Cavity optomechanics is a branch of physics which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of optics, quantum optics, solid-state physics and materials science. The motivation for research on cavity optomechanics comes from fundamental effects of quantum theory and gravity, as well as technological applications, such as quantum precision measurement.
The name of the field relates to the main effect of interest: the enhancement of radiation pressure interaction between light (photons) and matter using optical resonators (cavities). It first became relevant in the context of gravitational wave detection, since optomechanical effects must be taken into account in interferometric gravitational wave detectors. Furthermore, one may envision optomechanical structures to allow the realization of Schrödinger's cat. Macroscopic objects consisting of billions of atoms share collective degrees of freedom which may behave quantum mechanically (e.g. a sphere of micrometer diameter being in a spatial superposition between two different places). Such a quantum state of motion would allow researchers to experimentally investigate decoherence, which describes the transition of objects from states that are described by quantum mechanics to states that are described by Newtonian mechanics. Optomechanical structures provide new methods to test the predictions of quantum mechanics and decoherence models and could help find answers to some of the most fundamental questions in modern physics.
There is a broad range of experimental optomechanical systems which are almost equivalent in their description, but completely different in size, mass, and frequency. Cavity optomechanics was featured as the most recent "milestone of photon history" in nature photonics along well established concepts and technology like quantum information, Bell inequalities and the laser.
Concepts of cavity optomechanics
Physical processes
Stokes and anti-Stokes scattering The most elementary light–matter interaction is a light beam scattering off an arbitrary object (atom, molecule, nanobeam etc.). There is always elastic light scattering, with the outgoing light frequency identical to the incoming frequency . Inelastic scattering, in contrast, is accompanied by excitation or de-excitation of the material object (e.g. internal atomic transitions may be excited). However, it is always possible to have Brillouin scattering independent of the internal electronic details of atoms or molecules due to the object's mechanical vibrations:
where is the vibrational frequency. The vibrations gain or lose energy, respectively, for these Stokes/anti-Stokes processes, while optical sidebands are created around the incoming light frequency:
If Stokes and anti-Stokes scattering occur at an equal rate, the vibrations will only heat up the object. However, an optical cavity can be used to suppress the (anti-)Stokes process, which reveals the principle of the basic optomechanical setup: a laser-driven optical cavity is coupled to the mechanical vibrations of some object. The purpose of the cavity is to select optical frequencies (e.g. to suppress the Stokes process) that resonantly enhance the light intensity and to enhance the sensitivity to the mechanical vibrations. The setup displays features of a true two-way interaction between light and mechanics, which is in contrast to optical tweezers, optical lattices, or vibrational spectroscopy, where the light field controls the mechanics (or vice versa) but the loop is not closed.
Radiation pressure force
Another but equivalent way to interpret the principle of optomechanical cavities is by using the concept of radiation pressure. According to the quantum theory of light, every photon with wavenumber carries a momentum , where is the Planck constant. This means that a photon reflected off a mirror surface transfers a momentum onto the mirror due to the conservation of momentum. This effect is extremely small and cannot be observed on most everyday objects; it becomes more significant when the mass of the mirror is very small and/or the number of photons is very large (i.e. high intensity of the light). Since the momentum of photons is extremely small and not enough to change the position of a suspended mirror significantly, the interaction needs to be enhanced. One possible way to do this is by using optical cavities. If a photon is enclosed between two mirrors, where one is the oscillator and the other is a heavy fixed one, it will bounce off the mirrors many times and transfer its momentum every time it hits the mirrors. The number of times a photon can transfer its momentum is directly related to the finesse of the cavity, which can be improved with highly reflective mirror surfaces. The radiation pressure of the photons does not simply shift the suspended mirror further and further away as the effect on the cavity light field must be taken into account: if the mirror is displaced, the cavity's length changes, which also alters the cavity resonance frequency. Therefore, the detuning—which determines the light amplitude inside the cavity—between the changed cavity and the unchanged laser driving frequency is modified. It determines the light amplitude inside the cavity – at smaller levels of detuning more light actually enters the cavity because it is closer to the cavity resonance frequency. Since the light amplitude, i.e. the number of photons inside the cavity, causes the radiation pressure force and consequently the displacement of the mirror, the loop is closed: the radiation pressure force effectively depends on the mirror position. Another advantage of optical cavities is that the modulation of the cavity length through an oscillating mirror can directly be seen in the spectrum of the cavity.
Optical spring effect
Some first effects of the light on the mechanical resonator can be captured by converting the radiation pressure force into a potential,
and adding it to the intrinsic harmonic oscillator potential of the mechanical oscillator, where is the slope of the radiation pressure force. This combined potential reveals the possibility of static multi-stability in the system, i.e. the potential can feature several stable minima. In addition, can be understood to be a modification of the mechanical spring constant,
This effect is known as the optical spring effect (light-induced spring constant).
However, the model is incomplete as it neglects retardation effects due to the finite cavity photon decay rate . The force follows the motion of the mirror only with some time delay, which leads to effects like friction. For example, assume the equilibrium position sits somewhere on the rising slope of the resonance. In thermal equilibrium, there will be oscillations around this position that do not follow the shape of the resonance because of retardation. The consequence of this delayed radiation force during one cycle of oscillation is that work is performed, in this particular case it is negative,, i.e. the radiation force extracts mechanical energy (there is extra, light-induced damping). This can be used to cool down the mechanical motion and is referred to as optical or optomechanical cooling. It is important for reaching the quantum regime of the mechanical oscillator where thermal noise effects on the device become negligible. Similarly, if the equilibrium position sits on the falling slope of the cavity resonance, the work is positive and the mechanical motion is amplified. In this case the extra, light-induced damping is negative and leads to amplification of the mechanical motion (heating). Radiation-induced damping of this kind has first been observed in pioneering experiments by Braginsky and coworkers in 1970.Braginskii, V. B., Manukin, A. B., Tikhonov, M. Yu. (1970). Investigation of dissipative ponderomotive effects of electromagnetic radiation. Soviet Physics JETP Vol 31, 5 (original russian: Zh. Eksp. Teor. Fiz. 58, 1549 (1970))
Quantized energy transfer Another explanation for the basic optomechanical effects of cooling and amplification can be given in a quantized picture: by detuning the incoming light from the cavity resonance to the red sideband, the photons can only enter the cavity if they take phonons with energy from the mechanics; it effectively cools the device until a balance with heating mechanisms from the environment and laser noise is reached. Similarly, it is also possible to heat structures (amplify the mechanical motion) by detuning the driving laser to the blue side; in this case the laser photons scatter into a cavity photon and create an additional phonon in the mechanical oscillator.
The principle can be summarized as: phonons are converted into photons when cooled and vice versa in amplification.
Three regimes of operation: cooling, heating, resonance
The basic behaviour of the optomechanical system can generally be divided into different regimes, depending on the detuning between the laser frequency and the cavity resonance frequency : Red-detuned regime, (most prominent effects on the red sideband, ): In this regime state exchange between two resonant oscillators can occur (i.e. a beam-splitter in quantum optics language). This can be used for state transfer between phonons and photons (which requires the so-called "strong coupling regime") or the above-mentioned optical cooling. Blue-detuned regime, (most prominent effects on the blue sideband, ): This regime describes "two-mode squeezing". It can be used to achieve quantum entanglement, squeezing, and mechanical "lasing" (amplification of the mechanical motion to self-sustained optomechanical oscillations / limit cycle oscillations), if the growth of the mechanical energy overwhelms the intrinsic losses (mainly mechanical friction). On-resonance regime, : In this regime the cavity is simply operated as an interferometer to read the mechanical motion.
The optical spring effect also depends on the detuning. It can be observed for high levels of detuning () and its strength varies with detuning and the laser drive.
Mathematical treatment
Hamiltonian The standard optomechanical setup is a Fabry–Pérot cavity, where one mirror is movable and thus provides an additional mechanical degree of freedom. This system can be mathematically described by a single optical cavity mode coupled to a single mechanical mode. The coupling originates from the radiation pressure of the light field that eventually moves the mirror, which changes the cavity length and resonance frequency. The optical mode is driven by an external laser. This system can be described by the following effective Hamiltonian:
where and are the bosonic annihilation operators of the given cavity mode and the mechanical resonator respectively, is the frequency of the optical mode, is the position of the mechanical resonator, is the mechanical mode frequency, is the driving laser frequency, and is the amplitude. It satisfies the commutation relations
is now dependent on . The last term describes the driving, given by
where is the input power coupled to the optical mode under consideration and its linewidth. The system is coupled to the environment so the full treatment of the system would also include optical and mechanical dissipation (denoted by and respectively) and the corresponding noise entering the system.
The standard optomechanical Hamiltonian is obtained by getting rid of the explicit time dependence of the laser driving term and separating the optomechanical interaction from the free optical oscillator. This is done by switching into a reference frame rotating at the laser frequency (in which case the optical mode annihilation operator undergoes the transformation ) and applying a Taylor expansion on . Quadratic and higher-order coupling terms are usually neglected, such that the standard Hamiltonian becomes