Floquet theory for temporal correlations and spectra in time-periodic open quantum systems: Application to squeezed parametric oscillation beyond the rotating-wave approximation

Open quantum systems can display periodic dynamics at the classical level either due to external periodic modulations or to self-pulsing phenomena typically following a Hopf bifurcation. In both cases, the quantum fluctuations around classical solutions do not reach a quantum-statistical stationary state, which prevents adopting the simple and reliable methods used for stationary quantum systems. Here we put forward a general and efficient method to compute two-time correlations and corresponding spectral densities of time-periodic open quantum systems within the usual linearized (Gaussian) approximation for their dynamics. Using Floquet theory we show how the quantum Langevin equations for the fluctuations can be efficiently integrated by partitioning the time domain into one-period duration intervals, and relating the properties of each period to the first one. Spectral densities, like squeezing spectra, are computed similarly, now in a two-dimensional temporal domain that is treated as a chessboard with one-period x one-period cells. This technique avoids cumulative numerical errors as well as efficiently saves computational time. As an illustration of the method, we analyze the quantum fluctuations of a damped parametrically-driven oscillator (degenerate parametric oscillator) below threshold and far away from rotating-wave approximation conditions, which is a relevant scenario for modern low-frequency quantum oscillators. Our method reveals that the squeezing properties of such devices are quite robust against the amplitude of the modulation or the low quality of the oscillator, although optimal squeezing can appear for parameters that are far from the ones predicted within the rotating-wave approximation.Comment: Comments and constructive criticism are welcom

Spin-orbit-torque driven magnetoimpedance in Pt-layer/magnetic-ribbon heterostructures

When a flow of electron passes through a paramagnetic layer with strong spin-orbit-coupling such as platinum (Pt), a net spin current is produced via spin Hall effect (SHE). This spin current can exert a torque on the magnetization of an adjacent ferromagnetic layer which can be probed via magnetization dynamic response, e.g. spin-torque ferromagnetic resonance (ST-FMR). Nevertheless, that effect in lower frequency magnetization dynamic regime (MHz) where skin effect occurs in high permeability ferromagnetic conductors namely the magneto-impedance (MI) effect can be fundamentally important which has not been studied so far. Here, by utilizing the MI effect in magnetic-ribbon/Pt heterostructure with high magnetic permeability that allows the ac current effectively confined at the skin depth of ~100 nm thickness, the effect of spin-orbit-torque (SOT) induced by the SHE probed via MI measurement is investigated. We observed a systematic MI frequency shift that increases by increasing the applied current amplitude and thickness of the Pt layer (varying from 0 nm to 20 nm). In addition, the role of Pt layer in ribbon/Pt heterostructure is evaluated with ferromagnetic resonance (FMR) effect representing standard Gilbert damping increase as the result of presence of the SHE. Our results unveil the role of SOT in dynamic control of the transverse magnetic permeability probed with impedance spectroscopy as useful and valuable technique for detection of future SHE devices.Comment: 15 pages,6 figure

Super-harmonic injection locking of nano-contact spin-torque vortex oscillators

Super-harmonic injection locking of single nano-contact (NC) spin-torque vortex oscillators (STVOs) subject to a small microwave current has been explored. Frequency locking was observed up to the fourth harmonic of the STVO fundamental frequency $f_{0}$ in microwave magneto-electronic measurements. The large frequency tunability of the STVO with respect to $f_{0}$ allowed the device to be locked to multiple sub-harmonics of the microwave frequency $f_{RF}$, or to the same sub-harmonic over a wide range of $f_{RF}$ by tuning the DC current. In general, analysis of the locking range, linewidth, and amplitude showed that the locking efficiency decreased as the harmonic number increased, as expected for harmonic synchronization of a non-linear oscillator. Time-resolved scanning Kerr microscopy (TRSKM) revealed significant differences in the spatial character of the magnetization dynamics of states locked to the fundamental and harmonic frequencies, suggesting significant differences in the core trajectories within the same device. Super-harmonic injection locking of a NC-STVO may open up possibilities for devices such as nanoscale frequency dividers, while differences in the core trajectory may allow mutual synchronisation to be achieved in multi-oscillator networks by tuning the spatial character of the dynamics within shared magnetic layers.Comment: 21 pages, 8 figure