Effective Hamiltonians and Scale Separation

An effective Hamiltonian describes selected states or time scales while retaining the effects of degrees of freedom that are not followed explicitly. A remote level, for example, can shift an energy or mediate a transition even when its population remains small. The useful reduced model includes these effects and states the energy range, observation times, and observables it can reproduce.

This chapter connects several ways to make that reduction. Projection and block diagonalization separate subspaces; adiabatic elimination separates slow and fast motion; rotating-frame and high-frequency methods separate time scales. Their assumptions overlap, but they need not answer the same question.

Helpful background. Projectors describe retained and eliminated subspaces; small parameters and errors explain coupling-to-gap ratios. Time-dependent Hamiltonians introduce the dynamics needed for rotating frames and periodic driving.

A remote level can mediate a coupling

Consider two degenerate states of energy zero coupled to a third state of energy $\Delta>0$:

$$H=\begin{pmatrix} 0&0&v_1\\ 0&0&v_2\\ v_1^*&v_2^*&\Delta \end{pmatrix}.$$

The $v_j$ have energy units. Let $P$ retain the first two states and let $Q=I-P$ select the third. Simply taking $PHP$ gives zero and misses all dynamics within the retained pair. When $g/\Delta\ll1$, where $g=\sqrt{|v_1|^2+|v_2|^2}$, eliminating the remote level gives to second order

$$H_{\mathrm{eff}}\approx-\frac1\Delta \begin{pmatrix} |v_1|^2&v_1v_2^*\\ v_2v_1^*&|v_2|^2 \end{pmatrix}.$$

The diagonal entries are energy shifts. The off-diagonal entries couple states that had no direct matrix element in the original Hamiltonian. Their relative complex phase matters. The collective coupling $g$, rather than either individual $|v_j|$, controls the separation from the third state.

One superposition of the retained states couples to the remote level and shifts down by approximately $g^2/\Delta$. The orthogonal superposition is exactly uncoupled in this model. Adiabatic elimination develops this bright-and-dark-state picture for a detuned three-level system, including drive conventions, transients, and decay.

Select the reduction for your question

Projection methods begin with an exact block identity. Whenever the required inverse exists,

$$\begin{aligned} H_{\mathrm{eff}}(E)&=H_{PP}+H_{PQ}R_Q(E)H_{QP},\\ R_Q(E)&=(E-H_{QQ})^{-1}. \end{aligned}$$

The correction describes propagation into the eliminated space and back. The equation is exact but energy dependent; replacing its denominator by a remote energy scale introduces the approximation. Resolvent poles warn that the retained space may be too small. For outgoing continuum conditions, Feshbach projection explains how an imaginary part describes escape while the full closed evolution remains unitary.

Physical situation	Method and its purpose
Weak mixing between spectrally separated blocks	Schrieffer–Wolff transformation: a perturbative unitary transformation gives an energy-independent Hermitian operator for an isolated sector.
Several selected levels need one common interaction	Folded effective Hamiltonians: reorganize energy-dependent corrections into a model-space operator.
Slow amplitudes coupled to a fast response	Adiabatic elimination: approximate the fast response, retaining necessary transient or derivative corrections.
Heavy nuclei and light electrons	Born–Oppenheimer scale separation: electronic energies become nuclear potential surfaces, with derivative couplings between them.

Nearby states that mix strongly should be retained together. Quasi-degenerate perturbation theory explains why expanding in a coupling divided by an equally small detuning is inappropriate. A small electronic gap similarly makes a single-surface Born–Oppenheimer treatment unreliable.

Fast driving introduces another time scale

A rotating-frame transformation is exact. For $|\psi_R\rangle=R^\dagger|\psi\rangle$,

$$H_R=R^\dagger H R-i\hbar R^\dagger\dot R.$$

An approximation begins when terms in this transformed Hamiltonian are discarded. The rotating-wave approximation keeps slowly varying near-resonant terms. Average Hamiltonian theory instead organizes pulse sequences in a toggling frame, including corrections that depend on pulse order.

For general time dependence, the Magnus expansion expresses evolution as one exponential with commutator corrections. For periodic driving of period $T$, Floquet–Magnus theory constructs a Hamiltonian from one-period evolution. It describes times separated by whole periods; the motion within a period requires a micromotion operator. High-frequency expansions compares this construction with other inverse-frequency methods and explains resonances and limited heating times.

Check observables as well as energies

If a unitary change of basis produces $\widetilde H=e^SHe^{-S}$ with $S^\dagger=-S$, apply the same change to an observable:

$$O_{\mathrm{eff}}=Pe^SOe^{-S}P.$$

Using a corrected Hamiltonian with the bare observable $POP$ can reproduce energies yet miss transition strengths. Small eliminated components still contribute to physical matrix elements. In a decaying fast sector, the reduction also changes dissipative jump operators; a non-Hermitian Hamiltonian by itself does not describe the unconditional, trace-preserving evolution.

Small leakage likewise does not guarantee accurate phase evolution. If two bounded Hermitian generators on the same reduced space differ by $\delta H$, their propagators obey

$$\|U_1(t)-U_2(t)\|\leq\frac{|t|}{\hbar}\|\delta H\|.$$

Thus a small generator error can become important over a long time. The estimate compares generators on a common space; applying it to a subspace reduction first requires a consistent map between the reduced and full states.

For applications, effective Hamiltonians in quantum information discusses qubit interactions and leakage, while effective Hamiltonians in quantum matter develops exchange and band examples. The effective-field-theory bridge explains how matching and scale separation extend to field theory.

Exercises

Find the uncoupled state. Assume $g>0$ in the three-level model. Show that the normalized retained-space vector $(v_2^*,-v_1^*)/g$ has zero coupling to the third state and zero energy. Find the shifted direction.

Solution

The third component of $H(v_2^*,-v_1^*,0)^T$ is $v_1^*v_2^*-v_2^*v_1^*=0$; the other components also vanish. The normalized bright direction is $(v_1,v_2)/g$, orthogonal to the dark direction. Together with the third state it forms the block $\bigl(\begin{smallmatrix}0&g\\g&\Delta\end{smallmatrix}\bigr)$. Its eigenvalue near zero is $-g^2/\Delta+O(g^4/\Delta^3)$, matching the effective matrix.

Which states should remain? Three unperturbed levels have energies $0,\delta,\Delta$, with couplings of size $v$, and $\delta\sim v\ll\Delta$. Which belong in the retained space?

Solution

Keep both $0$ and $\delta$ and diagonalize their coupling together. The level at $\Delta$ can be eliminated perturbatively if the total coupling to it is small compared with the gap. Expanding in $v/\delta$ is uncontrolled because that ratio is of order one.

Spectrum versus matrix elements. Expand $O_{\mathrm{eff}}$ to first order in $S$. When is bare projection sufficient at that order?

Solution

For bounded operators in the perturbative regime, $O_{\mathrm{eff}}=POP+P[S,O]P+O(\|S\|^2\|O\|)$. Bare projection is sufficient through first order when the projected first-order commutator vanishes, for example by a symmetry or block-structure constraint. It is a condition on the observable, not a consequence of accurate energies alone.

References

• Blanes, S., 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. Time-dependent exponential expansions and their limitations.
• Bravyi, S., D. P. DiVincenzo, and D. Loss. “Schrieffer–Wolff transformation for quantum many-body systems.” Annals of Physics 326, 2793–2826 (2011). doi:10.1016/j.aop.2011.06.004. Perturbative block diagonalization and effective operators.
• Feshbach, H. “Unified theory of nuclear reactions.” Annals of Physics 5, 357–390 (1958). doi:10.1016/0003-4916(58)90007-1. Projection and energy-dependent interactions.
• Reiter, F., and A. S. Sørensen. “Effective operator formalism for open quantum systems.” Physical Review A 85, 032111 (2012). doi:10.1103/PhysRevA.85.032111. Coherent and dissipative terms generated by eliminating fast states.