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arXiv:2407.03928 · Physical review. B./Physical review. B

Quantum dynamics of frustrated Josephson junction arrays embedded in a transmission line: An effective X X spin chain with long-range interaction

Benedikt J. P. Pernack, M. V. Fistul, Ilya Eremin

Published Nov 1, 2024• 2 citations
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Abstract

We study theoretically a variety of collective quantum phases occurring in frustrated sawtooth chains of Josephson junctions embedded in a dissipationless transmission line. The basic element of a system, i.e., the triangular superconducting cell, contains two 0 and one $\ensuremath{\pi}$ Josephson junctions characterized by ${E}_{J}$ and $\ensuremath{\alpha}{E}_{J}$ Josephson energies, accordingly. In the frustrated regime the low-energy quantum dynamics of a single cell is determined by anticlockwise or clockwise flowing persistent currents (vortex or antivortex). The direct embedding of $\ensuremath{\pi}$ Josephson junctions in a transmission line allows us to establish a short or long-range interaction between (anti)vortices of well-separated cells. By making use of the variational approach, we map the superconducting circuit Hamiltonian to an effective $XX$ spin model with an exchange spin-spin interaction decaying with the distance $x$ as ${x}^{\ensuremath{-}\ensuremath{\beta}}$, and the local ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x,n}$ terms corresponding to the coherent quantum beats between vortex and antivortex in a single cell. We obtain that, in long arrays such as $N\ensuremath{\gg}{\ensuremath{\ell}}_{0}\ensuremath{\simeq}\sqrt{C/{C}_{0}}$, where $C$ and ${C}_{0}$ are capacitances of a 0 Josephson junction and a transmission line, respectively, the amplitude of quantum beats is strongly suppressed. By means of exact numerical diagonalization, we study the interplay between the coherent quantum beats and the exchange spin-spin interaction leading to the appearance of various collective quantum phases such as the paramagnetic ($P$), compressible superfluid ($CS$), and weakly compressible superfluid ($w\text{\ensuremath{-}}CS$) states.

DOI
10.1103/physrevb.110.184502

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We report a theoretical study of coherent collective quantum dynamic effects in an array of NNN qubits (two-level systems) incorporated into a low-dissipation resonant cavity. Individual qubits are characterized by energy level differences ${\ensuremath{\Delta}}_{i}$ and a spread of ${\ensuremath{\Delta}}_{i}$ is taken into account. Noninteracting qubits display coherent quantum beatings with NNN different frequencies, i.e., ${\ensuremath{\omega}}_{i}={\ensuremath{\Delta}}_{i}/\ensuremath{\hbar}$. Virtual emission and absorption of cavity photons provides a long-range interaction between qubits. In the presence of such interaction we analyze quantum correlation functions of individual qubits Ci(t){C}_{i}(t)Ci​(t) to obtain two collective quantum-mechanical coherent oscillations, characterized by frequencies ${\ensuremath{\omega}}_{1}=\overline{\ensuremath{\Delta}}/\ensuremath{\hbar}$ and ${\ensuremath{\omega}}_{2}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$, where ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$ is the resonant frequency of the cavity renormalized by interaction. The amplitude of these oscillations can be strongly enhanced in the resonant case when ${\ensuremath{\omega}}_{1}\ensuremath{\simeq}{\ensuremath{\omega}}_{2}$. These collective quantum oscillations can be directly observed, e.g., by measurements of frequency dependent transmission coefficient $D(\ensuremath{\omega})$ of electromagnetic field propagating in a transmission line coupled to the system.
We study theoretically the collective quantum dynamics occurring in various interacting superconducting qubit arrays (SQAs) in the presence of a spread of individual qubit frequencies. The interaction is provided by mutual inductive coupling between adjacent qubits (short-range Ising interaction) or inductive coupling to a low-dissipative resonator (long-range exchange interaction). In the absence of interaction, the Fourier transform of the temporal correlation function of the total polarization (zzz projection of the total spin), i.e., the dynamic susceptibility $C(\ensuremath{\omega})$, demonstrates a set of sharp small magnitude resonances corresponding to the transitions of individual superconducting qubits. We show that even a weak interaction between qubits can overcome the disorder with a simultaneous formation of the collective excited states. This collective behavior manifests itself by a single large resonance in $C(\ensuremath{\omega})$. In the presence of a weak nonresonant microwave photon field in the low-dissipative resonator, the positions of dominant resonances depend on the number of photons, i.e., the collective ac Stark effect. Coupling of an SQA to the transmission line allows a straightforward experimental access of the collective states in microwave transmission experiments and, at the same time, to employ SQAs as sensitive single-photon detectors.
We analyze the coherent dynamics of a fluxonium device (Manucharyan et al 2009 Science 326 113 ) formed by a superconducting ring of Josephson junctions in which strong quantum phase fluctuations are localized exclusively on a single weak element. In such a system, quantum phase tunnelling by occurring at the weak element couples the states of the ring with supercurrents circulating in opposite directions, while the rest of the ring provides an intrinsic electromagnetic environment of the qubit. Taking into account the capacitive coupling between nearest neighbors and the capacitance to the ground, we show that the homogeneous part of the ring can sustain electrodynamic modes which couple to the two levels of the flux qubit. In particular, when the number of Josephson junctions is increased, several low-energy modes can have frequencies lower than the qubit frequency. This gives rise to a quasiperiodic dynamics, which manifests itself as a decay of oscillations between the two counterpropagating current states at short times, followed by oscillation-like revivals at later times. We analyze how the system approaches such a dynamics as the ring's length is increased and discuss possible experimental implications of this non-adiabatic regime.
We study quantum phase-slip (QPS) processes in a superconducting ring containing NNN 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{E}_{J}EJ​ of the junctions is larger than the charging energy EC=e2/2C{E}_{C}={e}^{2}/2CEC​=e2/2C, where CCC is the junction capacitance, the quantum amplitude for the QPS process is exponentially small in the ratio EJ/EC{E}_{J}/{E}_{C}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 NNN, taking into account the effect of a finite capacitance C0{C}_{0}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 NNN. 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{E}_{0}={e}^{2}/2{C}_{0}E0​=e2/2C0​ and the Josephson coupling energy EJ{E}_{J}EJ​.
Superconductivity provides a canonical example of a quantum phase of matter. When superconducting islands are connected by Josephson junctions in a lattice, the low-temperature state of the system can map to the celebrated XY model and its associated universality classes. This has been used to experimentally implement realizations of Mott insulator and Berezinskii-Kosterlitz-Thouless transitions to vortex dynamics analogous to those in type-II superconductors. When an external magnetic field is added, the effective spins of the XY model become frustrated, leading to the formation of topological defects (vortices). Here, we observe the many-body dynamics of such an array, including frustration, via its coupling to a superconducting microwave cavity. We take the design of the transmon qubit, but replace the single junction between two antenna pads with the complete array. This allows us to probe the system at 10 mK with minimal self-heating by using weak coherent states at the single (microwave) photon level to probe the resonance frequency of the cavity. We observe signatures of an ordered vortex lattice at rational flux fillings of the array.
Conventional tunnel Josephson junctions form the basic building blocks of superconducting electronics. Their operation is based on comparatively simple dynamics where the superconducting phases across the junctions is the only dynamical degree of freedom. The goal of the project on which this thesis is based, is to understand the physics and develop a consistent theoretical description of the quantum dynamics of unconventional Josephson structures: spin active junctions, junctions containing low energy quasiparticles, and junctions composed of ddd-wave superconductors. Spin active Josephson junctions involve the additional degree of freedom of quasiparticle spin. Here we theoretically investigate the spin-structure of the current carrying interface bound states - Andreev Bound States (ABS), and induced ground state spin polarization of a short (shorter than the superconducting coherence length) nanowire junction containing a localized magnetic moment. We show that temporal variation of the superconducting phase difference and magnetic moment induce rotations in spin polarized and unpolarized subspaces respectively, which is manifested by temporal oscillation of the Josephson current.The interlevel transitions obey a selection rule that forbids manipulations in a certain region ofexternal parameters and results from specific properties of Andreev bound states in magnetic contacts where, in a certain regime of phase bias and strength of magnetic scatterer, ABS belong to different eigenstates of the combined charge and parity inversion symmetry operator.A general framework is developed to include low energy quasiparticle states into the dynamics of Josephson junctions. Such states exist in e.g. constriction type junctions with high transparency channels or resonant states, as well as in junctions of unconventional superconductors. The existence of such states prevent the separation of energy scales which in conventional tunnel junctions freezes out the quasiparticle degrees of freedom. We show that the resonant interaction with such low energy states rather than the Josephson potential defines the nonlinear Josephson dynamics at small amplitudes.The framework is applied to high temperature Josephson junctions, whose unconventional order parameter admits ABS with energies in the middle of the superconducting gap - Mid Gap States (MGS). In addition to the non-linear dynamics induced by the MGS, we identify a reentrance effect in the transition between thermal activation and macroscopic quantum tunneling associated with the MGS, and connect this phenomenon to experimental observations.