We study coherent quantum phase slips which lift the ground state degeneracy in a Josephson junction ring, pierced by a magnetic flux of the magnitude equal to half of a flux quantum. The quantum phase-slip amplitude is sensitive to the normal mode structure of superconducting phase oscillations in the ring (Mooij-Sch\"on modes). These, in turn, are affected by spatial inhomogeneities in the ring. We analyze the case of weak periodic modulations of the system parameters and calculate the corresponding modification of the quantum phase-slip amplitude.
We study coherent quantum phase slips which lift the ground state degeneracy in a Josephson junction ring, pierced by a magnetic flux of the magnitude equal to half of a flux quantum. The quantum phase-slip amplitude is sensitive to the normal mode structure of superconducting phase oscillations in the ring (Mooij-Sch"on modes). These, in turn, are affected by spatial inhomogeneities in the ring. We analyze the case of weak periodic modulations of the system parameters and calculate the corresponding modification of the quantum phase-slip amplitude.
We study coherent quantum phase slips in a Josephson junction chain, including two types of quenched disorder: random spatial modulation of the junction areas and random induced background charges. Usually, the quantum phase-slip amplitude is sensitive to the normal-mode structure of superconducting phase oscillations in the ring (Mooij-Sch\"on modes, which are all localized by the area disorder). However, we show that the modes' contribution to the disorder-induced phase-slip action fluctuations is small, and the fluctuations of the action on different junctions are mainly determined by the local junction parameters. We study the statistics of the total quantum phase-slip amplitude on the chain and show that it can be non-Gaussian for not sufficiently long chains.
We study quantum phase-slip (QPS) processes in a superconducting ring containing N Josephson junctions and threaded by an external static magnetic flux ${\ensuremath{\Phi}}_{B}$. In such a system, a QPS consists of a quantum tunneling event connecting two distinct classical states of the phases with different persistent currents [Matveev et al., Phys. Rev. Lett. 89, 096802 (2002)]. When the Josephson coupling energy EJ of the junctions is larger than the charging energy EC=e2/2C, where C is the junction capacitance, the quantum amplitude for the QPS process is exponentially small in the ratio EJ/EC. At given magnetic flux, each QPS can be described as the tunneling of the phase difference of a single junction of almost $2\ensuremath{\pi}$, accompanied by a small harmonic displacement of the phase difference of the other $N\ensuremath{-}1$ junctions. As a consequence, the total QPS amplitude ${\ensuremath{\nu}}_{\mathrm{ring}}$ is a global property of the ring. Here, we study the dependence of ${\ensuremath{\nu}}_{\mathrm{ring}}$ on the ring size N, taking into account the effect of a finite capacitance C0 to ground, which leads to the appearance of low-frequency dispersive modes. Josephson and charging effects compete and lead to a nonmonotonic dependence of the ring's critical current on N. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the system converges either towards a superconducting or an insulating state, depending on the ratio between the charging energy E0=e2/2C0 and the Josephson coupling energy EJ.
This paper investigates the physical nature of a series of step structures or oscillations present in a Josephson current within one magnetic flux quantum when either a Josephson junction, a double junction squId, or a single junction superconducting ring is placed inside a high Q cavity under the condition in which the Josephson frequency is equal to the resonance frequency of the cavity. The frequency effect of the feedback field produces a series of induced current steps in the Josephson junction or the two-junction squId when ..omega.. = ..omega../sub r/. The oscillating electromagnetic field excited by the junction current acts on the junction itself. A series of new current step structures occur within one magnetic flux phi/sub 0/ quantum, since the amplitude of the feedback field is modulated by an applied magnetic field as the modulation of the amplitude of the feedback field is coupled to that of the radiation. For the superconducting ring containing a single junction, as n..omega.. = ..omega../sub r/, the feedback radiation of the stationary electromagnetic wave will give rise to a constrained oscillation of the supercurrent with a period smaller than that of the superconducting current oscillation phi/sub 0/.
Superconducting order in a sufficiently narrow and infinitely long wire is destroyed at zero temperature by quantum fluctuations, which induce $2\ensuremath{\pi}$ slips of the phase of the order parameter. However, in a finite-length wire, coherent quantum phase slips would manifest themselves simply as shifts of energy levels in the excitation spectrum of an electrical circuit incorporating this wire. The higher the phase slips' probability amplitude, the larger are the shifts. Phase slips occurring at different locations along the wire interfere with each other. Due to the Aharonov-Casher effect, the resulting full amplitude of a phase slip depends on the offset charges surrounding the wire. Slow temporal fluctuations of the offset charges make the phase-slip amplitudes random functions of time, and therefore turn energy level shifts into linewidths. We experimentally observed this effect on a long Josephson junction array acting as a ``slippery'' wire. The slip-induced linewidths, despite being only of order 100kHz, were resolved from the flux-dependent dephasing of the fluxonium qubit.
This thesis describes a series of theoretical proposals of novel circuits that embed ultrathin superconducting nanowires with coherent quantum phase-slips (QPS). The motivation for our proposals is twofold: firstly, to facilitate unambiguous experimental verification of coherent phase-slips. Secondly, to suggest superconducting devices with new features and functionalities. The first circuit is a driven LC oscillator with the superconducting inductance subject to QPS. We have studied the effect of QPS in two limits: for small and for large phase-slip amplitudes. We expect that weakQPS (whereweakmeans that the QPS amplitude is small compared to the resonance frequency) have ameasurable effect when the oscillator is resonantly driven. The charge induced by the gate affects the quantum interference of QPS with opposite shifts and thus the effect of QPS is periodic in gate voltage. The experimental observation of such dependence would unambiguously identify the quantum coherence of phase-slips. The other crucial effect of QPS is that they produce uncommon non-linearities in the proposed device. The distinctive feature of these non-linearities is the oscillatory dependence on number of photons n with a local period of the order of pn. We analyzed the regime where the QPS correction becomes large by employing a semiclassical approximation as well as by solving the full quantum mechanical equations. At semiclassical level, the QPS-induced non-linearities persist until very high photon number and they result in amultitude ofmetastable states. This was illustrated in the responses of the oscillator: the QPS-induced non-linearities cause deviations fromthe otherwise Lorentzian curve that represents the number of photons N versus detuning. At sufficiently large phase-slip amplitudes this results in an impressive characteristic, a "corkscrew" shape. At any given detuning one finds a multitude of states that differ in the number of photons. About half of these states are stable. At quantum level, we have demonstrated that there is indeed a single quantum state corresponding to the semiclassical metastable states. We have found that these states are robust, their switching time is exponentially long, although they encompass only a few photons. The QPS oscillator is highly tunable: small changes of the driving force, detuning, or charge induced change the number of stable states, thereby enabling easy manipulation of the number of photons. All these features of the phase-slip oscillator make it useful for a wide range of applications, such as ultra-sensitive measurements, quantum manipula- tion and naturally, an unambiguous experimental verification of coherent QPS. In Chapter 3 we propose a series of devices that illustrate the emergence of Coulomb blockade from coherent QPS in thin superconducting wires. The basic ideas is that QPS create an isolation in a wire which is similar to the isolation required by Coulomb blockade. We exploit this idea and suggest setups that are derived from Cooper-pair box and Cooper-pair transistor, so we refer to them as QPS-box and QPS-transistor, respectively. Our main goal was to demonstrate that the devices exhibit sensitivity to a charge induced by a gate electrode, this being the main signature of Coulomb blockade. We analyze the emergence of discrete charging in the limit of strong phaseslips. We investigated six distinct regimes that are realized depending on the relation between the three characteristic energy scales: inductive and charging energy, and QPS amplitude. In both cases the charge sensitivity appears already for small QPS amplitudes as a perturbative correction to the ground-state energy. In both cases, the charge-sensitive part of the perturbative correction is exponentially suppressed in the limit of low impedance. In contrast to the QPS-box, the QPStransistor exhibits both flux and charge sensitivity that makes it potentially useful for measurements. However, if the QPS amplitude becomes of the order of either charging energy (large impedance regime) or inductive energy (small impedance regime), both devices showdiscrete charge states that followthe commonCoulomb blockade pattern of "crossing parabolas". The crossover to Coulomb blockade occurs differently in the limits of large and small impedance. For QPS-transistor, we have analyzed the flux sensitivity (superconducting current) as well as combined flux-charge sensitivity. For a symmetric QPS-transistor we have found a variable separation at specific values of induced charge that leads to double degeneracy of the states at q/e = 1 and half-integer external flux. We have calculated the superconducting current through QPS-transistor to show the non-triviality of the device. Experimental realization of these devices (achievable with the state-or-theart technology), will unambiguously prove the Coulomb blockade as an effect of coherent phase-slip processes. For completeness, we discuss the Josephson-based devices that are dual to QPS-box and QPS-transistor. In Chapter 4 we propose a way to realize quantum synchronization of Josephson and Bloch oscillations in a superconducting device. Essentially, this implies the synchronization of quantumconjugated variables: phase and charge. A circuit comprising of a QPS junction, series resistor and biased by a d.c. voltage exhibits Bloch physics of the charge variable. To ensure well-defined classical oscillations, the resistor needs to be large compared to the resistance quantum. We couple this Bloch subcircuit with a Josephson subcircuit in which the phase variable performs thewell-known Josephson oscillations. Themain effect of the coupling is the transfer of oscillating voltage/current from Josephson/Bloch to Bloch/Josephson part, whereby the voltage/current is multiplied with an amplification coefficient. This amplification coefficient is required to be high for stable synchronization. This is achieved by using an LC-resonator with high quality factorQ. The synchronization takes place in a rather broad interval of frequencies near the LC oscillator’s resonant value. Owing to this synchronization, the transresistance of our device exhibits a typical devil’s staircase curve, very similar to (fractional) Quantum Hall samples. In principle, fluctuations of phase and charge could could destroy the synchronization. Using the quantum description in the framework of Keldysh action formalism, we have investigated the effect of fluctuations on our circuit. The minimum synchronization error rate is shown to be exponential in Q. There is also significant practical interest in the field on metrology. The suggested device can be used to close the famous "metrological triangle" for threemetrological standards: those of resistance, voltage and current—within a single device.
Abstract Superconducting circuits represent a versatile playground to explore novel quantum phenomena and a platform to realize quantum bits (qubits). Superconducting qubits hinge on the coherent tunneling of Cooper pairs across the insulating barrier of a Josephson junction. This qubit implementation is, however, plagued with the variability of the fabricated junction, along with the need for magnetic tunability of the qubit transition frequency. Alternatively, a dual to the Josephson effect—referred to as quantum phase slips—allows for a coherent coupling of flux states and has been used to build phase-slip qubits. In this paper, we propose coupling flux states of a superconducting ring using a bias voltage. By solving the time-dependent Ginzburg–Landau equations, we demonstrate that the bias voltage locally suppresses the density of Cooper pairs, thereby forming a weak link where phase slips occur. Moreover, we show that a voltage-biased superconducting ring serves as an electrically-controlled magnetic memory. The work presented in this paper is an important step towards realizing electrically-tunable superconducting devices, such as qubits and SQUIDs.