Parametric resonance in tunable superconducting cavities

We develop a theory of parametric resonance in tunable superconducting cavities. The nonlinearity introduced by the 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.Comment: 25 pages, 17 figure

Parametric Effects in Circuit Quantum Electrodynamics (Review Article)

We review recent advances in the research on quantum parametric phenomena in superconducting circuits with Josephson junctions. We discuss physical processes in parametrically driven tunable cavity and outline theoretical foundations for their description. Amplification and frequency conversion are discussed in detail for degenerate and non-degenerate parametric resonance, including quantum noise squeezing and photon entanglement. Experimental advances in this area played decisive role in successful development of quantum limited parametric amplifiers for superconducting quantum information technology. We also discuss nonlinear down-conversion processes and experiments on self-sustained parametric and subharmonic oscillations

Boosting fluxons for ballistic-logic power using an Aharonov-Casher ring

Superconducting logic is fast and energy-efficient relative to CMOS, but also fundamental studies are needed to scale up circuits for greater utility. Recently, ballistic shift registers for single-flux quanta (SFQ) bits were shown in simulation to allow high-efficiency superconducting gates. However, these gates are unpowered such that the bits slow after each gate operation and only a short sequence of gates is possible without added power. Here we show that a circuit based on an Aharonov-Casher ring can power these shift registers by boosting the bit velocity to a constant value, despite their unusual bit states constituted by two polarities of SFQ. As a step in its operation, each bit state is forced into a different ring arm and then accelerated. The circuit dynamics depend on various circuit parameters and choices of how to merge the bit-state paths. One design from each merge design choice is proposed to enable scaling up to an array of gates by adding serial biasing in a relatively simple way. We find adequate performance for ballistic logic in terms of boosted velocity, energy efficiency, and parameter margins. We also discuss the circuit's classical barriers; in a different regime this relates to the Aharonov-Casher effect.Comment: 17 pages, 10 figures, 1 tabl

Generalized Bose-Einstein condensation into multiple states in driven-dissipative systems

Bose-Einstein condensation, the macroscopic occupation of a single quantum state, appears in equilibrium quantum statistical mechanics and persists also in the hydrodynamic regime close to equilibrium. Here we show that even when a degenerate Bose gas is driven into a steady state far from equilibrium, where the notion of a single-particle ground state becomes meaningless, Bose-Einstein condensation survives in a generalized form: the unambiguous selection of an odd number of states acquiring large occupations. Within mean-field theory we derive a criterion for when a single and when multiple states are Bose selected in a non-interacting gas. We study the effect in several driven-dissipative model systems, and propose a quantum switch for heat conductivity based on shifting between one and three selected states.Comment: 5+3 pages, 2+2 figure

Non-equilibrium steady states of ideal bosonic and fermionic quantum gases

We investigate non-equilibrium steady states of driven-dissipative ideal quantum gases of both bosons and fermions. We focus on systems of sharp particle number that are driven out of equilibrium either by the coupling to several heat baths of different temperature or by time-periodic driving in combination with the coupling to a heat bath. Within the framework of (Floquet-)Born-Markov theory, several analytical and numerical methods are described in detail. This includes a mean-field theory in terms of occupation numbers, an augmented mean-field theory taking into account also non-trivial two-particle correlations, and quantum-jump-type Monte-Carlo simulations. For the case of the ideal Fermi gas, these methods are applied to simple lattice models and the possibility of achieving exotic states via bath engineering is pointed out. The largest part of this work is devoted to bosonic quantum gases and the phenomenon of Bose selection, a non-equilibrium generalization of Bose condensation, where multiple single-particle states are selected to acquire a large occupation [Phys. Rev. Lett. 111, 240405 (2013)]. In this context, among others, we provide a theory for transitions where the set of selected states changes, describe an efficient algorithm for finding the set of selected states, investigate beyond-mean-field effects, and identify the dominant mechanisms for heat transport in the Bose selected state

Switching mechanism in periodically driven quantum systems with dissipation

We introduce a switching mechanism in the asymptotic occupations of quantum states induced by the combined effects of a periodic driving and a weak coupling to a heat bath. It exploits one of the ubiquitous avoided crossings in driven systems and works even if both involved Floquet states have small occupations. It is independent of the initial state and the duration of the driving. As a specific example of this general switching mechanism we show how an asymmetric double well potential can be switched between the lower and the upper well by a periodic driving that is much weaker than the asymmetry.Comment: 5 pages, 5 figure

Nondegenerate Parametric Resonance in a Tunable Superconducting Cavity

We develop a theory for nondegenerate parametric resonance in a tunable superconducting cavity. We focus on nonlinear effects that are caused by nonlinear Josephson elements connected to the cavity. We analyze parametric amplification in a strong nonlinear regime at the parametric-instability threshold, and we calculate maximum gain values. Above the threshold, in the parametric-oscillator regime, the cavity linear response diverges at the oscillator frequency at all pump strengths. We show that this divergence is related to the continuous degeneracy of the free oscillator state with respect to the phase. Applying on-resonance input lifts the degeneracy and removes the divergence. We also investigate quantum noise squeezing. It is shown that in the strong amplification regime, the noise undergoes four-mode squeezing, and that, in this regime, the output signal-to-noise ratio can significantly exceed the input value. We also analyze the intermode frequency conversion and identify the parameters at which full conversion is achieved