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 amplitude and phase noise of limit-cycle oscillations in a mechanical resonator coupled parametrically to an optical cavity driven above its resonant frequency. At a given temperature the limit-cycle oscillations have lower amplitude noise than states of the same average amplitude excited by a pure harmonic drive; for sufficiently low thermal noise a sub-Poissonian resonator state can be produced. We also calculate the linewidth narrowing that occurs in the limit-cycle states and show that while the minimum is set by direct phase diffusion, diffusion due to the optical spring effect can dominate if the cavity is not driven exactly at a sideband resonance.

We study the time evolution of N q two-level atoms (or qubits) interacting with a single mode of the quantised radiation field. In the case of two qubits, we show that for a set of initial conditions the reduced density matrix of the atomic system approaches that of a pure state at tr 4 , halfway between that start of the collapse and the first mini revival peak, where t r is the time of the main revival. The pure state approached is the same for a set of initial conditions and is thus termed an 'attractor state'. The set itself is termed the basin of attraction and the features are at the center of our attention. Extending to more qubits, we find that attractors are a generic feature of the multi qubit Jaynes Cummings model (JCM) and we therefore generalise the discovery by Gea-Banacloche for the one qubit case. We give the 'basin of attraction' for N q qubits and discuss the implications of the 'attractor' state in terms of the dynamics of N q -body entanglement. We observe both collapse and revival and sudden birth/death of entanglement depending on the initial conditions. I. INTRODUCTIONThe dynamics of two level quantum systems (also known as qubits) such as spins in a magnetic field, Rydberg atoms or superconducting qubits, coupled to a single mode of an electromagnetic cavity are of considerable interest in connection with NMR studies of atomic nuclei [1], cavity quantum electrodynamics [2] and the physics of quantum computing [3]. The simplest model that captures the salient features of the physics in these fields is the Jaynes-Cummings model (JCM) [4], which deals with only one qubit and its generalisation by Tavis and Cummings to the case of multiple qubits [5].These models have many interesting features [6], but perhaps the most surprising prediction is * Electronic address: catherine.jarvis@bristol.ac.uk

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 timescales 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.

We analyze the current and zero-frequency current noise properties of a superconducting single electron transistor (SSET) coupled to a resonator, focusing on the regime where the SSET is operated in the vicinity of the Josephson quasiparticle resonance. We consider a range of coupling strengths and resonator frequencies to reflect the fact that in practice the system can be tuned to quite a high degree with the resonator formed either by a nanomechanical oscillator or a superconducting stripline fabricated in close proximity to the SSET. For very weak couplings the SSET acts on the resonator like an effective thermal bath. In this regime the current characteristics of the SSET are only weakly modified by the resonator. Using a mean field approach, we show that the current noise is nevertheless very sensitive to the correlations between the resonator and the SSET charge. For stronger couplings, the SSET can drive the resonator into limit cycle states where self-sustained oscillation occurs and we find that regions of well-defined bistability exist. Dynamical transitions into and out of the limit cycle state are marked by strong fluctuations in the resonator energy, but these fluctuations are suppressed within the limit cycle state. We find that the current noise of the SSET is strongly influenced by the fluctuations in the resonator energy and hence should provide a useful indicator of the resonator's dynamics.

We explore the quantum dynamics of a mechanical resonator whose position is coupled to the frequency of an optical (or microwave) cavity mode. When the cavity is driven at a frequency above resonance the mechanical resonator can gain energy and for sufficiently strong coupling this results in limit-cycle oscillations. Using a truncated Wigner function approach, which captures the zero-point fluctuations in the system, we develop an approximate analytic treatment of the resonator dynamics in the limit-cycle regime. We find that the limit-cycle oscillations produced by the cavity are associated with rather low levels of energy fluctuations in the resonator. Compared to a resonator at the same temperature which is driven by a pure harmonic drive to a given average energy, the cavity-driven oscillations can have much lower energy fluctuations. Furthermore, at sufficiently low temperatures, the cavity can drive the resonator into a non-classical state which is number-squeezed.

We investigate the dynamical instabilities of a resonator coupled to a superconducting single-electron transistor (SSET) tuned to the Josephson quasiparticle (JQP) resonance. Starting from the quantum master equation of the system, we use a standard semiclassical approximation to derive a closed set of mean field equations which describe the average dynamics of the resonator and SSET charge. Using amplitude and phase coordinates for the resonator and assuming that the amplitude changes much more slowly than the phase, we explore the instabilities which arise in the resonator dynamics as a function of coupling to the SSET, detuning from the JQP resonance and the resonator frequency. We find that the locations (in parameter space) and sizes of the limit cycle states predicted by the mean field equations agree well with numerical solutions of the full master equation for sufficiently weak SSET-resonator coupling. The mean field equations also give a good qualitative description of the set of dynamical transitions in the resonator state that occur as the coupling is progressively increased.

We provide a broad outline of the requirements that should be met by components produced for a Quantum Information Technology (QIT) industry, and we identify electromagnetically induced transparency (EIT) as potentially key enabling science toward the goal of providing widely available few-qubit quantum information processing within the next decade. As a concrete example, we build on earlier work and discuss the implementation of a two-photon controlled phase gate (and, briefly, a one-photon phase gate) using the approximate Kerr nonlinearity provided by EIT. In this paper, we rigorously analyze the dependence of the performance of these gates on atomic dephasing and field detuning and intensity, and we calculate the optimum parameters needed to apply a π phase shift in a gate of a given fidelity. Although high-fidelity gate operation will be difficult to achieve with realistic system dephasing rates, the moderate fidelities that we believe will be needed for few-qubit QIT seem much more obtainable.