Nonergodic Measurements of Qubit Frequency Noise.

Slow fluctuations of a qubit frequency are one of the major problems faced by quantum computers. To understand their origin it is necessary to go beyond the analysis of their spectra. We show that characteristic features of the fluctuations can be revealed using comparatively short sequences of periodically repeated Ramsey measurements, with the sequence duration smaller than needed for the noise to approach the ergodic limit. The outcomes distribution and its dependence on the sequence duration are sensitive to the nature of the noise. The time needed for quantum measurements to display quasiergodic behavior can strongly depend on the measurement parameters.

Many-Electron System on Helium and Color Center Spectroscopy.

Electrons on the helium surface display sharp resonant absorption lines related to the transitions between the subbands of quantized motion transverse to the surface. A magnetic field parallel to the surface strongly affects the absorption spectrum. We show that the effect results from admixing the intersubband transitions to the in-plane quantum dynamics of the strongly correlated electron liquid or a Wigner crystal. This is similar to the admixing of electron transitions in color centers to phonons. The spectrum permits a direct characterization of the many-electron dynamics and also enables testing the theory of color centers in a system with controllable coupling.

Noise-induced switching from a symmetry-protected shallow metastable state.

We consider escape from a metastable state of a nonlinear oscillator driven close to triple its eigenfrequency. The oscillator can have three stable states of period-3 vibrations and a zero-amplitude state. Because of the symmetry of period-tripling, the zero-amplitude state remains stable as the driving increases. However, it becomes shallow in the sense that the rate of escape from this state exponentially increases, while the system still lacks detailed balance. We find the escape rate and show how it scales with the parameters of the oscillator and the driving. The results facilitate using nanomechanical, Josephson-junction based, and other mesoscopic vibrational systems for studying, in a well-controlled setting, the rates of rare events in systems lacking detailed balance. They also describe how fluctuations spontaneously break the time-translation symmetry of a driven oscillator.

Nonlocal random walk over Floquet states of a dissipative nonlinear oscillator.

We study transitions between the Floquet states of a periodically driven oscillator caused by the coupling of the oscillator to a thermal reservoir. The analysis refers to the oscillator that is driven close to triple its eigenfrequency and displays resonant period tripling. The interstate transitions result in a random "walk" over the states. We find the transition rates and show that the walk is nonlocal in the state space: the stationary distribution over the states is formed by the transitions between remote states. This is to be contrasted with systems in thermal equilibrium, where the distribution is usually formed by transitions between nearby states. The analysis of period tripling allows us to explore generic features of the dynamics of resonantly driven systems related to the multiplicity of the states, including those missing in the previously explored models of driven oscillators. We use the results to study switching between the period-3 states of the oscillator due to quantum fluctuations and find the scaling of the switching rates with the parameters.

Resonantly Induced Friction and Frequency Combs in Driven Nanomechanical Systems.

We propose a new mechanism of friction in resonantly driven vibrational systems. The form of the friction force follows from the time- and spatial-symmetry arguments. We consider a microscopic mechanism of this resonant force in nanomechanical systems. The friction can be negative, leading to the onset of self-sustained oscillations of the amplitude and phase of forced vibrations, which result in a frequency comb in the power spectrum.

Strong negative nonlinear friction from induced two-phonon processes in vibrational systems.

Self-sustained vibrations in systems ranging from lasers to clocks to biological systems are often associated with the coefficient of linear friction, which relates the friction force to the velocity, becoming negative. The runaway of the vibration amplitude is prevented by positive nonlinear friction that increases rapidly with the amplitude. Here we use a modulated electromechanical resonator to show that nonlinear friction can be made negative and sufficiently strong to overcome positive linear friction at large vibration amplitudes. The experiment involves applying a drive that simultaneously excites two phonons of the studied mode and a phonon of a faster decaying high-frequency mode. We study generic features of the oscillator dynamics with negative nonlinear friction. Remarkably, self-sustained vibrations of the oscillator require activation in this case. When, in addition, a resonant force is applied, a branch of large-amplitude forced vibrations can emerge, isolated from the branch of the ordinary small-amplitude response.

Driven nonlinear nanomechanical resonators as digital signal detectors.

Because of their nonlinearity, vibrational modes of resonantly driven nanomechanical systems have coexisting stable states of forced vibrations in a certain range of the amplitude of the driving force. Depending on its phase, which encodes binary information, a signal at the same frequency increases or decreases the force amplitude. The resulting force amplitude can be outside the range of bistability. The values of the mode amplitude differ significantly on the opposite sides of the bistability region. Therefore the mode amplitude is very sensitive to the signal phase. This suggests using a driven mode as a bi-directional bifurcation amplifier, which switches in the opposite directions depending on the signal phase and provides an essentially digital output. We study the operation of the amplifier near the critical point where the width of the bistability region goes to zero and thus the threshold of the signal amplitude is low. We also develop an analytical technique and study the error rate near the threshold. The results apply to a broad range of currently studied systems and extend to micromechanical systems and nonlinear electromagnetic cavities.

Anomalous Decay of Nanomechanical Modes Going Through Nonlinear Resonance.

Because of the small size of nanomechanical systems, their vibrations become nonlinear already for small amplitudes. Many nontrivial aspects of the vibration dynamics arise from the coexistence of several nonlinearly coupled modes. We show that such coupling can lead to anomalous decay of the modes where they go through nonlinear resonance, so that their amplitude-dependent frequencies become commensurate. We demonstrate the possibility of a strongly nonmonotonic dependence of the decay rate on the amplitude if one of the modes serves as a thermal reservoir for another mode. Where the decay of both modes is slow compared to the rate of resonant energy exchange, the decay is accompanied by amplitude oscillations. Depending on the initial conditions, with increasing time it can display an extremely sharp or a comparatively smooth crossover between different regimes. The results provide insight into recent experimental results by several groups and suggest new ways of characterizing and controlling nanomechanical systems.

Ripplonic Lamb Shift for Electrons on Liquid Helium.

We study the shift of the energy levels of electrons on a helium surface due to the coupling to the quantum field of surface vibrations. As in quantum electrodynamics, the coupling is known, and it is known to lead to an ultraviolet divergence of the level shifts. We show that there are diverging terms of different nature and use the Bethe-type approach to show that they cancel each other, to leading order. This resolves the long-standing theoretical controversy and explains the existing experiments. The results allow us to study the temperature dependence of the level shift. The predictions are in good agreement with the experimental data, with no adjustable parameters.

Correlated anomalous phase diffusion of coupled phononic modes in a sideband-driven resonator.

The dynamical backaction from a periodically driven optical cavity can reduce the damping of a mechanical resonator, leading to parametric instability accompanied by self-sustained oscillations. Here we study experimentally and theoretically new aspects of the backaction and the discrete time-translation symmetry of a driven system using a micromechanical resonator with two nonlinearly coupled vibrational modes with strongly differing frequencies and decay rates. We find self-sustained oscillations in both the low- and high-frequency modes. Their frequencies and amplitudes are determined by the nonlinearity, which also leads to bistability and hysteresis. The phase fluctuations of the two modes show near-perfect anti-correlation, a consequence of the discrete time-translation symmetry. Concurrently, the phase of each mode undergoes anomalous diffusion. The phase variance follows a power law time dependence, with an exponent determined by the 1/f-type resonator frequency noise. Our findings enable compensating for the fluctuations using a feedback scheme to achieve stable frequency downconversion.

Critical fluctuations and the rates of interstate switching near the excitation threshold of a quantum parametric oscillator.

We study the dynamics of a nonlinear oscillator near the critical point where period-two vibrations are first excited with the increasing amplitude of parametric driving. Above the threshold, quantum fluctuations induce transitions between the period-two states over the quasienergy barrier. We find the effective quantum activation energies for such transitions and their scaling with the difference of the driving amplitude from its critical value. We also find the scaling of the fluctuation correlation time with the quantum noise parameters in the critical region near the threshold. The results are extended to oscillators with nonlinear friction.

Noise-induced switching and extinction in systems with delay.

We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of interattractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.

Nanotube mechanical resonators with quality factors of up to 5 million.

Carbon nanotube mechanical resonators have attracted considerable interest because of their small mass, the high quality of their surfaces, and the pristine electronic states they host. However, their small dimensions result in fragile vibrational states that are difficult to measure. Here, we observe quality factors Q as high as 5 × 10(6) in ultra-clean nanotube resonators at a cryostat temperature of 30 mK, where we define Q as the ratio of the resonant frequency over the linewidth. Measuring such high quality factors requires the use of an ultra-low-noise method to rapidly detect minuscule vibrations, as well as careful reduction of the noise of the electrostatic environment. We observe that the measured quality factors fluctuate because of fluctuations of the resonant frequency. We measure record-high quality factors, which are comparable to the highest Q values reported in mechanical resonators of much larger size, a remarkable result considering that reducing the size of resonators is usually concomitant with decreasing quality factors. The combination of ultra-low mass and very large Q offers new opportunities for ultra-sensitive detection schemes and quantum optomechanical experiments.

Interplay of driving and frequency noise in the spectra of vibrational systems.

We study the spectral effect of the fluctuations of the vibration frequency. Such fluctuations play a major role in nanomechanical and other mesoscopic vibrational systems. We find that, for periodically driven systems, the interplay of the driving and frequency fluctuations results in specific spectral features. We present measurements on a carbon nanotube resonator and show that our theory allows not only the characterization of the frequency fluctuations but also the quantification of the decay rate without ring-down measurements. The results bear on identifying the decoherence of mesoscopic oscillators and on the general problem of resonance fluorescence and light scattering by oscillators.

Symmetry breaking in a mechanical resonator made from a carbon nanotube.

Nanotubes behave as semi-flexible polymers in that they can bend by a sizeable amount. When integrating a nanotube in a mechanical resonator, the bending is expected to break the symmetry of the restoring potential. Here we report on a new detection method that allows us to demonstrate such symmetry breaking. The method probes the motion of the nanotube resonator at nearly zero-frequency; this motion is the low-frequency counterpart of the second overtone of resonantly excited vibrations. We find that symmetry breaking leads to the spectral broadening of mechanical resonances, and to an apparent quality factor that drops below 100 at room temperature. The low quality factor at room temperature is a striking feature of nanotube resonators whose origin has remained elusive for many years. Our results shed light on the role played by symmetry breaking in the mechanics of nanotube resonators.

Ultrasensitive force detection with a nanotube mechanical resonator.

Since the advent of atomic force microscopy, mechanical resonators have been used to study a wide variety of phenomena, including the dynamics of individual electron spins, persistent currents in normal metal rings and the Casimir force. Key to these experiments is the ability to measure weak forces. Here, we report on force sensing experiments with a sensitivity of 12 zN Hz(-1/2) at a temperature of 1.2 K using a resonator made of a carbon nanotube. An ultrasensitive method based on cross-correlated electrical noise measurements, in combination with parametric downconversion, is used to detect the low-amplitude vibrations of the nanotube induced by weak forces. The force sensitivity is quantified by applying a known capacitive force. This detection method also allows us to measure the Brownian vibrations of the nanotube down to cryogenic temperatures. Force sensing with nanotube resonators offers new opportunities for detecting and manipulating individual nuclear spins as well as for magnetometry measurements.

Singular probability distribution of shot-noise driven systems.

We study the stationary probability distribution of a system driven by shot noise. We find that both in the overdamped and underdamped regime, the coordinate distribution displays power-law singularities in its central part. For sufficiently low rate of noise pulses they correspond to distribution peaks. We find the positions of the peaks and the corresponding exponents. In the underdamped regime the peak positions are given by a geometric progression. The energy distribution in this case also displays multiple peaks with positions given by a geometric progression. Such structure is a signature of the shot-noise induced fluctuations. The analytical results are in excellent agreement with numerical simulations.

Large rare fluctuations in systems with delayed dissipation.

We study the probability distribution and the escape rate in systems with delayed dissipation that comes from the coupling to a thermal bath. To logarithmic accuracy in the fluctuation intensity, the problem is reduced to a variational problem. It describes the most probable fluctuational paths, which are given by acausal equations due to the delay. In thermal equilibrium, the most probable path passing through a remote state has time-reversal symmetry, even though one cannot uniquely define a path that starts from a state with given system coordinate and momentum. The corrections to the distribution and the escape activation energy for small delay and small noise correlation time are obtained in explicit form.

Sharp tunneling peaks in a parametric oscillator: quantum resonances missing in the rotating wave approximation.

We describe a new mechanism of tunneling between period-two vibrational states of a weakly nonlinear, parametrically modulated oscillator. The tunneling results from resonant transitions induced by the fast oscillating terms conventionally disregarded in the rotating wave approximation (RWA). The tunneling amplitude displays resonant peaks as a function of the modulation frequency; near the maxima it is exponentially larger than the RWA tunneling amplitude.

Singular response of bistable systems driven by telegraph noise.

We show that weak periodic driving can exponentially strongly change the rate of escape from a potential well of a system driven by telegraph noise. The analysis refers to an overdamped system, where escape requires that the noise amplitude θ exceed a critical value θ(c). For θ close to θ(c), the exponent of the escape rate displays a nonanalytic dependence on the amplitude of an additional low-frequency modulation. This leads to giant nonlinearity of the response of a bistable system to periodic modulation. Also studied is the linear response to periodic modulation far from θ(c). We analyze the scaling of the logarithm of the escape rate with the distance to the saddle-node and pitchfork bifurcation points. The analytical results are in excellent agreement with numerical simulations.