B. Bonnard, D. Sugny
The objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two-level quantum systems whose dynamics is governed by the Lindblad equation. We focus our analysis on the case where the extremal Hamiltonian is integrable.
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We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers-Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutio…
The objective of this article is to complete preliminary results concerning the time-minimal control of dissipative two-level quantum systems whose dynamics is governed by Lindblad equations. The extremal system is described by a 3D-Hamiltonian depending upon three parameters. We combine geometric techniques with numerical simulations to deduce the optimal solutions.
We study a one-parameter generalization of the symmetric simple exclusion process on a one dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. Then, we derive the sp…
We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{ζζ}+g(v)v_ζ+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h_0(v) is a linear function, h_0(v)=λ^2v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipat…
We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.
We consider a particular class of equations of motion, generalizing to n degrees of freedom the "dissipative spin--orbit problem", commonly studied in Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian framework with action-angle coordinates; they contain a quasi-integrable conservative part and friction terms, assumed to be linear and isotropic with respect to the action variables. In such a context, we transfer two methods determining quasi-periodic solutions, which were originally designed to analyze purely Hamiltonian quasi-integrable problems. First, we show how t…
Liouvillian skin effects and fragmented condensates in an integrable dissipative Bose-Hubbard model
本文证明在损耗率匹配跃迁幅度时，耗散Bose-Hubbard模型在任意相互作用下均精确可解，且可积性在无序和一般边界条件下依然成立，揭示了包含非厄米Mott趋肤效应和碎片化凝聚体等复杂动力学相图。
We consider a dissipative vector field which is represented by a nearly-integrable Hamiltonian flow to which a non symplectic force is added, so that the phase space volume is not preserved. The vector field depends upon two parameters, namely the perturbing and dissipative parameters, and by a drift function. We study the general case of an l-dimensional, time-dependent vector field. Assuming to start with non-resonant initial conditions, we prove the stability of the variables which are actions of the conservative system (namely, when the dissipative parameter is set to zero) for exponential…
We associate a periodic two-dimensional Schrodinger operator to every Lagrangian torus in CP^2 and define the spectral curve of a torus as the Floquet spectrum of this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that Novikov-Veselov hierarchy of equations induces integrable deformations of minimal Lagrangian torus in CP^2 preserving the spectral curve. We also show that the highest flows on the space of smooth periodic solutions of the Tzizeica equation are given by the Novikov-Veselov hierarchy.
We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville gravity.
Integrable system on minimal nilpotent orbit
本文证明复单李代数的Schubert除子方程可给出极小幂零轨道的完全可积系统，利用Chevalley基和Heisenberg代数基显式计算哈密顿函数，并重新发现经典情形的Gelfand-Zeitlin系统低阶项。
We study the transition between integrable and chaotic behaviour in dissipative open quantum systems, exemplified by a boundary driven quantum spin-chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance $s$ is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature $β\in[0,2]$. Here, $β=0$ yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and $β=2$ equals the distribut…
We explicitly construct an integrable and strongly interacting dissipative quantum circuit via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneous transfer matrix, from which conserved superoperator charges can be derived, in particular, the circuit's dynamical generator. After showing the trace preservation and complete positivity of local maps, we reinterpret them as the Kraus representation of the local dynamics of free fermions with single-site dephasing. The integrability of the map is broken by adding interactions t…
We construct new Yang-Baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model WLM(1,p) giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit. We interpret these W-extended Kac representations as finitely-generated W-extended Feigin-Fuchs modules over the triplet W-algebra W(p). The W-extended fusion rules of these representations are inferred from the recently conjectured Virasoro fusion rules of the Kac representations in the underlying logarithmic minimal model LM(1,p). We also introduce the modules contrag…
The spin-boson model (or the dissipative two-state system) is a model for the study of dissipation and decoherence in quantum mechanics. The spin-boson model with Ohmic dissipation is an integrable theory, related to several other integrable systems including the anisotropic Kondo and resonant level models. Here we consider the problem of computing the overlaps between two ground states corresponding to different values of parameters of the Ohmic spin-boson Hamiltonian. We argue that this can be understood as a part of the problem of quantizing the mKdV/sine-Gordon integrable hierarchy. The ma…
We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states $u(z)$ in the spectral theory of soliton gases for the one dimensional integrable focusing Nonlinear Schrödinger Equation (fNLS) and for the Korteweg de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimizing (positive) measure for certain energy functional is absolutely continuous and its density $u(z)\geq 0$ solves the required integral equation. In a similar fashion we show that $v(z)$, the tempo…
A new family of Darboux integrable partial differential equations
本文构造了一类新的达布可积偏微分方程，其两个特征方向积分的最小阶数可同时任意高，揭示达布可积方程的完整列表或远超现有认知。
We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra. This is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in additio…
Complete Minimal Form Factors for Irrelevant Deformations of Integrable Quantum Field Theory
本文针对受广义$T\ar{T}$微扰的对角可积场论，提出无自由参数的最小形状因子计算方法，发现原有形状因子方程不变的假设过强，需修改相关核心假设。
Minimal nonintegrable models with three-site interactions
本研究通过构建五类具有真实三体相互作用的自旋-1/2模型，揭示了超越最近邻范式的可积性与非可积性之间的明确边界。
We develop a Hamiltonian theory of a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The proposed Hamiltonian couples the given system to auxiliary fields, in the universal form of a so-called canonical heat bath. After integrating out the heat bath the original dissipative evolution is exactly reproduced. Furthermore, we show that the dynamics associated to a minimal Hamiltonian are essentially unique, up to a natural class of isomorphisms. Using this formalism, we obtain closed form expressions for t…
We introduce families of one-dimensional Lindblad equations describing open many-particle quantum systems that are exactly solvable in the following sense: $(i)$ the space of operators splits into exponentially many (in system size) subspaces that are left invariant under the dissipative evolution; $(ii)$ the time evolution of the density matrix on each invariant subspace is described by an integrable Hamiltonian. The prototypical example is the quantum version of the asymmetric simple exclusion process (ASEP) which we analyze in some detail. We show that in each invariant subspace the dynamic…
What We Talk About When We Talk About Dissipative Quantum Chaos
本文探讨了耗散量子混沌理论的核心问题，介绍了其基于开放量子动力学的谱性质方法，并总结了该领域近年来的发展和基础理论。
With the XXZ spin chains as examples, we prove two theorems: (1) the functional relations derived from the off-diagonal Bethe Ansatz scheme are the sufficient and necessary conditions to characterize the complete spectrum of the corresponding transfer matrix; (2) each eigenvalue of the transfer matrix can be parameterized by a minimal inhomogeneous T-Q relation. These statements hold for both with and without inhomogeneity. The proof can be generalized to other finite-dimensional quantum integrable models.
Working in the dense loop representation, we use the planar Temperley-Lieb algebra to build integrable lattice models called logarithmic minimal models LM(p,p'). Specifically, we construct Yang-Baxter integrable Temperley-Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley-Lieb algebra are inherently non-local and not (time-reversal) symmetric. In the continuum scaling limit, they yield logarithmic conformal field theories with central charges c=1-6(p-p')^2/pp' where p,p'=1,2,... are…
Integrable Spherical Brane Model at Large $N$
研究一个最简单的具有边界的二维可积量子场论：bulk中N个非紧缩标量，被非线性约束在半径为1/√g的(N-1)球面上。大N情况下对应高自旋杂质或多通道情形，添加边界磁场保持可积性并验证了Lukyanov和Zamolodchikov的猜想。
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