Adiabatic Elimination

Adiabatic elimination replaces rapidly evolving or weakly populated quantum degrees of freedom by their approximate response to slow ones. The eliminated variables do not simply disappear: virtual excursions through them shift energies, induce couplings, and, in open systems, generate effective decay and dephasing.

The familiar instruction “set the fast derivative to zero” is useful only when read as the leading term of a slow–fast expansion. An exact elimination produces a transient and a memory kernel. A local effective Hamiltonian appears after showing that the slow state and the couplings change little over the fast response time.

Required background. Projection methods introduces the block decomposition, error estimates explains the scale ratios, and the interaction picture supplies the rotating-frame construction. Unlike the adiabatic theorem, elimination does not require following one slowly varying instantaneous eigenstate.

Slow and Fast Variables

Set $\hbar=1$ in the closed-system derivation; restore it in the atomic example. Split a state into slow and fast components,

$$\lvert\Psi(t)\rangle = \begin{pmatrix} \psi_s(t)\\ \psi_f(t) \end{pmatrix},$$

and write the Schrödinger equation in block form:

$$i\frac{d}{dt} \begin{pmatrix} \psi_s\\ \psi_f \end{pmatrix} = \begin{pmatrix} A & B\\ C & D \end{pmatrix} \begin{pmatrix} \psi_s\\ \psi_f \end{pmatrix}.$$

Thus

$$\begin{aligned} i\dot\psi_s &= A\psi_s+B\psi_f, \\ i\dot\psi_f &= C\psi_s+D\psi_f. \end{aligned}$$

Here $A$ and $D$ generate motion within the slow and fast sectors, while $B$ and $C$ transfer amplitude between them. For a Hermitian Hamiltonian,

$$A^\dagger=A, \qquad D^\dagger=D, \qquad C=B^\dagger.$$

“Fast” is a statement about frequencies in a chosen frame, not about a label attached to a basis state. After removing a convenient reference energy, the spectrum of $D$ should be far from the frequencies resolved by $\psi_s$. A large common energy added to both blocks has no dynamical significance and does not create scale separation.

Two common mechanisms make a sector fast:

• Large detuning: eigenfrequencies in $D$ lie far from the slow band.
• Rapid relaxation: eigenmodes in the fast sector decay on a timescale short compared with the retained dynamics.

The second case requires an open-system description; it cannot generally be represented by a Hermitian Hamiltonian alone.

Exact Elimination Produces Memory

Let $U_f(t,t')$ be the propagator generated by $D(t)$:

$$\begin{aligned} i\frac{\partial}{\partial t}U_f(t,t') &= D(t)U_f(t,t'), \\ U_f(t',t') &= I_f. \end{aligned}$$

Variation of constants gives the exact fast amplitude

$$\psi_f(t) = U_f(t,t_0)\psi_f(t_0) -i\int_{t_0}^{t} U_f(t,t') C(t')\psi_s(t')\,dt'.$$

Substitution into the slow equation yields

$$i\dot\psi_s(t) = A(t)\psi_s(t) + B(t)U_f(t,t_0)\psi_f(t_0) -iB(t) \int_{t_0}^{t}dt'\, U_f(t,t') \times C(t')\psi_s(t').$$

This identity separates two effects:

• The term containing $\psi_f(t_0)$ is an initial transient. It retains knowledge of any fast-sector population present at the starting time.
• The integral is a memory term. The present slow derivative depends on the slow state at earlier times.

Adiabatic elimination is the approximation that turns this nonlocal equation into a local one. It is justified when the kernel $U_f(t,t')$ oscillates or decays before $\psi_s(t')$ and the couplings change appreciably.

For constant $D$,

$$U_f(t,t') = e^{-iD(t-t')}.$$

For a finite closed fast sector, $e^{-iD(t-t')}$ oscillates but does not decay. Integration by parts isolates a slowly varying particular solution and an endpoint oscillation. For constant invertible $D,C$, to leading derivative order,

$$\psi_f(t)\simeq{}-D^{-1}C\psi_s(t) +e^{-iD(t-t_0)}\bigl[\psi_f(t_0)+D^{-1}C\psi_s(t_0)\bigr].$$

The oscillation is present even if the bare fast amplitude initially vanishes. Preparing a dressed slow state suppresses it; coarse graining can hide it in selected observables but does not make it vanish microscopically. A genuinely decaying fast propagator can instead erase the initial term after its relaxation time. These alternatives must not be conflated with an automatic loss of memory in a finite closed system.

Leading Algebraic Elimination

The rough prescription sets the fast derivative to zero:

$$i\dot\psi_f \simeq 0.$$

If $D$ is invertible, the fast amplitude is then slaved to the slow one:

$$\psi_f \simeq - D^{-1}C\psi_s.$$

The slow equation becomes

$$i\dot\psi_s \simeq H_{\mathrm{eff}}^{(0)}\psi_s,$$

with

$$H_{\mathrm{eff}}^{(0)} = A-BD^{-1}C.$$

The correction has a simple path interpretation:

$$s \xrightarrow{\,C\,} f \xrightarrow{\,-D^{-1}\,} f \xrightarrow{\,B\,} s.$$

It shifts retained levels and can connect two slow states that had no direct matrix element in $A$. For a Hermitian problem with $C=B^\dagger$, the leading Hamiltonian is Hermitian.

This expression is the zero-frequency Schur complement. It agrees with the leading stationary projection and Schrieffer–Wolff results when all methods use the same frame, model space, and perturbative order. Their higher-order state and observable conventions need not be identical.

The reconstructed fast population scales as

$$\lVert\psi_f\rVert^2 \lesssim \lVert D^{-1}C\rVert^2 \lVert\psi_s\rVert^2.$$

Small fast population therefore requires $\lVert D^{-1}C\rVert\ll1$. It does not imply a negligible effect on slow phases: a correction of order $g^2/\Delta$ can accumulate coherently for times of order $\Delta/g^2$.

Derivative Corrections

For time-independent $D$ and $C$, discard the homogeneous transient and formally solve

$$\psi_f = - \left( D-i\partial_t \right)^{-1} C\psi_s.$$

On slow, bandwidth-limited states, expand the inverse:

$$\psi_f = - D^{-1}C\psi_s -iD^{-2}C\dot\psi_s + D^{-3}C\ddot\psi_s +\cdots.$$

The algebraic substitution is the first term. Keeping the first derivative gives

$$\begin{aligned} N_s &= I_s+BD^{-2}C ,\\ iN_s\dot\psi_s &= \left( A-BD^{-1}C \right)\psi_s + \text{higher derivatives}. \end{aligned}$$

The matrix multiplying $\dot\psi_s$ is a reminder that the projected slow amplitude is not yet the fully normalized dressed state. In the Hermitian case, define

$$\begin{aligned} N &= I_s+BD^{-2}B^\dagger, \\ \chi &= N^{1/2}\psi_s. \end{aligned}$$

To this order,

$$\lVert\Psi\rVert^2 \simeq \langle\psi_s\rvert N\lvert\psi_s\rangle = \lVert\chi\rVert^2.$$

For constant blocks, a standard Hermitian generator for $\chi$ is

$$H_{\mathrm{eff}}^{(1)} = N^{-1/2} \left( A-BD^{-1}B^\dagger \right) N^{-1/2},$$

up to the order retained in the derivative expansion. This normalization step explains why an unsymmetrized higher-order substitution can appear non-Hermitian even when the full Hamiltonian is Hermitian.

If $B$, $C$, or $D$ depends on time, derivatives also act on those operators and their order matters. In that setting, either derive the local series from the exact propagator or use a systematic time-dependent block transformation. Blindly replacing every fast derivative by zero can miss ramp-induced terms.

Validity Conditions

For a finite-dimensional Hermitian fast block measured relative to the slow reference frequency, define

$$\Delta_f = \lVert D^{-1}\rVert^{-1}.$$

This is the distance to the closest dangerous fast-sector frequency. Let $g$ bound the coupling blocks, let $\omega_s$ characterize the retained bandwidth, and let $\tau_{\mathrm{ramp}}$ be the shortest timescale on which the blocks change. Useful diagnostics are

$$\begin{aligned} \epsilon_c &= \frac{g}{\Delta_f}, \\ \epsilon_s &= \frac{\omega_s}{\Delta_f}, \\ \epsilon_r &= \frac{1} {\Delta_f\tau_{\mathrm{ramp}}}. \end{aligned}$$

Leading adiabatic elimination requires all relevant ratios to be small. A responsible calculation checks more than one denominator:

1. Spectral or decay separation: the fast response time must be short compared with retained motion.
2. Weak occupation: $\lVert D^{-1}C\rVert\ll1$ for states in the retained sector.
3. Slow envelopes: drives and control parameters should not change appreciably during the fast response time.
4. Compatible initial data: fast transients must be absent, irrelevant after coarse graining, or included explicitly.
5. No hidden resonance: a near-resonant fast eigenstate belongs in the retained space.
6. Controlled time window: if the neglected generator is $\delta H$, require $\lVert\delta H\rVert t\ll1$ for phase-sensitive predictions.
7. Observable consistency: leakage and fast-sector observables require state reconstruction, not only $H_{\mathrm{eff}}$.

For a non-normal fast generator, eigenvalue distance alone can be misleading. The resolvent norm or pseudospectral response can be large even when every eigenvalue appears far away. In unbounded problems, domains and relative bounds require separate analysis.

Choosing the frame

The instruction $\dot\psi_f\simeq0$ is not invariant under an arbitrary rapidly rotating change of picture. A poor frame can make a slow amplitude look fast or shift a large frequency into the wrong block. Choose a frame in which:

• retained amplitudes vary on the physical slow scale;
• the fast detunings or decay rates remain explicit in $D$;
• the couplings have slowly varying envelopes.

Interaction Picture and Rotating-Wave Approximation develop the transformations often used before elimination. The rotating-wave approximation removes rapidly oscillating Hamiltonian terms; adiabatic elimination removes a fast dynamical variable. One does not automatically imply the other.

Worked Example: A Detuned Λ System

Consider two long-lived states $\lvert g_1\rangle$ and $\lvert g_2\rangle$ coupled to an excited state $\lvert e\rangle$. In a rotating frame, after any rotating-wave approximation has been justified, take

$$\frac{H}{\hbar} = \delta \lvert g_2\rangle\langle g_2\rvert + \Delta \lvert e\rangle\langle e\rvert + \frac{1}{2} \left( \Omega_1 \lvert e\rangle\langle g_1\rvert + \Omega_2 \lvert e\rangle\langle g_2\rvert + \mathrm{h.c.} \right).$$

Here $\Delta$ is the one-photon detuning and $\delta$ is the two-photon detuning in this convention.

When $\lvert\Delta\rvert$ dominates the Rabi frequencies and slow ground-manifold scales, the excited amplitude is weak and fast. Eliminating $\lvert e\rangle$ leaves a Raman coupling between $\lvert g_1\rangle$ and $\lvert g_2\rangle$ together with state-dependent ac Stark shifts.

For

$$\lvert\Psi\rangle = c_1\lvert g_1\rangle + c_2\lvert g_2\rangle + c_e\lvert e\rangle,$$

the amplitude equations are

$$\begin{aligned} i\dot c_1 &= \frac{\Omega_1^*}{2}c_e, \\ i\dot c_2 &= \delta c_2 + \frac{\Omega_2^*}{2}c_e, \\ i\dot c_e &= \Delta c_e + \frac{\Omega_1c_1+\Omega_2c_2}{2}. \end{aligned}$$

If

$$\lvert\Omega_1\rvert, \lvert\Omega_2\rvert, \lvert\delta\rvert \ll \lvert\Delta\rvert,$$

the leading fast response is

$$c_e \simeq - \frac{ \Omega_1c_1+\Omega_2c_2 } {2\Delta}.$$

Substitution gives the ground-manifold Hamiltonian

$$\frac{H_{\mathrm{eff}}}{\hbar} = \begin{pmatrix} -\dfrac{\lvert\Omega_1\rvert^2}{4\Delta} & -\dfrac{\Omega_1^*\Omega_2}{4\Delta} \\ -\dfrac{\Omega_2^*\Omega_1}{4\Delta} & \delta-\dfrac{\lvert\Omega_2\rvert^2}{4\Delta} \end{pmatrix}.$$

The diagonal terms are ac Stark shifts. Defining the effective two-state coupling by an off-diagonal matrix element $\hbar\Omega_{\mathrm R}/2$ gives

$$\Omega_{\mathrm R} = - \frac{\Omega_1^*\Omega_2} {2\Delta}.$$

The effective two-photon detuning is

$$\delta_{\mathrm{eff}} = \delta - \frac{ \lvert\Omega_2\rvert^2 - \lvert\Omega_1\rvert^2 } {4\Delta}.$$

Thus the Raman resonance is shifted by the differential ac Stark shift. Dropping the diagonal terms while keeping the Raman coupling is inconsistent because both arise at the same order.

Bright and dark combinations

For $\delta=0$, define

$$\Omega = \sqrt{ \lvert\Omega_1\rvert^2 + \lvert\Omega_2\rvert^2 }.$$

The normalized combinations

$$\begin{aligned} \lvert B\rangle &= \frac{ \Omega_1^*\lvert g_1\rangle + \Omega_2^*\lvert g_2\rangle }{\Omega}, \\ \lvert D\rangle &= \frac{ \Omega_2\lvert g_1\rangle - \Omega_1\lvert g_2\rangle }{\Omega} \end{aligned}$$

are bright and dark, respectively. The excitation operator annihilates $\lvert D\rangle$, while

$$\begin{aligned} H_{\mathrm{eff}}\lvert B\rangle &= - \frac{\hbar\Omega^2}{4\Delta} \lvert B\rangle, \\ H_{\mathrm{eff}}\lvert D\rangle &= 0. \end{aligned}$$

For the slowly varying dressed component, the leading excited-state population obeys

$$\lvert c_e\rvert^2 \lesssim \frac{\Omega^2}{4\Delta^2}.$$

An abruptly prepared bare bright state has an additional fast oscillation: its peak excited population is approximately $\Omega^2/\Delta^2$, four times the dressed-state estimate. The dark state is exactly uncoupled for the ideal Hamiltonian, whereas the bright state acquires a virtual light shift. Time-dependent dark-state transport is a separate adiabatic-following problem; it should not be inferred from elimination alone.

Open-System Elimination

Suppose the fast excited sector also decays. Begin with a Lindblad equation,

$$\dot\rho = - \frac{i}{\hbar}[H,\rho] + \sum_k \mathcal D[L_k]\rho,$$

where

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12 \left\{ L^\dagger L,\rho \right\}.$$

Write

$$H = H_g+H_e+V_++V_-, \qquad V_-=V_+^\dagger,$$

where $V_+$ excites the slow sector into the fast one. If each $L_k$ returns population from the excited sector to the ground sector, define the excited-sector non-Hermitian Hamiltonian

$$H_{\mathrm{NH}} = H_e - \frac{i\hbar}{2} \sum_k L_k^\dagger L_k.$$

For weak excitation, fast relaxation compared with retained dynamics, and a bounded inverse fast response, the leading effective operators are (Reiter and Sørensen, 2012)

$$\begin{aligned} H_{\mathrm{eff}} &= H_g - \frac12 V_- \left[ H_{\mathrm{NH}}^{-1} + \left( H_{\mathrm{NH}}^{-1} \right)^\dagger \right] V_+, \\ L_{\mathrm{eff}}^{(k)} &= L_k H_{\mathrm{NH}}^{-1} V_+. \end{aligned}$$

Here the ground-state splittings and couplings must be small on the fast response scale in the chosen frame. Large retained splittings require energy-resolved denominators or the generalized effective-operator construction; invertibility alone does not justify the displayed approximation. These operators generate a trace-preserving effective master equation entirely in the retained sector:

$$\dot\rho_g = - \frac{i}{\hbar} [H_{\mathrm{eff}},\rho_g] + \sum_k \mathcal D \left[ L_{\mathrm{eff}}^{(k)} \right]\rho_g.$$

The sequence encoded by an effective jump is

$$g \xrightarrow{\,V_+\,} e \xrightarrow{\,H_{\mathrm{NH}}^{-1}\,} e \xrightarrow{\,L_k\,} g.$$

For one excited state with detuning $\Delta$ and total linewidth $\Gamma$,

$$H_{\mathrm{NH}} = \hbar \left( \Delta-\frac{i\Gamma}{2} \right) \lvert e\rangle\langle e\rvert.$$

A weak Rabi coupling $\Omega$ then produces an effective scattering scale

$$\Gamma_{\mathrm{sc}} \sim \frac{ \Gamma\lvert\Omega\rvert^2 } { 4\left[ \Delta^2+(\Gamma/2)^2 \right] }.$$

Replacing $\Delta$ by $\Delta-i\Gamma/2$ in a Hamiltonian but omitting the effective jumps describes only no-jump loss. It does not preserve trace and generally misses where the emitted population goes. Lindblad–GKSL Equation explains the master-equation structure, while Quantum-Jump Trajectories develops its conditional interpretation.

Exercises

1. Derive an effective scattering operator

A ground state $\lvert g\rangle$ couples to a decaying excited state $\lvert e\rangle$ with

$$\begin{aligned} V_+ &= \frac{\hbar\Omega}{2} \lvert e\rangle\langle g\rvert, \\ L &= \sqrt{\Gamma} \lvert g\rangle\langle e\rvert. \end{aligned}$$

Use $H_{\mathrm{NH}}=\hbar(\Delta-i\Gamma/2)\lvert e\rangle\langle e\rvert$ to find $L_{\mathrm{eff}}$ and its rate scale.

Solution

The inverse fast propagator is

$$H_{\mathrm{NH}}^{-1} = \frac{ \lvert e\rangle\langle e\rvert } {\hbar(\Delta-i\Gamma/2)}.$$

Therefore

$$L_{\mathrm{eff}} = \frac{ \sqrt{\Gamma}\,\Omega } {2(\Delta-i\Gamma/2)} \lvert g\rangle\langle g\rvert.$$

Its squared coefficient is

$$\Gamma_{\mathrm{sc}} = \frac{ \Gamma\lvert\Omega\rvert^2 } { 4\left[ \Delta^2+(\Gamma/2)^2 \right] }.$$

If another ground state is unaffected by the drive, this projector-valued jump dephases superpositions relative to that state even though it returns population to $\lvert g\rangle$.

2. Scattering without a change of the internal state

For the jump $L_{\mathrm{eff}}=\sqrt{\gamma}\lvert g\rangle\langle g\rvert$, calculate the dissipator on $\rho=\lvert g\rangle\langle g\rvert$ and, when an undriven state $\lvert u\rangle$ exists, on the coherence $\rho_{gu}$.

Solution

The dissipator vanishes on $\lvert g\rangle\langle g\rvert$, although the mean photon count rate $\operatorname{Tr}(L^\dagger L\rho)$ is $\gamma$. In the two-state retained manifold, $\dot\rho_{gu}=-\gamma\rho_{gu}/2$. Emission can reveal which component was occupied and dephase a superposition while leaving each population unchanged. The scattering rate and the decay rate of a coherence are different observables.

3. A phase error with small leakage

In the constant two-state model $H=\left(\begin{smallmatrix}0&g\\g&\Delta\end{smallmatrix}\right)$ with real $g$ and $\Delta>0$, the retained eigenvalue is $-g^2/\Delta+g^4/\Delta^3+\cdots$. For a dressed initial state, how long can the leading Hamiltonian predict its phase accurately?

Solution

With $\hbar=1$, the omitted phase is $Tg^4/\Delta^3$. Requiring it to be much smaller than one gives $T\ll\Delta^3/g^4$. The eliminated population stays of order $g^2/\Delta^2$ throughout. Small leakage therefore does not guarantee an accurate phase for arbitrarily long evolution. The stricter tolerance $\varepsilon$ replaces the upper time scale by $\varepsilon\Delta^3/g^4$.

References

• L. Bouten and A. Silberfarb, “Adiabatic elimination in quantum stochastic models,” Communications in Mathematical Physics 283, 491–505 (2008).
• S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793–2826 (2011).
• E. Brion, L. H. Pedersen, and K. Mølmer, “Adiabatic elimination in a lambda system,” Journal of Physics A: Mathematical and Theoretical 40, 1033–1043 (2007).
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley (1992).
• I. L. Egusquiza, “Beyond adiabatic elimination: Systematic expansions,” arXiv:1309.0628 (2013).
• D. F. V. James and J. Jerke, “Effective Hamiltonian theory and its applications in quantum information,” Canadian Journal of Physics 85, 625–632 (2007).
• V. Paulisch, R. Han, H. K. Ng, and B.-G. Englert, “Beyond adiabatic elimination: A hierarchy of approximations for multi-photon processes,” European Physical Journal Plus 129, 12 (2014).
• F. Reiter and A. S. Sørensen, “Effective operator formalism for open quantum systems,” Physical Review A 85, 032111 (2012).