Numerics.FloquetStates

Floquet Theory: light-dressed states

If the electric fields F(t) in the time-dependent Schrödinger equation are periodically oscillating in time, it may be advantageous to use the Floquet picture of light-dressed states. Note that the rotating wave approximation represents a special case of light-dressed states, see below.

There are two equivalent ways of introducing Floquet theory. The easier one is the introduction of cavity-dressed states. For this, we combine the Hilbert space of a quantum system with basis |φ_i⟩ and the Fock space of photon states with a certain central frequency ω, where one basis are the photon number states |n⟩. The combined space has then a basis that contains all quantum states and photon number states, |n,φ_i⟩=|n⟩|φ_i⟩.
In this quantum-field picture, the Hamiltonian for the photon-system interaction becomes time-independent, and the size of its matrix elements depends on the number of photons n (i.e., on the intensity). We can describe for example the absorption of a photon by starting our system in a state |n₀;φ₀⟩, and waiting for it to occupy some state |n₀-1;φ₁⟩. We can trivially lift every operator  that measures only properties of the quantum system into the product state by the definition ⟨n,φ_i|Â|n';φ_j⟩ = ⟨φ_i|Â|φ_j⟩δ_nn', and calculate appropriate expectation values.

Alternatively, we can consider a purely classical electric field. We start with a periodic Hamiltonian, and want to propagate some quantum state through time for all possible initial phases θ₀ of the periodicity at once. For this, we first define an extended Hilbert space (Floquet space) as the direct product of the quantum system's Hilbert space (with basis |φ_i⟩), and the space of all possible initial phases θ₀∈(0,2π) where 0 and 2π are the same element (with a basis |n⟩∼exp(inθ₀), n∈(-∞,∞). Again, operators that are independent of the phase (i.e., of the electric field) can be trivially lifted by the same definition as before; the lifted operator has no dependence on the initial phase θ₀, and expectation values of the lifted operator are the expectation values of the original operator averaged over all possible initial phases of the Hamiltonian.

It can be shown, see this paper that both descriptions are equivalent; however, the classical Floquet picture is easier extended to describe also laser fields with a slowly varying envelope and/or central frequency. For these cases, we can introduce adiabatic Floquet theory, where we assume that all states in the Floquet space adapt to the instantaneous field strength and frequency without population changes. Corrections to the adiabatic Floquet theory yield superadiabatic Floquet theories in some sort of perturbation series, see this paper.

Advantages of using Floquet theory

• Imagine we want to study a specific process, e.g., one-photon absorption from a ground state φ₀ to some excited state φ₁. Normally, we would pick as basis these two states, and have a classical field with (near-)resonant frequency ω₀=E₁−E₀+Δ with Δ being a (small) detuning from the resonance or a typical width of the spectrum. For the propagation, we generally would need a time step, which matches the fastest time scale, usually the oscillation of the electric field 1/ω₀. If we use the cavity-dressed state picture, we have two states |n;φ₀⟩, |n-1;φ₁⟩ with a time-independent Hamiltonian (n being some large number of photons for the classical approximation). The energies of the states are then E₀+E_n and E₁+E_n−ħω₀=E₀+E_n−Δ, and the time stepping for propagation is then on the order of 1/Δ (a similar relation holds for the classical Floquet picture). Thus, staying in the larger Floquet space allows us to eliminate the fast oscillation of the laser field.

• If our states φ_i are coupled, then each resonant excitation becomes formally equivalent to a diabatic state crossing. If we adiabatize the dressed states, we obtain intensity-dependent potential energy surfaces, showing avoided crossings or conical intersections at each resonance. This can help with understanding some laser-induced processes, and Floquet theory has been applied to describe for example bond softening and hardening in photodissociation, see, e.g., this paper.

• Using the adiabatic and super-adiabatic Floquet states, one can construct a perturbation expansion in orders of T₀/τ, where T₀ is the period of the oscillation, and τ is the typical time on which the field parameters (intensity, frequency) vary.

Rotating Wave Approximation

The conventional way of deriving the RWA is by splitting up a real-valued electric field into a sum of fields with positive and negative frequency components, and identifying these with absorption and emission processes, respectively. Then, those elements of the Hamiltonian are neglected that correspond to the less likely process (e.g., excitation to a state with higher energy under emission of a photon), and obtains the RWA.

This is essentially identical to a Floquet calculation, where only the ground state dressed with n₀ photons and the excited state dressed with n₀-1 photons are included (n₀ being a large but arbitrary photon number). The formulation with complex electric fields is included specially because it is more widely used in the context of optimal control.