We experimentally demonstrate that the entanglement between Gaussian entangled states can be increased by non-Gaussian operations. Coherent subtraction of single photons from Gaussian quadrature-entangled light pulses, created by a nondegenerate parametric amplifier, produces delocalized states with negative Wigner functions and complex structures more entangled than the initial states in terms of negativity. The experimental results are in very good agreement with the theoretical predictions.

We investigate the collective optomechanics of an ensemble of scatterers inside a Fabry-Pérot resonator and identify an optimized configuration where the ensemble is transmissive, in contrast with the usual reflective optomechanics approach. In this configuration, the optomechanical coupling of a specific collective mechanical mode can be several orders of magnitude larger than the singleelement case, and long-range interactions can be generated between the different elements since light permeates throughout the array. This new regime should realistically allow for achieving strong single-photon optomechanical coupling with massive resonators, realizing hybrid quantum interfaces, and exploiting collective long-range interactions in arrays of atoms or mechanical oscillators.The field of optomechanics has made tremendous progress over the past decades [1], cooling of massive mechanical oscillators to the motional quantum ground state being but one of a series of achievements that demonstrate the power of coupling light to moving scatterers [2,3]. The control of mechanical motion in the quantum regime has many important applications, ranging from precision measurements [4], quantum information processing [5], and fundamental tests of quantum mechanics [6], to the photonics sciences [7]. Despite recent progress the coupling between a single photon and a single phonon remains typically very weak, therefore necessitating the use of many photons to amplify the interaction [1,8]. In this regime, which is useful for cooling and light-motion entanglement generation, a stronger coupling per photon is desirable to limit the negative effects of using large powers, e.g., bulk temperature increases or phase-noise heating [9]. Ultimately, reaching the strong (single-photon) coupling regime, in which a single quantum of light can appreciably affect the motion of the mechanical oscillator, is essential to exploiting fully the quantum nature of the optomechanical interaction, as exhibited by such effects as the optomechanical photon blockade [10] and non-Gaussian mechanical states [11].Among the various approaches currently followed to couple mechanical oscillators with optical resonators, a successful one involves positioning reflecting objects -dielectric membranes [12,13], atoms [14], or microspheres [15] -inside an optical cavity. With dielectric membranes the optomechanical interaction strength saturates to a fundamental limit g as the reflectivity of the membrane approaches unity [12]. For a highly reflective membrane placed to the center of a Fabry-Pérot (FP) resonator of length L and resonance frequency ω, the * Corresponding author. andre.xuereb@qub.ac.uk single-photon coupling strength is given by the shift in cavity frequency when the mirror moves through a distance equal to the spread x 0 of its zero-point fluctuations, g = 2ωx 0 /L, and is typically rather weak for a macroscopic cavity [12]. Several approaches can be followed to improve quantum motional control in single membrane systems, by e.g., tailoring of the opt...

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PACS. 42.50.Lc -Quantum fluctuations, quantum noise, and quantum jumps. PACS. 03.67.Mn -Entanglement production, characterization and manipulation. PACS. 05.40.Jc -Brownian motion.Abstract. -We propose a double-cavity set-up capable of generating a stationary entangled state of two movable mirrors at cryogenic temperatures. The scheme is based on the optimal transfer of squeezing of input optical fields to mechanical vibrational modes of the mirrors, realized by the radiation pressure of the intracavity light. We show that the presence of macroscopic entanglement can be demonstrated by an appropriate read out of the output light of the two cavities.Introduction. -Quantum entanglement is a physical phenomenon in which the quantum states of two or more systems can only be described with reference to each other. This leads to correlations between observables of the systems that cannot be understood on the basis of local realistic theories [1,2]. Its importance today exceeds the realm of the foundations of quantum physics and entanglement has become an important physical resource that allows performing communication and computation tasks with an efficiency which is not achievable classically [3]. In particular, it is important to investigate under which conditions entanglement between macroscopic objects, each containing a large number of the constituents, can arise. Entanglement between two atomic ensembles has been successfully demonstrated in Ref. [4] by sending pulses of coherent light through two atomic vapor cells. Then, other proposals suggested to entangle a nano-mechanical oscillator with a Cooper-pair box [5], arrays of nanomechanical oscillators [6], two mirrors of an optical ring cavity [7], or two mirrors of two different cavities illuminated with entangled light beams [8]. Here we elaborate on these two latter schemes and propose a new double-cavity set-up able to generate a stationary entangled state of two vibrating cavity mirrors, exploiting the radiation pressure of the intracavity fields. Entanglement between mechanical degrees of freedom is achieved if the input fields are squeezed and if this squeezing is efficiently transferred to the movable mirrors. We show that a stationary entangled state can be generated with state-of-the-art apparata at cryogenic temperatures, and that it can be detected with a non-stationary homodyne measurement of the output light [9,10].

The control of one light field by another, ultimately at the single photon level 1-6 , is a challenging task which has numerous interesting applications within nonlinear optics 4,5 and quantum information science 6,7 . Due to the extremely weak direct interactions between optical photons in vacuum, this type of control can in practice only be achieved through highly nonlinear interactions within a medium 1-9 . Electromagnetic induced transparency (EIT) 1,5 constitutes one such meansto obtain the extremely strong nonlinear coupling needed to facilitate interactions between two faint light fields 2-6,8-11 . Here, we demonstrate for the first time EIT as well as all-optical EIT-based light switching using ion Coulomb crystals situated in an optical cavity. Unprecedented narrow cavity EIT feature widths down to a few kHz and a change from essentially full transmission to full absorption of the probe field within a window of only ~100 kHz are achieved. By applying a weak switching field, we furthermore demonstrate nearly perfect switching of the transmission of the probe field. These results represent important milestones for future realizations of quantum information processing devices, such as high-efficiency quantum memories 12,13 , single-photon transistors 14,15 and single-photon gates 4,6,8 .Electromagnetically induced transparency is a quantum interference phenomenon appearing when two electromagnetic fields excite resonantly two different transitions

We show that a quasiperfect quantum-state transfer between an atomic ensemble and fields in an optical cavity can be achieved in electromagnetically induced transparency (EIT). A squeezed vacuum field state can be mapped onto the long-lived atomic spin associated to the ground-state sublevels of the ⌳-type atoms considered. The EIT on-resonance situation show interesting similarities with the Raman off-resonant configuration. We then show how to transfer the atomic squeezing back to the field exiting the cavity, thus realizing a quantum memory-type operation.

We study the interaction of a nearly resonant linearly polarized laser beam with a cloud of cold cesium atoms in a high finesse optical cavity. We show theoretically and experimentally that the cross-Kerr effect due to the saturation of the optical transition produces quadrature squeezing on both the mean field and the orthogonally polarized vacuum mode. An interpretation of this vacuum squeezing as polarization squeezing is given and a method for measuring quantum Stokes parameters for weak beams via a local oscillator is developed.PACS numbers: 42.50. Dv, 42.50.Lc, 03.67.Hk A great deal of attention has been recently given to the quantum features of the polarization states of the light, essentially because of their connections with quantum information technology. Several theoretical schemes to produce polarization squeezing using Kerr-like media have been proposed [1] and realized using optical fibers [2]. Other experimental realizations achieve polarization squeezing by mixing squeezed vacuum (generated by an OPO) with a strong coherent beam on a polarizing beam splitter [3] or mixing two independent quadrature squeezed beams (generated by an OPA) on a polarizing beam splitter [4]. Very recently it has been proposed to propagate a linearly polarized light beam through an atomic medium exhibiting self rotation to generate squeezed vacuum in the orthogonal polarization [5], which is equivalent to achieving polarization squeezing. In previous works [6] the interaction between a cloud of cold cesium atoms placed in a high finesse optical cavity and a circularly polarized laser beam nearly resonant with an atomic transition has been studied. Because of optical pumping, the atomic medium is conveniently modelled by an ensemble of two-level atoms. The saturation of the optical transition gives rise to an intensity-dependent refraction index. It is well known that the interaction of the light with a Kerr-like medium produces bistable behavior of the light transmitted by the cavity and that, at the turning point of the bistability curve, the quantum fluctuations of the light can be strongly modified and generate quadrature squeezing [7]. A noise reduction of 40% has thus been observed in our group [6]. In this paper we focus on the theoretical and experimental investigation of polarization squeezing via the interaction of a linearly polarized laser beam with cold cesium atoms. In this configuration, the two-level atom model is no longer applicable and the situation much more complicated. We describe the interaction between light and the atomic medium by means of an X-like four-level quantum model based on the linear input-output method. Our theoretical analysis shows clearly that competitive optical pumping may result in polarization switching, and polarization squeezing is predicted by the model [8]. In agreement with the model we observe quadrature squeezing in the probe laser mode and in the orthogonal vacuum mode. Experimentally, we obtain a polarization squeezing of 13% and we show for the first time that the ...

We present an experimental demonstration of both quadrature and polarization entanglement generated via the interaction between a coherent linearly polarized field and cold atoms in a high finesse optical cavity. The non linear atom-field interaction produces two squeezed modes with orthogonal polarizations which are used to generate a pair of non separable beams, the entanglement of which is demonstrated by checking the inseparability criterion for continuous variables recently derived by Duan et al. [Phys. Rev. Lett. 84, 2722] and calculating the entanglement of formation [Giedke et al., Phys. Rev. Lett. 91, 107901 (2003) Bowen et al. [6] by mixing two squeezed beams issued from independent OPAs. The Kerr non-linearity of fibers was also exploited by Glöckl et al. to generate a pulsed source of polarization entanglement [7].In this paper we show evidence for continuous variable entanglement generated using the interaction between a linearly polarized coherent field and a cloud of cold cesium atoms placed in a high finesse optical cavity. We demonstrate the entanglement using the inseparability criterion proposed by Duan et al. and Simon [8]. We generate two kinds of entanglement with the same system, quadrature entanglement and polarization entanglement. For this, we use the recently reported generation of polarization squeezing [9] in the field that has interacted with cold atoms; both the mean field mode and the vacuum mode with orthogonal polarization exiting the cavity can be squeezed. First, we show how a direct measurement of the quadrature entanglement of the beam exiting the cavity can be achieved using two balanced homodyne detections. We then give the form of the covariance matrix and the associated entanglement of formation (EOF), which, for Gaussian symmetric states, is directly related to the inseparability criterion value [10]. Last, we produce two non separable beams by mixing two parts of the previous outgoing beam with a strong field and achieve polarization entanglement by locking the relative phases between the strong field and the weak field exiting the cavity. (1) For states with Gaussian statistics, I a,b < 2 is a sufficient condition for entanglement and has already been used in several experiments to demonstrate continuous variable entanglement [5,6,7]. Moreover, Giedke et al. recently calculated the EOF of Gaussian symmetric states [10] and showed it to be directly related to the amount of EPR-type correlations given by (1).In our system, an x-polarized beam interacts with a cloud of cold cesium atoms in an optical cavity. The experimental set-up [9] is shown in Fig. 1. We probe the atoms with a linearly polarized laser beam detuned by about 50 MHz in the red of the 6S 1/2 , F=4 to 6P 3/2 , F=5 transition. The optical power of the probe beam ranges from 5 to 15 µW. After exiting the cavity, both the mean field mode A x and the orthogonally polarized vacuum mode A y are squeezed for frequencies ranging between 3 and 12 MHz. An interpretation of these results can be provided by modelli...