Electro-optomechanical equivalent circuits for quantum transduction

Using the techniques of optomechanics, a high-$Q$ mechanical oscillator may serve as a link between electromagnetic modes of vastly different frequencies. This approach has successfully been exploited for the frequency conversion of classical signals and has the potential of performing quantum state transfer between superconducting circuitry and a traveling optical signal. Such transducers are often operated in a linear regime, where the hybrid system can be described using linear response theory based on the Heisenberg-Langevin equations. While mathematically straightforward to solve, this approach yields little intuition about the dynamics of the hybrid system to aid the optimization of the transducer. As an analysis and design tool for such electro-optomechanical transducers, we introduce an equivalent circuit formalism, where the entire transducer is represented by an electrical circuit. Thereby we integrate the transduction functionality of optomechanical systems into the toolbox of electrical engineering allowing the use of its well-established design techniques. This unifying impedance description can be applied both for static (DC) and harmonically varying (AC) drive fields, accommodates arbitrary linear circuits, and is not restricted to the resolved-sideband regime. Furthermore, by establishing the quantized input-output formalism for the equivalent circuit, we obtain the scattering matrix for linear transducers using circuit analysis, and thereby have a complete quantum mechanical characterization of the transducer. Hence, this mapping of the entire transducer to the language of electrical engineering both sheds light on how the transducer performs and can at the same time be used to optimize its performance by aiding the design of a suitable electrical circuit.Comment: 30 pages, 9 figure

Optomechanically induced transparency

Coherent interaction of laser radiation with multilevel atoms and molecules can lead to quantum interference in the electronic excitation pathways. A prominent example observed in atomic three-level-systems is the phenomenon of electromagnetically induced transparency (EIT), in which a control laser induces a narrow spectral transparency window for a weak probe laser beam. The concomitant rapid variation of the refractive index in this spectral window can give rise to dramatic reduction of the group velocity of a propagating pulse of probe light. Dynamic control of EIT via the control laser enables even a complete stop, that is, storage, of probe light pulses in the atomic medium. Here, we demonstrate optomechanically induced transparency (OMIT)--formally equivalent to EIT--in a cavity optomechanical system operating in the resolved sideband regime. A control laser tuned to the lower motional sideband of the cavity resonance induces a dipole-like interaction of optical and mechanical degrees of freedom. Under these conditions, the destructive interference of excitation pathways for an intracavity probe field gives rise to a window of transparency when a two-photon resonance condition is met. As a salient feature of EIT, the power of the control laser determines the width and depth of the probe transparency window. OMIT could therefore provide a new approach for delaying, slowing and storing light pulses in long-lived mechanical excitations of optomechanical systems, whose optical and mechanical properties can be tailored in almost arbitrary ways in the micro- and nano-optomechanical platforms developed to date

Multimode optomechanical system in the quantum regime

We realise a simple and robust optomechanical system with a multitude of long-lived ($Q>10^7$) mechanical modes in a phononic-bandgap shielded membrane resonator. An optical mode of a compact Fabry-Perot resonator detects these modes' motion with a measurement rate ($96~\mathrm{kHz}$) that exceeds the mechanical decoherence rates already at moderate cryogenic temperatures ($10\,\mathrm{K}$). Reaching this quantum regime entails, i.~a., quantum measurement backaction exceeding thermal forces, and thus detectable optomechanical quantum correlations. In particular, we observe ponderomotive squeezing of the output light mediated by a multitude of mechanical resonator modes, with quantum noise suppression up to -2.4 dB (-3.6 dB if corrected for detection losses) and bandwidths $\lesssim 90\,\mathrm{ kHz}$. The multi-mode nature of the employed membrane and Fabry-Perot resonators lends itself to hybrid entanglement schemes involving multiple electromagnetic, mechanical, and spin degrees of freedom.Comment: 19 pages, 9 figure

Determination of the vacuum optomechanical coupling rate using frequency noise calibration

The strength of optomechanical interactions in a cavity optomechanical system can be quantified by a vacuum coupling rate \vcr analogous to cavity quantum electrodynamics. This single figure of merit removes the ambiguity in the frequently quoted coupling parameter defining the frequency shift for a given mechanical displacement, and the effective mass of the mechanical mode. Here we demonstrate and verify a straightforward experimental technique to derive the vacuum optomechanical coupling rate. It only requires applying a known frequency modulation of the employed electromagnetic probe field and knowledge of the mechanical oscillator's occupation. The method is experimentally verified for a micromechanical mode in a toroidal whispering-gallery-resonator and a nanomechanical oscillator coupled to a toroidal cavity via its near field.Comment: 11 pages, 2 figure

Tunable pulse delay and advancement in a coupled nanomechanical resonator-superconducting microwave cavity system

We theoretically study the transmission of a weak probe field under the influence of a strong pump field in a coupled nanomechanical resonator-superconducting microwave cavity system. Using the standard input-output theory, we find that both pulse delay (slow light effect) and advancement (fast light effect) of the probe field can appear in this coupled system provided that we choose the suitable detuning of the pump field from cavity resonance. The magnitude of the delay (advancement) can be tuned continuously by adjusting the power of the pump field. This technique demonstrates great potential in applications including microwave phase shifter and delay line.Comment: 12pages, 3 figure

Astronomical spectrograph calibration with broad-spectrum frequency combs

Broadband femtosecond-laser frequency combs are filtered to spectrographically resolvable frequency-mode spacing, and the limitations of using cavities for spectral filtering are considered. Data and theory are used to show implications to spectrographic calibration of high-resolution, astronomical spectrometers

Control of microwave signals using circuit nano-electromechanics

Waveguide resonators are crucial elements in sensitive astrophysical detectors [1] and circuit quantum electrodynamics (cQED) [2]. Coupled to artificial atoms in the form of superconducting qubits [3, 4], they now provide a technologically promising and scalable platform for quantum information processing tasks [2, 5-8]. Coupling these circuits, in situ, to other quantum systems, such as molecules [9, 10], spin ensembles [11, 12], quantum dots [13] or mechanical oscillators [14, 15] has been explored to realize hybrid systems with extended functionality. Here, we couple a superconducting coplanar waveguide resonator to a nano-coshmechanical oscillator, and demonstrate all-microwave field controlled slowing, advancing and switching of microwave signals. This is enabled by utilizing electromechanically induced transparency [16-18], an effect analogous to electromagnetically induced transparency (EIT) in atomic physics [19]. The exquisite temporal control gained over this phenomenon provides a route towards realizing advanced protocols for storage of both classical and quantum microwave signals [20-22], extending the toolbox of control techniques of the microwave field.Comment: 9 figure