Observation of a tripartite quantum phase for coexisting extended, localized, and critical states

Technical Summary: Observation of a Tripartite Quantum Phase for Coexisting Extended, Localized, and Critical States

Problem Statement
The quantum world of disordered systems is traditionally characterized by a dichotomy between extended (metallic) and localized (insulating) states, governed by Anderson localization. However, a third, more enigmatic class of states exists: critical states, which exhibit multifractal geometry and scale-invariant properties. While theoretical frameworks predict the existence of a "tripartite phase" where extended, localized, and critical states coexist within a single energy spectrum—separated by anomalous mobility edges—experimental realization has remained elusive. The primary challenges lie in the precise preparation of high-energy eigenstates in many-particle systems and the unambiguous detection of these distinct state types, particularly the critical states which are difficult to populate via standard adiabatic loading.

Methodology
The authors report the experimental realization of this tripartite phase using ultracold $^{87}\text{Rb}$ atoms in a one-dimensional, quasi-periodically driven orbital optical lattice.

• Physical System: The experiment employs a bichromatic optical lattice consisting of a deep primary static lattice ($\lambda_0 = 1064$ nm) and a weak secondary lattice ($\lambda_1 = 760$ nm). The secondary lattice is shaken at a frequency resonant with the energy gap between the $s$ and $p$ orbitals of the primary lattice. The spatial incommensurability between the two lattices ($\beta \approx 1.411$) induces a site-dependent inter-orbital coupling, creating a quasiperiodic Floquet system.
• Theoretical Model: The system is described by a tight-binding Hamiltonian where $s$ and $p$ orbitals are coupled via a quasiperiodic modulation. Theoretical analysis using renormalization group (RG) methods and exact diagonalization confirms that for specific parameter regimes, the spectrum hosts three distinct zones: extended states (dominated by $p$-orbital hopping), localized states (dominated by quasiperiodic potentials), and critical states (emerging from nontrivial $s$-$p$ coupling).
• Two-Stage Preparation Protocol: To overcome the difficulty of preparing specific high-energy eigenstates, the authors developed a novel two-stage protocol:
  1. Adiabatic Frequency Ramp: The shaking frequency is slowly ramped to the $s$-$p$ resonance ($f_f = 10.2$ kHz). Depending on the initial frequency and shaking amplitude, the system evolves into either a low-energy extended state (via Landau-Zener transition) or a high-energy extended state (via adiabatic evolution).
  2. Diabatic Amplitude Ramp: The shaking amplitude is rapidly reduced. This step projects the intermediate high-energy extended state into the energy window hosting critical states, while the low-energy state remains extended and the low-amplitude state remains localized.
• Detection: The nature of the prepared states is identified through:
  • Momentum Distributions: Measured via time-of-flight (TOF) imaging. Extended states show sharp Bragg peaks; localized states show broad distributions; critical states exhibit fragmented, multi-peak structures.
  • Characteristic Lengths: Spatial coherence length ($L_{\text{Coh}}$) and momentum-space density-density correlation length ($k_{\text{Cor}}$) are extracted to distinguish the three regimes.
  • Expansion Dynamics: The spatial width $\sigma(t)$ of the atomic cloud is monitored after trap release. The scaling exponent $\alpha$ in $\sigma(t) \sim t^\alpha$ serves as a dynamical signature: $\alpha \approx 1$ for ballistic (extended), $\alpha \approx 0$ for localized, and $\alpha \approx 0.5$ for anomalous diffusion (critical).

Key Contributions and Results

• Experimental Realization: This work presents the first experimental observation of a tripartite quantum phase where extended, localized, and critical states coexist in a single spectrum.
• State Preparation: The authors successfully demonstrated a controlled method to selectively prepare ultracold bosons into any of the three state types by tuning the initial shaking frequency and amplitude. This includes the first controlled population of critical states in this context.
• Verification of Criticality:
  • Fractal Dimension: Numerical simulations of the fractal dimension (FD) confirmed the existence of three distinct regions in the energy spectrum, separated by sharp transitions.
  • Dual Delocalization: Experimental data showed that critical states exhibit "dual delocalization," characterized by large values of both spatial coherence length and momentum correlation length, consistent with multifractal wavefunctions.
  • Universal Transport: The expansion dynamics measurements yielded scaling exponents of $\alpha \approx 1$ (extended), $\alpha \approx 0$ (localized), and $\alpha \approx 0.5$ (critical), providing direct dynamical evidence of the distinct transport properties of each phase.
• Anomalous Mobility Edges: The effective preparation phase diagram, mapping the final state against initial parameters, revealed the existence of anomalous mobility edges separating the three distinct localization regimes.

Significance
The paper claims that this work paves the way for a comprehensive exploration of unconventional critical phenomena and localization physics. By establishing a versatile platform based on quasiperiodic shaken lattices, the authors provide a powerful toolbox for synthesizing complex quasiperiodic models and circumventing the state-preparation challenges inherent in static systems. The ability to selectively populate specific eigenstates in targeted energy regions lays the foundation for investigating interaction-driven localization, many-body critical phases, and the elusive many-body mobility edges. The study confirms the theoretical prediction of the tripartite phase and offers a benchmark for exploring the seven fundamental localization phases predicted in quasiperiodic systems.