Figures 6.13 Graphical representation of the distribution of eigenvalues of the component of the angular momenta of two independent particles 228 6.14 Angle observable and step function 241 7.1 Interferometric example of indistinguishability 1223 Example of counting the number of possible configurations of bosons 25 Potential wells of a natural atom and of a quantum dot .1 Passive and active transformations 260 Representation of a measurement on the sphere of density matrices Two ways of tuning the coupling function 9.3 Decohering histories 290 9.4 Schrodinger cat 292 95 Experin ental realiz on of a Schrodinger cat with a trapped ion SOUID 9.7 Wigner function of an entangled state 296 9.8 Schematic representation of the experiment proposed by Scully and co-workers 301 9.9 Schematic representation of Mandel's experiment 309 9.10 Interference and visibility in Mandel's experiment 310 9.11 Scully.Englert.and Walther's proposed experiment 312 9.12 Interaction-free measurement 316 9.13 Repeated interaction-free measurements 9.14 Probability of success in repeated interaction-free measurements 318 9.15 Interaction-free mea ent with 9.16 Schematic representation of Mandel's experiment on empty waves 9.17 Depiction of Wheeler's experiment 320 9.18 Interferometry experiment for testing delaved-choice 321 919 Optical version of the Zeno effect 9.20 Example of POVMs 9.21 Plot of the estimate of the wave function 349 9.22 Another example of POVM 352 10.1 Stark effect 10.2 WKB approximation:forbidden regions outside a potential well 10.3 WKB and potential well 382 18 WKB and potential barrie The different paths 10.6 The analogy of path integral integration 391 10.7 The sum over paths 392 10.8 Two possible paths fromi to f both passing through the same central point c 394 10.9 Path integrals and scattering 397 111 Electron coordinates in the atomic system 406 11.2 Resulting potential in the hydrogen atom Grotrian scheme 114 Plot of the radial eigenfunctions of the hydrogenoid atom 412
xiii Figures 6.13 Graphical representation of the distribution of eigenvalues of the z component of the angular momenta of two independent particles 228 6.14 Angle observable and step function 241 7.1 Interferometric example of indistinguishability 246 7.2 Example of counting the number of possible configurations of bosons 253 7.3 Potential wells of a natural atom and of a quantum dot 257 8.1 Passive and active transformations 260 9.1 Representation of a measurement on the sphere of density matrices 280 9.2 Two ways of tuning the coupling function 283 9.3 Decohering histories 290 9.4 Schrödinger cat 292 9.5 Experimental realization of a Schrödinger cat with a trapped ion 293 9.6 SQUID 294 9.7 Wigner function of an entangled state 296 9.8 Schematic representation of the experiment proposed by Scully and co-workers 301 9.9 Schematic representation of Mandel’s experiment 309 9.10 Interference and visibility in Mandel’s experiment 310 9.11 Scully, Englert, and Walther’s proposed experiment 312 9.12 Interaction-free measurement 316 9.13 Repeated interaction-free measurements 317 9.14 Probability of success in repeated interaction-free measurements 318 9.15 Interaction-free measurement with two cavities 318 9.16 Schematic representation of Mandel’s experiment on empty waves 319 9.17 Depiction of Wheeler’s experiment 320 9.18 Interferometry experiment for testing delayed-choice 321 9.19 Optical version of the Zeno effect 324 9.20 Example of POVMs 334 9.21 Plot of the estimate of the wave function 349 9.22 Another example of POVM 352 10.1 Stark effect 364 10.2 WKB approximation: forbidden regions outside a potential well 381 10.3 WKB and potential well 382 10.4 WKB and potential barrier 383 10.5 The different paths 390 10.6 The analogy of path integral integration 391 10.7 The sum over paths 392 10.8 Two possible paths from i to f both passing through the same central point c 394 10.9 Path integrals and scattering 397 11.1 Electron coordinates in the atomic system 406 11.2 Resulting potential in the hydrogen atom 407 11.3 Grotrian scheme 411 11.4 Plot of the radial eigenfunctions of the hydrogenoid atom 412
xiv Figures 11.5 Plot of the radial probability densities 413 11.6 s-,p-,and d-states versus energy levels 414 11.7 Lande vectorial model for the Paschen-Bach effect s and p levels in presence of the Paschen-Bach effect 11.9 Paschen-Bach spectroscopical lines 419 1110 lande vectorial model for the zeeman effect 11.11 Energy levels for the Zeeman effect 11.12 Spectroscopical lines for the Zeeman effect 物 12.1 Spheroidal coordinates for the H ion 440 122 Molecular potential energy 446 12.3 Vibrational and rotational levels of two electronic states I and II in a diatomic molecule 447 12.4 Schematic diagram of the LCAo function f(E) 450 molecular ion H时molecule 15 13.1 The three directions of the electromagnetic field 460 13.2 Displacement operator for coherent states Phase-number certainty properties of cohe tates Phase convention for squeezed states 13.5 Generation of a squeezed state 473 13.6 Phase-space of amplitude-and phas squeezed states 4 13.7 Representation of the Q-function of coh erent,number,and squeezed states 475 13.8 Representation of the W-function of coherent,number.and squeezed states 13.9 Homodyne detection 483 13.10 Jaynes-Cummings energy levels 13.11 Rabi oscillations 13.12 Collapse and revival 13.13 Spontaneous and stimulated emission for a two-level atom 495 13.14 Spontaneous and stimulated emission for a three-level atom 13.15 Schematic diagram of a laser 13.16 Parametric down conversion 496 13.17 Magnetic and electric ab effects 497 13.18 Parallel transpor 14.1 Bloch-sphere representation of states 14.2 Techniques for integrating a function 531 14.3 B model for dissipation 144 Interference fringes and their sensitivity to losses in the Yurke-Stoler mode 14.5 Pictorial representation of a coherent state and separation between the two components 537 Haroche's experiment Interference fringes in Haroche's experiment 15.1 Tomographic method for reconstructing the W-function 15.2 Tomographic measurements of the state 560 16.1 Overview of the EPR-Bohm experimen 574
xiv Figures 11.5 Plot of the radial probability densities 413 11.6 s-, p-, and d-states versus energy levels 414 11.7 Landé vectorial model for the Paschen-Bach effect 417 11.8 s and p levels in presence of the Paschen-Bach effect 418 11.9 Paschen-Bach spectroscopical lines 419 11.10 Landé vectorial model for the Zeeman effect 420 11.11 Energy levels for the Zeeman effect 422 11.12 Spectroscopical lines for the Zeeman effect 422 12.1 Spheroidal coordinates for the H+ 2 ion 440 12.2 Molecular potential energy 446 12.3 Vibrational and rotational levels of two electronic states I and I I in a diatomic molecule 447 12.4 Schematic diagram of the LCAO function f (E) 450 12.5 LCAO energy solutions for the H+ 2 molecular ion 451 12.6 Symmetric and antisymmetric states of the ground level of the H+ 2 molecule 452 13.1 The three directions of the electromagnetic field 460 13.2 Displacement operator for coherent states 469 13.3 Phase-number uncertainty properties of coherent states 471 13.4 Phase convention for squeezed states 472 13.5 Generation of a squeezed state 473 13.6 Phase-space of amplitude- and phase-squeezed states 473 13.7 Representation of the Q-function of coherent, number, and squeezed states 475 13.8 Representation of the W-function of coherent, number, and squeezed states 480 13.9 Homodyne detection 483 13.10 Jaynes–Cummings energy levels 490 13.11 Rabi oscillations 491 13.12 Collapse and revival 492 13.13 Spontaneous and stimulated emission for a two-level atom 495 13.14 Spontaneous and stimulated emission for a three-level atom 495 13.15 Schematic diagram of a laser 495 13.16 Parametric down conversion 496 13.17 Magnetic and electric AB effects 497 13.18 Parallel transport 501 14.1 Bloch-sphere representation of states 527 14.2 Techniques for integrating a function 531 14.3 BS model for dissipation 534 14.4 Interference fringes and their sensitivity to losses in the Yurke–Stoler model 536 14.5 Pictorial representation of a coherent state and separation between the two components 537 14.6 Haroche’s experiment 538 14.7 Interference fringes in Haroche’s experiment 540 15.1 Tomographic method for reconstructing the W-function 559 15.2 Tomographic measurements of the state 560 16.1 Overview of the EPR – Bohm experiment 574
Figures 16.2 Preparation of a singlet state 576 16.3 Particle trajectories for two Gaussian slit systems after Bohm's model,and the corresponding quantum potential 580 16.4 Trajectories for a potential barrier(E=V/2)after Bohm's model,and the corresponding quantum potential 16.5 The three-dimensional Hilbert space proposed by Bell 16.6 Scheme of the experiment proposed for proving the second Bell theorem 589 Experiment proposed by CHSH Optimal orientation for a,a,b,and b'for testing the CHSH inequality 16.9 Typical dependence of f()upon ne for cases I-III 595 16.10 Partial Grotrian diagram of atomic calcium for Freedman and Clauser's experiment 599 16.11 Schematic diagram of apparatus and associated electronics of the experiment by Freedman and Clauser 599 16.12 Freedman-Clauser experiment and Aspect and co- -workers'experiment 600 16.13 Alley-Shih and Ou-Mandel's experiment 602 16.14 Measured coincidence counting rate as a function of the polarizer angle 01. with 62 fixed at 45 % 16.15 Experimental set-up in order to solve detection loopholes 16.16 "Entanglement"with vacuum 606 16.17 Yurke and Stoler's experimen 6 16.18 611 16.19 ent sw apping 612 16.20 Orientations for the proof of Stapp's theorem 614 16.21 The GHSZ proposed experiment 616 16.22 Conditional entanglement 619 16,23 Necessary criterion for separability 623 17.1 Information difference in bits versus angle 6 for the information-theoretic Bell inequality 635 17.2 Informational distance by quadrilateral inequality Schematic representation of quantum non-separability 17.4 Diagram for entangled and disentangled states 17.5 Representation of all density matrices De compression of information Teleportation 17.8 Realization of teleportation with photons 17.9 The CNOT gate 652 17.10 Implementation of a CNOT gate by means of a polarization interferometer 17.11 The quantum computation device as an equivalent of a Mach-Zender interferometer 653 17.12 Generation of Bell states by means of a Hadamard gate followed by a CNOT gate 17.13 Preparation of a GHSZ state 17.14 Toffoli gate 657
xv Figures 16.2 Preparation of a singlet state 576 16.3 Particle trajectories for two Gaussian slit systems after Bohm’s model, and the corresponding quantum potential 580 16.4 Trajectories for a potential barrier (E = V/2) after Bohm’s model, and the corresponding quantum potential 581 16.5 The three-dimensional Hilbert space proposed by Bell 585 16.6 Scheme of the experiment proposed for proving the second Bell theorem 589 16.7 Experiment proposed by CHSH 590 16.8 Optimal orientation for a, a , b, and b for testing the CHSH inequality 592 16.9 Typical dependence of f (θ) upon nθ for cases I-III 595 16.10 Partial Grotrian diagram of atomic calcium for Freedman and Clauser’s experiment 599 16.11 Schematic diagram of apparatus and associated electronics of the experiment by Freedman and Clauser 599 16.12 Freedman–Clauser experiment and Aspect and co-workers’ experiment 600 16.13 Alley-Shih and Ou-Mandel’s experiment 602 16.14 Measured coincidence counting rate as a function of the polarizer angle θ1, with θ2 fixed at 45◦ 604 16.15 Experimental set-up in order to solve detection loopholes 605 16.16 “Entanglement” with vacuum 606 16.17 Yurke and Stoler’s experiment 608 16.18 Entanglement swapping 611 16.19 Variation of entanglement swapping 612 16.20 Orientations for the proof of Stapp’s theorem 614 16.21 The GHSZ proposed experiment 616 16.22 Conditional entanglement 619 16.23 Necessary criterion for separability 623 17.1 Information difference in bits versus angle θ for the information-theoretic Bell inequality 635 17.2 Informational distance by quadrilateral inequality 635 17.3 Schematic representation of quantum non-separability 636 17.4 Diagram for entangled and disentangled states 637 17.5 Representation of all density matrices 638 17.6 Decompression of information 639 17.7 Teleportation 644 17.8 Realization of teleportation with photons 646 17.9 The CNOT gate 652 17.10 Implementation of a CNOT gate by means of a polarization interferometer 653 17.11 The quantum computation device as an equivalent of a Mach–Zender interferometer 653 17.12 Generation of Bell states by means of a Hadamard gate followed by a CNOT gate 655 17.13 Preparation of a GHSZ state 655 17.14 Toffoli gate 657
XVI Figures 17.15 Representation of computational complexity 660 17.16 Boolean transformation of an initial bit Device for solving Deutsch's problem 17.18 Device for solving Deutsch's problem for n+1 input states 17.19 Device for solving Deutsch's problem for n+m input states 17.20 Representation of Shor's theorem 17.21 Implementation of Grover's algorithm 17.22 Computational steps in Grover's algorithm 17.23 Classical error correction 667 17.24 Quantum circuit for error correction 669 17.25 Environmental wave functions and their overlapping as a function of time in quantum computation 670
xvi Figures 17.15 Representation of computational complexity 660 17.16 Boolean transformation of an initial bit 660 17.17 Device for solving Deutsch’s problem 661 17.18 Device for solving Deutsch’s problem for n + 1 input states 662 17.19 Device for solving Deutsch’s problem for n + m input states 662 17.20 Representation of Shor’s theorem 664 17.21 Implementation of Grover’s algorithm 666 17.22 Computational steps in Grover’s algorithm 666 17.23 Classical error correction 667 17.24 Quantum circuit for error correction 669 17.25 Environmental wave functions and their overlapping as a function of time in quantum computation 670
Tables 2.1 Different cases and ways of expressing the basic quantum formalism page 71 Eigenstates and eigenvalues of Eigenvalues,for the three-dimensional harmonic oscillator 6.3 Values of jand m and the corresponding number of possible states 233 64 Clebsch-Gordan coefficients fori==1/2 Clebsch-Gordan coefficients for j=1 and j2=1/2 66 Clebsch-Gordan coefficients for j=j2=1 7 Fermions and bosons 12 Fermionic distributions 252 7.3 Bo sonic distributions 11.1 Ground-state energy of helioid atoms 13.1 Electromagnetic spectrum 17.1 Sequence transmission 23 Toffoli truth table Fredkin truth table 8 17.4 Classical error correction 668
Tables 2.1 Different cases and ways of expressing the basic quantum formalism page 71 6.1 Eigenstates and eigenvalues of ˆlz 199 6.2 Eigenvalues, for the three-dimensional harmonic oscillator 215 6.3 Values of j and m and the corresponding number of possible states 233 6.4 Clebsch–Gordan coefficients for j1 = j2 = 1/2 238 6.5 Clebsch–Gordan coefficients for j1 = 1 and j2 = 1/2 238 6.6 Clebsch–Gordan coefficients for j1 = j2 = 1 238 7.1 Fermions and bosons 250 7.2 Fermionic distributions 252 7.3 Bosonic distributions 254 11.1 Ground-state energy of helioid atoms 429 13.1 Electromagnetic spectrum 456 17.1 Sequence transmission 648 17.2 Toffoli truth table 658 17.3 Fredkin truth table 658 17.4 Classical error correction 668