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Simple determination of dark states in a general multi-level system

Kaixuan Zhou, June Wu, Junheng Shi, Tim Byrnes

TL;DR

This work addresses the problem of identifying dark states in general multi-level quantum networks without full Hamiltonian diagonalization. It shows that dark states exist if and only if the ground-to-excited coupling block has a zero (right) singular value, which is equivalent to a vanishing determinant of the product B†B, and provides a determinant-based construction for the dark state from the same submatrix. The authors prove key results: excited-to-excited transitions not connected to ground states do not affect darkness, and a simple rank criterion (Rank(B) < N_g) guarantees a dark state; they illustrate the approach with Lambda, 3-layer pyramid, and a 12-level system, deriving explicit dark-state expressions. The method yields analytic, scalable means to obtain dark states in large networks, with potential applicability to STIRAP and related coherent control schemes, while ignoring dissipation in the core analysis.

Abstract

In a multi-level energy system with energy transitions, dark states are eigenstates of a Hamiltonian that consist entirely of ground states, with zero amplitude in the excited states. We present several criteria which allows one to deduce the presence of dark states in a general multi-level system based on the submatrices of the Hamiltonian. The dark states can be shown to be the right-singular vectors of the submatrix that connect the ground states to the excited states. Furthermore, we show a simple way of finding the dark state involving the determinant of a matrix constructed from the same submatrix.

Simple determination of dark states in a general multi-level system

TL;DR

This work addresses the problem of identifying dark states in general multi-level quantum networks without full Hamiltonian diagonalization. It shows that dark states exist if and only if the ground-to-excited coupling block has a zero (right) singular value, which is equivalent to a vanishing determinant of the product B†B, and provides a determinant-based construction for the dark state from the same submatrix. The authors prove key results: excited-to-excited transitions not connected to ground states do not affect darkness, and a simple rank criterion (Rank(B) < N_g) guarantees a dark state; they illustrate the approach with Lambda, 3-layer pyramid, and a 12-level system, deriving explicit dark-state expressions. The method yields analytic, scalable means to obtain dark states in large networks, with potential applicability to STIRAP and related coherent control schemes, while ignoring dissipation in the core analysis.

Abstract

In a multi-level energy system with energy transitions, dark states are eigenstates of a Hamiltonian that consist entirely of ground states, with zero amplitude in the excited states. We present several criteria which allows one to deduce the presence of dark states in a general multi-level system based on the submatrices of the Hamiltonian. The dark states can be shown to be the right-singular vectors of the submatrix that connect the ground states to the excited states. Furthermore, we show a simple way of finding the dark state involving the determinant of a matrix constructed from the same submatrix.
Paper Structure (9 sections, 9 theorems, 52 equations, 2 figures)

This paper contains 9 sections, 9 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Properties of the multi-level network
  4. Examples
  5. $\Lambda$ System
  6. 3 layer pyramid
  7. 12 level system
  8. Summary and conclusions
  9. Acknowledgements

Key Result

Lemma 1

Assuming that $\text{det}(A)\neq 0$, the determinant of the Hamiltonian (chosen_form) obeys

Figures (2)

  • Figure 1: An example multi-level energy network that is considered in this paper. Shown are the (a) physical energy level configuration corresponding to Hamiltonian (\ref{['genham']}); (b) the equivalent system after performing a unitary transformation to obtain (\ref{['detuningham']}). Arrows between energy level indicate non-zero transition amplitudes $\Omega_{ij}$ with frequencies $\omega_{ij}$ between energy levels $|i \rangle$ and $| j \rangle$. Dashed lines indicate the energy to which the transitions are detuned to, with the associated detunings $\Delta_1, \Delta_2$ as marked. Using the transformation (\ref{['unitarytrans']}), the time dependent Hamiltonian (\ref{['genham']}) is transformed to the time independent Hamiltonian (\ref{['detuningham']}) as shown in (b).
  • Figure 2: Two multi-level energy configurations considered in Sec. \ref{['sec:4']}. A (a) $N = 6$, $N_g = 3$, $N_e = 2$; (b) $N = 13$, $N_g = 5$, $N_e = 5$ level system. We use the Hamiltonian after performing the transformation (\ref{['unitarytrans']}) such that the transition amplitudes are time-inedependent and the associated detunings are marked.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 8 more