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 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

Mechanically Detecting and Avoiding the Quantum Fluctuations of a Microwave Field

During the theoretical investigation of the ultimate sensitivity of gravitational wave detectors through the 1970's and '80's, it was debated whether quantum fluctuations of the light field used for detection, also known as photon shot noise, would ultimately produce a force noise which would disturb the detector and limit the sensitivity. Carlton Caves famously answered this question with "They do." With this understanding came ideas how to avoid this limitation by giving up complete knowledge of the detector's motion. In these back-action evading (BAE) or quantum non-demolition (QND) schemes, one manipulates the required quantum measurement back-action by placing it into a component of the motion which is unobserved and dynamically isolated. Using a superconducting, electro-mechanical device, we realize a sensitive measurement of a single motional quadrature with imprecision below the zero-point fluctuations of motion, detect both the classical and quantum measurement back-action, and demonstrate BAE avoiding the quantum back-action from the microwave photons by 9 dB. Further improvements of these techniques are expected to provide a practical route to manipulate and prepare a squeezed state of motion with mechanical fluctuations below the quantum zero-point level, which is of interest both fundamentally and for the detection of very weak forces

ARTICLE Photon-assisted tunnelling with nonclassical light

International audienceAmong the most exciting recent advances in the field of superconducting quantum circuits is the ability to coherently couple microwave photons in low-loss cavities to quantum electronic conductors. These hybrid quantum systems hold great promise for quantum information-processing applications; even more strikingly, they enable exploration of new physical regimes. Here we study theoretically the new physics emerging when a quantum electronic conductor is exposed to nonclassical microwaves (for example, squeezed states, Fock states). We study this interplay in the experimentally relevant situation where a superconducting microwave cavity is coupled to a conductor in the tunnelling regime. We find that the conductor acts as a nontrivial probe of the microwave state: the emission and absorption of photons by the conductor is characterized by a nonpositive definite quasi-probability distribution, which is related to the Glauber-Sudarshan P-function of quantum optics. These negative quasi-probabilities have a direct influence on the conductance of the conductor

Quantum master equation descriptions of a nanomechanical resonator coupled to a single-electron transistor

We analyse the quantum dynamics of a nanomechanical resonator coupled to a normal-state single-electron transistor (SET). Starting from a microscopic description of the system, we derive a master equation for the SET island charge and resonator which is valid in the limit of weak electro-mechanical coupling. Using this master equation we show that, apart from brief transients, the resonator always behaves like a damped harmonic oscillator with a shifted frequency and relaxes into a thermal-like steady state. Although the behaviour remains qualitatively the same, we find that the magnitude of the resonator damping rate and frequency shift depend very sensitively on the relative magnitudes of the resonator period and the electron tunnelling time. Maximum damping occurs when the electrical and mechanical time-scales are the same, but the frequency shift is greatest when the resonator moves much more slowly than the island charge. We then derive reduced master equations which describe just the resonator dynamics. By making slightly different approximations, we obtain two different reduced master equations for the resonator. Apart from minor differences, the two reduced master equations give rise to a consistent picture of the resonator dynamics which matches that obtained from the master equation including the SET island charge.Comment: 22 pages, 4 figure

Quantum nano-electromechanics with electrons, quasiparticles and Cooper pairs: effective bath descriptions and strong feedback effects

Using a quantum noise approach, we discuss the physics of both normal metal and superconducting single electron transistors (SET) coupled to mechanical resonators. Particular attention is paid to the regime where transport occurs via incoherent Cooper-pair tunneling (either via the Josephson quasiparticle (JQP) or double Josephson quasiparticle (DJQP) process). We show that, surprisingly, the back-action of tunneling Cooper pairs (or superconducting quasiparticles) can be used to significantly cool the oscillator. We also discuss the physical origin of negative damping effects in this system, and how they can lead to a regime of strong electro-mechanical feedback, where despite a weak SET - oscillator coupling, the motion of the oscillator strongly effects the tunneling of the Cooper pairs. We show that in this regime, the oscillator is characterized by an energy-dependent effective temperature. Finally, we discuss the strong analogy between back-action effects of incoherent Cooper-pair tunneling and ponderomotive effects in an optical cavity with a moveable mirror; in our case, tunneling Cooper pairs play the role of the cavity photons.Comment: 27 pages, 7 figures; submitted to the New Journal of Physics focus issue on Nano-electromechanical Systems; error in references correcte

Andreev bound states and π\pi -junction transition in a superconductor / quantum-dot / superconductor system

We study Andreev bound states and $\pi$-junction transition in a superconductor / quantum-dot / superconductor (S-QD-S) system by Green function method. We derive an equation to describe the Andreev bound states in S-QD-S system, and provide a unified understanding of the $\pi$-junction transition caused by three different mechanisms: (1) {\it Zeeman splitting.} For QD with two spin levels $E_{\uparrow}$ and $E_{\downarrow}$, we find that the surface of the Josephson current $I(\phi =\frac \pi 2)$ vs the configuration of $(E_{\uparrow},E_{\downarrow})$ exhibits interesting profile: a sharp peak around $E_{\uparrow}=E_{\downarrow}=0$; a positive ridge in the region of $E_{\uparrow}\cdot E_{\downarrow}>0$; and a {\em % negative}, flat, shallow plain in the region of $E_{\uparrow}\cdot E_{\downarrow}<0$. (2){\it \ Intra-dot interaction.} We deal with the intra-dot Coulomb interaction by Hartree-Fock approximation, and find that the system behaves as a $\pi$-junction when QD becomes a magnetic dot due to the interaction. The conditions for $\pi$-junction transition are also discussed. (3) {\it \ Non-equilibrium distribution.} We replace the Fermi distribution $f(\omega)$ by a non-equilibrium one $\frac 12[ f(\omega -V_c)+f(\omega +V_c)]$, and allow Zeeman splitting in QD where $$ The curves of $I(\phi =\frac \pi 2)$ vs $$ show the novel effect of interplay of non-equilibrium distribution with magnetization in QD.Comment: 18 pages, 8 figures, Late

Recent Results on the Periodic Lorentz Gas

The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption -- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases -- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.Comment: 62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new versio