Ultrasensitive measurement of mechanical modes and Brownian motion
The high quality factor of the optical resonances in silica micotoroids enables ultra-sensitive measurements of changes in their size. Such changes can occur due to external influence like a weak force. According to the laws of thermodynamics, however, the size fluctuates already intrinsically for finite temperatures. All molecules constituting a toroid execute Bownian motion, i.e. they “shiver” around their equilibrium position. Using elasticity theory, the motion of the molecules can be described most easily be mechanical eigenmodes, which are characterized by a certain displacement profile and an associated eigenfrequency. Browian motion of the molecules can then be understood as a thermal excitation of these eigenmodes.
The mechanical modes which create a displacement of the material at the location of the optical mode modulate the optically monitored size of the toroid. These fluctuations are extremely small; at room temperature typical amplitudes are 10-15 meter. With an appropriate measurement setup, however, a sensitivity of 10-18 meter can be reached in only one second of averaging – this corresponds to the hundred-millionth fraction of the diameter of a hydrogen atom! In this manner, a detailed analysis of the eigenmodes of the structure becomes possible and allows a comparison with numerical simulations (Fig. 1).
Figure 1. a) Scanning electron micrograph of a microtoroid with ultrahigh optical and mechanical Q. Light circulates at the rim of the torus, which is held by a silica disc at the tip of a sharp silicon pillar. b) Optically measured spectrum of radius fluctuations of the resonator at room temperature (red line). Above the extremely low measurement background (grey), which is only due to quantum noise of the readout laser, the high-frequency mechanical modes are clearly discernible as sharp resonances. Using numerical simulations, the eigenfrequencies can be uniquely assigned to different mechanical modes. Three examples are shown in the insets.
It is particularly interesting to examine the validity of the approximation of completely orthogonal eigenmodes. Indeed, a variety of dissipation mechanisms leads to coupling of different modes and thus to a finite Q of the mechanical modes. Scientists are still far from a profound understanding of these mechanisms-in spite of their high relevance for the application of nano- or micromechanical systems in sensing and diagnostics. The combination of well-controlled, reproducible microfabrication, numerical simulations and ultra-sensitive measurements with toroidal microresonators can make a significant contribution to a deeper understanding. Also, these insights may be used to engineer structures with ultra-high mechanical quality factors.
Optomechanics: The mechanical properties of light
Moreover, the sensitivity reached enables for the first time the examination of a novel class of phenomena, namely effects which are caused by the transfer of light’s momentum to massive particles. To date, these “light forces” play a role virtually only in atomic and molecular physics, where they are used to control the motional state of these elementary particles. In the context of ultra-sensitive measurements of length they have actually been predicted for more than 20 years . Until recently, however, only one of them, an optomechanical bistability, could be demonstrated with great efforts. As an aside, this experiment was conducted already 1983 in the group of laser pioneer Herberth Walther at the Max-Planck-Institut für Quantenoptik in Garching.
In this context, optical microresonators benefit not only from a very high measurement sensitivity, but also from the very high quality factor, which gives rise to a strongly enhanced circulating optical power (x100000 and more), enhancing the light forces as well. Only in this manner, “optomechanical” effects become observable, and a variety of to date unseen phenomena experimentally accessible. Particularly interesting dynamics arise from the mutual coupling of optical and mechanical degrees of freedom: While the mechanical displacement changes the optical resonance condition, and thus more or less light is admitted into the resonator, the optical energy stored in the cavity determines the light force exerted on the mechanical degree of freedom, which in turn causes displacement (Fig. 2). From this “feedback” arises a non-linear dynamical behavior, which can e.g. lead to an optomechanical instability, in which the mechanical mode is oscillating [2, 3] -driven only by a light beam
Figure 2. Generic optomechanical system. A resonator, here illustrated with two mirrors, is pumped with (near-)resonant light. If the quality of the resonator is high enough, the circulating power is enhanced as compared to the irradiated power (in reality more than 100000 times). Then the light force exerted on the mirrors can become large enough to cause the displacement of a movable mirror, as indicated by the spring-mounted mirror. The concomitant change in the resonance condition between mirror separation and light wavelength leads to a change in coupling of light and thus in turn influences the light force itself. A complex non-linear dynamic arises, in which optical and mechanical modes interact.
Laser cooling of mechanical oscillators
Even more fascinating for researchers in the young field of optomechanics is the fact that this feedback can be negative in the sense that the motion of the mechanical mode is damped by light. In this case, the damping even reduces the thermal excitation of the mode. This is why one also speaks about laser “cooling”, since the modes are brought to a state which corresponds to a lower temperature.
As one of the first systems, this effect was demonstrated in silica microtoroids, and it was shown that in this system it is indeed solely caused by light forces  (Fig. 3). Together with experimental data, a conclusive theoretical model was introduced, which could anticipate the results accurately over a wide range of parameters. The achieved temperature reduction by a factor of 30 can be viewed as the successful demonstration of a new effect, but also the introduction of a new method: laser cooling of macroscopic objects.
Figure 3. Optical refrigeration of a mode at 57.8 MHz. By coupling laser light into a microtoroid, the thermal fluctuations of its radius are reduced (points), as predicted by theory (lines). The more power is coupled, the smaller are the fluctuations (power is increased by x7 from red to dark blue). For the highest optical power an effective temperature reduction to about 11 K is reached.
From a comparison with the field of atomic physics, in which laser cooling has enabled some of the most spectacular breakthroughs in the last decade, it is easy to understand the fast-growing interest in laser cooling of larger objects. Indeed, many groups across the world now pursue this subject, e.g. at Laboratoire Kastler Brossel of the École Normale Supérieure in Paris, the MIT in Boston or at Yale University.
Quantum effects in optomechanical systems attract particular attention. For example, the combination of powerful classic cryogenic techniques could allow cooling mechanical modes to the fundamental, quantum-mechanical limit of their excitation-the quantum ground state. According to the laws of quantum mechanics, the motional amplitude should still be different from zero even in this extreme case. This scenario has be scrutinized in collaboration with the group of Willhelm Zwerger at the Technical University in Munich . The results of this study are highly relevant for the implementation of the planned experiments: Laser cooling to the ground state requires that the photon storage time in the resonator is much longer than the oscillation period of the mechanical mode. This condition represents an experimental challenge. Recent experiments have shown that microtoroids can also enter this regime . They do therefore appear as interesting candidates for the quest of ground-state cooling.
While quantum-optomechanical effects have been theoretically considered very thoroughly for years, their observation represents a formidable challenge for experimental physicists from fields like gravitational-wave astronomy, quantum optics and nano- and micromechanics. Their efforts are driven by application specific questions like the one of the fundamental limit to the sensitivity of length measurement-but as well by the fascination of literal quantum mechanics.
Strong coupling and quantum-coherent effects in optomechanics
One of the great motivational challenges in the field of optomechanics has been the realization of a true macroscopic system which exhibits quantum mechanical behaviour. Following in the footsteps of the precedent set by the atomic physics community, the goal has been to achieve what is usually termed strong coupling.
Strong coupling is a necessary first step that needs to taken to realize full quantum control of a system. In optomechanics, and especially in the silica microtoroids we work with in our group, a strong coupling regime can be attained when the optomechanical coupling is larger than the relevant decay rates.
One such instance, which has been pioneered in the group, is that in which the dressed optomechanical coupling rate is larger than the bare mechanical and cavity decay rates. Under such a condition, it becomes possible to observe a very close analog of the so-called electromagnetically induced transparency (EIT) known from atomic physics. The manifestation of what we call optomechanically induced transparency (OMIT) , happens when the cavity is probed by a weak beam while a strong pump beam is simultaneously tuned into the cavity red-side, such that the relative detuning is exactly equal to the mechanical frequency. When the pump is red detuned from the cavity resonance, the beating intracavity field drives the mechanics so as to generate an optomechanical sideband of the pump, which destructively intereferes with the probe beam. The spectroscopic signature of this effect is a modification of the cavity resonance via the opening-up of a narrow transparency window, quite similar to the Autler-Townes effect in atomic physics.
|Figure 4(a) A strong pump (control) laser is red detuned by the mechanical frequency, while a weak laser probes the cavity resonance.||Figure 4(b) The resulting mechanical drive produces a phase-anticorrelated sideband that results in the narrow transparency window.|
But a much more stringent criteria has to be met if optomechanical systems are to be enabled for true quantum mechanical operation. What is of utmost importance in such a scenario is the ability to maintain a quantum mechanical state with all its coherence, in either the mechanical or optical modes. Obviously, mechanical state preparation and state transfer is the challenge in optomechanics.
The fundamental source of decoherence of the mechanical state is its coupling to a thermal environment. Furthermore, it is well understood that the decoherence rate is larger if the state has a higher mean phonon occupation. Intuitively then (and even otherwise), the relevant condition to be met to mitigate decoherence is that the optomechanical coupling rate is larger than the mechanical and optical decoherence rates; it is not just sufficient to overcome the bare decay rates, which are usually smaller by a factor corresponding to the respective occupations.
|Figure 5 spoke-anchored toroid
To reach this so-called quantum coherent coupling regime , we developed and fabricated a new generation of spoke-anchored silica microtoroids, which had a significantly lesser decay rate to begin with, and a larger optomechanical coupling, owing to a lesser effective mass.
With these systems, it became possible to laser cool the mechanical oscillator to an average occupancy of 1.7 phonons, low enough to have a mechanical decoherence rate lesser than the optomechanical coupling rate. In such a regime, we were able to witness the coherent exchange of energy between the optical field and the mechanical oscillator, thus opening the door to an interesting plethora of future experiments.
|Figure 6(a) a “cooling” beam is red detuned to the cavity, to effect continuous laser cooling to the mechanical oscillator to close to its ground state. A weak excitation pulse is superimposed on the cooling beam.||Figure 6(b) in the quantum coherent coupling regime, an excitation pulse carrying approximately 1 quantum on average is seen to exchange coherently between the optical and the mechanical modes.|
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