We derive an equation for the cooling dynamics of the quantum motion of an atom trapped by an external potential inside an optical resonator. This equation has broad validity and allows us to identify novel regimes where the motion can be efficiently cooled to the potential ground state. Our result shows that the motion is critically affected by quantum correlations induced by the mechanical coupling with the resonator, which may lead to selective suppression of certain transitions for the appropriate parameters regimes, thereby increasing the cooling efficiency. DOI: 10.1103/PhysRevLett.95.143001 PACS numbers: 32.80.Pj, 32.80.Lg, 42.50.Pq Cavity cooling is a recent expression, which stresses the role of the mechanical effects of a resonator on the atomic and molecular center-of-mass dynamics. Indeed, the coupling between resonator and atom gives rise to complex dynamics, one aspect of which is the substantial modification of the atom spectroscopic properties [1]. This property allows one to change the atom scattering cross section, thereby affecting, and eventually tailoring, the mechanical dynamics of the atomic center of mass [2]. Moreover, the motion of the atom changes the medium density, thereby affecting the resonator field itself. Several recent experiments have reported relevant features of these complex dynamics [3][4][5][6][7][8]. Experimental demonstrations of atom cooling in resonators [3][4][5][6][7] have shown, among others, that cavities are a promising tool for preparing and controlling cold atomic samples of scalable dimensions, which may find relevant applications, for instance, in quantum information processing [9].In this Letter, we present a study of the quantum dynamics of the center-of-mass motion of an atomic dipole, which couples to a resonator and to a laser field driving it from the side. The system is sketched in Fig. 1. Differently from recent theoretical works on cavity cooling of atomic clouds [10], here the center of mass is confined by a tight trap, in a configuration that may correspond to the experimental situations reported, for instance, in [4,11,12]. Starting from a master equation for the quantum variables of the dipole, cavity, and center of mass, we derive a closed equation for the center-of-mass dynamics. This equation generalizes previous theoretical studies [13,14] and allows us to identify novel parameter regimes, where cooling can be efficient. Moreover, its form permits us to identify the individual scattering contributions, thereby gaining insight into the role of the various physical parameters. We show that quantum correlations between atom and resonator may lead to the suppression of scattering transitions, thereby enhancing the cooling efficiency.The starting point is the master equation for an atom of mass M whose dipole transition between the ground and excited states jgi and jei couples (quasi)resonantly to a laser and the mode of an optical resonator with wave vector k L and k c , respectively (jk L j jk c j k). In the reference frame of the laser, the...

We investigate a general scheme for generating, either dynamically or in the steady state, continuous variable entanglement between two mechanical resonators with different frequencies. We employ an optomechanical system in which a single optical cavity mode driven by a suitably chosen two-tone field is coupled to the two resonators. Significantly large mechanical entanglement can be achieved, which is extremely robust with respect to temperature.

We present a quantum theory of cooling of a mechanical resonator using back-action with constant electron current. The resonator device is based on a doubly clamped nanotube, which mechanically vibrates and acts as a double quantum dot for electron transport. Mechanical vibrations and electrons are coupled electrostatically using an external gate. The fundamental eigenmode is cooled by absorbing phonons when electrons tunnel through the double quantum dot. We identify the regimes in which ground state cooling can be achieved for realistic experimental parameters.Cooling mechanical resonators has recently attracted considerable interest, as it allows ultrasensitive detection of mass [1,2,3,4], of mechanical displacements [5], and of spin [6]. An appealing prospect is to cool the mechanical resonator to its phononic ground state. This achievement would open the possibility to create and manipulate non classical states at the macroscopic scale and to study the transition from the classical to the quantum regime [7,8,9].The lowest phononic occupation number achieved so far has been experimentally realized by cooling down the resonator in a dilution fridge [10]. Another promising approach is to employ back-action, which consists of coupling mechanical oscillations to visible or microwave photons [11,12,13,14,15,16,17,18,19]. Recently, it has been theoretically proposed [20,21,22] and experimentally demonstrated [10] that back-action cooling can be achieved by coupling mechanical resonators to the constant electron current through electronic nano-devices, such as normal-metal and superconducting single-electron transistors. This approach is appealing because it is easy to implement in a dilution fridge as compared to techniques based on photons. Within this approach, however, modest occupations of the phononic ground state have been predicted [19,20,21,22]. In particular, using an analogy with laser cooling of atoms [23], back-action cooling by constant electron current in these systems is essentially analogous to Doppler cooling [21].In this Letter, we theoretically demonstrate ground state cooling of a mechanical nanotube resonator using constant electron current. Specifically, the nanotube is employed both as the mechanical resonator and the electronic device through which the current flows. In addition, we consider the device layout in which the nanotube acts as a double quantum dot (DQD). This setup allows us to access an analogous regime of sideband cooling of the oscillator [23]. Calculations are carried out by including the coupling of the resonator to the thermal noise of the electrodes and the effect of electronic dephasing inside the DQD. For realistic device parameters the temperature is lowered by a factor of about 100. Moreover, we identify the regime in which the oscillator ground state can reach more than 90% occupation.The device layout is sketched in Fig. 1(a). The DQD system is obtained by locally depleting a semiconducting nanotube with gate T [24,25,26,27]. The dot on the right is suspended, so it...

We investigate theoretically two-photon processes in a microcavity containing one quantum dot in the strong-coupling regime. The cavity mode can be tuned to resonantly drive the two-photon transition between the ground and the biexciton states while the exciton states are far-off resonance due to the biexciton binding energy. We study the steady state of the quantum dot and cavity field in presence of a continuous incoherent pumping. We identify the regime where the system acts as two-photon emitter and discuss the feasibility and performance of realistic single quantum-dot devices for two-photon lasing.

We realise a phase-sensitive closed-loop control scheme to engineer the fluctuations of the pump field which drives an optomechanical system, and show that the corresponding cooling dynamics can be significantly improved. In particular, operating in the counter-intuitive "anti-squashing" regime of positive feedback and increased field fluctuations, sideband cooling of a nanomechanical membrane within an optical cavity can be improved by 7.5 dB with respect to the case without feedback. Close to the quantum regime of reduced thermal noise, such feedback-controlled light would allow going well below the quantum backaction cooling limit.Feedback loops based on real-time continuous measurements [1] are commonly used for stabilisation purposes, and they have also been successfully applied to the stabilisation of quantum systems [2][3][4]. Typically a system is continuously monitored and the acquired signal drives the actuator which in turn drives the system to the desired target. Here we demonstrate a novel approach to closed-loop control in which the feedback acts on an additional control field which is used to drive the system of interest. In particular, the actuator acts on the control field in order to engineer its phase and amplitude fluctuations. The resulting feedback-controlled inloop field is then exploited to manipulate the system and improve its performance. In-loop optical fields have been studied for decades both theoretically [5][6][7][8] and experimentally [9, 10]. A lot of effort has been made to reduce (squash) the noise exhibited by the field fluctuations inside the loop. However, in-loop sub-shot-noise fluctuations cannot be recognised as squeezed below the vacuum noise level, for two different reasons: firstly, the free field commutation relations are no longer valid for time events separated by more than the loop delay-time, since in-loop fields are not free fields [6]; secondly, the corresponding out-of-loop fields exhibit supershot-noise fluctuations [7]. Nevertheless, useful applications of these fields have been proposed and realised, e.g. suppression of the radiation pressure noise [9], removal of classical intensity noise [10], and atomic line narrowing [8]. The common basis of these works is the negative feedback regime. Negative feedback has also been successfully employed in mechanical [11][12][13], and cavity optomechanical systems [4], where an electromagnetic field is used to probe a mechanical resonator, and in turn to control the feedback actuator, which acts directly on the mechanical oscillator. Engineered light fluctuations in the form of squeezed light have also been used in optomechanical systems to improve both the detection sensitivity [14][15][16][17] and the cooling efficiency [18][19][20]. In the present work we show that it is possible to manipulate, with a feedback system [see Figure 1 (a)], the fluctuations of the laser field that drives an optomechanical system to enhance optomechanical sideband cooling [21][22][23][24]. Our analysis demonstrates the effectiveness of t...

We investigate theoretically the mechanical effects of light on atoms trapped by an external potential, whose dipole transition couples to the mode of an optical resonator and is driven by a laser. We derive an analytical expression for the quantum center-of-mass dynamics, which is valid in presence of a tight external potential. This equation has broad validity and allows for a transparent interpretation of the individual scattering processes leading to cooling. We show that the dynamics are a competition of the mechanical effects of the cavity and of the laser photons, which may mutually interfere. We focus onto the good-cavity limit and identify novel cooling schemes, which are based on quantum interference effects and lead to efficient ground state cooling in experimentally accessible parameter regimes.

We study the optomechanical behaviour of a driven Fabry-Pérot cavity containing two vibrating dielectric membranes. We characterize the cavity mode frequency shift as a function of the twomembrane positions, and report a ∼2.47 gain in the optomechanical coupling strength of the membrane relative motion with respect to the single membrane case. This is achieved when the two membranes are properly positioned to form an inner cavity which is resonant with the driving field. We also show that this two-membrane system has the capability to tune the single-photon optomechanical coupling on demand, and represents a promising platform for implementing cavity optomechanics with distinct oscillators. Such a configuration has the potential to enable cavity optomechanics in the strong single-photon coupling regime, and to study synchronization in optically linked mechanical resonators.

It was shown [New J. Phys. 17, 103037 (2015)] that large and robust entanglement between two different mechanical resonators could be achieved, either dynamically or in the steady state, in an optomechanical system in which the two mechanical resonators are coupled to a single cavity mode driven by a suitably chosen two-tone field. An important limitation of the scheme is that the cavity decay rate must be much smaller than the two mechanical frequencies and their difference. Here we show that the entanglement can be remarkably enhanced, and the validity of the scheme can be largely extended, by adding a coherent feedback loop that effectively reduces the cavity decay rate