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.

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.

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.

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.

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.

Diffusion-induced bistability of driven nanomechanical resonators.

We study nanomechanical resonators with frequency fluctuations due to diffusion of absorbed particles. The diffusion depends on the vibration amplitude through inertial effect. We find that, if the diffusion coefficient D is sufficiently large, the resonator response to periodic driving displays bistability. The lifetime of the coexisting vibrational states exponentially increases with increasing D and displays a scaling dependence on the parameters close to bifurcation points.

Spectrum of an oscillator with jumping frequency and the interference of partial susceptibilities.

We study an underdamped oscillator with random frequency jumps. We describe the oscillator spectrum in terms of coupled susceptibilities for different-frequency states. Depending on the parameters, the spectrum has a fine structure or displays a single asymmetric peak. For nanomechanical resonators with a fluctuating number of attached molecules, it is found in a simple analytical form. The results bear on dephasing in various types of systems with jumping frequency.

Switching exponent scaling near bifurcation points for non-Gaussian noise.

We study noise-induced switching of a system close to bifurcation parameter values where the number of stable states changes. For non-Gaussian noise, the switching exponent, which gives the logarithm of the switching rate, displays a non-power-law dependence on the distance to the bifurcation point. This dependence is found for Poisson noise. Even weak additional Gaussian noise dominates switching sufficiently close to the bifurcation point, leading to a crossover in the behavior of the switching exponent.

Proposal for manipulating and detecting spin and orbital States of trapped electrons on helium using cavity quantum electrodynamics.

We propose a hybrid architecture in which an on-chip high finesse superconducting cavity is coupled to the lateral motion and spin state of a single electron trapped on the surface of superfluid helium. We estimate the motional coherence times to exceed 15 µs, while energy will be coherently exchanged with the cavity photons in less than 10 ns for charge states and faster than 1 µs for spin states, making the system attractive for quantum information processing and strong coupling cavity quantum electrodynamics experiments. The cavity is used for nondestructive readout and as a quantum bus mediating interactions between distant electrons or an electron and a superconducting qubit.

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.

Exponential peak and scaling of work fluctuations in modulated systems.

We show that steady-state work fluctuations in periodically modulated systems display universal features, which are not described by the standard fluctuation theorems. Modulated systems often have coexisting stable periodic states. We find that work fluctuations sharply increase near a kinetic phase transition where the state populations are close to each other. We also show that the work variance displays scaling with the distance to a bifurcation point where a stable state disappears and find the critical exponent for a saddle-node bifurcation.

Switching-path distribution in multidimensional systems.

We explore the distribution of paths followed in fluctuation-induced switching between coexisting stable states. We introduce a quantitative characteristic of the path distribution in phase space that does not require a priori knowledge of system dynamics. The theory of the distribution is developed and its direct measurement is performed in a micromechanical oscillator driven into parametric resonance. The experimental and theoretical results on the shape and position of the distribution are in excellent agreement, with no adjustable parameters. In addition, the experiment provides the first demonstration of the lack of time-reversal symmetry in switching of systems far from thermal equilibrium. The results open the possibility of efficient control of the switching probability based on the measured narrow path distribution.

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.

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.

Critical exponents in metastable decay via quantum activation.

We consider decay of metastable states of forced vibrations of a quantum oscillator close to the bifurcation points where the states disappear. Decay occurs via quantum activation over a quasienergy barrier, a mechanism that differs from both tunneling and thermal activation. The decay probability W scales with the distance eta to the bifurcation point as /ln W/ proportional to eta(xi). The exponent xi is found for a resonantly driven oscillator and an oscillator modulated at nearly twice its eigenfrequency.

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.

Pathways of activated escape in periodically modulated systems.

We investigate dynamics of activated escape in periodically modulated systems. The trajectories followed in escape form diffusion-broadened tubes, which are periodically repeated in time. We show that these tubes can be directly observed and find their shape. Quantitatively, the tubes are characterized by the distribution of trajectories that, after escape, pass through a given point in phase space for a given modulation phase. This distribution may display several peaks separated by the modulation period. Analytical results agree with the results of simulations of a Brownian particle in a model modulated potential.

Resonant symmetry lifting in a parametrically modulated oscillator.

We study a parametrically modulated oscillator that has two stable states of vibration at half the modulation frequency omega{F}. Fluctuations of the oscillator lead to interstate switching. A comparatively weak additional field can strongly affect the switching rates because it changes the switching activation energies. The change is linear in the field amplitude. When the additional field frequency omega{d} is omega{F}2 , the field makes the populations of the vibrational states different, thus lifting the states symmetry. If omega{d} differs from omega{F}2 , the field modulates the state populations at the difference frequency, leading to fluctuation-mediated wave mixing. For an underdamped oscillator, the change of the activation energy displays characteristic resonant peaks as a function of frequency.