We present the theory of a Josephson parametric amplifier employing two-pump sources. Our calculations are based on input-output theory, and can easily be generalized to any coupled system involving parametric interactions. We analyze the operation of the device, taking into account the feedback introduced by the reaction of the signal and noise on the pump power, and in this framework, compute the response functions of interest---signal and idler gains, internal gain of the amplifier, and self-oscillation signal amplitude. To account for this back action between signal and pump, we adopt a mean-field approach and self-consistently explore the boundary between amplification and self-oscillation. The coincidence of bifurcation and self-oscillation thresholds reveals that the origin of coherent emission of the amplifier lies in the multiwave mixing of the noise components. Incorporation of the back action leads the system to exhibit hysteresis, dependent on parameters such as temperature and detuning from resonance. Our analysis also shows that the resonance condition itself changes in the presence of back action and this can be understood in terms of the change in plasma frequency of the junction. The potential of the double-pump amplifier for quantum-limited measurements and as a squeezer is also discussed.
We present the theory of a Josephson parametric amplifier employing two-pump sources. Our calculations are based on input-output theory, and can easily be generalized to any coupled system involving parametric interactions. We analyze the operation of the device, taking into account the feedback introduced by the reaction of the signal and noise on the pump power, and in this framework, compute the response functions of interest---signal and idler gains, internal gain of the amplifier, and self-oscillation signal amplitude. To account for this back action between signal and pump, we adopt a mean-field approach and self-consistently explore the boundary between amplification and self-oscillation. The coincidence of bifurcation and self-oscillation thresholds reveals that the origin of coherent emission of the amplifier lies in the multiwave mixing of the noise components. Incorporation of the back action leads the system to exhibit hysteresis, dependent on parameters such as temperature and detuning from resonance. Our analysis also shows that the resonance condition itself changes in the presence of back action and this can be understood in terms of the change in plasma frequency of the junction. The potential of the double-pump amplifier for quantum-limited measurements and as a squeezer is also discussed.
A comprehensive account of the theory of the singly quasidegenerate Josephson junction parametric amplifier is given. In this mode the signal and idler frequencies are both approximately equal to half the pump frequency, and hence the signal and idler channels have a common termination. It is shown that the performance of the amplifier characterized by properties such as gain-bandwidth product, noise temperature, and saturation depends strongly on the embedding microwave circuit. In particular any uncompensated series reactance may deteriorate the overall performance. Internal shot and Johnson noise as well as incident thermal radiation is included in the discussion. It is shown that the noise temperature is virtually independent of noise injected from external sources and that minimum noise temperature is automatically achieved if the amplifier is tuned to maximize the gain-bandwidth product. The analytical results are illustrated by self-consistent numerical solutions of the circuit equations.
Crossover between thermal and quantum noise in the externally-pumped Josephson-junction parametric amplifiers is theoretically followed. Two different approaches are used for the analysis : i. Josephson junction is replaced with its RSJ model with the equilibrium source of fluctuations; ii. microscopic theory of the tunnel junctions is used, which takes into account not only thermal and quantum but also shot noise. Because of inconvenience of the noise temperature TNfor the amplifier sensitivity characterization in the quantum limit, we express our results in terms of the figure ΘNwhich is essentially the amplifier output noise energy, reduced to its input (in the classical limit,\Theta_{N}=k_{B}T_{N}). For the nondegenerate amplifier, the minimum value of ΘN. equals hω/2 and is due to quantum noise (zero-point oscillations) in the idle-frequency resonator of the amplifier. For the degenerate amplifier, sensitive to only one quadrature component of the input signal, ΘNcan be made much less than the above "quantum limit" even at operation temperatures much higher thanh\omega/k_{B}, if the Josephson junction characteristic frequency ωcis high enough,h\omega_{c}\gsimk_{B}T. Our analysis of the experimental situatlon shows that the values\Theta_{N}<h\omega/2can be achieved using two-dimensional arrays of a large number (N\simeq100) single-tunnel-junction interferometers with low inductances, inductively coupled to a microwave cavity.
Recent advancements in quantum technologies and advanced detection experiments have underscored the pressing need for the detection of exceedingly weak signals within the microwave frequency spectrum. Addressing this challenge, the Josephson Traveling Wave Parametric Amplifier (JTWPA) has been proposed as a cryogenic front-end amplifier capable of approaching the quantum noise limit while providing a relevant bandwidth. This research is centered on a comprehensive numerical investigation of the JTWPA, without resorting to simplifications regarding the nonlinearity of the essential components. Specifically, this study focuses on a thorough examination of the system, characterized by coupled nonlinear differential equations representing all components of the device. Proper input and output signals at the device's boundaries are considered. The analysis of the output signals undergoing the parametric amplification process involves a detailed exploration of phase-space dynamics and Fourier spectral analysis of the output voltage. This study is conducted while considering the parameters ruling the response of the device under pump and signal excitations. In addition to the expected signal amplification, the findings reveal that the nonlinear nature of the system can give rise to unforeseen phenomena, depending on the system's operational conditions, which include: the generation of pump tone harmonics, modulation of the signal gain, and incommensurate frequency generation—effects that are not easily accommodated by simplistic linearized approaches.
The recent developments in quantum technologies, as well as advanced detection experiments, have raised the need to detect extremely weak signals in the microwave frequency spectrum. To this aim, the Josephson travelling wave parametric amplifier, a device capable of reaching the quantum noise limit while providing a wide bandwidth, has been proposed as a suitable cryogenic front-end amplifier. This work deals with the numerical study of a Josephson travelling wave parametric amplifier, without approximations regarding the nonlinearity of the key elements. In particular, we focus on the investigation of the system of coupled nonlinear differential equations representing all the cells of the Josephson travelling wave parametric amplifier, with proper input and output signals at the boundaries. The investigation of the output signals generated by the parametric amplification process explores the phase-space and the Fourier spectral analysis of the output voltage, as a function of the parameters describing the pump and signal tones that excite the device. Beside the expected behavior, i.e., the signal amplification, we show that, depending on the system operation, unwanted effects (such as pump tone harmonics, incommensurate frequency generation, and noise rise), which are not accounted for in simple linearized approaches, can be generated in the whole nonlinear system.
The high-frequency amplification properties of transformer coupled, resistively shunted Josephson tunnel junctions have been investigated. The importance of the shunt loop inductance is stressed. It allows a high cutoff frequency, of significance for good high-frequency performance. The self-pumped parametric amplifier showed none of the excessive noise rise, which has hitherto plagued the development of externally pumped Josephson junction amplifiers. Around 10 GHz, we estimated a noise temperature less than 30 K for an amplifier pumped by a Josephson oscillation with a frequency well above twice the signal frequency. The corresponding gain of 5 dB may be increased in a better impedance matched circuit. The gain was very stable against variations in the bias conditions. A gain-bandwidth product as high as 0.3 was registered. The experimental results agreed well with the established theory for self-pumped parametric Josephson amplifiers. It should be possible to extend the low noise amplification by this device to mm wave frequencies. A relaxation oscillation occurred at a subharmonic of the Josephson frequency when the shunt loop inductance became large. The amplification in this mode followed closely the predictions of a simple model, where the signal modulated the switching of the sawtooth-like (relatively low frequency) relaxation current. Gains of about 15 dB were measured around 10 GHz, but the amplification was sensitive to bias conditions and noisy in this case where the relaxation frequency fell well below the signal frequency. Much improved properties were registered when the inductance was decreased so that the relaxation frequency approached the Josephson frequency and exceeded twice the signal frequency. The behavior then resembled that of a Josephson mode parametric amplifier, but the high content of harmonics of a relaxation oscillation meant that the amplifier became noisier due to converted noise from the many idler frequencies.
We study the dynamics of a driven atomic Josephson junction that we propose as a parametric amplifier. By periodically modulating the position of the barrier, we induce a small current across the junction, serving as our input signal. The pump field is implemented by modulating the barrier height at twice the Josephson plasma frequency. The resulting dynamics exhibit parametric amplification of the signal through nonlinear mixing between the signal and pump fields, which is encoded in a specific microscopic pattern of density waves and phase excitations that can be addressed within the experimental cold-atom capabilities. This work paves the way for tunable amplifiers in atomtronic circuits, with potential applications in several fields including precision measurements and quantum information processing. At the same time, our analysis provides the microscopic explanation of the general notion of parametric amplification occurring in nonlinear coherent devices.