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

Abstract

We study theoretically a variety of collective quantum phases occurring in frustrated saw-tooth 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 $\pi$- Josephson junctions characterized by $E_J$ and $\alpha E_J$ 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/antivortex). The direct embedding of $\pi$-Josephson junctions in a transmission line allows to establish a short/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 $x^{-\beta }$, and the local ${\hat{\sigma }}_{x,n}$-terms corresponding to the coherent quantum beats between vortex and antivortex in a single cell. We obtain that in long arrays as $N\gg {\ell }_0\simeq \sqrt{C/C_0}$, where $C$ and $C_0$ are capacitances of $0$-Josephson junction and transmission line, accordingly, 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 ($w$-$CS$) states.

13 pages, 12 figures

References in corpus (44)

• Fault-tolerant quantum computation by anyons
• Quantum Simulation
• Fractionalized excitations in the spin liquid state of a kagomé lattice antiferromagnet
• Nematic order in iron superconductors - who is in the driver's seat?
• Engineered 2D Ising interactions on a trapped-ion quantum simulator with hundreds of spins
• Probing Topological Spin Liquids on a Programmable Quantum Simulator
• Programmable Quantum Simulations of Spin Systems with Trapped Ions
• A Superconducting Persistent Current Qubit
• The European Quantum Technologies Roadmap
• Observation of bound states in Lieb photonic lattices
• Computational Studies of Quantum Spin Systems
• Discrete breathers in nonlinear lattices: Experimental detection in a Josephson array
• Observation of breathers in Josephson ladders
• Observation of topological phenomena in a programmable lattice of 1,800 qubits
• Introduction to Quantum Electromagnetic Circuits
• Implementation of low-loss superinductances for quantum circuits
• Electronic nature of chiral charge order in the kagome superconductor CsV3Sb5
• Continuous Symmetry Breaking in a Two-dimensional Rydberg Array
• The kagome antiferromagnet: a chiral topological spin liquid ?
• Magnon pairing in quantum spin nematic
• Pairing of Cooper Pairs in a Fully Frustrated Josephson Junction Chain
• Progress in Superconducting Metamaterials
• Scaling advantage in quantum simulation of geometrically frustrated magnets
• The Anderson Transition in Two-Dimensional Systems with Spin-Orbit Coupling
• First-Principles Design of a Half-Filled Flat Band of the Kagome Lattice in Two-Dimensional Metal-Organic Frameworks
• Transport in Sawtooth photonic lattices
• Quantum Selection of Order in an $XXZ$ Antiferromagnet on a Kagomé Lattice
• The anisotropic Heisenberg model in a transverse magnetic field
• Gap generation in the XXZ model in a transverse magnetic field
• Coherent Operation of a Gap-tunable Flux Qubit
• Measurement of the Current-Phase Relation in Josephson Junctions Rhombi Chains
• XX model on the circle
• Possible Nematic Spin Liquid in Spin-$1$ Antiferromagnetic System on the Square Lattice: Implication for the Nematic Paramagnetic State of FeSe
• 4e-condensation in a fully frustrated Josephson junction diamond chain
• Thermodynamic properties of Ba${}_2$CoSi${}_2$O${}_6$Cl${}_2$ in strong magnetic field: Realization of flat-band physics in a highly frustrated quantum magnet
• Fabrication and characterization of aluminum SQUID transmission lines
• Frustration induced highly anisotropic magnetic patterns in classical $XY$ model on kagome lattice
• Superconductivity Versus Phase Separation, Stripes, and Checkerboard Ordering: A Two-Dimensional Monte Carlo Study
• Quantum phase transitions in non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric transverse-field Ising spin chains
• Ion Trap Long-Range XY Model for Quantum State Transfer and Optimal Spatial Search
• Resonant frequencies and spatial correlations in frustrated arrays of Josephson type nonlinear oscillators
• Signatures of quantum phase transitions in the dynamic response of fluxonium qubit chains
• Resonances in one and two rows of triangular Josephson junction cells
• Long-range Ising spins models emerging from frustrated Josephson junctions arrays with topological constraints

Cited by in corpus (1)

• Continuously varying critical exponents in an exactly solvable long-range cluster XY mode