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Spontaneous Emission and Superradiance Spontaneous Emission and Superradiance Bachelorarbeit zur Erlangung des akademischen Grades Bachelor of Science (BSc) eingereicht an der Fakultät für Mathematik, Informatik und Physik der Universität Innsbruck von Andreas Kruckenhauser Betreuer: Univ. Prof. Dr. Helmut Ritsch Institut für Theoretische Physik Innsbruck, am 21. Juni 2015 Abstract An atom in an excited electronic state coupled to the electromagnetic field, will decay via spontaneous emission to a lower energy state. This thesis discusses spontaneous emission of several near atoms. We treat the quantisation of the free electromagnetic field and dipole interaction between the field and atom, the theoretical concepts of time evolution in quantum mechanics in both, open and closed systems. We study the Schrödinger, Heisenberg and Interaction picture for closed systems, define the density matrix and develop the master equation for open systems from its time evolution. As an example we establish the master equation in Lindblad form of an atom, where we find the spontaneous emission or Einstein A coefficient as a result. A central result is the superradiance, the enhanced collective emission of light from several atoms, which results from the established master equation for a system of N identical atoms. The emission is discussed in the example of two atoms. I Contents 1 Introduction 1 2 Theoretical Concepts 2 2.1 Quantisation of the free Electromagnetic Field . .2 2.1.1 Classical Field . .2 2.1.2 Quantisation . .3 2.2 System Dynamics . .5 2.2.1 Schrödinger Picture . .5 2.2.2 Heisenberg Picture . .5 2.2.3 Interaction Picture . .6 2.3 Open Quantum Systems . .7 2.4 Master Equation . .7 2.4.1 Density Operator . .8 2.4.2 Master Equation . .9 3 Spontaneous Emission of a Single Atom 12 3.1 Two Level Atom . 12 3.2 Dipole Approximation . 13 3.2.1 Minimal Coupling . 13 3.2.2 Approximation . 14 3.3 Spontaneous Emission . 15 3.3.1 Representation of an Operator . 15 3.3.2 Master Equation for a Two-Level Atom in Thermal Equilibrium 16 4 Collective Atom Dynamics 21 4.1 Superradiance Master Equation for N Two Level Atoms . 21 4.1.1 Hamiltonian . 21 4.1.2 Derivation of the Master Equation . 22 4.2 Example for Two Two-Level Atoms . 25 5 Conclusions 28 Bibliography 29 II Chapter 1 Introduction The fact that matter can spontaneously radiate light has been known for a long time in the form of Fluorescence and Luminescence. The fist theoretical description of spontaneous emission was done by Victor Weisskopf and Eugen Wigner in 1930. They used Dirac’s light-theory and calculated the natural linewidth of atomic crossovers. They have shown, that an excited electronic state is not an stationary state, as we know it from Schrödinger’s equation of the electron core system. The nature of this evolution is due to the coupling of the electromagnetic vacuum fluctuations to the atom [10]. By studying these light sources some other effects like superradiance, stimulated emission and absorption were discovered. At first glance superradiance and stimulated absorption look the same. But in fact they are two independent effects. The emitted radiation of a system with stimulated emission is proportional to the number of atoms and the light source’s density, for superradiance only the density of atoms is relevant. This was first discussed by the theory of Dicke. This theory describes the collective spontaneous decays of more than one atom. These decays are not independent and lead to a higher photon flux for short times as the same amount of independent decays. This short time photon flux is proportional to N2 for high numbers, where N is the number of atoms. Therefore Dicke called it an optical bomb [2]. The main reason why quantum optics was found, was a new kind of light source - the laser. Quantum optics is a field theory which discusses the interaction between electromagnetic fields and quantum mechanical systems. In quantum optics the theory of open systems has been a theme since its birth, because sources of light are open systems. In future superradiance may be used for lasers, which have a linewidth more than 1000× below the current standard. This can be used for a new generation of atomic clocks, optical lattices and laser cooling methods. But a lot of development and work is necessary to fullfill this goal.[6]. 1 Chapter 2 Theoretical Concepts In this chapter we present the theoretical fundament, which are needed for the under- standing of the next chapters. We start with a quantum model of the free electromagnetic field. Over the time evolution of closed and open systems to the master equation of an electron interacting with vacuum fluctuations. 2.1 Quantisation of the free Electromagnetic Field In this section the quantisation is achieved by an heuristic approach, where aspects as Lorentz covariance will not be discussed. 2.1.1 Classical Field First we state the classical field, which is described by Maxwell’s equations. If the region is free from charge (ρ(~r, t) = 0) and currents (~j(~r, t) = 0), like in the vacuum, the Maxwell equations for the electric field E~ (~r, t) and magnetic field B~ (~r, t) are given by ∂B~ (~r, t) divE~ (~r, t) = 0, rotE~ (~r, t) = − , (2.1a) ∂t 1 ∂E~ (~r, t) divB~ (~r, t) = 0, rotB~ (~r, t) = . (2.1b) c2 ∂t A usual way is to define the vector potential A~(~r, t) and electric potential Φ(~r, t). In vacuum it is very useful to use the Coulomb-gauge (divA~ = 0). These potentials have to fullfill the following equations, which can be deduced from the Maxwell equations 1 ∂2 A~(~r, t) − 4A(~r, t) = 0, (2.2a) c2 ∂t2 Z 1 ρ(~r, t) 3 0 Φ(~r, t) = 0 d r = 0. (2.2b) 4π0 |~r − ~r | 2 2 Theoretical Concepts The first equation, is a wave equation, which has the general result ~ X i~k~r A(~r, t) = Re A~k,λ~e~k,λe q~k,λ(t) . (2.3) ~k,λ Here A~k,λ are the amplitudes of the potential in direction of the polarisation vector ~e~k,λ, ~ k is the wave-vector and q~k,λ(t) the time dependent term with the dimension of a length. The electric and magnetic field can be calculated from the potentials ∂A~(~r, t) E~ (~r, t) = −grad Φ − = Re X A ~e ei~k~rq˙ (t) , (2.4a) ∂t ~k,λ ~k,λ ~k,λ ~k,λ ~ ~ X i~k~r B(~r, t) = rotA(~r, t) = Re A~k,1|k|~e~k,2 − A~k,2|k|~e~k,1 ie q~k,λ(t) . (2.4b) ~k ~ For the second equation we chose two orthogonal polarisation vectors and used ~e~k,λ·k = 0, which is a property of the e.m. field. Hence ~e = ~e × ~k and ~e = −~e × ~k . We ~k,1 ~k,2 |~k| ~k,2 ~k,1 |~k| now choose periodic boundaries appropriate for a cavity with side-length L, therefore ~ 2π only waves with a wave-vector k = (kx,nx , ky,ny , kz,nz ) with ki,ni = ni L , where i = x, y, z and ni = ±1, ±2, ±3,... are allowed. The Hamiltonian H for the total field energy in the Volume L3 is Z 2 2 L3 H = dV 0 E~ (~r, t) + c2 B~ (~r, t) = X 0 A2 q˙2 (t) + c2A2 |~k|2q2 (t) 2 2 ~k,λ ~k,λ ~k,λ ~k,λ V =L3 ~k,λ p2 (t) q2 (t) 1 X 2 2 2 X ~k,λ 2 ~k,λ = m~kq˙~k,λ(t) + m~kω~kq~k,λ(t) = + m~kω~k , (2.5) 2 2m~ 2 ~k,λ ~k,λ k with m = L3 A2 (which has the dimension of a mass), the angular velocity ω = c|~k| ~k,λ 0 ~k,λ ~k and the canonical momentum p~k,λ = q˙~k,λ(t)m~k. One sees that the energy of the system is a sum of independent harmonic oscillator energies, i.e. each mode and polarisation is equivalent to a harmonic oscillator. 2.1.2 Quantisation The quantisation can be made if we identify the p~k,λ and q~k,λ in equation (2.5) as operators which follow this commutator relations [8] h i q~k,λ, p~k0,λ0 = ihδ¯ ~k,~k0 δλ,λ0 , (2.6a) h i h i q~k,λ, q~k0,λ0 = 0, p~k,λ, p~k0,λ0 = 0. (2.6b) 3 2 Theoretical Concepts It is common to make a canonical transformation to a and a† ~k,λ ~k,λ 1 a~k,λ = q m~kω~kq~k,λ + ip~k,λ , (2.7a) 2m~khω¯ ~k 1 a† = m ω q − ip . (2.7b) ~k,λ q ~k ~k ~k,λ ~k,λ 2m~khω¯ ~k In terms of a and a† , equation (2.5) becomes ~k,λ ~k,λ 1 H =h ¯ X ω a† a + . (2.8) ~k ~k,λ ~k,λ 2 ~k,λ The operators a and a† follow the commutator relations derived from (2.6a) and ~k,λ ~k,λ (2.6b) h † i a , a = δ δ 0 , (2.9a) ~k,λ ~k0,λ0 ~k,~k0 λ,λ h i h i a , a = 0, a† , a† = 0. (2.9b) ~k,λ ~k0,λ0 ~k,λ ~k0,λ0 These operators are often called annihilation (a ) and creation (a† ) operator. Now, ~k,λ ~k,λ we can write the electric and magnetic field in terms of these operators E~ (~r, t) = X E ~e a + a† ei~k~r, (2.10a) ~k,λ ~k,λ ~k,λ ~k,λ ~k,λ 1 B~ (~r, t) = X E ~e a − a† ei~k~r, (2.10b) c ~k,λ ~k,λ ~k,λ ~k,λ ~k,λ 1 ¯hω~k 2 where E~ = has the dimenison of an electric field.
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