Chapter 1 Photons in a cavity Sketch of an optical cavity with a standing wave 1.1 Quantized electromagnetic fields Expansion of cavity fields E and B: ‘separation of variables’ 3 The amplitudes A1 and A2 are called quadratures in (quantum) optics. They play the same role as position and momentum coordinates in a harmonic oscillator. But which of them is ‘position’ and which one ‘momentum’ is less strictly defined and changes depending on the context. Maxwell equations relations between electric and magnetic mode ⇒ functions f(x) and g(x) and vector Helmholtz equation Equations of motion = dynamics (time evolution) of the quadratures A1 and A2 (non-conventional notation! λ is not a wavelength) A˙ = kA , A˙ = c2λA (1.1) 2 − 1 1 2 Translation table to harmonic oscillator of classical mechanics (1.2) Quantization of the electromagnetic field in the cavity: in the same way as for the harmonic oscillator – translation table yields pˆ2 kxˆ2 Hˆ = + osc 2m 2 c2λAˆ2 k Hˆ = 2 + Aˆ2 (1.3) 2 2 1 Key recipe for quantizing: correspondence principle = the Heisenberg equa- tions of motion must have the same form as the classical equations of mo- tion d i Aˆ = [Hˆ , Aˆ ] (1.4) dt 2 h¯ 2 4 Work this out: i i [Hˆ , Aˆ ] = Aˆ [kAˆ , Aˆ ] = kAˆ (1.5) h¯ 2 h¯ 1 1 2 − 1 provided the quadrature operators have the commutator [Aˆ1, Aˆ2] = i¯h (1.6) which is consistent with the translation table (1.2). stationary states (wave functions in the position representation) for the harmonic oscillator energy levels for the stationary quantum states of the cavity field The photon concept. Interprete the energy levels of the harmonic oscil- 1 lator, En = h¯ωc(n + 2 ) as states with n packets of energy per photon, each packet having h¯ωc (Einstein and de Broglie). This works only for the har- monic oscillator where the energy spectrum is equidistant. In other words: 5 A photon is an elementary excitation of a mode of the elec- tromagnetic field. (70% of ‘Photon licence’ awarded by W. E. Lamb.) Note the shift away from ‘wave functions’ ψn(x) towards the slightly more abstract Dirac notation n with ‘ket’ and ‘bra’. | # Creation and annihilation operators are built from the ‘ladder operators’ aˆ and aˆ† of the harmonic oscillator spectrum: Expression in terms of position and momentum coordinates aˆ k pˆ = xˆ i , (1.7) aˆ $2¯hω ± √2¯hω m † c c dimensionless operators with the commutator [a,ˆ aˆ†] = 1ˆ (unit operator!). Fix constants k and λ from the energy of the resonator mode. Corre- spondence principle: quantum-mechanical energy = sum of electric and magnetic field energy (both are integrals over energy densities). Electric energy (density) 2 2 ε0 2 ε0A1 2 ε0A1 Eel = dx E (x) = dx f(x) = (1.8) % 2 2 % | | 2 = 1 & '( ) where we have used a specific normalization for the mode function f(x). The integral is finite because the mode is localized in the cavity (it is nor- malizable). Hence the correspondence principle fixes the constant k: ε0 2 ! k ˆ2 dx E (x) = A1 k = ε0 (1.9) % 2 2 ⇒ 6 Magnetic energy 1 A2 dx B2(x) = 2 dx g(x) 2 (1.10) % 2µ0 2µ0 % | | This integral is linked to the previous one by the Maxwell equations 2 1 1 dx g(x) = 2 dx ( f) ( f) = 2 dx f [ ( f)] (1.11) % | | k % ∇ × · ∇ × k % · ∇ × ∇ × after a partial integration. The boundary terms involve the normal compo- nents of f ( f) and are zero for perfectly reflecting mirrors. Use the × ∇ × vector Helmholtz equation to get 2 kλ λ dx g(x) = 2 dx f f = (1.12) % | | k % · k Therefore, we get again 2 2 ˆ2 1 2 A2λ ! c λA2 dx B (x) = = k = ε0 (1.13) % 2µ0 2µ0k 2 ⇒ To improve the symmetry between electric and magnetic fields, we re-scale the magnetic mode g˜ = (k/λ)1/2g such that g˜ has the same normalization as f. We end up with the following expressions for the electromagnetic quadratures A1 and A2 in terms of the boson operators a and a†: 2 ε0 c λ a = A1 + iA2 (1.14) $2¯hωc $2¯hωc This yields the following expressions for the electric and magnetic field operators h¯ω Eˆ(x, t) = c (aˆ(t) f(x) + h.c.) (1.15) $ 2ε0 h¯ω µ Bˆ (x, t) = i c 0 (aˆ(t)g˜(x) h.c.) (1.16) − $ 2 − These very important formulas describe in a precise manner the way classi- cal electrodynamics and quantum mechanics are linked in quantum optics and, more generally, in quantum field theory. 7 Classical properties: encoded in the mode function f(x), like frequency ωc. For a standing wave, the mode has no definite momentum. But a plane wave with wave vector k would have that property. Similarly for the angular momentum which (already in classical electrodynamics) is a sum of orbital angular momentum and spin angular momentum.1 Quantum properties: one can only talk meaningfully about a photon cre- ated by a† in the mode functions f (and g˜). Its energy is h¯ωc, its momentum would be h¯k if we are dealing with a plane wave. See the exercises for an expansion of a standing wave in momentum eigenstates. Quantum field properties: the typical magnitude of the field ‘per photon’ is given by the square root h¯ωc 1ph = (1.17) E $2ε0V where V is the ‘mode volume’ (related to the ‘size’ of f). The electric field (operator) is fluctuating around a zero mean value, even when there are no photons, with a magnitude of order . These fluctuations are a con- E1ph sequence of an uncertainty relation that exists in quantum field theory be- tween field (electric and magnetic) at nearby positions. In other words: the fields are ‘grainy’ and are exchanged in energy packets (given by the famous h¯ωc). The relevant packet size is a classical quantity and may even have some spread if the field is not a monochromatic mode. In the following, no more hats on operators written down explicitly. Plane wave expansion For later use, we write down already here the expression for the field op- erators in free space, where an infinite number of modes contributes. A convenient basis is given by plane waves with wave vectors k and polar- ization vectors e orthogonal to k, k e = 0 and normalized e 2 = 1 kµ · kµ | kµ| 1Orbital angular momentum: relates to derivative of f, hence phase gradients. Exam- ples are helicoidal or vortex fields where the phase increases by 2π (or 4π when going − around a zero of the field intensity). Spin angular momentum: related to circular polar- ization, hence the vector components of the field. Both angular momenta have integer quantum numbers because photons are bosons (not half-integer like electrons/fermions have). 8 (two possible values for the polarization index µ). If we restrict the mode functions to be periodic in a volume V , we have the natural orthogonality relation ik x ik! x 3 e− · e · d x ek∗µ ek!µ! = δµµ! δkk! (1.18) %V √V · √V where the Kronecker symbol δkk! makes sense because the periodic bound- ary conditions make the allowed k-vectors discrete. The same orthogonality relation is transferred to the photon operators akµ: [a , a† ] = δ δ (1.19) kµ k!µ! µµ! kk! The frequency of a plane wave mode is of course given by ω = c k . k | | After these notations, we finally get the field operators h¯ωk ik x Eˆ(x, t) = aˆkµ(t) ekµ e · + h.c. (1.20) $2ε0V *kµ + , h¯ωkµ0 ik x Bˆ (x, t) = aˆkµ(t) bkµ e · + h.c. (1.21) $ 2V *kµ + , where the unit vectors for the magnetic field are given by (ω/c)b = k kµ × ekµ. In the following, we come back to a single mode and discuss its dynam- ics and quantum states. Dynamics Heisenberg equation of motion for boson mode operator da i 1 = [H , a] = iω [a†a + , a] = iω [a†a , a] (1.22) dt h¯ c 2 c dynamics does not depend on zero point energy. Product rule for commu- tator with product yields da iωct = iω a , a(t) = a(0) e− (1.23) dt − c ⇒ This equation of motion is just a complex combination of the two classical equations for the quadrature coordinates x and p. The form of the ex- iωct iEnt/¯h ponential e− is similar to e− for an energy eigenstate in quantum 9 mechanics. This has produced the name positive frequency operator for a(t). More generally, we call the complex field operator E(+)(x, t) = a(t) f(x) (1.24) E1ph the positive frequency part of the field. The notation is confusing, as the ( ) term with the dagger a† gives the negative frequency field E − (x, t). But the language has become established over the years. Negative frequency solutions and antiparticles. Not a problem in clas- sical field theory: any real-valued field has both positive and negative fre- quency components. Electrodynamics, hydrodynamics etc. Not a problem in quantum electrodynamics: in the negative frequency +iωct term a† e , the prefactor is interpreted as a creation operator of a photon, the same particle that is annihaled by a. This is generally true in elementary particle physics: a particle is its own antiparticle if its classical field is real (the quantized field becomes a hermitean operator). In the quantized Dirac theory, for example, where a relativistic wave equation for the electron is quantized, the field operators for the electron are not hermitean.