Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1606.07007 · Physical Review A

Quantum state reconstruction of an oscillator network in an optomechanical setting

Darren W. Moore, Tommaso Tufarelli, Mauro Paternostro, Alessandro Ferraro

Published Nov 7, 2016• 12 citations
arXivHTMLPDF

Abstract

We introduce a scheme to reconstruct an arbitrary quantum state of a mechanical oscillator network. We assume that a single element of the network is coupled to a cavity field via a linearized optomechanical interaction, the time dependence of which is controlled by a classical driving field. By designing a suitable interaction profile, we show how the statistics of an arbitrary mechanical quadrature can be encoded in the cavity field, which can then be measured. We discuss the important special case of Gaussian state reconstruction and study numerically the effectiveness of our scheme for a finite number of measurements. Finally, we speculate on possible routes to extend our ideas to the regime of single-photon optomechanics.

DOI
10.1103/physreva.94.053811

Related papers

Hybrid Atomic-Optomechanical Systems -- Observing Quantum Effects in Macroscopic Oscillators

Swati Singh

This thesis concentrates on generating and measuring non-classical states of mechanical oscillators by coupling them to atomic and molecular quantum systems. We start with a discussion of what novel physics can be explored by mechanical systems operating in the quantum regime. We then discuss one technique making it a possibility- cavity optomechanics, particularly optomechanical cooling. We investigate the limits of optomechanical cooling and review how the coupling of mechanical oscillators to external heat baths limits the minimum attainable phonon occupation number. As a possible alternative for circumventing clamping losses, we consider an all-optical approach where the mechanical element (in this case, a Bragg mirror) is suspended via optical forces, and discuss some limitations of this approach. We explore several schemes aimed at the generation of quantum states in mechanical oscillators. We consider specifically two examples: one in which the mechanical oscillator is coupled to polar molecules via dipole-dipole interaction, and another where it is magnetically coupled to a Bose condensate. The first example emphasizes that such an interaction can generate parametric squeezing and entanglement. The second scheme demonstrates that the back action of BEC spin measurements can be used to generate quantum states of motion of a mechanical oscillator. We then discuss possible methods for measuring the entire density matrix of a mechanical oscillator. The first method achieves the tomographic reconstruction of the mechanical Wigner function by coupling it simultaneously to a classical optical oscillator and a qubit. The second approach involves a state transfer scheme between momentum excitations of a bose-condensate in a cavity and a moving mirror of the cavity that is entirely mediated by the light field. We conclude with a discussion of the broader implications of this work, and some future research directions.

•Jan 1, 2012•0 citations

Two-mode back-action-evading measurements in cavity optomechanics

Matthew J. Woolley, Aashish A. Clerk

We study theoretically a three-mode optomechanical system where two mechanical oscillators are coupled to a single cavity mode. By using two-tone (i.e., amplitude-modulated) driving of the cavity, it is possible to couple the cavity to a single collective quadrature of the mechanical oscillators. In such a way, a back-action-evading measurement of the collective mechanical quadrature is possible. We discuss how this can allow one to measure both quadratures of a mechanical force beyond the full quantum limit, paying close attention to the role of dissipation and experimental imperfections. We also describe how this scheme allows one to generate steady-state mechanical entanglement; namely, one can conditionally prepare an entangled, two-mode squeezed mechanical state. This entanglement can be verified directly from the measurement record by applying a generalized version of Duan's inequality; we also discuss how feedback can be used to produce unconditional entanglement.

arXiv:1304.4059•Physical Review A•Jun 28, 2013•117 citations

Nonlinear cavity quantum optomechanics: from quantum measurement to quantum state engineering

Jack Clarke

In a nutshell, a cavity optomechanical system consists of a mechanical oscillator inside an optical cavity. Cavity optomechanics studies the interplay between the light-matter radiation-pressure interaction and the change in the cavity frequency induced by mechanical motion. By leveraging the tools of quantum optics and quantum measurement, cavity quantum optomechanics enables the engineering of nonclassical quantum states of bulk matter. In this thesis, I present protocols for engineering such nonclassical quantum states in Part II. First, I introduce a scheme to ‘grow’ a quantum kitten state of a mechanical oscillator into a macroscopic Schrödinger cat state using a series of nonlinear pulsed optomechanical interactions. Next, I detail our proposed schemes to generate and verify entanglement between two mechanical oscillators. Moving on from pulsed optomechanics, I describe the theory behind an experiment that achieved multi-phonon subtraction from a mechanical oscillator in a continuously driven Brillouin optomechanical system. In the final chapter of Part II, I provide a theoretical preview of an experiment that investigates the behaviour of the mechanical oscillator when no light is detected at the output. In Part III, I introduce a new framework for nonlinear cavity quantum optomechanics that incorporates the intrinsic nonlinearity of the radiation-pressure interaction, the nonlinearity from the cavity enhancement, and higher-order mechanical position terms in the expansion of the cavity frequency. I then propose a scheme for measuring mechanical position using general-dyne detection, applicable to pulsed and continuous measurements. This predicts a new regime of operation, the “stochastic Gaussian” regime. Finally, I discuss a protocol for deterministic mechanical squeezing that utilizes the cavity response.

•Jun 16, 2022•0 citations

Wigner Reconstruction in Levitated Optomechanics

Muddassar Rashid, Marko Toroš, Hendrik Ulbricht

We demonstrate the reconstruction of the Wigner function from marginal distributions of the motion of a single trapped particle using homodyne detection. We show that it is possible to generate quantum states of levitated optomechanical systems even under the effect of continuous measurement by the trapping laser light. We describe the opto-mechanical coupling for the case of the particle trapped by a free-space focused laser beam, explicitly for the case without an optical cavity. We use the scheme to reconstruct the Wigner function of experimental data in perfect agreement with the expected Gaussian distribution of a thermal state of motion. This opens a route for quantum state preparation in levitated optomechanics.

•Jul 25, 2017•0 citations

Microwave Electromechanics: Measuring and Manipulating the Quantum State of a Macroscopic Mechanical Oscillator

Jennifer Harlow

In the past several years, the field of optomechanics has progressed from proof-of-principle experiments to the realization of mechanical oscillators and measurements in the quantum regime. Mechanical oscillators are of great interest because they can have small dissipation rates, can couple to many different systems of interest, and are the fundamental elements of ultrasensitive force detectors. Coupling these mechanical oscillators to microwave or optical fields provides a twofold advantage. Firstly, information about mechanical position can be encoded in the interrogating field, enabling sensitive readout of the mechanical oscillator. Secondly, the radiation pressure force of that field can be used to control the state of the mechanical oscillator. Including a high-quality microwave or optical cavity enhances both of these effects, as the field strength is resonantly increased.\nThe major questions in the field of optomechanics in the last several years have dealt with using mechanical oscillators for ultrasensitive measurements and as tools for quantum information. Both of these goals have the prerequisite that we be able to read out the motion of the mechanical oscillator in a quantum efficient manner. To that end, we developed a nearly shot-noise limited microwave interferometer capable of measuring mechanical motion with an imprecision below that at the standard quantum limit. This achievement is not only a critical improvement for the electromechanical experiments we do, but is also an important tool for any experiment that encodes the information of interest in microwave fields. In order to use mechanical oscillators as tools for quantum information, the mechanical oscillator must also be cooled into the quantum regime and fully controllable by the interrogating fields. To this end, we used the radiation pressure of microwave fields to cool our macroscopic mechanical oscillator to less than one phonon. We also demonstrated coherent transfer between itinerant microwave states and the mechanical oscillator, even for incident fields with less than one photon of energy.\nThese accomplishments have set the foundation for further experiments to extend the quantum information abilities of optomechanical systems, couple diverse quantum systems via a mechanical intermediary, and potentially explore the foundations of quantum mechanics at macroscopic scales.

•Jan 1, 2013•4 citations

Towards optomechanical quantum state reconstruction of mechanical motion

Michael Ross Vanner, Igor Pikovski, M. S. Kim

Utilizing the tools of quantum optics to prepare and manipulate quantum states of motion of a mechanical resonator is currently one of the most promising routes to explore non‐classicality at a macroscopic scale. An important quantum optomechanical tool yet to be experimentally demonstrated is the ability to perform complete quantum state reconstruction. Here, after providing a brief introduction to quantum states in phase space, the current proposals for state reconstruction of mechanical motional states are reviewed and contrasted and experimental progress is discussed. Furthermore, it is shown that mechanical quadrature tomography using back‐action‐evading interactions gives an s‐parameterized Wigner function where the numerical parameter s is directly related to the optomechanical measurement strength. The effects of classical noise in the optical probe for both state reconstruction and state preparation by measurement are also discussed.

arXiv:1406.1013•Annalen der Physik•Aug 29, 2014•60 citations
We introduce a scheme to reconstruct an arbitrary quantum state of a mechanical oscillator network. We assume that a single element of the network is coupled to a cavity field via a linearized optomechanical interaction, the time dependence of which is controlled by a classical driving field. By designing a suitable interaction profile, we show how the statistics of an arbitrary mechanical quadrature can be encoded in the cavity field, which can then be measured. We discuss the important special case of Gaussian state reconstruction and study numerically the effectiveness of our scheme for a finite number of measurements. Finally, we speculate on possible routes to extend our ideas to the regime of single-photon optomechanics.
This thesis concentrates on generating and measuring non-classical states of mechanical oscillators by coupling them to atomic and molecular quantum systems. We start with a discussion of what novel physics can be explored by mechanical systems operating in the quantum regime. We then discuss one technique making it a possibility- cavity optomechanics, particularly optomechanical cooling. We investigate the limits of optomechanical cooling and review how the coupling of mechanical oscillators to external heat baths limits the minimum attainable phonon occupation number. As a possible alternative for circumventing clamping losses, we consider an all-optical approach where the mechanical element (in this case, a Bragg mirror) is suspended via optical forces, and discuss some limitations of this approach. We explore several schemes aimed at the generation of quantum states in mechanical oscillators. We consider specifically two examples: one in which the mechanical oscillator is coupled to polar molecules via dipole-dipole interaction, and another where it is magnetically coupled to a Bose condensate. The first example emphasizes that such an interaction can generate parametric squeezing and entanglement. The second scheme demonstrates that the back action of BEC spin measurements can be used to generate quantum states of motion of a mechanical oscillator. We then discuss possible methods for measuring the entire density matrix of a mechanical oscillator. The first method achieves the tomographic reconstruction of the mechanical Wigner function by coupling it simultaneously to a classical optical oscillator and a qubit. The second approach involves a state transfer scheme between momentum excitations of a bose-condensate in a cavity and a moving mirror of the cavity that is entirely mediated by the light field. We conclude with a discussion of the broader implications of this work, and some future research directions.
We study theoretically a three-mode optomechanical system where two mechanical oscillators are coupled to a single cavity mode. By using two-tone (i.e., amplitude-modulated) driving of the cavity, it is possible to couple the cavity to a single collective quadrature of the mechanical oscillators. In such a way, a back-action-evading measurement of the collective mechanical quadrature is possible. We discuss how this can allow one to measure both quadratures of a mechanical force beyond the full quantum limit, paying close attention to the role of dissipation and experimental imperfections. We also describe how this scheme allows one to generate steady-state mechanical entanglement; namely, one can conditionally prepare an entangled, two-mode squeezed mechanical state. This entanglement can be verified directly from the measurement record by applying a generalized version of Duan's inequality; we also discuss how feedback can be used to produce unconditional entanglement.
In a nutshell, a cavity optomechanical system consists of a mechanical oscillator inside an optical cavity. Cavity optomechanics studies the interplay between the light-matter radiation-pressure interaction and the change in the cavity frequency induced by mechanical motion. By leveraging the tools of quantum optics and quantum measurement, cavity quantum optomechanics enables the engineering of nonclassical quantum states of bulk matter. In this thesis, I present protocols for engineering such nonclassical quantum states in Part II. First, I introduce a scheme to ‘grow’ a quantum kitten state of a mechanical oscillator into a macroscopic Schrödinger cat state using a series of nonlinear pulsed optomechanical interactions. Next, I detail our proposed schemes to generate and verify entanglement between two mechanical oscillators. Moving on from pulsed optomechanics, I describe the theory behind an experiment that achieved multi-phonon subtraction from a mechanical oscillator in a continuously driven Brillouin optomechanical system. In the final chapter of Part II, I provide a theoretical preview of an experiment that investigates the behaviour of the mechanical oscillator when no light is detected at the output. In Part III, I introduce a new framework for nonlinear cavity quantum optomechanics that incorporates the intrinsic nonlinearity of the radiation-pressure interaction, the nonlinearity from the cavity enhancement, and higher-order mechanical position terms in the expansion of the cavity frequency. I then propose a scheme for measuring mechanical position using general-dyne detection, applicable to pulsed and continuous measurements. This predicts a new regime of operation, the “stochastic Gaussian” regime. Finally, I discuss a protocol for deterministic mechanical squeezing that utilizes the cavity response.
We demonstrate the reconstruction of the Wigner function from marginal distributions of the motion of a single trapped particle using homodyne detection. We show that it is possible to generate quantum states of levitated optomechanical systems even under the effect of continuous measurement by the trapping laser light. We describe the opto-mechanical coupling for the case of the particle trapped by a free-space focused laser beam, explicitly for the case without an optical cavity. We use the scheme to reconstruct the Wigner function of experimental data in perfect agreement with the expected Gaussian distribution of a thermal state of motion. This opens a route for quantum state preparation in levitated optomechanics.
In the past several years, the field of optomechanics has progressed from proof-of-principle experiments to the realization of mechanical oscillators and measurements in the quantum regime. Mechanical oscillators are of great interest because they can have small dissipation rates, can couple to many different systems of interest, and are the fundamental elements of ultrasensitive force detectors. Coupling these mechanical oscillators to microwave or optical fields provides a twofold advantage. Firstly, information about mechanical position can be encoded in the interrogating field, enabling sensitive readout of the mechanical oscillator. Secondly, the radiation pressure force of that field can be used to control the state of the mechanical oscillator. Including a high-quality microwave or optical cavity enhances both of these effects, as the field strength is resonantly increased.\nThe major questions in the field of optomechanics in the last several years have dealt with using mechanical oscillators for ultrasensitive measurements and as tools for quantum information. Both of these goals have the prerequisite that we be able to read out the motion of the mechanical oscillator in a quantum efficient manner. To that end, we developed a nearly shot-noise limited microwave interferometer capable of measuring mechanical motion with an imprecision below that at the standard quantum limit. This achievement is not only a critical improvement for the electromechanical experiments we do, but is also an important tool for any experiment that encodes the information of interest in microwave fields. In order to use mechanical oscillators as tools for quantum information, the mechanical oscillator must also be cooled into the quantum regime and fully controllable by the interrogating fields. To this end, we used the radiation pressure of microwave fields to cool our macroscopic mechanical oscillator to less than one phonon. We also demonstrated coherent transfer between itinerant microwave states and the mechanical oscillator, even for incident fields with less than one photon of energy.\nThese accomplishments have set the foundation for further experiments to extend the quantum information abilities of optomechanical systems, couple diverse quantum systems via a mechanical intermediary, and potentially explore the foundations of quantum mechanics at macroscopic scales.
Utilizing the tools of quantum optics to prepare and manipulate quantum states of motion of a mechanical resonator is currently one of the most promising routes to explore non‐classicality at a macroscopic scale. An important quantum optomechanical tool yet to be experimentally demonstrated is the ability to perform complete quantum state reconstruction. Here, after providing a brief introduction to quantum states in phase space, the current proposals for state reconstruction of mechanical motional states are reviewed and contrasted and experimental progress is discussed. Furthermore, it is shown that mechanical quadrature tomography using back‐action‐evading interactions gives an s‐parameterized Wigner function where the numerical parameter s is directly related to the optomechanical measurement strength. The effects of classical noise in the optical probe for both state reconstruction and state preparation by measurement are also discussed.