Quantum Optics” Friedrich-Schiller-Universit¨At Jena Summer Semester 2017

Lecture notes ”Quantum optics” Friedrich-Schiller-Universit¨at Jena Summer Semester 2017

Frank Setzpfandt

May 17, 2017 Chapter 1

Introduction

1.1 Some history

- usually, quantum mechanics based on axioms, then everything is derived - here we introduce main foundations in a more loose and compressed way - anyway: foundation of quantum mechanics no formal proof possible, these just have → to believed in

1.2.1 Wavefunction

The state of a system in a space of coordinates q at a given time is fully described by the complex-valued wavefunction ψ (q) , which is interpreted as a probability amplitude. probability density w (q) = ψ (q) 2 (1.1) | | probability to ﬁnd system in volume d q around coordinate q →

dW (q) = ψ (q) 2 dq = ψ∗ (q) ψ (q) d q (1.2) | |

normalization condition →

dq ψ (q) 2 = dqψ ∗ (q) ψ (q) = ψ ψ =1 (1.3) | | h | i Z Z where we introduced Dirac’s Bra-Ket notation with ψ ψ , ( ψ∗)⊤ ψ and ψ ψ ≡ | i ≡ h | h | i being the scalar product as deﬁned by Eq. (1.3) superposition principle holds: if ψ and φ are allowed wavefunctions for a given sys- | i | i tem, then θ = α ψ +β φ is also a meaningful wavefunction if normalization condition | i | i | i is fulﬁlled

1 1.2.2 Operators ⊤ Measurable physical quantities A are described by Hermitian operator Aˆ = Aˆ∗ = Aˆ† examples from mechanics: coordinatex ˆ = x, momentump ˆ = ih¯ ∂ x − ∂x but: not every operator corresponds to measurable quantity not every operator is Hermitian → most important operator: Hamiltonian Hˆ for energy ˆ 1 2 single particle in potential: H = V (ˆr) + 2m pˆ measuring a physical quantity in a system means to apply the corresponding operator to the systems wavefunction two diﬀerent types of measurements: single measurement and measurement of expectation value (classical average) single measurements: single measurements will always return an allowed value for a physical quantity (e.g. electron spin = 0.5) ± for each allowed single measurement outcome A , there is a wavefunction φ which is n | ni an eigenfunction of the corresponding operator Aˆ with the eigenvalue An

Aˆ φ = A φ (1.4) | ni n | ni these eigenfunctions form a complete and orthogonal function basis with φ φ = δ , h m| ni mn so that a general wavefunction can be expressed as

ψ = ψ φ (1.5) | i n | ni n X with ψ = ψ φ = dqψ ∗ (q) φ (q) (1.6) n h | ni n Z - projection of state ψ on eigenstate φ , i.e. how much of eigenstate is in state | i | ni ψn deﬁne probability to ﬁnd a certain result in single measurement as

W = ψ 2 = ψ∗ψ with ψ∗ψ =1 (1.7) n | n| n n n n n X measurement is changing system, projecting it into eigenstate φ of Aˆ with probabil- → | ni ity Wn outcome of single measurement is always diﬀerent, ruled by probabilities W → n

expectation value:

2 Figure 1.1: Sketch of an experiment, where a two-level system is prepared in the ground state (left), then excited with a certain probability by a laser beam (middle) and then anylzed (right). The analysis can only give allowed values for the energy Ea or Eb, with probabilities α 2 and β 2, respectively. After passing the laser beam, the system is in a superposition| of| eigenstates,| | ater the measurement, the two-level system is deﬁnitely in one of the eigenstates. more comparable to classical measurements: ensemble average A with h i

A = ψ Aˆ ψ = dqψ ∗ (q) Aψˆ (q) (1.8) h i h | | i Z where A is the expectation value of a measurement h i matrix representation of states and operators: in a chosen basis, every state vector ψ can be expressed as vector of the ψ -coeﬃcients → | i n

ψ1

 ψ2  ∗ ∗ ∗ ψ = and ψ = ( ψ ψ ... ψ ) (1.9) | i . h | 1 2 N  .     ψN     

3 - basis states of one speciﬁc operator also allow to express all operators in matrix form

A11 A21 ... A N1  A21  Aˆ with A = φ Aˆ φ (1.10) → . .. ij h j| | ii  . .     A N ANN   1    - A 2 is probability for transition between states φ and φ under action of operator | ij | | ii | ji Aˆ - transition probability zero if φ eigenstates of Aˆ | ii Example Hamiltonian of two-level system: eigenstates a and a with eigenenergies E and E | i | i a b Hˆ a = E a and Hˆ b = E b → | i a | i | i b | i then a Hˆ a = E , b Hˆ a = 0, ... h | | i a h | | i Hamiltonian in matrix form: E 0 Hˆ = a (1.11) 0 Eb !

1.2.3 measurement uncertainty

- average result of ensemble measurement is given by expectation value - as result of individual measurements can be any of the eigenvalues of measurement operators, another important quantity is the variance of the measurement

2 2 ∆Aˆ = Aˆ2 Aˆ (1.12) − D E D E with 2 ∆Aˆ = Aˆ Aˆ and ∆Aˆ 0 (1.13) − ≥ D E - using this one can derive the following important result

2 2 1 2 ∆Aˆ ∆Bˆ i A,ˆ Bˆ (1.14) ≥ 2 D h iE with commutator A,ˆ Bˆ = AˆBˆ BˆAˆ (1.15) − h i known as Heisenberg’s uncertainty relation - commutators are not zero in general - this means one in general cannot measure two quantities A and B with unlimited pre- cision - this is a property of any wave-like system (think of spectrum - time in optics) - quantum mechanics is wave theory →

4 example position-momentum uncertainty known from mechanics, example in 1D position operatorx ˆ = x, momentum operatorp ˆ = ih¯ ∂ x − ∂x we have to calculate the commutator, speciﬁcally, its action on spatially dependent function ∂ ∂ [ˆx, pˆx] f (x) = ihx¯ f (x) + ih¯ [x f (x)] − ∂x ∂x · (1.16) ∂ ∂ ∂ = ihx¯ f (x) + ihx¯ f (x) + ihf¯ (x) x = ihf¯ (x) − ∂x ∂x ∂x this means for commutator:

[ˆx, pˆx] = ih¯ (1.17) uncertainty relation: h¯ (∆ˆx)2 (∆ˆp )2 (1.18) x ≥ 2 q 1.2.4 Time evolution

- up to now no explicit time dependendence - time evolution of wavefunction governed by Schr¨odinger-equation

∂ ψ (t) ih¯ | i = Hˆ ψ (t) (1.19) ∂t | i

- that’s why Hamiltonian has a special role - special case: wavefunction is eigenstate of Hamiltonian ψ = φ | i | ni ∂ ψ (t) ∂ φ (t) ih¯ | i = ih¯ | n i = Hˆ φ (t) = E φ (t) (1.20) ∂t ∂t | n i n | n i with formal solution i φ (t) = e− h¯ Ent φ (0) (1.21) | n i | i stationary state, where just phase evolves - time evolution of expectation values

d 1 ∂Aˆ Aˆ = A,ˆ Hˆ + (1.22) dt h¯ * ∂t + D E Dh iE 1.2.5 Canonical quantization

- how to get from classical description to quantized description in a given physical system? - canonical quantization is based on mechanics, which uses the classical Hamilton formal-

5 ism to describe motion of system based on variables for coordinates r and impulses p - classical Hamiltonian for 1D-harmonic oscillator with mass m

mω 2 1 H = x2 (t) + p 2 (t) (1.23) 2 2m x

- classical Hamiltons’s equations

dx ∂H p dp ∂H d2x = = x ; x = = mω 2x = ω2x (1.24) dt ∂p m dt − ∂x − → dt2 − x

- x and px are called conjugate canonical variables of the system - quantization in mechanics case: replace x and px with operatorsx ˆ andp ˆ x to ﬁnd Hamilton operator

mω 2 1 Hˆ = xˆ2 + pˆ2 (1.25) 2 2m x impose canonical commutation relation

[ˆx, pˆx] = ih¯ (1.26)

- then specific forms of observables obeying these equations can be found, e.g. x forr ˆ x and ih¯ ∂ forp ˆ − ∂x x - generalized procedure 1) find canonical variables so that Hamiltonian fulfills equations of motion Eq. (1.24) 2) replace them with operators and impose canonical quantization 3) find explicit form of operators we will do this for EM fields in chapter 3

6 Chapter 2

Light-Matter Interaction in a semiclassical treatment

- historically, quantization was done in two steps: 1) first matter (atoms) where quantized, the fields were kept classical (first quantization) 2) both matter and fields were quantized (second quantization) - we will adopt this step- wise procedure for following reasons: 1) makes you familiar with quantum treatment for simpler system of quantized atom + classical fields 2) will lead to well know results 3) will be instructive to see for that quantum optics, e.g. second quantization, is really necessary

2.1 Interaction Hamiltonian

- up to now very general introduction of basic principles of quantum theory - now we also quickly review the equations govering classical electrodynamics

7 2.1.1 Classical Electrodynamics

- Maxwell’s equations:

1 E (r, t ) = ρ (r, t ) , ∇ · ǫ0 B (r, t ) = 0 , ∇ · ∂ (2.1) E (r, t ) = B (r, t ) , ∇ × −∂t 1 ∂ 1 B r, t E r, t j r, t . ( ) = 2 ( ) + 2 ( ) ∇ × −c ∂t ǫ0c with electric field E (r, t ), magnetic field B (r, t ), current density j (r, t ), charge density ρ (r, t ) - the vectorial fields E (r, t ) and B (r, t ) can be expressed using the vector and scalar potentials A (r, t ) and U (r, t ) - these define the fields by

B (r, t ) = A (r, t ) (2.2) ∇ × and ∂ E (r, t ) = A (r, t ) U (r, t ) (2.3) −∂t − ∇ - important: these definitions are not unique, there is an infinite number of pairs A (r, t ) , U (r, t ) { } that will describe the same fields E (r, t ) , B (r, t ) { } - these pairs are related by gauge transformation

A′ (r, t ) = A (r, t ) + F (r, t ) (2.4) ∇ and ∂ U ′ (r, t ) = U (r, t ) F (r, t ) (2.5) − ∂t with arbitrary scalar field F (r, t ) - the potentials can be fixed by an additional constraint, so-called gauge condition, example will be shown later - convenient tool to bring potentials to different mathematical forms, which may be especially intuitive or simple to treat mathematically - important: does not change physics!!! example Coulob gauge: - one particularly often used gauge condition in quantum optics is the so-called Coulomb gauge, defined by the additional condition

A (r, t ) = 0 (2.6) ∇ ·

8 - potential for plane wave in Coulomb gauge

E0 A⊥ (ˆr , t ) = sin ( ωt k r) , U (r, t ) = 0 with E k =0 (2.7) ω − · 0 · where the symbol denotes Coulomb gauge and k is the usual wavevector ⊥ plane-wave nature can be checked by calculating ﬁelds

- matter consists of moving particles with mass mα, charge qα, which are at the loca- tions rα and move with velocities vα - then, the charge and current densities in Maxwell’s equation can be expressed as

ρ (r, t ) = q δ (r r (t)) (2.8) α − α α X and j (r, t ) = q v δ (r r (t)) (2.9) α α − α α X - Maxwell’s equation describe the dynamics of the fields in dependence on the charges and currents - the dynamics of the particles, which constitute these charges and currents, under the influence of the fields is described by the Newton-Lorentz equations

dv m α = q [E (r (t) , t ) + v B (r (t) , t )] (2.10) α dt α α α × α

- Maxwell’s equations, the equations for charges and currents, and the Newton-Lorentz equations form a complete set of coupled equations to describe the state of ﬁelds and particles - all results of classical electrodynamics can be derived from these - next task: generalize to case were states of particles are quantized

2.1.2 General form of Hamiltonian for one particle

- we treat simplest atom, one electron in the field of a stationary nucleus (corresponding to hydrogen) - is basically a single electron, which motion is treated quantum mechanically, and moving in a stationary electric field - field is described by potentials A (r, t ) and U (r, t ), which contain Coulomb field of nucleus and external fields

9 - dynamics of this electron is governed by following Hamiltonian

1 Hˆ = (ˆp qA (ˆr , t )) 2 + qU (ˆr , t ) (2.11) 2m − with position operator ˆr = r and the momentum operator ˆp = ih¯ − ∇ - this Hamiltonian is not derived here, justiﬁcation may be obtained by deriving classical equations from it

2.1.3 Interaction Hamiltonian in G¨oppert-Mayer Gauge

- to achieve intuitive form of interaction Hamiltonian, we introduce the G¨oppert-Mayer gauge, often used in atomic physics - can be obtained from Coulomb gauge by transformation

F (r, t ) = (r r ) A⊥ (r , t ) (2.12) − − 0 · 0

-privileges special position of nucleus and leads to G¨oppert-Mayer potentials

′ A (r, t ) = A⊥ (r, t ) A⊥ (r , t ) (2.13) ⊥ − 0 and ′ ∂ U (r) = U (r) + ( r r ) A⊥ (r , t ) (2.14) Coul − 0 · ∂t 0 now, Hamiltonian becomes

1 ′ 2 ∂ Hˆ = (ˆp qA (ˆr , t )) + qU (ˆr ) + q (ˆr r ) A⊥ (r , t ) (2.15) 2m − ⊥ Coul − 0 · ∂t 0 now we use that electric ﬁeld is

∂ E (r , t ) = A⊥ (r , t ) (2.16) 0 −∂t 0 and introduce dipole operator of atom as

Dˆ = q (ˆr r ) (2.17) − 0 to ﬁnd 1 Hˆ = (ˆp qA′ (ˆr , t )) 2 + qU (ˆr ) Dˆ E (r , t ) (2.18) 2m − ⊥ Coul − · 0 - further simpliﬁcation possible by comparing atom size and typical wavelengths of light - for hydrogen, typical wavelengths for transitions are larger 100 nm, whereas atom is in the order of 0.05 nm (Bohr radius)

10 - that means, the external ﬁeld is constant over atom radius and

A′ (ˆr , t ) A′ (r , t )=0 (2.19) ⊥ ≈ ⊥ 0 to ﬁnd Hamiltonian

Hˆ = Hˆ0 + HˆI (2.20) with ˆp 2 ˆp 2 Hˆ = + qU (ˆr ) = + V (ˆr ) , (2.21) 0 2m Coul 2m Coul the usual Hamiltonian of charged particle in Coulomb potential, and

Hˆ = Dˆ E (r , t ) (2.22) I − · 0 which is called electric dipole Hamiltonian long wavelength approximation decouples movement of charge in potential and interaction with field - if long-wavelength approximation does not hold, dipole Hamiltonian cannot be used, then exact Hamiltonian or other gauge have to be used different gauges may lead to different mathematical form of Hamiltonian physics in exact case always the same!!

2.1.4 Two-level atom

- we now consider atom with just two states, a and b , where energy E = 0 and | i | i a

E E =hω ¯ (2.23) b − a 0

- now we have to ﬁnd explicit (matrix) form of Hamiltonian Hˆ = Hˆ0 + HˆI - free Hamiltonian 0 0 Hˆ0 =h ¯ (2.24) 0 ω0 ! which ensures that a and b are eigenstates of this Hamiltonian with correct energies | i | i 0 0 0 0 Hˆ0 b =h ¯ =hω ¯ 0 = Eb b (2.25) | i 0 ω0 ! 1 ! 1 ! | i

- interaction Hamiltonian in dipole form

H = b Dˆ E (r , t ) a (2.26) I,ba h | − · 0 | i

11 with plane wave excitation

E (r0, t ) = E (r0) cos ( ωt + ϕ (r0)) = eE (r0) cos ( ωt + ϕ (r0)) (2.27) with the unit vector e specifying the polarization then

b Dˆ E (r , t ) a = b Dˆ e a E (r ) cos ( ωt + ϕ (r )) − h | · 0 | i − h | · | i 0 0 = dE (r ) cos ( ωt + ϕ (r )) (2.28) − 0 0 =h ¯ Ω1 cos ( ωt + ϕ (r0)) with matrix element d of dipole moment in direction of polarization vector and Rabi −dE (r0) frequency Ω 1 = ¯h complete Hamiltonian is then

0 Ω 1 cos ( ωt + ϕ) Hˆ = Hˆ0 + HˆI =h ¯ (2.29) Ω1 cos ( ωt + ϕ) ω0 ! aim is to describe temporal dynamics of atom, which in general is described by wavefunction ψ (t) | i there we expand this state in the basis functions

ψ (t) = γ (t) a + γ (t) e−iω 0t b (2.30) | i a | i b | i where coefficients give probability to find atom in either eigenstate upon measurement now we use Schrödinger equation

∂ ψ (t) ih¯ | i = Hˆ ψ (t) (2.31) ∂t | i to ﬁnd equations for time evolutions of coeﬃcients

∂ γ (t) 0 Ω cos ( ωt + ϕ) γ (t) ih¯ a =h ¯ 1 a −iω 0t −iω 0t ∂t γb (t) e ! Ω1 cos ( ωt + ϕ) ω0 ! γb (t) e ! (2.32) this leads to following equations for coeﬃcients (using cos ( x) = 1 (exp ( ix ) + exp( ix ))) 2 − d Ω eiϕ Ω e−iϕ i γ = 1 ei(ω−ω0)tγ + 1 e−i(ω+ω0)tγ (2.33) dt a 2 b 2 b

12 and d d ih¯ γ e−iω 0t = ihe¯ −iω 0t γ +hω ¯ γ e−iω 0t dt b dt b 0 b   " =h ¯ Ω cos ( ωt + ϕ) γ +hω ¯ " γ e−iω 0t (2.34)  1 a 0 b     1 i(ωt +ϕ) −i(ωt +ϕ) −iω 0t =h ¯ Ω1 e + e +hω ¯ 0γbe  2    leading to  "  d Ω e−iϕ Ω eiϕ i γ = 1 e−i(ω−ω0)tγ + 1 ei(ω+ω0)tγ (2.35) dt b 2 a 2 a the terms with ei(ω+ω0)t oscillate very fast and give negligible average contribution, will be neglected this leads to following system of coupled diﬀerential equations { } d Ω eiϕ i γ = 1 ei(ω−ω0)tγ dt a 2 b  −iϕ  (2.36)  d Ω1e −i(ω−ω0)t   i γb = e γa dt 2   to simplify, we introduce detuning from resonance δ = ω ω and make the following − 0 variable transformation:

δ δ γ =γ ˜ exp i t and γ =γ ˜ exp i t (2.37) a a 2 b b − 2 system of equations is then transformed to system with constant coeﬃcients:

iϕ d δ Ω1e i γ˜a = γ˜a + γ˜b dt 2 2 (2.38) d Ω e−iϕ δ i γ˜ = 1 γ˜ γ˜ dt b 2 a − 2 b this system has two oscillating eigen-solutions of the form

γ˜ (t) α λ a = exp i t (2.39) γ t β − 2 # ˜b ( ) % # % with λ = Ω2 + δ2 (2.40) ± ± 1 q with corresponding ratio α Ω eiϕ = 1 (2.41) β δ Ω2 + δ2 ± ± 1 p

13 this basically are eigenstates of new Hamiltonian with ﬁelds general solution is then superposition of these

iϕ 2 2 iϕ 2 2 Ω1e Ω1 + δ Ω1e Ω1 + δ γ˜a = K exp i t + L exp i t ( Ω2 + δ2 δ − 2 ! Ω2 + δ2 + δ 2 !) 1 − p 1 p (2.42) p p we are looking for particular solution with initial conditions

γ˜a (t0) = 1 andγ ˜ b (t0)=0 (2.43)

meaning that the atom is in the ground state at t0 (probability to ﬁnd it there upon measurement is 1) solution is then

2 2 2 2 Ω1 + δ δ Ω1 + δ γã (t) = cos (t t0) i sin (t t0) 2 − − Ω2 + δ2 2 − p 1 p (2.44) −iϕ 2 2 Ω1e Ω1 + δp γ˜b (t) = i sin (t t0) − 2 2 2 − Ω1 + δ p p 2.1.5 Rabi oscillations the time dependence of the coefficients enables to determine the probability of finding atom in excited state at time t

Ω2 Ω2 + δ2 P Max (t , t ) = γ˜ (t) 2 = 1 sin 2 1 (t t ) (2.45) a→b 0 b 2 δ2 0 | | Ω1 + p 2 −

−dE remember Rabi frequency Ω 1 = ¯h probability oscillates with frequency proportional to excitation ﬁeld strength → always change between absorption and stimulated emission maximum achievable value of oscillation

Ω2 P t , t 1 a→b ( 0 ) = 2 2 (2.46) Ω1 + δ resonant with frequency detuning probability of 1 to ﬁnd excited state only achievable → with excitation frequency corresponding to energy diﬀerence between states lineshape of resonance is Lorentzian with width 2Ω increases with amplitude of EM 1 → wave at resonance atomic population can be completely excited known as π-pulse excitation → we derived this result for initial condition in ground state, hence it describes absorption

14 Figure 2.1: (Left) Rabi oscillations for different electric field strength and δ = 0, (right) Rabi oscillations for different detunings, solid δ = 1, dashed δ = Ω 1, dotted δ = 2Ω 1

Figure 2.2: Maximum probability of finding atom in excited state as function of excitation frequency same result can be obtained for atom initially in excited state stimulated emission → many effects can be described with semiclassical model, e.g. excitation depending on light frequency, rate depends on intensity (photoelectric effect) important: for E (r ) = 0 Ω = 0 and transition probability zero in both directions 0 → 1 spontaneous emission cannot be described in this formalism (practically it is described assuming a phenomenologic lifetime of excited state, after which it decays again) to fundamentally understand reason for spontaneous emission, EM field has to be quantized as well

15 Chapter 3

The quantization of electromagnetic radiation

aim: find quantum description of EM field in the absence of material i.e. charges and currents we use canonical quantization, that means: 1) find conjugate canonical variables to describe EM field 2) formulate Hamiltonian in terms of these variables 3) use canonical commutators to find radiation states of light

3.1 Free electromagnetic ﬁeld and transversality start with Maxwell’s equations in vacuum

E (r, t ) = 0 , ∇ · B (r, t ) = 0 , ∇ · ∂ (3.1) E (r, t ) = B (r, t ) , ∇ × −∂t 1 ∂ B (r, t ) = E (r, t ) . ∇ × −c2 ∂t state of system is fully described by six ﬁeld components, these for complete set of dynamical variables problem: continuous inﬁnity of coupled variables hard to quantize directly → therefore we simplify problem to countable set of decoupled variables which are also decoupled pairs of conjugate canonical variables

16 3.1.1 Expansion in polarized Fourier components to to this, we assume cubic empty volume with volume V , side length L, and periodic boundary conditions ﬁeld can be decomposed into plane waves

ikn·r E (r, t ) = E˜ n (t) e (3.2) n X with wavevector kn and vector n = n , n , n { x y z} ﬁelds are transverse to kn

E (r, t ) = 0 kn E˜ n =0 (3.3) ∇ · → ·

fields belong to two-dimensional space orthogonal to wavevector two orthogonal unit vectors can be chosen in this plane: en,1 and en,2 thus we can write ˜ En = Eñ,1en,1 + Eñ,2en,2 (3.4) this defines polarization components i.e. fields can be expanded in terms of polarized Fourier components with indices n , n , n ; s { x y z } with s being either 1 or 2 will be described by

l = ( nx, n y, n z; s) = ( n; s) (3.5) and we can write ikl·r E (r, t ) = elE˜l (t) e (3.6) l X with 1 E˜ (t) = d3re E (r, t ) e −ikl·r (3.7) l L3 l · ZV with vector kl 2π 2π 2π k = n , k = n , k = n (3.8) lx x L ly y L lz z L Finally, deﬁne l = ( n , n , n ; s), so that − − x − y − z

e− = e and k− = k (3.9) l l l − l

Furthermore, introduce unit vector to deﬁne right-handed triad with kl and el

′ kl el = el (3.10) kl ×

17 then we can also expand B-ﬁeld in plane waves

′ ˜ ikl·r B (r, t ) = elBl (t) e (3.11) l X in Coulomb gauge we can also expand vector potential

ikl·r A (r, t ) = elA˜l (t) e (3.12) l X with d E˜ (t) = A˜ (t) (3.13) l −dt l and

B˜l (t) = ik lA˜l (t) (3.14) now we can use either B-ﬁeld or vector potential

3.2 Expansion in normal modes

3.2.1 Dynamical equations of polarized Fourier components up to now we dealt with ﬁrst two of Maxwell’s equation and established transversality of ﬁelds inserting expansion into polarized Fourier components into other two equations leads to

d B˜ (t) = ik E˜ (t) (3.15) dt l − l l d E˜ (t) = ic 2k B˜ (t) (3.16) dt l − l l where we used that V (r, t ) k V˜ (t) ∇ × → n × n solutions to this have oscillatory character with frequencies ω = ck ± l ± l using this and substituting B-ﬁeld with vector potential leads to

d A˜ (t) = E˜ (t) (3.17) dt l − l d E˜ (t) = ω2A˜ (t) (3.18) dt l l l these tow equations form closed system, completely describing dynamics of system do not couple components with diﬀerent absolute values of l, but are not completely decoupled since ˜ ˜∗ ˜ ˜∗ E−l = El and A−l = Al (3.19)

18 3.2.2 Normal variables solution of dynamical equation for each l-component is characterized by two complex integration constants, i.e. four independent variables which describe a given situation these four parameters can be separated into two pairs of independent dynamical variables to do this, we define 1 αl = ωlA˜l iE˜l (3.20) 2ǫ(1) − l 1 βl = ωlA˜l + iE˜l (3.21) 2ǫ(1) l (1) here ǫl is a constant, will be fixed later we use now reality conditions for polarized Fourier components Eq. (3.19) which are inserted into first equation of definition comparison with second equation leads to

∗ βl = α−l (3.22) system of coupled equations for E˜l and A˜l becomes

dαl 1 d d 1 = ωl A˜l i E˜l = iω l iE˜l + ωlA˜l = iω lαl (3.23) dt 2ǫ(1) dt − dt − 2ǫ(1) − − l l dβ l = iω β (3.24) dt l l equations are decoupled, but coeﬃcients are linked and anything could be expressed using just one of the series of normal variables solutions for these equations are

−iω lt αl (t) = αl (0) e (3.25)

iω lt βl (t) = βl (0) e (3.26) using deﬁnition of αl and βl the general evolution of polarized Fourier components can be obtained (1) (1) ˜ ǫl ǫl ∗ Al = [αl (t) + βl (t)] = αl (t) + α−l (t) (3.27) ωl ωl E˜ = ǫ(1) [iα (t) iβ (t)] = ǫ(1) iα (t) iα ∗ (t) (3.28) l l l − l l l − −l evolution now still dependas on four variables, real and imaginary parts of αl and βl however, now these can be associated in pairs (Re α , Im α ) and (Re β , Im β ) { l} { l} { l} { l} with independent dynamics using these complex normal variables allows expression of ﬁeld as function of decoupled

19 pairs of real dynamical variables will later be identified with conjugate canonical variables first, we show how actual fields can be expressed using these variables

3.2.3 Expansion of free field in normal modes the found solutions to Maxwell’s equations can be used to express EM field in time domain as ǫ(1) ˜ ikl·r l ∗ ikl·r A (r, t ) = elAl (t) e = el αl (t) + α−l (t) e (3.29) ωl l l X X this equation contains two sums over index l, which can be exchanged by l in second − sum, leading to (1) ǫl ikl·r ∗ −ikl·r A (r, t ) = el αl (t) e + αl (t) e (3.30) ωl l X this form explicitly shows that vector potential is real similarly E (r, t ) = e ǫ(1) iα (t) e ikl·r iα ∗ (t) e −ikl·r (3.31) l l l − l l X and ǫ(1) B (r, t ) = e′ l iα (t) e ikl·r iα ∗ (t) e −ikl·r (3.32) l c l − l l X thus, EM field is expanded as a sum of real components Al, El, and Bl, each of which is monochromatic plane wave example: if we write explicitly α = α eiϕ l e−iω lt (3.33) l | l| we get (1) ǫl Al = el 2 αl cos ( kl r ωlt + ϕl) (3.34) ωl | | · − E = e ǫ(1) 2 α sin ( k r ω t + ϕ ) (3.35) l − l l | l| l · − l l ǫ(1) B = e′ l 2 α sin ( k r ω t + ϕ ) (3.36) l − l c | l| l · − l l each αl therefore corresponds to a component of field expressed relative to basis of monochromatic polarized plane waves this decomposition is called normal mode decomposition, the monochromatic polarized plane waves form a set of normal modes important: for other system than free space, other types of modes are normal modes here, choice of periodic boundary conditions on fictitious volume V lead to plane waves other possible expansions are e.g. Gaussian modes in paraxial problems, or resonator modes if real bounded resonators are considered

20 message: quantum optics as introduced below is always connected to certain set of modes, physically meaningless without knowledge of modes

3.3 Hamiltonian of free radiation

3.3.1 Radiation energy energy HR of free EM ﬁeld is integral over energy density

ε H = 0 d3r E2 (r, t ) + c2B2 (r, t ) (3.37) R 2 ZV using expansion in normal modes for electric ﬁeld we have

d3rE2 (r, t ) ZV (1) (1) 3 ikl·r ∗ −ikl·r ikl′ ·r ∗ −ikl′ ·r = e e ′ ǫ ǫ ′ d r α e α e α ′ e α ′ e − l l l l l − l l − l l l′ ZV X X (1) (1) 3 i(kl+kl′ )·r ∗ i(kl−kl′ )·r = e e ′ ǫ ǫ ′ d r α α ′ e α α ′ e (3.38) − l l l l l l − l l l l′ ZV X X " ∗ i(kl′ −kl)·r ∗ ∗ −i(kl+kl′ )·r α α ′ e + α α ′ e − l l l l 2 3 (1) ∗ ∗ ∗ ∗ =L ǫ α α + α α α α− α α l l l l l − l l − l −l l X h i " remember what index l actually is, l = n , n , n ; s { x y z } ′ el1 and el2 associated with same n are orthogonal and integral for kl and kl with diﬀerent n are zero, as harmonic functions are also orthogonal thus, integral only gives contribution for l′ = l or l′ = l − similarly, we get

d3rc 2B2 (r, t ) V 2 Z (3.39) 3 (1) ∗ ∗ ∗ = L ǫl 2αlαl + αlα−l + αl α−l l X h i " when added, the last terms cancel, and we obtain energy

2 H = 2 ε L3 ǫ(1) α 2 (3.40) R 0 l | l| l X h i simple sum of amplitudes of normal modes, no cross terms

21 3.3.2 Conjugate canonical variables for a single radiation mode will just treat single mode of radiation with index l recall evolution equation for αl dα l = iω α (3.41) dt − l l where real and imaginary part can be separated as

d Re α = ω Im α (3.42) dt { l} l { l} d Im α = ω Re α (3.43) dt { l} − l { l} now we deﬁne variables 3 4ε0L (1) Ql = ǫl Re αl (3.44) s ωl { }

3 4ε0L (1) Pl = ǫl Im αl (3.45) s ωl { } Hamiltonian then becomes

2 ω ω H = 2 ε L3 ǫ(1) α 2 = l (Q + iP ) ( Q iP ) = l Q2 + P 2 (3.46) R 0 l | l| 2 l l l − l 2 l l h i " now we can formulate evolution equations for canonical variables

3 3 dQl 4ε0L (1) d 4ε0L (1) ∂H l = ǫl Re αl = ǫl ωlIm αl = ωlPl = (3.47) dt s ωl dt { } s ωl { } ∂P l

dP ∂H l = l (3.48) dt −∂Q l these correspond to Hamiltons equations of motion for Ql and Pl thus, we have identiﬁed the conjugate canonical variables as the scaled real and imaginary parts of the normal modes

3.4 Quantization of Radiation

3.4.1 Canonical commutation relations steps of canonical qunatization:

1) identify conjugate canonical variables Ql (t) and Pl (t), identify them with time independent Hermitian operators Qˆl and Pˆl 2) impose canonical commutation relations on these operators of one mode 3) zero commutators are associated with operators having diﬀerent l as diﬀerent modes

22 are decoupled last two points in equations:

Qˆ , Pˆ ′ = Qˆ Pˆ ′ Pˆ ′ Qˆ = ihδ¯ ′ (3.49) l l l l − l l ll h i

Qˆl, Qˆl′ = Pˆl, Pˆl′ =0 (3.50) h i h i hence, normal mode amplitude αl (t) is associated with an operatorα ˆ l

3 ˆ ˆ 4ε0L (1) Ql + iPl = ǫl αˆl (3.51) s ωl commutator for this is derived as follows

† † † αˆ , αˆ ′ =α ˆ αˆ ′ αˆ ′ αˆ l l l l − l l h i √ωlωl′ 1 † † Qˆ iPˆ Qˆ ′ iPˆ ′ Qˆ ′ iPˆ ′ Qˆ iPˆ = 3 (1) (1) l + l l + l l + l l + l 4ε0L ǫ ǫ ′ − l l √ωlωl′ 1 Qˆ Qˆ ′ iQˆ Pˆ ′ iPˆ Qˆ ′ Pˆ Pˆ ′ Qˆ ′ Qˆ iQˆ ′ Pˆ iPˆ ′ Qˆ Pˆ ′ Pˆ = 3 (1) (1) l l l l + l l + l l l l l l + l l l l 4ε0L ǫ ǫ ′ − − − − l l h i √ωlωl′ 1 Qˆ , Qˆ ′ Pˆ , Pˆ ′ i Qˆ , Pˆ ′ i Qˆ ′ , Pˆ = 3 (1) (1) l l + l l l l l l 4ε0L ǫ ǫ ′ − − l l hh i h i h i h ii h¯√ωlωl′ 1 = δ 2ε L3 (1) (1) ll ′ 0 ǫl ǫl′ (3.52) where we used that canonical operators are Hermitian

[α ˆ l, αˆl′ ]=0 (3.53)

(1) now we ﬁx the up-to-now undeﬁned constant ǫl to

hω¯ ǫ(1) = l (3.54) l 2ε L3 r 0 and end up with the basic commutation relations of quantum optics

† αˆl, αˆl′ = δll ′ (3.55) h i [α ˆ l, αˆl′ ]=0 (3.56) furthermore, deﬁnition of constant yields

1 αˆl = Qˆl + iPˆl (3.57) √2¯h 23 † 1 αˆ = Qˆl iPˆl (3.58) l √2¯h − and 1 αˆ αˆ† +α ˆ †αˆ = Qˆ2 + Pˆ2 (3.59) l l l l h¯ l l 3.4.2 Hamiltonian of quantized radiation classical Hamiltonian in terms of conjugate variables

ω H = hω¯ α 2 = l Q2 + P 2 (3.60) R l | l| 2 l l l l X X " quantum operator is therefore

ω hω¯ 1 Hˆ = l Qˆ2 + Pˆ2 = l αˆ αˆ† +α ˆ †αˆ = hω¯ αˆ†αˆ + (3.61) R 2 l l 2 l l l l l l l 2 l l l X X X formally identical to Hamiltonian of assembly of decoupled quantum oscillators

3.4.3 Field operators

Hermitian operators of observable ﬁelds

(1) ǫl ikl·r † −ikl·r (+) (−) Aˆ (r) = el αˆle +α ˆ l e = Aˆ (r) + Aˆ (r) (3.62) ωl l X Eˆ (r) = ie ǫ(1) αˆ eikl·r αˆ †e−ikl·r = Eˆ (+) (r) + Eˆ (−) (r) (3.63) l l l − l l X k e ˆ l l (1) ikl·r † −ikl·r ˆ (+) ˆ (−) B (r) = i × ǫl αˆle αˆl e = B (r) + B (r) (3.64) ωl − l X h i - can be decomposed into pair of non-Hermitian operators Xˆ (+) (r) and Xˆ (−), which are Hermitian conjugates of each other - positive parts associated with complex ﬁelds

3.5 Quantized radiation state and photons quantum properties of system are described using Hermitian operators for measurable quantities called observables and their commutation relations and state vectors describing speciﬁc state of system we have constructed operators, now we have to introduce quantum states basis for expansion of these states will be found in eigenstates of Hamiltonian, which we derive now

24 3.5.1 Eigenstates and Eigenvalues of radiation Hamiltonian we use procedure that has been developed for mechanical oscillator there, Hamiltonian is expressed as

1 Hˆ = hω¯ Nˆ + (3.65) R l l 2 l X with operator ˆ † Nl =α ˆ l αˆl (3.66) commutator † αˆl, αˆl =1 (3.67) h i can be used to show that eigenvalues of Nˆl are non-negative integers thus, there exist eigenvectors n such that | li

Nˆ n = n n , with n = 0 , 1, 2, ... (3.68) l | li l | li l these number states (Fock states) form basis for radiation states in mode l main properties αˆ n = √n n 1 if n > 1 (3.69) l | li l | l − i l αˆ 0 =0 (3.70) l | li αˆ† n = √n + 1 n + 1 (3.71) l | li l | l i lowest-energy states 0 has special role, discussed later | li all number states can be derived from 0 by | li n † aˆl nl = 0l (3.72) | i √nl! | i now we have established eigenstates of operator Nˆ, what about Hamiltonian? eigenstate of Hˆ are tensor products of states n , i.e. have the form n n ... n ... R | li | 1i⊗| 2i⊗ ⊗| li reason is that Nˆl commute, i.e. Nˆl, Nˆl′ = 0 we will use abbreviated form n1h, n 2, ...,i n l, .. | i thus 1 Hˆ n , n , ..., n , .. = n + hω¯ n , n , ..., n , .. (3.73) R | 1 2 l i l 2 l | 1 2 l i l X ground state of radiation, called vacuum state, is were all nl are zero and will be simple written as 0 | i 0 = n = 0 , n = 0 , ..., n = 0 , .. (3.74) | i | 1 2 l i

25 important: this vacuum state has energy diﬀerent from zero, which is

1 E = hω¯ (3.75) V 2 l l X note that for infinite volume V , this energy actually goes to infinity is often dealt with by renormalizing energy, which is only defined up to a constant i.e. vacuum energy is defined as zero, and Hamiltonian becomes

HˆR = hω¯ lNˆl (3.76) l X can be also explained by ambiguity in deﬁnition of operators, which appears because classically ordering does not matter

3.5.2 The notion of a photon

Hamiltonian measures energy hence, equation for eigenstates shows, that stationary state n , n , ..., n , .. , compared | 1 2 l i to vacuum state, has extra energy

E E = n hω¯ (3.77) n1,n 2,...,n l,.. − V l l l X that means, state n , n , ..., n , .. contains n particles of energyhω ¯ , n particles of | 1 2 l i 1 1 2 energyhω ¯ 2, and so on ground state does not contain any particles, therefore called vacuum we will call these particle-like things photons they are elementary excitations of the quantized EM ﬁeld operatora ˆ reduces number of photons in mode l by one annihilation operator l → operatora ˆ † increases number of photons in mode l by one creation operator l → operator N is the observable characterizing number of photons in mode l number l → operator operator

Nˆ = Nˆl (3.78) l X gives total number of photons in volume v these number and total number operators could be measured with detectors sensitive to photon number (and mode) based on discussion up to now we can deﬁne the general form of the radiation state

26 wavefunction, which is

∞ ∞ ∞ Ψ = ...... C n , n , ..., n , .. (3.79) | i n1,n 2,...,n l,.. | 1 2 l i n1=0 n2=0 nl=0 X X X where Cn1,n 2,...,n l,.. are arbitrary complex numbers, with only restriction being normalization condition Ψ Ψ = 1 h | i enourmous number of possible states, as each of inﬁnitely many indices nl can have in-

ﬁnitely many values Cn1,n 2,...,n l,.. this is a lot more variety than possible with classical radiation, which is described by one complex number for each mode example: suppose system with just M modes classical radiation is completely described by just M complex numbers in quantum radiation, with restriction to N photons per mode, there exist ( N 1) M − possible states conclusion: many things in quantum optics are probably not possible with classical light / description science on this just beginning, many interesting things already found, many more still to be discovered will discuss a few examples of interesting states later in the lecture, ﬁrst, we discuss ground state

3.5.3 The vacuum: ground state of quantum radiation we discuss here in more detail properties of ground state 0 = n = 0 , n = 0 , ..., n = 0 , .. | i | 1 2 l i radiation state without source of light traditionally called vacuum, although this is not ideal name vacuum is not nothing, as we have already seen it has energy recall general Heisenberg relation

2 2 1 2 ∆Aˆ ∆Bˆ i A,ˆ Bˆ (3.80) ≥ 2 D h iE with 2 2 ∆Aˆ = Aˆ2 Aˆ (3.81) − D E D E recall canonical commutation relation

Qˆl, Pˆl = ih¯ (3.82) h i

27 with this we can ﬁnd Heisenberg relation for quantized ﬁeld

2 2 h¯ ∆Qˆl ∆Pˆl (3.83) s ≥ 2 valid for any state of quantized radiation (because form of quantum optics independent of choice of modes) implies that real and imaginary parts components of vector potential cannot be zero simultaneously this is why even for 0 the energy and ﬁeld variances cannot be zero | i now we calculate variances for any mode l, the following holds

αˆ 0 = 0 ; 0 αˆ† =0 (3.84) l | i h | l i.e. no photon can be extracted from vacuum then we can calculate expectation values of ﬁelds in vacuum

0 Eˆ (r) 0 = ie ǫ(1) 0 αˆ 0 eikl·r 0 αˆ† 0 e−ikl·r =0 (3.85) h | | i l l h | l | i − h | l | i l X h i and 0 Bˆ (r) 0 = 0 Aˆ (r) 0 =0 (3.86) h | | i h | | i these are zero as expected for vacuum now we calculate variances, where we additionally use

0 αˆ αˆ† 0 = δ and 0 αˆ αˆ 0 = 0 αˆ†αˆ† 0 =0 (3.87) h | l m | i lm h | l m | i h | l m | i we ﬁnd

2 2 2 ∆Aˆ = 0 Aˆ (r) 0 0 Aˆ (r) 0 h | | i − h | | i (1) (1) ǫl ǫm ikl·r † −ikl·r ikm·r † −ikm·r = elem 0 αˆle +α ˆ l e αˆme +α ˆ me 0 ωlωm h | | i l m X X h i 1 2 = ǫ(1) ω2 l l l X h i (3.88)

2 2 2 hω¯ ∆Eˆ = 0 Eˆ (r) 0 = ǫ(1) = l (3.89) h | | i l 2ε L3 l l 0 X h i X

28 2 2 1 2 1 2 ∆Bˆ = 0 Bˆ (r) 0 = ǫ(1) = ∆Eˆ (3.90) h | | i c2 l c2 l l X h i expressions are time independent, since vacuum is eigenstate of field Hamiltonian, i.e. stationary state this tells us, that even in ground state the field is seat of fluctuations, vacuum fluctuations fundamental property of quantum fields, which cannot be just reduced to existence of light quanta (photons) leads to several phenomena atoms can change their state even if no photon is present in field spontaneous emis- → sion Lamb shift can only be explained by interaction of Hydrogen with vacuum fluctuations, describes energy difference in state with same main quantum number and same overall angular momentum Casimir force is direct consequence of vacuum fluctuations, force exerted by radiation pressure if mode structure on two sides of an object is different, e.g. two parallel mirrors will go back to light matter interaction to show that quantization solves problem of spontaneous emission