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Nonlinearity and parametric driving in a SQUID-tunable microchip resonator

Yarema Reshitnyk

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

Advances in the field of nanoscale quantum engineered systems depend crucially on the ability to make quantum limited measurements. Current capabilities are limited by technical and quantum noise introduced by devices used for transduction and amplification of the inherently small signals of interest. Many experimental implementations, including nano-mechanical systems, solid-state quantum bits (qubits), and circuit quantum-electrodynamics (c-QED) devices, utilize transmitted or reflected microwave signals to measure the field quadratures of signals composed of few photons. Traditional low-noise cryogenic HEMT amplifiers provide an unsatisfactory solution to this quantum measurement problem because the targeted signal is often much smaller than the added noise. In principle, c-QED devices can be used for quantum-limited amplification, particularly as a parametric amplifier (PA). An ideal parametric amplifier should amplify signals without introducing additional noise, which would allow for the attainment of quantum feedback algorithms. In this thesis, I describe the characterization of a c-QED device, known as a tunable resonator, with the goal of quantifying its limits when used as a parametric amplifier. Tunable resonators are currently being used as PAs in quantum limited experiments, however, some of the underlying dynamics of these devices have not yet been fully characterized. I investigated the frequency, internal loss, power dependent frequency shift (nonlinearity) of a tunable microwave co-planar waveguide (CPW) resonator as a function of external magnetic flux biasing, which changes the effective length of the resonator. By looking at multiple samples with different design parameters, I observed that nonlinear damping is inherent to this device. Nonlinear damping had previously been observed but not described in sufficient detail to provide insight into its origins, which may explain some underlying dynamics of the tunable resonator. My measurements of the nonlinear damping shows that at larger flux biases, the nonlinear damping dominates linear internal losses and represent the main contribution to the total internal loss rate. I then operated this device as a parametric oscillator (PO) and found that the nonlinear damping and large Kerr (or quartic) nonlinearity likely cause an increase in the above-threshold switching rates between π-phase shifted steady states, which in turn may be associated with anomalous above-threshold squeezing in these devices. The results of my thesis show that nonlinear damping is present in these devices and affects performance in them dearly. This result is important for future work in PAs as improved understanding of the nonlinear damping will enable engineering of a device which can act as a PA with maximal gain and minimum noise.

Related papers

Parametrically pumped superconducting circuits

Philip Krantz

In this licentiate thesis, I present the design, fabrication, and characterization of superconducting parametric resonators, for use in quantum information processing. These devices are quarter-wavelength coplanar waveguide resonators (~5 GHz), terminated at one end by a non-linear inductance provided by a SQUID (superconducting quantum interference device). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using on-chip transmission lines. The modulation occurs on a timescale much faster than the photon loss out of the resonator. The non-linearity provides several different regimes for operating this device. I focused on the parametric regime, in which the non-linear element was being modulated (pumped) at or near twice the resonant frequency. This pumping can lead to amplification and frequency conversion of an incoming signal. It can also give rise to instabilities and self-oscillations "parametric oscillations" - that is, the generation of an intense electric field in the resonator (the creation of photons). Parametric oscillations set in when the pumping strength exceeds a threshold value, and also depends on the pump-frequency detuning from twice the resonance and on the static flux through the SQUID. I characterized the devices over a wide parameter range by doing homodyne detection of the reflected signal or of the emitted field oscillations. In particular, I investigated the damping and two leading nonlinearities, dominating in different operating regimes, which influence the dynamics of the parametric resonator. First, I extracted the Duffing nonlinearity by studying the line width of the resonator as a function of the input signal (without pumping the SQUID). Second, with high pump amplitude, I extracted the pump-induced nonlinearity by determining the threshold parameters (pump amplitude and detuning) for the onset of parametric oscillations; this nonlinearity leads to the generation of higher-order pump terms that complicate the system dynamics, but can be mitigated now that they have been understood. My results validate a recent theoretical model for the classical, non-linear dynamics of parametric resonance, and helps determine workable design parameters. Finally, I helped develop a linearized model $-$ the "pumpistor" - describing this device in terms of a flux-controlled impedance. This will be useful when designing more complex circuits. One goal of this work is to demonstrate single-shot measurements on superconducting qubits (quantum bits of information). We will use the parametric resonator as a threshold discriminator, associating the qubit's two energy eigenstates with the parametric oscillations either turning on or not. This would be a very useful device in the quantum engineer's toolbox when designing systems for quantum information processing and communication.

•Jan 1, 2013•1 citation

Parametric Microwave Amplification using a Tunable Superconducting Resonator

C. W. Sandbo Chang

In this Master thesis, I present a theoretical description and experimental measurements of a tunable microwave resonator operated as a parametric amplifier. Superconducting parametric amplifiers have attracted great interest in recent years as it has been demonstrated that they can operate with a noise performance at the standard quantum limit. \nParametric amplification is achieved in the devices by modulating the electrical length of the resonator at twice the resonance frequency. The device measured in this thesis consists of a quarter-wavelength microwave resonator fabricated in a thin-film aluminum coplanar waveguide geometry. With a termination to ground through a Superconducting Quantum Interference Device (SQUID), the boundary condition of the resonator becomes sensitive to an applied magnetic field. This enables the tuning of the resonance frequencies, using a DC magnetic field, over a wide frequency range of 670MHz. Similarly, an AC magnetic field then allows for the modulation of the resonator electrical length at microwave frequencies. \nWe characterized the device by first measuring the DC tuning curves of the 2nd and 3rd harmonics of the resonator, around 4 GHz and 6 GHz respectively. We then studied parametric amplification in the device when the electrical length was modulated at approximately twice the resonance frequency of one of the modes. We scanned the pump frequency and pump power over a wide range to study the region in parameters which gives parametric gain. We performed an in-depth parameter sweep over this region, identifying pump parameters that provided maximum gain and maximum system noise improvements compared the following HEMT amplifier. \nWe find that the use of the parametric amplifier can improve the system noise temperature by more than 10 dB compared to a start-of-the-art commercial HEMT amplifier. This implies a performance near the standard quantum limit.

•Sep 2, 2015•0 citations

The Josephson parametric oscillator - From microscopic studies to single-shot qubit readout

Philip Krantz

Circuit quantum electrodynamics (cQED) is a prominent platform for quantum information processing, in which microwave photons are confined into resonant cavities coupled to superconducting quantum bits (qubits). The large effective dipole moment of the qubit, in combination with the high energy density of the quasi 1-D resonator allow these systems to enter the so-called strong coupling regime. The quantum state of the qubit can then be assessed by probing the frequency of the resonator, protecting the qubit from otherwise losing its energy to the environment. However, eventhough this so-called dispersive readout technique has proven useful, it is often in itself insufficient to render single-shot readout performance | one of the crucial tasks required for realizing a quantum computer. This thesis describes the demonstration of a single-shot readout technique forsuperconducting quantum bits, based on coupling the qubit to a frequency-tunable resonator. The backbone of our device is a 5 GHz quarter-wavelength coplanar waveguide resonator, terminated at one end by a non-linear inductance provided by a superconducting quantum interference device (SQUID). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using an on-chip transmission line. The modulation occurs on a timescale much faster than the photon loss out of the resonator and if the pump strength exceeds a threshold value set by the damping rate of the resonator, an intense photon field will build up inside the resonator | known as "parametric oscillations". By heterodyne detection of the output signal from the Josephson parametric oscillator (JPO), we first extracted two leading nonlinear effects of the system (dominating in different limits of applied magnetic flux). Next, we couple a qubit to the JPO and demonstrate that we can encode its quantum state onto the strong output field of the parametric oscillator (or lack thereof), rendering a signal-to-noise ratio sufficient for single-shot state detection and therefore also obviating a quantum limited parametric amplifier. The thesis also contains results from microscopic studies of the Josephson junctions, which we use to provide the necessary nonlinearities in the above mentioned devices. In particular, we show how the microstructure of the barrier layer and its atomic composition can be used to infer important electrical properties of the junctions. This knowledge allows us to better tailor the properties of Josephson-based devices.

•Jan 1, 2016•3 citations

Parametric interactions with signals and the vacuum

Michaël Simoen

In this thesis I present different experiments on superconducting circuits exploring parametric interactions with external signals and the vacuum in the microwave regime. These parametric processes are the result of the periodic modulation of a property of a system which results in different interactions. The systems used in this thesis are circuits where the nonlinear inductance of a superconducting quantum interference device (SQUID) is parametrically driven.I present the first experimental observation of the dynamical Casimir effect (DCE), since it was predicted in 1970. The DCE is an interaction between the vacuum and a periodically modulated boundary condition of the electromagnetic field, here implemented by a flux-tuned SQUID. In essence the modulated boundary will parametrically amplify the vacuum fluctuations which results in the pairwise generation of photons over a broad frequency range. I have characterized the system and measured the statistical properties of the emitted radiation to show that the radiation exhibits two-mode squeezing.Next, I present measurements on a superconducting multimode resonator containing a SQUID. I show that it is possible to get parametric amplification by driving the SQUID at either twice the frequency of one of the modes or by driving it at the sum of two mode frequencies. In both cases I show that it is possible to reach quantum-limited noise performance.In the same system I also demonstrate frequency conversion which occurs when the system is pumped at the difference frequency between two modes. Microwave photons are coherently transferred from one mode to the other. I show that the coupling strength depends linearly on the applied pump amplitude.The thesis also contains a linearized theoretical model to describe and analyze the flux-pumped SQUID. The model describes an equivalent circuit element called the pumpistor, with an impedance which depends on the pump phase. I show that under specific conditions the impedance becomes real and negative allowing pump power to be injected into the circuit, providing gain.Finally I also present a demonstration of an on-chip Mach-Zehnder interferometer. This experiment uses the tunability of the SQUID to provide a controllable phase shift in one of the interferometer arms. The transmission through the device can be modulated with a maximum change of 45 dB.

•Jan 1, 2015•4 citations

Graphene-based superconducting quantum circuits : the Josephson parametric amplifier

Guilliam Butseraen

With a large portfolio of elemental quantum components, superconducting quantum circuits have contributed to dramatic advances in microwave quantum optics. Of these elements, quantum-limited parametric amplifiers have proven to be essential for low noise readout of quantum systems whose energy range is intrinsically low (tens of µeV). They are also used to generate non classical states of light that can be a resource for quantum enhanced detection. Superconducting parametric amplifiers, like quantum bits, typically utilize a Josephson junction as a source of magnetically tunable and dissipation-free nonlinearity. The magnetic control is not an industry standard for devices and starts already to be an issue in large scale circuits. In recent years, efforts have been made to introduce semiconductor weak links as electrically tunable nonlinear elements, with demonstrations of microwave resonators and quantum bits using semiconductor nanowires, a two dimensional electron gas, carbon nanotubes and graphene. However, given the challenge of balancing nonlinearity, dissipation, participation, and energy scale, parametric amplifiers have not yet been implemented with a semiconductor weak link. The work presented in this PhD thesis demonstrates the design, fabrication and performances of a parametric amplifier leveraging a graphene Josephson junction.The graphene is encapsulated in between hBN flakes in order to improve its quality. Using such high quality junctions, we demonstrate amplification with 20 dB gain which is the milestone for overcoming noise of the classical amplifiers in the rest of the amplification chain. The Josephson parametric amplifier being based on a resonant structure (around 6 GHz) suffers from the gain bandwidth product limiting the amplification to a few MegaHertz frequency range. The use of a side gate to electrically tune the graphene gives control of the Josephson junction inductance and thus the resonant frequency of the amplifier. We demonstrate a near 1 GHz tunability of the amplification frequency with the use of a side gate which is as good as the tunability offered by using a magnetic flux on a SQUID for similar amplifier designs. Moreover, graphene has shown to exhibit nonlinear loss under a strong microwave irradiation arising because of dynamics in the Andreev states. Dissipation is known to add intrinsic noise on quantum amplifiers and could be a problem for quantum limited parametric amplification. We demonstrate that despite the presence of nonlinear loss, graphene based Josephson parametric amplifiers can reach near quantum limited amplification. A two photon loss model was used to describe the behaviour of the device but we show that a more complex model is required in order to take into account the nonlinear dissipation and the non-sinusoidal current phase relation. We also studied the dynamic range and showed that the graphene based Josephson parametric amplifier can reach a 1 dB compression point of -123 dBm which is as good as single tunnel junctions based parametric amplifiers. Our results expand the toolset for electrically tunable superconducting quantum circuits and offer new opportunities for the development of quantum technologies such as quantum computing, quantum sensing and fundamental science.

•Nov 8, 2022•0 citations

Parametric resonance in tunable superconducting cavities

Waltraut Wustmann, Vitaly S. Shumeiko

We develop a theory of parametric resonance in tunable superconducting cavities. The nonlinearity introduced by the superconducting quantum interference device (SQUID) attached to the cavity and damping due to connection of the cavity to a transmission line are taken into consideration. We study in detail the nonlinear classical dynamics of the cavity field below and above the parametric threshold for the degenerate parametric resonance, featuring regimes of multistability and parametric radiation. We investigate the phase-sensitive amplification of external signals on resonance, as well as amplification of detuned signals, and relate the amplifier performance to that of linear parametric amplifiers. We also discuss applications of the device for dispersive qubit readout. Beyond the classical response of the cavity, we investigate small quantum fluctuations around the amplified classical signals. We evaluate the noise power spectrum both for the internal field in the cavity and the output field. Other quantum-statistical properties of the noise are addressed such as squeezing spectra, second-order coherence, and two-mode entanglement.

arXiv:1302.3484•Physical Review B•May 1, 2013•121 citations
Advances in the field of nanoscale quantum engineered systems depend crucially on the ability to make quantum limited measurements. Current capabilities are limited by technical and quantum noise introduced by devices used for transduction and amplification of the inherently small signals of interest. Many experimental implementations, including nano-mechanical systems, solid-state quantum bits (qubits), and circuit quantum-electrodynamics (c-QED) devices, utilize transmitted or reflected microwave signals to measure the field quadratures of signals composed of few photons. Traditional low-noise cryogenic HEMT amplifiers provide an unsatisfactory solution to this quantum measurement problem because the targeted signal is often much smaller than the added noise. In principle, c-QED devices can be used for quantum-limited amplification, particularly as a parametric amplifier (PA). An ideal parametric amplifier should amplify signals without introducing additional noise, which would allow for the attainment of quantum feedback algorithms. In this thesis, I describe the characterization of a c-QED device, known as a tunable resonator, with the goal of quantifying its limits when used as a parametric amplifier. Tunable resonators are currently being used as PAs in quantum limited experiments, however, some of the underlying dynamics of these devices have not yet been fully characterized. I investigated the frequency, internal loss, power dependent frequency shift (nonlinearity) of a tunable microwave co-planar waveguide (CPW) resonator as a function of external magnetic flux biasing, which changes the effective length of the resonator. By looking at multiple samples with different design parameters, I observed that nonlinear damping is inherent to this device. Nonlinear damping had previously been observed but not described in sufficient detail to provide insight into its origins, which may explain some underlying dynamics of the tunable resonator. My measurements of the nonlinear damping shows that at larger flux biases, the nonlinear damping dominates linear internal losses and represent the main contribution to the total internal loss rate. I then operated this device as a parametric oscillator (PO) and found that the nonlinear damping and large Kerr (or quartic) nonlinearity likely cause an increase in the above-threshold switching rates between π-phase shifted steady states, which in turn may be associated with anomalous above-threshold squeezing in these devices. The results of my thesis show that nonlinear damping is present in these devices and affects performance in them dearly. This result is important for future work in PAs as improved understanding of the nonlinear damping will enable engineering of a device which can act as a PA with maximal gain and minimum noise.
In this licentiate thesis, I present the design, fabrication, and characterization of superconducting parametric resonators, for use in quantum information processing. These devices are quarter-wavelength coplanar waveguide resonators (~5 GHz), terminated at one end by a non-linear inductance provided by a SQUID (superconducting quantum interference device). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using on-chip transmission lines. The modulation occurs on a timescale much faster than the photon loss out of the resonator. The non-linearity provides several different regimes for operating this device. I focused on the parametric regime, in which the non-linear element was being modulated (pumped) at or near twice the resonant frequency. This pumping can lead to amplification and frequency conversion of an incoming signal. It can also give rise to instabilities and self-oscillations "parametric oscillations" - that is, the generation of an intense electric field in the resonator (the creation of photons). Parametric oscillations set in when the pumping strength exceeds a threshold value, and also depends on the pump-frequency detuning from twice the resonance and on the static flux through the SQUID. I characterized the devices over a wide parameter range by doing homodyne detection of the reflected signal or of the emitted field oscillations. In particular, I investigated the damping and two leading nonlinearities, dominating in different operating regimes, which influence the dynamics of the parametric resonator. First, I extracted the Duffing nonlinearity by studying the line width of the resonator as a function of the input signal (without pumping the SQUID). Second, with high pump amplitude, I extracted the pump-induced nonlinearity by determining the threshold parameters (pump amplitude and detuning) for the onset of parametric oscillations; this nonlinearity leads to the generation of higher-order pump terms that complicate the system dynamics, but can be mitigated now that they have been understood. My results validate a recent theoretical model for the classical, non-linear dynamics of parametric resonance, and helps determine workable design parameters. Finally, I helped develop a linearized model −-− the "pumpistor" - describing this device in terms of a flux-controlled impedance. This will be useful when designing more complex circuits. One goal of this work is to demonstrate single-shot measurements on superconducting qubits (quantum bits of information). We will use the parametric resonator as a threshold discriminator, associating the qubit's two energy eigenstates with the parametric oscillations either turning on or not. This would be a very useful device in the quantum engineer's toolbox when designing systems for quantum information processing and communication.
In this Master thesis, I present a theoretical description and experimental measurements of a tunable microwave resonator operated as a parametric amplifier. Superconducting parametric amplifiers have attracted great interest in recent years as it has been demonstrated that they can operate with a noise performance at the standard quantum limit. \nParametric amplification is achieved in the devices by modulating the electrical length of the resonator at twice the resonance frequency. The device measured in this thesis consists of a quarter-wavelength microwave resonator fabricated in a thin-film aluminum coplanar waveguide geometry. With a termination to ground through a Superconducting Quantum Interference Device (SQUID), the boundary condition of the resonator becomes sensitive to an applied magnetic field. This enables the tuning of the resonance frequencies, using a DC magnetic field, over a wide frequency range of 670MHz. Similarly, an AC magnetic field then allows for the modulation of the resonator electrical length at microwave frequencies. \nWe characterized the device by first measuring the DC tuning curves of the 2nd and 3rd harmonics of the resonator, around 4 GHz and 6 GHz respectively. We then studied parametric amplification in the device when the electrical length was modulated at approximately twice the resonance frequency of one of the modes. We scanned the pump frequency and pump power over a wide range to study the region in parameters which gives parametric gain. We performed an in-depth parameter sweep over this region, identifying pump parameters that provided maximum gain and maximum system noise improvements compared the following HEMT amplifier. \nWe find that the use of the parametric amplifier can improve the system noise temperature by more than 10 dB compared to a start-of-the-art commercial HEMT amplifier. This implies a performance near the standard quantum limit.
Circuit quantum electrodynamics (cQED) is a prominent platform for quantum information processing, in which microwave photons are confined into resonant cavities coupled to superconducting quantum bits (qubits). The large effective dipole moment of the qubit, in combination with the high energy density of the quasi 1-D resonator allow these systems to enter the so-called strong coupling regime. The quantum state of the qubit can then be assessed by probing the frequency of the resonator, protecting the qubit from otherwise losing its energy to the environment. However, eventhough this so-called dispersive readout technique has proven useful, it is often in itself insufficient to render single-shot readout performance | one of the crucial tasks required for realizing a quantum computer. This thesis describes the demonstration of a single-shot readout technique forsuperconducting quantum bits, based on coupling the qubit to a frequency-tunable resonator. The backbone of our device is a 5 GHz quarter-wavelength coplanar waveguide resonator, terminated at one end by a non-linear inductance provided by a superconducting quantum interference device (SQUID). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using an on-chip transmission line. The modulation occurs on a timescale much faster than the photon loss out of the resonator and if the pump strength exceeds a threshold value set by the damping rate of the resonator, an intense photon field will build up inside the resonator | known as "parametric oscillations". By heterodyne detection of the output signal from the Josephson parametric oscillator (JPO), we first extracted two leading nonlinear effects of the system (dominating in different limits of applied magnetic flux). Next, we couple a qubit to the JPO and demonstrate that we can encode its quantum state onto the strong output field of the parametric oscillator (or lack thereof), rendering a signal-to-noise ratio sufficient for single-shot state detection and therefore also obviating a quantum limited parametric amplifier. The thesis also contains results from microscopic studies of the Josephson junctions, which we use to provide the necessary nonlinearities in the above mentioned devices. In particular, we show how the microstructure of the barrier layer and its atomic composition can be used to infer important electrical properties of the junctions. This knowledge allows us to better tailor the properties of Josephson-based devices.
In this thesis I present different experiments on superconducting circuits exploring parametric interactions with external signals and the vacuum in the microwave regime. These parametric processes are the result of the periodic modulation of a property of a system which results in different interactions. The systems used in this thesis are circuits where the nonlinear inductance of a superconducting quantum interference device (SQUID) is parametrically driven.I present the first experimental observation of the dynamical Casimir effect (DCE), since it was predicted in 1970. The DCE is an interaction between the vacuum and a periodically modulated boundary condition of the electromagnetic field, here implemented by a flux-tuned SQUID. In essence the modulated boundary will parametrically amplify the vacuum fluctuations which results in the pairwise generation of photons over a broad frequency range. I have characterized the system and measured the statistical properties of the emitted radiation to show that the radiation exhibits two-mode squeezing.Next, I present measurements on a superconducting multimode resonator containing a SQUID. I show that it is possible to get parametric amplification by driving the SQUID at either twice the frequency of one of the modes or by driving it at the sum of two mode frequencies. In both cases I show that it is possible to reach quantum-limited noise performance.In the same system I also demonstrate frequency conversion which occurs when the system is pumped at the difference frequency between two modes. Microwave photons are coherently transferred from one mode to the other. I show that the coupling strength depends linearly on the applied pump amplitude.The thesis also contains a linearized theoretical model to describe and analyze the flux-pumped SQUID. The model describes an equivalent circuit element called the pumpistor, with an impedance which depends on the pump phase. I show that under specific conditions the impedance becomes real and negative allowing pump power to be injected into the circuit, providing gain.Finally I also present a demonstration of an on-chip Mach-Zehnder interferometer. This experiment uses the tunability of the SQUID to provide a controllable phase shift in one of the interferometer arms. The transmission through the device can be modulated with a maximum change of 45 dB.
With a large portfolio of elemental quantum components, superconducting quantum circuits have contributed to dramatic advances in microwave quantum optics. Of these elements, quantum-limited parametric amplifiers have proven to be essential for low noise readout of quantum systems whose energy range is intrinsically low (tens of µeV). They are also used to generate non classical states of light that can be a resource for quantum enhanced detection. Superconducting parametric amplifiers, like quantum bits, typically utilize a Josephson junction as a source of magnetically tunable and dissipation-free nonlinearity. The magnetic control is not an industry standard for devices and starts already to be an issue in large scale circuits. In recent years, efforts have been made to introduce semiconductor weak links as electrically tunable nonlinear elements, with demonstrations of microwave resonators and quantum bits using semiconductor nanowires, a two dimensional electron gas, carbon nanotubes and graphene. However, given the challenge of balancing nonlinearity, dissipation, participation, and energy scale, parametric amplifiers have not yet been implemented with a semiconductor weak link. The work presented in this PhD thesis demonstrates the design, fabrication and performances of a parametric amplifier leveraging a graphene Josephson junction.The graphene is encapsulated in between hBN flakes in order to improve its quality. Using such high quality junctions, we demonstrate amplification with 20 dB gain which is the milestone for overcoming noise of the classical amplifiers in the rest of the amplification chain. The Josephson parametric amplifier being based on a resonant structure (around 6 GHz) suffers from the gain bandwidth product limiting the amplification to a few MegaHertz frequency range. The use of a side gate to electrically tune the graphene gives control of the Josephson junction inductance and thus the resonant frequency of the amplifier. We demonstrate a near 1 GHz tunability of the amplification frequency with the use of a side gate which is as good as the tunability offered by using a magnetic flux on a SQUID for similar amplifier designs. Moreover, graphene has shown to exhibit nonlinear loss under a strong microwave irradiation arising because of dynamics in the Andreev states. Dissipation is known to add intrinsic noise on quantum amplifiers and could be a problem for quantum limited parametric amplification. We demonstrate that despite the presence of nonlinear loss, graphene based Josephson parametric amplifiers can reach near quantum limited amplification. A two photon loss model was used to describe the behaviour of the device but we show that a more complex model is required in order to take into account the nonlinear dissipation and the non-sinusoidal current phase relation. We also studied the dynamic range and showed that the graphene based Josephson parametric amplifier can reach a 1 dB compression point of -123 dBm which is as good as single tunnel junctions based parametric amplifiers. Our results expand the toolset for electrically tunable superconducting quantum circuits and offer new opportunities for the development of quantum technologies such as quantum computing, quantum sensing and fundamental science.
We develop a theory of parametric resonance in tunable superconducting cavities. The nonlinearity introduced by the superconducting quantum interference device (SQUID) attached to the cavity and damping due to connection of the cavity to a transmission line are taken into consideration. We study in detail the nonlinear classical dynamics of the cavity field below and above the parametric threshold for the degenerate parametric resonance, featuring regimes of multistability and parametric radiation. We investigate the phase-sensitive amplification of external signals on resonance, as well as amplification of detuned signals, and relate the amplifier performance to that of linear parametric amplifiers. We also discuss applications of the device for dispersive qubit readout. Beyond the classical response of the cavity, we investigate small quantum fluctuations around the amplified classical signals. We evaluate the noise power spectrum both for the internal field in the cavity and the output field. Other quantum-statistical properties of the noise are addressed such as squeezing spectra, second-order coherence, and two-mode entanglement.