Frequency Control and Coherent Excitation Transfer in a Nanostring-resonator Network

Matthias Pernpeintner, Philip Schmidt, Daniel Schwienbacher, Rudolf Gross, and Hans Huebl

Phys. Rev. Applied 10, 034007 – Published 5 September 2018

Abstract

A key ingredient for mechanical information processing with resonator networks is the possibility to tune the relevant mode frequencies independently and to operate the resonators in the strong coupling regime. We investigate a proof-of-principle realization of such a network based on two coupled highly tensile-stressed silicon nitride nanostring resonators. We demonstrate that the fundamental mode frequencies of both nanostrings can be tuned individually by a strong drive tone resonant with one of their higher-harmonic modes. Using this frequency tuning concept, we investigate the coherent dynamics of the two strongly coupled nanostring resonators acting as an effective classical two-level system. We show classical Rabi oscillations as well as Landau-Zener dynamics, enabling the controlled spacial transfer of mechanical excitation.

Received 8 August 2017
Revised 5 June 2018

DOI:https://doi.org/10.1103/PhysRevApplied.10.034007

Physics Subject Headings (PhySH)

Collective dynamics Continuum mechanics Coupled oscillators Mechanical & acoustical properties Mechanical effects of light on material media

Nanowires Synchronization

Rheology techniques

Condensed Matter, Materials & Applied PhysicsNonlinear DynamicsQuantum Information, Science & Technology

Authors & Affiliations

Matthias Pernpeintner^1,2,3, Philip Schmidt^1,2,3, Daniel Schwienbacher^1,2,3, Rudolf Gross^1,2,3, and Hans Huebl^1,2,3,*

¹Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meißner-Str. 8, Garching D-85748, Germany
²Nanosystems Initiative Munich, Schellingstraße 4, München D-80799, Germany
³Physik-Department, Technische Universität München, James-Franck-Str. 1, Garching D-85748, Germany

^*huebl@wmi.badw.de

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 10, Iss. 3 — September 2018

Subject Areas

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Experimental setup and thermal motion spectra. (a) False-colored scanning electron microscope image of the sample (tilt angle: $60^{\circ}$ ) and schematic illustration of the experimental setup. The two nanostrings are mechanically coupled by a shared support which is partially suspended. The whole chip is mounted on top of a piezoelectric actuator, which is driven with two radiofrequency tones: a strong continuous auxiliary drive (which is used to tune the eigenfrequency of nanostring $A$ ) and a pulsed drive (used to excite nanostring $A$ or $B$ ). For readout of the mechanical motion, a laser ( $λ = 633$ nm) is focused either on nanostring $A$ or $B$ . The reflected laser light is guided to a photodetector, filtered, amplified, and detected with a spectrum analyzer or a digitizer card. (b) Thermal motion spectra obtained by focusing the laser on nanostring $A$ (red circles) or nanostring $B$ (black circles) without an auxiliary drive applied.
Reuse & Permissions
Figure 2
Frequency tuning and mode splitting. (a) Amplitude spectrum of the second-order mode (auxiliary mode) of nanostring $A$ for the drive amplitudes 30, 125, 300, 400, 530, 710, 940, and 1.3 $V_{rms}$ shown as solid circles colored from light to dark blue. The data are offset by 5 nm for clarity. (b) Thermal motion spectrum of nanostring $A$ as a function of the auxiliary drive frequency. The auxiliary drive amplitude is $V_{aux} = 3.1 V_{rms}$ . The dashed green line indicates the expected eigenfrequency shift given by Eq. (1). The inset defines the color coding and the frequency labels. (c) Amplitude and (d) linewidth of the thermal motion spectra shown in (b). The magenta circles indicate the upper mode, the orange circles denote the lower mode, and the shaded areas indicate the statistical error taken over the 15 neighboring data points.
Reuse & Permissions
Figure 3
Rabi oscillations. (a) Measurement scheme: After a short excitation pulse (length $t_{p} = 400 μ s$ , amplitude $V_{p} = 0.22 V_{rms}$ ) at the lower mode’s frequency $Ω_{-}$ , the displacement of nanostring $A$ is measured as a function of time. A constant drive tone (drive voltage $V_{aux} = 3.1 V_{rms}$ ) at the auxiliary frequency $Ω_{aux}$ is used to tune the eigenfrequency $Ω_{A}$ of nanostring $A$ . (b) Squared displacement amplitude of the fundamental mode of nanostring $A$ as a function of time $t$ and eigenfrequency $Ω_{A}$ . (c) Model results, obtained by solving the equations of motion as detailed in the Supplemental Material [30]. Experimental observations and the model calculations show quantitative agreement.
Reuse & Permissions
Figure 4
Landau-Zener transitions. (a) Measurement scheme: After a short excitation pulse (length $t_{p} = 400 μ s$ , amplitude $V_{p} = 0.22 V_{rms}$ ) with frequency $Ω_{A}^{0}$ , the displacement of the fundamental mode of nanostring $A$ (or $B$ ) is measured as a function of time. An auxiliary drive tone (drive voltage $V_{aux} = 4.2 V_{rms}$ ) with frequency $Ω_{aux}$ is used to ramp the eigenfrequency $Ω_{A}$ of nanostring $A$ from $Ω_{A}^{0} < Ω_{B}$ to $Ω_{A}^{0} + Δ Ω_{A} > Ω_{B}$ with ramp rate $ζ = Δ Ω_{A} / τ$ . (b) Schematic illustration of adiabatic and diabatic transitions: For slow tuning of nanostring $A$ (adiabatic transition), the system follows the lower branch of the mode splitting and ends up in state (1), i.e., the excitation of nanostring $A$ is transferred to nanostring $B$ . For fast frequency ramps, the system passes the mode crossing diabatically without energy transfer between the nanostrings [final state: (2)]. (c),(d) Squared displacement amplitude of the fundamental mode of nanostring $A$ (c) and (d) nanostring $B$ as a function of time $t$ and frequency ramp rate $ζ$ . (e),(f) Model results, obtained by solving the equations of motion as detailed in the Supplemental Material [30]. Dashed green lines in (c)–(f) indicate the time $t_{s}$ at which nanostrings $A$ and $B$ are resonant, $Ω_{A} = Ω_{B}$ . Experimental observations and the model calculations show quantitative agreement.
Reuse & Permissions

Physical Review Applied