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arXiv:2102.03114 · Physical review. B./Physical review. B

Voltage staircase in a current-biased quantum-dot Josephson junction

D. O. Oriekhov, Yevheniia Cheipesh, C. W. J. Beenakker

Published Mar 29, 2021• 12 citations
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Abstract

We calculate the current-voltage ($I\ensuremath{-}V$) characteristic of a Josephson junction containing a resonant level in the weakly coupled regime (resonance width small compared to the superconducting gap). The phase $\ensuremath{\phi}$ across the junction becomes time dependent in response to a DC current bias. Rabi oscillations in the Andreev levels produce a staircase $I\ensuremath{-}V$ characteristic. The number of voltage steps counts the number of Rabi oscillations per $2\ensuremath{\pi}$ increment of $\ensuremath{\phi}$, providing a way to probe the coherence of the qubit in the absence of any external AC driving. The phenomenology is the same as the Majorana-induced DC Shapiro steps in topological Josephson junctions of Choi et al. [Phys. Rev. B 102, 140501(R) (2020)]---but now for a nontopological Andreev qubit.

DOI
10.1103/physrevb.103.094518

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We calculate the current-voltage (I\ensuremath−VI\ensuremath{-}VI\ensuremath−V) characteristic of a Josephson junction containing a resonant level in the weakly coupled regime (resonance width small compared to the superconducting gap). The phase \ensuremathϕ\ensuremath{\phi}\ensuremathϕ across the junction becomes time dependent in response to a DC current bias. Rabi oscillations in the Andreev levels produce a staircase I\ensuremath−VI\ensuremath{-}VI\ensuremath−V characteristic. The number of voltage steps counts the number of Rabi oscillations per 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ increment of \ensuremathϕ\ensuremath{\phi}\ensuremathϕ, providing a way to probe the coherence of the qubit in the absence of any external AC driving. The phenomenology is the same as the Majorana-induced DC Shapiro steps in topological Josephson junctions of Choi et al. [Phys. Rev. B 102, 140501(R) (2020)]---but now for a nontopological Andreev qubit.
Deep theoretical understanding of the electrical response of Josephson junctions is indispensable regarding both recent discoveries of new kinds of superconductivity and technological advances such as superconducting quantum computers. Here, we study the microscopic theory of the dc current-biased I-V characteristics of Josephson tunnel junctions. We derive an analytical formula of the I-V characteristics of generic junctions. We identify subharmonics of the I-V characteristics and their underlying mechanism as the feedback effect of intrinsic ac currents generated by voltage pulses in the past. We apply our theory to analytically solve the Werthamer equation and describe various dc current-biased I-V characteristics as a function of softening of the superconducting gap. Strikingly, we identify voltage staircases of the I-V characteristics in a genuine Josephson junction without ac current bias or qubit dynamics. Our general analytical formalism opens new avenues for a microscopic understanding of I-V characteristics of Josephson junctions that have been limited to phenomenological models so far.
A study of the Andreev level current through a weakly voltage-biased single-mode superconducting quantum point contact under microwave irradiation is presented. The applied microwave field induces resonant transitions between the current carrying Andreev states localized to the contact region, which results in an increase of the dc subgap current by several orders of magnitude. The applied voltage-bias couples to the superconductor phase difference across the junction, which will evolve in time. In turn the Andreev levels will oscillate adiabatically within the superconductor energy gap. Depending on the frequency of the applied field there can be either one or two resonances per oscillation period. When considering the case of two resonances the calculated dc-current exhibits an interference pattern as a function of inverse voltage. A measurement of the oscillation period would make it possible to reconstruct the Andreev level spectrum. Further, a detailed study is conducted concerning the microwave-induced current as a function of voltage bias, and the amplitude and frequency of the applied field. In realistic experimental situations there are equilibrium fluctuations of the applied voltage bias and sometimes impurity induced noise in the contact transparency. The amount of dephasing induced on the interference pattern by these fluctuations is estimated. In most cases the interference pattern will be subject to some amount of decay. However, there are parameter values for which this decay will be small. The decay originating from equilibrium fluctuations is predicted to have a non-monotonic dependence on the position of the resonances, while the decay induced by impurities should increase linearly when the separation between the resonances is increased. This difference could be used to distinguish between the two sources. The results also indicate that it should be possible to experimentally investigate the coupling between Andreev states and their environment, explicitly the impedance which the contact "sees" and the coupling strength between impurities and Andreev states.
We investigate nonequilibrium transport through superconducting nanojunctions using a Liouville space approach. The formalism allows us to study finite-gap effects, and to account for both quasiparticle and Cooper-pair tunneling. With focus on the weak-tunneling limit, we study the stationary dc and ac current up to second order (cotunneling) in the hybridization energy. For the particular case of a strongly interacting quantum dot sandwiched between two superconductors, we identify the characteristic virtual processes that yield the Andreev and Josephson current and obtain the dependence on the gate and bias voltage for the dc current, the critical current, and the phase-dependent dissipative current. In particular, the critical current is characterized by regions in the stability diagram in which its sign changes from positive to negative, resulting in a multitude of $0\ensuremath{-}\ensuremath{\pi}$ transitions. The latter signal the interplay between strong interactions and tunneling at finite bias.
We study the dynamics of a quantum superconducting circuit which consists of a Josephson charge qubit, coupled capacitively to a current biased Josephson junction. Under certain conditions, the eigenstates of the qubit and the junction become entangled. We obtain the time evolution of these states in the limit of weak coupling. Rabi oscillations occur, as a result of the spontaneous emission and re-absorption of a single oscillation quantum in the junction. We discuss a possible way to experimentally determine the quantum state of the junction and hence observe the Rabi oscillations.
The zero-voltage state of a Josephson junction biased with constant current consists of a set of metastable quantum energy levels. We probe the spacings of these levels by using microwaves to induce transitions and thereby enhance the escape rate to the voltage state. The widths of the resonances give a measurement of the spectroscopic coherence time of the two metastable states involved in the transitions. We observe a decoherence time shorter than that expected from dissipation alone in resonantly isolated $20\ifmmode\times\else\texttimes\fi{}5\ensuremath{\mu}{\mathrm{m}}^{2}$ area Al/AlOx/Al\mathrm{Al}/\mathrm{AlO}x/\mathrm{Al}Al/AlOx/Al junctions at 60 mK. The data are well fit by a model that includes the dephasing effects of both low-frequency current noise and the escape rate to the voltage state. We discuss implications for quantum computation using current-biased Josephson-junction qubits, including limits on the minimum number of levels needed in the well.
A transmon qubit embedded in a high-impedance environment acts in a way dual to a conventional Josephson junction. In analogy to the AC Josephson effect, biasing of the transmon by a direct current leads to the oscillations of voltage across it. These oscillations are known as the Bloch oscillations. We find the Bloch oscillations spectrum, and show that the zero-point fluctuations of charge make it broadband. Despite having a broad-band spectrum, Bloch oscillations can be brought in resonance with an external microwave radiation. The resonances lead to steps in the voltage-current relation, which are dual to the conventional Shapiro steps. We find how the shape of the steps depends on the environment impedance R, parameters of the transmon, and the microwave amplitude. The Bloch oscillations rely on the insulating state of the transmon which is realized at impedances exceeding the Schmid transition point, R > RQ = h/(2e)2. When a Josephson junction is embedded into a highly-resistive environment, it loses its superconducting properties and starts to behave as an insulator. This results in voltage oscillations across the current-biased junction - the Bloch oscillations. Here the authors develop a fully quantum theory of this effect.