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Introduction to Quantum Optics: an Amateur's View Lecture Notes Introduction to Quantum Optics: an amateur's view Lecture notes M.I. Petrov, D.F. Kornovan, I.V. Toftul ITMO University, Department of Physics and Mathematics Autumn, 2019 Preface I am very grateful to Andrey Bogdanov who helped me to organize this course, and to Kristina Frizyuk for her enormous help in preparing this manuscript. 1 Contents Recommended literature 5 1 Atom-field interaction. Semiclassical theory 6 Homework . 10 2 Density matrix of two energy level system 12 2.1 Density matrix of a subsystem . 13 2.2 Density matrix of a mixed state . 14 2.3 Density matrix of a two-level system . 15 2.4 Bloch sphere . 16 2.5 Dissipations . 17 2.5.1 Spontaneous emission of TLS . 17 2.6 Dielectric constant of media . 18 2.7 Homework . 20 Homework . 20 3 Secondary quantization 21 3.1 Vector potential of the electromagnetic field . 21 3.2 Field in the box, harmonics expansion, and the energy of the electromag- netic field . 22 3.3 Field quantization . 24 3.4 Ladder operators. Fock state. Second quantization . 25 3.5 Fields' fluctuation . 26 3.6 Homework . 27 4 Coherent states 28 4.1 Eigenstates of anihilation operator . 28 4.2 Basic properties of coherent states . 29 Homework . 30 4.3 Classical field . 30 Homework . 31 4.4 Fluctuations . 31 4.5 Squeezed states or getting the maximum accuracy! . 34 Homework . 35 5 The coherence of light 37 5.1 Michelson stellar interferometer . 37 5.2 Quantum theory of photodetection . 40 6 Atom–field interaction. Quantum approach 42 6.1 Jaynes{Cummings model (RWA) . 42 Homework . 46 6.2 Collapse and revival . 46 6.3 Energy spectrum. Dispersion relation . 46 7 Spontaneous relaxation. Weisskopf-Wigner theory 51 2 8 Dipole radiation. Dyadic Green's function. The Purcell effect: classical approach 55 8.1 Dipole radiation and dyadic Green's function . 55 8.1.1 Derivation of the Green's function for Maxwell equations . 56 8.1.2 Near-, intermediate- and far-field parts of Green's function . 57 8.2 Spontaneous relaxation and local density-of-state (IN A MIXED UNITS) . 57 8.2.1 An expression for spontaneous decay . 57 8.2.2 Spontaneous decay and Green's dyadics . 59 8.3 The Purcell factor . 60 Homework . 61 9 Theory of relaxation of electromagnetic filed. Heisenberg{Langevin method 62 9.1 In previous series . 62 9.2 the Heisenberg{Langevin equation . 62 Homework . 64 9.3 Properties of the stochastic operator . 64 9.4 Equation of motion for the field correlation functions. Wiener{Khintchine theorem . 65 10 Atom in a damped cavity 68 10.1 The Purcell factor for a closed cavity . 68 10.2 Rigorous derivation of the atomic decay . 69 11 Casimir force and his close friends 72 11.1 Casimir force between two perfectly conducting plates . 74 11.1.1 Case D =3 ............................... 74 11.1.2 Case D = 1 and philosophy about divergent sums . 76 11.2 Casimir{Polder force . 77 11.3 Orders of forces . 79 11.4 The latest advances . 80 11.4.1 The dynamical Casimir effect . 80 11.4.2 Quantum levitation or repulsive Casimir{Lifshitz forces . 80 3 Notation is the energy of the system •E Eb(r; t), and Hb (r; t) are the calssical electric and magnetic field vectors • Eb(r; t), and Hb (r; t) are the quantum electric and magnetic field operators • Hcis the quantum hamiltonian of the system • By !0 we normally denote the atomic transition frequency, while the frequency of • the field we denote as !; TLS =def Two-Level System • 4 Recommended literature 1. Scully, M. O., & Zubairy, M. S. (1999). Quantum optics. 2. Novotny, L., & Hecht, B. (2012). Principles of nano-optics. Cambridge university press. 3. Mandel, L., & Wolf, E. (1995). Optical coherence and quantum optics. Cambridge university press. 4. Fox, M. (2006). Quantum optics: an introduction (Vol. 15). OUP Oxford. 5. Loudon, R., & von Foerster, T. (1974). The quantum theory of light. American Journal of Physics, 42(11), 1041-1042. 5 1 Atom-field interaction. Semiclassical theory We start with consideration of a basic problem in our course: the interaction of elec- tromagnetic field with quantum system. In the following we will refer to such system as an "atom". The word semiclassical in this context means that we treat electromagnetic field classical, but describe atom as a quantum system. We start with a two-level system (see Fig. 1) as a simplest but very rich example of light-matter interaction. E k ω L |a> |b r0 > r Figure 1: The energy system of two-level atom. We assume that the electron in the system is initially in the ground state, and at time moment t = 0 it is excited with a plane wave with polarization E and wavevector k = !=c. Let us find the probability of atom to be in the excited state at time moment t. In the suggested formulation the problem is a typical example from quantum mechanics course. One should start with writing down the Hamiltonian Hc0 of an electron in the system without electric field : 2 pb Hc0 = + Vb(r): (1.1) 2m Here pb is the momentum operator, and Vb is the potential energy of the electron. One can find the eigen states and energy levels of the atom: Hc0 = : (1.2) j i E j i In the following we will assume that there are two eigen states, which we define as a j i and b (ground state) with energies Ea and Eb correspondingly. Their difference gives j i the energy of atomic transition ~!0 = a b. The Hamiltonian of the system after introducing the electromagnetic field is asE follows:− E (p^ e A)2 Hc= − c + V^ (r) e': (1.3) 2m − NB: Potentials are not determined uniquely, and gauge transformation may take place: c c @χ A A + ~ χ, ' ' ~ ; (1.4) ! e r ! − e @t wehre χ(r; t) is a real-valued function of coordinate and time. Then the wave function should also be transformed. eiχ(r;t): (1.5) ! 6 Next, we write down a vector potential describing the incident plane wave: ikr−i!t ikr A = A0e = A0(t)e : (1.6) Assuming that characteristic scale of the system L is much smaller than the wavelength L λ (1.7) one can expanding A near r0 (the radius-vector of atom center) in Taylor series: ikr0 ikr0 A(r; t) A0(t)e (1 + ikρ) A0e : (1.8) ≈ ≈ So the vector potential does not change in space, but in time. By choosing the coor- dinate so that r0 = 0 one can simplify the system even further A A0(t): ≈ 2 ^ 1 e H = p^ A0(t) + V (r) + e': (1.9) 2m − c The standard choice it to use Coulomb gauge: div A = 0;' = 0: (1.10) After that, using p^ = i~ , we obtain − r 2 2 ^ ~ ie H = A0 + V (r): (1.11) −2m r − ~c Our goal is to find out the temporal evolution of the system. This is can be done by solving the Schr¨odingerequation: @ Hc = i ~ @t First of all, we simplify the Hamiltonian by introducing a new wave function e = e −i A0r y e c~ . This substitution and related unitary transformation Hc u Hcu will make | {z } ! ,!=u an momentum shift p^ e=cA p^. Indeed, if one recalls that − ! ( ig)( ig) eeigr = = ( ig) e eigr = 2 e eigr; (1.12) r − r − r − r r def e where g = A0(t), the Schr¨oedingerequation will give us c~ 2 @ ~ 2 @ e @g H^ = i~ + V e(r; t) = i~ ~r e: (1.13) @t ) −2mr @t − · @t | {z } H^0 Lets pay attention to the second term in right and side @g e @A (t) e = 0 = E(t): (1.14) @t ~c @t −~ Using that we can rewrite (1.13) as ^ @ e H0 e erE e = i~ : (1.15) − | {z } @t atom-field interaction 7 The second component in the lefthand side is the interaction term, which appeared after transformation of the Hamiltonian. By its form one can treat is as a dipole energy in the ^ def def electric field, and we will define H1 = d E, and d = er is the dipole moment. This, finally, leads us to a simplified form of− the· Schr¨odingerequation− (we omit the tilde sign here to avoid additional idle symbols): ~ ~ @ Hc0 + Hc1 = i : (1.16) ~ @t In order to solve this equation one can apply the expansion of over the eigenstates of non-perturbed system: j i ~E = Ca(t) a + Cb(t) b ; (1.17) j i j i where ^ ^ H0 a = a a ; H0 b = b b : (1.18) j i E j i j i E j i NB: Here we switch to "bra" and "ket" respresentation. We recall that a b = ~!0 is the transition frequency. We assume that the incident E − E ^ field has following time dependence E = E0 cos !t, so H1 = d E. Let us assume that initially the system is in the ground state in accordance with− the· formulation of the problem: E C (0) = 0; ~ = b a (1.19) t=0 j i ! Cb(0) = 1: Then one can rewrite the equation in the form: _ _ (EaCa a + EbCb b ) d E Ca a d E Cb b = i~Ca a + i~Cb b : (1.20) j i j i − · j i − · j i j i j i Projecting it over a and b one can get: j i j i 8 _ a (1.20) : EaCa Ca a d " a Cb a d " b = i~Ca; <>h j · − h j · j i − h j · j i (1.21) > _ : b (1.20) : EbCb Cb b d " b Ca b d " a = i~Cb: h j · − h j · j i − h j · j i In the dipole approximation the field E does not change in space, so we can write a d E a = a d a E; a d E b = a d b E: (1.22) h j · j i h j j i · h j · j i h j j i · This allows one to introduce dipole matrix elements: Z def ∗ dαβ = α d β = dV (r)er β(r) α; β = a; b: (1.23) h j j i α By symmetry considerations it follows dαα dαβ , since normally neighbouring j j j jα6=β states have opposite parity, and er is the odd function.
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