The Quantum Mechanics Solver 27 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 26 Laser Cooling and Trapping By shining laser light onto an assembly of neutral atoms or ions it is possible to cool and trap these particles. In this chapter we study a simple cooling mechanism Doppler cooling and we derive the corresponding equilibrium temperature. We then show that the cooled atoms can be confined in the potential well created by a focused laser beam. We consider a two state atom whose levels are denoted g ground state and e excited state with respective energies 0 and hw0. This atom interacts with a classical electromagnetic wave of frequency wl 2k. For an atom located at r the Hamiltonian is H hw0 e e - d E r t e g E r t g e where d which is assumed to be real represents the matrix element of the atomic electric dipole operator between the states g and e . d e D g g D e . The quantity E E represents the electric field. We set E r t E0 r exp -i Lt . In all the chapter we assume that the detuning A wL w0 is small compared with wL and w0. We treat classically the motion r t of the atomic center of mass. Optical Bloch Equations for an Atom at Rest . Write the evolution equations for the four components of the density operator of the atom pgg peg pge and pee under the effect of the Hamiltonian H. . We take into account the coupling of the atom with the empty modes of the radiation field which are in particular responsible for the spontaneous emission of the atom when it is in the excited state e . We shall assume that this boils down to adding to the above evolution equations relaxation terms 268 26 Laser Cooling and Trapping 1 1 -_r H d Upo d _ r d _ r dtpeg . - 1peg dtpge . -Ipge relax relax where F 1 is the radiative lifetime of the excited state. Justify qualitatively these terms. . Check that for times much larger than F-1 these equations have the following stationary solutions s d E r t h 1 Pee 2 s 1 Peg A ir 2 1 s 2 s d E r t h 1 Pgg 2 s 1 Pge A - ir 2 1 s where we have set 2 d E0 r 2 h2 s A2 r2 4 . .
