Resonators coupled to voltage-biased Josephson junctions: From linear response to strongly driven nonlinear oscillations

Abstract

Motivated by recent experiments, where a voltage biased Josephson junction is placed in series with a resonator, the classical dynamics of the circuit is studied in various domains of parameter space. This problem can be mapped onto the dissipative motion of a single degree of freedom in a nonlinear time-dependent potential, where in contrast to conventional settings the nonlinearity appears in the driving while the static potential is purely harmonic. For long times the system approaches steady states which are analyzed in the underdamped regime over the full range of driving parameters including the fundamental resonance as well as higher and sub-harmonics. Observables such as the dc-Josephson current and the radiated microwave power give direct information about the underlying dynamics covering phenomena as bifurcations, irregular motion, up-and down conversion. Due to their tunability, present and future set-ups provide versatile platforms to explore the changeover from linear response to strongly nonlinear behavior in driven dissipative systems under well defined conditions. I. INTRODUCTIONThe nonlinear properties of Josephson junctions (JJs) have made such devices a key circuit element for classical and quantum electronics. Accordingly, there has been a long tradition of studying non-linear phenomena in driven superconducting circuits, starting as early as the 1960s with the discovery of Shapiro steps 1 . While Shapiro-steps have remained a tool in exploring new directions in Josephson physics, for instance in atomic point contacts 2-6 , other nonlinear phenomena, like synchronization, have been investigated in arrays of JJs: as a test-bed for generic theory models to capture synchronization phenomena 7,8 , but as importantly with the prospect of applications as sources of more intense coherent radiation 9 , cf. also new developments using intrinsic arrays 10-12 .More recently, the nonlinearity of the JJ was exploited as the crucial factor in enabling the high sensitivity of Josephson bifurcation amplifiers, achieving substantial improvements towards reaching quantum-limited measurement processes [13][14][15][16] . Most of the features of Josephson bifurcation amplifiers, in fact, only rely on (and can consequently be described by) any type of nonlinearity 17-22 , e.g., Duffing-type models, so that only recently the full nonlinear potential of the JJ has become of interest in this field 23 .A recent addition to the field of driven nonlinear JJs 24-26 are experiments on a dc-biased JJ connected to a resonator 27-29 . In this sort of setup, charge transfer through the JJ leads to excitations in the resonator, and therefore allows to convert energy carried by charge quanta into quantum microwave photons. In these devices measurement of both the Josephson current and the emitted microwave radiation is possible, a distinct advantage in comparison to other recently proposed transport setups 30,31 , which show similar nonlinear features like bifurcations, period multiplication and up-and downconversion 32-34 . S...

Josephson emission with frequency span 1–11 THz from small Bi2Sr2CaCu2O8+δ mesa structures

“…On the other hand, a similar threshold behavior was observed for junction arrays 9 . The threshold appears when the synchronizing cavity mode is excited collectively by all junctions 40–42 . In this case the photon number in the cavity grows in a cascade manner 33,37 with the number of active junctions, like in a quantum cascade laser 5–8 .…”

Efficient Activation of Nanomechanical Resonators

“…[29] To facilitate radiation detection in the terahertz band, in a recent study, [30] a coupled system comprised of an LC tank and a nanomechanical resonator was shown to be working as a microwave amplifier, capable of amplifying weak electrical signals. [32,33] LC tank circuit coupled to Josephson junction has been utilized to model the electrical activities in the neurons [35] and to understand the complex dynamics of particles moving in nonlinear time-dependent potential field. [34] Note that the utilization of similar LC tank resonant circuits to enhance the detection bandwidth is well known in the electron transport community.…”

Theory of the Josephson Junction Laser

“…[6] (see also Ref. [12]) and also for a single mode in [13] we give a re-derivation of it here for completeness. For simplicity we will perform the derivation in the absence of loss.…”

“…[6], a theory to describe the newly observed phenomena was proposed (see also Refs. [12,13] where similar theoretical work is developed). This theory gives a set of many simultaneous differential equations describing the many excitation modes of the resonator cavity.…”

“…The first term on the right represents the integration from −∞ to t − T . The interpretation is that a signal Ψ(t − T ) [6,12,13]). The cavity can be thought of as an LC chain (See Supplemental Material [14]).…”