A Josephson junction subject to an ohmic dissipative environment undergoes a zero-temperature quantum phase transition called Schmid-Bulgadaev transition form an insulating state to a superconducting state upon increasing the impedance of the environment above the resistance quantum. Superconducting qubits based on Josephson junction chains represent a system where it is possible to dynamically explore both sides of this transition by probing the scattering of individual plasma modes of the junction chain on a SQUID at the boundary which was achieved in recent experiments. Here, we provide a theoretical analysis of the scattering in the system paying special attention to the non-perturbative superconducting regime. We find that in this regime, a specific diagrammatic resummation must be performed in order to correctly account for fluctuation renormalization of the SQUID's potential. We find analytic expressions for the modes' frequency shifts and quality factors which together provide a full description of scattering. We argue that the non-perturbative nature of the superconducting phase manifests itself through a power-law dependence of the frequency shifts on the SQUID flux and through non-monotonous dependence of the quality factors on the mode frequency. Our predictions may be readily verified in modern low-temperature experiments.
A Josephson junction subject to an ohmic dissipative environment undergoes a zero-temperature quantum phase transition called Schmid-Bulgadaev transition form an insulating state to a superconducting state upon increasing the impedance of the environment above the resistance quantum. Superconducting qubits based on Josephson junction chains represent a system where it is possible to dynamically explore both sides of this transition by probing the scattering of individual plasma modes of the junction chain on a SQUID at the boundary which was achieved in recent experiments. Here, we provide a theoretical analysis of the scattering in the system paying special attention to the non-perturbative superconducting regime. We find that in this regime, a specific diagrammatic resummation must be performed in order to correctly account for fluctuation renormalization of the SQUID's potential. We find analytic expressions for the modes' frequency shifts and quality factors which together provide a full description of scattering. We argue that the non-perturbative nature of the superconducting phase manifests itself through a power-law dependence of the frequency shifts on the SQUID flux and through non-monotonous dependence of the quality factors on the mode frequency. Our predictions may be readily verified in modern low-temperature experiments.
Schmid transition was introduced first as a superconductor-insulator transition in the zero-frequency response of a shunted Josephson junction in equilibrium at zero temperature. As it is typical for a quantum impurity problem, at finite frequencies the transition is broadened to a crossover. Modern attempts to find Schmid transition rely on finite-frequency measurements of a quantum circuit. We predict the frequency dependence of the admittance and reflection phase shift for a high-impedance transmission line terminated by a Josephson junction for a wide variety of devices, from a charge qubit to a transmon. Our results identify the circuit parameters needed for the experimental observation of universal scaling of the responses with frequency. On the insulating side of the transition, the full crossover from weak to strong coupling can only be observed in a transmon. On the contrary, observation of such crossover on the superconducting side of the transition is possible only with the charge qubit. The frequency dependence gets weaker and vanishes upon approaching the Schmid transition from either side.
Schmid transition was introduced first as a superconductor-insulator transition in the zero-frequency response of a shunted Josephson junction in equilibrium at zero temperature. As it is typical for a quantum impurity problem, at finite frequencies the transition is broadened to a crossover. Modern attempts to find Schmid transition rely on finite-frequency measurements of a quantum circuit. We predict the frequency dependence of the admittance and reflection phase shift for a high-impedance transmission line terminated by a Josephson junction for a wide variety of devices, from a charge qubit to a transmon. Our results identify the circuit parameters allowing for the universal scaling of the responses with frequency, thus helping to identify the Schmid transition from the finite-frequency measurements.
Using state-of-the-art numerical techniques, we investigate how quantum phase fluctuations and quasiparticle tunneling shape the behavior of a small-capacitance Josephson junction across Ohmic, sub-Ohmic, and super-Ohmic dissipation regimes. We show that increasing the Ohmic dissipation strength drives a Berezinskii-Kosterlitz-Thouless quantum phase transition at thermodynamic equilibrium. Deviations from Ohmic behavior profoundly alter this scenario: the super-Ohmic regime exhibits no phase transition, whereas the sub-Ohmic regime displays a continuous second-order transition, consistent with the universality classes of the spin-boson model. Within the Ohmic regime, real-frequency linear-response calculations reveal that the phase particle does not undergo the commonly assumed diffusive-to-localized crossover. Instead, finite resistance progressively suppresses the singular zero-frequency response, producing a ballistic-to-localized change in the dynamics. At finite frequencies, coupling to the environment generates a long-lived excitation in the charge response, which evolves into a resonance as the subgap and shunt resistances are reduced.
We have studied the temperature dependence of escape phenomena in various underdamped Josephson junctions (JJs). The junctions had different Josephson coupling energies E J which were relatively small, but larger than the charging energy E C . Upon increasing the temperature T , we first observe the usual cross-over between macroscopic quantum tunnelling and thermally activated (TA) behaviour at temperatures k B T ∼ ℏω p , where ω p is the plasma frequency of the junction. Increasing T further, the width of the switching current distribution has, counterintuitively, a non-monotonic temperature dependence. This can be explained by the novel cross-over from TA behaviour to underdamped phase diffusion. We show that this cross-over is expected to occur at temperatures such that k B T ∼ E J (1 − 4/π Q ) 3 /2, where Q is the quality factor of the junction at the plasma frequency, in agreement with experiment. Our findings can be compared with detailed model calculations which take into account dissipation and level quantization in a metastable well. Particular attention is paid to the sample with the smallest E J , which shows extensive phase diffusion even at the lowest temperatures. This sample consists of a dc-SQUID and a single JJ close to each other, such that the SQUID acts as a tunable inductive protection for the single junction from fluctuations of a dissipative environment. By varying the flux through the dc-SQUID, we present, for the first time, experimental evidence of the escape of a JJ from the phase diffusion regime to the free running state in a tunable environment. We also show that in the zero voltage state the losses mainly occur at frequencies near the plasma resonance.
In this talk I will address the problem of supercurrent instability in highly transparent weak links. I will demonstrate that at subgap energies quantum behavior of superconducting weak links can be exactly accounted for by an effective Hamiltonian for a Josephson particle in a quantum dissipative environment formed by Andreev levels. I will identify several macroscopic quantum tunneling (MQT) regimes and predict a non-trivial non-monotonous dependence of the switching current distribution on temperature which can be observed in MQT experiments with transparent superconducting weak links. I will also argue that intrinsic dissipation in such weak links can be studied by means of qubit spectroscopy. Varying either the bias current or magnetic flux through a superconducting ring in the rf-SQUID setup, one can tune the level splitting value close to the bottom of the Josephson potential well. Monitoring the qubit energy relaxation time one can probe the spectral density of the effective dissipative bath and unambiguously identify the contribution emerging from Andreev levels.
We propose a scheme to implement a tunable, wide frequency-band dissipative environment using a double chain of Josephson junctions. The two parallel chains consist of identical superconducting quantum interference devices (SQUIDs), with magnetic-flux tunable inductance, coupled to each other at each node via a capacitance much larger than the junction capacitance. Thanks to this capacitive coupling, the system sustains electromagnetic modes with a wide frequency dispersion. The internal quality factor of the modes is maintained as high as possible, and the damping is introduced by a uniform coupling of the modes to a transmission line, itself connected to an amplification and readout circuit. For sufficiently long chains, containing several thousands of junctions, the resulting admittance is a smooth function versus frequency in the microwave domain, and its effective dissipation can be continuously monitored by recording the emitted radiation in the transmission line. We show that by varying in situ the SQUIDs' inductance, the double chain can operate as a tunable ohmic resistor in a frequency band spanning up to 1 GHz, with a resistance that can be swept through values comparable to the resistance quantum ${R}_{q}=h/(4{e}^{2})\ensuremath{\simeq}6.5\phantom{\rule{4pt}{0ex}}\text{k}\mathrm{\ensuremath{\Omega}}$. We argue that the circuit complexity is within reach using current Josephson junction technology.