Liouvillian gaps, Jordan blocks, and observable relaxation

Key claim

A nonzero Liouvillian spectral gap supplies an asymptotic exponential scale. It does not, by itself, specify the relaxation law or the time required to reach a given accuracy.

Jordan blocks add polynomial factors. Nonorthogonal left and right eigenmodes alter prefactors. An initial state and an observable may also have zero overlap with the slowest mode.

Theme

Liouvillian spectra, exceptional points, and finite-accuracy mixing.

Guiding question

If every eigenvalue of a Markovian generator is known, what information is still missing from a prediction of observable relaxation?

Setup

Consider a resonantly driven qubit with Markovian pure dephasing:

This is a time-independent GKSL equation for the assumed effective model. It is not an exact statement about every microscopic bath that could produce dephasing.

Write

The Bloch equations are

and

For $\mathrm{\Omega }\ne 0$, the unique stationary state is ${\rho }_{\mathrm{s}\mathrm{s}}=I/2$.

Gap convention

For a finite-dimensional primitive semigroup, we use ${\mathrm{\Delta }}_{\mathcal{L}}=\underset{\lambda \ne 0}{min}[-\mathrm{Re}\lambda ]$. Primitivity excludes additional stationary or purely oscillatory sectors.

The Liouvillian spectrum contains $0$, the $x$-sector eigenvalue $-2\kappa$, and the two eigenvalues of $M$.

Analysis

Derivation: a finite-gap exceptional point

The characteristic polynomial of $M$ is

Hence

This gives three regimes:

At $\mathrm{\Omega }=\kappa$,

The repeated eigenvalue has only one eigenvector in this sector. The propagator is therefore

The gap is still

Yet a generic component in this sector contains

not a single exponential. Exceptional points of two-level Lindblad generators and their defective eigenmodes are analyzed explicitly by Hatano.

For the initial condition $z(0)=1$, $y(0)=0$, eliminate $y$ using $\dot{z}=\mathrm{\Omega }y$. This gives

At the exceptional point,

The logarithmic decay rate still approaches the gap:

The polynomial changes the relaxation curve and produces a subleading $\ln \ln (1/\epsilon )$ correction to the time needed to reach accuracy $\epsilon$.

Scope and assumptions: gap, mixing time, and mode geometry

For a diagonalizable finite-dimensional generator, write its decaying part as

where

for biorthogonally normalized right and left eigenmatrices. An observable deviation is

The eigenvalues fix the available exponential factors. The two overlaps decide which factors appear in a particular experiment.

If the coefficient of every mode with real part $-{\mathrm{\Delta }}_{\mathcal{L}}$ vanishes, the observable relaxes faster than the global gap suggests. Oscillatory eigenvalues add phases, and Jordan chains add powers of $t$.

Observable gap

One may define an effective decay rate for a chosen pair $(O,{\rho }_0)$ from the slowest mode with nonzero overlap. This is not a new spectral gap of $\mathcal{L}$.

Non-normality matters even away from an exact exceptional point. The norms of the projectors ${\mathcal{P}}_n$ can be large when left and right eigenvectors are nearly parallel.

Large modal contributions may then cancel at one time and reinforce at another. This changes intermediate-time behavior and sensitivity to perturbations, although a completely positive trace-preserving map remains contractive in trace distance between states.

Define the worst-case trace-distance mixing time by

A bound of the schematic form

requires more than ${\mathrm{\Delta }}_{\mathcal{L}}$. The constant $C$ depends on eigenmode geometry and norm conversion; $p$ records the largest relevant Jordan block.

Consequently,

The gap fixes the leading exponential scale at fixed finite dimension. It does not determine $C$, $p$, or the finite-accuracy time.

Many-body scaling

For a family ${\mathcal{L}}_L$, a size-independent gap need not give size-independent trace-norm mixing. Prefactors can grow with Hilbert-space dimension or with properties of ${\rho }_{\mathrm{s}\mathrm{s}}$. Logarithmic Sobolev constants provide stronger convergence control in many settings.

Quantum logarithmic Sobolev inequalities and their relation to convergence bounds are developed by Kastoryano and Temme. A 2026 preprint further proposes mode trace norms as predictors beyond the gap Zhou.

The last source is recent and should be read as a current proposal, not as the basis of the finite-dimensional Jordan-block result above.

Physical interpretation: critical damping, slow sectors, and memory

The $y$-$z$ equations are identical to a classical damped oscillator:

The Liouvillian exceptional point is the critical-damping point. The term $t{\mathrm{e}}^{-\kappa t}$ comes from a repeated defective root, not from a closing gap.

In the strong-dephasing regime $\mathrm{\Omega }\ll \kappa$,

Thus

Increasing the microscopic dephasing rate can slow the population dynamics. Frequent destruction of phase coherence suppresses the coherent transfer that changes $z$, producing a quantum-Zeno-like slow mode.

A cluster of slow eigenvalues separated from the rest can create a metastable window. The system first approaches a low-dimensional manifold and only later reaches its unique stationary state.

Identifying that manifold requires the slow eigenmatrices, not only the smallest nonzero real part. This spectral construction is developed by Macieszczak, Guță, Lesanovsky, and Garrahan.

Markovian boundary

If the reduced evolution contains a memory kernel, a single constant $\mathcal{L}$ need not organize the full dynamics. Exceptional points of an approximate Bloch generator may then miss features of the microscopic evolution.

For example, a time-nonlocal equation may take the form

The comparison by Seshadri, Li, and Galperin shows how the approximations leading to a Bloch master equation can limit Liouvillian-exceptional-point reasoning in a driven two-level model.

False claim to diagnose

If a primitive finite-dimensional Lindbladian has gap $\mathrm{\Delta }>0$, every observable deviation must equal $C{\mathrm{e}}^{-\mathrm{\Delta }t}$ at late times.

This statement contains three errors.

First, a Jordan block at the gap edge gives $t^m{\mathrm{e}}^{-\mathrm{\Delta }t}$. Second, complex eigenvalues add oscillations. Third, the initial state or observable may have zero overlap with all gap-edge modes.

The defensible statement is narrower:

At fixed finite dimension, the gap identifies the slowest available asymptotic exponential rate. The realized decay law also depends on generalized eigenspaces and experimental overlaps.

What follows — and what does not

Statement	Status
The qubit model has ${\lambda }_{\pm }=-\kappa \pm \sqrt{{\kappa }^2-{\mathrm{\Omega }}^2}$.	Exact within the stated GKSL model
$\mathrm{\Omega }=\kappa$ is a second-order exceptional point.	Exact matrix result
The gap at that point is $\kappa$.	Exact with the stated gap convention
Every component decays as a pure ${\mathrm{e}}^{-\kappa t}$.	False; a Jordan chain gives $t{\mathrm{e}}^{-\kappa t}$
A finite gap guarantees a unique stationary state.	False without primitivity or equivalent assumptions
The gap alone fixes $t_{\mathrm{m}\mathrm{i}\mathrm{x}}(\epsilon )$.	False; prefactors, Jordan order, and norms enter
A chosen observable must decay at the global gap.	False when its slow-mode overlap vanishes
A size-independent many-body gap implies uniform rapid mixing.	Not without further bounds
The same spectral analysis applies unchanged to non-Markovian dynamics.	False in general

Exercise

For

complete the following tasks.

1. Derive ${\lambda }_{\pm }$ and classify the three damping regimes.
2. At $\mathrm{\Omega }=\kappa$, derive $z(t)$ for $z(0)=1$, $y(0)=0$ without diagonalizing $M$.
3. Compute $y(t)$ and the trace distance from $I/2$.
4. Estimate the small-$\epsilon$ time at which $|z(t)|\le \epsilon$.
5. For $\mathrm{\Omega }\ll \kappa$, derive the slow rate and explain its dependence on $\kappa$.

Hint 1

Use $\dot{z}=\mathrm{\Omega }y$ to obtain a second-order equation. Critical damping gives a repeated root.

Hint 2

At the exceptional point, set $u=\kappa t$. Solving $(1+u){\mathrm{e}}^{-u}=\epsilon$ iteratively gives $u=\ln (1/\epsilon )+\ln u+o(1)$.

Oral check 1. Why can a polynomial prefactor change the mixing time without closing the Liouvillian gap?

Oral check 2. What two overlaps decide whether a slow Liouvillian mode appears in a measured signal?

Solution

The characteristic equation is

$${\lambda }^2+2\kappa \lambda +{\mathrm{\Omega }}^2=0,$$Copy LaTeX

so

$${\lambda }_{\pm }=-\kappa \pm \sqrt{{\kappa }^2-{\mathrm{\Omega }}^2}.$$Copy LaTeX

The roots are distinct and real for $\mathrm{\Omega }<\kappa$, repeated and defective for $\mathrm{\Omega }=\kappa$, and complex conjugates for $\mathrm{\Omega }>\kappa$.

Eliminating $y$ gives

$$\ddot{z}+2\kappa \dot{z}+{\mathrm{\Omega }}^{2}z=0.$$Copy LaTeX

At $\mathrm{\Omega }=\kappa$,

$$z(t)=(A+Bt){\mathrm{e}}^{-\kappa t}.$$Copy LaTeX

The conditions $z(0)=1$ and $\dot{z}(0)=\kappa y(0)=0$ give $A=1$ and $B=\kappa$:

$$z(t)=(1+\kappa t){\mathrm{e}}^{-\kappa t}.$$Copy LaTeX

Since $y=\dot{z}/\kappa$,

$$y(t)=-\kappa t{\mathrm{e}}^{-\kappa t}.$$Copy LaTeX

Here $x(t)=0$. The trace distance from $I/2$ is

$$D(\rho (t),\frac{I}{2})=\frac{1}{2}\sqrt{y(t{)}^2+z(t{)}^2}=\frac{{\mathrm{e}}^{-\kappa t}}{2}\sqrt{(\kappa t{)}^2+(1+\kappa t{)}^2}.$$Copy LaTeX

For the simpler condition $|z(t)|\le \epsilon$, let $u=\kappa t$. At small $\epsilon$,

$$u=\ln \frac{1}{\epsilon }+\ln \ln \frac{1}{\epsilon }+o(1).$$Copy LaTeX

Thus the gap supplies the coefficient $1/\kappa$, while the Jordan block adds the double-logarithmic correction.

Finally,

$${\lambda }_{+}=-\kappa +\kappa \sqrt{1-\frac{{\mathrm{\Omega }}^2}{{\kappa }^2}}\simeq -\frac{{\mathrm{\Omega }}^2}{2\kappa }.$$Copy LaTeX

The slow time is $2\kappa /{\mathrm{\Omega }}^2$. Strong dephasing suppresses the coherent process that transfers population between the ${\sigma }_z$ states.

Check your understanding

Two primitive Lindbladians have the same spectrum, including multiplicities, but one is diagonalizable and the other has a Jordan block at the gap edge.

Which asymptotic quantity must agree, and which features of a finite-time relaxation experiment can differ?

You may also reply with “deeper,” “too easy,” “too hard,” or your derivation.

Further Reading

• A short introduction to the Lindblad Master Equation
• Exceptional points of the Lindblad operator of a two-level system
• Quantum logarithmic Sobolev inequalities and rapid mixing
• Towards a theory of metastability in open quantum dynamics
• Liouvillian exceptional points of an open driven two-level system
• Universal Predictors for Mixing Time more than Liouvillian Gap

Connections and next step

• Generator level: the spectrum supplies exponential rates in operator space.
• Geometric level: projectors, Jordan chains, and pseudospectral sensitivity shape finite-time dynamics.
• Experimental level: state preparation and observable choice select which modes are visible.
• Many-body level: rapid mixing requires uniform control beyond a finite-size gap.
• Approximation level: a constant Lindbladian presupposes time-local Markovian reduction.
• Next step: compare spectral-gap bounds with logarithmic Sobolev and pseudospectral bounds for local Lindbladians.
• Revisit: connect the slow strong-dephasing mode to the quantum-Zeno limit and to metastable manifolds.