A. D. Armour

Cooling a nanomechanical resonator with quantum back-action

Quantum mechanics demands that the act of measurement must affect the measured object. When a linear amplifier is used to continuously monitor the position of an object, the Heisenberg uncertainty relationship requires that the object be driven by force impulses, called back-action. Here we measure the back-action of a superconducting single-electron transistor (SSET) on a radio-frequency nanomechanical resonator. The conductance of the SSET, which is capacitively coupled to the resonator, provides a sensitive probe of the latters position; back-action effects manifest themselves as an effective thermal bath, the properties of which depend sensitively on SSET bias conditions. Surprisingly, when the SSET is biased near a transport resonance, we observe cooling of the nanomechanical mode from 550 mK to 300 mK—an effect that is analogous to laser cooling in atomic physics. Our measurements have implications for nanomechanical readout of quantum information devices and the limits of ultrasensitive force microscopy (such as single-nuclear-spin magnetic resonance force microscopy). Furthermore, we anticipate the use of these back-action effects to prepare ultracold and quantum states of mechanical structures, which would not be accessible with existing technology.

Entanglement and Decoherence of a Micromechanical Resonator via Coupling to a Cooper-Pair Box

We analyze the quantum dynamics of a micromechanical resonator capacitively coupled to a Cooper-pair box. With appropriate quantum state control of the Cooper box, the resonator can be driven into a superposition of spatially separated states. The Cooper box can also be used to probe the decay of the resonator superposition state due to environmental decoherence.

Classical dynamics of a nanomechanical resonator coupled to a single-electron transistor

We analyze the dynamics of a nanomechanical resonator coupled to a single-electron transistor (SET) in the regime where the resonator behaves classically. A master equation is derived describing the dynamics of the coupled system which is then used to obtain equations of motion for the average charge state of the SET and the average position of the resonator. We show that the action of the SET on the resonator is very similar to that of a thermal bath, as it leads to a steady-state probability distribution for the resonator which can be described by mean values of the resonator position, a renormalized frequency, an effective temperature, and an intrinsic damping constant. Including the effects of extrinsic damping and finite temperature, we find that there remain experimentally accessible regimes where the intrinsic damping of the resonator still dominates its behavior. We also obtain the average current through the SET as a function of the coupling to the resonator.

Dynamics of a nanomechanical resonator coupled to a superconducting single-electron transistor

We present an analysis of the dynamics of a nanomechanical resonator coupled to a superconducting single-electron transistor (SSET) in the vicinity of Josephson quasi-particle (JQP) and double Josephson quasi-particle (DJQP) resonances. For weak coupling and wide separation of dynamical timescales, we find that for either superconducting resonances the dynamics of the resonator are given by a Fokker–Planck equation, i.e. the SSET behaves effectively as an equilibrium heat bath, characterized by an effective temperature, which also damps the resonator and renormalizes its frequency. Depending on the gate and drain–source voltage bias points with respect to the superconducting resonance, the SSET can also give rise to an instability in the mechanical resonator marked by negative damping and temperature within the appropriate Fokker–Planck equation. Furthermore, sufficiently close to a resonance, we find that the Fokker–Planck description breaks down. We also point out that there is a close analogy between coupling of a nanomechanical resonator to an SSET in the vicinity of the JQP resonance and Doppler cooling of atoms by means of lasers.

Quantum dynamics of a resonator driven by a superconducting single-electron transistor : A solid-state analogue of the micromaser

We investigate the behavior of a quantum resonator coupled to a superconducting single-electron transistor (SSET) tuned to the Josephson quasiparticle resonance and show that the dynamics is similar in many ways to that found in a micromaser. Coupling to the SSET can drive the resonator into nonclassical states of self-sustained oscillation via either continuous or discontinuous transitions. Increasing the coupling further leads to a sequence of transitions and regions of multistability.

Probing the quantum coherence of a nanomechanical resonator using a superconducting qubit: I. Echo scheme

We propose a scheme in which the quantum coherence of a nanomechanical resonator can be probed using a superconducting qubit. We consider a mechanical resonator coupled capacitively to a Cooper pair box and assume that the superconducting qubit is tuned to the degeneracy point so that its coherence time is maximized and the electro-mechanical coupling can be approximated by a dispersive Hamiltonian. When the qubit is prepared in a superposition of states, this drives the mechanical resonator progressively into a superposition which in turn leads to apparent decoherence of the qubit. Applying a suitable control pulse to the qubit allows its population to be inverted resulting in a reversal of the resonator dynamics. However, the resonators interactions with its environment mean that the dynamics is not completely reversible. We show that this irreversibility is largely due to the decoherence of the mechanical resonator and can be inferred from appropriate measurements on the qubit alone. Using estimates for the parameters involved based on a specific realization of the system, we show that it should be possible to carry out this scheme with existing device technology.

Nonlinear modal coupling in a high-stress doubly-clamped nanomechanical resonator

We present results from a study of the nonlinear inter-modal coupling between different flexural vibrational modes of a single high-stress, doubly-clamped silicon nitride nanomechanical beam. Using the magnetomotive technique and working at 100 mK we explored the nonlinear behaviour and modal couplings of the first, third and fifth modes of a 25.5 μm long beam. We find very good agreement between our results and a simple analytical model which assumes that the different modes of the resonator are coupled to each other by displacement induced tension in the beam. The small size of our resonator leads to relatively strong nonlinear couplings, for example we find a shift of about 7 Hz in the third mode for a 1 nm displacement in the first mode and frequency shifts ~ 20 times larger than the linewidth (130 Hz) are readily observed.

Dissipation due to tunneling two-level systems in gold nanomechanical resonators

We present measurements of the dissipation and frequency shift in gold nanomechanical resonators at temperatures down to 10 mK. The resonators were fabricated as doubly clamped beams above a GaAs substrate and actuated magnetomotively. Measurements on beams with frequencies 7.95 and 3.87 MHz revealed that from 30 to 500 mK the dissipation increases with temperature as T 0.5 , with saturation occurring at higher temperatures. The relative frequency shift of the resonators increases logarithmically with temperature up to at least 400 mK. Similarities with the behavior of bulk amorphous solids suggest that the dissipation in our resonators is dominated by two-level systems.

Possibility of an electromechanical which-path interferometer

eling through an Aharonov-Bohm ring incorporating a quantum dot in one of the arms are dephased by an interaction with the fundamental flexural mode of a radio-frequency cantilever. The cantilever is positioned so that its tip lies just above the dot and a bias is applied so that an electric field exists between the dot and the tip. This electric field is modified when an additional electron hops onto the dot, coupling the flexural mode of the cantilever and the microscopic electronic degrees of freedom. We analyze the transmission properties of this system and the dependence of interference fringe visibility on the cantilever-dot coupling and on the mechanical properties of the cantilever. The fringes are progressively destroyed as the interaction with the cantilever is turned up, in part due to dephasing arising from the entanglement of the electron and cantilever states and also due to the thermal smearing that results from fluctuations in the state of the cantilever. When the dwell time of the electron on the dot is comparable to or longer than the cantilever period, we find coherent features in the transmission amplitude. These features are washed out when the cantilever is decohered by its coupling to the environment.

Transport via a quantum shuttle

We present preliminary results for a model of electron transport through an electromechanical system in the extreme quantum mechanical and Coulomb blockade limits. The system consists of three quantum dots in which the outer dots are fixed and the central dot is mounted on a quantum harmonic oscillator, thus forming a quantum shuttle. The current through the system as a function of the energy level shift between the outer dots has marked resonances which are associated with avoided level-crossings in the eigenvalue spectrum.