Bidégaray-Fesquet, Brigitte Hierarchy of models in quantum optics. From Maxwell-Bloch to nonlinear Schrödinger. (Hiérarchie de modèles en optique quantique. De Maxwell-Bloch à Schrödinger non-linéaire.) (French) Zbl 1108.81055 Mathématiques & Applications (Berlin) 49. Berlin: Springer (ISBN 3-540-27238-0/pbk). xiii, 171 p. (2006). This book deals with mathematical and numerical analysis of models of quantum optics given by the Maxwell-Bloch equation \[ \varepsilon_0\varepsilon_\infty \partial_t{\mathbf E}= {1\over\mu_0} \text{rot\,}{\mathbf B}-\partial_t{\mathbf P}, \]\[ \partial_t{\mathbf B}= -\text{rot\,}{\mathbf E},\quad{\mathbf P}= N_a \text{Tr}({\mathbf p}\rho), \]\[ \partial_t\rho_{jk}= -i\omega_{jk}\rho_{jk}+ {i\over\hslash}{\mathbf E}\cdot[{\mathbf p},\rho]_{jk}+ Q(\rho)_{jk}. \]Here \(\rho =|\psi\rangle\langle\psi|\), \(\psi\) is the state vector. The first three equations of this system are the Maxwell equation with minor modifications (§2.2). The last equation is the Bloch equation with the relaxation term \(Q\). The vacuum expectation value of its solution describes the correlation function of the quantum nonlinear Schrödinger equation at the small mass limit. Physical meanings and derivations of this equation are explained in §2 of Part I. In §3 first the effect of the relaxation term is discussed by using the Trotter-Kato formula \[ \rho(t)= e^{t(L+ Q)}\rho(0)= \lim_{n\to\infty} (e^{tL/n} e^{tQ/n}\rho(0))^n, \] and showing the conservation of the trace for some special \(Q\) (Th.3.2). Then rewriting the Maxwell-Bloch equation as \[ \partial_t U= \sum_\mu A_\mu\partial_\mu U+ F(U),\quad U=^t({\mathbf E},{\mathbf B},D,R,I), \]\[ D_j= \sigma_{jj},\quad R_{s(j,k)}= {\mathfrak R}\sigma_{jk},\quad I_{s(j,k)}={\mathfrak I}\sigma_{jk}, \] its Cauchy problem is shown to be well posed and to have a unique local solution (Th.3.7). The proof is done applying the results and methods in [T. Cazenave and A. Haraux, Introduction aux problèmes d’evolution semi-linéaires. Paris: Ellipses (1990; Zbl 0786.35070); English translation in Oxford, Clarendon Press (1998; Zbl 0926.35049)]. Then introducing the physical energy \({\mathcal E}(t)\) by \[ {1\over 2}\int_{\mathbb{R}^d}(\varepsilon_0 \varepsilon_\infty|{\mathbf E}(t, x)|^2+{1\over \mu_0} |{\mathbf B}(t, x)|^2+ N_a\hslash\omega_c \text{Tr}(\sigma^2)(t,{\mathbf x}))\,d{\mathbf x}, \] the existence of a constant \(C\) which satisfies the estimate \[ {\mathcal E}(t)\leq e^{C\omega_c t}{\mathcal E}(0), \] is shown (Estimation 3.11). §3 is concluded with the statement of global existence and uniqueness theorems for some simplified equations [cf. P. Donnat and J. Rauch, Arch. Ration. Mech. Anal. 136, 291–303 (1996; Zbl 0873.35093); J. L. Joly, G. Métivier and J. Rauch, Ann. Henri Poincaré 1, No. 2, 307–340 (2000; Zbl 0964.35155)]. Part I is concluded by showing some numerical simulations of Maxwell-Bloch equations [§4. cf. R. W. Ziolkowski, J. M. Arnold and D. M. Cogny, Ultrafast pulse interaction with two-level atoms, Phys. Rev. A 52, 3082–3094 (1995)].Part II introduces the rate equation, a hierarchy of the models. It starts from the one-dimensional one level model (§2.1.6, (2.7)) and assumes \(E(t)= e^{i\omega t}\). Setting \(V = E_0 p_{12}/\hslash\) and \(V= \varepsilon v\), the solution \(\rho\) of the form \(\rho^0+ \varepsilon\rho^1+ \varepsilon^2\rho^2+\cdots\) is sought. The rate equation is the equation of \(\rho^i\). For \(\rho^2\), the equation is \[ \partial_t \rho^{(2)}_{22}=- \gamma\rho^{(2)}_{22}+ 2|v|^2{\gamma'\over (\omega_{12}+ \omega)^2+ \gamma^{\prime 2}}, \] (§5.3). It is shown that the equilibrium state of the Cauchy problem of the rate equation is uniquely determined (Lemma 5.3). Numerical studies of the rate equation for some classical models, such as the Debye model and the Lorentz model, are done in §6. In §7, setting \[ {\mathbf E}= A(t,{\mathbf x})\exp(i(k_z z-\omega t)){\mathbf u}+ c.c.,\;{\mathbf P}= \Pi(t,{\mathbf x}\exp(i(k_zz-\omega t)){\mathbf u}+ c.c., \] the equation of \(A\) and \(\Pi\) (enveloping equation) is treated. Then dimensionless models are considered. The author says that the dimensionless model is the leaf in the hierarchy of the theory whose localization is the field (§7.2).The last part (Part III) of the book is devoted to the numerical study. In §8 a precise discretization of the Bloch equation is done by using the Crank-Nicolson scheme; \[ {\rho^{n+1}- \rho^n\over\delta t}= \text{Rn}\Biggl({\rho^{n+1}+ \rho^n\over 2}\Biggr)+ i\Biggl[V^{n+1/2},{\rho^{n+2}+ \rho^n\over 2}\Biggr], \] [cf. R. W. Ziolkowski, The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations, IEEE Trans. Autom. Propag. 45, 3750–3791 (1997)]. In §9, discretization of the Maxwell equation is done by the scheme of K. S. Yee [Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic median, IEEE Trans. Autom. Propag. 14, 302–307 (1966)]. Combining these results, the discretization of the Maxwell-Bloch equation is done in §10, together with numerical examples. Applications to classical models, such as the Debye model and the Lorentz model, are explained in §11.Relations and hierarchies of models in this book are illustrated as Fig.11.1 in “Trends pour Avenir”. The book is concluded with discussing the enrichment of models discussed in this book. Reviewer: Akira Asada (Takarazuka) Cited in 3 Documents MSC: 81V80 Quantum optics 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 35L45 Initial value problems for first-order hyperbolic systems 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q60 PDEs in connection with optics and electromagnetic theory 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 81V10 Electromagnetic interaction; quantum electrodynamics Keywords:Maxwell-Bloch equation; rate equation; enveloping equation; Cauchy problem; Yee scheme; Crank-Nicholson scheme; Debye model; Lorentz model Citations:Zbl 0786.35070; Zbl 0873.35093; Zbl 0964.35155; Zbl 0926.35049 × Cite Format Result Cite Review PDF Full Text: DOI