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Formulation and Numerical Solution of Quantum Control Problems

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Formulation and Numerical Solution of Quantum Control Problems Index adjoint equation, 86, 113 Coulomb potential, 44, 49, 347 Gâteaux differentiability, 79 Armijo condition, 223, 231 Crank–Nicolson scheme, 182 Galerkin projection, 282 Arzelà–Ascoli theorem, 140 generalized Bloch vectors, 67 azimuthal quantum number, 24 de Broglie’s wave hypothesis, 8 generalized Jacobian, 245 density functional theory, 43 generalized Laguerre BFGS method, 228 density functional theory polynomials, 348 black-box methods, 222 (DFT), 43 global phase, 44 Bloch sphere, 65 density matrix, 27, 29, 31 group velocity, 9 Bochner space, 359 density operator as a gyromagnetic ratio, 18 Bohr radius, 348 rank-one-projector, 29 Born–Oppenheimer DFT’s function sets, 44 Hamilton operator of a approximation, 42 differentiability spherical symmetric Bose–Einstein condensate, 39 real-Fréchet, 133 problem, 345 bosons, 17, 39 real-Gâteaux, 133 Hamiltonian dynamics, 62 bra and ket vectors, 28 dipole approximation, 331, 343 Hartree potential, 50, 128 Dirac formalism, 28 Hermitian operator, 11 canonical momentum, 16 discretize-before-optimize, 236, Hessian, 245, 256 Carathéodory’s theorem, 76 254 heteronuclear spins, 33 cascadic black-box schemes, 230 Duhamel form, 140 hierarchy of objectives, 281 charged particle, 16 Hohenberg–Kohn theorem, 47 coercivity of the Hessian, 98 Ehrenfest’s theorem, 12, 15 homodyne measurements, 64 commutation relation, 15 eigenvalue problem, 14 homotopy continuation commutator, 12, 15 electric scalar potential, 16 scheme, 275 conical eigenvalue intersection, electron density, 43 169, 179 ensemble of quantum states, 27 implicit midpoint scheme, 182, connectivity graph of the exact and approximate 235 control Hamiltonian, controllability, 163 implicit-function theorem, 77 168 exact-controllability problem, infinite-barrier well potential, continuation for initializing 291 155 optimization schemes, exchange potential, 129 infinite-well potential, 14 274 exponential midpoint scheme, intrinsic angular momentum, continuation technique, 292 183 17 continuity equation, 51 control constraints, 231 fermions, 17 KKT system, 111 control Hamiltonian, 164, 354 fidelity, 318 Kohn–Sham (KS) approach, 49 control-to-state map, 77 finite-dimensional Krotov method, 232 controlled Fokker–Planck Schrödinger–Pauli Krylov–Newton methods, 242 equation, 66 equation, 27 KS potential, 50, 128 convolution theorem, 132 Fokker–Planck equation, 65 correlation potential, 129 Fréchet differentiability, 79, Lagrange framework, 88 Coulomb gauge condition, 16 104, 133, 253 Landau levels, 16 389 390 Index Laplacian in spherical Pauli matrices, 20 spin-down, 17 coordinates, 345 Pauli principle, 38, 50 spin quantum number, 18, 25 Larmor frequency, 19, 33 Planck’s constant, 8 spin-spin interaction graph, 172 Legendre polynomials and posterior state vector, 64 spin-up, 17 Rodrigues formula, 347 precession, 19 spinor, 19 Lindblad operator, 63 principal quantum number, 24 stability in the sense of Lindblad–Kossakowski probability density function, 65 Hadamard, 76 equation, 31, 63 projection method, 231 steepest descent method, 223 linearized constraint equation, pure quantum state, 28 stimulated Raman adiabatic 86 passage (STIRAP), 73 Liouville–von Neumann quadratic and superlinear stochastic Belavkin model, 65 equation, 31, 59 convergence, 244 stochastic diffusion process, 64 Lorenz force, 16 quadratic model of the stochastic Schrödinger equation objective, 281 (SE), 64 magnetic moment of a particle, quantum filtering equation, 161 subdifferential, 119, 245 18 quantum Liouville equation, 57 subgradients, 262 magnetic quantum number, 25 quantum nondemolition sufficient optimality magnetic resonance imaging, 19 principle, 161 conditions, 87 mixed quantum state, 28 quantum numbers, 24 symmetric or antisymmetric momentum-stress tensor, 52 quasi-Newton methods, 228 wavefunction, 38 monotonic scheme, 232 symplectic matrix, 183 multigrid optimization radial quantum number, 24 (MGOPT) method, 277 radiofrequency pulse or time reversibility, 182 multiparticle quantum system, control, 19, 37 time-dependent density 37 reachable set, 163, 293 functional theory multiple controls, 164 real-linear, 133 (TDDFT), 50 reduced cost functional, 84 time-dependent Kohn–Sham necessary optimality reduced generalized Jacobian, (TDKS) model, 56 conditions, 87, 119 265 two-level system, 167 nonlinear complementarity reduced gradient, 84 problem (NCP) reduced Hessian, 248 uncertainty principle, 16 functions, 259 reduced Planck’s constant, 8 unitary orbit, 171 nonlinear conjugate gradient rotating-frame transformation, unitary propagator, 182 (NCG) method, 224 33 nonlinear multigrid methods, van Leeuwen, theorem of, 55 276 Schrödinger equation (SE), 9, vector potential, 20 nuclear magnetic resonance 12, 14 (NMR), 19 Schur reduction, 257 wave packet, 9 nuclear magneton, 18 semiclassical limit, 61 weak sequential continuity, 80 open quantum system, 65 Sherman–Morrison–Woodbury Wigner function, 57, 58 operator controllability, 164 formula, 228 Wirtinger calculus, 253 optimality condition, 86 shooting-type method, 299 Wolfe condition, 224, 231 optimality system, 93, 109 Slater determinant, 38, 46 optimize-before-discretize, 236, Slater permanent, 38 Young’s inequality, 132 253 spherical Bessel functions, 349 orbital angular momentum, 17 spherical harmonics, 23, 346, Zeeman effect, 18, 25 350 Zowe–Kurcyusz constraint parity operator, 38 spin, 17 qualification, 90.
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