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Geometric Floquet Condition for Quantum Adiabaticity

Jie Gu, X. -G. Zhang

Abstract

Quantum adiabaticity is the evolution of a quantum system that remains close to an instantaneous eigenstate of a time-dependent Hamiltonian. Using Floquet formalism, we derive a rigorous sufficient condition for adiabaticity in periodically driven systems that is valid for arbitrarily many driving periods. The condition is stroboscopic and geometric, depending only on single-cycle information: the Fubini--Study length of the instantaneous eigenray and a quasienergy-separation measure extracted from the Floquet operator. We illustrate the criterion and contrast it with conventional instantaneous-gap conditions in three representative examples.

Geometric Floquet Condition for Quantum Adiabaticity

Abstract

Quantum adiabaticity is the evolution of a quantum system that remains close to an instantaneous eigenstate of a time-dependent Hamiltonian. Using Floquet formalism, we derive a rigorous sufficient condition for adiabaticity in periodically driven systems that is valid for arbitrarily many driving periods. The condition is stroboscopic and geometric, depending only on single-cycle information: the Fubini--Study length of the instantaneous eigenray and a quasienergy-separation measure extracted from the Floquet operator. We illustrate the criterion and contrast it with conventional instantaneous-gap conditions in three representative examples.
Paper Structure (12 sections, 2 theorems, 43 equations, 3 figures)

This paper contains 12 sections, 2 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction.---
  2. Setup and main results.---
  3. Remarks and connection with existing conditions.---
  4. Experimental considerations.---
  5. Example 1: Schwinger-Rabi model.---
  6. Example 2.---
  7. Example 3: Many-body collective Ising model.---
  8. Discussion and outlook.---
  9. A geometric bound in the instantaneous eigenbasis
  10. A stroboscopic Floquet mixing bound
  11. Dominant Floquet component and uniform-in-time fidelity
  12. State-targeted variant

Key Result

Theorem 1

The loss of fidelity amplitude, $1-|d_n(t)|$, is bounded uniformly in time as if $\mathcal{L}_n \le {\pi}/{2}$, where the one-period Fubini--Study length ProvostVallee1980AnandanAharonov1990 with the gauge-invariant Fubini--Study speed and the (mod-$\omega$) quasienergy separation measure

Figures (3)

  • Figure 1: Maximum loss of fidelity amplitude (dashed lines) and the upper bound (solid lines) as a function of $\omega/\omega_0$ for different $\theta$ values (different colors). The inset shows $\omega/\omega_0 \in [0.8,1.2]$.
  • Figure 2: Parameter regions in which each criterion ensures that the maximum loss of fidelity amplitude does not exceed $\varepsilon=0.05$.
  • Figure 3: Mitigation of the $N$-scaling issue in the geometric Floquet condition. Parameters are $\Omega_0=15$, $\kappa=1$, $A=0.002$, and $\omega=\Omega_0(N_s+1)$.

Theorems & Definitions (2)

  • Theorem 1
  • Lemma 1: Fubini--Study bound