Kicked fluxonium with quantum strange attractor

1. Problem Statement

The paper investigates the quantum dissipative time evolution of a fluxonium qubit subjected to a periodic pulsed field (kicks). While the classical dynamics of such "kicked harmonic oscillators" (specifically the Zaslavsky web map) are well-understood to converge to strange chaotic attractors with fractal structures in phase space, the behavior of the corresponding quantum system in the presence of dissipation remains less explored.

The core problem addresses the interplay between:

• Quantum Chaos: The tendency of quantum wave packets to spread (Ehrenfest explosion) due to exponential instability in chaotic systems.
• Dissipation: The environmental coupling (modeled by a Lindblad equation) that tends to localize the system.
• The Transition: Determining under what conditions the quantum system converges to a "quantum strange attractor" that mirrors its classical counterpart, and how the eigenstates of the density matrix behave (localized vs. delocalized) depending on the dissipation strength.

2. Methodology

The authors employ a combination of analytical derivations and large-scale numerical simulations.

• Model System:

  • The system is a fluxonium circuit described by a time-dependent Hamiltonian involving a kinetic term ($4E_C \hat{N}^2$), an inductive term ($E_L \hat{\phi}^2/2$), and a Josephson junction term modulated by periodic $\delta$-function kicks.
  • In dimensionless units, this maps to a kicked harmonic oscillator (Zaslavsky web map) with Hamiltonian $\hat{H}_f = (\hat{p}^2 + \hat{x}^2)/2 - K \cos(q\hat{x}) \sum_m \delta(t-mT)$.
  • The study focuses on the case where the kick period $T$ relates to the oscillator period by $R=4$ (i.e., $T = \pi/2$).

• Dissipative Dynamics:

  • The evolution is governed by the Lindblad equation for the density matrix $\hat{\rho}$:
    $$\frac{\partial \hat{\rho}}{\partial t} = -i[\hat{H}, \hat{\rho}] + 2\gamma \left( \hat{a}\hat{\rho}\hat{a}^\dagger - \frac{1}{2}\{\hat{a}^\dagger\hat{a}, \hat{\rho}\} \right)$$
    where $\gamma$ is the dissipation rate.
  • The evolution is split into two steps per period:
    1. Free Propagation: Analytical solution of the damped harmonic oscillator between kicks.
    2. Kick: Unitary transformation applied to the density matrix using matrix elements of the cosine operator in the harmonic oscillator basis (involving Hermite and Laguerre polynomials).

• Numerical Implementation:

  • Simulations were performed using a basis of up to $N=2000$ oscillator eigenstates, resulting in a density matrix with up to $4 \times 10^6$ components.
  • The code was adapted from previous work on dissipative qubits to handle the specific kicked fluxonium Hamiltonian.
  • Metrics: The authors analyzed the Husimi function (phase-space distribution), eigenvalues/eigenvectors of the density matrix, entanglement entropy ($S_E$), and quantum negativity ($G_N$) to characterize the quantum state.

3. Key Contributions

• Demonstration of Quantum Strange Attractors: The paper provides numerical evidence that a dissipative quantum kicked fluxonium converges to a steady-state density matrix that forms a "quantum strange attractor," structurally similar to the classical strange attractor.
• Localization vs. Delocalization Regimes: The authors identify distinct regimes based on dissipation strength ($\gamma$):
  • Strong/Moderate Dissipation: The eigenstates of the density matrix are localized in phase space. The system exhibits a "collapse" of the wave packet, forming Schrödinger cat-like eigenstates (symmetric pairs).
  • Weak Dissipation: The authors argue (based on the Ehrenfest time scaling) that eigenstates become delocalized, linked to the "Ehrenfest explosion" of the wave packet, though this regime was not fully simulated due to computational limits.
• Decoherence Timescales: The study clarifies the relationship between the Ehrenfest time ($t_E \sim |\ln \hbar|/\Lambda$) and the dissipative time ($t_\gamma = 1/\gamma$). It confirms that when $t_\gamma < t_E$, the system collapses to a localized attractor; when $t_\gamma > t_E$, quantum interference leads to delocalization.

4. Key Results

• Steady-State Convergence: For parameters $K_{cl} = 6.4$ and $\gamma = 0.05$, the quantum Husimi function converges to a steady state after approximately 40 kicks. This distribution closely matches the classical strange attractor obtained from $4 \times 10^6$ classical trajectories.
• Fractal Dimension: The fractal information dimension of the attractor is estimated as $d_1 \approx 1.95$, consistent with the classical prediction $d_1 = 2 - \gamma/\Lambda$.
• Eigenstate Structure:
  • The dominant eigenstate of the density matrix localizes in phase space.
  • Over time, this localized packet splits into two symmetric packets (a "Schrödinger cat" state) due to the system's $x \to -x$ symmetry.
  • This splitting occurs on a timescale $t_{cat} \approx 420$, which is significantly longer than the global relaxation time $t_\gamma \approx 20$, due to quasi-degeneracy in the eigenvalue spectrum.
• Entanglement and Negativity:
  • Entanglement Entropy ($S_E$): Grows with time, reaching a maximum determined by the size of the strange attractor in the available Hilbert space.
  • Quantum Negativity ($G_N$): Drops rapidly to near zero for times $t > t_\gamma$. This indicates that for times longer than the dissipation time, the quantum features (non-classical correlations) are washed out, and the system behaves effectively classically, despite the underlying quantum formalism.

5. Significance and Future Outlook

• Experimental Realizability: The paper argues that current superconducting fluxonium qubits, which have achieved long coherence times and high fidelity, are ideal candidates for experimentally observing these quantum strange attractors. The required parameters (kick strength, dissipation rates) are within reach of existing technology.
• Fundamental Physics: The work bridges the gap between classical chaotic dynamics and open quantum systems, providing a concrete framework for understanding how quantum coherence survives (or is destroyed) in chaotic environments with dissipation.
• Implications for Quantum Control: Understanding the transition between localized (cat-like) and delocalized states is crucial for error correction and the stability of quantum information in chaotic systems. The results suggest that while the global density matrix mimics classical chaos, the internal structure (eigenstates) retains specific quantum signatures (like cat states) that could be targeted for quantum control protocols.

In summary, the paper establishes that dissipative quantum chaos in a kicked fluxonium system leads to a quantum strange attractor, where the interplay between dissipation and chaos determines whether the system's eigenstates remain localized (classical-like) or delocalized (quantum explosion), with significant implications for the experimental study of open quantum chaotic systems.