THEORETICAL PROBLEM 2 SOLUTION DOPPLER LASER COOLING AND OPTICAL MOLASSES. v 1 c

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1 THEOETICA POBEM SOUTION DOPPE ASE COOING AND OPTICA MOASSES The key to this problem is the Doppler effet (to be preise, the longitudinal Doppler effet): The frequeny of a monohromi beam of light deteted by an observer depends on its ste of motion relive to the emitter, i.e. the observed frequeny is 1 v / 1 v / v 1 where v is the relive speed of emitter and observer and the frequeny of the emitter. The upper-lower signs orrespond, respetively, when soure and observer move towards or away from eah other. The seond equality holds in the limit of low veloities (non-relivisti limit). The frequeny of the laser in the lab is ; 0 is the transition frequeny of the om; the om moves with speed v towards the inident diretion of the laser: It is important to point out th the results must be given to first signifiant order in or q / mv. PAT I: BASICS OF ASE COOING 1. Absorption. v / 1a Write down the resonane ondition for the absorption of the oton. v b Write down the momentum p of the om after absorption, as seen in the laborory p p q mv 1 Write down the energy of the om after absorption, as seen in the laborory p mv 0 m

2 . Spontaneous emission in the x diretion. First, one alules the energy of the emitted oton, as seen in the lab referene frame. One must be areful to keep the orret order; this is beause the veloity of the om hanges after the absorption, however, this is seond order orretion for the emitted frequeny: v q 0 1 with v v m thus, v q 01 m v v q 1 1 m q 1 m q v 1 mv a Write down the energy of the emitted oton, proess in the x diretion, as seen in the laborory. 0. b Write down the momentum of the emitted oton p proess in the x diretion, as seen in the laborory. p / 0. Use onservion of momentum (see 1b): p p p q Write down the momentum of the om p proess in the x diretion, as seen in the laborory. p p mv 0. d Write down the energy of the om proess in the x diretion, as seen in the laborory. 0. p m mv

3 3. Spontaneous emission in the x diretion. The same as in the previous questions, keeping the right order 3a Write down the energy of the emitted oton, proess in the x diretion, as seen in the laborory. v v v v b Write down the momentum of the emitted oton p proess in the x diretion, as seen in the laborory. v p Write down the momentum of the om p proess in the x diretion, as seen in the laborory. v p p q p p q 1 mv 0. 3d Write down the energy of the om proess in the x diretion, as seen in the laborory. p mv q 1 m mv Average emission after absorption. The spontaneous emission proesses our with equal probabilities in both diretions. 4a Write down the average energy of an emitted oton,, after the emission proess. 1 1 v b Write down the average momentum of an emitted oton p, after the emission proess. 1 1 v q v p p p mv 0 seond order mv 0. 4 Write down the average energy of the om proess. 1 1 mv q 1 mv 0.

4 4d Write down the average momentum of the om proess. 1 1 p p p p p Energy and momentum transfer. Assuming a omplete one-oton absorption-emission proess only, as desribed above, there is a net average momentum and energy transfer between the laser and the om. 5a Write down the average energy hange of the om after a omplete one-oton absorption-emission proess. after before 1 1 v qv 0. 5b Write down the average momentum hange p of the om after a omplete one-oton absorption-emission proess. after before p p p q 6. Energy and momentum transfer by a laser beam along the x diretion. 0. 6a Write down the average energy hange of the om after a omplete one-oton absorption-emission proess. after before 1 1 v qv 0.3 6b Write down the average momentum hange p of the om after a omplete one-oton absorption-emission proess. after before p p p q 0.3 PAT II: DISSIPATION AND THE FUNDAMENTAS OF OPTICA MOASSES Two ounterpropaging laser beams with the same but arbitrary frequeny are inident on a beam of N oms th move in the x diretion with (average) veloity v.

5 7. Fore on the omi beam by the lasers. On the average, the fration of oms found in the exited ste is given by, P ex N ex N 0 4 where 0 is the resonane frequeny of the oms and is the so-alled abi frequeny; is proportional to the intensity of the laser beam. The lifetime of the exited energy level of the om is 1. The fore is aluled as the number of absorption-emission yles, times the momentum exhange in eah event, divided by the time of eah event. CAEFU! One must take into aount the Doppler shift of eah laser, as seen by the oms: 7a With the informion found so far, find the fore th the lasers exert on the omi beam. You must assume th mv q. F Np P ex Np v 0 P ex 4 v 0 4 Nq ow veloity limit. Assume now the veloity to be small enough in order to expand the fore to first order in v. 8a Find an expression for the fore found in Question (7a), in this limit. 4Nq F ( 0 ) v b Write down the ondition to obtain a positive fore (speeding up the om) Write down the ondition to obtain a zero fore

6 8d Write down the ondition to obtain a negive fore (slowing down the om). 0 this is the famous rule tune below resonane for ooling down 0.5 8e Consider now th the oms are moving with a veloity v (in the x diretion). Write down the ondition to obtain a slowing down fore on the oms. 0 i.e. independent of the diretion motion of the om Optial molasses In the ase of a negive fore, one obtains a fritional dissipive fore. Assume th initially, t 0, the gas of oms has veloity v 0. 9a In the limit of low veloities, find the veloity of the oms after the laser beams have been on for a time. F v m dv dt v an be read from (8a) v v 0 e t / m 1.5 9b Assume now th the gas of oms is in thermal equilibrium a temperure T 0. Find the temperure T after the laser beams have been on for a time. 0.5 ealling th 1 mv 1 kt in 1 dimension, and using v as the average thermal veloity in the equion of (9a), we an write down T T 0 e t / m