Source-linked AI summary
Information Retrieval and Criticality in Parity-Time-Symmetric Systems
Kohei Kawabata, Yuto Ashida, Masahito Ueda
TL;DR
The paper examines recurrence and distinguishability near exceptional points in PT-symmetric dynamics using spectral and eigenstate analyses. It derives power-law behavior for recurrence time and distinguishability, alongside exponential distinguishability decay in the PT-broken phase.
Problem
The paper investigates how recurrence time and quantum-state distinguishability behave near exceptional points in PT-symmetric dynamics.
Method
It uses spectral decomposition and biorthogonal eigenstate analysis to derive recurrence, relaxation, and distinguishability behavior across PT phases.
Results
The analysis derives power-law recurrence time in the PT-unbroken phase, exponential distinguishability decay in the PT-broken phase, and power-law distinguishability behavior at the exceptional point.
Takeaways & Limitations
These results characterize distinct dynamical scaling regimes associated with PT phases and the exceptional point.
Takeaways & Limitations
The broken-phase analysis focuses on the eigenstates with the largest and second-largest imaginary energy components and their coalescence at the exceptional point.
Abstract
from arXiv · showhide
By investigating information flow between a general parity-time (PT) -symmetric non-Hermitian system and an environment, we find that the complete information retrieval from the environment can be achieved in the PT-unbroken phase, whereas no information can be retrieved in the PT-broken phase. The PT-transition point thus marks the reversible-irreversible criticality of information flow, around which many physical quantities such as the recurrence time and the distinguishability between quantum states exhibit power-law behavior. Moreover, by embedding a PT-symmetric system into a larger Hilbert space so that the entire system obeys unitary dynamics, we reveal that behind the information retrieval lies a hidden entangled partner protected by PT symmetry. Possible experimental situations are also discussed.
Supplemental Material
The supplemental material derives the recurrence-time criticality in the PT-unbroken phase and the distinguishability and relaxation-time behavior in the PT-broken phase. It shows that exceptional-point power laws depend on eigenstate coalescence and system-specific coefficients.
- Recurrence-time criticality: The recurrence time is the least common multiple of 2π/ω_mn and diverges at the exceptional point as T ∼ 2π/∆ω.The vanishing energy difference ∆ω between coalescing states controls the divergence.
- PT-broken-phase dynamics: In the PT-broken phase, the distinguishability decays exponentially while the relaxation time follows a power law.The asymptotic dynamics is dominated by the two eigenstates with the largest imaginary parts.
- Exceptional-point scaling: Near the exceptional point, the imaginary-part splitting satisfies ∆Γ ∼ (∆λ)^(1/p).The exponent follows from the power-law expansion of the coalescing eigenenergies.
- Distinguishability criticality: The critical exponent for |C_ρ − C_σ| is e_1 = 1 when |ϕ_2⟩ coalesces at the exceptional point and e_1 = p − 1 otherwise.The two cases correspond to whether |ϕ_2⟩ converges to |EP⟩ or to a distinct state.
- Distinguishability criticality: For 1 − |⟨ϕ_1|ϕ_2⟩|^2, the critical exponent is e_2 = 1 when |ϕ_2⟩ coalesces and e_2 = 0 otherwise.The result follows from the different zeroth- and second-order terms in the eigenstate-overlap expansion.
Explicit form of the extended Hermitian Hamiltonian
The extended Hermitian Hamiltonian is constructed so that its projected dynamics reproduce the original PT-symmetric system and its Hermitian counterpart. The resulting operators are Hermitian, with the interaction vanishing for Hermitian H_PT, while the construction also yields the two-level distinguishability dynamics.
- Explicit construction: The extended Hamiltonian H_tot is explicitly constructed from an ansatz with H_S and V required to be Hermitian.This follows from the Hermiticity of H_tot.
- Projected dynamics: Projection onto the |↑⟩ sector reproduces H_PT, whereas projection onto the |↓⟩ sector reproduces ζ^1/2 H_PT ζ^-1/2.The corresponding operator relations are given in Eq. (S25).
- Hermiticity: The solved operators H_S and V are indeed Hermitian, consistent with the Hermiticity of η, ζ, and h and the reality of c.The construction uses h := η^1/2 H_PT η^-1/2 as a Hermitian operator.
- Special cases: The interaction H_I := σ_y ⊗ V vanishes when H_PT is Hermitian, and for N = 2, ζ^1/2 reduces to η.The two-level reduction is identified as reproducing the results of Ref..
- Two-level dynamics: For the two-level system H_PT = s(σ_x + iaσ_z), the time-evolution operator and evolved states yield the distinguishability dynamics in Eq. (11).The distinguishability is calculated from the time-evolved states initialized in |↑⟩ and |↓⟩.
Criticality at the exceptional point in the PT-symmetric optics
At the exceptional point of a PT-symmetric periodic potential, the norm grows linearly with propagation distance for k0 = −1, while distinguishability decays by power laws determined by the initial states. Different centers yield a 1/z2 approach to a nonzero asymptote, whereas different widths yield 1/z decay.
- Setup: At the exceptional point, a Gaussian input profile is used, with k0 and w denoting the initial beam’s offset and width.The setup relies on an exact solution of the Schrödinger equation for the PT-symmetric periodic potential.
- Norm criticality: For k0 = −1, the norm |ψ(x, z)| grows linearly with propagation distance z.This linear norm growth occurs in the specified exceptional-point setup.
- Distinguishability criticality: For initial states with different centers, |D(z) − D(z = ∞)| ∼ 1/z2, with D(z = ∞) = 0.430531 ≠ 0.The distinguishability therefore approaches a nonzero asymptotic value rather than decaying to zero.
- Distinguishability criticality: For initial states with different widths, the distinguishability asymptotically behaves as D ∼ 1/z at sufficiently long propagation distance.The two initial-state choices thus produce distinct critical exponents for distinguishability.