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Conservative Quantum Dynamical Semigroups for mean-field quantum diffusion models

Anton Arnold, Christof Sparber

Published Sep 23, 2003• 6 citations
OpenAlex

Abstract

We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup.

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The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator. AMS 1991 Subject classification: 35Q40, 35S10, 81Q99, 81V99 Key words: Open quantum system, Wigner equation, large-time behavior, self-similarity. 1. Introduction This paper is concerned with the analysis of a class of dissipative quantum models that arise when considering the motio...

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Quantum Dynamical Semigroups

Kalyan B. Sinha

Feller [1] and Kato [2] proved the existence of a unique minimal semigroup associated with the classical Fokker-Planck equation: (1.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaada % WcaaqaaiaadsgacaWGWbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqa % aiaadsgacaWG0baaaiabg2da9maaqafabaGaamiCamaaBaaaleaaca % WGPbGaam4AaaqabaGccqqHPoWvdaWgaaWcbaGaam4AaiaadQgaaeqa % aOGaaiilaiaaywW7caWG0bGaeyyzImRaaGimaiaaywW7caWG3bGaam % yAaiaadshacaWGObGaaGPaVlaadMgacaWGUbGaamyAaiaadshacaWG % PbGaamyyaiaadYgacaaMc8Uaam4yaiaad+gacaWGUbGaamizaiaadM % gacaWG0bGaamyAaiaad+gacaWGUbGaaGPaVlaadchadaWgaaWcbaGa % amyAaiaadQgaaeqaaOGaaiikaiaaicdacaGGPaGaeyypa0JaeqiTdq % 2aaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGRbaabeqdcqGHris5 % aaGccaGL9baaaaa!703E! $$ \left. {\frac{{d{p_{ij}}}}{{dt}} = \sum\limits_k {{p_{ik}}{\Omega _{kj}},\quad t \geqslant 0\quad with\,initial\,condition\,{p_{ij}}(0) = {\delta _{ij}}} } \right\} $$ and subject to the Markov condition: (1.2) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqUbcqqHPo % WvdaWgaaWcbaGaam4AaiaadQgaaeqaaOGaeyyzImRaaGimaiaadAga % caWGVbGaamOCaiaaykW7caWGRbGaeyiyIKRaamOAaiaaykW7caWGHb % GaamOBaiaadsgacaaMc8UaeyOhIu6aaabuaeaacqqHPoWvdaWgaaWc % baGaam4AaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaeuyQdC1aaSbaaS % qaaiaadUgacaWGRbaabeaatCvAUfeBSn0BKvguHDwzZbqegeezVjwz % GyuyUD2CV52zGmfDKbIuaGqbaOGaa8hpaiabg6HiLcWcbaGaamOAai % abgcMi5kaadUgaaeqaniabggHiLdGccaGGUaaaaa!6B15! $$ {\Omega _{kj}} \geqslant 0for\,k \ne j\,and\,\infty \sum\limits_{j \ne k} {{\Omega _{kj}} = - {\Omega _{kk}} < \infty } . $$

Operator theory•Jan 1, 1994•10 citations

QUANTUM FOKKER-PLANCK MODELS: THE LINDBLAD AND WIGNER APPROACHES

Anton Arnold, Franco Fagnola, Lukas Neumann

In this article we try to bridge the gap between the quantum dynamical semigroup and Wigner function approaches to quantum open systems. In particular we study stationary states and the long time asymptotics for the quantum Fokker-Planck equation. Our new results apply to open quantum systems in a harmonic confinement potential, perturbed by a (large) sub-quadratic term.

arXiv:0806.2984•Oct 1, 2008•10 citations
We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup.
We formally derive two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium. These models mainly consist of a quantum, dissipative O(Planck 3) hydrodynamic/O(Planck 4) stochastic correction to the frictional (Caldeira-Leggett-)Schrödinger equation. The main ingredient lies in the use of the hydrodynamic/stochastic fluid model approach associated with the quantum Fokker-Planck equation and the identification of the associated pressure field. Then, Madelung transformations set the problem in the Schrödinger picture of dissipative quantum mechanics. We also describe the stationary dynamics associated with both systems.
We investigate what kind of Markovian quantum master equation (QME) in the Gorini-Kossakowski-Sudarshan-Lindblad form is realized under Poincar\'e symmetry. The solution of the Markovian QME is given by a quantum dynamical semigroup, for which we introduce invariance under Poincar\'e transformations. Using the invariance of the dynamical semigroup and applying the unitary representation of the Poincar\'e group, we derive the Markovian QME for a relativistic massive spin-0 particle. Introducing the field operator of the massive particle and examining its evolution, we find that the field follows a dissipative Klein-Gordon equation. In addition, we show that any two local operators for spacelike separated regions commute with each other. This means that the microcausality condition is satisfied for the dissipative model of the massive particle.
We consider quantum particles coupled to local and collective thermal quantum environments. The coupling is energy conserving, and the collective coupling is scaled in the mean-field way. There is no direct interaction between the particles. We show that an initially factorized state of the particles remains factorized at all times, in the limit of large particle number. Each single-particle factor evolves according to an explicit, nonlinear, dissipative and time-dependent Hartree–Lindblad equation. The model is exactly solvable; we do not make any weak coupling or any Markovian approximations, and our results are mathematically rigorous.
The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator. AMS 1991 Subject classification: 35Q40, 35S10, 81Q99, 81V99 Key words: Open quantum system, Wigner equation, large-time behavior, self-similarity. 1. Introduction This paper is concerned with the analysis of a class of dissipative quantum models that arise when considering the motio...
Feller [1] and Kato [2] proved the existence of a unique minimal semigroup associated with the classical Fokker-Planck equation: (1.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaada % WcaaqaaiaadsgacaWGWbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqa % aiaadsgacaWG0baaaiabg2da9maaqafabaGaamiCamaaBaaaleaaca % WGPbGaam4AaaqabaGccqqHPoWvdaWgaaWcbaGaam4AaiaadQgaaeqa % aOGaaiilaiaaywW7caWG0bGaeyyzImRaaGimaiaaywW7caWG3bGaam % yAaiaadshacaWGObGaaGPaVlaadMgacaWGUbGaamyAaiaadshacaWG % PbGaamyyaiaadYgacaaMc8Uaam4yaiaad+gacaWGUbGaamizaiaadM % gacaWG0bGaamyAaiaad+gacaWGUbGaaGPaVlaadchadaWgaaWcbaGa % amyAaiaadQgaaeqaaOGaaiikaiaaicdacaGGPaGaeyypa0JaeqiTdq % 2aaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGRbaabeqdcqGHris5 % aaGccaGL9baaaaa!703E! dpijdt=∑kpikΩkj,t⩾0with initial condition pij(0)=δij} \left. {\frac{{d{p_{ij}}}}{{dt}} = \sum\limits_k {{p_{ik}}{\Omega _{kj}},\quad t \geqslant 0\quad with\,initial\,condition\,{p_{ij}}(0) = {\delta _{ij}}} } \right\} dtdpij​​=k∑​pik​Ωkj​,t⩾0withinitialconditionpij​(0)=δij​} and subject to the Markov condition: (1.2) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqUbcqqHPo % WvdaWgaaWcbaGaam4AaiaadQgaaeqaaOGaeyyzImRaaGimaiaadAga % caWGVbGaamOCaiaaykW7caWGRbGaeyiyIKRaamOAaiaaykW7caWGHb % GaamOBaiaadsgacaaMc8UaeyOhIu6aaabuaeaacqqHPoWvdaWgaaWc % baGaam4AaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaeuyQdC1aaSbaaS % qaaiaadUgacaWGRbaabeaatCvAUfeBSn0BKvguHDwzZbqegeezVjwz % GyuyUD2CV52zGmfDKbIuaGqbaOGaa8hpaiabg6HiLcWcbaGaamOAai % abgcMi5kaadUgaaeqaniabggHiLdGccaGGUaaaaa!6B15! Ωkj⩾0for k≠j and ∞∑j≠kΩkj=−Ωkk<∞. {\Omega _{kj}} \geqslant 0for\,k \ne j\,and\,\infty \sum\limits_{j \ne k} {{\Omega _{kj}} = - {\Omega _{kk}} < \infty } . Ωkj​⩾0fork=jand∞j=k∑​Ωkj​=−Ωkk​<∞.