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 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 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 k carries a momentum p=\hbar k, where \hbar is the
Planck constant. 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. 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 . Laser light interacts with a
glass sphere: the radiation pressure force causes it to vibrate. The presence of a single molecule on the sphere disturbs that (thermal) vibration, and causes its resonance frequency to shift: the molecule, via the light, induces an optical spring effect. The resonance frequency shift can be read out as a displacement of the
oscillator spectrum displayed on the left monitor. Some first effects of the light on the mechanical resonator can be captured by converting the radiation pressure force into a potential, \frac{d}{dx}V_\text{rad}(x) = -F(x), and adding it to the intrinsic
harmonic oscillator 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_0 - \frac{dF}{dx}. 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 F \, dx , 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 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_\text{cav}: H_\text{tot} = \hbar \omega_\text{cav}(x) a^\dagger a + \hbar \omega_m b^\dagger b + i \hbar E \left( a e^{i\omega_L t} - a^\dagger e^{-i\omega_L t}\right) where a and b are the bosonic annihilation operators of the given cavity mode and the mechanical resonator respectively, \omega_\text{cav} 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, a^\dagger] = [b, b^\dagger] = 1. \omega_{cav} is now dependent on x. The last term describes the driving, given by E = \sqrt{\frac{P \kappa}{\hbar \omega_L}} 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^{-i\omega_L t}) and applying a
Taylor expansion on \omega_\text{cav}. Quadratic and higher-order coupling terms are usually neglected, such that the standard Hamiltonian becomes H_\text{tot} = -\hbar \Delta a^\dagger a + \hbar \omega_m b^\dagger b - \hbar g_0 a^\dagger a \frac{x}{x_\text{zpf}}+ i\hbar E \left( a - a^\dagger \right) where \Delta = \omega_L - \omega_\text{cav} the laser detuning and the
position operator x = x_\text{zpf} (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{d\omega_\text{cav}}{dx}\right|_{x=0} x_\text{zpf} 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_\text{zpf} = \sqrt{\hbar / 2m_\text{eff} \omega_m}, where m_\text{eff} is the effective mass of the mechanical oscillator. It is sometimes more convenient to use the frequency pull parameter, or G = \frac{g_0}{x_\text{zpf}}, 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{\omega_\text{cav}(0) x_\text{zpf}}{L}. • \Delta\approx-\omega_m: a
rotating wave approximation of the linearized Hamiltonian, where one omits all non-resonant terms, reduces the coupling Hamiltonian to a beamsplitter operator, H_\text{int} = \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 of the linearized Hamiltonian leads to other resonant terms. The coupling Hamiltonian takes the form H_\text{int} = \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_\text{int} = \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 to detect gravitational waves.
Equations of motion From the linearized Hamiltonian, the so-called linearized quantum
Langevin equations, which govern the dynamics of the optomechanical system, can be derived when dissipation and noise terms to the
Heisenberg equations of motion are added. \begin{align} \delta \dot{a} &= (i \Delta-\kappa/2) \delta a + i g (b+b^\dagger) - \sqrt{\kappa} a_\text{in} \\[1ex] \dot b &= -(i\omega_m+\Gamma/2)b +i g (\delta a+\delta a^\dagger) - \sqrt{\Gamma}b_\text{in} \end{align} Here a_\text{in} and b_\text{in} 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 can be solved easily when they are rewritten in
frequency space (i.e. a
Fourier transform 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{\Delta-\omega_m}{\kappa^2/4+(\Delta-\omega_m)^2}+\frac{\Delta+\omega_m}{\kappa^2/4+(\Delta+\omega_m)^2}\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. \Gamma^\text{eff} = \Gamma + g^2\left(\frac{\kappa}{\kappa^2/4+(\Delta+\omega_m)^2} - \frac{\kappa}{\kappa^2/4+(\Delta-\omega_m)^2}\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^\text{eff} ), 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 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. • Levitated system: an
optically levitated nanoparticle 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 can be either coupled to a mechanical mode of the
toroid or evanescently to a
nanobeam that is brought in proximity. • Optomechanical crystal structures: patterned dielectrics or
metamaterials can confine optical and/or mechanical (acoustic) modes. If the patterned material is designed to confine light, it is called a
photonic crystal cavity. If it is designed to confine sound, it is called a
phononic crystal 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 circuits 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 frequency range; therefore, LC circuits are also termed
microwave 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 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. 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==