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arXiv:cond-mat/0612288 · Physical Review B

Quantum chaos in the mesoscopic device for the Josephson flux qubit

Ezequiel N. Pozzo, Daniel Domı́nguez, María José Sánchez

Published Jan 22, 2008• 7 citations
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Abstract

We study the quantum spectra and eigenfunctions of the three-junction superconducting quantum interference device designed for the Josephson flux qubit at high energies. We analyze the spectral statistics of the parameter region where the system has a mixed classical phase space where regular and chaotic orbits can be found at the same classical energy. We perform a numerical calculation of eigenvalues and eigenstates for different values of the ratio of the Josephson and charging energies, ${E}_{J}∕{E}_{C}$, which is directly related to an effective $\ensuremath{\hbar}$ parameter. We find that the nearest-neighbor distributions $P(s)$ of the energy-level spacings are well fitted by the Berry-Robnik theory employing as free parameters the pure classical measures of the chaotic and regular regions of phase space in the different energy regions in the semiclassical case. The phase-space representation of the wave functions is obtained via the Husimi distributions, and the localization of the states of classical structures is analyzed. We discuss for which values of ${E}_{J}∕{E}_{C}$ it can be possible to perform experiments that could be sensitive to the structure of a mixed classical phase space.

DOI
10.1103/physrevb.77.024518

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We study the quantum spectra and eigenfunctions of the three-junction superconducting quantum interference device designed for the Josephson flux qubit at high energies. We analyze the spectral statistics of the parameter region where the system has a mixed classical phase space where regular and chaotic orbits can be found at the same classical energy. We perform a numerical calculation of eigenvalues and eigenstates for different values of the ratio of the Josephson and charging energies, EJ∕EC{E}_{J}∕{E}_{C}EJ​∕EC​, which is directly related to an effective \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ parameter. We find that the nearest-neighbor distributions P(s)P(s)P(s) of the energy-level spacings are well fitted by the Berry-Robnik theory employing as free parameters the pure classical measures of the chaotic and regular regions of phase space in the different energy regions in the semiclassical case. The phase-space representation of the wave functions is obtained via the Husimi distributions, and the localization of the states of classical structures is analyzed. We discuss for which values of EJ∕EC{E}_{J}∕{E}_{C}EJ​∕EC​ it can be possible to perform experiments that could be sensitive to the structure of a mixed classical phase space.
We show that the three-junction SQUID device designed for the Josephson flux qubit can be used to study the dynamics of quantum chaos when operated at high energies. We determine the parameter region where the system is classically chaotic. We calculate numerically the fidelity or Loschmidt echo (LE) in the quantum dynamics under perturbations in the magnetic field and in the critical currents, and study different regimes of the LE. We discuss how the LE could be observed experimentally considering both the preparation of the initial state and the measurement procedure.
Abstract The dynamics of chaos have been widely used in nonlinear science, such as neural networks, extreme event statistics, and the biophysics of chaos self-organization. Superconducting qubits are artificial atoms based on the Josephson junction structure of nonlinear superconducting devices, offering high design flexibility and ease of coupling and control. In this paper, the generation of chaos through the coupling between superconducting qubits and LC resonant circuits is investigated. By varying the ratio of Josephson energy to charge energy, the coupling coefficient, and the energy of the external driving field, the generation and control of chaos within the system are numerically analyzed. This study provides theoretical support for parameter selection ensuring the confidentiality and fidelity of quantum information transmission based on the chaotic dynamics of superconducting qubits.
Solid-state devices for quantum-bit computation (qubits) are not perfect isolated two-level systems since additional higher energy levels always exist. One example is the Josephson flux qubit, which consists on a mesoscopic superconducting quantum interference device loop with three Josephson junctions operated at or near a magnetic flux of half quantum. We study intrinsic leakage effects, i.e., direct transitions from the allowed qubit states to higher excited states of the system during the application of pulses for quantum computation operations. The system is started in the ground state and rf-magnetic field pulses are applied at the qubit resonant frequency with pulse intensity fp{f}_{p}fp​. A perturbative calculation of the average leakage for small fp{f}_{p}fp​ is performed for this case, obtaining that the leakage is quadratic in fp{f}_{p}fp​, and that it depends mainly on the matrix elements of the supercurrent. Numerical simulations of the time-dependent Schr\"odinger equation corresponding to the full Hamiltonian of this device were also performed. From the simulations we obtain the value of fp{f}_{p}fp​ above which the two-level approximation breaks down, and we estimate the maximum Rabi frequency that can be achieved. We study the leakage as a function of the ratio $\ensuremath{\alpha}$ among the Josephson energies of the junctions of the device, obtaining the best value for minimum leakage $(\ensuremath{\alpha}\ensuremath{\approx}0.85)$. The effects of flux noise on the leakage are also discussed.
The Husimi distribution for the eigenstates of a classically nonintegrable two-dimensional system with mixed phase space was calculated in order to analyze its structure in terms of classical trajectories. Besides eigenfunctions concentrated on invariant tori or with support on chaotic regions of phase space, we also found others that combine different classical invariant sets (like a stable and an unstable periodic orbit) and families of eigenstates scarred by classically unstable periodic orbits.
In quantum information processing, a tension between two different tasks occurs: while qubits' states can be preserved by isolating them, quantum gates can be realized only through qubit-qubit interactions. In arrays of qubits, weak coupling leads to states being spatially localized and strong coupling to delocalized states. Here, we study the average energy level spacing and the relative entropy of the distribution of the level spacings (Kullback-Leibler divergence from Poisson and Gaussian Orthogonal Ensemble) to analyze the crossover between localized and delocalized (chaotic) regimes in linear arrays of superconducting qubits. We consider both transmons as well as capacitively shunted flux qubits, which enables us to tune the qubit anharmonicity. Arrays with uniform anharmonicity, comprising only transmons or flux qubits, display remarkably similar dependencies of level statistics on the coupling strength. In systems with alternating anharmonicity, for typical disorder in the qubit frequencies the localized regime is found to be more resilient to the increase in qubit-qubit coupling strength in comparison to arrays with a single qubit type. Our results, which we also confirm using generalized Bose-Hubbard models, support designing devices that incorporate different qubit types to achieve higher performances.