Quasienergies

Quasienergies arise in quantum systems with a time-periodic Hamiltonian H(t) = H(t + T), where T is the driving period. According to Floquet theory, solutions to the Schrödinger equation can be written as psi_alpha(t) = exp(-i epsilon_alpha t / hbar) phi_alpha(t), where phi_alpha(t) is periodic with period T and epsilon_alpha are the quasienergies. The quasienergies are defined modulo hbar omega, with omega = 2 pi / T, so adding multiples of hbar omega yields physically equivalent states.

Quasienergies are not simply eigenvalues of H(t) at any moment. They are the eigenvalues of the one-period evolution operator U(T,0): U(T,0) |phi_alpha(0)> = exp(-i epsilon_alpha T / hbar) |phi_alpha(0)>. Equivalently, they are the eigenvalues of the Floquet operator or of the Floquet Hamiltonian H_F = H(t) - i hbar d/dt in the enlarged, time-periodic Hilbert space. In practice, one often diagonalizes U(T,0) to obtain the quasienergies.

Interpretation and implications: the phase accumulated per period is governed by epsilon_alpha, and energy is not conserved in a driven system. Physics is invariant under shifts epsilon_alpha -> epsilon_alpha + m hbar omega, so quasienergies form a Brillouin-like zone of width hbar omega. The concept mirrors Bloch theory for spatial periodicity, but in the time domain.

Applications and methods: quasienergies underpin Floquet engineering, where high-frequency driving creates effective static Hamiltonians with tailored properties. They are central to periodically driven two-level systems, cold atoms in modulated lattices, and topological phases of driven matter. Practical analyses often use approximate effective Hamiltonians or truncated Floquet expansions, with attention to heating and resonance effects.