We analyze the quantum dynamics of a micromechanical resonator capacitively coupled to a Cooperpair 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. DOI: 10.1103/PhysRevLett.88.148301 PACS numbers: 85.85. +j, 03.65.Yz Micromechanical resonators with fundamental vibrational mode frequencies in the range 10 MHz -1 GHz can now be fabricated [1,2]. Applications include fast, ultrasensitive force and displacement detectors [3], electrometers [4,5], and radio frequency signal processors [6]. Advances in the development of micromechanical devices also raise the fundamental question of whether mechanical systems containing macroscopic numbers of atoms will exhibit quantum behavior. Because of their size, quantum behavior in micromechanical systems will be strongly influenced by interactions with the environment and the existence of an experimentally accessible quantum regime will depend on the rate at which decoherence occurs [7].In this Letter, we analyze an experimentally implementable scheme to create and detect superpositions of macroscopically distinct quantum states in a micromechanical resonator, and furthermore measure their decoherence rates, by entangling the resonator with a Cooper box [8][9][10]. The key advantage over optomechanical schemes [7,11] is the demonstrated coherent control of the Cooper box quantum charge state [9], together with the strong (controllable) coupling which can be achieved between the Cooper box state and the motional degree of freedom of a micron-sized mechanical oscillator. Cooper box-based schemes have also been proposed for creating macroscopic quantum state superpositions in superconducting islands [12] and superconducting resonators [13].A Cooper box consists of a small superconducting island weakly linked to a superconducting reservoir [8 -10]. The state of the Cooper box is determined by the balance between its Coulomb charging energy, and the strength of the Cooper-pair tunneling between the island and reservoir. Using an external gate, the Cooper box can be driven into either of two states of definite Cooper-pair number or a linear superposition of the two states [9]. Cooper boxes are being explored as possible candidates for qubits in future quantum computing devices since they act as readily controllable two-level quantum systems [10,14].The electrostatic interaction between a conducting cantilever and a nearby Cooper box causes a displacement in the cantilever whose sign depends on which of the two charge states the Cooper box is in. When the Cooper box is prepared in a superposition of charge states, it and the cantilever become entangled and the cantilever is driven into a superposition of spatially separated states. If the coupling is strong enough, then the separation between the states in the superposition can become larger than their quantum positio...

Surprisingly, when biasing near a transport resonance, we observe cooling of the nanomechanical mode from 550 mK to 300 mK. These measurements have implications for nanomechanical readout of quantum information devices and the limits of ultra-sensitive force microscopy, e.g. single nuclear spin magnetic resonance force microscopy. Furthermore, we anticipate the use of these backaction effects to prepare ultra-cold and quantum states of mechanical structures, which would not be accessible with existing technology.In practice, these back-action impulses arise from the quantized and stochastic nature of the fundamental particles utilized in the measuring device. For example, in high precision optical interferometers such as the LIGO gravitational wave detector 4 or in the single-spin force microscope 5 , the position of a test mass is monitored by reflecting laser-light off of the measured object and interfering this light with a reference beam at a detector. The measured signal is the arrival rate of photons, and one might say that the optical "conductance" of the interferometer is modulated by the position of the measured object. Back-action forces which stochastically drive the measured object result from the random impact and momentum transfer of the discrete photons. This mechanical effect of light is thought to provide the ultimate limit to the position and force sensitivity of an optical interferometer. Although this photon "ponderomotive" noise has not yet been detected during the measurement of a macroscopic object 6 , these back-action effects are clearly observed and carefully utilized in the cooling of dilute atomic vapors to nanoKelvin temperatures.In the experiments reported here, we study an SSET which is capacitively coupled to a voltage-biased (V NR ), doubly-clamped nanomechanical resonator (Fig. 1). Like the interferometer, the conductance of the SSET is a very sensitive probe of the resonator's position, whereas the particles transported in this case are a mixture of single andCooper-paired electrons. We have recently shown the SSET to be nearly a quantumlimited position detector 7 , however reaching the best sensitivity will ultimately be limited by the back-action of the charged particles 3 , which could not be observed in previous experiments because of insufficient SSET-resonator coupling.The back-action force of the SSET results in three measurable effects on the resonator: a frequency shift, a damping rate, and position fluctuations. The frequency shift and damping rate are caused by the in-phase and small out-of phase response in the average electrostatic force between the SSET and resonator, as the resonator oscillates. .MHz is clearly visible, and accurately fits a simple harmonic oscillator response function, on top of a white power spectrum due to an ultra-low noise microwave preamplifier used to read out the SSET with microwave reflectometry 8 .For low SSET-nanoresonator coupling strengths, and the SSET biased close to the Josephson Quasiparticle Peak (JQP) 9 , T NR simply follows T ...

We analyze the dynamics of a nanomechanical resonator coupled to a single-electron transistor ͑SET͒ 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.

We investigate the behavior of a quantum resonator coupled to a superconducting single-electron transistor 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 non-classical 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.PACS numbers: 85.85.+j, 85.35.Gv, 74.78.Na Systems where a mesoscopic conductor such as a single-electron transistor is coupled to a nanomechanical resonator have been studied intensively because the current through the conductor can be extremely sensitive to the motion of the resonator and hence may be used to monitor its position with almost quantum-limited precision [1,2,3,4]. Furthermore, where either the coupling between the electrons and the resonator is non-linear [5] or the electronic transport occurs via a resonance [4], dynamic instabilities in the resonator can occur leading to self-sustained oscillations. The way a nanomechanical resonator can be driven into states of finite amplitude oscillation by successive interactions with a current of electrons in a conductor parallels the behavior of quantum optical systems, such as the micromaser, in which an electromagnetic cavity is pumped by interactions with a steady stream of individual two-level atoms [6]. This contrasts with a standard laser (a nanomechanical version of which was envisioned in [7]) where an oscillator interacts simultaneously with many two-level systems.In a superconducting single-electron transistor (SSET) transport can occur via resonant processes involving both coherent motion of Cooper pairs and incoherent quasiparticle tunneling, the simplest of which is the Josephson quasiparticle (JQP) resonance [8]. In the vicinity of a JQP resonance, the dynamics of a resonator coupled linearly to the SSET is very sensitive to the bias point [3,4,9]. For bias points on one side of the resonance, the SSET acts on the resonator like a thermal bath and its current can monitor the position of the resonator with exquisite sensitivity. In contrast, biasing on the opposite side of the JQP resonance can drive the resonator into states of self-sustained oscillation [4].In this Letter we explore the quantum dynamics of a resonator coupled to a SSET and show that it is analogous to that of a micromaser. Less noisy than a laser, a micromaser [6,10] can generate number-squeezed states of the cavity and exhibits not a single threshold transition, but a series of transitions between different dynamical states. Although the SSET-resonator system and micromaser differ in the details of the interactions between their respective sub-components, we find a num- ber of important similarities in their dynamics, many of which first arise when the resonator is sufficiently fast to match the time-scale of the electrical transport. Previous theoretical studies of this system have concentrat...

We analyze the quantum dynamics of a superconducting cavity coupled to a voltage-biased Josephson junction. The cavity is strongly excited at resonances where the voltage energy lost by a Cooper pair traversing the circuit is a multiple of the cavity photon energy. We find that the resonances are accompanied by substantial squeezing of the quantum fluctuations of the cavity over a broad range of parameters and are able to identify regimes where the fluctuations in the system take on universal values.

We investigate the effect of a quantized vibrational mode on electron tunneling through a chain of three quantum dots. The outer dots are coupled to voltage leads, but the position of the central dot is not rigidly fixed. Motion of the central dot modulates the size of the tunneling barriers in opposite ways so that electron tunneling is correlated with the position of the oscillator. We treat the electronic part of the problem using a simple Coulomb-blockade picture, and model the vibration of the central dot as a quantum oscillator. We calculate the eigenspectrum of the system as a function of the energy level shift between the outer dots. Using a density matrix method, we include couplings to external leads and calculate the steady-state current through the device. The current shows marked resonances that correspond to avoided-level crossings in the eigenvalue spectrum. When the tunneling length of the electrons is of order the zero-point position uncertainty of the quantum oscillator, current far from the electronic resonance is dominated by electrons hopping on and off the central dot sequentially; the oscillator can be regarded as shuttling electrons across the system ͓L.Y. Gorelik et al., Phys. Rev. Lett. 80, 4526 ͑1998͔͒. Damping of the oscillator can increase the current by preventing electrons from hopping ''backwards.''

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We present an analysis of the dynamics of a nanomechanical resonator coupled to a superconducting single electron transistor (SSET) in the vicinity of the Josephson quasiparticle (JQP) and double Josephson quasiparticle (DJQP) resonances. For weak coupling and wide separation of dynamical timescales, we find that for either superconducting resonance the dynamics of the resonator is given by a Fokker-Planck equation, i.e., the SSET behaves effectively as an equilibrium heat bath, characterised 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 a nanomechanical resonator to a SSET in the vicinity of the JQP resonance and Doppler cooling of atoms by means of lasers.