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arXiv:1009.4115 · Physical Review Letters

Tunable Backaction of a DC SQUID on an Integrated Micromechanical Resonator

Menno Poot, Samir Etaki, Imran Mahboob, Koji Onomitsu, Hiroshi Yamaguchi, Yaroslav M. Blanter, Herre S. J. van der Zant

Published Nov 10, 2010• 30 citations
arXivHTMLPDF

Abstract

We have measured the backaction of a dc superconducting quantum interference device (SQUID) position detector on an integrated 1 MHz flexural resonator. The frequency and quality factor of the micromechanical resonator can be tuned with bias current and applied magnetic flux. The backaction is caused by the Lorentz force due to the change in circulating current when the resonator displaces. The experimental features are reproduced by numerical calculations using the resistively and capacitively shunted junction model.

DOI
10.1103/physrevlett.105.207203

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We have measured the backaction of a dc superconducting quantum interference device (SQUID) position detector on an integrated 1 MHz flexural resonator. The frequency and quality factor of the micromechanical resonator can be tuned with bias current and applied magnetic flux. The backaction is caused by the Lorentz force due to the change in circulating current when the resonator displaces. The experimental features are reproduced by numerical calculations using the resistively and capacitively shunted junction model.
We present a study of the controllable nonlinear dynamics of a micromechanical beam coupled to a dc-SQUID (superconducting quantum interference device). The coupling between these systems places the modes of the beam in a highly nonlinear potential, whose shape can be altered by varying the bias current and applied flux of the SQUID. We detect the position of the beam by placing it in an optical cavity, which sets free the SQUID to be used solely for actuation. This enables us to probe the previously unexplored full parameter space of this device. We measure the frequency response of the beam and find that it displays a Duffing oscillator behavior which is periodic in the applied magnetic flux. To account for this, we develop a model based on the standard theory for SQUID dynamics. In addition, with the aim of understanding if the device can reach nonlinearity at the single phonon level, we use this model to show that the responsivity of the current circulating in the SQUID to the position of the beam can become divergent, with its magnitude limited only by noise. This suggests a direction for the generation of macroscopically distinguishable superposition states of the beam.
We present a study of the controllable nonlinear dynamics of a micromechanical beam coupled to a dc-SQUID (superconducting quantum interference device). The coupling between these systems places the modes of the beam in a highly nonlinear potential, whose shape can be altered by varying the control parameters of the SQUID. We detect the position of the beam by placing it in an optical cavity, which frees the SQUID to be used solely for actuation. This enables us to probe the previously unexplored full parameter space of this device. We measure the frequency response of the beam and find that it displays a periodic dependence on applied magnetic flux. To account for this, we develop a model based on the standard theory for SQUID dynamics. In addition, with the aim of understanding if the device can reach nonlinearity at the single phonon level, we use this model to show that the responsivity of the current circulating in the SQUID to the position of the beam can become divergent, with its magnitude limited only by noise. This suggests a direction for the generation of macroscopically distinguishable superposition states of the beam.
Recent experiments on superconducting flux qubits, consisting of a superconducting loop interrupted by Josephson junctions, have demonstrated quantum coherence between two different quantum states. The state of the qubit is measured with a superconducting quantum interference device (SQUID). Such measurements require the SQUID to have high resolution while exerting minimal backaction on the qubit. By designing shunts across the SQUID junctions appropriately, one can improve the measurement resolution without increasing the backaction significantly. Using a path-integral approach to analyze the Caldeira-Leggett model, we calculate the narrowing of the distribution of the switching events from the zero-voltage state of the SQUID for arbitrary shunt admittances, focusing on shunts consisting of a capacitance Cs{C}_{s}Cs​ and resistance Rs{R}_{s}Rs​ in series. To test this model, we fabricated a dc SQUID in which each junction is shunted with a thin-film interdigitated capacitor in series with a resistor, and measured the switching distribution as a function of temperature and applied magnetic flux. After accounting for the damping due to the SQUID leads, we found good agreement between the measured escape rates and the predictions of our model. We analyze the backaction of a shunted symmetric SQUID on a flux qubit. For the given parameters of our SQUID and realistic parameters for a flux qubit, at the degeneracy point we find a relaxation time of $113\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, which limits the decoherence time to $226\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$. Based on our analysis of the escape process, we determine that a SQUID with purely capacitive shunts should have narrow switching distributions and no dissipation.
The dynamics and noise of a dc superconducting quantum interference device (SQUID) with the McCumber parameter βc=2πR2IcC/Φ0 close to the unity (where Ic, R, C are the critical current, the shunt resistance, and the capacitance of the Josephson junctions comprising the SQUID, respectively, and Φ0=2.07×10−15 Wb is the magnetic flux quantum) integrated with a planar spiral input coil have been experimentally studied. The length of the spiral input coil was chosen to match its λ/4 microwave resonance frequency to the plasma resonance frequency of the SQUID. The input coil resonance enhances the overall quality factor Q of the Josephson oscillations in the SQUID and, as a result, increases the dynamic resistance Rd and the gradient of the flux-to-voltage characteristics ∂V/∂Φ without hysteresis. This relaxes the tolerance for the βc parameter, simplifies the technological process, and improves the yield of devices. A dc SQUID with loop inductance L=31.4 pH, βc=0.72, and a six turn input coil has demonstrated a nondistorted quasisinusoidal flux-to-voltage transfer function with an exceptionally large modulation depth of approximately 140 μV peak-to-peak. A spectral density of the intrinsic magnetic flux noise as low as 3.5×10−7 Φ0/Hz1/2 has been measured in the double stage configuration at a temperature of 4.2 K using direct read-out electronics. This corresponds to the intrinsic energy resolution of ε=12.5h. In combination with an intermediary transformer, the current resolution of the SQUID is as low as 1.25 pA/Hz1/2 with an input coil inductance of 58 nH. The coupled energy resolution is εc=45h in the white noise region.
At an extremely low temperature of 20 mK, we measured the loop current in a tunable rf superconducting quantum interference device (SQUID) with a dc-SQUID. By adjusting the magnetic flux applied to the rf-SQUID loop (Φ f ) and the small dc-SQUID (Φ cjj f ), respectively, the potential shape of the system can be fully controlled in situ. Variation in the transition step and overlap size in the switching current with a barrier flux bias are analyzed, from which we can obtain some relevant device parameters and build a model to explain the experimental phenomenon.