath{-}19}\phantom{\rule{0.28em}{0ex}}\mathrm{N}/\sqrt{\mathrm{Hz}}$, about 7070%70 enhancement compared to the standard case. Moreover, we show that at nonzero cavity detuning, the signal response can be amplified at a level three to five times larger than that in the standard case without variational homodyne readout, improving the signal-to-noise ratio. Finally, the variational readout CQNC developed in this paper may be applied to other optomechanical-like platforms such as levitated systems and multimode optomechanical arrays or crystals as well as Josephson-based optomechanical systems.
Measurements in quantum mechanics cannot perfectly distinguish all states and necessarily disturb the measured system. We present and analyze a proposal to demonstrate fundamental limits on quantum control of a single qubit arising from these properties of quantum measurements. We consider a qubit prepared in one of two nonorthogonal states and subsequently subjected to dephasing noise. The task is to use measurement and feedback control to attempt to correct the state of the qubit. We demonstrate that projective measurements are not optimal for this task, and that there exists a nonprojective measurement with an optimum measurement strength which achieves the best trade-off between gaining information about the system and disturbing it through measurement backaction. We study the performance of a quantum control scheme that makes use of this weak measurement followed by feedback control, and demonstrate that it realizes the optimal recovery from noise for this system. We contrast this approach with various classically inspired control schemes.