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The optomechanical instability in the quantum regime

Max Ludwig, Björn Kubala, Florian Marquardt

Published Jan 1, 2008• 94 citations
OpenAlex

Abstract

We consider a generic optomechanical system, consisting of a driven optical cavity and a movable mirror attached to a cantilever. Systems of this kind (and analogues) have been realized in many recent experiments. It is well known that those systems can exhibit an instability towards a regime where the cantilever settles into self-sustained oscillations. In this paper, we briefly review the classical theory of the optomechanical instability, and then discuss the features arising in the quantum regime. We solve numerically a full quantum master equation for the coupled system, and use it to analyze the photon number, the cantilever’s mechanical energy, the phonon probability distribution and the mechanical Wigner density, as a function of experimentally accessible control parameters. We observe and discuss the quantum-to-classical transition as a function of a suitable dimensionless quantum parameter. 1

Related papers

The optomechanical instability in the quantum regime

Max Ludwig, Björn Kubala, Florian Marquardt

We consider a generic optomechanical system, consisting of a driven optical cavity and a movable mirror attached to a cantilever. Systems of this kind (and analogues) have been realized in many recent experiments. It is well known that these systems can exhibit an instability towards a regime where the cantilever settles into self-sustained oscillations. In this paper, we briefly review the classical theory of the optomechanical instability, and then discuss the features arising in the quantum regime. We solve numerically a full quantum master equation for the coupled system, and use it to analyze the photon number, the cantilever's mechanical energy, the phonon probability distribution and the mechanical Wigner density, as a function of experimentally accessible control parameters. When a suitable dimensionless 'quantum parameter' is sent to zero, the results of the quantum mechanical model converge towards the classical predictions. We discuss this quantum-to-classical transition in some detail.

arXiv:0803.3714•New Journal of Physics•Sep 30, 2008•171 citations

Quantum mechanical study of a generic quadratically coupled optomechanical system

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Typical optomechanical systems involving optical cavities and mechanical oscillators rely on a coupling that varies linearly with the oscillator displacement. However, recently a coupling varying instead as the square of the mechanical displacement has been realized, presenting new possibilities for nondemolition measurements and mechanical squeezing. In this article we present a quantum mechanical study of a generic quadratic-coupling optomechanical Hamiltonian. First, neglecting dissipation, we provide analytical results for the dressed states, spectrum, phonon statistics and entanglement. Subsequently, accounting for dissipation, we supply a numerical treatment using a master equation approach. We expect our results to be of use to optomechanical spectroscopy, state transfer, wave-function engineering, and entanglement generation.

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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

A Microscopic Model for Quantum Optomechanics

Kanu Sinha

We study a microscopic model, the Mirror-Oscillator-Field (MOF) model proposed in [1], for describing optomechanical interactions. In contrast with the conventional approach where the mirror-field interaction is understood as arising from the radiation pressure of an optical field inducing the motion of the mirror’s CoM, the MOF model incorporates the dynamics of the internal degrees of freedom of the mirror that couple to the optical field directly. Considering the mirror’s internal and mechanical degrees of freedom as two separate degrees of freedom we derive the optomechanical properties of the coupled mirror and field system. The major advantage in this approach is that it provides a self-consistent treatment of the three relevant subsystems (the mirror’s motion, its internal degrees of freedom and the field) including their back-actions on each other, thereby giving a more accurate account of the coupled internal and external dynamics. The optical and the mechanical properties of a mirror arising from its dynamical interaction with the field are obtained without imposing any boundary conditions on the field additionally, as is done in the conventional way. We find that our results agree with those from the boundary condition approach in the appropriate limits and more generally the model provides a framework within which one can study optomechanical elements with different internal structures and mechanical properties, which makes it suited for studying hybrid systems. Considering the quantum dynamics of the coupled subsystems we look at the entanglement between the mirror’s motion and the field, showing that the internal degrees of the mirror, in the appropriate parameter regimes, can act as a means to coherently transfer quantum correlations between the field and the mechanics thus leading to a larger optomechanical entanglement. We then use the MOF model to study the entanglement between the motion of an atom and a field for the setup in [95] and find a larger optomechanical entanglement when the field is closer to the internal resonance. We also study the interaction between two mirrors as described by the MOF model, specifically looking at the entanglement between the motion of their centers of mass. Conclusively, we see that including the dynamics of the internal degrees of freedom of a mirror, which is the quintessential mediator of interaction between the mirror center of mass and the field, leads to qualitatively different physics, specifically in the quantum regime, thus giving a physically more complete treatment of mirror-field interactions.

•Jan 1, 2015•1 citation

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Rina Kanamoto, Pierre Meystre

We explore theoretically the optomechanical interaction between a light field and a mechanical mode of ultracold fermionic atoms inside a Fabry-Pérot cavity. The low-lying phonon mode of the fermionic ensemble is a collective density oscillation associated with particle-hole excitations, and is mathematically analogous to the momentum side-mode excitations of a bosonic condensate. The mechanical motion of the fermionic particle-hole system behaves hence as a "moving mirror." We derive an effective system Hamiltonian that has the form of generic optomechanical systems. We also discuss the experimental consequences the optomechanical coupling in optical bistability and in the noise spectrum of the system.

arXiv:0911.4748•Physical Review Letters•Feb 9, 2010•84 citations

Inducing a Single Vibration

Ian S. Osborne

Applied Physics Optomechanical systems typically consist of a mechanical oscillator or cantilever that can be driven by the optical radiation pressure of light. The light can either amplify the motion of the cantilever or be used to reduce the amount of vibration and effectively cool it. Recent work has demonstrated cooling to the extent that the number of vibrations, or phonons, in an oscillator can be quantized, even reaching the quantum-mechanical ground. Galland et al. introduce a method to introduce a single phonon into their mechanical oscillator. Starting off in the quantum-mechanical ground, they pump the system with a laser pulse that generates an entangled phonon-photon pair. Detection of the photon then heralds that a single-phonon state has been induced into the oscillator. The ability to deterministically control the number of vibrational quanta in such an optomechanical system has important implications for the likes of quantum information processing and quantum enhanced sensing applications. Phys. Rev. Lett. 112 , 143602 (2014).

Science•May 1, 2014•0 citations
We consider a generic optomechanical system, consisting of a driven optical cavity and a movable mirror attached to a cantilever. Systems of this kind (and analogues) have been realized in many recent experiments. It is well known that those systems can exhibit an instability towards a regime where the cantilever settles into self-sustained oscillations. In this paper, we briefly review the classical theory of the optomechanical instability, and then discuss the features arising in the quantum regime. We solve numerically a full quantum master equation for the coupled system, and use it to analyze the photon number, the cantilever’s mechanical energy, the phonon probability distribution and the mechanical Wigner density, as a function of experimentally accessible control parameters. We observe and discuss the quantum-to-classical transition as a function of a suitable dimensionless quantum parameter. 1
We consider a generic optomechanical system, consisting of a driven optical cavity and a movable mirror attached to a cantilever. Systems of this kind (and analogues) have been realized in many recent experiments. It is well known that these systems can exhibit an instability towards a regime where the cantilever settles into self-sustained oscillations. In this paper, we briefly review the classical theory of the optomechanical instability, and then discuss the features arising in the quantum regime. We solve numerically a full quantum master equation for the coupled system, and use it to analyze the photon number, the cantilever's mechanical energy, the phonon probability distribution and the mechanical Wigner density, as a function of experimentally accessible control parameters. When a suitable dimensionless 'quantum parameter' is sent to zero, the results of the quantum mechanical model converge towards the classical predictions. We discuss this quantum-to-classical transition in some detail.
Typical optomechanical systems involving optical cavities and mechanical oscillators rely on a coupling that varies linearly with the oscillator displacement. However, recently a coupling varying instead as the square of the mechanical displacement has been realized, presenting new possibilities for nondemolition measurements and mechanical squeezing. In this article we present a quantum mechanical study of a generic quadratic-coupling optomechanical Hamiltonian. First, neglecting dissipation, we provide analytical results for the dressed states, spectrum, phonon statistics and entanglement. Subsequently, accounting for dissipation, we supply a numerical treatment using a master equation approach. We expect our results to be of use to optomechanical spectroscopy, state transfer, wave-function engineering, and entanglement generation.
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 a microscopic model, the Mirror-Oscillator-Field (MOF) model proposed in [1], for describing optomechanical interactions. In contrast with the conventional approach where the mirror-field interaction is understood as arising from the radiation pressure of an optical field inducing the motion of the mirror’s CoM, the MOF model incorporates the dynamics of the internal degrees of freedom of the mirror that couple to the optical field directly. Considering the mirror’s internal and mechanical degrees of freedom as two separate degrees of freedom we derive the optomechanical properties of the coupled mirror and field system. The major advantage in this approach is that it provides a self-consistent treatment of the three relevant subsystems (the mirror’s motion, its internal degrees of freedom and the field) including their back-actions on each other, thereby giving a more accurate account of the coupled internal and external dynamics. The optical and the mechanical properties of a mirror arising from its dynamical interaction with the field are obtained without imposing any boundary conditions on the field additionally, as is done in the conventional way. We find that our results agree with those from the boundary condition approach in the appropriate limits and more generally the model provides a framework within which one can study optomechanical elements with different internal structures and mechanical properties, which makes it suited for studying hybrid systems. Considering the quantum dynamics of the coupled subsystems we look at the entanglement between the mirror’s motion and the field, showing that the internal degrees of the mirror, in the appropriate parameter regimes, can act as a means to coherently transfer quantum correlations between the field and the mechanics thus leading to a larger optomechanical entanglement. We then use the MOF model to study the entanglement between the motion of an atom and a field for the setup in [95] and find a larger optomechanical entanglement when the field is closer to the internal resonance. We also study the interaction between two mirrors as described by the MOF model, specifically looking at the entanglement between the motion of their centers of mass. Conclusively, we see that including the dynamics of the internal degrees of freedom of a mirror, which is the quintessential mediator of interaction between the mirror center of mass and the field, leads to qualitatively different physics, specifically in the quantum regime, thus giving a physically more complete treatment of mirror-field interactions.
We explore theoretically the optomechanical interaction between a light field and a mechanical mode of ultracold fermionic atoms inside a Fabry-Pérot cavity. The low-lying phonon mode of the fermionic ensemble is a collective density oscillation associated with particle-hole excitations, and is mathematically analogous to the momentum side-mode excitations of a bosonic condensate. The mechanical motion of the fermionic particle-hole system behaves hence as a "moving mirror." We derive an effective system Hamiltonian that has the form of generic optomechanical systems. We also discuss the experimental consequences the optomechanical coupling in optical bistability and in the noise spectrum of the system.
Applied Physics Optomechanical systems typically consist of a mechanical oscillator or cantilever that can be driven by the optical radiation pressure of light. The light can either amplify the motion of the cantilever or be used to reduce the amount of vibration and effectively cool it. Recent work has demonstrated cooling to the extent that the number of vibrations, or phonons, in an oscillator can be quantized, even reaching the quantum-mechanical ground. Galland et al. introduce a method to introduce a single phonon into their mechanical oscillator. Starting off in the quantum-mechanical ground, they pump the system with a laser pulse that generates an entangled phonon-photon pair. Detection of the photon then heralds that a single-phonon state has been induced into the oscillator. The ability to deterministically control the number of vibrational quanta in such an optomechanical system has important implications for the likes of quantum information processing and quantum enhanced sensing applications. Phys. Rev. Lett. 112 , 143602 (2014).