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arXiv:0905.3729 · Physical Review A

Probing macroscopic quantum states with a sub-Heisenberg accuracy

Haixing Miao, S. L. Danilishin, H. Müller‐Ebhardt, H. Rehbein, Kentaro Somiya, Yanbei Chen

Published Jan 25, 2010• 47 citations
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Abstract

Significant achievements in high-sensitivity measurements will soon allow us to probe quantum behaviors of macroscopic mechanical oscillators. In a recent work [Phys. Rev. A 80, 043802 (2009)], we formulated a general framework for treating preparation of Gaussian quantum states of macroscopic oscillators through linear position measurements. To outline a complete procedure for testing macroscopic quantum mechanics, here we consider a subsequent verification stage which probes the prepared macroscopic quantum state and verifies the quantum dynamics. By adopting an optimal time-dependent homodyne detection in which the phase of the local oscillator varies in time, the conditional quantum state can be characterized below the Heisenberg limit, thereby achieving a quantum tomography. In the limiting case of no readout loss, such a scheme evades measurement-induced back action, which is identical to the variational-type measurement scheme invented by Vyatchanin et al. [JETP 77, 218 (1993)] but in the context for detecting gravitational waves. To motivate macroscopic quantum mechanics experiments with future gravitational-wave detectors, we mostly focus on the parameter regime where the characteristic measurement frequency is much higher than the oscillator frequency and the classical noises are Markovian, which captures the main feature of a broadband gravitational-wave detector. In addition, we discuss verifications of Einstein-Podolsky-Rosen-type entanglement between macroscopic test masses in future gravitational-wave detectors, which enables us to test one particular version of gravity decoherence conjectured by Di\'osi [Phys. Lett. A120, 377 (1987)] and Penrose [Gen. Rel. Grav. 28, 581 (1996)].

DOI
10.1103/physreva.81.012114

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Significant achievements in high-sensitivity measurements will soon allow us to probe quantum behaviors of macroscopic mechanical oscillators. In a recent work [Phys. Rev. A 80, 043802 (2009)], we formulated a general framework for treating preparation of Gaussian quantum states of macroscopic oscillators through linear position measurements. To outline a complete procedure for testing macroscopic quantum mechanics, here we consider a subsequent verification stage which probes the prepared macroscopic quantum state and verifies the quantum dynamics. By adopting an optimal time-dependent homodyne detection in which the phase of the local oscillator varies in time, the conditional quantum state can be characterized below the Heisenberg limit, thereby achieving a quantum tomography. In the limiting case of no readout loss, such a scheme evades measurement-induced back action, which is identical to the variational-type measurement scheme invented by Vyatchanin et al. [JETP 77, 218 (1993)] but in the context for detecting gravitational waves. To motivate macroscopic quantum mechanics experiments with future gravitational-wave detectors, we mostly focus on the parameter regime where the characteristic measurement frequency is much higher than the oscillator frequency and the classical noises are Markovian, which captures the main feature of a broadband gravitational-wave detector. In addition, we discuss verifications of Einstein-Podolsky-Rosen-type entanglement between macroscopic test masses in future gravitational-wave detectors, which enables us to test one particular version of gravity decoherence conjectured by Di'osi [Phys. Lett. A120, 377 (1987)] and Penrose [Gen. Rel. Grav. 28, 581 (1996)].
Long-baseline laser-interferometer gravitational-wave (GW) detectors are operating at a factor of $\ensuremath{\sim}10$ (in amplitude) above the standard quantum limit (SQL) within a broad frequency band (in the sense that $\ensuremath{\Delta}f\ensuremath{\sim}f$). Such a low-noise budget has already allowed the creation of a controlled 2.7 kg macroscopic oscillator with an effective eigenfrequency of 150 Hz and an occupation number of $\ensuremath{\sim}200$. This result, along with the prospect for further improvements, heralds the possibility of experimentally probing macroscopic quantum mechanics (MQM)---quantum mechanical behavior of objects in the realm of everyday experience---using GW detectors. In this paper, we provide the mathematical foundation for the first step of a MQM experiment: the preparation of a macroscopic test mass into a nearly minimum-Heisenberg-limited Gaussian quantum state, which is possible if the interferometer's classical noise beats the SQL in a broad frequency band. Our formalism, based on Wiener filtering, allows a straightforward conversion from the noise budget of a laser interferometer, in terms of noise spectra, into the strategy for quantum-state preparation and the quality of the prepared state. Using this formalism, we consider how Gaussian entanglement can be built among two macroscopic test masses and the performance of the planned Advanced LIGO interferometers in quantum-state preparation.
The quantum measurement process connects the quantum world and the classical world. The phrase ‘quantum measurement’ can have two meanings: measurement of a weak classical force, with the impact of quatum fluctuations on the measurement sensitivity, and the quantum mechanics of macroscopic objects: to try to prepare, manipulate and characterize the quantum state of a macroscopic quantum object through quantum measurement. Quantum noise leads to the Standard Quantum Limit (SQL), which provides the magnitude in which we must consider both measurement precision and measurement-induced back-action. The beginning of the chapter will be devoted to this thread of thought. The free-mass SQL actually provides a benchmark for the ‘quantum-ness’ of the system. We will show that a sub-SQL device can be used to prepare nearly pure quantum states and mechanical entanglement, as well as non-Gaussian quantum states that have no classical counterparts.
Macroscopic optomechanical systems offer a promising testbed for distinguishing whether gravity acts as a quantum interaction or as a classical field. Schrodinger-Newton (SN) theory is the nonrelativistic limit of semi-classical gravity where quantum matter couples to classical gravity. Based on SN theory, this work investigates how classical self-gravity affects continuous quantum state tomography of a macroscopic mechanical oscillator monitored by variable-angle homodyne detection. In the Schrodinger-Newton (SN) theory, the measurement record arises from a different conditional test mass dynamics from that in quantum-gravity (QG)/standard quantum mechanics, consequently, applying the QG-optimised reconstruction map introduces an additional state-dependent contribution. We show that this contribution makes the reconstructed covariance depend on the chosen set of tomography angles and can drive the SN covariance--after QG filtering--outside the standard Gaussian-covariance domain set by the Heisenberg uncertainty principle. We quantify the resulting QG-SN distinguishability via the Hellinger distance and analyse its dependence on measurement strength and temperature. We then formulate the same issue in the broader setting of nonlinear quantum mechanics: when the system's conditional dynamics during the readout process depends on the state being inferred, the tomographic map acquires nonlinear, model-dependent corrections to the usual Radon or Gaussian reconstruction map.
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Continuous measurements can prepare macroscopic mechanical oscillators in conditional quantum states, but their covariance is difficult to verify. The conventional retrodictive estimator assumes a forward--backward covariance symmetry and can be biased, because physical dynamics such as feedback damping reduces the observability of the state from future records. Here, we derive an exact linear-Gaussian estimator from causal, retrodictive, and smoothed trajectories. For a milligram-scale mirror, it agrees with a Riccati prediction based on parameters fixed independently, while the conventional estimate exhibits a covariance-space bias of dM≃3.5d_M\simeq3.5dM​≃3.5. Our method paves the way toward unbiased testing of macroscopic entanglement within a calibrated linear-Gaussian model, applicable to both tabletop mirrors and kg-scale gravitational-wave test masses.
Probing quantum entanglement with macroscopic objects allows us to test quantum mechanics in new regimes. One way to realize such behavior is to couple a macroscopic mechanical oscillator to a continuous light field via radiation pressure. In view of this, the system that is discussed comprises an optomechanical cavity driven by a coherent optical field in the unresolved sideband regime where we assume Gaussian states and dynamics. We develop a framework to quantify the amount of entanglement in the system numerically. Different from previous work, we treat non-Markovian noise and take into account both the continuous optical field and the cavity mode. We apply our framework to the case of the Advanced Laser Interferometer Gravitational-Wave Observatory and discuss the parameter regimes where entanglement exists, even in the presence of quantum and classical noises. Published by the American Physical Society 2024