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Reversible logic gates were previously implemented in superconducting circuits as adiabatic-reversible gates, which are powered with a sufficiently slow clock. In contrast, we are studying ballistic-reversible gates, where fluxons serve to both encode the information and power the gates. No power is applied to the gate apart from the energy of the input fluxons, and the two possible flux polarities represent the bit states. Undamped long Josephson junctions (LJJs), where fluxons move at practically constant speed from inertia, form the input and output channels of the gates. LJJs are connected in the gates by circuit interfaces, which are designed to allow the ballistic scattering from input to output fluxon states, using the temporary excitation of a localized mode. The duration of the resonant scattering determines the operation time of the gate, approximately a few Josephson plasma periods. Due to the coherent conversions between fluxon and localized modes the ballistic gates can be very efficient: in our simulations only a few percent of the fluxon's energy are dissipated in the gate operation. Ballistic-reversible gates can be combined with other, non-ballistic gate circuits to extend the range of gate functionalities. Here we describe how the CNOT can be built as a structure that includes the IDSN (Identity-else-Same-gives-NOT) and Store-and-Launch (SNL) gates. The IDSN is a 2-bit ballistic gate, which we describe and analyze in terms of equivalent 1-bit circuits. The SNL is a clocking gate, that allows the storage of a bit and the clocked launch of a fluxon on a bit-state dependent output path. In the CNOT the SNL gates provide the necessary routing and fluxon synchronization for the input to the IDSN gate., 13 pages, 8 figures, 4 tables. Accepted for publication in IEEE - Transactions on Applied Superconductivity

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., 22 pages, 25 figures

Today's industrial digital logic gates are reaching a technological limit and meanwhile, superconducting circuits produce fundamentally different technologies for the future of digital computing. The standard logic is irreversible and yet superconducting circuits allow digital reversible logic with a much higher energy efficiency per gate operation. Previous reversible gate circuits were “adiabatic,” meaning that they used adiabatic-clocking waveforms for their operation power. However, we are studying logic starting from a ballistic model, where ftuxons enable gates using only energy from their inertia. Our ftuxons are defined in Long Josephson Junctions (LJJs) and may travel ballistically, similar to a particle with negligible damping. Once a ftuxon's energy approaches close enough to the gate, a resonance is induced at the gate and the ftuxon loses its definite topology. Gates are comprised of the (few Josephson-penetration-depth long) ends of LJJs and a connecting circuit interface. Only after the resonance does a ftuxon get formed and yield the gate result without external power: a ftuxon for bit-state 0 or an antiftuxon for bit-state 1. Through earlier analysis of the numerically discovered phenomena we find that dynamics can be described with ftuxon- and antiftuxon-like excitations at the ends of LJJs within the gate. The bit-switching action in our gates is resonant indicating fundamentally different dynamics than the classic model of adiabatic reversible circuits. Our ballistic Reversible Fluxon Logic (RFL) gates have no added damping and calculated energy efficiencies of over 97%. Thus in our dynamical process the “bit energy” is preserved. However, irreversible logic completely dissipates this at each operation (e.g., charging energy in CMOS or SFQ energy in irreversible SFQ logic). An RFL gate can achieve a fast gate operation since its resonance is only few JJ plasma periods. We also describe the CNOT in our technology. It is enabled by a couple of vital gates: A IDSN logic gate, similar to our other ballistic gates, and a Store-aNd-Launch (SNL) timing gate to ensure proper synchronization of the bits. The latter allows bit storage followed by launching of a data ftuxon using an adiabatic pulse from a clock ftuxon with lower energy than the data ftuxon for good CNOT efficiency.

Compared to irreversible gate operations, reversible digital logic gates can provide a fundamental advantage in energy efficiency. Here we discuss previously discovered 1-bit ballistic reversible logic gates and new experimental plans to measure it. The gates consist of long Josephson junctions (LJJs) connected by a circuit interface. The gate dynamics evanescently extends into the LJJs to realize the physically resonant gate operation. It differs from known adiabatic superconducting gates because it does not work in the adiabatic limit. Depending on the gate type, that is the Identity or NOT gate, the polarity of the outgoing fluxon should be preserved or inverted, respectively. The dynamics of the scattering process, originally discovered in full numerical simulation, is understood using collective coordinate analysis. We plan to experimentally study how well a fluxon can travel ballistically towards the interface, and scatter with the designed gate operation. Here we present gate and SQUID-sensing layout designs for a NOT gate. The LJJ circuit layout uses niobium trilayer short junctions with connecting wiring for inductors. Part of the design includes the shunting capacitors at the gate interface which is key for the resonant dynamics in the gates.

Nonlinear amplification, and nonlinear frequency conversion between cavity eigen modes is theoretically investigated. Parametric oscillation, and gains below and above instability threshold are analyzed. Vacuum noise squeezing is evaluated and output noise-tosignal ratio is calculated.

We have developed and tested a doubly tunable resonator, with the intention to simulate fast motion of the resonator boundaries in real space. Our device is a superconducting coplanar-waveguide half-wavelength microwave resonator, with fundamental resonant frequency ~5 GHz. Both of its ends are terminated by dc-SQUIDs, which serve as magnetic-flux-controlled inductances. Applying a flux to either SQUID allows tuning of the resonant frequency by approximately 700 MHz. By using two separate on-chip magnetic-flux lines, we modulate the SQUIDs with two tones of equal frequency, close to twice that of the resonator's fundamental mode. We observe photon generation, at the fundamental frequency, above a certain pump amplitude threshold. By varying the relative phase of the two pumps we are able to control the photon generation threshold, in good agreement with a theoretical model for the modulation of the boundary conditions. At the same time, some of our observations deviate from the theoretical predictions, which we attribute to parasitic couplings, resulting in current driving of the SQUIDs., Comment: 7 pages, 4 figures

We develop a theory for non-degenerate 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 calculate maximum gain values. Above the threshold, in the parametric oscillator regime the linear cavity 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 the 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 parameters at which full conversion is achieved., Comment: 21 pages, 15 figs

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.

We have developed and measured a high-gain quantum-limited microwave parametric amplifier based on a superconducting lumped LC resonator with the inductor L including an array of 8 superconducting quantum interference devices (SQUIDs). This amplifier is parametrically pumped by modulating the flux threading the SQUIDs at twice the resonator frequency. Around 5 GHz, a maximum gain of 31 dB, a product amplitude-gain x bandwidth above 60 MHz, and a 1 dB compression point of -123 dBm at 20 dB gain are obtained in the non-degenerate mode of operation. Phase sensitive amplification-deamplification is also measured in the degenerate mode and yields a maximum gain of 37 dB. The compression point obtained is 18 dB above what would be obtained with a single SQUID of the same inductance, due to the smaller nonlinearity of the SQUID array., 7 pages, 4 figures, 23 references

We characterize a novel Josephson parametric amplifier based on a flux-tunable quarter-wavelength resonator. The fundamental resonance frequency is ~1GHz, but we use higher modes of the resonator for our measurements. An on-chip tuning line allows for magnetic flux pumping of the amplifier. We investigate and compare degenerate parametric amplification, involving a single mode, and nondegenerate parametric amplification, using a pair of modes. We show that we reach quantum-limited noise performance in both cases, and we show that the added noise can be less than 0.5 added photons in the case of low gain.

We present detailed theory of parametric resonance in high quality tunable superconducting cavity. Nonlinear amplification of classical signals and quantum noise characteristics are calculated for broad range of device parameters below and above parametric threshold. © OSA 2014.

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., 5+3 pages, 2+2 figures

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 figures

We experimentally study the behavior of a parametrically pumped nonlinear oscillator, which is based on a superconducting quarter wavelength resonator, and is terminated by a flux-tunable superconducting quantum interference device. We extract parameters for two devices. In particular, we study the effect of the nonlinearities in the system and compare to theory. The Duffing nonlinearity is determined from the probe-power dependent frequency shift of the oscillator, and the nonlinearity related to the parametric flux pumping, is determined from the pump amplitude for the onset of parametric oscillations. Both nonlinearities depend on the parameters of the device and can be tuned in situ by the applied dc flux. We also suggest how to cancel the effect of by adding a small dc flux and a pump tone at twice the pump frequency.

We investigate the asymptotic state of time-periodic quantum systems with regular and chaotic Floquet states weakly coupled to a heat bath. The asymptotic occupation probabilities of these two types of states follow fundamentally different distributions. Among regular states the probability decreases from the state in the center of a regular island to the outermost state by orders of magnitude, while chaotic states have almost equal probabilities. We derive an analytical expression for the occupations of regular states of kicked systems, which depends on the winding numbers of the regular tori and the parameters temperature and driving frequency. For a constant winding number within a regular island it simplifies to Boltzmann-like weights $\exp(-\betaeff \Ereg_m)$, similar to time-independent systems. For this we introduce the regular energies $\Ereg_m$ of the quantizing tori and an effective winding-number-dependent temperature $1/\betaeff$, different from the actual bath temperature. Furthermore, the occupations of other typical Floquet states in a mixed phase space are studied, i.e. regular states on nonlinear resonances, beach states, and hierarchical states, giving rise to distinct features in the occupation distribution. Avoided crossings involving a regular state lead to drastic consequences for the entire set of occupations. We introduce a simplified rate model whose analytical solutions describe the occupations quite accurately., 18 pages, 11 figures

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., 5 pages, 5 figures

The dc field Stark effect is studied theoretically for atoms in high intensity laser fields. We prove that the first-order perturbation corrections for the energy and photoionization rate vanish when the dc field strength serves as a perturbational strength parameter. Our calculations show that by applying a dc field in the same direction as the polarization direction of the ac field, the photoinduced ionization rate is almost entirely suppressed. This suppression is attributed to changes in the phase shift of the continuum atomic wave functions which can be controlled by the dc field.