Symbols G Greenfunction Go(r,:r,)= free Green function group generator of aa continuos transformation or of a h=6.626069×10-34-s Planck constant h=h/2m Ih) state of horizontal polarization H Hilbert space HA Hilbert space of the apparatus Hilbert space of the system A Hamiltonian operator Ho Ar field Hamiltonian H interaction Hamiltonian interaction Hamiltonian in Dirac picture Hic Javnes-Cummings Hamiltonian planar part of the Hamiltonian Ha()=(-1)"esfe-n-th Hermite polynomial,for all n0 imaginary unity Cartesian verso initial state vector intensity (of radiation) moment of inertia identity operator )=罗 imaginary part of a complex number z Jy Cartesian versor arbitrary ket,element of a continuous or discrete basis j.m) eigenket of density of the probability h incidental current density reflected current density transmitted current density total angular momentum jump superoperator k.k wave vector Cartesian versor Boltzmann constant basis
xxiii Symbols G Green function G0(r , t ; r, t) = −ı m 2πıh¯(t −t) 3 2 e ım|r −r| 2 2h¯(t−t) free Green function G group Gˆ generator of a a continuos transformation or of a group h = 6.626069 × 10−34J · s Planck constant h¯ = h/2π | h state of horizontal polarization H Hilbert space HA Hilbert space of the apparatus HS Hilbert space of the system Hˆ Hamiltonian operator Hˆ0 unperturbed Hamiltonian Hˆ A Hamiltonian of a free atom HˆF field Hamiltonian HˆI interaction Hamiltonian Hˆ I I interaction Hamiltonian in Dirac picture Hˆ JC Jaynes–Cummings Hamiltonian Hˆr planar part of the Hamiltonian Hn(ζ ) = (−1) n eζ 2 dn dζ n e−ζ 2 n-th Hermite polynomial, for all n = 0 ı imaginary unity ı x Cartesian versor |i initial state vector I intensity (of radiation) I moment of inertia ˆI identity operator (z) = z−z∗ 2ı imaginary part of a complex number z j y Cartesian versor jˆ = Jˆ /h¯ = (jˆx , jˆy , jˆz) | j arbitrary ket, element of a continuous or discrete basis | j, m eigenket of Jˆ z J density of the probability current JI incidental current density JR reflected current density JT transmitted current density Jˆ = Lˆ + Sˆ = (Jˆ x , Jˆ y , Jˆ z) total angular momentum ˆ Jˆ jump superoperator k, k wave vector k z Cartesian versor kB Boltzmann constant | k generic ket, element of a continuous or discrete basis
XXiv Symbols arbitrary length i=(i,iv.1.)L/h raising and lowering operators for the levels of the angular momentum 10 generic ket t.m eigenket of l Lq1,,9m1,,9m） classical Lagrangian function Lagrangian multiplier operator L=(Lx,Ly,L:) orbital angular momentum Lindblad superoperator me mn mass of the nucleus mp mass of the proton m magnetic quantum number m: eigenvalue ofj spin magnetic quantum number or secondary spin quantum number generic ket,eigenket of the energy measure of purity M meter arbitrary matrix direction vector eigenvector of the harmonic oscillator Hamiltonian or of the number operator N number of elements of a given set normalization constant number operator Nk=atdk 0万 j-th eigenvalue of the observable lo) eigenket of the observable O j-th eigenket of the observable o generic operators.generic observables observable in the Heisenberg picture 6 observable in the Dirac picture os observable in the Schrodinger picture apparatus'pointer 8 observable of the obiect system non-demolition observable super-ket (or S-ket) classical generalized momentum component =(px.py.pz) three-dimensional momentum operator one-dimensional momentum operator
xxiv Symbols l arbitrary length ˆl = (ˆlx , ˆly , ˆlz) = Lˆ /h¯ ˆl± = ˆlx ± ıˆly raising and lowering operators for the levels of the angular momentum |l generic ket l, ml eigenket of ˆlz L(q1, ... , qn; q˙1, ... , q˙n) classical Lagrangian function Lˆ Lagrangian multiplier operator Lˆ = (Lx , Ly , Lz) orbital angular momentum ˆ Lˆ Lindblad superoperator m mass of a particle me mass of the electron mn mass of the nucleus m p mass of the proton ml magnetic quantum number m j eigenvalue of jˆz ms spin magnetic quantum number or secondary spin quantum number |m generic ket, eigenket of the energy M measure of purity M meter Mˆ arbitrary matrix n, n direction vectors |n eigenvector of the harmonic oscillator Hamiltonian or of the number operator N number of elements of a given set N normalization constant Nˆ = ˆa† aˆ number operator Nˆk = ˆa† kaˆk oj j-th eigenvalue of the observable Oˆ |o eigenket of the observable Oˆ oj j-th eigenket of the observable Oˆ Oˆ, Oˆ , Oˆ generic operators, generic observables Oˆ H observable in the Heisenberg picture OˆI observable in the Dirac picture Oˆ S observable in the Schrödinger picture OˆA apparatus’ pointer OˆS observable of the object system OˆND non-demolition observable Oˆ super-ket (or S-ket) pk classical generalized momentum component pˆ = (pˆx , pˆ y , pˆz) three-dimensional momentum operator pˆx one-dimensional momentum operator
Symbols time derivative ofp 序，=-1h}是r radial part of the momentum operator P(a.a*) P-function projection onto the state path predictability path predicability operator jor(j) probability of the eventj (D)=(D]Hg) probability density function that the particular set D of data is observed when the system is actually in state k p(但H)=Tr(EH）】 conditional probability that one chooses the hypothesis Hi when He is true classical generalized position component charge density Qa,a*=是(alla Q-function Q quantum algebra field of rational numbers r.r rk k-th eigenvalue of a density matrix Bohr's radius =(优，，) three-dimensional position operator R reflection coefficient R(r) radial part of the eigenfunctions ofin speherical coordinates RR' reference frames R reservoir R field of real numbers real part of a complex quantity z RR(B.中.) rotation operator,generator of rotations Ro resolvent of the operator =∑1ACjk risk operator for the j-th hypothesis IR) initial state of the reservoir spin quantum number 8=(6x,y,)=S/h spin vector operator 社=x士1y raising and lowering spin operators action generic quantum system entropy s=(x,5y,) spin observable time time operato eigenket of the time operator
xxv Symbols ˆ p˙x time derivative of pˆx pˆr = −ıh¯ 1 r ∂ ∂r r radial part of the momentum operator P(α, α∗) P-function Pˆj projection onto the state | j or b j P path predictability Pˆ path predicability operator ℘j or ℘(j) probability of the event j ℘k (D) = ℘ (D|Hk ) probability density function that the particular set D of data is observed when the system is actually in state k ℘ Hj|Hk = Tr ρˆk EˆHj conditional probability that one chooses the hypothesis Hj when Hk is true qk classical generalized position component Q charge density Q(α, α∗) = 1 π α| ˆρ|α Q-function Q quantum algebra Q( field of rational numbers r spherical coordinate r · r scalar product between vectors r and r rk k-th eigenvalue of a density matrix r0 = h¯ 2 me2 Bohr’s radius rˆ = (xˆ, yˆ,zˆ) three-dimensional position operator R reflection coefficient R(r) radial part of the eigenfunctions of ˆlz in speherical coordinates R, R reference frames R reservoir IR field of real numbers (z) = z+z∗ 2 real part of a complex quantity z Rˆ, Rˆ (β, φ, θ) rotation operator, generator of rotations Rˆ Oˆ resolvent of the operator Oˆ Rˆ j = N k=1 ℘A k Cjkρˆk risk operator for the j-th hypothesis | R initial state of the reservoir s spin quantum number sˆ = (sˆx ,sˆy ,sˆz) = Sˆ/h¯ spin vector operator sˆ± = ˆsx ± ısˆy raising and lowering spin operators S action S generic quantum system S entropy Sˆ = (Sˆx , Sˆy , Sˆz) spin observable t time t ˆ time operator |t eigenket of the time operator
xxvi Symbols 7 transmission coefficient temperature,classical kinetic energy kinetic energy operator time reversal operator generic transformation (v.T) energy density uo=音r k-th mode function of the electromagnetic field scalar potential unitary operator UBs beam splitting unitary operator UPBS polarization beam-splitting unitary operator unitary controlled-not operator Uf Boolean unitary transformation UE Fourier unitary transformation unitary Hadamard operato 0p(w) unitary momentum translation Up permutation operator 0R(p) rotation operator space-reflection operator 0 time translation unitary operator Ux(a) one-dimensional space translation unitary operator Ur(a) three-dimensional space translation unitary operator unitary rotation operator Ve unitary phase operator OSA=ekffisau) unitary operator coupling system and apparatus for time interval OsAe=e一isAt unitary operator which couples the environment Eto the system and apparatus S+Aat time antiunitary operato time reversal generic transformation that can be either unitary or antiunitary 1) state of vertical polarization element of a discrete basis potential energy scalar potential of the electromagnetic field V)=Ψ centrifugal-barrier potential energy classical potential energy potential energy operator
xxvi Symbols T transmission coefficient T temperature, classical kinetic energy Tˆ kinetic energy operator Tˆ time reversal operator ˆ Tˆ , T generic transformation u(ν, T ) energy density uk(r) = e L 3 2 eık·r k-th mode function of the electromagnetic field U scalar potential Uˆ unitary operator UˆBS beam splitting unitary operator UˆPBS polarization beam-splitting unitary operator UˆCNOT unitary controlled-not operator Uˆ f Boolean unitary transformation Uˆ F Fourier unitary transformation Uˆ H unitary Hadamard operator Uˆp(v) unitary momentum translation Uˆ P permutation operator UˆR(φ) rotation operator UˆR space-reflection operator Uˆt time translation unitary operator Uˆ x (a) one-dimensional space translation unitary operator Uˆr(a) three-dimensional space translation unitary operator Uˆθ unitary rotation operator Uˆφ unitary phase operator Uˆ SA τ = e − ı h¯ τ 0 dt Hˆ SA(t) unitary operator coupling system and apparatus for time interval τ Uˆ SA,E t = e− ı h¯ t HˆSA,E unitary operator which couples the environment E to the system and apparatus S + A at time t ˜ Uˆ antiunitary operator ˜ UˆT time reversal Uˆ generic transformation that can be either unitary or antiunitary |v state of vertical polarization |vn element of a discrete basis V potential energy Ve scalar potential of the electromagnetic field Vc(r) = h¯ 2l(l+1) 2mr 2 centrifugal-barrier potential energy Vc classical potential energy Vˆ potential energy operator
xxvii Symbols volume generic vector visibility of interference,generic vectorial space visibility of interference operator k-thprobability weight Iwn） element of a discrete basis vector W(a.c*)三 Wigner function ∫dae-o*+axw.n first Cartesian axis,coordinate 1x) eigenket of one-dimensional position operator time derivative of =2(t+) quadrature ,=方-创 quadrature set second Cartesian axis,coordinate Yim(.) spherical harmonics third Cartesian axis,coordinate atomic number Z(B)=T(e-BA） partition function parameter space field of integer numbers Greek letters a angle,(complex)number 1o=ey∑会m coherent state B variable =(kBT) ) coherent state 21 damping constant Euler gamma function phase space i reservoir operator Kronecker 4=2=品+票+ Laplacian △b uncertainty in the state) small quantity Levi-Civita tensor coupling consta nt o=()）del vacuum Rabi frequency
xxvii Symbols V volume V generic vector V visibility of interference, generic vectorial space Vˆ visibility of interference operator wk k-th probability weight |wn element of a discrete basis vector W(α, α∗) = 1 π2 d2αe−ηα∗+η∗αχW (η, η∗) Wigner function x first Cartesian axis, coordinate | x eigenket of xˆ xˆ one-dimensional position operator ˆ x˙ time derivative of xˆ Xˆ 1 = √ 1 2 aˆ † + ˆa quadrature Xˆ 2 = √ı 2 aˆ † − ˆa quadrature X set y second Cartesian axis, coordinate Ylm(θ, φ) spherical harmonics z third Cartesian axis, coordinate Z atomic number Z(β) = Tr e−β Hˆ partition function Z parameter space ZZ field of integer numbers Greek letters α angle, (complex) number |α = e− |α| 2 2 ∞ n=0 αn √n! |n coherent state β angle, (complex) number, thermodynamic variable = (kBT ) −1 |β coherent state γ damping constant Euler gamma function phase space ˆ k reservoir operator δ jk Kronecker symbol δ(x) Dirac delta function = ∇2 = ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 Laplacian ψ uncertainty in the state |ψ small quantity jkn Levi–Civita tensor ε coupling constant ε0 = ω 20h¯l 3 1 2 |d · e| vacuum Rabi frequency