Superconducting resonators with voltage-controlled frequency and nonlinearity
William M. Strickland, Bassel Heiba Elfeky, Joseph O'Connell Yuan, William F. Schiela, Peng Yu, Dylan Langone, Maxim G. Vavilov, Vladimir E. Manucharyan, Javad Shabani
Voltage-tunable superconductor-semiconductor devices offer a unique platform to realize dynamic tunability in superconducting quantum circuits. By galvanically connecting a gated InAs-Al Josephson junction to a coplanar waveguide resonator, we demonstrate the use of a wide-range gate-tunable superconducting element. We show that the resonant frequency is controlled via a gate-tunable Josephson inductance and that the non-linearity of the voltage-controlled InAs-Al junction is non-dissipative as is the case with conventional Al-AlO$_{x}$ junctions. As the gate voltage is decreased, the inductive participation of the junction increases up to $44\%$, resulting in the resonant frequency being tuned by over 2 GHz. Utilizing the wide tunability of the device, we demonstrate that two resonant modes can be adjusted such that they strongly hybridize, exhibiting an avoided level crossing with a coupling strength of 51 MHz. Implementing such voltage-tunable resonators is the first step toward realizing wafer-scale continuous voltage control in superconducting circuits for qubit-qubit coupling, quantum-limited amplifiers, and quantum memory platforms.
We address quantitatively the relationship between the nonlinearity of a mechanical resonator and the nonclassicality of its ground state. In particular, we analyze the nonclassical properties of the nonlinear Duffing oscillator (being driven or not) as a paradigmatic example of a nonlinear nanomechanical resonator. We first discuss how to quantify the nonlinearity of this system and then show that the nonclassicality of the ground state, as measured by the volume occupied by the negative part of the Wigner function, monotonically increases with the nonlinearity in all the working regimes addr…
Nonlinearity in macroscopic mechanical system plays a crucial role in a wide variety of applications, including signal transduction and processing, synchronization, and building logical devices. However, it is difficult to generate nonlinearity due to the fact that macroscopic mechanical systems follow the Hooke's law and response linearly to external force, unless strong drive is used. Here we propose and experimentally realize a record-high nonlinear response in macroscopic mechanical system by exploring the anharmonicity in deforming a single chemical bond. We then demonstrate the tunabilit…
We show a purely electronic cooling scheme to cool a charged mechanical resonator (MR) down to nearly the vibrational ground state by elaborately tuning bias gate voltages on the electrodes, which couple the MR by Coulomb interaction. The key step is the modification of time-dependent effective eigen-frequency of the MR based on the Lewis-Riesenfeld invariant. With respect to a relevant idea proposed previously [Li et al., Phys. Rev. A 83, 043803 (2011)], our scheme is simpler, more practical and completely within the reach of current technology.
We present a scheme for the deterministic generation of entangled photon pairs in a superconducting resonator array. The resonators form a Jaynes-Cummings lattice via the coupling to superconducting qubits, and the Kerr-like nonlinearity arises due to the coupling.We show that entangled photons can be generated on demand by applying spectroscopic techniques and exploiting the nonlinearity and symmetry in the resonators. The scheme is robust against small parameter spreads due to fabrication errors. Our findings can be used as a key element for quantum information processing in superconducting …
We propose a scheme for enhancing the optomechanical coupling between microwave and mechanical resonators by up to seven orders of magnitude to the ultrastrong coupling limit in a circuit optomechanical setting. The tripartite system considered here consists of a Josephson junction Cooper-pair box that mediates the coupling between the microwave cavity and the mechanical resonator. The optomechanical coupling can be modified by tuning the gate charge and the magnetic flux bias of the Cooper-pair box which in turn affect the Josephson capacitance of the Cooper-pair box. We additionally show tha…
When the nonlinearity of nanomechanical resonator is not negligible, the quantum decoherence of charge qubit is studied analytically. Using nonlinear Jaynes-Cummings model, one explores the possibility of being quantum data bus for nonlinear nanomechanical resonator, the nonlinearity destroys the dynamical quantum information-storage and maintains the revival of quantum coherence of charge qubit. With the calculation of decoherence factor, we demonstrate the influence of the nonlinearity of nanomechanical resonator on engineered decoherence of charge qubit.
We show that highly confined superfluid films are extremely nonlinear mechanical resonators, offering the prospect to realize a mechanical qubit. Specifically, we consider third-sound surface waves, with nonlinearities introduced by the van der Waals interaction with the substrate. Confining these waves to a disk, we derive analytic expressions for the cubic and quartic nonlinearities and determine the resonance frequency shifts they introduce. We predict single-phonon shifts that are three orders of magnitude larger than in current state-of-the-art nonlinear resonators. Combined with the exqu…
The absence of the single-photon nonlinearity has been a major roadblock in developing quantum photonic circuits at optical frequencies. In this paper, we demonstrate a periodically-poled thin film lithium niobate microring resonator (PPLNMR) that reaches 5,000,000%/W second harmonic conversion efficiency---almost 20-fold enhancement over the state-of-the-art---by accessing its largest $χ^{(2)}$ tensor component $d_{33}$ via quasi-phase matching. The corresponding single photon coupling rate $g/2π$ is estimated to be 1.2 MHz, which is an important milestone as it approaches the dissipation rat…
We study controllable single photon scattering in a one-dimensional waveguide coupling with an additional cavity by second order nonlinear materials in a non-cascading configuration, where the additional cavity is embedded with two-level atom and filled with Kerr-nonlinear materials. Considering the second order nonlinear coupling, we analyze the transmission properties of the three different coupling forms as follows: (i) The two-level atom is excited without the Kerr-nonlinearity. (ii)The Kerr-nonlinearity is excited without the two-level atom. (iii) Both of the two-level atom and Kerr-nonli…
Considering the intrinsic nonlinearity in a nanomechanical resonator coupled to a charge qubit, vacuum Rabi splitting effect is studied in a nanomechanical QED (qubit-resonator) system. A driven nonlinear Jaynes-Cummings model describes the dynamics of this qubit-resonator system. Using quantum regression theorem and master equation approach, we have calculated the two-time correlation spectrum analytically. In the weak driving limit, these analytical results clarify the influence of the driving strength and nonlinearity parameter on the correlation spectrum. Also, numerical calculations confi…
The addition of nonlinearity to an harmonic resonator provides a route to complex dynamical behaviour of resonant modes, including coupling between them. We present a superconducting device that makes use of the nonlinearity of Josephson junctions to introduce a controlled, tunable, nonlinear inductance to a thin film coplanar waveguide resonator. Considering the device as a potential quantum optical component in the microwave regime, we create a sensitive bifurcation amplifier and then demonstrate spectroscopy of other resonant modes via the intermode coupling. We find that the sensitivity of…
In this work, we propose a method to investigate controllable qubit-resonator interactions in a Dicke model with driven biased term. The nonlinearity of spectrum, which can be induced by qubit-resonator interactions, plays an important role in such controllable interactions. To gain insight into mechanism of the nonlinearity, we perform a unitary transformation to the Hamiltonian. The results show that the nonlinearity of the transformed Hamiltonian depends on the qubit-resonator coupling strength. The general forms of the effective Hamiltonians are discussed in detail based on the frequency m…
We provide a general method to find the Hamiltonian of a linear circuit in the presence of a nonlinearity. Focussing on the case of a Josephson junction embedded in a transmission-line resonator, we solve for the normal modes of the system by taking into account exactly the effect of the quadratic (i.e. inductive) part of the Josephson potential. The nonlinearity is then found to lead to self and cross-Kerr effect, as well as beam-splitter type interactions between modes. By adjusting the parameters of the circuit, the Kerr coefficient K can be made to reach values that are weak (K < κ), stron…
Ions confined using a Paul trap require a stable, high voltage and low noise radio frequency (RF) potential. We present a guide for the design and construction of a helical coil resonator for a desired frequency that maximises the quality factor for a set of experimental constraints. We provide an in-depth analysis of the system formed from a shielded helical coil and an ion trap by treating the system as a lumped element model. This allows us to predict the resonant frequency and quality factor in terms of the physical parameters of the resonator and the properties of the ion trap. We also co…
We seek the first indications that a nanoelectromechanical system (NEMS) is entering the quantum domain as its mass and temperature are decreased. We find them by studying the transition from classical to quantum behavior of a driven nonlinear Duffing resonator. Numerical solutions of the equations of motion, operating in the bistable regime of the resonator, demonstrate that the quantum Wigner function gradually deviates from the corresponding classical phase-space probability density. These clear differences that develop due to nonlinearity can serve as experimental evidence, in the near fut…
We investigate a network of coupled superconducting transmission line resonators, each of them made nonlinear with a capacitively shunted Josephson junction coupling to the odd flux modes of the resonator. The resulting eigenmode spectrum shows anticrossings between the plasma mode of the shunted junction and the odd resonator modes. Notably, we find that the combined device can inherit the complete nonlinearity of the junction, allowing for a description as a harmonic oscillator with a Kerr nonlinearity. Using a dc SQUID instead of a single junction, the nonlinearity can be tuned between 10 k…
We consider a nanomechanical analogue of a nonlinear interferometer, consisting of two parallel, flexural nanomechanical resonators, each with an intrinsic Duffing nonlinearity and with a switchable beamsplitter-like coupling between them. We calculate the precision with which the strength of the nonlinearity can be estimated and show that it scales as $1/n^{3/2}$, where $n$ is the mean phonon number of the initial state. This result holds even in the presence of dissipation, but assumes the ability to make measurements of the quadrature components of the nanoresonators.
We explore theoretically the physics of dynamic hysteresis for driven-dissipative nonlinear photonic resonators. In the regime where the semiclassical mean-field theory predicts bistability, the exact steady-state density matrix is known to be unique, being a statistical mixture of two states: in particular, no static hysteresis cycle of the excited population occurs as a function of the driving intensity. Here, we predict that in the quantum regime a dynamic hysteresis with a rich phenomenology does appear when sweeping the driving amplitude in a finite time. The hysteresis area as a function…
We have performed spectroscopic measurements of a superconducting qubit dispersively coupled to a nonlinear resonator driven by a pump microwave field. Measurements of the qubit frequency shift provide a sensitive probe of the intracavity field, yielding a precise characterization of the resonator nonlinearity. The qubit linewidth has a complex dependence on the pump frequency and amplitude, which is correlated with the gain of the nonlinear resonator operated as a small-signal amplifier. The corresponding dephasing rate is found to be close to the quantum limit in the low-gain limit of the am…
We study the full field and frequency filtered output photon statistics of a resonator in thermal equilibrium with a bath and containing an arbitrarily large quartic nonlinearity. According to the general theory of photodetection, we derive general input-output relations valid for the ultra-anharmonic regime, where the nonlinearity becomes comparable to the energy of the resonator, and show how the emission properties are modified as compared to the generally assumed simple anharmonic regime. We analyse the impact of the nonlinearity on the full statistics of the emission and its spectral prop…
We consider a transmission line resonator which is driven by electrons tunneling through a voltage-biased tunnel junction. Using the Born-Markovian quantum master equation in the polaron basis we investigate the nonequilibrium photon state and emission spectrum of the resonator as well as properties of the transport current across the tunnel junction and its noise spectrum. The electroluminescence is optimized, with maximum peak height and narrow linewidth, when the back-action of the tunnel junction on the resonator and the decay rate of the resonator are similar in strength. For strong coupl…
We study the quantum dynamics of a symmetric nanomechanical graphene resonator with degenerate flexural modes. Applying voltage pulses to two back gates, flexural vibrations of the membrane can be selectively actuated and manipulated. For graphene, nonlinear response becomes important already for amplitudes comparable to the magnitude of zero point fluctuations. We show, using analytical and numerical methods, that this allows for creation of cat-like superpositions of coherent states as well as superpositions of coherent cat-like non-product states.
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…
We report on the investigation and implementation of a lumped-component, radio-frequency resonator used in a cryogenic vacuum environment to drive an ion trap. The resonator was required to achieve the voltages necessary to trap (about 100 V), while dissipating as little power as possible (< 250 mW). Ultimately a voltage gain of 100 was measured at 5.7 K. Single calcium ions were confined in a trap driven by this device, providing proof of successful resonator operation at low temperature.
Superconducting circuits present a promising platform with which to realize a quantum refrigerator. Motivated by this, we fabricate and perform spectroscopy of a gated Cooper-pair box, capacitively coupled to two superconducting coplanar waveguide resonators with different frequencies. We experimentally demonstrate the strong coupling of a charge qubit to two superconducting resonators, with the ability to perform voltage driving of the qubit at GHz frequencies. We go on to discuss how the measured device could be modified to operate as a cyclic quantum refrigerator by terminating the resonato…
We have designed and fabricated a half-wavelength reflection line resonator (RLR) that consists of a pair of two coupled microstrip lines on a GaAs/AlGaAs heterostructure. By changing the top gate voltage on a square of two dimensional electron gas under the resonator, a large range of the quality factors can be obtained. Energy loss in the two-dimensional electron gas can be minimized, thus realizing a versatile resonator suitable for integration with semiconductor quantum circuits.
We have observed period-tripling subharmonic oscillations, in a superconducting coplanar waveguide resonator operated in the quantum regime, $k_B T \ll \hbarω$. The resonator is terminated by a tunable inductance that provides a Kerr-type nonlinearity. We detected the output field quadratures at frequencies near the fundamental mode, $ω/2π\sim 5\,$GHz, when the resonator was driven by a current at $3ω$ with an amplitude exceeding an instability threshold. The output radiation was red-detuned from the fundamental mode. We observed three stable radiative states with equal amplitudes and phase-sh…
In quantum computing architectures, one important factor is the trade-off between the need to couple qubits to each other and to an external drive and the need to isolate them well enough in order to protect the information for an extended period of time. In the case of superconducting circuits, one approach is to utilize fixed frequency qubits coupled to coplanar waveguide resonators such that the system can be kept in a configuration that is relatively insensitive to noise. Here, we propose a scalable voltage-tunable quantum memory (QuMem) design concept compatible with superconducting qubit…
We considered a nanoelectromechanical system consisting of a movable Cooper-pair box qubit which is subject to an electrostatic field, and coupled to the two bulk superconductors via tunneling processes. We suggest that qubit dynamics is related to the one of a quantum oscillator and demonstrate that a bias voltage applied between superconductors generates states represented by the entanglement of qubit states and coherent states of the oscillator if certain resonant conditions are fulfilled. It is shown that a structure of this entanglement may be controlled by the bias voltage in a way that …
We propose a spectroscopic approach to probe tiny vibrations of a nanomechanical resonator (NAMR), which may reveal classical or quantum behavior depending on the decoherence-inducing environment. Our proposal is based on the detection of the voltage-fluctuation spectrum in a superconducting transmission line resonator (TLR), which is {\it indirectly} coupled to the NAMR via a controllable Josephson qubit acting as a quantum transducer. The classical (quantum mechanical) vibrations of the NAMR induce symmetric (asymmetric) Stark shifts of the qubit levels, which can be measured by the voltage …
The nonlinearity exhibited by the kinetic inductance of a superconducting stripline couples stripline resonator modes together in a manner suitable for quantum non-demolition measurement of the number of photons in a given resonator mode. Quantum non-demolition measurement is accomplished by coherently driving another resonator mode, referred to as the detector mode, and measuring its response. We show that the sensitivity of such a detection scheme is directly related to the dephasing rate induced by such an intermode coupling. We show that high sensitivity is expected when the detector mode …
Voltage current characteristics of a PN-junction diode are intrinsically nonlinear in nature. It is shown in this paper that a mathematical form of nonlinearity of a PN-junction diode resembles the nonlinear response of electric polarization of a dielectric medium to the electric field. Nonlinearity of a PN-junction can be expressed in a series of successively increasing orders of the nonlinearity. For a PN-junction diode, higher order nonlinear terms become significant as a voltage across the diode is increased. In this paper, a gradual emergence of a nonlinear regime with the amplitude of a …
The realization of cross-Kerr nonlinearity is an important task for many applications in quantum information processing. In this work, we propose a method for realizing cross-Kerr nonlinearity interaction between two superconducting coplanar waveguide resonators coupled by a three-level superconducting flux qutrit (coupler). By employing the qutrit-resonator dispersive interaction, we derive an effective Hamiltonian involving two-photon number operators and a coupler operator. This Hamiltonian can be used to describe a cross-Kerr nonlinearity interaction between two resonators when the coupler…
We demonstrate ultra-sensitive measurement of fluctuations in a surface-acoustic-wave~(SAW) resonator using a hybrid quantum system consisting of the SAW resonator, a microwave (MW) resonator and a superconducting qubit. The nonlinearity of the driven qubit induces parametric coupling, which up-converts the excitation in the SAW resonator to that in the MW resonator. Thermal fluctuations of the SAW resonator near the quantum limit are observed in the noise spectroscopy in the MW domain.
We propose to simulate quantum gates by \textit{LC} resonators, where the amplitude and the phase of the voltage describe the quantum state. By controlling capacitance or inductance of resonators, it is possible to control the phase of the voltage arbitrarily. A set of resonators acts as the phase-shift, the Hadamard and the CNOT gates. They constitute a set of universal quantum gates. We also discuss an application to an artificial neuron. As an example, we study a pattern recognition of numbers and alphabets by evaluating the similarity between an input and the reference pattern. We also stu…
Superconducting resonators provide a convenient way to measure loss tangents of various dielectrics at low temperature. For the purpose of examining the microscopic loss mechanisms in dielectrics, precise measurements of the internal quality factor at different values of energy stored in the resonators are required. Here, we present a consistent method to analyze a LC superconducting resonator coupled to a transmission line. We first derive an approximate expression for the transmission S-parameter $S_{21}$ based on a complete circuit model. In the weak coupling limit, we show that the interna…
Entangled-photon pairs are an essential resource for quantum information technologies. Chip-scale sources of entangled pairs have been integrated with various photonic platforms, including silicon, nitrides, indium phosphide, and lithium niobate, but each has fundamental limitations that restrict the photon-pair brightness and quality, including weak optical nonlinearity or high waveguide loss. Here, we demonstrate a novel, ultra-low-loss AlGaAs-on-insulator platform capable of generating time-energy entangled photons in a $Q$ $>1$ million microring resonator with nearly 1,000-fold improvement…
Cat states of the microwave field stored in high-Q resonators show great promise for robust encoding and manipulation of quantum information. Here we propose an approach to efficiently prepare such cat states in a Kerr-nonlinear resonator by the use of a two-photon drive. We show that this preparation is robust against single-photon loss. We moreover find that it is possible to remove undesirable phase evolution induced by a Kerr nonlinearity using a two-photon drive of appropriate amplitude and phase. Finally, we present a universal set of quantum logical gates that can be performed on the en…
We study the dynamics of a transmon qubit that is capacitively coupled to an open multimode superconducting resonator. Our effective equations are derived by eliminating resonator degrees of freedom while encoding their effect in the Green's function of the electromagnetic background. We account for the dissipation of the resonator exactly by employing a spectral representation for the Green's function in terms of a set of non-Hermitian modes and show that it is possible to derive effective Heisenberg-Langevin equations without resorting to the rotating wave, two level or Markov approximations…
We propose the use of a superconducting charge qubit capacitively coupled to two resonant nanomechanical resonators to generate Yurke-Stoler states, i.e. quantum superpositions of pairs of distinguishable coherent states 180$^\circ$ out of phase with each other. This is achieved by effectively implementing Kerr nonlinearities induced through application of a strong external driving field in one of the resonators. A simple study of the effect of dissipation on our scheme is also presented, and lower bounds of fidelity and purity of the generated state are calculated. Our procedure to implement …
We explore applications of nonlinear circuit QED with a charge qubit inductively coupled to a microwave LC resonator in the photonic engineering and ultrastrong-coupling multiphoton quantum optics. Simply sweeping the gate-voltage bias achieves arbitrary Fock-state pulsed maser, where the single qubit plays the role of artificial gain medium. Resonantly pumping the parametric qubit-resonator interface leads to the squeezing of resonator field, which is utilizable to the quantum-limited microwave amplification. Moreover, upwards and downwards multiphoton quantum jumps may be observed in the ste…
We study coherent phonon oscillations and tunneling between two coupled nonlinear nanomechanical resonators. We show that the coupling between two nanomechanical resonators creates an effective phonon Josephson junction which exhibits two different dynamical behaviors: Josephson oscillation (phonon-Rabi oscillation) and macroscopic self-trapping (phonon blockade). Self-trapping originates from mechanical nonlinearities, meaning that when the nonlinearity exceeds its critical value, the energy exchange between the two resonators is suppressed, and phonon-Josephson oscillations between them are …
We consider a generalization of recently proposed non-Hermitian model for resonant cavities coupled by a chiral mirror by taking into account number non-conservation and nonlinear interactions. We analyze non-Hermitian quantum dynamics of populations and entanglement of the cavity modes. We find that an interplay of initial coherence and non-Hermitian coupling leads to a counterintuitive population transfer. While an initially coherent cavity mode is depleted, the other empty cavity can be populated more or less than the initially filled one. Moreover, presence of nonlinearity yields populatio…
Suppressing Cavity Frequency Noise Using a Kerr Nonlinearity
Kavli Institute of NanoScience, Delft University of Technology（代尔夫特理工大学卡弗里纳米科学研究所）；Department of Electrical and Software Engineering, University of Calgary（卡尔加里大学电气与软件工程系）
Creation of correlated photon pairs is one of the key topics in contemporary quantum optics. Here, we theoretically describe the generation of photon pairs in the process of spontaneous parametric down-conversion in a resonant spherical nanoparticle made of a dielectric material with bulk $χ^{(2)}$ nonlinearity. We pick the nanoparticle size that satisfies the condition of resonant eigenmodes described by Mie theory. We reveal that highly directional photon-pair generation can be observed utilising the nonlinear Kerker-type effect, and that this regime provides useful polarisation correlations…
