Magnus Expansion

The Magnus expansion rewrites time-dependent evolution as a single exponential whose exponent is built from integrals and nested commutators of the Hamiltonian. It is a systematic way to replace time-ordered evolution by an effective generator while preserving unitarity order by order.

For a time-dependent Hamiltonian, the exact evolution operator is

$$U(t,t_0) = \mathcal T \exp \left[ - \frac{i}{\hbar} \int_{t_0}^{t}H(t')\,dt' \right].$$

The Magnus expansion writes the same operator as

$$U(t,t_0)=e^{\Omega(t,t_0)}.$$

The exponent $\Omega$ is not usually just the integral of $H(t)$. Noncommutativity at different times produces commutator corrections.

Required background. Time-dependent Hamiltonians and time ordering define the propagator; unitary operators explains the structure preserved by exponentiating an anti-Hermitian generator.

The Magnus Generator

Begin with finite matrices or bounded norm-integrable Hermitian Hamiltonians. Unbounded operators require a common-domain argument and cannot inherit the norm estimates below automatically. Write the evolution equation as

$$\frac{dU}{dt}=A(t)U(t), \qquad A(t)=-\frac{i}{\hbar}H(t).$$

Then the Magnus series is

$$\Omega(t,t_0) = \Omega_1+\Omega_2+\Omega_3+\cdots.$$

The first term is

$$\Omega_1 = \int_{t_0}^{t}dt_1\,A(t_1).$$

The second term is

$$\Omega_2 = \frac12 \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2\, [A(t_1),A(t_2)].$$

In Hamiltonian language,

$$\Omega_1 = - \frac{i}{\hbar} \int_{t_0}^{t}dt_1\,H(t_1),$$

and

$$\Omega_2 = - \frac{1}{2\hbar^2} \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2\, [H(t_1),H(t_2)].$$

The second term vanishes if the Hamiltonian commutes with itself at all pairs of times.

Effective Hamiltonian Over an Interval

Over a finite interval of length

$$T=t-t_0,$$

define an effective Hamiltonian by

$$U(t_0+T,t_0) = \exp \left[ - \frac{i}{\hbar} H_{\mathrm{eff}}T \right].$$

Then

$$H_{\mathrm{eff}} = \frac{i\hbar}{T} \Omega(t_0+T,t_0).$$

Keeping only $\Omega_1$ gives the average Hamiltonian

$$H_{\mathrm{eff}}^{(1)} = \frac1T \int_{t_0}^{t_0+T}dt\,H(t).$$

Write $H_{\mathrm{eff}}=H_{\mathrm{eff}}^{(1)}+H_{\mathrm{eff}}^{(2)}+\cdots$, where superscripts label contributions, not cumulative truncations. Keeping $\Omega_2$ adds the leading commutator correction:

$$H_{\mathrm{eff}}^{(2)} = - \frac{i}{2\hbar T} \int_{t_0}^{t_0+T}dt_1 \int_{t_0}^{t_1}dt_2\, [H(t_1),H(t_2)].$$

Because $[H(t_1),H(t_2)]$ is anti-Hermitian for Hermitian Hamiltonians, the factor $-i$ makes this correction Hermitian.

Relation to the Dyson Series

The Dyson series expands the time-ordered exponential directly:

$$U=I+U_1+U_2+\cdots.$$

The Magnus expansion instead expands the logarithm of the same evolution:

$$U=e^{\Omega_1+\Omega_2+\cdots}.$$

Both contain the same physics when summed exactly. Their truncations behave differently. A finite Dyson truncation is not exactly unitary. A finite Magnus truncation is unitary when the retained $\Omega$ is anti-Hermitian and exponentiated exactly.

This makes Magnus methods attractive in control, numerical time propagation, periodically driven systems, and semiclassical approximations where preserving unitarity is structurally important.

For the complementary use of a finite Dyson truncation to organize transition paths, intermediate states, and probability bookkeeping, see Dyson Expansion for Transition Amplitudes.

Commuting Hamiltonians

If

$$[H(t_1),H(t_2)]=0$$

for all times, every commutator correction vanishes. The exact result reduces to the ordinary exponential

$$U(t,t_0) = \exp \left[ - \frac{i}{\hbar} \int_{t_0}^{t}H(t')\,dt' \right].$$

This is the special case in which time ordering is unnecessary.

Periodic Driving

For a periodic Hamiltonian

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

the one-period evolution operator is

$$U(T,0) = \exp \left[ - \frac{i}{\hbar} H_FT \right],$$

where $H_F$ is a Floquet effective Hamiltonian, up to branch choices in the logarithm. The Magnus expansion gives one systematic route to approximating $H_F$ in high-frequency or weak-driving regimes.

At leading order, $H_F$ is the time average of $H(t)$. Higher orders encode the noncommutativity of different parts of the drive cycle. This is why pulse order matters even when the time-averaged Hamiltonian is the same.

Convergence and Use

The Magnus expansion is not guaranteed to converge for arbitrary time intervals and Hamiltonians. For bounded operators on a Hilbert space, a standard sufficient condition uses the operator norm and the precise threshold $\pi$ (Blanes et al., 2009):

$$\int_{t_0}^{t} dt'\,\|A(t')\| < \pi.$$

Since $\|A(t)\|=\|H(t)\|/\hbar$, the identical condition is

$$\frac1\hbar \int_{t_0}^{t} dt'\,\|H(t')\| < \pi.$$

This condition is sufficient, not necessary, and is not itself an estimate of a chosen truncation error. Subtracting a known scalar term $c(t)I$ and integrating its global phase exactly can improve the bound without altering any transition probability. In applications, one often combines it with physical checks: high drive frequency, small commutator corrections, comparison with exact numerics for small systems, or stability under adding the next term.

Practical Interpretation

The first Magnus term is what one would write if all Hamiltonians at different times commuted. The second term measures the leading failure of that simplification. It is sensitive not just to how much Hamiltonian was applied, but to the order in which noncommuting pieces were applied.

For two short pulses generated by $H_A$ and $H_B$, reversing the order changes the sign of the leading commutator contribution. Average Hamiltonian Theory applies this algebra to toggling-frame control cycles, NMR selective averaging, sequence symmetry, and dynamical-decoupling corrections.

Magnus Expansion Error tests this two-pulse algebra against an exact propagator, measures the first four truncation orders, and shows separately how unitarity, convergence bounds, long-time accuracy, and logarithm branches can fail to answer the same question.

Exercises

1. Show that the second Magnus term vanishes when $H(t)=f(t)H_0$ for a fixed Hamiltonian $H_0$.

Solution

For $H(t)=f(t)H_0$,

$$[H(t_1),H(t_2)] = f(t_1)f(t_2)[H_0,H_0] =0.$$

Therefore $\Omega_2=0$. All higher nested commutators also vanish, so the exact evolution is the ordinary exponential of the time integral of $H(t)$.

2. Consider two constant Hamiltonians applied in sequence: $H_A$ for time $\tau$ and then $H_B$ for time $\tau$. Use the second Magnus term to identify the leading commutator correction to the effective Hamiltonian over the full time $2\tau$.

Solution

The first term gives

$$H_{\mathrm{eff}}^{(1)} = \frac{H_A+H_B}{2}.$$

The commutator contributes only when $t_1$ lies in the $H_B$ interval and $t_2$ lies in the earlier $H_A$ interval. Thus

$$\int_0^{2\tau}dt_1 \int_0^{t_1}dt_2\, [H(t_1),H(t_2)] = \tau^2[H_B,H_A].$$

Using $T=2\tau$,

$$H_{\mathrm{eff}}^{(2)} = - \frac{i\tau}{4\hbar} [H_B,H_A].$$

Reversing the pulse order changes the sign of this correction.

3. Unitary does not mean accurate

For $H=\hbar\omega\sigma_z/2$, compare the exact propagator with the deliberately incorrect approximation $U_{\mathrm{bad}}=I$. What do a unitarity test and an operator-error test report?

Solution

Both operators satisfy $U^\dagger U=I$ exactly. Yet

$$\|U(T)-I\|=2\left|\sin\frac{\omega T}{4}\right|,$$

which can be order one. Exact unitarity of a truncated Magnus exponential prevents artificial norm loss but does not certify its phase, rotation axis, or convergence. Independent comparisons of observables or the propagator are still needed.

References

• S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications”, Physics Reports 470, 151–238 (2009), doi:10.1016/j.physrep.2008.11.001.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley, 1992.
• U. Haeberlen, High Resolution NMR in Solids: Selective Averaging, Academic Press, 1976.
• W. Magnus, “On the exponential solution of differential equations for a linear operator,” Communications on Pure and Applied Mathematics 7, 649-673, 1954.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.