Quasienergies

Quasienergies are the spectral labels of one-period time evolution in a periodically driven quantum system. They play the role that energy eigenvalues play for time-independent Hamiltonians, but only stroboscopically and only modulo the drive quantum $\hbar\Omega$.

The Floquet theorem explains the solution form, and Floquet Operators explains the one-period unitary. This page focuses on the quasienergy spectrum itself: modularity, Floquet zones, branch choices, resonances, and avoided crossings. Floquet Systems Preview owns exponential quasienergy crowding, Floquet ETH, heating, and interacting periodic phases.

Why Energy Is Not Conserved

For a time-independent Hamiltonian, continuous time-translation symmetry gives energy conservation. If

$$H(t)=H_0,$$

then $H_0$ commutes with the evolution it generates, and stationary states accumulate phases

$$e^{-iE_nt/\hbar}.$$

For a periodically driven system,

$$H(t+T)=H(t), \qquad \Omega=\frac{2\pi}{T},$$

continuous time-translation symmetry is broken to discrete time translation by $T$. The system can exchange energy with the drive. Ordinary energy is not generally conserved.

The remaining exact symmetry is stroboscopic:

$$t\mapsto t+T.$$

The spectral object associated with this discrete translation is the eigenphase of the one-period evolution operator, not an instantaneous energy eigenvalue.

Definition from the Floquet Operator

Choose a reference phase $t_0$ of the drive and define

$$U_F(t_0) = U(t_0+T,t_0).$$

For a closed system, $U_F(t_0)$ is unitary. Its eigenvalues can be written as phases:

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

The quasienergy representative $\varepsilon_\alpha$ is defined by

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

so

$$U_F(t_0)\lvert u_\alpha(t_0)\rangle = e^{-i\varepsilon_\alpha T/\hbar} \lvert u_\alpha(t_0)\rangle.$$

The invariant datum is the eigenvalue $e^{-i\theta_\alpha}$. The number $\varepsilon_\alpha$ is a representative chosen from a modular equivalence class.

Modulo Drive Quantum

Because $\Omega T=2\pi$,

$$e^{-i(\varepsilon_\alpha+m\hbar\Omega)T/\hbar} = e^{-i\varepsilon_\alpha T/\hbar} e^{-i2\pi m} = e^{-i\varepsilon_\alpha T/\hbar}$$

for every $m\in\mathbb Z$. Therefore

$$\varepsilon_\alpha \sim \varepsilon_\alpha+m\hbar\Omega.$$

This is analogous to crystal momentum being defined modulo a reciprocal lattice vector in Bloch theory. The analogy is useful:

Spatial crystal	Periodically driven system
lattice spacing $a$	period $T$
reciprocal vector $G=2\pi/a$	drive frequency $\Omega=2\pi/T$
quasimomentum $k\sim k+G$	quasienergy $\varepsilon\sim\varepsilon+\hbar\Omega$
Brillouin zone	Floquet zone

The analogy should not be overread. Quasienergy is not momentum; it is an eigenphase of time translation by one period.

Floquet Gauge and Equivalent Representatives

The same physical Floquet solution can be written with different quasienergy representatives. If

$$\lvert\psi_\alpha(t)\rangle = e^{-i\varepsilon_\alpha(t-t_0)/\hbar} \lvert u_\alpha(t)\rangle, \qquad \lvert u_\alpha(t+T)\rangle=\lvert u_\alpha(t)\rangle,$$

then for any integer $m$,

$$\varepsilon_\alpha' = \varepsilon_\alpha+m\hbar\Omega,$$

and

$$\lvert u_\alpha'(t)\rangle = e^{im\Omega(t-t_0)} \lvert u_\alpha(t)\rangle$$

give the same physical state:

$$e^{-i\varepsilon_\alpha'(t-t_0)/\hbar} \lvert u_\alpha'(t)\rangle = e^{-i\varepsilon_\alpha(t-t_0)/\hbar} \lvert u_\alpha(t)\rangle.$$

This integer relabeling is often called a Floquet gauge choice. It is not an approximation and not a physical change of state.

Floquet Zones

To plot or compare quasienergies, one usually chooses a Floquet zone, for example

$$-\frac{\hbar\Omega}{2} < \varepsilon \le \frac{\hbar\Omega}{2},$$

or

$$0 \le \varepsilon < \hbar\Omega.$$

Both are conventions. A quasienergy outside the chosen interval is shifted by an integer multiple of $\hbar\Omega$ back into the interval.

There are two common plotting schemes:

Scheme	What it shows	Main risk
extended-zone	continuous branches $\varepsilon+m\hbar\Omega$	may show many equivalent copies
reduced-zone	all branches folded into one interval	artificial jumps at zone edges

When a branch crosses a zone boundary in a reduced-zone plot, it may appear to jump discontinuously. Often the underlying eigenphase on the unit circle is perfectly continuous; the jump is caused by the branch cut used to define $\log U_F$.

Static Spectrum Folded into a Floquet Zone

A time-independent Hamiltonian can be treated as a trivial Floquet problem for any chosen period $T$. If

$$H_0\lvert E_n\rangle = E_n\lvert E_n\rangle,$$

then

$$U_F\lvert E_n\rangle = e^{-iE_nT/\hbar}\lvert E_n\rangle.$$

Thus $E_n$ is a quasienergy representative, but only modulo $\hbar\Omega$:

$$\varepsilon_n = E_n \quad \text{mod }\hbar\Omega.$$

This example is useful because it separates the modular structure from the physics of driving. The modularity comes from looking at one-period phases. The drive determines how different folded sectors couple.

Extended Floquet Space

The Floquet eigenvalue equation is

$$\bigl(H(t)-i\hbar\partial_t\bigr) \lvert u_\alpha(t)\rangle = \varepsilon_\alpha \lvert u_\alpha(t)\rangle,$$

with $\lvert u_\alpha(t+T)\rangle=\lvert u_\alpha(t)\rangle$. Expand

$$\lvert u_\alpha(t)\rangle = \sum_{m\in\mathbb Z} e^{im\Omega t} \lvert u_{\alpha m}\rangle$$

and

$$H(t) = \sum_{q\in\mathbb Z} e^{iq\Omega t}H_q.$$

Then the Fourier components satisfy

$$\sum_{m\in\mathbb Z} H_{k-m}\lvert u_{\alpha m}\rangle + k\hbar\Omega\,\lvert u_{\alpha k}\rangle = \varepsilon_\alpha \lvert u_{\alpha k}\rangle.$$

This is the Sambe-space viewpoint. The label $k$ marks copies shifted by $k\hbar\Omega$. Periodic driving couples these copies through the Fourier components $H_q$.

In an undriven system, the extended-space energies are

$$E_n+m\hbar\Omega.$$

A drive can hybridize states whose extended energies nearly coincide. This is the origin of many avoided crossings in quasienergy spectra.

Resonances and Avoided Crossings

Suppose two undriven levels satisfy an approximate resonance condition

$$E_b-E_a \approx n\hbar\Omega.$$

In extended-zone language, the copies

$$E_a \quad \text{and} \quad E_b-n\hbar\Omega$$

come close. A drive Fourier component that connects the two states produces an effective two-level problem near the crossing:

$$H_{\rm eff} = \begin{pmatrix} E_a & V\\ V^* & E_b-n\hbar\Omega \end{pmatrix}.$$

The two quasienergy branches are locally

$$\varepsilon_\pm = \frac{E_a+E_b-n\hbar\Omega}{2} \pm \sqrt{ \left( \frac{E_a-E_b+n\hbar\Omega}{2} \right)^2 + |V|^2 },$$

again understood modulo $\hbar\Omega$. At exact resonance, the gap is $2|V|$ in this effective model. If the coupling is forbidden by a symmetry, the crossing may remain exact.

Driven Two-Level Example

Consider a near-resonantly driven two-level system,

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

The drive period is $T=2\pi/\omega$. In a rotating-wave approximation near resonance, with detuning

$$\delta=\omega_0-\omega,$$

one obtains an effective rotating-frame Hamiltonian of the form

$$H_{\rm RWA} = \frac{\hbar}{2} \left( \delta\sigma_z + g\sigma_x \right),$$

up to convention-dependent factors in the drive amplitude. The approximate quasienergy splitting is therefore

$$\Delta\varepsilon \approx \hbar\sqrt{\delta^2+g^2},$$

modulo $\hbar\omega$. At resonance, the folded levels avoid crossing and the gap is controlled by the drive matrix element. The exact Floquet spectrum is obtained from $U_F$, while the rotating-wave approximation gives a useful near-resonant model.

Reference Phase and Spectral Invariance

Changing the reference time $t_0$ changes the Floquet operator by unitary conjugation:

$$U_F(t_0+s) = V(s)U_F(t_0)V^{-1}(s), \qquad V(s)=U(t_0+s,t_0).$$

Therefore the quasienergy eigenvalues are independent of the reference phase. The Floquet modes at the snapshot time do change, because the eigenvectors are conjugated.

This distinction is practical. Spectra are drive-phase invariant, but stroboscopic observables and micromotion depend on when in the period the system is observed.

No Absolute Quasienergy Ground State

Because quasienergy is defined modulo $\hbar\Omega$, there is generally no absolute lowest quasienergy. A branch can always be shifted by $-\hbar\Omega$ without changing the one-period eigenvalue.

This has several consequences:

• a “filled lowest quasienergy band” is not automatically meaningful without additional structure;
• thermal occupation by quasienergy alone is not the same as thermal occupation by energy;
• reduced-zone spectra can hide which branch is continuously connected to an undriven state;
• many-body Floquet systems can absorb energy from the drive even when a stroboscopic effective Hamiltonian exists.

When a high-frequency effective Hamiltonian is valid, it may provide an approximate energy-like ordering over a finite time window. That is an approximation, not part of the exact modular definition of quasienergy.

Common Mistakes

• Calling instantaneous eigenvalues of $H(t)$ quasienergies.
• Forgetting that $\varepsilon$ and $\varepsilon+\hbar\Omega$ label the same one-period eigenphase.
• Treating a reduced-zone discontinuity as a physical jump.
• Looking for an absolute quasienergy ground state without specifying extra structure.
• Comparing quasienergy spectra from different branch cuts without unfolding them consistently.
• Mistaking a reference-phase change of Floquet modes for a change in quasienergy spectrum.
• Assuming every apparent crossing becomes an avoided crossing; symmetries can protect exact crossings.
• Interpreting high-frequency effective-Hamiltonian spectra as exact quasienergy spectra outside their regime.

Cross-Links

• Floquet Theorem in Quantum Mechanics gives the solution theorem and micromotion.
• Floquet Operators gives the one-period unitary and logarithm branches.
• Time-Dependent Hamiltonians sets up driven closed-system dynamics.
• Time Ordering explains why noncommuting drives cannot be replaced by a simple averaged Hamiltonian.
• Floquet–Magnus Expansion develops the direct one-period Magnus representative.
• High-Frequency Expansions develops van Vleck effective Hamiltonians, resonant-sector tests, and prethermal validity.
• Rotating-Wave Approximation gives the near-resonant two-level approximation used above.
• Bloch Theorem gives the spatial analogue of a spectrum defined modulo a reciprocal period.
• Formula Sheet collects the core Floquet formulas.

References

• J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Physical Review 138, B979, 1965.
• H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Physical Review A 7, 2203, 1973.
• M. Grifoni and P. Hänggi, “Driven quantum tunneling,” Physics Reports 304, 229, 1998.
• A. Eckardt, “Colloquium: Atomic quantum gases in periodically driven optical lattices,” Reviews of Modern Physics 89, 011004, 2017.
• 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, 2015.
• 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 modular equivalence of quasienergies.

Solution

The one-period eigenvalue is

$$\lambda = e^{-i\varepsilon T/\hbar}.$$

For $\varepsilon'=\varepsilon+m\hbar\Omega$,

$$e^{-i\varepsilon'T/\hbar} = e^{-i\varepsilon T/\hbar} e^{-im\Omega T} = e^{-i\varepsilon T/\hbar} e^{-i2\pi m} = e^{-i\varepsilon T/\hbar}.$$

Thus both representatives give the same eigenvalue of $U_F$.

2. Fold a static spectrum into a Floquet zone.

Suppose a time-independent Hamiltonian has energies

$$E_1=0, \qquad E_2=0.8\,\hbar\Omega, \qquad E_3=1.3\,\hbar\Omega.$$

Choose the zone $-\hbar\Omega/2\lt\varepsilon\leq\hbar\Omega/2$. Find the quasienergy representatives.

Solution

The first energy is already in the zone:

$$\varepsilon_1=0.$$

The second is shifted down by $\hbar\Omega$:

$$\varepsilon_2 = 0.8\,\hbar\Omega-\hbar\Omega = -0.2\,\hbar\Omega.$$

The third is also shifted down by $\hbar\Omega$:

$$\varepsilon_3 = 1.3\,\hbar\Omega-\hbar\Omega = 0.3\,\hbar\Omega.$$

Thus the reduced-zone representatives are $0$, $-0.2\,\hbar\Omega$, and $0.3\,\hbar\Omega$.

3. Show that a Floquet gauge shift leaves the physical solution unchanged.

Solution

Start with

$$\lvert\psi(t)\rangle = e^{-i\varepsilon(t-t_0)/\hbar} \lvert u(t)\rangle.$$

Define

$$\varepsilon' = \varepsilon+m\hbar\Omega, \qquad \lvert u'(t)\rangle = e^{im\Omega(t-t_0)} \lvert u(t)\rangle.$$

Then

$$\begin{aligned} e^{-i\varepsilon'(t-t_0)/\hbar} \lvert u'(t)\rangle &= e^{-i(\varepsilon+m\hbar\Omega)(t-t_0)/\hbar} e^{im\Omega(t-t_0)} \lvert u(t)\rangle \\ &= e^{-i\varepsilon(t-t_0)/\hbar} \lvert u(t)\rangle. \end{aligned}$$

The physical state is unchanged.

4. Diagonalize the two-level avoided-crossing model.

Let

$$H_{\rm eff} = \begin{pmatrix} \Delta/2 & V\\ V^* & -\Delta/2 \end{pmatrix}.$$

Find its quasienergy splitting in the effective model.

Solution

The characteristic equation is

$$\lambda^2 - \left( \frac{\Delta^2}{4}+|V|^2 \right) = 0.$$

Thus

$$\lambda_\pm = \pm \sqrt{ \frac{\Delta^2}{4}+|V|^2 }.$$

The splitting is

$$\lambda_+-\lambda_- = 2\sqrt{ \frac{\Delta^2}{4}+|V|^2 }.$$

At exact resonance, $\Delta=0$, the gap is $2|V|$.

5. Explain why a quasienergy spectrum does not define thermal equilibrium by itself.

Solution

Thermal equilibrium for a time-independent closed system is organized by energy, which has an absolute ordering once the Hamiltonian is fixed. Quasienergy is modular:

$$\varepsilon \sim \varepsilon+m\hbar\Omega.$$

Therefore there is no unique lowest quasienergy and no unique Boltzmann factor based only on $\varepsilon$. A driven system can exchange energy with the drive, and many-body Floquet systems may heat unless additional mechanisms or approximations intervene. Any thermal or prethermal description needs extra assumptions beyond the exact quasienergy spectrum.