Cavity optomechanics

441:. 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 5306: 450:—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. 459: 747:. 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. 707:. 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, 4673: 3788: 20: 3136:. 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. 3259:, 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. 4804:

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.

4788:. 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. 3649: 461: 466: 464: 460: 465: 5193:). 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. 3405: 3996: 463: 4213: 1968: 1274: 478:: 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 4646:

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Three types of dispersively coupled cavity optomechanical systems are shown. (a) A high-stress silicon nitride nanobeam coupled to a whispering gallery mode microdisk by dipole interaction. (b) An optomechanical crystal with colocalized mechanical and optical modes. (c) A mechanically compliant

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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

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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

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Simon Rips, Martin Kiffner, Ignacio Wilson-Rae, & Michael Hartmann. (2011). Cavity Optomechanics with Nonlinear Mechanical Resonators in the Quantum Regime - OSA Technical Digest (CD). CLEO/Europe and EQEC 2011 Conference Digest (p. JSI2_3). Optical Society of America. Retrieved from

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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

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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

924:): 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. 3797: 5153: 4041: 1808: 5166:

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

1098: 3644:{\displaystyle {\begin{aligned}\delta {\dot {a}}&=(i\Delta -\kappa /2)\delta a+ig(b+b^{\dagger })-{\sqrt {\kappa }}a_{\text{in}}\\{\dot {b}}&=-(i\omega _{m}+\Gamma /2)b+ig(\delta a+\delta a^{\dagger })-{\sqrt {\Gamma }}b_{\text{in}}\end{aligned}}} 755:

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

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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

2771: 5258:, 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. 6456:

Corbitt, Thomas; Chen, Yanbei; Innerhofer, Edith; Müller-Ebhardt, Helge; Ottaway, David; Rehbein, Henning; Sigg, Daniel; Whitcomb, Stanley; Wipf, Christopher; Mavalvala, Nergis (2007-04-13). "An All-Optical Trap for a Gram-Scale Mirror".

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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

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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

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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

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Corbitt, Thomas; Wipf, Christopher; Bodiya, Timothy; Ottaway, David; Sigg, Daniel; Smith, Nicolas; Whitcomb, Stanley; Mavalvala, Nergis (2007-10-18). "Optical Dilution and Feedback Cooling of a Gram-Scale Oscillator to 6.9 mK".

3257: 3134: 2564:, 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. 6874:

Teufel, J. D.; Donner, T.; Li, Dale; Harlow, J. W.; Allman, M. S.; Cicak, K.; Sirois, A. J.; Whittaker, J. D.; Lehnert, K. W.; Simmonds, R. W. (2011). "Sideband cooling of micromechanical motion to the quantum ground state".

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Braginskii, V. B., Manukin, A. B., Tikhonov, M. Yu. (1970). Investigation of dissipative ponderomotive effects of electromagnetic radiation. Soviet Physics JETP Vol 31, 5 (original russian: Zh. Eksp. Teor. Fiz. 58, 1549

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Anetsberger, G.; Arcizet, O.; Unterreithmeier, Q. P.; Rivière, R.; Schliesser, A.; Weig, E. M.; Kotthaus, J. P.; Kippenberg, T. J. (2009-10-11). "Near-field cavity optomechanics with nanomechanical oscillators".

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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

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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

5290:. These systems share very similar Hamiltonians, but have fewer particles (about 10 for ion traps and 10–10 for Bose–Einstein condensates) interacting with the field of light. It is also related to the field of 300:

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

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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.

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Nunnenkamp, A.; Sudhir, V.; Feofanov, A. K.; Roulet, A.; Kippenberg, T. J. (2014-07-11). "Quantum-Limited Amplification and Parametric Instability in the Reversed Dissipation Regime of Cavity Optomechanics".

97:. 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. 1609: 4599:

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

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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,

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Thompson, J. D.; Zwickl, B. M.; Jayich, A. M.; Marquardt, Florian; Girvin, S. M.; Harris, J. G. E. (2008-03-06). "Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane".

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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 (

556: 293: 215: 166:. 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 7022:

Safavi-Naeini, A. H., Chan, J., Hill, J. T., Alegre, T. P. M., Krause, A., & Painter, O. (2011). Measurement of the quantum zero-point motion of a nanomechanical resonator, 6. Retrieved from

3991:{\displaystyle \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)} 6411:

Genes, C.; Vitali, D.; Tombesi, P.; Gigan, S.; Aspelmeyer, M. (2008-03-03). "Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes".

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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

4310:) 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 3046: 2108: 4299:

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.

3169: 85:. 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 4269: 678: 4843:) 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). 4461: 4420: 4208:{\displaystyle \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)} 164: 4032: 1803: 1341: 987: 922: 7254:

Zhang, Mian; Shah, Shreyas; Cardenas, Jaime; Lipson, Michal (2015-10-16). "Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light".

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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)

4334:. 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, 5191: 2597: 1963:{\displaystyle 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)} 4532: 788: 435: 4631: 3763: 3734: 1269:{\displaystyle H_{\text{tot}}=\hbar \omega _{\text{cav}}(x)a^{\dagger }a+\hbar \omega _{m}b^{\dagger }b+i\hbar E\left(ae^{i\omega _{L}t}-a^{\dagger }e^{-i\omega _{L}t}\right)} 7312:(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 3766:

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.

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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

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Pulsed optomechanics: the continuous laser driving is replaced by pulsed laser driving. It is useful for creating entanglement and allows backaction-evading measurements.

4379: 1720: 1415: 1388: 242: 5223: 743:, 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 3285: 1028: 376: 4467:

Another distinction can be made depending on the optomechanical coupling strength. If the (enhanced) optomechanical coupling becomes larger than the cavity linewidth (

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can be omitted as it leads to a constant radiation pressure force which simply shifts the resonator's equilibrium position. The linearized optomechanical Hamiltonian

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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

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Verhagen, E.; Deléglise, S.; Weis, S.; Schliesser, A.; Kippenberg, T. J. (2012). "Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode".

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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

2907:{\displaystyle H_{\text{lin}}=-\hbar \Delta \delta a^{\dagger }\delta a+\hbar \omega _{m}b^{\dagger }b-\hbar g(\delta a+\delta a^{\dagger })(b+b^{\dagger })} 3178: 3055: 5976:

Chan, Jasper; Alegre, T. P. Mayer; Safavi-Naeini, Amir H.; Hill, Jeff T.; Krause, Alex; Gröblacher, Simon; Aspelmeyer, Markus; Painter, Oskar (Oct 2011).

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Krause, Alexander G.; Winger, Martin; Blasius, Tim D.; Lin, Qiang; Painter, Oskar (2012). "A high-resolution microchip optomechanical accelerometer".

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Bochmann, Joerg; Vainsencher, Amit; Awschalom, David D.; Cleland, Andrew N. (2013). "Nanomechanical coupling between microwave and optical photons".

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Safavi-Naeini, Amir H; Chan, Jasper; Hill, Jeff T; Gröblacher, Simon; Miao, Haixing; Chen, Yanbei; Aspelmeyer, Markus; Painter, Oskar (2013-03-06).

5155:. 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 5845: 5572: 131:

The most elementary light-matter interaction is a light beam scattering off an arbitrary object (atom, molecule, nanobeam etc.). There is always

6267: 2408: 489: 4959: 4651:, kilometers. (although LIGO is dedicated to the detection of gravitational waves and not the investigation of optomechanics specifically). 2246: 23:

The typical model for many structures in cavity optomechanics is an optical cavity consisting of a fixed mirror and a mechanical oscillator.

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Law, C. K. (1994-01-01). "Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium".

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with a mechanically compliant capacitance like a membrane with metallic coating or a tiny capacitor plate glued onto it. By using movable

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Some first effects of the light on the mechanical resonator can be captured by converting the radiation pressure force into a potential,

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Elste, Florian; Girvin, S. M.; Clerk, A. A. (2009-05-22). "Quantum Noise Interference and Backaction Cooling in Cavity Nanomechanics".

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Years before cavity optomechanics gained the status of an independent field of research, many of its techniques were already used in

2954:, it is considered "linearized" because it leads to linear equations of motion. It is a valid description of many experiments, where 3052:

of the linearized Hamiltonian, where one omits all non-resonant terms, reduces the coupling Hamiltonian to a beamsplitter operator,

5148:{\displaystyle g_{\text{sq}}={\frac {1}{2}}\left.{\tfrac {d^{2}\omega _{\text{cav}}(x)}{dx^{2}}}\right|_{x=0}x_{\text{zpf}}^{2}} 93:, which describes the transition of objects from states that are described by quantum mechanics to states that are described by 4715:

can confine optical and/or mechanical (acoustic) modes. If the patterned material is designed to confine light, it is called a

2153:) are the free optical and mechanical Hamiltonians respectively. The third term contains the optomechanical interaction, where 624: 4537: 1973: 815: 6243: 5388: 4815:

using squeezed states of light, or the asymmetry of the sidebands in the spectrum of a cavity near the quantum ground state.

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is the effective mass of the mechanical oscillator. It is sometimes more convenient to use the frequency pull parameter, or

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Dissipative coupling: the coupling between optics and mechanics arises from a position-dependent optical dissipation rate

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Quadratic coupling: a system with quadratic optomechanical coupling can be investigated beyond the linear coupling term

308:, or vibrational spectroscopy, where the light field controls the mechanics (or vice versa) but the loop is not closed. 6829:

Eichenfield, Matt; Chan, Jasper; Camacho, Ryan M.; Vahala, Kerry J.; Painter, Oskar (2009). "Optomechanical crystals".

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If Stokes and anti-Stokes scattering occur at an equal rate, the vibrations will only heat up the object. However, an

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Sheard, Benjamin S.; Gray, Malcolm B.; Mow-Lowry, Conor M.; McClelland, David E.; Whitcomb, Stanley E. (2004-05-07).

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between two different places). Such a quantum state of motion would allow researchers to experimentally investigate

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oscillations), if the growth of the mechanical energy overwhelms the intrinsic losses (mainly mechanical friction).

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The principle can be summarized as: phonons are converted into photons when cooled and vice versa in amplification.

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which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of

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The optical output field can also be measured with single photon detectors to achieve photon counting statistics.

2732: 997:, and mechanical "lasing" (amplification of the mechanical motion to self-sustained optomechanical oscillations / 479: 3398:, which govern the dynamics of the optomechanical system, can be derived when dissipation and noise terms to the 3015: 2074: 1092: 170:

independent of the internal electronic details of atoms or molecules due to the object's mechanical vibrations:

3141: 6327:"Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation" 5291: 4271:), which means that it turns into an overall amplification rather than a damping of the mechanical oscillator. 4241: 687:

However, the model is incomplete as it neglects retardation effects due to the finite cavity photon decay rate

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In addition to the standard cavity optomechanics explained above, there are variations of the simplest model:

4425: 4384: 1805:. Quadratic and higher-order coupling terms are usually neglected, such that the standard Hamiltonian becomes 6920:

Bose, S.; Jacobs, K.; Knight, P. L. (1999-05-01). "Scheme to probe the decoherence of a macroscopic object".

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One of the questions which are still subject to current debate is the exact mechanism of decoherence. In the

4755:, which transforms mechanical oscillation into electrical oscillation. LC oscillators have resonances in the 5262:

Extensions to the standard optomechanical system include coupling to more and physically different systems:

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The most basic regimes in which the optomechanical system can be operated are defined by the laser detuning

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Another but equivalent way to interpret the principle of optomechanical cavities is by using the concept of

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Hybrid systems: an optomechanical system can be coupled to a system of a different nature (e.g. a cloud of

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are the bosonic annihilation operators of the given cavity mode and the mechanical resonator respectively,

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Demir, Dilek,"A table-top demonstration of radiation pressure", 2011, Diplomathesis, E-Theses univie. doi:

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Cavities with a moving mirror: the archetype of an optomechanical system. The light is reflected from the

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interaction reduces effective damping. Instability can occur when the effective damping drops below zero (

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The optically induced damping of the mechanical oscillator that adds to the intrinsic mechanical damping.

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The optical spring effect also depends on the detuning. It can be observed for high levels of detuning (

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Two main effects of the light on the mechanical oscillator can then be expressed in the following ways:

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With a particular choice of detuning, different phenomena can be observed (see also the section about

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of the linearized Hamiltonian leads to other resonant terms. The coupling Hamiltonian takes the form

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cooling of the mechanical oscillator, i.e. cooling to an average mechanical occupation number below

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optomechanical array (with more than two coupled systems) consists of seven optomechanical systems.

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and transfers momentum onto the movable one, which in turn changes the cavity resonance frequency.

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Clerk, A. A.; Devoret, M. H.; Girvin, S. M.; Marquardt, Florian; Schoelkopf, R. J. (2010-04-15).

5849: 4697: 4357: 1698: 1393: 1366: 994: 220: 86: 5199: 4800: 82: 6374:"Squeezing of intracavity and traveling-wave light fields produced in parametric amplification" 4835:) to enhance the precision. Further applications include the development of quantum memory for 3264: 1007: 352: 4949:{\displaystyle g_{0}=\left.{\tfrac {d\omega _{\text{cav}}(x)}{dx}}\right|_{x=0}x_{\text{zpf}}} 2622: 7303: 4735:, mechanical motion (physical displacement) of the plate or membrane changes the capacitance 4720: 4337: 4282: 4220: 2985: 2602: 2547: 1674: 1654: 1634: 690: 470:

In this optomechanical system, the radiation pressure force is leveraged to detect a single

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Kiesel, N.; Blaser, F.; Delic, U.; Grass, D.; Kaltenbaek, R.; Aspelmeyer, M. (2013-08-12).

6571: 6385: 6338: 6297: 6190: 6072: 5999: 5926: 5871: 5819: 5752: 5687: 5620: 5494: 5427: 2957: 2520: 2235:{\displaystyle g_{0}=\left.{\tfrac {d\omega _{\text{cav}}}{dx}}\right|_{x=0}x_{\text{zpf}}} 990: 791:

scatter into a cavity photon and create an additional phonon in the mechanical oscillator.

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is the vibrational frequency. The vibrations gain or lose energy, respectively, for these

8: 7309: 6978:

Jaekel, M. T; Reynaud, S (1990-10-15). "Quantum Limits in Interferometric Measurements".

5472: 2505: 102: 94: 43: 6688: 6575: 6389: 6342: 6301: 6203: 6194: 6168: 6076: 6003: 5930: 5875: 5823: 5756: 5691: 5624: 5498: 5431: 3373:{\displaystyle H_{\text{int}}=\hbar g_{0}(\delta a+\delta a^{\dagger })(b+b^{\dagger })} 2504:

the laser in such a way that it only populates a single optical mode (implying that the

50:. The motivation for research on cavity optomechanics comes from fundamental effects of 7263: 7218: 7173: 7079: 6987: 6929: 6884: 6838: 6793: 6747: 6715: 6674: 6662: 6625: 6595: 6561: 6512: 6466: 6420: 6261: 6180: 6104: 6062: 6031: 5989: 5958: 5895: 5861: 5783: 5742: 5730: 5711: 5677: 5610: 5518: 5484: 5453: 5417: 4785: 4738: 4501:, where the bare optomechanical coupling becomes of the order of the cavity linewidth, 4317: 3399: 2951: 2388: 1614: 1542: 1420: 1346: 1299: 1279: 332: 323: 317: 132: 74: 62: 6549: 5807: 5665: 5278:), which can lead to new effects on both the optomechanical and the additional system. 81:. Furthermore, one may envision optomechanical structures to allow the realization of 7345: 7333: 7281: 7236: 7191: 7132: 7097: 7052: 7005: 7001: 6947: 6902: 6856: 6811: 6765: 6720: 6702: 6643: 6599: 6591: 6530: 6484: 6438: 6354: 6249: 6239: 6208: 6149: 6096: 6088: 6023: 6015: 5950: 5942: 5899: 5887: 5788: 5770: 5715: 5703: 5646: 5638: 5510: 5384: 4807:

Some easier to check predictions of quantum mechanics are the prediction of negative

4666: 4659: 3778: 3395: 3252:{\displaystyle H_{\text{int}}=\hbar g_{0}(\delta ab+\delta a^{\dagger }b^{\dagger })} 3129:{\displaystyle H_{\text{int}}=\hbar g_{0}(\delta a^{\dagger }b+\delta ab^{\dagger })} 106: 51: 47: 6373: 6326: 5522: 5471:

Marshall, William; Simon, Christoph; Penrose, Roger; Bouwmeester, Dik (2003-09-23).

5405: 2385:

For example, the optomechanical coupling strength of a Fabry–Pérot cavity of length

7325: 7277: 7273: 7232: 7228: 7187: 7183: 7124: 7089: 7044: 6997: 6939: 6894: 6848: 6803: 6757: 6710: 6692: 6635: 6587: 6579: 6522: 6476: 6430: 6393: 6346: 6305: 6198: 6141: 6108: 6080: 6049:

Arcizet, O.; Cohadon, P.-F.; Briant, T.; Pinard, M.; Heidmann, A. (November 2006).

6035: 6007: 5962: 5934: 5879: 5827: 5778: 5760: 5695: 5664:

Metzger, Constanze; Favero, Ivan; Ortlieb, Alexander; Karrai, Khaled (2008-07-09).

5628: 5549: 5502: 5457: 5443: 5435: 5376: 5334: 5275: 5159:

of the standing wave inside the cavity. One possible application is to carry out a

4836: 4764: 4716: 4683: 3774: 1722:(in which case the optical mode annihilation operator undergoes the transformation 301: 6526: 6480: 5506: 4727:

Electromechanical implementations of an optomechanical system use superconducting

7035:

Cole, Garrett D.; Aspelmeyer, Markus (2011). "Mechanical memory sees the light".

6310: 6285: 5271: 399: 305: 61:

The name of the field relates to the main effect of interest: the enhancement of

5666:"Optical self cooling of a deformable Fabry-Perot cavity in the classical limit" 4672: 4217:

The equation above shows optical damping, i.e. the intrinsic mechanical damping

6583: 6434: 5831: 5699: 5324: 5311: 4464:

one phonon, which leads to greater amplification of the mechanical oscillator.

4381:). This requirement leads to a condition for the so-called sideband parameter: 1031: 447: 402:. This means that a photon reflected off a mirror surface transfers a momentum 296: 70: 39: 6050: 5977: 5914: 5553: 5536:

Khalili, Farid Ya; Danilishin, Stefan L. (2016-01-01), Visser, Taco D. (ed.),

5380: 4828: 7354: 7337: 7285: 7240: 7195: 7136: 7101: 7093: 7056: 7009: 6951: 6943: 6906: 6860: 6815: 6769: 6706: 6647: 6534: 6488: 6442: 6397: 6350: 6253: 6229: 6212: 6153: 6092: 6019: 5946: 5891: 5774: 5707: 5642: 5439: 5329: 4840: 4812: 1775: 744: 248:, while optical sidebands are created around the incoming light frequency: 6697: 6145: 6051:"Radiation-pressure cooling and optomechanical instability of a micromirror" 5978:"Laser cooling of a nanomechanical oscillator into its quantum ground state" 5371:

Aspelmeyer, Markus; Kippenberg, Tobias J.; Marquardt, Florian, eds. (2014).

2405:

with a moving end-mirror can be directly determined from the geometry to be

7048: 6724: 6100: 6027: 5954: 5883: 5792: 5650: 5514: 4712: 4686: 4311: 3009:). The clearest distinction can be made between the following three cases: 621:

can be understood to be a modification of the mechanical spring constant,

245: 6358: 6233: 4354:, is smaller than the distance from the cavity resonance to the sideband ( 807: 446:

changes, which also alters the cavity resonance frequency. Therefore, the

7329: 7316:

Kippenberg, Tobias J.; Vahala, Kerry J. (2007). "Cavity Opto-Mechanics".

6992: 6934: 6471: 6067: 5633: 5489: 5448: 5422: 998: 989:): This regime describes "two-mode squeezing". It can be used to achieve 475: 90: 6898: 6852: 6782: 6761: 6639: 6084: 6011: 5938: 5765: 3787: 2466:{\displaystyle g_{0}={\frac {\omega _{\text{cav}}(0)x_{\text{zpf}}}{L}}} 135:, with the outgoing light frequency identical to the incoming frequency 7149:

Palomaki, T.A., Teufel, J. D., Simmonds, R. W., Lehnert, K. W. (2013).

4728: 4705: 2301:{\textstyle x_{\text{zpf}}={\sqrt {\hbar /2m_{\text{eff}}\omega _{m}}}} 1631:

is the input power coupled to the optical mode under consideration and

327: 19: 7128: 6807: 5021:{\displaystyle \hbar g_{\text{quad}}a^{\dagger }a(b+b^{\dagger })^{2}} 2544:

is usually a small frequency, much smaller than the cavity decay rate

77:

detection, since optomechanical effects must be taken into account in

5406:"Preparation of nonclassical states in cavities with a moving mirror" 4760: 4756: 3707:

are the input noise operators (either quantum or thermal noise) and

2382:, to determine the frequency change per displacement of the mirror. 7268: 7114: 6455: 5747: 5156: 4768: 7223: 7178: 7084: 7023: 6889: 6843: 6798: 6752: 6679: 6630: 6566: 6517: 6425: 6185: 5994: 5866: 5682: 5615: 3394:

From the linearized Hamiltonian, the so-called linearized quantum

4784:

The optomechanical system can be measured by using a scheme like

3381:

used to measure very small displacements. This setup has enabled

802: 798: 757: 471: 55: 31: 6286:"Nobel Lecture: LIGO and the discovery of gravitational waves I" 4677:

aluminum capacitor used to form a superconducting LC oscillator.

6967:

http://www.opticsinfobase.org/abstract.cfm?URI=EQEC-2011-JSI2_3

6737: 6550:"Introduction to quantum noise, measurement, and amplification" 6166: 5470: 1691:

respectively) and the corresponding noise entering the system.

1604:{\displaystyle E={\sqrt {\frac {P\kappa }{\hbar \omega _{L}}}}} 66: 35: 7162: 6169:"Laser noise in cavity-optomechanical cooling and thermometry" 4633:) and only investigate effects of the linearized Hamiltonian. 326:. According to the quantum theory of light, every photon with 6663:"Cavity cooling of an optically levitated submicron particle" 6615: 5805: 5729:

Yu, Wenyan; Jiang, Wei C.; Lin, Qiang; Lu, Tao (2016-07-27).

5370: 4811:

for certain quantum states, measurement precision beyond the

1077:) and its strength varies with detuning and the laser drive. 110: 6828: 4711:

Optomechanical crystal structures: patterned dielectrics or

6547: 5975: 5225:

instead of a position-dependent cavity resonance frequency

5060: 4880: 4719:

cavity. If it is designed to confine sound, it is called a

4648: 3382: 2175: 6048: 5731:"Cavity optomechanical spring sensing of single molecules" 5663: 4534:. Effects of the full non-linear interaction described by 1437:

is the amplitude. It satisfies the commutation relations

5913:

Metzger, Constanze Höhberger; Karrai, Khaled (Dec 2004).

3287:: In this case of driving on-resonance, all the terms in 6792:(12). Springer Science and Business Media LLC: 909–914. 6501: 4956:. The interaction Hamiltonian would then feature a term 4759:

frequency range; therefore, LC circuits are also termed

4592:{\displaystyle \hbar g_{0}a^{\dagger }a(b+b^{\dagger })} 2009:{\displaystyle \Delta =\omega _{L}-\omega _{\text{cav}}} 851:{\displaystyle \Delta =\omega _{L}-\omega _{\text{cav}}} 7253: 6611: 6609: 6410: 5808:"Observation and characterization of an optical spring" 808:

Three regimes of operation: cooling, heating, resonance

551:{\displaystyle {\frac {d}{dx}}V_{\text{rad}}(x)=-F(x),} 7069: 6660: 5063: 4883: 2642:

denotes the fluctuations. Expanding the photon number

2249: 2178: 713: 5231: 5202: 5173: 5034: 4962: 4864: 4741: 4654:

Examples of real optomechanical implementations are:

4606: 4540: 4507: 4473: 4428: 4387: 4360: 4340: 4320: 4285: 4244: 4223: 4044: 4007: 3800: 3742: 3713: 3686: 3659: 3408: 3293: 3267: 3181: 3144: 3058: 3018: 2988: 2960: 2920: 2774: 2735: 2708: 2678: 2648: 2625: 2605: 2570: 2550: 2523: 2482: 2411: 2391: 2341: 2314: 2159: 2116: 2077: 2022: 1976: 1811: 1784: 1728: 1701: 1677: 1657: 1637: 1617: 1565: 1545: 1512: 1443: 1423: 1396: 1369: 1349: 1322: 1302: 1282: 1101: 1044: 1030:: In this regime the cavity is simply operated as an 1010: 959: 933: 894: 868: 818: 766: 693: 627: 598: 568: 492: 408: 384: 355: 335: 254: 223: 176: 141: 6873: 6606: 5301: 2729:

can be obtained by neglecting the second order term

126: 4794: 2517:The single-photon optomechanical coupling strength 797:Further information on elementary excitations: 116: 7308:Michel Deverot, Bejamin Huard, Robert Schoelkopf, 5404:Bose, S.; Jacobs, K.; Knight, P. L. (1997-11-01). 5250: 5217: 5185: 5147: 5020: 4948: 4747: 4700:can be either coupled to a mechanical mode of the 4625: 4591: 4526: 4485: 4455: 4414: 4373: 4346: 4326: 4291: 4263: 4229: 4207: 4026: 3990: 3757: 3728: 3699: 3672: 3643: 3372: 3279: 3251: 3163: 3128: 3040: 2994: 2973: 2942: 2906: 2760: 2721: 2694: 2664: 2634: 2611: 2591: 2556: 2536: 2495: 2465: 2397: 2374: 2327: 2300: 2234: 2145: 2102: 2063: 2008: 1962: 1797: 1766: 1714: 1683: 1663: 1643: 1623: 1603: 1551: 1531: 1499: 1429: 1409: 1382: 1355: 1335: 1308: 1288: 1268: 1069: 1022: 981: 945: 916: 880: 850: 782: 735: 699: 672: 613: 583: 550: 429: 390: 370: 341: 287: 236: 209: 158: 5535: 2375:{\displaystyle G={\frac {g_{0}}{x_{\text{zpf}}}}} 1559:. The last term describes the driving, given by 7352: 7315: 7208: 6919: 5596: 5573:"Milestone 23 : Nature Milestones: Photons" 5403: 4846: 3773:can be solved easily when they are rewritten in 2064:{\displaystyle x=x_{\text{zpf}}(b+b^{\dagger })} 1767:{\displaystyle a\rightarrow ae^{-i\omega _{L}t}} 288:{\displaystyle \omega '=\omega \mp \omega _{m}.} 210:{\displaystyle \omega '=\omega \pm \omega _{m},} 6667:Proceedings of the National Academy of Sciences 6371: 6324: 5597:Kippenberg, T. J.; Vahala, K. J. (2007-12-10). 7034: 6372:Collett, M. J.; Gardiner, C. W. (1984-09-01). 6325:Gardiner, C. W.; Collett, M. J. (1985-06-01). 4274: 2146:{\displaystyle \hbar \omega _{m}b^{\dagger }b} 1363:is the position of the mechanical resonator, 1070:{\displaystyle \Delta \gg \omega _{m},\kappa } 953:(most prominent effects on the blue sideband, 562:potential of the mechanical oscillator, where 6977: 5912: 3002:around which the linearization was executed. 2016:the laser detuning and the position operator 888:(most prominent effects on the red sideband, 73:. It first became relevant in the context of 5850:"A short walk through quantum optomechanics" 5728: 5473:"Towards Quantum Superpositions of a Mirror" 4636: 2761:{\displaystyle ~\delta a^{\dagger }\delta a} 750: 79:interferometric gravitational wave detectors 5282:Cavity optomechanics is closely related to 3041:{\displaystyle \Delta \approx -\omega _{m}} 2103:{\displaystyle -\hbar \Delta a^{\dagger }a} 311: 6266:: CS1 maint: location missing publisher ( 6238:. Milburn, G. J. (Gerard J.). Boca Raton. 3164:{\displaystyle \Delta \approx \omega _{m}} 7267: 7222: 7177: 7083: 6991: 6933: 6888: 6842: 6797: 6751: 6714: 6696: 6678: 6629: 6565: 6516: 6470: 6424: 6309: 6202: 6184: 6066: 5993: 5865: 5782: 5764: 5746: 5681: 5632: 5614: 5488: 5447: 5421: 4264:{\displaystyle \Gamma ^{\text{eff}}<0} 1080: 720: 673:{\displaystyle D=D_{0}-{\frac {dF}{dx}}.} 58:, as well as technological applications. 4671: 4456:{\displaystyle \omega _{m}/\kappa \ll 1} 4415:{\displaystyle \omega _{m}/\kappa \gg 1} 3786: 457: 453: 18: 5844: 5538:"Chapter Three - Quantum Optomechanics" 3006: 7353: 3389: 2619:is the mean light field amplitude and 1343:is the frequency of the optical mode, 6283: 6279: 6277: 6228: 6224: 6222: 121: 5366: 5364: 5362: 5360: 5358: 5356: 5354: 5352: 5350: 4499:single-photon strong-coupling regime 4027:{\displaystyle \omega _{m}\gtrsim 1} 1798:{\displaystyle \omega _{\text{cav}}} 1417:is the driving laser frequency, and 1336:{\displaystyle \omega _{\text{cav}}} 982:{\displaystyle \Delta =+\omega _{m}} 917:{\displaystyle \Delta =-\omega _{m}} 6131: 4308:resolved/unresolved sideband regime 13: 7296: 6274: 6219: 6042: 5186:{\displaystyle \kappa \ll \Gamma } 5180: 4286: 4246: 4224: 4171: 4109: 4058: 4046: 3954: 3912: 3877: 3835: 3771:first order differential equations 3746: 3622: 3563: 3441: 3268: 3145: 3019: 2794: 2592:{\displaystyle a=\alpha +\delta a} 2084: 1977: 1831: 1678: 1390:is the mechanical mode frequency, 1045: 1011: 960: 934: 895: 869: 819: 409: 14: 7372: 5347: 5161:quantum nondemolition measurement 4665:Membrane-in-the-middle system: a 4527:{\displaystyle g_{0}\geq \kappa } 3307: 3195: 3072: 2848: 2819: 2791: 2081: 1928: 1879: 1850: 1828: 1185: 1153: 1115: 783:{\displaystyle \hbar \omega _{m}} 745:optical or optomechanical cooling 684:(light-induced spring constant). 430:{\displaystyle \Delta p=2\hbar k} 421: 362: 127:Stokes and anti-Stokes scattering 5915:"Cavity cooling of a microlever" 5304: 4795:Relation to fundamental research 4626:{\displaystyle g_{0}\ll \kappa } 3758:{\displaystyle -\Gamma \delta p} 3729:{\displaystyle -\kappa \delta a} 2512: 2506:spacing between the cavity modes 159:{\displaystyle \omega '=\omega } 117:Concepts of cavity optomechanics 7247: 7202: 7156: 7143: 7108: 7063: 7028: 7024:https://arxiv.org/abs/1108.4680 7016: 6971: 6958: 6913: 6867: 6822: 6776: 6731: 6654: 6541: 6495: 6449: 6404: 6365: 6318: 6160: 6125: 6115: 5969: 5906: 5838: 4839:, high precision sensors (e.g. 4818: 558:and adding it to the intrinsic 474:. Laser light interacts with a 7278:10.1103/PhysRevLett.115.163902 7233:10.1103/PhysRevLett.102.207209 7188:10.1103/PhysRevLett.113.023604 5799: 5722: 5657: 5590: 5565: 5529: 5464: 5397: 5292:cavity quantum electrodynamics 5212: 5206: 5092: 5086: 5009: 4989: 4905: 4899: 4779: 4586: 4567: 4188: 4168: 4126: 4106: 3971: 3951: 3894: 3874: 3614: 3589: 3574: 3544: 3495: 3476: 3458: 3435: 3400:Heisenberg equations of motion 3385:to detect gravitational waves. 3367: 3348: 3345: 3320: 3246: 3208: 3123: 3085: 2950:. While this Hamiltonian is a 2943:{\displaystyle g=g_{0}\alpha } 2901: 2882: 2879: 2854: 2722:{\displaystyle H_{\text{lin}}} 2496:{\displaystyle H_{\text{tot}}} 2444: 2438: 2328:{\displaystyle m_{\text{eff}}} 2058: 2039: 1732: 1488: 1469: 1463: 1444: 1134: 1128: 1085: 1034:to read the mechanical motion. 608: 602: 578: 572: 542: 536: 524: 518: 482:displayed on the left monitor. 63:radiation pressure interaction 1: 6527:10.1103/PhysRevLett.99.160801 6481:10.1103/PhysRevLett.98.150802 6204:10.1088/1367-2630/15/3/035007 5507:10.1103/PhysRevLett.91.130401 5340: 5251:{\displaystyle \omega _{cav}} 4847:Related fields and expansions 4708:that is brought in proximity. 4486:{\displaystyle g\geq \kappa } 3700:{\displaystyle b_{\text{in}}} 3673:{\displaystyle a_{\text{in}}} 2665:{\displaystyle a^{\dagger }a} 1532:{\displaystyle \omega _{cav}} 736:{\textstyle \oint F\,dx<0} 71:optical resonators (cavities) 6311:10.1103/RevModPhys.90.040501 6284:Weiss, Rainer (2018-12-18). 4825:gravitational wave detectors 2695:{\displaystyle ~\alpha ^{2}} 946:{\displaystyle \Delta >0} 881:{\displaystyle \Delta <0} 680:This effect is known as the 246:Stokes/anti-Stokes processes 7: 7304:Classical and Modern Optics 5320:Quantum harmonic oscillator 5297: 4374:{\displaystyle \omega _{m}} 4275:Important parameter regimes 3173:rotating wave approximation 3050:rotating wave approximation 1715:{\displaystyle \omega _{L}} 1410:{\displaystyle \omega _{L}} 1383:{\displaystyle \omega _{m}} 437:onto the mirror due to the 237:{\displaystyle \omega _{m}} 10: 7377: 7002:10.1209/0295-5075/13/4/003 6584:10.1103/RevModPhys.82.1155 6435:10.1103/PhysRevA.77.033804 5832:10.1103/PhysRevA.69.051801 5700:10.1103/PhysRevB.78.035309 5218:{\displaystyle \kappa (x)} 4831:) and use similar tricks ( 2982:classical light amplitude 2476:This standard Hamiltonian 796: 315: 6980:Europhysics Letters (EPL) 6592:21.11116/0000-0001-D7A2-5 6554:Reviews of Modern Physics 6290:Reviews of Modern Physics 5554:10.1016/bs.po.2015.09.001 5381:10.1007/978-3-642-55312-7 5288:Bose–Einstein condensates 4637:Experimental realizations 3280:{\displaystyle \Delta =0} 1023:{\displaystyle \Delta =0} 751:Quantized energy transfer 371:{\displaystyle p=\hbar k} 7094:10.1038/nphoton.2012.245 6944:10.1103/PhysRevA.59.3204 6398:10.1103/PhysRevA.30.1386 6351:10.1103/PhysRevA.31.3761 5440:10.1103/PhysRevA.56.4175 4833:squeezed coherent states 4696:that support an optical 4667:micromechanical membrane 4641: 2635:{\displaystyle \delta a} 439:conservation of momentum 312:Radiation pressure force 133:elastic light scattering 7256:Physical Review Letters 7211:Physical Review Letters 7166:Physical Review Letters 6698:10.1073/pnas.1309167110 6505:Physical Review Letters 6459:Physical Review Letters 6146:10.1103/PhysRevA.49.433 5599:"Cavity Opto-Mechanics" 5477:Physical Review Letters 4698:whispering gallery mode 4347:{\displaystyle \kappa } 4292:{\displaystyle \Delta } 4230:{\displaystyle \Gamma } 2995:{\displaystyle \alpha } 2612:{\displaystyle \alpha } 2557:{\displaystyle \kappa } 2071:. The first two terms ( 1684:{\displaystyle \Gamma } 1664:{\displaystyle \kappa } 1644:{\displaystyle \kappa } 700:{\displaystyle \kappa } 7310:Leticia F. Cugliandolo 7049:10.1038/nnano.2011.199 6173:New Journal of Physics 5884:10.1002/andp.201200226 5252: 5219: 5187: 5149: 5022: 4950: 4813:standard quantum limit 4749: 4678: 4627: 4593: 4528: 4495:strong-coupling regime 4487: 4457: 4416: 4375: 4348: 4328: 4304:good/bad cavity regime 4293: 4265: 4231: 4209: 4028: 3992: 3792: 3759: 3730: 3701: 3674: 3645: 3374: 3281: 3253: 3165: 3130: 3042: 2996: 2975: 2944: 2908: 2762: 2723: 2696: 2666: 2636: 2613: 2593: 2558: 2538: 2509:of the optical modes. 2497: 2467: 2399: 2376: 2329: 2302: 2236: 2147: 2104: 2065: 2010: 1964: 1799: 1768: 1716: 1685: 1665: 1645: 1625: 1605: 1553: 1533: 1501: 1431: 1411: 1384: 1357: 1337: 1310: 1290: 1270: 1081:Mathematical treatment 1071: 1024: 983: 947: 918: 882: 852: 784: 737: 701: 674: 615: 585: 552: 483: 431: 392: 391:{\displaystyle \hbar } 372: 343: 289: 238: 211: 160: 24: 7346:10.25365/thesis.16381 7037:Nature Nanotechnology 6235:Quantum optomechanics 5735:Nature Communications 5548:, Elsevier: 113–236, 5253: 5220: 5188: 5163:of the phonon number. 5150: 5023: 4951: 4750: 4704:or evanescently to a 4682:Levitated system: an 4675: 4628: 4594: 4529: 4488: 4458: 4417: 4376: 4349: 4329: 4294: 4266: 4232: 4210: 4029: 3993: 3790: 3760: 3731: 3702: 3675: 3646: 3375: 3282: 3254: 3166: 3131: 3043: 2997: 2976: 2974:{\displaystyle g_{0}} 2945: 2909: 2763: 2724: 2697: 2667: 2637: 2614: 2594: 2559: 2539: 2537:{\displaystyle g_{0}} 2498: 2468: 2400: 2377: 2330: 2303: 2237: 2148: 2105: 2066: 2011: 1965: 1800: 1769: 1717: 1686: 1666: 1646: 1626: 1606: 1554: 1534: 1502: 1432: 1412: 1385: 1358: 1338: 1311: 1291: 1271: 1072: 1025: 1004:On-resonance regime, 984: 948: 927:Blue-detuned regime, 919: 883: 853: 785: 738: 702: 682:optical spring effect 675: 616: 586: 553: 469: 454:Optical spring effect 432: 393: 373: 344: 290: 239: 212: 161: 22: 7330:10.1364/OE.15.017172 6232:(18 November 2015). 5634:10.1364/OE.15.017172 5373:Cavity Optomechanics 5229: 5200: 5171: 5032: 4960: 4862: 4841:acceleration sensors 4739: 4604: 4538: 4505: 4471: 4426: 4385: 4358: 4338: 4318: 4283: 4242: 4221: 4042: 4005: 3798: 3740: 3711: 3684: 3657: 3406: 3291: 3265: 3179: 3142: 3056: 3016: 2986: 2958: 2918: 2772: 2733: 2706: 2676: 2646: 2623: 2603: 2568: 2548: 2521: 2480: 2409: 2389: 2339: 2312: 2247: 2157: 2114: 2075: 2020: 1974: 1809: 1782: 1726: 1699: 1675: 1655: 1635: 1615: 1563: 1543: 1539:is now dependent on 1510: 1500:{\displaystyle ==1.} 1441: 1421: 1394: 1367: 1347: 1320: 1300: 1280: 1099: 1042: 1008: 991:quantum entanglement 957: 931: 892: 866: 862:Red-detuned regime, 816: 764: 711: 691: 625: 614:{\displaystyle F(x)} 596: 584:{\displaystyle F(x)} 566: 490: 406: 382: 353: 333: 252: 221: 174: 168:Brillouin scattering 139: 101:and technology like 28:Cavity optomechanics 6899:10.1038/nature10261 6853:10.1038/nature08524 6762:10.1038/nature10787 6689:2013PNAS..11014180K 6673:(35): 14180–14185. 6640:10.1038/nature06715 6576:2010RvMP...82.1155C 6390:1984PhRvA..30.1386C 6343:1985PhRvA..31.3761G 6302:2018RvMP...90d0501W 6195:2013NJPh...15c5007S 6085:10.1038/nature05244 6077:2006Natur.444...71A 6012:10.1038/nature10461 6004:2011Natur.478...89C 5939:10.1038/nature03118 5931:2004Natur.432.1002M 5925:(7020): 1002–1005. 5876:2013AnP...525..215M 5824:2004PhRvA..69e1801S 5766:10.1038/ncomms12311 5757:2016NatCo...712311Y 5692:2008PhRvB..78c5309M 5625:2007OExpr..1517172K 5609:(25): 17172–17205. 5499:2003PhRvL..91m0401M 5432:1997PhRvA..56.4175B 5144: 4684:optically levitated 3390:Equations of motion 560:harmonic oscillator 480:oscillator spectrum 349:carries a momentum 103:quantum information 95:Newtonian mechanics 69:) and matter using 44:solid-state physics 7153:342, 6159, 710-713 5854:Annalen der Physik 5542:Progress in Optics 5248: 5215: 5183: 5145: 5130: 5112: 5018: 4946: 4918: 4786:homodyne detection 4745: 4679: 4623: 4589: 4524: 4483: 4453: 4412: 4371: 4344: 4324: 4289: 4261: 4227: 4205: 4024: 3988: 3793: 3755: 3726: 3697: 3670: 3641: 3639: 3396:Langevin equations 3370: 3277: 3249: 3161: 3126: 3038: 3007:physical processes 2992: 2971: 2952:quadratic function 2940: 2904: 2758: 2719: 2692: 2662: 2632: 2609: 2589: 2554: 2534: 2493: 2463: 2395: 2372: 2325: 2298: 2232: 2204: 2143: 2100: 2061: 2006: 1960: 1795: 1764: 1712: 1681: 1661: 1641: 1621: 1601: 1549: 1529: 1497: 1427: 1407: 1380: 1353: 1333: 1306: 1286: 1266: 1067: 1020: 979: 943: 914: 878: 848: 780: 733: 697: 670: 611: 581: 548: 484: 427: 388: 368: 339: 324:radiation pressure 318:Radiation pressure 285: 234: 207: 156: 122:Physical processes 75:gravitational wave 25: 7129:10.1038/nphys2748 6922:Physical Review A 6883:(7356): 359–363. 6808:10.1038/nphys1425 6413:Physical Review A 6378:Physical Review A 6331:Physical Review A 6245:978-1-4822-5916-2 6230:Bowen, Warwick P. 6134:Physical Review A 5812:Physical Review A 5670:Physical Review B 5410:Physical Review A 5390:978-3-642-55311-0 5137: 5111: 5083: 5056: 5042: 4973: 4943: 4917: 4896: 4837:quantum computers 4829:photon shot noise 4801:Schrödinger's cat 4748:{\displaystyle C} 4327:{\displaystyle 1} 4306:(also called the 4252: 4198: 4136: 4052: 3981: 3904: 3779:Fourier transform 3694: 3667: 3634: 3625: 3531: 3515: 3506: 3425: 3301: 3189: 3066: 2782: 2738: 2716: 2681: 2490: 2461: 2454: 2435: 2398:{\displaystyle L} 2370: 2367: 2322: 2296: 2283: 2257: 2229: 2203: 2191: 2036: 2003: 1920: 1917: 1819: 1792: 1774:) and applying a 1624:{\displaystyle P} 1599: 1598: 1552:{\displaystyle x} 1430:{\displaystyle E} 1356:{\displaystyle x} 1330: 1309:{\displaystyle b} 1289:{\displaystyle a} 1125: 1109: 845: 665: 515: 506: 467: 342:{\displaystyle k} 107:Bell inequalities 83:Schrödinger's cat 48:materials science 16:Branch of physics 7368: 7341: 7290: 7289: 7271: 7251: 7245: 7244: 7226: 7206: 7200: 7199: 7181: 7160: 7154: 7147: 7141: 7140: 7112: 7106: 7105: 7087: 7072:Nature Photonics 7067: 7061: 7060: 7032: 7026: 7020: 7014: 7013: 6995: 6993:quant-ph/0101104 6975: 6969: 6962: 6956: 6955: 6937: 6935:quant-ph/9712017 6928:(5): 3204–3210. 6917: 6911: 6910: 6892: 6871: 6865: 6864: 6846: 6826: 6820: 6819: 6801: 6780: 6774: 6773: 6755: 6735: 6729: 6728: 6718: 6700: 6682: 6658: 6652: 6651: 6633: 6613: 6604: 6603: 6569: 6560:(2): 1155–1208. 6545: 6539: 6538: 6520: 6499: 6493: 6492: 6474: 6472:quant-ph/0612188 6453: 6447: 6446: 6428: 6408: 6402: 6401: 6384:(3): 1386–1391. 6369: 6363: 6362: 6337:(6): 3761–3774. 6322: 6316: 6315: 6313: 6281: 6272: 6271: 6265: 6257: 6226: 6217: 6216: 6206: 6188: 6164: 6158: 6157: 6129: 6123: 6119: 6113: 6112: 6070: 6068:quant-ph/0607205 6046: 6040: 6039: 5997: 5973: 5967: 5966: 5910: 5904: 5903: 5869: 5842: 5836: 5835: 5803: 5797: 5796: 5786: 5768: 5750: 5726: 5720: 5719: 5685: 5661: 5655: 5654: 5636: 5618: 5594: 5588: 5587: 5585: 5584: 5575:. Archived from 5569: 5563: 5562: 5561: 5560: 5533: 5527: 5526: 5492: 5490:quant-ph/0210001 5468: 5462: 5461: 5451: 5425: 5423:quant-ph/9708002 5416:(5): 4175–4186. 5401: 5395: 5394: 5368: 5335:Coherent control 5314: 5309: 5308: 5276:two-level system 5257: 5255: 5254: 5249: 5247: 5246: 5224: 5222: 5221: 5216: 5192: 5190: 5189: 5184: 5154: 5152: 5151: 5146: 5143: 5138: 5135: 5129: 5128: 5117: 5113: 5110: 5109: 5108: 5095: 5085: 5084: 5081: 5075: 5074: 5064: 5057: 5049: 5044: 5043: 5040: 5027: 5025: 5024: 5019: 5017: 5016: 5007: 5006: 4985: 4984: 4975: 4974: 4971: 4955: 4953: 4952: 4947: 4945: 4944: 4941: 4935: 4934: 4923: 4919: 4916: 4908: 4898: 4897: 4894: 4884: 4874: 4873: 4809:Wigner functions 4754: 4752: 4751: 4746: 4733:capacitor plates 4721:phononic crystal 4717:photonic crystal 4632: 4630: 4629: 4624: 4616: 4615: 4598: 4596: 4595: 4590: 4585: 4584: 4563: 4562: 4553: 4552: 4533: 4531: 4530: 4525: 4517: 4516: 4492: 4490: 4489: 4484: 4462: 4460: 4459: 4454: 4443: 4438: 4437: 4421: 4419: 4418: 4413: 4402: 4397: 4396: 4380: 4378: 4377: 4372: 4370: 4369: 4353: 4351: 4350: 4345: 4333: 4331: 4330: 4325: 4298: 4296: 4295: 4290: 4270: 4268: 4267: 4262: 4254: 4253: 4250: 4236: 4234: 4233: 4228: 4214: 4212: 4211: 4206: 4204: 4200: 4199: 4197: 4196: 4195: 4186: 4185: 4161: 4156: 4155: 4142: 4137: 4135: 4134: 4133: 4124: 4123: 4099: 4094: 4093: 4080: 4073: 4072: 4054: 4053: 4050: 4033: 4031: 4030: 4025: 4017: 4016: 3997: 3995: 3994: 3989: 3987: 3983: 3982: 3980: 3979: 3978: 3969: 3968: 3944: 3939: 3938: 3928: 3927: 3926: 3910: 3905: 3903: 3902: 3901: 3892: 3891: 3867: 3862: 3861: 3851: 3850: 3849: 3833: 3826: 3825: 3813: 3812: 3764: 3762: 3761: 3756: 3735: 3733: 3732: 3727: 3706: 3704: 3703: 3698: 3696: 3695: 3692: 3679: 3677: 3676: 3671: 3669: 3668: 3665: 3650: 3648: 3647: 3642: 3640: 3636: 3635: 3632: 3626: 3621: 3613: 3612: 3570: 3559: 3558: 3533: 3532: 3524: 3517: 3516: 3513: 3507: 3502: 3494: 3493: 3454: 3427: 3426: 3418: 3379: 3377: 3376: 3371: 3366: 3365: 3344: 3343: 3319: 3318: 3303: 3302: 3299: 3286: 3284: 3283: 3278: 3258: 3256: 3255: 3250: 3245: 3244: 3235: 3234: 3207: 3206: 3191: 3190: 3187: 3170: 3168: 3167: 3162: 3160: 3159: 3135: 3133: 3132: 3127: 3122: 3121: 3100: 3099: 3084: 3083: 3068: 3067: 3064: 3047: 3045: 3044: 3039: 3037: 3036: 3001: 2999: 2998: 2993: 2980: 2978: 2977: 2972: 2970: 2969: 2949: 2947: 2946: 2941: 2936: 2935: 2913: 2911: 2910: 2905: 2900: 2899: 2878: 2877: 2841: 2840: 2831: 2830: 2809: 2808: 2784: 2783: 2780: 2767: 2765: 2764: 2759: 2751: 2750: 2736: 2728: 2726: 2725: 2720: 2718: 2717: 2714: 2701: 2699: 2698: 2693: 2691: 2690: 2679: 2671: 2669: 2668: 2663: 2658: 2657: 2641: 2639: 2638: 2633: 2618: 2616: 2615: 2610: 2598: 2596: 2595: 2590: 2563: 2561: 2560: 2555: 2543: 2541: 2540: 2535: 2533: 2532: 2502: 2500: 2499: 2494: 2492: 2491: 2488: 2472: 2470: 2469: 2464: 2462: 2457: 2456: 2455: 2452: 2437: 2436: 2433: 2426: 2421: 2420: 2404: 2402: 2401: 2396: 2381: 2379: 2378: 2373: 2371: 2369: 2368: 2365: 2359: 2358: 2349: 2334: 2332: 2331: 2326: 2324: 2323: 2320: 2307: 2305: 2304: 2299: 2297: 2295: 2294: 2285: 2284: 2281: 2272: 2264: 2259: 2258: 2255: 2241: 2239: 2238: 2233: 2231: 2230: 2227: 2221: 2220: 2209: 2205: 2202: 2194: 2193: 2192: 2189: 2179: 2169: 2168: 2152: 2150: 2149: 2144: 2139: 2138: 2129: 2128: 2109: 2107: 2106: 2101: 2096: 2095: 2070: 2068: 2067: 2062: 2057: 2056: 2038: 2037: 2034: 2015: 2013: 2012: 2007: 2005: 2004: 2001: 1992: 1991: 1969: 1967: 1966: 1961: 1959: 1955: 1954: 1953: 1921: 1919: 1918: 1915: 1906: 1901: 1900: 1891: 1890: 1872: 1871: 1862: 1861: 1843: 1842: 1821: 1820: 1817: 1804: 1802: 1801: 1796: 1794: 1793: 1790: 1776:Taylor expansion 1773: 1771: 1770: 1765: 1763: 1762: 1758: 1757: 1721: 1719: 1718: 1713: 1711: 1710: 1690: 1688: 1687: 1682: 1670: 1668: 1667: 1662: 1650: 1648: 1647: 1642: 1630: 1628: 1627: 1622: 1610: 1608: 1607: 1602: 1600: 1597: 1596: 1595: 1582: 1574: 1573: 1558: 1556: 1555: 1550: 1538: 1536: 1535: 1530: 1528: 1527: 1506: 1504: 1503: 1498: 1487: 1486: 1462: 1461: 1436: 1434: 1433: 1428: 1416: 1414: 1413: 1408: 1406: 1405: 1389: 1387: 1386: 1381: 1379: 1378: 1362: 1360: 1359: 1354: 1342: 1340: 1339: 1334: 1332: 1331: 1328: 1315: 1313: 1312: 1307: 1295: 1293: 1292: 1287: 1275: 1273: 1272: 1267: 1265: 1261: 1260: 1259: 1255: 1254: 1234: 1233: 1221: 1220: 1216: 1215: 1175: 1174: 1165: 1164: 1146: 1145: 1127: 1126: 1123: 1111: 1110: 1107: 1076: 1074: 1073: 1068: 1060: 1059: 1029: 1027: 1026: 1021: 988: 986: 985: 980: 978: 977: 952: 950: 949: 944: 923: 921: 920: 915: 913: 912: 887: 885: 884: 879: 857: 855: 854: 849: 847: 846: 843: 834: 833: 789: 787: 786: 781: 779: 778: 742: 740: 739: 734: 706: 704: 703: 698: 679: 677: 676: 671: 666: 664: 656: 648: 643: 642: 620: 618: 617: 612: 590: 588: 587: 582: 557: 555: 554: 549: 517: 516: 513: 507: 505: 494: 472:protein molecule 468: 436: 434: 433: 428: 397: 395: 394: 389: 377: 375: 374: 369: 348: 346: 345: 340: 306:optical lattices 302:optical tweezers 294: 292: 291: 286: 281: 280: 262: 243: 241: 240: 235: 233: 232: 216: 214: 213: 208: 203: 202: 184: 165: 163: 162: 157: 149: 7376: 7375: 7371: 7370: 7369: 7367: 7366: 7365: 7351: 7350: 7299: 7297:Further reading 7294: 7293: 7252: 7248: 7207: 7203: 7161: 7157: 7148: 7144: 7123:(11): 712–716. 7113: 7109: 7078:(11): 768–772. 7068: 7064: 7043:(11): 690–691. 7033: 7029: 7021: 7017: 6976: 6972: 6963: 6959: 6918: 6914: 6872: 6868: 6837:(7269): 78–82. 6827: 6823: 6781: 6777: 6746:(7383): 63–67. 6736: 6732: 6659: 6655: 6624:(7183): 72–75. 6614: 6607: 6546: 6542: 6500: 6496: 6454: 6450: 6409: 6405: 6370: 6366: 6323: 6319: 6282: 6275: 6259: 6258: 6246: 6227: 6220: 6165: 6161: 6130: 6126: 6120: 6116: 6061:(7115): 71–74. 6047: 6043: 5988:(7367): 89–92. 5974: 5970: 5911: 5907: 5846:Meystre, Pierre 5843: 5839: 5804: 5800: 5727: 5723: 5662: 5658: 5595: 5591: 5582: 5580: 5571: 5570: 5566: 5558: 5556: 5534: 5530: 5469: 5465: 5402: 5398: 5391: 5369: 5348: 5343: 5310: 5303: 5300: 5272:ultracold atoms 5236: 5232: 5230: 5227: 5226: 5201: 5198: 5197: 5172: 5169: 5168: 5139: 5134: 5118: 5104: 5100: 5096: 5080: 5076: 5070: 5066: 5065: 5062: 5059: 5058: 5048: 5039: 5035: 5033: 5030: 5029: 5012: 5008: 5002: 4998: 4980: 4976: 4970: 4966: 4961: 4958: 4957: 4940: 4936: 4924: 4909: 4893: 4889: 4885: 4882: 4879: 4878: 4869: 4865: 4863: 4860: 4859: 4849: 4821: 4797: 4782: 4740: 4737: 4736: 4644: 4639: 4611: 4607: 4605: 4602: 4601: 4580: 4576: 4558: 4554: 4548: 4544: 4539: 4536: 4535: 4512: 4508: 4506: 4503: 4502: 4472: 4469: 4468: 4439: 4433: 4429: 4427: 4424: 4423: 4398: 4392: 4388: 4386: 4383: 4382: 4365: 4361: 4359: 4356: 4355: 4339: 4336: 4335: 4319: 4316: 4315: 4284: 4281: 4280: 4277: 4249: 4245: 4243: 4240: 4239: 4222: 4219: 4218: 4191: 4187: 4181: 4177: 4157: 4151: 4147: 4146: 4141: 4129: 4125: 4119: 4115: 4095: 4089: 4085: 4084: 4079: 4078: 4074: 4068: 4064: 4049: 4045: 4043: 4040: 4039: 4012: 4008: 4006: 4003: 4002: 3974: 3970: 3964: 3960: 3940: 3934: 3930: 3929: 3922: 3918: 3911: 3909: 3897: 3893: 3887: 3883: 3863: 3857: 3853: 3852: 3845: 3841: 3834: 3832: 3831: 3827: 3821: 3817: 3808: 3804: 3799: 3796: 3795: 3775:frequency space 3741: 3738: 3737: 3712: 3709: 3708: 3691: 3687: 3685: 3682: 3681: 3664: 3660: 3658: 3655: 3654: 3638: 3637: 3631: 3627: 3620: 3608: 3604: 3566: 3554: 3550: 3534: 3523: 3522: 3519: 3518: 3512: 3508: 3501: 3489: 3485: 3450: 3428: 3417: 3416: 3409: 3407: 3404: 3403: 3392: 3361: 3357: 3339: 3335: 3314: 3310: 3298: 3294: 3292: 3289: 3288: 3266: 3263: 3262: 3240: 3236: 3230: 3226: 3202: 3198: 3186: 3182: 3180: 3177: 3176: 3155: 3151: 3143: 3140: 3139: 3117: 3113: 3095: 3091: 3079: 3075: 3063: 3059: 3057: 3054: 3053: 3032: 3028: 3017: 3014: 3013: 2987: 2984: 2983: 2965: 2961: 2959: 2956: 2955: 2931: 2927: 2919: 2916: 2915: 2895: 2891: 2873: 2869: 2836: 2832: 2826: 2822: 2804: 2800: 2779: 2775: 2773: 2770: 2769: 2746: 2742: 2734: 2731: 2730: 2713: 2709: 2707: 2704: 2703: 2686: 2682: 2677: 2674: 2673: 2653: 2649: 2647: 2644: 2643: 2624: 2621: 2620: 2604: 2601: 2600: 2569: 2566: 2565: 2549: 2546: 2545: 2528: 2524: 2522: 2519: 2518: 2515: 2487: 2483: 2481: 2478: 2477: 2451: 2447: 2432: 2428: 2427: 2425: 2416: 2412: 2410: 2407: 2406: 2390: 2387: 2386: 2364: 2360: 2354: 2350: 2348: 2340: 2337: 2336: 2319: 2315: 2313: 2310: 2309: 2290: 2286: 2280: 2276: 2268: 2263: 2254: 2250: 2248: 2245: 2244: 2226: 2222: 2210: 2195: 2188: 2184: 2180: 2177: 2174: 2173: 2164: 2160: 2158: 2155: 2154: 2134: 2130: 2124: 2120: 2115: 2112: 2111: 2091: 2087: 2076: 2073: 2072: 2052: 2048: 2033: 2029: 2021: 2018: 2017: 2000: 1996: 1987: 1983: 1975: 1972: 1971: 1949: 1945: 1938: 1934: 1914: 1910: 1905: 1896: 1892: 1886: 1882: 1867: 1863: 1857: 1853: 1838: 1834: 1816: 1812: 1810: 1807: 1806: 1789: 1785: 1783: 1780: 1779: 1753: 1749: 1742: 1738: 1727: 1724: 1723: 1706: 1702: 1700: 1697: 1696: 1676: 1673: 1672: 1656: 1653: 1652: 1636: 1633: 1632: 1616: 1613: 1612: 1591: 1587: 1583: 1575: 1572: 1564: 1561: 1560: 1544: 1541: 1540: 1517: 1513: 1511: 1508: 1507: 1482: 1478: 1457: 1453: 1442: 1439: 1438: 1422: 1419: 1418: 1401: 1397: 1395: 1392: 1391: 1374: 1370: 1368: 1365: 1364: 1348: 1345: 1344: 1327: 1323: 1321: 1318: 1317: 1301: 1298: 1297: 1281: 1278: 1277: 1250: 1246: 1239: 1235: 1229: 1225: 1211: 1207: 1203: 1199: 1195: 1191: 1170: 1166: 1160: 1156: 1141: 1137: 1122: 1118: 1106: 1102: 1100: 1097: 1096: 1088: 1083: 1055: 1051: 1043: 1040: 1039: 1009: 1006: 1005: 973: 969: 958: 955: 954: 932: 929: 928: 908: 904: 893: 890: 889: 867: 864: 863: 842: 838: 829: 825: 817: 814: 813: 810: 805: 774: 770: 765: 762: 761: 753: 712: 709: 708: 692: 689: 688: 657: 649: 647: 638: 634: 626: 623: 622: 597: 594: 593: 567: 564: 563: 512: 508: 498: 493: 491: 488: 487: 458: 456: 407: 404: 403: 400:Planck constant 383: 380: 379: 354: 351: 350: 334: 331: 330: 320: 314: 276: 272: 255: 253: 250: 249: 228: 224: 222: 219: 218: 198: 194: 177: 175: 172: 171: 142: 140: 137: 136: 129: 124: 119: 65:between light ( 30:is a branch of 17: 12: 11: 5: 7374: 7364: 7363: 7361:Quantum optics 7349: 7348: 7342: 7318:Optics Express 7313: 7306: 7302:Daniel Steck, 7298: 7295: 7292: 7291: 7262:(16): 163902. 7246: 7217:(20): 207209. 7201: 7155: 7142: 7117:Nature Physics 7107: 7062: 7027: 7015: 6986:(4): 301–306. 6970: 6957: 6912: 6866: 6821: 6786:Nature Physics 6775: 6730: 6653: 6605: 6540: 6511:(16): 160801. 6494: 6465:(15): 150802. 6448: 6403: 6364: 6317: 6273: 6244: 6218: 6159: 6140:(1): 433–437. 6124: 6114: 6041: 5968: 5905: 5860:(3): 215–233. 5837: 5798: 5721: 5656: 5603:Optics Express 5589: 5564: 5528: 5483:(13): 130401. 5463: 5396: 5389: 5345: 5344: 5342: 5339: 5338: 5337: 5332: 5327: 5325:Optical cavity 5322: 5316: 5315: 5312:Physics portal 5299: 5296: 5280: 5279: 5268: 5260: 5259: 5245: 5242: 5239: 5235: 5214: 5211: 5208: 5205: 5194: 5182: 5179: 5176: 5164: 5142: 5133: 5127: 5124: 5121: 5116: 5107: 5103: 5099: 5094: 5091: 5088: 5079: 5073: 5069: 5061: 5055: 5052: 5047: 5038: 5015: 5011: 5005: 5001: 4997: 4994: 4991: 4988: 4983: 4979: 4969: 4965: 4939: 4933: 4930: 4927: 4922: 4915: 4912: 4907: 4904: 4901: 4892: 4888: 4881: 4877: 4872: 4868: 4856: 4848: 4845: 4820: 4817: 4796: 4793: 4781: 4778: 4773: 4772: 4744: 4725: 4709: 4691: 4680: 4663: 4643: 4640: 4638: 4635: 4622: 4619: 4614: 4610: 4588: 4583: 4579: 4575: 4572: 4569: 4566: 4561: 4557: 4551: 4547: 4543: 4523: 4520: 4515: 4511: 4482: 4479: 4476: 4452: 4449: 4446: 4442: 4436: 4432: 4411: 4408: 4405: 4401: 4395: 4391: 4368: 4364: 4343: 4323: 4288: 4276: 4273: 4260: 4257: 4248: 4226: 4203: 4194: 4190: 4184: 4180: 4176: 4173: 4170: 4167: 4164: 4160: 4154: 4150: 4145: 4140: 4132: 4128: 4122: 4118: 4114: 4111: 4108: 4105: 4102: 4098: 4092: 4088: 4083: 4077: 4071: 4067: 4063: 4060: 4057: 4048: 4023: 4020: 4015: 4011: 3986: 3977: 3973: 3967: 3963: 3959: 3956: 3953: 3950: 3947: 3943: 3937: 3933: 3925: 3921: 3917: 3914: 3908: 3900: 3896: 3890: 3886: 3882: 3879: 3876: 3873: 3870: 3866: 3860: 3856: 3848: 3844: 3840: 3837: 3830: 3824: 3820: 3816: 3811: 3807: 3803: 3754: 3751: 3748: 3745: 3725: 3722: 3719: 3716: 3690: 3663: 3630: 3624: 3619: 3616: 3611: 3607: 3603: 3600: 3597: 3594: 3591: 3588: 3585: 3582: 3579: 3576: 3573: 3569: 3565: 3562: 3557: 3553: 3549: 3546: 3543: 3540: 3537: 3535: 3530: 3527: 3521: 3520: 3511: 3505: 3500: 3497: 3492: 3488: 3484: 3481: 3478: 3475: 3472: 3469: 3466: 3463: 3460: 3457: 3453: 3449: 3446: 3443: 3440: 3437: 3434: 3431: 3429: 3424: 3421: 3415: 3412: 3411: 3391: 3388: 3387: 3386: 3369: 3364: 3360: 3356: 3353: 3350: 3347: 3342: 3338: 3334: 3331: 3328: 3325: 3322: 3317: 3313: 3309: 3306: 3297: 3276: 3273: 3270: 3260: 3248: 3243: 3239: 3233: 3229: 3225: 3222: 3219: 3216: 3213: 3210: 3205: 3201: 3197: 3194: 3185: 3158: 3154: 3150: 3147: 3137: 3125: 3120: 3116: 3112: 3109: 3106: 3103: 3098: 3094: 3090: 3087: 3082: 3078: 3074: 3071: 3062: 3035: 3031: 3027: 3024: 3021: 2991: 2968: 2964: 2939: 2934: 2930: 2926: 2923: 2903: 2898: 2894: 2890: 2887: 2884: 2881: 2876: 2872: 2868: 2865: 2862: 2859: 2856: 2853: 2850: 2847: 2844: 2839: 2835: 2829: 2825: 2821: 2818: 2815: 2812: 2807: 2803: 2799: 2796: 2793: 2790: 2787: 2778: 2757: 2754: 2749: 2745: 2741: 2712: 2689: 2685: 2661: 2656: 2652: 2631: 2628: 2608: 2588: 2585: 2582: 2579: 2576: 2573: 2553: 2531: 2527: 2514: 2511: 2486: 2460: 2450: 2446: 2443: 2440: 2431: 2424: 2419: 2415: 2394: 2363: 2357: 2353: 2347: 2344: 2318: 2293: 2289: 2279: 2275: 2271: 2267: 2262: 2253: 2225: 2219: 2216: 2213: 2208: 2201: 2198: 2187: 2183: 2176: 2172: 2167: 2163: 2142: 2137: 2133: 2127: 2123: 2119: 2099: 2094: 2090: 2086: 2083: 2080: 2060: 2055: 2051: 2047: 2044: 2041: 2032: 2028: 2025: 1999: 1995: 1990: 1986: 1982: 1979: 1958: 1952: 1948: 1944: 1941: 1937: 1933: 1930: 1927: 1924: 1913: 1909: 1904: 1899: 1895: 1889: 1885: 1881: 1878: 1875: 1870: 1866: 1860: 1856: 1852: 1849: 1846: 1841: 1837: 1833: 1830: 1827: 1824: 1815: 1788: 1761: 1756: 1752: 1748: 1745: 1741: 1737: 1734: 1731: 1709: 1705: 1680: 1660: 1640: 1620: 1594: 1590: 1586: 1581: 1578: 1571: 1568: 1548: 1526: 1523: 1520: 1516: 1496: 1493: 1490: 1485: 1481: 1477: 1474: 1471: 1468: 1465: 1460: 1456: 1452: 1449: 1446: 1426: 1404: 1400: 1377: 1373: 1352: 1326: 1305: 1285: 1264: 1258: 1253: 1249: 1245: 1242: 1238: 1232: 1228: 1224: 1219: 1214: 1210: 1206: 1202: 1198: 1194: 1190: 1187: 1184: 1181: 1178: 1173: 1169: 1163: 1159: 1155: 1152: 1149: 1144: 1140: 1136: 1133: 1130: 1121: 1117: 1114: 1105: 1087: 1084: 1082: 1079: 1066: 1063: 1058: 1054: 1050: 1047: 1036: 1035: 1032:interferometer 1019: 1016: 1013: 1002: 976: 972: 968: 965: 962: 942: 939: 936: 925: 911: 907: 903: 900: 897: 877: 874: 871: 841: 837: 832: 828: 824: 821: 809: 806: 777: 773: 769: 752: 749: 732: 729: 726: 723: 719: 716: 696: 669: 663: 660: 655: 652: 646: 641: 637: 633: 630: 610: 607: 604: 601: 580: 577: 574: 571: 547: 544: 541: 538: 535: 532: 529: 526: 523: 520: 511: 504: 501: 497: 455: 452: 426: 423: 420: 417: 414: 411: 387: 367: 364: 361: 358: 338: 316:Main article: 313: 310: 297:optical cavity 284: 279: 275: 271: 268: 265: 261: 258: 231: 227: 206: 201: 197: 193: 190: 187: 183: 180: 155: 152: 148: 145: 128: 125: 123: 120: 118: 115: 52:quantum theory 40:quantum optics 15: 9: 6: 4: 3: 2: 7373: 7362: 7359: 7358: 7356: 7347: 7343: 7339: 7335: 7331: 7327: 7324:(25): 17172. 7323: 7319: 7314: 7311: 7307: 7305: 7301: 7300: 7287: 7283: 7279: 7275: 7270: 7265: 7261: 7257: 7250: 7242: 7238: 7234: 7230: 7225: 7220: 7216: 7212: 7205: 7197: 7193: 7189: 7185: 7180: 7175: 7172:(2): 023604. 7171: 7167: 7159: 7152: 7146: 7138: 7134: 7130: 7126: 7122: 7118: 7111: 7103: 7099: 7095: 7091: 7086: 7081: 7077: 7073: 7066: 7058: 7054: 7050: 7046: 7042: 7038: 7031: 7025: 7019: 7011: 7007: 7003: 6999: 6994: 6989: 6985: 6981: 6974: 6968: 6961: 6953: 6949: 6945: 6941: 6936: 6931: 6927: 6923: 6916: 6908: 6904: 6900: 6896: 6891: 6886: 6882: 6878: 6870: 6862: 6858: 6854: 6850: 6845: 6840: 6836: 6832: 6825: 6817: 6813: 6809: 6805: 6800: 6795: 6791: 6787: 6779: 6771: 6767: 6763: 6759: 6754: 6749: 6745: 6741: 6734: 6726: 6722: 6717: 6712: 6708: 6704: 6699: 6694: 6690: 6686: 6681: 6676: 6672: 6668: 6664: 6657: 6649: 6645: 6641: 6637: 6632: 6627: 6623: 6619: 6612: 6610: 6601: 6597: 6593: 6589: 6585: 6581: 6577: 6573: 6568: 6563: 6559: 6555: 6551: 6544: 6536: 6532: 6528: 6524: 6519: 6514: 6510: 6506: 6498: 6490: 6486: 6482: 6478: 6473: 6468: 6464: 6460: 6452: 6444: 6440: 6436: 6432: 6427: 6422: 6419:(3): 033804. 6418: 6414: 6407: 6399: 6395: 6391: 6387: 6383: 6379: 6375: 6368: 6360: 6356: 6352: 6348: 6344: 6340: 6336: 6332: 6328: 6321: 6312: 6307: 6303: 6299: 6296:(4): 040501. 6295: 6291: 6287: 6280: 6278: 6269: 6263: 6255: 6251: 6247: 6241: 6237: 6236: 6231: 6225: 6223: 6214: 6210: 6205: 6200: 6196: 6192: 6187: 6182: 6179:(3): 035007. 6178: 6174: 6170: 6163: 6155: 6151: 6147: 6143: 6139: 6135: 6128: 6118: 6110: 6106: 6102: 6098: 6094: 6090: 6086: 6082: 6078: 6074: 6069: 6064: 6060: 6056: 6052: 6045: 6037: 6033: 6029: 6025: 6021: 6017: 6013: 6009: 6005: 6001: 5996: 5991: 5987: 5983: 5979: 5972: 5964: 5960: 5956: 5952: 5948: 5944: 5940: 5936: 5932: 5928: 5924: 5920: 5916: 5909: 5901: 5897: 5893: 5889: 5885: 5881: 5877: 5873: 5868: 5863: 5859: 5855: 5851: 5847: 5841: 5833: 5829: 5825: 5821: 5818:(5): 051801. 5817: 5813: 5809: 5802: 5794: 5790: 5785: 5780: 5776: 5772: 5767: 5762: 5758: 5754: 5749: 5744: 5740: 5736: 5732: 5725: 5717: 5713: 5709: 5705: 5701: 5697: 5693: 5689: 5684: 5679: 5676:(3): 035309. 5675: 5671: 5667: 5660: 5652: 5648: 5644: 5640: 5635: 5630: 5626: 5622: 5617: 5612: 5608: 5604: 5600: 5593: 5579:on 2011-10-21 5578: 5574: 5568: 5555: 5551: 5547: 5543: 5539: 5532: 5524: 5520: 5516: 5512: 5508: 5504: 5500: 5496: 5491: 5486: 5482: 5478: 5474: 5467: 5459: 5455: 5450: 5445: 5441: 5437: 5433: 5429: 5424: 5419: 5415: 5411: 5407: 5400: 5392: 5386: 5382: 5378: 5374: 5367: 5365: 5363: 5361: 5359: 5357: 5355: 5353: 5351: 5346: 5336: 5333: 5331: 5330:Laser cooling 5328: 5326: 5323: 5321: 5318: 5317: 5313: 5307: 5302: 5295: 5293: 5289: 5285: 5277: 5273: 5269: 5265: 5264: 5263: 5243: 5240: 5237: 5233: 5209: 5203: 5195: 5177: 5174: 5165: 5162: 5158: 5140: 5131: 5125: 5122: 5119: 5114: 5105: 5101: 5097: 5089: 5077: 5071: 5067: 5053: 5050: 5045: 5036: 5013: 5003: 4999: 4995: 4992: 4986: 4981: 4977: 4967: 4963: 4937: 4931: 4928: 4925: 4920: 4913: 4910: 4902: 4890: 4886: 4875: 4870: 4866: 4857: 4854: 4853: 4852: 4844: 4842: 4838: 4834: 4830: 4826: 4816: 4814: 4810: 4805: 4802: 4792: 4789: 4787: 4777: 4770: 4766: 4765:optical light 4762: 4758: 4742: 4734: 4730: 4726: 4722: 4718: 4714: 4713:metamaterials 4710: 4707: 4703: 4699: 4695: 4692: 4688: 4685: 4681: 4674: 4668: 4664: 4661: 4657: 4656: 4655: 4652: 4650: 4634: 4620: 4617: 4612: 4608: 4581: 4577: 4573: 4570: 4564: 4559: 4555: 4549: 4545: 4541: 4521: 4518: 4513: 4509: 4500: 4496: 4480: 4477: 4474: 4465: 4450: 4447: 4444: 4440: 4434: 4430: 4409: 4406: 4403: 4399: 4393: 4389: 4366: 4362: 4341: 4321: 4313: 4309: 4305: 4300: 4272: 4258: 4255: 4215: 4201: 4192: 4182: 4178: 4174: 4165: 4162: 4158: 4152: 4148: 4143: 4138: 4130: 4120: 4116: 4112: 4103: 4100: 4096: 4090: 4086: 4081: 4075: 4069: 4065: 4061: 4055: 4037: 4021: 4018: 4013: 4009: 3998: 3984: 3975: 3965: 3961: 3957: 3948: 3945: 3941: 3935: 3931: 3923: 3919: 3915: 3906: 3898: 3888: 3884: 3880: 3871: 3868: 3864: 3858: 3854: 3846: 3842: 3838: 3828: 3822: 3818: 3814: 3809: 3805: 3801: 3789: 3785: 3782: 3781:is applied). 3780: 3776: 3772: 3767: 3752: 3749: 3743: 3723: 3720: 3717: 3714: 3688: 3661: 3651: 3628: 3617: 3609: 3605: 3601: 3598: 3595: 3592: 3586: 3583: 3580: 3577: 3571: 3567: 3560: 3555: 3551: 3547: 3541: 3538: 3536: 3528: 3525: 3509: 3503: 3498: 3490: 3486: 3482: 3479: 3473: 3470: 3467: 3464: 3461: 3455: 3451: 3447: 3444: 3438: 3432: 3430: 3422: 3419: 3413: 3401: 3397: 3384: 3362: 3358: 3354: 3351: 3340: 3336: 3332: 3329: 3326: 3323: 3315: 3311: 3304: 3295: 3274: 3271: 3261: 3241: 3237: 3231: 3227: 3223: 3220: 3217: 3214: 3211: 3203: 3199: 3192: 3183: 3174: 3156: 3152: 3148: 3138: 3118: 3114: 3110: 3107: 3104: 3101: 3096: 3092: 3088: 3080: 3076: 3069: 3060: 3051: 3033: 3029: 3025: 3022: 3012: 3011: 3010: 3008: 3003: 2989: 2966: 2962: 2953: 2937: 2932: 2928: 2924: 2921: 2896: 2892: 2888: 2885: 2874: 2870: 2866: 2863: 2860: 2857: 2851: 2845: 2842: 2837: 2833: 2827: 2823: 2816: 2813: 2810: 2805: 2801: 2797: 2788: 2785: 2776: 2755: 2752: 2747: 2743: 2739: 2710: 2687: 2683: 2659: 2654: 2650: 2629: 2626: 2606: 2586: 2583: 2580: 2577: 2574: 2571: 2551: 2529: 2525: 2513:Linearization 2510: 2507: 2484: 2474: 2458: 2448: 2441: 2429: 2422: 2417: 2413: 2392: 2383: 2361: 2355: 2351: 2345: 2342: 2316: 2291: 2287: 2277: 2273: 2269: 2265: 2260: 2251: 2223: 2217: 2214: 2211: 2206: 2199: 2196: 2185: 2181: 2170: 2165: 2161: 2140: 2135: 2131: 2125: 2121: 2117: 2097: 2092: 2088: 2078: 2053: 2049: 2045: 2042: 2030: 2026: 2023: 1997: 1993: 1988: 1984: 1980: 1956: 1950: 1946: 1942: 1939: 1935: 1931: 1925: 1922: 1911: 1907: 1902: 1897: 1893: 1887: 1883: 1876: 1873: 1868: 1864: 1858: 1854: 1847: 1844: 1839: 1835: 1825: 1822: 1813: 1786: 1777: 1759: 1754: 1750: 1746: 1743: 1739: 1735: 1729: 1707: 1703: 1692: 1658: 1638: 1618: 1592: 1588: 1584: 1579: 1576: 1569: 1566: 1546: 1524: 1521: 1518: 1514: 1494: 1491: 1483: 1479: 1475: 1472: 1466: 1458: 1454: 1450: 1447: 1424: 1402: 1398: 1375: 1371: 1350: 1324: 1303: 1283: 1262: 1256: 1251: 1247: 1243: 1240: 1236: 1230: 1226: 1222: 1217: 1212: 1208: 1204: 1200: 1196: 1192: 1188: 1182: 1179: 1176: 1171: 1167: 1161: 1157: 1150: 1147: 1142: 1138: 1131: 1119: 1112: 1103: 1094: 1078: 1064: 1061: 1056: 1052: 1048: 1033: 1017: 1014: 1003: 1000: 996: 992: 974: 970: 966: 963: 940: 937: 926: 909: 905: 901: 898: 875: 872: 861: 860: 859: 839: 835: 830: 826: 822: 804: 800: 795: 792: 775: 771: 767: 759: 748: 746: 730: 727: 724: 721: 717: 714: 694: 685: 683: 667: 661: 658: 653: 650: 644: 639: 635: 631: 628: 605: 599: 575: 569: 561: 545: 539: 533: 530: 527: 521: 509: 502: 499: 495: 481: 477: 473: 451: 449: 444: 440: 424: 418: 415: 412: 401: 385: 365: 359: 356: 336: 329: 325: 319: 309: 307: 303: 298: 282: 277: 273: 269: 266: 263: 259: 256: 247: 229: 225: 204: 199: 195: 191: 188: 185: 181: 178: 169: 153: 150: 146: 143: 134: 114: 112: 108: 104: 98: 96: 92: 88: 87:superposition 84: 80: 76: 72: 68: 64: 59: 57: 53: 49: 45: 41: 37: 33: 29: 21: 7321: 7317: 7259: 7255: 7249: 7214: 7210: 7204: 7169: 7165: 7158: 7150: 7145: 7120: 7116: 7110: 7075: 7071: 7065: 7040: 7036: 7030: 7018: 6983: 6979: 6973: 6960: 6925: 6921: 6915: 6880: 6876: 6869: 6834: 6830: 6824: 6789: 6785: 6778: 6743: 6739: 6733: 6670: 6666: 6656: 6621: 6617: 6557: 6553: 6543: 6508: 6504: 6497: 6462: 6458: 6451: 6416: 6412: 6406: 6381: 6377: 6367: 6334: 6330: 6320: 6293: 6289: 6234: 6176: 6172: 6162: 6137: 6133: 6127: 6117: 6058: 6054: 6044: 5985: 5981: 5971: 5922: 5918: 5908: 5857: 5853: 5840: 5815: 5811: 5801: 5741:(1): 12311. 5738: 5734: 5724: 5673: 5669: 5659: 5606: 5602: 5592: 5581:. Retrieved 5577:the original 5567: 5557:, retrieved 5545: 5541: 5531: 5480: 5476: 5466: 5413: 5409: 5399: 5372: 5286:physics and 5281: 5261: 4850: 4822: 4819:Applications 4806: 4798: 4790: 4783: 4774: 4771:laser light. 4694:Microtoroids 4693: 4687:nanoparticle 4653: 4645: 4498: 4494: 4466: 4312:ground state 4307: 4303: 4301: 4278: 4216: 3999: 3794: 3783: 3768: 3652: 3393: 3004: 2516: 2475: 2384: 1693: 1089: 1037: 811: 793: 760:with energy 754: 686: 681: 485: 476:glass sphere 321: 130: 99: 60: 27: 26: 5449:10044/1/312 5284:trapped ion 4780:Measurement 4729:LC circuits 4036:Hooke's law 3402:are added. 2672:, the term 1093:Hamiltonian 1086:Hamiltonian 999:limit cycle 91:decoherence 7269:1505.02009 5748:1504.03727 5583:2011-12-26 5559:2020-08-06 5341:References 328:wavenumber 7338:1094-4087 7286:0031-9007 7241:0031-9007 7224:0903.2242 7196:0031-9007 7179:1312.5867 7137:1745-2473 7102:1749-4885 7085:1203.5730 7057:1748-3387 7010:0295-5075 6952:1050-2947 6907:0028-0836 6890:1103.2144 6861:0028-0836 6844:0906.1236 6816:1745-2473 6799:0904.4051 6770:0028-0836 6753:1107.3761 6707:0027-8424 6680:1304.6679 6648:0028-0836 6631:0707.1724 6600:119200464 6567:0810.4729 6535:0031-9007 6518:0705.1018 6489:0031-9007 6443:1050-2947 6426:0705.1728 6262:cite book 6254:929952165 6213:1367-2630 6186:1210.2671 6154:1050-2947 6093:1476-4687 6020:1476-4687 5995:1106.3614 5947:1476-4687 5900:118388281 5892:1521-3889 5867:1210.3619 5775:2041-1723 5716:119121252 5708:1098-0121 5683:0707.4153 5643:1094-4087 5616:0712.1618 5234:ω 5204:κ 5181:Γ 5178:≪ 5175:κ 5078:ω 5004:† 4982:† 4964:ℏ 4891:ω 4761:microwave 4757:microwave 4621:κ 4618:≪ 4582:† 4560:† 4542:ℏ 4522:κ 4519:≥ 4481:κ 4478:≥ 4448:≪ 4445:κ 4431:ω 4407:≫ 4404:κ 4390:ω 4363:ω 4342:κ 4287:Δ 4247:Γ 4225:Γ 4179:ω 4175:− 4172:Δ 4149:κ 4144:κ 4139:− 4117:ω 4110:Δ 4087:κ 4082:κ 4059:Γ 4047:Γ 4019:≳ 4010:ω 3962:ω 3955:Δ 3932:κ 3920:ω 3913:Δ 3885:ω 3881:− 3878:Δ 3855:κ 3843:ω 3839:− 3836:Δ 3806:ω 3802:δ 3750:δ 3747:Γ 3744:− 3721:δ 3718:κ 3715:− 3623:Γ 3618:− 3610:† 3602:δ 3593:δ 3564:Γ 3552:ω 3542:− 3529:˙ 3504:κ 3499:− 3491:† 3462:δ 3448:κ 3445:− 3442:Δ 3423:˙ 3414:δ 3363:† 3341:† 3333:δ 3324:δ 3308:ℏ 3269:Δ 3242:† 3232:† 3224:δ 3212:δ 3196:ℏ 3153:ω 3149:≈ 3146:Δ 3119:† 3108:δ 3097:† 3089:δ 3073:ℏ 3030:ω 3026:− 3023:≈ 3020:Δ 2990:α 2938:α 2897:† 2875:† 2867:δ 2858:δ 2849:ℏ 2846:− 2838:† 2824:ω 2820:ℏ 2811:δ 2806:† 2798:δ 2795:Δ 2792:ℏ 2789:− 2753:δ 2748:† 2740:δ 2684:α 2655:† 2627:δ 2607:α 2584:δ 2578:α 2552:κ 2430:ω 2288:ω 2266:ℏ 2186:ω 2136:† 2122:ω 2118:ℏ 2093:† 2085:Δ 2082:ℏ 2079:− 2054:† 1998:ω 1994:− 1985:ω 1978:Δ 1951:† 1943:− 1929:ℏ 1898:† 1880:ℏ 1877:− 1869:† 1855:ω 1851:ℏ 1840:† 1832:Δ 1829:ℏ 1826:− 1787:ω 1751:ω 1744:− 1733:→ 1704:ω 1679:Γ 1659:κ 1639:κ 1589:ω 1585:ℏ 1580:κ 1515:ω 1484:† 1459:† 1399:ω 1372:ω 1325:ω 1248:ω 1241:− 1231:† 1223:− 1209:ω 1186:ℏ 1172:† 1158:ω 1154:ℏ 1143:† 1120:ω 1116:ℏ 1065:κ 1053:ω 1049:≫ 1046:Δ 1012:Δ 995:squeezing 971:ω 961:Δ 935:Δ 906:ω 902:− 896:Δ 870:Δ 840:ω 836:− 827:ω 820:Δ 772:ω 768:ℏ 715:∮ 695:κ 645:− 531:− 422:ℏ 410:Δ 386:ℏ 363:ℏ 274:ω 270:∓ 267:ω 257:ω 226:ω 196:ω 192:± 189:ω 179:ω 154:ω 144:ω 7355:Category 6725:23940352 6101:17080085 6028:21979049 5955:15616555 5848:(2013). 5793:27460277 5651:19551012 5523:16651036 5515:14525288 5298:See also 5157:extremum 4769:infrared 4706:nanobeam 3777:(i.e. a 2599:, where 2308:, where 448:detuning 378:, where 260:′ 182:′ 147:′ 109:and the 7151:Science 6716:3761640 6685:Bibcode 6572:Bibcode 6386:Bibcode 6359:9895956 6339:Bibcode 6298:Bibcode 6191:Bibcode 6122:(1970)) 6109:1449162 6073:Bibcode 6036:4382148 6000:Bibcode 5963:4304653 5927:Bibcode 5872:Bibcode 5820:Bibcode 5784:4974467 5753:Bibcode 5688:Bibcode 5621:Bibcode 5495:Bibcode 5458:6572957 5428:Bibcode 4724:system. 4690:system. 4670:system. 4660:mirrors 803:Phonons 799:Photons 758:phonons 443:finesse 398:is the 67:photons 56:gravity 32:physics 7336: 7284: 7239: 7194: 7135: 7100: 7055: 7008: 6950: 6905: 6877:Nature 6859: 6831:Nature 6814: 6768: 6740:Nature 6723: 6713: 6705: 6646: 6618:Nature 6598: 6533: 6487: 6441: 6357: 6252: 6242: 6211: 6152: 6107: 6099: 6091: 6055:Nature 6034: 6026: 6018: 5982:Nature 5961: 5953: 5945: 5919:Nature 5898: 5890: 5791: 5781: 5773: 5714: 5706: 5649: 5641: 5521: 5513: 5456: 5387: 5274:and a 4702:toroid 3769:These 2914:where 2737: 2680: 1970:where 1611:where 1276:where 217:where 36:optics 7264:arXiv 7219:arXiv 7174:arXiv 7080:arXiv 6988:arXiv 6930:arXiv 6885:arXiv 6839:arXiv 6794:arXiv 6748:arXiv 6675:arXiv 6626:arXiv 6596:S2CID 6562:arXiv 6513:arXiv 6467:arXiv 6421:arXiv 6181:arXiv 6105:S2CID 6063:arXiv 6032:S2CID 5990:arXiv 5959:S2CID 5896:S2CID 5862:arXiv 5743:arXiv 5712:S2CID 5678:arXiv 5611:arXiv 5519:S2CID 5485:arXiv 5454:S2CID 5418:arXiv 5028:with 4642:Setup 4493:), a 4422:. 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