Quasienergy

Quasienergy is a concept in the physics of periodically driven quantum systems, arising from Floquet theory. When a system is governed by a time-periodic Hamiltonian H(t) with period T, the Schrödinger equation admits solutions that can be written as a product of a phase factor and a time-periodic function. These solutions define quasienergies as energy-like labels for the stationary states under periodic driving.

Mathematically, for H(t) = H(t+T), the Floquet theorem guarantees solutions of the form |ψα(t)> = e^{-iεα t/ħ} |φα(t)>, where |φα(t)> = |φα(t+T)>. The quasienergies εα are the eigenvalues of the Floquet operator, equivalently the eigenvalues of the extended (Floquet) Hamiltonian H(t) − iħ ∂/∂t in the enlarged Hilbert space, or, equivalently, the eigenphases of the one-period evolution operator U(T) = T exp[−(i/ħ) ∫0^T H(t') dt']. The quasienergies are defined modulo ħΩ, with Ω = 2π/T, so the spectrum is periodic and can be represented in a Brillouin-zone-like interval of width ħΩ.

Properties and interpretation: quasienergies act as conserved quantities modulo ħΩ for the stroboscopic evolution but are not strict energy eigenvalues of the instantaneous Hamiltonian, since energy is not generally conserved in a time-dependent drive. The concept is analogous to Bloch’s theorem in space, reflecting time-translation symmetry in periodically driven systems. The quasienergy spectrum enables efficient descriptions via effective Hamiltonians, especially in the high-frequency regime, where U(T) ≈ e^{−iH_eff T/ħ}.

Applications include coherent destruction of tunneling, dynamical localization, and the study of heating and prethermalization in interacting Floquet systems. Computation typically involves diagonalizing the Floquet operator or analyzing H_eff through Magnus or high-frequency expansions.

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