We study noise induced switching in systems far from equilibrium by using an underdamped micromechanical torsional oscillator driven into the nonlinear regime. Within a certain range of driving frequencies, the oscillator possesses two stable dynamical states with different oscillation amplitudes. We induce the oscillator to escape from one dynamical state into the other by introducing noise in the excitation. By measuring the rate of random transitions as a function of noise intensity, we deduce the activation energy as a function of frequency detuning. Close to the critical point, the activation energy is expected to display system-independent scaling. The measured critical exponent is in good agreement with variational calculations and asymptotic scaling theory. Fluctuation-induced escape from a metastable state is an important problem that is relevant to many phenomena, such as protein folding and nucleation in phase transitions. For systems in thermal equilibrium, the escape rate can be deduced from the height of the free-energy barrier.1 The barrier decreases as the control parameter approaches a critical ͑bifurcational͒ value c where the metastable state disappears. It has been established theoretically and experimentally 2,3 that, in the simplest and arguably most common case of the saddle-node ͑spinodal͒ bifurcation, 4 the barrier height scales as ͑ − c ͒ 3/2 . Much less is known about escape in systems far from thermal equilibrium.5-7 Such systems are not characterized by free energy, and the scaling behavior of the escape rate near a saddle-node bifurcation has not been studied experimentally until recently. 8,9 In particular, the problem of escape far from equilibrium has attracted significant experimental attention in the context of systems where multistability itself arises as a result of strong periodic modulation. Escape was studied in parametrically driven electrons in a Penning trap, 10 doubly clamped nanomechanical oscillators, 11,12 and radio frequency driven Josephson junctions. 13 We report here our investigation of noise-activated switching in systems far from equilibrium. By using a wellcharacterized system, an underdamped micromechanical torsional oscillator periodically driven into nonlinear oscillations, we study the dependence of the escape rate on the control parameter as it approaches the critical value and reveal the scaling of the activation energy of escape in a system far from thermal equilibrium. The strongly driven micromechanical oscillator has two stable dynamical states with different oscillation amplitude within a certain range of driving frequencies. We induce the oscillator to escape from one state into the other by injecting noise in the driving force. By measuring the rate of random transitions as a function of noise intensity, we demonstrate the activated behavior for switching and deduce the activation energy as a function of frequency detuning. Close to the bifurcation frequency where the high-amplitude state disappears, the activation energy is predicted by variati...

We demonstrate that the paths followed by a system in fluctuation-activated switching form a narrow tube in phase space. A theory of the path distribution is developed and its direct measurement is performed in a micromechanical oscillator. The experimental and theoretical results are in excellent agreement, with no adjustable parameters. We also demonstrate the lack of time-reversal symmetry in switching of systems far from thermal equilibrium. [6,7] in modulated traps and rf-driven Josephson junctions [8,9] and nano-and micromechanical resonators [10 -12]. Nonequilibrium systems generally lack detailed balance, and the switching rates may not be found by a simple extension of the Kramers approach.The analysis of switching in both nonequilibrium systems and complex equilibrium systems, including biomolecules, relies on the idea that, even though the system motion is random, the switching paths form narrow tubes in phase space. The tube is centered at the most probable switching path (MPSP). For low fluctuation intensity, the MPSP is obtained from a variational problem, which also determines the switching activation barrier [13][14][15][16][17][18][19][20][21][22]. Despite its fundamental role, the concept of the narrow tube of switching paths has not been tested experimentally nor has this tube been characterized quantitatively. Prior experimental studies [23] and simulations [24,25] focused on the distribution in space and time of fluctuational paths to a certain space-time point [26]. The methods [23][24][25][26] do not apply to the switching paths distribution for systems with more than one dynamical variable because of the motion slowing down near the stable states and the related loss of path synchronization.In this Letter, we present a theory of the switching paths distribution and report its direct observation in a wellcharacterized system, a micromechanical oscillator driven into parametric resonance. The observed distribution is shown in Figs. 1(a) and 1(b). Our experimental and theoretical results are in excellent agreement, with no adjustable parameters. In addition to proving the concept of the narrow tube of switching paths and putting it on a quantitative basis, the results provide the first demonstration of the lack of time-reversal symmetry in switching of systems far from thermal equilibrium. The results also open the possibility of efficient control of the switching probability based on the measured narrow path distribution.We consider a bistable system with several dynamical variables q q 1 ; . . . ; q N . The stable states A 1 and A 2 are located at q A 1 and q A 2 , respectively. For low fluctuation intensity, the physical picture of switching is as follows. The system prepared initially near state A 1 , for example,

We explore fluctuation-induced switching in parametrically driven micromechanical torsional oscillators. The oscillators possess one, two, or three stable attractors depending on the modulation frequency. Noise induces transitions between the coexisting attractors. Near the bifurcation points, the activation barriers are found to have a power law dependence on frequency detuning with critical exponents that are in agreement with predicted universal scaling relationships. At large detuning, we observe a crossover to a different power law dependence with an exponent that is device specific. [4 -6], and atoms in magneto-optical traps [7,8], develop multistability under sufficiently strong periodic driving, but are monostable otherwise. These systems are not characterized by free energy and the transition rate must be calculated from system dynamics [9][10][11][12].In both equilibrium and nonequilibrium systems, as a system parameter approaches a bifurcation value b , the activation barrier decreases to zero and the number of stable states of the system changes. In general, the activation barrier is determined by the device parameters and depends on the specifics of the system under study. However, at parameter values close to the bifurcation point, the activation barrier is expected to exhibit universal scaling. The activation barrier varies as kj ÿ b j with a critical exponent that is system independent. While the prefactor k might be different for each system, is universal for all systems and depends only on the type of bifurcation [9]. For instance, in a Duffing oscillator resonantly driven into bistability, spinodal bifurcations occur at the boundaries of the bistable region. One stable state merges with the unstable state while the other stable state remains far away in phase space. Recent experiments in micromechanical oscillators [5] and rf-driven Josephson junctions [13] have confirmed the theoretical prediction [9,14] that the activation barrier scales with critical exponent 3=2 near spinodal bifurcations in driven systems. On the other hand, a different critical exponent of 2 is expected at a pitchfork bifurcation in systems where all three states merge [15]. Such bifurcation commonly takes place in parametrically driven systems where period doubling occurs. For instance, fluctuation-induced phase slips were observed in parametrically driven electrons in Penning traps [2] between two coexisting attractors, and transitions between three attractors were studied in modulated magneto-optical traps [7]. To our knowledge, the activation barriers have not been measured over a wide enough parameter range in these parametrically driven systems to demonstrate the universal scaling at driving frequencies near the two critical points and the crossover to systemspecific dependence at large frequency detuning.In this Letter, we report measurements of the activation barrier for fluctuation-induced switching in parametrically driven micromechanical torsional oscillators, a system that is far from thermal equilibrium. Th...

We demonstrate a "membrane in the middle" optomechanical system using a silicon nitride membrane patterned as a subwavelength grating. The grating has a reflectivity of over 99.8%, effectively creating two sub-cavities, with free spectral ranges of 6 GHz, optically coupled via photon tunneling. Measurements of the transmission and reflection spectra show an avoided crossing where the two sub-cavities simultaneously come into resonance, with a frequency splitting of 54 MHz. We derive expressions for the lineshapes of the symmetric and antisymmetric modes at the avoided crossing, and infer the grating reflection, transmission, absorption, and scattering through comparison with the experimental data.

We measure the spectral densities of fluctuations of an underdamped nonlinear micromechanical oscillator. By applying a sufficiently large periodic excitation, two stable dynamical states are obtained within a particular range of driving frequency. White noise is injected into the excitation, allowing the system to overcome the activation barrier and switch between the two states. While the oscillator predominately resides in one of the two states for most frequencies, a narrow range of frequencies exist where the occupations of the two states are approximately equal. At these frequencies, the oscillator undergoes a kinetic phase transition that resembles the phase transition of thermal equilibrium systems. We observe a supernarrow peak in the spectral densities of fluctuations of the oscillator. This peak is centered at the excitation frequency and arises as a result of noise-induced transitions between the two dynamical states. DOI: 10.1103/PhysRevLett.97.110602 PACS numbers: 05.40.Ca, 05.45.ÿa, 89.75.Da Periodically driven nonlinear systems often display multiple coexisting dynamical states under sufficiently strong driving fields. The presence of fluctuations enables these systems to occasionally overcome the activation barrier in phase space, resulting in transitions between the dynamical states [1]. Noise-induced switching has been studied experimentally in a number of dynamical nonlinear systems that are far from equilibrium, including parametrically driven electrons in a Penning trap [2], atoms in a magneto-optical trap [3], radio frequency driven Josephson junctions [4] and micromechanical and nanomechanical oscillators [5][6][7]. When the noise is weak, the escape rate ÿ i out of state i (i 1 or 2) depends exponentially on the ratio of an activation barrier R i and the noise intensity I N [1]:R i typically depends on system parameters such as the driving frequency and amplitude, as well as the shape of the power spectrum of the noise. The ratio of the populations of the two dynamical states is given by [1]As a result of the exponential dependence of the population ratio on the difference in the activation barriers, the system will be found in either state 1 or state 2 with overwhelmingly large probability over most of the parameter space [1,8]. The occupations of the two states are comparable only over a very narrow range of parameters. This behavior bears close resemblance to systems in thermal equilibrium with two phases such as liquid and vapor. Such thermodynamic systems are usually in either one of the two phases and only at the phase transition will the two phases coexist. Even though driven, nonlinear systems are in general far from equilibrium, theoretical works predicted that a similar kinetic phase transition would occur under the appropriate conditions [1]. Like thermodynamic systems, fluctuations increase significantly when these nonequilibrium systems undergo kinetic phase transitions. A range of generic, system-independent phenomena, including the appearance of a supernarrow peak in b...