Majorana bound states have been proposed as building blocks for qubits on which certain operations can be performed in a topologically protected way using braiding. However, the set of these protected operations is not sufficient to realize universal quantum computing. We show that the electric field in a microwave cavity can induce Rabi oscillations between adjacent Majorana bound states. These oscillations can be used to implement an additional single-qubit gate. Supplemented with one braiding operation, this gate allows to perform arbitrary single-qubit operations.
We find a superradiant quantum phase transition in the model of triangular molecular magnets coupled to the electric component of a microwave cavity field. The transition occurs when the coupling strength exceeds a critical value which, in sharp contrast to the standard two-level emitters, can be tuned by an external magnetic field. In addition to emitted radiation, the molecules develop an in-plane electric dipole moment at the transition. We estimate that the transition can be detected in state of the art microwave strip-line cavities containing $10^{15}$ molecules.
We propose microwave-controlled rotations for qubits realized as Majorana bound states. To this end we study an inhomogeneous Kitaev chain in a microwave cavity. The chain consists of two topologically nontrivial regions separated by a topologically trivial, gapped region. The Majorana bound states at the interfaces between the left (right) regions and the central region are coupled, and their energies are split by virtual cotunneling processes. The amplitude for these cotunneling processes decreases exponentially in the number of sites of the gapped region, and the decay length diverges as th…
We study the interplay between Andreev (Majorana) bound states that form at the boundary of a (topological) superconductor and a train of microwave pulses. We find that the extra dynamical phase coming from the pulses can shift the phase of the Andreev reflection, resulting in the appear- ance of dynamical Andreev states. As an application we study the presence of the zero bias peak in the differential conductance of a normal-topological superconductor junction - the simplest, yet somehow ambiguous, experimental signature for Majorana states. Adding microwave radiation to the measuring electro…
We present an experimental study of missing level statistics of three-dimensional chaotic microwave cavities. The investigation is reinforced by the power spectrum of level fluctuations analysis which also takes into account the missing levels. On the basis of our data sets we demonstrate that the power spectrum of level fluctuations in combination with short- and long-range spectral fluctuations provides a powerful tool for the determination of the fraction of randomly missing levels in systems that display wave chaos such as the three-dimensional chaotic microwave cavities. The experimental …
We study a voltage biased Josephson junction coupled to two resonators of incommensurate frequencies. Using a density approach to analyze the cavity fields and an input-output description to analyze the emitted photonic fluxes and their correlation functions, we have shown, both for infinite and finite bandwidth detectors, that the emitted radiation is non-classical in the sense that the correlators violates Cauchy-Schwarz inequalities. We have also studied the time dependence of the photonic correlations and showed that their line-width becomes narrower with the increase of the emission rate …
We apply Mie scattering theory to study the interaction of magnetic spheres with microwaves in cavities beyond the magnetostatic and rotating wave approximations. We demonstrate that both strong and ultra-strong coupling can be realized for a stand alone magnetic spheres made from yttrium iron garnet (YIG), acting as an efficient microwave antenna. The eigenmodes of YIG spheres with radii of the order mm's display distinct higher angular momentum character that has been observed in experiments.
We formulate a scattering theory to study magnetic films in microwave cavities beyond the independent-spin and rotating wave approximations of the Tavis-Cummings model. We demonstrate that strong coupling can be realized not only for the ferromagnetic resonance (FMR) mode, but also for spin wave resonances (SWRs); the coupling strengths are mode dependent and decrease with increasing mode index. The strong coupling regime can be also accessed electrically by spin pumping into a metal contact.
Graphene is an attractive material for nanomechanical devices because it allows for exceptional properties, such as high frequencies and quality factors, and low mass. An outstanding challenge, however, has been to obtain large coupling between the motion and external systems for efficient readout and manipulation. Here, we report on a novel approach, in which we capacitively couple a high-Q graphene mechanical resonator ($Q \sim 10^5$) to a superconducting microwave cavity. The initial devices exhibit a large single-photon coupling of $\sim 10$ Hz. Remarkably, we can electrostatically change …
Low-loss cavities are important in building high-coherence superconducting quantum computers. Generating high quality joints between parts is crucial to the realization of a scalable quantum computer using the circuit quantum electrodynamics (cQED) framework. In this paper, we adapt the technique of indium bump bonding to the cQED architecture to realize high quality superconducting microwave joints between chips. We use this technique to fabricate compact superconducting cavities in the multilayer microwave integrated quantum circuits (MMIQC) architecture and achieve single photon quality fac…
Distinguishing Majorana bound states from accidental zero-energy modes with a microwave cavity
International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所马格托国际研究中心）；CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎理工学院综合理工学院中心）；Department of Physics, University of Basel（巴塞尔大学物理系）
We consider an interferometer based on artificially induced topological superconductivity and chiral 1D Majorana fermions. The (non-topological) superconducting island inducing the superconducting correlations in the topological substrate is assumed to be floating. This allows probing the physics of interfering Majorana modes via microwave response, i.e., the frequency dependent impedance between the island and the earth. Namely, charging and discharging of the island is controlled by the time-delayed interference of chiral Majorana excitations in both normal and Andreev channels. We argue tha…
We find the admittance of a topological Josephson junction $Y(ω,φ_0, T)$ as a function of frequency $ω$, the static phase bias $φ_0$ applied to the superconducting leads, and temperature $T$. The dissipative part of $Y$ allows for spectroscopy of the sub-gap states in the junction. The resonant frequencies $ω_{\mathcal{M}, n} (φ_0)$ for transitions involving the Majorana ($\mathcal{M}$) doublet exhibit characteristic kinks in the $φ_0$-dependence at $φ_0=π$. The kinks -- associated with decoupled Majorana states -- remain sharp and the corresponding spectroscopic lines are bright at any temper…
Microwaves couple to magnetic moments in both ferromagnets and antiferromagnets. Although the magnons in ferromagnets and antiferromagnets radically differ, they can become entangled via strong coupling to the same microwave mode in a cavity. The equilibrium configuration of the magnetic moments crucially governs the coupling between the different magnons because the antiferromagnetic and ferromagnetic magnons have the opposite spins when their dispersion relations cross. We derive analytical expressions for the coupling strengths and find that the coupling between antiferromagnets and ferroma…
We present a comprehensive study of internal quality factors in superconducting stub-geometry 3-dimensional cavities made of aluminum. We use wet etching, annealing and electrochemichal polishing to improve the as machined quality factor. We find that the dominant loss channel is split between two-level system loss and an unknown source with 60:40 proportion. A total of 17 cavities of different purity, resonance frequency and size were studied. Our treatment results in reproducible cavities, with ten of them showing internal quality factors above 80 million at a power corresponding to an avera…
Majorana qubits offer a promising way to store and manipulate quantum information by encoding it into the state of Majorana zero modes. As the information is stored in a topological property of the system, local noise cannot lead to decoherence. Manipulation of the information is achieved by braiding the zero modes. The measurement, however, is challenging as the information is well hidden and thus inherently hard to access. Here, we discuss a setup for measuring the state of a Majorana qubit by employing standard tools of microwave engineering. The basic physical effect that we employ is the …
By driving a 3D transmon with microwave fields, we generate an effective avoided energy-level crossing. Then we chirp microwave frequency, which is equivalent to driving the system through the avoided energy-level crossing by sweeping the avoided crossing. A double-passage chirp produces Landau-Zener-Stückelberg-Majorana interference that agree well with the numerical results. Our method is fully applicable to other quantum systems that contain no intrinsic avoided level crossing, providing an alternative approach for quantum control and quantum simulation.
Microwave-to-optical conversion via ferromagnetic magnons has so-far been limited by the optical coupling rates achieved in mm-scale whispering gallery mode resonators. Towards overcoming this limitation, we propose and demonstrate an open magneto-optical cavity containing a thin-film of yttrium iron garnet (YIG). We achieve a 0.1 pL (100 $μ$m$^{3}$) optical mode volume, $\sim$50 times smaller than previous devices. From this, we estimate the magnon single-photon coupling rate is $G\approx50$ Hz. This open cavity design offers the prospect of wavelength scale mode volumes, small polarization s…
We study the spectrum of a one-dimensional Kitaev chain placed in a microwave cavity. In the off-resonant regime, the frequency shift of the cavity can be used to identify the topological phase transition of the coupled system. In the resonant regime, the topology of the system can be controlled via the microwave cavity occupation and, moreover, for a large number of photons (classical limit), the physics becomes similar to that of periodically-driven systems (Floquet insulators). We also analyze numerically a finite chain and show the existence of a degenerate subspace in the presence of the …
A trijunction made of three topological semiconducting wires, each supporting a Majorana bound state at its two extremities, appears as one of the simplest geometry in order to perform braiding of Majorana fermions. By embedding the trijunction into a microwave cavity allows to study the intricate dynamics of the low-energy Majorana bound states (MBSs) coupled to the cavity electric field under a braiding operation. Extending a previous work (Phys. Rev. Lett. 2019, 122, 236803), the full time evolution of the density matrix of the low-energy states, including various relaxation channels, is co…
We describe how microwave spectroscopy of cold fermions in quasi-1D traps can be used to detect, manipulate, and entangle exotic non-local qbits associated with "Majorana" edge modes. We present different approaches to generate the p-wave superfluidity which is responsible for these topological zero-energy edge modes. We find that the edge modes have clear signatures in the microwave spectrum, and that the line shape distinguishes between the degenerate states of a qbit encoded in these edge modes. Moreover, the microwaves rotate the system in its degenerate ground-state manifold. We use these…
No experiment could directly test the particle/antiparticle duality of Majorana fermions, so far. However, this property represents a necessary ingredient towards the realization of topological quantum computing schemes. Here, we show how to complete this task by using microwave techniques. The direct coupling between a pair of overlapping Majorana bound states and the electric field from a microwave cavity is extremely difficult to detect due to the self-adjoint character of Majorana fermions which forbids direct energy exchanges with the cavity. We show theoretically how this problem can be …
Coupling Majorana fermion excitations to coherent external fields is an important stage towards their manipulation and detection. We analyse the charge and transmon regimes of a topological nano-wire embedded within a Cooper-Pair-Box, where the superconducting phase difference is coupled to the zero energy parity states that arise from Majorana quasi-particles. We show that at special gate bias points, the photon-qubit coupling can be switched off via quantum interference, and in other points it is exponentially dependent on the control parameter $E_J/E_C$. As well as a probe for topological-s…
We present a theoretical approach to use ferro- or ferrimagnetic nanoparticles as microwave nanomagnonic cavities to concentrate microwave magnetic fields into deeply subwavelength volumes $\sim 10^{-13}$ mm$^3$. We show that the field in such nanocavities can efficiently couple to isolated spin emitters (spin qubits) positioned close to the nanoparticle surface reaching the single magnon-spin strong-coupling regime and mediate efficient long-range quantum state transfer between isolated spin emitters. Nanomagnonic cavities thus pave the way towards magnon-based quantum networks and magnon-med…
We present the design of a reflective stop-band filter based on quasi-lumped elements that can be utilized to introduce large dc and low-frequency voltage biases into a low-loss superconducting coplanar waveguide (CPW) cavity. Transmission measurements of the filter are seen to be in good agreement with simulations and demonstrate insertion losses greater than $20\,{\rm dB}$ in the range of ${\rm 3\,to\,10\,GHz}$. Moreover, transmission measurements of the CPW's fundamental mode demonstrate that loaded quality factors exceeding $10^5$ can be achieved with this design for dc voltages as large a…
Experiments with superconducting microwave cavities have been performed in our laboratory for more than two decades. The purpose of the present article is to recapitulate some of the highlights achieved. We briefly review (i) results obtained with flat, cylindrical microwave resonators, so-called microwave billiards, concerning the universal fluctuation properties of the eigenvalues of classically chaotic systems with no, a threefold and a broken symmetry; (ii) summarize our findings concerning the wave-dynamical chaos in three-dimensional microwave cavities; (iii) present a new approach for t…
We present a scalable architecture for the exploration of interacting topological phases of photons in arrays of microwave cavities, using established techniques from cavity and circuit quantum electrodynamics. A time-reversal symmetry breaking (non-reciprocal) flux is induced by coupling the microwave cavities to ferrites, allowing for the production of a variety of topological band structures including the $α=1/4$ Hofstadter model. Effective photon-photon interactions are included by coupling the cavities to superconducting qubits, and are sufficient to produce a $ν=1/2$ bosonic Laughlin pud…
We propose a method of encoding a topologically-protected qubit using Majorana fermions in a trapped-ion chain. This qubit is protected against major sources of decoherence, while local operations and measurements can be realized. Furthermore, we show that an efficient quantum interface and memory for arbitrary multiqubit photonic states can be built, encoding them into a set of entangled Majorana-fermion qubits inside cavities.
We present a model for how frequency fluctuations comparable to the total cavity linewidth may arise in tunable and nonlinear microwave cavities, and how these fluctuations affect the measurement of scattering matrix elements. Applying this model to the specific case of a two-sided cavity, we obtain closed-form expressions for the average scattering matrix elements in several important cases. A key signature of our model is the subtle deformation of the trajectories swept out by scattering matrix elements in the complex plane. Despite this signature, the fluctuating and non-fluctuating models …
We successfully demonstrate low-frequency noise suppression in the resonant frequency fluctuations of a cavity-embedded Cooper pair transistor (cCPT) driven at single-photon occupancy. In particular, we report a reduction in the resonant frequency fluctuations caused by the internal charge noise over a bandwidth of $\sim$1.4 kHz when the cavity is driven at an average photon number $n=10$, and a bandwidth of 11 Hz for average $n=1$. The gate-dependent tunability of the cCPT allows us to implement a feedback-scheme, derived from the Pound-Drever-Hall locking technique. This reduces fluctuations…
We investigate the microwave magnetic field confinement in several microwave 3D-cavities, using 3D finite-element analysis to determine the best design and achieve strong coupling between microwave resonant cavity photons and solid state spins. Specifically, we design cavities for achieving strong coupling of electromagnetic modes with an ensemble of nitrogen vacancy (NV) defects in diamond. We report here a novel and practical cavity design with a magnetic filling factor of up to 4 times (2 times higher collective coupling) than previously achieved using 1D superconducting cavities with small…
We investigate the low-energy theory of a one-dimensional finite capacitance topological Josephson junction. Charge fluctuations across the junction couple to resonant microwave fields and can be used to probe microscopic excitations such as Majorana and Andreev bound states. This marriage between localized microscopic degrees of freedom and macroscopic dynamics of the superconducting phase, leads to unique spectroscopic patterns which allow us to reveal the presence of Majorana fermions among the low-lying excitations.
We propose a quantum non-demolition Majorana qubit readout protocol based on parametricmodulation of a longitudinal interaction between a pair of Majorana bound states and a resonator. The interaction can be modulated through microwave frequency gate-voltage or flux control of a tunable tunnel barrier. The qubit-resonator coupling is quantum non-demolition to exponential accuracy, a property inherited from the topological nature of the Majorana zero modes. The same mechanism as used for single-qubit readout can also be extended to perform multi-qubit measurements and be used to enact long rang…
The ability to detect single photons has become increasingly essential due to the rise of photon-based quantum computing. In this theoretical work, we propose a system consisting of a quantum dot (QD) side-coupled to a superconducting nanowire. The coupling opens a gap in both the QD mode and the Majorana zero mode (MZM) at the nanowire edge, enabling photon absorption in the system. We show that the absorbed photoelectron decays via rapid (sub-nanosecond to nanosecond) nonradiative heat transfer to the nanowire phonon modes rather than by spontaneous emission. Furthermore, we calculate the te…
We propose and analyze a hybrid device by integrating a microscale diamond beam with a single built-in nitrogen-vacancy (NV) center spin to a superconducting coplanar waveguide (CPW) cavity. We find that under an ac electric field the quantized motion of the diamond beam can strongly couple to the single cavity photons via dielectric interaction. Together with the strong spin-motion interaction via a large magnetic field gradient, it provides a hybrid quantum device where the dia- mond resonator can strongly couple both to the single microwave cavity photons and to the single NV center spin. T…
Topological excitations, such as Majorana zero modes, are a promising route for encoding quantum information. Topologically protected gates of Majorana qubits, based on their braiding, will require some form of network. Here, we propose to build such a network by entangling Majorana matter with light in a microwave cavity QED setup. Our scheme exploits a light-induced interaction which is universal to all the Majorana nanoscale circuit platforms. This effect stems from a parametric drive of the light-matter coupling in a one-dimensional chain of physical Majorana modes. Our setup enables all t…
The interaction between two magnetic spheres in microwave cavities is studied by Mie scattering theory beyond the magnetostatic and rotating wave approximations. We demonstrate that two spatially separated dielectric and magnetic spheres can be strongly coupled over a long distance by standing cavity modes. The interactions splits acoustical (dark) and optical (bright) modes in a way that can be mapped on a molecular orbital theory of the hydrogen molecule. Breaking the symmetry by assigning different radii to the two spheres introduces "ionic" character to the magnonic bonds. These results il…
Coupling a semiconducting nanowire to a microwave cavity provides a powerfull means to assess the presence or absence of isolated Majorana fermions in the nanowire. These exotic bound states can cause a significant cavity frequency shift but also a strong cavity nonlinearity leading for instance to light squeezing. The dependence of these effects on the nanowire gate voltages gives direct signatures of the unique properties of Majorana fermions, such as their self-adjoint character and their exponential confinement.
Significant improvements in superconducting qubit coherence times have been achieved recently with three-dimensional microwave waveguide cavities coupled to transmon qubits. While many of the measurements in this direction have utilized superconducting aluminum cavities, other recent work has involved qubits coupled to copper cavities with coherence times approaching 0.1 ms. The copper provides a good path for thermalizing the cavity walls and qubit chip, although the substantial cavity loss makes conventional dispersive qubit measurements challenging. We are exploring various approaches for i…
When connecting a voltage-biased Josephson junction in series to several microwave cavities, a Cooper-pair current across the junction gives rise to a continuous emission of strongly correlated photons into the cavity modes. Tuning the bias voltage to the resonance where a single Cooper pair provides the energy to create an additional photon in each of the cavities, we demonstrate the entangling nature of these creation processes by simple witnesses in terms of experimentally accessible observables. To characterize the entanglement properties of the such created quantum states of light to the …
