Equivalence of dynamics of disordered quantum ensembles and semi-infinite lattices

1. Problem Statement

The paper addresses the challenge of calculating the exact dynamical evolution of disordered quantum ensembles.

• Context: A disordered ensemble consists of many non-interacting quantum systems where parameters (e.g., energy levels) vary randomly according to a specific probability distribution $p(\lambda)$.
• The Issue: To predict observable properties, one must average over all possible disorder realizations. While individual systems evolve unitarily, the ensemble average exhibits dephasing (loss of coherence), a classical mechanism distinct from open-system environmental decoherence.
• Current Limitations: The standard approach involves Monte Carlo sampling or numerical quadrature (sampling the disorder space). This is computationally expensive, yields approximate results dependent on the number of samples, and struggles to capture exact long-time dynamics or specific features like coherence revivals without massive sampling.

2. Methodology

The authors propose a formalism that maps the continuous integral representation of the disordered ensemble onto a single, discrete, semi-infinite lattice. This is achieved through a unitary change of basis using orthogonal polynomials.

Key Steps in the Mapping:

1. Pure State Formalism: The ensemble is described by a pure state $|\Psi_0\rangle = \int d\lambda \sqrt{p(\lambda)} |\psi_0, \lambda\rangle$ and a Hamiltonian $\hat{H}_{Ens} = \int d\lambda \hat{H}_\lambda$.
2. Orthogonal Polynomial Basis: A set of orthonormal polynomials $\{\phi_k(\lambda)\}$ is introduced, which are mutually orthonormal with respect to the disorder measure $p(\lambda)d\lambda$.
3. Basis Transformation: The continuous disorder label $\lambda$ is transformed into a discrete lattice index $k$ via:
   $$|n, \lambda\rangle = \sum_{k=0}^{\infty} \sqrt{p(\lambda)}\phi_k(\lambda) |n, k\rangle$$
   where $|n, k\rangle$ represents the $n$-th system state on the $k$-th lattice site.
4. Hamiltonian Transformation:
   • The disorder-independent part of the Hamiltonian projects onto the diagonal of the lattice.
   • The disorder-dependent part (e.g., linear disorder $V_\lambda \propto \lambda$) transforms into hopping terms (couplings) between lattice sites.
   • Crucially, if the disorder enters linearly, the resulting lattice Hamiltonian involves only nearest-neighbor interactions. The coupling strengths are determined by the recurrence coefficients ($\alpha_k, \beta_k$) of the orthogonal polynomials associated with the distribution $p(\lambda)$.

Recovering Dynamics:

The exact ensemble-averaged density matrix $\bar{\rho}(t)$ is recovered by performing a partial trace over the lattice degree of freedom (the index $k$) of the evolved pure state on the lattice.

3. Key Contributions

• Exact Equivalence: The paper establishes a rigorous, unitary equivalence between the dynamics of a disordered quantum ensemble and a single particle propagating on a semi-infinite lattice. This bypasses the need for statistical sampling.
• Geometric Interpretation of Dephasing: The loss of coherence in the ensemble average is geometrically interpreted as the spreading (delocalization) of the wavefunction along the lattice. As the probability amplitude spreads to higher lattice sites ($k$), the partial trace over $k$ results in a loss of off-diagonal elements (coherence) in the system's reduced density matrix.
• Reverse Mapping: The authors demonstrate the duality: one can also generate lattice dynamics by averaging over disorder realizations of a unit cell.
• Handling Specific Distributions: The method explicitly handles various distributions (Gaussian, Cauchy, Semicircle, Uniform) by utilizing their specific recurrence coefficients.
• Energy Cutoff Strategy: For distributions with undefined moments (e.g., Cauchy) or to improve numerical efficiency for Gaussian distributions, the authors introduce an energy cutoff to ensure well-defined recurrence coefficients and asymptotic convergence of couplings.

4. Results and Examples

The authors validate the formalism through several examples:

• Disordered Qubit Dephasing:

  • Setup: An ensemble of qubits with excited state energies sampled from different distributions (Gaussian, Cauchy, Semicircle, Uniform).
  • Result: The lattice simulation reproduces exact analytical solutions for dephasing.
  • Observation: Distributions with finite support (Semicircle, Uniform) exhibit coherence revivals (oscillations returning to high coherence), whereas infinite support distributions (Gaussian) show monotonic decay. The Cauchy distribution requires an energy cutoff for numerical stability but still captures the correct physics.
  • Accuracy: Errors are found to be on the order of machine precision for well-behaved distributions.

• Disordered Dimer (Light-Harvesting Complex LH2):

  • Setup: An ensemble of dimers with site energies sampled from Gaussian distributions.
  • Result: The ensemble average shows non-unitary dynamics where populations relax initially but settle into a new oscillatory stationary state rather than a static equilibrium. This demonstrates the method's ability to capture complex many-body effects in disordered systems.

• Wigner Semicircle and Constant Couplings:

  • Insight: A Wigner semicircle distribution leads to constant recurrence coefficients, mapping the ensemble to a translationally invariant semi-infinite lattice with constant nearest-neighbor coupling. This provides a new physical interpretation of lattice dynamics as an ensemble average of disordered systems.

5. Significance

• Computational Efficiency: The method allows for the calculation of exact ensemble dynamics in a single simulation, avoiding the statistical noise and high computational cost of Monte Carlo sampling. It is particularly efficient for low-dimensional disorder (few random variables).
• Physical Insight: It bridges two seemingly distinct fields: disordered quantum systems and lattice dynamics. It offers a geometric intuition for decoherence, framing it as wavefunction propagation in an auxiliary space.
• Versatility: The approach is applicable to a wide range of fields including condensed matter, quantum optics, quantum chemistry, and quantum biology, wherever ensemble averaging over disorder is required.
• Theoretical Foundation: It connects the theory of orthogonal polynomials (specifically the three-term recurrence relation) directly to the physical dynamics of quantum ensembles, offering a new tool for analyzing operator complexity and spectral properties.

In summary, the paper provides a powerful mathematical framework that transforms the stochastic problem of disorder averaging into a deterministic problem of unitary evolution on a lattice, offering both computational advantages and deep physical insights into the nature of decoherence in disordered systems.