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Register a new userRescaling of Applied Oscillating Voltages in Small Josephson Junctions
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Godwill Mbiti Kanyolo PhD
Publication date
2019
fields
Physics
and research's language is
English
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The standard theory of dynamical Coulomb blockade [$P(E)$ theory] in ultra-small tunnel junctions has been formulated on the basis of phase-phase correlations by several authors. It was recently extended by several experimental and theoretical works to account for novel features such as electromagnetic environment-based renormalization effects. Despite this progress, aspects of the theory remain elusive especially in the case of linear arrays. Here, we apply path integral formalism to re-derive the Cooper-pair current and the BCS quasi-particle current in single small Josephson junctions and extend it to include long Josephson junction arrays as effective single junctions. We consider renormalization effects of applied oscillating voltages due to the impedance environment of a single junction as well as its implication to the array. As is the case in the single junction, we find that the amplitude of applied oscillating electromagnetic fields is renormalized by the same complex-valued weight $Xi(omega) = |Xi(omega)|exp ieta(omega)$ that rescales the environmental impedance in the $P(E)$ function. This weight acts as a linear response function for applied oscillating electromagnetic fields driving the quantum circuit, leading to a mass gap in the thermal spectrum of the electromagnetic field. The mass gap can be modeled as a pair of exotic `particle excitation with quantum statistics determined by the argument $eta(omega)$. In the case of the array, this pair corresponds to a bosonic charge soliton/anti-soliton pair injected into the array by the electromagnetic field. Possible application of these results is in dynamical Coulomb blockade experiments where long arrays are used as electromagnetic power detectors.
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This doctorate thesis focuses on the effects of the electromagnetic environment on applied electromagnetic fields in single small junctions as well as arrays. We apply radio-frequency (RF) microwaves in the sub-gigahertz frequency range on a one-dimensional array of small Josephson junctions exhibiting distinct Coulomb blockade characteristics. We observed a gradual lifting of Coulomb blockade with increase in the microwave power which we interpret is due to photon-assisted tunneling of Cooper pairs in the classical (multi-photon absorption) regime. We observe that, due to its high sensitivity to microwave power, the array is well-suited for in situ microwave detection applications in low temperature environments. A detailed analysis of the characteristics in the classical (multi-photon absorption) limit reveals that the microwave amplitude is rescaled (renormalized), which we attribute to the difference in dc and ac voltage response of the array. We proceed to rigorously consider the origin of the aforementioned renormalization effect by considering the effect of the electromagnetic environment of the Josephson junction on applied oscillating voltages. We theoretically demonstrate that its effect is simply to renormalize the amplitude of oscillation in a predictable manner traced to the physics of wave function renormalization (Lehmann weights) consistent with circuit-QED. We also introduce Einsteins A and B coefficients for small Josephson junctions, in a bid to relate the renormalization effect to the modification of photon absorption and emission amplitudes. Such renormalization implies that the sensitivity of the single junction and the array to oscillating electromagnetic fields (e.g. microwaves) is modulated and depends on the environmental impedance. The renormalization effect can be exploited to configure `opaque, `translucent or `transparent quantum circuits to microwaves.
The influence of radio frequency microwaves on the Coulomb blockade characteristics in small Josephson junctions was studied using a one-dimensional array of ten small Al tunnel junctions in the frequency range from 1 MHz to 1000 MHz. Coulomb blockade voltage ($V_{rm th}$) is diminished with increasing microwave power ($V_{rm ac}$), where the $V_{rm th}$-$V_{rm ac}$ plots for varied frequencies fall on a single curve. We observed and theoretically analyzed a magnetic field $dependent$ renormalization of the applied microwave power, in addition to a magnetic-field $independent$ renormalization effect explained using an effective circuit approach of the array. Due to its high sensitivity to microwave power, the array is well-suited for on-chip detection applications in low temperature environments.
We study the spin transport through a 1D quantum Ising-XY-Ising spin link that emulates a topological superconducting-normal-superconducting structure via Jordan-Wigner (JW) transformation. We calculate, both analytically and numerically, the spectrum of spin Andreev bound states and the resulting $mathbb{Z}_2$ fractional spin Josephson effect (JE) pertaining to the emerging Majorana JW fermions. Deep in the topological regime, we identify an effective time-reversal symmetry that leads to $mathbb{Z}_4$ fractional spin JE in the $textit{presence}$ of interactions within the junction. Moreover, we uncover a hidden inversion time-reversal symmetry that protects the $mathbb{Z}_4$ periodicity in chains with an odd number of spins, even in the $textit{absence}$ of interactions. We also analyze the entanglement between pairs of spins by evaluating the concurrence in the presence of spin current and highlight the effects of the JW Majorana states. We propose to use a microwave cavity setup for detecting the aforementioned JEs by dispersive readout methods and show that, surprisingly, the $mathbb{Z}_2$ periodicity is immune to $textit{any}$ local magnetic perturbations. Our results are relevant for a plethora of spin systems, such as trapped ions, photonic lattices, electron spins in quantum dots, or magnetic impurities on surfaces.
Topological superconductivity holds promise for fault-tolerant quantum computing. While planar Josephson junctions are attractive candidates to realize this exotic state, direct phase-measurements as the fingerprint of the topological transition are missing. By embedding two gate-tunable Al/InAs Josephson junctions in a loop geometry, we measure a $pi$-jump in the junction phase with increasing in-plane magnetic field, ${bf B}_|$. This jump is accompanied by a minimum of the critical current, indicating a closing and reopening of the superconducting gap, strongly anisotropic in ${bf B}_|$. Our theory confirms that these signatures of a topological transition are compatible with the emergence of Majorana states.
Majorana zero modes are quasiparticle states localized at the boundaries of topological superconductors that are expected to be ideal building blocks for fault-tolerant quantum computing. Several observations of zero-bias conductance peaks measured in tunneling spectroscopy above a critical magnetic field have been reported as experimental indications of Majorana zero modes in superconductor/semiconductor nanowires. On the other hand, two dimensional systems offer the alternative approach to confine Ma jorana channels within planar Josephson junctions, in which the phase difference {phi} between the superconducting leads represents an additional tuning knob predicted to drive the system into the topological phase at lower magnetic fields. Here, we report the observation of phase-dependent zero-bias conductance peaks measured by tunneling spectroscopy at the end of Josephson junctions realized on a InAs/Al heterostructure. Biasing the junction to {phi} ~ {pi} significantly reduces the critical field at which the zero-bias peak appears, with respect to {phi} = 0. The phase and magnetic field dependence of the zero-energy states is consistent with a model of Majorana zero modes in finite-size Josephson junctions. Besides providing experimental evidence of phase-tuned topological superconductivity, our devices are compatible with superconducting quantum electrodynamics architectures and scalable to complex geometries needed for topological quantum computing.
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