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"id": "PHYS-2026-08-01-01",
"canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-01-01/",
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"id": "PHYS-2026-08-01-01",
"date": "2026-08-01",
"updated_at": "2026-08-01",
"title": "Liouvillian gaps, Jordan blocks, and observable relaxation",
"summary": "Why a finite Liouvillian gap fixes an asymptotic exponential scale but not the full relaxation curve, mixing time, or decay seen by a chosen observable.",
"language": "en",
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"content_markdown": "\n## Key claim\n\n**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.**\n\nJordan 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.\n\n## Theme\n\n**Liouvillian spectra, exceptional points, and finite-accuracy mixing.**\n\n## Guiding question\n\nIf every eigenvalue of a Markovian generator is known, what information is still missing from a prediction of observable relaxation?\n\n## Setup\n\nConsider a resonantly driven qubit with Markovian pure dephasing:\n\n$$\n\\dot\\rho=\\mathcal L(\\rho)\n=-\\ii\\left[\\frac{\\Omega}{2}\\sigma_x,\\rho\\right]\n+\\kappa(\\sigma_z\\rho\\sigma_z-\\rho),\n\\qquad \\Omega,\\kappa>0.\n$$\n\nThis 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.\n\nWrite\n\n$$\n\\rho=\\frac12(I+x\\sigma_x+y\\sigma_y+z\\sigma_z).\n$$\n\nThe Bloch equations are\n\n$$\n\\dot x=-2\\kappa x,\n$$\n\nand\n\n$$\n\\frac{\\dd}{\\dd t}\n\\begin{pmatrix}y\\\\z\\end{pmatrix}\n=M\\begin{pmatrix}y\\\\z\\end{pmatrix},\n\\qquad\nM=\n\\begin{pmatrix}\n-2\\kappa&-\\Omega\\\\\n\\Omega&0\n\\end{pmatrix}.\n$$\n\nFor $\\Omega\\ne0$, the unique stationary state is $\\rho_{\\rm ss}=I/2$.\n\n> [!margin: Gap convention]\n> For a finite-dimensional primitive semigroup, we use $\\Delta_{\\mathcal L}=\\min_{\\lambda\\ne0}[-\\operatorname{Re}\\lambda]$. Primitivity excludes additional stationary or purely oscillatory sectors.\n\nThe Liouvillian spectrum contains $0$, the $x$-sector eigenvalue $-2\\kappa$, and the two eigenvalues of $M$.\n\n## Analysis\n\n### Derivation: a finite-gap exceptional point\n\nThe characteristic polynomial of $M$ is\n\n$$\n\\det(M-\\lambda I)\n=\\lambda^2+2\\kappa\\lambda+\\Omega^2.\n$$\n\nHence\n\n$$\n\\lambda_\\pm=-\\kappa\\pm\\sqrt{\\kappa^2-\\Omega^2}.\n$$\n\nThis gives three regimes:\n\n$$\n\\begin{array}{c|c}\n\\Omega<\\kappa & \\lambda_\\pm\\text{ real: overdamped}\\\\\n\\Omega=\\kappa & \\lambda_+=\\lambda_-=-\\kappa\\text{: exceptional point}\\\\\n\\Omega>\\kappa & \\lambda_\\pm=-\\kappa\\pm\\ii\\sqrt{\\Omega^2-\\kappa^2}\n\\end{array}\n$$\n\nAt $\\Omega=\\kappa$,\n\n$$\nM=-\\kappa I+N,\n\\qquad\nN=\\kappa\n\\begin{pmatrix}\n-1&-1\\\\\n1&1\n\\end{pmatrix},\n\\qquad\nN^2=0,quad N\\ne0.\n$$\n\nThe repeated eigenvalue has only one eigenvector in this sector. The propagator is therefore\n\n$$\n\\ee^{Mt}=\\ee^{-\\kappa t}\\ee^{Nt}\n=\\ee^{-\\kappa t}(I+tN).\n$$\n\nThe gap is still\n\n$$\n\\Delta_{\\mathcal L}=\\kappa.\n$$\n\nYet a generic component in this sector contains\n\n$$\n(A+Bt)\\ee^{-\\kappa t},\n$$\n\nnot a single exponential. Exceptional points of two-level Lindblad generators and their defective eigenmodes are analyzed explicitly by [Hatano](https://arxiv.org/abs/1903.04676).\n\nFor the initial condition $z(0)=1$, $y(0)=0$, eliminate $y$ using $\\dot z=\\Omega y$. This gives\n\n$$\n\\ddot z+2\\kappa\\dot z+\\Omega^2z=0.\n$$\n\nAt the exceptional point,\n\n$$\n\\boxed{z(t)=(1+\\kappa t)\\ee^{-\\kappa t}}.\n$$\n\nThe logarithmic decay rate still approaches the gap:\n\n$$\n-\\lim_{t\\to\\infty}\\frac{1}{t}\\ln|z(t)|=\\kappa.\n$$\n\nThe polynomial changes the relaxation curve and produces a subleading $\\ln\\ln(1/\\varepsilon)$ correction to the time needed to reach accuracy $\\varepsilon$.\n\n### Scope and assumptions: gap, mixing time, and mode geometry\n\nFor a diagonalizable finite-dimensional generator, write its decaying part as\n\n$$\n\\ee^{t\\mathcal L}-\\mathcal P_{\\rm ss}\n=\\sum_{n\\ge1}\\ee^{\\lambda_nt}\\mathcal P_n,\n$$\n\nwhere\n\n$$\n\\mathcal P_n=|R_n\\rangle\\!\\rangle\\langle\\!\\langle L_n|\n$$\n\nfor biorthogonally normalized right and left eigenmatrices. An observable deviation is\n\n$$\n\\delta\\langle O(t)\\rangle\n=\\sum_{n\\ge1}\\ee^{\\lambda_nt}\n\\langle\\!\\langle O|R_n\\rangle\\!\\rangle\n\\langle\\!\\langle L_n|\\delta\\rho(0)\\rangle\\!\\rangle.\n$$\n\nThe eigenvalues fix the available exponential factors. The two overlaps decide which factors appear in a particular experiment.\n\nIf the coefficient of every mode with real part $-\\Delta_{\\mathcal L}$ vanishes, the observable relaxes faster than the global gap suggests. Oscillatory eigenvalues add phases, and Jordan chains add powers of $t$.\n\n> [!margin: Observable gap]\n> 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 $\\mathcal L$.\n\nNon-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.\n\nLarge 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.\n\nDefine the worst-case trace-distance mixing time by\n\n$$\nt_{\\rm mix}(\\varepsilon)\n=\\inf\\left\\{t:\\sup_{\\rho}\n\\frac12\\left\\|\\ee^{t\\mathcal L}(\\rho)-\\rho_{\\rm ss}\\right\\|_1\n\\le\\varepsilon\\right\\}.\n$$\n\nA bound of the schematic form\n\n$$\n\\sup_\\rho\\frac12\\left\\|\\ee^{t\\mathcal L}(\\rho)-\\rho_{\\rm ss}\\right\\|_1\n\\le C(1+t^{p})\\ee^{-\\Delta_{\\mathcal L}t}\n$$\n\nrequires more than $\\Delta_{\\mathcal L}$. The constant $C$ depends on eigenmode geometry and norm conversion; $p$ records the largest relevant Jordan block.\n\nConsequently,\n\n$$\nt_{\\rm mix}(\\varepsilon)\n\\lesssim\n\\frac{1}{\\Delta_{\\mathcal L}}\n\\left[\\ln C+\\ln\\frac1\\varepsilon+p\\ln t_{\\rm mix}\\right].\n$$\n\nThe gap fixes the leading exponential scale at fixed finite dimension. It does not determine $C$, $p$, or the finite-accuracy time.\n\n> [!margin: Many-body scaling]\n> 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 $\\rho_{\\rm ss}$. Logarithmic Sobolev constants provide stronger convergence control in many settings.\n\nQuantum logarithmic Sobolev inequalities and their relation to convergence bounds are developed by [Kastoryano and Temme](https://arxiv.org/abs/1207.3261). A 2026 preprint further proposes mode trace norms as predictors beyond the gap [Zhou](https://arxiv.org/abs/2601.06256).\n\nThe last source is recent and should be read as a current proposal, not as the basis of the finite-dimensional Jordan-block result above.\n\n### Physical interpretation: critical damping, slow sectors, and memory\n\nThe $y$-$z$ equations are identical to a classical damped oscillator:\n\n$$\n\\ddot z+2\\kappa\\dot z+\\Omega^2z=0.\n$$\n\nThe Liouvillian exceptional point is the critical-damping point. The term $t\\ee^{-\\kappa t}$ comes from a repeated defective root, not from a closing gap.\n\nIn the strong-dephasing regime $\\Omega\\ll\\kappa$,\n\n$$\n\\lambda_+\n=-\\kappa+\\kappa\\sqrt{1-\\frac{\\Omega^2}{\\kappa^2}}\n\\simeq-\\frac{\\Omega^2}{2\\kappa}.\n$$\n\nThus\n\n$$\n\\tau_{\\rm slow}\\simeq\\frac{2\\kappa}{\\Omega^2}.\n$$\n\nIncreasing 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.\n\nA 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.\n\nIdentifying that manifold requires the slow eigenmatrices, not only the smallest nonzero real part. This spectral construction is developed by [Macieszczak, Guță, Lesanovsky, and Garrahan](https://arxiv.org/abs/1512.05801).\n\n> [!margin: Markovian boundary]\n> 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.\n\nFor example, a time-nonlocal equation may take the form\n\n$$\n\\dot\\rho(t)=\\int_0^t\\dd t'\\,K(t-t')\\rho(t').\n$$\n\nThe comparison by [Seshadri, Li, and Galperin](https://arxiv.org/abs/2401.04011) shows how the approximations leading to a Bloch master equation can limit Liouvillian-exceptional-point reasoning in a driven two-level model.\n\n## False claim to diagnose\n\n> If a primitive finite-dimensional Lindbladian has gap $\\Delta>0$, every observable deviation must equal $C\\ee^{-\\Delta t}$ at late times.\n\nThis statement contains three errors.\n\nFirst, a Jordan block at the gap edge gives $t^m\\ee^{-\\Delta t}$. Second, complex eigenvalues add oscillations. Third, the initial state or observable may have zero overlap with all gap-edge modes.\n\nThe defensible statement is narrower:\n\n> 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.\n\n## What follows — and what does not\n\n| Statement | Status |\n| --- | --- |\n| The qubit model has $\\lambda_\\pm=-\\kappa\\pm\\sqrt{\\kappa^2-\\Omega^2}$. | Exact within the stated GKSL model |\n| $\\Omega=\\kappa$ is a second-order exceptional point. | Exact matrix result |\n| The gap at that point is $\\kappa$. | Exact with the stated gap convention |\n| Every component decays as a pure $\\ee^{-\\kappa t}$. | False; a Jordan chain gives $t\\ee^{-\\kappa t}$ |\n| A finite gap guarantees a unique stationary state. | False without primitivity or equivalent assumptions |\n| The gap alone fixes $t_{\\rm mix}(\\varepsilon)$. | False; prefactors, Jordan order, and norms enter |\n| A chosen observable must decay at the global gap. | False when its slow-mode overlap vanishes |\n| A size-independent many-body gap implies uniform rapid mixing. | Not without further bounds |\n| The same spectral analysis applies unchanged to non-Markovian dynamics. | False in general |\n\n## Exercise\n\nFor\n\n$$\nM=\n\\begin{pmatrix}\n-2\\kappa&-\\Omega\\\\\n\\Omega&0\n\\end{pmatrix},\n$$\n\ncomplete the following tasks.\n\n1. Derive $\\lambda_\\pm$ and classify the three damping regimes.\n2. At $\\Omega=\\kappa$, derive $z(t)$ for $z(0)=1$, $y(0)=0$ without diagonalizing $M$.\n3. Compute $y(t)$ and the trace distance from $I/2$.\n4. Estimate the small-$\\varepsilon$ time at which $|z(t)|\\le\\varepsilon$.\n5. For $\\Omega\\ll\\kappa$, derive the slow rate and explain its dependence on $\\kappa$.\n\n\nHint 1
\n\nUse $\\dot z=\\Omega y$ to obtain a second-order equation. Critical damping gives a repeated root.\n\n \n\n\nHint 2
\n\nAt the exceptional point, set $u=\\kappa t$. Solving $(1+u)\\ee^{-u}=\\varepsilon$ iteratively gives $u=\\ln(1/\\varepsilon)+\\ln u+o(1)$.\n\n \n\n**Oral check 1.** Why can a polynomial prefactor change the mixing time without closing the Liouvillian gap?\n\n**Oral check 2.** What two overlaps decide whether a slow Liouvillian mode appears in a measured signal?\n\n\nSolution
\n\nThe characteristic equation is\n\n$$\n\\lambda^2+2\\kappa\\lambda+\\Omega^2=0,\n$$\n\nso\n\n$$\n\\lambda_\\pm=-\\kappa\\pm\\sqrt{\\kappa^2-\\Omega^2}.\n$$\n\nThe roots are distinct and real for $\\Omega<\\kappa$, repeated and defective for $\\Omega=\\kappa$, and complex conjugates for $\\Omega>\\kappa$.\n\nEliminating $y$ gives\n\n$$\n\\ddot z+2\\kappa\\dot z+\\Omega^2z=0.\n$$\n\nAt $\\Omega=\\kappa$,\n\n$$\nz(t)=(A+Bt)\\ee^{-\\kappa t}.\n$$\n\nThe conditions $z(0)=1$ and $\\dot z(0)=\\kappa y(0)=0$ give $A=1$ and $B=\\kappa$:\n\n$$\nz(t)=(1+\\kappa t)\\ee^{-\\kappa t}.\n$$\n\nSince $y=\\dot z/\\kappa$,\n\n$$\ny(t)=-\\kappa t\\ee^{-\\kappa t}.\n$$\n\nHere $x(t)=0$. The trace distance from $I/2$ is\n\n$$\nD\\!\\left(\\rho(t),\\frac I2\\right)\n=\\frac12\\sqrt{y(t)^2+z(t)^2}\n=\\frac{\\ee^{-\\kappa t}}{2}\n\\sqrt{(\\kappa t)^2+(1+\\kappa t)^2}.\n$$\n\nFor the simpler condition $|z(t)|\\le\\varepsilon$, let $u=\\kappa t$. At small $\\varepsilon$,\n\n$$\nu=\\ln\\frac1\\varepsilon+\\ln\\ln\\frac1\\varepsilon+o(1).\n$$\n\nThus the gap supplies the coefficient $1/\\kappa$, while the Jordan block adds the double-logarithmic correction.\n\nFinally,\n\n$$\n\\lambda_+\n=-\\kappa+\\kappa\\sqrt{1-\\frac{\\Omega^2}{\\kappa^2}}\n\\simeq-\\frac{\\Omega^2}{2\\kappa}.\n$$\n\nThe slow time is $2\\kappa/\\Omega^2$. Strong dephasing suppresses the coherent process that transfers population between the $\\sigma_z$ states.\n\n \n\n## Check your understanding\n\nTwo primitive Lindbladians have the same spectrum, including multiplicities, but one is diagonalizable and the other has a Jordan block at the gap edge.\n\nWhich asymptotic quantity must agree, and which features of a finite-time relaxation experiment can differ?\n\nYou may also reply with “deeper,” “too easy,” “too hard,” or your derivation.\n\n## Further Reading\n\n- [A short introduction to the Lindblad Master Equation](https://arxiv.org/abs/1906.04478)\n- [Exceptional points of the Lindblad operator of a two-level system](https://arxiv.org/abs/1903.04676)\n- [Quantum logarithmic Sobolev inequalities and rapid mixing](https://arxiv.org/abs/1207.3261)\n- [Towards a theory of metastability in open quantum dynamics](https://arxiv.org/abs/1512.05801)\n- [Liouvillian exceptional points of an open driven two-level system](https://arxiv.org/abs/2401.04011)\n- [Universal Predictors for Mixing Time more than Liouvillian Gap](https://arxiv.org/abs/2601.06256)\n\n## Connections and next step\n\n- Generator level: the spectrum supplies exponential rates in operator space.\n- Geometric level: projectors, Jordan chains, and pseudospectral sensitivity shape finite-time dynamics.\n- Experimental level: state preparation and observable choice select which modes are visible.\n- Many-body level: rapid mixing requires uniform control beyond a finite-size gap.\n- Approximation level: a constant Lindbladian presupposes time-local Markovian reduction.\n- Next step: compare spectral-gap bounds with logarithmic Sobolev and pseudospectral bounds for local Lindbladians.\n- Revisit: connect the slow strong-dephasing mode to the quantum-Zeno limit and to metastable manifolds.\n",
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"tex": "\\lambda_+\n=-\\kappa+\\kappa\\sqrt{1-\\frac{\\Omega^2}{\\kappa^2}}\n\\simeq-\\frac{\\Omega^2}{2\\kappa}.",
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"tex": "M=\n\\begin{pmatrix}\n-2\\kappa&-\\Omega\\\\\n\\Omega&0\n\\end{pmatrix},",
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{
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"tex": "D\\!\\left(\\rho(t),\\frac I2\\right)\n=\\frac12\\sqrt{y(t)^2+z(t)^2}\n=\\frac{\\ee^{-\\kappa t}}{2}\n\\sqrt{(\\kappa t)^2+(1+\\kappa t)^2}.",
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"tex": "u=\\ln\\frac1\\varepsilon+\\ln\\ln\\frac1\\varepsilon+o(1).",
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{
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"tex": "\\lambda_+\n=-\\kappa+\\kappa\\sqrt{1-\\frac{\\Omega^2}{\\kappa^2}}\n\\simeq-\\frac{\\Omega^2}{2\\kappa}.",
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]
}