Coherent conversion between microwave and optical photons - an overview of physical implementations Nicholas J. Lambert,1, 2 Alfredo Rueda,3, 4 Florian Sedlmeir,5, 6 and Harald G. L. Schwefel1,2, ∗ 1Department of Physics, University of Otago, Dunedin, New Zealand 2The Dodd-Walls Centre for Photonic and Quantum Technologies, New Zealand 3The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin, New Zealand 4Institute of Science and Technology Austria, am Campus 1, 3600 Klosterneuburg, Austria 5Max Planck Institute for the Science of Light, Staudtstr. 2, 90158 Erlangen, Germany 6Institute for Optics, Information and Photonics, University Erlangen-Nuernberg, Staudtstr. 7/B2, 91058 Erlangen, Germany (Dated: June 26, 2019) Quantum information technology based on solid state qubits has created much interest in convert- ing quantum states from the microwave to the optical domain. Optical photons, unlike microwave photons, can be transmitted by fiber, making them suitable for long distance quantum communi- cation. Moreover, the optical domain offers access to a large set of very well developed quantum optical tools, such as highly efficient single-photon detectors and long-lived quantum memories. For a high fidelity microwave to optical transducer, efficient conversion at single photon level and low added noise is needed. Currently, the most promising approaches to build such systems are based on second order nonlinear phenomena such as optomechanical and electro-optic interactions. Alterna- tive approaches, although not yet as efficient, include magneto-optical coupling and schemes based on isolated quantum systems like atoms, ions or quantum dots. In this Progress Report, we pro- vide the necessary theoretical foundations for the most important microwave-to-optical conversion experiments, describe their implementations and discuss current limitations and future prospects. CONTENTS charge qubits in various material systems [6, 7]. Typical characteristic energy scales for these systems correspond I. Introduction 1 to radiation frequencies of order 10 GHz, allowing them to be readily manipulated using commercial microwave II. Cavities 2 technology. A. Cavity properties 2 Proposals for novel quantum information processing B. Coupled systems 3 techniques often rely on a quantum network [8–12], link- C. Weak and strong coupling 4 ing together multiple qubits or groups of qubits to enable quantum-secure communication [13, 14], novel metrol- III. Experimental approaches 4 ogy techniques [15, 16], or distributed quantum comput- A. Electro-optic coupling 5 ing [17]. However, microwave frequency photons are dif- B. Magneto-optically mediated coupling 7 ficult to transmit over long distances -typical attenua- C. Λ-systems and Rydberg atoms 9 tion in low-loss microwave cables at 10 GHz is more than D. Optomechanically mediated coupling 10 1 dB m−1, which compares very poorly with optical fibres with losses below 0.2 dB km−1 at telecom wavelengths IV. Conclusion and future perspectives 11 (λ 1550 nm, f 193THz). The advantages of trans- mitting≈ quantum≈ information over fibers is immediately Acknowledgments 13 apparent. References 13 Furthermore, the thermal occupancy of optical fre- quency channels is close to zero at room temperature, in arXiv:1906.10255v1 [physics.optics] 24 Jun 2019 contrast to microwave modes; a mode with a frequency I. INTRODUCTION of 10GHz must be cooled below 100mK to reduce the average photon occupancy below 1%. The availability The building block for quantum information technolo- of telecom band single photon detectors, quantum mem- gies is the qubit - a two level system the quantum ories [18, 19] and other technologies common in quan- state of which can be prepared, manipulated and mea- tum optics experiments [20] also suggests the need for the sured [1]. Many promising solid state implementations of development of techniques for the bi-directional transfer qubits have been demonstrated, including superconduct- of quantum information between microwave and optical ing qubits of diverse flavours [2, 3], spin qubits [4, 5], and photons. Such a transducer must have a high fidelity, or quan- tum capacity. Although error correcting quantum algo- rithms exist [21, 22], they typically require an error rate ∗ [email protected] less than 1% [23] with the precise figure depending on 2 both the scheme to be implemented and the nature of quency f0. The mode is subject to loss, which might be the errors. A transducer must also have a high quantum due to radiative losses, absorption by scattering centers, efficiency - close to one output photon must be produced ohmic losses, dielectric losses, etc; these are termed dissi- for every input photon. Quantum capacity is finite only pative losses, and we denote the dissipative intensity loss if the conversion efficiency is greater than 50% [24], al- rate due to all these effects by κ′. (Alternatively, the field though indirect schemes involving heralded entanglement loss rate γ′ = κ′/2 can be used.) Typically the mode is of photons may avoid this limit [25]. probed via its coupling to travelling waves that are prop- Qubit implementations relying on microwave excita- agating either in free space, or in waveguides such as a tions are typically operated in dilution fridges with base coax cable or optical fibre, through one or more ports temperatures of around 10 mK. The transducer must with coupling rates κj . The mode can be excited via therefore operate in a cryogenic environment. This also these ports, but also loses energy through them. The to- prevents inadvertent up-conversion of thermal microwave tal loss rate is therefore the sum of dissipative losses, and ′ photons, but places stringent restrictions on the power external losses through ports, κ = κ + j κj . This gives dissipation in the device; the cooling power of a dilu- the linewidth of the mode. Narrow linewidthsP correspond tion fridge at 100 mK is typically only a few hundred to long photon confinement times τp = 1/κ, allowing microwatts. longer interaction lengths for weak non-linear effects to Finally, quantum systems rapidly lose information to become significant. The quality factor of a mode of an- their environment due to decoherence. The device must gular frequency ω0 is defined as Q = ω0τp, with a higher therefore have enough bandwidth for sufficient infor- quality factor (Q) corresponding to a longer lifetime for mation to be transmitted before it is lost - the best photons in the cavity at ω0, but also a slower response of decoherence times for superconducting qubits approach the system to a change in stimulus and hence a smaller 0.1 ms [26], corresponding to a bandwidth of 10 kHz. bandwidth. These requirements combine to make the task a chal- The cavity mode can be characterized by exciting the lenging one. Efficient frequency mixing cannot occur mode via a port, and measuring the reflected power. unless a significant non-linearity is introduced. This The ratio of reflected to incident power is termed S11. can come from the susceptibility of a transparent ma- (Modes can also be probed in transmission by measuring terial such as lithium niobate (LiNbO3), leading to an the power emitted from a second port, but this is unde- electro-optic non-linearity. Alternatively, more extreme sirable because the absence of a baseline makes analy- non-linearities are found near the resonances of three (or sis of resonator loss impossible.) The interplay between more) level systems, such as rare earth ions in crystals, the coupling rate at the measured port κ1 and the other ′ or rubidium vapors. The non-linearity can also emerge loss rates κ + j≥2 κj allows three different regimes to due to indirect coupling mediated by another mode, such ′ be defined; under-coupledP (κ1 < κ + j≥2 κj ), over- as mechanical vibrations or magnetostatic modes. ′ coupled (κ1 > κ + j≥2 κj ), and criticallyP coupled The effect of non-linearities can be increased by plac- ′ (κ1 = κ + κj ).P They can be distinguished by ex- ing the material in resonant cavities, where they expe- j≥2 amining bothP real and imaginary parts (or, equivalently, rience both an enhanced photon interaction time, and a amplitude and phase) of the reflected signal as a function modified density of optical states. Because of the im- of frequency. Typical data for a 3D microwave cavity are portance of cavities to experimental implementations of shown in Fig 1, and the analysis can be made robust microwave-to-optical transducers, we start this Progress against noise and reflections by including terms describ- Report by giving an overview of the physics of cavity ing the environment to which it is coupled [27]. modes. We then detail current experimental approaches, Besides coherent excitation at the input port, at finite before summarizing progress to date and outlining pos- temperatures a cavity mode is thermally occupied. For sible future directions. a mode of temperature T and angular frequency ω, the mean thermal occupancy is nth = kBT/~ω. The tem- II. CAVITIES perature of the mode can be reduced by reducing the temperature of the internal cavity environment, Ti, but is also dependent on the external temperatures, T , to A. Cavity properties j which the mode is coupled via the jth port. The mode temperature is given by the weighted average of all cou- 1 Electromagnetic cavities support long-lived localized pled temperatures, electromagnetic modes, characterized by a resonant fre- κ′ κ T = T + (j) T . (1) κ i κ (j) 1 The terms resonator and cavity are sometimes used interchange- Xj ably in the literature. Here, for brevity, we use the term cavity to describe all structures supporting an electromagnetic mode, Thermal occupancy is negligible for optical frequency −13 including metallic cavities, transmission line resonators, Fabry- modes even at room temperature (nth < 10 ), but P´erot type optical cavities formed from parallel mirrors, photonic is significant for microwave modes (Ω 10 GHz unless crystal cavities, and dielectric resonators. ∼ −2 cooled to cryogenic temperatures (nth = 10 at 104 mK.