The pumpistor: A linearized model of a flux-pumped superconducting quantum interference device for use as a negative-resistance parametric amplifier

Abstract

We describe a circuit model for a flux-driven SQUID. This is useful for developing insight into how these devices perform as active elements in parametric amplifiers. The key concept is that frequency mixing in a flux-pumped SQUID allows for the appearance of an effective negative resistance. In the three-wave, degenerate case treated here, a negative resistance appears only over a certain range of allowed input signal phase. This model readily lends itself to testable predictions of more complicated circuits. Systems closely related to these amplifiers are also providing new physics, such as photon measurements of fast-tunable resonators 13 and the observation of the dynamical Casimir effect 14 . Since topics related to the manipulation of coherent states of light have traditionally been associated with quantum optics, a quantum-optics formalism dominates the commonly encountered explanations of these systems. However, under suitably small-signal limits, a truly nonlinear reactance may be modeled simply as a timevarying reactance. Under this approximation, the principle of superposition holds and we have the standard lexicon of linear analytical techniques available to us, such as Fourier analysis. In fact, "classical" parametric amplifiers were often treated in this linearized manner in literature 15-17 generated during the 1960s and 70s. While this literature was commonly depicting circuits utilizing varactor diodes as active elements, it remains a general premise that a parametrically driven nonlinear reactance leads to frequency mixing.In this work, a linearized method of analysis allows us to depict the effects of amplification using simple, intuitive models of equivalent electrical circuit elements. We examine the case of a three-wave degenerate parametric amplifier based on a dc Superconducting QUantum Interference Device (SQUID). Although outside the scope a) Electronic mail: kyle.sundqvist@gmail.com of this work it is also possible to consider the nondegenerate case, were an idler tone is introduced and considered separately from a signal tone.The parametric interaction is supplied by the SQUID, acting as a tunable, nonlinear inductance. By way of a mutual inductance to a control line, a time-varying magnetic flux, Φ ac , is applied to the SQUID and acts as our pump.We will show how the application of a dc and an ac pump flux allows us to treat the SQUID electrically as the well-known Josephson inductance, in parallel to a special circuit element which we introduce as "the pumpistor." We find that the pumpistor defined under these conditions leads to a phase sensitive impedance, where the phase angle between the pump and signal tones becomes important. In particular the pumpistor can act as a negative resistance, producing gain. Thus, our treatment presents a simple, analytical, albeit classical understanding of the phase sensitivity associated with degenerate parametric amplification.Our circuit model of a flux-pumped SQUID allows us to analyze much more complicated circuits in a straightforward way. ...

A quantum engineer's guide to superconducting qubits

“…These are illustrated in Fig. 27(b)-(c) and are referred to as currentpumping 351,[353][354][355][356] and flux-pumping 95,349,352,[357][358][359][360][361][362] , respectively. The type of mixing process that takes place depends on the leading order of the nonlinearity of the system, as reflected in its Hamiltonian.…”

Investigation of nonlinear effects in Josephson parametric oscillators used in circuit quantum electrodynamics

“…The flux modulation enters into the Josephson energy term of the resonator's boundary condition in the form of a mixing product with the field inside the resonator [9], 2E J |cos(π (t)/ 0 )| sin(φ(t)), where E J is the Josephson energy, (t) and 0 are the magnetic flux and flux quantum, respectively, and φ(t) denotes the phase across the Josephson junctions directly related to the field in the resonator. Considering the Taylor expansions of the flux-and phase contributions to the mixing product [13], the number of terms entering into the dynamics is set by the microwave pump strength and the number of photons in the resonator.…”

Thermal and Residual Excited-State Population in a 3D Transmon Qubit

“…This assertion reflects, in part, several successes over the past decade addressing the fundamental operability of this qubit modality [1][2][3]. A partial list includes a 5-orders-of-magnitude increase in the coherence time T 2 [4], the active initialization of qubits in their ground state [1,5], the demonstration of low-noise parametric amplifiers [6][7][8][9][10][11][12] enabling high-fidelity readout [13][14][15][16], and the implementation of a universal set of high-fidelity gates [17]. In addition, prototypical quantum algorithms [18][19][20] and simulations [21,22] have been demonstrated with few-qubit systems, and the basic parity measurements underlying certain error detection protocols are now being realized with qubit stabilizers [23][24][25][26][27][28] and photonic memories [29].…”