Cavity optomechanics

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 ${\displaystyle {\omega }^{\prime }=\omega }$. 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: $${\displaystyle {\omega }^{\prime }=\omega \pm {\omega }_m,}$$ where ${\displaystyle {\omega }_m}$ 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: $${\displaystyle {\omega }^{\prime }=\omega \mp {\omega }_m.}$$ 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 ${\displaystyle k}$ carries a momentum ${\displaystyle p=\hslash k}$, where ${\displaystyle \hslash }$ is the Planck constant. This means that a photon reflected off a mirror surface transfers a momentum ${\displaystyle \mathrm{\Delta }p=2\hslash k}$ 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

Some first effects of the light on the mechanical resonator can be captured by converting the radiation pressure force into a potential, $${\displaystyle \frac{d}{dx}{V}_{\text{rad}}(x)=-F(x),}$$ and adding it to the intrinsic harmonic oscillator potential of the mechanical oscillator, where ${\displaystyle F(x)}$ 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, ${\displaystyle F(x)}$ can be understood to be a modification of the mechanical spring constant, $${\displaystyle D=D_0-\frac{dF}{dx}.}$$ 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 ${\displaystyle \kappa }$. 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 with energy ${\displaystyle \hslash {\omega }_m}$ 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 ${\displaystyle \mathrm{\Delta }={\omega }_L-{\omega }_{\text{cav}}}$:

• Red-detuned regime, ${\displaystyle \mathrm{\Delta }<0}$ (most prominent effects on the red sideband, ${\displaystyle \mathrm{\Delta }=-{\omega }_m}$): 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, ${\displaystyle \mathrm{\Delta }>0}$ (most prominent effects on the blue sideband, ${\displaystyle \mathrm{\Delta }=+{\omega }_m}$): 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, ${\displaystyle \mathrm{\Delta }=0}$: 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 (${\displaystyle \mathrm{\Delta }\gg {\omega }_m,\kappa }$) 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: $${\displaystyle H_{\text{tot}}=\hslash {\omega }_{\text{cav}}(x)a^{†}a+\hslash {\omega }_m{b}^{†}b+i\hslash E\left(a{e}^{i{\omega }_{L}t}-a^{†}{e}^{-i{\omega }_{L}t}\right)}$$ where ${\displaystyle a}$ and ${\displaystyle b}$ are the bosonic annihilation operators of the given cavity mode and the mechanical resonator respectively, ${\displaystyle {\omega }_{\text{cav}}}$ is the frequency of the optical mode, ${\displaystyle x}$ is the position of the mechanical resonator, ${\displaystyle {\omega }_m}$ is the mechanical mode frequency, ${\displaystyle {\omega }_L}$ is the driving laser frequency, and ${\displaystyle E}$ is the amplitude. It satisfies the commutation relations $${\displaystyle [a,a^{†}]=[b,b^{†}]=1.}$$ ${\displaystyle {\omega }_{cav}}$ is now dependent on ${\displaystyle x}$. The last term describes the driving, given by $${\displaystyle E=\sqrt{\frac{P\kappa }{\hslash {\omega }_L}}}$$ where ${\displaystyle P}$ is the input power coupled to the optical mode under consideration and ${\displaystyle \kappa }$ 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 ${\displaystyle \kappa }$ and ${\displaystyle \mathrm{\Gamma }}$ 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 ${\displaystyle {\omega }_L}$ (in which case the optical mode annihilation operator undergoes the transformation ${\displaystyle a\to a{e}^{-i{\omega }_{L}t}}$) and applying a Taylor expansion on ${\displaystyle {\omega }_{\text{cav}}}$. Quadratic and higher-order coupling terms are usually neglected, such that the standard Hamiltonian becomes $${\displaystyle H_{\text{tot}}=-\hslash \mathrm{\Delta }{a}^{†}a+\hslash {\omega }_m{b}^{†}b-\hslash g_0{a}^{†}a\frac{x}{x_{\text{zpf}}}+i\hslash E\left(a-a^{†}\right)}$$ where ${\displaystyle \mathrm{\Delta }={\omega }_L-{\omega }_{\text{cav}}}$ the laser detuning and the position operator ${\displaystyle x=x_{\text{zpf}}(b+b^{†})}$. The first two terms (${\displaystyle -\hslash \mathrm{\Delta }{a}^{†}a}$ and ${\displaystyle \hslash {\omega }_m{b}^{†}b}$) are the free optical and mechanical Hamiltonians respectively. The third term contains the optomechanical interaction, where ${\displaystyle g_0={\frac{d{\omega }_{\text{cav}}}{dx}|}_{x=0}{x}_{\text{zpf}}}$ 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 $x_{\text{zpf}}=\sqrt{\hslash /2m_{\text{eff}}{\omega }_m}$, where ${\displaystyle m_{\text{eff}}}$ is the effective mass of the mechanical oscillator. It is sometimes more convenient to use the frequency pull parameter, or ${\displaystyle G=\frac{g_0}{x_{\text{zpf}}}}$, to determine the frequency change per displacement of the mirror.

For example, the optomechanical coupling strength of a Fabry–Pérot cavity of length ${\displaystyle L}$ with a moving end-mirror can be directly determined from the geometry to be ${\displaystyle g_0=\frac{{\omega }_{\text{cav}}(0)x_{\text{zpf}}}{L}}$.

This standard Hamiltonian ${\displaystyle H_{\text{tot}}}$ 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 ${\displaystyle g_0}$ is usually a small frequency, much smaller than the cavity decay rate ${\displaystyle \kappa }$, 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. ${\displaystyle a=\alpha +\delta a}$, where ${\displaystyle \alpha }$ is the mean light field amplitude and ${\displaystyle \delta a}$ denotes the fluctuations. Expanding the photon number ${\displaystyle a^{†}a}$, the term ${\displaystyle {\alpha }^2}$ can be omitted as it leads to a constant radiation pressure force which simply shifts the resonator's equilibrium position. The linearized optomechanical Hamiltonian ${\displaystyle H_{\text{lin}}}$ can be obtained by neglecting the second order term ${\displaystyle \delta a^{†}\delta a}$: $${\displaystyle H_{\text{lin}}=-\hslash \mathrm{\Delta }\delta a^{†}\delta a+\hslash {\omega }_m{b}^{†}b-\hslash g(\delta a+\delta a^{†})(b+b^{†})}$$ where ${\displaystyle g=g_0\alpha }$. While this Hamiltonian is a quadratic function, it is considered "linearized" because it leads to linear equations of motion. It is a valid description of many experiments, where ${\displaystyle g_0}$ 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 ${\displaystyle \alpha }$ 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:

• ${\displaystyle \mathrm{\Delta }\approx -{\omega }_m}$: a rotating wave approximation of the linearized Hamiltonian, where one omits all non-resonant terms, reduces the coupling Hamiltonian to a beamsplitter operator, ${\displaystyle H_{\text{int}}=\hslash g_0(\delta a^{†}b+\delta a{b}^{†})}$. 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.
• ${\displaystyle \mathrm{\Delta }\approx {\omega }_m}$: a rotating wave approximation of the linearized Hamiltonian leads to other resonant terms. The coupling Hamiltonian takes the form ${\displaystyle H_{\text{int}}=\hslash g_0(\delta ab+\delta a^{†}{b}^{†})}$, 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.
• ${\displaystyle \mathrm{\Delta }=0}$: In this case of driving on-resonance, all the terms in ${\displaystyle H_{\text{int}}=\hslash g_0(\delta a+\delta a^{†})(b+b^{†})}$ 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. $${\displaystyle \begin{array}{cc}\delta \dot{a}& =(i\mathrm{\Delta }-\kappa /2)\delta a+ig(b+b^{†})-\sqrt{\kappa }{a}_{\text{in}}\\ \dot{b}& =-(i{\omega }_m+\mathrm{\Gamma }/2)b+ig(\delta a+\delta a^{†})-\sqrt{\mathrm{\Gamma }}{b}_{\text{in}}\end{array}}$$

Here ${\displaystyle a_{\text{in}}}$ and ${\displaystyle b_{\text{in}}}$ are the input noise operators (either quantum or thermal noise) and ${\displaystyle -\kappa \delta a}$ and ${\displaystyle -\mathrm{\Gamma }\delta p}$ 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:

$${\displaystyle \delta {\omega }_m=g^2\left(\frac{\mathrm{\Delta }-{\omega }_m}{{\kappa }^2/4+(\mathrm{\Delta }-{\omega }_m{)}^2}+\frac{\mathrm{\Delta }+{\omega }_m}{{\kappa }^2/4+(\mathrm{\Delta }+{\omega }_m{)}^2}\right)}$$

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 (${\displaystyle {\omega }_m\gtrsim 1}$ 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. $${\displaystyle {\mathrm{\Gamma }}^{\text{eff}}=\mathrm{\Gamma }+g^2\left(\frac{\kappa }{{\kappa }^2/4+(\mathrm{\Delta }+{\omega }_m{)}^2}-\frac{\kappa }{{\kappa }^2/4+(\mathrm{\Delta }-{\omega }_m{)}^2}\right)}$$

The equation above shows optical damping, i.e. the intrinsic mechanical damping ${\displaystyle \mathrm{\Gamma }}$ 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 (${\displaystyle {\mathrm{\Gamma }}^{\text{eff}}<0}$), 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 ${\displaystyle \mathrm{\Delta }}$ 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 ${\displaystyle 1}$. 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, ${\displaystyle \kappa }$, is smaller than the distance from the cavity resonance to the sideband (${\displaystyle {\omega }_m}$). This requirement leads to a condition for the so-called sideband parameter: ${\displaystyle {\omega }_m/\kappa \gg 1}$. If ${\displaystyle {\omega }_m/\kappa \ll 1}$ 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 (${\displaystyle g\ge \kappa }$), 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, ${\displaystyle g_0\ge \kappa }$. Effects of the full non-linear interaction described by ${\displaystyle \hslash g_0{a}^{†}a(b+b^{†})}$ 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 ${\displaystyle g_0\ll \kappa }$) and only investigate effects of the linearized Hamiltonian.