APPENDIX A. Lukas Novotny. University ofrochester. Rochester, NY Bert Hecht. Institute of Physical Chemistry

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1 OPTICS ON THE NANOMETER SCALE 1 : APPENDIX A Lukas Novotny Te Institute of Optics University ofrocester Rocester, NY Bert Hect Institute of Pysical Cemistry Swiss Federal Institute of Tecnology, ETH 8092 Zuric, Switzerland January 10, DO NOT DISTRIBUTE!!

2 Appendix A Semianalytical Derivation of te Atomic Polarizability Te purpose of tis section is to derive te linear polarizabilityof atwo-level quantum system in te dipole approximation. Te quantum system migt be an atom, a molecule, or a quantum dot. For simplicity, we will denote te system as atom. Once te atomic polarizability is known, te interaction between atom and radiation eld can be treated in many applications classically. A generally valid analytical expression for te polarizability cannot be derived. Instead, one as to distinguis between several approximate expressions wic depend on te relative spectral properties of atom and eld. Te two most important regimes are o-resonance and near-resonance excitation. In te former case, te atom resides mostly in its ground state wereas in te latter case, saturation of te excited level becomes signicant. According to quantum mecanics, te beavior of a system of N particles is described by te wavefunction (r;t) = (r 1 ;:::;r N ;t) ; (A.1) were r i denotes te spatial coordinate of particle i and t represents te time variable. To make te notation simpler, te entire set of particle coordinates is represented by te single coordinate r wic also includes spin. However, it sould be kept in mind tat operations on r are operations on all particle coordinates r 1 ;::;r 2. Te wavefunction is a solution of te Scrodinger equation ^H (r;t) = i d dt (r;t) : (A.2) ^H denotes te Hamilton operator, also called Hamiltonian. Its form depends on te considered system. For an isolated atom wit no external perturbation te Hamiltonian is time independent and it as te general form ^H o = X i;j, 2 2m i r 2 i + V (r i ; r j ) 1 i : (A.3)

3 2 APPENDIX A. ATOMIC POLARIZABILITY Te sum runs over all particles involved in te system. Te index in r i species operation on te coordinate r i. V (r i ; r j ) is te potential interaction energy of te it and jt particle. In general, V as contributions from all four fundamental interactions so far known, namely strong, electromagnetic, weak, and gravitational interaction. For te beaviour of electrons only te electromagnetic contribution is of importance, and witin te electromagnetic interaction te electrostatic potential is of dominance. Since te masses of nuclei are muc greater tan te mass of an electron, te nuclei move muc slower tan te electrons. Tis allows te electrons to practically instantaneously follow te nuclear motion. For an electron, te nucleus appears to be at rest. Tis is te essence of te Born-Oppeneimer approximation wic allows us to separate te nuclear wavefunction from te electronic one. We terefore consider a nucleus of total carge qz, Z being te atomic number. We assume tat te nucleus is located at te origin of coordinates (r = 0) surrounded by Z electrons eac of carge,q. We can restrict te index i in Eq. A.3 to run only over electron coordinates. In te case of a time independent Hamiltonian we can separate te t and r dependence as (r;t) = 1X n=1 e,i = E n t 'n (r) : (A.4) Inserting tis wavefunction into Eq. A.2 and using ^H = ^Ho we obtain te energy eigenvalue equation (time-independent Scrodinger equation) ^H o ' n (r) = E n ' n (r) ; (A.5) were E n are te energy eigenvalues of te stationary states jni. In te following we restrict ourselfs to te case of a two-level atom (n=[1; 2]) wit te two stationary wavefunctions 1(r;t) = e,i = E 1 t '1 (r) (A.6) 2(r;t) = e,i = E 2 t '2 (r) : In a next step, we expose te atomic system to te radiation eld. Te system ten experiences an external, time dependent perturbation represented by te interaction Hamiltonian ^H 0 (t). We obtain for te total Hamiltonian ^H = ^H o + ^H 0 (t) ; (A.7) were ^Ho represents te unperturbed system according to Eq. A.5. Te size of an atom is on te order of a couple of Bor radii, a B 0:05 nm. Since a B, being te wavelengt of te radiation eld, we can assume tat te electric eld E is constant across te dimensions of te atomic system. Assuming time-armonic elds, we can write E(r;t) = Re E(r)e,i!t E o cos(!t) ; (A.8) were we ave set te pase of te eld equal to zero or, equivalently, weave cosen te complex eld amplitude to be real. Eac electron in te system experiences te same eld strengt E o and te same time dependence cos(!t). Using te total electric dipole moment of te atom a (r) = a (r 1 ;:::;r Z ) = q ZX i=1 r i (A.9)

4 3 we nd for te interaction Hamiltonian in te dipole approximation ^H 0 =, a (r) E o cos(!t) : (A.10) Te dipolar interaction Hamiltonian is real and as odd parity, i.e. inversion operation r i =,r i is applied to all r i. ^H0 canges sign if te To solve te Scrodinger equation A.2 for te perturbed system we make a time dependent superposition of te stationary atomic wavefunctions in Eq. A.6 as (r;t) = c 1 (t) 1 (r;t)+c 2 (t) 2 (r;t) : (A.11) We coose te time dependent coecients c 1 and c 2 suc tat te normalization condition j i = R dv = jc 1 j 2 + jc 2 j 2 = 1 is fullled. For clarity, we will drop te arguments in te wavefunctions. After inserting tis wavefunction into Eq. A.2, rearranging terms and making use of Eq. A.3 and Eqs. A.6 we obtain ^H 0 (c c 2 2 ) = i [ 1 _c _c 2 ] ; (A.12) were te dots denote dierentiation wit respect to time. It sould be kept in mind, tat te arguments of and ' are (r; t) and (r), respectively. To eliminate te spatial dependence we multiply Eq. A.12 from te left by 1 on bot sides, introduce expressions A.6 for te wavefunctions and integrate over all space. After repeating te procedure wit instead 2 of 1 we obtain a set of two time-dependent coupled dierential equations _c 1 (t) = c 2 (t) i = 12 E o cos(!t)e,i = (E 2,E 1)t (A.13) _c 2 (t) = c 1 (t) i = 21 E o cos(!t)e +i = (E 2,E 1)t : (A.14) We ave introduced te denition of te dipole matrix element between te states jii and jji as ij = ij a jji = Z ' i (r) a (r) ' j (r) dv ; (A.15) It as to be empasized again tat te integration runs over all electron coordinates r = r 1 ;:::;r Z. In Eqs. A.13 and A.14 we ave used te fact tat p ii = 0. Tis follows from te odd parity of ^H0 wic makes te integrands of ii odd functions of r. Integration over r =[,1 :::0] leads to te opposite result tan integration over r =[0:::1]. Upon integration over all space te two contributions cancel. Te dipole matrix elements satisfy 12 = 21 because a is a Hermitian operator. However, it is convenient tocoose te pases of te eigenfunctions ' 1 and ' 2 suc tat te dipole matrix elements are real, i.e. 12 = 21 : (A.16) In te following, we will assume tat E = E 2,E 1 > 0, and we introduce te transition frequency! o = E= : (A.17) for te sake of simpler notation. Te state j1i is te ground state and te state j2i te excited state.

5 4 APPENDIX A. ATOMIC POLARIZABILITY Semiclassical teory does not account for spontaneous emission. Te spontaneous emission process can only be found by use of a quantized radiation eld wose Hamiltonian as to be included in Eq. A.7. To be in accordance wit quantum electrodynamics we ave to include te eects of spontaneous emission by introducing a penomenological damping term in Eq. A.14. Te coupled dierential equations ave ten te form _c 1 (t) = c 2 (t) i = 12 E o cos(!t)e,i!ot _c 2 (t) + / 2 c 2 (t) = c 1 (t) i = 21 E o cos(!t)e +i!ot : (A.18) Te introduction of te damping term asserts tat an excited atom must ultimately decay to its ground state by spontaneous emission. In te absence of te radiation eld, E o =0, Eq. A.18 can be integrated at once and we obtain c 2 (t) = c 2 (0) e, = 2 t : (A.19) Te average lifetime of te excited state is =1=, being te spontaneous decay rate. Since tere is no direct analytical solution of Eqs. A.18 we ave to nd approximate solutions for dierent types of excitations. Steady state polarizability for o-resonance excitation We assume tat te interaction between atom and radiation eld is weak. Te solution for c 1 (t) and c 2 (t) can ten be represented as a power series in 21 E o.toderive te rst order term in tis series we set c 1 (t)=1 and c 2 (t) = 0 on te rigt and side of Eqs. A.18. Once we ave found te rst order solution we can insert it again to te rigt and side to nd te sceond order solution and so on. However, we will restrict ourselves to te rst order term. Te solution for c 1 is c 1 (t) = 1 indicating tat te atom resides always in its ground state. Tis solution is te zero order solution, i.e. tere is no rst order solution for c 1. Te next iger term would be of second order. Te rst order solution for c 2 is obtained by a superposition of te omogeneous solution in Eq. A.19 and a particular solution. Te latter is easily found by writing te cosine term as a sum of two exponentials. We ten obtain for te rst order solution of c 2 c 2 (t) = 21 E o 1 2 e i(! o+!) t! o +!,i/ 2 + ei(!o,!) t! o,!,i/ 2 + c 2 (0)e, = 2 t : (A.20) We are interested in calculating te steady state beavior for wic te atom as been subjected to te electric eld E o cos(!t) for an innitely long period of time. In tis situation te inomogeneous term disappears and te solution is given by te omogeneous solution alone. Te expectation value of te dipole moment is dened as (t) = j a j i = Z (r) a (r) (r) dv ; (A.21) Te integration runs again over all coordinates r i. Using te wavefunction of Eq. A.11 te expression for becomes (t) = c 1 c 2 12 e,i!o t + c 1 c 2 21 e i!o t ; (A.22)

6 5 were we used te denition of te dipole matrix elements of Eq. A.15 and te property ii =0. Using te rst order solutions for c 1 and c 2 we obtain (t) = [ E o] e i! t e +,i! t e +,i! t e + i! t : (A.23) 2! o +!,i/ 2! o,!,i/ 2! o +!+i/ 2! o,!+i/ 2 Since te exciting electric eld is given as E=(1=2)E o [exp(i!t)+exp(,i!t)] we rewrite te dipole moment above as (t) = 1 2 were $ is te atomic polarizability $ (!)e i! t + $ (!)e,i! t i E o = Ren $(!)e,i! t $ (!) = tensor 1! o,!,i/ 2 + 1! o +!+i/ 2 o E o ; (A.24) : (A.25) denotes te matrix formed by te outer product between te (real) transition dipole moments. It is convenient to write te polarizability in terms of a single denominator. Furtermore, we realize tat te damping term is muc smaller tan! o wic allows us to drop terms in 2. Finally, weave to generalize te result to a system wit more tan two states. Besides te dierent matrix elements, eac state dierent from te ground state beaves in a similar way tan our previous state j2i. Tus, eac new level is caracterized by its natural frequency! n, its damping term n and te transition dipole moments 1n, n1. Ten, te polarizability takes on te form $ (!) = X n $ f n e 2 =m ;! n 2,! 2, i! n $ f n = 2 m! n e 2 1n n1 ; (A.26) were $ f n is te so-called oscillator strengt 1 and e and m denote te electron carge and mass, respectively. It is for istorical reasons tat we ave cast te polarizabilityinto te form of Eq. A.26. Before te advent of quantum mecanics, H. A. Lorentz developed a classical model for te atomic polarizability wic, besides te expression for $ f n, is identical wit our result. Te model considered by Lorentz consists of a collection of armonic oscillators for te electrons of an atom. Eac electron responds to te driving incident eld according to te equation of motion + _ +! 2 o = (q 2 =m) $ f E(t) : (A.27) In tis teory, te oscillator strengt is a tting parameter since tere is no direct way to know ow muc an electron contributes to a particular atomic mode. On te oter and, te semiclassical teory directly relates te oscillator strengt to te transition dipole matrix elements and tus to te atomic wavefunctions. Furtermore, te f-sum rule tells us tat te sum of all oscillator strengts is equal to one. 1 Te average over all polarizations reduces te oscillator strengt to a scalar quantity wit an extra factor of 1=3.

7 6 APPENDIX A. ATOMIC POLARIZABILITY If te energy! of te exciting eld is close to te energy dierence E between two atomic states, te rst term in Eq. A.25 is muc larger tan te second one. In tis case we can discard te second term (rotating wave approximation) and te imaginary part of te polarizability becomes a perfect Lorentzian function. It is important to notice tat tere is a linear relationsip between te exciting electric eld E and te induced dipole moment. Terefore, a monocromatic eld wit angular frequency! produces a armonically oscillating dipole wit te same frequency. Tis allows us to use te complex notation for and E and write = $ E ; (A.28) from wic we obtain te time dependence of E and by simply multiplying by exp(,i!t) and taking te real part. Near-resonance excitation in absence of damping In te previous section we required tat te interaction between excitation beam and te atom is weak and tat te atom resides mostly in its ground state. Tis condition can be relaxed if we consider an exciting eld wose energy! is close to te energy dierence E between two atomic states. As mentioned before, tere is no direct analytical solution to te coupled dierential equations in Eqs. A.18. However, a quite accurate solution can be found if we drop te damping term and if te energy of te radiation eld is close to te energy dierence between excited and ground state, i.e. j!,ej! +E: (A.29) In tis case, we can apply te so-called rotating wave approximation. After rewriting te cosines in Eqs. A.18 in terms of exponentials we nd exponents wit (! E). In te rotating wave approximation we only retain terms wit (!, E) because of teir dominating contributions. Eqs. A.18 ten become 2 were we introduced te Rabi frequency! R dened as i 2! R e,i(!o,!)t c 2 (t) = _c 1 (t) (A.30) i 2! R e i(!o,!)t c 1 (t) = _c 2 (t) ; (A.31)! R = j 12 E oj = j 21 E oj : (A.32)! R is a measure for te strengt of te time-varying external eld. Inserting te trial solution c 1 (t) = exp(it) into te rst equation A.30 we nd c 2 (t) =(2=! R ) exp(i[! o,! + ]t). Substituting bot c 1 and c 2 into te second equation A.31 we nd a quadratic equation for 2 We again coose te pases of te atomic wavefunctions suc tat te transition dipole matrix elements are real.

8 7 te unknown parameter leading to te two solutions 1 and 2. Te general solutions for te amplitudes c 1 and c 2 can ten be written as c 1 (t) = A e i1t + B e i2t (A.33) c 2 (t) = (2=! R )e i(!o,!)t A 1 e i1t + B 2 e i2t : (A.34) To determine te constants A and B we ave to require appropriate boundary conditions. Te probability for nding te atomic system in te excited state j2i is jc 2 j 2. Similarily, te probability for nding te atom in its ground state j1i is jc 1 j 2. By using te boundary conditions for te atom being initially in its ground state jc 1 (t=0)j 2 = 1 (A.35) jc 2 (t=0)j 2 = 0 ; te unknown constants A and B can be determined. Using te expressions for 1, 2, A, and B, we nally nd te solution c 1 (t) = e,i = 2 (! o,!)t cos(t=2), i(!,! o) sin(t=2) (A.36) c 2 (t) = i! R ei = 2 (! o,!)t sin(t=2) ; (A.37) were denotes te Rabi-opping frequency dened as q = (! o,!) 2 +! R 2 : (A.38) It can be easily sown tat jc 1 j 2 +jc 2 j 2 =1. Te probability for nding te atom in its excited state becomes sin 2 (t=2) jc 2 (t)j 2 =! R 2 : (A.39) 2 Te transition probability is a periodic function of time. Te system oscillates between te levels E 1 and E 2 at te frequency =2 wic depends on te detuning! o,! and te eld strengt represented by! R. If! R is small we ave (! o,!) and, in te absence of damping, te results become identical wit te results of te previous section. Te expectation value of te dipole moment is dened by Eqs. A.21 and Eq. A.22. Inserting te solutions for c 1 and c 2 and using Eq. A.16 we obtain! (t) = R (!,!o ) 12 [1,cos(t)] cos(!t) + sin(t) sin(!t) : (A.40) We see tat te induced dipole moment oscillates at te frequency of te radiation eld. However, it does not instantaneously follow te driving eld: it as in-pase and quadrature components. Let us write in te complex representation as (t) = Re e,i!t : (A.41) We ten nd for te complex dipole moment! = R (!,!o ) 12 [1,cos(t)] + i sin(t) : (A.42)

9 8 APPENDIX A. ATOMIC POLARIZABILITY To determine te atomic polarizability, dened as = $ E ; (A.43) we ave to express te Rabi frequency! R by its denition Eq. A.32 and obtain $ (!) = (!,!o ) 2 [1,cos(t)] + i sin(t) : (A.44) Te most remarkable property of te polarizability is its dependence on eld strengt (troug! R ) and its time-dependence. Tis is dierent to te polarizability derived in te previous section. In te present case, te time-beavior is determined by te Rabi opping frequency. In practical situations te time-dependence disappears witin tens of nanoseconds because of te damping term wic as been neglected in our derivation. For te case of exact resonance (! =! o ) te polarizability reduces to a sinusoidal function of! R t. Tis oscillation is muc slower tan te oscillation of te optical resonance. For weak interactions! R is small and te polarizability becomes a linear function of t. Steady state polarizability for near-resonance excitation Te damping term does not allow for a purely oscillatory solution as derived in te previous section. Instead, after a suciently long time, te system will settle down into a steady-state. In order to calculate te steady state beavior it is sucient to solve for te term c 1 c 2 wic, togeter wit its complex conjugate, denes te expectation value of te dipole moment (c.f. Eq. A.22). In te steady-state, te probability of nding te atom in its excited state will be time-independent, i.e. d dt [c 2c 2] = 0 (steady, state) : (A.45) Furtermore, in te rotating-wave approximation, it can be expected tat te time-dependence of te o-diagonal matrix element c 1 c 2 will be solely dened by te factor exp(,i[! o,!]t). Tus, d dt [c 1c 2] =,i(! o,!)[c 1 c 2] (steady, state) ; (A.46) wit a similar equation for c 2 c 1. Using d ci c j = ci _c j dt + c j _c i ; (A.47) inserting Eqs. A.18, applying te rotating-wave approximation, and making use of te steadystate conditions above, we obtain! R exp(,i[! o,!]t)[c 2 c 1],! R exp(i[! o,!]t)[c 1 c 2], 2i [c 2 c 2] = 0 (A.48)! R ([c 1 c 1], [c 2 c 2]), (2[! o,!]+i) exp(i[! o,!]t)[c 1 c 2] = 0 (A.49)! R ([c 1 c ], 1 [c 2c ]), 2 (2[! o,!], i) exp(i[! o,!]t)[c 2 c ] 1 = 0 (A.50)

10 9 Tis set of equations can be solved for [c 1 c 2] and gives [c 1 c 2] = e,i(!o,!)t 1 = 2! R (! o,!, i/ 2 ) (! o,!) / = 2! 2 R ; (A.51) wit te complex conjugate solution for [c 2 c 1]. Te expectation value of te dipole moment can now be calculated by using Eq. A.22 and te steady-state solution for te atomic polarizability for near-resonance excitation (!! o ) can be determined as $ (!) = (! o,! + i/ 2 ) (! o,!) / = 2! 2 R : (A.52) Te most remarkable dierence to te o-resonant case is te appearance of te term! 2 R in te denominator. Tis term accounts for saturation of te excited state tereby reducing te absorption rate and increasing te linewidt from to, +2! 2 R 1=2 wic is denoted as saturation broadening. Tus, te damping constant becomes dependent on te acting electric eld strengt. Saturation is not a nonlinear beavior in te usual sense since te dipole moment as always te same armonic time-dependence as te driving electric eld. Saturation in te steady-state gives only rise to a nonlinear relationsip between te amplitudes of dipole moment and electric eld. For! R!0, te polarizability reduces to $ (!) = wic is identical wit te rotating-wave term of Eq. A.25. 1! o,!,i/ 2 ; (A.53) Te polarizability can be calculated once te energy levels E 1 and E 2 and te dipole matrix element 12 are known. Te latter is dened by Eq. A.15 troug te wavefunctions ' 1 and ' 2.Itistus necessary to solve te energy eigenvalue equation A.5 for te considered quantum system in order to accurately determine te energy levels and te dipole matrix element. However, Eq. A.5 can be solved analytically only for simple systems often restricted to two interacting particles. Systems wit more tan two interacting particles ave to be treated wit approximate metods suc as te Hartree-Fock metod or numerically.