Nonreciprocal microwave devices are ubiquitous in radar and radio communication and indispensable in the readout chains of superconducting quantum circuits. Since they commonly rely on ferrite materials requiring large magnetic fields that make them bulky and lossy, there has been significant interest in magnetic-field-free on-chip alternatives, such as those recently implemented using the Josephson nonlinearity. Here, we realize reconfigurable nonreciprocal transmission between two microwave modes using purely optomechanical interactions in a superconducting electromechanical circuit. The scheme relies on the interference in two mechanical modes that mediate coupling between the microwave cavities and requires no magnetic field. We analyse the isolation, transmission and the noise properties of this nonreciprocal circuit. Finally, we show how quantum-limited circulators can be realized with the same principle. All-optomechanically mediated nonreciprocity demonstrated here can also be extended to directional amplifiers, and it forms the basis towards realizing topological states of light and sound.

Directional amplifiers are an important resource in quantum information processing, as they protect sensitive quantum systems from excess noise. Here, we propose an implementation of phase-preserving and phase-sensitive directional amplifiers for microwave signals in an electromechanical setup comprising two microwave cavities and two mechanical resonators. We show that both can reach their respective quantum limits on added noise. In the reverse direction, they emit thermal noise stemming from the mechanical resonators and we discuss how this noise can be suppressed, a crucial aspect for technological applications. The isolation bandwidth in both is of the order of the mechanical linewidth divided by the amplitude gain. We derive the bandwidth and gain-bandwidth product for both and find that the phase-sensitive amplifier has an unlimited gain-bandwidth product. Our study represents an important step toward flexible, on-chip integrated nonreciprocal amplifiers of microwave signals.Introduction.-Nonreciprocal transmission and amplification of signals are essential in communication and signal processing, as they protect the signal source from extraneous noise. Conventional ferrite-based devices rely on magnetic fields and are challenging to integrate in superconducting circuits. Hence, there exists strong incentive to find more suitable implementations [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In the microwave domain, the strong Josephson nonlinearity and parametric pumping can achieve both photon gain and conversion processes, which have been exploited to realize circulators and directional amplifiers [5,[13][14][15]]. Another promising platform is optomechanics, where nonreciprocal devices [16][17][18][19][20][21][22][23][24][25][26][27], phase-preserving amplifiers [28][29][30][31][32], and phase-sensitive amplifiers [33][34][35][36] have been proposed and realized.In recent theoretical work, Ranzani and Aumentado [16,17] analyzed general conditions for nonreciprocity in parametrically coupled systems, and showed that nonreciprocity arises due to dissipation in ancillary modes and multi-path interference. Metelmann and Clerk [18] have shown that any coherent interaction can be made directional by balancing it with a dissipative process. Indeed, this insight led to a demonstration of nonreciprocity using optomechanics in the optical domain [19], and theoretical investigations into minimal implementations of directional amplifiers [20].While implementing the balance between a direct coherent coupling between the cavities and a dissipative interaction is challenging experimentally, Refs. [25][26][27] have recently studied and demonstrated nonreciprocal transmission between two cavity modes where two mechanical resonators each mediate both coherent and dissipative coupling. Here, building on this concept, we propose directional amplifiers using exclusively optomechanical interactions. Microwave tones on the red and blue sidebands enable so-called beam-splitter and two-mode squeezing interactions (cf. Fig. ...

We develop a Floquet approach to solve time-periodic quantum Langevin equations in steady state. We show that two-time correlation functions of system operators can be expanded in a Fourier series and that a generalized Wiener-Khinchin theorem relates the Fourier transform of their zeroth Fourier component to the measured spectrum. We apply our framework to bichromatically driven cavity optomechanical systems, a setting in which mechanical oscillators have recently been prepared in quantum-squeezed states. Our method provides an intuitive way to calculate the power spectral densities for time-periodic quantum Langevin equations in arbitrary rotating frames. arXiv:1605.04749v2 [cond-mat.mes-hall] 1 Aug 2016

Sources of entangled electromagnetic radiation are a cornerstone in quantum information processing and offer unique opportunities for the study of quantum many-body physics in a controlled experimental setting. Generation of multi-mode entangled states of radiation with a large entanglement length, that is neither probabilistic nor restricted to generate specific types of states, remains challenging. Here, we demonstrate the fully deterministic generation of purely photonic entangled states such as the cluster, GHZ, and W state by sequentially emitting microwave photons from a controlled auxiliary system into a waveguide. We tomographically reconstruct the entire quantum many-body state for up to N = 4 photonic modes and infer the quantum state for even larger N from process tomography. We estimate that localizable entanglement persists over a distance of approximately ten photonic qubits.

Quantum mechanics imposes a limit on the precision of a continuous position measurement of a harmonic oscillator, due to backaction arising from quantum fluctuations in the measurement field. This standard quantum limit can be surpassed by monitoring only one of the two non-commuting quadratures of the motion, known as backaction-evading measurement. This technique has not been implemented using optical interferometers to date. Here we demonstrate, in a cavity optomechanical system operating in the optical domain, a continuous two-tone backaction-evading measurement of a localized gigahertz-frequency mechanical mode of a photonic-crystal nanobeam cryogenically and optomechanically cooled close to the ground state. Employing quantum-limited optical heterodyne detection, we explicitly show the transition from conventional to backaction-evading measurement. We observe up to 0.67 dB (14%) reduction of total measurement noise, thereby demonstrating the viability of backaction-evading measurements in nanomechanical resonators for optical ultrasensitive measurements of motion and force.