Periodic Hamiltonians

A periodic Hamiltonian repeats after a fixed drive period $T$:

$$H(t+T)=H(t).$$

The corresponding angular frequency is

$$\Omega=\frac{2\pi}{T}.$$

This is a discrete time-translation symmetry. The Hamiltonian is invariant under shifts by integer multiples of $T$, not under arbitrary continuous time translations. Ordinary energy therefore need not be conserved, but evolution over one full period becomes a distinguished unitary operation.

This page sets up that structure. Floquet Theorem in Quantum Mechanics owns the solution theorem, Floquet Operators owns the one-period spectral and numerical analysis, and Quasienergies owns the modular spectrum.

Periodic Driving

A common model is

$$H(t) = H_0+V(t), \qquad V(t+T)=V(t).$$

Examples include:

• sinusoidal electric, magnetic, microwave, or optical driving;
• periodic modulation of trap frequency, lattice depth, or tunneling;
• repeated pulse sequences;
• piecewise-constant switching protocols;
• impulsive kicks repeated every period.

If the periodic dependence is sufficiently regular, it can be expanded as

$$H(t) = \sum_{m\in\mathbb Z} H_m e^{im\Omega t},$$

with Hermiticity requiring

$$H_{-m}=H_m^*.$$

The harmonics $m\Omega$ provide frequencies that can connect levels separated by corresponding Bohr frequencies. The Fourier series organizes the drive, but it does not by itself solve the dynamics: different Fourier components and Hamiltonians at different times may not commute.

Exact periodicity is an idealization. A finite pulse train, a slowly drifting carrier, or a noisy drive is not globally periodic. Floquet reasoning can still be useful over controlled windows, but an approximation or enlarged description is then required.

Propagator Covariance

Let $U(t,s)$ be the propagator:

$$i\hbar \frac{\partial}{\partial t} U(t,s) = H(t)U(t,s), \qquad U(s,s)=I.$$

Periodicity implies

$$U(t+T,s+T) = U(t,s).$$

To see this, define

$$V(t,s) = U(t+T,s+T).$$

It obeys the same differential equation and initial condition as $U(t,s)$ because $H(t+T)=H(t)$. Uniqueness of the propagator then gives $V(t,s)=U(t,s)$.

This covariance is the dynamical expression of discrete time-translation symmetry. It is weaker than dependence only on $t-s$. For a time-independent Hamiltonian,

$$U(t,s)=U(t-s),$$

but a driven propagator generally depends on both times because the phase of the drive matters.

One-Period Evolution

Choose a reference time $t_0$. The one-period propagator is

$$U_F(t_0) \equiv U(t_0+T,t_0).$$

This unitary maps a state from one chosen phase of the drive to the same phase one period later. It is often called the Floquet operator.

Using composition and periodicity,

$$U(t_0+nT,t_0) = U_F(t_0)^n$$

for every nonnegative integer $n$. For negative $n$, unitarity supplies the inverse.

Thus a continuous time-dependent problem induces a discrete quantum map:

$$\lvert\psi_{n+1}\rangle = U_F(t_0)\lvert\psi_n\rangle, \qquad \lvert\psi_n\rangle = \lvert\psi(t_0+nT)\rangle.$$

The continuous motion within each period is not discarded physically; it is simply absent from the stroboscopic sequence.

Reference Phase of the Drive

Changing $t_0$ changes the phase at which each period is sampled. Let $t_1$ be another reference time. Then

$$U_F(t_1) = U(t_1,t_0) U_F(t_0) U(t_1,t_0)^*.$$

The two one-period operators are unitarily conjugate. Consequently, their eigenvalues agree, although their eigenvectors are related by the intervening micromotion.

This is why one should record the reference phase when quoting a Floquet operator or an effective Hamiltonian. Spectral eigenphases are invariant under the shift, but operator representatives and intra-period observables can change.

For a time-symmetric drive, a judicious reference phase can also make symmetries or approximation formulas more transparent. It does not alter the underlying physical protocol.

Stroboscopic Dynamics

The sampling protocol, observables, micromotion reconstruction, aliasing, and simulation interpretation are developed in Stroboscopic Dynamics.

Stroboscopic observation samples the system only at

$$t_n=t_0+nT.$$

At these times, the evolution is generated by repeated powers of one unitary. This can simplify:

• long-time recurrence analysis;
• expectation values sampled once per cycle;
• stability and resonance diagnostics;
• comparison with discrete quantum maps;
• numerical propagation over many periods.

For an observable $A$ in the Heisenberg picture,

$$A_n = \left( U_F^* \right)^n A U_F^n.$$

The sequence is exact at the sampling times.

Stroboscopic simplicity can hide large micromotion. Two protocols can produce the same $U_F$ while following different trajectories inside each period. Measurements performed at intermediate phases can therefore distinguish drives that look identical once per cycle.

The full one-period operator and micromotion decomposition are developed in Floquet Operators. The general discrete-map viewpoint is developed in Quantum Maps and Discrete-Time Evolution.

Why the Period Average Is Usually Insufficient

Define the average Hamiltonian

$$\overline H = \frac{1}{T} \int_{t_0}^{t_0+T} H(t)\,dt.$$

It is tempting to write

$$U_F \stackrel{?}{=} e^{-i\overline H T/\hbar}.$$

This is exact when

$$[H(t),H(t')]=0$$

for all times in the period. It is not generally exact when the Hamiltonian changes direction in operator space.

The first corrections involve nested time commutators. Their direct one-period organization belongs to the Floquet–Magnus Expansion; van Vleck organization, micromotion dressing, resonances, and prethermal validity belong to High-Frequency Expansions.

The warning is physical as well as algebraic. Two drives with the same time average can implement different unitaries because the order of noncommuting operations matters.

Quasienergy Motivation

Because $U_F$ is unitary, an eigenvalue has the form

$$U_F(t_0) \lvert\phi_\alpha(t_0)\rangle = e^{-i\theta_\alpha} \lvert\phi_\alpha(t_0)\rangle.$$

It is useful to write

$$\theta_\alpha = \frac{\varepsilon_\alpha T}{\hbar}.$$

The quantity $\varepsilon_\alpha$ is a quasienergy. The eigenphase does not change under

$$\varepsilon_\alpha \longmapsto \varepsilon_\alpha+m\hbar\Omega, \qquad m\in\mathbb Z,$$

because $\Omega T=2\pi$. Quasienergy is therefore defined modulo the drive quantum $\hbar\Omega$.

If an initial state is an eigenvector of $U_F$, it returns to the same ray after every period:

$$\lvert\psi(t_0+nT)\rangle = e^{-in\varepsilon_\alpha T/\hbar} \lvert\psi(t_0)\rangle.$$

This observation motivates the Floquet solution form: a quasienergy phase multiplied by periodic micromotion. The theorem and modular spectral interpretation belong to their dedicated pages.

Quasienergy is not ordinary conserved energy. The drive can exchange energy with the system, and no absolute lowest quasienergy exists without extra structure.

Example: Commuting Modulation

Suppose

$$H(t) = H_0+f(t)A,$$

where $f(t+T)=f(t)$ and

$$[H_0,A]=0.$$

Then all Hamiltonians commute at different times, and

$$U_F = \exp \left[ -\frac{iT}{\hbar} \left( H_0+\overline f A \right) \right],$$

where

$$\overline f = \frac{1}{T} \int_{t_0}^{t_0+T} f(t)\,dt.$$

In this special case the average Hamiltonian is exact. The drive changes phases in the common eigenbasis but does not create transitions between those eigenvectors.

Example: Two-Step Drive

Let

$$H(t) = \begin{cases} H_1, & 0\leq t\lt\tau, \\ H_2, & \tau\leq t\lt T, \end{cases}$$

and repeat this protocol every period. Sampling immediately before the $H_1$ segment gives

$$U_F = e^{-iH_2(T-\tau)/\hbar} e^{-iH_1\tau/\hbar}.$$

The rightmost factor acts first. If

$$[H_1,H_2]\neq0,$$

then reversing the two steps changes $U_F$ even though the period average

$$\overline H = \frac{ \tau H_1+(T-\tau)H_2 }{T}$$

is unchanged.

This is the simplest exact demonstration that ordering, not only averaging, controls periodic dynamics.

Example: Repeated Kicks

An ideal kicked Hamiltonian has the schematic form

$$H(t) = H_0 + K V \sum_{n\in\mathbb Z} \delta(t-nT).$$

Sampling immediately after one kick and ending immediately after the next gives

$$U_F = e^{-iKV/\hbar} e^{-iH_0T/\hbar}.$$

The state first evolves freely and then receives the next kick. Sampling on the other side of the kick changes the factorization by unitary conjugation.

Ideal delta kicks are approximations to pulses short compared with the other dynamical time scales. Their value is that the one-period map is explicit and can display rich stroboscopic behavior.

Example: Sinusoidally Driven Two-Level System

A standard periodic model is

$$H(t) = \frac{\hbar\omega_0}{2}\sigma_z + \frac{\hbar\Omega_R}{2} \cos(\Omega t)\sigma_x.$$

Near resonance, a rotating frame and rotating-wave approximation can yield a simple effective Hamiltonian. Exact Floquet theory instead constructs $U_F$ for the full periodic drive.

The two methods answer related but distinct questions. A rotating-wave treatment is an approximation controlled by detuning and drive scales. Floquet theory is an exact structural theorem for the periodic model, although computing its spectrum may still require approximation or numerics.

Scope and Validity

The closed-system periodic framework assumes:

• a self-adjoint time-dependent Hamiltonian with well-defined unitary propagation;
• a fixed period $T$;
• no unmodeled dissipation or stochastic drive noise;
• a consistent reference phase for stroboscopic quantities.

Modifications are required for:

• finite pulse trains and slow envelope changes;
• quasiperiodic drives with incommensurate frequencies;
• open systems described by periodic channels or Liouvillians;
• many-body heating and prethermal time scales;
• unbounded Hamiltonians with nontrivial common domains;
• numerical time steps that do not resolve the intra-period dynamics.

Periodicity supplies a powerful organization principle. It does not guarantee a simple effective Hamiltonian, weak heating, adiabatic following, or convergence of a high-frequency expansion.

Common Mistakes

• Assuming $H(t+T)=H(t)$ implies the state itself is periodic.
• Confusing discrete time-translation symmetry with ordinary energy conservation.
• Replacing $U_F$ by the exponential of the average Hamiltonian without checking commutators.
• Ignoring the reference phase $t_0$ of the drive.
• Treating stroboscopic agreement as agreement throughout the period.
• Forgetting that quasienergy is modular.
• Calling a finite pulse train exactly periodic.
• Applying closed-system Floquet theory unchanged to dissipative evolution.
• Treating a rotating-wave approximation as the Floquet theorem.
• Reversing operator order in a piecewise or kicked drive.

References

• J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Physical Review 138, B979–B987, 1965.
• H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Physical Review A 7, 2203–2213, 1973.
• M. Grifoni and P. Hänggi, “Driven quantum tunneling,” Physics Reports 304, 229–354, 1998.
• M. Bukov, L. D’Alessio, and A. Polkovnikov, “Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering,” Advances in Physics 64, 139–226, 2015.
• A. Eckardt, “Colloquium: Atomic quantum gases in periodically driven optical lattices,” Reviews of Modern Physics 89, 011004, 2017.
• N. Goldman and J. Dalibard, “Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields,” Physical Review X 4, 031027, 2014.

Exercises

1. Prove the propagator covariance

$$U(t+T,s+T)=U(t,s).$$

Solution

Define

$$V(t,s)=U(t+T,s+T).$$

Differentiating with respect to $t$ gives

$$i\hbar\partial_tV(t,s) = H(t+T)V(t,s) = H(t)V(t,s).$$

At $t=s$,

$$V(s,s) = U(s+T,s+T) =I.$$

Thus $V(t,s)$ and $U(t,s)$ satisfy the same initial-value problem. Uniqueness gives $V(t,s)=U(t,s)$.

2. Show that stroboscopic evolution is generated by powers of one unitary.

Solution

For two periods,

$$\begin{aligned} U(t_0+2T,t_0) &= U(t_0+2T,t_0+T) U(t_0+T,t_0)\\ &= U_F(t_0)U_F(t_0), \end{aligned}$$

where periodicity makes the first factor equal to the same one-period operator. Repeating the argument gives

$$U(t_0+nT,t_0) = U_F(t_0)^n.$$

3. When is the period-average Hamiltonian exact?

Solution

If

$$[H(t),H(t')]=0$$

for every pair of times in the period, time ordering is irrelevant. Then

$$\begin{aligned} U_F &= \exp \left[ -\frac{i}{\hbar} \int_{t_0}^{t_0+T} H(t)\,dt \right]\\ &= \exp \left[ -\frac{i}{\hbar} \overline H T \right]. \end{aligned}$$

Without pairwise commutation, nested commutators generally produce corrections.

4. Compare two two-step drives with the same average Hamiltonian but opposite order.

Solution

For equal half periods, the first ordering gives

$$U_F^{(21)} = e^{-iH_2T/(2\hbar)} e^{-iH_1T/(2\hbar)},$$

while the reversed ordering gives

$$U_F^{(12)} = e^{-iH_1T/(2\hbar)} e^{-iH_2T/(2\hbar)}.$$

Both have

$$\overline H = \frac{H_1+H_2}{2}.$$

If $[H_1,H_2]\neq0$, the two products are generally unequal. The average loses the temporal ordering information.

5. Derive the conjugacy relation between Floquet operators at two reference phases.

Solution

Let

$$U_F(t_1) = U(t_1+T,t_1).$$

Insert $t_0+T$ and $t_0$:

$$\begin{aligned} U_F(t_1) &= U(t_1+T,t_0+T) U(t_0+T,t_0) U(t_0,t_1). \end{aligned}$$

Periodicity gives

$$U(t_1+T,t_0+T) = U(t_1,t_0),$$

and unitarity gives

$$U(t_0,t_1) = U(t_1,t_0)^*.$$

Therefore

$$U_F(t_1) = U(t_1,t_0) U_F(t_0) U(t_1,t_0)^*.$$

The eigenvalues agree because unitary conjugation preserves the spectrum.