Cavity Optomechanics
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Cavity optomechanics is a branch of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
,
quantum optics Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...
,
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
and materials science. The motivation for research on cavity optomechanics comes from fundamental effects of
quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...
and
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, as well as technological applications. The name of the field relates to the main effect of interest: the enhancement of radiation pressure interaction between light (
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s) and matter using optical resonators (cavities). It first became relevant in the context of
gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
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 In quantum mechanics, Schrödinger's cat is a thought experiment that illustrates a paradox of quantum superposition. In the thought experiment, a hypothetical cat may be considered simultaneously both alive and dead, while it is unobserved in ...
. 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 Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
, which describes the transition of objects from states that are described by quantum mechanics to states that are described by
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
. Optomechanical structures provide new methods to test the predictions of quantum mechanics and decoherence models and thereby might allow to answer 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 Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
, Bell inequalities and the
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
.


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 \omega'=\omega. 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 Brillouin scattering (also known as Brillouin light scattering or BLS), named after Léon Brillouin, refers to the interaction of light with the material waves in a medium (e.g. electrostriction and magnetostriction). It is mediated by the refractiv ...
independent of the internal electronic details of atoms or molecules due to the object's mechanical vibrations: :\omega'=\omega\pm\omega_m, where \omega_m 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: :\omega'=\omega\mp\omega_m. If Stokes and anti-Stokes scattering occur at an equal rate, the vibrations will only heat up the object. However, an
optical cavity An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that forms a cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and provi ...
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 tweezers (originally called single-beam gradient force trap) are scientific instruments that use a highly focused laser beam to hold and move microscopic and sub-microscopic objects like atoms, nanoparticles and droplets, in a manner simila ...
,
optical lattice An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregat ...
s, 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 Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
. According to the quantum theory of light, every photon with
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
k carries a momentum p=\hbar k, where \hbar is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. This means that a photon reflected off a mirror surface transfers a momentum \Delta p=2\hbar k onto the mirror due to the
conservation of momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
. 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 In contract bridge and similar games, a finesse is a type of card play technique which will enable a player to win an additional trick or tricks should there be a favorable position of one or more cards in the hands of the opponents. The player a ...
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, :\fracV_(x)=-F(x), and adding it to the intrinsic
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
potential of the mechanical oscillator, where F(x) 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, F(x) can be understood to be a modification of the mechanical spring constant, :D=D_-\frac. 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 \kappa. 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,\oint Fdx<0, 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.


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 In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
with energy \hbar\omega_m 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 \Delta=\omega_L-\omega_: * Red-detuned regime, \Delta<0 (most prominent effects on the red sideband, \Delta=-\omega_m): 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, \Delta>0 (most prominent effects on the blue sideband, \Delta=+\omega_m): This regime describes "two-mode squeezing". It can be used to achieve
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
, squeezing, and mechanical "lasing" (amplification of the mechanical motion to self-sustained optomechanical oscillations /
limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
oscillations), if the growth of the mechanical energy overwhelms the intrinsic losses (mainly mechanical friction). * On-resonance regime, \Delta=0: In this regime the cavity is simply operated as an
interferometer Interferometry is a technique which uses the ''interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber op ...
to read the mechanical motion. The optical spring effect also depends on the detuning. It can be observed for high levels of detuning (\Delta\gg\omega_m,\kappa) 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
: : H_ = \hbar \omega_(x) a^\dagger a + \hbar \omega_m b^\dagger b + i \hbar E \left( a e^ - a^\dagger e^\right) where a and b are the bosonic annihilation operators of the given cavity mode and the mechanical resonator respectively, \omega_ is the frequency of the optical mode, x is the position of the mechanical resonator, \omega_m is the mechanical mode frequency, \omega_L is the driving laser frequency, and E is the amplitude. It satisfies the commutation relations : , a^\dagger= , b^\dagger= 1. \omega_ is now dependent on x. The last term describes the driving, given by :E = \sqrt where P is the input power coupled to the optical mode under consideration and \kappa 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 \kappa and \Gamma 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 \omega_L (in which case the optical mode annihilation operator undergoes the transformation a \rightarrow a e^) and applying a
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
on \omega_. Quadratic and higher-order coupling terms are usually neglected, such that the standard Hamiltonian becomes :H_ = -\hbar \Delta a^\dagger a + \hbar \omega_m b^\dagger b - \hbar g_0 a^\dagger a \frac+ i\hbar E \left( a - a^\dagger \right) where \Delta = \omega_L - \omega_ the laser detuning and the position operator x = x_(b + b^\dagger). The first two terms (-\hbar \Delta a^\dagger a and \hbar \omega_m b^\dagger b) are the free optical and mechanical Hamiltonians respectively. The third term contains the optomechanical interaction, where g_0 = \left.\tfrac\_x_ is the single-photon optomechanical coupling strength (also known as the bare optomechanical coupling). It determines the amount of cavity resonance frequency shift if the mechanical oscillator is displaced by the zero point uncertainty x_=\sqrt, where m_ is the effective mass of the mechanical oscillator. It is sometimes more convenient to use the frequency pull parameter, or G=\frac, to determine the frequency change per displacement of the mirror. For example, the optomechanical coupling strength of a Fabry–Pérot cavity of length L with a moving end-mirror can be directly determined from the geometry to be g_0 = \frac. This standard Hamiltonian H_ is based on the assumption that only one optical and mechanical mode interact. In principle, each optical cavity supports an infinite number of modes and mechanical oscillators which have more than a single oscillation/vibration mode. The validity of this approach relies on the possibility to tune the laser in such a way that it only populates a single optical mode (implying that the spacing between the cavity modes needs to be sufficiently large). Furthermore, scattering of photons to other modes is supposed to be negligible, which holds if the mechanical (motional) sidebands of the driven mode do not overlap with other cavity modes; i.e. if the mechanical mode frequency is smaller than the typical separation of the optical modes.


Linearization

The single-photon optomechanical coupling strength g_0 is usually a small frequency, much smaller than the cavity decay rate \kappa, but the effective optomechanical coupling can be enhanced by increasing the drive power. With a strong enough drive, the dynamics of the system can be considered as quantum fluctuations around a classical steady state, i.e. a = \alpha + \delta a, where \alpha is the mean light field amplitude and \delta a denotes the fluctuations. Expanding the photon number a^\dagger a, the term ~\alpha^2 can be omitted as it leads to a constant radiation pressure force which simply shifts the resonator's equilibrium position. The linearized optomechanical Hamiltonian H_ can be obtained by neglecting the second order term ~\delta a^\dagger \delta a: H_ = -\hbar\Delta \delta a^\dagger \delta a + \hbar\omega_m b^\dagger b - \hbar g (\delta a + \delta a^\dagger)(b + b^\dagger) where g = g_0 \alpha. While this Hamiltonian is a
quadratic function In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
, it is considered "linearized" because it leads to linear equations of motion. It is a valid description of many experiments, where g_0 is typically very small and needs to be enhanced by the driving laser. For a realistic description, dissipation should be added to both the optical and the mechanical oscillator. The driving term from the standard Hamiltonian is not part of the linearized Hamiltonian, since it is the source of the classical light amplitude \alpha around which the linearization was executed. With a particular choice of detuning, different phenomena can be observed (see also the section about physical processes). The clearest distinction can be made between the following three cases: * \Delta\approx-\omega_m: a
rotating wave approximation The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radi ...
of the linearized Hamiltonian, where one omits all non-resonant terms, reduces the coupling Hamiltonian to a beamsplitter operator, H_=\hbar g_0(\delta a^\dagger b + \delta a b^\dagger). This approximation works best on resonance; i.e. if the detuning becomes exactly equal to the negative mechanical frequency. Negative detuning (red detuning of the laser from the cavity resonance) by an amount equal to the mechanical mode frequency favors the anti-Stokes sideband and leads to a net cooling of the resonator. Laser photons absorb energy from the mechanical oscillator by annihilating phonons in order to become resonant with the cavity. * \Delta \approx \omega_m: a
rotating wave approximation The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radi ...
of the linearized Hamiltonian leads to other resonant terms. The coupling Hamiltonian takes the form H_= \hbar g_0(\delta a b + \delta a^\dagger b^\dagger), which is proportional to the two-mode squeezing operator. Therefore, two-mode squeezing and entanglement between the mechanical and optical modes can be observed with this parameter choice. Positive detuning (blue detuning of the laser from the cavity resonance) can also lead to instability. The Stokes sideband is enhanced, i.e. the laser photons shed energy, increasing the number of phonons and becoming resonant with the cavity in the process. * \Delta = 0: In this case of driving on-resonance, all the terms in H_ =\hbar g_0 (\delta a + \delta a^\dagger) (b+b^\dagger) must be considered. The optical mode experiences a shift proportional to the mechanical displacement, which translates into a phase shift of the light transmitted through (or reflected off) the cavity. The cavity serves as an interferometer augmented by the factor of the optical finesse and can be used to measure very small displacements. This setup has enabled
LIGO The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory designed to detect cosmic gravitational waves and to develop gravitational-wave observations as an astronomical tool. Two large ...
to detect gravitational waves.


Equations of motion

From the linearized Hamiltonian, the so-called linearized quantum
Langevin equation In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lange ...
s, which govern the dynamics of the optomechanical system, can be derived when dissipation and noise terms to the Heisenberg equations of motion are added. \delta \dot = (i \Delta-\kappa/2) \delta a + i g (b+b^\dagger) - \sqrt a_ \dot b = -(i\omega_m+\Gamma/2)b +i g (\delta a+\delta a^\dagger)-\sqrtb_ Here a_ and b_ are the input noise operators (either quantum or thermal noise) and -\kappa \delta a and -\Gamma \delta p are the corresponding dissipative terms. For optical photons, thermal noise can be neglected due to the high frequencies, such that the optical input noise can be described by quantum noise only; this does not apply to microwave implementations of the optomechanical system. For the mechanical oscillator thermal noise has to be taken into account and is the reason why many experiments are placed in additional cooling environments to lower the ambient temperature. These
first order differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
can be solved easily when they are rewritten in
frequency space In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a sig ...
(i.e. a
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
is applied). Two main effects of the light on the mechanical oscillator can then be expressed in the following ways: \delta\omega_m=g^2\left(\frac+\frac\right) The equation above is termed the optical-spring effect and may lead to significant frequency shifts in the case of low-frequency oscillators, such as pendulum mirrors. In the case of higher resonance frequencies (\omega_m \gtrsim 1 MHz), it does not significantly alter the frequency. For a harmonic oscillator, the relation between a frequency shift and a change in the spring constant originates from
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
. \Gamma^ = \Gamma + g^2\left(\frac-\frac\right) The equation above shows optical damping, i.e. the intrinsic mechanical damping \Gamma becomes stronger (or weaker) due to the optomechanical interaction. From the formula, in the case of negative detuning and large coupling, mechanical damping can be greatly increased, which corresponds to the cooling of the mechanical oscillator. In the case of positive detuning the optomechanical interaction reduces effective damping. Instability can occur when the effective damping drops below zero (\Gamma^<0), which means that it turns into an overall amplification rather than a damping of the mechanical oscillator.


Important parameter regimes

The most basic regimes in which the optomechanical system can be operated are defined by the laser detuning \Delta and described above. The resulting phenomena are either cooling or heating of the mechanical oscillator. However, additional parameters determine what effects can actually be observed. The ''good/bad cavity regime'' (also called the ''resolved/unresolved sideband regime'') relates the mechanical frequency to the optical linewidth. The good cavity regime (resolved sideband limit) is of experimental relevance since it is a necessary requirement to achieve
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
cooling of the mechanical oscillator, i.e. cooling to an average mechanical occupation number below 1. The term "resolved sideband regime" refers to the possibility of distinguishing the motional sidebands from the cavity resonance, which is true if the linewidth of the cavity, \kappa, is smaller than the distance from the cavity resonance to the sideband (\omega_m). This requirement leads to a condition for the so-called sideband parameter: \omega_m/\kappa\gg1. If \omega_m/\kappa\ll1 the system resides in the bad cavity regime (unresolved sideband limit), where the motional sideband lies within the peak of the cavity resonance. In the unresolved sideband regime, many motional sidebands can be included in the broad cavity linewidth, which allows a single photon to create more than one phonon, which leads to greater amplification of the mechanical oscillator. Another distinction can be made depending on the optomechanical coupling strength. If the (enhanced) optomechanical coupling becomes larger than the cavity linewidth (g\geq\kappa), a ''strong-coupling regime'' is achieved. There the optical and mechanical modes hybridize and normal-mode splitting occurs. This regime must be distinguished from the (experimentally much more challenging) ''single-photon strong-coupling regime'', where the bare optomechanical coupling becomes of the order of the cavity linewidth, g_0\geq\kappa. Effects of the full non-linear interaction described by \hbar g_0 a^\dagger a (b+b^\dagger) only become observable in this regime. For example, it is a precondition to create non-Gaussian states with the optomechanical system. Typical experiments currently operate in the linearized regime (small g_0\ll\kappa) and only investigate effects of the linearized Hamiltonian.


Experimental realizations


Setup

The strength of the optomechanical Hamiltonian is the large range of experimental implementations to which it can be applied, which results in wide parameter ranges for the optomechanical parameters. For example, the size of optomechanical systems can be on the order of micrometers or in the case for
LIGO The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory designed to detect cosmic gravitational waves and to develop gravitational-wave observations as an astronomical tool. Two large ...
, kilometers. (although LIGO is dedicated to the detection of gravitational waves and not the investigation of optomechanics specifically). Examples of real optomechanical implementations are: * Cavities with a moving mirror: the archetype of an optomechanical system. The light is reflected from the
mirrors A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
and transfers momentum onto the movable one, which in turn changes the cavity resonance frequency. * Membrane-in-the-middle system: a micromechanical membrane is brought into a cavity consisting of fixed massive mirrors. The membrane takes the role of the mechanical oscillator. Depending on the positioning of the membrane inside the cavity, this system behaves like the standard optomechanical system.Thompson, J. D., Zwickl, B. M., Jayich, A. M., Marquardt, F., Girvin, S. M., & Harris, J. G. E. (2008). Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane. ''Nature'', 452(7183), 72-5. Nature Publishing Group. *Levitated system: an optically levitated
nanoparticle A nanoparticle or ultrafine particle is usually defined as a particle of matter that is between 1 and 100 nanometres (nm) in diameter. The term is sometimes used for larger particles, up to 500 nm, or fibers and tubes that are less than 1 ...
is brought into a cavity consisting of fixed massive mirrors. The levitated nanoparticle takes the role of the mechanical oscillator. Depending on the positioning of the particle inside the cavity, this system behaves like the standard optomechanical system. * ''Microtoroids'' that support an optical
whispering gallery mode Whispering-gallery waves, or whispering-gallery modes, are a type of wave that can travel around a concave surface. Originally discovered for sound waves in the whispering gallery of St Paul's Cathedral, they can exist for light and for other waves ...
can be either coupled to a mechanical mode of the
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
or evanescently to a nanobeam that is brought in proximity. *Optomechanical crystal structures: patterned dielectrics or
metamaterial A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
s can confine optical and/or mechanical (acoustic) modes. If the patterned material is designed to confine light, it is called a
photonic crystal A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
cavity. If it is designed to confine sound, it is called a
phononic crystal An acoustic metamaterial, sonic crystal, or phononic crystal, is a material designed to control, direct, and manipulate sound waves or phonons in gases, liquids, and solids (crystal lattices). Sound wave control is accomplished through manipulating ...
cavity. Either can be used respectively as the optical or mechanical component. Hybrid crystals, which confine both sound and light to the same area, are especially useful, as they form a complete optomechanical system. * Electromechanical implementations of an optomechanical system use superconducting
LC circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
s with a mechanically compliant capacitance like a membrane with metallic coating or a tiny capacitor plate glued onto it. By using movable capacitor plates, mechanical motion (physical displacement) of the plate or membrane changes the capacitance C, which transforms mechanical oscillation into electrical oscillation. LC oscillators have resonances in the
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
frequency range; therefore, LC circuits are also termed
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
resonators. The physics is exactly the same as in optical cavities but the range of parameters is different because microwave radiation has a larger wavelength than optical light or
infrared Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
laser light. A purpose of studying different designs of the same system is the different parameter regimes that are accessible by different setups and their different potential to be converted into tools of commercial use.


Measurement

The optomechanical system can be measured by using a scheme like
homodyne detection In electrical engineering, homodyne detection is a method of extracting information encoded as modulation of the phase and/or frequency of an oscillating signal, by comparing that signal with a standard oscillation that would be identical to the si ...
. Either the light of the driving laser is measured, or a two-mode scheme is followed where a strong laser is used to drive the optomechanical system into the state of interest and a second laser is used for the read-out of the state of the system. This second “probe” laser is typically weak, i.e. its optomechanical interaction can be neglected compared to the effects caused by the strong “pump” laser. The optical output field can also be measured with single photon detectors to achieve photon counting statistics.


Relation to fundamental research

One of the questions which are still subject to current debate is the exact mechanism of decoherence. In the
Schrödinger's cat In quantum mechanics, Schrödinger's cat is a thought experiment that illustrates a paradox of quantum superposition. In the thought experiment, a hypothetical cat may be considered simultaneously both alive and dead, while it is unobserved in ...
thought experiment, the cat would never be seen in a quantum state: there needs to be something like a collapse of the quantum wave functions, which brings it from a quantum state to a pure classical state. The question is where the boundary lies between objects with quantum properties and classical objects. Taking spatial superpositions as an example, there might be a size limit to objects which can be brought into superpositions, there might be a limit to the spatial separation of the centers of mass of a superposition or even a limit to the superposition of gravitational fields and its impact on small test masses. Those predictions can be checked with large mechanical structures that can be manipulated at the quantum level. Some easier to check predictions of quantum mechanics are the prediction of negative Wigner functions for certain quantum states, measurement precision beyond the
standard quantum limit A quantum limit in physics is a limit on measurement accuracy at quantum scales. Depending on the context, the limit may be absolute (such as the Heisenberg limit), or it may only apply when the experiment is conducted with naturally occurring qua ...
using squeezed states of light, or the asymmetry of the sidebands in the spectrum of a cavity near the quantum ground state.


Applications

Years before cavity optomechanics gained the status of an independent field of research, many of its techniques were already used in
gravitational wave detector A gravitational-wave detector (used in a gravitational-wave observatory) is any device designed to measure tiny distortions of spacetime called gravitational waves. Since the 1960s, various kinds of gravitational-wave detectors have been built ...
s where it is necessary to measure displacements of mirrors on the order of the Planck scale. Even if these detectors do not address the measurement of quantum effects, they encounter related issues ( photon shot noise) and use similar tricks (
squeezed coherent states In physics, a squeezed coherent state is a quantum state that is usually described by two Commutator, non-commuting Observable, observables having continuous spectra of Eigenvalues and eigenvectors, eigenvalues. Examples are position x and momen ...
) to enhance the precision. Further applications include the development of quantum memory for
quantum computer Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
s, high precision sensors (e.g. acceleration sensors) and quantum transducers e.g. between the optical and the microwave domain (taking advantage of the fact that the mechanical oscillator can easily couple to both frequency regimes).


Related fields and expansions

In addition to the standard cavity optomechanics explained above, there are variations of the simplest model: * Pulsed optomechanics: the continuous laser driving is replaced by pulsed laser driving. It is useful for creating entanglement and allows backaction-evading measurements. * Quadratic coupling: a system with quadratic optomechanical coupling can be investigated beyond the linear coupling term g_0= \left.\tfrac\_x_. The interaction Hamiltonian would then feature a term \hbar g_a^\dagger a(b+b^\dagger)^2 with g_=\frac \left.\tfrac\_x_^2. In membrane-in-the-middle setups it is possible to achieve quadratic coupling in the absence of linear coupling by positioning the membrane at an
extremum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
of the standing wave inside the cavity. One possible application is to carry out a quantum nondemolition measurement of the phonon number. * Reversed dissipation regime: in the standard optomechanical system the mechanical damping is much smaller than the optical damping. A system where this hierarchy is reversed can be engineered; i.e. the optical damping is much smaller than the mechanical damping (\kappa\ll\Gamma). Within the linearized regime, symmetry implies an inversion of the above described effects; For example, cooling of the mechanical oscillator in the standard optomechanical system is replaced by cooling of the optical oscillator in a system with reversed dissipation hierarchy. This effect was also seen in optical fiber loops in the 1970s. * Dissipative coupling: the coupling between optics and mechanics arises from a position-dependent optical dissipation rate \kappa(x) instead of a position-dependent cavity resonance frequency \omega_, which changes the interaction Hamiltonian and alters many effects of the standard optomechanical system. For example, this scheme allows the mechanical resonator to cool to its ground state without the requirement of the good cavity regime. Extensions to the standard optomechanical system include coupling to more and physically different systems: * Optomechanical arrays: coupling several optomechanical systems to each other (e.g. using evanescent coupling of the optical modes) allows multi-mode phenomena like synchronization to be studied. So far many theoretical predictions have been made, but only few experiments exist. The first optomechanical array (with more than two coupled systems) consists of seven optomechanical systems.Zhang, M., Shah, S., Cardenas, J, Lipson, M. (2015). Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light ''Physical Review Letters'', 115, 163902. * Hybrid systems: an optomechanical system can be coupled to a system of a different nature (e.g. a cloud of
ultracold atoms Ultracold atoms are atoms that are maintained at temperatures close to 0 kelvin (absolute zero), typically below several tens of microkelvin (µK). At these temperatures the atom's quantum-mechanical properties become important. To reach such low ...
and a
two-level system In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a syst ...
), which can lead to new effects on both the optomechanical and the additional system. Cavity optomechanics is closely related to
trapped ion An ion trap is a combination of electric and/or magnetic fields used to capture charged particles — known as ions — often in a system isolated from an external environment. Atomic and molecular ion traps have a number of applications in phys ...
physics and
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
s. These systems share very similar Hamiltonians, but have fewer particles (about 10 for ion traps and 10^5-10^8 for Bose–Einstein condensates) interacting with the field of light. It is also related to the field of
cavity quantum electrodynamics Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. It could in principle be us ...
.


See also

*
Quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...
*
Optical cavity An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that forms a cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and provi ...
*
Laser cooling Laser cooling includes a number of techniques in which atoms, molecules, and small mechanical systems are cooled, often approaching temperatures near absolute zero. Laser cooling techniques rely on the fact that when an object (usually an atom) ...
*
Coherent control Coherent control is a quantum mechanics-based method for controlling dynamic processes by light. The basic principle is to control quantum interference phenomena, typically by shaping the phase of laser pulses. The basic ideas have proliferated, f ...


References


Further reading

* Daniel Steck
Classical and Modern Optics
* Michel Deverot, Bejamin Huard, Robert Schoelkopf, Leticia F. Cugliandolo (2014). Quantum Machines: Measurement and Control of Engineered Quantum Systems. Lecture Notes of the Les Houches Summer School: Volume 96, July 2011. Oxford University Press * Kippenberg, T. J., & Vahala, K. J. (2007). Cavity Opto-Mechanics. Optics Express, 15(25), 17172. OSA. {{doi, 10.1364/OE.15.017172 * Demir, Dilek,"A table-top demonstration of radiation pressure", 2011, Diplomathesis, E-Theses univie. doi: 10.25365/thesis.16381 Quantum optics